LMFDB - Embedded newform 1512.2.k.a.377.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(377,1512)] 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("1512.377"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 $ x^16 - 24x^14 + 230x^12 - 1052x^10 + 2139x^8 - 1244x^6 + 1134x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.11
Root			$2.32849 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1512.377
Dual form			1512.2.k.a.377.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.10598 q^{5} +(-2.64465 - 0.0763047i) q^{7} -3.42810i q^{11} -4.98427i q^{13} -6.38903 q^{17} +4.65697i q^{19} +8.98172i q^{23} -3.77681 q^{25} -1.51249i q^{29} +6.71632i q^{31} +(-2.92492 - 0.0843913i) q^{35} -2.83122 q^{37} -10.1147 q^{41} +10.8973 q^{43} -12.8935 q^{47} +(6.98836 + 0.403599i) q^{49} -3.02497i q^{53} -3.79140i q^{55} -9.54791 q^{59} -9.16134i q^{61} -5.51249i q^{65} -5.07938 q^{67} -8.94058i q^{71} -7.79378i q^{73} +(-0.261580 + 9.06612i) q^{77} -2.05440 q^{79} -5.73444 q^{83} -7.06612 q^{85} +6.47681 q^{89} +(-0.380323 + 13.1816i) q^{91} +5.15051i q^{95} +8.04062i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{7} - 12 q^{25} - 8 q^{37} + 8 q^{43} + 2 q^{49} - 28 q^{67} - 44 q^{79} + 16 q^{85} + 18 q^{91}+O(q^{100}) $ 16 * q - 2 * q^7 - 12 * q^25 - 8 * q^37 + 8 * q^43 + 2 * q^49 - 28 * q^67 - 44 * q^79 + 16 * q^85 + 18 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	1.10598			0.494608			0.247304	−	0.968938i	$-0.420455\pi$
$5$	1.10598			0.494608			0.247304	+	0.968938i	$0.420455\pi$
$6$		0			0
$7$	−2.64465	−	0.0763047i	−0.999584	−	0.0288405i
$7$	−2.64465	−	0.0763047i	−0.999584	−	0.0288405i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.42810i		−	1.03361i	−0.856103	−	0.516805i	$-0.827121\pi$
$11$		−	3.42810i		−	1.03361i	0.856103	−	0.516805i	$-0.172879\pi$
$12$		0			0
$13$		−	4.98427i		−	1.38239i	−0.722670	−	0.691194i	$-0.757085\pi$
$13$		−	4.98427i		−	1.38239i	0.722670	−	0.691194i	$-0.242915\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.38903			−1.54957			−0.774783	−	0.632227i	$-0.782141\pi$
$17$	−6.38903			−1.54957			−0.774783	+	0.632227i	$0.782141\pi$
$18$		0			0
$19$			4.65697i			1.06838i	0.845363	+	0.534192i	$0.179384\pi$
$19$			4.65697i			1.06838i	−0.845363	+	0.534192i	$0.820616\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.98172i			1.87282i	0.350909	+	0.936410i	$0.385873\pi$
$23$			8.98172i			1.87282i	−0.350909	+	0.936410i	$0.614127\pi$
$24$		0			0
$25$	−3.77681			−0.755363
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1.51249i		−	0.280862i	−0.990090	−	0.140431i	$-0.955151\pi$
$29$		−	1.51249i		−	0.280862i	0.990090	−	0.140431i	$-0.0448488\pi$
$30$		0			0
$31$			6.71632i			1.20629i	0.797633	+	0.603143i	$0.206085\pi$
$31$			6.71632i			1.20629i	−0.797633	+	0.603143i	$0.793915\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.92492	−	0.0843913i	−0.494402	−	0.0142647i
$36$		0			0
$37$	−2.83122			−0.465449			−0.232725	−	0.972543i	$-0.574764\pi$
$37$	−2.83122			−0.465449			−0.232725	+	0.972543i	$0.574764\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.1147			−1.57965			−0.789826	−	0.613331i	$-0.789829\pi$
$41$	−10.1147			−1.57965			−0.789826	+	0.613331i	$0.789829\pi$
$42$		0			0
$43$	10.8973			1.66183			0.830914	−	0.556401i	$-0.187818\pi$
$43$	10.8973			1.66183			0.830914	+	0.556401i	$0.187818\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12.8935			−1.88070			−0.940352	−	0.340202i	$-0.889504\pi$
$47$	−12.8935			−1.88070			−0.940352	+	0.340202i	$0.889504\pi$
$48$		0			0
$49$	6.98836	+	0.403599i	0.998336	+	0.0576570i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.02497i		−	0.415512i	−0.978181	−	0.207756i	$-0.933384\pi$
$53$		−	3.02497i		−	0.415512i	0.978181	−	0.207756i	$-0.0666160\pi$
$54$		0			0
$55$		−	3.79140i		−	0.511232i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.54791			−1.24303			−0.621516	−	0.783402i	$-0.713483\pi$
$59$	−9.54791			−1.24303			−0.621516	+	0.783402i	$0.713483\pi$
$60$		0			0
$61$		−	9.16134i		−	1.17299i	−0.809953	−	0.586495i	$-0.800508\pi$
$61$		−	9.16134i		−	1.17299i	0.809953	−	0.586495i	$-0.199492\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5.51249i		−	0.683740i
$66$		0			0
$67$	−5.07938			−0.620545			−0.310272	−	0.950648i	$-0.600420\pi$
$67$	−5.07938			−0.620545			−0.310272	+	0.950648i	$0.600420\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.94058i		−	1.06105i	−0.847669	−	0.530526i	$-0.821994\pi$
$71$		−	8.94058i		−	1.06105i	0.847669	−	0.530526i	$-0.178006\pi$
$72$		0			0
$73$		−	7.79378i		−	0.912193i	−0.889930	−	0.456097i	$-0.849247\pi$
$73$		−	7.79378i		−	0.912193i	0.889930	−	0.456097i	$-0.150753\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.261580	+	9.06612i	−0.0298098	+	1.03318i
$78$		0			0
$79$	−2.05440			−0.231138			−0.115569	−	0.993299i	$-0.536869\pi$
$79$	−2.05440			−0.231138			−0.115569	+	0.993299i	$0.536869\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.73444			−0.629436			−0.314718	−	0.949185i	$-0.601910\pi$
