LMFDB - Embedded newform 1584.3.j.k.1297.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(1297,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 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("1584.1297"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1584.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,-52,0,22] 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:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1750426112.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 4x^{3} + 228x^{2} + 368x + 548 $ x^6 - 2x^5 - 21x^4 + 4x^3 + 228x^2 + 368*x + 548
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.1
Root			$-2.88824 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.k.1297.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.11838 q^{5} -12.7373i q^{7} +(10.0066 + 4.56811i) q^{11} +14.9148i q^{13} -1.42357i q^{17} +29.0756i q^{19} -11.8882 q^{23} +1.19781 q^{25} -16.3383i q^{29} +16.8088 q^{31} +65.1942i q^{35} +56.2507 q^{37} -16.8489i q^{41} +36.1186i q^{43} +21.8544 q^{47} -113.238 q^{49} +0.185315 q^{53} +(-51.2177 - 23.3813i) q^{55} +17.9264 q^{59} -77.0185i q^{61} -76.3394i q^{65} +10.9412 q^{67} +139.271 q^{71} -82.2022i q^{73} +(58.1853 - 127.457i) q^{77} -30.8506i q^{79} -74.1713i q^{83} +7.28637i q^{85} -9.98603 q^{89} +189.973 q^{91} -148.820i q^{95} +82.3831 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{11} - 52 q^{23} + 22 q^{25} + 36 q^{31} - 48 q^{37} - 60 q^{47} - 170 q^{49} - 108 q^{53} - 172 q^{55} + 236 q^{59} + 292 q^{67} + 300 q^{71} + 240 q^{77} - 384 q^{89} + 48 q^{91} + 400 q^{97}+O(q^{100}) $ 6 * q + 10 * q^11 - 52 * q^23 + 22 * q^25 + 36 * q^31 - 48 * q^37 - 60 * q^47 - 170 * q^49 - 108 * q^53 - 172 * q^55 + 236 * q^59 + 292 * q^67 + 300 * q^71 + 240 * q^77 - 384 * q^89 + 48 * q^91 + 400 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−5.11838			−1.02368			−0.511838	−	0.859082i	$-0.671035\pi$
$5$	−5.11838			−1.02368			−0.511838	+	0.859082i	$0.671035\pi$
$6$		0			0
$7$		−	12.7373i		−	1.81961i	−0.415035	−	0.909806i	$-0.636231\pi$
$7$		−	12.7373i		−	1.81961i	0.415035	−	0.909806i	$-0.363769\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.0066	+	4.56811i	0.909692	+	0.415283i
$11$	10.0066	+	4.56811i	0.909692	+	0.415283i
$12$		0			0
$13$			14.9148i			1.14729i	0.819104	+	0.573645i	$0.194471\pi$
$13$			14.9148i			1.14729i	−0.819104	+	0.573645i	$0.805529\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.42357i		−	0.0837394i	−0.999123	−	0.0418697i	$-0.986669\pi$
$17$		−	1.42357i		−	0.0837394i	0.999123	−	0.0418697i	$-0.0133314\pi$
$18$		0			0
$19$			29.0756i			1.53030i	0.643855	+	0.765148i	$0.277334\pi$
$19$			29.0756i			1.53030i	−0.643855	+	0.765148i	$0.722666\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−11.8882			−0.516880			−0.258440	−	0.966027i	$-0.583208\pi$
$23$	−11.8882			−0.516880			−0.258440	+	0.966027i	$0.583208\pi$
$24$		0			0
$25$	1.19781			0.0479123
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	16.3383i		−	0.563391i	−0.959504	−	0.281695i	$-0.909103\pi$
$29$		−	16.3383i		−	0.563391i	0.959504	−	0.281695i	$-0.0908967\pi$
$30$		0			0
$31$	16.8088			0.542220			0.271110	−	0.962548i	$-0.412609\pi$
$31$	16.8088			0.542220			0.271110	+	0.962548i	$0.412609\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			65.1942i			1.86269i
$36$		0			0
$37$	56.2507			1.52029			0.760145	−	0.649754i	$-0.225128\pi$
$37$	56.2507			1.52029			0.760145	+	0.649754i	$0.225128\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	16.8489i		−	0.410948i	−0.978663	−	0.205474i	$-0.934126\pi$
$41$		−	16.8489i		−	0.410948i	0.978663	−	0.205474i	$-0.0658736\pi$
$42$		0			0
$43$			36.1186i			0.839968i	0.907532	+	0.419984i	$0.137964\pi$
$43$			36.1186i			0.839968i	−0.907532	+	0.419984i	$0.862036\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	21.8544			0.464987			0.232493	−	0.972598i	$-0.425312\pi$
$47$	21.8544			0.464987			0.232493	+	0.972598i	$0.425312\pi$
$48$		0			0
$49$	−113.238			−2.31098
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.185315			0.00349651			0.00174826	−	0.999998i	$-0.499444\pi$
$53$	0.185315			0.00349651			0.00174826	+	0.999998i	$0.499444\pi$
$54$		0			0
$55$	−51.2177	−	23.3813i	−0.931230	−	0.425115i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	17.9264			0.303838			0.151919	−	0.988393i	$-0.451455\pi$
$59$	17.9264			0.303838			0.151919	+	0.988393i	$0.451455\pi$
$60$		0			0
$61$		−	77.0185i		−	1.26260i	−0.775540	−	0.631299i	$-0.782522\pi$
$61$		−	77.0185i		−	1.26260i	0.775540	−	0.631299i	$-0.217478\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	76.3394i		−	1.17445i
$66$		0			0
$67$	10.9412			0.163302			0.0816508	−	0.996661i	$-0.473981\pi$
$67$	10.9412			0.163302			0.0816508	+	0.996661i	$0.473981\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	139.271			1.96156			0.980779	−	0.195123i	$-0.0625107\pi$
$71$	139.271			1.96156			0.980779	+	0.195123i	$0.0625107\pi$
$72$		0			0
$73$		−	82.2022i		−	1.12606i	−0.826437	−	0.563029i	$-0.809636\pi$
