LMFDB - Embedded newform 3168.2.o.e.703.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(703,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3168.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.o (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$25.2966073603$
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:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.4
Root			$1.35489 - 0.405301i$ of defining polynomial
Character	$\chi$	$=$	3168.703
Dual form			3168.2.o.e.703.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48929 q^{5} +1.62121 q^{7} +O(q^{10})$	$q-2.48929 q^{5} +1.62121 q^{7} +(3.31339 + 0.146365i) q^{11} +1.20722i q^{13} +5.41957i q^{17} +2.59114 q^{19} -3.63565i q^{23} +1.19656 q^{25} -9.86920i q^{29} -0.949808i q^{31} -4.03565 q^{35} +2.48929 q^{37} +7.83401i q^{41} -6.62679 q^{43} +5.66442i q^{47} -4.37169 q^{49} +7.37169 q^{53} +(-8.24799 - 0.364346i) q^{55} +12.0288i q^{59} -0.969933i q^{61} -3.00512i q^{65} -8.61423i q^{67} +14.3215i q^{71} +7.83401i q^{73} +(5.37169 + 0.237289i) q^{77} +11.3137 q^{79} -3.38438 q^{83} -13.4909i q^{85} +14.5468 q^{89} +1.95715i q^{91} -6.45009 q^{95} +6.15371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4 q^{25} + 44 q^{49} - 8 q^{53} - 32 q^{77} - 64 q^{97}+O(q^{100}) $ 12 * q - 4 * q^25 + 44 * q^49 - 8 * q^53 - 32 * q^77 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.48929			−1.11324			−0.556622	−	0.830766i	$-0.687903\pi$
$5$	−2.48929			−1.11324			−0.556622	+	0.830766i	$0.687903\pi$
$6$		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
$9$		0			0
$10$		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
$15$		0			0
$16$		0			0
$17$			5.41957i			1.31444i	0.753700	+	0.657219i	$0.228267\pi$
$17$			5.41957i			1.31444i	−0.753700	+	0.657219i	$0.771733\pi$
$18$		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
$21$		0			0
$22$		0			0
$23$		−	3.63565i		−	0.758086i	−0.925379	−	0.379043i	$-0.876253\pi$
$23$		−	3.63565i		−	0.758086i	0.925379	−	0.379043i	$-0.123747\pi$
$24$		0			0
$25$	1.19656			0.239312
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	9.86920i		−	1.83266i	−0.400419	−	0.916332i	$-0.631135\pi$
$29$		−	9.86920i		−	1.83266i	0.400419	−	0.916332i	$-0.368865\pi$
$30$		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			0
$34$		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
$39$		0			0
$40$		0			0
$41$			7.83401i			1.22347i	0.791064	+	0.611733i	$0.209527\pi$
$41$			7.83401i			1.22347i	−0.791064	+	0.611733i	$0.790473\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$			5.66442i			0.826241i	0.910676	+	0.413121i	$0.135561\pi$
$47$			5.66442i			0.826241i	−0.910676	+	0.413121i	$0.864439\pi$
$48$		0			0
$49$	−4.37169			−0.624527
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.37169			1.01258			0.506290	−	0.862363i	$-0.331017\pi$
$53$	7.37169			1.01258			0.506290	+	0.862363i	$0.331017\pi$
$54$		0			0
$55$	−8.24799	−	0.364346i	−1.11216	−	0.0491284i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.0288i			1.56601i	0.622014	+	0.783006i	$0.286315\pi$
$59$			12.0288i			1.56601i	−0.622014	+	0.783006i	$0.713685\pi$
$60$		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
$63$		0			0
$64$		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
$69$		0			0
$70$		0			0
$71$			14.3215i			1.69965i	0.527065	+	0.849825i	$0.323292\pi$
$71$			14.3215i			1.69965i	−0.527065	+	0.849825i	$0.676708\pi$
$72$		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
$75$		0			0
$76$		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
$81$		0			0
