LMFDB - Embedded newform 5040.2.d.b.4591.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5040,2,Mod(4591,5040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5040.4591"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,-4,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1680)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4591.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	5040.4591
Dual form			5040.2.d.b.4591.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +(-2.44949 + 1.00000i) q^{7} +2.44949i q^{11} -4.44949i q^{13} +6.89898i q^{17} +4.44949 q^{19} -2.00000i q^{23} -1.00000 q^{25} +6.89898 q^{29} -8.44949 q^{31} +(1.00000 + 2.44949i) q^{35} +6.89898 q^{37} +0.898979i q^{41} -8.00000i q^{43} -10.8990 q^{47} +(5.00000 - 4.89898i) q^{49} -10.4495 q^{53} +2.44949 q^{55} -4.89898 q^{59} -12.8990i q^{61} -4.44949 q^{65} +15.7980i q^{67} +6.44949i q^{71} -5.34847i q^{73} +(-2.44949 - 6.00000i) q^{77} +9.79796i q^{79} +2.89898 q^{83} +6.89898 q^{85} +15.7980i q^{89} +(4.44949 + 10.8990i) q^{91} -4.44949i q^{95} +2.65153i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{19} - 4 q^{25} + 8 q^{29} - 24 q^{31} + 4 q^{35} + 8 q^{37} - 24 q^{47} + 20 q^{49} - 32 q^{53} - 8 q^{65} - 8 q^{83} + 8 q^{85} + 8 q^{91}+O(q^{100}) $ 4 * q + 8 * q^19 - 4 * q^25 + 8 * q^29 - 24 * q^31 + 4 * q^35 + 8 * q^37 - 24 * q^47 + 20 * q^49 - 32 * q^53 - 8 * q^65 - 8 * q^83 + 8 * q^85 + 8 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$-1$	$1$

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
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.44949i			0.738549i	0.929320	+	0.369274i	$0.120394\pi$
$11$			2.44949i			0.738549i	−0.929320	+	0.369274i	$0.879606\pi$
$12$		0			0
$13$		−	4.44949i		−	1.23407i	−0.786937	−	0.617033i	$-0.788334\pi$
$13$		−	4.44949i		−	1.23407i	0.786937	−	0.617033i	$-0.211666\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.89898i			1.67325i	0.547777	+	0.836624i	$0.315474\pi$
$17$			6.89898i			1.67325i	−0.547777	+	0.836624i	$0.684526\pi$
$18$		0			0
$19$	4.44949			1.02078			0.510391	−	0.859942i	$-0.329501\pi$
$19$	4.44949			1.02078			0.510391	+	0.859942i	$0.329501\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.89898			1.28111			0.640554	−	0.767913i	$-0.278705\pi$
$29$	6.89898			1.28111			0.640554	+	0.767913i	$0.278705\pi$
$30$		0			0
$31$	−8.44949			−1.51757			−0.758787	−	0.651339i	$-0.774207\pi$
$31$	−8.44949			−1.51757			−0.758787	+	0.651339i	$0.774207\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	+	2.44949i	0.169031	+	0.414039i
$36$		0			0
$37$	6.89898			1.13419			0.567093	−	0.823654i	$-0.308068\pi$
$37$	6.89898			1.13419			0.567093	+	0.823654i	$0.308068\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.898979i			0.140397i	0.997533	+	0.0701985i	$0.0223633\pi$
$41$			0.898979i			0.140397i	−0.997533	+	0.0701985i	$0.977637\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.8990			−1.58978			−0.794890	−	0.606754i	$-0.792471\pi$
$47$	−10.8990			−1.58978			−0.794890	+	0.606754i	$0.792471\pi$
$48$		0			0
$49$	5.00000	−	4.89898i	0.714286	−	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.4495			−1.43535			−0.717674	−	0.696379i	$-0.754793\pi$
$53$	−10.4495			−1.43535			−0.717674	+	0.696379i	$0.754793\pi$
$54$		0			0
$55$	2.44949			0.330289
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.89898			−0.637793			−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898			−0.637793			−0.318896	+	0.947790i	$0.603312\pi$
$60$		0			0
$61$		−	12.8990i		−	1.65155i	−0.564003	−	0.825773i	$-0.690739\pi$
$61$		−	12.8990i		−	1.65155i	0.564003	−	0.825773i	$-0.309261\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.44949			−0.551891
$66$		0			0
$67$			15.7980i			1.93003i	0.262199	+	0.965014i	$0.415552\pi$
$67$			15.7980i			1.93003i	−0.262199	+	0.965014i	$0.584448\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.44949i			0.765414i	0.923870	+	0.382707i	$0.125008\pi$
$71$			6.44949i			0.765414i	−0.923870	+	0.382707i	$0.874992\pi$
$72$		0			0
$73$		−	5.34847i		−	0.625991i	−0.949755	−	0.312995i	$-0.898668\pi$
$73$		−	5.34847i		−	0.625991i	0.949755	−	0.312995i	$-0.101332\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.44949	−	6.00000i	−0.279145	−	0.683763i
$78$		0			0
$79$			9.79796i			1.10236i	0.834388	+	0.551178i	$0.185822\pi$
$79$			9.79796i			1.10236i	−0.834388	+	0.551178i	$0.814178\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.89898			0.318204			0.159102	−	0.987262i	$-0.449140\pi$
$83$	2.89898			0.318204			0.159102	+	0.987262i	$0.449140\pi$
$84$		0			0
$85$	6.89898			0.748299
$86$		0			0
$87$		0			0
$88$		0			0
$89$			15.7980i			1.67458i	0.546759	+	0.837290i	$0.315861\pi$
$89$			15.7980i			1.67458i	−0.546759	+	0.837290i	$0.684139\pi$
$90$		0			0