$83$	−5.73444			−0.629436			−0.314718	+	0.949185i	$0.601910\pi$
$84$		0			0
$85$	−7.06612			−0.766428
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.47681			0.686541			0.343270	−	0.939237i	$-0.388465\pi$
$89$	6.47681			0.686541			0.343270	+	0.939237i	$0.388465\pi$
$90$		0			0
$91$	−0.380323	+	13.1816i	−0.0398687	+	1.38181i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5.15051i			0.528431i
$96$		0			0
$97$			8.04062i			0.816401i	0.912892	+	0.408201i	$0.133844\pi$
$97$			8.04062i			0.816401i	−0.912892	+	0.408201i	$0.866156\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.3172			−1.32511			−0.662557	−	0.749012i	$-0.730529\pi$
$101$	−13.3172			−1.32511			−0.662557	+	0.749012i	$0.730529\pi$
$102$		0			0
$103$		−	6.86893i		−	0.676816i	−0.941000	−	0.338408i	$-0.890112\pi$
$103$		−	6.86893i		−	0.676816i	0.941000	−	0.338408i	$-0.109888\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.09109i		−	0.588848i	−0.955675	−	0.294424i	$-0.904872\pi$
$107$		−	6.09109i		−	0.588848i	0.955675	−	0.294424i	$-0.0951278\pi$
$108$		0			0
$109$	14.6580			1.40398			0.701990	−	0.712187i	$-0.252295\pi$
$109$	14.6580			1.40398			0.701990	+	0.712187i	$0.252295\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			18.1322i			1.70574i	0.522126	+	0.852868i	$0.325139\pi$
$113$			18.1322i			1.70574i	−0.522126	+	0.852868i	$0.674861\pi$
$114$		0			0
$115$			9.93358i			0.926311i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	16.8967	+	0.487513i	1.54892	+	0.0446902i
$120$		0			0
$121$	−0.751840			−0.0683491
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.70696			−0.868217
$126$		0			0
$127$	−7.23490			−0.641993			−0.320997	−	0.947080i	$-0.604018\pi$
$127$	−7.23490			−0.641993			−0.320997	+	0.947080i	$0.604018\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	14.5823			1.27406			0.637029	−	0.770840i	$-0.280163\pi$
$131$	14.5823			1.27406			0.637029	+	0.770840i	$0.280163\pi$
$132$		0			0
$133$	0.355349	−	12.3161i	0.0308127	−	1.06794i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			5.17492i			0.442123i	0.975260	+	0.221062i	$0.0709522\pi$
$137$			5.17492i			0.442123i	−0.975260	+	0.221062i	$0.929048\pi$
$138$		0			0
$139$			11.2419i			0.953523i	0.879033	+	0.476761i	$0.158189\pi$
$139$			11.2419i			0.953523i	−0.879033	+	0.476761i	$0.841811\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−17.0865			−1.42885
$144$		0			0
$145$		−	1.67278i		−	0.138917i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	7.66243i		−	0.627731i	−0.949467	−	0.313866i	$-0.898376\pi$
$149$		−	7.66243i		−	0.627731i	0.949467	−	0.313866i	$-0.101624\pi$
$150$		0			0
$151$	9.28930			0.755953			0.377976	−	0.925815i	$-0.376620\pi$
$151$	9.28930			0.755953			0.377976	+	0.925815i	$0.376620\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			7.42810i			0.596639i
$156$		0			0
$157$		−	13.8404i		−	1.10458i	−0.833651	−	0.552292i	$-0.813754\pi$
$157$		−	13.8404i		−	1.10458i	0.833651	−	0.552292i	$-0.186246\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.685348	−	23.7535i	0.0540130	−	1.87204i
$162$		0			0
$163$	−15.6580			−1.22643			−0.613214	−	0.789917i	$-0.710124\pi$
$163$	−15.6580			−1.22643			−0.613214	+	0.789917i	$0.710124\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.81347			−0.295095			−0.147548	−	0.989055i	$-0.547138\pi$
$167$	−3.81347			−0.295095			−0.147548	+	0.989055i	$0.547138\pi$
$168$		0			0
$169$	−11.8429			−0.910995
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.36140			0.483649			0.241824	−	0.970320i	$-0.422254\pi$
$173$	6.36140			0.483649			0.241824	+	0.970320i	$0.422254\pi$
$174$		0			0
$175$	9.98836	+	0.288189i	0.755049	+	0.0217850i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	4.04114i		−	0.302049i	−0.988530	−	0.151025i	$-0.951743\pi$
$179$		−	4.04114i		−	0.302049i	0.988530	−	0.151025i	$-0.0482573\pi$
$180$		0			0
$181$		−	2.84357i		−	0.211361i	−0.994400	−	0.105681i	$-0.966298\pi$
$181$		−	2.84357i		−	0.211361i	0.994400	−	0.105681i	$-0.0337021\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.13126			−0.230215
$186$		0			0
$187$			21.9022i			1.60165i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.1505i		−	0.951537i	−0.879570	−	0.475769i	$-0.842170\pi$
$191$		−	13.1505i		−	0.951537i	0.879570	−	0.475769i	$-0.157830\pi$
$192$		0			0
$193$	−9.97671			−0.718139			−0.359070	−	0.933311i	$-0.616906\pi$
$193$	−9.97671			−0.718139			−0.359070	+	0.933311i	$0.616906\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	15.4448i		−	1.10040i	−0.835034	−	0.550199i	$-0.814552\pi$
$197$		−	15.4448i		−	1.10040i	0.835034	−	0.550199i	$-0.185448\pi$
$198$		0			0
$199$		−	2.25521i		−	0.159868i	−0.996800	−	0.0799338i	$-0.974529\pi$
$199$		−	2.25521i		−	0.159868i	0.996800	−	0.0799338i	$-0.0254709\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.115410	+	4.00000i	−0.00810019	+	0.280745i
$204$		0			0
$205$	−11.1866			−0.781308
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15.9646			1.10429
$210$		0			0
