LMFDB - Embedded newform 2736.3.o.n.721.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (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:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.184143974400.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 $ x^8 - 4x^7 - 22x^6 + 80x^5 + 215x^4 - 568x^3 - 1022x^2 + 1320*x + 2628
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.5
Root			$-1.51885 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.n.721.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.01452 q^{5} +10.9135 q^{7} +O(q^{10})$	$q+4.01452 q^{5} +10.9135 q^{7} +9.36952 q^{11} -15.4135i q^{13} +13.8125 q^{17} +(18.2685 + 5.22130i) q^{19} +35.0090 q^{23} -8.88362 q^{25} +25.0889i q^{29} +20.4850i q^{31} +43.8125 q^{35} -51.6116i q^{37} -52.7791i q^{41} -23.7704 q^{43} -4.83252 q^{47} +70.1045 q^{49} +70.7300i q^{53} +37.6141 q^{55} +81.7613i q^{59} -12.6366 q^{61} -61.8778i q^{65} -74.5334i q^{67} -52.2889i q^{71} -119.104 q^{73} +102.254 q^{77} -118.003i q^{79} +0.221245 q^{83} +55.4505 q^{85} +77.3777i q^{89} -168.215i q^{91} +(73.3393 + 20.9610i) q^{95} -104.226i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} + 12 q^{7}+O(q^{10}) $ 8 * q - 4 * q^5 + 12 * q^7	$ 8 q - 4 q^{5} + 12 q^{7} + 4 q^{11} - 4 q^{17} + 36 q^{19} - 56 q^{23} + 140 q^{25} + 236 q^{35} - 100 q^{43} - 188 q^{47} - 36 q^{49} - 28 q^{55} - 180 q^{61} - 356 q^{73} - 68 q^{77} + 136 q^{83} + 148 q^{85} - 140 q^{95}+O(q^{100}) $ 8 * q - 4 * q^5 + 12 * q^7 + 4 * q^11 - 4 * q^17 + 36 * q^19 - 56 * q^23 + 140 * q^25 + 236 * q^35 - 100 * q^43 - 188 * q^47 - 36 * q^49 - 28 * q^55 - 180 * q^61 - 356 * q^73 - 68 * q^77 + 136 * q^83 + 148 * q^85 - 140 * q^95

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
$5$	4.01452			0.802904			0.401452	−	0.915880i	$-0.368506\pi$
$5$	4.01452			0.802904			0.401452	+	0.915880i	$0.368506\pi$
$6$		0			0
$7$	10.9135			1.55907			0.779536	−	0.626358i	$-0.215455\pi$
$7$	10.9135			1.55907			0.779536	+	0.626358i	$0.215455\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.36952			0.851774			0.425887	−	0.904776i	$-0.359962\pi$
$11$	9.36952			0.851774			0.425887	+	0.904776i	$0.359962\pi$
$12$		0			0
$13$		−	15.4135i		−	1.18565i	−0.805330	−	0.592826i	$-0.798012\pi$
$13$		−	15.4135i		−	1.18565i	0.805330	−	0.592826i	$-0.201988\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	13.8125			0.812499			0.406249	−	0.913762i	$-0.366836\pi$
$17$	13.8125			0.812499			0.406249	+	0.913762i	$0.366836\pi$
$18$		0			0
$19$	18.2685	+	5.22130i	0.961500	+	0.274805i
$19$	18.2685	+	5.22130i	0.961500	+	0.274805i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	35.0090			1.52213			0.761065	−	0.648675i	$-0.224677\pi$
$23$	35.0090			1.52213			0.761065	+	0.648675i	$0.224677\pi$
$24$		0			0
$25$	−8.88362			−0.355345
$26$		0			0
$27$		0			0
$28$		0			0
$29$			25.0889i			0.865133i	0.901602	+	0.432567i	$0.142392\pi$
$29$			25.0889i			0.865133i	−0.901602	+	0.432567i	$0.857608\pi$
$30$		0			0
$31$			20.4850i			0.660805i	0.943840	+	0.330403i	$0.107185\pi$
$31$			20.4850i			0.660805i	−0.943840	+	0.330403i	$0.892815\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	43.8125			1.25179
$36$		0			0
$37$		−	51.6116i		−	1.39491i	−0.716630	−	0.697454i	$-0.754316\pi$
$37$		−	51.6116i		−	1.39491i	0.716630	−	0.697454i	$-0.245684\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	52.7791i		−	1.28729i	−0.765322	−	0.643647i	$-0.777420\pi$
$41$		−	52.7791i		−	1.28729i	0.765322	−	0.643647i	$-0.222580\pi$
$42$		0			0
$43$	−23.7704			−0.552800			−0.276400	−	0.961043i	$-0.589141\pi$
$43$	−23.7704			−0.552800			−0.276400	+	0.961043i	$0.589141\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.83252			−0.102820			−0.0514098	−	0.998678i	$-0.516371\pi$
$47$	−4.83252			−0.102820			−0.0514098	+	0.998678i	$0.516371\pi$
$48$		0			0
$49$	70.1045			1.43070
$50$		0			0
$51$		0			0
$52$		0			0
$53$			70.7300i			1.33453i	0.744821	+	0.667264i	$0.232535\pi$
$53$			70.7300i			1.33453i	−0.744821	+	0.667264i	$0.767465\pi$
$54$		0			0
$55$	37.6141			0.683893
$56$		0			0
$57$		0			0
$58$		0			0
$59$			81.7613i			1.38578i	0.721041	+	0.692892i	$0.243664\pi$
$59$			81.7613i			1.38578i	−0.721041	+	0.692892i	$0.756336\pi$
$60$		0			0
$61$	−12.6366			−0.207158			−0.103579	−	0.994621i	$-0.533029\pi$
$61$	−12.6366			−0.207158			−0.103579	+	0.994621i	$0.533029\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	61.8778i		−	0.951966i
$66$		0			0
$67$		−	74.5334i		−	1.11244i	−0.831035	−	0.556220i	$-0.812251\pi$
$67$		−	74.5334i		−	1.11244i	0.831035	−	0.556220i	$-0.187749\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	52.2889i		−	0.736463i	−0.929734	−	0.368231i	$-0.879963\pi$
$71$		−	52.2889i		−	0.736463i	0.929734	−	0.368231i	$-0.120037\pi$
$72$		0			0
$73$	−119.104			−1.63157			−0.815784	−	0.578356i	$-0.803694\pi$
$73$	−119.104			−1.63157			−0.815784	+	0.578356i	$0.803694\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	102.254			1.32798
$78$		0			0
