LMFDB - Embedded newform 1152.3.h.f.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(449,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1152.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.5473632256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 49x^{4} + 256 $ x^8 + 49*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$1.81129 + 1.81129i$ of defining polynomial
Character	$\chi$	$=$	1152.449
Dual form			1152.3.h.f.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.83095 q^{5} +2.82843 q^{7} +O(q^{10})$	$q-5.83095 q^{5} +2.82843 q^{7} -16.4924 q^{11} +8.24621i q^{13} -7.07107i q^{17} +23.3238i q^{19} -20.0000i q^{23} +9.00000 q^{25} +29.1548 q^{29} +42.4264 q^{31} -16.4924 q^{35} -49.4773i q^{37} +26.8701i q^{41} -44.0000i q^{47} -41.0000 q^{49} -29.1548 q^{53} +96.1665 q^{55} +65.9697 q^{59} +82.4621i q^{61} -48.0833i q^{65} -116.619i q^{67} -100.000i q^{71} +40.0000 q^{73} -46.6476 q^{77} +127.279 q^{79} +82.4621 q^{83} +41.2311i q^{85} +114.551i q^{89} +23.3238i q^{91} -136.000i q^{95} -40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 72 q^{25} - 328 q^{49} + 320 q^{73} - 320 q^{97}+O(q^{100}) $ 8 * q + 72 * q^25 - 328 * q^49 + 320 * q^73 - 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−5.83095	−1.16619	−0.583095	−	0.812404i	$-0.698159\pi$
$5$	−5.83095	−1.16619	−0.583095	+	0.812404i	$0.698159\pi$
$6$	0	0
$7$	2.82843	0.404061	0.202031	−	0.979379i	$-0.435246\pi$
$7$	2.82843	0.404061	0.202031	+	0.979379i	$0.435246\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−16.4924	−1.49931	−0.749656	−	0.661828i	$-0.769781\pi$
$11$	−16.4924	−1.49931	−0.749656	+	0.661828i	$0.769781\pi$
$12$	0	0
$13$	8.24621i	0.634324i	0.948371	+	0.317162i	$0.102730\pi$
$13$	8.24621i	0.634324i	−0.948371	+	0.317162i	$0.897270\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.07107i	− 0.415945i	−0.978135	−	0.207973i	$-0.933314\pi$
$17$	− 7.07107i	− 0.415945i	0.978135	−	0.207973i	$-0.0666865\pi$
$18$	0	0
$19$	23.3238i	1.22757i	0.789474	+	0.613784i	$0.210354\pi$
$19$	23.3238i	1.22757i	−0.789474	+	0.613784i	$0.789646\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 20.0000i	− 0.869565i	−0.900535	−	0.434783i	$-0.856825\pi$
$23$	− 20.0000i	− 0.869565i	0.900535	−	0.434783i	$-0.143175\pi$
$24$	0	0
$25$	9.00000	0.360000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29.1548	1.00534	0.502668	−	0.864479i	$-0.332352\pi$
$29$	29.1548	1.00534	0.502668	+	0.864479i	$0.332352\pi$
$30$	0	0
$31$	42.4264	1.36859	0.684297	−	0.729204i	$-0.260109\pi$
$31$	42.4264	1.36859	0.684297	+	0.729204i	$0.260109\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−16.4924	−0.471212
$36$	0	0
$37$	− 49.4773i	− 1.33722i	−0.743612	−	0.668612i	$-0.766889\pi$
$37$	− 49.4773i	− 1.33722i	0.743612	−	0.668612i	$-0.233111\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	26.8701i	0.655367i	0.944788	+	0.327684i	$0.106268\pi$
$41$	26.8701i	0.655367i	−0.944788	+	0.327684i	$0.893732\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 44.0000i	− 0.936170i	−0.883683	−	0.468085i	$-0.844944\pi$
$47$	− 44.0000i	− 0.936170i	0.883683	−	0.468085i	$-0.155056\pi$
$48$	0	0
$49$	−41.0000	−0.836735
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−29.1548	−0.550090	−0.275045	−	0.961431i	$-0.588693\pi$
$53$	−29.1548	−0.550090	−0.275045	+	0.961431i	$0.588693\pi$
$54$	0	0
$55$	96.1665	1.74848
$56$	0	0
$57$	0	0
$58$	0	0
$59$	65.9697	1.11813	0.559065	−	0.829124i	$-0.311160\pi$
$59$	65.9697	1.11813	0.559065	+	0.829124i	$0.311160\pi$
$60$	0	0
$61$	82.4621i	1.35184i	0.736976	+	0.675919i	$0.236253\pi$
$61$	82.4621i	1.35184i	−0.736976	+	0.675919i	$0.763747\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 48.0833i	− 0.739742i
$66$	0	0
$67$	− 116.619i	− 1.74058i	−0.492537	−	0.870291i	$-0.663930\pi$
$67$	− 116.619i	− 1.74058i	0.492537	−	0.870291i	$-0.336070\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 100.000i	− 1.40845i	−0.709977	−	0.704225i	$-0.751294\pi$
$71$	− 100.000i	− 1.40845i	0.709977	−	0.704225i	$-0.248706\pi$
$72$	0	0
$73$	40.0000	0.547945	0.273973	−	0.961737i	$-0.411662\pi$
$73$	40.0000	0.547945	0.273973	+	0.961737i	$0.411662\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−46.6476	−0.605813
$78$	0	0
$79$	127.279	1.61113	0.805565	−	0.592508i	$-0.201862\pi$
$79$	127.279	1.61113	0.805565	+	0.592508i	$0.201862\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	82.4621	0.993519	0.496760	−	0.867888i	$-0.334523\pi$
$83$	82.4621	0.993519	0.496760	+	0.867888i	$0.334523\pi$
$84$	0	0
