LMFDB - Embedded newform 6840.2.a.w.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{5} +1.64575 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +1.64575 q^{7} +0.354249 q^{11} +3.64575 q^{13} -1.00000 q^{19} -2.00000 q^{23} +1.00000 q^{25} -9.64575 q^{29} -2.00000 q^{31} -1.64575 q^{35} -6.93725 q^{37} -1.64575 q^{41} -4.93725 q^{43} -6.00000 q^{47} -4.29150 q^{49} -4.00000 q^{53} -0.354249 q^{55} -3.29150 q^{59} +11.2915 q^{61} -3.64575 q^{65} +4.00000 q^{67} +3.29150 q^{71} -2.00000 q^{73} +0.583005 q^{77} -8.00000 q^{79} +15.8745 q^{83} +12.2288 q^{89} +6.00000 q^{91} +1.00000 q^{95} +18.2288 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.64575	0.622036	0.311018	−	0.950404i	$-0.399330\pi$
$7$	1.64575	0.622036	0.311018	+	0.950404i	$0.399330\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.354249	0.106810	0.0534050	−	0.998573i	$-0.482993\pi$
$11$	0.354249	0.106810	0.0534050	+	0.998573i	$0.482993\pi$
$12$	0	0
$13$	3.64575	1.01115	0.505575	−	0.862783i	$-0.331280\pi$
$13$	3.64575	1.01115	0.505575	+	0.862783i	$0.331280\pi$
$14$	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
$29$	−9.64575	−1.79117	−0.895586	−	0.444889i	$-0.853243\pi$
$29$	−9.64575	−1.79117	−0.895586	+	0.444889i	$0.853243\pi$
$30$	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
$35$	−1.64575	−0.278183
$36$	0	0
$37$	−6.93725	−1.14048	−0.570239	−	0.821479i	$-0.693149\pi$
$37$	−6.93725	−1.14048	−0.570239	+	0.821479i	$0.693149\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.64575	−0.257023	−0.128512	−	0.991708i	$-0.541020\pi$
$41$	−1.64575	−0.257023	−0.128512	+	0.991708i	$0.541020\pi$
$42$	0	0
$43$	−4.93725	−0.752924	−0.376462	−	0.926432i	$-0.622859\pi$
$43$	−4.93725	−0.752924	−0.376462	+	0.926432i	$0.622859\pi$
$44$	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
$49$	−4.29150	−0.613072
$50$	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
$55$	−0.354249	−0.0477669
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.29150	−0.428517	−0.214259	−	0.976777i	$-0.568734\pi$
$59$	−3.29150	−0.428517	−0.214259	+	0.976777i	$0.568734\pi$
$60$	0	0
$61$	11.2915	1.44573	0.722864	−	0.690990i	$-0.242825\pi$
$61$	11.2915	1.44573	0.722864	+	0.690990i	$0.242825\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.64575	−0.452200
$66$	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
$71$	3.29150	0.390629	0.195315	−	0.980741i	$-0.437427\pi$
$71$	3.29150	0.390629	0.195315	+	0.980741i	$0.437427\pi$
$72$	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
$77$	0.583005	0.0664396
$78$	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	$-0.163325\pi$
$83$	15.8745	1.74245	0.871227	+	0.490881i	$0.163325\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.2288	1.29625	0.648123	−	0.761536i	$-0.275554\pi$
$89$	12.2288	1.29625	0.648123	+	0.761536i	$0.275554\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	18.2288	1.85085	0.925425	−	0.378931i	$-0.123708\pi$
$97$	18.2288	1.85085	0.925425	+	0.378931i	$0.123708\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.8745	−1.57957	−0.789786	−	0.613382i	$-0.789809\pi$
$101$	−15.8745	−1.57957	−0.789786	+	0.613382i	$0.789809\pi$
$102$	0	0
$103$	14.5830	1.43691	0.718453	−	0.695575i	$-0.244850\pi$
$103$	14.5830	1.43691	0.718453	+	0.695575i	$0.244850\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.5830	−1.40979	−0.704896	−	0.709311i	$-0.749006\pi$
$107$	−14.5830	−1.40979	−0.704896	+	0.709311i	$0.749006\pi$
$108$	0	0
$109$	5.29150	0.506834	0.253417	−	0.967357i	$-0.418446\pi$
$109$	5.29150	0.506834	0.253417	+	0.967357i	$0.418446\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.8745	−1.68149	−0.840746	−	0.541430i	$-0.817883\pi$