$79$		−	118.003i		−	1.49371i	−0.664986	−	0.746856i	$-0.731562\pi$
$79$		−	118.003i		−	1.49371i	0.664986	−	0.746856i	$-0.268438\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.221245			0.00266561			0.00133280	−	0.999999i	$-0.499576\pi$
$83$	0.221245			0.00266561			0.00133280	+	0.999999i	$0.499576\pi$
$84$		0			0
$85$	55.4505			0.652359
$86$		0			0
$87$		0			0
$88$		0			0
$89$			77.3777i			0.869413i	0.900572	+	0.434706i	$0.143148\pi$
$89$			77.3777i			0.869413i	−0.900572	+	0.434706i	$0.856852\pi$
$90$		0			0
$91$		−	168.215i		−	1.84852i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	73.3393	+	20.9610i	0.771992	+	0.220642i
$96$		0			0
$97$		−	104.226i		−	1.07450i	−0.843424	−	0.537248i	$-0.819464\pi$
$97$		−	104.226i		−	1.07450i	0.843424	−	0.537248i	$-0.180536\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−73.8617			−0.731304			−0.365652	−	0.930752i	$-0.619154\pi$
$101$	−73.8617			−0.731304			−0.365652	+	0.930752i	$0.619154\pi$
$102$		0			0
$103$			37.9010i			0.367971i	0.982929	+	0.183985i	$0.0588999\pi$
$103$			37.9010i			0.367971i	−0.982929	+	0.183985i	$0.941100\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			51.7574i			0.483714i	0.970312	+	0.241857i	$0.0777565\pi$
$107$			51.7574i			0.483714i	−0.970312	+	0.241857i	$0.922243\pi$
$108$		0			0
$109$			107.995i			0.990780i	0.868671	+	0.495390i	$0.164975\pi$
$109$			107.995i			0.990780i	−0.868671	+	0.495390i	$0.835025\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			168.728i			1.49317i	0.665292	+	0.746584i	$0.268307\pi$
$113$			168.728i			1.49317i	−0.665292	+	0.746584i	$0.731693\pi$
$114$		0			0
$115$	140.544			1.22212
$116$		0			0
$117$		0			0
$118$		0			0
$119$	150.743			1.26674
$120$		0			0
$121$	−33.2121			−0.274480
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−136.027			−1.08821
$126$		0			0
$127$			228.668i			1.80054i	0.435335	+	0.900269i	$0.356630\pi$
$127$			228.668i			1.80054i	−0.435335	+	0.900269i	$0.643370\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−122.408			−0.934412			−0.467206	−	0.884148i	$-0.654739\pi$
$131$	−122.408			−0.934412			−0.467206	+	0.884148i	$0.654739\pi$
$132$		0			0
$133$	199.373	+	56.9827i	1.49905	+	0.428441i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−47.0670			−0.343555			−0.171777	−	0.985136i	$-0.554951\pi$
$137$	−47.0670			−0.343555			−0.171777	+	0.985136i	$0.554951\pi$
$138$		0			0
$139$	−187.564			−1.34938			−0.674692	−	0.738100i	$-0.735723\pi$
$139$	−187.564			−1.34938			−0.674692	+	0.738100i	$0.735723\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	144.417i		−	1.00991i
$144$		0			0
$145$			100.720i			0.694619i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−45.5143			−0.305465			−0.152732	−	0.988268i	$-0.548807\pi$
$149$	−45.5143			−0.305465			−0.152732	+	0.988268i	$0.548807\pi$
$150$		0			0
$151$		−	115.304i		−	0.763605i	−0.924244	−	0.381802i	$-0.875303\pi$
$151$		−	115.304i		−	0.763605i	0.924244	−	0.381802i	$-0.124697\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			82.2373i			0.530563i
$156$		0			0
$157$	−52.5813			−0.334913			−0.167456	−	0.985880i	$-0.553555\pi$
$157$	−52.5813			−0.334913			−0.167456	+	0.985880i	$0.553555\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	382.071			2.37311
$162$		0			0
$163$	−139.826			−0.857830			−0.428915	−	0.903345i	$-0.641104\pi$
$163$	−139.826			−0.857830			−0.428915	+	0.903345i	$0.641104\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	81.7059i		−	0.489257i	−0.969617	−	0.244628i	$-0.921334\pi$
$167$		−	81.7059i		−	0.489257i	0.969617	−	0.244628i	$-0.0786660\pi$
$168$		0			0
$169$	−68.5755			−0.405772
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	28.4822i		−	0.164637i	−0.996606	−	0.0823186i	$-0.973768\pi$
$173$		−	28.4822i		−	0.164637i	0.996606	−	0.0823186i	$-0.0262325\pi$
$174$		0			0
$175$	−96.9514			−0.554008
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	188.954i		−	1.05561i	−0.849365	−	0.527806i	$-0.823015\pi$
$179$		−	188.954i		−	1.05561i	0.849365	−	0.527806i	$-0.176985\pi$
$180$		0			0
$181$		−	83.0719i		−	0.458961i	−0.973313	−	0.229480i	$-0.926297\pi$
$181$		−	83.0719i		−	0.458961i	0.973313	−	0.229480i	$-0.0737027\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	207.196i		−	1.11998i
$186$		0			0
$187$	129.416			0.692066
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−195.604			−1.02411			−0.512053	−	0.858954i	$-0.671115\pi$
$191$	−195.604			−1.02411			−0.512053	+	0.858954i	$0.671115\pi$
$192$		0			0
$193$		−	207.085i		−	1.07298i	−0.843906	−	0.536491i	$-0.819750\pi$
$193$		−	207.085i		−	1.07298i	0.843906	−	0.536491i	$-0.180250\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	357.723			1.81586			0.907928	−	0.419127i	$-0.137664\pi$
$197$	357.723			1.81586			0.907928	+	0.419127i	$0.137664\pi$
$198$		0			0
$199$	226.328			1.13733			0.568663	−	0.822571i	$-0.307461\pi$
$199$	226.328			1.13733			0.568663	+	0.822571i	$0.307461\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			273.807i			1.34880i
$204$		0			0
