LMFDB - Embedded newform 352.2.e.a.351.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [352,2,Mod(351,352)] 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("352.351"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	352.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.81073415115$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.10
Root			$1.35489 + 0.405301i$ of defining polynomial
Character	$\chi$	$=$	352.351
Dual form			352.2.e.a.351.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.34292i q^{3} +2.48929 q^{5} +1.62121 q^{7} +1.19656 q^{9} +(-3.31339 - 0.146365i) q^{11} +1.20722i q^{13} +3.34292i q^{15} -5.41957i q^{17} +2.59114 q^{19} +2.17715i q^{21} +3.63565i q^{23} +1.19656 q^{25} +5.63565i q^{27} +9.86920i q^{29} -0.949808i q^{31} +(0.196558 - 4.44963i) q^{33} +4.03565 q^{35} +2.48929 q^{37} -1.62121 q^{39} -7.83401i q^{41} -6.62679 q^{43} +2.97858 q^{45} -5.66442i q^{47} -4.37169 q^{49} +7.27806 q^{51} -7.37169 q^{53} +(-8.24799 - 0.364346i) q^{55} +3.47970i q^{57} -12.0288i q^{59} -0.969933i q^{61} +1.93987 q^{63} +3.00512i q^{65} -8.61423i q^{67} -4.88240 q^{69} -14.3215i q^{71} +7.83401i q^{73} +1.60688i q^{75} +(-5.37169 - 0.237289i) q^{77} +11.3137 q^{79} -3.97858 q^{81} +3.38438 q^{83} -13.4909i q^{85} -13.2536 q^{87} -14.5468 q^{89} +1.95715i q^{91} +1.27552 q^{93} +6.45009 q^{95} +6.15371 q^{97} +(-3.96467 - 0.175135i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9} - 4 q^{25} - 16 q^{33} - 24 q^{45} + 44 q^{49} + 8 q^{53} + 8 q^{69} + 32 q^{77} + 12 q^{81} - 88 q^{93} - 64 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 4 * q^25 - 16 * q^33 - 24 * q^45 + 44 * q^49 + 8 * q^53 + 8 * q^69 + 32 * q^77 + 12 * q^81 - 88 * q^93 - 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/352\mathbb{Z}\right)^\times$.

$n$	$133$	$287$	$321$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.34292i			0.775337i	0.921799	+	0.387669i	$0.126719\pi$
$3$			1.34292i			0.775337i	−0.921799	+	0.387669i	$0.873281\pi$
$4$		0			0
$5$	2.48929			1.11324			0.556622	−	0.830766i	$-0.312097\pi$
$5$	2.48929			1.11324			0.556622	+	0.830766i	$0.312097\pi$
$6$		0			0
$7$	1.62121			0.612758			0.306379	−	0.951910i	$-0.400882\pi$
$7$	1.62121			0.612758			0.306379	+	0.951910i	$0.400882\pi$
$8$		0			0
$9$	1.19656			0.398853
$10$		0			0
$11$	−3.31339	−	0.146365i	−0.999026	−	0.0441309i
$11$	−3.31339	−	0.146365i	−0.999026	−	0.0441309i
$12$		0			0
$13$			1.20722i			0.334823i	0.985887	+	0.167412i	$0.0535408\pi$
$13$			1.20722i			0.334823i	−0.985887	+	0.167412i	$0.946459\pi$
$14$		0			0
$15$			3.34292i			0.863139i
$16$		0			0
$17$		−	5.41957i		−	1.31444i	−0.753700	−	0.657219i	$-0.771733\pi$
$17$		−	5.41957i		−	1.31444i	0.753700	−	0.657219i	$-0.228267\pi$
$18$		0			0
$19$	2.59114			0.594448			0.297224	−	0.954808i	$-0.403939\pi$
$19$	2.59114			0.594448			0.297224	+	0.954808i	$0.403939\pi$
$20$		0			0
$21$			2.17715i			0.475094i
$22$		0			0
$23$			3.63565i			0.758086i	0.925379	+	0.379043i	$0.123747\pi$
$23$			3.63565i			0.758086i	−0.925379	+	0.379043i	$0.876253\pi$
$24$		0			0
$25$	1.19656			0.239312
$26$		0			0
$27$			5.63565i			1.08458i
$28$		0			0
$29$			9.86920i			1.83266i	0.400419	+	0.916332i	$0.368865\pi$
$29$			9.86920i			1.83266i	−0.400419	+	0.916332i	$0.631135\pi$
$30$		0			0
$31$		−	0.949808i		−	0.170591i	−0.996356	−	0.0852953i	$-0.972817\pi$
$31$		−	0.949808i		−	0.170591i	0.996356	−	0.0852953i	$-0.0271834\pi$
$32$		0			0
$33$	0.196558	−	4.44963i	0.0342163	−	0.774582i
$34$		0			0
$35$	4.03565			0.682149
$36$		0			0
$37$	2.48929			0.409237			0.204618	−	0.978842i	$-0.434405\pi$
$37$	2.48929			0.409237			0.204618	+	0.978842i	$0.434405\pi$
$38$		0			0
$39$	−1.62121			−0.259601
$40$		0			0
$41$		−	7.83401i		−	1.22347i	−0.791064	−	0.611733i	$-0.790473\pi$
$41$		−	7.83401i		−	1.22347i	0.791064	−	0.611733i	$-0.209527\pi$
$42$		0			0
$43$	−6.62679			−1.01058			−0.505288	−	0.862951i	$-0.668614\pi$
$43$	−6.62679			−1.01058			−0.505288	+	0.862951i	$0.668614\pi$
$44$		0			0
$45$	2.97858			0.444020
$46$		0			0
$47$		−	5.66442i		−	0.826241i	−0.910676	−	0.413121i	$-0.864439\pi$
$47$		−	5.66442i		−	0.826241i	0.910676	−	0.413121i	$-0.135561\pi$
$48$		0			0
$49$	−4.37169			−0.624527
$50$		0			0
$51$	7.27806			1.01913
$52$		0			0
$53$	−7.37169			−1.01258			−0.506290	−	0.862363i	$-0.668983\pi$
$53$	−7.37169			−1.01258			−0.506290	+	0.862363i	$0.668983\pi$
$54$		0			0
$55$	−8.24799	−	0.364346i	−1.11216	−	0.0491284i
$56$		0			0
$57$			3.47970i			0.460897i
$58$		0			0
$59$		−	12.0288i		−	1.56601i	−0.622014	−	0.783006i	$-0.713685\pi$
$59$		−	12.0288i		−	1.56601i	0.622014	−	0.783006i	$-0.286315\pi$
$60$		0			0
$61$		−	0.969933i		−	0.124187i	−0.998070	−	0.0620936i	$-0.980222\pi$
$61$		−	0.969933i		−	0.124187i	0.998070	−	0.0620936i	$-0.0197777\pi$
$62$		0			0
$63$	1.93987			0.244400
$64$		0			0
$65$			3.00512i			0.372740i
$66$		0			0
$67$		−	8.61423i		−	1.05240i	−0.850362	−	0.526198i	$-0.823617\pi$
$67$		−	8.61423i		−	1.05240i	0.850362	−	0.526198i	$-0.176383\pi$
$68$		0			0
$69$	−4.88240			−0.587772
$70$		0			0
$71$		−	14.3215i		−	1.69965i	−0.527065	−	0.849825i	$-0.676708\pi$
$71$		−	14.3215i		−	1.69965i	0.527065	−	0.849825i	$-0.323292\pi$
$72$		0			0
$73$			7.83401i			0.916901i	0.888720	+	0.458451i	$0.151595\pi$
$73$			7.83401i			0.916901i	−0.888720	+	0.458451i	$0.848405\pi$
$74$		0			0
$75$			1.60688i			0.185547i
$76$		0			0
$77$	−5.37169	−	0.237289i	−0.612161	−	0.0270415i