$91$	4.44949	+	10.8990i	0.466433	+	1.14252i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	4.44949i		−	0.456508i
$96$		0			0
$97$			2.65153i			0.269222i	0.990899	+	0.134611i	$0.0429785\pi$
$97$			2.65153i			0.269222i	−0.990899	+	0.134611i	$0.957021\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.89898i			0.885482i	0.896650	+	0.442741i	$0.145994\pi$
$101$			8.89898i			0.885482i	−0.896650	+	0.442741i	$0.854006\pi$
$102$		0			0
$103$	−8.89898			−0.876843			−0.438421	−	0.898770i	$-0.644462\pi$
$103$	−8.89898			−0.876843			−0.438421	+	0.898770i	$0.644462\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.7980i			1.52725i	0.645662	+	0.763623i	$0.276581\pi$
$107$			15.7980i			1.52725i	−0.645662	+	0.763623i	$0.723419\pi$
$108$		0			0
$109$	−13.7980			−1.32160			−0.660802	−	0.750560i	$-0.729784\pi$
$109$	−13.7980			−1.32160			−0.660802	+	0.750560i	$0.729784\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.34847			0.314997			0.157499	−	0.987519i	$-0.449657\pi$
$113$	3.34847			0.314997			0.157499	+	0.987519i	$0.449657\pi$
$114$		0			0
$115$	−2.00000			−0.186501
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.89898	−	16.8990i	−0.632428	−	1.54913i
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			1.79796i			0.159543i	0.996813	+	0.0797715i	$0.0254191\pi$
$127$			1.79796i			0.159543i	−0.996813	+	0.0797715i	$0.974581\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18.6969			1.63356			0.816780	−	0.576950i	$-0.195757\pi$
$131$	18.6969			1.63356			0.816780	+	0.576950i	$0.195757\pi$
$132$		0			0
$133$	−10.8990	+	4.44949i	−0.945061	+	0.385820i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−21.1464			−1.80666			−0.903331	−	0.428945i	$-0.858885\pi$
$137$	−21.1464			−1.80666			−0.903331	+	0.428945i	$0.858885\pi$
$138$		0			0
$139$	−7.55051			−0.640426			−0.320213	−	0.947346i	$-0.603754\pi$
$139$	−7.55051			−0.640426			−0.320213	+	0.947346i	$0.603754\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.8990			0.911418
$144$		0			0
$145$		−	6.89898i		−	0.572929i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.79796			0.311141			0.155570	−	0.987825i	$-0.450278\pi$
$149$	3.79796			0.311141			0.155570	+	0.987825i	$0.450278\pi$
$150$		0			0
$151$			8.89898i			0.724189i	0.932141	+	0.362094i	$0.117938\pi$
$151$			8.89898i			0.724189i	−0.932141	+	0.362094i	$0.882062\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			8.44949i			0.678679i
$156$		0			0
$157$			17.3485i			1.38456i	0.721630	+	0.692279i	$0.243393\pi$
$157$			17.3485i			1.38456i	−0.721630	+	0.692279i	$0.756607\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.00000	+	4.89898i	0.157622	+	0.386094i
$162$		0			0
$163$			23.5959i			1.84817i	0.382181	+	0.924087i	$0.375173\pi$
$163$			23.5959i			1.84817i	−0.382181	+	0.924087i	$0.624827\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.00000			0.619059			0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000			0.619059			0.309529	+	0.950890i	$0.399829\pi$
$168$		0			0
$169$	−6.79796			−0.522920
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	3.79796i		−	0.288753i	−0.989523	−	0.144377i	$-0.953882\pi$
$173$		−	3.79796i		−	0.288753i	0.989523	−	0.144377i	$-0.0461177\pi$
$174$		0			0
$175$	2.44949	−	1.00000i	0.185164	−	0.0755929i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			0.247449i			0.0184952i	0.999957	+	0.00924759i	$0.00294364\pi$
$179$			0.247449i			0.0184952i	−0.999957	+	0.00924759i	$0.997056\pi$
$180$		0			0
$181$			13.7980i			1.02559i	0.858510	+	0.512797i	$0.171391\pi$
$181$			13.7980i			1.02559i	−0.858510	+	0.512797i	$0.828609\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.89898i		−	0.507223i
$186$		0			0
$187$	−16.8990			−1.23578
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.4495i			0.756099i	0.925785	+	0.378049i	$0.123405\pi$
$191$			10.4495i			0.756099i	−0.925785	+	0.378049i	$0.876595\pi$
$192$		0			0
$193$	−12.6969			−0.913946			−0.456973	−	0.889481i	$-0.651066\pi$
$193$	−12.6969			−0.913946			−0.456973	+	0.889481i	$0.651066\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.247449			−0.0176300			−0.00881500	−	0.999961i	$-0.502806\pi$
$197$	−0.247449			−0.0176300			−0.00881500	+	0.999961i	$0.502806\pi$
$198$		0			0
$199$	4.44949			0.315416			0.157708	−	0.987486i	$-0.449590\pi$
$199$	4.44949			0.315416			0.157708	+	0.987486i	$0.449590\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16.8990	+	6.89898i	−1.18608	+	0.484213i
$204$		0			0
$205$	0.898979			0.0627875
$206$		0			0
$207$		0			0
$208$		0			0
$209$			10.8990i			0.753898i
$210$		0			0
$211$			6.69694i			0.461036i	0.973068	+	0.230518i	$0.0740421\pi$
$211$			6.69694i			0.461036i	−0.973068	+	0.230518i	$0.925958\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000			−0.545595
$216$		0			0
