LMFDB - Embedded newform 441.3.d.b.244.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,3,Mod(244,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.244");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	441.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.0163796583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.244
Dual form			441.3.d.b.244.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.00000 q^{4} -5.19615i q^{5} +7.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} -3.00000 q^{4} -5.19615i q^{5} +7.00000 q^{8} +5.19615i q^{10} +11.0000 q^{11} +6.92820i q^{13} +5.00000 q^{16} -24.2487i q^{17} +3.46410i q^{19} +15.5885i q^{20} -11.0000 q^{22} -28.0000 q^{23} -2.00000 q^{25} -6.92820i q^{26} -25.0000 q^{29} -32.9090i q^{31} -33.0000 q^{32} +24.2487i q^{34} -58.0000 q^{37} -3.46410i q^{38} -36.3731i q^{40} -3.46410i q^{41} +26.0000 q^{43} -33.0000 q^{44} +28.0000 q^{46} -76.2102i q^{47} +2.00000 q^{50} -20.7846i q^{52} -31.0000 q^{53} -57.1577i q^{55} +25.0000 q^{58} +8.66025i q^{59} -13.8564i q^{61} +32.9090i q^{62} +13.0000 q^{64} +36.0000 q^{65} -52.0000 q^{67} +72.7461i q^{68} -64.0000 q^{71} +6.92820i q^{73} +58.0000 q^{74} -10.3923i q^{76} +17.0000 q^{79} -25.9808i q^{80} +3.46410i q^{82} -53.6936i q^{83} -126.000 q^{85} -26.0000 q^{86} +77.0000 q^{88} -79.6743i q^{89} +84.0000 q^{92} +76.2102i q^{94} +18.0000 q^{95} +91.7987i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 6 q^{4} + 14 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 6 * q^4 + 14 * q^8	$ 2 q - 2 q^{2} - 6 q^{4} + 14 q^{8} + 22 q^{11} + 10 q^{16} - 22 q^{22} - 56 q^{23} - 4 q^{25} - 50 q^{29} - 66 q^{32} - 116 q^{37} + 52 q^{43} - 66 q^{44} + 56 q^{46} + 4 q^{50} - 62 q^{53} + 50 q^{58} + 26 q^{64} + 72 q^{65} - 104 q^{67} - 128 q^{71} + 116 q^{74} + 34 q^{79} - 252 q^{85} - 52 q^{86} + 154 q^{88} + 168 q^{92} + 36 q^{95}+O(q^{100}) $ 2 * q - 2 * q^2 - 6 * q^4 + 14 * q^8 + 22 * q^11 + 10 * q^16 - 22 * q^22 - 56 * q^23 - 4 * q^25 - 50 * q^29 - 66 * q^32 - 116 * q^37 + 52 * q^43 - 66 * q^44 + 56 * q^46 + 4 * q^50 - 62 * q^53 + 50 * q^58 + 26 * q^64 + 72 * q^65 - 104 * q^67 - 128 * q^71 + 116 * q^74 + 34 * q^79 - 252 * q^85 - 52 * q^86 + 154 * q^88 + 168 * q^92 + 36 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$1$

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 $

$2$	−1.00000	−0.500000	−0.250000	−	0.968246i	$-0.580431\pi$
$2$	−1.00000	−0.500000	−0.250000	+	0.968246i	$0.580431\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−0.750000
$5$	− 5.19615i	− 1.03923i	−0.854400	−	0.519615i	$-0.826075\pi$
$5$	− 5.19615i	− 1.03923i	0.854400	−	0.519615i	$-0.173925\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	7.00000	0.875000
$9$	0	0
$10$	5.19615i	0.519615i
$11$	11.0000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$11$	11.0000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$12$	0	0