$73$		−	82.2022i		−	1.12606i	0.826437	−	0.563029i	$-0.190364\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	58.1853	−	127.457i	0.755653	−	1.65529i
$78$		0			0
$79$		−	30.8506i		−	0.390514i	−0.980752	−	0.195257i	$-0.937446\pi$
$79$		−	30.8506i		−	0.390514i	0.980752	−	0.195257i	$-0.0625541\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	74.1713i		−	0.893631i	−0.894626	−	0.446815i	$-0.852558\pi$
$83$		−	74.1713i		−	0.893631i	0.894626	−	0.446815i	$-0.147442\pi$
$84$		0			0
$85$			7.28637i			0.0857220i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.98603			−0.112203			−0.0561013	−	0.998425i	$-0.517867\pi$
$89$	−9.98603			−0.112203			−0.0561013	+	0.998425i	$0.517867\pi$
$90$		0			0
$91$	189.973			2.08762
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	148.820i		−	1.56653i
$96$		0			0
$97$	82.3831			0.849311			0.424655	−	0.905355i	$-0.360395\pi$
$97$	82.3831			0.849311			0.424655	+	0.905355i	$0.360395\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			128.202i			1.26932i	0.772791	+	0.634661i	$0.218860\pi$
$101$			128.202i			1.26932i	−0.772791	+	0.634661i	$0.781140\pi$
$102$		0			0
$103$	48.0662			0.466662			0.233331	−	0.972397i	$-0.425037\pi$
$103$	48.0662			0.466662			0.233331	+	0.972397i	$0.425037\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			201.703i			1.88508i	0.334098	+	0.942538i	$0.391568\pi$
$107$			201.703i			1.88508i	−0.334098	+	0.942538i	$0.608432\pi$
$108$		0			0
$109$		−	164.512i		−	1.50929i	−0.656134	−	0.754644i	$-0.727809\pi$
$109$		−	164.512i		−	1.50929i	0.656134	−	0.754644i	$-0.272191\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−85.1713			−0.753728			−0.376864	−	0.926269i	$-0.622998\pi$
$113$	−85.1713			−0.753728			−0.376864	+	0.926269i	$0.622998\pi$
$114$		0			0
$115$	60.8485			0.529117
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−18.1324			−0.152373
$120$		0			0
$121$	79.2647	+	91.4227i	0.655080	+	0.755559i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	121.829			0.974629
$126$		0			0
$127$		−	62.1786i		−	0.489595i	−0.969574	−	0.244797i	$-0.921279\pi$
$127$		−	62.1786i		−	0.489595i	0.969574	−	0.244797i	$-0.0787215\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	65.6205i		−	0.500920i	−0.968127	−	0.250460i	$-0.919418\pi$
$131$		−	65.6205i		−	0.500920i	0.968127	−	0.250460i	$-0.0805818\pi$
$132$		0			0
$133$	370.344			2.78454
$134$		0			0
$135$		0			0
$136$		0			0
$137$	123.279			0.899845			0.449922	−	0.893068i	$-0.351452\pi$
$137$	123.279			0.899845			0.449922	+	0.893068i	$0.351452\pi$
$138$		0			0
$139$		−	76.6909i		−	0.551733i	−0.961196	−	0.275866i	$-0.911035\pi$
$139$		−	76.6909i		−	0.551733i	0.961196	−	0.275866i	$-0.0889647\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−68.1323	+	149.246i	−0.476450	+	1.04368i
$144$		0			0
$145$			83.6258i			0.576729i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	203.286i		−	1.36433i	−0.731196	−	0.682167i	$-0.761038\pi$
$149$		−	203.286i		−	1.36433i	0.731196	−	0.682167i	$-0.238962\pi$
$150$		0			0
$151$			109.100i			0.722519i	0.932465	+	0.361259i	$0.117653\pi$
$151$			109.100i			0.722519i	−0.932465	+	0.361259i	$0.882347\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−86.0339			−0.555057
$156$		0			0
$157$	−72.5435			−0.462060			−0.231030	−	0.972947i	$-0.574210\pi$
$157$	−72.5435			−0.462060			−0.231030	+	0.972947i	$0.574210\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			151.424i			0.940520i
$162$		0			0
$163$	144.357			0.885628			0.442814	−	0.896613i	$-0.353980\pi$
$163$	144.357			0.885628			0.442814	+	0.896613i	$0.353980\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			120.985i			0.724461i	0.932089	+	0.362231i	$0.117985\pi$
$167$			120.985i			0.724461i	−0.932089	+	0.362231i	$0.882015\pi$
$168$		0			0
$169$	−53.4501			−0.316273
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.3329i			0.105970i	0.998595	+	0.0529852i	$0.0168736\pi$
$173$			18.3329i			0.105970i	−0.998595	+	0.0529852i	$0.983126\pi$
$174$		0			0
$175$		−	15.2568i		−	0.0871817i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	257.827			1.44037			0.720186	−	0.693781i	$-0.244057\pi$
$179$	257.827			1.44037			0.720186	+	0.693781i	$0.244057\pi$
$180$		0			0
$181$	83.1244			0.459251			0.229625	−	0.973279i	$-0.426250\pi$
$181$	83.1244			0.459251			0.229625	+	0.973279i	$0.426250\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−287.913			−1.55628
$186$		0			0
$187$	6.50302	−	14.2451i	0.0347755	−	0.0761770i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	158.747			0.831137			0.415568	−	0.909562i	$-0.363583\pi$
$191$	158.747			0.831137			0.415568	+	0.909562i	$0.363583\pi$
$192$		0			0
$193$			212.675i			1.10194i	0.834524	+	0.550971i	$0.185743\pi$
$193$			212.675i			1.10194i	−0.834524	+	0.550971i	$0.814257\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	36.6530i		−	0.186056i	−0.995664	−	0.0930279i	$-0.970345\pi$
$197$		−	36.6530i		−	0.186056i	0.995664	−	0.0930279i	$-0.0296546\pi$
$198$		0			0
$199$	141.828			0.712703			0.356352	−	0.934352i	$-0.384021\pi$
