LMFDB - Embedded newform 1680.2.t.k.1009.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(1009,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.t (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,2,0,0,0,-6,0,-12,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.3
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	1680.1009
Dual form			1680.2.t.k.1009.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.17009 + 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +0.921622i q^{13} +(0.539189 - 2.17009i) q^{15} -1.07838i q^{17} +3.07838 q^{19} +1.00000 q^{21} -2.34017i q^{23} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +6.68035 q^{29} +7.75872 q^{31} +2.00000i q^{33} +(-0.539189 + 2.17009i) q^{35} -10.8371i q^{37} +0.921622 q^{39} +6.49693 q^{41} +6.52359i q^{43} +(-2.17009 - 0.539189i) q^{45} +4.68035i q^{47} -1.00000 q^{49} -1.07838 q^{51} +3.75872i q^{53} +(-4.34017 - 1.07838i) q^{55} -3.07838i q^{57} +10.5236 q^{59} -4.15676 q^{61} -1.00000i q^{63} +(-0.496928 + 2.00000i) q^{65} +4.68035i q^{67} -2.34017 q^{69} -2.00000 q^{71} +7.07838i q^{73} +(2.34017 - 4.41855i) q^{75} -2.00000i q^{77} +6.15676 q^{79} +1.00000 q^{81} -6.83710i q^{83} +(0.581449 - 2.34017i) q^{85} -6.68035i q^{87} -8.34017 q^{89} -0.921622 q^{91} -7.75872i q^{93} +(6.68035 + 1.65983i) q^{95} +8.43907i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 6 q^{9} - 12 q^{11} + 12 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} - 4 q^{31} + 12 q^{39} + 4 q^{41} - 2 q^{45} - 6 q^{49} - 4 q^{55} + 32 q^{59} - 12 q^{61} + 32 q^{65} + 8 q^{69} - 12 q^{71}+ \cdots + 12 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 - 6 * q^9 - 12 * q^11 + 12 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 - 4 * q^31 + 12 * q^39 + 4 * q^41 - 2 * q^45 - 6 * q^49 - 4 * q^55 + 32 * q^59 - 12 * q^61 + 32 * q^65 + 8 * q^69 - 12 * q^71 - 8 * q^75 + 24 * q^79 + 6 * q^81 + 32 * q^85 - 28 * q^89 - 12 * q^91 - 4 * q^95 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$-1$	$1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	2.17009	+	0.539189i	0.970492	+	0.241133i
$5$	2.17009	+	0.539189i	0.970492	+	0.241133i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			0.921622i			0.255612i	0.991799	+	0.127806i	$0.0407935\pi$
$13$			0.921622i			0.255612i	−0.991799	+	0.127806i	$0.959207\pi$
$14$		0			0
$15$	0.539189	−	2.17009i	0.139218	−	0.560314i
$16$		0			0
$17$		−	1.07838i		−	0.261545i	−0.991412	−	0.130773i	$-0.958254\pi$
$17$		−	1.07838i		−	0.261545i	0.991412	−	0.130773i	$-0.0417457\pi$
$18$		0			0
$19$	3.07838			0.706228			0.353114	−	0.935580i	$-0.385123\pi$
$19$	3.07838			0.706228			0.353114	+	0.935580i	$0.385123\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$		−	2.34017i		−	0.487960i	−0.969780	−	0.243980i	$-0.921547\pi$
$23$		−	2.34017i		−	0.487960i	0.969780	−	0.243980i	$-0.0784531\pi$
$24$		0			0
$25$	4.41855	+	2.34017i	0.883710	+	0.468035i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	6.68035			1.24051			0.620255	−	0.784401i	$-0.287029\pi$
$29$	6.68035			1.24051			0.620255	+	0.784401i	$0.287029\pi$
$30$		0			0
$31$	7.75872			1.39351			0.696754	−	0.717310i	$-0.254627\pi$
$31$	7.75872			1.39351			0.696754	+	0.717310i	$0.254627\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	−0.539189	+	2.17009i	−0.0911396	+	0.366812i
$36$		0			0
$37$		−	10.8371i		−	1.78161i	−0.454387	−	0.890804i	$-0.650142\pi$
$37$		−	10.8371i		−	1.78161i	0.454387	−	0.890804i	$-0.349858\pi$
$38$		0			0
$39$	0.921622			0.147578
$40$		0			0
$41$	6.49693			1.01465			0.507325	−	0.861755i	$-0.330634\pi$
$41$	6.49693			1.01465			0.507325	+	0.861755i	$0.330634\pi$
$42$		0			0
$43$			6.52359i			0.994838i	0.867510	+	0.497419i	$0.165719\pi$
$43$			6.52359i			0.994838i	−0.867510	+	0.497419i	$0.834281\pi$
$44$		0			0
$45$	−2.17009	−	0.539189i	−0.323497	−	0.0803775i
$46$		0			0
$47$			4.68035i			0.682699i	0.939937	+	0.341349i	$0.110884\pi$
$47$			4.68035i			0.682699i	−0.939937	+	0.341349i	$0.889116\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.07838			−0.151003
$52$		0			0
$53$			3.75872i			0.516300i	0.966105	+	0.258150i	$0.0831129\pi$
$53$			3.75872i			0.516300i	−0.966105	+	0.258150i	$0.916887\pi$
$54$		0			0
$55$	−4.34017	−	1.07838i	−0.585229	−	0.145408i
$56$		0			0
$57$		−	3.07838i		−	0.407741i
$58$		0			0
$59$	10.5236			1.37005			0.685027	−	0.728517i	$-0.259790\pi$
$59$	10.5236			1.37005			0.685027	+	0.728517i	$0.259790\pi$
$60$		0			0
$61$	−4.15676			−0.532218			−0.266109	−	0.963943i	$-0.585738\pi$
$61$	−4.15676			−0.532218			−0.266109	+	0.963943i	$0.585738\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−0.496928	+	2.00000i	−0.0616364	+	0.248069i
$66$		0			0
$67$			4.68035i			0.571795i	0.958260	+	0.285898i	$0.0922917\pi$
$67$			4.68035i			0.571795i	−0.958260	+	0.285898i	$0.907708\pi$
$68$		0			0
$69$	−2.34017			−0.281724
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$			7.07838i			0.828461i	0.910172	+	0.414231i	$0.135949\pi$
$73$			7.07838i			0.828461i	−0.910172	+	0.414231i	$0.864051\pi$
$74$		0			0
$75$	2.34017	−	4.41855i	0.270220	−	0.510210i
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	6.15676			0.692689			0.346345	−	0.938107i	$-0.387423\pi$
$79$	6.15676			0.692689			0.346345	+	0.938107i	$0.387423\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.83710i		−	0.750469i	−0.926930	−	0.375235i	$-0.877562\pi$
$83$		−	6.83710i		−	0.750469i	0.926930	−	0.375235i	$-0.122438\pi$
$84$		0			0
$85$	0.581449	−	2.34017i	0.0630670	−	0.253827i
$86$		0			0
$87$		−	6.68035i		−	0.716208i
$88$		0			0
