LMFDB - Embedded newform 1584.3.j.g.1297.2

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)

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);

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")

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 = [4,0,0,0,-4,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,52,0,-84] 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:	$4$
Coefficient field:	4.0.131904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 16x^{2} - 6x + 69 $ x^4 - 2x^3 + 16x^2 - 6*x + 69
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.2
Root			$1.36603 + 3.25547i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.g.1297.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{5} +11.2773i q^{7} +(10.1962 + 4.12777i) q^{11} +19.5328i q^{13} +30.8101i q^{17} -8.25554i q^{19} +18.1962 q^{23} -17.5359 q^{25} -45.1091i q^{29} +28.2487 q^{31} -30.8101i q^{35} -61.0333 q^{37} -6.04347i q^{41} -8.25554i q^{43} +11.7654 q^{47} -78.1769 q^{49} +74.1577 q^{53} +(-27.8564 - 11.2773i) q^{55} -4.10512 q^{59} +81.1530i q^{61} -53.3646i q^{65} -69.4256 q^{67} +35.2679 q^{71} -6.04347i q^{73} +(-46.5500 + 114.985i) q^{77} +66.8540i q^{79} -34.6415i q^{83} -84.1747i q^{85} -103.962 q^{89} -220.277 q^{91} +22.5545i q^{95} -33.3872 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 20 q^{11} + 52 q^{23} - 84 q^{25} + 16 q^{31} - 64 q^{37} - 140 q^{47} - 188 q^{49} + 68 q^{53} - 56 q^{55} + 136 q^{59} - 56 q^{67} + 148 q^{71} + 84 q^{77} - 208 q^{89} - 216 q^{91} + 296 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 20 * q^11 + 52 * q^23 - 84 * q^25 + 16 * q^31 - 64 * q^37 - 140 * q^47 - 188 * q^49 + 68 * q^53 - 56 * q^55 + 136 * q^59 - 56 * q^67 + 148 * q^71 + 84 * q^77 - 208 * q^89 - 216 * q^91 + 296 * 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
$5$	−2.73205			−0.546410			−0.273205	−	0.961956i	$-0.588084\pi$
$5$	−2.73205			−0.546410			−0.273205	+	0.961956i	$0.588084\pi$
$6$		0			0
$7$			11.2773i			1.61104i	0.592569	+	0.805520i	$0.298114\pi$
$7$			11.2773i			1.61104i	−0.592569	+	0.805520i	$0.701886\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.1962	+	4.12777i	0.926923	+	0.375252i
$11$	10.1962	+	4.12777i	0.926923	+	0.375252i
$12$		0			0
$13$			19.5328i			1.50252i	0.660004	+	0.751262i	$0.270555\pi$
$13$			19.5328i			1.50252i	−0.660004	+	0.751262i	$0.729445\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			30.8101i			1.81236i	0.422894	+	0.906179i	$0.361014\pi$
$17$			30.8101i			1.81236i	−0.422894	+	0.906179i	$0.638986\pi$
$18$		0			0
$19$		−	8.25554i		−	0.434502i	−0.976116	−	0.217251i	$-0.930291\pi$
$19$		−	8.25554i		−	0.434502i	0.976116	−	0.217251i	$-0.0697090\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	18.1962			0.791137			0.395569	−	0.918436i	$-0.370548\pi$
$23$	18.1962			0.791137			0.395569	+	0.918436i	$0.370548\pi$
$24$		0			0
$25$	−17.5359			−0.701436
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	45.1091i		−	1.55549i	−0.628582	−	0.777743i	$-0.716364\pi$
$29$		−	45.1091i		−	1.55549i	0.628582	−	0.777743i	$-0.283636\pi$
$30$		0			0
$31$	28.2487			0.911249			0.455624	−	0.890172i	$-0.349416\pi$
$31$	28.2487			0.911249			0.455624	+	0.890172i	$0.349416\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	30.8101i		−	0.880288i
$36$		0			0
$37$	−61.0333			−1.64955			−0.824775	−	0.565462i	$-0.808698\pi$
$37$	−61.0333			−1.64955			−0.824775	+	0.565462i	$0.808698\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.04347i		−	0.147402i	−0.997280	−	0.0737009i	$-0.976519\pi$
$41$		−	6.04347i		−	0.147402i	0.997280	−	0.0737009i	$-0.0234810\pi$
$42$		0			0
$43$		−	8.25554i		−	0.191989i	−0.995382	−	0.0959946i	$-0.969397\pi$
$43$		−	8.25554i		−	0.191989i	0.995382	−	0.0959946i	$-0.0306032\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	11.7654			0.250327			0.125164	−	0.992136i	$-0.460054\pi$
$47$	11.7654			0.250327			0.125164	+	0.992136i	$0.460054\pi$
$48$		0			0
$49$	−78.1769			−1.59545
$50$		0			0
$51$		0			0
$52$		0			0
$53$	74.1577			1.39920			0.699601	−	0.714534i	$-0.253361\pi$
$53$	74.1577			1.39920			0.699601	+	0.714534i	$0.253361\pi$
$54$		0			0
$55$	−27.8564	−	11.2773i	−0.506480	−	0.205041i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.10512			−0.0695783			−0.0347891	−	0.999395i	$-0.511076\pi$
$59$	−4.10512			−0.0695783			−0.0347891	+	0.999395i	$0.511076\pi$
$60$		0			0
$61$			81.1530i			1.33038i	0.746676	+	0.665188i	$0.231649\pi$
$61$			81.1530i			1.33038i	−0.746676	+	0.665188i	$0.768351\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	53.3646i		−	0.820994i
$66$		0			0
$67$	−69.4256			−1.03620			−0.518102	−	0.855319i	$-0.673361\pi$
$67$	−69.4256			−1.03620			−0.518102	+	0.855319i	$0.673361\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	35.2679			0.496732			0.248366	−	0.968666i	$-0.420106\pi$
$71$	35.2679			0.496732			0.248366	+	0.968666i	$0.420106\pi$
$72$		0			0
$73$		−	6.04347i		−	0.0827873i	−0.999143	−	0.0413937i	$-0.986820\pi$
$73$		−	6.04347i		−	0.0827873i	0.999143	−	0.0413937i	$-0.0131798\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−46.5500	+	114.985i	−0.604545	+	1.49331i