$205$		−	211.883i		−	1.03357i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	171.167	+	48.9211i	0.818981	+	0.234072i
$210$		0			0
$211$			329.588i			1.56203i	0.624513	+	0.781014i	$0.285297\pi$
$211$			329.588i			1.56203i	−0.624513	+	0.781014i	$0.714703\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−95.4267			−0.443845
$216$		0			0
$217$			223.563i			1.03024i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	212.898i		−	0.963341i
$222$		0			0
$223$			349.473i			1.56714i	0.621301	+	0.783572i	$0.286605\pi$
$223$			349.473i			1.56714i	−0.621301	+	0.783572i	$0.713395\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			185.986i			0.819322i	0.912238	+	0.409661i	$0.134353\pi$
$227$			185.986i			0.819322i	−0.912238	+	0.409661i	$0.865647\pi$
$228$		0			0
$229$	−83.5787			−0.364972			−0.182486	−	0.983208i	$-0.558414\pi$
$229$	−83.5787			−0.364972			−0.182486	+	0.983208i	$0.558414\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−98.8298			−0.424162			−0.212081	−	0.977252i	$-0.568024\pi$
$233$	−98.8298			−0.424162			−0.212081	+	0.977252i	$0.568024\pi$
$234$		0			0
$235$	−19.4003			−0.0825543
$236$		0			0
$237$		0			0
$238$		0			0
$239$	428.563			1.79315			0.896575	−	0.442891i	$-0.146047\pi$
$239$	428.563			1.79315			0.896575	+	0.442891i	$0.146047\pi$
$240$		0			0
$241$		−	252.120i		−	1.04614i	−0.852289	−	0.523071i	$-0.824786\pi$
$241$		−	252.120i		−	1.04614i	0.852289	−	0.523071i	$-0.175214\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	281.436			1.14872
$246$		0			0
$247$	80.4785	−	281.581i	0.325824	−	1.14000i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	349.189			1.39119			0.695597	−	0.718433i	$-0.255140\pi$
$251$	349.189			1.39119			0.695597	+	0.718433i	$0.255140\pi$
$252$		0			0
$253$	328.017			1.29651
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	225.053i		−	0.875694i	−0.899050	−	0.437847i	$-0.855741\pi$
$257$		−	225.053i		−	0.875694i	0.899050	−	0.437847i	$-0.144259\pi$
$258$		0			0
$259$		−	563.263i		−	2.17476i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	325.838			1.23893			0.619463	−	0.785026i	$-0.287350\pi$
$263$	325.838			1.23893			0.619463	+	0.785026i	$0.287350\pi$
$264$		0			0
$265$			283.947i			1.07150i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	109.763i		−	0.408041i	−0.978967	−	0.204021i	$-0.934599\pi$
$269$		−	109.763i		−	0.408041i	0.978967	−	0.204021i	$-0.0654009\pi$
$270$		0			0
$271$	195.476			0.721314			0.360657	−	0.932699i	$-0.382553\pi$
$271$	195.476			0.721314			0.360657	+	0.932699i	$0.382553\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−83.2352			−0.302674
$276$		0			0
$277$	353.671			1.27679			0.638395	−	0.769709i	$-0.279599\pi$
$277$	353.671			1.27679			0.638395	+	0.769709i	$0.279599\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	48.7708i		−	0.173561i	−0.996227	−	0.0867807i	$-0.972342\pi$
$281$		−	48.7708i		−	0.173561i	0.996227	−	0.0867807i	$-0.0276580\pi$
$282$		0			0
$283$	−49.6959			−0.175604			−0.0878019	−	0.996138i	$-0.527984\pi$
$283$	−49.6959			−0.175604			−0.0878019	+	0.996138i	$0.527984\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	576.004i		−	2.00698i
$288$		0			0
$289$	−98.2154			−0.339846
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	557.273i		−	1.90195i	−0.309260	−	0.950977i	$-0.600081\pi$
$293$		−	557.273i		−	1.90195i	0.309260	−	0.950977i	$-0.399919\pi$
$294$		0			0
$295$			328.232i			1.11265i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	539.611i		−	1.80472i
$300$		0			0
$301$	−259.418			−0.861854
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−50.7300			−0.166328
$306$		0			0
$307$			605.567i			1.97253i	0.165163	+	0.986266i	$0.447185\pi$
$307$			605.567i			1.97253i	−0.165163	+	0.986266i	$0.552815\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−36.9963			−0.118959			−0.0594796	−	0.998230i	$-0.518944\pi$
$311$	−36.9963			−0.118959			−0.0594796	+	0.998230i	$0.518944\pi$
$312$		0			0
$313$	518.918			1.65789			0.828943	−	0.559333i	$-0.188943\pi$
$313$	518.918			1.65789			0.828943	+	0.559333i	$0.188943\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	16.6943i		−	0.0526636i	−0.999653	−	0.0263318i	$-0.991617\pi$
$317$		−	16.6943i		−	0.0526636i	0.999653	−	0.0263318i	$-0.00838263\pi$
$318$		0			0
$319$			235.071i			0.736898i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	252.333	+	72.1192i	0.781218	+	0.223279i
$324$		0			0
$325$			136.928i			0.421315i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−52.7397			−0.160303
$330$		0			0
$331$		−	57.7517i		−	0.174477i	−0.996187	−	0.0872383i	$-0.972196\pi$
$331$		−	57.7517i		−	0.174477i	0.996187	−	0.0872383i	$-0.0278041\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	299.216i		−	0.893182i
$336$		0			0
$337$			443.995i			1.31749i	0.752366	+	0.658746i	$0.228913\pi$
$337$			443.995i			1.31749i	−0.752366	+	0.658746i	$0.771087\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			191.934i			0.562857i
$342$		0			0
$343$	230.324			0.671498