$89$	−8.34017			−0.884057			−0.442028	−	0.897001i	$-0.645741\pi$
$89$	−8.34017			−0.884057			−0.442028	+	0.897001i	$0.645741\pi$
$90$		0			0
$91$	−0.921622			−0.0966123
$92$		0			0
$93$		−	7.75872i		−	0.804542i
$94$		0			0
$95$	6.68035	+	1.65983i	0.685389	+	0.170295i
$96$		0			0
$97$			8.43907i			0.856858i	0.903576	+	0.428429i	$0.140933\pi$
$97$			8.43907i			0.856858i	−0.903576	+	0.428429i	$0.859067\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−5.81658			−0.578772			−0.289386	−	0.957213i	$-0.593451\pi$
$101$	−5.81658			−0.578772			−0.289386	+	0.957213i	$0.593451\pi$
$102$		0			0
$103$			2.15676i			0.212511i	0.994339	+	0.106256i	$0.0338862\pi$
$103$			2.15676i			0.212511i	−0.994339	+	0.106256i	$0.966114\pi$
$104$		0			0
$105$	2.17009	+	0.539189i	0.211779	+	0.0526194i
$106$		0			0
$107$		−	16.4969i		−	1.59482i	−0.603439	−	0.797409i	$-0.706203\pi$
$107$		−	16.4969i		−	1.59482i	0.603439	−	0.797409i	$-0.293797\pi$
$108$		0			0
$109$	12.8371			1.22957			0.614786	−	0.788694i	$-0.289243\pi$
$109$	12.8371			1.22957			0.614786	+	0.788694i	$0.289243\pi$
$110$		0			0
$111$	−10.8371			−1.02861
$112$		0			0
$113$		−	5.23513i		−	0.492480i	−0.969209	−	0.246240i	$-0.920805\pi$
$113$		−	5.23513i		−	0.492480i	0.969209	−	0.246240i	$-0.0791951\pi$
$114$		0			0
$115$	1.26180	−	5.07838i	0.117663	−	0.473561i
$116$		0			0
$117$		−	0.921622i		−	0.0852040i
$118$		0			0
$119$	1.07838			0.0988547
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		−	6.49693i		−	0.585808i
$124$		0			0
$125$	8.32684	+	7.46081i	0.744775	+	0.667315i
$126$		0			0
$127$			1.84324i			0.163562i	0.996650	+	0.0817808i	$0.0260607\pi$
$127$			1.84324i			0.163562i	−0.996650	+	0.0817808i	$0.973939\pi$
$128$		0			0
$129$	6.52359			0.574370
$130$		0			0
$131$	−1.47641			−0.128995			−0.0644973	−	0.997918i	$-0.520544\pi$
$131$	−1.47641			−0.128995			−0.0644973	+	0.997918i	$0.520544\pi$
$132$		0			0
$133$			3.07838i			0.266929i
$134$		0			0
$135$	−0.539189	+	2.17009i	−0.0464060	+	0.186771i
$136$		0			0
$137$		−	4.43907i		−	0.379255i	−0.981856	−	0.189628i	$-0.939272\pi$
$137$		−	4.43907i		−	0.379255i	0.981856	−	0.189628i	$-0.0607281\pi$
$138$		0			0
$139$	13.6020			1.15370			0.576852	−	0.816849i	$-0.304281\pi$
$139$	13.6020			1.15370			0.576852	+	0.816849i	$0.304281\pi$
$140$		0			0
$141$	4.68035			0.394156
$142$		0			0
$143$		−	1.84324i		−	0.154140i
$144$		0			0
$145$	14.4969	+	3.60197i	1.20390	+	0.299127i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	−15.6742			−1.28408			−0.642040	−	0.766671i	$-0.721912\pi$
$149$	−15.6742			−1.28408			−0.642040	+	0.766671i	$0.721912\pi$
$150$		0			0
$151$	−5.84324			−0.475516			−0.237758	−	0.971324i	$-0.576413\pi$
$151$	−5.84324			−0.475516			−0.237758	+	0.971324i	$0.576413\pi$
$152$		0			0
$153$			1.07838i			0.0871817i
$154$		0			0
$155$	16.8371	+	4.18342i	1.35239	+	0.336020i
$156$		0			0
$157$		−	4.92162i		−	0.392788i	−0.980525	−	0.196394i	$-0.937077\pi$
$157$		−	4.92162i		−	0.392788i	0.980525	−	0.196394i	$-0.0629232\pi$
$158$		0			0
$159$	3.75872			0.298086
$160$		0			0
$161$	2.34017			0.184431
$162$		0			0
$163$			9.84324i			0.770982i	0.922712	+	0.385491i	$0.125968\pi$
$163$			9.84324i			0.770982i	−0.922712	+	0.385491i	$0.874032\pi$
$164$		0			0
$165$	−1.07838	+	4.34017i	−0.0839516	+	0.337882i
$166$		0			0
$167$		−	19.2039i		−	1.48605i	−0.669266	−	0.743023i	$-0.733391\pi$
$167$		−	19.2039i		−	1.48605i	0.669266	−	0.743023i	$-0.266609\pi$
$168$		0			0
$169$	12.1506			0.934662
$170$		0			0
$171$	−3.07838			−0.235409
$172$		0			0
$173$		−	22.4391i		−	1.70601i	−0.521902	−	0.853005i	$-0.674777\pi$
$173$		−	22.4391i		−	1.70601i	0.521902	−	0.853005i	$-0.325223\pi$
$174$		0			0
$175$	−2.34017	+	4.41855i	−0.176900	+	0.334011i
$176$		0			0
$177$		−	10.5236i		−	0.791001i
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	−8.52359			−0.633553			−0.316777	−	0.948500i	$-0.602601\pi$
$181$	−8.52359			−0.633553			−0.316777	+	0.948500i	$0.602601\pi$
$182$		0			0
$183$			4.15676i			0.307276i
$184$		0			0
$185$	5.84324	−	23.5174i	0.429604	−	1.72904i
$186$		0			0
$187$			2.15676i			0.157718i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−15.3607			−1.11146			−0.555730	−	0.831363i	$-0.687561\pi$
$191$	−15.3607			−1.11146			−0.555730	+	0.831363i	$0.687561\pi$
$192$		0			0
$193$			8.36683i			0.602258i	0.953583	+	0.301129i	$0.0973635\pi$
$193$			8.36683i			0.602258i	−0.953583	+	0.301129i	$0.902637\pi$
$194$		0			0
$195$	2.00000	+	0.496928i	0.143223	+	0.0355858i
$196$		0			0
$197$			11.7587i			0.837774i	0.908038	+	0.418887i	$0.137580\pi$
$197$			11.7587i			0.837774i	−0.908038	+	0.418887i	$0.862420\pi$
$198$		0			0
$199$	−22.5958			−1.60178			−0.800888	−	0.598814i	$-0.795639\pi$
$199$	−22.5958			−1.60178			−0.800888	+	0.598814i	$0.795639\pi$
$200$		0			0
$201$	4.68035			0.330126
$202$		0			0
$203$			6.68035i			0.468868i
$204$		0			0
$205$	14.0989	+	3.50307i	0.984710	+	0.244665i
$206$		0			0
$207$			2.34017i			0.162653i
$208$		0			0
$209$	−6.15676			−0.425872
$210$		0			0
$211$	13.6742			0.941371			0.470685	−	0.882301i	$-0.344007\pi$
$211$	13.6742			0.941371			0.470685	+	0.882301i	$0.344007\pi$
$212$		0			0
$213$			2.00000i			0.137038i
$214$		0			0
$215$	−3.51745	+	14.1568i	−0.239888	+	0.965483i
$216$		0			0
$217$			7.75872i			0.526696i
$218$		0			0
$219$	7.07838			0.478312
$220$		0			0
$221$	0.993857			0.0668541
$222$		0			0
$223$			21.6742i			1.45141i	0.688005	+	0.725706i	$0.258487\pi$
$223$			21.6742i			1.45141i	−0.688005	+	0.725706i	$0.741513\pi$
$224$		0			0
$225$	−4.41855	−	2.34017i	−0.294570	−	0.156012i
