LMFDB - Embedded newform 2736.2.f.f.1025.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.f (of order $2$, degree $1$, not 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 171)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.4
Root			$2.28825i$ of defining polynomial
Character	$\chi$	$=$	2736.1025
Dual form			2736.2.f.f.1025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{5} +2.00000 q^{7} +O(q^{10})$	$q+1.41421i q^{5} +2.00000 q^{7} +1.41421i q^{11} +6.32456i q^{13} +4.24264i q^{17} +(3.00000 - 3.16228i) q^{19} -7.07107i q^{23} +3.00000 q^{25} -4.47214 q^{29} +6.32456i q^{31} +2.82843i q^{35} +4.47214 q^{41} -4.00000 q^{43} +7.07107i q^{47} -3.00000 q^{49} -13.4164 q^{53} -2.00000 q^{55} -8.94427 q^{59} +8.00000 q^{61} -8.94427 q^{65} +6.32456i q^{67} +6.00000 q^{73} +2.82843i q^{77} -12.6491i q^{79} +1.41421i q^{83} -6.00000 q^{85} +13.4164 q^{89} +12.6491i q^{91} +(4.47214 + 4.24264i) q^{95} +6.32456i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} + 12 q^{19} + 12 q^{25} - 16 q^{43} - 12 q^{49} - 8 q^{55} + 32 q^{61} + 24 q^{73} - 24 q^{85}+O(q^{100}) $ 4 * q + 8 * q^7 + 12 * q^19 + 12 * q^25 - 16 * q^43 - 12 * q^49 - 8 * q^55 + 32 * q^61 + 24 * q^73 - 24 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$			6.32456i			1.75412i	0.480384	+	0.877058i	$0.340497\pi$
$13$			6.32456i			1.75412i	−0.480384	+	0.877058i	$0.659503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.24264i			1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$			4.24264i			1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$		0			0
$19$	3.00000	−	3.16228i	0.688247	−	0.725476i
$19$	3.00000	−	3.16228i	0.688247	−	0.725476i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.07107i		−	1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$		−	7.07107i		−	1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.47214			−0.830455			−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214			−0.830455			−0.415227	+	0.909718i	$0.636298\pi$
$30$		0			0
$31$			6.32456i			1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$31$			6.32456i			1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.82843i			0.478091i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.47214			0.698430			0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214			0.698430			0.349215	+	0.937043i	$0.386448\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.07107i			1.03142i	0.856763	+	0.515711i	$0.172472\pi$
$47$			7.07107i			1.03142i	−0.856763	+	0.515711i	$0.827528\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13.4164			−1.84289			−0.921443	−	0.388514i	$-0.872988\pi$
$53$	−13.4164			−1.84289			−0.921443	+	0.388514i	$0.872988\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.94427			−1.10940
$66$		0			0
$67$			6.32456i			0.772667i	0.922359	+	0.386334i	$0.126259\pi$
$67$			6.32456i			0.772667i	−0.922359	+	0.386334i	$0.873741\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.82843i			0.322329i
$78$		0			0
$79$		−	12.6491i		−	1.42314i	−0.702617	−	0.711568i	$-0.747985\pi$
$79$		−	12.6491i		−	1.42314i	0.702617	−	0.711568i	$-0.252015\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.41421i			0.155230i	0.996983	+	0.0776151i	$0.0247305\pi$
$83$			1.41421i			0.155230i	−0.996983	+	0.0776151i	$0.975269\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.4164			1.42214			0.711068	−	0.703123i	$-0.248212\pi$
$89$	13.4164			1.42214			0.711068	+	0.703123i	$0.248212\pi$
$90$		0			0
$91$			12.6491i			1.32599i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.47214	+	4.24264i	0.458831	+	0.435286i
$96$		0			0
$97$			6.32456i			0.642161i	0.947052	+	0.321081i	$0.104046\pi$
$97$			6.32456i			0.642161i	−0.947052	+	0.321081i	$0.895954\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1.41421i			0.140720i	0.997522	+	0.0703598i	$0.0224147\pi$
$101$			1.41421i			0.140720i	−0.997522	+	0.0703598i	$0.977585\pi$
$102$		0			0
$103$			12.6491i			1.24635i	0.782081	+	0.623177i	$0.214158\pi$
$103$			12.6491i			1.24635i	−0.782081	+	0.623177i	$0.785842\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	6.32456i		−	0.605783i	−0.953025	−	0.302891i	$-0.902048\pi$
$109$		−	6.32456i		−	0.605783i	0.953025	−	0.302891i	$-0.0979519\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.47214			−0.420703			−0.210352	−	0.977626i	$-0.567461\pi$
