LMFDB - Embedded newform 1008.4.b.i.559.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,4,Mod(559,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.559"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,-56,0,0,0,0,0,-656] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 $ x^8 + 158x^6 + 8461x^4 + 180672*x^2 + 1232100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.5
Root			$3.58293i$ of defining polynomial
Character	$\chi$	$=$	1008.559
Dual form			1008.4.b.i.559.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.33129i q^{5} +(18.4617 - 1.47188i) q^{7} +21.1315i q^{11} +36.7607i q^{13} -46.5942i q^{17} -89.7520 q^{19} +190.308i q^{23} +106.240 q^{25} -196.881 q^{29} +87.5824 q^{31} +(6.37513 + 79.9629i) q^{35} -17.9215 q^{37} +374.897i q^{41} -504.442i q^{43} -399.286 q^{47} +(338.667 - 54.3467i) q^{49} +19.8855 q^{53} -91.5265 q^{55} +74.2866 q^{59} -5.16488i q^{61} -159.221 q^{65} -276.852i q^{67} +474.674i q^{71} +653.765i q^{73} +(31.1029 + 390.122i) q^{77} -586.924i q^{79} -441.028 q^{83} +201.813 q^{85} +862.965i q^{89} +(54.1072 + 678.664i) q^{91} -388.742i q^{95} +890.287i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 56 q^{19} - 656 q^{25} - 240 q^{29} + 320 q^{31} - 600 q^{35} + 392 q^{37} + 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 1824 q^{59} + 816 q^{65} + 2064 q^{77} - 1680 q^{83} + 2568 q^{85}+ \cdots - 864 q^{91}+O(q^{100}) $ 8 * q - 4 * q^7 - 56 * q^19 - 656 * q^25 - 240 * q^29 + 320 * q^31 - 600 * q^35 + 392 * q^37 + 816 * q^47 - 16 * q^49 - 288 * q^53 - 456 * q^55 + 1824 * q^59 + 816 * q^65 + 2064 * q^77 - 1680 * q^83 + 2568 * q^85 - 864 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$			4.33129i			0.387403i	0.981061	+	0.193701i	$0.0620493\pi$
$5$			4.33129i			0.387403i	−0.981061	+	0.193701i	$0.937951\pi$
$6$		0			0
$7$	18.4617	−	1.47188i	0.996837	−	0.0794739i
$7$	18.4617	−	1.47188i	0.996837	−	0.0794739i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			21.1315i			0.579216i	0.957145	+	0.289608i	$0.0935249\pi$
$11$			21.1315i			0.579216i	−0.957145	+	0.289608i	$0.906475\pi$
$12$		0			0
$13$			36.7607i			0.784275i	0.919907	+	0.392137i	$0.128264\pi$
$13$			36.7607i			0.784275i	−0.919907	+	0.392137i	$0.871736\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	46.5942i		−	0.664750i	−0.943147	−	0.332375i	$-0.892150\pi$
$17$		−	46.5942i		−	0.664750i	0.943147	−	0.332375i	$-0.107850\pi$
$18$		0			0
$19$	−89.7520			−1.08371			−0.541856	−	0.840471i	$-0.682278\pi$
$19$	−89.7520			−1.08371			−0.541856	+	0.840471i	$0.682278\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			190.308i			1.72530i	0.505800	+	0.862651i	$0.331197\pi$
$23$			190.308i			1.72530i	−0.505800	+	0.862651i	$0.668803\pi$
$24$		0			0
$25$	106.240			0.849919
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−196.881			−1.26069			−0.630343	−	0.776317i	$-0.717086\pi$
$29$	−196.881			−1.26069			−0.630343	+	0.776317i	$0.717086\pi$
$30$		0			0
$31$	87.5824			0.507428			0.253714	−	0.967279i	$-0.418348\pi$
$31$	87.5824			0.507428			0.253714	+	0.967279i	$0.418348\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.37513	+	79.9629i	0.0307884	+	0.386177i
$36$		0			0
$37$	−17.9215			−0.0796289			−0.0398145	−	0.999207i	$-0.512677\pi$
$37$	−17.9215			−0.0796289			−0.0398145	+	0.999207i	$0.512677\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			374.897i			1.42803i	0.700132	+	0.714013i	$0.253124\pi$
$41$			374.897i			1.42803i	−0.700132	+	0.714013i	$0.746876\pi$
$42$		0			0
$43$		−	504.442i		−	1.78899i	−0.447075	−	0.894496i	$-0.647534\pi$
$43$		−	504.442i		−	1.78899i	0.447075	−	0.894496i	$-0.352466\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−399.286			−1.23919			−0.619594	−	0.784923i	$-0.712703\pi$
$47$	−399.286			−1.23919			−0.619594	+	0.784923i	$0.712703\pi$
$48$		0			0
$49$	338.667	−	54.3467i	0.987368	−	0.158445i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	19.8855			0.0515374			0.0257687	−	0.999668i	$-0.491797\pi$
$53$	19.8855			0.0515374			0.0257687	+	0.999668i	$0.491797\pi$
$54$		0			0
$55$	−91.5265			−0.224390
$56$		0			0
$57$		0			0
$58$		0			0
$59$	74.2866			0.163920			0.0819601	−	0.996636i	$-0.473882\pi$
$59$	74.2866			0.163920			0.0819601	+	0.996636i	$0.473882\pi$
$60$		0			0
$61$		−	5.16488i		−	0.0108409i	−0.999985	−	0.00542046i	$-0.998275\pi$
$61$		−	5.16488i		−	0.0108409i	0.999985	−	0.00542046i	$-0.00172539\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−159.221			−0.303830
$66$		0			0
$67$		−	276.852i		−	0.504818i	−0.967621	−	0.252409i	$-0.918777\pi$
$67$		−	276.852i		−	0.504818i	0.967621	−	0.252409i	$-0.0812229\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			474.674i			0.793428i	0.917942	+	0.396714i	$0.129850\pi$
$71$			474.674i			0.793428i	−0.917942	+	0.396714i	$0.870150\pi$
$72$		0			0
$73$			653.765i			1.04818i	0.851662	+	0.524092i	$0.175595\pi$
$73$			653.765i			1.04818i	−0.851662	+	0.524092i	$0.824405\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	31.1029	+	390.122i	0.0460326	+	0.577384i
$78$		0			0
$79$		−	586.924i		−	0.835875i	−0.908476	−	0.417937i	$-0.862753\pi$