$226$		0			0
$227$		−	11.5174i		−	0.764440i	−0.924071	−	0.382220i	$-0.875160\pi$
$227$		−	11.5174i		−	0.764440i	0.924071	−	0.382220i	$-0.124840\pi$
$228$		0			0
$229$	−12.8371			−0.848300			−0.424150	−	0.905592i	$-0.639427\pi$
$229$	−12.8371			−0.848300			−0.424150	+	0.905592i	$0.639427\pi$
$230$		0			0
$231$	−2.00000			−0.131590
$232$		0			0
$233$		−	6.76487i		−	0.443181i	−0.975140	−	0.221591i	$-0.928875\pi$
$233$		−	6.76487i		−	0.443181i	0.975140	−	0.221591i	$-0.0711248\pi$
$234$		0			0
$235$	−2.52359	+	10.1568i	−0.164621	+	0.662554i
$236$		0			0
$237$		−	6.15676i		−	0.399924i
$238$		0			0
$239$	−23.3607			−1.51108			−0.755539	−	0.655104i	$-0.772625\pi$
$239$	−23.3607			−1.51108			−0.755539	+	0.655104i	$0.772625\pi$
$240$		0			0
$241$	−14.6803			−0.945644			−0.472822	−	0.881158i	$-0.656765\pi$
$241$	−14.6803			−0.945644			−0.472822	+	0.881158i	$0.656765\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−2.17009	−	0.539189i	−0.138642	−	0.0344475i
$246$		0			0
$247$			2.83710i			0.180520i
$248$		0			0
$249$	−6.83710			−0.433284
$250$		0			0
$251$	−9.16290			−0.578357			−0.289179	−	0.957275i	$-0.593382\pi$
$251$	−9.16290			−0.578357			−0.289179	+	0.957275i	$0.593382\pi$
$252$		0			0
$253$			4.68035i			0.294251i
$254$		0			0
$255$	−2.34017	−	0.581449i	−0.146547	−	0.0364118i
$256$		0			0
$257$			5.07838i			0.316781i	0.987377	+	0.158390i	$0.0506304\pi$
$257$			5.07838i			0.316781i	−0.987377	+	0.158390i	$0.949370\pi$
$258$		0			0
$259$	10.8371			0.673385
$260$		0			0
$261$	−6.68035			−0.413503
$262$		0			0
$263$		−	5.65983i		−	0.349000i	−0.984657	−	0.174500i	$-0.944169\pi$
$263$		−	5.65983i		−	0.349000i	0.984657	−	0.174500i	$-0.0558309\pi$
$264$		0			0
$265$	−2.02666	+	8.15676i	−0.124497	+	0.501066i
$266$		0			0
$267$			8.34017i			0.510410i
$268$		0			0
$269$	27.8576			1.69851			0.849255	−	0.527984i	$-0.177052\pi$
$269$	27.8576			1.69851			0.849255	+	0.527984i	$0.177052\pi$
$270$		0			0
$271$	−25.1194			−1.52590			−0.762948	−	0.646460i	$-0.776249\pi$
$271$	−25.1194			−1.52590			−0.762948	+	0.646460i	$0.776249\pi$
$272$		0			0
$273$			0.921622i			0.0557791i
$274$		0			0
$275$	−8.83710	−	4.68035i	−0.532897	−	0.282235i
$276$		0			0
$277$			28.1978i			1.69424i	0.531401	+	0.847121i	$0.321666\pi$
$277$			28.1978i			1.69424i	−0.531401	+	0.847121i	$0.678334\pi$
$278$		0			0
$279$	−7.75872			−0.464503
$280$		0			0
$281$	−20.3545			−1.21425			−0.607125	−	0.794606i	$-0.707677\pi$
$281$	−20.3545			−1.21425			−0.607125	+	0.794606i	$0.707677\pi$
$282$		0			0
$283$		−	23.5174i		−	1.39797i	−0.715138	−	0.698984i	$-0.753636\pi$
$283$		−	23.5174i		−	1.39797i	0.715138	−	0.698984i	$-0.246364\pi$
$284$		0			0
$285$	1.65983	−	6.68035i	0.0983197	−	0.395710i
$286$		0			0
$287$			6.49693i			0.383502i
$288$		0			0
$289$	15.8371			0.931594
$290$		0			0
$291$	8.43907			0.494707
$292$		0			0
$293$		−	2.92162i		−	0.170683i	−0.996352	−	0.0853415i	$-0.972802\pi$
$293$		−	2.92162i		−	0.170683i	0.996352	−	0.0853415i	$-0.0271981\pi$
$294$		0			0
$295$	22.8371	+	5.67420i	1.32963	+	0.330365i
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$	2.15676			0.124728
$300$		0			0
$301$	−6.52359			−0.376014
$302$		0			0
$303$			5.81658i			0.334154i
$304$		0			0
$305$	−9.02052	−	2.24128i	−0.516513	−	0.128335i
$306$		0			0
$307$		−	10.4703i		−	0.597570i	−0.954321	−	0.298785i	$-0.903419\pi$
$307$		−	10.4703i		−	0.597570i	0.954321	−	0.298785i	$-0.0965813\pi$
$308$		0			0
$309$	2.15676			0.122694
$310$		0			0
$311$	−23.8310			−1.35133			−0.675665	−	0.737209i	$-0.736143\pi$
$311$	−23.8310			−1.35133			−0.675665	+	0.737209i	$0.736143\pi$
$312$		0			0
$313$			32.7526i			1.85129i	0.378399	+	0.925643i	$0.376475\pi$
$313$			32.7526i			1.85129i	−0.378399	+	0.925643i	$0.623525\pi$
$314$		0			0
$315$	0.539189	−	2.17009i	0.0303799	−	0.122271i
$316$		0			0
$317$			17.9155i			1.00623i	0.864218	+	0.503117i	$0.167813\pi$
$317$			17.9155i			1.00623i	−0.864218	+	0.503117i	$0.832187\pi$
$318$		0			0
$319$	−13.3607			−0.748055
$320$		0			0
$321$	−16.4969			−0.920769
$322$		0			0
$323$		−	3.31965i		−	0.184710i
$324$		0			0
$325$	−2.15676	+	4.07223i	−0.119635	+	0.225887i
$326$		0			0
$327$		−	12.8371i		−	0.709893i
$328$		0			0
$329$	−4.68035			−0.258036
$330$		0			0
$331$	1.36069			0.0747904			0.0373952	−	0.999301i	$-0.488094\pi$
$331$	1.36069			0.0747904			0.0373952	+	0.999301i	$0.488094\pi$
$332$		0			0
$333$			10.8371i			0.593870i
$334$		0			0
$335$	−2.52359	+	10.1568i	−0.137878	+	0.554923i
$336$		0			0
$337$			25.3607i			1.38148i	0.723101	+	0.690742i	$0.242716\pi$
$337$			25.3607i			1.38148i	−0.723101	+	0.690742i	$0.757284\pi$
$338$		0			0
$339$	−5.23513			−0.284333
$340$		0			0
$341$	−15.5174			−0.840317
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−5.07838	−	1.26180i	−0.273411	−	0.0679328i
$346$		0			0
$347$			16.8638i			0.905294i	0.891690	+	0.452647i	$0.149520\pi$
$347$			16.8638i			0.905294i	−0.891690	+	0.452647i	$0.850480\pi$
$348$		0			0
$349$	−9.51745			−0.509457			−0.254729	−	0.967013i	$-0.581986\pi$
$349$	−9.51745			−0.509457			−0.254729	+	0.967013i	$0.581986\pi$
$350$		0			0
$351$	−0.921622			−0.0491926
$352$		0			0
$353$		−	35.7998i		−	1.90543i	−0.303867	−	0.952715i	$-0.598278\pi$
$353$		−	35.7998i		−	1.90543i	0.303867	−	0.952715i	$-0.401722\pi$
$354$		0			0
$355$	−4.34017	−	1.07838i	−0.230352	−	0.0572343i
$356$		0			0
$357$		−	1.07838i		−	0.0570738i
$358$		0			0
$359$	−22.3135			−1.17766			−0.588831	−	0.808256i	$-0.700412\pi$
$359$	−22.3135			−1.17766			−0.588831	+	0.808256i	$0.700412\pi$
$360$		0			0
$361$	−9.52359			−0.501242
