LMFDB - Embedded newform 2016.3.e.c.1007.24

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,3,Mod(1007,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.1007"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-336] 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:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1007.24
Character	$\chi$	$=$	2016.1007
Dual form			2016.3.e.c.1007.23

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.10939i q^{5} +(-0.958628 - 6.93405i) q^{7} -9.66169i q^{11} +2.31922 q^{13} +21.8883 q^{17} +26.9198i q^{19} -15.4172 q^{23} -57.9809 q^{25} -18.1366 q^{29} -45.9762 q^{31} +(63.1649 - 8.73251i) q^{35} +32.3318i q^{37} +33.0463 q^{41} +60.5091 q^{43} +29.4932i q^{47} +(-47.1621 + 13.2943i) q^{49} +65.7621 q^{53} +88.0121 q^{55} +8.47268 q^{59} -74.9438 q^{61} +21.1266i q^{65} -20.2120 q^{67} -21.7110 q^{71} +100.202i q^{73} +(-66.9947 + 9.26197i) q^{77} +78.9852i q^{79} -67.6265 q^{83} +199.389i q^{85} -130.927 q^{89} +(-2.22326 - 16.0816i) q^{91} -245.223 q^{95} -175.267i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 336 q^{25} + 384 q^{43} + 368 q^{49} + 896 q^{67} + 576 q^{91}+O(q^{100}) $ 48 * q - 336 * q^25 + 384 * q^43 + 368 * q^49 + 896 * q^67 + 576 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$5$			9.10939i			1.82188i	0.412542	+	0.910939i	$0.364641\pi$
$5$			9.10939i			1.82188i	−0.412542	+	0.910939i	$0.635359\pi$
$6$		0			0
$7$	−0.958628	−	6.93405i	−0.136947	−	0.990578i
$7$	−0.958628	−	6.93405i	−0.136947	−	0.990578i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	9.66169i		−	0.878336i	−0.898405	−	0.439168i	$-0.855273\pi$
$11$		−	9.66169i		−	0.878336i	0.898405	−	0.439168i	$-0.144727\pi$
$12$		0			0
$13$	2.31922			0.178401			0.0892006	−	0.996014i	$-0.471569\pi$
$13$	2.31922			0.178401			0.0892006	+	0.996014i	$0.471569\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	21.8883			1.28755			0.643773	−	0.765217i	$-0.277368\pi$
$17$	21.8883			1.28755			0.643773	+	0.765217i	$0.277368\pi$
$18$		0			0
$19$			26.9198i			1.41683i	0.705795	+	0.708416i	$0.250590\pi$
$19$			26.9198i			1.41683i	−0.705795	+	0.708416i	$0.749410\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−15.4172			−0.670311			−0.335156	−	0.942163i	$-0.608789\pi$
$23$	−15.4172			−0.670311			−0.335156	+	0.942163i	$0.608789\pi$
$24$		0			0
$25$	−57.9809			−2.31924
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−18.1366			−0.625400			−0.312700	−	0.949852i	$-0.601233\pi$
$29$	−18.1366			−0.625400			−0.312700	+	0.949852i	$0.601233\pi$
$30$		0			0
$31$	−45.9762			−1.48310			−0.741552	−	0.670896i	$-0.765910\pi$
$31$	−45.9762			−1.48310			−0.741552	+	0.670896i	$0.765910\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	63.1649	−	8.73251i	1.80471	−	0.249500i
$36$		0			0
$37$			32.3318i			0.873832i	0.899502	+	0.436916i	$0.143929\pi$
$37$			32.3318i			0.873832i	−0.899502	+	0.436916i	$0.856071\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	33.0463			0.806007			0.403004	−	0.915198i	$-0.367966\pi$
$41$	33.0463			0.806007			0.403004	+	0.915198i	$0.367966\pi$
$42$		0			0
$43$	60.5091			1.40719			0.703594	−	0.710602i	$-0.251577\pi$
$43$	60.5091			1.40719			0.703594	+	0.710602i	$0.251577\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			29.4932i			0.627514i	0.949503	+	0.313757i	$0.101588\pi$
$47$			29.4932i			0.627514i	−0.949503	+	0.313757i	$0.898412\pi$
$48$		0			0
$49$	−47.1621	+	13.2943i	−0.962491	+	0.271313i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	65.7621			1.24079			0.620397	−	0.784288i	$-0.286971\pi$
$53$	65.7621			1.24079			0.620397	+	0.784288i	$0.286971\pi$
$54$		0			0
$55$	88.0121			1.60022
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.47268			0.143605			0.0718024	−	0.997419i	$-0.477125\pi$
$59$	8.47268			0.143605			0.0718024	+	0.997419i	$0.477125\pi$
$60$		0			0
$61$	−74.9438			−1.22859			−0.614294	−	0.789077i	$-0.710559\pi$
$61$	−74.9438			−1.22859			−0.614294	+	0.789077i	$0.710559\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			21.1266i			0.325025i
$66$		0			0
$67$	−20.2120			−0.301671			−0.150836	−	0.988559i	$-0.548196\pi$
$67$	−20.2120			−0.301671			−0.150836	+	0.988559i	$0.548196\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−21.7110			−0.305789			−0.152894	−	0.988243i	$-0.548859\pi$
$71$	−21.7110			−0.305789			−0.152894	+	0.988243i	$0.548859\pi$
$72$		0			0
$73$			100.202i			1.37263i	0.727304	+	0.686315i	$0.240773\pi$
$73$			100.202i			1.37263i	−0.727304	+	0.686315i	$0.759227\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−66.9947	+	9.26197i	−0.870061	+	0.120285i
$78$		0			0
$79$			78.9852i			0.999813i	0.866079	+	0.499906i	$0.166632\pi$
$79$			78.9852i			0.999813i	−0.866079	+	0.499906i	$0.833368\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−67.6265			−0.814777			−0.407388	−	0.913255i	$-0.633560\pi$
$83$	−67.6265			−0.814777			−0.407388	+	0.913255i	$0.633560\pi$
$84$		0			0
$85$			199.389i			2.34575i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−130.927			−1.47109			−0.735546	−	0.677474i	$-0.763074\pi$