$13$	6.92820i	0.532939i	0.963843	+	0.266469i	$0.0858571\pi$
$13$	6.92820i	0.532939i	−0.963843	+	0.266469i	$0.914143\pi$
$14$	0	0
$15$	0	0
$16$	5.00000	0.312500
$17$	− 24.2487i	− 1.42639i	−0.700963	−	0.713197i	$-0.747246\pi$
$17$	− 24.2487i	− 1.42639i	0.700963	−	0.713197i	$-0.252754\pi$
$18$	0	0
$19$	3.46410i	0.182321i	0.995836	+	0.0911606i	$0.0290577\pi$
$19$	3.46410i	0.182321i	−0.995836	+	0.0911606i	$0.970942\pi$
$20$	15.5885i	0.779423i
$21$	0	0
$22$	−11.0000	−0.500000
$23$	−28.0000	−1.21739	−0.608696	−	0.793404i	$-0.708307\pi$
$23$	−28.0000	−1.21739	−0.608696	+	0.793404i	$0.708307\pi$
$24$	0	0
$25$	−2.00000	−0.0800000
$26$	− 6.92820i	− 0.266469i
$27$	0	0
$28$	0	0
$29$	−25.0000	−0.862069	−0.431034	−	0.902335i	$-0.641851\pi$
$29$	−25.0000	−0.862069	−0.431034	+	0.902335i	$0.641851\pi$
$30$	0	0
$31$	− 32.9090i	− 1.06158i	−0.847504	−	0.530790i	$-0.821895\pi$
$31$	− 32.9090i	− 1.06158i	0.847504	−	0.530790i	$-0.178105\pi$
$32$	−33.0000	−1.03125
$33$	0	0
$34$	24.2487i	0.713197i
$35$	0	0
$36$	0	0
$37$	−58.0000	−1.56757	−0.783784	−	0.621034i	$-0.786713\pi$
$37$	−58.0000	−1.56757	−0.783784	+	0.621034i	$0.786713\pi$
$38$	− 3.46410i	− 0.0911606i
$39$	0	0
$40$	− 36.3731i	− 0.909327i
$41$	− 3.46410i	− 0.0844903i	−0.999107	−	0.0422451i	$-0.986549\pi$
$41$	− 3.46410i	− 0.0844903i	0.999107	−	0.0422451i	$-0.0134510\pi$
$42$	0	0
$43$	26.0000	0.604651	0.302326	−	0.953205i	$-0.402237\pi$
$43$	26.0000	0.604651	0.302326	+	0.953205i	$0.402237\pi$
$44$	−33.0000	−0.750000
$45$	0	0
$46$	28.0000	0.608696
$47$	− 76.2102i	− 1.62149i	−0.585396	−	0.810747i	$-0.699061\pi$
$47$	− 76.2102i	− 1.62149i	0.585396	−	0.810747i	$-0.300939\pi$
$48$	0	0
$49$	0	0
$50$	2.00000	0.0400000
$51$	0	0
$52$	− 20.7846i	− 0.399704i
$53$	−31.0000	−0.584906	−0.292453	−	0.956280i	$-0.594471\pi$
$53$	−31.0000	−0.584906	−0.292453	+	0.956280i	$0.594471\pi$
$54$	0	0
$55$	− 57.1577i	− 1.03923i
$56$	0	0
$57$	0	0
$58$	25.0000	0.431034
$59$	8.66025i	0.146784i	0.997303	+	0.0733920i	$0.0233824\pi$
$59$	8.66025i	0.146784i	−0.997303	+	0.0733920i	$0.976618\pi$
$60$	0	0
$61$	− 13.8564i	− 0.227154i	−0.993529	−	0.113577i	$-0.963769\pi$
$61$	− 13.8564i	− 0.227154i	0.993529	−	0.113577i	$-0.0362309\pi$
$62$	32.9090i	0.530790i
$63$	0	0
$64$	13.0000	0.203125
$65$	36.0000	0.553846
$66$	0	0
$67$	−52.0000	−0.776119	−0.388060	−	0.921634i	$-0.626855\pi$
$67$	−52.0000	−0.776119	−0.388060	+	0.921634i	$0.626855\pi$
$68$	72.7461i	1.06980i
$69$	0	0
$70$	0	0
$71$	−64.0000	−0.901408	−0.450704	−	0.892673i	$-0.648827\pi$
$71$	−64.0000	−0.901408	−0.450704	+	0.892673i	$0.648827\pi$
$72$	0	0
$73$	6.92820i	0.0949069i	0.998873	+	0.0474534i	$0.0151106\pi$
$73$	6.92820i	0.0949069i	−0.998873	+	0.0474534i	$0.984889\pi$
$74$	58.0000	0.783784
$75$	0	0
$76$	− 10.3923i	− 0.136741i
$77$	0	0
$78$	0	0
$79$	17.0000	0.215190	0.107595	−	0.994195i	$-0.465685\pi$
