LMFDB - Embedded newform 1584.3.j.m.1297.4

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 = [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,68] 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62x^{10} + 1168x^{8} + 7496x^{6} + 15956x^{4} + 10664x^{2} + 64 $ x^12 + 62x^10 + 1168x^8 + 7496x^6 + 15956x^4 + 10664*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.4
Root			$4.34570i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.m.1297.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38336 q^{5} +10.1056i q^{7} +(-6.00816 - 9.21423i) q^{11} -1.76179i q^{13} -6.88875i q^{17} +5.12625i q^{19} -14.1833 q^{23} -19.3196 q^{25} -17.1966i q^{29} +37.8039 q^{31} -24.0853i q^{35} -33.8039 q^{37} -11.5397i q^{41} -1.30892i q^{43} +73.8322 q^{47} -53.1235 q^{49} +2.81609 q^{53} +(14.3196 + 21.9608i) q^{55} -30.6505 q^{59} -76.1716i q^{61} +4.19896i q^{65} +70.4431 q^{67} +102.865 q^{71} -64.4755i q^{73} +(93.1155 - 60.7162i) q^{77} +128.486i q^{79} -47.4413i q^{83} +16.4184i q^{85} +157.198 q^{89} +17.8039 q^{91} -12.2177i q^{95} -36.1235 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 68 q^{25} + 48 q^{37} + 116 q^{49} - 128 q^{55} - 208 q^{67} - 240 q^{91} + 320 q^{97}+O(q^{100}) $ 12 * q + 68 * q^25 + 48 * q^37 + 116 * q^49 - 128 * q^55 - 208 * q^67 - 240 * q^91 + 320 * 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
$5$	−2.38336			−0.476672			−0.238336	−	0.971183i	$-0.576602\pi$
$5$	−2.38336			−0.476672			−0.238336	+	0.971183i	$0.576602\pi$
$6$		0			0
$7$			10.1056i			1.44366i	0.692070	+	0.721830i	$0.256699\pi$
$7$			10.1056i			1.44366i	−0.692070	+	0.721830i	$0.743301\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.00816	−	9.21423i	−0.546197	−	0.837657i
$11$	−6.00816	−	9.21423i	−0.546197	−	0.837657i
$12$		0			0
$13$		−	1.76179i		−	0.135522i	−0.997702	−	0.0677610i	$-0.978414\pi$
$13$		−	1.76179i		−	0.135522i	0.997702	−	0.0677610i	$-0.0215855\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.88875i		−	0.405221i	−0.979259	−	0.202610i	$-0.935058\pi$
$17$		−	6.88875i		−	0.405221i	0.979259	−	0.202610i	$-0.0649425\pi$
$18$		0			0
$19$			5.12625i			0.269803i	0.990859	+	0.134901i	$0.0430717\pi$
$19$			5.12625i			0.269803i	−0.990859	+	0.134901i	$0.956928\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−14.1833			−0.616666			−0.308333	−	0.951278i	$-0.599771\pi$
$23$	−14.1833			−0.616666			−0.308333	+	0.951278i	$0.599771\pi$
$24$		0			0
$25$	−19.3196			−0.772784
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	17.1966i		−	0.592985i	−0.955035	−	0.296492i	$-0.904183\pi$
$29$		−	17.1966i		−	0.592985i	0.955035	−	0.296492i	$-0.0958169\pi$
$30$		0			0
$31$	37.8039			1.21948			0.609741	−	0.792601i	$-0.291274\pi$
$31$	37.8039			1.21948			0.609741	+	0.792601i	$0.291274\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	24.0853i		−	0.688152i
$36$		0			0
$37$	−33.8039			−0.913620			−0.456810	−	0.889564i	$-0.651008\pi$
$37$	−33.8039			−0.913620			−0.456810	+	0.889564i	$0.651008\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.5397i		−	0.281456i	−0.990048	−	0.140728i	$-0.955056\pi$
$41$		−	11.5397i		−	0.281456i	0.990048	−	0.140728i	$-0.0449443\pi$
$42$		0			0
$43$		−	1.30892i		−	0.0304399i	−0.999884	−	0.0152200i	$-0.995155\pi$
$43$		−	1.30892i		−	0.0304399i	0.999884	−	0.0152200i	$-0.00484485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	73.8322			1.57090			0.785449	−	0.618926i	$-0.212432\pi$
$47$	73.8322			1.57090			0.785449	+	0.618926i	$0.212432\pi$
$48$		0			0
$49$	−53.1235			−1.08415
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.81609			0.0531337			0.0265669	−	0.999647i	$-0.491543\pi$
$53$	2.81609			0.0531337			0.0265669	+	0.999647i	$0.491543\pi$
$54$		0			0
$55$	14.3196	+	21.9608i	0.260356	+	0.399287i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−30.6505			−0.519499			−0.259750	−	0.965676i	$-0.583640\pi$
$59$	−30.6505			−0.519499			−0.259750	+	0.965676i	$0.583640\pi$
$60$		0			0
$61$		−	76.1716i		−	1.24871i	−0.781139	−	0.624357i	$-0.785361\pi$
$61$		−	76.1716i		−	1.24871i	0.781139	−	0.624357i	$-0.214639\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			4.19896i			0.0645995i
$66$		0			0
$67$	70.4431			1.05139			0.525695	−	0.850673i	$-0.323805\pi$
$67$	70.4431			1.05139			0.525695	+	0.850673i	$0.323805\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	102.865			1.44881			0.724403	−	0.689377i	$-0.242115\pi$
$71$	102.865			1.44881			0.724403	+	0.689377i	$0.242115\pi$
$72$		0			0
$73$		−	64.4755i		−	0.883226i	−0.897206	−	0.441613i	$-0.854406\pi$
$73$		−	64.4755i		−	0.883226i	0.897206	−	0.441613i	$-0.145594\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	93.1155	−	60.7162i	1.20929	−	0.788522i
$78$		0			0
$79$			128.486i			1.62640i	0.581982	+	0.813202i	$0.302278\pi$
$79$			128.486i			1.62640i	−0.581982	+	0.813202i	$0.697722\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	47.4413i		−	0.571582i	−0.958292	−	0.285791i	$-0.907744\pi$