$79$		−	586.924i		−	0.835875i	0.908476	−	0.417937i	$-0.137247\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−441.028			−0.583242			−0.291621	−	0.956534i	$-0.594195\pi$
$83$	−441.028			−0.583242			−0.291621	+	0.956534i	$0.594195\pi$
$84$		0			0
$85$	201.813			0.257526
$86$		0			0
$87$		0			0
$88$		0			0
$89$			862.965i			1.02780i	0.857850	+	0.513899i	$0.171800\pi$
$89$			862.965i			1.02780i	−0.857850	+	0.513899i	$0.828200\pi$
$90$		0			0
$91$	54.1072	+	678.664i	0.0623294	+	0.781794i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	388.742i		−	0.419833i
$96$		0			0
$97$			890.287i			0.931907i	0.884809	+	0.465953i	$0.154289\pi$
$97$			890.287i			0.931907i	−0.884809	+	0.465953i	$0.845711\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			684.148i			0.674013i	0.941502	+	0.337006i	$0.109414\pi$
$101$			684.148i			0.674013i	−0.941502	+	0.337006i	$0.890586\pi$
$102$		0			0
$103$	−385.671			−0.368944			−0.184472	−	0.982838i	$-0.559058\pi$
$103$	−385.671			−0.368944			−0.184472	+	0.982838i	$0.559058\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	341.995i		−	0.308990i	−0.987994	−	0.154495i	$-0.950625\pi$
$107$		−	341.995i		−	0.308990i	0.987994	−	0.154495i	$-0.0493751\pi$
$108$		0			0
$109$	−1807.72			−1.58852			−0.794259	−	0.607579i	$-0.792141\pi$
$109$	−1807.72			−1.58852			−0.794259	+	0.607579i	$0.792141\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−707.257			−0.588789			−0.294394	−	0.955684i	$-0.595118\pi$
$113$	−707.257			−0.588789			−0.294394	+	0.955684i	$0.595118\pi$
$114$		0			0
$115$	−824.279			−0.668386
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−68.5810	−	860.207i	−0.0528303	−	0.662648i
$120$		0			0
$121$	884.462			0.664509
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1001.57i			0.716664i
$126$		0			0
$127$			1282.50i			0.896092i	0.894010	+	0.448046i	$0.147880\pi$
$127$			1282.50i			0.896092i	−0.894010	+	0.448046i	$0.852120\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2295.10			−1.53071			−0.765357	−	0.643606i	$-0.777438\pi$
$131$	−2295.10			−1.53071			−0.765357	+	0.643606i	$0.777438\pi$
$132$		0			0
$133$	−1656.97	+	132.104i	−1.08028	+	0.0861268i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−375.692			−0.234288			−0.117144	−	0.993115i	$-0.537374\pi$
$137$	−375.692			−0.234288			−0.117144	+	0.993115i	$0.537374\pi$
$138$		0			0
$139$	−1346.64			−0.821728			−0.410864	−	0.911697i	$-0.634773\pi$
$139$	−1346.64			−0.821728			−0.410864	+	0.911697i	$0.634773\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−776.806			−0.454264
$144$		0			0
$145$		−	852.749i		−	0.488393i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2440.27			1.34171			0.670855	−	0.741588i	$-0.265927\pi$
$149$	2440.27			1.34171			0.670855	+	0.741588i	$0.265927\pi$
$150$		0			0
$151$			1896.11i			1.02188i	0.859618	+	0.510938i	$0.170702\pi$
$151$			1896.11i			1.02188i	−0.859618	+	0.510938i	$0.829298\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			379.345i			0.196579i
$156$		0			0
$157$			3198.03i			1.62567i	0.582493	+	0.812836i	$0.302077\pi$
$157$			3198.03i			1.62567i	−0.582493	+	0.812836i	$0.697923\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	280.110	+	3513.40i	0.137116	+	1.71984i
$162$		0			0
$163$			2106.63i			1.01229i	0.862447	+	0.506147i	$0.168931\pi$
$163$			2106.63i			1.01229i	−0.862447	+	0.506147i	$0.831069\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−82.8763			−0.0384022			−0.0192011	−	0.999816i	$-0.506112\pi$
$167$	−82.8763			−0.0384022			−0.0192011	+	0.999816i	$0.506112\pi$
$168$		0			0
$169$	845.654			0.384913
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4383.57i			1.92646i	0.268687	+	0.963228i	$0.413410\pi$
$173$			4383.57i			1.92646i	−0.268687	+	0.963228i	$0.586590\pi$
$174$		0			0
$175$	1961.37	−	156.372i	0.847231	−	0.0675464i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2994.66i		−	1.25045i	−0.780443	−	0.625227i	$-0.785006\pi$
$179$		−	2994.66i		−	1.25045i	0.780443	−	0.625227i	$-0.214994\pi$
$180$		0			0
$181$		−	518.569i		−	0.212956i	−0.994315	−	0.106478i	$-0.966043\pi$
$181$		−	518.569i		−	0.212956i	0.994315	−	0.106478i	$-0.0339573\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	77.6231i		−	0.0308485i
$186$		0			0
$187$	984.603			0.385034
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	2580.23i		−	0.977482i	−0.872429	−	0.488741i	$-0.837456\pi$
$191$		−	2580.23i		−	0.977482i	0.872429	−	0.488741i	$-0.162544\pi$
$192$		0			0
$193$	861.276			0.321223			0.160611	−	0.987018i	$-0.448653\pi$
$193$	861.276			0.321223			0.160611	+	0.987018i	$0.448653\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2206.28			0.797922			0.398961	−	0.916968i	$-0.369371\pi$
$197$	2206.28			0.797922			0.398961	+	0.916968i	$0.369371\pi$
$198$		0			0
$199$	−2855.66			−1.01725			−0.508624	−	0.860989i	$-0.669846\pi$
$199$	−2855.66			−1.01725			−0.508624	+	0.860989i	$0.669846\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3634.75	+	289.785i	−1.25670	+	0.100192i
$204$		0			0
$205$	−1623.79			−0.553221