$362$		0			0
$363$			7.00000i			0.367405i
$364$		0			0
$365$	−3.81658	+	15.3607i	−0.199769	+	0.804015i
$366$		0			0
$367$			20.3135i			1.06036i	0.847886	+	0.530178i	$0.177875\pi$
$367$			20.3135i			1.06036i	−0.847886	+	0.530178i	$0.822125\pi$
$368$		0			0
$369$	−6.49693			−0.338217
$370$		0			0
$371$	−3.75872			−0.195143
$372$		0			0
$373$		−	16.0000i		−	0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$		−	16.0000i		−	0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$		0			0
$375$	7.46081	−	8.32684i	0.385275	−	0.429996i
$376$		0			0
$377$			6.15676i			0.317089i
$378$		0			0
$379$	6.15676			0.316251			0.158126	−	0.987419i	$-0.449455\pi$
$379$	6.15676			0.316251			0.158126	+	0.987419i	$0.449455\pi$
$380$		0			0
$381$	1.84324			0.0944323
$382$		0			0
$383$		−	26.8371i		−	1.37131i	−0.727926	−	0.685656i	$-0.759515\pi$
$383$		−	26.8371i		−	1.37131i	0.727926	−	0.685656i	$-0.240485\pi$
$384$		0			0
$385$	1.07838	−	4.34017i	0.0549592	−	0.221196i
$386$		0			0
$387$		−	6.52359i		−	0.331613i
$388$		0			0
$389$	5.63317			0.285613			0.142806	−	0.989751i	$-0.454387\pi$
$389$	5.63317			0.285613			0.142806	+	0.989751i	$0.454387\pi$
$390$		0			0
$391$	−2.52359			−0.127623
$392$		0			0
$393$			1.47641i			0.0744750i
$394$		0			0
$395$	13.3607	+	3.31965i	0.672249	+	0.167030i
$396$		0			0
$397$		−	37.7998i		−	1.89712i	−0.316604	−	0.948558i	$-0.602543\pi$
$397$		−	37.7998i		−	1.89712i	0.316604	−	0.948558i	$-0.397457\pi$
$398$		0			0
$399$	3.07838			0.154112
$400$		0			0
$401$	−13.6332			−0.680808			−0.340404	−	0.940279i	$-0.610564\pi$
$401$	−13.6332			−0.680808			−0.340404	+	0.940279i	$0.610564\pi$
$402$		0			0
$403$			7.15061i			0.356197i
$404$		0			0
$405$	2.17009	+	0.539189i	0.107832	+	0.0267925i
$406$		0			0
$407$			21.6742i			1.07435i
$408$		0			0
$409$	−12.3545			−0.610893			−0.305447	−	0.952209i	$-0.598806\pi$
$409$	−12.3545			−0.610893			−0.305447	+	0.952209i	$0.598806\pi$
$410$		0			0
$411$	−4.43907			−0.218963
$412$		0			0
$413$			10.5236i			0.517832i
$414$		0			0
$415$	3.68649	−	14.8371i	0.180963	−	0.728325i
$416$		0			0
$417$		−	13.6020i		−	0.666091i
$418$		0			0
$419$	28.9939			1.41644			0.708221	−	0.705991i	$-0.249498\pi$
$419$	28.9939			1.41644			0.708221	+	0.705991i	$0.249498\pi$
$420$		0			0
$421$	−15.1629			−0.738994			−0.369497	−	0.929232i	$-0.620470\pi$
$421$	−15.1629			−0.738994			−0.369497	+	0.929232i	$0.620470\pi$
$422$		0			0
$423$		−	4.68035i		−	0.227566i
$424$		0			0
$425$	2.52359	−	4.76487i	0.122412	−	0.231130i
$426$		0			0
$427$		−	4.15676i		−	0.201159i
$428$		0			0
$429$	−1.84324			−0.0889927
$430$		0			0
$431$	10.3135			0.496784			0.248392	−	0.968660i	$-0.420098\pi$
$431$	10.3135			0.496784			0.248392	+	0.968660i	$0.420098\pi$
$432$		0			0
$433$			20.4391i			0.982239i	0.871092	+	0.491120i	$0.163412\pi$
$433$			20.4391i			0.982239i	−0.871092	+	0.491120i	$0.836588\pi$
$434$		0			0
$435$	3.60197	−	14.4969i	0.172701	−	0.695075i
$436$		0			0
$437$		−	7.20394i		−	0.344611i
$438$		0			0
$439$	−16.9216			−0.807625			−0.403812	−	0.914842i	$-0.632315\pi$
$439$	−16.9216			−0.807625			−0.403812	+	0.914842i	$0.632315\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			12.8104i			0.608642i	0.952569	+	0.304321i	$0.0984296\pi$
$443$			12.8104i			0.608642i	−0.952569	+	0.304321i	$0.901570\pi$
$444$		0			0
$445$	−18.0989	−	4.49693i	−0.857970	−	0.213175i
$446$		0			0
$447$			15.6742i			0.741364i
$448$		0			0
$449$	14.6270			0.690292			0.345146	−	0.938549i	$-0.387829\pi$
$449$	14.6270			0.690292			0.345146	+	0.938549i	$0.387829\pi$
$450$		0			0
$451$	−12.9939			−0.611857
$452$		0			0
$453$			5.84324i			0.274540i
$454$		0			0
$455$	−2.00000	−	0.496928i	−0.0937614	−	0.0232964i
$456$		0			0
$457$		−	14.1568i		−	0.662225i	−0.943591	−	0.331113i	$-0.892576\pi$
$457$		−	14.1568i		−	0.662225i	0.943591	−	0.331113i	$-0.107424\pi$
$458$		0			0
$459$	1.07838			0.0503344
$460$		0			0
$461$	0.340173			0.0158434			0.00792172	−	0.999969i	$-0.497478\pi$
$461$	0.340173			0.0158434			0.00792172	+	0.999969i	$0.497478\pi$
$462$		0			0
$463$			9.84324i			0.457454i	0.973491	+	0.228727i	$0.0734564\pi$
$463$			9.84324i			0.457454i	−0.973491	+	0.228727i	$0.926544\pi$
$464$		0			0
$465$	4.18342	−	16.8371i	0.194001	−	0.780802i
$466$		0			0
$467$		−	11.5174i		−	0.532964i	−0.963840	−	0.266482i	$-0.914139\pi$
$467$		−	11.5174i		−	0.532964i	0.963840	−	0.266482i	$-0.0858613\pi$
$468$		0			0
$469$	−4.68035			−0.216118
$470$		0			0
$471$	−4.92162			−0.226776
$472$		0			0
$473$		−	13.0472i		−	0.599910i
$474$		0			0
$475$	13.6020	+	7.20394i	0.624101	+	0.330539i
$476$		0			0
$477$		−	3.75872i		−	0.172100i
$478$		0			0
$479$	−19.5174			−0.891775			−0.445887	−	0.895089i	$-0.647112\pi$
$479$	−19.5174			−0.891775			−0.445887	+	0.895089i	$0.647112\pi$
$480$		0			0
$481$	9.98771			0.455401
$482$		0			0
$483$		−	2.34017i		−	0.106482i
$484$		0			0
$485$	−4.55025	+	18.3135i	−0.206616	+	0.831574i
$486$		0			0
$487$			23.1506i			1.04905i	0.851394	+	0.524527i	$0.175758\pi$
$487$			23.1506i			1.04905i	−0.851394	+	0.524527i	$0.824242\pi$
$488$		0			0
$489$	9.84324			0.445127
$490$		0			0
$491$	−2.00000			−0.0902587			−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000			−0.0902587			−0.0451294	+	0.998981i	$0.514370\pi$
$492$		0			0
$493$		−	7.20394i		−	0.324449i
$494$		0			0
$495$	4.34017	+	1.07838i	0.195076	+	0.0484695i
$496$		0			0
$497$		−	2.00000i		−	0.0897123i
$498$		0			0
$499$	27.2039			1.21782			0.608908	−	0.793241i	$-0.291608\pi$
$499$	27.2039			1.21782			0.608908	+	0.793241i	$0.291608\pi$
$500$		0			0