$85$	41.2311i	0.485071i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	114.551i	1.28709i	0.765407	+	0.643547i	$0.222538\pi$
$89$	114.551i	1.28709i	−0.765407	+	0.643547i	$0.777462\pi$
$90$	0	0
$91$	23.3238i	0.256306i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 136.000i	− 1.43158i
$96$	0	0
$97$	−40.0000	−0.412371	−0.206186	−	0.978513i	$-0.566105\pi$
$97$	−40.0000	−0.412371	−0.206186	+	0.978513i	$0.566105\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	29.1548	0.288661	0.144330	−	0.989530i	$-0.453897\pi$
$101$	29.1548	0.288661	0.144330	+	0.989530i	$0.453897\pi$
$102$	0	0
$103$	166.877	1.62017	0.810083	−	0.586315i	$-0.199422\pi$
$103$	166.877	1.62017	0.810083	+	0.586315i	$0.199422\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	− 206.155i	− 1.89133i	−0.325138	−	0.945666i	$-0.605411\pi$
$109$	− 206.155i	− 1.89133i	0.325138	−	0.945666i	$-0.394589\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 205.061i	− 1.81470i	−0.420377	−	0.907349i	$-0.638102\pi$
$113$	− 205.061i	− 1.81470i	0.420377	−	0.907349i	$-0.361898\pi$
$114$	0	0
$115$	116.619i	1.01408i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 20.0000i	− 0.168067i
$120$	0	0
$121$	151.000	1.24793
$122$	0	0
$123$	0	0
$124$	0	0
$125$	93.2952	0.746362
$126$	0	0
$127$	144.250	1.13583	0.567913	−	0.823089i	$-0.307751\pi$
$127$	144.250	1.13583	0.567913	+	0.823089i	$0.307751\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−98.9545	−0.755378	−0.377689	−	0.925932i	$-0.623281\pi$
$131$	−98.9545	−0.755378	−0.377689	+	0.925932i	$0.623281\pi$
$132$	0	0
$133$	65.9697i	0.496013i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	120.208i	0.877432i	0.898626	+	0.438716i	$0.144567\pi$
$137$	120.208i	0.877432i	−0.898626	+	0.438716i	$0.855433\pi$
$138$	0	0
$139$	− 116.619i	− 0.838986i	−0.907759	−	0.419493i	$-0.862208\pi$
$139$	− 116.619i	− 0.838986i	0.907759	−	0.419493i	$-0.137792\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 136.000i	− 0.951049i
$144$	0	0
$145$	−170.000	−1.17241
$146$	0	0
$147$	0	0
$148$	0	0
$149$	145.774	0.978348	0.489174	−	0.872186i	$-0.337298\pi$
$149$	145.774	0.978348	0.489174	+	0.872186i	$0.337298\pi$
$150$	0	0
$151$	127.279	0.842909	0.421454	−	0.906850i	$-0.361520\pi$
$151$	127.279	0.842909	0.421454	+	0.906850i	$0.361520\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−247.386	−1.59604
$156$	0	0
$157$	49.4773i	0.315142i	0.987508	+	0.157571i	$0.0503663\pi$
$157$	49.4773i	0.315142i	−0.987508	+	0.157571i	$0.949634\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 56.5685i	− 0.351357i
$162$	0	0
$163$	233.238i	1.43091i	0.698660	+	0.715454i	$0.253780\pi$
$163$	233.238i	1.43091i	−0.698660	+	0.715454i	$0.746220\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 160.000i	− 0.958084i	−0.877792	−	0.479042i	$-0.840984\pi$
$167$	− 160.000i	− 0.958084i	0.877792	−	0.479042i	$-0.159016\pi$
$168$	0	0
$169$	101.000	0.597633
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−320.702	−1.85377	−0.926885	−	0.375344i	$-0.877524\pi$
$173$	−320.702	−1.85377	−0.926885	+	0.375344i	$0.877524\pi$
$174$	0	0
$175$	25.4558	0.145462
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−98.9545	−0.552819	−0.276409	−	0.961040i	$-0.589145\pi$
$179$	−98.9545	−0.552819	−0.276409	+	0.961040i	$0.589145\pi$
$180$	0	0
$181$	206.155i	1.13898i	0.821998	+	0.569490i	$0.192859\pi$
$181$	206.155i	1.13898i	−0.821998	+	0.569490i	$0.807141\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	288.500i	1.55946i
$186$	0	0
$187$	116.619i	0.623631i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	160.000i	0.837696i	0.908056	+	0.418848i	$0.137566\pi$
$191$	160.000i	0.837696i	−0.908056	+	0.418848i	$0.862434\pi$
$192$	0	0
$193$	−150.000	−0.777202	−0.388601	−	0.921406i	$-0.627042\pi$
$193$	−150.000	−0.777202	−0.388601	+	0.921406i	$0.627042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−157.436	−0.799166	−0.399583	−	0.916697i	$-0.630845\pi$
$197$	−157.436	−0.799166	−0.399583	+	0.916697i	$0.630845\pi$
$198$	0	0
$199$	−70.7107	−0.355330	−0.177665	−	0.984091i	$-0.556854\pi$
$199$	−70.7107	−0.355330	−0.177665	+	0.984091i	$0.556854\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	82.4621	0.406217
$204$	0	0
$205$	− 156.678i	− 0.764283i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 384.666i	− 1.84051i
$210$	0	0
$211$	− 349.857i	− 1.65809i	−0.559181	−	0.829045i	$-0.688884\pi$
$211$	− 349.857i	− 1.65809i	0.559181	−	0.829045i	$-0.311116\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	120.000	0.552995
$218$	0	0
$219$	0	0
$220$	0	0
$221$	58.3095	0.263844