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−397.045			−1.14422			−0.572111	−	0.820176i	$-0.693875\pi$
$347$	−397.045			−1.14422			−0.572111	+	0.820176i	$0.693875\pi$
$348$		0			0
$349$	−221.288			−0.634062			−0.317031	−	0.948415i	$-0.602686\pi$
$349$	−221.288			−0.634062			−0.317031	+	0.948415i	$0.602686\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	483.388			1.36937			0.684685	−	0.728839i	$-0.259940\pi$
$353$	483.388			1.36937			0.684685	+	0.728839i	$0.259940\pi$
$354$		0			0
$355$		−	209.915i		−	0.591309i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−305.090			−0.849832			−0.424916	−	0.905233i	$-0.639696\pi$
$359$	−305.090			−0.849832			−0.424916	+	0.905233i	$0.639696\pi$
$360$		0			0
$361$	306.476	+	190.771i	0.848964	+	0.528451i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−478.148			−1.30999
$366$		0			0
$367$	−10.8647			−0.0296040			−0.0148020	−	0.999890i	$-0.504712\pi$
$367$	−10.8647			−0.0296040			−0.0148020	+	0.999890i	$0.504712\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			771.912i			2.08063i
$372$		0			0
$373$		−	0.596672i		−	0.00159966i	−1.00000		0.000799829i	$-0.999745\pi$
$373$		−	0.596672i		−	0.00159966i	1.00000		0.000799829i	$-0.000254593\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	386.707			1.02575
$378$		0			0
$379$		−	500.170i		−	1.31971i	−0.751393	−	0.659855i	$-0.770618\pi$
$379$		−	500.170i		−	1.31971i	0.751393	−	0.659855i	$-0.229382\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			446.252i			1.16515i	0.812778	+	0.582574i	$0.197954\pi$
$383$			446.252i			1.16515i	−0.812778	+	0.582574i	$0.802046\pi$
$384$		0			0
$385$	410.502			1.06624
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−277.045			−0.712197			−0.356099	−	0.934448i	$-0.615893\pi$
$389$	−277.045			−0.712197			−0.356099	+	0.934448i	$0.615893\pi$
$390$		0			0
$391$	483.561			1.23673
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	473.727i		−	1.19931i
$396$		0			0
$397$	−391.139			−0.985237			−0.492619	−	0.870245i	$-0.663960\pi$
$397$	−391.139			−0.985237			−0.492619	+	0.870245i	$0.663960\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			13.0516i			0.0325477i	0.999868	+	0.0162738i	$0.00518035\pi$
$401$			13.0516i			0.0325477i	−0.999868	+	0.0162738i	$0.994820\pi$
$402$		0			0
$403$	315.745			0.783486
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	483.576i		−	1.18815i
$408$		0			0
$409$		−	36.1329i		−	0.0883444i	−0.999024	−	0.0441722i	$-0.985935\pi$
$409$		−	36.1329i		−	0.0883444i	0.999024	−	0.0441722i	$-0.0140650\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			892.302i			2.16054i
$414$		0			0
$415$	0.888194			0.00214023
$416$		0			0
$417$		0			0
$418$		0			0
$419$	388.540			0.927304			0.463652	−	0.886017i	$-0.346539\pi$
$419$	388.540			0.927304			0.463652	+	0.886017i	$0.346539\pi$
$420$		0			0
$421$			281.608i			0.668903i	0.942413	+	0.334451i	$0.108551\pi$
$421$			281.608i			0.668903i	−0.942413	+	0.334451i	$0.891449\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−122.705			−0.288717
$426$		0			0
$427$	−137.910			−0.322974
$428$		0			0
$429$		0			0
$430$		0			0
$431$			384.648i			0.892456i	0.894919	+	0.446228i	$0.147233\pi$
$431$			384.648i			0.892456i	−0.894919	+	0.446228i	$0.852767\pi$
$432$		0			0
$433$		−	530.903i		−	1.22610i	−0.790043	−	0.613052i	$-0.789942\pi$
$433$		−	530.903i		−	1.22610i	0.790043	−	0.613052i	$-0.210058\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	639.562	+	182.793i	1.46353	+	0.418290i
$438$		0			0
$439$			204.751i			0.466404i	0.972428	+	0.233202i	$0.0749203\pi$
$439$			204.751i			0.466404i	−0.972428	+	0.233202i	$0.925080\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	135.745			0.306422			0.153211	−	0.988193i	$-0.451039\pi$
$443$	135.745			0.306422			0.153211	+	0.988193i	$0.451039\pi$
$444$		0			0
$445$			310.634i			0.698055i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			358.310i			0.798018i	0.916947	+	0.399009i	$0.130646\pi$
$449$			358.310i			0.798018i	−0.916947	+	0.399009i	$0.869354\pi$
$450$		0			0
$451$		−	494.514i		−	1.09648i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	675.303i		−	1.48418i
$456$		0			0
$457$	490.886			1.07415			0.537075	−	0.843535i	$-0.319529\pi$
$457$	490.886			1.07415			0.537075	+	0.843535i	$0.319529\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−460.649			−0.999239			−0.499619	−	0.866245i	$-0.666527\pi$
$461$	−460.649			−0.999239			−0.499619	+	0.866245i	$0.666527\pi$
$462$		0			0
$463$	109.757			0.237055			0.118528	−	0.992951i	$-0.462183\pi$
$463$	109.757			0.237055			0.118528	+	0.992951i	$0.462183\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	411.822			0.881846			0.440923	−	0.897545i	$-0.354651\pi$
$467$	411.822			0.881846			0.440923	+	0.897545i	$0.354651\pi$
$468$		0			0
$469$		−	813.421i		−	1.73437i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−222.717			−0.470861
$474$		0			0
$475$	−162.290	−	46.3841i	−0.341664	−	0.0976507i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−504.643			−1.05353			−0.526767	−	0.850010i	$-0.676596\pi$