$78$		0			0
$79$	11.3137			1.27289			0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137			1.27289			0.636446	+	0.771321i	$0.280404\pi$
$80$		0			0
$81$	−3.97858			−0.442064
$82$		0			0
$83$	3.38438			0.371484			0.185742	−	0.982599i	$-0.440531\pi$
$83$	3.38438			0.371484			0.185742	+	0.982599i	$0.440531\pi$
$84$		0			0
$85$		−	13.4909i		−	1.46329i
$86$		0			0
$87$	−13.2536			−1.42093
$88$		0			0
$89$	−14.5468			−1.54196			−0.770980	−	0.636859i	$-0.780233\pi$
$89$	−14.5468			−1.54196			−0.770980	+	0.636859i	$0.780233\pi$
$90$		0			0
$91$			1.95715i			0.205166i
$92$		0			0
$93$	1.27552			0.132265
$94$		0			0
$95$	6.45009			0.661765
$96$		0			0
$97$	6.15371			0.624815			0.312407	−	0.949948i	$-0.398865\pi$
$97$	6.15371			0.624815			0.312407	+	0.949948i	$0.398865\pi$
$98$		0			0
$99$	−3.96467	−	0.175135i	−0.398464	−	0.0176017i
$100$		0			0
$101$			10.1065i			1.00563i	0.864393	+	0.502817i	$0.167703\pi$
$101$			10.1065i			1.00563i	−0.864393	+	0.502817i	$0.832297\pi$
$102$		0			0
$103$			12.2927i			1.21124i	0.795754	+	0.605619i	$0.207075\pi$
$103$			12.2927i			1.21124i	−0.795754	+	0.605619i	$0.792925\pi$
$104$		0			0
$105$			5.41957i			0.528895i
$106$		0			0
$107$	−17.1473			−1.65769			−0.828844	−	0.559480i	$-0.811001\pi$
$107$	−17.1473			−1.65769			−0.828844	+	0.559480i	$0.811001\pi$
$108$		0			0
$109$		−	12.2836i		−	1.17656i	−0.808658	−	0.588280i	$-0.799805\pi$
$109$		−	12.2836i		−	1.17656i	0.808658	−	0.588280i	$-0.200195\pi$
$110$		0			0
$111$			3.34292i			0.317296i
$112$		0			0
$113$	0.782020			0.0735662			0.0367831	−	0.999323i	$-0.488289\pi$
$113$	0.782020			0.0735662			0.0367831	+	0.999323i	$0.488289\pi$
$114$		0			0
$115$			9.05019i			0.843935i
$116$		0			0
$117$			1.44451i			0.133545i
$118$		0			0
$119$		−	8.78623i		−	0.805432i
$120$		0			0
$121$	10.9572	+	0.969933i	0.996105	+	0.0881757i
$122$		0			0
$123$	10.5205			0.948599
$124$		0			0
$125$	−9.46787			−0.846832
$126$		0			0
$127$	18.1172			1.60764			0.803820	−	0.594872i	$-0.202797\pi$
$127$	18.1172			1.60764			0.803820	+	0.594872i	$0.202797\pi$
$128$		0			0
$129$		−	8.89927i		−	0.783537i
$130$		0			0
$131$	2.59114			0.226389			0.113194	−	0.993573i	$-0.463892\pi$
$131$	2.59114			0.226389			0.113194	+	0.993573i	$0.463892\pi$
$132$		0			0
$133$	4.20077			0.364253
$134$		0			0
$135$			14.0288i			1.20740i
$136$		0			0
$137$	−9.17513			−0.783885			−0.391942	−	0.919990i	$-0.628197\pi$
$137$	−9.17513			−0.783885			−0.391942	+	0.919990i	$0.628197\pi$
$138$		0			0
$139$	−5.83355			−0.494795			−0.247398	−	0.968914i	$-0.579575\pi$
$139$	−5.83355			−0.494795			−0.247398	+	0.968914i	$0.579575\pi$
$140$		0			0
$141$	7.60688			0.640615
$142$		0			0
$143$	0.176696	−	4.00000i	0.0147760	−	0.334497i
$144$		0			0
$145$			24.5673i			2.04020i
$146$		0			0
$147$		−	5.87085i		−	0.484219i
$148$		0			0
$149$		−	1.20722i		−	0.0988994i	−0.998777	−	0.0494497i	$-0.984253\pi$
$149$		−	1.20722i		−	0.0988994i	0.998777	−	0.0494497i	$-0.0157468\pi$
$150$		0			0
$151$	−21.3249			−1.73539			−0.867697	−	0.497094i	$-0.834400\pi$
$151$	−21.3249			−1.73539			−0.867697	+	0.497094i	$0.834400\pi$
$152$		0			0
$153$		−	6.48482i		−	0.524267i
$154$		0			0
$155$		−	2.36435i		−	0.189909i
$156$		0			0
$157$	5.90383			0.471177			0.235588	−	0.971853i	$-0.424298\pi$
$157$	5.90383			0.471177			0.235588	+	0.971853i	$0.424298\pi$
$158$		0			0
$159$		−	9.89962i		−	0.785091i
$160$		0			0
$161$			5.89414i			0.464524i
$162$		0			0
$163$			18.4507i			1.44517i	0.691284	+	0.722583i	$0.257045\pi$
$163$			18.4507i			1.44517i	−0.691284	+	0.722583i	$0.742955\pi$
$164$		0			0
$165$	0.489289	−	11.0764i	0.0380911	−	0.862298i
$166$		0			0
$167$	9.69250			0.750028			0.375014	−	0.927019i	$-0.377638\pi$
$167$	9.69250			0.750028			0.375014	+	0.927019i	$0.377638\pi$
$168$		0			0
$169$	11.5426			0.887894
$170$		0			0
$171$	3.10045			0.237097
$172$		0			0
$173$		−	18.7685i		−	1.42694i	−0.700686	−	0.713470i	$-0.747123\pi$
$173$		−	18.7685i		−	1.42694i	0.700686	−	0.713470i	$-0.252877\pi$
$174$		0			0
$175$	1.93987			0.146640
$176$		0			0
$177$	16.1537			1.21419
$178$		0			0
$179$		−	17.4005i		−	1.30057i	−0.759690	−	0.650286i	$-0.774649\pi$
$179$		−	17.4005i		−	1.30057i	0.759690	−	0.650286i	$-0.225351\pi$
$180$		0			0
$181$	15.8610			1.17894			0.589469	−	0.807791i	$-0.299337\pi$
$181$	15.8610			1.17894			0.589469	+	0.807791i	$0.299337\pi$
$182$		0			0
$183$	1.30255			0.0962869
$184$		0			0
$185$	6.19656			0.455580
$186$		0			0
$187$	−0.793237	+	17.9572i	−0.0580073	+	1.31316i
$188$		0			0
$189$			9.13655i			0.664587i
$190$		0			0
$191$			15.0502i			1.08899i	0.838763	+	0.544497i	$0.183279\pi$
$191$			15.0502i			1.08899i	−0.838763	+	0.544497i	$0.816721\pi$
$192$		0			0
$193$			22.1528i			1.59460i	0.603586	+	0.797298i	$0.293738\pi$
$193$			22.1528i			1.59460i	−0.603586	+	0.797298i	$0.706262\pi$
$194$		0			0
$195$	−4.03565			−0.288999
$196$		0			0
$197$		−	3.38438i		−	0.241127i	−0.992706	−	0.120563i	$-0.961530\pi$
$197$		−	3.38438i		−	0.241127i	0.992706	−	0.120563i	$-0.0384701\pi$
$198$		0			0
$199$			5.07896i			0.360038i	0.983663	+	0.180019i	$0.0576159\pi$
$199$			5.07896i			0.360038i	−0.983663	+	0.180019i	$0.942384\pi$
$200$		0			0
$201$	11.5682			0.815961
$202$		0			0
$203$			16.0000i			1.12298i
$204$		0			0
$205$		−	19.5011i		−	1.36202i
$206$		0			0
$207$			4.35027i			0.302365i
$208$		0			0
$209$	−8.58546	−	0.379253i	−0.593869	−	0.0262335i
$210$		0			0
$211$	17.9405			1.23507			0.617537	−	0.786542i	$-0.288130\pi$