$501$	−19.2039			−0.857969
$502$		0			0
$503$			18.8371i			0.839905i	0.907546	+	0.419952i	$0.137953\pi$
$503$			18.8371i			0.839905i	−0.907546	+	0.419952i	$0.862047\pi$
$504$		0			0
$505$	−12.6225	−	3.13624i	−0.561693	−	0.139561i
$506$		0			0
$507$		−	12.1506i		−	0.539628i
$508$		0			0
$509$	−6.81044			−0.301867			−0.150934	−	0.988544i	$-0.548228\pi$
$509$	−6.81044			−0.301867			−0.150934	+	0.988544i	$0.548228\pi$
$510$		0			0
$511$	−7.07838			−0.313129
$512$		0			0
$513$			3.07838i			0.135914i
$514$		0			0
$515$	−1.16290	+	4.68035i	−0.0512434	+	0.206241i
$516$		0			0
$517$		−	9.36069i		−	0.411683i
$518$		0			0
$519$	−22.4391			−0.984966
$520$		0			0
$521$	−25.8166			−1.13105			−0.565523	−	0.824733i	$-0.691325\pi$
$521$	−25.8166			−1.13105			−0.565523	+	0.824733i	$0.691325\pi$
$522$		0			0
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$		0			0
$525$	4.41855	+	2.34017i	0.192841	+	0.102134i
$526$		0			0
$527$		−	8.36683i		−	0.364465i
$528$		0			0
$529$	17.5236			0.761895
$530$		0			0
$531$	−10.5236			−0.456685
$532$		0			0
$533$			5.98771i			0.259357i
$534$		0			0
$535$	8.89496	−	35.7998i	0.384563	−	1.54776i
$536$		0			0
$537$		−	10.0000i		−	0.431532i
$538$		0			0
$539$	2.00000			0.0861461
$540$		0			0
$541$	25.8843			1.11285			0.556426	−	0.830897i	$-0.312172\pi$
$541$	25.8843			1.11285			0.556426	+	0.830897i	$0.312172\pi$
$542$		0			0
$543$			8.52359i			0.365782i
$544$		0			0
$545$	27.8576	+	6.92162i	1.19329	+	0.296490i
$546$		0			0
$547$			11.3197i			0.483993i	0.970277	+	0.241997i	$0.0778023\pi$
$547$			11.3197i			0.483993i	−0.970277	+	0.241997i	$0.922198\pi$
$548$		0			0
$549$	4.15676			0.177406
$550$		0			0
$551$	20.5646			0.876083
$552$		0			0
$553$			6.15676i			0.261812i
$554$		0			0
$555$	−23.5174	−	5.84324i	−0.998260	−	0.248032i
$556$		0			0
$557$			26.6491i			1.12916i	0.825378	+	0.564580i	$0.190962\pi$
$557$			26.6491i			1.12916i	−0.825378	+	0.564580i	$0.809038\pi$
$558$		0			0
$559$	−6.01229			−0.254293
$560$		0			0
$561$	2.15676			0.0910583
$562$		0			0
$563$		−	46.3545i		−	1.95361i	−0.214128	−	0.976806i	$-0.568691\pi$
$563$		−	46.3545i		−	1.95361i	0.214128	−	0.976806i	$-0.431309\pi$
$564$		0			0
$565$	2.82273	−	11.3607i	0.118753	−	0.477948i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	14.3668			0.602289			0.301145	−	0.953579i	$-0.402631\pi$
$569$	14.3668			0.602289			0.301145	+	0.953579i	$0.402631\pi$
$570$		0			0
$571$	−38.7214			−1.62044			−0.810220	−	0.586126i	$-0.800652\pi$
$571$	−38.7214			−1.62044			−0.810220	+	0.586126i	$0.800652\pi$
$572$		0			0
$573$			15.3607i			0.641702i
$574$		0			0
$575$	5.47641	−	10.3402i	0.228382	−	0.431215i
$576$		0			0
$577$		−	43.4740i		−	1.80984i	−0.425577	−	0.904922i	$-0.639929\pi$
$577$		−	43.4740i		−	1.80984i	0.425577	−	0.904922i	$-0.360071\pi$
$578$		0			0
$579$	8.36683			0.347714
$580$		0			0
$581$	6.83710			0.283651
$582$		0			0
$583$		−	7.51745i		−	0.311341i
$584$		0			0
$585$	0.496928	−	2.00000i	0.0205455	−	0.0826898i
$586$		0			0
$587$			36.0288i			1.48707i	0.668699	+	0.743533i	$0.266851\pi$
$587$			36.0288i			1.48707i	−0.668699	+	0.743533i	$0.733149\pi$
$588$		0			0
$589$	23.8843			0.984135
$590$		0			0
$591$	11.7587			0.483689
$592$		0			0
$593$			31.4863i			1.29299i	0.762920	+	0.646493i	$0.223765\pi$
$593$			31.4863i			1.29299i	−0.762920	+	0.646493i	$0.776235\pi$
$594$		0			0
$595$	2.34017	+	0.581449i	0.0959377	+	0.0238371i
$596$		0			0
$597$			22.5958i			0.924786i
$598$		0			0
$599$	29.0349			1.18633			0.593167	−	0.805080i	$-0.297877\pi$
$599$	29.0349			1.18633			0.593167	+	0.805080i	$0.297877\pi$
$600$		0			0
$601$	15.3607			0.626576			0.313288	−	0.949658i	$-0.398570\pi$
$601$	15.3607			0.626576			0.313288	+	0.949658i	$0.398570\pi$
$602$		0			0
$603$		−	4.68035i		−	0.190598i
$604$		0			0
$605$	−15.1906	−	3.77432i	−0.617586	−	0.153448i
$606$		0			0
$607$		−	13.0472i		−	0.529569i	−0.964308	−	0.264784i	$-0.914699\pi$
$607$		−	13.0472i		−	0.529569i	0.964308	−	0.264784i	$-0.0853008\pi$
$608$		0			0
$609$	6.68035			0.270701
$610$		0			0
$611$	−4.31351			−0.174506
$612$		0			0
$613$		−	15.5174i		−	0.626744i	−0.949630	−	0.313372i	$-0.898541\pi$
$613$		−	15.5174i		−	0.626744i	0.949630	−	0.313372i	$-0.101459\pi$
$614$		0			0
$615$	3.50307	−	14.0989i	0.141257	−	0.568522i
$616$		0			0
$617$			22.7649i			0.916479i	0.888829	+	0.458240i	$0.151520\pi$
$617$			22.7649i			0.916479i	−0.888829	+	0.458240i	$0.848480\pi$
$618$		0			0
$619$	7.92777			0.318644			0.159322	−	0.987227i	$-0.449069\pi$
$619$	7.92777			0.318644			0.159322	+	0.987227i	$0.449069\pi$
$620$		0			0
$621$	2.34017			0.0939079
$622$		0			0
$623$		−	8.34017i		−	0.334142i
$624$		0			0
$625$	14.0472	+	20.6803i	0.561887	+	0.827214i
$626$		0			0
$627$			6.15676i			0.245877i
$628$		0			0
$629$	−11.6865			−0.465971
$630$		0			0
$631$	−19.2039			−0.764497			−0.382248	−	0.924060i	$-0.624850\pi$
$631$	−19.2039			−0.764497			−0.382248	+	0.924060i	$0.624850\pi$
$632$		0			0
$633$		−	13.6742i		−	0.543501i
$634$		0			0
$635$	−0.993857	+	4.00000i	−0.0394400	+	0.158735i
$636$		0			0
$637$		−	0.921622i		−	0.0365160i
$638$		0			0
$639$	2.00000			0.0791188
$640$		0			0
$641$	−5.94668			−0.234880			−0.117440	−	0.993080i	$-0.537469\pi$
$641$	−5.94668			−0.234880			−0.117440	+	0.993080i	$0.537469\pi$
$642$		0			0
$643$			30.8904i			1.21820i	0.793094	+	0.609100i	$0.208469\pi$
$643$			30.8904i			1.21820i	−0.793094	+	0.609100i	$0.791531\pi$
$644$		0			0
$645$	14.1568	+	3.51745i	0.557422	+	0.138499i
$646$		0			0
$647$		−	19.2039i		−	0.754985i	−0.926013	−	0.377492i	$-0.876786\pi$