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1896.59i		−	0.627703i
$210$		0			0
$211$		−	2314.89i		−	0.755279i	−0.925953	−	0.377640i	$-0.876736\pi$
$211$		−	2314.89i		−	0.755279i	0.925953	−	0.377640i	$-0.123264\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2184.89			0.693060
$216$		0			0
$217$	1616.92	−	128.911i	0.505823	−	0.0403273i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1712.83			0.521347
$222$		0			0
$223$	−2776.63			−0.833799			−0.416899	−	0.908953i	$-0.636883\pi$
$223$	−2776.63			−0.833799			−0.416899	+	0.908953i	$0.636883\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2451.68			0.716846			0.358423	−	0.933559i	$-0.383315\pi$
$227$	2451.68			0.716846			0.358423	+	0.933559i	$0.383315\pi$
$228$		0			0
$229$			362.328i			0.104556i	0.998633	+	0.0522780i	$0.0166482\pi$
$229$			362.328i			0.104556i	−0.998633	+	0.0522780i	$0.983352\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6493.96			−1.82589			−0.912947	−	0.408078i	$-0.866199\pi$
$233$	−6493.96			−1.82589			−0.912947	+	0.408078i	$0.866199\pi$
$234$		0			0
$235$		−	1729.42i		−	0.480065i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			272.296i			0.0736962i	0.999321	+	0.0368481i	$0.0117318\pi$
$239$			272.296i			0.0736962i	−0.999321	+	0.0368481i	$0.988268\pi$
$240$		0			0
$241$			5941.82i			1.58816i	0.607814	+	0.794079i	$0.292047\pi$
$241$			5941.82i			1.58816i	−0.607814	+	0.794079i	$0.707953\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	235.391	+	1466.87i	0.0613820	+	0.382509i
$246$		0			0
$247$		−	3299.34i		−	0.849928i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3810.08			−0.958129			−0.479065	−	0.877780i	$-0.659024\pi$
$251$	−3810.08			−0.958129			−0.479065	+	0.877780i	$0.659024\pi$
$252$		0			0
$253$	−4021.48			−0.999322
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3955.09i			0.959967i	0.877278	+	0.479983i	$0.159357\pi$
$257$			3955.09i			0.959967i	−0.877278	+	0.479983i	$0.840643\pi$
$258$		0			0
$259$	−330.860	+	26.3782i	−0.0793771	+	0.00632842i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5506.53i			1.29105i	0.763737	+	0.645527i	$0.223362\pi$
$263$			5506.53i			1.29105i	−0.763737	+	0.645527i	$0.776638\pi$
$264$		0			0
$265$			86.1299i			0.0199657i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	7132.82i		−	1.61671i	−0.588694	−	0.808356i	$-0.700358\pi$
$269$		−	7132.82i		−	1.61671i	0.588694	−	0.808356i	$-0.299642\pi$
$270$		0			0
$271$	5017.04			1.12459			0.562294	−	0.826937i	$-0.309919\pi$
$271$	5017.04			1.12459			0.562294	+	0.826937i	$0.309919\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2245.00i			0.492287i
$276$		0			0
$277$	6117.95			1.32705			0.663524	−	0.748155i	$-0.269060\pi$
$277$	6117.95			1.32705			0.663524	+	0.748155i	$0.269060\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5293.44			−1.12377			−0.561886	−	0.827214i	$-0.689924\pi$
$281$	−5293.44			−1.12377			−0.561886	+	0.827214i	$0.689924\pi$
$282$		0			0
$283$	−402.940			−0.0846371			−0.0423185	−	0.999104i	$-0.513474\pi$
$283$	−402.940			−0.0846371			−0.0423185	+	0.999104i	$0.513474\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	551.803	+	6921.23i	0.113491	+	1.42351i
$288$		0			0
$289$	2741.98			0.558107
$290$		0			0
$291$		0			0
$292$		0			0
$293$			900.733i			0.179595i	0.995960	+	0.0897976i	$0.0286220\pi$
$293$			900.733i			0.179595i	−0.995960	+	0.0897976i	$0.971378\pi$
$294$		0			0
$295$			321.757i			0.0635031i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6995.84			−1.35311
$300$		0			0
$301$	−742.477	−	9312.85i	−0.142178	−	1.78333i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	22.3706			0.00419980
$306$		0			0
$307$	−2809.28			−0.522262			−0.261131	−	0.965303i	$-0.584095\pi$
$307$	−2809.28			−0.522262			−0.261131	+	0.965303i	$0.584095\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8837.72			1.61139			0.805693	−	0.592333i	$-0.201793\pi$
$311$	8837.72			1.61139			0.805693	+	0.592333i	$0.201793\pi$
$312$		0			0
$313$		−	7844.03i		−	1.41652i	−0.705952	−	0.708260i	$-0.749481\pi$
$313$		−	7844.03i		−	1.41652i	0.705952	−	0.708260i	$-0.250519\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5733.24			1.01581			0.507904	−	0.861414i	$-0.330421\pi$
$317$	5733.24			1.01581			0.507904	+	0.861414i	$0.330421\pi$
$318$		0			0
$319$		−	4160.38i		−	0.730209i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4181.92i			0.720398i
$324$		0			0
$325$			3905.45i			0.666570i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7371.49	+	587.700i	−1.23527	+	0.0984831i
$330$		0			0
$331$		−	6075.19i		−	1.00883i	−0.863461	−	0.504415i	$-0.831708\pi$
$331$		−	6075.19i		−	1.00883i	0.863461	−	0.504415i	$-0.168292\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1199.13			0.195568
$336$		0			0
$337$	−1154.68			−0.186646			−0.0933230	−	0.995636i	$-0.529749\pi$
$337$	−1154.68			−0.186646			−0.0933230	+	0.995636i	$0.529749\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1850.74i			0.293910i
$342$		0			0
$343$	6172.37	−	1501.81i	0.971652	−	0.236414i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			3063.66i			0.473965i	0.971514	+	0.236982i	$0.0761583\pi$