$113$	−17.8745	−1.68149	−0.840746	+	0.541430i	$0.817883\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8745	−0.988592
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.58301	−0.229205	−0.114602	−	0.993411i	$-0.536559\pi$
$127$	−2.58301	−0.229205	−0.114602	+	0.993411i	$0.536559\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.64575	−0.668012	−0.334006	−	0.942571i	$-0.608401\pi$
$131$	−7.64575	−0.668012	−0.334006	+	0.942571i	$0.608401\pi$
$132$	0	0
$133$	−1.64575	−0.142705
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.583005	0.0498095	0.0249047	−	0.999690i	$-0.492072\pi$
$137$	0.583005	0.0498095	0.0249047	+	0.999690i	$0.492072\pi$
$138$	0	0
$139$	−17.8745	−1.51610	−0.758048	−	0.652199i	$-0.773847\pi$
$139$	−17.8745	−1.51610	−0.758048	+	0.652199i	$0.773847\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.29150	0.108001
$144$	0	0
$145$	9.64575	0.801036
$146$	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	$-0.281633\pi$
$157$	15.8745	1.26692	0.633462	+	0.773774i	$0.281633\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.29150	−0.259407
$162$	0	0
$163$	9.64575	0.755514	0.377757	−	0.925905i	$-0.376696\pi$
$163$	9.64575	0.755514	0.377757	+	0.925905i	$0.376696\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.29150	0.0999395	0.0499697	−	0.998751i	$-0.484088\pi$
$167$	1.29150	0.0999395	0.0499697	+	0.998751i	$0.484088\pi$
$168$	0	0
$169$	0.291503	0.0224233
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.70850	−0.662095	−0.331047	−	0.943614i	$-0.607402\pi$
$173$	−8.70850	−0.662095	−0.331047	+	0.943614i	$0.607402\pi$
$174$	0	0
$175$	1.64575	0.124407
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.29150	0.544992	0.272496	−	0.962157i	$-0.412151\pi$
$179$	7.29150	0.544992	0.272496	+	0.962157i	$0.412151\pi$
$180$	0	0
$181$	−5.29150	−0.393314	−0.196657	−	0.980472i	$-0.563009\pi$
$181$	−5.29150	−0.393314	−0.196657	+	0.980472i	$0.563009\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.93725	0.510037
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.9373	−0.791392	−0.395696	−	0.918382i	$-0.629497\pi$
$191$	−10.9373	−0.791392	−0.395696	+	0.918382i	$0.629497\pi$
$192$	0	0
$193$	2.93725	0.211428	0.105714	−	0.994397i	$-0.466287\pi$
$193$	2.93725	0.211428	0.105714	+	0.994397i	$0.466287\pi$
$194$	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
$199$	−23.2915	−1.65109	−0.825545	−	0.564336i	$-0.809132\pi$
$199$	−23.2915	−1.65109	−0.825545	+	0.564336i	$0.809132\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−15.8745	−1.11417
$204$	0	0
$205$	1.64575	0.114944
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.354249	−0.0245039
$210$	0	0
$211$	−10.5830	−0.728564	−0.364282	−	0.931289i	$-0.618686\pi$
$211$	−10.5830	−0.728564	−0.364282	+	0.931289i	$0.618686\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.93725	0.336718
$216$	0	0
$217$	−3.29150	−0.223442
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.41699	−0.0948890	−0.0474445	−	0.998874i	$-0.515108\pi$
$223$	−1.41699	−0.0948890	−0.0474445	+	0.998874i	$0.515108\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.2915	−1.67866	−0.839328	−	0.543625i	$-0.817051\pi$
$227$	−25.2915	−1.67866	−0.839328	+	0.543625i	$0.817051\pi$
$228$	0	0
$229$	7.29150	0.481836	0.240918	−	0.970545i	$-0.422552\pi$
$229$	7.29150	0.481836	0.240918	+	0.970545i	$0.422552\pi$
$230$	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
$239$	13.0627	0.844959	0.422479	−	0.906373i	$-0.361160\pi$
$239$	13.0627	0.844959	0.422479	+	0.906373i	$0.361160\pi$
$240$	0	0
$241$	−4.58301	−0.295217	−0.147609	−	0.989046i	$-0.547158\pi$