$479$	−504.643			−1.05353			−0.526767	+	0.850010i	$0.676596\pi$
$480$		0			0
$481$	−795.514			−1.65388
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	418.418i		−	0.862718i
$486$		0			0
$487$		−	275.399i		−	0.565501i	−0.959193	−	0.282751i	$-0.908753\pi$
$487$		−	275.399i		−	0.565501i	0.959193	−	0.282751i	$-0.0912469\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	752.633			1.53286			0.766429	−	0.642329i	$-0.222032\pi$
$491$	752.633			1.53286			0.766429	+	0.642329i	$0.222032\pi$
$492$		0			0
$493$			346.539i			0.702920i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	570.654i		−	1.14820i
$498$		0			0
$499$	−26.8053			−0.0537180			−0.0268590	−	0.999639i	$-0.508551\pi$
$499$	−26.8053			−0.0537180			−0.0268590	+	0.999639i	$0.508551\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−650.252			−1.29275			−0.646374	−	0.763021i	$-0.723715\pi$
$503$	−650.252			−1.29275			−0.646374	+	0.763021i	$0.723715\pi$
$504$		0			0
$505$	−296.520			−0.587167
$506$		0			0
$507$		0			0
$508$		0			0
$509$			634.900i			1.24735i	0.781685	+	0.623673i	$0.214361\pi$
$509$			634.900i			1.24735i	−0.781685	+	0.623673i	$0.785639\pi$
$510$		0			0
$511$	−1299.85			−2.54373
$512$		0			0
$513$		0			0
$514$		0			0
$515$			152.154i			0.295445i
$516$		0			0
$517$	−45.2784			−0.0875791
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	296.431i		−	0.568965i	−0.958681	−	0.284483i	$-0.908178\pi$
$521$		−	296.431i		−	0.568965i	0.958681	−	0.284483i	$-0.0918218\pi$
$522$		0			0
$523$		−	607.973i		−	1.16247i	−0.813735	−	0.581237i	$-0.802569\pi$
$523$		−	607.973i		−	1.16247i	0.813735	−	0.581237i	$-0.197431\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			282.948i			0.536904i
$528$		0			0
$529$	696.630			1.31688
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−813.509			−1.52628
$534$		0			0
$535$			207.781i			0.388376i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	656.845			1.21864
$540$		0			0
$541$	−926.384			−1.71236			−0.856178	−	0.516681i	$-0.827167\pi$
$541$	−926.384			−1.71236			−0.856178	+	0.516681i	$0.827167\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			433.548i			0.795501i
$546$		0			0
$547$		−	158.814i		−	0.290336i	−0.989407	−	0.145168i	$-0.953628\pi$
$547$		−	158.814i		−	0.290336i	0.989407	−	0.145168i	$-0.0463723\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−130.997	+	458.336i	−0.237743	+	0.831825i
$552$		0			0
$553$		−	1287.83i		−	2.32880i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−475.603			−0.853866			−0.426933	−	0.904283i	$-0.640406\pi$
$557$	−475.603			−0.853866			−0.426933	+	0.904283i	$0.640406\pi$
$558$		0			0
$559$			366.384i			0.655428i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			547.620i			0.972682i	0.873769	+	0.486341i	$0.161669\pi$
$563$			547.620i			0.972682i	−0.873769	+	0.486341i	$0.838331\pi$
$564$		0			0
$565$			677.362i			1.19887i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	873.227i		−	1.53467i	−0.641247	−	0.767335i	$-0.721583\pi$
$569$		−	873.227i		−	1.53467i	0.641247	−	0.767335i	$-0.278417\pi$
$570$		0			0
$571$	−39.3341			−0.0688863			−0.0344431	−	0.999407i	$-0.510966\pi$
$571$	−39.3341			−0.0688863			−0.0344431	+	0.999407i	$0.510966\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−311.007			−0.540881
$576$		0			0
$577$	29.7849			0.0516203			0.0258102	−	0.999667i	$-0.491783\pi$
$577$	29.7849			0.0516203			0.0258102	+	0.999667i	$0.491783\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.41456			0.00415587
$582$		0			0
$583$			662.706i			1.13672i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−424.980			−0.723986			−0.361993	−	0.932181i	$-0.617904\pi$
$587$	−424.980			−0.723986			−0.361993	+	0.932181i	$0.617904\pi$
$588$		0			0
$589$	−106.958	+	374.230i	−0.181593	+	0.635364i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	551.577			0.930146			0.465073	−	0.885272i	$-0.346028\pi$
$593$	551.577			0.930146			0.465073	+	0.885272i	$0.346028\pi$
$594$		0			0
$595$	605.159			1.01707
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1048.92i		−	1.75111i	−0.483117	−	0.875556i	$-0.660496\pi$
$599$		−	1048.92i		−	1.75111i	0.483117	−	0.875556i	$-0.339504\pi$
$600$		0			0
$601$			177.370i			0.295125i	0.989053	+	0.147563i	$0.0471428\pi$
$601$			177.370i			0.295125i	−0.989053	+	0.147563i	$0.952857\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−133.331			−0.220382
$606$		0			0
$607$			560.053i			0.922658i	0.887229	+	0.461329i	$0.152627\pi$
$607$			560.053i			0.922658i	−0.887229	+	0.461329i	$0.847373\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			74.4860i			0.121908i
$612$		0			0
$613$	−707.650			−1.15441			−0.577203	−	0.816601i	$-0.695856\pi$
$613$	−707.650			−1.15441			−0.577203	+	0.816601i	$0.695856\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−352.563			−0.571415			−0.285707	−	0.958317i	$-0.592229\pi$
$617$	−352.563			−0.571415			−0.285707	+	0.958317i	$0.592229\pi$
$618$		0			0