$347$			3063.66i			0.473965i	−0.971514	+	0.236982i	$0.923842\pi$
$348$		0			0
$349$			7903.85i			1.21227i	0.795361	+	0.606136i	$0.207281\pi$
$349$			7903.85i			1.21227i	−0.795361	+	0.606136i	$0.792719\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	10440.0i		−	1.57411i	−0.616880	−	0.787057i	$-0.711603\pi$
$353$		−	10440.0i		−	1.57411i	0.616880	−	0.787057i	$-0.288397\pi$
$354$		0			0
$355$	−2055.95			−0.307376
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	7891.79i		−	1.16020i	−0.814544	−	0.580101i	$-0.803013\pi$
$359$		−	7891.79i		−	1.16020i	0.814544	−	0.580101i	$-0.196987\pi$
$360$		0			0
$361$	1196.42			0.174431
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2831.65			−0.406069
$366$		0			0
$367$	69.3175			0.00985924			0.00492962	−	0.999988i	$-0.498431\pi$
$367$	69.3175			0.00985924			0.00492962	+	0.999988i	$0.498431\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	367.120	−	29.2690i	0.0513744	−	0.00409588i
$372$		0			0
$373$	−1221.71			−0.169591			−0.0847956	−	0.996398i	$-0.527024\pi$
$373$	−1221.71			−0.169591			−0.0847956	+	0.996398i	$0.527024\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7237.47i		−	0.988724i
$378$		0			0
$379$			9565.80i			1.29647i	0.761440	+	0.648235i	$0.224493\pi$
$379$			9565.80i			1.29647i	−0.761440	+	0.648235i	$0.775507\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5413.33			0.722215			0.361107	−	0.932524i	$-0.382399\pi$
$383$	5413.33			0.722215			0.361107	+	0.932524i	$0.382399\pi$
$384$		0			0
$385$	−1689.73	+	134.716i	−0.223680	+	0.0178331i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5222.11			−0.680647			−0.340323	−	0.940308i	$-0.610537\pi$
$389$	−5222.11			−0.680647			−0.340323	+	0.940308i	$0.610537\pi$
$390$		0			0
$391$	8867.24			1.14689
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2542.14			0.323820
$396$		0			0
$397$		−	13550.8i		−	1.71308i	−0.516079	−	0.856541i	$-0.672609\pi$
$397$		−	13550.8i		−	1.71308i	0.516079	−	0.856541i	$-0.327391\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14908.0			1.85654			0.928268	−	0.371912i	$-0.121298\pi$
$401$	14908.0			1.85654			0.928268	+	0.371912i	$0.121298\pi$
$402$		0			0
$403$			3219.59i			0.397963i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	378.707i		−	0.0461223i
$408$		0			0
$409$			5137.85i			0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$409$			5137.85i			0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1371.46	−	109.341i	0.163402	−	0.0130274i
$414$		0			0
$415$		−	1910.22i		−	0.225949i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11861.5			1.38299			0.691494	−	0.722382i	$-0.256953\pi$
$419$	11861.5			1.38299			0.691494	+	0.722382i	$0.256953\pi$
$420$		0			0
$421$	5965.71			0.690619			0.345310	−	0.938489i	$-0.387774\pi$
$421$	5965.71			0.690619			0.345310	+	0.938489i	$0.387774\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	4950.16i		−	0.564984i
$426$		0			0
$427$	−7.60208	−	95.3524i	−0.000861570	−	0.0108066i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	9360.31i		−	1.04610i	−0.852301	−	0.523051i	$-0.824794\pi$
$431$		−	9360.31i		−	1.04610i	0.852301	−	0.523051i	$-0.175206\pi$
$432$		0			0
$433$		−	9022.58i		−	1.00138i	−0.865627	−	0.500690i	$-0.833080\pi$
$433$		−	9022.58i		−	1.00138i	0.865627	−	0.500690i	$-0.166920\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	17080.5i		−	1.86973i
$438$		0			0
$439$	2551.97			0.277446			0.138723	−	0.990331i	$-0.455700\pi$
$439$	2551.97			0.277446			0.138723	+	0.990331i	$0.455700\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	14077.7i		−	1.50983i	−0.655824	−	0.754914i	$-0.727679\pi$
$443$		−	14077.7i		−	1.50983i	0.655824	−	0.754914i	$-0.272321\pi$
$444$		0			0
$445$	−3737.75			−0.398172
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15244.7			1.60232			0.801162	−	0.598448i	$-0.204216\pi$
$449$	15244.7			1.60232			0.801162	+	0.598448i	$0.204216\pi$
$450$		0			0
$451$	−7922.12			−0.827135
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2939.49	+	234.354i	−0.302869	+	0.0241466i
$456$		0			0
$457$	2533.83			0.259360			0.129680	−	0.991556i	$-0.458605\pi$
$457$	2533.83			0.259360			0.129680	+	0.991556i	$0.458605\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	9315.46i		−	0.941137i	−0.882363	−	0.470569i	$-0.844049\pi$
$461$		−	9315.46i		−	0.941137i	0.882363	−	0.470569i	$-0.155951\pi$
$462$		0			0
$463$			15998.0i			1.60581i	0.596104	+	0.802907i	$0.296714\pi$
$463$			15998.0i			1.60581i	−0.596104	+	0.802907i	$0.703286\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13869.3			−1.37430			−0.687148	−	0.726518i	$-0.741138\pi$
$467$	−13869.3			−1.37430			−0.687148	+	0.726518i	$0.741138\pi$
$468$		0			0
$469$	−407.492	−	5111.15i	−0.0401199	−	0.503221i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10659.6			1.03621
$474$		0			0
$475$	−9535.25			−0.921068
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16338.3			1.55849			0.779245	−	0.626719i	$-0.215603\pi$
$479$	16338.3			1.55849			0.779245	+	0.626719i	$0.215603\pi$
$480$		0			0