$78$		0			0
$79$			66.8540i			0.846253i	0.906071	+	0.423126i	$0.139067\pi$
$79$			66.8540i			0.846253i	−0.906071	+	0.423126i	$0.860933\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	34.6415i		−	0.417367i	−0.977983	−	0.208684i	$-0.933082\pi$
$83$		−	34.6415i		−	0.417367i	0.977983	−	0.208684i	$-0.0669179\pi$
$84$		0			0
$85$		−	84.1747i		−	0.990291i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−103.962			−1.16811			−0.584054	−	0.811715i	$-0.698534\pi$
$89$	−103.962			−1.16811			−0.584054	+	0.811715i	$0.698534\pi$
$90$		0			0
$91$	−220.277			−2.42063
$92$		0			0
$93$		0			0
$94$		0			0
$95$			22.5545i			0.237416i
$96$		0			0
$97$	−33.3872			−0.344197			−0.172099	−	0.985080i	$-0.555055\pi$
$97$	−33.3872			−0.344197			−0.172099	+	0.985080i	$0.555055\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	10.4676i		−	0.103640i	−0.998656	−	0.0518198i	$-0.983498\pi$
$101$		−	10.4676i		−	0.103640i	0.998656	−	0.0518198i	$-0.0165022\pi$
$102$		0			0
$103$	28.7461			0.279089			0.139544	−	0.990216i	$-0.455436\pi$
$103$	28.7461			0.279089			0.139544	+	0.990216i	$0.455436\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			148.007i			1.38324i	0.722260	+	0.691621i	$0.243103\pi$
$107$			148.007i			1.38324i	−0.722260	+	0.691621i	$0.756897\pi$
$108$		0			0
$109$		−	81.1530i		−	0.744523i	−0.928128	−	0.372261i	$-0.878582\pi$
$109$		−	81.1530i		−	0.744523i	0.928128	−	0.372261i	$-0.121418\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−135.387			−1.19812			−0.599058	−	0.800705i	$-0.704458\pi$
$113$	−135.387			−1.19812			−0.599058	+	0.800705i	$0.704458\pi$
$114$		0			0
$115$	−49.7128			−0.432285
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−347.454			−2.91978
$120$		0			0
$121$	86.9230	+	84.1747i	0.718372	+	0.695659i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	116.210			0.929682
$126$		0			0
$127$			155.453i			1.22404i	0.790843	+	0.612019i	$0.209642\pi$
$127$			155.453i			1.22404i	−0.790843	+	0.612019i	$0.790358\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	207.415i		−	1.58332i	−0.610961	−	0.791661i	$-0.709217\pi$
$131$		−	207.415i		−	1.58332i	0.610961	−	0.791661i	$-0.290783\pi$
$132$		0			0
$133$	93.1000			0.700000
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−27.9334			−0.203893			−0.101947	−	0.994790i	$-0.532507\pi$
$137$	−27.9334			−0.203893			−0.101947	+	0.994790i	$0.532507\pi$
$138$		0			0
$139$		−	9.87488i		−	0.0710423i	−0.999369	−	0.0355212i	$-0.988691\pi$
$139$		−	9.87488i		−	0.0710423i	0.999369	−	0.0355212i	$-0.0113091\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−80.6269	+	199.160i	−0.563825	+	1.39272i
$144$		0			0
$145$			123.240i			0.849933i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	169.969i		−	1.14073i	−0.821391	−	0.570365i	$-0.806802\pi$
$149$		−	169.969i		−	1.14073i	0.821391	−	0.570365i	$-0.193198\pi$
$150$		0			0
$151$		−	51.9622i		−	0.344121i	−0.985086	−	0.172060i	$-0.944958\pi$
$151$		−	51.9622i		−	0.344121i	0.985086	−	0.172060i	$-0.0550424\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−77.1769			−0.497916
$156$		0			0
$157$	153.426			0.977233			0.488617	−	0.872499i	$-0.337502\pi$
$157$	153.426			0.977233			0.488617	+	0.872499i	$0.337502\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			205.203i			1.27455i
$162$		0			0
$163$	35.9230			0.220387			0.110193	−	0.993910i	$-0.464853\pi$
$163$	35.9230			0.220387			0.110193	+	0.993910i	$0.464853\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			186.480i			1.11665i	0.829624	+	0.558323i	$0.188555\pi$
$167$			186.480i			1.11665i	−0.829624	+	0.558323i	$0.811445\pi$
$168$		0			0
$169$	−212.531			−1.25758
$170$		0			0
$171$		0			0
$172$		0			0
$173$			209.034i			1.20829i	0.796874	+	0.604146i	$0.206486\pi$
$173$			209.034i			1.20829i	−0.796874	+	0.604146i	$0.793514\pi$
$174$		0			0
$175$		−	197.757i		−	1.13004i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.0282			0.111889			0.0559446	−	0.998434i	$-0.482183\pi$
$179$	20.0282			0.111889			0.0559446	+	0.998434i	$0.482183\pi$
$180$		0			0
$181$	−110.890			−0.612650			−0.306325	−	0.951927i	$-0.599099\pi$
$181$	−110.890			−0.612650			−0.306325	+	0.951927i	$0.599099\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	166.746			0.901330
$186$		0			0
$187$	−127.177	+	314.144i	−0.680090	+	1.67992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−219.899			−1.15130			−0.575651	−	0.817696i	$-0.695251\pi$
$191$	−219.899			−1.15130			−0.575651	+	0.817696i	$0.695251\pi$
$192$		0			0
$193$		−	221.121i		−	1.14571i	−0.819658	−	0.572853i	$-0.805836\pi$
$193$		−	221.121i		−	1.14571i	0.819658	−	0.572853i	$-0.194164\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	223.926i		−	1.13668i	−0.822793	−	0.568340i	$-0.807586\pi$
$197$		−	223.926i		−	1.13668i	0.822793	−	0.568340i	$-0.192414\pi$
$198$		0			0
$199$	252.354			1.26811			0.634055	−	0.773288i	$-0.281389\pi$
$199$	252.354			1.26811			0.634055	+	0.773288i	$0.281389\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	508.708			2.50595
$204$		0			0