$83$		−	47.4413i		−	0.571582i	0.958292	−	0.285791i	$-0.0922564\pi$
$84$		0			0
$85$			16.4184i			0.193157i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	157.198			1.76627			0.883134	−	0.469120i	$-0.155429\pi$
$89$	157.198			1.76627			0.883134	+	0.469120i	$0.155429\pi$
$90$		0			0
$91$	17.8039			0.195648
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	12.2177i		−	0.128607i
$96$		0			0
$97$	−36.1235			−0.372408			−0.186204	−	0.982511i	$-0.559618\pi$
$97$	−36.1235			−0.372408			−0.186204	+	0.982511i	$0.559618\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	70.7467i		−	0.700463i	−0.936663	−	0.350231i	$-0.886103\pi$
$101$		−	70.7467i		−	0.700463i	0.936663	−	0.350231i	$-0.113897\pi$
$102$		0			0
$103$	82.4431			0.800419			0.400209	−	0.916424i	$-0.368937\pi$
$103$	82.4431			0.800419			0.400209	+	0.916424i	$0.368937\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	75.9010i		−	0.709355i	−0.934989	−	0.354677i	$-0.884591\pi$
$107$		−	75.9010i		−	0.709355i	0.934989	−	0.354677i	$-0.115409\pi$
$108$		0			0
$109$		−	85.4939i		−	0.784348i	−0.919891	−	0.392174i	$-0.871723\pi$
$109$		−	85.4939i		−	0.784348i	0.919891	−	0.392174i	$-0.128277\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−147.266			−1.30324			−0.651621	−	0.758545i	$-0.725911\pi$
$113$	−147.266			−1.30324			−0.651621	+	0.758545i	$0.725911\pi$
$114$		0			0
$115$	33.8039			0.293947
$116$		0			0
$117$		0			0
$118$		0			0
$119$	69.6151			0.585001
$120$		0			0
$121$	−48.8039	+	110.721i	−0.403338	+	0.915051i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	105.629			0.845036
$126$		0			0
$127$		−	138.102i		−	1.08742i	−0.839274	−	0.543709i	$-0.817020\pi$
$127$		−	138.102i		−	1.08742i	0.839274	−	0.543709i	$-0.182980\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	195.424i		−	1.49178i	−0.666068	−	0.745891i	$-0.732024\pi$
$131$		−	195.424i		−	1.49178i	0.666068	−	0.745891i	$-0.267976\pi$
$132$		0			0
$133$	−51.8039			−0.389503
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−66.2321			−0.483446			−0.241723	−	0.970345i	$-0.577713\pi$
$137$	−66.2321			−0.483446			−0.241723	+	0.970345i	$0.577713\pi$
$138$		0			0
$139$			25.3619i			0.182460i	0.995830	+	0.0912300i	$0.0290798\pi$
$139$			25.3619i			0.182460i	−0.995830	+	0.0912300i	$0.970920\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−16.2335	+	10.5851i	−0.113521	+	0.0740216i
$144$		0			0
$145$			40.9856i			0.282659i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	223.657i		−	1.50105i	−0.660842	−	0.750525i	$-0.729801\pi$
$149$		−	223.657i		−	1.50105i	0.660842	−	0.750525i	$-0.270199\pi$
$150$		0			0
$151$			9.76296i			0.0646554i	0.999477	+	0.0323277i	$0.0102920\pi$
$151$			9.76296i			0.0646554i	−0.999477	+	0.0323277i	$0.989708\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−90.1003			−0.581292
$156$		0			0
$157$	−76.9686			−0.490246			−0.245123	−	0.969492i	$-0.578828\pi$
$157$	−76.9686			−0.490246			−0.245123	+	0.969492i	$0.578828\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	143.331i		−	0.890256i
$162$		0			0
$163$	−58.5765			−0.359365			−0.179683	−	0.983725i	$-0.557507\pi$
$163$	−58.5765			−0.359365			−0.179683	+	0.983725i	$0.557507\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	228.635i		−	1.36907i	−0.728978	−	0.684537i	$-0.760004\pi$
$167$		−	228.635i		−	1.36907i	0.728978	−	0.684537i	$-0.239996\pi$
$168$		0			0
$169$	165.896			0.981634
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	93.8261i		−	0.542348i	−0.962530	−	0.271174i	$-0.912588\pi$
$173$		−	93.8261i		−	0.542348i	0.962530	−	0.271174i	$-0.0874118\pi$
$174$		0			0
$175$		−	195.237i		−	1.11564i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	129.463			0.723257			0.361629	−	0.932322i	$-0.382221\pi$
$179$	129.463			0.723257			0.361629	+	0.932322i	$0.382221\pi$
$180$		0			0
$181$	189.855			1.04892			0.524461	−	0.851434i	$-0.324267\pi$
$181$	189.855			1.04892			0.524461	+	0.851434i	$0.324267\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	80.5669			0.435497
$186$		0			0
$187$	−63.4745	+	41.3887i	−0.339436	+	0.221330i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−54.8345			−0.287092			−0.143546	−	0.989644i	$-0.545850\pi$
$191$	−54.8345			−0.287092			−0.143546	+	0.989644i	$0.545850\pi$
$192$		0			0
$193$		−	235.512i		−	1.22027i	−0.792297	−	0.610135i	$-0.791115\pi$
$193$		−	235.512i		−	1.22027i	0.792297	−	0.610135i	$-0.208885\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	43.0892i		−	0.218727i	−0.994002	−	0.109363i	$-0.965119\pi$
$197$		−	43.0892i		−	0.218727i	0.994002	−	0.109363i	$-0.0348812\pi$
$198$		0			0
$199$	−299.969			−1.50738			−0.753690	−	0.657230i	$-0.771728\pi$
$199$	−299.969			−1.50738			−0.753690	+	0.657230i	$0.771728\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	173.782			0.856068
$204$		0			0
$205$			27.5032i			0.134162i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	47.2344	−	30.7994i	0.226002	−	0.147365i
$210$		0			0