$241$	−4.58301	−0.295217	−0.147609	+	0.989046i	$0.547158\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.29150	0.274174
$246$	0	0
$247$	−3.64575	−0.231974
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.22876	0.393156	0.196578	−	0.980488i	$-0.437017\pi$
$251$	6.22876	0.393156	0.196578	+	0.980488i	$0.437017\pi$
$252$	0	0
$253$	−0.708497	−0.0445428
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.87451	0.615955	0.307977	−	0.951394i	$-0.400348\pi$
$257$	9.87451	0.615955	0.307977	+	0.951394i	$0.400348\pi$
$258$	0	0
$259$	−11.4170	−0.709418
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.29150	0.0796375	0.0398187	−	0.999207i	$-0.487322\pi$
$263$	1.29150	0.0796375	0.0398187	+	0.999207i	$0.487322\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.93725	−0.301030	−0.150515	−	0.988608i	$-0.548093\pi$
$269$	−4.93725	−0.301030	−0.150515	+	0.988608i	$0.548093\pi$
$270$	0	0
$271$	12.7085	0.771986	0.385993	−	0.922502i	$-0.373859\pi$
$271$	12.7085	0.771986	0.385993	+	0.922502i	$0.373859\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.354249	0.0213620
$276$	0	0
$277$	−24.5830	−1.47705	−0.738525	−	0.674226i	$-0.764477\pi$
$277$	−24.5830	−1.47705	−0.738525	+	0.674226i	$0.764477\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.93725	0.294532	0.147266	−	0.989097i	$-0.452953\pi$
$281$	4.93725	0.294532	0.147266	+	0.989097i	$0.452953\pi$
$282$	0	0
$283$	−3.77124	−0.224177	−0.112089	−	0.993698i	$-0.535754\pi$
$283$	−3.77124	−0.224177	−0.112089	+	0.993698i	$0.535754\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.70850	−0.159878
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.8745	1.04424	0.522120	−	0.852872i	$-0.325141\pi$
$293$	17.8745	1.04424	0.522120	+	0.852872i	$0.325141\pi$
$294$	0	0
$295$	3.29150	0.191639
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.29150	−0.421678
$300$	0	0
$301$	−8.12549	−0.468346
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.2915	−0.646550
$306$	0	0
$307$	12.7085	0.725312	0.362656	−	0.931923i	$-0.381870\pi$
$307$	12.7085	0.725312	0.362656	+	0.931923i	$0.381870\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.35425	0.246907	0.123453	−	0.992350i	$-0.460603\pi$
$311$	4.35425	0.246907	0.123453	+	0.992350i	$0.460603\pi$
$312$	0	0
$313$	−18.7085	−1.05747	−0.528733	−	0.848788i	$-0.677333\pi$
$313$	−18.7085	−1.05747	−0.528733	+	0.848788i	$0.677333\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.5830	−1.26839	−0.634194	−	0.773174i	$-0.718668\pi$
$317$	−22.5830	−1.26839	−0.634194	+	0.773174i	$0.718668\pi$
$318$	0	0
$319$	−3.41699	−0.191315
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.64575	0.202230
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.87451	−0.544399
$330$	0	0
$331$	16.5830	0.911484	0.455742	−	0.890112i	$-0.349374\pi$
$331$	16.5830	0.911484	0.455742	+	0.890112i	$0.349374\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−22.2288	−1.21088	−0.605439	−	0.795892i	$-0.707002\pi$
$337$	−22.2288	−1.21088	−0.605439	+	0.795892i	$0.707002\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.708497	−0.0383673
$342$	0	0
$343$	−18.5830	−1.00339
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.29150	−0.498794	−0.249397	−	0.968401i	$-0.580232\pi$
$347$	−9.29150	−0.498794	−0.249397	+	0.968401i	$0.580232\pi$
$348$	0	0
$349$	−27.1660	−1.45416	−0.727082	−	0.686551i	$-0.759124\pi$
$349$	−27.1660	−1.45416	−0.727082	+	0.686551i	$0.759124\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.1660	1.87170	0.935849	−	0.352401i	$-0.114635\pi$
$353$	35.1660	1.87170	0.935849	+	0.352401i	$0.114635\pi$
$354$	0	0
$355$	−3.29150	−0.174695