$199$	141.828			0.712703			0.356352	+	0.934352i	$0.384021\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−208.106			−1.02515
$204$		0			0
$205$			86.2390i			0.420678i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−132.821	+	290.948i	−0.635505	+	1.39210i
$210$		0			0
$211$		−	28.0451i		−	0.132915i	−0.997789	−	0.0664576i	$-0.978830\pi$
$211$		−	28.0451i		−	0.132915i	0.997789	−	0.0664576i	$-0.0211697\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	184.869i		−	0.859855i
$216$		0			0
$217$		−	214.098i		−	0.986629i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	21.2322			0.0960733
$222$		0			0
$223$	93.6765			0.420074			0.210037	−	0.977693i	$-0.432642\pi$
$223$	93.6765			0.420074			0.210037	+	0.977693i	$0.432642\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	325.742i		−	1.43499i	−0.696565	−	0.717494i	$-0.745289\pi$
$227$		−	325.742i		−	1.43499i	0.696565	−	0.717494i	$-0.254711\pi$
$228$		0			0
$229$	−31.5964			−0.137976			−0.0689878	−	0.997618i	$-0.521977\pi$
$229$	−31.5964			−0.137976			−0.0689878	+	0.997618i	$0.521977\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			85.6680i			0.367674i	0.982957	+	0.183837i	$0.0588518\pi$
$233$			85.6680i			0.367674i	−0.982957	+	0.183837i	$0.941148\pi$
$234$		0			0
$235$	−111.859			−0.475996
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	264.111i		−	1.10507i	−0.833491	−	0.552533i	$-0.813661\pi$
$239$		−	264.111i		−	1.10507i	0.833491	−	0.552533i	$-0.186339\pi$
$240$		0			0
$241$		−	209.659i		−	0.869955i	−0.900441	−	0.434978i	$-0.856756\pi$
$241$		−	209.659i		−	0.869955i	0.900441	−	0.434978i	$-0.143244\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	579.596			2.36570
$246$		0			0
$247$	−433.656			−1.75569
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−406.438			−1.61928			−0.809638	−	0.586930i	$-0.800336\pi$
$251$	−406.438			−1.61928			−0.809638	+	0.586930i	$0.800336\pi$
$252$		0			0
$253$	−118.961	−	54.3068i	−0.470202	−	0.214651i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	200.291			0.779343			0.389672	−	0.920954i	$-0.372589\pi$
$257$	200.291			0.779343			0.389672	+	0.920954i	$0.372589\pi$
$258$		0			0
$259$		−	716.481i		−	2.76634i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			86.1641i			0.327620i	0.986492	+	0.163810i	$0.0523784\pi$
$263$			86.1641i			0.327620i	−0.986492	+	0.163810i	$0.947622\pi$
$264$		0			0
$265$	−0.948513			−0.00357929
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−312.768			−1.16270			−0.581352	−	0.813652i	$-0.697476\pi$
$269$	−312.768			−1.16270			−0.581352	+	0.813652i	$0.697476\pi$
$270$		0			0
$271$		−	64.5578i		−	0.238221i	−0.992881	−	0.119110i	$-0.961996\pi$
$271$		−	64.5578i		−	0.238221i	0.992881	−	0.119110i	$-0.0380042\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.9860	+	5.47171i	0.0435854	+	0.0198971i
$276$		0			0
$277$			145.687i			0.525945i	0.964803	+	0.262973i	$0.0847029\pi$
$277$			145.687i			0.525945i	−0.964803	+	0.262973i	$0.915297\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	335.693i		−	1.19464i	−0.802005	−	0.597318i	$-0.796233\pi$
$281$		−	335.693i		−	1.19464i	0.802005	−	0.597318i	$-0.203767\pi$
$282$		0			0
$283$		−	73.8343i		−	0.260899i	−0.991455	−	0.130449i	$-0.958358\pi$
$283$		−	73.8343i		−	0.260899i	0.991455	−	0.130449i	$-0.0416420\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−214.609			−0.747766
$288$		0			0
$289$	286.973			0.992988
$290$		0			0
$291$		0			0
$292$		0			0
$293$			335.182i			1.14397i	0.820265	+	0.571983i	$0.193826\pi$
$293$			335.182i			1.14397i	−0.820265	+	0.571983i	$0.806174\pi$
$294$		0			0
$295$	−91.7544			−0.311032
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	177.310i		−	0.593011i
$300$		0			0
$301$	460.053			1.52841
$302$		0			0
$303$		0			0
$304$		0			0
$305$			394.210i			1.29249i
$306$		0			0
$307$			289.197i			0.942011i	0.882130	+	0.471005i	$0.156109\pi$
$307$			289.197i			0.942011i	−0.882130	+	0.471005i	$0.843891\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	86.6486			0.278613			0.139306	−	0.990249i	$-0.455513\pi$
$311$	86.6486			0.278613			0.139306	+	0.990249i	$0.455513\pi$
$312$		0			0
$313$	−394.046			−1.25893			−0.629467	−	0.777027i	$-0.716727\pi$
$313$	−394.046			−1.25893			−0.629467	+	0.777027i	$0.716727\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	207.808			0.655546			0.327773	−	0.944756i	$-0.393702\pi$
$317$	207.808			0.655546			0.327773	+	0.944756i	$0.393702\pi$
$318$		0			0
$319$	74.6353	−	163.491i	0.233967	−	0.512512i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	41.3911			0.128146
$324$		0			0
$325$			17.8650i			0.0549692i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	278.365i		−	0.846095i
$330$		0			0
$331$	487.682			1.47336			0.736680	−	0.676241i	$-0.236392\pi$
$331$	487.682			1.47336			0.736680	+	0.676241i	$0.236392\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−56.0012			−0.167168
$336$		0			0
$337$			388.113i			1.15167i	0.817566	+	0.575835i	$0.195323\pi$
$337$			388.113i			1.15167i	−0.817566	+	0.575835i	$0.804677\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	168.199	+	76.7845i	0.493253	+	0.225175i