$647$		−	19.2039i		−	0.754985i	0.926013	−	0.377492i	$-0.123214\pi$
$648$		0			0
$649$	−21.0472			−0.826174
$650$		0			0
$651$	7.75872			0.304088
$652$		0			0
$653$		−	28.5548i		−	1.11744i	−0.829358	−	0.558718i	$-0.811294\pi$
$653$		−	28.5548i		−	1.11744i	0.829358	−	0.558718i	$-0.188706\pi$
$654$		0			0
$655$	−3.20394	−	0.796064i	−0.125188	−	0.0311048i
$656$		0			0
$657$		−	7.07838i		−	0.276154i
$658$		0			0
$659$	−27.9877			−1.09025			−0.545123	−	0.838356i	$-0.683517\pi$
$659$	−27.9877			−1.09025			−0.545123	+	0.838356i	$0.683517\pi$
$660$		0			0
$661$	−22.1445			−0.861320			−0.430660	−	0.902514i	$-0.641719\pi$
$661$	−22.1445			−0.861320			−0.430660	+	0.902514i	$0.641719\pi$
$662$		0			0
$663$		−	0.993857i		−	0.0385982i
$664$		0			0
$665$	−1.65983	+	6.68035i	−0.0643653	+	0.259053i
$666$		0			0
$667$		−	15.6332i		−	0.605319i
$668$		0			0
$669$	21.6742			0.837973
$670$		0			0
$671$	8.31351			0.320940
$672$		0			0
$673$			2.21008i			0.0851923i	0.999092	+	0.0425962i	$0.0135629\pi$
$673$			2.21008i			0.0851923i	−0.999092	+	0.0425962i	$0.986437\pi$
$674$		0			0
$675$	−2.34017	+	4.41855i	−0.0900733	+	0.170070i
$676$		0			0
$677$		−	19.5486i		−	0.751315i	−0.926758	−	0.375658i	$-0.877417\pi$
$677$		−	19.5486i		−	0.751315i	0.926758	−	0.375658i	$-0.122583\pi$
$678$		0			0
$679$	−8.43907			−0.323862
$680$		0			0
$681$	−11.5174			−0.441350
$682$		0			0
$683$		−	11.8166i		−	0.452149i	−0.974110	−	0.226074i	$-0.927411\pi$
$683$		−	11.8166i		−	0.452149i	0.974110	−	0.226074i	$-0.0725893\pi$
$684$		0			0
$685$	2.39350	−	9.63317i	0.0914508	−	0.368064i
$686$		0			0
$687$			12.8371i			0.489766i
$688$		0			0
$689$	−3.46412			−0.131973
$690$		0			0
$691$	−11.7587			−0.447323			−0.223661	−	0.974667i	$-0.571801\pi$
$691$	−11.7587			−0.447323			−0.223661	+	0.974667i	$0.571801\pi$
$692$		0			0
$693$			2.00000i			0.0759737i
$694$		0			0
$695$	29.5174	+	7.33403i	1.11966	+	0.278196i
$696$		0			0
$697$		−	7.00614i		−	0.265377i
$698$		0			0
$699$	−6.76487			−0.255871
$700$		0			0
$701$	9.94668			0.375681			0.187840	−	0.982200i	$-0.439851\pi$
$701$	9.94668			0.375681			0.187840	+	0.982200i	$0.439851\pi$
$702$		0			0
$703$		−	33.3607i		−	1.25822i
$704$		0			0
$705$	10.1568	+	2.52359i	0.382526	+	0.0950439i
$706$		0			0
$707$		−	5.81658i		−	0.218755i
$708$		0			0
$709$	11.0472			0.414886			0.207443	−	0.978247i	$-0.433486\pi$
$709$	11.0472			0.414886			0.207443	+	0.978247i	$0.433486\pi$
$710$		0			0
$711$	−6.15676			−0.230896
$712$		0			0
$713$		−	18.1568i		−	0.679976i
$714$		0			0
$715$	0.993857	−	4.00000i	0.0371681	−	0.149592i
$716$		0			0
$717$			23.3607i			0.872421i
$718$		0			0
$719$	−6.15676			−0.229608			−0.114804	−	0.993388i	$-0.536624\pi$
$719$	−6.15676			−0.229608			−0.114804	+	0.993388i	$0.536624\pi$
$720$		0			0
$721$	−2.15676			−0.0803218
$722$		0			0
$723$			14.6803i			0.545968i
$724$		0			0
$725$	29.5174	+	15.6332i	1.09625	+	0.580601i
$726$		0			0
$727$			2.89043i			0.107200i	0.998562	+	0.0536000i	$0.0170696\pi$
$727$			2.89043i			0.107200i	−0.998562	+	0.0536000i	$0.982930\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	7.03489			0.260195
$732$		0			0
$733$		−	25.7998i		−	0.952936i	−0.879192	−	0.476468i	$-0.841917\pi$
$733$		−	25.7998i		−	0.952936i	0.879192	−	0.476468i	$-0.158083\pi$
$734$		0			0
$735$	−0.539189	+	2.17009i	−0.0198883	+	0.0800448i
$736$		0			0
$737$		−	9.36069i		−	0.344806i
$738$		0			0
$739$	−1.04718			−0.0385212			−0.0192606	−	0.999814i	$-0.506131\pi$
$739$	−1.04718			−0.0385212			−0.0192606	+	0.999814i	$0.506131\pi$
$740$		0			0
$741$	2.83710			0.104224
$742$		0			0
$743$			9.97334i			0.365886i	0.983123	+	0.182943i	$0.0585624\pi$
$743$			9.97334i			0.365886i	−0.983123	+	0.182943i	$0.941438\pi$
$744$		0			0
$745$	−34.0144	−	8.45136i	−1.24619	−	0.309634i
$746$		0			0
$747$			6.83710i			0.250156i
$748$		0			0
$749$	16.4969			0.602785
$750$		0			0
$751$	−3.26633			−0.119190			−0.0595950	−	0.998223i	$-0.518981\pi$
$751$	−3.26633			−0.119190			−0.0595950	+	0.998223i	$0.518981\pi$
$752$		0			0
$753$			9.16290i			0.333915i
$754$		0			0
$755$	−12.6803	−	3.15061i	−0.461485	−	0.114663i
$756$		0			0
$757$			49.9877i			1.81683i	0.418065	+	0.908417i	$0.362708\pi$
$757$			49.9877i			1.81683i	−0.418065	+	0.908417i	$0.637292\pi$
$758$		0			0
$759$	4.68035			0.169886
$760$		0			0
$761$	2.61265			0.0947083			0.0473542	−	0.998878i	$-0.484921\pi$
$761$	2.61265			0.0947083			0.0473542	+	0.998878i	$0.484921\pi$
$762$		0			0
$763$			12.8371i			0.464734i
$764$		0			0
$765$	−0.581449	+	2.34017i	−0.0210223	+	0.0846091i
$766$		0			0
$767$			9.69878i			0.350202i
$768$		0			0
$769$	15.6742			0.565226			0.282613	−	0.959234i	$-0.408799\pi$
$769$	15.6742			0.565226			0.282613	+	0.959234i	$0.408799\pi$
$770$		0			0
$771$	5.07838			0.182893
$772$		0			0
$773$			5.81205i			0.209045i	0.994523	+	0.104522i	$0.0333314\pi$
$773$			5.81205i			0.209045i	−0.994523	+	0.104522i	$0.966669\pi$
$774$		0			0
$775$	34.2823	+	18.1568i	1.23146	+	0.652210i
$776$		0			0
$777$		−	10.8371i		−	0.388779i
$778$		0			0
$779$	20.0000			0.716574
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$			6.68035i			0.238736i
$784$		0			0
$785$	2.65368	−	10.6803i	0.0947140	−	0.381198i
$786$		0			0
$787$			39.3484i			1.40262i	0.712857	+	0.701310i	$0.247401\pi$
$787$			39.3484i			1.40262i	−0.712857	+	0.701310i	$0.752599\pi$
$788$		0			0
$789$	−5.65983			−0.201495
$790$		0			0
$791$	5.23513			0.186140
$792$		0			0
$793$		−	3.83096i		−	0.136041i
$794$		0			0
$795$	8.15676	+	2.02666i	0.289290	+	0.0718783i
$796$		0			0
$797$		−	28.2823i		−	1.00181i	−0.865502	−	0.500905i	$-0.833000\pi$