$211$	17.9405			1.23507			0.617537	+	0.786542i	$0.288130\pi$
$212$		0			0
$213$	19.2327			1.31780
$214$		0			0
$215$	−16.4960			−1.12502
$216$		0			0
$217$		−	1.53983i		−	0.104531i
$218$		0			0
$219$	−10.5205			−0.710908
$220$		0			0
$221$	6.54262			0.440104
$222$		0			0
$223$			15.0502i			1.00784i	0.863752	+	0.503918i	$0.168109\pi$
$223$			15.0502i			1.00784i	−0.863752	+	0.503918i	$0.831891\pi$
$224$		0			0
$225$	1.43175			0.0954500
$226$		0			0
$227$	17.1473			1.13810			0.569052	−	0.822302i	$-0.307310\pi$
$227$	17.1473			1.13810			0.569052	+	0.822302i	$0.307310\pi$
$228$		0			0
$229$	−20.0533			−1.32516			−0.662581	−	0.748991i	$-0.730539\pi$
$229$	−20.0533			−1.32516			−0.662581	+	0.748991i	$0.730539\pi$
$230$		0			0
$231$	0.318660	−	7.21377i	0.0209663	−	0.474631i
$232$		0			0
$233$			6.76875i			0.443436i	0.975111	+	0.221718i	$0.0711664\pi$
$233$			6.76875i			0.443436i	−0.975111	+	0.221718i	$0.928834\pi$
$234$		0			0
$235$		−	14.1004i		−	0.919808i
$236$		0			0
$237$			15.1934i			0.986920i
$238$		0			0
$239$	−24.9207			−1.61198			−0.805992	−	0.591926i	$-0.798368\pi$
$239$	−24.9207			−1.61198			−0.805992	+	0.591926i	$0.798368\pi$
$240$		0			0
$241$		−	17.3240i		−	1.11593i	−0.829863	−	0.557967i	$-0.811582\pi$
$241$		−	17.3240i		−	1.11593i	0.829863	−	0.557967i	$-0.188418\pi$
$242$		0			0
$243$			11.5640i			0.741833i
$244$		0			0
$245$	−10.8824			−0.695251
$246$		0			0
$247$			3.12808i			0.199035i
$248$		0			0
$249$			4.54496i			0.288025i
$250$		0			0
$251$		−	5.98592i		−	0.377828i	−0.981994	−	0.188914i	$-0.939503\pi$
$251$		−	5.98592i		−	0.377828i	0.981994	−	0.188914i	$-0.0604968\pi$
$252$		0			0
$253$	0.532134	−	12.0464i	0.0334550	−	0.757348i
$254$		0			0
$255$	18.1172			1.13454
$256$		0			0
$257$	−12.5426			−0.782387			−0.391193	−	0.920308i	$-0.627938\pi$
$257$	−12.5426			−0.782387			−0.391193	+	0.920308i	$0.627938\pi$
$258$		0			0
$259$	4.03565			0.250763
$260$		0			0
$261$			11.8091i			0.730963i
$262$		0			0
$263$	16.8146			1.03684			0.518418	−	0.855128i	$-0.326521\pi$
$263$	16.8146			1.03684			0.518418	+	0.855128i	$0.326521\pi$
$264$		0			0
$265$	−18.3503			−1.12725
$266$		0			0
$267$		−	19.5353i		−	1.19554i
$268$		0			0
$269$	−15.3717			−0.937229			−0.468614	−	0.883403i	$-0.655247\pi$
$269$	−15.3717			−0.937229			−0.468614	+	0.883403i	$0.655247\pi$
$270$		0			0
$271$	0.318660			0.0193572			0.00967862	−	0.999953i	$-0.496919\pi$
$271$	0.318660			0.0193572			0.00967862	+	0.999953i	$0.496919\pi$
$272$		0			0
$273$	−2.62831			−0.159072
$274$		0			0
$275$	−3.96467	−	0.175135i	−0.239078	−	0.0105610i
$276$		0			0
$277$			12.2836i			0.738052i	0.929419	+	0.369026i	$0.120309\pi$
$277$			12.2836i			0.738052i	−0.929419	+	0.369026i	$0.879691\pi$
$278$		0			0
$279$		−	1.13650i		−	0.0680405i
$280$		0			0
$281$		−	13.2536i		−	0.790642i	−0.918543	−	0.395321i	$-0.870633\pi$
$281$		−	13.2536i		−	0.790642i	0.918543	−	0.395321i	$-0.129367\pi$
$282$		0			0
$283$	−14.6981			−0.873710			−0.436855	−	0.899532i	$-0.643908\pi$
$283$	−14.6981			−0.873710			−0.436855	+	0.899532i	$0.643908\pi$
$284$		0			0
$285$			8.66198i			0.513091i
$286$		0			0
$287$		−	12.7005i		−	0.749689i
$288$		0			0
$289$	−12.3717			−0.727747
$290$		0			0
$291$			8.26396i			0.484442i
$292$		0			0
$293$			20.9456i			1.22366i	0.790991	+	0.611828i	$0.209566\pi$
$293$			20.9456i			1.22366i	−0.790991	+	0.611828i	$0.790434\pi$
$294$		0			0
$295$		−	29.9431i		−	1.74335i
$296$		0			0
$297$	0.824865	−	18.6731i	0.0478635	−	1.08353i
$298$		0			0
$299$	−4.38904			−0.253825
$300$		0			0
$301$	−10.7434			−0.619238
$302$		0			0
$303$	−13.5722			−0.779705
$304$		0			0
$305$		−	2.41444i		−	0.138251i
$306$		0			0
$307$	−0.651273			−0.0371701			−0.0185850	−	0.999827i	$-0.505916\pi$
$307$	−0.651273			−0.0371701			−0.0185850	+	0.999827i	$0.505916\pi$
$308$		0			0
$309$	−16.5082			−0.939118
$310$		0			0
$311$			24.9933i			1.41724i	0.705591	+	0.708619i	$0.250682\pi$
$311$			24.9933i			1.41724i	−0.705591	+	0.708619i	$0.749318\pi$
$312$		0			0
$313$	8.11087			0.458453			0.229227	−	0.973373i	$-0.426380\pi$
$313$	8.11087			0.458453			0.229227	+	0.973373i	$0.426380\pi$
$314$		0			0
$315$	4.82889			0.272077
$316$		0			0
$317$	12.4464			0.699062			0.349531	−	0.936925i	$-0.386341\pi$
$317$	12.4464			0.699062			0.349531	+	0.936925i	$0.386341\pi$
$318$		0			0
$319$	1.44451	−	32.7005i	0.0808770	−	1.83088i
$320$		0			0
$321$		−	23.0274i		−	1.28527i
$322$		0			0
$323$		−	14.0428i		−	0.781365i
$324$		0			0
$325$			1.44451i			0.0801270i
$326$		0			0
$327$	16.4960			0.912230
$328$		0			0
$329$		−	9.18319i		−	0.506286i
$330$		0			0
$331$			6.71462i			0.369069i	0.982826	+	0.184534i	$0.0590777\pi$
$331$			6.71462i			0.369069i	−0.982826	+	0.184534i	$0.940922\pi$
$332$		0			0
$333$	2.97858			0.163225
$334$		0			0
$335$		−	21.4433i		−	1.17157i
$336$		0			0
$337$			0.590680i			0.0321764i	0.999871	+	0.0160882i	$0.00512125\pi$
$337$			0.590680i			0.0321764i	−0.999871	+	0.0160882i	$0.994879\pi$
$338$		0			0
$339$			1.05019i			0.0570386i
$340$		0			0
$341$	−0.139019	+	3.14709i	−0.00752831	+	0.170424i
$342$		0			0
$343$	−18.4359			−0.995442
$344$		0			0
$345$	−12.1537			−0.654334
$346$		0			0
$347$	−12.3184			−0.661285			−0.330642	−	0.943756i	$-0.607265\pi$
$347$	−12.3184			−0.661285			−0.330642	+	0.943756i	$0.607265\pi$
$348$		0			0
$349$			18.0566i			0.966548i	0.875469	+	0.483274i	$0.160552\pi$
$349$			18.0566i			0.966548i	−0.875469	+	0.483274i	$0.839448\pi$