$211$	5.81796			0.400525			0.200262	−	0.979742i	$-0.435821\pi$
$211$	5.81796			0.400525			0.200262	+	0.979742i	$0.435821\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	12.0522			0.821953
$216$		0			0
$217$	0.512487	−	17.7623i	0.0347899	−	1.20578i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			31.8446i			2.14210i
$222$		0			0
$223$		−	1.52241i		−	0.101948i	−0.998700	−	0.0509741i	$-0.983767\pi$
$223$		−	1.52241i		−	0.101948i	0.998700	−	0.0509741i	$-0.0162326\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.2803			−1.21331			−0.606654	−	0.794966i	$-0.707488\pi$
$227$	−18.2803			−1.21331			−0.606654	+	0.794966i	$0.707488\pi$
$228$		0			0
$229$			10.2738i			0.678909i	0.940622	+	0.339454i	$0.110242\pi$
$229$			10.2738i			0.678909i	−0.940622	+	0.339454i	$0.889758\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	−14.2599			−0.930212
$236$		0			0
$237$		0			0
$238$		0			0
$239$			21.1161i			1.36588i	0.730472	+	0.682942i	$0.239300\pi$
$239$			21.1161i			1.36588i	−0.730472	+	0.682942i	$0.760700\pi$
$240$		0			0
$241$		−	22.9594i		−	1.47894i	−0.673188	−	0.739471i	$-0.735076\pi$
$241$		−	22.9594i		−	1.47894i	0.673188	−	0.739471i	$-0.264924\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.72896	+	0.446371i	0.493785	+	0.0285176i
$246$		0			0
$247$	23.2116			1.47692
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.66332			0.609943			0.304972	−	0.952361i	$-0.401353\pi$
$251$	9.66332			0.609943			0.304972	+	0.952361i	$0.401353\pi$
$252$		0			0
$253$	30.7902			1.93576
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.34471			0.582907			0.291453	−	0.956585i	$-0.405861\pi$
$257$	9.34471			0.582907			0.291453	+	0.956585i	$0.405861\pi$
$258$		0			0
$259$	7.48758	+	0.216035i	0.465256	+	0.0134238i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.8129i		−	1.03673i	−0.855159	−	0.518365i	$-0.826541\pi$
$263$		−	16.8129i		−	1.03673i	0.855159	−	0.518365i	$-0.173459\pi$
$264$		0			0
$265$		−	3.34555i		−	0.205516i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.93981			0.423127			0.211564	−	0.977364i	$-0.432144\pi$
$269$	6.93981			0.423127			0.211564	+	0.977364i	$0.432144\pi$
$270$		0			0
$271$		−	2.35168i		−	0.142855i	−0.997446	−	0.0714273i	$-0.977245\pi$
$271$		−	2.35168i		−	0.142855i	0.997446	−	0.0714273i	$-0.0227554\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			12.9473i			0.780750i
$276$		0			0
$277$	−1.87948			−0.112927			−0.0564635	−	0.998405i	$-0.517982\pi$
$277$	−1.87948			−0.112927			−0.0564635	+	0.998405i	$0.517982\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	15.3072i		−	0.913148i	−0.889685	−	0.456574i	$-0.849076\pi$
$281$		−	15.3072i		−	0.913148i	0.889685	−	0.456574i	$-0.150924\pi$
$282$		0			0
$283$			30.3294i			1.80289i	0.432889	+	0.901447i	$0.357494\pi$
$283$			30.3294i			1.80289i	−0.432889	+	0.901447i	$0.642506\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	26.7499	+	0.771800i	1.57899	+	0.0455579i
$288$		0			0
$289$	23.8196			1.40116
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.1585			0.827148			0.413574	−	0.910471i	$-0.364280\pi$
$293$	14.1585			0.827148			0.413574	+	0.910471i	$0.364280\pi$
$294$		0			0
$295$	−10.5598			−0.614813
$296$		0			0
$297$		0			0
$298$		0			0
$299$	44.7673			2.58896
$300$		0			0
$301$	−28.8196	−	0.831518i	−1.66114	−	0.0479279i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	10.1322i		−	0.580170i
$306$		0			0
$307$			0.267126i			0.0152457i	0.999971	+	0.00762283i	$0.00242645\pi$
$307$			0.267126i			0.0152457i	−0.999971	+	0.00762283i	$0.997574\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.962946			−0.0546037			−0.0273018	−	0.999627i	$-0.508692\pi$
$311$	−0.962946			−0.0546037			−0.0273018	+	0.999627i	$0.508692\pi$
$312$		0			0
$313$			6.67306i			0.377184i	0.982056	+	0.188592i	$0.0603923\pi$
$313$			6.67306i			0.377184i	−0.982056	+	0.188592i	$0.939608\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	14.1499i		−	0.794740i	−0.917658	−	0.397370i	$-0.869923\pi$
$317$		−	14.1499i		−	0.794740i	0.917658	−	0.397370i	$-0.130077\pi$
$318$		0			0
$319$	−5.18495			−0.290301
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	29.7535i		−	1.65553i
$324$		0			0
$325$			18.8247i			1.04420i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	34.0987	+	0.983832i	1.87992	+	0.0542404i
$330$		0			0
$331$	12.5054			0.687357			0.343679	−	0.939087i	$-0.388327\pi$
$331$	12.5054			0.687357			0.343679	+	0.939087i	$0.388327\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.61768			−0.306926
$336$		0			0
$337$	17.1322			0.933252			0.466626	−	0.884455i	$-0.345469\pi$
$337$	17.1322			0.933252			0.466626	+	0.884455i	$0.345469\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	23.0242			1.24683
$342$		0			0
$343$	−18.4510	−	1.60062i	−0.996258	−	0.0864255i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6.83178i			0.366749i	0.983043	+	0.183375i	$0.0587021\pi$
$347$			6.83178i			0.366749i	−0.983043	+	0.183375i	$0.941298\pi$
$348$		0			0