$82$		0			0
$83$	−3.38438			−0.371484			−0.185742	−	0.982599i	$-0.559469\pi$
$83$	−3.38438			−0.371484			−0.185742	+	0.982599i	$0.559469\pi$
$84$		0			0
$85$		−	13.4909i		−	1.46329i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.5468			1.54196			0.770980	−	0.636859i	$-0.219767\pi$
$89$	14.5468			1.54196			0.770980	+	0.636859i	$0.219767\pi$
$90$		0			0
$91$			1.95715i			0.205166i
$92$		0			0
$93$		0			0
$94$		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
$99$		0			0
$100$		0			0
$101$		−	10.1065i		−	1.00563i	−0.864393	−	0.502817i	$-0.832297\pi$
$101$		−	10.1065i		−	1.00563i	0.864393	−	0.502817i	$-0.167703\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$	17.1473			1.65769			0.828844	−	0.559480i	$-0.188999\pi$
$107$	17.1473			1.65769			0.828844	+	0.559480i	$0.188999\pi$
$108$		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
$111$		0			0
$112$		0			0
$113$	−0.782020			−0.0735662			−0.0367831	−	0.999323i	$-0.511711\pi$
$113$	−0.782020			−0.0735662			−0.0367831	+	0.999323i	$0.511711\pi$
$114$		0			0
$115$			9.05019i			0.843935i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			8.78623i			0.805432i
$120$		0			0
$121$	10.9572	+	0.969933i	0.996105	+	0.0881757i
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		0			0
$131$	−2.59114			−0.226389			−0.113194	−	0.993573i	$-0.536108\pi$
$131$	−2.59114			−0.226389			−0.113194	+	0.993573i	$0.536108\pi$
$132$		0			0
$133$	4.20077			0.364253
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.17513			0.783885			0.391942	−	0.919990i	$-0.371803\pi$
$137$	9.17513			0.783885			0.391942	+	0.919990i	$0.371803\pi$
$138$		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
$141$		0			0
$142$		0			0
$143$	−0.176696	+	4.00000i	−0.0147760	+	0.334497i
$144$		0			0
$145$			24.5673i			2.04020i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			1.20722i			0.0988994i	0.998777	+	0.0494497i	$0.0157468\pi$
$149$			1.20722i			0.0988994i	−0.998777	+	0.0494497i	$0.984253\pi$
$150$		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
$153$		0			0
$154$		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
$159$		0			0
$160$		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			0
$166$		0			0
$167$	−9.69250			−0.750028			−0.375014	−	0.927019i	$-0.622362\pi$
$167$	−9.69250			−0.750028			−0.375014	+	0.927019i	$0.622362\pi$
$168$		0			0
$169$	11.5426			0.887894
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.7685i			1.42694i	0.700686	+	0.713470i	$0.252877\pi$
$173$			18.7685i			1.42694i	−0.700686	+	0.713470i	$0.747123\pi$
$174$		0			0
$175$	1.93987			0.146640
$176$		0			0
$177$		0			0
$178$		0			0
$179$			17.4005i			1.30057i	0.759690	+	0.650286i	$0.225351\pi$
$179$			17.4005i			1.30057i	−0.759690	+	0.650286i	$0.774649\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	−6.19656			−0.455580
$186$		0			0
$187$	−0.793237	+	17.9572i	−0.0580073	+	1.31316i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	15.0502i		−	1.08899i	−0.838763	−	0.544497i	$-0.816721\pi$
$191$		−	15.0502i		−	1.08899i	0.838763	−	0.544497i	$-0.183279\pi$
$192$		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
$195$		0			0
$196$		0			0
$197$			3.38438i			0.241127i	0.992706	+	0.120563i	$0.0384701\pi$
$197$			3.38438i			0.241127i	−0.992706	+	0.120563i	$0.961530\pi$
$198$		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
$201$		0			0
$202$		0			0
$203$		−	16.0000i		−	1.12298i
$204$		0			0
$205$		−	19.5011i		−	1.36202i
$206$		0			0
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	16.4960			1.12502
$216$		0			0