$619$	1099.71			1.77660			0.888298	−	0.459267i	$-0.151888\pi$
$619$	1099.71			1.77660			0.888298	+	0.459267i	$0.151888\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			844.462i			1.35548i
$624$		0			0
$625$	−323.991			−0.518385
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	712.884i		−	1.13336i
$630$		0			0
$631$	−574.876			−0.911056			−0.455528	−	0.890221i	$-0.650549\pi$
$631$	−574.876			−0.911056			−0.455528	+	0.890221i	$0.650549\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			917.994i			1.44566i
$636$		0			0
$637$		−	1080.55i		−	1.69632i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			690.617i			1.07741i	0.842496	+	0.538703i	$0.181085\pi$
$641$			690.617i			1.07741i	−0.842496	+	0.538703i	$0.818915\pi$
$642$		0			0
$643$	−367.510			−0.571556			−0.285778	−	0.958296i	$-0.592252\pi$
$643$	−367.510			−0.571556			−0.285778	+	0.958296i	$0.592252\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1099.22			1.69895			0.849475	−	0.527630i	$-0.176919\pi$
$647$	1099.22			1.69895			0.849475	+	0.527630i	$0.176919\pi$
$648$		0			0
$649$			766.064i			1.18038i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−125.694			−0.192487			−0.0962434	−	0.995358i	$-0.530683\pi$
$653$	−125.694			−0.192487			−0.0962434	+	0.995358i	$0.530683\pi$
$654$		0			0
$655$	−491.410			−0.750244
$656$		0			0
$657$		0			0
$658$		0			0
$659$			186.946i			0.283681i	0.989890	+	0.141840i	$0.0453020\pi$
$659$			186.946i			0.283681i	−0.989890	+	0.141840i	$0.954698\pi$
$660$		0			0
$661$			358.916i			0.542990i	0.962440	+	0.271495i	$0.0875180\pi$
$661$			358.916i			0.542990i	−0.962440	+	0.271495i	$0.912482\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	800.388	+	228.758i	1.20359	+	0.343997i
$666$		0			0
$667$			878.336i			1.31685i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−118.399			−0.176452
$672$		0			0
$673$		−	1193.93i		−	1.77404i	−0.461735	−	0.887018i	$-0.652773\pi$
$673$		−	1193.93i		−	1.77404i	0.461735	−	0.887018i	$-0.347227\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1262.80i			1.86528i	0.360807	+	0.932640i	$0.382501\pi$
$677$			1262.80i			1.86528i	−0.360807	+	0.932640i	$0.617499\pi$
$678$		0			0
$679$		−	1137.47i		−	1.67522i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	210.739i		−	0.308549i	−0.988028	−	0.154275i	$-0.950696\pi$
$683$		−	210.739i		−	0.308549i	0.988028	−	0.154275i	$-0.0493040\pi$
$684$		0			0
$685$	−188.952			−0.275842
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1090.20			1.58229
$690$		0			0
$691$	−284.971			−0.412404			−0.206202	−	0.978509i	$-0.566110\pi$
$691$	−284.971			−0.412404			−0.206202	+	0.978509i	$0.566110\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−752.981			−1.08343
$696$		0			0
$697$		−	729.010i		−	1.04593i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1034.70			−1.47604			−0.738018	−	0.674781i	$-0.764238\pi$
$701$	−1034.70			−1.47604			−0.738018	+	0.674781i	$0.764238\pi$
$702$		0			0
$703$	269.480	−	942.866i	0.383328	−	1.34120i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−806.090			−1.14016
$708$		0			0
$709$	220.328			0.310759			0.155379	−	0.987855i	$-0.450340\pi$
$709$	220.328			0.310759			0.155379	+	0.987855i	$0.450340\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			717.158i			1.00583i
$714$		0			0
$715$		−	579.765i		−	0.810860i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−827.065			−1.15030			−0.575150	−	0.818048i	$-0.695056\pi$
$719$	−827.065			−1.15030			−0.575150	+	0.818048i	$0.695056\pi$
$720$		0			0
$721$			413.633i			0.573693i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	222.880i		−	0.307421i
$726$		0			0
$727$	−390.000			−0.536451			−0.268225	−	0.963356i	$-0.586437\pi$
$727$	−390.000			−0.536451			−0.268225	+	0.963356i	$0.586437\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−328.328			−0.449149
$732$		0			0
$733$	300.656			0.410172			0.205086	−	0.978744i	$-0.434253\pi$
$733$	300.656			0.410172			0.205086	+	0.978744i	$0.434253\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	698.342i		−	0.947547i
$738$		0			0
$739$	−1281.11			−1.73357			−0.866787	−	0.498679i	$-0.833819\pi$
$739$	−1281.11			−1.73357			−0.866787	+	0.498679i	$0.833819\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1121.60i			1.50956i	0.655980	+	0.754779i	$0.272256\pi$
$743$			1121.60i			1.50956i	−0.655980	+	0.754779i	$0.727744\pi$
$744$		0			0
$745$	−182.718			−0.245259
$746$		0			0
$747$		0			0
$748$		0			0
$749$			564.854i			0.754145i
$750$		0			0
$751$			96.4185i			0.128387i	0.997937	+	0.0641934i	$0.0204475\pi$
$751$			96.4185i			0.128387i	−0.997937	+	0.0641934i	$0.979553\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	462.892i		−	0.613102i
$756$		0			0
$757$	−1151.12			−1.52063			−0.760317	−	0.649552i	$-0.774956\pi$
$757$	−1151.12			−1.52063			−0.760317	+	0.649552i	$0.774956\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−402.552			−0.528978			−0.264489	−	0.964389i	$-0.585203\pi$
$761$	−402.552			−0.528978			−0.264489	+	0.964389i	$0.585203\pi$