$349$			32.1708i			1.72206i	0.508552	+	0.861031i	$0.330181\pi$
$349$			32.1708i			1.72206i	−0.508552	+	0.861031i	$0.669819\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.7317			−0.996988			−0.498494	−	0.866893i	$-0.666113\pi$
$353$	−18.7317			−0.996988			−0.498494	+	0.866893i	$0.666113\pi$
$354$		0			0
$355$		−	9.88808i		−	0.524805i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			15.9589i			0.842276i	0.906996	+	0.421138i	$0.138369\pi$
$359$			15.9589i			0.842276i	−0.906996	+	0.421138i	$0.861631\pi$
$360$		0			0
$361$	−2.68741			−0.141443
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	8.61974i		−	0.451178i
$366$		0			0
$367$		−	25.9175i		−	1.35288i	−0.736498	−	0.676440i	$-0.763522\pi$
$367$		−	25.9175i		−	1.35288i	0.736498	−	0.676440i	$-0.236478\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.230820	+	8.00000i	−0.0119836	+	0.415339i
$372$		0			0
$373$	−18.2848			−0.946753			−0.473377	−	0.880860i	$-0.656965\pi$
$373$	−18.2848			−0.946753			−0.473377	+	0.880860i	$0.656965\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.53864			−0.388260
$378$		0			0
$379$	−22.3454			−1.14781			−0.573903	−	0.818923i	$-0.694571\pi$
$379$	−22.3454			−1.14781			−0.573903	+	0.818923i	$0.694571\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	22.7042			1.16013			0.580066	−	0.814570i	$-0.303027\pi$
$383$	22.7042			1.16013			0.580066	+	0.814570i	$0.303027\pi$
$384$		0			0
$385$	−0.289301	+	10.0269i	−0.0147442	+	0.511019i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	17.7947i		−	0.902225i	−0.892467	−	0.451113i	$-0.851027\pi$
$389$		−	17.7947i		−	0.902225i	0.892467	−	0.451113i	$-0.148973\pi$
$390$		0			0
$391$		−	57.3845i		−	2.90206i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.27212			−0.114323
$396$		0			0
$397$		−	17.8124i		−	0.893979i	−0.894539	−	0.446989i	$-0.852496\pi$
$397$		−	17.8124i		−	0.893979i	0.894539	−	0.446989i	$-0.147504\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			29.0195i			1.44917i	0.689187	+	0.724583i	$0.257968\pi$
$401$			29.0195i			1.44917i	−0.689187	+	0.724583i	$0.742032\pi$
$402$		0			0
$403$	33.4759			1.66756
$404$		0			0
$405$		0			0
$406$		0			0
$407$			9.70568i			0.481093i
$408$		0			0
$409$		−	21.6502i		−	1.07053i	−0.844683	−	0.535266i	$-0.820211\pi$
$409$		−	21.6502i		−	1.07053i	0.844683	−	0.535266i	$-0.179789\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	25.2509	+	0.728551i	1.24251	+	0.0358496i
$414$		0			0
$415$	−6.34216			−0.311324
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.66535			0.325624			0.162812	−	0.986657i	$-0.447944\pi$
$419$	6.66535			0.325624			0.162812	+	0.986657i	$0.447944\pi$
$420$		0			0
$421$	−3.16878			−0.154437			−0.0772185	−	0.997014i	$-0.524604\pi$
$421$	−3.16878			−0.154437			−0.0772185	+	0.997014i	$0.524604\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	24.1302			1.17048
$426$		0			0
$427$	−0.699054	+	24.2285i	−0.0338296	+	1.17250i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2.84949i			0.137255i	0.997642	+	0.0686276i	$0.0218620\pi$
$431$			2.84949i			0.137255i	−0.997642	+	0.0686276i	$0.978138\pi$
$432$		0			0
$433$			32.2375i			1.54923i	0.632431	+	0.774617i	$0.282057\pi$
$433$			32.2375i			1.54923i	−0.632431	+	0.774617i	$0.717943\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−41.8277			−2.00089
$438$		0			0
$439$		−	37.6052i		−	1.79480i	−0.441221	−	0.897398i	$-0.645455\pi$
$439$		−	37.6052i		−	1.79480i	0.441221	−	0.897398i	$-0.354545\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	12.7407i		−	0.605328i	−0.953097	−	0.302664i	$-0.902124\pi$
$443$		−	12.7407i		−	0.605328i	0.953097	−	0.302664i	$-0.0978760\pi$
$444$		0			0
$445$	7.16321			0.339569
$446$		0			0
$447$		0			0
$448$		0			0
$449$			8.18218i			0.386141i	0.981185	+	0.193070i	$0.0618446\pi$
$449$			8.18218i			0.386141i	−0.981185	+	0.193070i	$0.938155\pi$
$450$		0			0
$451$			34.6742i			1.63274i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.420629	+	14.5786i	−0.0197194	+	0.683455i
$456$		0			0
$457$	9.37313			0.438457			0.219228	−	0.975674i	$-0.429646\pi$
$457$	9.37313			0.438457			0.219228	+	0.975674i	$0.429646\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.7688			−0.827574			−0.413787	−	0.910374i	$-0.635794\pi$
$461$	−17.7688			−0.827574			−0.413787	+	0.910374i	$0.635794\pi$
$462$		0			0
$463$	0.231992			0.0107816			0.00539079	−	0.999985i	$-0.498284\pi$
$463$	0.231992			0.0107816			0.00539079	+	0.999985i	$0.498284\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.57569			−0.304287			−0.152143	−	0.988358i	$-0.548618\pi$
$467$	−6.57569			−0.304287			−0.152143	+	0.988358i	$0.548618\pi$
$468$		0			0
$469$	13.4332	+	0.387581i	0.620287	+	0.0178968i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	37.3571i		−	1.71768i
$474$		0			0
$475$		−	17.5885i		−	0.807017i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16.3544			0.747253			0.373626	−	0.927579i	$-0.378114\pi$
$479$	16.3544			0.747253			0.373626	+	0.927579i	$0.378114\pi$
$480$		0			0