$797$		−	28.2823i		−	1.00181i	0.865502	−	0.500905i	$-0.167000\pi$
$798$		0			0
$799$	5.04718			0.178556
$800$		0			0
$801$	8.34017			0.294686
$802$		0			0
$803$		−	14.1568i		−	0.499581i
$804$		0			0
$805$	5.07838	+	1.26180i	0.178989	+	0.0444724i
$806$		0			0
$807$		−	27.8576i		−	0.980635i
$808$		0			0
$809$	15.6742			0.551076			0.275538	−	0.961290i	$-0.411144\pi$
$809$	15.6742			0.551076			0.275538	+	0.961290i	$0.411144\pi$
$810$		0			0
$811$	42.1666			1.48067			0.740335	−	0.672238i	$-0.234667\pi$
$811$	42.1666			1.48067			0.740335	+	0.672238i	$0.234667\pi$
$812$		0			0
$813$			25.1194i			0.880976i
$814$		0			0
$815$	−5.30737	+	21.3607i	−0.185909	+	0.748232i
$816$		0			0
$817$			20.0821i			0.702583i
$818$		0			0
$819$	0.921622			0.0322041
$820$		0			0
$821$	−39.0472			−1.36276			−0.681378	−	0.731932i	$-0.738619\pi$
$821$	−39.0472			−1.36276			−0.681378	+	0.731932i	$0.738619\pi$
$822$		0			0
$823$			36.5646i			1.27456i	0.770631	+	0.637281i	$0.219941\pi$
$823$			36.5646i			1.27456i	−0.770631	+	0.637281i	$0.780059\pi$
$824$		0			0
$825$	−4.68035	+	8.83710i	−0.162949	+	0.307668i
$826$		0			0
$827$			50.2245i			1.74648i	0.487294	+	0.873238i	$0.337984\pi$
$827$			50.2245i			1.74648i	−0.487294	+	0.873238i	$0.662016\pi$
$828$		0			0
$829$	−32.8371			−1.14048			−0.570240	−	0.821478i	$-0.693150\pi$
$829$	−32.8371			−1.14048			−0.570240	+	0.821478i	$0.693150\pi$
$830$		0			0
$831$	28.1978			0.978171
$832$		0			0
$833$			1.07838i			0.0373636i
$834$		0			0
$835$	10.3545	−	41.6742i	0.358334	−	1.44220i
$836$		0			0
$837$			7.75872i			0.268181i
$838$		0			0
$839$	−13.3607			−0.461262			−0.230631	−	0.973041i	$-0.574079\pi$
$839$	−13.3607			−0.461262			−0.230631	+	0.973041i	$0.574079\pi$
$840$		0			0
$841$	15.6270			0.538863
$842$		0			0
$843$			20.3545i			0.701048i
$844$		0			0
$845$	26.3679	+	6.55148i	0.907083	+	0.225378i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$	−23.5174			−0.807117
$850$		0			0
$851$	−25.3607			−0.869353
$852$		0			0
$853$		−	39.6430i		−	1.35735i	−0.734438	−	0.678675i	$-0.762554\pi$
$853$		−	39.6430i		−	1.35735i	0.734438	−	0.678675i	$-0.237446\pi$
$854$		0			0
$855$	−6.68035	−	1.65983i	−0.228463	−	0.0567649i
$856$		0			0
$857$			29.7054i			1.01472i	0.861735	+	0.507359i	$0.169378\pi$
$857$			29.7054i			1.01472i	−0.861735	+	0.507359i	$0.830622\pi$
$858$		0			0
$859$	−3.07838			−0.105033			−0.0525164	−	0.998620i	$-0.516724\pi$
$859$	−3.07838			−0.105033			−0.0525164	+	0.998620i	$0.516724\pi$
$860$		0			0
$861$	6.49693			0.221415
$862$		0			0
$863$			6.39350i			0.217637i	0.994062	+	0.108819i	$0.0347068\pi$
$863$			6.39350i			0.217637i	−0.994062	+	0.108819i	$0.965293\pi$
$864$		0			0
$865$	12.0989	−	48.6947i	0.411375	−	1.65567i
$866$		0			0
$867$		−	15.8371i		−	0.537856i
$868$		0			0
$869$	−12.3135			−0.417707
$870$		0			0
$871$	−4.31351			−0.146158
$872$		0			0
$873$		−	8.43907i		−	0.285619i
$874$		0			0
$875$	−7.46081	+	8.32684i	−0.252221	+	0.281499i
$876$		0			0
$877$			1.21622i			0.0410689i	0.999789	+	0.0205345i	$0.00653678\pi$
$877$			1.21622i			0.0410689i	−0.999789	+	0.0205345i	$0.993463\pi$
$878$		0			0
$879$	−2.92162			−0.0985439
$880$		0			0
$881$	15.9733			0.538155			0.269078	−	0.963118i	$-0.413281\pi$
$881$	15.9733			0.538155			0.269078	+	0.963118i	$0.413281\pi$
$882$		0			0
$883$		−	11.6865i		−	0.393282i	−0.980476	−	0.196641i	$-0.936997\pi$
$883$		−	11.6865i		−	0.393282i	0.980476	−	0.196641i	$-0.0630033\pi$
$884$		0			0
$885$	5.67420	−	22.8371i	0.190736	−	0.767661i
$886$		0			0
$887$		−	25.6209i		−	0.860265i	−0.902766	−	0.430132i	$-0.858467\pi$
$887$		−	25.6209i		−	0.860265i	0.902766	−	0.430132i	$-0.141533\pi$
$888$		0			0
$889$	−1.84324			−0.0618204
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$			14.4079i			0.482141i
$894$		0			0
$895$	21.7009	+	5.39189i	0.725380	+	0.180231i
$896$		0			0
$897$		−	2.15676i		−	0.0720120i
$898$		0			0
$899$	51.8310			1.72866
$900$		0			0
$901$	4.05332			0.135036
$902$		0			0
$903$			6.52359i			0.217091i
$904$		0			0
$905$	−18.4969	−	4.59583i	−0.614859	−	0.152770i
$906$		0			0
$907$		−	57.7563i		−	1.91777i	−0.283802	−	0.958883i	$-0.591596\pi$
$907$		−	57.7563i		−	1.91777i	0.283802	−	0.958883i	$-0.408404\pi$
$908$		0			0
$909$	5.81658			0.192924
$910$		0			0
$911$	35.9877			1.19233			0.596163	−	0.802863i	$-0.296691\pi$
$911$	35.9877			1.19233			0.596163	+	0.802863i	$0.296691\pi$
$912$		0			0
$913$			13.6742i			0.452550i
$914$		0			0
$915$	−2.24128	+	9.02052i	−0.0740943	+	0.298209i
$916$		0			0
$917$		−	1.47641i		−	0.0487553i
$918$		0			0
$919$	46.7214			1.54120			0.770598	−	0.637321i	$-0.219958\pi$
$919$	46.7214			1.54120			0.770598	+	0.637321i	$0.219958\pi$
$920$		0			0
$921$	−10.4703			−0.345007
$922$		0			0
$923$		−	1.84324i		−	0.0606711i
$924$		0			0
$925$	25.3607	−	47.8843i	0.833854	−	1.57443i
$926$		0			0
$927$		−	2.15676i		−	0.0708371i
$928$		0			0
$929$	53.0493			1.74049			0.870245	−	0.492619i	$-0.163960\pi$
$929$	53.0493			1.74049			0.870245	+	0.492619i	$0.163960\pi$
$930$		0			0
$931$	−3.07838			−0.100890
$932$		0			0
$933$			23.8310i			0.780191i
$934$		0			0
$935$	−1.16290	+	4.68035i	−0.0380308	+	0.153064i
$936$		0			0
$937$			16.1256i			0.526799i	0.964687	+	0.263400i	$0.0848437\pi$
$937$			16.1256i			0.526799i	−0.964687	+	0.263400i	$0.915156\pi$
$938$		0			0
$939$	32.7526			1.06884
$940$		0			0
$941$	24.7070			0.805425			0.402713	−	0.915326i	$-0.368067\pi$
$941$	24.7070			0.805425			0.402713	+	0.915326i	$0.368067\pi$
$942$		0			0
$943$		−	15.2039i		−	0.495108i
$944$		0			0