$342$		0			0
$343$			818.220i			2.38548i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	107.644i		−	0.310212i	−0.987898	−	0.155106i	$-0.950428\pi$
$347$		−	107.644i		−	0.310212i	0.987898	−	0.155106i	$-0.0495719\pi$
$348$		0			0
$349$		−	304.548i		−	0.872629i	−0.899794	−	0.436315i	$-0.856283\pi$
$349$		−	304.548i		−	0.872629i	0.899794	−	0.436315i	$-0.143717\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−300.542			−0.851393			−0.425697	−	0.904866i	$-0.639971\pi$
$353$	−300.542			−0.851393			−0.425697	+	0.904866i	$0.639971\pi$
$354$		0			0
$355$	−712.840			−2.00800
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	254.389i		−	0.708605i	−0.935131	−	0.354302i	$-0.884718\pi$
$359$		−	254.389i		−	0.708605i	0.935131	−	0.354302i	$-0.115282\pi$
$360$		0			0
$361$	−484.391			−1.34180
$362$		0			0
$363$		0			0
$364$		0			0
$365$			420.742i			1.15272i
$366$		0			0
$367$	421.179			1.14763			0.573814	−	0.818986i	$-0.305463\pi$
$367$	421.179			1.14763			0.573814	+	0.818986i	$0.305463\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	2.36041i		−	0.00636229i
$372$		0			0
$373$			147.279i			0.394850i	0.980318	+	0.197425i	$0.0632578\pi$
$373$			147.279i			0.394850i	−0.980318	+	0.197425i	$0.936742\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	243.682			0.646372
$378$		0			0
$379$	−76.7735			−0.202569			−0.101284	−	0.994858i	$-0.532295\pi$
$379$	−76.7735			−0.202569			−0.101284	+	0.994858i	$0.532295\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	434.879			1.13546			0.567728	−	0.823216i	$-0.307823\pi$
$383$	434.879			1.13546			0.567728	+	0.823216i	$0.307823\pi$
$384$		0			0
$385$	−297.815	+	652.373i	−0.773544	+	1.69448i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−506.077			−1.30097			−0.650485	−	0.759519i	$-0.725434\pi$
$389$	−506.077			−1.30097			−0.650485	+	0.759519i	$0.725434\pi$
$390$		0			0
$391$			16.9237i			0.0432832i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			157.905i			0.399760i
$396$		0			0
$397$	−275.285			−0.693413			−0.346707	−	0.937974i	$-0.612700\pi$
$397$	−275.285			−0.693413			−0.346707	+	0.937974i	$0.612700\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	469.509			1.17085			0.585423	−	0.810728i	$-0.300929\pi$
$401$	469.509			1.17085			0.585423	+	0.810728i	$0.300929\pi$
$402$		0			0
$403$			250.699i			0.622083i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	562.879	+	256.960i	1.38300	+	0.631350i
$408$		0			0
$409$		−	682.090i		−	1.66770i	−0.551990	−	0.833851i	$-0.686131\pi$
$409$		−	682.090i		−	1.66770i	0.551990	−	0.833851i	$-0.313869\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	228.334i		−	0.552867i
$414$		0			0
$415$			379.637i			0.914788i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	212.596			0.507388			0.253694	−	0.967284i	$-0.418354\pi$
$419$	212.596			0.507388			0.253694	+	0.967284i	$0.418354\pi$
$420$		0			0
$421$	347.762			0.826037			0.413019	−	0.910723i	$-0.364474\pi$
$421$	347.762			0.826037			0.413019	+	0.910723i	$0.364474\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.70516i		−	0.00401214i
$426$		0			0
$427$	−981.006			−2.29744
$428$		0			0
$429$		0			0
$430$		0			0
$431$			392.374i			0.910381i	0.890394	+	0.455191i	$0.150429\pi$
$431$			392.374i			0.910381i	−0.890394	+	0.455191i	$0.849571\pi$
$432$		0			0
$433$	−554.325			−1.28020			−0.640099	−	0.768293i	$-0.721107\pi$
$433$	−554.325			−1.28020			−0.640099	+	0.768293i	$0.721107\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	345.658i		−	0.790979i
$438$		0			0
$439$		−	130.153i		−	0.296477i	−0.988952	−	0.148239i	$-0.952640\pi$
$439$		−	130.153i		−	0.296477i	0.988952	−	0.148239i	$-0.0473603\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	107.762			0.243254			0.121627	−	0.992576i	$-0.461189\pi$
$443$	107.762			0.243254			0.121627	+	0.992576i	$0.461189\pi$
$444$		0			0
$445$	51.1123			0.114859
$446$		0			0
$447$		0			0
$448$		0			0
$449$	86.8082			0.193337			0.0966683	−	0.995317i	$-0.469181\pi$
$449$	86.8082			0.193337			0.0966683	+	0.995317i	$0.469181\pi$
$450$		0			0
$451$	76.9676	−	168.600i	0.170660	−	0.373837i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−972.356			−2.13705
$456$		0			0
$457$			358.902i			0.785344i	0.919679	+	0.392672i	$0.128449\pi$
$457$			358.902i			0.785344i	−0.919679	+	0.392672i	$0.871551\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	463.660i		−	1.00577i	−0.864353	−	0.502885i	$-0.832272\pi$
$461$		−	463.660i		−	1.00577i	0.864353	−	0.502885i	$-0.167728\pi$
$462$		0			0
$463$	21.6620			0.0467862			0.0233931	−	0.999726i	$-0.492553\pi$
$463$	21.6620			0.0467862			0.0233931	+	0.999726i	$0.492553\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	383.773			0.821784			0.410892	−	0.911684i	$-0.365217\pi$
$467$	383.773			0.821784			0.410892	+	0.911684i	$0.365217\pi$
$468$		0			0
$469$		−	139.361i		−	0.297145i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−164.994	+	361.425i	−0.348824	+	0.764112i
$474$		0			0