$762$		0			0
$763$			1178.60i			1.54470i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1260.23			1.64306
$768$		0			0
$769$	1205.07			1.56706			0.783529	−	0.621355i	$-0.213418\pi$
$769$	1205.07			1.56706			0.783529	+	0.621355i	$0.213418\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			123.706i			0.160033i	0.996794	+	0.0800167i	$0.0254974\pi$
$773$			123.706i			0.160033i	−0.996794	+	0.0800167i	$0.974503\pi$
$774$		0			0
$775$		−	181.981i		−	0.234814i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	275.576	−	964.194i	0.353755	−	1.23773i
$780$		0			0
$781$		−	489.921i		−	0.627300i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−211.089			−0.268903
$786$		0			0
$787$		−	495.008i		−	0.628981i	−0.949261	−	0.314490i	$-0.898166\pi$
$787$		−	495.008i		−	0.628981i	0.949261	−	0.314490i	$-0.101834\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1841.41i			2.32795i
$792$		0			0
$793$			194.775i			0.245617i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			478.479i			0.600350i	0.953884	+	0.300175i	$0.0970450\pi$
$797$			478.479i			0.600350i	−0.953884	+	0.300175i	$0.902955\pi$
$798$		0			0
$799$	−66.7491			−0.0835408
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1115.95			−1.38973
$804$		0			0
$805$	1533.83			1.90538
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1364.41			1.68654			0.843268	−	0.537493i	$-0.180628\pi$
$809$	1364.41			1.68654			0.843268	+	0.537493i	$0.180628\pi$
$810$		0			0
$811$			743.540i			0.916819i	0.888741	+	0.458410i	$0.151581\pi$
$811$			743.540i			0.916819i	−0.888741	+	0.458410i	$0.848419\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−561.336			−0.688755
$816$		0			0
$817$	−434.249	−	124.112i	−0.531517	−	0.151912i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−882.499			−1.07491			−0.537454	−	0.843293i	$-0.680614\pi$
$821$	−882.499			−1.07491			−0.537454	+	0.843293i	$0.680614\pi$
$822$		0			0
$823$	1481.17			1.79973			0.899863	−	0.436173i	$-0.143666\pi$
$823$	1481.17			1.79973			0.899863	+	0.436173i	$0.143666\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			976.130i			1.18033i	0.807284	+	0.590163i	$0.200937\pi$
$827$			976.130i			1.18033i	−0.807284	+	0.590163i	$0.799063\pi$
$828$		0			0
$829$		−	264.576i		−	0.319150i	−0.987186	−	0.159575i	$-0.948988\pi$
$829$		−	264.576i		−	0.319150i	0.987186	−	0.159575i	$-0.0510124\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	968.317			1.16245
$834$		0			0
$835$		−	328.010i		−	0.392826i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	775.780i		−	0.924649i	−0.886711	−	0.462324i	$-0.847016\pi$
$839$		−	775.780i		−	0.924649i	0.886711	−	0.462324i	$-0.152984\pi$
$840$		0			0
$841$	211.549			0.251545
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−275.298			−0.325796
$846$		0			0
$847$	−362.461			−0.427935
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1806.87i		−	2.12323i
$852$		0			0
$853$	−103.897			−0.121802			−0.0609012	−	0.998144i	$-0.519397\pi$
$853$	−103.897			−0.121802			−0.0609012	+	0.998144i	$0.519397\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			928.401i			1.08331i	0.840599	+	0.541657i	$0.182203\pi$
$857$			928.401i			1.08331i	−0.840599	+	0.541657i	$0.817797\pi$
$858$		0			0
$859$	253.490			0.295099			0.147549	−	0.989055i	$-0.452861\pi$
$859$	253.490			0.295099			0.147549	+	0.989055i	$0.452861\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	621.081i		−	0.719677i	−0.933015	−	0.359839i	$-0.882832\pi$
$863$		−	621.081i		−	0.719677i	0.933015	−	0.359839i	$-0.117168\pi$
$864$		0			0
$865$		−	114.342i		−	0.132188i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1105.63i		−	1.27231i
$870$		0			0
$871$	−1148.82			−1.31897
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1484.53			−1.69660
$876$		0			0
$877$		−	680.068i		−	0.775448i	−0.921775	−	0.387724i	$-0.873261\pi$
$877$		−	680.068i		−	0.775448i	0.921775	−	0.387724i	$-0.126739\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−422.081			−0.479093			−0.239546	−	0.970885i	$-0.576999\pi$
$881$	−422.081			−0.479093			−0.239546	+	0.970885i	$0.576999\pi$
$882$		0			0
$883$	−765.658			−0.867110			−0.433555	−	0.901127i	$-0.642741\pi$
$883$	−765.658			−0.867110			−0.433555	+	0.901127i	$0.642741\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	111.292i		−	0.125470i	−0.998030	−	0.0627352i	$-0.980018\pi$
$887$		−	111.292i		−	0.125470i	0.998030	−	0.0627352i	$-0.0199824\pi$
$888$		0			0
$889$			2495.57i			2.80717i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−88.2829	−	25.2321i	−0.0988611	−	0.0282554i
$894$		0			0
$895$		−	758.562i		−	0.847555i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−513.944			−0.571685
$900$		0			0
$901$			976.957i			1.08430i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	333.494i		−	0.368502i
$906$		0			0
$907$			10.6316i			0.0117217i	0.999983	+	0.00586086i	$0.00186558\pi$
$907$			10.6316i			0.0117217i	−0.999983	+	0.00586086i	$0.998134\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1239.98i		−	1.36112i	−0.732694	−	0.680558i	$-0.761737\pi$