$350$		0			0
$351$	−6.80348			−0.363143
$352$		0			0
$353$	22.1537			1.17912			0.589562	−	0.807723i	$-0.299301\pi$
$353$	22.1537			1.17912			0.589562	+	0.807723i	$0.299301\pi$
$354$		0			0
$355$		−	35.6503i		−	1.89212i
$356$		0			0
$357$	11.7992			0.624482
$358$		0			0
$359$	31.3708			1.65569			0.827843	−	0.560960i	$-0.189568\pi$
$359$	31.3708			1.65569			0.827843	+	0.560960i	$0.189568\pi$
$360$		0			0
$361$	−12.2860			−0.646632
$362$		0			0
$363$	−1.30255	+	14.7146i	−0.0683659	+	0.772317i
$364$		0			0
$365$			19.5011i			1.02073i
$366$		0			0
$367$			25.7936i			1.34641i	0.739454	+	0.673207i	$0.235084\pi$
$367$			25.7936i			1.34641i	−0.739454	+	0.673207i	$0.764916\pi$
$368$		0			0
$369$		−	9.37384i		−	0.487983i
$370$		0			0
$371$	−11.9510			−0.620467
$372$		0			0
$373$		−	16.4007i		−	0.849194i	−0.905382	−	0.424597i	$-0.860416\pi$
$373$		−	16.4007i		−	0.849194i	0.905382	−	0.424597i	$-0.139584\pi$
$374$		0			0
$375$		−	12.7146i		−	0.656580i
$376$		0			0
$377$	−11.9143			−0.613618
$378$		0			0
$379$		−	15.9431i		−	0.818941i	−0.912323	−	0.409470i	$-0.865713\pi$
$379$		−	15.9431i		−	0.818941i	0.912323	−	0.409470i	$-0.134287\pi$
$380$		0			0
$381$			24.3300i			1.24646i
$382$		0			0
$383$			11.7507i			0.600434i	0.953871	+	0.300217i	$0.0970591\pi$
$383$			11.7507i			0.600434i	−0.953871	+	0.300217i	$0.902941\pi$
$384$		0			0
$385$	−13.3717	−	0.590680i	−0.681485	−	0.0301038i
$386$		0			0
$387$	−7.92933			−0.403071
$388$		0			0
$389$	−1.71148			−0.0867755			−0.0433877	−	0.999058i	$-0.513815\pi$
$389$	−1.71148			−0.0867755			−0.0433877	+	0.999058i	$0.513815\pi$
$390$		0			0
$391$	19.7037			0.996457
$392$		0			0
$393$			3.47970i			0.175528i
$394$		0			0
$395$	28.1631			1.41704
$396$		0			0
$397$	−4.62831			−0.232288			−0.116144	−	0.993232i	$-0.537053\pi$
$397$	−4.62831			−0.232288			−0.116144	+	0.993232i	$0.537053\pi$
$398$		0			0
$399$			5.64131i			0.282419i
$400$		0			0
$401$	14.2008			0.709153			0.354576	−	0.935027i	$-0.384625\pi$
$401$	14.2008			0.709153			0.354576	+	0.935027i	$0.384625\pi$
$402$		0			0
$403$	1.14663			0.0571176
$404$		0			0
$405$	−9.90383			−0.492125
$406$		0			0
$407$	−8.24799	−	0.364346i	−0.408838	−	0.0180600i
$408$		0			0
$409$			8.89927i			0.440040i	0.975495	+	0.220020i	$0.0706123\pi$
$409$			8.89927i			0.440040i	−0.975495	+	0.220020i	$0.929388\pi$
$410$		0			0
$411$		−	12.3215i		−	0.607775i
$412$		0			0
$413$		−	19.5011i		−	0.959587i
$414$		0			0
$415$	8.42469			0.413552
$416$		0			0
$417$		−	7.83401i		−	0.383633i
$418$		0			0
$419$		−	7.12181i		−	0.347923i	−0.984752	−	0.173962i	$-0.944343\pi$
$419$		−	7.12181i		−	0.347923i	0.984752	−	0.173962i	$-0.0556568\pi$
$420$		0			0
$421$	3.37169			0.164326			0.0821631	−	0.996619i	$-0.473817\pi$
$421$	3.37169			0.164326			0.0821631	+	0.996619i	$0.473817\pi$
$422$		0			0
$423$		−	6.77781i		−	0.329548i
$424$		0			0
$425$		−	6.48482i		−	0.314560i
$426$		0			0
$427$		−	1.57246i		−	0.0760967i
$428$		0			0
$429$	5.37169	+	0.237289i	0.259348	+	0.0114564i
$430$		0			0
$431$	−22.9461			−1.10527			−0.552637	−	0.833422i	$-0.686378\pi$
$431$	−22.9461			−1.10527			−0.552637	+	0.833422i	$0.686378\pi$
$432$		0			0
$433$	14.1537			0.680184			0.340092	−	0.940392i	$-0.389542\pi$
$433$	14.1537			0.680184			0.340092	+	0.940392i	$0.389542\pi$
$434$		0			0
$435$	−32.9920			−1.58184
$436$		0			0
$437$			9.42048i			0.450643i
$438$		0			0
$439$	−13.2536			−0.632559			−0.316279	−	0.948666i	$-0.602434\pi$
$439$	−13.2536			−0.632559			−0.316279	+	0.948666i	$0.602434\pi$
$440$		0			0
$441$	−5.23098			−0.249094
$442$		0			0
$443$			15.5008i			0.736467i	0.929733	+	0.368234i	$0.120037\pi$
$443$			15.5008i			0.736467i	−0.929733	+	0.368234i	$0.879963\pi$
$444$		0			0
$445$	−36.2113			−1.71658
$446$		0			0
$447$	1.62121			0.0766804
$448$		0			0
$449$	−17.9614			−0.847649			−0.423825	−	0.905744i	$-0.639313\pi$
$449$	−17.9614			−0.847649			−0.423825	+	0.905744i	$0.639313\pi$
$450$		0			0
$451$	−1.14663	+	25.9572i	−0.0539926	+	1.22227i
$452$		0			0
$453$		−	28.6377i		−	1.34551i
$454$		0			0
$455$			4.87192i			0.228399i
$456$		0			0
$457$			2.88902i			0.135143i	0.997714	+	0.0675713i	$0.0215250\pi$
$457$			2.88902i			0.135143i	−0.997714	+	0.0675713i	$0.978475\pi$
$458$		0			0
$459$	30.5428			1.42562
$460$		0			0
$461$			19.0524i			0.887358i	0.896186	+	0.443679i	$0.146327\pi$
$461$			19.0524i			0.887358i	−0.896186	+	0.443679i	$0.853673\pi$
$462$		0			0
$463$			1.00735i			0.0468154i	0.999726	+	0.0234077i	$0.00745158\pi$
$463$			1.00735i			0.0468154i	−0.999726	+	0.0234077i	$0.992548\pi$
$464$		0			0
$465$	3.17513			0.147243
$466$		0			0
$467$			36.0863i			1.66988i	0.550345	+	0.834938i	$0.314496\pi$
$467$			36.0863i			1.66988i	−0.550345	+	0.834938i	$0.685504\pi$
$468$		0			0
$469$		−	13.9654i		−	0.644864i
$470$		0			0
$471$			7.92839i			0.365321i
$472$		0			0
$473$	21.9572	+	0.969933i	1.00959	+	0.0445976i
$474$		0			0
$475$	3.10045			0.142258
$476$		0			0
$477$	−8.82065			−0.403870
$478$		0			0
$479$	−10.9950			−0.502377			−0.251188	−	0.967938i	$-0.580821\pi$
$479$	−10.9950			−0.502377			−0.251188	+	0.967938i	$0.580821\pi$
$480$		0			0
$481$			3.00512i			0.137022i
$482$		0			0
$483$	−7.91538			−0.360162
$484$		0			0
$485$	15.3184			0.695571
$486$		0			0
$487$			10.4647i			0.474202i	0.971485	+	0.237101i	$0.0761973\pi$
$487$			10.4647i			0.474202i	−0.971485	+	0.237101i	$0.923803\pi$
$488$		0			0
$489$	−24.7778			−1.12049
$490$		0			0