$205$			16.5111i			0.0805418i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	34.0770	−	84.1747i	0.163048	−	0.402750i
$210$		0			0
$211$			88.0061i			0.417091i	0.978013	+	0.208545i	$0.0668729\pi$
$211$			88.0061i			0.417091i	−0.978013	+	0.208545i	$0.933127\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			22.5545i			0.104905i
$216$		0			0
$217$			318.568i			1.46806i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−601.808			−2.72311
$222$		0			0
$223$	−228.277			−1.02366			−0.511832	−	0.859086i	$-0.671033\pi$
$223$	−228.277			−1.02366			−0.511832	+	0.859086i	$0.671033\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	28.0053i		−	0.123371i	−0.998096	−	0.0616857i	$-0.980352\pi$
$227$		−	28.0053i		−	0.123371i	0.998096	−	0.0616857i	$-0.0196476\pi$
$228$		0			0
$229$	9.29234			0.0405779			0.0202890	−	0.999794i	$-0.493541\pi$
$229$	9.29234			0.0405779			0.0202890	+	0.999794i	$0.493541\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			330.655i			1.41912i	0.704644	+	0.709561i	$0.251107\pi$
$233$			330.655i			1.41912i	−0.704644	+	0.709561i	$0.748893\pi$
$234$		0			0
$235$	−32.1436			−0.136781
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	462.744i		−	1.93617i	−0.250627	−	0.968084i	$-0.580637\pi$
$239$		−	462.744i		−	1.93617i	0.250627	−	0.968084i	$-0.419363\pi$
$240$		0			0
$241$		−	64.8589i		−	0.269124i	−0.990905	−	0.134562i	$-0.957037\pi$
$241$		−	64.8589i		−	0.269124i	0.990905	−	0.134562i	$-0.0429627\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	213.583			0.871769
$246$		0			0
$247$	161.254			0.652850
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−51.2923			−0.204352			−0.102176	−	0.994766i	$-0.532580\pi$
$251$	−51.2923			−0.204352			−0.102176	+	0.994766i	$0.532580\pi$
$252$		0			0
$253$	185.531	+	75.1095i	0.733323	+	0.296876i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−327.023			−1.27246			−0.636232	−	0.771498i	$-0.719508\pi$
$257$	−327.023			−1.27246			−0.636232	+	0.771498i	$0.719508\pi$
$258$		0			0
$259$		−	688.289i		−	2.65749i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			76.5119i			0.290920i	0.989364	+	0.145460i	$0.0464662\pi$
$263$			76.5119i			0.290920i	−0.989364	+	0.145460i	$0.953534\pi$
$264$		0			0
$265$	−202.603			−0.764538
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−103.488			−0.384716			−0.192358	−	0.981325i	$-0.561613\pi$
$269$	−103.488			−0.384716			−0.192358	+	0.981325i	$0.561613\pi$
$270$		0			0
$271$		−	138.942i		−	0.512700i	−0.966584	−	0.256350i	$-0.917480\pi$
$271$		−	138.942i		−	0.512700i	0.966584	−	0.256350i	$-0.0825200\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−178.799	−	72.3841i	−0.650177	−	0.263215i
$276$		0			0
$277$		−	253.927i		−	0.916702i	−0.888771	−	0.458351i	$-0.848440\pi$
$277$		−	253.927i		−	0.916702i	0.888771	−	0.458351i	$-0.151560\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	227.758i		−	0.810525i	−0.914200	−	0.405263i	$-0.867180\pi$
$281$		−	227.758i		−	0.810525i	0.914200	−	0.405263i	$-0.132820\pi$
$282$		0			0
$283$			403.770i			1.42675i	0.700783	+	0.713374i	$0.252834\pi$
$283$			403.770i			1.42675i	−0.700783	+	0.713374i	$0.747166\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	68.1539			0.237470
$288$		0			0
$289$	−660.261			−2.28464
$290$		0			0
$291$		0			0
$292$		0			0
$293$			8.84826i			0.0301988i	0.999886	+	0.0150994i	$0.00480648\pi$
$293$			8.84826i			0.0301988i	−0.999886	+	0.0150994i	$0.995194\pi$
$294$		0			0
$295$	11.2154			0.0380183
$296$		0			0
$297$		0			0
$298$		0			0
$299$			355.422i			1.18870i
$300$		0			0
$301$	93.1000			0.309302
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	221.714i		−	0.726931i
$306$		0			0
$307$			338.911i			1.10394i	0.833862	+	0.551972i	$0.186125\pi$
$307$			338.911i			1.10394i	−0.833862	+	0.551972i	$0.813875\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.52697			−0.0242025			−0.0121012	−	0.999927i	$-0.503852\pi$
$311$	−7.52697			−0.0242025			−0.0121012	+	0.999927i	$0.503852\pi$
$312$		0			0
$313$	−422.708			−1.35050			−0.675252	−	0.737587i	$-0.735965\pi$
$313$	−422.708			−1.35050			−0.675252	+	0.737587i	$0.735965\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	595.412			1.87827			0.939135	−	0.343549i	$-0.111629\pi$
$317$	595.412			1.87827			0.939135	+	0.343549i	$0.111629\pi$
$318$		0			0
$319$	186.200	−	459.939i	0.583699	−	1.44182i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	254.354			0.787473
$324$		0			0
$325$		−	342.525i		−	1.05392i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			132.681i			0.403287i
$330$		0			0
$331$	453.836			1.37111			0.685553	−	0.728023i	$-0.259561\pi$
$331$	453.836			1.37111			0.685553	+	0.728023i	$0.259561\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	189.674			0.566192
$336$		0			0
$337$			281.122i			0.834190i	0.908863	+	0.417095i	$0.136952\pi$
$337$			281.122i			0.834190i	−0.908863	+	0.417095i	$0.863048\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	288.028	+	116.604i	0.844657	+	0.341948i
$342$		0			0
$343$		−	329.036i		−	0.959289i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	683.865i		−	1.97079i	−0.170274	−	0.985397i	$-0.554465\pi$