$89$	−130.927			−1.47109			−0.735546	+	0.677474i	$0.763074\pi$
$90$		0			0
$91$	−2.22326	−	16.0816i	−0.0244315	−	0.176720i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−245.223			−2.58129
$96$		0			0
$97$		−	175.267i		−	1.80688i	−0.428719	−	0.903438i	$-0.641035\pi$
$97$		−	175.267i		−	1.80688i	0.428719	−	0.903438i	$-0.358965\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			71.2685i			0.705629i	0.935693	+	0.352815i	$0.114775\pi$
$101$			71.2685i			0.705629i	−0.935693	+	0.352815i	$0.885225\pi$
$102$		0			0
$103$	−84.3756			−0.819181			−0.409590	−	0.912270i	$-0.634328\pi$
$103$	−84.3756			−0.819181			−0.409590	+	0.912270i	$0.634328\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	80.7290i		−	0.754476i	−0.926116	−	0.377238i	$-0.876874\pi$
$107$		−	80.7290i		−	0.754476i	0.926116	−	0.377238i	$-0.123126\pi$
$108$		0			0
$109$			100.232i			0.919557i	0.888034	+	0.459778i	$0.152071\pi$
$109$			100.232i			0.919557i	−0.888034	+	0.459778i	$0.847929\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	14.6523i		−	0.129666i	−0.997896	−	0.0648330i	$-0.979349\pi$
$113$		−	14.6523i		−	0.129666i	0.997896	−	0.0648330i	$-0.0206515\pi$
$114$		0			0
$115$		−	140.441i		−	1.22122i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−20.9827	−	151.774i	−0.176325	−	1.27541i
$120$		0			0
$121$	27.6517			0.228526
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	300.436i		−	2.40349i
$126$		0			0
$127$		−	22.9919i		−	0.181039i	−0.995895	−	0.0905193i	$-0.971147\pi$
$127$		−	22.9919i		−	0.181039i	0.995895	−	0.0905193i	$-0.0288527\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−155.033			−1.18345			−0.591727	−	0.806138i	$-0.701554\pi$
$131$	−155.033			−1.18345			−0.591727	+	0.806138i	$0.701554\pi$
$132$		0			0
$133$	186.663	−	25.8061i	1.40348	−	0.194031i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			65.5667i			0.478589i	0.970947	+	0.239294i	$0.0769161\pi$
$137$			65.5667i			0.478589i	−0.970947	+	0.239294i	$0.923084\pi$
$138$		0			0
$139$			149.873i			1.07823i	0.842233	+	0.539113i	$0.181240\pi$
$139$			149.873i			1.07823i	−0.842233	+	0.539113i	$0.818760\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	22.4076i		−	0.156696i
$144$		0			0
$145$		−	165.213i		−	1.13940i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−63.7340			−0.427745			−0.213873	−	0.976862i	$-0.568608\pi$
$149$	−63.7340			−0.427745			−0.213873	+	0.976862i	$0.568608\pi$
$150$		0			0
$151$			87.7691i			0.581252i	0.956837	+	0.290626i	$0.0938636\pi$
$151$			87.7691i			0.581252i	−0.956837	+	0.290626i	$0.906136\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	418.815i		−	2.70203i
$156$		0			0
$157$	12.9625			0.0825635			0.0412818	−	0.999148i	$-0.486856\pi$
$157$	12.9625			0.0825635			0.0412818	+	0.999148i	$0.486856\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	14.7793	+	106.903i	0.0917969	+	0.663996i
$162$		0			0
$163$	−129.380			−0.793745			−0.396872	−	0.917874i	$-0.629904\pi$
$163$	−129.380			−0.793745			−0.396872	+	0.917874i	$0.629904\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	189.489i		−	1.13466i	−0.823489	−	0.567332i	$-0.807976\pi$
$167$		−	189.489i		−	1.13466i	0.823489	−	0.567332i	$-0.192024\pi$
$168$		0			0
$169$	−163.621			−0.968173
$170$		0			0
$171$		0			0
$172$		0			0
$173$			309.161i			1.78706i	0.449008	+	0.893528i	$0.351778\pi$
$173$			309.161i			1.78706i	−0.449008	+	0.893528i	$0.648222\pi$
$174$		0			0
$175$	55.5821	+	402.043i	0.317612	+	2.29739i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	109.315i		−	0.610696i	−0.952241	−	0.305348i	$-0.901227\pi$
$179$		−	109.315i		−	0.610696i	0.952241	−	0.305348i	$-0.0987728\pi$
$180$		0			0
$181$	−75.4317			−0.416750			−0.208375	−	0.978049i	$-0.566817\pi$
$181$	−75.4317			−0.416750			−0.208375	+	0.978049i	$0.566817\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−294.523			−1.59201
$186$		0			0
$187$		−	211.478i		−	1.13090i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−204.357			−1.06993			−0.534966	−	0.844874i	$-0.679676\pi$
$191$	−204.357			−1.06993			−0.534966	+	0.844874i	$0.679676\pi$
$192$		0			0
$193$	−114.368			−0.592579			−0.296290	−	0.955098i	$-0.595749\pi$
$193$	−114.368			−0.592579			−0.296290	+	0.955098i	$0.595749\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−227.241			−1.15351			−0.576753	−	0.816919i	$-0.695680\pi$
$197$	−227.241			−1.15351			−0.576753	+	0.816919i	$0.695680\pi$
$198$		0			0
$199$	−113.715			−0.571434			−0.285717	−	0.958314i	$-0.592232\pi$
$199$	−113.715			−0.571434			−0.285717	+	0.958314i	$0.592232\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	17.3863	+	125.760i	0.0856466	+	0.619508i
$204$		0			0
$205$			301.032i			1.46845i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	260.091			1.24445
$210$		0			0
$211$	−389.006			−1.84363			−0.921815	−	0.387630i	$-0.873294\pi$
$211$	−389.006			−1.84363			−0.921815	+	0.387630i	$0.873294\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			551.201i			2.56373i
$216$		0			0