$491$	18.2939			0.825592			0.412796	−	0.910824i	$-0.364552\pi$
$491$	18.2939			0.825592			0.412796	+	0.910824i	$0.364552\pi$
$492$		0			0
$493$	53.4868			2.40892
$494$		0			0
$495$	−9.86920	−	0.435961i	−0.443587	−	0.0195950i
$496$		0			0
$497$		−	23.2181i		−	1.04147i
$498$		0			0
$499$		−	7.12181i		−	0.318816i	−0.987213	−	0.159408i	$-0.949041\pi$
$499$		−	7.12181i		−	0.318816i	0.987213	−	0.159408i	$-0.0509585\pi$
$500$		0			0
$501$			13.0163i			0.581525i
$502$		0			0
$503$	16.4960			0.735520			0.367760	−	0.929921i	$-0.380125\pi$
$503$	16.4960			0.735520			0.367760	+	0.929921i	$0.380125\pi$
$504$		0			0
$505$			25.1580i			1.11951i
$506$		0			0
$507$			15.5008i			0.688417i
$508$		0			0
$509$	28.4464			1.26087			0.630433	−	0.776244i	$-0.282877\pi$
$509$	28.4464			1.26087			0.630433	+	0.776244i	$0.282877\pi$
$510$		0			0
$511$			12.7005i			0.561839i
$512$		0			0
$513$			14.6028i			0.644728i
$514$		0			0
$515$			30.6002i			1.34840i
$516$		0			0
$517$	−0.829076	+	18.7685i	−0.0364627	+	0.825436i
$518$		0			0
$519$	25.2046			1.10636
$520$		0			0
$521$	40.2259			1.76233			0.881165	−	0.472808i	$-0.156760\pi$
$521$	40.2259			1.76233			0.881165	+	0.472808i	$0.156760\pi$
$522$		0			0
$523$	36.8857			1.61290			0.806449	−	0.591304i	$-0.201387\pi$
$523$	36.8857			1.61290			0.806449	+	0.591304i	$0.201387\pi$
$524$		0			0
$525$			2.60509i			0.113695i
$526$		0			0
$527$	−5.14755			−0.224231
$528$		0			0
$529$	9.78202			0.425305
$530$		0			0
$531$		−	14.3931i		−	0.624608i
$532$		0			0
$533$	9.45738			0.409645
$534$		0			0
$535$	−42.6845			−1.84541
$536$		0			0
$537$	23.3675			1.00838
$538$		0			0
$539$	14.4851	+	0.639865i	0.623919	+	0.0275609i
$540$		0			0
$541$		−	27.4304i		−	1.17933i	−0.807649	−	0.589663i	$-0.799260\pi$
$541$		−	27.4304i		−	1.17933i	0.807649	−	0.589663i	$-0.200740\pi$
$542$		0			0
$543$			21.3001i			0.914074i
$544$		0			0
$545$		−	30.5775i		−	1.30980i
$546$		0			0
$547$	0.141965			0.00606997			0.00303499	−	0.999995i	$-0.499034\pi$
$547$	0.141965			0.00606997			0.00303499	+	0.999995i	$0.499034\pi$
$548$		0			0
$549$		−	1.16058i		−	0.0495324i
$550$		0			0
$551$			25.5725i			1.08942i
$552$		0			0
$553$	18.3418			0.779975
$554$		0			0
$555$			8.32150i			0.353228i
$556$		0			0
$557$			3.33774i			0.141424i	0.997497	+	0.0707122i	$0.0225272\pi$
$557$			3.33774i			0.141424i	−0.997497	+	0.0707122i	$0.977473\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$	−24.1151	−	1.06526i	−1.01814	−	0.0449752i
$562$		0			0
$563$	−32.7805			−1.38154			−0.690768	−	0.723077i	$-0.742727\pi$
$563$	−32.7805			−1.38154			−0.690768	+	0.723077i	$0.742727\pi$
$564$		0			0
$565$	1.94667			0.0818971
$566$		0			0
$567$	−6.45009			−0.270878
$568$		0			0
$569$			14.2027i			0.595409i	0.954658	+	0.297705i	$0.0962211\pi$
$569$			14.2027i			0.595409i	−0.954658	+	0.297705i	$0.903779\pi$
$570$		0			0
$571$	−8.28272			−0.346621			−0.173311	−	0.984867i	$-0.555446\pi$
$571$	−8.28272			−0.346621			−0.173311	+	0.984867i	$0.555446\pi$
$572$		0			0
$573$	−20.2113			−0.844337
$574$		0			0
$575$			4.35027i			0.181419i
$576$		0			0
$577$	46.2688			1.92619			0.963097	−	0.269153i	$-0.0867437\pi$
$577$	46.2688			1.92619			0.963097	+	0.269153i	$0.0867437\pi$
$578$		0			0
$579$	−29.7496			−1.23635
$580$		0			0
$581$	5.48677			0.227630
$582$		0			0
$583$	24.4253	+	1.07896i	1.01159	+	0.0446860i
$584$		0			0
$585$			3.59580i			0.148668i
$586$		0			0
$587$		−	18.2499i		−	0.753253i	−0.926365	−	0.376627i	$-0.877084\pi$
$587$		−	18.2499i		−	0.753253i	0.926365	−	0.376627i	$-0.122916\pi$
$588$		0			0
$589$		−	2.46108i		−	0.101407i
$590$		0			0
$591$	4.54496			0.186955
$592$		0			0
$593$			38.2954i			1.57261i	0.617842	+	0.786303i	$0.288007\pi$
$593$			38.2954i			1.57261i	−0.617842	+	0.786303i	$0.711993\pi$
$594$		0			0
$595$		−	21.8715i		−	0.896643i
$596$		0			0
$597$	−6.82065			−0.279151
$598$		0			0
$599$		−	30.8353i		−	1.25990i	−0.776636	−	0.629949i	$-0.783076\pi$
$599$		−	30.8353i		−	1.25990i	0.776636	−	0.629949i	$-0.216924\pi$
$600$		0			0
$601$		−	36.9462i		−	1.50707i	−0.657408	−	0.753534i	$-0.728347\pi$
$601$		−	36.9462i		−	1.50707i	0.657408	−	0.753534i	$-0.271653\pi$
$602$		0			0
$603$		−	10.3074i		−	0.419751i
$604$		0			0
$605$	27.2755	+	2.41444i	1.10891	+	0.0981611i
$606$		0			0
$607$	−6.48482			−0.263211			−0.131605	−	0.991302i	$-0.542013\pi$
$607$	−6.48482			−0.263211			−0.131605	+	0.991302i	$0.542013\pi$
$608$		0			0
$609$	−21.4868			−0.870688
$610$		0			0
$611$	6.83821			0.276645
$612$		0			0
$613$		−	35.9018i		−	1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$613$		−	35.9018i		−	1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$614$		0			0
$615$	26.1885			1.05602
$616$		0			0
$617$	−8.62831			−0.347363			−0.173681	−	0.984802i	$-0.555566\pi$
$617$	−8.62831			−0.347363			−0.173681	+	0.984802i	$0.555566\pi$
$618$		0			0
$619$			18.6289i			0.748760i	0.927275	+	0.374380i	$0.122144\pi$
$619$			18.6289i			0.748760i	−0.927275	+	0.374380i	$0.877856\pi$
$620$		0			0
$621$	−20.4893			−0.822207
$622$		0			0
$623$	−23.5834			−0.944849
$624$		0			0
$625$	−29.5510			−1.18204
$626$		0			0
$627$	0.509308	−	11.5296i	0.0203398	−	0.460448i
$628$		0			0
$629$		−	13.4909i		−	0.537916i
$630$		0			0
$631$		−	5.15058i		−	0.205041i	−0.994731	−	0.102521i	$-0.967309\pi$
$631$		−	5.15058i		−	0.205041i	0.994731	−	0.102521i	$-0.0326908\pi$
$632$		0			0
$633$			24.0927i			0.957599i
$634$		0			0
$635$	45.0989			1.78970
$636$		0			0