$217$		−	1.53983i		−	0.104531i
$218$		0			0
$219$		0			0
$220$		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
$225$		0			0
$226$		0			0
$227$	−17.1473			−1.13810			−0.569052	−	0.822302i	$-0.692690\pi$
$227$	−17.1473			−1.13810			−0.569052	+	0.822302i	$0.692690\pi$
$228$		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			0
$232$		0			0
$233$		−	6.76875i		−	0.443436i	−0.975111	−	0.221718i	$-0.928834\pi$
$233$		−	6.76875i		−	0.443436i	0.975111	−	0.221718i	$-0.0711664\pi$
$234$		0			0
$235$		−	14.1004i		−	0.919808i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.9207			1.61198			0.805992	−	0.591926i	$-0.201632\pi$
$239$	24.9207			1.61198			0.805992	+	0.591926i	$0.201632\pi$
$240$		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
$243$		0			0
$244$		0			0
$245$	10.8824			0.695251
$246$		0			0
$247$			3.12808i			0.199035i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5.98592i			0.377828i	0.981994	+	0.188914i	$0.0604968\pi$
$251$			5.98592i			0.377828i	−0.981994	+	0.188914i	$0.939503\pi$
$252$		0			0
$253$	0.532134	−	12.0464i	0.0334550	−	0.757348i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.5426			0.782387			0.391193	−	0.920308i	$-0.372062\pi$
$257$	12.5426			0.782387			0.391193	+	0.920308i	$0.372062\pi$
$258$		0			0
$259$	4.03565			0.250763
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.8146			−1.03684			−0.518418	−	0.855128i	$-0.673479\pi$
$263$	−16.8146			−1.03684			−0.518418	+	0.855128i	$0.673479\pi$
$264$		0			0
$265$	−18.3503			−1.12725
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.3717			0.937229			0.468614	−	0.883403i	$-0.344753\pi$
$269$	15.3717			0.937229			0.468614	+	0.883403i	$0.344753\pi$
$270$		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
$273$		0			0
$274$		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
$279$		0			0
$280$		0			0
$281$			13.2536i			0.790642i	0.918543	+	0.395321i	$0.129367\pi$
$281$			13.2536i			0.790642i	−0.918543	+	0.395321i	$0.870633\pi$
$282$		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
$285$		0			0
$286$		0			0
$287$			12.7005i			0.749689i
$288$		0			0
$289$	−12.3717			−0.727747
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	20.9456i		−	1.22366i	−0.790991	−	0.611828i	$-0.790434\pi$
$293$		−	20.9456i		−	1.22366i	0.790991	−	0.611828i	$-0.209566\pi$
$294$		0			0
$295$		−	29.9431i		−	1.74335i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.38904			0.253825
$300$		0			0
$301$	−10.7434			−0.619238
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		0			0
$311$		−	24.9933i		−	1.41724i	−0.705591	−	0.708619i	$-0.749318\pi$
$311$		−	24.9933i		−	1.41724i	0.705591	−	0.708619i	$-0.250682\pi$
$312$		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
$315$		0			0
$316$		0			0
$317$	−12.4464			−0.699062			−0.349531	−	0.936925i	$-0.613659\pi$
$317$	−12.4464			−0.699062			−0.349531	+	0.936925i	$0.613659\pi$
$318$		0			0
$319$	1.44451	−	32.7005i	0.0808770	−	1.83088i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			14.0428i			0.781365i
$324$		0			0
$325$			1.44451i			0.0801270i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		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
$339$		0			0
$340$		0			0
$341$	0.139019	−	3.14709i	0.00752831	−	0.170424i
$342$		0			0
$343$	−18.4359			−0.995442
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.3184			0.661285			0.330642	−	0.943756i	$-0.392735\pi$
$347$	12.3184			0.661285			0.330642	+	0.943756i	$0.392735\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$	−22.1537			−1.17912			−0.589562	−	0.807723i	$-0.700699\pi$