$475$			34.8269i			0.0733199i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			437.216i			0.912768i	0.889783	+	0.456384i	$0.150856\pi$
$479$			437.216i			0.912768i	−0.889783	+	0.456384i	$0.849144\pi$
$480$		0			0
$481$			838.966i			1.74421i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−421.668			−0.869419
$486$		0			0
$487$	158.744			0.325963			0.162981	−	0.986629i	$-0.447889\pi$
$487$	158.744			0.325963			0.162981	+	0.986629i	$0.447889\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			899.040i			1.83104i	0.402273	+	0.915520i	$0.368220\pi$
$491$			899.040i			1.83104i	−0.402273	+	0.915520i	$0.631780\pi$
$492$		0			0
$493$	−23.2587			−0.0471780
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1773.93i		−	3.56927i
$498$		0			0
$499$	501.946			1.00590			0.502951	−	0.864315i	$-0.332247\pi$
$499$	501.946			1.00590			0.502951	+	0.864315i	$0.332247\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	245.501i		−	0.488074i	−0.969766	−	0.244037i	$-0.921528\pi$
$503$		−	245.501i		−	0.488074i	0.969766	−	0.244037i	$-0.0784718\pi$
$504$		0			0
$505$		−	656.184i		−	1.29937i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.33172			0.0183334			0.00916672	−	0.999958i	$-0.497082\pi$
$509$	9.33172			0.0183334			0.00916672	+	0.999958i	$0.497082\pi$
$510$		0			0
$511$	−1047.03			−2.04899
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−246.021			−0.477710
$516$		0			0
$517$	218.688	+	99.8332i	0.422995	+	0.193101i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−862.330			−1.65514			−0.827572	−	0.561360i	$-0.810278\pi$
$521$	−862.330			−1.65514			−0.827572	+	0.561360i	$0.810278\pi$
$522$		0			0
$523$			391.948i			0.749423i	0.927142	+	0.374711i	$0.122258\pi$
$523$			391.948i			0.749423i	−0.927142	+	0.374711i	$0.877742\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	23.9285i		−	0.0454051i
$528$		0			0
$529$	−387.670			−0.732835
$530$		0			0
$531$		0			0
$532$		0			0
$533$	251.297			0.471477
$534$		0			0
$535$		−	1032.39i		−	1.92971i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1133.13	−	517.285i	−2.10228	−	0.959712i
$540$		0			0
$541$		−	238.052i		−	0.440023i	−0.975497	−	0.220011i	$-0.929391\pi$
$541$		−	238.052i		−	0.440023i	0.975497	−	0.220011i	$-0.0706095\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			842.037i			1.54502i
$546$		0			0
$547$			417.066i			0.762461i	0.924480	+	0.381230i	$0.124500\pi$
$547$			417.066i			0.762461i	−0.924480	+	0.381230i	$0.875500\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	475.047			0.862154
$552$		0			0
$553$	−392.953			−0.710584
$554$		0			0
$555$		0			0
$556$		0			0
$557$			151.232i			0.271513i	0.990742	+	0.135756i	$0.0433464\pi$
$557$			151.232i			0.271513i	−0.990742	+	0.135756i	$0.956654\pi$
$558$		0			0
$559$	−538.700			−0.963686
$560$		0			0
$561$		0			0
$562$		0			0
$563$			875.506i			1.55507i	0.628838	+	0.777536i	$0.283531\pi$
$563$			875.506i			1.55507i	−0.628838	+	0.777536i	$0.716469\pi$
$564$		0			0
$565$	435.939			0.771573
$566$		0			0
$567$		0			0
$568$		0			0
$569$			706.404i			1.24148i	0.784015	+	0.620741i	$0.213168\pi$
$569$			706.404i			1.24148i	−0.784015	+	0.620741i	$0.786832\pi$
$570$		0			0
$571$			944.530i			1.65417i	0.562078	+	0.827084i	$0.310002\pi$
$571$			944.530i			1.65417i	−0.562078	+	0.827084i	$0.689998\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14.2398			−0.0247649
$576$		0			0
$577$	−638.796			−1.10710			−0.553549	−	0.832816i	$-0.686727\pi$
$577$	−638.796			−1.10710			−0.553549	+	0.832816i	$0.686727\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−944.741			−1.62606
$582$		0			0
$583$	1.85438	+	0.846540i	0.00318075	+	0.00145204i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	499.575			0.851065			0.425532	−	0.904943i	$-0.360087\pi$
$587$	499.575			0.851065			0.425532	+	0.904943i	$0.360087\pi$
$588$		0			0
$589$			488.726i			0.829756i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			71.9478i			0.121329i	0.998158	+	0.0606643i	$0.0193219\pi$
$593$			71.9478i			0.121329i	−0.998158	+	0.0606643i	$0.980678\pi$
$594$		0			0
$595$	92.8085			0.155981
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−770.834			−1.28687			−0.643434	−	0.765502i	$-0.722491\pi$
$599$	−770.834			−1.28687			−0.643434	+	0.765502i	$0.722491\pi$
$600$		0			0
$601$			545.840i			0.908219i	0.890946	+	0.454110i	$0.150043\pi$
$601$			545.840i			0.908219i	−0.890946	+	0.454110i	$0.849957\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−405.707	−	467.936i	−0.670590	−	0.773448i
$606$		0			0
$607$		−	659.902i		−	1.08715i	−0.839360	−	0.543576i	$-0.817070\pi$
$607$		−	659.902i		−	1.08715i	0.839360	−	0.543576i	$-0.182930\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			325.953i			0.533474i
$612$		0			0
$613$		−	752.909i		−	1.22824i	−0.789214	−	0.614118i	$-0.789512\pi$
$613$		−	752.909i		−	1.22824i	0.789214	−	0.614118i	$-0.210488\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	49.7497			0.0806316			0.0403158	−	0.999187i	$-0.487164\pi$
$617$	49.7497			0.0806316			0.0403158	+	0.999187i	$0.487164\pi$
$618$		0			0
$619$	−501.926			−0.810866			−0.405433	−	0.914125i	$-0.632879\pi$