$945$	−2.17009	−	0.539189i	−0.0705929	−	0.0175398i
$946$		0			0
$947$		−	6.53797i		−	0.212455i	−0.994342	−	0.106228i	$-0.966123\pi$
$947$		−	6.53797i		−	0.212455i	0.994342	−	0.106228i	$-0.0338772\pi$
$948$		0			0
$949$	−6.52359			−0.211765
$950$		0			0
$951$	17.9155			0.580949
$952$		0			0
$953$			6.11327i			0.198028i	0.995086	+	0.0990142i	$0.0315689\pi$
$953$			6.11327i			0.198028i	−0.995086	+	0.0990142i	$0.968431\pi$
$954$		0			0
$955$	−33.3340	−	8.28231i	−1.07866	−	0.268009i
$956$		0			0
$957$			13.3607i			0.431890i
$958$		0			0
$959$	4.43907			0.143345
$960$		0			0
$961$	29.1978			0.941864
$962$		0			0
$963$			16.4969i			0.531606i
$964$		0			0
$965$	−4.51130	+	18.1568i	−0.145224	+	0.584487i
$966$		0			0
$967$		−	25.6209i		−	0.823912i	−0.911204	−	0.411956i	$-0.864846\pi$
$967$		−	25.6209i		−	0.823912i	0.911204	−	0.411956i	$-0.135154\pi$
$968$		0			0
$969$	−3.31965			−0.106643
$970$		0			0
$971$	−4.05332			−0.130077			−0.0650387	−	0.997883i	$-0.520717\pi$
$971$	−4.05332			−0.130077			−0.0650387	+	0.997883i	$0.520717\pi$
$972$		0			0
$973$			13.6020i			0.436059i
$974$		0			0
$975$	4.07223	+	2.15676i	0.130416	+	0.0690715i
$976$		0			0
$977$			3.81205i			0.121958i	0.998139	+	0.0609791i	$0.0194223\pi$
$977$			3.81205i			0.121958i	−0.998139	+	0.0609791i	$0.980578\pi$
$978$		0			0
$979$	16.6803			0.533106
$980$		0			0
$981$	−12.8371			−0.409857
$982$		0			0
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$		0			0
$985$	−6.34017	+	25.5174i	−0.202015	+	0.813053i
$986$		0			0
$987$			4.68035i			0.148977i
$988$		0			0
$989$	15.2663			0.485441
$990$		0			0
$991$	42.4079			1.34713			0.673565	−	0.739128i	$-0.264762\pi$
$991$	42.4079			1.34713			0.673565	+	0.739128i	$0.264762\pi$
$992$		0			0
$993$		−	1.36069i		−	0.0431803i
$994$		0			0
$995$	−49.0349	−	12.1834i	−1.55451	−	0.386240i
$996$		0			0
$997$		−	43.4740i		−	1.37683i	−0.725315	−	0.688417i	$-0.758306\pi$
$997$		−	43.4740i		−	1.37683i	0.725315	−	0.688417i	$-0.241694\pi$
$998$		0			0
$999$	10.8371			0.342871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.k.1009.3		6
3.2	odd	2		5040.2.t.v.1009.1		6
4.3	odd	2		105.2.d.b.64.6	yes	6
5.2	odd	4		8400.2.a.dg.1.2		3
5.3	odd	4		8400.2.a.dj.1.2		3
5.4	even	2	inner	1680.2.t.k.1009.6		6
12.11	even	2		315.2.d.e.64.1		6
15.14	odd	2		5040.2.t.v.1009.2		6
20.3	even	4		525.2.a.k.1.3		3
20.7	even	4		525.2.a.j.1.1		3
20.19	odd	2		105.2.d.b.64.1	✓	6
28.3	even	6		735.2.q.f.79.1		12
28.11	odd	6		735.2.q.e.79.1		12
28.19	even	6		735.2.q.f.214.6		12
28.23	odd	6		735.2.q.e.214.6		12
28.27	even	2		735.2.d.b.589.6		6
60.23	odd	4		1575.2.a.w.1.1		3
60.47	odd	4		1575.2.a.x.1.3		3
60.59	even	2		315.2.d.e.64.6		6
84.83	odd	2		2205.2.d.l.1324.1		6
140.19	even	6		735.2.q.f.214.1		12
140.27	odd	4		3675.2.a.bi.1.1		3
140.39	odd	6		735.2.q.e.79.6		12
140.59	even	6		735.2.q.f.79.6		12
140.79	odd	6		735.2.q.e.214.1		12
140.83	odd	4		3675.2.a.bj.1.3		3
140.139	even	2		735.2.d.b.589.1		6
420.419	odd	2		2205.2.d.l.1324.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.1	✓	6	20.19	odd	2
105.2.d.b.64.6	yes	6	4.3	odd	2
315.2.d.e.64.1		6	12.11	even	2
315.2.d.e.64.6		6	60.59	even	2
525.2.a.j.1.1		3	20.7	even	4
525.2.a.k.1.3		3	20.3	even	4
735.2.d.b.589.1		6	140.139	even	2
735.2.d.b.589.6		6	28.27	even	2
735.2.q.e.79.1		12	28.11	odd	6
735.2.q.e.79.6		12	140.39	odd	6
735.2.q.e.214.1		12	140.79	odd	6
735.2.q.e.214.6		12	28.23	odd	6
735.2.q.f.79.1		12	28.3	even	6
735.2.q.f.79.6		12	140.59	even	6
735.2.q.f.214.1		12	140.19	even	6
735.2.q.f.214.6		12	28.19	even	6
1575.2.a.w.1.1		3	60.23	odd	4
1575.2.a.x.1.3		3	60.47	odd	4
1680.2.t.k.1009.3		6	1.1	even	1	trivial
1680.2.t.k.1009.6		6	5.4	even	2	inner
2205.2.d.l.1324.1		6	84.83	odd	2
2205.2.d.l.1324.6		6	420.419	odd	2
3675.2.a.bi.1.1		3	140.27	odd	4
3675.2.a.bj.1.3		3	140.83	odd	4
5040.2.t.v.1009.1		6	3.2	odd	2
5040.2.t.v.1009.2		6	15.14	odd	2
8400.2.a.dg.1.2		3	5.2	odd	4
8400.2.a.dj.1.2		3	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.17009 + 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +0.921622i q^{13} +(0.539189 - 2.17009i) q^{15} -1.07838i q^{17} +3.07838 q^{19} +1.00000 q^{21} -2.34017i q^{23} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +6.68035 q^{29} +7.75872 q^{31} +2.00000i q^{33} +(-0.539189 + 2.17009i) q^{35} -10.8371i q^{37} +0.921622 q^{39} +6.49693 q^{41} +6.52359i q^{43} +(-2.17009 - 0.539189i) q^{45} +4.68035i q^{47} -1.00000 q^{49} -1.07838 q^{51} +3.75872i q^{53} +(-4.34017 - 1.07838i) q^{55} -3.07838i q^{57} +10.5236 q^{59} -4.15676 q^{61} -1.00000i q^{63} +(-0.496928 + 2.00000i) q^{65} +4.68035i q^{67} -2.34017 q^{69} -2.00000 q^{71} +7.07838i q^{73} +(2.34017 - 4.41855i) q^{75} -2.00000i q^{77} +6.15676 q^{79} +1.00000 q^{81} -6.83710i q^{83} +(0.581449 - 2.34017i) q^{85} -6.68035i q^{87} -8.34017 q^{89} -0.921622 q^{91} -7.75872i q^{93} +(6.68035 + 1.65983i) q^{95} +8.43907i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 6 q^{9} - 12 q^{11} + 12 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} - 4 q^{31} + 12 q^{39} + 4 q^{41} - 2 q^{45} - 6 q^{49} - 4 q^{55} + 32 q^{59} - 12 q^{61} + 32 q^{65} + 8 q^{69} - 12 q^{71}+ \cdots + 12 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 - 6 * q^9 - 12 * q^11 + 12 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 - 4 * q^31 + 12 * q^39 + 4 * q^41 - 2 * q^45 - 6 * q^49 - 4 * q^55 + 32 * q^59 - 12 * q^61 + 32 * q^65 + 8 * q^69 - 12 * q^71 - 8 * q^75 + 24 * q^79 + 6 * q^81 + 32 * q^85 - 28 * q^89 - 12 * q^91 - 4 * q^95 + 12 * q^99

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