$353$	−22.1537			−1.17912			−0.589562	+	0.807723i	$0.700699\pi$
$354$		0			0
$355$		−	35.6503i		−	1.89212i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−31.3708			−1.65569			−0.827843	−	0.560960i	$-0.810432\pi$
$359$	−31.3708			−1.65569			−0.827843	+	0.560960i	$0.810432\pi$
$360$		0			0
$361$	−12.2860			−0.646632
$362$		0			0
$363$		0			0
$364$		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
$369$		0			0
$370$		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
$375$		0			0
$376$		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
$381$		0			0
$382$		0			0
$383$		−	11.7507i		−	0.600434i	−0.953871	−	0.300217i	$-0.902941\pi$
$383$		−	11.7507i		−	0.600434i	0.953871	−	0.300217i	$-0.0970591\pi$
$384$		0			0
$385$	−13.3717	−	0.590680i	−0.681485	−	0.0301038i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.71148			0.0867755			0.0433877	−	0.999058i	$-0.486185\pi$
$389$	1.71148			0.0867755			0.0433877	+	0.999058i	$0.486185\pi$
$390$		0			0
$391$	19.7037			0.996457
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	−14.2008			−0.709153			−0.354576	−	0.935027i	$-0.615375\pi$
$401$	−14.2008			−0.709153			−0.354576	+	0.935027i	$0.615375\pi$
$402$		0			0
$403$	1.14663			0.0571176
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$			19.5011i			0.959587i
$414$		0			0
$415$	8.42469			0.413552
$416$		0			0
$417$		0			0
$418$		0			0
$419$			7.12181i			0.347923i	0.984752	+	0.173962i	$0.0556568\pi$
$419$			7.12181i			0.347923i	−0.984752	+	0.173962i	$0.944343\pi$
$420$		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
$423$		0			0
$424$		0			0
$425$			6.48482i			0.314560i
$426$		0			0
$427$		−	1.57246i		−	0.0760967i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.9461			1.10527			0.552637	−	0.833422i	$-0.313622\pi$
$431$	22.9461			1.10527			0.552637	+	0.833422i	$0.313622\pi$
$432$		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
$435$		0			0
$436$		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
$441$		0			0
$442$		0			0
$443$		−	15.5008i		−	0.736467i	−0.929733	−	0.368234i	$-0.879963\pi$
$443$		−	15.5008i		−	0.736467i	0.929733	−	0.368234i	$-0.120037\pi$
$444$		0			0
$445$	−36.2113			−1.71658
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.9614			0.847649			0.423825	−	0.905744i	$-0.360687\pi$
$449$	17.9614			0.847649			0.423825	+	0.905744i	$0.360687\pi$
$450$		0			0
$451$	−1.14663	+	25.9572i	−0.0539926	+	1.22227i
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$		−	19.0524i		−	0.887358i	−0.896186	−	0.443679i	$-0.853673\pi$
$461$		−	19.0524i		−	0.887358i	0.896186	−	0.443679i	$-0.146327\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$		−	36.0863i		−	1.66988i	−0.550345	−	0.834938i	$-0.685504\pi$
$467$		−	36.0863i		−	1.66988i	0.550345	−	0.834938i	$-0.314496\pi$
$468$		0			0
$469$		−	13.9654i		−	0.644864i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−21.9572	−	0.969933i	−1.00959	−	0.0445976i
$474$		0			0
$475$	3.10045			0.142258
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.9950			0.502377			0.251188	−	0.967938i	$-0.419179\pi$
$479$	10.9950			0.502377			0.251188	+	0.967938i	$0.419179\pi$
$480$		0			0
$481$			3.00512i			0.137022i
$482$		0			0
$483$		0			0
$484$		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
$489$		0			0
$490$		0			0
$491$	−18.2939			−0.825592			−0.412796	−	0.910824i	$-0.635448\pi$
$491$	−18.2939			−0.825592			−0.412796	+	0.910824i	$0.635448\pi$
$492$		0			0
$493$	53.4868			2.40892
$494$		0			0
$495$		0			0
$496$		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