$619$	−501.926			−0.810866			−0.405433	+	0.914125i	$0.632879\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			127.195i			0.204165i
$624$		0			0
$625$	−653.510			−1.04562
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	80.0768i		−	0.127308i
$630$		0			0
$631$	545.597			0.864655			0.432327	−	0.901717i	$-0.357693\pi$
$631$	545.597			0.864655			0.432327	+	0.901717i	$0.357693\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			318.253i			0.501187i
$636$		0			0
$637$		−	1688.92i		−	2.65137i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	450.576			0.702927			0.351463	−	0.936202i	$-0.385684\pi$
$641$	450.576			0.702927			0.351463	+	0.936202i	$0.385684\pi$
$642$		0			0
$643$	864.073			1.34382			0.671908	−	0.740635i	$-0.265475\pi$
$643$	864.073			1.34382			0.671908	+	0.740635i	$0.265475\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1200.90			−1.85610			−0.928052	−	0.372450i	$-0.878518\pi$
$647$	−1200.90			−1.85610			−0.928052	+	0.372450i	$0.878518\pi$
$648$		0			0
$649$	179.383	+	81.8900i	0.276399	+	0.126179i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−272.815			−0.417787			−0.208894	−	0.977938i	$-0.566986\pi$
$653$	−272.815			−0.417787			−0.208894	+	0.977938i	$0.566986\pi$
$654$		0			0
$655$			335.871i			0.512780i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	328.580i		−	0.498604i	−0.968426	−	0.249302i	$-0.919799\pi$
$659$		−	328.580i		−	0.498604i	0.968426	−	0.249302i	$-0.0802011\pi$
$660$		0			0
$661$	123.900			0.187444			0.0937220	−	0.995598i	$-0.470124\pi$
$661$	123.900			0.187444			0.0937220	+	0.995598i	$0.470124\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1895.56			−2.85047
$666$		0			0
$667$			194.234i			0.291205i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	351.829	−	770.694i	0.524335	−	1.14858i
$672$		0			0
$673$		−	782.630i		−	1.16290i	−0.813583	−	0.581449i	$-0.802486\pi$
$673$		−	782.630i		−	1.16290i	0.813583	−	0.581449i	$-0.197514\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1209.30i			1.78626i	0.449803	+	0.893128i	$0.351494\pi$
$677$			1209.30i			1.78626i	−0.449803	+	0.893128i	$0.648506\pi$
$678$		0			0
$679$		−	1049.34i		−	1.54541i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	113.737			0.166525			0.0832625	−	0.996528i	$-0.473466\pi$
$683$	113.737			0.166525			0.0832625	+	0.996528i	$0.473466\pi$
$684$		0			0
$685$	−630.987			−0.921149
$686$		0			0
$687$		0			0
$688$		0			0
$689$			2.76393i			0.00401151i
$690$		0			0
$691$	460.880			0.666975			0.333487	−	0.942755i	$-0.391775\pi$
$691$	460.880			0.666975			0.333487	+	0.942755i	$0.391775\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			392.533i			0.564795i
$696$		0			0
$697$	−23.9855			−0.0344125
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1209.73i		−	1.72572i	−0.505446	−	0.862858i	$-0.668672\pi$
$701$		−	1209.73i		−	1.72572i	0.505446	−	0.862858i	$-0.331328\pi$
$702$		0			0
$703$			1635.52i			2.32649i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1632.94			2.30967
$708$		0			0
$709$	346.827			0.489178			0.244589	−	0.969627i	$-0.421347\pi$
$709$	346.827			0.489178			0.244589	+	0.969627i	$0.421347\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−199.827			−0.280262
$714$		0			0
$715$	348.727	−	763.899i	0.487730	−	1.06839i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1053.75			−1.46558			−0.732788	−	0.680457i	$-0.761781\pi$
$719$	−1053.75			−1.46558			−0.732788	+	0.680457i	$0.761781\pi$
$720$		0			0
$721$		−	612.232i		−	0.849143i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	19.5702i		−	0.0269933i
$726$		0			0
$727$	−649.906			−0.893956			−0.446978	−	0.894545i	$-0.647500\pi$
$727$	−649.906			−0.893956			−0.446978	+	0.894545i	$0.647500\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	51.4173			0.0703383
$732$		0			0
$733$			1115.68i			1.52207i	0.648710	+	0.761036i	$0.275309\pi$
$733$			1115.68i			1.52207i	−0.648710	+	0.761036i	$0.724691\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	109.484	+	49.9806i	0.148554	+	0.0678163i
$738$		0			0
$739$			1274.73i			1.72494i	0.506106	+	0.862471i	$0.331085\pi$
$739$			1274.73i			1.72494i	−0.506106	+	0.862471i	$0.668915\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			347.381i			0.467538i	0.972292	+	0.233769i	$0.0751059\pi$
$743$			347.381i			0.467538i	−0.972292	+	0.233769i	$0.924894\pi$
$744$		0			0
$745$			1040.49i			1.39664i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2569.15			3.43011
$750$		0			0
$751$	863.715			1.15009			0.575043	−	0.818123i	$-0.304985\pi$
$751$	863.715			1.15009			0.575043	+	0.818123i	$0.304985\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	558.417i		−	0.739625i
$756$		0			0
$757$	−6.91483			−0.00913452			−0.00456726	−	0.999990i	$-0.501454\pi$
$757$	−6.91483			−0.00913452			−0.00456726	+	0.999990i	$0.501454\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	734.052i		−	0.964589i	−0.876009	−	0.482295i	$-0.839803\pi$
$761$		−	734.052i		−	0.964589i	0.876009	−	0.482295i	$-0.160197\pi$
$762$		0			0
$763$	−2095.44			−2.74632