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.2288	0.539853	0.269927	−	0.962881i	$-0.413001\pi$
$359$	10.2288	0.539853	0.269927	+	0.962881i	$0.413001\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	0.228757	0.0119410	0.00597050	−	0.999982i	$-0.498100\pi$
$367$	0.228757	0.0119410	0.00597050	+	0.999982i	$0.498100\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.58301	−0.341773
$372$	0	0
$373$	−8.35425	−0.432567	−0.216283	−	0.976331i	$-0.569393\pi$
$373$	−8.35425	−0.432567	−0.216283	+	0.976331i	$0.569393\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−35.1660	−1.81114
$378$	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
$383$	5.41699	0.276796	0.138398	−	0.990377i	$-0.455805\pi$
$383$	5.41699	0.276796	0.138398	+	0.990377i	$0.455805\pi$
$384$	0	0
$385$	−0.583005	−0.0297127
$386$	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
$401$	−6.35425	−0.317316	−0.158658	−	0.987334i	$-0.550717\pi$
$401$	−6.35425	−0.317316	−0.158658	+	0.987334i	$0.550717\pi$
$402$	0	0
$403$	−7.29150	−0.363216
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.45751	−0.121814
$408$	0	0
$409$	−6.70850	−0.331714	−0.165857	−	0.986150i	$-0.553039\pi$
$409$	−6.70850	−0.331714	−0.165857	+	0.986150i	$0.553039\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.41699	−0.266553
$414$	0	0
$415$	−15.8745	−0.779249
$416$	0	0
$417$	0	0
$418$	0	0
$419$	38.9373	1.90221	0.951105	−	0.308869i	$-0.0999504\pi$
$419$	38.9373	1.90221	0.951105	+	0.308869i	$0.0999504\pi$
$420$	0	0
$421$	−0.583005	−0.0284139	−0.0142070	−	0.999899i	$-0.504522\pi$
$421$	−0.583005	−0.0284139	−0.0142070	+	0.999899i	$0.504522\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.5830	0.899295
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.4575	1.56342	0.781712	−	0.623640i	$-0.214347\pi$
$431$	32.4575	1.56342	0.781712	+	0.623640i	$0.214347\pi$
$432$	0	0
$433$	−26.2288	−1.26047	−0.630237	−	0.776403i	$-0.717042\pi$
$433$	−26.2288	−1.26047	−0.630237	+	0.776403i	$0.717042\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	5.16601	0.246560	0.123280	−	0.992372i	$-0.460659\pi$
$439$	5.16601	0.246560	0.123280	+	0.992372i	$0.460659\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.58301	−0.407791	−0.203895	−	0.978993i	$-0.565360\pi$
$443$	−8.58301	−0.407791	−0.203895	+	0.978993i	$0.565360\pi$
$444$	0	0
$445$	−12.2288	−0.579699
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.35425	0.111104	0.0555519	−	0.998456i	$-0.482308\pi$
$449$	2.35425	0.111104	0.0555519	+	0.998456i	$0.482308\pi$
$450$	0	0
$451$	−0.583005	−0.0274526
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	9.29150	0.434638	0.217319	−	0.976101i	$-0.430269\pi$
$457$	9.29150	0.434638	0.217319	+	0.976101i	$0.430269\pi$
$458$	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
$463$	−7.77124	−0.361160	−0.180580	−	0.983560i	$-0.557798\pi$
$463$	−7.77124	−0.361160	−0.180580	+	0.983560i	$0.557798\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.583005	0.0269783	0.0134891	−	0.999909i	$-0.495706\pi$
$467$	0.583005	0.0269783	0.0134891	+	0.999909i	$0.495706\pi$
$468$	0	0
$469$	6.58301	0.303975
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.74902	−0.0804198
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.5203	0.617756	0.308878	−	0.951102i	$-0.400046\pi$
$479$	13.5203	0.617756	0.308878	+	0.951102i	$0.400046\pi$
$480$	0	0
$481$	−25.2915	−1.15319
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18.2288	−0.827725
$486$	0	0
$487$	3.29150	0.149152	0.0745761	−	0.997215i	$-0.476240\pi$
$487$	3.29150	0.149152	0.0745761	+	0.997215i	$0.476240\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.8118	1.48077	0.740387	−	0.672181i	$-0.234642\pi$