$113$	−4.47214			−0.420703			−0.210352	+	0.977626i	$0.567461\pi$
$114$		0			0
$115$	10.0000			0.932505
$116$		0			0
$117$		0			0
$118$		0			0
$119$			8.48528i			0.777844i
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$		−	18.9737i		−	1.68364i	−0.539758	−	0.841820i	$-0.681484\pi$
$127$		−	18.9737i		−	1.68364i	0.539758	−	0.841820i	$-0.318516\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			15.5563i			1.35916i	0.733599	+	0.679582i	$0.237839\pi$
$131$			15.5563i			1.35916i	−0.733599	+	0.679582i	$0.762161\pi$
$132$		0			0
$133$	6.00000	−	6.32456i	0.520266	−	0.548408i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	9.89949i		−	0.845771i	−0.906183	−	0.422885i	$-0.861017\pi$
$137$		−	9.89949i		−	0.845771i	0.906183	−	0.422885i	$-0.138983\pi$
$138$		0			0
$139$	10.0000			0.848189			0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000			0.848189			0.424094	+	0.905618i	$0.360592\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.94427			−0.747958
$144$		0			0
$145$		−	6.32456i		−	0.525226i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			18.3848i			1.50614i	0.657941	+	0.753070i	$0.271428\pi$
$149$			18.3848i			1.50614i	−0.657941	+	0.753070i	$0.728572\pi$
$150$		0			0
$151$			12.6491i			1.02937i	0.857379	+	0.514685i	$0.172091\pi$
$151$			12.6491i			1.02937i	−0.857379	+	0.514685i	$0.827909\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.94427			−0.718421
$156$		0			0
$157$	−8.00000			−0.638470			−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000			−0.638470			−0.319235	+	0.947676i	$0.603426\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	14.1421i		−	1.11456i
$162$		0			0
$163$	6.00000			0.469956			0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000			0.469956			0.234978	+	0.972001i	$0.424498\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.8885			−1.38426			−0.692129	−	0.721774i	$-0.743327\pi$
$167$	−17.8885			−1.38426			−0.692129	+	0.721774i	$0.743327\pi$
$168$		0			0
$169$	−27.0000			−2.07692
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.47214			−0.340010			−0.170005	−	0.985443i	$-0.554378\pi$
$173$	−4.47214			−0.340010			−0.170005	+	0.985443i	$0.554378\pi$
$174$		0			0
$175$	6.00000			0.453557
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.8885			−1.33705			−0.668526	−	0.743689i	$-0.733075\pi$
$179$	−17.8885			−1.33705			−0.668526	+	0.743689i	$0.733075\pi$
$180$		0			0
$181$			18.9737i			1.41030i	0.709057	+	0.705151i	$0.249121\pi$
$181$			18.9737i			1.41030i	−0.709057	+	0.705151i	$0.750879\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.00000			−0.438763
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1.41421i		−	0.102329i	−0.998690	−	0.0511645i	$-0.983707\pi$
$191$		−	1.41421i		−	0.102329i	0.998690	−	0.0511645i	$-0.0162933\pi$
$192$		0			0
$193$			12.6491i			0.910503i	0.890363	+	0.455251i	$0.150451\pi$
$193$			12.6491i			0.910503i	−0.890363	+	0.455251i	$0.849549\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.2132i			1.51138i	0.654931	+	0.755689i	$0.272698\pi$
$197$			21.2132i			1.51138i	−0.654931	+	0.755689i	$0.727302\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.94427			−0.627765
$204$		0			0
$205$			6.32456i			0.441726i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.47214	+	4.24264i	0.309344	+	0.293470i
$210$		0			0
$211$		−	6.32456i		−	0.435400i	−0.976016	−	0.217700i	$-0.930145\pi$
$211$		−	6.32456i		−	0.435400i	0.976016	−	0.217700i	$-0.0698555\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	5.65685i		−	0.385794i
$216$		0			0
$217$			12.6491i			0.858678i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−26.8328			−1.80497
$222$		0			0
$223$			12.6491i			0.847047i	0.905885	+	0.423524i	$0.139207\pi$
$223$			12.6491i			0.847047i	−0.905885	+	0.423524i	$0.860793\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.8885			1.18730			0.593652	−	0.804722i	$-0.297686\pi$
$227$	17.8885			1.18730			0.593652	+	0.804722i	$0.297686\pi$
$228$		0			0
$229$	20.0000			1.32164			0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000			1.32164			0.660819	+	0.750546i	$0.270209\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	21.2132i		−	1.38972i	−0.719144	−	0.694862i	$-0.755466\pi$