$79$	17.0000	0.215190	0.107595	+	0.994195i	$0.465685\pi$
$80$	− 25.9808i	− 0.324760i
$81$	0	0
$82$	3.46410i	0.0422451i
$83$	− 53.6936i	− 0.646911i	−0.946243	−	0.323455i	$-0.895155\pi$
$83$	− 53.6936i	− 0.646911i	0.946243	−	0.323455i	$-0.104845\pi$
$84$	0	0
$85$	−126.000	−1.48235
$86$	−26.0000	−0.302326
$87$	0	0
$88$	77.0000	0.875000
$89$	− 79.6743i	− 0.895217i	−0.894230	−	0.447609i	$-0.852276\pi$
$89$	− 79.6743i	− 0.895217i	0.894230	−	0.447609i	$-0.147724\pi$
$90$	0	0
$91$	0	0
$92$	84.0000	0.913043
$93$	0	0
$94$	76.2102i	0.810747i
$95$	18.0000	0.189474
$96$	0	0
$97$	91.7987i	0.946378i	0.880961	+	0.473189i	$0.156897\pi$
$97$	91.7987i	0.946378i	−0.880961	+	0.473189i	$0.843103\pi$
$98$	0	0
$99$	0	0
$100$	6.00000	0.0600000
$101$	20.7846i	0.205788i	0.994692	+	0.102894i	$0.0328103\pi$
$101$	20.7846i	0.205788i	−0.994692	+	0.102894i	$0.967190\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	48.4974i	0.466321i
$105$	0	0
$106$	31.0000	0.292453
$107$	−31.0000	−0.289720	−0.144860	−	0.989452i	$-0.546273\pi$
$107$	−31.0000	−0.289720	−0.144860	+	0.989452i	$0.546273\pi$
$108$	0	0
$109$	−136.000	−1.24771	−0.623853	−	0.781542i	$-0.714434\pi$
$109$	−136.000	−1.24771	−0.623853	+	0.781542i	$0.714434\pi$
$110$	57.1577i	0.519615i
$111$	0	0
$112$	0	0
$113$	74.0000	0.654867	0.327434	−	0.944874i	$-0.393816\pi$
$113$	74.0000	0.654867	0.327434	+	0.944874i	$0.393816\pi$
$114$	0	0
$115$	145.492i	1.26515i
$116$	75.0000	0.646552
$117$	0	0
$118$	− 8.66025i	− 0.0733920i
$119$	0	0
$120$	0	0
$121$	0	0
$122$	13.8564i	0.113577i
$123$	0	0
$124$	98.7269i	0.796185i
$125$	− 119.512i	− 0.956092i
$126$	0	0
$127$	−1.00000	−0.00787402	−0.00393701	−	0.999992i	$-0.501253\pi$
$127$	−1.00000	−0.00787402	−0.00393701	+	0.999992i	$0.501253\pi$
$128$	119.000	0.929688
$129$	0	0
$130$	−36.0000	−0.276923
$131$	181.865i	1.38828i	0.719838	+	0.694142i	$0.244216\pi$
$131$	181.865i	1.38828i	−0.719838	+	0.694142i	$0.755784\pi$
$132$	0	0
$133$	0	0
$134$	52.0000	0.388060
$135$	0	0
$136$	− 169.741i	− 1.24810i
$137$	−88.0000	−0.642336	−0.321168	−	0.947022i	$-0.604075\pi$
$137$	−88.0000	−0.642336	−0.321168	+	0.947022i	$0.604075\pi$
$138$	0	0
$139$	− 190.526i	− 1.37069i	−0.728220	−	0.685344i	$-0.759652\pi$
$139$	− 190.526i	− 1.37069i	0.728220	−	0.685344i	$-0.240348\pi$
$140$	0	0
$141$	0	0
$142$	64.0000	0.450704
$143$	76.2102i	0.532939i
$144$	0	0
$145$	129.904i	0.895888i
$146$	− 6.92820i	− 0.0474534i
$147$	0	0
$148$	174.000	1.17568
$149$	230.000	1.54362	0.771812	−	0.635851i	$-0.219351\pi$
$149$	230.000	1.54362	0.771812	+	0.635851i	$0.219351\pi$
$150$	0	0
$151$	227.000	1.50331	0.751656	−	0.659556i	$-0.229256\pi$
$151$	227.000	1.50331	0.751656	+	0.659556i	$0.229256\pi$
$152$	24.2487i	0.159531i
$153$	0	0
$154$	0	0
$155$	−171.000	−1.10323
$156$	0	0
$157$	− 48.4974i	− 0.308901i	−0.988001	−	0.154450i	$-0.950639\pi$