$637$		−	5.27760i		−	0.209106i
$638$		0			0
$639$		−	17.1365i		−	0.677909i
$640$		0			0
$641$	−8.38890			−0.331342			−0.165671	−	0.986181i	$-0.552979\pi$
$641$	−8.38890			−0.331342			−0.165671	+	0.986181i	$0.552979\pi$
$642$		0			0
$643$		−	16.6142i		−	0.655201i	−0.944816	−	0.327601i	$-0.893760\pi$
$643$		−	16.6142i		−	0.655201i	0.944816	−	0.327601i	$-0.106240\pi$
$644$		0			0
$645$		−	22.1528i		−	0.872267i
$646$		0			0
$647$			1.67850i			0.0659886i	0.999456	+	0.0329943i	$0.0105043\pi$
$647$			1.67850i			0.0659886i	−0.999456	+	0.0329943i	$0.989496\pi$
$648$		0			0
$649$	−1.76060	+	39.8560i	−0.0691095	+	1.56449i
$650$		0			0
$651$	2.06788			0.0810465
$652$		0			0
$653$	8.53213			0.333888			0.166944	−	0.985966i	$-0.446610\pi$
$653$	8.53213			0.333888			0.166944	+	0.985966i	$0.446610\pi$
$654$		0			0
$655$	6.45009			0.252026
$656$		0			0
$657$			9.37384i			0.365708i
$658$		0			0
$659$	−18.2939			−0.712629			−0.356314	−	0.934366i	$-0.615967\pi$
$659$	−18.2939			−0.712629			−0.356314	+	0.934366i	$0.615967\pi$
$660$		0			0
$661$	0.916828			0.0356605			0.0178302	−	0.999841i	$-0.494324\pi$
$661$	0.916828			0.0356605			0.0178302	+	0.999841i	$0.494324\pi$
$662$		0			0
$663$			8.78623i			0.341229i
$664$		0			0
$665$	10.4569			0.405502
$666$		0			0
$667$	−35.8810			−1.38932
$668$		0			0
$669$	−20.2113			−0.781412
$670$		0			0
$671$	−0.141965	+	3.21377i	−0.00548049	+	0.124066i
$672$		0			0
$673$			32.0428i			1.23516i	0.786508	+	0.617580i	$0.211887\pi$
$673$			32.0428i			1.23516i	−0.786508	+	0.617580i	$0.788113\pi$
$674$		0			0
$675$			6.74338i			0.259553i
$676$		0			0
$677$		−	16.8752i		−	0.648568i	−0.945960	−	0.324284i	$-0.894877\pi$
$677$		−	16.8752i		−	0.648568i	0.945960	−	0.324284i	$-0.105123\pi$
$678$		0			0
$679$	9.97643			0.382860
$680$		0			0
$681$			23.0274i			0.882414i
$682$		0			0
$683$		−	8.29273i		−	0.317313i	−0.987334	−	0.158656i	$-0.949284\pi$
$683$		−	8.29273i		−	0.317313i	0.987334	−	0.158656i	$-0.0507162\pi$
$684$		0			0
$685$	−22.8396			−0.872655
$686$		0			0
$687$		−	26.9301i		−	1.02745i
$688$		0			0
$689$		−	8.89927i		−	0.339035i
$690$		0			0
$691$			14.0435i			0.534238i	0.963663	+	0.267119i	$0.0860718\pi$
$691$			14.0435i			0.534238i	−0.963663	+	0.267119i	$0.913928\pi$
$692$		0			0
$693$	−6.42754	−	0.283929i	−0.244162	−	0.0107856i
$694$		0			0
$695$	−14.5214			−0.550828
$696$		0			0
$697$	−42.4569			−1.60817
$698$		0			0
$699$	−9.08991			−0.343812
$700$		0			0
$701$		−	4.80302i		−	0.181408i	−0.995878	−	0.0907039i	$-0.971088\pi$
$701$		−	4.80302i		−	0.181408i	0.995878	−	0.0907039i	$-0.0289117\pi$
$702$		0			0
$703$	6.45009			0.243270
$704$		0			0
$705$	18.9357			0.713161
$706$		0			0
$707$			16.3847i			0.616210i
$708$		0			0
$709$	1.20329			0.0451904			0.0225952	−	0.999745i	$-0.492807\pi$
$709$	1.20329			0.0451904			0.0225952	+	0.999745i	$0.492807\pi$
$710$		0			0
$711$	13.5375			0.507696
$712$		0			0
$713$	3.45317			0.129322
$714$		0			0
$715$	0.439846	−	9.95715i	0.0164493	−	0.372376i
$716$		0			0
$717$		−	33.4665i		−	1.24983i
$718$		0			0
$719$			28.1354i			1.04927i	0.851326	+	0.524637i	$0.175799\pi$
$719$			28.1354i			1.04927i	−0.851326	+	0.524637i	$0.824201\pi$
$720$		0			0
$721$			19.9290i			0.742196i
$722$		0			0
$723$	23.2647			0.865225
$724$		0			0
$725$			11.8091i			0.438578i
$726$		0			0
$727$			14.7637i			0.547557i	0.961793	+	0.273778i	$0.0882734\pi$
$727$			14.7637i			0.547557i	−0.961793	+	0.273778i	$0.911727\pi$
$728$		0			0
$729$	−27.4653			−1.01724
$730$		0			0
$731$			35.9143i			1.32834i
$732$		0			0
$733$		−	46.2248i		−	1.70735i	−0.520806	−	0.853675i	$-0.674368\pi$
$733$		−	46.2248i		−	1.70735i	0.520806	−	0.853675i	$-0.325632\pi$
$734$		0			0
$735$		−	14.6142i		−	0.539054i
$736$		0			0
$737$	−1.26083	+	28.5423i	−0.0464431	+	1.05137i
$738$		0			0
$739$	−15.0515			−0.553677			−0.276839	−	0.960916i	$-0.589287\pi$
$739$	−15.0515			−0.553677			−0.276839	+	0.960916i	$0.589287\pi$
$740$		0			0
$741$	−4.20077			−0.154319
$742$		0			0
$743$	41.4167			1.51943			0.759715	−	0.650256i	$-0.225338\pi$
$743$	41.4167			1.51943			0.759715	+	0.650256i	$0.225338\pi$
$744$		0			0
$745$		−	3.00512i		−	0.110099i
$746$		0			0
$747$	4.04960			0.148167
$748$		0			0
$749$	−27.7992			−1.01576
$750$		0			0
$751$		−	44.3080i		−	1.61682i	−0.588618	−	0.808412i	$-0.700328\pi$
$751$		−	44.3080i		−	1.61682i	0.588618	−	0.808412i	$-0.299672\pi$
$752$		0			0
$753$	8.03863			0.292944
$754$		0			0
$755$	−53.0838			−1.93192
$756$		0			0
$757$	−23.3717			−0.849459			−0.424729	−	0.905320i	$-0.639631\pi$
$757$	−23.3717			−0.849459			−0.424729	+	0.905320i	$0.639631\pi$
$758$		0			0
$759$	16.1773	+	0.714615i	0.587200	+	0.0259389i
$760$		0			0
$761$		−	36.4717i		−	1.32210i	−0.750343	−	0.661049i	$-0.770112\pi$
$761$		−	36.4717i		−	1.32210i	0.750343	−	0.661049i	$-0.229888\pi$
$762$		0			0
$763$		−	19.9143i		−	0.720946i
$764$		0			0
$765$		−	16.1426i		−	0.583637i
$766$		0			0
$767$	14.5214			0.524337
$768$		0			0
$769$			0.590680i			0.0213005i	0.999943	+	0.0106502i	$0.00339014\pi$
$769$			0.590680i			0.0213005i	−0.999943	+	0.0106502i	$0.996610\pi$
$770$		0			0
$771$		−	16.8438i		−	0.606614i
$772$		0			0
$773$	7.57246			0.272362			0.136181	−	0.990684i	$-0.456517\pi$
$773$	7.57246			0.272362			0.136181	+	0.990684i	$0.456517\pi$
$774$		0			0
$775$		−	1.13650i		−	0.0408243i
$776$		0			0
$777$			5.41957i			0.194426i
$778$		0			0
$779$		−	20.2990i		−	0.727287i
$780$		0			0
$781$	−2.09617	+	47.4528i	−0.0750070	+	1.69799i