$233$		−	21.2132i		−	1.38972i	0.719144	−	0.694862i	$-0.244534\pi$
$234$		0			0
$235$	−10.0000			−0.652328
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.89949i			0.640345i	0.947359	+	0.320173i	$0.103741\pi$
$239$			9.89949i			0.640345i	−0.947359	+	0.320173i	$0.896259\pi$
$240$		0			0
$241$		−	6.32456i		−	0.407400i	−0.979033	−	0.203700i	$-0.934703\pi$
$241$		−	6.32456i		−	0.407400i	0.979033	−	0.203700i	$-0.0652968\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	4.24264i		−	0.271052i
$246$		0			0
$247$	20.0000	+	18.9737i	1.27257	+	1.20727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	12.7279i		−	0.803379i	−0.915776	−	0.401690i	$-0.868423\pi$
$251$		−	12.7279i		−	0.803379i	0.915776	−	0.401690i	$-0.131577\pi$
$252$		0			0
$253$	10.0000			0.628695
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.47214			0.278964			0.139482	−	0.990225i	$-0.455456\pi$
$257$	4.47214			0.278964			0.139482	+	0.990225i	$0.455456\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.41421i		−	0.0872041i	−0.999049	−	0.0436021i	$-0.986117\pi$
$263$		−	1.41421i		−	0.0872041i	0.999049	−	0.0436021i	$-0.0138834\pi$
$264$		0			0
$265$		−	18.9737i		−	1.16554i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.4164			0.818013			0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164			0.818013			0.409006	+	0.912532i	$0.365875\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			4.24264i			0.255841i
$276$		0			0
$277$	28.0000			1.68236			0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000			1.68236			0.841178	+	0.540758i	$0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.47214			0.266785			0.133393	−	0.991063i	$-0.457413\pi$
$281$	4.47214			0.266785			0.133393	+	0.991063i	$0.457413\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.94427			0.527964
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.47214			−0.261265			−0.130632	−	0.991431i	$-0.541701\pi$
$293$	−4.47214			−0.261265			−0.130632	+	0.991431i	$0.541701\pi$
$294$		0			0
$295$		−	12.6491i		−	0.736460i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	44.7214			2.58630
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$		0			0
$304$		0			0
$305$			11.3137i			0.647821i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.41421i			0.0801927i	0.999196	+	0.0400963i	$0.0127665\pi$
$311$			1.41421i			0.0801927i	−0.999196	+	0.0400963i	$0.987234\pi$
$312$		0			0
$313$	−4.00000			−0.226093			−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000			−0.226093			−0.113047	+	0.993590i	$0.536061\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.47214			0.251180			0.125590	−	0.992082i	$-0.459918\pi$
$317$	4.47214			0.251180			0.125590	+	0.992082i	$0.459918\pi$
$318$		0			0
$319$		−	6.32456i		−	0.354107i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13.4164	+	12.7279i	0.746509	+	0.708201i
$324$		0			0
$325$			18.9737i			1.05247i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			14.1421i			0.779681i
$330$		0			0
$331$		−	12.6491i		−	0.695258i	−0.937632	−	0.347629i	$-0.886987\pi$
$331$		−	12.6491i		−	0.695258i	0.937632	−	0.347629i	$-0.113013\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.94427			−0.488678
$336$		0			0
$337$		−	31.6228i		−	1.72260i	−0.508095	−	0.861301i	$-0.669650\pi$
$337$		−	31.6228i		−	1.72260i	0.508095	−	0.861301i	$-0.330350\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.94427			−0.484359
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	32.5269i		−	1.74614i	−0.487598	−	0.873068i	$-0.662127\pi$
$347$		−	32.5269i		−	1.74614i	0.487598	−	0.873068i	$-0.337873\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	7.07107i		−	0.376355i	−0.982135	−	0.188177i	$-0.939742\pi$
$353$		−	7.07107i		−	0.376355i	0.982135	−	0.188177i	$-0.0602580\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.24264i			0.223918i	0.993713	+	0.111959i	$0.0357125\pi$
$359$			4.24264i			0.223918i	−0.993713	+	0.111959i	$0.964287\pi$
$360$		0			0
$361$	−1.00000	−	18.9737i	−0.0526316	−	0.998614i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			8.48528i			0.444140i
$366$		0			0
$367$	−32.0000			−1.67039			−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000			−1.67039			−0.835193	+	0.549957i	$0.814644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−26.8328			−1.39309