$501$		0			0
$502$		0			0
$503$	−16.4960			−0.735520			−0.367760	−	0.929921i	$-0.619875\pi$
$503$	−16.4960			−0.735520			−0.367760	+	0.929921i	$0.619875\pi$
$504$		0			0
$505$			25.1580i			1.11951i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−28.4464			−1.26087			−0.630433	−	0.776244i	$-0.717123\pi$
$509$	−28.4464			−1.26087			−0.630433	+	0.776244i	$0.717123\pi$
$510$		0			0
$511$			12.7005i			0.561839i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	30.6002i		−	1.34840i
$516$		0			0
$517$	−0.829076	+	18.7685i	−0.0364627	+	0.825436i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−40.2259			−1.76233			−0.881165	−	0.472808i	$-0.843240\pi$
$521$	−40.2259			−1.76233			−0.881165	+	0.472808i	$0.843240\pi$
$522$		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
$525$		0			0
$526$		0			0
$527$	5.14755			0.224231
$528$		0			0
$529$	9.78202			0.425305
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.45738			−0.409645
$534$		0			0
$535$	−42.6845			−1.84541
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		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
$549$		0			0
$550$		0			0
$551$		−	25.5725i		−	1.08942i
$552$		0			0
$553$	18.3418			0.779975
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	3.33774i		−	0.141424i	−0.997497	−	0.0707122i	$-0.977473\pi$
$557$		−	3.33774i		−	0.141424i	0.997497	−	0.0707122i	$-0.0225272\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	32.7805			1.38154			0.690768	−	0.723077i	$-0.257273\pi$
$563$	32.7805			1.38154			0.690768	+	0.723077i	$0.257273\pi$
$564$		0			0
$565$	1.94667			0.0818971
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	14.2027i		−	0.595409i	−0.954658	−	0.297705i	$-0.903779\pi$
$569$		−	14.2027i		−	0.595409i	0.954658	−	0.297705i	$-0.0962211\pi$
$570$		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
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$	−5.48677			−0.227630
$582$		0			0
$583$	24.4253	+	1.07896i	1.01159	+	0.0446860i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18.2499i			0.753253i	0.926365	+	0.376627i	$0.122916\pi$
$587$			18.2499i			0.753253i	−0.926365	+	0.376627i	$0.877084\pi$
$588$		0			0
$589$		−	2.46108i		−	0.101407i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	38.2954i		−	1.57261i	−0.617842	−	0.786303i	$-0.711993\pi$
$593$		−	38.2954i		−	1.57261i	0.617842	−	0.786303i	$-0.288007\pi$
$594$		0			0
$595$		−	21.8715i		−	0.896643i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			30.8353i			1.25990i	0.776636	+	0.629949i	$0.216924\pi$
$599$			30.8353i			1.25990i	−0.776636	+	0.629949i	$0.783076\pi$
$600$		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
$603$		0			0
$604$		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
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$	8.62831			0.347363			0.173681	−	0.984802i	$-0.444434\pi$
$617$	8.62831			0.347363			0.173681	+	0.984802i	$0.444434\pi$
$618$		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
$621$		0			0
$622$		0			0
$623$	23.5834			0.944849
$624$		0			0
$625$	−29.5510			−1.18204
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	−45.0989			−1.78970
$636$		0			0
$637$		−	5.27760i		−	0.209106i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.38890			0.331342			0.165671	−	0.986181i	$-0.447021\pi$
$641$	8.38890			0.331342			0.165671	+	0.986181i	$0.447021\pi$
$642$		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
$645$		0			0
$646$		0			0
$647$		−	1.67850i		−	0.0659886i	−0.999456	−	0.0329943i	$-0.989496\pi$
$647$		−	1.67850i		−	0.0659886i	0.999456	−	0.0329943i	$-0.0105043\pi$