$217$	44.0741	+	318.801i	0.203106	+	1.46913i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	50.7636			0.229700
$222$		0			0
$223$	228.516			1.02474			0.512368	−	0.858766i	$-0.328768\pi$
$223$	228.516			1.02474			0.512368	+	0.858766i	$0.328768\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	182.563			0.804242			0.402121	−	0.915587i	$-0.368273\pi$
$227$	182.563			0.804242			0.402121	+	0.915587i	$0.368273\pi$
$228$		0			0
$229$	−331.984			−1.44971			−0.724857	−	0.688900i	$-0.758094\pi$
$229$	−331.984			−1.44971			−0.724857	+	0.688900i	$0.758094\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	360.147i		−	1.54570i	−0.634592	−	0.772848i	$-0.718832\pi$
$233$		−	360.147i		−	1.54570i	0.634592	−	0.772848i	$-0.281168\pi$
$234$		0			0
$235$	−268.665			−1.14325
$236$		0			0
$237$		0			0
$238$		0			0
$239$	333.265			1.39442			0.697208	−	0.716869i	$-0.254425\pi$
$239$	333.265			1.39442			0.697208	+	0.716869i	$0.254425\pi$
$240$		0			0
$241$		−	331.691i		−	1.37631i	−0.725564	−	0.688155i	$-0.758421\pi$
$241$		−	331.691i		−	1.37631i	0.725564	−	0.688155i	$-0.241579\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−121.103	−	429.618i	−0.494299	−	1.75354i
$246$		0			0
$247$			62.4329i			0.252765i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	350.730			1.39733			0.698666	−	0.715448i	$-0.253777\pi$
$251$	350.730			1.39733			0.698666	+	0.715448i	$0.253777\pi$
$252$		0			0
$253$			148.956i			0.588758i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	361.582			1.40694			0.703468	−	0.710727i	$-0.251634\pi$
$257$	361.582			1.40694			0.703468	+	0.710727i	$0.251634\pi$
$258$		0			0
$259$	224.190	−	30.9941i	0.865599	−	0.119668i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−255.224			−0.970434			−0.485217	−	0.874394i	$-0.661259\pi$
$263$	−255.224			−0.970434			−0.485217	+	0.874394i	$0.661259\pi$
$264$		0			0
$265$			599.053i			2.26058i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			141.207i			0.524934i	0.964941	+	0.262467i	$0.0845360\pi$
$269$			141.207i			0.524934i	−0.964941	+	0.262467i	$0.915464\pi$
$270$		0			0
$271$	−405.606			−1.49670			−0.748350	−	0.663304i	$-0.769154\pi$
$271$	−405.606			−1.49670			−0.748350	+	0.663304i	$0.769154\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			560.194i			2.03707i
$276$		0			0
$277$			31.1395i			0.112417i	0.998419	+	0.0562084i	$0.0179011\pi$
$277$			31.1395i			0.112417i	−0.998419	+	0.0562084i	$0.982099\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			15.2327i			0.0542087i	0.999633	+	0.0271044i	$0.00862864\pi$
$281$			15.2327i			0.0542087i	−0.999633	+	0.0271044i	$0.991371\pi$
$282$		0			0
$283$			222.494i			0.786199i	0.919496	+	0.393100i	$0.128597\pi$
$283$			222.494i			0.786199i	−0.919496	+	0.393100i	$0.871403\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−31.6791	−	229.145i	−0.110380	−	0.798414i
$288$		0			0
$289$	190.097			0.657773
$290$		0			0
$291$		0			0
$292$		0			0
$293$			64.2745i			0.219367i	0.993967	+	0.109683i	$0.0349837\pi$
$293$			64.2745i			0.219367i	−0.993967	+	0.109683i	$0.965016\pi$
$294$		0			0
$295$			77.1809i			0.261630i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−35.7557			−0.119584
$300$		0			0
$301$	−58.0057	−	419.573i	−0.192710	−	1.39393i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	682.692i		−	2.23834i
$306$		0			0
$307$			409.069i			1.33247i	0.745740	+	0.666237i	$0.232096\pi$
$307$			409.069i			1.33247i	−0.745740	+	0.666237i	$0.767904\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			117.035i			0.376318i	0.982139	+	0.188159i	$0.0602520\pi$
$311$			117.035i			0.376318i	−0.982139	+	0.188159i	$0.939748\pi$
$312$		0			0
$313$		−	6.47974i		−	0.0207021i	−0.999946	−	0.0103510i	$-0.996705\pi$
$313$		−	6.47974i		−	0.0207021i	0.999946	−	0.0103510i	$-0.00329489\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−403.391			−1.27253			−0.636263	−	0.771472i	$-0.719521\pi$
$317$	−403.391			−1.27253			−0.636263	+	0.771472i	$0.719521\pi$
$318$		0			0
$319$			175.230i			0.549311i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			589.228i			1.82424i
$324$		0			0
$325$	−134.470			−0.413755
$326$		0			0
$327$		0			0
$328$		0			0
$329$	204.507	−	28.2730i	0.621602	−	0.0859360i
$330$		0			0
$331$	87.5977			0.264646			0.132323	−	0.991207i	$-0.457756\pi$
$331$	87.5977			0.264646			0.132323	+	0.991207i	$0.457756\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	184.119i		−	0.549608i
$336$		0			0
$337$	254.014			0.753752			0.376876	−	0.926264i	$-0.376998\pi$
$337$	254.014			0.753752			0.376876	+	0.926264i	$0.376998\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			444.208i			1.30266i
$342$		0			0
$343$	137.394	+	314.280i	0.400567	+	0.916267i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			206.734i			0.595777i	0.954601	+	0.297888i	$0.0962823\pi$
$347$			206.734i			0.595777i	−0.954601	+	0.297888i	$0.903718\pi$
$348$		0			0
$349$	618.655			1.77265			0.886325	−	0.463063i	$-0.153250\pi$
$349$	618.655			1.77265			0.886325	+	0.463063i	$0.153250\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	357.742			1.01343			0.506717	−	0.862113i	$-0.330859\pi$