$372$		0			0
$373$			12.6491i			0.654946i	0.944861	+	0.327473i	$0.106197\pi$
$373$			12.6491i			0.654946i	−0.944861	+	0.327473i	$0.893803\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	28.2843i		−	1.45671i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.8885			0.914062			0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885			0.914062			0.457031	+	0.889451i	$0.348913\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	9.89949i		−	0.501924i	−0.967997	−	0.250962i	$-0.919253\pi$
$389$		−	9.89949i		−	0.501924i	0.967997	−	0.250962i	$-0.0807470\pi$
$390$		0			0
$391$	30.0000			1.51717
$392$		0			0
$393$		0			0
$394$		0			0
$395$	17.8885			0.900070
$396$		0			0
$397$	18.0000			0.903394			0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000			0.903394			0.451697	+	0.892171i	$0.350819\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	31.3050			1.56329			0.781647	−	0.623721i	$-0.214380\pi$
$401$	31.3050			1.56329			0.781647	+	0.623721i	$0.214380\pi$
$402$		0			0
$403$	−40.0000			−1.99254
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	6.32456i		−	0.312729i	−0.987699	−	0.156365i	$-0.950023\pi$
$409$		−	6.32456i		−	0.312729i	0.987699	−	0.156365i	$-0.0499775\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.8885			−0.880238
$414$		0			0
$415$	−2.00000			−0.0981761
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	9.89949i		−	0.483622i	−0.970323	−	0.241811i	$-0.922259\pi$
$419$		−	9.89949i		−	0.483622i	0.970323	−	0.241811i	$-0.0777414\pi$
$420$		0			0
$421$			6.32456i			0.308240i	0.988052	+	0.154120i	$0.0492542\pi$
$421$			6.32456i			0.308240i	−0.988052	+	0.154120i	$0.950746\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			12.7279i			0.617395i
$426$		0			0
$427$	16.0000			0.774294
$428$		0			0
$429$		0			0
$430$		0			0
$431$	26.8328			1.29249			0.646246	−	0.763129i	$-0.276338\pi$
$431$	26.8328			1.29249			0.646246	+	0.763129i	$0.276338\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.3607	−	21.2132i	−1.06966	−	1.01477i
$438$		0			0
$439$		−	25.2982i		−	1.20742i	−0.797205	−	0.603709i	$-0.793689\pi$
$439$		−	25.2982i		−	1.20742i	0.797205	−	0.603709i	$-0.206311\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	35.3553i		−	1.67978i	−0.542754	−	0.839891i	$-0.682619\pi$
$443$		−	35.3553i		−	1.67978i	0.542754	−	0.839891i	$-0.317381\pi$
$444$		0			0
$445$			18.9737i			0.899438i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	22.3607			1.05527			0.527633	−	0.849473i	$-0.323080\pi$
$449$	22.3607			1.05527			0.527633	+	0.849473i	$0.323080\pi$
$450$		0			0
$451$			6.32456i			0.297812i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−17.8885			−0.838628
$456$		0			0
$457$	−38.0000			−1.77757			−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000			−1.77757			−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	15.5563i		−	0.724531i	−0.932075	−	0.362266i	$-0.882003\pi$
$461$		−	15.5563i		−	0.724531i	0.932075	−	0.362266i	$-0.117997\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			7.07107i			0.327210i	0.986526	+	0.163605i	$0.0523123\pi$
$467$			7.07107i			0.327210i	−0.986526	+	0.163605i	$0.947688\pi$
$468$		0			0
$469$			12.6491i			0.584082i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5.65685i		−	0.260102i
$474$		0			0
$475$	9.00000	−	9.48683i	0.412948	−	0.435286i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	32.5269i		−	1.48619i	−0.669185	−	0.743096i	$-0.733356\pi$
$479$		−	32.5269i		−	1.48619i	0.669185	−	0.743096i	$-0.266644\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.94427			−0.406138
$486$		0			0
$487$			18.9737i			0.859779i	0.902882	+	0.429889i	$0.141447\pi$
$487$			18.9737i			0.859779i	−0.902882	+	0.429889i	$0.858553\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			15.5563i			0.702048i	0.936366	+	0.351024i	$0.114166\pi$
$491$			15.5563i			0.702048i	−0.936366	+	0.351024i	$0.885834\pi$
$492$		0			0
$493$		−	18.9737i		−	0.854531i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	10.0000			0.447661			0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000			0.447661			0.223831	+	0.974628i	$0.428144\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12.7279i		−	0.567510i	−0.958897	−	0.283755i	$-0.908420\pi$
$503$		−	12.7279i		−	0.567510i	0.958897	−	0.283755i	$-0.0915802\pi$