$782$		0			0
$783$	−55.6194			−1.98767
$784$		0			0
$785$	14.6963			0.524534
$786$		0			0
$787$	44.4476			1.58439			0.792194	−	0.610270i	$-0.208939\pi$
$787$	44.4476			1.58439			0.792194	+	0.610270i	$0.208939\pi$
$788$		0			0
$789$			22.5808i			0.803897i
$790$		0			0
$791$	1.26781			0.0450783
$792$		0			0
$793$	1.17092			0.0415807
$794$		0			0
$795$		−	24.6430i		−	0.873997i
$796$		0			0
$797$	−18.4977			−0.655223			−0.327611	−	0.944813i	$-0.606244\pi$
$797$	−18.4977			−0.655223			−0.327611	+	0.944813i	$0.606244\pi$
$798$		0			0
$799$	−30.6987			−1.08604
$800$		0			0
$801$	−17.4061			−0.615015
$802$		0			0
$803$	1.14663	−	25.9572i	0.0404636	−	0.916008i
$804$		0			0
$805$			14.6722i			0.517128i
$806$		0			0
$807$		−	20.6430i		−	0.726668i
$808$		0			0
$809$		−	15.2680i		−	0.536794i	−0.963308	−	0.268397i	$-0.913506\pi$
$809$		−	15.2680i		−	0.536794i	0.963308	−	0.268397i	$-0.0864939\pi$
$810$		0			0
$811$	−21.8202			−0.766212			−0.383106	−	0.923704i	$-0.625146\pi$
$811$	−21.8202			−0.766212			−0.383106	+	0.923704i	$0.625146\pi$
$812$		0			0
$813$			0.427936i			0.0150084i
$814$		0			0
$815$			45.9290i			1.60882i
$816$		0			0
$817$	−17.1709			−0.600735
$818$		0			0
$819$			2.34185i			0.0818308i
$820$		0			0
$821$		−	9.39462i		−	0.327875i	−0.986471	−	0.163937i	$-0.947581\pi$
$821$		−	9.39462i		−	0.327875i	0.986471	−	0.163937i	$-0.0524195\pi$
$822$		0			0
$823$		−	33.8511i		−	1.17998i	−0.807412	−	0.589988i	$-0.799133\pi$
$823$		−	33.8511i		−	1.17998i	0.807412	−	0.589988i	$-0.200867\pi$
$824$		0			0
$825$	0.235192	−	5.32424i	0.00818835	−	0.185366i
$826$		0			0
$827$	2.94453			0.102391			0.0511957	−	0.998689i	$-0.483697\pi$
$827$	2.94453			0.102391			0.0511957	+	0.998689i	$0.483697\pi$
$828$		0			0
$829$	−3.55356			−0.123420			−0.0617101	−	0.998094i	$-0.519655\pi$
$829$	−3.55356			−0.123420			−0.0617101	+	0.998094i	$0.519655\pi$
$830$		0			0
$831$	−16.4960			−0.572239
$832$		0			0
$833$			23.6927i			0.820902i
$834$		0			0
$835$	24.1274			0.834964
$836$		0			0
$837$	5.35279			0.185019
$838$		0			0
$839$		−	41.4496i		−	1.43100i	−0.698614	−	0.715499i	$-0.746199\pi$
$839$		−	41.4496i		−	1.43100i	0.698614	−	0.715499i	$-0.253801\pi$
$840$		0			0
$841$	−68.4011			−2.35866
$842$		0			0
$843$	17.7985			0.613014
$844$		0			0
$845$	28.7329			0.988442
$846$		0			0
$847$	17.7638	+	1.57246i	0.610371	+	0.0540304i
$848$		0			0
$849$		−	19.7384i		−	0.677420i
$850$		0			0
$851$			9.05019i			0.310237i
$852$		0			0
$853$			17.5871i			0.602171i	0.953597	+	0.301085i	$0.0973490\pi$
$853$			17.5871i			0.602171i	−0.953597	+	0.301085i	$0.902651\pi$
$854$		0			0
$855$	7.71791			0.263947
$856$		0			0
$857$		−	19.2638i		−	0.658040i	−0.944323	−	0.329020i	$-0.893282\pi$
$857$		−	19.2638i		−	0.658040i	0.944323	−	0.329020i	$-0.106718\pi$
$858$		0			0
$859$		−	15.5584i		−	0.530845i	−0.964132	−	0.265423i	$-0.914489\pi$
$859$		−	15.5584i		−	0.530845i	0.964132	−	0.265423i	$-0.0855115\pi$
$860$		0			0
$861$	17.0558			0.581262
$862$		0			0
$863$			15.8223i			0.538599i	0.963056	+	0.269299i	$0.0867921\pi$
$863$			15.8223i			0.538599i	−0.963056	+	0.269299i	$0.913208\pi$
$864$		0			0
$865$		−	46.7201i		−	1.58853i
$866$		0			0
$867$		−	16.6142i		−	0.564249i
$868$		0			0
$869$	−37.4868	−	1.65594i	−1.27165	−	0.0561738i
$870$		0			0
$871$	10.3993			0.352366
$872$		0			0
$873$	7.36327			0.249209
$874$		0			0
$875$	−15.3494			−0.518903
$876$		0			0
$877$			16.4007i			0.553811i	0.960897	+	0.276905i	$0.0893089\pi$
$877$			16.4007i			0.553811i	−0.960897	+	0.276905i	$0.910691\pi$
$878$		0			0
$879$	−28.1284			−0.948746
$880$		0			0
$881$	7.49602			0.252547			0.126274	−	0.991995i	$-0.459698\pi$
$881$	7.49602			0.252547			0.126274	+	0.991995i	$0.459698\pi$
$882$		0			0
$883$			10.8353i			0.364638i	0.983239	+	0.182319i	$0.0583604\pi$
$883$			10.8353i			0.364638i	−0.983239	+	0.182319i	$0.941640\pi$
$884$		0			0
$885$	40.2113			1.35169
$886$		0			0
$887$	−12.6163			−0.423612			−0.211806	−	0.977312i	$-0.567935\pi$
$887$	−12.6163			−0.423612			−0.211806	+	0.977312i	$0.567935\pi$
$888$		0			0
$889$	29.3717			0.985095
$890$		0			0
$891$	13.1826	+	0.582326i	0.441633	+	0.0195087i
$892$		0			0
$893$		−	14.6773i		−	0.491157i
$894$		0			0
$895$		−	43.3148i		−	1.44785i
$896$		0			0
$897$		−	5.89414i		−	0.196800i
$898$		0			0
$899$	9.37384			0.312635
$900$		0			0
$901$			39.9514i			1.33097i
$902$		0			0
$903$		−	14.4275i		−	0.480118i
$904$		0			0
$905$	39.4826			1.31244
$906$		0			0
$907$		−	8.29273i		−	0.275356i	−0.990477	−	0.137678i	$-0.956036\pi$
$907$		−	8.29273i		−	0.275356i	0.990477	−	0.137678i	$-0.0439639\pi$
$908$		0			0
$909$			12.0930i			0.401099i
$910$		0			0
$911$			14.2499i			0.472120i	0.971739	+	0.236060i	$0.0758562\pi$
$911$			14.2499i			0.472120i	−0.971739	+	0.236060i	$0.924144\pi$
$912$		0			0
$913$	−11.2138	−	0.495356i	−0.371122	−	0.0163939i
$914$		0			0
$915$	3.24241			0.107191
$916$		0			0
$917$	4.20077			0.138722
$918$		0			0
$919$	−6.76875			−0.223281			−0.111640	−	0.993749i	$-0.535610\pi$
$919$	−6.76875			−0.223281			−0.111640	+	0.993749i	$0.535610\pi$
$920$		0			0
$921$		−	0.874609i		−	0.0288193i
$922$		0			0
$923$	17.2892			0.569082
$924$		0			0
$925$	2.97858			0.0979350
$926$		0			0
$927$			14.7090i			0.483106i
$928$		0			0
$929$	42.1151			1.38175			0.690875	−	0.722974i	$-0.257225\pi$
$929$	42.1151			1.38175			0.690875	+	0.722974i	$0.257225\pi$
$930$		0			0
$931$	−11.3277			−0.371249
$932$		0			0