$157$	− 48.4974i	− 0.308901i	0.988001	−	0.154450i	$-0.0493607\pi$
$158$	−17.0000	−0.107595
$159$	0	0
$160$	171.473i	1.07171i
$161$	0	0
$162$	0	0
$163$	212.000	1.30061	0.650307	−	0.759672i	$-0.274640\pi$
$163$	212.000	1.30061	0.650307	+	0.759672i	$0.274640\pi$
$164$	10.3923i	0.0633677i
$165$	0	0
$166$	53.6936i	0.323455i
$167$	− 96.9948i	− 0.580807i	−0.956904	−	0.290404i	$-0.906210\pi$
$167$	− 96.9948i	− 0.580807i	0.956904	−	0.290404i	$-0.0937896\pi$
$168$	0	0
$169$	121.000	0.715976
$170$	126.000	0.741176
$171$	0	0
$172$	−78.0000	−0.453488
$173$	214.774i	1.24147i	0.784020	+	0.620735i	$0.213166\pi$
$173$	214.774i	1.24147i	−0.784020	+	0.620735i	$0.786834\pi$
$174$	0	0
$175$	0	0
$176$	55.0000	0.312500
$177$	0	0
$178$	79.6743i	0.447609i
$179$	−46.0000	−0.256983	−0.128492	−	0.991711i	$-0.541014\pi$
$179$	−46.0000	−0.256983	−0.128492	+	0.991711i	$0.541014\pi$
$180$	0	0
$181$	− 31.1769i	− 0.172248i	−0.996284	−	0.0861241i	$-0.972552\pi$
$181$	− 31.1769i	− 0.172248i	0.996284	−	0.0861241i	$-0.0274481\pi$
$182$	0	0
$183$	0	0
$184$	−196.000	−1.06522
$185$	301.377i	1.62906i
$186$	0	0
$187$	− 266.736i	− 1.42639i
$188$	228.631i	1.21612i
$189$	0	0
$190$	−18.0000	−0.0947368
$191$	−208.000	−1.08901	−0.544503	−	0.838759i	$-0.683282\pi$
$191$	−208.000	−1.08901	−0.544503	+	0.838759i	$0.683282\pi$
$192$	0	0
$193$	239.000	1.23834	0.619171	−	0.785256i	$-0.287469\pi$
$193$	239.000	1.23834	0.619171	+	0.785256i	$0.287469\pi$
$194$	− 91.7987i	− 0.473189i
$195$	0	0
$196$	0	0
$197$	26.0000	0.131980	0.0659898	−	0.997820i	$-0.478980\pi$
$197$	26.0000	0.131980	0.0659898	+	0.997820i	$0.478980\pi$
$198$	0	0
$199$	− 242.487i	− 1.21853i	−0.792967	−	0.609264i	$-0.791465\pi$
$199$	− 242.487i	− 1.21853i	0.792967	−	0.609264i	$-0.208535\pi$
$200$	−14.0000	−0.0700000
$201$	0	0
$202$	− 20.7846i	− 0.102894i
$203$	0	0
$204$	0	0
$205$	−18.0000	−0.0878049
$206$	0	0
$207$	0	0
$208$	34.6410i	0.166543i
$209$	38.1051i	0.182321i
$210$	0	0
$211$	−52.0000	−0.246445	−0.123223	−	0.992379i	$-0.539323\pi$
$211$	−52.0000	−0.246445	−0.123223	+	0.992379i	$0.539323\pi$
$212$	93.0000	0.438679
$213$	0	0
$214$	31.0000	0.144860
$215$	− 135.100i	− 0.628372i
$216$	0	0
$217$	0	0
$218$	136.000	0.623853
$219$	0	0
$220$	171.473i	0.779423i
$221$	168.000	0.760181
$222$	0	0
$223$	− 22.5167i	− 0.100972i	−0.998725	−	0.0504858i	$-0.983923\pi$
$223$	− 22.5167i	− 0.100972i	0.998725	−	0.0504858i	$-0.0160770\pi$
$224$	0	0
$225$	0	0
$226$	−74.0000	−0.327434
$227$	− 67.5500i	− 0.297577i	−0.988869	−	0.148789i	$-0.952463\pi$
$227$	− 67.5500i	− 0.297577i	0.988869	−	0.148789i	$-0.0475374\pi$
$228$	0	0
$229$	31.1769i	0.136144i	0.997680	+	0.0680719i	$0.0216847\pi$
$229$	31.1769i	0.136144i	−0.997680	+	0.0680719i	$0.978315\pi$
$230$	− 145.492i	− 0.632575i
$231$	0	0
$232$	−175.000	−0.754310
$233$	−262.000	−1.12446	−0.562232	−	0.826980i	$-0.690057\pi$
$233$	−262.000	−1.12446	−0.562232	+	0.826980i	$0.690057\pi$