$481$		−	658.805i		−	0.0624510i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3856.09			−0.361023
$486$		0			0
$487$		−	5149.77i		−	0.479175i	−0.970875	−	0.239588i	$-0.922988\pi$
$487$		−	5149.77i		−	0.479175i	0.970875	−	0.239588i	$-0.0770122\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	1425.88i		−	0.131057i	−0.997851	−	0.0655287i	$-0.979127\pi$
$491$		−	1425.88i		−	0.131057i	0.997851	−	0.0655287i	$-0.0208734\pi$
$492$		0			0
$493$			9173.51i			0.838041i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	698.662	+	8763.28i	0.0630569	+	0.790919i
$498$		0			0
$499$			9498.21i			0.852101i	0.904699	+	0.426050i	$0.140095\pi$
$499$			9498.21i			0.852101i	−0.904699	+	0.426050i	$0.859905\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16057.1			1.42336			0.711680	−	0.702503i	$-0.247934\pi$
$503$	16057.1			1.42336			0.711680	+	0.702503i	$0.247934\pi$
$504$		0			0
$505$	−2963.25			−0.261114
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	3313.42i		−	0.288536i	−0.989539	−	0.144268i	$-0.953917\pi$
$509$		−	3313.42i		−	0.288536i	0.989539	−	0.144268i	$-0.0460828\pi$
$510$		0			0
$511$	962.262	+	12069.6i	0.0833032	+	1.04487i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	1670.45i		−	0.142930i
$516$		0			0
$517$		−	8437.49i		−	0.717757i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	7862.04i		−	0.661118i	−0.943785	−	0.330559i	$-0.892763\pi$
$521$		−	7862.04i		−	0.661118i	0.943785	−	0.330559i	$-0.107237\pi$
$522$		0			0
$523$	17648.7			1.47557			0.737784	−	0.675037i	$-0.235873\pi$
$523$	17648.7			1.47557			0.737784	+	0.675037i	$0.235873\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	4080.83i		−	0.337313i
$528$		0			0
$529$	−24050.1			−1.97666
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13781.5			−1.11997
$534$		0			0
$535$	1481.28			0.119704
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1148.42	+	7156.53i	0.0917739	+	0.571899i
$540$		0			0
$541$	744.076			0.0591319			0.0295659	−	0.999563i	$-0.490587\pi$
$541$	744.076			0.0591319			0.0295659	+	0.999563i	$0.490587\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	7829.78i		−	0.615396i
$546$		0			0
$547$		−	9139.33i		−	0.714387i	−0.934031	−	0.357193i	$-0.883734\pi$
$547$		−	9139.33i		−	0.714387i	0.934031	−	0.357193i	$-0.116266\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17670.5			1.36622
$552$		0			0
$553$	−863.880	−	10835.6i	−0.0664303	−	0.833231i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6925.09			−0.526796			−0.263398	−	0.964687i	$-0.584843\pi$
$557$	−6925.09			−0.526796			−0.263398	+	0.964687i	$0.584843\pi$
$558$		0			0
$559$	18543.6			1.40306
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22776.2			−1.70497			−0.852487	−	0.522748i	$-0.824907\pi$
$563$	−22776.2			−1.70497			−0.852487	+	0.522748i	$0.824907\pi$
$564$		0			0
$565$		−	3063.34i		−	0.228098i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9189.15			0.677028			0.338514	−	0.940961i	$-0.390076\pi$
$569$	9189.15			0.677028			0.338514	+	0.940961i	$0.390076\pi$
$570$		0			0
$571$			7124.04i			0.522122i	0.965322	+	0.261061i	$0.0840724\pi$
$571$			7124.04i			0.522122i	−0.965322	+	0.261061i	$0.915928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			20218.3i			1.46637i
$576$		0			0
$577$		−	20110.0i		−	1.45094i	−0.688255	−	0.725469i	$-0.741623\pi$
$577$		−	20110.0i		−	1.45094i	0.688255	−	0.725469i	$-0.258377\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8142.11	+	649.139i	−0.581397	+	0.0463525i
$582$		0			0
$583$			420.210i			0.0298513i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11857.5			−0.833753			−0.416876	−	0.908963i	$-0.636875\pi$
$587$	−11857.5			−0.833753			−0.416876	+	0.908963i	$0.636875\pi$
$588$		0			0
$589$	−7860.70			−0.549906
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	195.777i		−	0.0135575i	−0.999977	−	0.00677876i	$-0.997842\pi$
$593$		−	195.777i		−	0.0135575i	0.999977	−	0.00677876i	$-0.00215776\pi$
$594$		0			0
$595$	3725.81	−	297.044i	0.256711	−	0.0204666i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	12375.6i		−	0.844161i	−0.906558	−	0.422081i	$-0.861300\pi$
$599$		−	12375.6i		−	0.844161i	0.906558	−	0.422081i	$-0.138700\pi$
$600$		0			0
$601$			20034.3i			1.35976i	0.733322	+	0.679882i	$0.237969\pi$
$601$			20034.3i			1.35976i	−0.733322	+	0.679882i	$0.762031\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			3830.86i			0.257433i
$606$		0			0
$607$	−14743.8			−0.985883			−0.492941	−	0.870062i	$-0.664078\pi$
$607$	−14743.8			−0.985883			−0.492941	+	0.870062i	$0.664078\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	14678.0i		−	0.971864i
$612$		0			0
$613$	25967.4			1.71095			0.855476	−	0.517843i	$-0.173265\pi$
$613$	25967.4			1.71095			0.855476	+	0.517843i	$0.173265\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23573.3			−1.53813			−0.769064	−	0.639172i	$-0.779277\pi$
$617$	−23573.3			−1.53813			−0.769064	+	0.639172i	$0.779277\pi$
$618$		0			0
$619$	−19905.1			−1.29250			−0.646248	−	0.763127i	$-0.723663\pi$
$619$	−19905.1			−1.29250			−0.646248	+	0.763127i	$0.723663\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1270.18	+	15931.8i	0.0816832	+	1.02455i