$347$		−	683.865i		−	1.97079i	0.170274	−	0.985397i	$-0.445535\pi$
$348$		0			0
$349$			297.850i			0.853439i	0.904384	+	0.426719i	$0.140331\pi$
$349$			297.850i			0.853439i	−0.904384	+	0.426719i	$0.859669\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−382.000			−1.08215			−0.541076	−	0.840973i	$-0.681983\pi$
$353$	−382.000			−1.08215			−0.541076	+	0.840973i	$0.681983\pi$
$354$		0			0
$355$	−96.3538			−0.271419
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	25.3593i		−	0.0706388i	−0.999376	−	0.0353194i	$-0.988755\pi$
$359$		−	25.3593i		−	0.0706388i	0.999376	−	0.0353194i	$-0.0112449\pi$
$360$		0			0
$361$	292.846			0.811208
$362$		0			0
$363$		0			0
$364$		0			0
$365$			16.5111i			0.0452358i
$366$		0			0
$367$	−331.703			−0.903822			−0.451911	−	0.892063i	$-0.649257\pi$
$367$	−331.703			−0.903822			−0.451911	+	0.892063i	$0.649257\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			836.296i			2.25417i
$372$		0			0
$373$			427.134i			1.14513i	0.819859	+	0.572566i	$0.194052\pi$
$373$			427.134i			1.14513i	−0.819859	+	0.572566i	$0.805948\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	881.108			2.33716
$378$		0			0
$379$	122.508			0.323239			0.161620	−	0.986853i	$-0.448328\pi$
$379$	122.508			0.323239			0.161620	+	0.986853i	$0.448328\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	566.701			1.47964			0.739819	−	0.672806i	$-0.234911\pi$
$383$	566.701			1.47964			0.739819	+	0.672806i	$0.234911\pi$
$384$		0			0
$385$	127.177	−	314.144i	0.330330	−	0.815959i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	83.9859			0.215902			0.107951	−	0.994156i	$-0.465571\pi$
$389$	83.9859			0.215902			0.107951	+	0.994156i	$0.465571\pi$
$390$		0			0
$391$			560.625i			1.43382i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	182.648i		−	0.462401i
$396$		0			0
$397$	320.515			0.807343			0.403672	−	0.914904i	$-0.367734\pi$
$397$	320.515			0.807343			0.403672	+	0.914904i	$0.367734\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−428.823			−1.06938			−0.534692	−	0.845047i	$-0.679572\pi$
$401$	−428.823			−1.06938			−0.534692	+	0.845047i	$0.679572\pi$
$402$		0			0
$403$			551.777i			1.36917i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−622.305	−	251.931i	−1.52901	−	0.618996i
$408$		0			0
$409$			252.524i			0.617418i	0.951156	+	0.308709i	$0.0998970\pi$
$409$			252.524i			0.617418i	−0.951156	+	0.308709i	$0.900103\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	46.2945i		−	0.112093i
$414$		0			0
$415$			94.6423i			0.228054i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.80007			0.0233892			0.0116946	−	0.999932i	$-0.496277\pi$
$419$	9.80007			0.0233892			0.0116946	+	0.999932i	$0.496277\pi$
$420$		0			0
$421$	−190.084			−0.451507			−0.225754	−	0.974184i	$-0.572484\pi$
$421$	−190.084			−0.451507			−0.225754	+	0.974184i	$0.572484\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	540.283i		−	1.27125i
$426$		0			0
$427$	−915.184			−2.14329
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	276.698i		−	0.641991i	−0.947081	−	0.320995i	$-0.895983\pi$
$431$		−	276.698i		−	0.641991i	0.947081	−	0.320995i	$-0.104017\pi$
$432$		0			0
$433$	780.046			1.80149			0.900746	−	0.434346i	$-0.143021\pi$
$433$	780.046			1.80149			0.900746	+	0.434346i	$0.143021\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	150.219i		−	0.343751i
$438$		0			0
$439$			166.354i			0.378939i	0.981887	+	0.189470i	$0.0606769\pi$
$439$			166.354i			0.378939i	−0.981887	+	0.189470i	$0.939323\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−491.713			−1.10996			−0.554981	−	0.831863i	$-0.687274\pi$
$443$	−491.713			−1.10996			−0.554981	+	0.831863i	$0.687274\pi$
$444$		0			0
$445$	284.028			0.638266
$446$		0			0
$447$		0			0
$448$		0			0
$449$	323.513			0.720519			0.360259	−	0.932852i	$-0.382688\pi$
$449$	323.513			0.720519			0.360259	+	0.932852i	$0.382688\pi$
$450$		0			0
$451$	24.9461	−	61.6202i	0.0553128	−	0.136630i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	601.808			1.32265
$456$		0			0
$457$			588.038i			1.28673i	0.765558	+	0.643367i	$0.222463\pi$
$457$			588.038i			1.28673i	−0.765558	+	0.643367i	$0.777537\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			385.047i			0.835242i	0.908621	+	0.417621i	$0.137136\pi$
$461$			385.047i			0.835242i	−0.908621	+	0.417621i	$0.862864\pi$
$462$		0			0
$463$	262.946			0.567918			0.283959	−	0.958836i	$-0.408352\pi$
$463$	262.946			0.567918			0.283959	+	0.958836i	$0.408352\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	885.520			1.89619			0.948095	−	0.317988i	$-0.103007\pi$
$467$	885.520			1.89619			0.948095	+	0.317988i	$0.103007\pi$
$468$		0			0
$469$		−	782.932i		−	1.66936i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	34.0770	−	84.1747i	0.0720443	−	0.177959i
$474$		0			0
$475$			144.768i			0.304775i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			175.578i			0.366552i	0.983062	+	0.183276i	$0.0586702\pi$
$479$			175.578i			0.366552i	−0.983062	+	0.183276i	$0.941330\pi$
$480$		0			0
$481$		−	1192.15i		−	2.47849i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	91.2154			0.188073
$486$		0			0