$911$		−	1239.98i		−	1.36112i	0.732694	−	0.680558i	$-0.238263\pi$
$912$		0			0
$913$	2.07296			0.00227050
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1335.90			−1.45682
$918$		0			0
$919$	1027.98			1.11859			0.559293	−	0.828970i	$-0.311073\pi$
$919$	1027.98			1.11859			0.559293	+	0.828970i	$0.311073\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−805.954			−0.873189
$924$		0			0
$925$			458.498i			0.495673i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−605.109			−0.651355			−0.325678	−	0.945481i	$-0.605592\pi$
$929$	−605.109			−0.651355			−0.325678	+	0.945481i	$0.605592\pi$
$930$		0			0
$931$	1280.70	+	366.037i	1.37562	+	0.393165i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	519.544			0.555662
$936$		0			0
$937$	85.0573			0.0907762			0.0453881	−	0.998969i	$-0.485548\pi$
$937$	85.0573			0.0907762			0.0453881	+	0.998969i	$0.485548\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1280.52i			1.36081i	0.732837	+	0.680404i	$0.238196\pi$
$941$			1280.52i			1.36081i	−0.732837	+	0.680404i	$0.761804\pi$
$942$		0			0
$943$		−	1847.74i		−	1.95943i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−254.792			−0.269052			−0.134526	−	0.990910i	$-0.542951\pi$
$947$	−254.792			−0.269052			−0.134526	+	0.990910i	$0.542951\pi$
$948$		0			0
$949$			1835.82i			1.93447i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1689.68i		−	1.77301i	−0.462714	−	0.886507i	$-0.653124\pi$
$953$		−	1689.68i		−	1.77301i	0.462714	−	0.886507i	$-0.346876\pi$
$954$		0			0
$955$	−785.257			−0.822258
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−513.666			−0.535627
$960$		0			0
$961$	541.366			0.563336
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	831.348i		−	0.861501i
$966$		0			0
$967$	421.463			0.435845			0.217923	−	0.975966i	$-0.430072\pi$
$967$	421.463			0.435845			0.217923	+	0.975966i	$0.430072\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			548.719i			0.565108i	0.959251	+	0.282554i	$0.0911816\pi$
$971$			548.719i			0.565108i	−0.959251	+	0.282554i	$0.908818\pi$
$972$		0			0
$973$	−2046.98			−2.10378
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	78.2303i		−	0.0800719i	−0.999198	−	0.0400360i	$-0.987253\pi$
$977$		−	78.2303i		−	0.0800719i	0.999198	−	0.0400360i	$-0.0127473\pi$
$978$		0			0
$979$			724.992i			0.740543i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			269.408i			0.274067i	0.990566	+	0.137033i	$0.0437567\pi$
$983$			269.408i			0.274067i	−0.990566	+	0.137033i	$0.956243\pi$
$984$		0			0
$985$	1436.09			1.45796
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−832.177			−0.841433
$990$		0			0
$991$		−	35.9067i		−	0.0362328i	−0.999836	−	0.0181164i	$-0.994233\pi$
$991$		−	35.9067i		−	0.0362328i	0.999836	−	0.0181164i	$-0.00576695\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	908.598			0.913164
$996$		0			0
$997$	−1174.88			−1.17841			−0.589205	−	0.807983i	$-0.700559\pi$
$997$	−1174.88			−1.17841			−0.589205	+	0.807983i	$0.700559\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.n.721.5		8
3.2	odd	2		912.3.o.d.721.6		8
4.3	odd	2		342.3.d.b.37.7		8
12.11	even	2		114.3.d.a.37.1	✓	8
19.18	odd	2	inner	2736.3.o.n.721.6		8
57.56	even	2		912.3.o.d.721.2		8
76.75	even	2		342.3.d.b.37.3		8
228.227	odd	2		114.3.d.a.37.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.3.d.a.37.1	✓	8	12.11	even	2
114.3.d.a.37.7	yes	8	228.227	odd	2
342.3.d.b.37.3		8	76.75	even	2
342.3.d.b.37.7		8	4.3	odd	2
912.3.o.d.721.2		8	57.56	even	2
912.3.o.d.721.6		8	3.2	odd	2
2736.3.o.n.721.5		8	1.1	even	1	trivial
2736.3.o.n.721.6		8	19.18	odd	2	inner

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.01452 q^{5} +10.9135 q^{7} +O(q^{10})\)	\(q+4.01452 q^{5} +10.9135 q^{7} +9.36952 q^{11} -15.4135i q^{13} +13.8125 q^{17} +(18.2685 + 5.22130i) q^{19} +35.0090 q^{23} -8.88362 q^{25} +25.0889i q^{29} +20.4850i q^{31} +43.8125 q^{35} -51.6116i q^{37} -52.7791i q^{41} -23.7704 q^{43} -4.83252 q^{47} +70.1045 q^{49} +70.7300i q^{53} +37.6141 q^{55} +81.7613i q^{59} -12.6366 q^{61} -61.8778i q^{65} -74.5334i q^{67} -52.2889i q^{71} -119.104 q^{73} +102.254 q^{77} -118.003i q^{79} +0.221245 q^{83} +55.4505 q^{85} +77.3777i q^{89} -168.215i q^{91} +(73.3393 + 20.9610i) q^{95} -104.226i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} + 12 q^{7}+O(q^{10}) \) 8 * q - 4 * q^5 + 12 * q^7	\( 8 q - 4 q^{5} + 12 q^{7} + 4 q^{11} - 4 q^{17} + 36 q^{19} - 56 q^{23} + 140 q^{25} + 236 q^{35} - 100 q^{43} - 188 q^{47} - 36 q^{49} - 28 q^{55} - 180 q^{61} - 356 q^{73} - 68 q^{77} + 136 q^{83} + 148 q^{85} - 140 q^{95}+O(q^{100}) \) 8 * q - 4 * q^5 + 12 * q^7 + 4 * q^11 - 4 * q^17 + 36 * q^19 - 56 * q^23 + 140 * q^25 + 236 * q^35 - 100 * q^43 - 188 * q^47 - 36 * q^49 - 28 * q^55 - 180 * q^61 - 356 * q^73 - 68 * q^77 + 136 * q^83 + 148 * q^85 - 140 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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