$211$			240.369i			1.13919i	0.821925	+	0.569595i	$0.192900\pi$
$211$			240.369i			1.13919i	−0.821925	+	0.569595i	$0.807100\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.11962i			0.0145099i
$216$		0			0
$217$			382.032i			1.76052i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.1365			−0.0549163
$222$		0			0
$223$	−153.412			−0.687945			−0.343973	−	0.938980i	$-0.611773\pi$
$223$	−153.412			−0.687945			−0.343973	+	0.938980i	$0.611773\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			166.965i			0.735530i	0.929919	+	0.367765i	$0.119877\pi$
$227$			166.965i			0.735530i	−0.929919	+	0.367765i	$0.880123\pi$
$228$		0			0
$229$	234.051			1.02206			0.511028	−	0.859564i	$-0.329265\pi$
$229$	234.051			1.02206			0.511028	+	0.859564i	$0.329265\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			254.959i			1.09425i	0.837052	+	0.547123i	$0.184277\pi$
$233$			254.959i			1.09425i	−0.837052	+	0.547123i	$0.815723\pi$
$234$		0			0
$235$	−175.969			−0.748803
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	57.1218i		−	0.239003i	−0.992834	−	0.119502i	$-0.961870\pi$
$239$		−	57.1218i		−	0.239003i	0.992834	−	0.119502i	$-0.0381297\pi$
$240$		0			0
$241$			41.0344i			0.170267i	0.996370	+	0.0851337i	$0.0271317\pi$
$241$			41.0344i			0.170267i	−0.996370	+	0.0851337i	$0.972868\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	126.612			0.516785
$246$		0			0
$247$	9.03135			0.0365642
$248$		0			0
$249$		0			0
$250$		0			0
$251$	210.099			0.837048			0.418524	−	0.908206i	$-0.362548\pi$
$251$	210.099			0.837048			0.418524	+	0.908206i	$0.362548\pi$
$252$		0			0
$253$	85.2157	+	130.688i	0.336821	+	0.516555i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−196.664			−0.765231			−0.382616	−	0.923908i	$-0.624977\pi$
$257$	−196.664			−0.765231			−0.382616	+	0.923908i	$0.624977\pi$
$258$		0			0
$259$		−	341.610i		−	1.31896i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	243.974i		−	0.927658i	−0.885925	−	0.463829i	$-0.846475\pi$
$263$		−	243.974i		−	0.927658i	0.885925	−	0.463829i	$-0.153525\pi$
$264$		0			0
$265$	−6.71175			−0.0253273
$266$		0			0
$267$		0			0
$268$		0			0
$269$	480.111			1.78480			0.892400	−	0.451246i	$-0.149020\pi$
$269$	480.111			1.78480			0.892400	+	0.451246i	$0.149020\pi$
$270$		0			0
$271$		−	120.314i		−	0.443964i	−0.975051	−	0.221982i	$-0.928747\pi$
$271$		−	120.314i		−	0.443964i	0.975051	−	0.221982i	$-0.0712526\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	116.075	+	178.015i	0.422092	+	0.647328i
$276$		0			0
$277$			506.725i			1.82933i	0.404212	+	0.914665i	$0.367546\pi$
$277$			506.725i			1.82933i	−0.404212	+	0.914665i	$0.632454\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			144.912i			0.515701i	0.966185	+	0.257850i	$0.0830141\pi$
$281$			144.912i			0.515701i	−0.966185	+	0.257850i	$0.916986\pi$
$282$		0			0
$283$			25.9250i			0.0916078i	0.998950	+	0.0458039i	$0.0145849\pi$
$283$			25.9250i			0.0916078i	−0.998950	+	0.0458039i	$0.985415\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	116.616			0.406327
$288$		0			0
$289$	241.545			0.835796
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	491.263i		−	1.67667i	−0.545158	−	0.838333i	$-0.683530\pi$
$293$		−	491.263i		−	1.67667i	0.545158	−	0.838333i	$-0.316470\pi$
$294$		0			0
$295$	73.0510			0.247631
$296$		0			0
$297$		0			0
$298$		0			0
$299$			24.9880i			0.0835718i
$300$		0			0
$301$	13.2274			0.0439449
$302$		0			0
$303$		0			0
$304$		0			0
$305$			181.544i			0.595227i
$306$		0			0
$307$		−	367.216i		−	1.19614i	−0.801442	−	0.598072i	$-0.795934\pi$
$307$		−	367.216i		−	1.19614i	0.801442	−	0.598072i	$-0.204066\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−490.215			−1.57625			−0.788127	−	0.615513i	$-0.788949\pi$
$311$	−490.215			−1.57625			−0.788127	+	0.615513i	$0.788949\pi$
$312$		0			0
$313$	16.5058			0.0527343			0.0263672	−	0.999652i	$-0.491606\pi$
$313$	16.5058			0.0527343			0.0263672	+	0.999652i	$0.491606\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	245.348			0.773967			0.386984	−	0.922087i	$-0.373517\pi$
$317$	245.348			0.773967			0.386984	+	0.922087i	$0.373517\pi$
$318$		0			0
$319$	−158.453	+	103.320i	−0.496718	+	0.323886i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	35.3134			0.109330
$324$		0			0
$325$			34.0370i			0.104729i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			746.120i			2.26784i
$330$		0			0
$331$	18.3098			0.0553165			0.0276583	−	0.999617i	$-0.491195\pi$
$331$	18.3098			0.0553165			0.0276583	+	0.999617i	$0.491195\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−167.891			−0.501168
$336$		0			0
$337$		−	211.973i		−	0.629001i	−0.949257	−	0.314500i	$-0.898163\pi$
$337$		−	211.973i		−	0.629001i	0.949257	−	0.314500i	$-0.101837\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−227.132	−	348.334i	−0.666077	−	1.02151i
$342$		0			0
$343$		−	41.6709i		−	0.121489i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			288.296i			0.830823i	0.909634	+	0.415412i	$0.136362\pi$