$222$	0	0
$223$	−2.82843	−0.0126835	−0.00634176	−	0.999980i	$-0.502019\pi$
$223$	−2.82843	−0.0126835	−0.00634176	+	0.999980i	$0.502019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	247.386	1.08981	0.544904	−	0.838499i	$-0.316566\pi$
$227$	247.386	1.08981	0.544904	+	0.838499i	$0.316566\pi$
$228$	0	0
$229$	− 206.155i	− 0.900241i	−0.892968	−	0.450121i	$-0.851381\pi$
$229$	− 206.155i	− 0.900241i	0.892968	−	0.450121i	$-0.148619\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	190.919i	0.819394i	0.912222	+	0.409697i	$0.134366\pi$
$233$	190.919i	0.819394i	−0.912222	+	0.409697i	$0.865634\pi$
$234$	0	0
$235$	256.562i	1.09175i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	40.0000i	0.167364i	0.996493	+	0.0836820i	$0.0266680\pi$
$239$	40.0000i	0.167364i	−0.996493	+	0.0836820i	$0.973332\pi$
$240$	0	0
$241$	−240.000	−0.995851	−0.497925	−	0.867220i	$-0.665905\pi$
$241$	−240.000	−0.995851	−0.497925	+	0.867220i	$0.665905\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	239.069	0.975792
$246$	0	0
$247$	−192.333	−0.778676
$248$	0	0
$249$	0	0
$250$	0	0
$251$	346.341	1.37984	0.689922	−	0.723884i	$-0.257645\pi$
$251$	346.341	1.37984	0.689922	+	0.723884i	$0.257645\pi$
$252$	0	0
$253$	329.848i	1.30375i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.2132i	− 0.0825416i	−0.999148	−	0.0412708i	$-0.986859\pi$
$257$	− 21.2132i	− 0.0825416i	0.999148	−	0.0412708i	$-0.0131406\pi$
$258$	0	0
$259$	− 139.943i	− 0.540320i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	360.000i	1.36882i	0.729097	+	0.684411i	$0.239940\pi$
$263$	360.000i	1.36882i	−0.729097	+	0.684411i	$0.760060\pi$
$264$	0	0
$265$	170.000	0.641509
$266$	0	0
$267$	0	0
$268$	0	0
$269$	437.321	1.62573	0.812865	−	0.582452i	$-0.197907\pi$
$269$	437.321	1.62573	0.812865	+	0.582452i	$0.197907\pi$
$270$	0	0
$271$	−240.416	−0.887145	−0.443573	−	0.896238i	$-0.646289\pi$
$271$	−240.416	−0.887145	−0.443573	+	0.896238i	$0.646289\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−148.432	−0.539752
$276$	0	0
$277$	− 321.602i	− 1.16102i	−0.814254	−	0.580509i	$-0.802853\pi$
$277$	− 321.602i	− 1.16102i	0.814254	−	0.580509i	$-0.197147\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	227.688i	0.810279i	0.914255	+	0.405139i	$0.132777\pi$
$281$	227.688i	0.810279i	−0.914255	+	0.405139i	$0.867223\pi$
$282$	0	0
$283$	− 466.476i	− 1.64833i	−0.566353	−	0.824163i	$-0.691646\pi$
$283$	− 466.476i	− 1.64833i	0.566353	−	0.824163i	$-0.308354\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	76.0000i	0.264808i
$288$	0	0
$289$	239.000	0.826990
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−215.745	−0.736332	−0.368166	−	0.929760i	$-0.620014\pi$
$293$	−215.745	−0.736332	−0.368166	+	0.929760i	$0.620014\pi$
$294$	0	0
$295$	−384.666	−1.30395
$296$	0	0
$297$	0	0
$298$	0	0
$299$	164.924	0.551586
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 480.833i	− 1.57650i
$306$	0	0
$307$	− 116.619i	− 0.379867i	−0.981797	−	0.189933i	$-0.939173\pi$
$307$	− 116.619i	− 0.379867i	0.981797	−	0.189933i	$-0.0608272\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 40.0000i	− 0.128617i	−0.997930	−	0.0643087i	$-0.979516\pi$
$311$	− 40.0000i	− 0.128617i	0.997930	−	0.0643087i	$-0.0204842\pi$
$312$	0	0
$313$	250.000	0.798722	0.399361	−	0.916794i	$-0.369232\pi$
$313$	250.000	0.798722	0.399361	+	0.916794i	$0.369232\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.4929	0.0551825	0.0275913	−	0.999619i	$-0.491216\pi$
$317$	17.4929	0.0551825	0.0275913	+	0.999619i	$0.491216\pi$
$318$	0	0
$319$	−480.833	−1.50731
$320$	0	0
$321$	0	0
$322$	0	0
$323$	164.924	0.510601
$324$	0	0
$325$	74.2159i	0.228357i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 124.451i	− 0.378270i
$330$	0	0
$331$	− 23.3238i	− 0.0704647i	−0.999379	−	0.0352323i	$-0.988783\pi$
$331$	− 23.3238i	− 0.0704647i	0.999379	−	0.0352323i	$-0.0112171\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	680.000i	2.02985i
$336$	0	0
$337$	−520.000	−1.54303	−0.771513	−	0.636213i	$-0.780500\pi$
$337$	−520.000	−1.54303	−0.771513	+	0.636213i	$0.780500\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−699.714	−2.05195
$342$	0	0
$343$	−254.558	−0.742153
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−247.386	−0.712929	−0.356464	−	0.934309i	$-0.616018\pi$
$347$	−247.386	−0.712929	−0.356464	+	0.934309i	$0.616018\pi$
$348$	0	0
$349$	82.4621i	0.236281i	0.992997	+	0.118141i	$0.0376933\pi$
$349$	82.4621i	0.236281i	−0.992997	+	0.118141i	$0.962307\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	360.624i	1.02160i	0.859700	+	0.510800i	$0.170651\pi$