$491$	32.8118	1.48077	0.740387	+	0.672181i	$0.234642\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.41699	0.242985
$498$	0	0
$499$	19.2915	0.863606	0.431803	−	0.901968i	$-0.357878\pi$
$499$	19.2915	0.863606	0.431803	+	0.901968i	$0.357878\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.29150	0.0575853	0.0287926	−	0.999585i	$-0.490834\pi$
$503$	1.29150	0.0575853	0.0287926	+	0.999585i	$0.490834\pi$
$504$	0	0
$505$	15.8745	0.706406
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.3542	−1.52273	−0.761363	−	0.648326i	$-0.775469\pi$
$509$	−34.3542	−1.52273	−0.761363	+	0.648326i	$0.775469\pi$
$510$	0	0
$511$	−3.29150	−0.145608
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.5830	−0.642604
$516$	0	0
$517$	−2.12549	−0.0934790
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.8118	−1.17464	−0.587322	−	0.809353i	$-0.699818\pi$
$521$	−26.8118	−1.17464	−0.587322	+	0.809353i	$0.699818\pi$
$522$	0	0
$523$	6.12549	0.267849	0.133925	−	0.990992i	$-0.457242\pi$
$523$	6.12549	0.267849	0.133925	+	0.990992i	$0.457242\pi$
$524$	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
$535$	14.5830	0.630478
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.52026	−0.0654822
$540$	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
$547$	−40.4575	−1.72984	−0.864919	−	0.501911i	$-0.832630\pi$
$547$	−40.4575	−1.72984	−0.864919	+	0.501911i	$0.832630\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.64575	0.410923
$552$	0	0
$553$	−13.1660	−0.559876
$554$	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
$563$	3.87451	0.163291	0.0816455	−	0.996661i	$-0.473982\pi$
$563$	3.87451	0.163291	0.0816455	+	0.996661i	$0.473982\pi$
$564$	0	0
$565$	17.8745	0.751986
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.479741	−0.0201118	−0.0100559	−	0.999949i	$-0.503201\pi$
$569$	−0.479741	−0.0201118	−0.0100559	+	0.999949i	$0.503201\pi$
$570$	0	0
$571$	13.8745	0.580630	0.290315	−	0.956931i	$-0.406240\pi$
$571$	13.8745	0.580630	0.290315	+	0.956931i	$0.406240\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	27.8745	1.16043	0.580215	−	0.814463i	$-0.302968\pi$
$577$	27.8745	1.16043	0.580215	+	0.814463i	$0.302968\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	26.1255	1.08387
$582$	0	0
$583$	−1.41699	−0.0586859
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−36.5830	−1.50994	−0.754971	−	0.655758i	$-0.772349\pi$
$587$	−36.5830	−1.50994	−0.754971	+	0.655758i	$0.772349\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.4170	0.633100	0.316550	−	0.948576i	$-0.397475\pi$
$593$	15.4170	0.633100	0.316550	+	0.948576i	$0.397475\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.41699	0.221332	0.110666	−	0.993858i	$-0.464702\pi$
$599$	5.41699	0.221332	0.110666	+	0.993858i	$0.464702\pi$
$600$	0	0
$601$	−15.8745	−0.647535	−0.323767	−	0.946137i	$-0.604950\pi$
$601$	−15.8745	−0.647535	−0.323767	+	0.946137i	$0.604950\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.8745	0.442112
$606$	0	0
$607$	−12.4575	−0.505635	−0.252817	−	0.967514i	$-0.581357\pi$
$607$	−12.4575	−0.505635	−0.252817	+	0.967514i	$0.581357\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.8745	−0.884948
$612$	0	0
$613$	−14.4575	−0.583933	−0.291967	−	0.956428i	$-0.594310\pi$
$613$	−14.4575	−0.583933	−0.291967	+	0.956428i	$0.594310\pi$
$614$	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
$619$	−37.8745	−1.52231	−0.761153	−	0.648573i	$-0.775366\pi$
$619$	−37.8745	−1.52231	−0.761153	+	0.648573i	$0.775366\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	20.1255	0.806311
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	37.1660	1.47956	0.739778	−	0.672851i	$-0.234931\pi$