$353$	357.742			1.01343			0.506717	+	0.862113i	$0.330859\pi$
$354$		0			0
$355$		−	197.774i		−	0.557110i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	54.1360			0.150797			0.0753983	−	0.997153i	$-0.475977\pi$
$359$	54.1360			0.150797			0.0753983	+	0.997153i	$0.475977\pi$
$360$		0			0
$361$	−363.676			−1.00741
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−912.779			−2.50077
$366$		0			0
$367$	45.1294			0.122968			0.0614842	−	0.998108i	$-0.480417\pi$
$367$	45.1294			0.122968			0.0614842	+	0.998108i	$0.480417\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−63.0414	−	455.998i	−0.169923	−	1.22910i
$372$		0			0
$373$			495.886i			1.32945i	0.747087	+	0.664726i	$0.231452\pi$
$373$			495.886i			1.32945i	−0.747087	+	0.664726i	$0.768548\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−42.0627			−0.111572
$378$		0			0
$379$	427.550			1.12810			0.564050	−	0.825741i	$-0.309243\pi$
$379$	427.550			1.12810			0.564050	+	0.825741i	$0.309243\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			77.1824i			0.201521i	0.994911	+	0.100760i	$0.0321275\pi$
$383$			77.1824i			0.201521i	−0.994911	+	0.100760i	$0.967872\pi$
$384$		0			0
$385$	−84.3708	−	610.280i	−0.219145	−	1.58514i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	510.843			1.31322			0.656610	−	0.754230i	$-0.271990\pi$
$389$	510.843			1.31322			0.656610	+	0.754230i	$0.271990\pi$
$390$		0			0
$391$	−337.455			−0.863056
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−719.507			−1.82154
$396$		0			0
$397$	−14.9294			−0.0376056			−0.0188028	−	0.999823i	$-0.505985\pi$
$397$	−14.9294			−0.0376056			−0.0188028	+	0.999823i	$0.505985\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	12.5890i		−	0.0313940i	−0.999877	−	0.0156970i	$-0.995003\pi$
$401$		−	12.5890i		−	0.0313940i	0.999877	−	0.0156970i	$-0.00499671\pi$
$402$		0			0
$403$	−106.629			−0.264588
$404$		0			0
$405$		0			0
$406$		0			0
$407$	312.380			0.767518
$408$		0			0
$409$			122.775i			0.300183i	0.988672	+	0.150091i	$0.0479568\pi$
$409$			122.775i			0.300183i	−0.988672	+	0.150091i	$0.952043\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.12215	−	58.7500i	−0.0196662	−	0.142252i
$414$		0			0
$415$		−	616.036i		−	1.48442i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	85.9785			0.205199			0.102600	−	0.994723i	$-0.467284\pi$
$419$	85.9785			0.205199			0.102600	+	0.994723i	$0.467284\pi$
$420$		0			0
$421$		−	32.5132i		−	0.0772285i	−0.999254	−	0.0386143i	$-0.987706\pi$
$421$		−	32.5132i		−	0.0772285i	0.999254	−	0.0386143i	$-0.0122944\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1269.10			−2.98612
$426$		0			0
$427$	71.8432	+	519.664i	0.168251	+	1.21701i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	769.964			1.78646			0.893230	−	0.449600i	$-0.148434\pi$
$431$	769.964			1.78646			0.893230	+	0.449600i	$0.148434\pi$
$432$		0			0
$433$		−	245.082i		−	0.566008i	−0.959119	−	0.283004i	$-0.908669\pi$
$433$		−	245.082i		−	0.566008i	0.959119	−	0.283004i	$-0.0913310\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	415.027i		−	0.949718i
$438$		0			0
$439$	69.4826			0.158275			0.0791374	−	0.996864i	$-0.474783\pi$
$439$	69.4826			0.158275			0.0791374	+	0.996864i	$0.474783\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			621.899i			1.40384i	0.712258	+	0.701918i	$0.247673\pi$
$443$			621.899i			1.40384i	−0.712258	+	0.701918i	$0.752327\pi$
$444$		0			0
$445$		−	1192.67i		−	2.68015i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			777.572i			1.73179i	0.500229	+	0.865893i	$0.333249\pi$
$449$			777.572i			1.73179i	−0.500229	+	0.865893i	$0.666751\pi$
$450$		0			0
$451$		−	319.283i		−	0.707945i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	146.493	−	20.2526i	0.321963	−	0.0445112i
$456$		0			0
$457$	746.303			1.63305			0.816524	−	0.577312i	$-0.195898\pi$
$457$	746.303			1.63305			0.816524	+	0.577312i	$0.195898\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	289.301i		−	0.627550i	−0.949497	−	0.313775i	$-0.898406\pi$
$461$		−	289.301i		−	0.627550i	0.949497	−	0.313775i	$-0.101594\pi$
$462$		0			0
$463$		−	317.745i		−	0.686275i	−0.939285	−	0.343137i	$-0.888510\pi$
$463$		−	317.745i		−	0.686275i	0.939285	−	0.343137i	$-0.111490\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−152.473			−0.326495			−0.163248	−	0.986585i	$-0.552197\pi$
$467$	−152.473			−0.326495			−0.163248	+	0.986585i	$0.552197\pi$
$468$		0			0
$469$	19.3758	+	140.151i	0.0413129	+	0.298829i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	584.621i		−	1.23598i
$474$		0			0
$475$		−	1560.84i		−	3.28597i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			413.409i			0.863067i	0.902097	+	0.431533i	$0.142027\pi$
$479$			413.409i			0.863067i	−0.902097	+	0.431533i	$0.857973\pi$
$480$		0			0
$481$			74.9844i			0.155893i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1596.57			3.29191
$486$		0			0
$487$			135.886i			0.279027i	0.990220	+	0.139513i	$0.0445538\pi$