$504$		0			0
$505$	−2.00000			−0.0889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.3607			0.991120			0.495560	−	0.868574i	$-0.334963\pi$
$509$	22.3607			0.991120			0.495560	+	0.868574i	$0.334963\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−17.8885			−0.788263
$516$		0			0
$517$	−10.0000			−0.439799
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22.3607			0.979639			0.489820	−	0.871824i	$-0.337063\pi$
$521$	22.3607			0.979639			0.489820	+	0.871824i	$0.337063\pi$
$522$		0			0
$523$		−	18.9737i		−	0.829660i	−0.909899	−	0.414830i	$-0.863841\pi$
$523$		−	18.9737i		−	0.829660i	0.909899	−	0.414830i	$-0.136159\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−26.8328			−1.16886
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$		0			0
$532$		0			0
$533$			28.2843i			1.22513i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	4.24264i		−	0.182743i
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.94427			0.383131
$546$		0			0
$547$		−	12.6491i		−	0.540837i	−0.962743	−	0.270418i	$-0.912838\pi$
$547$		−	12.6491i		−	0.540837i	0.962743	−	0.270418i	$-0.0871621\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−13.4164	+	14.1421i	−0.571558	+	0.602475i
$552$		0			0
$553$		−	25.2982i		−	1.07579i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			7.07107i			0.299611i	0.988716	+	0.149805i	$0.0478647\pi$
$557$			7.07107i			0.299611i	−0.988716	+	0.149805i	$0.952135\pi$
$558$		0			0
$559$		−	25.2982i		−	1.07000i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	35.7771			1.50782			0.753912	−	0.656975i	$-0.228164\pi$
$563$	35.7771			1.50782			0.753912	+	0.656975i	$0.228164\pi$
$564$		0			0
$565$		−	6.32456i		−	0.266076i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.47214			0.187482			0.0937408	−	0.995597i	$-0.470117\pi$
$569$	4.47214			0.187482			0.0937408	+	0.995597i	$0.470117\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	21.2132i		−	0.884652i
$576$		0			0
$577$	12.0000			0.499567			0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000			0.499567			0.249783	+	0.968302i	$0.419641\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			2.82843i			0.117343i
$582$		0			0
$583$		−	18.9737i		−	0.785809i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			35.3553i			1.45927i	0.683836	+	0.729636i	$0.260310\pi$
$587$			35.3553i			1.45927i	−0.683836	+	0.729636i	$0.739690\pi$
$588$		0			0
$589$	20.0000	+	18.9737i	0.824086	+	0.781796i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	12.7279i		−	0.522673i	−0.965248	−	0.261337i	$-0.915837\pi$
$593$		−	12.7279i		−	0.522673i	0.965248	−	0.261337i	$-0.0841632\pi$
$594$		0			0
$595$	−12.0000			−0.491952
$596$		0			0
$597$		0			0
$598$		0			0
$599$	44.7214			1.82727			0.913633	−	0.406541i	$-0.133265\pi$
$599$	44.7214			1.82727			0.913633	+	0.406541i	$0.133265\pi$
$600$		0			0
$601$		−	25.2982i		−	1.03194i	−0.856608	−	0.515968i	$-0.827432\pi$
$601$		−	25.2982i		−	1.03194i	0.856608	−	0.515968i	$-0.172568\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			12.7279i			0.517464i
$606$		0			0
$607$			6.32456i			0.256706i	0.991729	+	0.128353i	$0.0409690\pi$
$607$			6.32456i			0.256706i	−0.991729	+	0.128353i	$0.959031\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−44.7214			−1.80923
$612$		0			0
$613$	−24.0000			−0.969351			−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000			−0.969351			−0.484675	+	0.874694i	$0.661062\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.5269i		−	1.30948i	−0.755852	−	0.654742i	$-0.772777\pi$
$617$		−	32.5269i		−	1.30948i	0.755852	−	0.654742i	$-0.227223\pi$
$618$		0			0
$619$	10.0000			0.401934			0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000			0.401934			0.200967	+	0.979598i	$0.435592\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	26.8328			1.07503
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	18.0000			0.716569			0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000			0.716569			0.358284	+	0.933613i	$0.383362\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	26.8328			1.06483
$636$		0			0
$637$		−	18.9737i		−	0.751764i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.47214			0.176639			0.0883194	−	0.996092i	$-0.471850\pi$
$641$	4.47214			0.176639			0.0883194	+	0.996092i	$0.471850\pi$