$487$	−587.282			−1.20592			−0.602959	−	0.797772i	$-0.706012\pi$
$487$	−587.282			−1.20592			−0.602959	+	0.797772i	$0.706012\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			360.280i			0.733768i	0.930267	+	0.366884i	$0.119575\pi$
$491$			360.280i			0.733768i	−0.930267	+	0.366884i	$0.880425\pi$
$492$		0			0
$493$	1389.82			2.81910
$494$		0			0
$495$		0			0
$496$		0			0
$497$			397.726i			0.800254i
$498$		0			0
$499$	772.000			1.54709			0.773547	−	0.633739i	$-0.218481\pi$
$499$	772.000			1.54709			0.773547	+	0.633739i	$0.218481\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	183.241i		−	0.364297i	−0.983271	−	0.182148i	$-0.941695\pi$
$503$		−	183.241i		−	0.364297i	0.983271	−	0.182148i	$-0.0583051\pi$
$504$		0			0
$505$			28.5980i			0.0566297i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	417.304			0.819850			0.409925	−	0.912119i	$-0.365555\pi$
$509$	417.304			0.819850			0.409925	+	0.912119i	$0.365555\pi$
$510$		0			0
$511$	68.1539			0.133374
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−78.5359			−0.152497
$516$		0			0
$517$	119.962	+	48.5647i	0.232034	+	0.0939357i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−421.433			−0.808893			−0.404446	−	0.914562i	$-0.632536\pi$
$521$	−421.433			−0.808893			−0.404446	+	0.914562i	$0.632536\pi$
$522$		0			0
$523$		−	550.750i		−	1.05306i	−0.850157	−	0.526530i	$-0.823493\pi$
$523$		−	550.750i		−	1.05306i	0.850157	−	0.526530i	$-0.176507\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			870.345i			1.65151i
$528$		0			0
$529$	−197.900			−0.374102
$530$		0			0
$531$		0			0
$532$		0			0
$533$	118.046			0.221475
$534$		0			0
$535$		−	404.363i		−	0.755818i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−797.104	−	322.696i	−1.47886	−	0.598694i
$540$		0			0
$541$		−	180.219i		−	0.333123i	−0.986031	−	0.166561i	$-0.946734\pi$
$541$		−	180.219i		−	0.333123i	0.986031	−	0.166561i	$-0.0532664\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			221.714i			0.406815i
$546$		0			0
$547$		−	947.884i		−	1.73288i	−0.499284	−	0.866439i	$-0.666403\pi$
$547$		−	947.884i		−	1.73288i	0.499284	−	0.866439i	$-0.333597\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−372.400			−0.675862
$552$		0			0
$553$	−753.931			−1.36335
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	633.898i		−	1.13806i	−0.822317	−	0.569029i	$-0.807319\pi$
$557$		−	633.898i		−	1.13806i	0.822317	−	0.569029i	$-0.192681\pi$
$558$		0			0
$559$	161.254			0.288468
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	571.685i		−	1.01543i	−0.861526	−	0.507713i	$-0.830491\pi$
$563$		−	571.685i		−	1.01543i	0.861526	−	0.507713i	$-0.169509\pi$
$564$		0			0
$565$	369.885			0.654663
$566$		0			0
$567$		0			0
$568$		0			0
$569$			439.163i			0.771815i	0.922537	+	0.385908i	$0.126112\pi$
$569$			439.163i			0.771815i	−0.922537	+	0.385908i	$0.873888\pi$
$570$		0			0
$571$		−	802.089i		−	1.40471i	−0.711827	−	0.702355i	$-0.752132\pi$
$571$		−	802.089i		−	1.40471i	0.711827	−	0.702355i	$-0.247868\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−319.086			−0.554932
$576$		0			0
$577$	129.541			0.224508			0.112254	−	0.993680i	$-0.464193\pi$
$577$	129.541			0.224508			0.112254	+	0.993680i	$0.464193\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	390.662			0.672395
$582$		0			0
$583$	756.123	+	306.106i	1.29695	+	0.525053i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−676.238			−1.15202			−0.576012	−	0.817441i	$-0.695392\pi$
$587$	−676.238			−1.15202			−0.576012	+	0.817441i	$0.695392\pi$
$588$		0			0
$589$		−	233.208i		−	0.395939i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	265.204i		−	0.447224i	−0.974678	−	0.223612i	$-0.928215\pi$
$593$		−	265.204i		−	0.447224i	0.974678	−	0.223612i	$-0.0717848\pi$
$594$		0			0
$595$	949.261			1.59540
$596$		0			0
$597$		0			0
$598$		0			0
$599$	847.773			1.41531			0.707657	−	0.706556i	$-0.249752\pi$
$599$	847.773			1.41531			0.707657	+	0.706556i	$0.249752\pi$
$600$		0			0
$601$			256.948i			0.427535i	0.976885	+	0.213767i	$0.0685734\pi$
$601$			256.948i			0.427535i	−0.976885	+	0.213767i	$0.931427\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−237.478	−	229.970i	−0.392526	−	0.380115i
$606$		0			0
$607$			510.282i			0.840662i	0.907371	+	0.420331i	$0.138086\pi$
$607$			510.282i			0.840662i	−0.907371	+	0.420331i	$0.861914\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			229.811i			0.376122i
$612$		0			0
$613$			336.916i			0.549618i	0.961499	+	0.274809i	$0.0886146\pi$
$613$			336.916i			0.549618i	−0.961499	+	0.274809i	$0.911385\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−29.6077			−0.0479865			−0.0239933	−	0.999712i	$-0.507638\pi$
$617$	−29.6077			−0.0479865			−0.0239933	+	0.999712i	$0.507638\pi$
$618$		0			0
$619$	98.2102			0.158660			0.0793298	−	0.996848i	$-0.474722\pi$
$619$	98.2102			0.158660			0.0793298	+	0.996848i	$0.474722\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1172.40i		−	1.88187i
$624$		0			0
$625$	120.905			0.193448
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1880.44i		−	2.98957i
$630$		0			0
$631$	524.228			0.830789			0.415395	−	0.909641i	$-0.363643\pi$