$631$	37.1660	1.47956	0.739778	+	0.672851i	$0.234931\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.58301	0.102503
$636$	0	0
$637$	−15.6458	−0.619907
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.2288	0.798988	0.399494	−	0.916736i	$-0.369186\pi$
$641$	20.2288	0.798988	0.399494	+	0.916736i	$0.369186\pi$
$642$	0	0
$643$	44.9373	1.77215	0.886076	−	0.463540i	$-0.153421\pi$
$643$	44.9373	1.77215	0.886076	+	0.463540i	$0.153421\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.2915	0.994312	0.497156	−	0.867661i	$-0.334378\pi$
$647$	25.2915	0.994312	0.497156	+	0.867661i	$0.334378\pi$
$648$	0	0
$649$	−1.16601	−0.0457699
$650$	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
$655$	7.64575	0.298744
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.1660	−0.512875	−0.256437	−	0.966561i	$-0.582549\pi$
$659$	−13.1660	−0.512875	−0.256437	+	0.966561i	$0.582549\pi$
$660$	0	0
$661$	−5.29150	−0.205816	−0.102908	−	0.994691i	$-0.532815\pi$
$661$	−5.29150	−0.205816	−0.102908	+	0.994691i	$0.532815\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.64575	0.0638195
$666$	0	0
$667$	19.2915	0.746970
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	9.52026	0.366979	0.183490	−	0.983022i	$-0.441261\pi$
$673$	9.52026	0.366979	0.183490	+	0.983022i	$0.441261\pi$
$674$	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
$685$	−0.583005	−0.0222755
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.5830	−0.555568
$690$	0	0
$691$	31.2915	1.19038	0.595192	−	0.803583i	$-0.297076\pi$
$691$	31.2915	1.19038	0.595192	+	0.803583i	$0.297076\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.8745	0.678019
$696$	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
$703$	6.93725	0.261643
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.1255	−0.982550
$708$	0	0
$709$	−32.4575	−1.21897	−0.609484	−	0.792799i	$-0.708623\pi$
$709$	−32.4575	−1.21897	−0.609484	+	0.792799i	$0.708623\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	−1.29150	−0.0482995
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.64575	0.285138	0.142569	−	0.989785i	$-0.454464\pi$
$719$	7.64575	0.285138	0.142569	+	0.989785i	$0.454464\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.64575	−0.358234
$726$	0	0
$727$	−39.9778	−1.48269	−0.741347	−	0.671122i	$-0.765813\pi$
$727$	−39.9778	−1.48269	−0.741347	+	0.671122i	$0.765813\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−32.5830	−1.20348	−0.601740	−	0.798692i	$-0.705526\pi$
$733$	−32.5830	−1.20348	−0.601740	+	0.798692i	$0.705526\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.41699	0.0521957
$738$	0	0
$739$	−10.5830	−0.389302	−0.194651	−	0.980873i	$-0.562357\pi$
$739$	−10.5830	−0.389302	−0.194651	+	0.980873i	$0.562357\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.7490	1.16476	0.582379	−	0.812917i	$-0.302122\pi$
$743$	31.7490	1.16476	0.582379	+	0.812917i	$0.302122\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	36.3320	1.32577	0.662887	−	0.748719i	$-0.269331\pi$
$751$	36.3320	1.32577	0.662887	+	0.748719i	$0.269331\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	−19.8745	−0.722351	−0.361176	−	0.932498i	$-0.617625\pi$
$757$	−19.8745	−0.722351	−0.361176	+	0.932498i	$0.617625\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.1660	0.404768	0.202384	−	0.979306i	$-0.435131\pi$
$761$	11.1660	0.404768	0.202384	+	0.979306i	$0.435131\pi$
$762$	0	0
$763$	8.70850	0.315269
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	21.7490	0.784290	0.392145	−	0.919904i	$-0.371733\pi$
$769$	21.7490	0.784290	0.392145	+	0.919904i	$0.371733\pi$