$642$		0			0
$643$	4.00000			0.157745			0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000			0.157745			0.0788723	+	0.996885i	$0.474868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			46.6690i			1.83475i	0.398024	+	0.917375i	$0.369696\pi$
$647$			46.6690i			1.83475i	−0.398024	+	0.917375i	$0.630304\pi$
$648$		0			0
$649$		−	12.6491i		−	0.496521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			15.5563i			0.608767i	0.952550	+	0.304383i	$0.0984504\pi$
$653$			15.5563i			0.608767i	−0.952550	+	0.304383i	$0.901550\pi$
$654$		0			0
$655$	−22.0000			−0.859611
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−44.7214			−1.74210			−0.871048	−	0.491197i	$-0.836559\pi$
$659$	−44.7214			−1.74210			−0.871048	+	0.491197i	$0.836559\pi$
$660$		0			0
$661$		−	12.6491i		−	0.491993i	−0.969271	−	0.245997i	$-0.920885\pi$
$661$		−	12.6491i		−	0.491993i	0.969271	−	0.245997i	$-0.0791152\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.94427	+	8.48528i	0.346844	+	0.329045i
$666$		0			0
$667$			31.6228i			1.22444i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			11.3137i			0.436761i
$672$		0			0
$673$		−	50.5964i		−	1.95035i	−0.221437	−	0.975175i	$-0.571075\pi$
$673$		−	50.5964i		−	1.95035i	0.221437	−	0.975175i	$-0.428925\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.3050			1.20315			0.601574	−	0.798817i	$-0.294541\pi$
$677$	31.3050			1.20315			0.601574	+	0.798817i	$0.294541\pi$
$678$		0			0
$679$			12.6491i			0.485428i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.8328			1.02673			0.513365	−	0.858171i	$-0.328399\pi$
$683$	26.8328			1.02673			0.513365	+	0.858171i	$0.328399\pi$
$684$		0			0
$685$	14.0000			0.534913
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	84.8528i		−	3.23263i
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			14.1421i			0.536442i
$696$		0			0
$697$			18.9737i			0.718679i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.7279i		−	0.480727i	−0.970683	−	0.240363i	$-0.922733\pi$
$701$		−	12.7279i		−	0.480727i	0.970683	−	0.240363i	$-0.0772666\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			2.82843i			0.106374i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	44.7214			1.67483
$714$		0			0
$715$		−	12.6491i		−	0.473050i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	46.6690i		−	1.74046i	−0.492644	−	0.870231i	$-0.663970\pi$
$719$		−	46.6690i		−	1.74046i	0.492644	−	0.870231i	$-0.336030\pi$
$720$		0			0
$721$			25.2982i			0.942155i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13.4164			−0.498273
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	16.9706i		−	0.627679i
$732$		0			0
$733$	34.0000			1.25582			0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000			1.25582			0.627909	+	0.778287i	$0.283911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.94427			−0.329466
$738$		0			0
$739$	34.0000			1.25071			0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000			1.25071			0.625355	+	0.780340i	$0.284954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.94427			0.328134			0.164067	−	0.986449i	$-0.447539\pi$
$743$	8.94427			0.328134			0.164067	+	0.986449i	$0.447539\pi$
$744$		0			0
$745$	−26.0000			−0.952566
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			12.6491i			0.461573i	0.973004	+	0.230786i	$0.0741298\pi$
$751$			12.6491i			0.461573i	−0.973004	+	0.230786i	$0.925870\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17.8885			−0.651031
$756$		0			0
$757$	12.0000			0.436147			0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000			0.436147			0.218074	+	0.975932i	$0.430023\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.41421i		−	0.0512652i	−0.999671	−	0.0256326i	$-0.991840\pi$
$761$		−	1.41421i		−	0.0512652i	0.999671	−	0.0256326i	$-0.00816000\pi$
$762$		0			0
$763$		−	12.6491i		−	0.457929i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	56.5685i		−	2.04257i
$768$		0			0
$769$	−20.0000			−0.721218			−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000			−0.721218			−0.360609	+	0.932717i	$0.617431\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22.3607			−0.804258			−0.402129	−	0.915583i	$-0.631730\pi$
$773$	−22.3607			−0.804258			−0.402129	+	0.915583i	$0.631730\pi$
$774$		0			0