$631$	524.228			0.830789			0.415395	+	0.909641i	$0.363643\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	424.705i		−	0.668827i
$636$		0			0
$637$		−	1527.01i		−	2.39720i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−596.823			−0.931081			−0.465541	−	0.885027i	$-0.654140\pi$
$641$	−596.823			−0.931081			−0.465541	+	0.885027i	$0.654140\pi$
$642$		0			0
$643$	645.769			1.00431			0.502153	−	0.864779i	$-0.332541\pi$
$643$	645.769			1.00431			0.502153	+	0.864779i	$0.332541\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	679.142			1.04968			0.524839	−	0.851201i	$-0.324125\pi$
$647$	679.142			1.04968			0.524839	+	0.851201i	$0.324125\pi$
$648$		0			0
$649$	−41.8564	−	16.9450i	−0.0644937	−	0.0261094i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−52.0910			−0.0797719			−0.0398859	−	0.999204i	$-0.512699\pi$
$653$	−52.0910			−0.0797719			−0.0398859	+	0.999204i	$0.512699\pi$
$654$		0			0
$655$			566.668i			0.865143i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			270.062i			0.409806i	0.978782	+	0.204903i	$0.0656878\pi$
$659$			270.062i			0.409806i	−0.978782	+	0.204903i	$0.934312\pi$
$660$		0			0
$661$	874.820			1.32348			0.661740	−	0.749733i	$-0.269818\pi$
$661$	874.820			1.32348			0.661740	+	0.749733i	$0.269818\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−254.354			−0.382487
$666$		0			0
$667$		−	820.812i		−	1.23060i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−334.981	+	827.448i	−0.499226	+	1.23316i
$672$		0			0
$673$			665.301i			0.988560i	0.869303	+	0.494280i	$0.164568\pi$
$673$			665.301i			0.988560i	−0.869303	+	0.494280i	$0.835432\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			664.116i			0.980968i	0.871450	+	0.490484i	$0.163180\pi$
$677$			664.116i			0.980968i	−0.871450	+	0.490484i	$0.836820\pi$
$678$		0			0
$679$		−	376.516i		−	0.554516i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−263.846			−0.386305			−0.193152	−	0.981169i	$-0.561871\pi$
$683$	−263.846			−0.386305			−0.193152	+	0.981169i	$0.561871\pi$
$684$		0			0
$685$	76.3154			0.111409
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1448.51i			2.10233i
$690$		0			0
$691$	477.636			0.691224			0.345612	−	0.938377i	$-0.387671\pi$
$691$	477.636			0.691224			0.345612	+	0.938377i	$0.387671\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			26.9787i			0.0388182i
$696$		0			0
$697$	186.200			0.267145
$698$		0			0
$699$		0			0
$700$		0			0
$701$			807.540i			1.15198i	0.817456	+	0.575991i	$0.195384\pi$
$701$			807.540i			1.15198i	−0.817456	+	0.575991i	$0.804616\pi$
$702$		0			0
$703$			503.863i			0.716732i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	118.046			0.166968
$708$		0			0
$709$	−643.051			−0.906983			−0.453492	−	0.891261i	$-0.649822\pi$
$709$	−643.051			−0.906983			−0.453492	+	0.891261i	$0.649822\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	514.018			0.720923
$714$		0			0
$715$	220.277	−	544.114i	0.308080	−	0.760999i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1236.58			1.71986			0.859929	−	0.510413i	$-0.170508\pi$
$719$	1236.58			1.71986			0.859929	+	0.510413i	$0.170508\pi$
$720$		0			0
$721$			324.178i			0.449623i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			791.029i			1.09107i
$726$		0			0
$727$	−629.713			−0.866180			−0.433090	−	0.901351i	$-0.642577\pi$
$727$	−629.713			−0.866180			−0.433090	+	0.901351i	$0.642577\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	254.354			0.347953
$732$		0			0
$733$			568.071i			0.774994i	0.921871	+	0.387497i	$0.126660\pi$
$733$			568.071i			0.774994i	−0.921871	+	0.387497i	$0.873340\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−707.874	−	286.573i	−0.960481	−	0.388837i
$738$		0			0
$739$			25.2005i			0.0341008i	0.999855	+	0.0170504i	$0.00542758\pi$
$739$			25.2005i			0.0341008i	−0.999855	+	0.0170504i	$0.994572\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1321.44i			1.77851i	0.457408	+	0.889257i	$0.348778\pi$
$743$			1321.44i			1.77851i	−0.457408	+	0.889257i	$0.651222\pi$
$744$		0			0
$745$			464.363i			0.623307i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1669.12			−2.22846
$750$		0			0
$751$	−544.305			−0.724774			−0.362387	−	0.932028i	$-0.618038\pi$
$751$	−544.305			−0.724774			−0.362387	+	0.932028i	$0.618038\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			141.963i			0.188031i
$756$		0			0
$757$	−231.254			−0.305487			−0.152744	−	0.988266i	$-0.548811\pi$
$757$	−231.254			−0.305487			−0.152744	+	0.988266i	$0.548811\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			737.230i			0.968765i	0.874856	+	0.484382i	$0.160956\pi$
$761$			737.230i			0.968765i	−0.874856	+	0.484382i	$0.839044\pi$
$762$		0			0
$763$	915.184			1.19946
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	80.1845i		−	0.104543i
$768$		0			0
$769$			973.836i			1.26637i	0.774002	+	0.633183i	$0.218252\pi$
$769$			973.836i			1.26637i	−0.774002	+	0.633183i	$0.781748\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−50.8653			−0.0658025			−0.0329013	−	0.999459i	$-0.510475\pi$
$773$	−50.8653			−0.0658025			−0.0329013	+	0.999459i	$0.510475\pi$