$217$	20.6969	−	8.44949i	1.40500	−	0.573589i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	30.6969			2.06490
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.6969			−1.37370			−0.686852	−	0.726797i	$-0.741008\pi$
$227$	−20.6969			−1.37370			−0.686852	+	0.726797i	$0.741008\pi$
$228$		0			0
$229$		−	7.10102i		−	0.469249i	−0.972086	−	0.234624i	$-0.924614\pi$
$229$		−	7.10102i		−	0.469249i	0.972086	−	0.234624i	$-0.0753860\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.2474			0.802357			0.401179	−	0.916000i	$-0.368601\pi$
$233$	12.2474			0.802357			0.401179	+	0.916000i	$0.368601\pi$
$234$		0			0
$235$			10.8990i			0.710971i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			26.4495i			1.71088i	0.517906	+	0.855438i	$0.326712\pi$
$239$			26.4495i			1.71088i	−0.517906	+	0.855438i	$0.673288\pi$
$240$		0			0
$241$		−	0.898979i		−	0.0579084i	−0.999581	−	0.0289542i	$-0.990782\pi$
$241$		−	0.898979i		−	0.0579084i	0.999581	−	0.0289542i	$-0.00921769\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.89898	−	5.00000i	−0.312984	−	0.319438i
$246$		0			0
$247$		−	19.7980i		−	1.25971i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.30306			−0.0822485			−0.0411243	−	0.999154i	$-0.513094\pi$
$251$	−1.30306			−0.0822485			−0.0411243	+	0.999154i	$0.513094\pi$
$252$		0			0
$253$	4.89898			0.307996
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.10102i		−	0.0686798i	−0.999410	−	0.0343399i	$-0.989067\pi$
$257$		−	1.10102i		−	0.0686798i	0.999410	−	0.0343399i	$-0.0109329\pi$
$258$		0			0
$259$	−16.8990	+	6.89898i	−1.05005	+	0.428682i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.10102i		−	0.0678918i	−0.999424	−	0.0339459i	$-0.989193\pi$
$263$		−	1.10102i		−	0.0678918i	0.999424	−	0.0339459i	$-0.0108074\pi$
$264$		0			0
$265$			10.4495i			0.641907i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	20.4949i		−	1.24960i	−0.780786	−	0.624798i	$-0.785181\pi$
$269$		−	20.4949i		−	1.24960i	0.780786	−	0.624798i	$-0.214819\pi$
$270$		0			0
$271$	19.1464			1.16306			0.581531	−	0.813524i	$-0.302454\pi$
$271$	19.1464			1.16306			0.581531	+	0.813524i	$0.302454\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	2.44949i		−	0.147710i
$276$		0			0
$277$	13.5959			0.816900			0.408450	−	0.912781i	$-0.366070\pi$
$277$	13.5959			0.816900			0.408450	+	0.912781i	$0.366070\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.10102			0.0656814			0.0328407	−	0.999461i	$-0.489545\pi$
$281$	1.10102			0.0656814			0.0328407	+	0.999461i	$0.489545\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.898979	−	2.20204i	−0.0530651	−	0.129982i
$288$		0			0
$289$	−30.5959			−1.79976
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	24.6969i		−	1.44281i	−0.692513	−	0.721405i	$-0.743497\pi$
$293$		−	24.6969i		−	1.44281i	0.692513	−	0.721405i	$-0.256503\pi$
$294$		0			0
$295$			4.89898i			0.285230i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.89898			−0.514641
$300$		0			0
$301$	8.00000	+	19.5959i	0.461112	+	1.12949i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.8990			−0.738593
$306$		0			0
$307$	−8.89898			−0.507892			−0.253946	−	0.967218i	$-0.581728\pi$
$307$	−8.89898			−0.507892			−0.253946	+	0.967218i	$0.581728\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.10102			−0.175843			−0.0879214	−	0.996127i	$-0.528022\pi$
$311$	−3.10102			−0.175843			−0.0879214	+	0.996127i	$0.528022\pi$
$312$		0			0
$313$		−	14.6515i		−	0.828153i	−0.910242	−	0.414077i	$-0.864105\pi$
$313$		−	14.6515i		−	0.828153i	0.910242	−	0.414077i	$-0.135895\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.1464			−0.963039			−0.481520	−	0.876435i	$-0.659915\pi$
$317$	−17.1464			−0.963039			−0.481520	+	0.876435i	$0.659915\pi$
$318$		0			0
$319$			16.8990i			0.946161i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			30.6969i			1.70802i
$324$		0			0
$325$			4.44949i			0.246813i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	26.6969	−	10.8990i	1.47185	−	0.600880i
$330$		0			0
$331$		−	7.10102i		−	0.390307i	−0.980773	−	0.195154i	$-0.937479\pi$
$331$		−	7.10102i		−	0.390307i	0.980773	−	0.195154i	$-0.0625206\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.7980			0.863135
$336$		0			0
$337$	12.6969			0.691646			0.345823	−	0.938300i	$-0.387600\pi$
$337$	12.6969			0.691646			0.345823	+	0.938300i	$0.387600\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	20.6969i		−	1.12080i
$342$		0			0
$343$	−7.34847	+	17.0000i	−0.396780	+	0.917914i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	34.8990i		−	1.87348i	−0.350032	−	0.936738i	$-0.613829\pi$
$347$		−	34.8990i		−	1.87348i	0.350032	−	0.936738i	$-0.386171\pi$
$348$		0			0
$349$			10.6969i			0.572594i	0.958141	+	0.286297i	$0.0924244\pi$