$648$		0			0
$649$	−1.76060	+	39.8560i	−0.0691095	+	1.56449i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.53213			−0.333888			−0.166944	−	0.985966i	$-0.553390\pi$
$653$	−8.53213			−0.333888			−0.166944	+	0.985966i	$0.553390\pi$
$654$		0			0
$655$	6.45009			0.252026
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.2939			0.712629			0.356314	−	0.934366i	$-0.384033\pi$
$659$	18.2939			0.712629			0.356314	+	0.934366i	$0.384033\pi$
$660$		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
$663$		0			0
$664$		0			0
$665$	−10.4569			−0.405502
$666$		0			0
$667$	−35.8810			−1.38932
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$			16.8752i			0.648568i	0.945960	+	0.324284i	$0.105123\pi$
$677$			16.8752i			0.648568i	−0.945960	+	0.324284i	$0.894877\pi$
$678$		0			0
$679$	9.97643			0.382860
$680$		0			0
$681$		0			0
$682$		0			0
$683$			8.29273i			0.317313i	0.987334	+	0.158656i	$0.0507162\pi$
$683$			8.29273i			0.317313i	−0.987334	+	0.158656i	$0.949284\pi$
$684$		0			0
$685$	−22.8396			−0.872655
$686$		0			0
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$	14.5214			0.550828
$696$		0			0
$697$	−42.4569			−1.60817
$698$		0			0
$699$		0			0
$700$		0			0
$701$			4.80302i			0.181408i	0.995878	+	0.0907039i	$0.0289117\pi$
$701$			4.80302i			0.181408i	−0.995878	+	0.0907039i	$0.971088\pi$
$702$		0			0
$703$	6.45009			0.243270
$704$		0			0
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$	−3.45317			−0.129322
$714$		0			0
$715$	0.439846	−	9.95715i	0.0164493	−	0.372376i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	28.1354i		−	1.04927i	−0.851326	−	0.524637i	$-0.824201\pi$
$719$		−	28.1354i		−	1.04927i	0.851326	−	0.524637i	$-0.175799\pi$
$720$		0			0
$721$			19.9290i			0.742196i
$722$		0			0
$723$		0			0
$724$		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
$729$		0			0
$730$		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
$735$		0			0
$736$		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
$741$		0			0
$742$		0			0
$743$	−41.4167			−1.51943			−0.759715	−	0.650256i	$-0.774662\pi$
$743$	−41.4167			−1.51943			−0.759715	+	0.650256i	$0.774662\pi$
$744$		0			0
$745$		−	3.00512i		−	0.110099i
$746$		0			0
$747$		0			0
$748$		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
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$			36.4717i			1.32210i	0.750343	+	0.661049i	$0.229888\pi$
$761$			36.4717i			1.32210i	−0.750343	+	0.661049i	$0.770112\pi$
$762$		0			0
$763$		−	19.9143i		−	0.720946i
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$	−7.57246			−0.272362			−0.136181	−	0.990684i	$-0.543483\pi$
$773$	−7.57246			−0.272362			−0.136181	+	0.990684i	$0.543483\pi$
$774$		0			0
$775$		−	1.13650i		−	0.0408243i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			20.2990i			0.727287i
$780$		0			0
$781$	−2.09617	+	47.4528i	−0.0750070	+	1.69799i
$782$		0			0
$783$		0			0
$784$		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
$789$		0			0
$790$		0			0
$791$	−1.26781			−0.0450783
$792$		0			0
$793$	1.17092			0.0415807
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.4977			0.655223			0.327611	−	0.944813i	$-0.393756\pi$
$797$	18.4977			0.655223			0.327611	+	0.944813i	$0.393756\pi$
$798$		0			0
$799$	−30.6987			−1.08604
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.14663	+	25.9572i	−0.0404636	+	0.916008i
$804$		0			0
$805$			14.6722i			0.517128i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			15.2680i			0.536794i	0.963308	+	0.268397i	$0.0864939\pi$
$809$			15.2680i			0.536794i	−0.963308	+	0.268397i	$0.913506\pi$