$624$		0			0
$625$	8941.90			0.572282
$626$		0			0
$627$		0			0
$628$		0			0
$629$			835.036i			0.0529333i
$630$		0			0
$631$		−	17257.1i		−	1.08874i	−0.838845	−	0.544370i	$-0.816769\pi$
$631$		−	17257.1i		−	1.08874i	0.838845	−	0.544370i	$-0.183231\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5554.89			−0.347148
$636$		0			0
$637$	1997.82	+	12449.6i	0.124265	+	0.774368i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22540.0			1.38889			0.694443	−	0.719547i	$-0.255651\pi$
$641$	22540.0			1.38889			0.694443	+	0.719547i	$0.255651\pi$
$642$		0			0
$643$	2084.87			0.127868			0.0639341	−	0.997954i	$-0.479635\pi$
$643$	2084.87			0.127868			0.0639341	+	0.997954i	$0.479635\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19323.6			1.17417			0.587087	−	0.809524i	$-0.300275\pi$
$647$	19323.6			1.17417			0.587087	+	0.809524i	$0.300275\pi$
$648$		0			0
$649$			1569.78i			0.0949452i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10267.1			0.615287			0.307643	−	0.951502i	$-0.400460\pi$
$653$	10267.1			0.615287			0.307643	+	0.951502i	$0.400460\pi$
$654$		0			0
$655$		−	9940.74i		−	0.593003i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16592.3i			0.980797i	0.871498	+	0.490398i	$0.163149\pi$
$659$			16592.3i			0.980797i	−0.871498	+	0.490398i	$0.836851\pi$
$660$		0			0
$661$			6908.93i			0.406545i	0.979122	+	0.203272i	$0.0651577\pi$
$661$			6908.93i			0.406545i	−0.979122	+	0.203272i	$0.934842\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−572.181	−	7176.84i	−0.0333658	−	0.418505i
$666$		0			0
$667$		−	37468.0i		−	2.17506i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	109.142			0.00627923
$672$		0			0
$673$	−26133.0			−1.49681			−0.748404	−	0.663243i	$-0.769180\pi$
$673$	−26133.0			−1.49681			−0.748404	+	0.663243i	$0.769180\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	5448.04i		−	0.309284i	−0.987971	−	0.154642i	$-0.950578\pi$
$677$		−	5448.04i		−	0.309284i	0.987971	−	0.154642i	$-0.0494224\pi$
$678$		0			0
$679$	1310.39	+	16436.2i	0.0740623	+	0.928959i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	3401.50i		−	0.190564i	−0.995450	−	0.0952818i	$-0.969625\pi$
$683$		−	3401.50i		−	0.190564i	0.995450	−	0.0952818i	$-0.0303752\pi$
$684$		0			0
$685$		−	1627.23i		−	0.0907639i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			731.004i			0.0404195i
$690$		0			0
$691$	2782.64			0.153193			0.0765967	−	0.997062i	$-0.475595\pi$
$691$	2782.64			0.153193			0.0765967	+	0.997062i	$0.475595\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	5832.67i		−	0.318339i
$696$		0			0
$697$	17468.0			0.949281
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7163.73			−0.385978			−0.192989	−	0.981201i	$-0.561818\pi$
$701$	−7163.73			−0.385978			−0.192989	+	0.981201i	$0.561818\pi$
$702$		0			0
$703$	1608.49			0.0862948
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1006.98	+	12630.5i	0.0535664	+	0.671881i
$708$		0			0
$709$	30301.5			1.60507			0.802537	−	0.596602i	$-0.203483\pi$
$709$	30301.5			1.60507			0.802537	+	0.596602i	$0.203483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			16667.6i			0.875466i
$714$		0			0
$715$		−	3364.58i		−	0.175983i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4035.13			0.209298			0.104649	−	0.994509i	$-0.466628\pi$
$719$	4035.13			0.209298			0.104649	+	0.994509i	$0.466628\pi$
$720$		0			0
$721$	−7120.13	+	567.660i	−0.367777	+	0.0293215i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−20916.6			−1.07148
$726$		0			0
$727$	390.945			0.0199441			0.00997205	−	0.999950i	$-0.496826\pi$
$727$	390.945			0.0199441			0.00997205	+	0.999950i	$0.496826\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−23504.1			−1.18923
$732$		0			0
$733$			7225.92i			0.364114i	0.983288	+	0.182057i	$0.0582755\pi$
$733$			7225.92i			0.364114i	−0.983288	+	0.182057i	$0.941724\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5850.28			0.292399
$738$		0			0
$739$		−	3342.36i		−	0.166374i	−0.996534	−	0.0831872i	$-0.973490\pi$
$739$		−	3342.36i		−	0.166374i	0.996534	−	0.0831872i	$-0.0265100\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			33993.8i			1.67848i	0.543761	+	0.839240i	$0.317000\pi$
$743$			33993.8i			1.67848i	−0.543761	+	0.839240i	$0.683000\pi$
$744$		0			0
$745$			10569.5i			0.519782i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−503.375	−	6313.81i	−0.0245567	−	0.308013i
$750$		0			0
$751$		−	30957.8i		−	1.50422i	−0.659039	−	0.752109i	$-0.729037\pi$
$751$		−	30957.8i		−	1.50422i	0.659039	−	0.752109i	$-0.270963\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8212.61			−0.395877
$756$		0			0
$757$	25919.3			1.24446			0.622228	−	0.782836i	$-0.286228\pi$
$757$	25919.3			1.24446			0.622228	+	0.782836i	$0.286228\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			29091.1i			1.38575i	0.721059	+	0.692873i	$0.243656\pi$
$761$			29091.1i			1.38575i	−0.721059	+	0.692873i	$0.756344\pi$
$762$		0			0
$763$	−33373.6	+	2660.75i	−1.58349	+	0.126246i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			2730.83i			0.128559i