$487$			135.886i			0.279027i	−0.990220	+	0.139513i	$0.955446\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	813.773i		−	1.65738i	−0.559708	−	0.828690i	$-0.689087\pi$
$491$		−	813.773i		−	1.65738i	0.559708	−	0.828690i	$-0.310913\pi$
$492$		0			0
$493$	−396.979			−0.805231
$494$		0			0
$495$		0			0
$496$		0			0
$497$	20.8128	+	150.545i	0.0418768	+	0.302908i
$498$		0			0
$499$	−91.2098			−0.182785			−0.0913925	−	0.995815i	$-0.529132\pi$
$499$	−91.2098			−0.182785			−0.0913925	+	0.995815i	$0.529132\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	546.674i		−	1.08683i	−0.839465	−	0.543414i	$-0.817131\pi$
$503$		−	546.674i		−	1.08683i	0.839465	−	0.543414i	$-0.182869\pi$
$504$		0			0
$505$	−649.213			−1.28557
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	366.160i		−	0.719372i	−0.933073	−	0.359686i	$-0.882884\pi$
$509$		−	366.160i		−	0.719372i	0.933073	−	0.359686i	$-0.117116\pi$
$510$		0			0
$511$	694.806	−	96.0564i	1.35970	−	0.187977i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	768.610i		−	1.49245i
$516$		0			0
$517$	284.954			0.551168
$518$		0			0
$519$		0			0
$520$		0			0
$521$	252.033			0.483748			0.241874	−	0.970308i	$-0.422238\pi$
$521$	252.033			0.483748			0.241874	+	0.970308i	$0.422238\pi$
$522$		0			0
$523$			335.060i			0.640651i	0.947308	+	0.320325i	$0.103792\pi$
$523$			335.060i			0.640651i	−0.947308	+	0.320325i	$0.896208\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1006.34			−1.90956
$528$		0			0
$529$	−291.311			−0.550683
$530$		0			0
$531$		0			0
$532$		0			0
$533$	76.6415			0.143793
$534$		0			0
$535$	735.392			1.37456
$536$		0			0
$537$		0			0
$538$		0			0
$539$	128.446	+	455.665i	0.238304	+	0.845390i
$540$		0			0
$541$			197.089i			0.364305i	0.983270	+	0.182152i	$0.0583065\pi$
$541$			197.089i			0.364305i	−0.983270	+	0.182152i	$0.941694\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−913.049			−1.67532
$546$		0			0
$547$	−296.822			−0.542637			−0.271318	−	0.962490i	$-0.587460\pi$
$547$	−296.822			−0.542637			−0.271318	+	0.962490i	$0.587460\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	488.234i		−	0.886087i
$552$		0			0
$553$	547.687	−	75.7174i	0.990393	−	0.136921i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−64.7524			−0.116252			−0.0581261	−	0.998309i	$-0.518513\pi$
$557$	−64.7524			−0.116252			−0.0581261	+	0.998309i	$0.518513\pi$
$558$		0			0
$559$	140.334			0.251044
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−143.148			−0.254260			−0.127130	−	0.991886i	$-0.540576\pi$
$563$	−143.148			−0.254260			−0.127130	+	0.991886i	$0.540576\pi$
$564$		0			0
$565$	133.473			0.236236
$566$		0			0
$567$		0			0
$568$		0			0
$569$			566.510i			0.995625i	0.867285	+	0.497812i	$0.165863\pi$
$569$			566.510i			0.995625i	−0.867285	+	0.497812i	$0.834137\pi$
$570$		0			0
$571$	−788.303			−1.38057			−0.690283	−	0.723539i	$-0.742514\pi$
$571$	−788.303			−1.38057			−0.690283	+	0.723539i	$0.742514\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	893.901			1.55461
$576$		0			0
$577$			273.614i			0.474200i	0.971485	+	0.237100i	$0.0761970\pi$
$577$			273.614i			0.474200i	−0.971485	+	0.237100i	$0.923803\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	64.8286	+	468.925i	0.111581	+	0.807100i
$582$		0			0
$583$		−	635.374i		−	1.08983i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.97914			0.00507520			0.00253760	−	0.999997i	$-0.499192\pi$
$587$	2.97914			0.00507520			0.00253760	+	0.999997i	$0.499192\pi$
$588$		0			0
$589$		−	1237.67i		−	2.10131i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	586.465			0.988979			0.494490	−	0.869184i	$-0.335355\pi$
$593$	586.465			0.988979			0.494490	+	0.869184i	$0.335355\pi$
$594$		0			0
$595$	1382.57	−	191.140i	2.32365	−	0.321243i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	44.6708			0.0745757			0.0372878	−	0.999305i	$-0.488128\pi$
$599$	44.6708			0.0745757			0.0372878	+	0.999305i	$0.488128\pi$
$600$		0			0
$601$			185.280i			0.308286i	0.988049	+	0.154143i	$0.0492617\pi$
$601$			185.280i			0.308286i	−0.988049	+	0.154143i	$0.950738\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			251.890i			0.416347i
$606$		0			0
$607$	−485.191			−0.799326			−0.399663	−	0.916662i	$-0.630873\pi$
$607$	−485.191			−0.799326			−0.399663	+	0.916662i	$0.630873\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			68.4010i			0.111949i
$612$		0			0
$613$		−	504.370i		−	0.822790i	−0.911457	−	0.411395i	$-0.865042\pi$
$613$		−	504.370i		−	0.822790i	0.911457	−	0.411395i	$-0.134958\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	20.6090i		−	0.0334019i	−0.999861	−	0.0167010i	$-0.994684\pi$
$617$		−	20.6090i		−	0.0334019i	0.999861	−	0.0167010i	$-0.00531633\pi$
$618$		0			0
$619$			925.741i			1.49554i	0.663957	+	0.747771i	$0.268876\pi$
$619$			925.741i			1.49554i	−0.663957	+	0.747771i	$0.731124\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	125.510	+	907.856i	0.201461	+	1.45723i
$624$		0			0
$625$	1287.27			2.05962
$626$		0			0
$627$		0			0
$628$		0			0
$629$			707.687i			1.12510i
$630$		0			0