$775$			18.9737i			0.681554i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.4164	−	14.1421i	0.480693	−	0.506695i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	11.3137i		−	0.403804i
$786$		0			0
$787$			31.6228i			1.12723i	0.826038	+	0.563615i	$0.190590\pi$
$787$			31.6228i			1.12723i	−0.826038	+	0.563615i	$0.809410\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.94427			−0.318022
$792$		0			0
$793$			50.5964i			1.79673i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.47214			0.158411			0.0792056	−	0.996858i	$-0.474762\pi$
$797$	4.47214			0.158411			0.0792056	+	0.996858i	$0.474762\pi$
$798$		0			0
$799$	−30.0000			−1.06132
$800$		0			0
$801$		0			0
$802$		0			0
$803$			8.48528i			0.299439i
$804$		0			0
$805$	20.0000			0.704907
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	18.3848i		−	0.646374i	−0.946335	−	0.323187i	$-0.895246\pi$
$809$		−	18.3848i		−	0.646374i	0.946335	−	0.323187i	$-0.104754\pi$
$810$		0			0
$811$			37.9473i			1.33251i	0.745724	+	0.666256i	$0.232104\pi$
$811$			37.9473i			1.33251i	−0.745724	+	0.666256i	$0.767896\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			8.48528i			0.297226i
$816$		0			0
$817$	−12.0000	+	12.6491i	−0.419827	+	0.442536i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1.41421i		−	0.0493564i	−0.999695	−	0.0246782i	$-0.992144\pi$
$821$		−	1.41421i		−	0.0493564i	0.999695	−	0.0246782i	$-0.00785611\pi$
$822$		0			0
$823$	26.0000			0.906303			0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000			0.906303			0.453152	+	0.891434i	$0.350300\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	26.8328			0.933068			0.466534	−	0.884503i	$-0.345502\pi$
$827$	26.8328			0.933068			0.466534	+	0.884503i	$0.345502\pi$
$828$		0			0
$829$		−	18.9737i		−	0.658983i	−0.944159	−	0.329491i	$-0.893123\pi$
$829$		−	18.9737i		−	0.658983i	0.944159	−	0.329491i	$-0.106877\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	12.7279i		−	0.440996i
$834$		0			0
$835$		−	25.2982i		−	0.875481i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	35.7771			1.23516			0.617581	−	0.786507i	$-0.288113\pi$
$839$	35.7771			1.23516			0.617581	+	0.786507i	$0.288113\pi$
$840$		0			0
$841$	−9.00000			−0.310345
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	38.1838i		−	1.31356i
$846$		0			0
$847$	18.0000			0.618487
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−31.3050			−1.06936			−0.534678	−	0.845056i	$-0.679567\pi$
$857$	−31.3050			−1.06936			−0.534678	+	0.845056i	$0.679567\pi$
$858$		0			0
$859$	−34.0000			−1.16007			−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000			−1.16007			−0.580033	+	0.814593i	$0.696960\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.8328			0.913400			0.456700	−	0.889621i	$-0.349031\pi$
$863$	26.8328			0.913400			0.456700	+	0.889621i	$0.349031\pi$
$864$		0			0
$865$		−	6.32456i		−	0.215041i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17.8885			0.606827
$870$		0			0
$871$	−40.0000			−1.35535
$872$		0			0
$873$		0			0
$874$		0			0
$875$			22.6274i			0.764946i
$876$		0			0
$877$			25.2982i			0.854260i	0.904190	+	0.427130i	$0.140475\pi$
$877$			25.2982i			0.854260i	−0.904190	+	0.427130i	$0.859525\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			29.6985i			1.00057i	0.865862	+	0.500284i	$0.166771\pi$
$881$			29.6985i			1.00057i	−0.865862	+	0.500284i	$0.833229\pi$
$882$		0			0
$883$	−6.00000			−0.201916			−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000			−0.201916			−0.100958	+	0.994891i	$0.532191\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.94427			−0.300319			−0.150160	−	0.988662i	$-0.547979\pi$
$887$	−8.94427			−0.300319			−0.150160	+	0.988662i	$0.547979\pi$
$888$		0			0
$889$		−	37.9473i		−	1.27271i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	22.3607	+	21.2132i	0.748272	+	0.709873i
$894$		0			0
$895$		−	25.2982i		−	0.845626i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	28.2843i		−	0.943333i
$900$		0			0
$901$		−	56.9210i		−	1.89631i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26.8328			−0.891953
$906$		0			0
$907$		−	44.2719i		−	1.47002i	−0.678054	−	0.735012i	$-0.737177\pi$
$907$		−	44.2719i		−	1.47002i	0.678054	−	0.735012i	$-0.262823\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.94427			−0.296337			−0.148168	−	0.988962i	$-0.547338\pi$