$764$		0			0
$765$		0			0
$766$		0			0
$767$			267.369i			0.348590i
$768$		0			0
$769$			1277.23i			1.66090i	0.557096	+	0.830448i	$0.311915\pi$
$769$			1277.23i			1.66090i	−0.557096	+	0.830448i	$0.688085\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1138.74			1.47315			0.736573	−	0.676359i	$-0.236443\pi$
$773$	1138.74			1.47315			0.736573	+	0.676359i	$0.236443\pi$
$774$		0			0
$775$	20.1337			0.0259790
$776$		0			0
$777$		0			0
$778$		0			0
$779$	489.891			0.628872
$780$		0			0
$781$	1393.63	+	636.204i	1.78441	+	0.814601i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	371.305			0.473000
$786$		0			0
$787$			715.623i			0.909305i	0.890669	+	0.454653i	$0.150237\pi$
$787$			715.623i			0.909305i	−0.890669	+	0.454653i	$0.849763\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1084.85i			1.37149i
$792$		0			0
$793$	1148.71			1.44857
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1348.15			1.69153			0.845767	−	0.533553i	$-0.179143\pi$
$797$	1348.15			1.69153			0.845767	+	0.533553i	$0.179143\pi$
$798$		0			0
$799$		−	31.1112i		−	0.0389377i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	375.509	−	822.566i	0.467632	−	1.02437i
$804$		0			0
$805$		−	775.044i		−	0.962788i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			779.307i			0.963297i	0.876365	+	0.481648i	$0.159962\pi$
$809$			779.307i			0.963297i	−0.876365	+	0.481648i	$0.840038\pi$
$810$		0			0
$811$		−	407.625i		−	0.502620i	−0.967907	−	0.251310i	$-0.919139\pi$
$811$		−	407.625i		−	0.502620i	0.967907	−	0.251310i	$-0.0808613\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−738.876			−0.906596
$816$		0			0
$817$	−1050.17			−1.28540
$818$		0			0
$819$		0			0
$820$		0			0
$821$			144.497i			0.176001i	0.996120	+	0.0880007i	$0.0280478\pi$
$821$			144.497i			0.176001i	−0.996120	+	0.0880007i	$0.971952\pi$
$822$		0			0
$823$	1257.65			1.52813			0.764063	−	0.645142i	$-0.223202\pi$
$823$	1257.65			1.52813			0.764063	+	0.645142i	$0.223202\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1058.91i		−	1.28043i	−0.768198	−	0.640213i	$-0.778846\pi$
$827$		−	1058.91i		−	1.28043i	0.768198	−	0.640213i	$-0.221154\pi$
$828$		0			0
$829$	237.316			0.286267			0.143134	−	0.989703i	$-0.454282\pi$
$829$	237.316			0.286267			0.143134	+	0.989703i	$0.454282\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			161.202i			0.193520i
$834$		0			0
$835$		−	619.247i		−	0.741613i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	920.950			1.09768			0.548838	−	0.835929i	$-0.315070\pi$
$839$	920.950			1.09768			0.548838	+	0.835929i	$0.315070\pi$
$840$		0			0
$841$	574.059			0.682591
$842$		0			0
$843$		0			0
$844$		0			0
$845$	273.578			0.323761
$846$		0			0
$847$	1164.48	−	1009.62i	1.37482	−	1.19199i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−668.722			−0.785807
$852$		0			0
$853$			751.983i			0.881574i	0.897612	+	0.440787i	$0.145301\pi$
$853$			751.983i			0.881574i	−0.897612	+	0.440787i	$0.854699\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1447.30i			1.68880i	0.535715	+	0.844399i	$0.320042\pi$
$857$			1447.30i			1.68880i	−0.535715	+	0.844399i	$0.679958\pi$
$858$		0			0
$859$	−1153.12			−1.34240			−0.671198	−	0.741278i	$-0.734220\pi$
$859$	−1153.12			−1.34240			−0.671198	+	0.741278i	$0.734220\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1231.34			1.42681			0.713405	−	0.700752i	$-0.247152\pi$
$863$	1231.34			1.42681			0.713405	+	0.700752i	$0.247152\pi$
$864$		0			0
$865$		−	93.8347i		−	0.108479i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	140.929	−	308.710i	0.162174	−	0.355248i
$870$		0			0
$871$			163.185i			0.187354i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	1551.77i		−	1.77345i
$876$		0			0
$877$			148.205i			0.168991i	0.996424	+	0.0844953i	$0.0269278\pi$
$877$			148.205i			0.168991i	−0.996424	+	0.0844953i	$0.973072\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1371.29			−1.55652			−0.778260	−	0.627942i	$-0.783898\pi$
$881$	−1371.29			−1.55652			−0.778260	+	0.627942i	$0.783898\pi$
$882$		0			0
$883$	183.895			0.208262			0.104131	−	0.994564i	$-0.466794\pi$
$883$	183.895			0.208262			0.104131	+	0.994564i	$0.466794\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1524.66i		−	1.71890i	−0.511222	−	0.859448i	$-0.670807\pi$
$887$		−	1524.66i		−	1.71890i	0.511222	−	0.859448i	$-0.329193\pi$
$888$		0			0
$889$	−791.986			−0.890872
$890$		0			0
$891$		0			0
$892$		0			0
$893$			635.429i			0.711567i
$894$		0			0
$895$	−1319.65			−1.47447
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	274.628i		−	0.305482i
$900$		0			0
$901$		−	0.263809i		−	0.000292796i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−425.462			−0.470124
$906$		0			0
$907$	1274.87			1.40559			0.702797	−	0.711390i	$-0.251934\pi$
$907$	1274.87			1.40559			0.702797	+	0.711390i	$0.251934\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	659.601			0.724041			0.362020	−	0.932170i	$-0.382087\pi$
$911$	659.601			0.724041			0.362020	+	0.932170i	$0.382087\pi$
$912$		0			0