$770$	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
$779$	1.64575	0.0589652
$780$	0	0
$781$	1.16601	0.0417231
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15.8745	−0.566585
$786$	0	0
$787$	11.2915	0.402499	0.201249	−	0.979540i	$-0.435500\pi$
$787$	11.2915	0.402499	0.201249	+	0.979540i	$0.435500\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−29.4170	−1.04595
$792$	0	0
$793$	41.1660	1.46185
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.2915	−1.39178	−0.695888	−	0.718150i	$-0.744989\pi$
$797$	−39.2915	−1.39178	−0.695888	+	0.718150i	$0.744989\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.708497	−0.0250023
$804$	0	0
$805$	3.29150	0.116010
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.58301	0.301762	0.150881	−	0.988552i	$-0.451789\pi$
$809$	8.58301	0.301762	0.150881	+	0.988552i	$0.451789\pi$
$810$	0	0
$811$	5.41699	0.190216	0.0951082	−	0.995467i	$-0.469680\pi$
$811$	5.41699	0.190216	0.0951082	+	0.995467i	$0.469680\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.64575	−0.337876
$816$	0	0
$817$	4.93725	0.172733
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.7085	−0.513330	−0.256665	−	0.966500i	$-0.582624\pi$
$821$	−14.7085	−0.513330	−0.256665	+	0.966500i	$0.582624\pi$
$822$	0	0
$823$	−43.2693	−1.50827	−0.754136	−	0.656718i	$-0.771944\pi$
$823$	−43.2693	−1.50827	−0.754136	+	0.656718i	$0.771944\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.7085	−0.789652	−0.394826	−	0.918756i	$-0.629195\pi$
$827$	−22.7085	−0.789652	−0.394826	+	0.918756i	$0.629195\pi$
$828$	0	0
$829$	−27.8745	−0.968122	−0.484061	−	0.875034i	$-0.660839\pi$
$829$	−27.8745	−0.968122	−0.484061	+	0.875034i	$0.660839\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.29150	−0.0446943
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28.7085	0.991127	0.495564	−	0.868572i	$-0.334962\pi$
$839$	28.7085	0.991127	0.495564	+	0.868572i	$0.334962\pi$
$840$	0	0
$841$	64.0405	2.20829
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.291503	−0.0100280
$846$	0	0
$847$	−17.8967	−0.614939
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.8745	0.475612
$852$	0	0
$853$	−45.2915	−1.55075	−0.775376	−	0.631500i	$-0.782440\pi$
$853$	−45.2915	−1.55075	−0.775376	+	0.631500i	$0.782440\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.7490	0.811251	0.405625	−	0.914039i	$-0.367054\pi$
$857$	23.7490	0.811251	0.405625	+	0.914039i	$0.367054\pi$
$858$	0	0
$859$	2.83399	0.0966945	0.0483472	−	0.998831i	$-0.484605\pi$
$859$	2.83399	0.0966945	0.0483472	+	0.998831i	$0.484605\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.16601	−0.0396915	−0.0198457	−	0.999803i	$-0.506318\pi$
$863$	−1.16601	−0.0396915	−0.0198457	+	0.999803i	$0.506318\pi$
$864$	0	0
$865$	8.70850	0.296098
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.83399	−0.0961365
$870$	0	0
$871$	14.5830	0.494126
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.64575	−0.0556365
$876$	0	0
$877$	−16.8118	−0.567693	−0.283846	−	0.958870i	$-0.591611\pi$
$877$	−16.8118	−0.567693	−0.283846	+	0.958870i	$0.591611\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.7085	1.30412	0.652061	−	0.758166i	$-0.273905\pi$
$881$	38.7085	1.30412	0.652061	+	0.758166i	$0.273905\pi$
$882$	0	0
$883$	25.6458	0.863048	0.431524	−	0.902101i	$-0.357976\pi$
$883$	25.6458	0.863048	0.431524	+	0.902101i	$0.357976\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.7490	−1.46895	−0.734474	−	0.678637i	$-0.762571\pi$
$887$	−43.7490	−1.46895	−0.734474	+	0.678637i	$0.762571\pi$
$888$	0	0
$889$	−4.25098	−0.142573
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	−7.29150	−0.243728