$631$		−	902.920i		−	1.43094i	−0.698646	−	0.715468i	$-0.746214\pi$
$631$		−	902.920i		−	1.43094i	0.698646	−	0.715468i	$-0.253786\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	209.442			0.329830
$636$		0			0
$637$	−109.379	+	30.8325i	−0.171710	+	0.0484026i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			235.354i			0.367167i	0.983004	+	0.183584i	$0.0587698\pi$
$641$			235.354i			0.367167i	−0.983004	+	0.183584i	$0.941230\pi$
$642$		0			0
$643$			189.811i			0.295197i	0.989047	+	0.147598i	$0.0471543\pi$
$643$			189.811i			0.295197i	−0.989047	+	0.147598i	$0.952846\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9.78084i			0.0151172i	0.999971	+	0.00755861i	$0.00240600\pi$
$647$			9.78084i			0.0151172i	−0.999971	+	0.00755861i	$0.997594\pi$
$648$		0			0
$649$		−	81.8605i		−	0.126133i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	700.168			1.07223			0.536117	−	0.844144i	$-0.319891\pi$
$653$	700.168			1.07223			0.536117	+	0.844144i	$0.319891\pi$
$654$		0			0
$655$		−	1412.25i		−	2.15611i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	584.716i		−	0.887277i	−0.896206	−	0.443639i	$-0.853687\pi$
$659$		−	584.716i		−	0.887277i	0.896206	−	0.443639i	$-0.146313\pi$
$660$		0			0
$661$	687.188			1.03962			0.519809	−	0.854282i	$-0.326003\pi$
$661$	687.188			1.03962			0.519809	+	0.854282i	$0.326003\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	235.078	+	1700.39i	0.353500	+	2.55697i
$666$		0			0
$667$	279.615			0.419213
$668$		0			0
$669$		0			0
$670$		0			0
$671$			724.084i			1.07911i
$672$		0			0
$673$	−394.412			−0.586050			−0.293025	−	0.956105i	$-0.594662\pi$
$673$	−394.412			−0.586050			−0.293025	+	0.956105i	$0.594662\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	115.342i		−	0.170372i	−0.996365	−	0.0851858i	$-0.972852\pi$
$677$		−	115.342i		−	0.170372i	0.996365	−	0.0851858i	$-0.0271484\pi$
$678$		0			0
$679$	−1215.31	+	168.016i	−1.78985	+	0.247446i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			208.661i			0.305506i	0.988264	+	0.152753i	$0.0488139\pi$
$683$			208.661i			0.305506i	−0.988264	+	0.152753i	$0.951186\pi$
$684$		0			0
$685$	−597.272			−0.871930
$686$		0			0
$687$		0			0
$688$		0			0
$689$	152.517			0.221359
$690$		0			0
$691$			755.615i			1.09351i	0.837293	+	0.546755i	$0.184137\pi$
$691$			755.615i			1.09351i	−0.837293	+	0.546755i	$0.815863\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1365.26			−1.96440
$696$		0			0
$697$	723.327			1.03777
$698$		0			0
$699$		0			0
$700$		0			0
$701$	611.239			0.871953			0.435977	−	0.899958i	$-0.356403\pi$
$701$	611.239			0.871953			0.435977	+	0.899958i	$0.356403\pi$
$702$		0			0
$703$	−870.365			−1.23807
$704$		0			0
$705$		0			0
$706$		0			0
$707$	494.179	−	68.3200i	0.698981	−	0.0966336i
$708$		0			0
$709$			182.040i			0.256756i	0.991725	+	0.128378i	$0.0409771\pi$
$709$			182.040i			0.256756i	−0.991725	+	0.128378i	$0.959023\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	708.822			0.994141
$714$		0			0
$715$	204.119			0.285481
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	927.993i		−	1.29067i	−0.763899	−	0.645336i	$-0.776717\pi$
$719$		−	927.993i		−	1.29067i	0.763899	−	0.645336i	$-0.223283\pi$
$720$		0			0
$721$	80.8848	+	585.065i	0.112184	+	0.811463i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1051.58			1.45045
$726$		0			0
$727$	−715.698			−0.984453			−0.492227	−	0.870467i	$-0.663817\pi$
$727$	−715.698			−0.984453			−0.492227	+	0.870467i	$0.663817\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1324.44			1.81182
$732$		0			0
$733$	565.483			0.771464			0.385732	−	0.922611i	$-0.373949\pi$
$733$	565.483			0.771464			0.385732	+	0.922611i	$0.373949\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			195.282i			0.264969i
$738$		0			0
$739$	−104.108			−0.140876			−0.0704382	−	0.997516i	$-0.522440\pi$
$739$	−104.108			−0.140876			−0.0704382	+	0.997516i	$0.522440\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1159.59			1.56068			0.780341	−	0.625354i	$-0.215045\pi$
$743$	1159.59			1.56068			0.780341	+	0.625354i	$0.215045\pi$
$744$		0			0
$745$		−	580.578i		−	0.779299i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−559.779	+	77.3890i	−0.747368	+	0.103323i
$750$		0			0
$751$			8.05706i			0.0107284i	0.999986	+	0.00536422i	$0.00170749\pi$
$751$			8.05706i			0.0107284i	−0.999986	+	0.00536422i	$0.998293\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−799.522			−1.05897
$756$		0			0
$757$			203.142i			0.268352i	0.990958	+	0.134176i	$0.0428387\pi$
$757$			203.142i			0.268352i	−0.990958	+	0.134176i	$0.957161\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1325.16			1.74134			0.870670	−	0.491867i	$-0.163685\pi$
$761$	1325.16			1.74134			0.870670	+	0.491867i	$0.163685\pi$
$762$		0			0
$763$	695.012	−	96.0849i	0.910893	−	0.125930i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	19.6500			0.0256193
$768$		0			0
$769$		−	27.4357i		−	0.0356772i	−0.999841	−	0.0178386i	$-0.994322\pi$
$769$		−	27.4357i		−	0.0356772i	0.999841	−	0.0178386i	$-0.00567850\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			127.533i			0.164985i	0.996592	+	0.0824925i	$0.0262880\pi$