$353$	360.624i	1.02160i	−0.859700	+	0.510800i	$0.829349\pi$
$354$	0	0
$355$	583.095i	1.64252i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	340.000i	0.947075i	0.880774	+	0.473538i	$0.157023\pi$
$359$	340.000i	0.947075i	−0.880774	+	0.473538i	$0.842977\pi$
$360$	0	0
$361$	−183.000	−0.506925
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−233.238	−0.639008
$366$	0	0
$367$	−364.867	−0.994188	−0.497094	−	0.867697i	$-0.665600\pi$
$367$	−364.867	−0.994188	−0.497094	+	0.867697i	$0.665600\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−82.4621	−0.222270
$372$	0	0
$373$	49.4773i	0.132647i	0.997798	+	0.0663234i	$0.0211269\pi$
$373$	49.4773i	0.132647i	−0.997798	+	0.0663234i	$0.978873\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	240.416i	0.637709i
$378$	0	0
$379$	116.619i	0.307702i	0.988094	+	0.153851i	$0.0491676\pi$
$379$	116.619i	0.307702i	−0.988094	+	0.153851i	$0.950832\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 184.000i	− 0.480418i	−0.970721	−	0.240209i	$-0.922784\pi$
$383$	− 184.000i	− 0.480418i	0.970721	−	0.240209i	$-0.0772159\pi$
$384$	0	0
$385$	272.000	0.706494
$386$	0	0
$387$	0	0
$388$	0	0
$389$	379.012	0.974324	0.487162	−	0.873312i	$-0.338032\pi$
$389$	379.012	0.974324	0.487162	+	0.873312i	$0.338032\pi$
$390$	0	0
$391$	−141.421	−0.361691
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−742.159	−1.87888
$396$	0	0
$397$	544.250i	1.37091i	0.728117	+	0.685453i	$0.240396\pi$
$397$	544.250i	1.37091i	−0.728117	+	0.685453i	$0.759604\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 267.286i	− 0.666550i	−0.942830	−	0.333275i	$-0.891846\pi$
$401$	− 267.286i	− 0.666550i	0.942830	−	0.333275i	$-0.108154\pi$
$402$	0	0
$403$	349.857i	0.868132i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	816.000i	2.00491i
$408$	0	0
$409$	232.000	0.567237	0.283619	−	0.958937i	$-0.408465\pi$
$409$	232.000	0.567237	0.283619	+	0.958937i	$0.408465\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	186.590	0.451793
$414$	0	0
$415$	−480.833	−1.15863
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.4924	0.0393614	0.0196807	−	0.999806i	$-0.493735\pi$
$419$	16.4924	0.0393614	0.0196807	+	0.999806i	$0.493735\pi$
$420$	0	0
$421$	− 123.693i	− 0.293808i	−0.989151	−	0.146904i	$-0.953069\pi$
$421$	− 123.693i	− 0.293808i	0.989151	−	0.146904i	$-0.0469308\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 63.6396i	− 0.149740i
$426$	0	0
$427$	233.238i	0.546225i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 140.000i	− 0.324826i	−0.986723	−	0.162413i	$-0.948072\pi$
$431$	− 140.000i	− 0.324826i	0.986723	−	0.162413i	$-0.0519277\pi$
$432$	0	0
$433$	−190.000	−0.438799	−0.219400	−	0.975635i	$-0.570410\pi$
$433$	−190.000	−0.438799	−0.219400	+	0.975635i	$0.570410\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	466.476	1.06745
$438$	0	0
$439$	−664.680	−1.51408	−0.757039	−	0.653370i	$-0.773355\pi$
$439$	−664.680	−1.51408	−0.757039	+	0.653370i	$0.773355\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−82.4621	−0.186145	−0.0930724	−	0.995659i	$-0.529669\pi$
$443$	−82.4621	−0.186145	−0.0930724	+	0.995659i	$0.529669\pi$
$444$	0	0
$445$	− 667.943i	− 1.50100i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	436.992i	0.973256i	0.873609	+	0.486628i	$0.161773\pi$
$449$	436.992i	0.973256i	−0.873609	+	0.486628i	$0.838227\pi$
$450$	0	0
$451$	− 443.152i	− 0.982599i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 136.000i	− 0.298901i
$456$	0	0
$457$	−600.000	−1.31291	−0.656455	−	0.754365i	$-0.727945\pi$
$457$	−600.000	−1.31291	−0.656455	+	0.754365i	$0.727945\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	87.4643	0.189727	0.0948636	−	0.995490i	$-0.469758\pi$
$461$	87.4643	0.189727	0.0948636	+	0.995490i	$0.469758\pi$
$462$	0	0
$463$	−53.7401	−0.116069	−0.0580347	−	0.998315i	$-0.518483\pi$
$463$	−53.7401	−0.116069	−0.0580347	+	0.998315i	$0.518483\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−82.4621	−0.176578	−0.0882892	−	0.996095i	$-0.528140\pi$
$467$	−82.4621	−0.176578	−0.0882892	+	0.996095i	$0.528140\pi$
$468$	0	0
$469$	− 329.848i	− 0.703302i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	209.914i	0.441925i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	860.000i	1.79541i	0.440600	+	0.897704i	$0.354766\pi$
$479$	860.000i	1.79541i	−0.440600	+	0.897704i	$0.645234\pi$
$480$	0	0
$481$	408.000	0.848233
$482$	0	0
$483$	0	0
$484$	0	0
$485$	233.238	0.480903
$486$	0	0
$487$	31.1127	0.0638864	0.0319432	−	0.999490i	$-0.489830\pi$