$768$		0			0
$769$		−	11075.2i		−	0.519352i	−0.965696	−	0.259676i	$-0.916384\pi$
$769$		−	11075.2i		−	0.519352i	0.965696	−	0.259676i	$-0.0836157\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			36140.2i			1.68160i	0.541350	+	0.840798i	$0.317914\pi$
$773$			36140.2i			1.68160i	−0.541350	+	0.840798i	$0.682086\pi$
$774$		0			0
$775$	9304.75			0.431273
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	33647.8i		−	1.54757i
$780$		0			0
$781$	−10030.5			−0.459566
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13851.6			−0.629789
$786$		0			0
$787$	28453.9			1.28878			0.644392	−	0.764695i	$-0.277110\pi$
$787$	28453.9			1.28878			0.644392	+	0.764695i	$0.277110\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−13057.1	+	1041.00i	−0.586926	+	0.0467933i
$792$		0			0
$793$	189.865			0.00850226
$794$		0			0
$795$		0			0
$796$		0			0
$797$			22987.4i			1.02165i	0.859684	+	0.510826i	$0.170660\pi$
$797$			22987.4i			1.02165i	−0.859684	+	0.510826i	$0.829340\pi$
$798$		0			0
$799$			18604.4i			0.823750i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13815.0			−0.607124
$804$		0			0
$805$	−15217.6	+	1213.24i	−0.666272	+	0.0531193i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4688.53			−0.203758			−0.101879	−	0.994797i	$-0.532485\pi$
$809$	−4688.53			−0.203758			−0.101879	+	0.994797i	$0.532485\pi$
$810$		0			0
$811$	−26381.0			−1.14225			−0.571124	−	0.820864i	$-0.693492\pi$
$811$	−26381.0			−1.14225			−0.571124	+	0.820864i	$0.693492\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9124.43			−0.392166
$816$		0			0
$817$			45274.7i			1.93875i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1244.52			0.0529039			0.0264520	−	0.999650i	$-0.491579\pi$
$821$	1244.52			0.0529039			0.0264520	+	0.999650i	$0.491579\pi$
$822$		0			0
$823$			7926.32i			0.335716i	0.985811	+	0.167858i	$0.0536850\pi$
$823$			7926.32i			0.335716i	−0.985811	+	0.167858i	$0.946315\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			32063.4i			1.34819i	0.738644	+	0.674095i	$0.235466\pi$
$827$			32063.4i			1.34819i	−0.738644	+	0.674095i	$0.764534\pi$
$828$		0			0
$829$		−	41224.4i		−	1.72712i	−0.504246	−	0.863560i	$-0.668230\pi$
$829$		−	41224.4i		−	1.72712i	0.504246	−	0.863560i	$-0.331770\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2532.24	−	15779.9i	−0.105326	−	0.656353i
$834$		0			0
$835$		−	358.962i		−	0.0148771i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25040.7			−1.03039			−0.515197	−	0.857072i	$-0.672281\pi$
$839$	−25040.7			−1.03039			−0.515197	+	0.857072i	$0.672281\pi$
$840$		0			0
$841$	14373.1			0.589327
$842$		0			0
$843$		0			0
$844$		0			0
$845$			3662.77i			0.149116i
$846$		0			0
$847$	16328.6	−	1301.82i	0.662407	−	0.0528112i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	3410.59i		−	0.137384i
$852$		0			0
$853$			35611.4i			1.42944i	0.699411	+	0.714720i	$0.253446\pi$
$853$			35611.4i			1.42944i	−0.699411	+	0.714720i	$0.746554\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			20425.0i			0.814125i	0.913400	+	0.407063i	$0.133447\pi$
$857$			20425.0i			0.814125i	−0.913400	+	0.407063i	$0.866553\pi$
$858$		0			0
$859$	−7823.05			−0.310732			−0.155366	−	0.987857i	$-0.549656\pi$
$859$	−7823.05			−0.310732			−0.155366	+	0.987857i	$0.549656\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	41712.4i		−	1.64532i	−0.568536	−	0.822658i	$-0.692490\pi$
$863$		−	41712.4i		−	1.64532i	0.568536	−	0.822658i	$-0.307510\pi$
$864$		0			0
$865$	−18986.5			−0.746314
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12402.6			0.484152
$870$		0			0
$871$	10177.3			0.395916
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1474.19	+	18490.6i	0.0569561	+	0.714397i
$876$		0			0
$877$	−10968.2			−0.422316			−0.211158	−	0.977452i	$-0.567723\pi$
$877$	−10968.2			−0.422316			−0.211158	+	0.977452i	$0.567723\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			29482.5i			1.12746i	0.825960	+	0.563729i	$0.190634\pi$
$881$			29482.5i			1.12746i	−0.825960	+	0.563729i	$0.809366\pi$
$882$		0			0
$883$			39081.9i			1.48948i	0.667356	+	0.744739i	$0.267426\pi$
$883$			39081.9i			1.48948i	−0.667356	+	0.744739i	$0.732574\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	44545.5			1.68624			0.843118	−	0.537728i	$-0.180717\pi$
$887$	44545.5			1.68624			0.843118	+	0.537728i	$0.180717\pi$
$888$		0			0
$889$	1887.69	+	23677.1i	0.0712160	+	0.893258i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	35836.7			1.34292
$894$		0			0
$895$	12970.7			0.484429
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17243.3			−0.639707
$900$		0			0
$901$		−	926.549i		−	0.0342595i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2246.08			0.0824995
$906$		0			0
$907$		−	12446.7i		−	0.455662i	−0.973701	−	0.227831i	$-0.926837\pi$
$907$		−	12446.7i		−	0.455662i	0.973701	−	0.227831i	$-0.0731634\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	5652.41i		−	0.205568i	−0.994704	−	0.102784i	$-0.967225\pi$
$911$		−	5652.41i		−	0.205568i	0.994704	−	0.102784i	$-0.0327751\pi$
$912$		0			0