$349$			10.6969i			0.572594i	−0.958141	+	0.286297i	$0.907576\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			31.7980i			1.69243i	0.532838	+	0.846217i	$0.321126\pi$
$353$			31.7980i			1.69243i	−0.532838	+	0.846217i	$0.678874\pi$
$354$		0			0
$355$	6.44949			0.342303
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12.2474i			0.646396i	0.946331	+	0.323198i	$0.104758\pi$
$359$			12.2474i			0.646396i	−0.946331	+	0.323198i	$0.895242\pi$
$360$		0			0
$361$	0.797959			0.0419978
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.34847			−0.279952
$366$		0			0
$367$	−9.79796			−0.511449			−0.255725	−	0.966750i	$-0.582314\pi$
$367$	−9.79796			−0.511449			−0.255725	+	0.966750i	$0.582314\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	25.5959	−	10.4495i	1.32887	−	0.542510i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	30.6969i		−	1.58097i
$378$		0			0
$379$		−	11.1010i		−	0.570221i	−0.958495	−	0.285111i	$-0.907970\pi$
$379$		−	11.1010i		−	0.570221i	0.958495	−	0.285111i	$-0.0920303\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.20204			0.316909			0.158455	−	0.987366i	$-0.449349\pi$
$383$	6.20204			0.316909			0.158455	+	0.987366i	$0.449349\pi$
$384$		0			0
$385$	−6.00000	+	2.44949i	−0.305788	+	0.124838i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	21.5959			1.09496			0.547478	−	0.836820i	$-0.315588\pi$
$389$	21.5959			1.09496			0.547478	+	0.836820i	$0.315588\pi$
$390$		0			0
$391$	13.7980			0.697793
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.79796			0.492989
$396$		0			0
$397$			37.8434i			1.89930i	0.313306	+	0.949652i	$0.398563\pi$
$397$			37.8434i			1.89930i	−0.313306	+	0.949652i	$0.601437\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−27.7980			−1.38816			−0.694082	−	0.719896i	$-0.744189\pi$
$401$	−27.7980			−1.38816			−0.694082	+	0.719896i	$0.744189\pi$
$402$		0			0
$403$			37.5959i			1.87279i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			16.8990i			0.837651i
$408$		0			0
$409$			17.7980i			0.880052i	0.897985	+	0.440026i	$0.145031\pi$
$409$			17.7980i			0.880052i	−0.897985	+	0.440026i	$0.854969\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.0000	−	4.89898i	0.590481	−	0.241063i
$414$		0			0
$415$		−	2.89898i		−	0.142305i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.5959			−0.761910			−0.380955	−	0.924593i	$-0.624405\pi$
$419$	−15.5959			−0.761910			−0.380955	+	0.924593i	$0.624405\pi$
$420$		0			0
$421$	−29.5959			−1.44242			−0.721208	−	0.692718i	$-0.756413\pi$
$421$	−29.5959			−1.44242			−0.721208	+	0.692718i	$0.756413\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	6.89898i		−	0.334650i
$426$		0			0
$427$	12.8990	+	31.5959i	0.624225	+	1.52903i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.55051i		−	0.0746855i	−0.999303	−	0.0373427i	$-0.988111\pi$
$431$		−	1.55051i		−	0.0746855i	0.999303	−	0.0373427i	$-0.0118893\pi$
$432$		0			0
$433$		−	0.449490i		−	0.0216011i	−0.999942	−	0.0108005i	$-0.996562\pi$
$433$		−	0.449490i		−	0.0216011i	0.999942	−	0.0108005i	$-0.00343799\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	8.89898i		−	0.425696i
$438$		0			0
$439$	9.75255			0.465464			0.232732	−	0.972541i	$-0.425234\pi$
$439$	9.75255			0.465464			0.232732	+	0.972541i	$0.425234\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			1.10102i			0.0523111i	0.999658	+	0.0261555i	$0.00832651\pi$
$443$			1.10102i			0.0523111i	−0.999658	+	0.0261555i	$0.991673\pi$
$444$		0			0
$445$	15.7980			0.748895
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.202041			−0.00953491			−0.00476745	−	0.999989i	$-0.501518\pi$
$449$	−0.202041			−0.00953491			−0.00476745	+	0.999989i	$0.501518\pi$
$450$		0			0
$451$	−2.20204			−0.103690
$452$		0			0
$453$		0			0
$454$		0			0
$455$	10.8990	−	4.44949i	0.510952	−	0.208595i
$456$		0			0
$457$	22.4949			1.05227			0.526133	−	0.850402i	$-0.323641\pi$
$457$	22.4949			1.05227			0.526133	+	0.850402i	$0.323641\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8.20204i			0.382007i	0.981589	+	0.191004i	$0.0611742\pi$
$461$			8.20204i			0.382007i	−0.981589	+	0.191004i	$0.938826\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.6969			−0.587544			−0.293772	−	0.955875i	$-0.594911\pi$
$467$	−12.6969			−0.587544			−0.293772	+	0.955875i	$0.594911\pi$
$468$		0			0
$469$	−15.7980	−	38.6969i	−0.729482	−	1.78686i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	19.5959			0.901021
$474$		0			0
$475$	−4.44949			−0.204157
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.10102			−0.141689			−0.0708446	−	0.997487i	$-0.522569\pi$
$479$	−3.10102			−0.141689			−0.0708446	+	0.997487i	$0.522569\pi$
$480$		0			0
$481$		−	30.6969i		−	1.39966i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.65153			0.120400
$486$		0			0
$487$		−	23.5959i		−	1.06923i	−0.845095	−	0.534617i	$-0.820456\pi$