$481$			14.1115i			0.643431i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			8.89274i			0.403799i
$486$		0			0
$487$	−13.1617			−0.596412			−0.298206	−	0.954502i	$-0.596388\pi$
$487$	−13.1617			−0.596412			−0.298206	+	0.954502i	$0.596388\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			15.4692i			0.698117i	0.937101	+	0.349058i	$0.113499\pi$
$491$			15.4692i			0.698117i	−0.937101	+	0.349058i	$0.886501\pi$
$492$		0			0
$493$			9.66332i			0.435214i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.682209	+	23.6447i	−0.0306013	+	1.06061i
$498$		0			0
$499$	−8.94728			−0.400535			−0.200268	−	0.979741i	$-0.564181\pi$
$499$	−8.94728			−0.400535			−0.200268	+	0.979741i	$0.564181\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−27.5654			−1.22908			−0.614539	−	0.788886i	$-0.710658\pi$
$503$	−27.5654			−1.22908			−0.614539	+	0.788886i	$0.710658\pi$
$504$		0			0
$505$	−14.7286			−0.655412
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.68961			−0.119215			−0.0596075	−	0.998222i	$-0.518985\pi$
$509$	−2.68961			−0.119215			−0.0596075	+	0.998222i	$0.518985\pi$
$510$		0			0
$511$	−0.594703	+	20.6118i	−0.0263081	+	0.911814i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	7.59688i		−	0.334758i
$516$		0			0
$517$			44.2000i			1.94391i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−32.5721			−1.42701			−0.713505	−	0.700651i	$-0.752893\pi$
$521$	−32.5721			−1.42701			−0.713505	+	0.700651i	$0.752893\pi$
$522$		0			0
$523$		−	9.00962i		−	0.393963i	−0.980407	−	0.196982i	$-0.936886\pi$
$523$		−	9.00962i		−	0.393963i	0.980407	−	0.196982i	$-0.0631139\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	42.9107i		−	1.86922i
$528$		0			0
$529$	−57.6714			−2.50745
$530$		0			0
$531$		0			0
$532$		0			0
$533$			50.4144i			2.18369i
$534$		0			0
$535$		−	6.73661i		−	0.291249i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.38358	−	23.9568i	0.0595948	−	1.03189i
$540$		0			0
$541$	21.9884			0.945356			0.472678	−	0.881235i	$-0.343287\pi$
$541$	21.9884			0.945356			0.472678	+	0.881235i	$0.343287\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.2114			0.694420
$546$		0			0
$547$	1.28930			0.0551266			0.0275633	−	0.999620i	$-0.491225\pi$
$547$	1.28930			0.0551266			0.0275633	+	0.999620i	$0.491225\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.04361			0.300068
$552$		0			0
$553$	5.43318	+	0.156761i	0.231042	+	0.00666614i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			7.44482i			0.315447i	0.987483	+	0.157724i	$0.0504155\pi$
$557$			7.44482i			0.315447i	−0.987483	+	0.157724i	$0.949584\pi$
$558$		0			0
$559$		−	54.3152i		−	2.29729i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−29.2542			−1.23292			−0.616458	−	0.787388i	$-0.711433\pi$
$563$	−29.2542			−1.23292			−0.616458	+	0.787388i	$0.711433\pi$
$564$		0			0
$565$			20.0538i			0.843671i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.4698i		−	1.10967i	−0.831960	−	0.554836i	$-0.812781\pi$
$569$		−	26.4698i		−	1.10967i	0.831960	−	0.554836i	$-0.187219\pi$
$570$		0			0
$571$	29.6626			1.24134			0.620670	−	0.784072i	$-0.286861\pi$
$571$	29.6626			1.24134			0.620670	+	0.784072i	$0.286861\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	33.9223i		−	1.41466i
$576$		0			0
$577$		−	1.65473i		−	0.0688875i	−0.999407	−	0.0344437i	$-0.989034\pi$
$577$		−	1.65473i		−	0.0688875i	0.999407	−	0.0344437i	$-0.0109660\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.1656	+	0.437565i	0.629174	+	0.0181532i
$582$		0			0
$583$	−10.3699			−0.429477
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.45826			0.390384			0.195192	−	0.980765i	$-0.437467\pi$
$587$	9.45826			0.390384			0.195192	+	0.980765i	$0.437467\pi$
$588$		0			0
$589$	−31.2777			−1.28878
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.17049			−0.130196			−0.0650981	−	0.997879i	$-0.520736\pi$
$593$	−3.17049			−0.130196			−0.0650981	+	0.997879i	$0.520736\pi$
$594$		0			0
$595$	18.6874	+	0.539178i	0.766109	+	0.0221042i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	23.2827i		−	0.951307i	−0.879633	−	0.475653i	$-0.842212\pi$
$599$		−	23.2827i		−	0.951307i	0.879633	−	0.475653i	$-0.157788\pi$
$600$		0			0
$601$		−	23.0358i		−	0.939649i	−0.882760	−	0.469825i	$-0.844317\pi$
$601$		−	23.0358i		−	0.939649i	0.882760	−	0.469825i	$-0.155683\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.831518			−0.0338060
$606$		0			0
$607$		−	0.737290i		−	0.0299257i	−0.999888	−	0.0149628i	$-0.995237\pi$
$607$		−	0.737290i		−	0.0299257i	0.999888	−	0.0149628i	$-0.00476300\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			64.2645i			2.59986i
$612$		0			0
$613$	36.4494			1.47218			0.736089	−	0.676885i	$-0.236670\pi$
$613$	36.4494			1.47218			0.736089	+	0.676885i	$0.236670\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.19990i		−	0.330115i	−0.986284	−	0.165058i	$-0.947219\pi$
$617$		−	8.19990i		−	0.330115i	0.986284	−	0.165058i	$-0.0527810\pi$
$618$		0			0
$619$			18.3236i			0.736487i	0.929729	+	0.368243i	$0.120041\pi$