$487$	31.1127	0.0638864	0.0319432	+	0.999490i	$0.489830\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	428.803	0.873326	0.436663	−	0.899625i	$-0.356160\pi$
$491$	428.803	0.873326	0.436663	+	0.899625i	$0.356160\pi$
$492$	0	0
$493$	− 206.155i	− 0.418165i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 282.843i	− 0.569100i
$498$	0	0
$499$	93.2952i	0.186964i	0.995621	+	0.0934822i	$0.0297998\pi$
$499$	93.2952i	0.186964i	−0.995621	+	0.0934822i	$0.970200\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 556.000i	− 1.10537i	−0.833391	−	0.552684i	$-0.813604\pi$
$503$	− 556.000i	− 1.10537i	0.833391	−	0.552684i	$-0.186396\pi$
$504$	0	0
$505$	−170.000	−0.336634
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−670.559	−1.31741	−0.658703	−	0.752403i	$-0.728895\pi$
$509$	−670.559	−1.31741	−0.658703	+	0.752403i	$0.728895\pi$
$510$	0	0
$511$	113.137	0.221403
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−973.053	−1.88942
$516$	0	0
$517$	725.667i	1.40361i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 920.653i	− 1.76709i	−0.468348	−	0.883544i	$-0.655151\pi$
$521$	− 920.653i	− 1.76709i	0.468348	−	0.883544i	$-0.344849\pi$
$522$	0	0
$523$	− 583.095i	− 1.11490i	−0.830209	−	0.557452i	$-0.811779\pi$
$523$	− 583.095i	− 1.11490i	0.830209	−	0.557452i	$-0.188221\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 300.000i	− 0.569260i
$528$	0	0
$529$	129.000	0.243856
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−221.576	−0.415715
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	676.189	1.25453
$540$	0	0
$541$	− 865.852i	− 1.60047i	−0.599689	−	0.800233i	$-0.704709\pi$
$541$	− 865.852i	− 1.60047i	0.599689	−	0.800233i	$-0.295291\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1202.08i	2.20565i
$546$	0	0
$547$	− 932.952i	− 1.70558i	−0.522254	−	0.852790i	$-0.674909\pi$
$547$	− 932.952i	− 1.70558i	0.522254	−	0.852790i	$-0.325091\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	680.000i	1.23412i
$552$	0	0
$553$	360.000	0.650995
$554$	0	0
$555$	0	0
$556$	0	0
$557$	87.4643	0.157027	0.0785137	−	0.996913i	$-0.474983\pi$
$557$	87.4643	0.157027	0.0785137	+	0.996913i	$0.474983\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1072.01	1.90410	0.952049	−	0.305945i	$-0.0989723\pi$
$563$	1072.01	1.90410	0.952049	+	0.305945i	$0.0989723\pi$
$564$	0	0
$565$	1195.70i	2.11628i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 72.1249i	− 0.126757i	−0.997990	−	0.0633786i	$-0.979812\pi$
$569$	− 72.1249i	− 0.126757i	0.997990	−	0.0633786i	$-0.0201876\pi$
$570$	0	0
$571$	793.009i	1.38881i	0.719585	+	0.694404i	$0.244332\pi$
$571$	793.009i	1.38881i	−0.719585	+	0.694404i	$0.755668\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 180.000i	− 0.313043i
$576$	0	0
$577$	890.000	1.54246	0.771231	−	0.636556i	$-0.219642\pi$
$577$	890.000	1.54246	0.771231	+	0.636556i	$0.219642\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	233.238	0.401442
$582$	0	0
$583$	480.833	0.824756
$584$	0	0
$585$	0	0
$586$	0	0
$587$	989.545	1.68577	0.842884	−	0.538096i	$-0.180856\pi$
$587$	989.545	1.68577	0.842884	+	0.538096i	$0.180856\pi$
$588$	0	0
$589$	989.545i	1.68004i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 473.762i	− 0.798923i	−0.916750	−	0.399462i	$-0.869197\pi$
$593$	− 473.762i	− 0.798923i	0.916750	−	0.399462i	$-0.130803\pi$
$594$	0	0
$595$	116.619i	0.195998i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 740.000i	− 1.23539i	−0.786417	−	0.617696i	$-0.788066\pi$
$599$	− 740.000i	− 1.23539i	0.786417	−	0.617696i	$-0.211934\pi$
$600$	0	0
$601$	−310.000	−0.515807	−0.257903	−	0.966171i	$-0.583032\pi$
$601$	−310.000	−0.515807	−0.257903	+	0.966171i	$0.583032\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−880.474	−1.45533
$606$	0	0
$607$	370.524	0.610418	0.305209	−	0.952285i	$-0.401274\pi$
$607$	370.524	0.610418	0.305209	+	0.952285i	$0.401274\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	362.833	0.593835
$612$	0	0
$613$	− 280.371i	− 0.457376i	−0.973500	−	0.228688i	$-0.926557\pi$
$613$	− 280.371i	− 0.457376i	0.973500	−	0.228688i	$-0.0734435\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 586.899i	− 0.951213i	−0.879658	−	0.475607i	$-0.842229\pi$
$617$	− 586.899i	− 0.951213i	0.879658	−	0.475607i	$-0.157771\pi$
$618$	0	0
$619$	− 699.714i	− 1.13039i	−0.824956	−	0.565197i	$-0.808800\pi$
$619$	− 699.714i	− 1.13039i	0.824956	−	0.565197i	$-0.191200\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	324.000i	0.520064i
$624$	0	0
$625$	−769.000	−1.23040
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−349.857	−0.556212