$487$		−	23.5959i		−	1.06923i	0.845095	−	0.534617i	$-0.179544\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			20.6515i			0.931991i	0.884787	+	0.465995i	$0.154304\pi$
$491$			20.6515i			0.931991i	−0.884787	+	0.465995i	$0.845696\pi$
$492$		0			0
$493$			47.5959i			2.14361i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.44949	−	15.7980i	−0.289299	−	0.708635i
$498$		0			0
$499$		−	5.30306i		−	0.237398i	−0.992930	−	0.118699i	$-0.962128\pi$
$499$		−	5.30306i		−	0.237398i	0.992930	−	0.118699i	$-0.0378723\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−43.5959			−1.94385			−0.971923	−	0.235299i	$-0.924393\pi$
$503$	−43.5959			−1.94385			−0.971923	+	0.235299i	$0.924393\pi$
$504$		0			0
$505$	8.89898			0.395999
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	4.89898i		−	0.217143i	−0.994089	−	0.108572i	$-0.965372\pi$
$509$		−	4.89898i		−	0.217143i	0.994089	−	0.108572i	$-0.0346277\pi$
$510$		0			0
$511$	5.34847	+	13.1010i	0.236602	+	0.579555i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			8.89898i			0.392136i
$516$		0			0
$517$		−	26.6969i		−	1.17413i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			3.10102i			0.135858i	0.997690	+	0.0679291i	$0.0216392\pi$
$521$			3.10102i			0.135858i	−0.997690	+	0.0679291i	$0.978361\pi$
$522$		0			0
$523$	−27.1010			−1.18504			−0.592522	−	0.805554i	$-0.701868\pi$
$523$	−27.1010			−1.18504			−0.592522	+	0.805554i	$0.701868\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	58.2929i		−	2.53928i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.00000			0.173259
$534$		0			0
$535$	15.7980			0.683005
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.0000	+	12.2474i	0.516877	+	0.527535i
$540$		0			0
$541$	−25.5959			−1.10045			−0.550227	−	0.835015i	$-0.685459\pi$
$541$	−25.5959			−1.10045			−0.550227	+	0.835015i	$0.685459\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			13.7980i			0.591040i
$546$		0			0
$547$		−	6.40408i		−	0.273819i	−0.990584	−	0.136909i	$-0.956283\pi$
$547$		−	6.40408i		−	0.273819i	0.990584	−	0.136909i	$-0.0437169\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	30.6969			1.30773
$552$		0			0
$553$	−9.79796	−	24.0000i	−0.416652	−	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.4495			0.442759			0.221380	−	0.975188i	$-0.428944\pi$
$557$	10.4495			0.442759			0.221380	+	0.975188i	$0.428944\pi$
$558$		0			0
$559$	−35.5959			−1.50555
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$		−	3.34847i		−	0.140871i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.7980			0.494596			0.247298	−	0.968939i	$-0.420457\pi$
$569$	11.7980			0.494596			0.247298	+	0.968939i	$0.420457\pi$
$570$		0			0
$571$			28.0000i			1.17176i	0.810397	+	0.585882i	$0.199252\pi$
$571$			28.0000i			1.17176i	−0.810397	+	0.585882i	$0.800748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			2.00000i			0.0834058i
$576$		0			0
$577$			23.5505i			0.980421i	0.871604	+	0.490210i	$0.163080\pi$
$577$			23.5505i			0.980421i	−0.871604	+	0.490210i	$0.836920\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.10102	+	2.89898i	−0.294600	+	0.120270i
$582$		0			0
$583$		−	25.5959i		−	1.06007i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−43.1918			−1.78272			−0.891359	−	0.453298i	$-0.850247\pi$
$587$	−43.1918			−1.78272			−0.891359	+	0.453298i	$0.850247\pi$
$588$		0			0
$589$	−37.5959			−1.54911
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	20.2020i		−	0.829598i	−0.909913	−	0.414799i	$-0.863852\pi$
$593$		−	20.2020i		−	0.829598i	0.909913	−	0.414799i	$-0.136148\pi$
$594$		0			0
$595$	−16.8990	+	6.89898i	−0.692791	+	0.282831i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	22.0454i		−	0.900751i	−0.892839	−	0.450375i	$-0.851290\pi$
$599$		−	22.0454i		−	0.900751i	0.892839	−	0.450375i	$-0.148710\pi$
$600$		0			0
$601$			20.8990i			0.852487i	0.904608	+	0.426244i	$0.140163\pi$
$601$			20.8990i			0.852487i	−0.904608	+	0.426244i	$0.859837\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	5.00000i		−	0.203279i
$606$		0			0
$607$	−15.5959			−0.633019			−0.316509	−	0.948589i	$-0.602511\pi$
$607$	−15.5959			−0.633019			−0.316509	+	0.948589i	$0.602511\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			48.4949i			1.96189i
$612$		0			0
$613$	31.7980			1.28431			0.642154	−	0.766576i	$-0.278041\pi$
$613$	31.7980			1.28431			0.642154	+	0.766576i	$0.278041\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−31.8434			−1.28197			−0.640983	−	0.767555i	$-0.721473\pi$
$617$	−31.8434			−1.28197			−0.640983	+	0.767555i	$0.721473\pi$
$618$		0			0
$619$	−12.0454			−0.484146			−0.242073	−	0.970258i	$-0.577827\pi$
$619$	−12.0454			−0.484146			−0.242073	+	0.970258i	$0.577827\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15.7980	−	38.6969i	−0.632932	−	1.55036i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			47.5959i			1.89777i