$911$	−8.94427			−0.296337			−0.148168	+	0.988962i	$0.547338\pi$
$912$		0			0
$913$	−2.00000			−0.0661903
$914$		0			0
$915$		0			0
$916$		0			0
$917$			31.1127i			1.02743i
$918$		0			0
$919$	6.00000			0.197922			0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000			0.197922			0.0989609	+	0.995091i	$0.468448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			18.3848i			0.603185i	0.953437	+	0.301592i	$0.0975182\pi$
$929$			18.3848i			0.603185i	−0.953437	+	0.301592i	$0.902482\pi$
$930$		0			0
$931$	−9.00000	+	9.48683i	−0.294963	+	0.310918i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	8.48528i		−	0.277498i
$936$		0			0
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.47214			−0.145787			−0.0728937	−	0.997340i	$-0.523223\pi$
$941$	−4.47214			−0.145787			−0.0728937	+	0.997340i	$0.523223\pi$
$942$		0			0
$943$		−	31.6228i		−	1.02978i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	18.3848i		−	0.597425i	−0.954343	−	0.298712i	$-0.903443\pi$
$947$		−	18.3848i		−	0.597425i	0.954343	−	0.298712i	$-0.0965572\pi$
$948$		0			0
$949$			37.9473i			1.23182i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.4164			0.434600			0.217300	−	0.976105i	$-0.430275\pi$
$953$	13.4164			0.434600			0.217300	+	0.976105i	$0.430275\pi$
$954$		0			0
$955$	2.00000			0.0647185
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	19.7990i		−	0.639343i
$960$		0			0
$961$	−9.00000			−0.290323
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−17.8885			−0.575853
$966$		0			0
$967$	2.00000			0.0643157			0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000			0.0643157			0.0321578	+	0.999483i	$0.489762\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	26.8328			0.861106			0.430553	−	0.902565i	$-0.358319\pi$
$971$	26.8328			0.861106			0.430553	+	0.902565i	$0.358319\pi$
$972$		0			0
$973$	20.0000			0.641171
$974$		0			0
$975$		0			0
$976$		0			0
$977$	31.3050			1.00153			0.500767	−	0.865582i	$-0.333051\pi$
$977$	31.3050			1.00153			0.500767	+	0.865582i	$0.333051\pi$
$978$		0			0
$979$			18.9737i			0.606401i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−44.7214			−1.42639			−0.713195	−	0.700966i	$-0.752753\pi$
$983$	−44.7214			−1.42639			−0.713195	+	0.700966i	$0.752753\pi$
$984$		0			0
$985$	−30.0000			−0.955879
$986$		0			0
$987$		0			0
$988$		0			0
$989$			28.2843i			0.899388i
$990$		0			0
$991$		−	18.9737i		−	0.602718i	−0.953511	−	0.301359i	$-0.902560\pi$
$991$		−	18.9737i		−	0.602718i	0.953511	−	0.301359i	$-0.0974403\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−32.0000			−1.01345			−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000			−1.01345			−0.506725	+	0.862108i	$0.669144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.f.f.1025.4		4
3.2	odd	2	inner	2736.2.f.f.1025.2		4
4.3	odd	2		171.2.d.b.170.2	yes	4
12.11	even	2		171.2.d.b.170.3	yes	4
19.18	odd	2	inner	2736.2.f.f.1025.3		4
57.56	even	2	inner	2736.2.f.f.1025.1		4
76.75	even	2		171.2.d.b.170.4	yes	4
228.227	odd	2		171.2.d.b.170.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.d.b.170.1	✓	4	228.227	odd	2
171.2.d.b.170.2	yes	4	4.3	odd	2
171.2.d.b.170.3	yes	4	12.11	even	2
171.2.d.b.170.4	yes	4	76.75	even	2
2736.2.f.f.1025.1		4	57.56	even	2	inner
2736.2.f.f.1025.2		4	3.2	odd	2	inner
2736.2.f.f.1025.3		4	19.18	odd	2	inner
2736.2.f.f.1025.4		4	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{5} +2.00000 q^{7} +O(q^{10})\)	\(q+1.41421i q^{5} +2.00000 q^{7} +1.41421i q^{11} +6.32456i q^{13} +4.24264i q^{17} +(3.00000 - 3.16228i) q^{19} -7.07107i q^{23} +3.00000 q^{25} -4.47214 q^{29} +6.32456i q^{31} +2.82843i q^{35} +4.47214 q^{41} -4.00000 q^{43} +7.07107i q^{47} -3.00000 q^{49} -13.4164 q^{53} -2.00000 q^{55} -8.94427 q^{59} +8.00000 q^{61} -8.94427 q^{65} +6.32456i q^{67} +6.00000 q^{73} +2.82843i q^{77} -12.6491i q^{79} +1.41421i q^{83} -6.00000 q^{85} +13.4164 q^{89} +12.6491i q^{91} +(4.47214 + 4.24264i) q^{95} +6.32456i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} + 12 q^{19} + 12 q^{25} - 16 q^{43} - 12 q^{49} - 8 q^{55} + 32 q^{61} + 24 q^{73} - 24 q^{85}+O(q^{100}) \) 4 * q + 8 * q^7 + 12 * q^19 + 12 * q^25 - 16 * q^43 - 12 * q^49 - 8 * q^55 + 32 * q^61 + 24 * q^73 - 24 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more