$630$	0	0
$631$	381.838	0.605131	0.302566	−	0.953129i	$-0.402157\pi$
$631$	381.838	0.605131	0.302566	+	0.953129i	$0.402157\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−841.114	−1.32459
$636$	0	0
$637$	− 338.095i	− 0.530761i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1203.50i	1.87753i	0.344560	+	0.938764i	$0.388028\pi$
$641$	1203.50i	1.87753i	−0.344560	+	0.938764i	$0.611972\pi$
$642$	0	0
$643$	233.238i	0.362734i	0.983415	+	0.181367i	$0.0580522\pi$
$643$	233.238i	0.362734i	−0.983415	+	0.181367i	$0.941948\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 844.000i	− 1.30448i	−0.758012	−	0.652241i	$-0.773829\pi$
$647$	− 844.000i	− 1.30448i	0.758012	−	0.652241i	$-0.226171\pi$
$648$	0	0
$649$	−1088.00	−1.67643
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1032.08	1.58052	0.790259	−	0.612773i	$-0.209946\pi$
$653$	1032.08	1.58052	0.790259	+	0.612773i	$0.209946\pi$
$654$	0	0
$655$	576.999	0.880915
$656$	0	0
$657$	0	0
$658$	0	0
$659$	263.879	0.400423	0.200212	−	0.979753i	$-0.435837\pi$
$659$	263.879	0.400423	0.200212	+	0.979753i	$0.435837\pi$
$660$	0	0
$661$	− 907.083i	− 1.37229i	−0.727465	−	0.686145i	$-0.759302\pi$
$661$	− 907.083i	− 1.37229i	0.727465	−	0.686145i	$-0.240698\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 384.666i	− 0.578445i
$666$	0	0
$667$	− 583.095i	− 0.874206i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1360.00i	− 2.02683i
$672$	0	0
$673$	1070.00	1.58990	0.794948	−	0.606678i	$-0.207498\pi$
$673$	1070.00	1.58990	0.794948	+	0.606678i	$0.207498\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.8167	0.0602905	0.0301452	−	0.999546i	$-0.490403\pi$
$677$	40.8167	0.0602905	0.0301452	+	0.999546i	$0.490403\pi$
$678$	0	0
$679$	−113.137	−0.166623
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1072.01	−1.56956	−0.784779	−	0.619776i	$-0.787223\pi$
$683$	−1072.01	−1.56956	−0.784779	+	0.619776i	$0.787223\pi$
$684$	0	0
$685$	− 700.928i	− 1.02325i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 240.416i	− 0.348935i
$690$	0	0
$691$	326.533i	0.472552i	0.971686	+	0.236276i	$0.0759269\pi$
$691$	326.533i	0.472552i	−0.971686	+	0.236276i	$0.924073\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	680.000i	0.978417i
$696$	0	0
$697$	190.000	0.272597
$698$	0	0
$699$	0	0
$700$	0	0
$701$	495.631	0.707034	0.353517	−	0.935428i	$-0.384986\pi$
$701$	495.631	0.707034	0.353517	+	0.935428i	$0.384986\pi$
$702$	0	0
$703$	1154.00	1.64153
$704$	0	0
$705$	0	0
$706$	0	0
$707$	82.4621	0.116637
$708$	0	0
$709$	123.693i	0.174461i	0.996188	+	0.0872307i	$0.0278017\pi$
$709$	123.693i	0.174461i	−0.996188	+	0.0872307i	$0.972198\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 848.528i	− 1.19008i
$714$	0	0
$715$	793.009i	1.10910i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 300.000i	− 0.417246i	−0.977996	−	0.208623i	$-0.933102\pi$
$719$	− 300.000i	− 0.417246i	0.977996	−	0.208623i	$-0.0668982\pi$
$720$	0	0
$721$	472.000	0.654646
$722$	0	0
$723$	0	0
$724$	0	0
$725$	262.393	0.361921
$726$	0	0
$727$	1298.25	1.78576	0.892880	−	0.450294i	$-0.148681\pi$
$727$	1298.25	1.78576	0.892880	+	0.450294i	$0.148681\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	74.2159i	0.101250i	0.998718	+	0.0506248i	$0.0161213\pi$
$733$	74.2159i	0.101250i	−0.998718	+	0.0506248i	$0.983879\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1923.33i	2.60967i
$738$	0	0
$739$	139.943i	0.189368i	0.995507	+	0.0946839i	$0.0301840\pi$
$739$	139.943i	0.189368i	−0.995507	+	0.0946839i	$0.969816\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1264.00i	− 1.70121i	−0.525804	−	0.850606i	$-0.676236\pi$
$743$	− 1264.00i	− 1.70121i	0.525804	−	0.850606i	$-0.323764\pi$
$744$	0	0
$745$	−850.000	−1.14094
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	664.680	0.885060	0.442530	−	0.896754i	$-0.354081\pi$
$751$	664.680	0.885060	0.442530	+	0.896754i	$0.354081\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−742.159	−0.982992
$756$	0	0
$757$	− 733.913i	− 0.969502i	−0.874652	−	0.484751i	$-0.838910\pi$
$757$	− 733.913i	− 0.969502i	0.874652	−	0.484751i	$-0.161090\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	72.1249i	0.0947765i	0.998877	+	0.0473882i	$0.0150898\pi$
$761$	72.1249i	0.0947765i	−0.998877	+	0.0473882i	$0.984910\pi$
$762$	0	0
$763$	− 583.095i	− 0.764214i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	544.000i	0.709257i
$768$	0	0
$769$	−10.0000	−0.0130039	−0.00650195	−	0.999979i	$-0.502070\pi$