$933$	−33.5640			−1.09884
$934$		0			0
$935$	−1.97460	+	44.7005i	−0.0645762	+	1.46186i
$936$		0			0
$937$		−	56.0194i		−	1.83007i	−0.403369	−	0.915037i	$-0.632161\pi$
$937$		−	56.0194i		−	1.83007i	0.403369	−	0.915037i	$-0.367839\pi$
$938$		0			0
$939$			10.8923i			0.355456i
$940$		0			0
$941$			8.21326i			0.267745i	0.990999	+	0.133872i	$0.0427412\pi$
$941$			8.21326i			0.267745i	−0.990999	+	0.133872i	$0.957259\pi$
$942$		0			0
$943$	28.4817			0.927493
$944$		0			0
$945$			22.7435i			0.739847i
$946$		0			0
$947$		−	3.91369i		−	0.127178i	−0.997976	−	0.0635890i	$-0.979745\pi$
$947$		−	3.91369i		−	0.127178i	0.997976	−	0.0635890i	$-0.0202547\pi$
$948$		0			0
$949$	−9.45738			−0.307000
$950$		0			0
$951$			16.7146i			0.542009i
$952$		0			0
$953$			40.3514i			1.30711i	0.756879	+	0.653555i	$0.226723\pi$
$953$			40.3514i			1.30711i	−0.756879	+	0.653555i	$0.773277\pi$
$954$		0			0
$955$			37.4643i			1.21232i
$956$		0			0
$957$	43.9143	+	1.93987i	1.41955	+	0.0627070i
$958$		0			0
$959$	−14.8748			−0.480332
$960$		0			0
$961$	30.0979			0.970899
$962$		0			0
$963$	−20.5177			−0.661173
$964$		0			0
$965$			55.1448i			1.77517i
$966$		0			0
$967$	−1.93987			−0.0623819			−0.0311909	−	0.999513i	$-0.509930\pi$
$967$	−1.93987			−0.0623819			−0.0311909	+	0.999513i	$0.509930\pi$
$968$		0			0
$969$	18.8585			0.605821
$970$		0			0
$971$			38.7146i			1.24241i	0.783648	+	0.621206i	$0.213357\pi$
$971$			38.7146i			1.24241i	−0.783648	+	0.621206i	$0.786643\pi$
$972$		0			0
$973$	−9.45738			−0.303190
$974$		0			0
$975$	−1.93987			−0.0621254
$976$		0			0
$977$	−30.9315			−0.989587			−0.494794	−	0.869011i	$-0.664756\pi$
$977$	−30.9315			−0.989587			−0.494794	+	0.869011i	$0.664756\pi$
$978$		0			0
$979$	48.1994	+	2.12915i	1.54046	+	0.0680480i
$980$		0			0
$981$		−	14.6981i		−	0.469274i
$982$		0			0
$983$			39.5500i			1.26145i	0.776007	+	0.630724i	$0.217242\pi$
$983$			39.5500i			1.26145i	−0.776007	+	0.630724i	$0.782758\pi$
$984$		0			0
$985$		−	8.42469i		−	0.268433i
$986$		0			0
$987$	12.3323			0.392542
$988$		0			0
$989$		−	24.0927i		−	0.766103i
$990$		0			0
$991$		−	7.23688i		−	0.229887i	−0.993372	−	0.114944i	$-0.963331\pi$
$991$		−	7.23688i		−	0.229887i	0.993372	−	0.114944i	$-0.0366687\pi$
$992$		0			0
$993$	−9.01721			−0.286153
$994$		0			0
$995$			12.6430i			0.400810i
$996$		0			0
$997$		−	3.81231i		−	0.120737i	−0.998176	−	0.0603686i	$-0.980772\pi$
$997$		−	3.81231i		−	0.120737i	0.998176	−	0.0603686i	$-0.0192276\pi$
$998$		0			0
$999$			14.0288i			0.443851i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.e.a.351.10	yes	12
3.2	odd	2		3168.2.o.e.703.4		12
4.3	odd	2	inner	352.2.e.a.351.3	✓	12
8.3	odd	2		704.2.e.d.703.9		12
8.5	even	2		704.2.e.d.703.4		12
11.10	odd	2	inner	352.2.e.a.351.9	yes	12
12.11	even	2		3168.2.o.e.703.1		12
16.3	odd	4		2816.2.g.d.1407.10		12
16.5	even	4		2816.2.g.d.1407.11		12
16.11	odd	4		2816.2.g.i.1407.4		12
16.13	even	4		2816.2.g.i.1407.1		12
33.32	even	2		3168.2.o.e.703.2		12
44.43	even	2	inner	352.2.e.a.351.4	yes	12
88.21	odd	2		704.2.e.d.703.3		12
88.43	even	2		704.2.e.d.703.10		12
132.131	odd	2		3168.2.o.e.703.3		12
176.21	odd	4		2816.2.g.d.1407.12		12
176.43	even	4		2816.2.g.i.1407.3		12
176.109	odd	4		2816.2.g.i.1407.2		12
176.131	even	4		2816.2.g.d.1407.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.e.a.351.3	✓	12	4.3	odd	2	inner
352.2.e.a.351.4	yes	12	44.43	even	2	inner
352.2.e.a.351.9	yes	12	11.10	odd	2	inner
352.2.e.a.351.10	yes	12	1.1	even	1	trivial
704.2.e.d.703.3		12	88.21	odd	2
704.2.e.d.703.4		12	8.5	even	2
704.2.e.d.703.9		12	8.3	odd	2
704.2.e.d.703.10		12	88.43	even	2
2816.2.g.d.1407.9		12	176.131	even	4
2816.2.g.d.1407.10		12	16.3	odd	4
2816.2.g.d.1407.11		12	16.5	even	4
2816.2.g.d.1407.12		12	176.21	odd	4
2816.2.g.i.1407.1		12	16.13	even	4
2816.2.g.i.1407.2		12	176.109	odd	4
2816.2.g.i.1407.3		12	176.43	even	4
2816.2.g.i.1407.4		12	16.11	odd	4
3168.2.o.e.703.1		12	12.11	even	2
3168.2.o.e.703.2		12	33.32	even	2
3168.2.o.e.703.3		12	132.131	odd	2
3168.2.o.e.703.4		12	3.2	odd	2

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:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.34292i q^{3} +2.48929 q^{5} +1.62121 q^{7} +1.19656 q^{9} +(-3.31339 - 0.146365i) q^{11} +1.20722i q^{13} +3.34292i q^{15} -5.41957i q^{17} +2.59114 q^{19} +2.17715i q^{21} +3.63565i q^{23} +1.19656 q^{25} +5.63565i q^{27} +9.86920i q^{29} -0.949808i q^{31} +(0.196558 - 4.44963i) q^{33} +4.03565 q^{35} +2.48929 q^{37} -1.62121 q^{39} -7.83401i q^{41} -6.62679 q^{43} +2.97858 q^{45} -5.66442i q^{47} -4.37169 q^{49} +7.27806 q^{51} -7.37169 q^{53} +(-8.24799 - 0.364346i) q^{55} +3.47970i q^{57} -12.0288i q^{59} -0.969933i q^{61} +1.93987 q^{63} +3.00512i q^{65} -8.61423i q^{67} -4.88240 q^{69} -14.3215i q^{71} +7.83401i q^{73} +1.60688i q^{75} +(-5.37169 - 0.237289i) q^{77} +11.3137 q^{79} -3.97858 q^{81} +3.38438 q^{83} -13.4909i q^{85} -13.2536 q^{87} -14.5468 q^{89} +1.95715i q^{91} +1.27552 q^{93} +6.45009 q^{95} +6.15371 q^{97} +(-3.96467 - 0.175135i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9} - 4 q^{25} - 16 q^{33} - 24 q^{45} + 44 q^{49} + 8 q^{53} + 8 q^{69} + 32 q^{77} + 12 q^{81} - 88 q^{93} - 64 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 4 * q^25 - 16 * q^33 - 24 * q^45 + 44 * q^49 + 8 * q^53 + 8 * q^69 + 32 * q^77 + 12 * q^81 - 88 * q^93 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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