$630$		0			0
$631$		−	49.3939i		−	1.96634i	−0.182695	−	0.983170i	$-0.558482\pi$
$631$		−	49.3939i		−	1.96634i	0.182695	−	0.983170i	$-0.441518\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.79796			0.0713498
$636$		0			0
$637$	−21.7980	−	22.2474i	−0.863667	−	0.881476i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.6969			−0.501499			−0.250749	−	0.968052i	$-0.580677\pi$
$641$	−12.6969			−0.501499			−0.250749	+	0.968052i	$0.580677\pi$
$642$		0			0
$643$	28.8990			1.13966			0.569832	−	0.821761i	$-0.307008\pi$
$643$	28.8990			1.13966			0.569832	+	0.821761i	$0.307008\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−27.5959			−1.08491			−0.542454	−	0.840086i	$-0.682505\pi$
$647$	−27.5959			−1.08491			−0.542454	+	0.840086i	$0.682505\pi$
$648$		0			0
$649$		−	12.0000i		−	0.471041i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.14643			−0.357927			−0.178964	−	0.983856i	$-0.557274\pi$
$653$	−9.14643			−0.357927			−0.178964	+	0.983856i	$0.557274\pi$
$654$		0			0
$655$		−	18.6969i		−	0.730550i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	24.2474i		−	0.944546i	−0.881452	−	0.472273i	$-0.843434\pi$
$659$		−	24.2474i		−	0.944546i	0.881452	−	0.472273i	$-0.156566\pi$
$660$		0			0
$661$			35.1010i			1.36527i	0.730759	+	0.682636i	$0.239166\pi$
$661$			35.1010i			1.36527i	−0.730759	+	0.682636i	$0.760834\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.44949	+	10.8990i	0.172544	+	0.422644i
$666$		0			0
$667$		−	13.7980i		−	0.534259i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	31.5959			1.21975
$672$		0			0
$673$	3.30306			0.127324			0.0636618	−	0.997972i	$-0.479722\pi$
$673$	3.30306			0.127324			0.0636618	+	0.997972i	$0.479722\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			8.69694i			0.334250i	0.985936	+	0.167125i	$0.0534484\pi$
$677$			8.69694i			0.334250i	−0.985936	+	0.167125i	$0.946552\pi$
$678$		0			0
$679$	−2.65153	−	6.49490i	−0.101756	−	0.249251i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	29.1010i		−	1.11352i	−0.830674	−	0.556760i	$-0.812044\pi$
$683$		−	29.1010i		−	1.11352i	0.830674	−	0.556760i	$-0.187956\pi$
$684$		0			0
$685$			21.1464i			0.807963i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			46.4949i			1.77131i
$690$		0			0
$691$	−23.5505			−0.895904			−0.447952	−	0.894058i	$-0.647846\pi$
$691$	−23.5505			−0.895904			−0.447952	+	0.894058i	$0.647846\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.55051i			0.286407i
$696$		0			0
$697$	−6.20204			−0.234919
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.6969			0.479557			0.239778	−	0.970828i	$-0.422925\pi$
$701$	12.6969			0.479557			0.239778	+	0.970828i	$0.422925\pi$
$702$		0			0
$703$	30.6969			1.15776
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.89898	−	21.7980i	−0.334681	−	0.819797i
$708$		0			0
$709$	35.7980			1.34442			0.672210	−	0.740360i	$-0.265345\pi$
$709$	35.7980			1.34442			0.672210	+	0.740360i	$0.265345\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			16.8990i			0.632872i
$714$		0			0
$715$		−	10.8990i		−	0.407599i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−31.1010			−1.15987			−0.579936	−	0.814662i	$-0.696923\pi$
$719$	−31.1010			−1.15987			−0.579936	+	0.814662i	$0.696923\pi$
$720$		0			0
$721$	21.7980	−	8.89898i	0.811798	−	0.331415i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.89898			−0.256222
$726$		0			0
$727$	6.20204			0.230021			0.115010	−	0.993364i	$-0.463310\pi$
$727$	6.20204			0.230021			0.115010	+	0.993364i	$0.463310\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	55.1918			2.04134
$732$		0			0
$733$			2.65153i			0.0979365i	0.998800	+	0.0489683i	$0.0155933\pi$
$733$			2.65153i			0.0979365i	−0.998800	+	0.0489683i	$0.984407\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−38.6969			−1.42542
$738$		0			0
$739$			47.5959i			1.75084i	0.483359	+	0.875422i	$0.339416\pi$
$739$			47.5959i			1.75084i	−0.483359	+	0.875422i	$0.660584\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	20.2929i		−	0.744473i	−0.928138	−	0.372236i	$-0.878591\pi$
$743$		−	20.2929i		−	0.744473i	0.928138	−	0.372236i	$-0.121409\pi$
$744$		0			0
$745$		−	3.79796i		−	0.139146i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15.7980	−	38.6969i	−0.577245	−	1.41396i
$750$		0			0
$751$			48.8990i			1.78435i	0.451691	+	0.892175i	$0.350821\pi$
$751$			48.8990i			1.78435i	−0.451691	+	0.892175i	$0.649179\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.89898			0.323867
$756$		0			0
$757$	48.6969			1.76992			0.884960	−	0.465667i	$-0.154185\pi$
$757$	48.6969			1.76992			0.884960	+	0.465667i	$0.154185\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	27.7980i		−	1.00768i	−0.863798	−	0.503838i	$-0.831921\pi$