$769$	−10.0000	−0.0130039	−0.00650195	+	0.999979i	$0.502070\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	250.731	0.324361	0.162180	−	0.986761i	$-0.448147\pi$
$773$	250.731	0.324361	0.162180	+	0.986761i	$0.448147\pi$
$774$	0	0
$775$	381.838	0.492694
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−626.712	−0.804508
$780$	0	0
$781$	1649.24i	2.11171i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 288.500i	− 0.367515i
$786$	0	0
$787$	816.333i	1.03727i	0.854995	+	0.518636i	$0.173560\pi$
$787$	816.333i	1.03727i	−0.854995	+	0.518636i	$0.826440\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 580.000i	− 0.733249i
$792$	0	0
$793$	−680.000	−0.857503
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−320.702	−0.402387	−0.201193	−	0.979552i	$-0.564482\pi$
$797$	−320.702	−0.402387	−0.201193	+	0.979552i	$0.564482\pi$
$798$	0	0
$799$	−311.127	−0.389395
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−659.697	−0.821540
$804$	0	0
$805$	329.848i	0.409750i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 550.129i	− 0.680011i	−0.940423	−	0.340006i	$-0.889571\pi$
$809$	− 550.129i	− 0.680011i	0.940423	−	0.340006i	$-0.110429\pi$
$810$	0	0
$811$	116.619i	0.143797i	0.997412	+	0.0718983i	$0.0229057\pi$
$811$	116.619i	0.143797i	−0.997412	+	0.0718983i	$0.977094\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1360.00i	− 1.66871i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−204.083	−0.248579	−0.124289	−	0.992246i	$-0.539665\pi$
$821$	−204.083	−0.248579	−0.124289	+	0.992246i	$0.539665\pi$
$822$	0	0
$823$	1439.67	1.74929	0.874647	−	0.484760i	$-0.161093\pi$
$823$	1439.67	1.74929	0.874647	+	0.484760i	$0.161093\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	494.773	0.598274	0.299137	−	0.954210i	$-0.403301\pi$
$827$	494.773	0.598274	0.299137	+	0.954210i	$0.403301\pi$
$828$	0	0
$829$	1113.24i	1.34287i	0.741064	+	0.671435i	$0.234322\pi$
$829$	1113.24i	1.34287i	−0.741064	+	0.671435i	$0.765678\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	289.914i	0.348036i
$834$	0	0
$835$	932.952i	1.11731i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	820.000i	0.977354i	0.872465	+	0.488677i	$0.162520\pi$
$839$	820.000i	0.977354i	−0.872465	+	0.488677i	$0.837480\pi$
$840$	0	0
$841$	9.00000	0.0107015
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−588.926	−0.696954
$846$	0	0
$847$	427.092	0.504241
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−989.545	−1.16280
$852$	0	0
$853$	− 1104.99i	− 1.29542i	−0.761887	−	0.647709i	$-0.775727\pi$
$853$	− 1104.99i	− 1.29542i	0.761887	−	0.647709i	$-0.224273\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 742.462i	− 0.866350i	−0.901310	−	0.433175i	$-0.857393\pi$
$857$	− 742.462i	− 0.866350i	0.901310	−	0.433175i	$-0.142607\pi$
$858$	0	0
$859$	932.952i	1.08609i	0.839703	+	0.543046i	$0.182729\pi$
$859$	932.952i	1.08609i	−0.839703	+	0.543046i	$0.817271\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1360.00i	− 1.57590i	−0.615740	−	0.787949i	$-0.711143\pi$
$863$	− 1360.00i	− 1.57590i	0.615740	−	0.787949i	$-0.288857\pi$
$864$	0	0
$865$	1870.00	2.16185
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2099.14	−2.41558
$870$	0	0
$871$	961.665	1.10409
$872$	0	0
$873$	0	0
$874$	0	0
$875$	263.879	0.301576
$876$	0	0
$877$	− 379.326i	− 0.432526i	−0.976335	−	0.216263i	$-0.930613\pi$
$877$	− 379.326i	− 0.432526i	0.976335	−	0.216263i	$-0.0693869\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 111.723i	− 0.126814i	−0.997988	−	0.0634069i	$-0.979803\pi$
$881$	− 111.723i	− 0.126814i	0.997988	−	0.0634069i	$-0.0201966\pi$
$882$	0	0
$883$	− 1399.43i	− 1.58486i	−0.609965	−	0.792428i	$-0.708817\pi$
$883$	− 1399.43i	− 1.58486i	0.609965	−	0.792428i	$-0.291183\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 840.000i	− 0.947012i	−0.880790	−	0.473506i	$-0.842988\pi$
$887$	− 840.000i	− 0.947012i	0.880790	−	0.473506i	$-0.157012\pi$
$888$	0	0
$889$	408.000	0.458943
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1026.25	1.14921
$894$	0	0
$895$	576.999	0.644692
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1236.93	1.37590
$900$	0	0
$901$	206.155i	0.228807i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1202.08i	− 1.32827i
$906$	0	0
$907$	932.952i	1.02861i	0.857606	+	0.514307i	$0.171951\pi$
$907$	932.952i	1.02861i	−0.857606	+	0.514307i	$0.828049\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	560.000i	0.614709i	0.951595	+	0.307355i	$0.0994438\pi$
$911$	560.000i	0.614709i	−0.951595	+	0.307355i	$0.900556\pi$
$912$	0	0
$913$	−1360.00	−1.48959
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−279.886	−0.305219