$773$			127.533i			0.164985i	−0.996592	+	0.0824925i	$0.973712\pi$
$774$		0			0
$775$	2665.74			3.43967
$776$		0			0
$777$		0			0
$778$		0			0
$779$			889.600i			1.14198i
$780$		0			0
$781$			209.765i			0.268585i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			118.080i			0.150421i
$786$		0			0
$787$			151.856i			0.192956i	0.995335	+	0.0964781i	$0.0307578\pi$
$787$			151.856i			0.192956i	−0.995335	+	0.0964781i	$0.969242\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−101.599	+	14.0461i	−0.128444	+	0.0177573i
$792$		0			0
$793$	−173.811			−0.219182
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1136.09i			1.42545i	0.701443	+	0.712726i	$0.252540\pi$
$797$			1136.09i			1.42545i	−0.701443	+	0.712726i	$0.747460\pi$
$798$		0			0
$799$			645.554i			0.807953i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	968.122			1.20563
$804$		0			0
$805$	−973.823	+	134.630i	−1.20972	+	0.167243i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			438.239i			0.541705i	0.962621	+	0.270852i	$0.0873055\pi$
$809$			438.239i			0.541705i	−0.962621	+	0.270852i	$0.912695\pi$
$810$		0			0
$811$			1406.16i			1.73386i	0.498427	+	0.866931i	$0.333911\pi$
$811$			1406.16i			1.73386i	−0.498427	+	0.866931i	$0.666089\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1178.58i		−	1.44611i
$816$		0			0
$817$			1628.89i			1.99375i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	592.105			0.721200			0.360600	−	0.932721i	$-0.382572\pi$
$821$	592.105			0.721200			0.360600	+	0.932721i	$0.382572\pi$
$822$		0			0
$823$		−	1078.89i		−	1.31092i	−0.755229	−	0.655461i	$-0.772474\pi$
$823$		−	1078.89i		−	1.31092i	0.755229	−	0.655461i	$-0.227526\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	111.468i		−	0.134785i	−0.997727	−	0.0673927i	$-0.978532\pi$
$827$		−	111.468i		−	0.134785i	0.997727	−	0.0673927i	$-0.0214680\pi$
$828$		0			0
$829$	−109.080			−0.131580			−0.0657900	−	0.997833i	$-0.520957\pi$
$829$	−109.080			−0.131580			−0.0657900	+	0.997833i	$0.520957\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1032.30	+	290.990i	−1.23925	+	0.349328i
$834$		0			0
$835$	1726.13			2.06722
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1297.65i			1.54667i	0.633999	+	0.773334i	$0.281412\pi$
$839$			1297.65i			1.54667i	−0.633999	+	0.773334i	$0.718588\pi$
$840$		0			0
$841$	−512.064			−0.608875
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1490.49i		−	1.76389i
$846$		0			0
$847$	−26.5076	−	191.738i	−0.0312959	−	0.226373i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	498.464i		−	0.585739i
$852$		0			0
$853$	−1099.79			−1.28932			−0.644658	−	0.764471i	$-0.723000\pi$
$853$	−1099.79			−1.28932			−0.644658	+	0.764471i	$0.723000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−616.955			−0.719900			−0.359950	−	0.932972i	$-0.617206\pi$
$857$	−616.955			−0.719900			−0.359950	+	0.932972i	$0.617206\pi$
$858$		0			0
$859$		−	1202.72i		−	1.40014i	−0.714075	−	0.700069i	$-0.753152\pi$
$859$		−	1202.72i		−	1.40014i	0.714075	−	0.700069i	$-0.246848\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	481.815			0.558303			0.279151	−	0.960247i	$-0.409947\pi$
$863$	481.815			0.558303			0.279151	+	0.960247i	$0.409947\pi$
$864$		0			0
$865$	−2816.26			−3.25580
$866$		0			0
$867$		0			0
$868$		0			0
$869$	763.131			0.878171
$870$		0			0
$871$	−46.8760			−0.0538186
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2083.24	+	288.006i	−2.38084	+	0.329150i
$876$		0			0
$877$		−	275.163i		−	0.313754i	−0.987618	−	0.156877i	$-0.949857\pi$
$877$		−	275.163i		−	0.313754i	0.987618	−	0.156877i	$-0.0501427\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−718.744			−0.815827			−0.407914	−	0.913020i	$-0.633744\pi$
$881$	−718.744			−0.815827			−0.407914	+	0.913020i	$0.633744\pi$
$882$		0			0
$883$	1561.31			1.76819			0.884095	−	0.467308i	$-0.154776\pi$
$883$	1561.31			1.76819			0.884095	+	0.467308i	$0.154776\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			983.157i			1.10841i	0.832381	+	0.554204i	$0.186977\pi$
$887$			983.157i			1.10841i	−0.832381	+	0.554204i	$0.813023\pi$
$888$		0			0
$889$	−159.427	+	22.0407i	−0.179333	+	0.0247927i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−793.950			−0.889082
$894$		0			0
$895$	995.789			1.11261
$896$		0			0
$897$		0			0
$898$		0			0
$899$	833.852			0.927533
$900$		0			0
$901$	1439.42			1.59758
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	687.137i		−	0.759267i
$906$		0			0
$907$	−1121.27			−1.23624			−0.618119	−	0.786085i	$-0.712105\pi$
$907$	−1121.27			−1.23624			−0.618119	+	0.786085i	$0.712105\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1308.23			−1.43604			−0.718020	−	0.696022i	$-0.754952\pi$
$911$	−1308.23			−1.43604			−0.718020	+	0.696022i	$0.754952\pi$
$912$		0			0
$913$			653.386i			0.715647i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	148.618	+	1075.00i	0.162070	+	1.17230i
$918$		0			0
$919$		−	992.984i		−	1.08050i	−0.841503	−	0.540252i	$-0.818329\pi$