$347$			288.296i			0.830823i	−0.909634	+	0.415412i	$0.863638\pi$
$348$		0			0
$349$		−	236.711i		−	0.678255i	−0.940740	−	0.339127i	$-0.889868\pi$
$349$		−	236.711i		−	0.678255i	0.940740	−	0.339127i	$-0.110132\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−594.587			−1.68438			−0.842191	−	0.539179i	$-0.818735\pi$
$353$	−594.587			−1.68438			−0.842191	+	0.539179i	$0.818735\pi$
$354$		0			0
$355$	−245.165			−0.690605
$356$		0			0
$357$		0			0
$358$		0			0
$359$			263.686i			0.734500i	0.930122	+	0.367250i	$0.119701\pi$
$359$			263.686i			0.734500i	−0.930122	+	0.367250i	$0.880299\pi$
$360$		0			0
$361$	334.722			0.927207
$362$		0			0
$363$		0			0
$364$		0			0
$365$			153.668i			0.421009i
$366$		0			0
$367$	200.114			0.545269			0.272634	−	0.962118i	$-0.412105\pi$
$367$	200.114			0.545269			0.272634	+	0.962118i	$0.412105\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			28.4583i			0.0767070i
$372$		0			0
$373$		−	425.416i		−	1.14053i	−0.821462	−	0.570263i	$-0.806841\pi$
$373$		−	425.416i		−	1.14053i	0.821462	−	0.570263i	$-0.193159\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−30.2966			−0.0803624
$378$		0			0
$379$	−381.380			−1.00628			−0.503140	−	0.864205i	$-0.667822\pi$
$379$	−381.380			−1.00628			−0.503140	+	0.864205i	$0.667822\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−378.644			−0.988626			−0.494313	−	0.869284i	$-0.664580\pi$
$383$	−378.644			−0.988626			−0.494313	+	0.869284i	$0.664580\pi$
$384$		0			0
$385$	−221.927	+	144.708i	−0.576435	+	0.375866i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−363.615			−0.934744			−0.467372	−	0.884061i	$-0.654799\pi$
$389$	−363.615			−0.934744			−0.467372	+	0.884061i	$0.654799\pi$
$390$		0			0
$391$			97.7053i			0.249886i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	306.228i		−	0.775260i
$396$		0			0
$397$	125.659			0.316521			0.158261	−	0.987397i	$-0.449411\pi$
$397$	125.659			0.316521			0.158261	+	0.987397i	$0.449411\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−122.491			−0.305465			−0.152732	−	0.988268i	$-0.548807\pi$
$401$	−122.491			−0.305465			−0.152732	+	0.988268i	$0.548807\pi$
$402$		0			0
$403$		−	66.6024i		−	0.165267i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	203.100	+	311.477i	0.499016	+	0.765300i
$408$		0			0
$409$		−	468.627i		−	1.14579i	−0.819630	−	0.572894i	$-0.805821\pi$
$409$		−	468.627i		−	1.14579i	0.819630	−	0.572894i	$-0.194179\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	309.742i		−	0.749980i
$414$		0			0
$415$			113.070i			0.272457i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	389.296			0.929107			0.464554	−	0.885545i	$-0.346215\pi$
$419$	389.296			0.929107			0.464554	+	0.885545i	$0.346215\pi$
$420$		0			0
$421$	−441.835			−1.04949			−0.524745	−	0.851259i	$-0.675839\pi$
$421$	−441.835			−1.04949			−0.524745	+	0.851259i	$0.675839\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			133.088i			0.313148i
$426$		0			0
$427$	769.761			1.80272
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	363.146i		−	0.842567i	−0.906929	−	0.421284i	$-0.861580\pi$
$431$		−	363.146i		−	0.842567i	0.906929	−	0.421284i	$-0.138420\pi$
$432$		0			0
$433$	108.898			0.251496			0.125748	−	0.992062i	$-0.459867\pi$
$433$	108.898			0.251496			0.125748	+	0.992062i	$0.459867\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	72.7072i		−	0.166378i
$438$		0			0
$439$			365.661i			0.832941i	0.909149	+	0.416471i	$0.136733\pi$
$439$			365.661i			0.832941i	−0.909149	+	0.416471i	$0.863267\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−696.669			−1.57262			−0.786308	−	0.617834i	$-0.788010\pi$
$443$	−696.669			−1.57262			−0.786308	+	0.617834i	$0.788010\pi$
$444$		0			0
$445$	−374.659			−0.841930
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−591.489			−1.31735			−0.658673	−	0.752429i	$-0.728882\pi$
$449$	−591.489			−1.31735			−0.658673	+	0.752429i	$0.728882\pi$
$450$		0			0
$451$	−106.329	+	69.3324i	−0.235764	+	0.153730i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−42.4331			−0.0932596
$456$		0			0
$457$		−	133.552i		−	0.292236i	−0.989267	−	0.146118i	$-0.953322\pi$
$457$		−	133.552i		−	0.292236i	0.989267	−	0.146118i	$-0.0466779\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			550.399i			1.19392i	0.802269	+	0.596962i	$0.203626\pi$
$461$			550.399i			1.19392i	−0.802269	+	0.596962i	$0.796374\pi$
$462$		0			0
$463$	306.400			0.661771			0.330886	−	0.943671i	$-0.392653\pi$
$463$	306.400			0.661771			0.330886	+	0.943671i	$0.392653\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−719.912			−1.54157			−0.770784	−	0.637097i	$-0.780135\pi$
$467$	−719.912			−1.54157			−0.770784	+	0.637097i	$0.780135\pi$
$468$		0			0
$469$			711.872i			1.51785i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.0607	+	7.86419i	−0.0254982	+	0.0166262i
$474$		0			0
$475$		−	99.0371i		−	0.208499i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			931.687i			1.94507i	0.232764	+	0.972533i	$0.425223\pi$