$619$			18.3236i			0.736487i	−0.929729	+	0.368243i	$0.879959\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−17.1289	−	0.494212i	−0.686255	−	0.0198002i
$624$		0			0
$625$	8.14840			0.325936
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18.0887			0.721244
$630$		0			0
$631$	−6.47425			−0.257736			−0.128868	−	0.991662i	$-0.541134\pi$
$631$	−6.47425			−0.257736			−0.128868	+	0.991662i	$0.541134\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.00163			−0.317535
$636$		0			0
$637$	2.01164	−	34.8318i	0.0797043	−	1.38009i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			30.9519i			1.22253i	0.791428	+	0.611263i	$0.209338\pi$
$641$			30.9519i			1.22253i	−0.791428	+	0.611263i	$0.790662\pi$
$642$		0			0
$643$		−	11.2017i		−	0.441754i	−0.975302	−	0.220877i	$-0.929108\pi$
$643$		−	11.2017i		−	0.441754i	0.975302	−	0.220877i	$-0.0708919\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−32.3689			−1.27255			−0.636276	−	0.771461i	$-0.719526\pi$
$647$	−32.3689			−1.27255			−0.636276	+	0.771461i	$0.719526\pi$
$648$		0			0
$649$			32.7311i			1.28481i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	44.9373i		−	1.75853i	−0.476332	−	0.879265i	$-0.658034\pi$
$653$		−	44.9373i		−	1.75853i	0.476332	−	0.879265i	$-0.341966\pi$
$654$		0			0
$655$	16.1276			0.630159
$656$		0			0
$657$		0			0
$658$		0			0
$659$			11.5969i			0.451750i	0.974156	+	0.225875i	$0.0725241\pi$
$659$			11.5969i			0.451750i	−0.974156	+	0.225875i	$0.927476\pi$
$660$		0			0
$661$			13.9488i			0.542547i	0.962502	+	0.271274i	$0.0874448\pi$
$661$			13.9488i			0.542547i	−0.962502	+	0.271274i	$0.912555\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.393008	−	13.6213i	0.0152402	−	0.528211i
$666$		0			0
$667$	13.5847			0.526003
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−31.4059			−1.21241
$672$		0			0
$673$	−16.9194			−0.652195			−0.326097	−	0.945336i	$-0.605734\pi$
$673$	−16.9194			−0.652195			−0.326097	+	0.945336i	$0.605734\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.6171			0.753947			0.376973	−	0.926224i	$-0.376965\pi$
$677$	19.6171			0.753947			0.376973	+	0.926224i	$0.376965\pi$
$678$		0			0
$679$	0.613537	−	21.2646i	0.0235454	−	0.816062i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			41.4738i			1.58695i	0.608602	+	0.793476i	$0.291731\pi$
$683$			41.4738i			1.58695i	−0.608602	+	0.793476i	$0.708269\pi$
$684$		0			0
$685$			5.72335i			0.218678i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.0773			−0.574399
$690$		0			0
$691$			20.2864i			0.771733i	0.922555	+	0.385866i	$0.126097\pi$
$691$			20.2864i			0.771733i	−0.922555	+	0.385866i	$0.873903\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			12.4332i			0.471620i
$696$		0			0
$697$	64.6231			2.44777
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.4302i		−	0.507252i	−0.967302	−	0.253626i	$-0.918377\pi$
$701$		−	13.4302i		−	0.507252i	0.967302	−	0.253626i	$-0.0816232\pi$
$702$		0			0
$703$		−	13.1849i		−	0.497278i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	35.2194	+	1.01617i	1.32456	+	0.0382169i
$708$		0			0
$709$	37.0046			1.38974			0.694868	−	0.719137i	$-0.255463\pi$
$709$	37.0046			1.38974			0.694868	+	0.719137i	$0.255463\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−60.3241			−2.25916
$714$		0			0
$715$	−18.8973			−0.706720
$716$		0			0
$717$		0			0
$718$		0			0
$719$	29.3967			1.09631			0.548156	−	0.836376i	$-0.315330\pi$
$719$	29.3967			1.09631			0.548156	+	0.836376i	$0.315330\pi$
$720$		0			0
$721$	−0.524132	+	18.1659i	−0.0195197	+	0.676534i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			5.71238i			0.212153i
$726$		0			0
$727$		−	19.8248i		−	0.735261i	−0.929972	−	0.367631i	$-0.880169\pi$
$727$		−	19.8248i		−	0.735261i	0.929972	−	0.367631i	$-0.119831\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−69.6233			−2.57511
$732$		0			0
$733$		−	11.8040i		−	0.435991i	−0.975950	−	0.217996i	$-0.930048\pi$
$733$		−	11.8040i		−	0.435991i	0.975950	−	0.217996i	$-0.0699518\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			17.4126i			0.641401i
$738$		0			0
$739$	−43.8446			−1.61285			−0.806425	−	0.591336i	$-0.798601\pi$
$739$	−43.8446			−1.61285			−0.806425	+	0.591336i	$0.798601\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	25.4603i		−	0.934049i	−0.884244	−	0.467024i	$-0.845326\pi$
$743$		−	25.4603i		−	0.934049i	0.884244	−	0.467024i	$-0.154674\pi$
$744$		0			0
$745$		−	8.47448i		−	0.310481i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.464779	+	16.1088i	−0.0169827	+	0.588603i
$750$		0			0
$751$	−22.4277			−0.818397			−0.409199	−	0.912445i	$-0.634192\pi$
$751$	−22.4277			−0.818397			−0.409199	+	0.912445i	$0.634192\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.2738			0.373900
$756$		0			0
$757$	−9.44806			−0.343395			−0.171698	−	0.985150i	$-0.554925\pi$
$757$	−9.44806			−0.343395			−0.171698	+	0.985150i	$0.554925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.6274			−0.928991			−0.464496	−	0.885575i	$-0.653764\pi$