$810$		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			0
$814$		0			0
$815$		−	45.9290i		−	1.60882i
$816$		0			0
$817$	−17.1709			−0.600735
$818$		0			0
$819$		0			0
$820$		0			0
$821$			9.39462i			0.327875i	0.986471	+	0.163937i	$0.0524195\pi$
$821$			9.39462i			0.327875i	−0.986471	+	0.163937i	$0.947581\pi$
$822$		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			0
$826$		0			0
$827$	−2.94453			−0.102391			−0.0511957	−	0.998689i	$-0.516303\pi$
$827$	−2.94453			−0.102391			−0.0511957	+	0.998689i	$0.516303\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$		−	23.6927i		−	0.820902i
$834$		0			0
$835$	24.1274			0.834964
$836$		0			0
$837$		0			0
$838$		0			0
$839$			41.4496i			1.43100i	0.698614	+	0.715499i	$0.253801\pi$
$839$			41.4496i			1.43100i	−0.698614	+	0.715499i	$0.746199\pi$
$840$		0			0
$841$	−68.4011			−2.35866
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−28.7329			−0.988442
$846$		0			0
$847$	17.7638	+	1.57246i	0.610371	+	0.0540304i
$848$		0			0
$849$		0			0
$850$		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
$855$		0			0
$856$		0			0
$857$			19.2638i			0.658040i	0.944323	+	0.329020i	$0.106718\pi$
$857$			19.2638i			0.658040i	−0.944323	+	0.329020i	$0.893282\pi$
$858$		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
$861$		0			0
$862$		0			0
$863$		−	15.8223i		−	0.538599i	−0.963056	−	0.269299i	$-0.913208\pi$
$863$		−	15.8223i		−	0.538599i	0.963056	−	0.269299i	$-0.0867921\pi$
$864$		0			0
$865$		−	46.7201i		−	1.58853i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	37.4868	+	1.65594i	1.27165	+	0.0561738i
$870$		0			0
$871$	10.3993			0.352366
$872$		0			0
$873$		0			0
$874$		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
$879$		0			0
$880$		0			0
$881$	−7.49602			−0.252547			−0.126274	−	0.991995i	$-0.540302\pi$
$881$	−7.49602			−0.252547			−0.126274	+	0.991995i	$0.540302\pi$
$882$		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
$885$		0			0
$886$		0			0
$887$	12.6163			0.423612			0.211806	−	0.977312i	$-0.432065\pi$
$887$	12.6163			0.423612			0.211806	+	0.977312i	$0.432065\pi$
$888$		0			0
$889$	29.3717			0.985095
$890$		0			0
$891$		0			0
$892$		0			0
$893$			14.6773i			0.491157i
$894$		0			0
$895$		−	43.3148i		−	1.44785i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.37384			−0.312635
$900$		0			0
$901$			39.9514i			1.33097i
$902$		0			0
$903$		0			0
$904$		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
$909$		0			0
$910$		0			0
$911$		−	14.2499i		−	0.472120i	−0.971739	−	0.236060i	$-0.924144\pi$
$911$		−	14.2499i		−	0.472120i	0.971739	−	0.236060i	$-0.0758562\pi$
$912$		0			0
$913$	−11.2138	−	0.495356i	−0.371122	−	0.0163939i
$914$		0			0
$915$		0			0
$916$		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			0
$922$		0			0
$923$	−17.2892			−0.569082
$924$		0			0
$925$	2.97858			0.0979350
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−42.1151			−1.38175			−0.690875	−	0.722974i	$-0.742775\pi$
$929$	−42.1151			−1.38175			−0.690875	+	0.722974i	$0.742775\pi$
$930$		0			0
$931$	−11.3277			−0.371249
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$		−	8.21326i		−	0.267745i	−0.990999	−	0.133872i	$-0.957259\pi$
$941$		−	8.21326i		−	0.267745i	0.990999	−	0.133872i	$-0.0427412\pi$
$942$		0			0
$943$	28.4817			0.927493
$944$		0			0
$945$		0			0
$946$		0			0
$947$			3.91369i			0.127178i	0.997976	+	0.0635890i	$0.0202547\pi$
$947$			3.91369i			0.127178i	−0.997976	+	0.0635890i	$0.979745\pi$
$948$		0			0
$949$	−9.45738			−0.307000