$479$			931.687i			1.94507i	−0.232764	+	0.972533i	$0.574777\pi$
$480$		0			0
$481$			59.5553i			0.123816i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	86.0953			0.177516
$486$		0			0
$487$	−37.2901			−0.0765711			−0.0382855	−	0.999267i	$-0.512190\pi$
$487$	−37.2901			−0.0765711			−0.0382855	+	0.999267i	$0.512190\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			798.869i			1.62702i	0.581548	+	0.813512i	$0.302447\pi$
$491$			798.869i			1.62702i	−0.581548	+	0.813512i	$0.697553\pi$
$492$		0			0
$493$	−118.463			−0.240290
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1039.52i			2.09158i
$498$		0			0
$499$	680.988			1.36471			0.682353	−	0.731023i	$-0.260957\pi$
$499$	680.988			1.36471			0.682353	+	0.731023i	$0.260957\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	19.7116i		−	0.0391881i	−0.999808	−	0.0195941i	$-0.993763\pi$
$503$		−	19.7116i		−	0.0391881i	0.999808	−	0.0195941i	$-0.00623739\pi$
$504$		0			0
$505$			168.615i			0.333891i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	551.041			1.08259			0.541297	−	0.840831i	$-0.317933\pi$
$509$	551.041			1.08259			0.541297	+	0.840831i	$0.317933\pi$
$510$		0			0
$511$	651.565			1.27508
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−196.492			−0.381537
$516$		0			0
$517$	−443.596	−	680.307i	−0.858020	−	1.31587i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−302.761			−0.581115			−0.290558	−	0.956857i	$-0.593841\pi$
$521$	−302.761			−0.581115			−0.290558	+	0.956857i	$0.593841\pi$
$522$		0			0
$523$			222.653i			0.425724i	0.977082	+	0.212862i	$0.0682784\pi$
$523$			222.653i			0.425724i	−0.977082	+	0.212862i	$0.931722\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	260.422i		−	0.494159i
$528$		0			0
$529$	−327.833			−0.619723
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−20.3305			−0.0381435
$534$		0			0
$535$			180.899i			0.338129i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	319.175	+	489.492i	0.592161	+	0.908149i
$540$		0			0
$541$			404.273i			0.747270i	0.927576	+	0.373635i	$0.121889\pi$
$541$			404.273i			0.747270i	−0.927576	+	0.373635i	$0.878111\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			203.763i			0.373876i
$546$		0			0
$547$			438.910i			0.802395i	0.915992	+	0.401198i	$0.131406\pi$
$547$			438.910i			0.802395i	−0.915992	+	0.401198i	$0.868594\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	88.1539			0.159989
$552$		0			0
$553$	−1298.43			−2.34797
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	852.545i		−	1.53060i	−0.643673	−	0.765300i	$-0.722590\pi$
$557$		−	852.545i		−	1.53060i	0.643673	−	0.765300i	$-0.277410\pi$
$558$		0			0
$559$	−2.30603			−0.00412528
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	410.035i		−	0.728304i	−0.931339	−	0.364152i	$-0.881359\pi$
$563$		−	410.035i		−	0.728304i	0.931339	−	0.364152i	$-0.118641\pi$
$564$		0			0
$565$	350.988			0.621218
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1030.89i			1.81176i	0.423534	+	0.905880i	$0.360790\pi$
$569$			1030.89i			1.81176i	−0.423534	+	0.905880i	$0.639210\pi$
$570$		0			0
$571$			702.660i			1.23058i	0.788302	+	0.615289i	$0.210961\pi$
$571$			702.660i			1.23058i	−0.788302	+	0.615289i	$0.789039\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	274.016			0.476550
$576$		0			0
$577$	169.884			0.294427			0.147214	−	0.989105i	$-0.452970\pi$
$577$	169.884			0.294427			0.147214	+	0.989105i	$0.452970\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	479.424			0.825171
$582$		0			0
$583$	−16.9195	−	25.9481i	−0.0290215	−	0.0445078i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−515.871			−0.878827			−0.439413	−	0.898285i	$-0.644814\pi$
$587$	−515.871			−0.878827			−0.439413	+	0.898285i	$0.644814\pi$
$588$		0			0
$589$			193.792i			0.329019i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1042.97i			1.75880i	0.476086	+	0.879398i	$0.342055\pi$
$593$			1042.97i			1.75880i	−0.476086	+	0.879398i	$0.657945\pi$
$594$		0			0
$595$	−165.918			−0.278853
$596$		0			0
$597$		0			0
$598$		0			0
$599$	781.045			1.30392			0.651958	−	0.758255i	$-0.273948\pi$
$599$	781.045			1.30392			0.651958	+	0.758255i	$0.273948\pi$
$600$		0			0
$601$		−	879.923i		−	1.46410i	−0.681252	−	0.732049i	$-0.738564\pi$
$601$		−	879.923i		−	1.46410i	0.681252	−	0.732049i	$-0.261436\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	116.317	−	263.888i	0.192260	−	0.436179i
$606$		0			0
$607$		−	268.935i		−	0.443056i	−0.975154	−	0.221528i	$-0.928896\pi$
$607$		−	268.935i		−	0.443056i	0.975154	−	0.221528i	$-0.0711044\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	130.077i		−	0.212891i
$612$		0			0
$613$		−	1183.43i		−	1.93056i	−0.261216	−	0.965280i	$-0.584124\pi$
$613$		−	1183.43i		−	1.93056i	0.261216	−	0.965280i	$-0.415876\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−491.855			−0.797172			−0.398586	−	0.917131i	$-0.630499\pi$