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.2915	0.643408
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.29150	0.175895
$906$	0	0
$907$	53.8745	1.78887	0.894437	−	0.447194i	$-0.147577\pi$
$907$	53.8745	1.78887	0.894437	+	0.447194i	$0.147577\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.7490	−0.786840	−0.393420	−	0.919359i	$-0.628708\pi$
$911$	−23.7490	−0.786840	−0.393420	+	0.919359i	$0.628708\pi$
$912$	0	0
$913$	5.62352	0.186111
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.5830	−0.415527
$918$	0	0
$919$	18.5830	0.612997	0.306498	−	0.951871i	$-0.400843\pi$
$919$	18.5830	0.612997	0.306498	+	0.951871i	$0.400843\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	−6.93725	−0.228096
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.8745	0.520826	0.260413	−	0.965497i	$-0.416141\pi$
$929$	15.8745	0.520826	0.260413	+	0.965497i	$0.416141\pi$
$930$	0	0
$931$	4.29150	0.140648
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.2915	0.564889	0.282444	−	0.959284i	$-0.408855\pi$
$937$	17.2915	0.564889	0.282444	+	0.959284i	$0.408855\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16.2288	0.529042	0.264521	−	0.964380i	$-0.414786\pi$
$941$	16.2288	0.529042	0.264521	+	0.964380i	$0.414786\pi$
$942$	0	0
$943$	3.29150	0.107186
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.7085	−0.867910	−0.433955	−	0.900935i	$-0.642882\pi$
$947$	−26.7085	−0.867910	−0.433955	+	0.900935i	$0.642882\pi$
$948$	0	0
$949$	−7.29150	−0.236692
$950$	0	0
$951$	0	0
$952$	0	0
$953$	28.7085	0.929959	0.464980	−	0.885321i	$-0.346062\pi$
$953$	28.7085	0.929959	0.464980	+	0.885321i	$0.346062\pi$
$954$	0	0
$955$	10.9373	0.353921
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.959482	0.0309833
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.93725	−0.0945535
$966$	0	0
$967$	51.5203	1.65678	0.828390	−	0.560152i	$-0.189257\pi$
$967$	51.5203	1.65678	0.828390	+	0.560152i	$0.189257\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.7490	0.377044	0.188522	−	0.982069i	$-0.439630\pi$
$971$	11.7490	0.377044	0.188522	+	0.982069i	$0.439630\pi$
$972$	0	0
$973$	−29.4170	−0.943066
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.3320	1.61026	0.805132	−	0.593096i	$-0.202094\pi$
$977$	50.3320	1.61026	0.805132	+	0.593096i	$0.202094\pi$
$978$	0	0
$979$	4.33202	0.138452
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.2915	−1.06183	−0.530917	−	0.847424i	$-0.678152\pi$
$983$	−33.2915	−1.06183	−0.530917	+	0.847424i	$0.678152\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.87451	0.313991
$990$	0	0
$991$	−3.74902	−0.119091	−0.0595457	−	0.998226i	$-0.518965\pi$
$991$	−3.74902	−0.119091	−0.0595457	+	0.998226i	$0.518965\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	23.2915	0.738390
$996$	0	0
$997$	−53.2915	−1.68776	−0.843879	−	0.536533i	$-0.819734\pi$
$997$	−53.2915	−1.68776	−0.843879	+	0.536533i	$0.819734\pi$
$998$	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.1.2		2
3.2	odd	2		2280.2.a.n.1.2	✓	2
12.11	even	2		4560.2.a.bp.1.1		2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.64575 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +1.64575 q^{7} +0.354249 q^{11} +3.64575 q^{13} -1.00000 q^{19} -2.00000 q^{23} +1.00000 q^{25} -9.64575 q^{29} -2.00000 q^{31} -1.64575 q^{35} -6.93725 q^{37} -1.64575 q^{41} -4.93725 q^{43} -6.00000 q^{47} -4.29150 q^{49} -4.00000 q^{53} -0.354249 q^{55} -3.29150 q^{59} +11.2915 q^{61} -3.64575 q^{65} +4.00000 q^{67} +3.29150 q^{71} -2.00000 q^{73} +0.583005 q^{77} -8.00000 q^{79} +15.8745 q^{83} +12.2288 q^{89} +6.00000 q^{91} +1.00000 q^{95} +18.2288 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more