$919$		−	992.984i		−	1.08050i	0.841503	−	0.540252i	$-0.181671\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−50.3525			−0.0545531
$924$		0			0
$925$		−	1874.63i		−	2.02662i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	47.0391			0.0506342			0.0253171	−	0.999679i	$-0.491940\pi$
$929$	47.0391			0.0506342			0.0253171	+	0.999679i	$0.491940\pi$
$930$		0			0
$931$	−357.881	−	1269.59i	−0.384405	−	1.36369i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1926.43			2.06036
$936$		0			0
$937$		−	1808.57i		−	1.93017i	−0.261932	−	0.965086i	$-0.584360\pi$
$937$		−	1808.57i		−	1.93017i	0.261932	−	0.965086i	$-0.415640\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	628.185i		−	0.667572i	−0.942649	−	0.333786i	$-0.891674\pi$
$941$		−	628.185i		−	0.667572i	0.942649	−	0.333786i	$-0.108326\pi$
$942$		0			0
$943$	−509.480			−0.540276
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1130.40i			1.19366i	0.802367	+	0.596831i	$0.203574\pi$
$947$			1130.40i			1.19366i	−0.802367	+	0.596831i	$0.796426\pi$
$948$		0			0
$949$			232.390i			0.244879i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	781.337i		−	0.819871i	−0.912114	−	0.409936i	$-0.865551\pi$
$953$		−	781.337i		−	0.819871i	0.912114	−	0.409936i	$-0.134449\pi$
$954$		0			0
$955$		−	1861.57i		−	1.94929i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	454.643	−	62.8540i	0.474080	−	0.0655412i
$960$		0			0
$961$	1152.81			1.19960
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1041.82i		−	1.07961i
$966$		0			0
$967$			1674.13i			1.73126i	0.500685	+	0.865629i	$0.333081\pi$
$967$			1674.13i			1.73126i	−0.500685	+	0.865629i	$0.666919\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−271.907			−0.280028			−0.140014	−	0.990150i	$-0.544715\pi$
$971$	−271.907			−0.280028			−0.140014	+	0.990150i	$0.544715\pi$
$972$		0			0
$973$	1039.23	−	143.673i	1.06807	−	0.147660i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1639.40i		−	1.67799i	−0.544138	−	0.838995i	$-0.683143\pi$
$977$		−	1639.40i		−	1.67799i	0.544138	−	0.838995i	$-0.316857\pi$
$978$		0			0
$979$			1264.98i			1.29211i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	824.584i		−	0.838844i	−0.907791	−	0.419422i	$-0.862233\pi$
$983$		−	824.584i		−	0.838844i	0.907791	−	0.419422i	$-0.137767\pi$
$984$		0			0
$985$		−	2070.02i		−	2.10155i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−932.878			−0.943254
$990$		0			0
$991$		−	888.843i		−	0.896916i	−0.893804	−	0.448458i	$-0.851973\pi$
$991$		−	888.843i		−	0.896916i	0.893804	−	0.448458i	$-0.148027\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1035.88i		−	1.04108i
$996$		0			0
$997$	−43.3671			−0.0434976			−0.0217488	−	0.999763i	$-0.506923\pi$
$997$	−43.3671			−0.0434976			−0.0217488	+	0.999763i	$0.506923\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.e.c.1007.24		48
3.2	odd	2	inner	2016.3.e.c.1007.33		48
4.3	odd	2		504.3.e.c.251.32	yes	48
7.6	odd	2	inner	2016.3.e.c.1007.7		48
8.3	odd	2	inner	2016.3.e.c.1007.5		48
8.5	even	2		504.3.e.c.251.19	yes	48
12.11	even	2		504.3.e.c.251.18	yes	48
21.20	even	2	inner	2016.3.e.c.1007.6		48
24.5	odd	2		504.3.e.c.251.29	yes	48
24.11	even	2	inner	2016.3.e.c.1007.8		48
28.27	even	2		504.3.e.c.251.31	yes	48
56.13	odd	2		504.3.e.c.251.20	yes	48
56.27	even	2	inner	2016.3.e.c.1007.34		48
84.83	odd	2		504.3.e.c.251.17	✓	48
168.83	odd	2	inner	2016.3.e.c.1007.23		48
168.125	even	2		504.3.e.c.251.30	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.e.c.251.17	✓	48	84.83	odd	2
504.3.e.c.251.18	yes	48	12.11	even	2
504.3.e.c.251.19	yes	48	8.5	even	2
504.3.e.c.251.20	yes	48	56.13	odd	2
504.3.e.c.251.29	yes	48	24.5	odd	2
504.3.e.c.251.30	yes	48	168.125	even	2
504.3.e.c.251.31	yes	48	28.27	even	2
504.3.e.c.251.32	yes	48	4.3	odd	2
2016.3.e.c.1007.5		48	8.3	odd	2	inner
2016.3.e.c.1007.6		48	21.20	even	2	inner
2016.3.e.c.1007.7		48	7.6	odd	2	inner
2016.3.e.c.1007.8		48	24.11	even	2	inner
2016.3.e.c.1007.23		48	168.83	odd	2	inner
2016.3.e.c.1007.24		48	1.1	even	1	trivial
2016.3.e.c.1007.33		48	3.2	odd	2	inner
2016.3.e.c.1007.34		48	56.27	even	2	inner

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.10939i q^{5} +(-0.958628 - 6.93405i) q^{7} -9.66169i q^{11} +2.31922 q^{13} +21.8883 q^{17} +26.9198i q^{19} -15.4172 q^{23} -57.9809 q^{25} -18.1366 q^{29} -45.9762 q^{31} +(63.1649 - 8.73251i) q^{35} +32.3318i q^{37} +33.0463 q^{41} +60.5091 q^{43} +29.4932i q^{47} +(-47.1621 + 13.2943i) q^{49} +65.7621 q^{53} +88.0121 q^{55} +8.47268 q^{59} -74.9438 q^{61} +21.1266i q^{65} -20.2120 q^{67} -21.7110 q^{71} +100.202i q^{73} +(-66.9947 + 9.26197i) q^{77} +78.9852i q^{79} -67.6265 q^{83} +199.389i q^{85} -130.927 q^{89} +(-2.22326 - 16.0816i) q^{91} -245.223 q^{95} -175.267i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 336 q^{25} + 384 q^{43} + 368 q^{49} + 896 q^{67} + 576 q^{91}+O(q^{100}) \) 48 * q - 336 * q^25 + 384 * q^43 + 368 * q^49 + 896 * q^67 + 576 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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