$617$	−491.855			−0.797172			−0.398586	+	0.917131i	$0.630499\pi$
$618$		0			0
$619$	−570.835			−0.922189			−0.461095	−	0.887351i	$-0.652543\pi$
$619$	−570.835			−0.922189			−0.461095	+	0.887351i	$0.652543\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1588.58i			2.54989i
$624$		0			0
$625$	231.237			0.369980
$626$		0			0
$627$		0			0
$628$		0			0
$629$			232.867i			0.370217i
$630$		0			0
$631$	56.6706			0.0898107			0.0449054	−	0.998991i	$-0.485701\pi$
$631$	56.6706			0.0898107			0.0449054	+	0.998991i	$0.485701\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			329.146i			0.518341i
$636$		0			0
$637$			93.5923i			0.146927i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−609.753			−0.951252			−0.475626	−	0.879648i	$-0.657778\pi$
$641$	−609.753			−0.951252			−0.475626	+	0.879648i	$0.657778\pi$
$642$		0			0
$643$	622.647			0.968347			0.484174	−	0.874972i	$-0.339120\pi$
$643$	622.647			0.968347			0.484174	+	0.874972i	$0.339120\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	217.138			0.335607			0.167804	−	0.985820i	$-0.446333\pi$
$647$	217.138			0.335607			0.167804	+	0.985820i	$0.446333\pi$
$648$		0			0
$649$	184.153	+	282.420i	0.283749	+	0.435162i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	123.213			0.188688			0.0943438	−	0.995540i	$-0.469925\pi$
$653$	123.213			0.188688			0.0943438	+	0.995540i	$0.469925\pi$
$654$		0			0
$655$			465.764i			0.711090i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	943.790i		−	1.43215i	−0.698021	−	0.716077i	$-0.745936\pi$
$659$		−	943.790i		−	1.43215i	0.698021	−	0.716077i	$-0.254064\pi$
$660$		0			0
$661$	−946.451			−1.43185			−0.715924	−	0.698179i	$-0.753994\pi$
$661$	−946.451			−1.43185			−0.715924	+	0.698179i	$0.753994\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	123.467			0.185665
$666$		0			0
$667$			243.904i			0.365674i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−701.862	+	457.651i	−1.04599	+	0.682044i
$672$		0			0
$673$		−	920.860i		−	1.36829i	−0.729346	−	0.684146i	$-0.760175\pi$
$673$		−	920.860i		−	1.36829i	0.729346	−	0.684146i	$-0.239825\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	377.325i		−	0.557348i	−0.960386	−	0.278674i	$-0.910105\pi$
$677$		−	377.325i		−	0.557348i	0.960386	−	0.278674i	$-0.0898949\pi$
$678$		0			0
$679$		−	365.051i		−	0.537630i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1277.29			−1.87012			−0.935059	−	0.354493i	$-0.884654\pi$
$683$	−1277.29			−1.87012			−0.935059	+	0.354493i	$0.884654\pi$
$684$		0			0
$685$	157.855			0.230445
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	4.96134i		−	0.00720079i
$690$		0			0
$691$	320.278			0.463500			0.231750	−	0.972775i	$-0.425555\pi$
$691$	320.278			0.463500			0.231750	+	0.972775i	$0.425555\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	60.4466i		−	0.0869735i
$696$		0			0
$697$	−79.4941			−0.114052
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	639.288i		−	0.911966i	−0.889988	−	0.455983i	$-0.849288\pi$
$701$		−	639.288i		−	0.911966i	0.889988	−	0.455983i	$-0.150712\pi$
$702$		0			0
$703$		−	173.287i		−	0.246497i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	714.940			1.01123
$708$		0			0
$709$	282.012			0.397760			0.198880	−	0.980024i	$-0.436270\pi$
$709$	282.012			0.397760			0.198880	+	0.980024i	$0.436270\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−536.185			−0.752013
$714$		0			0
$715$	38.6902	−	25.2281i	0.0541122	−	0.0352840i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.0644			−0.0237335			−0.0118667	−	0.999930i	$-0.503777\pi$
$719$	−17.0644			−0.0237335			−0.0118667	+	0.999930i	$0.503777\pi$
$720$		0			0
$721$			833.139i			1.15553i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			332.231i			0.458249i
$726$		0			0
$727$	−641.679			−0.882639			−0.441319	−	0.897350i	$-0.645489\pi$
$727$	−641.679			−0.882639			−0.441319	+	0.897350i	$0.645489\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.01680			−0.0123349
$732$		0			0
$733$			828.464i			1.13024i	0.825010	+	0.565119i	$0.191170\pi$
$733$			828.464i			1.13024i	−0.825010	+	0.565119i	$0.808830\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−423.234	−	649.079i	−0.574266	−	0.880704i
$738$		0			0
$739$		−	1270.46i		−	1.71916i	−0.511001	−	0.859580i	$-0.670725\pi$
$739$		−	1270.46i		−	1.71916i	0.511001	−	0.859580i	$-0.329275\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	214.199i		−	0.288290i	−0.989557	−	0.144145i	$-0.953957\pi$
$743$		−	214.199i		−	0.288290i	0.989557	−	0.144145i	$-0.0460432\pi$
$744$		0			0
$745$			533.054i			0.715508i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	767.026			1.02407
$750$		0			0
$751$	1.88626			0.00251167			0.00125583	−	0.999999i	$-0.499600\pi$
$751$	1.88626			0.00251167			0.00125583	+	0.999999i	$0.499600\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	23.2686i		−	0.0308194i
$756$		0			0
$757$	1044.64			1.37997			0.689986	−	0.723822i	$-0.257617\pi$