$761$	−25.6274			−0.928991			−0.464496	+	0.885575i	$0.653764\pi$
$762$		0			0
$763$	−38.7652	−	1.11847i	−1.40340	−	0.0404915i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			47.5893i			1.71835i
$768$		0			0
$769$			33.4382i			1.20581i	0.797812	+	0.602906i	$0.205991\pi$
$769$			33.4382i			1.20581i	−0.797812	+	0.602906i	$0.794009\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−51.5205			−1.85306			−0.926531	−	0.376218i	$-0.877224\pi$
$773$	−51.5205			−1.85306			−0.926531	+	0.376218i	$0.877224\pi$
$774$		0			0
$775$		−	25.3663i		−	0.911184i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	47.1039i		−	1.68767i
$780$		0			0
$781$	−30.6492			−1.09671
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	15.3072i		−	0.546336i
$786$		0			0
$787$		−	14.7759i		−	0.526703i	−0.964700	−	0.263352i	$-0.915172\pi$
$787$		−	14.7759i		−	0.526703i	0.964700	−	0.263352i	$-0.0848279\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.38358	−	47.9534i	0.0491943	−	1.70503i
$792$		0			0
$793$	−45.6626			−1.62153
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−23.9822			−0.849493			−0.424747	−	0.905312i	$-0.639637\pi$
$797$	−23.9822			−0.849493			−0.424747	+	0.905312i	$0.639637\pi$
$798$		0			0
$799$	82.3766			2.91428
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−26.7178			−0.942852
$804$		0			0
$805$	0.757980	−	26.2709i	0.0267153	−	0.925926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			48.3821i			1.70102i	0.525956	+	0.850512i	$0.323708\pi$
$809$			48.3821i			1.70102i	−0.525956	+	0.850512i	$0.676292\pi$
$810$		0			0
$811$			6.70030i			0.235279i	0.993056	+	0.117640i	$0.0375328\pi$
$811$			6.70030i			0.235279i	−0.993056	+	0.117640i	$0.962467\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17.3174			−0.606601
$816$		0			0
$817$			50.7486i			1.77547i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	15.3803i		−	0.536775i	−0.963311	−	0.268387i	$-0.913509\pi$
$821$		−	15.3803i		−	0.536775i	0.963311	−	0.268387i	$-0.0864907\pi$
$822$		0			0
$823$	−22.3466			−0.778954			−0.389477	−	0.921036i	$-0.627344\pi$
$823$	−22.3466			−0.778954			−0.389477	+	0.921036i	$0.627344\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22.0013i			0.765059i	0.923943	+	0.382529i	$0.124947\pi$
$827$			22.0013i			0.765059i	−0.923943	+	0.382529i	$0.875053\pi$
$828$		0			0
$829$		−	35.8055i		−	1.24358i	−0.783185	−	0.621789i	$-0.786406\pi$
$829$		−	35.8055i		−	1.24358i	0.783185	−	0.621789i	$-0.213594\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−44.6488	−	2.57860i	−1.54699	−	0.0893433i
$834$		0			0
$835$	−4.21761			−0.145956
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.9278			−0.618937			−0.309468	−	0.950910i	$-0.600151\pi$
$839$	−17.9278			−0.618937			−0.309468	+	0.950910i	$0.600151\pi$
$840$		0			0
$841$	26.7124			0.921117
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.0980			−0.450585
$846$		0			0
$847$	1.98836	+	0.0573690i	0.0683207	+	0.00197122i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	25.4292i		−	0.871702i
$852$		0			0
$853$			56.3175i			1.92828i	0.265401	+	0.964138i	$0.414496\pi$
$853$			56.3175i			1.92828i	−0.265401	+	0.964138i	$0.585504\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.4692			0.425939			0.212970	−	0.977059i	$-0.431686\pi$
$857$	12.4692			0.425939			0.212970	+	0.977059i	$0.431686\pi$
$858$		0			0
$859$		−	6.69514i		−	0.228435i	−0.993456	−	0.114218i	$-0.963564\pi$
$859$		−	6.69514i		−	0.228435i	0.993456	−	0.114218i	$-0.0364361\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			46.1734i			1.57176i	0.618379	+	0.785880i	$0.287790\pi$
$863$			46.1734i			1.57176i	−0.618379	+	0.785880i	$0.712210\pi$
$864$		0			0
$865$	7.03557			0.239216
$866$		0			0
$867$		0			0
$868$		0			0
$869$			7.04269i			0.238907i
$870$		0			0
$871$			25.3170i			0.857833i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	25.6715	+	0.740687i	0.867855	+	0.0250398i
$876$		0			0
$877$	−55.3250			−1.86819			−0.934096	−	0.357021i	$-0.883792\pi$
$877$	−55.3250			−1.86819			−0.934096	+	0.357021i	$0.883792\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6.38954			−0.215269			−0.107634	−	0.994191i	$-0.534328\pi$
$881$	−6.38954			−0.215269			−0.107634	+	0.994191i	$0.534328\pi$
$882$		0			0
$883$	20.0855			0.675931			0.337966	−	0.941158i	$-0.390261\pi$
$883$	20.0855			0.675931			0.337966	+	0.941158i	$0.390261\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.80420			0.0605793			0.0302896	−	0.999541i	$-0.490357\pi$
$887$	1.80420			0.0605793			0.0302896	+	0.999541i	$0.490357\pi$
$888$		0			0
$889$	19.1338	+	0.552057i	0.641726	+	0.0185154i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	60.0445i		−	2.00931i
$894$		0			0
$895$		−	4.46941i		−	0.149396i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10.1583			0.338800
$900$		0			0
$901$			19.3266i			0.643864i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	3.14493i		−	0.104541i
$906$		0			0
$907$	−9.24272			−0.306900			−0.153450	−	0.988156i	$-0.549038\pi$