$774$		0			0
$775$	−495.367			−0.639183
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−49.8921			−0.0640464
$780$		0			0
$781$	359.597	+	145.578i	0.460432	+	0.186399i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−419.167			−0.533970
$786$		0			0
$787$			1129.51i			1.43520i	0.696453	+	0.717602i	$0.254760\pi$
$787$			1129.51i			1.43520i	−0.696453	+	0.717602i	$0.745240\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1526.80i		−	1.93021i
$792$		0			0
$793$	−1585.15			−1.99892
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1305.80			1.63840			0.819198	−	0.573511i	$-0.194419\pi$
$797$	1305.80			1.63840			0.819198	+	0.573511i	$0.194419\pi$
$798$		0			0
$799$			362.492i			0.453682i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24.9461	−	61.6202i	0.0310661	−	0.0767375i
$804$		0			0
$805$		−	560.625i		−	0.696429i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			153.458i			0.189688i	0.995492	+	0.0948441i	$0.0302353\pi$
$809$			153.458i			0.189688i	−0.995492	+	0.0948441i	$0.969765\pi$
$810$		0			0
$811$			807.265i			0.995394i	0.867351	+	0.497697i	$0.165821\pi$
$811$			807.265i			0.995394i	−0.867351	+	0.497697i	$0.834179\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−98.1436			−0.120422
$816$		0			0
$817$	−68.1539			−0.0834197
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	200.620i		−	0.244361i	−0.992508	−	0.122180i	$-0.961011\pi$
$821$		−	200.620i		−	0.244361i	0.992508	−	0.122180i	$-0.0389886\pi$
$822$		0			0
$823$	302.477			0.367530			0.183765	−	0.982970i	$-0.441172\pi$
$823$	302.477			0.367530			0.183765	+	0.982970i	$0.441172\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			91.8375i			0.111049i	0.998457	+	0.0555245i	$0.0176831\pi$
$827$			91.8375i			0.111049i	−0.998457	+	0.0555245i	$0.982317\pi$
$828$		0			0
$829$	231.457			0.279200			0.139600	−	0.990208i	$-0.455418\pi$
$829$	231.457			0.279200			0.139600	+	0.990208i	$0.455418\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2408.64i		−	2.89152i
$834$		0			0
$835$		−	509.472i		−	0.610147i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	609.581			0.726556			0.363278	−	0.931681i	$-0.381658\pi$
$839$	609.581			0.726556			0.363278	+	0.931681i	$0.381658\pi$
$840$		0			0
$841$	−1193.83			−1.41954
$842$		0			0
$843$		0			0
$844$		0			0
$845$	580.645			0.687154
$846$		0			0
$847$	−949.261	+	980.255i	−1.12073	+	1.15733i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1110.57			−1.30502
$852$		0			0
$853$			95.2932i			0.111715i	0.998439	+	0.0558577i	$0.0177893\pi$
$853$			95.2932i			0.111715i	−0.998439	+	0.0558577i	$0.982211\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	694.767i		−	0.810696i	−0.914162	−	0.405348i	$-0.867150\pi$
$857$		−	694.767i		−	0.810696i	0.914162	−	0.405348i	$-0.132850\pi$
$858$		0			0
$859$	−1073.78			−1.25003			−0.625017	−	0.780611i	$-0.714908\pi$
$859$	−1073.78			−1.25003			−0.625017	+	0.780611i	$0.714908\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−908.788			−1.05306			−0.526529	−	0.850157i	$-0.676507\pi$
$863$	−908.788			−1.05306			−0.526529	+	0.850157i	$0.676507\pi$
$864$		0			0
$865$		−	571.093i		−	0.660223i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−275.958	+	681.653i	−0.317558	+	0.784411i
$870$		0			0
$871$		−	1356.08i		−	1.55692i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			1310.53i			1.49775i
$876$		0			0
$877$		−	688.073i		−	0.784575i	−0.919843	−	0.392288i	$-0.871684\pi$
$877$		−	688.073i		−	0.784575i	0.919843	−	0.392288i	$-0.128316\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	229.902			0.260956			0.130478	−	0.991451i	$-0.458349\pi$
$881$	229.902			0.260956			0.130478	+	0.991451i	$0.458349\pi$
$882$		0			0
$883$	−11.8152			−0.0133807			−0.00669036	−	0.999978i	$-0.502130\pi$
$883$	−11.8152			−0.0133807			−0.00669036	+	0.999978i	$0.502130\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			109.100i			0.122999i	0.998107	+	0.0614995i	$0.0195883\pi$
$887$			109.100i			0.122999i	−0.998107	+	0.0614995i	$0.980412\pi$
$888$		0			0
$889$	−1753.08			−1.97197
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	97.1295i		−	0.108768i
$894$		0			0
$895$	−54.7180			−0.0611374
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1274.27i		−	1.41743i
$900$		0			0
$901$			2284.80i			2.53585i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	302.956			0.334758
$906$		0			0
$907$	−1116.05			−1.23048			−0.615240	−	0.788340i	$-0.710941\pi$
$907$	−1116.05			−1.23048			−0.615240	+	0.788340i	$0.710941\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−388.165			−0.426087			−0.213043	−	0.977043i	$-0.568338\pi$
$911$	−388.165			−0.426087			−0.213043	+	0.977043i	$0.568338\pi$
$912$		0			0
$913$	142.992	−	353.210i	0.156618	−	0.386867i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2339.08			2.55079
$918$		0			0
$919$		−	932.341i		−	1.01452i	−0.861794	−	0.507259i	$-0.830659\pi$
$919$		−	932.341i		−	1.01452i	0.861794	−	0.507259i	$-0.169341\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			688.882i			0.746351i
$924$		0			0
$925$	1070.27			1.15705