$757$	1044.64			1.37997			0.689986	+	0.723822i	$0.257617\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			659.863i			0.867100i	0.901130	+	0.433550i	$0.142739\pi$
$761$			659.863i			0.867100i	−0.901130	+	0.433550i	$0.857261\pi$
$762$		0			0
$763$	863.969			1.13233
$764$		0			0
$765$		0			0
$766$		0			0
$767$			53.9995i			0.0704036i
$768$		0			0
$769$			932.777i			1.21297i	0.795093	+	0.606487i	$0.207422\pi$
$769$			932.777i			1.21297i	−0.795093	+	0.606487i	$0.792578\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−608.179			−0.786777			−0.393389	−	0.919372i	$-0.628697\pi$
$773$	−608.179			−0.786777			−0.393389	+	0.919372i	$0.628697\pi$
$774$		0			0
$775$	−730.357			−0.942396
$776$		0			0
$777$		0			0
$778$		0			0
$779$	59.1554			0.0759376
$780$		0			0
$781$	−618.031	−	947.824i	−0.791333	−	1.21360i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	183.444			0.233686
$786$		0			0
$787$			1041.50i			1.32338i	0.749776	+	0.661691i	$0.230161\pi$
$787$			1041.50i			1.32338i	−0.749776	+	0.661691i	$0.769839\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1488.22i		−	1.88144i
$792$		0			0
$793$	−134.198			−0.169228
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−891.689			−1.11881			−0.559404	−	0.828895i	$-0.688970\pi$
$797$	−891.689			−1.11881			−0.559404	+	0.828895i	$0.688970\pi$
$798$		0			0
$799$		−	508.612i		−	0.636560i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−594.092	+	387.379i	−0.739840	+	0.482415i
$804$		0			0
$805$			341.610i			0.424360i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	913.209i		−	1.12881i	−0.825497	−	0.564406i	$-0.809105\pi$
$809$		−	913.209i		−	1.12881i	0.825497	−	0.564406i	$-0.190895\pi$
$810$		0			0
$811$		−	975.780i		−	1.20318i	−0.798805	−	0.601591i	$-0.794534\pi$
$811$		−	975.780i		−	1.20318i	0.798805	−	0.601591i	$-0.205466\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	139.609			0.171299
$816$		0			0
$817$	6.70984			0.00821278
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1290.51i		−	1.57188i	−0.618304	−	0.785939i	$-0.712180\pi$
$821$		−	1290.51i		−	1.57188i	0.618304	−	0.785939i	$-0.287820\pi$
$822$		0			0
$823$	1474.65			1.79179			0.895897	−	0.444261i	$-0.146534\pi$
$823$	1474.65			1.79179			0.895897	+	0.444261i	$0.146534\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1150.50i		−	1.39117i	−0.718444	−	0.695585i	$-0.755145\pi$
$827$		−	1150.50i		−	1.39117i	0.718444	−	0.695585i	$-0.244855\pi$
$828$		0			0
$829$	1478.34			1.78328			0.891639	−	0.452747i	$-0.149556\pi$
$829$	1478.34			1.78328			0.891639	+	0.452747i	$0.149556\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			365.955i			0.439321i
$834$		0			0
$835$			544.920i			0.652599i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	365.373			0.435487			0.217743	−	0.976006i	$-0.430130\pi$
$839$	365.373			0.435487			0.217743	+	0.976006i	$0.430130\pi$
$840$		0			0
$841$	545.278			0.648369
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−395.390			−0.467917
$846$		0			0
$847$	−1118.91	−	493.194i	−1.32102	−	0.582283i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	479.452			0.563398
$852$		0			0
$853$			820.953i			0.962430i	0.876602	+	0.481215i	$0.159804\pi$
$853$			820.953i			0.962430i	−0.876602	+	0.481215i	$0.840196\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1411.01i			1.64646i	0.567711	+	0.823228i	$0.307829\pi$
$857$			1411.01i			1.64646i	−0.567711	+	0.823228i	$0.692171\pi$
$858$		0			0
$859$	−938.988			−1.09312			−0.546559	−	0.837421i	$-0.684063\pi$
$859$	−938.988			−1.09312			−0.546559	+	0.837421i	$0.684063\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−198.277			−0.229753			−0.114876	−	0.993380i	$-0.536647\pi$
$863$	−198.277			−0.229753			−0.114876	+	0.993380i	$0.536647\pi$
$864$		0			0
$865$			223.621i			0.258522i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1183.90	−	771.964i	1.36237	−	0.888336i
$870$		0			0
$871$		−	124.106i		−	0.142486i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			1067.45i			1.21994i
$876$		0			0
$877$		−	816.451i		−	0.930959i	−0.885059	−	0.465479i	$-0.845882\pi$
$877$		−	816.451i		−	0.930959i	0.885059	−	0.465479i	$-0.154118\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1319.44			1.49766			0.748831	−	0.662761i	$-0.230616\pi$
$881$	1319.44			1.49766			0.748831	+	0.662761i	$0.230616\pi$
$882$		0			0
$883$	−1621.64			−1.83651			−0.918253	−	0.395993i	$-0.870400\pi$
$883$	−1621.64			−1.83651			−0.918253	+	0.395993i	$0.870400\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	51.3762i		−	0.0579213i	−0.999581	−	0.0289607i	$-0.990780\pi$
$887$		−	51.3762i		−	0.0579213i	0.999581	−	0.0289607i	$-0.00921975\pi$
$888$		0			0
$889$	1395.61			1.56986
$890$		0			0
$891$		0			0
$892$		0			0
$893$			378.482i			0.423833i
$894$		0			0
$895$	−308.557			−0.344756
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	650.097i		−	0.723134i
$900$		0			0
$901$		−	19.3993i		−	0.0215309i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−452.492			−0.499991