$918$	0	0
$919$	−777.817	−0.846374	−0.423187	−	0.906042i	$-0.639089\pi$
$919$	−777.817	−0.846374	−0.423187	+	0.906042i	$0.639089\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	824.621	0.893414
$924$	0	0
$925$	− 445.295i	− 0.481400i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1497.65i	1.61211i	0.591839	+	0.806056i	$0.298402\pi$
$929$	1497.65i	1.61211i	−0.591839	+	0.806056i	$0.701598\pi$
$930$	0	0
$931$	− 956.276i	− 1.02715i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 680.000i	− 0.727273i
$936$	0	0
$937$	−510.000	−0.544290	−0.272145	−	0.962256i	$-0.587733\pi$
$937$	−510.000	−0.544290	−0.272145	+	0.962256i	$0.587733\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	553.940	0.588672	0.294336	−	0.955702i	$-0.404902\pi$
$941$	553.940	0.588672	0.294336	+	0.955702i	$0.404902\pi$
$942$	0	0
$943$	537.401	0.569885
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	329.848i	0.347575i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	912.168i	0.957154i	0.878046	+	0.478577i	$0.158847\pi$
$953$	912.168i	0.957154i	−0.878046	+	0.478577i	$0.841153\pi$
$954$	0	0
$955$	− 932.952i	− 0.976913i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	340.000i	0.354536i
$960$	0	0
$961$	839.000	0.873049
$962$	0	0
$963$	0	0
$964$	0	0
$965$	874.643	0.906366
$966$	0	0
$967$	−285.671	−0.295420	−0.147710	−	0.989031i	$-0.547190\pi$
$967$	−285.671	−0.295420	−0.147710	+	0.989031i	$0.547190\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	643.204	0.662414	0.331207	−	0.943558i	$-0.392544\pi$
$971$	643.204	0.662414	0.331207	+	0.943558i	$0.392544\pi$
$972$	0	0
$973$	− 329.848i	− 0.339001i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 388.909i	− 0.398064i	−0.979993	−	0.199032i	$-0.936220\pi$
$977$	− 388.909i	− 0.398064i	0.979993	−	0.199032i	$-0.0637798\pi$
$978$	0	0
$979$	− 1889.23i	− 1.92975i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	984.000i	1.00102i	0.865732	+	0.500509i	$0.166854\pi$
$983$	984.000i	1.00102i	−0.865732	+	0.500509i	$0.833146\pi$
$984$	0	0
$985$	918.000	0.931980
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	890.955	0.899046	0.449523	−	0.893269i	$-0.351594\pi$
$991$	890.955	0.899046	0.449523	+	0.893269i	$0.351594\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	412.311	0.414382
$996$	0	0
$997$	− 940.068i	− 0.942897i	−0.881894	−	0.471448i	$-0.843731\pi$
$997$	− 940.068i	− 0.942897i	0.881894	−	0.471448i	$-0.156269\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.h.f.449.4	yes	8
3.2	odd	2	inner	1152.3.h.f.449.8	yes	8
4.3	odd	2	inner	1152.3.h.f.449.2	yes	8
8.3	odd	2	inner	1152.3.h.f.449.5	yes	8
8.5	even	2	inner	1152.3.h.f.449.7	yes	8
12.11	even	2	inner	1152.3.h.f.449.6	yes	8
16.3	odd	4		2304.3.e.m.1025.7		8
16.5	even	4		2304.3.e.m.1025.2		8
16.11	odd	4		2304.3.e.m.1025.4		8
16.13	even	4		2304.3.e.m.1025.5		8
24.5	odd	2	inner	1152.3.h.f.449.3	yes	8
24.11	even	2	inner	1152.3.h.f.449.1	✓	8
48.5	odd	4		2304.3.e.m.1025.6		8
48.11	even	4		2304.3.e.m.1025.8		8
48.29	odd	4		2304.3.e.m.1025.1		8
48.35	even	4		2304.3.e.m.1025.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.h.f.449.1	✓	8	24.11	even	2	inner
1152.3.h.f.449.2	yes	8	4.3	odd	2	inner
1152.3.h.f.449.3	yes	8	24.5	odd	2	inner
1152.3.h.f.449.4	yes	8	1.1	even	1	trivial
1152.3.h.f.449.5	yes	8	8.3	odd	2	inner
1152.3.h.f.449.6	yes	8	12.11	even	2	inner
1152.3.h.f.449.7	yes	8	8.5	even	2	inner
1152.3.h.f.449.8	yes	8	3.2	odd	2	inner
2304.3.e.m.1025.1		8	48.29	odd	4
2304.3.e.m.1025.2		8	16.5	even	4
2304.3.e.m.1025.3		8	48.35	even	4
2304.3.e.m.1025.4		8	16.11	odd	4
2304.3.e.m.1025.5		8	16.13	even	4
2304.3.e.m.1025.6		8	48.5	odd	4
2304.3.e.m.1025.7		8	16.3	odd	4
2304.3.e.m.1025.8		8	48.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.83095 q^{5} +2.82843 q^{7} +O(q^{10})\)	\(q-5.83095 q^{5} +2.82843 q^{7} -16.4924 q^{11} +8.24621i q^{13} -7.07107i q^{17} +23.3238i q^{19} -20.0000i q^{23} +9.00000 q^{25} +29.1548 q^{29} +42.4264 q^{31} -16.4924 q^{35} -49.4773i q^{37} +26.8701i q^{41} -44.0000i q^{47} -41.0000 q^{49} -29.1548 q^{53} +96.1665 q^{55} +65.9697 q^{59} +82.4621i q^{61} -48.0833i q^{65} -116.619i q^{67} -100.000i q^{71} +40.0000 q^{73} -46.6476 q^{77} +127.279 q^{79} +82.4621 q^{83} +41.2311i q^{85} +114.551i q^{89} +23.3238i q^{91} -136.000i q^{95} -40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 72 q^{25} - 328 q^{49} + 320 q^{73} - 320 q^{97}+O(q^{100}) \) 8 * q + 72 * q^25 - 328 * q^49 + 320 * q^73 - 320 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more