$761$		−	27.7980i		−	1.00768i	0.863798	−	0.503838i	$-0.168079\pi$
$762$		0			0
$763$	33.7980	−	13.7980i	1.22357	−	0.499520i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			21.7980i			0.787079i
$768$		0			0
$769$		−	8.49490i		−	0.306334i	−0.988200	−	0.153167i	$-0.951053\pi$
$769$		−	8.49490i		−	0.306334i	0.988200	−	0.153167i	$-0.0489472\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			43.3939i			1.56077i	0.625300	+	0.780385i	$0.284977\pi$
$773$			43.3939i			1.56077i	−0.625300	+	0.780385i	$0.715023\pi$
$774$		0			0
$775$	8.44949			0.303515
$776$		0			0
$777$		0			0
$778$		0			0
$779$			4.00000i			0.143315i
$780$		0			0
$781$	−15.7980			−0.565295
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.3485			0.619193
$786$		0			0
$787$	−18.6969			−0.666474			−0.333237	−	0.942843i	$-0.608141\pi$
$787$	−18.6969			−0.666474			−0.333237	+	0.942843i	$0.608141\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.20204	+	3.34847i	−0.291631	+	0.119058i
$792$		0			0
$793$	−57.3939			−2.03812
$794$		0			0
$795$		0			0
$796$		0			0
$797$			15.7980i			0.559592i	0.960059	+	0.279796i	$0.0902669\pi$
$797$			15.7980i			0.559592i	−0.960059	+	0.279796i	$0.909733\pi$
$798$		0			0
$799$		−	75.1918i		−	2.66010i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.1010			0.462325
$804$		0			0
$805$	4.89898	−	2.00000i	0.172666	−	0.0704907i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	34.8990			1.22698			0.613491	−	0.789701i	$-0.289765\pi$
$809$	34.8990			1.22698			0.613491	+	0.789701i	$0.289765\pi$
$810$		0			0
$811$	35.1464			1.23416			0.617079	−	0.786901i	$-0.288316\pi$
$811$	35.1464			1.23416			0.617079	+	0.786901i	$0.288316\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.5959			0.826529
$816$		0			0
$817$		−	35.5959i		−	1.24534i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.49490			0.226673			0.113337	−	0.993557i	$-0.463846\pi$
$821$	6.49490			0.226673			0.113337	+	0.993557i	$0.463846\pi$
$822$		0			0
$823$		−	45.3939i		−	1.58233i	−0.611602	−	0.791166i	$-0.709475\pi$
$823$		−	45.3939i		−	1.58233i	0.611602	−	0.791166i	$-0.290525\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			43.3939i			1.50895i	0.656327	+	0.754476i	$0.272109\pi$
$827$			43.3939i			1.50895i	−0.656327	+	0.754476i	$0.727891\pi$
$828$		0			0
$829$		−	5.39388i		−	0.187337i	−0.995603	−	0.0936685i	$-0.970141\pi$
$829$		−	5.39388i		−	0.187337i	0.995603	−	0.0936685i	$-0.0298594\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33.7980	+	34.4949i	1.17103	+	1.19518i
$834$		0			0
$835$		−	8.00000i		−	0.276851i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	53.7980			1.85731			0.928656	−	0.370942i	$-0.120965\pi$
$839$	53.7980			1.85731			0.928656	+	0.370942i	$0.120965\pi$
$840$		0			0
$841$	18.5959			0.641239
$842$		0			0
$843$		0			0
$844$		0			0
$845$			6.79796i			0.233857i
$846$		0			0
$847$	−12.2474	+	5.00000i	−0.420827	+	0.171802i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	13.7980i		−	0.472988i
$852$		0			0
$853$		−	36.0454i		−	1.23417i	−0.786896	−	0.617086i	$-0.788313\pi$
$853$		−	36.0454i		−	1.23417i	0.786896	−	0.617086i	$-0.211687\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$	35.5505			1.21297			0.606484	−	0.795096i	$-0.292579\pi$
$859$	35.5505			1.21297			0.606484	+	0.795096i	$0.292579\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			31.3939i			1.06866i	0.845276	+	0.534330i	$0.179436\pi$
$863$			31.3939i			1.06866i	−0.845276	+	0.534330i	$0.820564\pi$
$864$		0			0
$865$	−3.79796			−0.129134
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	70.2929			2.38178
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.00000	−	2.44949i	−0.0338062	−	0.0828079i
$876$		0			0
$877$	−13.5959			−0.459102			−0.229551	−	0.973297i	$-0.573726\pi$
$877$	−13.5959			−0.459102			−0.229551	+	0.973297i	$0.573726\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	8.20204i		−	0.276334i	−0.990409	−	0.138167i	$-0.955879\pi$
$881$		−	8.20204i		−	0.276334i	0.990409	−	0.138167i	$-0.0441210\pi$
$882$		0			0
$883$			0.202041i			0.00679922i	0.999994	+	0.00339961i	$0.00108213\pi$
$883$			0.202041i			0.00679922i	−0.999994	+	0.00339961i	$0.998918\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.2929			−0.412754			−0.206377	−	0.978473i	$-0.566167\pi$
$887$	−12.2929			−0.412754			−0.206377	+	0.978473i	$0.566167\pi$
$888$		0			0
$889$	−1.79796	−	4.40408i	−0.0603016	−	0.147708i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−48.4949			−1.62282
$894$		0			0
$895$	0.247449			0.00827130
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−58.2929			−1.94418
$900$		0			0
$901$		−	72.0908i		−	2.40169i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	13.7980			0.458660
$906$		0			0
$907$		−	1.59592i		−	0.0529916i	−0.999649	−	0.0264958i	$-0.991565\pi$