$906$		0			0
$907$	−518.192			−0.571326			−0.285663	−	0.958330i	$-0.592214\pi$
$907$	−518.192			−0.571326			−0.285663	+	0.958330i	$0.592214\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−429.166			−0.471093			−0.235546	−	0.971863i	$-0.575688\pi$
$911$	−429.166			−0.471093			−0.235546	+	0.971863i	$0.575688\pi$
$912$		0			0
$913$	−437.135	+	285.035i	−0.478790	+	0.312196i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1974.88			2.15363
$918$		0			0
$919$		−	397.278i		−	0.432294i	−0.976361	−	0.216147i	$-0.930651\pi$
$919$		−	397.278i		−	0.432294i	0.976361	−	0.216147i	$-0.0693491\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	181.226i		−	0.196345i
$924$		0			0
$925$	653.079			0.706031
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−926.024			−0.996797			−0.498398	−	0.866948i	$-0.666078\pi$
$929$	−926.024			−0.996797			−0.498398	+	0.866948i	$0.666078\pi$
$930$		0			0
$931$		−	272.325i		−	0.292508i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	151.282	−	98.6442i	0.161799	−	0.105502i
$936$		0			0
$937$		−	1630.89i		−	1.74055i	−0.492569	−	0.870273i	$-0.663942\pi$
$937$		−	1630.89i		−	1.74055i	0.492569	−	0.870273i	$-0.336058\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1211.26i		−	1.28720i	−0.765362	−	0.643600i	$-0.777440\pi$
$941$		−	1211.26i		−	1.28720i	0.765362	−	0.643600i	$-0.222560\pi$
$942$		0			0
$943$			163.671i			0.173564i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−35.6756			−0.0376723			−0.0188361	−	0.999823i	$-0.505996\pi$
$947$	−35.6756			−0.0376723			−0.0188361	+	0.999823i	$0.505996\pi$
$948$		0			0
$949$	−113.592			−0.119696
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1113.69i		−	1.16861i	−0.811533	−	0.584307i	$-0.801366\pi$
$953$		−	1113.69i		−	1.16861i	0.811533	−	0.584307i	$-0.198634\pi$
$954$		0			0
$955$	130.690			0.136848
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	669.317i		−	0.697932i
$960$		0			0
$961$	468.137			0.487135
$962$		0			0
$963$		0			0
$964$		0			0
$965$			561.310i			0.581668i
$966$		0			0
$967$			381.345i			0.394359i	0.980367	+	0.197180i	$0.0631782\pi$
$967$			381.345i			0.394359i	−0.980367	+	0.197180i	$0.936822\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1536.34			1.58222			0.791111	−	0.611672i	$-0.209503\pi$
$971$	1536.34			1.58222			0.791111	+	0.611672i	$0.209503\pi$
$972$		0			0
$973$	−256.298			−0.263410
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1530.48			1.56651			0.783255	−	0.621700i	$-0.213558\pi$
$977$	1530.48			1.56651			0.783255	+	0.621700i	$0.213558\pi$
$978$		0			0
$979$	−944.471	−	1448.46i	−0.964730	−	1.47953i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	487.056			0.495479			0.247739	−	0.968827i	$-0.420312\pi$
$983$	487.056			0.495479			0.247739	+	0.968827i	$0.420312\pi$
$984$		0			0
$985$			102.697i			0.104261i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			18.5648i			0.0187713i
$990$		0			0
$991$	−785.412			−0.792545			−0.396272	−	0.918133i	$-0.629696\pi$
$991$	−785.412			−0.792545			−0.396272	+	0.918133i	$0.629696\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	714.933			0.718525
$996$		0			0
$997$		−	299.670i		−	0.300572i	−0.988643	−	0.150286i	$-0.951981\pi$
$997$		−	299.670i		−	0.300572i	0.988643	−	0.150286i	$-0.0480194\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.m.1297.4		12
3.2	odd	2	inner	1584.3.j.m.1297.10		12
4.3	odd	2		792.3.j.b.505.3	✓	12
11.10	odd	2	inner	1584.3.j.m.1297.3		12
12.11	even	2		792.3.j.b.505.9	yes	12
33.32	even	2	inner	1584.3.j.m.1297.9		12
44.43	even	2		792.3.j.b.505.4	yes	12
132.131	odd	2		792.3.j.b.505.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.3.j.b.505.3	✓	12	4.3	odd	2
792.3.j.b.505.4	yes	12	44.43	even	2
792.3.j.b.505.9	yes	12	12.11	even	2
792.3.j.b.505.10	yes	12	132.131	odd	2
1584.3.j.m.1297.3		12	11.10	odd	2	inner
1584.3.j.m.1297.4		12	1.1	even	1	trivial
1584.3.j.m.1297.9		12	33.32	even	2	inner
1584.3.j.m.1297.10		12	3.2	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)

\(f(q)\)	\(=\)	\(q-2.38336 q^{5} +10.1056i q^{7} +(-6.00816 - 9.21423i) q^{11} -1.76179i q^{13} -6.88875i q^{17} +5.12625i q^{19} -14.1833 q^{23} -19.3196 q^{25} -17.1966i q^{29} +37.8039 q^{31} -24.0853i q^{35} -33.8039 q^{37} -11.5397i q^{41} -1.30892i q^{43} +73.8322 q^{47} -53.1235 q^{49} +2.81609 q^{53} +(14.3196 + 21.9608i) q^{55} -30.6505 q^{59} -76.1716i q^{61} +4.19896i q^{65} +70.4431 q^{67} +102.865 q^{71} -64.4755i q^{73} +(93.1155 - 60.7162i) q^{77} +128.486i q^{79} -47.4413i q^{83} +16.4184i q^{85} +157.198 q^{89} +17.8039 q^{91} -12.2177i q^{95} -36.1235 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 68 q^{25} + 48 q^{37} + 116 q^{49} - 128 q^{55} - 208 q^{67} - 240 q^{91} + 320 q^{97}+O(q^{100}) \) 12 * q + 68 * q^25 + 48 * q^37 + 116 * q^49 - 128 * q^55 - 208 * q^67 - 240 * q^91 + 320 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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