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	40.3514i		−	1.30711i	−0.756879	−	0.653555i	$-0.773277\pi$
$953$		−	40.3514i		−	1.30711i	0.756879	−	0.653555i	$-0.226723\pi$
$954$		0			0
$955$			37.4643i			1.21232i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.8748			0.480332
$960$		0			0
$961$	30.0979			0.970899
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		0			0
$971$		−	38.7146i		−	1.24241i	−0.783648	−	0.621206i	$-0.786643\pi$
$971$		−	38.7146i		−	1.24241i	0.783648	−	0.621206i	$-0.213357\pi$
$972$		0			0
$973$	−9.45738			−0.303190
$974$		0			0
$975$		0			0
$976$		0			0
$977$	30.9315			0.989587			0.494794	−	0.869011i	$-0.335244\pi$
$977$	30.9315			0.989587			0.494794	+	0.869011i	$0.335244\pi$
$978$		0			0
$979$	48.1994	+	2.12915i	1.54046	+	0.0680480i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	39.5500i		−	1.26145i	−0.776007	−	0.630724i	$-0.782758\pi$
$983$		−	39.5500i		−	1.26145i	0.776007	−	0.630724i	$-0.217242\pi$
$984$		0			0
$985$		−	8.42469i		−	0.268433i
$986$		0			0
$987$		0			0
$988$		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
$993$		0			0
$994$		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
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.o.e.703.4		12
3.2	odd	2		352.2.e.a.351.10	yes	12
4.3	odd	2	inner	3168.2.o.e.703.1		12
11.10	odd	2	inner	3168.2.o.e.703.2		12
12.11	even	2		352.2.e.a.351.3	✓	12
24.5	odd	2		704.2.e.d.703.4		12
24.11	even	2		704.2.e.d.703.9		12
33.32	even	2		352.2.e.a.351.9	yes	12
44.43	even	2	inner	3168.2.o.e.703.3		12
48.5	odd	4		2816.2.g.d.1407.11		12
48.11	even	4		2816.2.g.i.1407.4		12
48.29	odd	4		2816.2.g.i.1407.1		12
48.35	even	4		2816.2.g.d.1407.10		12
132.131	odd	2		352.2.e.a.351.4	yes	12
264.131	odd	2		704.2.e.d.703.10		12
264.197	even	2		704.2.e.d.703.3		12
528.131	odd	4		2816.2.g.d.1407.9		12
528.197	even	4		2816.2.g.d.1407.12		12
528.395	odd	4		2816.2.g.i.1407.3		12
528.461	even	4		2816.2.g.i.1407.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.e.a.351.3	✓	12	12.11	even	2
352.2.e.a.351.4	yes	12	132.131	odd	2
352.2.e.a.351.9	yes	12	33.32	even	2
352.2.e.a.351.10	yes	12	3.2	odd	2
704.2.e.d.703.3		12	264.197	even	2
704.2.e.d.703.4		12	24.5	odd	2
704.2.e.d.703.9		12	24.11	even	2
704.2.e.d.703.10		12	264.131	odd	2
2816.2.g.d.1407.9		12	528.131	odd	4
2816.2.g.d.1407.10		12	48.35	even	4
2816.2.g.d.1407.11		12	48.5	odd	4
2816.2.g.d.1407.12		12	528.197	even	4
2816.2.g.i.1407.1		12	48.29	odd	4
2816.2.g.i.1407.2		12	528.461	even	4
2816.2.g.i.1407.3		12	528.395	odd	4
2816.2.g.i.1407.4		12	48.11	even	4
3168.2.o.e.703.1		12	4.3	odd	2	inner
3168.2.o.e.703.2		12	11.10	odd	2	inner
3168.2.o.e.703.3		12	44.43	even	2	inner
3168.2.o.e.703.4		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.48929 q^{5} +1.62121 q^{7} +O(q^{10})\)	\(q-2.48929 q^{5} +1.62121 q^{7} +(3.31339 + 0.146365i) q^{11} +1.20722i q^{13} +5.41957i q^{17} +2.59114 q^{19} -3.63565i q^{23} +1.19656 q^{25} -9.86920i q^{29} -0.949808i q^{31} -4.03565 q^{35} +2.48929 q^{37} +7.83401i q^{41} -6.62679 q^{43} +5.66442i q^{47} -4.37169 q^{49} +7.37169 q^{53} +(-8.24799 - 0.364346i) q^{55} +12.0288i q^{59} -0.969933i q^{61} -3.00512i q^{65} -8.61423i q^{67} +14.3215i q^{71} +7.83401i q^{73} +(5.37169 + 0.237289i) q^{77} +11.3137 q^{79} -3.38438 q^{83} -13.4909i q^{85} +14.5468 q^{89} +1.95715i q^{91} -6.45009 q^{95} +6.15371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4 q^{25} + 44 q^{49} - 8 q^{53} - 32 q^{77} - 64 q^{97}+O(q^{100}) \) 12 * q - 4 * q^25 + 44 * q^49 - 8 * q^53 - 32 * q^77 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more