$907$		−	1.59592i		−	0.0529916i	0.999649	−	0.0264958i	$-0.00843486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			32.6515i			1.08179i	0.841089	+	0.540897i	$0.181915\pi$
$911$			32.6515i			1.08179i	−0.841089	+	0.540897i	$0.818085\pi$
$912$		0			0
$913$			7.10102i			0.235009i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−45.7980	+	18.6969i	−1.51238	+	0.617427i
$918$		0			0
$919$		−	28.4949i		−	0.939960i	−0.882677	−	0.469980i	$-0.844261\pi$
$919$		−	28.4949i		−	0.939960i	0.882677	−	0.469980i	$-0.155739\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	28.6969			0.944571
$924$		0			0
$925$	−6.89898			−0.226837
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	30.0000i		−	0.984268i	−0.870519	−	0.492134i	$-0.836217\pi$
$929$		−	30.0000i		−	0.984268i	0.870519	−	0.492134i	$-0.163783\pi$
$930$		0			0
$931$	22.2474	−	21.7980i	0.729131	−	0.714399i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			16.8990i			0.552656i
$936$		0			0
$937$			18.2474i			0.596118i	0.954547	+	0.298059i	$0.0963392\pi$
$937$			18.2474i			0.596118i	−0.954547	+	0.298059i	$0.903661\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$	1.79796			0.0585496
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	4.69694i		−	0.152630i	−0.997084	−	0.0763150i	$-0.975685\pi$
$947$		−	4.69694i		−	0.152630i	0.997084	−	0.0763150i	$-0.0243155\pi$
$948$		0			0
$949$	−23.7980			−0.772514
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.247449			0.00801565			0.00400783	−	0.999992i	$-0.498724\pi$
$953$	0.247449			0.00801565			0.00400783	+	0.999992i	$0.498724\pi$
$954$		0			0
$955$	10.4495			0.338138
$956$		0			0
$957$		0			0
$958$		0			0
$959$	51.7980	−	21.1464i	1.67264	−	0.682854i
$960$		0			0
$961$	40.3939			1.30303
$962$		0			0
$963$		0			0
$964$		0			0
$965$			12.6969i			0.408729i
$966$		0			0
$967$		−	31.3939i		−	1.00956i	−0.863248	−	0.504780i	$-0.831574\pi$
$967$		−	31.3939i		−	1.00956i	0.863248	−	0.504780i	$-0.168426\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	39.1918			1.25773			0.628863	−	0.777516i	$-0.283521\pi$
$971$	39.1918			1.25773			0.628863	+	0.777516i	$0.283521\pi$
$972$		0			0
$973$	18.4949	−	7.55051i	0.592919	−	0.242058i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−16.6515			−0.532730			−0.266365	−	0.963872i	$-0.585823\pi$
$977$	−16.6515			−0.532730			−0.266365	+	0.963872i	$0.585823\pi$
$978$		0			0
$979$	−38.6969			−1.23676
$980$		0			0
$981$		0			0
$982$		0			0
$983$	30.8990			0.985524			0.492762	−	0.870164i	$-0.335987\pi$
$983$	30.8990			0.985524			0.492762	+	0.870164i	$0.335987\pi$
$984$		0			0
$985$			0.247449i			0.00788437i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000			−0.508770
$990$		0			0
$991$		−	22.6969i		−	0.720992i	−0.932761	−	0.360496i	$-0.882607\pi$
$991$		−	22.6969i		−	0.720992i	0.932761	−	0.360496i	$-0.117393\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4.44949i		−	0.141058i
$996$		0			0
$997$			11.9546i			0.378606i	0.981919	+	0.189303i	$0.0606228\pi$
$997$			11.9546i			0.378606i	−0.981919	+	0.189303i	$0.939377\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.d.b.4591.1		4
3.2	odd	2		1680.2.d.b.1231.3	yes	4
4.3	odd	2		5040.2.d.a.4591.2		4
7.6	odd	2		5040.2.d.a.4591.4		4
12.11	even	2		1680.2.d.a.1231.4	yes	4
21.20	even	2		1680.2.d.a.1231.2	✓	4
28.27	even	2	inner	5040.2.d.b.4591.3		4
84.83	odd	2		1680.2.d.b.1231.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.d.a.1231.2	✓	4	21.20	even	2
1680.2.d.a.1231.4	yes	4	12.11	even	2
1680.2.d.b.1231.1	yes	4	84.83	odd	2
1680.2.d.b.1231.3	yes	4	3.2	odd	2
5040.2.d.a.4591.2		4	4.3	odd	2
5040.2.d.a.4591.4		4	7.6	odd	2
5040.2.d.b.4591.1		4	1.1	even	1	trivial
5040.2.d.b.4591.3		4	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +(-2.44949 + 1.00000i) q^{7} +2.44949i q^{11} -4.44949i q^{13} +6.89898i q^{17} +4.44949 q^{19} -2.00000i q^{23} -1.00000 q^{25} +6.89898 q^{29} -8.44949 q^{31} +(1.00000 + 2.44949i) q^{35} +6.89898 q^{37} +0.898979i q^{41} -8.00000i q^{43} -10.8990 q^{47} +(5.00000 - 4.89898i) q^{49} -10.4495 q^{53} +2.44949 q^{55} -4.89898 q^{59} -12.8990i q^{61} -4.44949 q^{65} +15.7980i q^{67} +6.44949i q^{71} -5.34847i q^{73} +(-2.44949 - 6.00000i) q^{77} +9.79796i q^{79} +2.89898 q^{83} +6.89898 q^{85} +15.7980i q^{89} +(4.44949 + 10.8990i) q^{91} -4.44949i q^{95} +2.65153i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{19} - 4 q^{25} + 8 q^{29} - 24 q^{31} + 4 q^{35} + 8 q^{37} - 24 q^{47} + 20 q^{49} - 32 q^{53} - 8 q^{65} - 8 q^{83} + 8 q^{85} + 8 q^{91}+O(q^{100}) \) 4 * q + 8 * q^19 - 4 * q^25 + 8 * q^29 - 24 * q^31 + 4 * q^35 + 8 * q^37 - 24 * q^47 + 20 * q^49 - 32 * q^53 - 8 * q^65 - 8 * q^83 + 8 * q^85 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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