$907$	−9.24272			−0.306900			−0.153450	+	0.988156i	$0.549038\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			20.7986i			0.689087i	0.938770	+	0.344543i	$0.111966\pi$
$911$			20.7986i			0.689087i	−0.938770	+	0.344543i	$0.888034\pi$
$912$		0			0
$913$			19.6582i			0.650591i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−38.5650	−	1.11270i	−1.27353	−	0.0367444i
$918$		0			0
$919$	26.0499			0.859309			0.429654	−	0.902993i	$-0.358635\pi$
$919$	26.0499			0.859309			0.429654	+	0.902993i	$0.358635\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−44.5623			−1.46678
$924$		0			0
$925$	10.6930			0.351583
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.539178			0.0176899			0.00884493	−	0.999961i	$-0.497185\pi$
$929$	0.539178			0.0176899			0.00884493	+	0.999961i	$0.497185\pi$
$930$		0			0
$931$	−1.87955	+	32.5446i	−0.0615997	+	1.06661i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			24.2233i			0.792187i
$936$		0			0
$937$			43.7537i			1.42937i	0.699446	+	0.714686i	$0.253430\pi$
$937$			43.7537i			1.42937i	−0.699446	+	0.714686i	$0.746570\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	28.2021			0.919361			0.459681	−	0.888084i	$-0.347964\pi$
$941$	28.2021			0.919361			0.459681	+	0.888084i	$0.347964\pi$
$942$		0			0
$943$		−	90.8475i		−	2.95840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	54.7252i		−	1.77833i	−0.457586	−	0.889165i	$-0.651286\pi$
$947$		−	54.7252i		−	1.77833i	0.457586	−	0.889165i	$-0.348714\pi$
$948$		0			0
$949$	−38.8463			−1.26100
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	40.1999i		−	1.30220i	−0.758991	−	0.651101i	$-0.774307\pi$
$953$		−	40.1999i		−	1.30220i	0.758991	−	0.651101i	$-0.225693\pi$
$954$		0			0
$955$		−	14.5442i		−	0.470638i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.394871	−	13.6859i	0.0127511	−	0.441939i
$960$		0			0
$961$	−14.1089			−0.455127
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11.0340			−0.355197
$966$		0			0
$967$	13.8180			0.444356			0.222178	−	0.975006i	$-0.428683\pi$
$967$	13.8180			0.444356			0.222178	+	0.975006i	$0.428683\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−53.6151			−1.72059			−0.860296	−	0.509795i	$-0.829721\pi$
$971$	−53.6151			−1.72059			−0.860296	+	0.509795i	$0.829721\pi$
$972$		0			0
$973$	0.857808	−	29.7308i	0.0275001	−	0.953126i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	30.1122i		−	0.963374i	−0.876343	−	0.481687i	$-0.840024\pi$
$977$		−	30.1122i		−	0.963374i	0.876343	−	0.481687i	$-0.159976\pi$
$978$		0			0
$979$		−	22.2031i		−	0.709615i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.1329			−0.578349			−0.289174	−	0.957276i	$-0.593381\pi$
$983$	−18.1329			−0.578349			−0.289174	+	0.957276i	$0.593381\pi$
$984$		0			0
$985$		−	17.0816i		−	0.544266i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			97.8768i			3.11230i
$990$		0			0
$991$	−23.2116			−0.737341			−0.368671	−	0.929560i	$-0.620187\pi$
$991$	−23.2116			−0.737341			−0.368671	+	0.929560i	$0.620187\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	2.49421i		−	0.0790718i
$996$		0			0
$997$		−	6.08067i		−	0.192577i	−0.995353	−	0.0962883i	$-0.969303\pi$
$997$		−	6.08067i		−	0.192577i	0.995353	−	0.0962883i	$-0.0306971\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.k.a.377.11	yes	16
3.2	odd	2	inner	1512.2.k.a.377.5	✓	16
4.3	odd	2		3024.2.k.k.1889.12		16
7.6	odd	2	inner	1512.2.k.a.377.6	yes	16
12.11	even	2		3024.2.k.k.1889.6		16
21.20	even	2	inner	1512.2.k.a.377.12	yes	16
28.27	even	2		3024.2.k.k.1889.5		16
84.83	odd	2		3024.2.k.k.1889.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.k.a.377.5	✓	16	3.2	odd	2	inner
1512.2.k.a.377.6	yes	16	7.6	odd	2	inner
1512.2.k.a.377.11	yes	16	1.1	even	1	trivial
1512.2.k.a.377.12	yes	16	21.20	even	2	inner
3024.2.k.k.1889.5		16	28.27	even	2
3024.2.k.k.1889.6		16	12.11	even	2
3024.2.k.k.1889.11		16	84.83	odd	2
3024.2.k.k.1889.12		16	4.3	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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.10598 q^{5} +(-2.64465 - 0.0763047i) q^{7} -3.42810i q^{11} -4.98427i q^{13} -6.38903 q^{17} +4.65697i q^{19} +8.98172i q^{23} -3.77681 q^{25} -1.51249i q^{29} +6.71632i q^{31} +(-2.92492 - 0.0843913i) q^{35} -2.83122 q^{37} -10.1147 q^{41} +10.8973 q^{43} -12.8935 q^{47} +(6.98836 + 0.403599i) q^{49} -3.02497i q^{53} -3.79140i q^{55} -9.54791 q^{59} -9.16134i q^{61} -5.51249i q^{65} -5.07938 q^{67} -8.94058i q^{71} -7.79378i q^{73} +(-0.261580 + 9.06612i) q^{77} -2.05440 q^{79} -5.73444 q^{83} -7.06612 q^{85} +6.47681 q^{89} +(-0.380323 + 13.1816i) q^{91} +5.15051i q^{95} +8.04062i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{7} - 12 q^{25} - 8 q^{37} + 8 q^{43} + 2 q^{49} - 28 q^{67} - 44 q^{79} + 16 q^{85} + 18 q^{91}+O(q^{100}) \) 16 * q - 2 * q^7 - 12 * q^25 - 8 * q^37 + 8 * q^43 + 2 * q^49 - 28 * q^67 - 44 * q^79 + 16 * q^85 + 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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