$913$	338.823	−	742.204i	0.371110	−	0.812929i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−835.827			−0.911479
$918$		0			0
$919$		−	1094.37i		−	1.19082i	−0.803420	−	0.595412i	$-0.796989\pi$
$919$		−	1094.37i		−	1.19082i	0.803420	−	0.595412i	$-0.203011\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2077.19i			2.25047i
$924$		0			0
$925$	67.3775			0.0728405
$926$		0			0
$927$		0			0
$928$		0			0
$929$	693.167			0.746143			0.373072	−	0.927803i	$-0.378305\pi$
$929$	693.167			0.746143			0.373072	+	0.927803i	$0.378305\pi$
$930$		0			0
$931$		−	3292.47i		−	3.53649i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−33.2849	+	72.9119i	−0.0355989	+	0.0779806i
$936$		0			0
$937$			394.132i			0.420632i	0.977633	+	0.210316i	$0.0674493\pi$
$937$			394.132i			0.420632i	−0.977633	+	0.210316i	$0.932551\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			387.265i			0.411547i	0.978600	+	0.205773i	$0.0659710\pi$
$941$			387.265i			0.411547i	−0.978600	+	0.205773i	$0.934029\pi$
$942$		0			0
$943$			200.303i			0.212411i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.2644			0.0435738			0.0217869	−	0.999763i	$-0.493064\pi$
$947$	41.2644			0.0435738			0.0217869	+	0.999763i	$0.493064\pi$
$948$		0			0
$949$	1226.03			1.29191
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1082.94i		−	1.13635i	−0.822909	−	0.568173i	$-0.807650\pi$
$953$		−	1082.94i		−	1.13635i	0.822909	−	0.568173i	$-0.192350\pi$
$954$		0			0
$955$	−812.528			−0.850814
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1570.24i		−	1.63737i
$960$		0			0
$961$	−678.464			−0.705998
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1088.55i		−	1.12803i
$966$		0			0
$967$			1004.94i			1.03924i	0.854399	+	0.519618i	$0.173926\pi$
$967$			1004.94i			1.03924i	−0.854399	+	0.519618i	$0.826074\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−123.788			−0.127485			−0.0637425	−	0.997966i	$-0.520304\pi$
$971$	−123.788			−0.127485			−0.0637425	+	0.997966i	$0.520304\pi$
$972$		0			0
$973$	−976.833			−1.00394
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1471.46			−1.50610			−0.753051	−	0.657962i	$-0.771419\pi$
$977$	−1471.46			−1.50610			−0.753051	+	0.657962i	$0.771419\pi$
$978$		0			0
$979$	−99.9264	−	45.6173i	−0.102070	−	0.0465958i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−524.027			−0.533089			−0.266545	−	0.963823i	$-0.585882\pi$
$983$	−524.027			−0.533089			−0.266545	+	0.963823i	$0.585882\pi$
$984$		0			0
$985$			187.604i			0.190461i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	429.387i		−	0.434162i
$990$		0			0
$991$	392.423			0.395986			0.197993	−	0.980203i	$-0.436558\pi$
$991$	392.423			0.395986			0.197993	+	0.980203i	$0.436558\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−725.929			−0.729577
$996$		0			0
$997$			1166.17i			1.16967i	0.811151	+	0.584837i	$0.198842\pi$
$997$			1166.17i			1.16967i	−0.811151	+	0.584837i	$0.801158\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.k.1297.1		6
3.2	odd	2		176.3.h.d.65.5		6
4.3	odd	2		792.3.j.a.505.2		6
11.10	odd	2	inner	1584.3.j.k.1297.2		6
12.11	even	2		88.3.h.a.65.2	yes	6
24.5	odd	2		704.3.h.g.65.1		6
24.11	even	2		704.3.h.h.65.6		6
33.32	even	2		176.3.h.d.65.6		6
44.43	even	2		792.3.j.a.505.1		6
132.131	odd	2		88.3.h.a.65.1	✓	6
264.131	odd	2		704.3.h.h.65.5		6
264.197	even	2		704.3.h.g.65.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.3.h.a.65.1	✓	6	132.131	odd	2
88.3.h.a.65.2	yes	6	12.11	even	2
176.3.h.d.65.5		6	3.2	odd	2
176.3.h.d.65.6		6	33.32	even	2
704.3.h.g.65.1		6	24.5	odd	2
704.3.h.g.65.2		6	264.197	even	2
704.3.h.h.65.5		6	264.131	odd	2
704.3.h.h.65.6		6	24.11	even	2
792.3.j.a.505.1		6	44.43	even	2
792.3.j.a.505.2		6	4.3	odd	2
1584.3.j.k.1297.1		6	1.1	even	1	trivial
1584.3.j.k.1297.2		6	11.10	odd	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:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1584.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.11838 q^{5} -12.7373i q^{7} +(10.0066 + 4.56811i) q^{11} +14.9148i q^{13} -1.42357i q^{17} +29.0756i q^{19} -11.8882 q^{23} +1.19781 q^{25} -16.3383i q^{29} +16.8088 q^{31} +65.1942i q^{35} +56.2507 q^{37} -16.8489i q^{41} +36.1186i q^{43} +21.8544 q^{47} -113.238 q^{49} +0.185315 q^{53} +(-51.2177 - 23.3813i) q^{55} +17.9264 q^{59} -77.0185i q^{61} -76.3394i q^{65} +10.9412 q^{67} +139.271 q^{71} -82.2022i q^{73} +(58.1853 - 127.457i) q^{77} -30.8506i q^{79} -74.1713i q^{83} +7.28637i q^{85} -9.98603 q^{89} +189.973 q^{91} -148.820i q^{95} +82.3831 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{11} - 52 q^{23} + 22 q^{25} + 36 q^{31} - 48 q^{37} - 60 q^{47} - 170 q^{49} - 108 q^{53} - 172 q^{55} + 236 q^{59} + 292 q^{67} + 300 q^{71} + 240 q^{77} - 384 q^{89} + 48 q^{91} + 400 q^{97}+O(q^{100}) \) 6 * q + 10 * q^11 - 52 * q^23 + 22 * q^25 + 36 * q^31 - 48 * q^37 - 60 * q^47 - 170 * q^49 - 108 * q^53 - 172 * q^55 + 236 * q^59 + 292 * q^67 + 300 * q^71 + 240 * q^77 - 384 * q^89 + 48 * q^91 + 400 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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