$913$		−	9319.56i		−	0.337823i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−42371.3	+	3378.10i	−1.52587	+	0.121652i
$918$		0			0
$919$			27140.5i			0.974191i	0.873349	+	0.487096i	$0.161944\pi$
$919$			27140.5i			0.974191i	−0.873349	+	0.487096i	$0.838056\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−17449.3			−0.622266
$924$		0			0
$925$	−1903.97			−0.0676782
$926$		0			0
$927$		0			0
$928$		0			0
$929$			3257.14i			0.115031i	0.998345	+	0.0575153i	$0.0183178\pi$
$929$			3257.14i			0.115031i	−0.998345	+	0.0575153i	$0.981682\pi$
$930$		0			0
$931$	−30396.1	+	4877.72i	−1.07002	+	0.171709i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			4264.61i			0.149163i
$936$		0			0
$937$		−	20532.1i		−	0.715852i	−0.933750	−	0.357926i	$-0.883484\pi$
$937$		−	20532.1i		−	0.715852i	0.933750	−	0.357926i	$-0.116516\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			39190.7i			1.35768i	0.734284	+	0.678842i	$0.237518\pi$
$941$			39190.7i			1.35768i	−0.734284	+	0.678842i	$0.762482\pi$
$942$		0			0
$943$	−71345.8			−2.46378
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20071.9i		−	0.688754i	−0.938831	−	0.344377i	$-0.888090\pi$
$947$		−	20071.9i		−	0.688754i	0.938831	−	0.344377i	$-0.111910\pi$
$948$		0			0
$949$	−24032.8			−0.822064
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21364.2			−0.726186			−0.363093	−	0.931753i	$-0.618279\pi$
$953$	−21364.2			−0.726186			−0.363093	+	0.931753i	$0.618279\pi$
$954$		0			0
$955$	11175.7			0.378679
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6935.90	+	552.972i	−0.233547	+	0.0186198i
$960$		0			0
$961$	−22120.3			−0.742517
$962$		0			0
$963$		0			0
$964$		0			0
$965$			3730.44i			0.124443i
$966$		0			0
$967$		−	437.986i		−	0.0145654i	−0.999973	−	0.00728268i	$-0.997682\pi$
$967$		−	437.986i		−	0.0145654i	0.999973	−	0.00728268i	$-0.00231817\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−43517.0			−1.43824			−0.719118	−	0.694888i	$-0.755454\pi$
$971$	−43517.0			−1.43824			−0.719118	+	0.694888i	$0.755454\pi$
$972$		0			0
$973$	−24861.2	+	1982.08i	−0.819129	+	0.0653059i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3333.85			−0.109170			−0.0545851	−	0.998509i	$-0.517384\pi$
$977$	−3333.85			−0.109170			−0.0545851	+	0.998509i	$0.517384\pi$
$978$		0			0
$979$	−18235.7			−0.595317
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−10148.5			−0.329284			−0.164642	−	0.986353i	$-0.552647\pi$
$983$	−10148.5			−0.329284			−0.164642	+	0.986353i	$0.552647\pi$
$984$		0			0
$985$			9556.03i			0.309117i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	95999.3			3.08655
$990$		0			0
$991$			38849.6i			1.24531i	0.782498	+	0.622653i	$0.213945\pi$
$991$			38849.6i			1.24531i	−0.782498	+	0.622653i	$0.786055\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	12368.7i		−	0.394085i
$996$		0			0
$997$			32853.3i			1.04360i	0.853066	+	0.521802i	$0.174740\pi$
$997$			32853.3i			1.04360i	−0.853066	+	0.521802i	$0.825260\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.b.i.559.5		8
3.2	odd	2		336.4.b.e.223.4	✓	8
4.3	odd	2		1008.4.b.k.559.5		8
7.6	odd	2		1008.4.b.k.559.4		8
12.11	even	2		336.4.b.f.223.4	yes	8
21.20	even	2		336.4.b.f.223.5	yes	8
24.5	odd	2		1344.4.b.f.895.5		8
24.11	even	2		1344.4.b.e.895.5		8
28.27	even	2	inner	1008.4.b.i.559.4		8
84.83	odd	2		336.4.b.e.223.5	yes	8
168.83	odd	2		1344.4.b.f.895.4		8
168.125	even	2		1344.4.b.e.895.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.b.e.223.4	✓	8	3.2	odd	2
336.4.b.e.223.5	yes	8	84.83	odd	2
336.4.b.f.223.4	yes	8	12.11	even	2
336.4.b.f.223.5	yes	8	21.20	even	2
1008.4.b.i.559.4		8	28.27	even	2	inner
1008.4.b.i.559.5		8	1.1	even	1	trivial
1008.4.b.k.559.4		8	7.6	odd	2
1008.4.b.k.559.5		8	4.3	odd	2
1344.4.b.e.895.4		8	168.125	even	2
1344.4.b.e.895.5		8	24.11	even	2
1344.4.b.f.895.4		8	168.83	odd	2
1344.4.b.f.895.5		8	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.33129i q^{5} +(18.4617 - 1.47188i) q^{7} +21.1315i q^{11} +36.7607i q^{13} -46.5942i q^{17} -89.7520 q^{19} +190.308i q^{23} +106.240 q^{25} -196.881 q^{29} +87.5824 q^{31} +(6.37513 + 79.9629i) q^{35} -17.9215 q^{37} +374.897i q^{41} -504.442i q^{43} -399.286 q^{47} +(338.667 - 54.3467i) q^{49} +19.8855 q^{53} -91.5265 q^{55} +74.2866 q^{59} -5.16488i q^{61} -159.221 q^{65} -276.852i q^{67} +474.674i q^{71} +653.765i q^{73} +(31.1029 + 390.122i) q^{77} -586.924i q^{79} -441.028 q^{83} +201.813 q^{85} +862.965i q^{89} +(54.1072 + 678.664i) q^{91} -388.742i q^{95} +890.287i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 56 q^{19} - 656 q^{25} - 240 q^{29} + 320 q^{31} - 600 q^{35} + 392 q^{37} + 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 1824 q^{59} + 816 q^{65} + 2064 q^{77} - 1680 q^{83} + 2568 q^{85}+ \cdots - 864 q^{91}+O(q^{100}) \) 8 * q - 4 * q^7 - 56 * q^19 - 656 * q^25 - 240 * q^29 + 320 * q^31 - 600 * q^35 + 392 * q^37 + 816 * q^47 - 16 * q^49 - 288 * q^53 - 456 * q^55 + 1824 * q^59 + 816 * q^65 + 2064 * q^77 - 1680 * q^83 + 2568 * q^85 - 864 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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