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−533.230			−0.573983			−0.286992	−	0.957933i	$-0.592655\pi$
$929$	−533.230			−0.573983			−0.286992	+	0.957933i	$0.592655\pi$
$930$		0			0
$931$			645.392i			0.693225i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	347.454	−	858.258i	0.371608	−	0.917923i
$936$		0			0
$937$		−	1272.22i		−	1.35776i	−0.734249	−	0.678880i	$-0.762466\pi$
$937$		−	1272.22i		−	1.35776i	0.734249	−	0.678880i	$-0.237534\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	558.572i		−	0.593594i	−0.954941	−	0.296797i	$-0.904082\pi$
$941$		−	558.572i		−	0.593594i	0.954941	−	0.296797i	$-0.0959184\pi$
$942$		0			0
$943$		−	109.968i		−	0.116615i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−334.087			−0.352785			−0.176392	−	0.984320i	$-0.556443\pi$
$947$	−334.087			−0.352785			−0.176392	+	0.984320i	$0.556443\pi$
$948$		0			0
$949$	118.046			0.124390
$950$		0			0
$951$		0			0
$952$		0			0
$953$			540.875i			0.567550i	0.958891	+	0.283775i	$0.0915869\pi$
$953$			540.875i			0.567550i	−0.958891	+	0.283775i	$0.908413\pi$
$954$		0			0
$955$	600.774			0.629083
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	315.012i		−	0.328480i
$960$		0			0
$961$	−163.010			−0.169626
$962$		0			0
$963$		0			0
$964$		0			0
$965$			604.115i			0.626026i
$966$		0			0
$967$			848.166i			0.877111i	0.898704	+	0.438556i	$0.144510\pi$
$967$			848.166i			0.877111i	−0.898704	+	0.438556i	$0.855490\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	366.123			0.377058			0.188529	−	0.982068i	$-0.439628\pi$
$971$	366.123			0.377058			0.188529	+	0.982068i	$0.439628\pi$
$972$		0			0
$973$	111.362			0.114452
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1482.33			1.51723			0.758613	−	0.651541i	$-0.225877\pi$
$977$	1482.33			1.51723			0.758613	+	0.651541i	$0.225877\pi$
$978$		0			0
$979$	−1060.01	−	429.129i	−1.08275	−	0.438334i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−64.1164			−0.0652253			−0.0326126	−	0.999468i	$-0.510383\pi$
$983$	−64.1164			−0.0652253			−0.0326126	+	0.999468i	$0.510383\pi$
$984$		0			0
$985$			611.778i			0.621094i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	150.219i		−	0.151890i
$990$		0			0
$991$	−545.931			−0.550889			−0.275444	−	0.961317i	$-0.588825\pi$
$991$	−545.931			−0.550889			−0.275444	+	0.961317i	$0.588825\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−689.443			−0.692908
$996$		0			0
$997$		−	85.1432i		−	0.0853994i	−0.999088	−	0.0426997i	$-0.986404\pi$
$997$		−	85.1432i		−	0.0853994i	0.999088	−	0.0426997i	$-0.0135959\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.g.1297.2		4
3.2	odd	2		528.3.j.b.241.2		4
4.3	odd	2		396.3.f.b.109.1		4
11.10	odd	2	inner	1584.3.j.g.1297.1		4
12.11	even	2		132.3.f.a.109.3	✓	4
24.5	odd	2		2112.3.j.c.769.4		4
24.11	even	2		2112.3.j.b.769.1		4
33.32	even	2		528.3.j.b.241.1		4
44.43	even	2		396.3.f.b.109.2		4
60.23	odd	4		3300.3.p.a.2749.5		8
60.47	odd	4		3300.3.p.a.2749.4		8
60.59	even	2		3300.3.b.a.901.2		4
132.131	odd	2		132.3.f.a.109.4	yes	4
264.131	odd	2		2112.3.j.b.769.2		4
264.197	even	2		2112.3.j.c.769.3		4
660.263	even	4		3300.3.p.a.2749.8		8
660.527	even	4		3300.3.p.a.2749.1		8
660.659	odd	2		3300.3.b.a.901.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.f.a.109.3	✓	4	12.11	even	2
132.3.f.a.109.4	yes	4	132.131	odd	2
396.3.f.b.109.1		4	4.3	odd	2
396.3.f.b.109.2		4	44.43	even	2
528.3.j.b.241.1		4	33.32	even	2
528.3.j.b.241.2		4	3.2	odd	2
1584.3.j.g.1297.1		4	11.10	odd	2	inner
1584.3.j.g.1297.2		4	1.1	even	1	trivial
2112.3.j.b.769.1		4	24.11	even	2
2112.3.j.b.769.2		4	264.131	odd	2
2112.3.j.c.769.3		4	264.197	even	2
2112.3.j.c.769.4		4	24.5	odd	2
3300.3.b.a.901.1		4	660.659	odd	2
3300.3.b.a.901.2		4	60.59	even	2
3300.3.p.a.2749.1		8	660.527	even	4
3300.3.p.a.2749.4		8	60.47	odd	4
3300.3.p.a.2749.5		8	60.23	odd	4
3300.3.p.a.2749.8		8	660.263	even	4

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)

\(f(q)\)	\(=\)	\(q-2.73205 q^{5} +11.2773i q^{7} +(10.1962 + 4.12777i) q^{11} +19.5328i q^{13} +30.8101i q^{17} -8.25554i q^{19} +18.1962 q^{23} -17.5359 q^{25} -45.1091i q^{29} +28.2487 q^{31} -30.8101i q^{35} -61.0333 q^{37} -6.04347i q^{41} -8.25554i q^{43} +11.7654 q^{47} -78.1769 q^{49} +74.1577 q^{53} +(-27.8564 - 11.2773i) q^{55} -4.10512 q^{59} +81.1530i q^{61} -53.3646i q^{65} -69.4256 q^{67} +35.2679 q^{71} -6.04347i q^{73} +(-46.5500 + 114.985i) q^{77} +66.8540i q^{79} -34.6415i q^{83} -84.1747i q^{85} -103.962 q^{89} -220.277 q^{91} +22.5545i q^{95} -33.3872 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 20 q^{11} + 52 q^{23} - 84 q^{25} + 16 q^{31} - 64 q^{37} - 140 q^{47} - 188 q^{49} + 68 q^{53} - 56 q^{55} + 136 q^{59} - 56 q^{67} + 148 q^{71} + 84 q^{77} - 208 q^{89} - 216 q^{91} + 296 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 20 * q^11 + 52 * q^23 - 84 * q^25 + 16 * q^31 - 64 * q^37 - 140 * q^47 - 188 * q^49 + 68 * q^53 - 56 * q^55 + 136 * q^59 - 56 * q^67 + 148 * q^71 + 84 * q^77 - 208 * q^89 - 216 * q^91 + 296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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