$234$	0	0
$235$	−396.000	−1.68511
$236$	− 25.9808i	− 0.110088i
$237$	0	0
$238$	0	0
$239$	−160.000	−0.669456	−0.334728	−	0.942315i	$-0.608644\pi$
$239$	−160.000	−0.669456	−0.334728	+	0.942315i	$0.608644\pi$
$240$	0	0
$241$	472.850i	1.96203i	0.193925	+	0.981016i	$0.437878\pi$
$241$	472.850i	1.96203i	−0.193925	+	0.981016i	$0.562122\pi$
$242$	0	0
$243$	0	0
$244$	41.5692i	0.170366i
$245$	0	0
$246$	0	0
$247$	−24.0000	−0.0971660
$248$	− 230.363i	− 0.928882i
$249$	0	0
$250$	119.512i	0.478046i
$251$	67.5500i	0.269123i	0.990905	+	0.134562i	$0.0429626\pi$
$251$	67.5500i	0.269123i	−0.990905	+	0.134562i	$0.957037\pi$
$252$	0	0
$253$	−308.000	−1.21739
$254$	1.00000	0.00393701
$255$	0	0
$256$	−171.000	−0.667969
$257$	− 405.300i	− 1.57704i	−0.615008	−	0.788521i	$-0.710847\pi$
$257$	− 405.300i	− 1.57704i	0.615008	−	0.788521i	$-0.289153\pi$
$258$	0	0
$259$	0	0
$260$	−108.000	−0.415385
$261$	0	0
$262$	− 181.865i	− 0.694142i
$263$	−106.000	−0.403042	−0.201521	−	0.979484i	$-0.564588\pi$
$263$	−106.000	−0.403042	−0.201521	+	0.979484i	$0.564588\pi$
$264$	0	0
$265$	161.081i	0.607852i
$266$	0	0
$267$	0	0
$268$	156.000	0.582090
$269$	− 434.745i	− 1.61615i	−0.589079	−	0.808076i	$-0.700509\pi$
$269$	− 434.745i	− 1.61615i	0.589079	−	0.808076i	$-0.299491\pi$
$270$	0	0
$271$	− 126.440i	− 0.466567i	−0.972409	−	0.233284i	$-0.925053\pi$
$271$	− 126.440i	− 0.466567i	0.972409	−	0.233284i	$-0.0749470\pi$
$272$	− 121.244i	− 0.445748i
$273$	0	0
$274$	88.0000	0.321168
$275$	−22.0000	−0.0800000
$276$	0	0
$277$	236.000	0.851986	0.425993	−	0.904727i	$-0.359925\pi$
$277$	236.000	0.851986	0.425993	+	0.904727i	$0.359925\pi$
$278$	190.526i	0.685344i
$279$	0	0
$280$	0	0
$281$	116.000	0.412811	0.206406	−	0.978466i	$-0.433823\pi$
$281$	116.000	0.412811	0.206406	+	0.978466i	$0.433823\pi$
$282$	0	0
$283$	− 370.659i	− 1.30975i	−0.755738	−	0.654874i	$-0.772722\pi$
$283$	− 370.659i	− 1.30975i	0.755738	−	0.654874i	$-0.227278\pi$
$284$	192.000	0.676056
$285$	0	0
$286$	− 76.2102i	− 0.266469i
$287$	0	0
$288$	0	0
$289$	−299.000	−1.03460
$290$	− 129.904i	− 0.447944i
$291$	0	0
$292$	− 20.7846i	− 0.0711802i
$293$	− 19.0526i	− 0.0650258i	−0.999471	−	0.0325129i	$-0.989649\pi$
$293$	− 19.0526i	− 0.0650258i	0.999471	−	0.0325129i	$-0.0103510\pi$
$294$	0	0
$295$	45.0000	0.152542
$296$	−406.000	−1.37162
$297$	0	0
$298$	−230.000	−0.771812
$299$	− 193.990i	− 0.648795i
$300$	0	0
$301$	0	0
$302$	−227.000	−0.751656
$303$	0	0
$304$	17.3205i	0.0569754i
$305$	−72.0000	−0.236066
$306$	0	0
$307$	− 457.261i	− 1.48945i	−0.667371	−	0.744725i	$-0.732580\pi$
$307$	− 457.261i	− 1.48945i	0.667371	−	0.744725i	$-0.267420\pi$
$308$	0	0
$309$	0	0
$310$	171.000	0.551613
$311$	− 329.090i	− 1.05817i	−0.848570	−	0.529083i	$-0.822536\pi$
$311$	− 329.090i	− 1.05817i	0.848570	−	0.529083i	$-0.177464\pi$
$312$	0	0
$313$	− 549.060i	− 1.75419i	−0.480321	−	0.877093i	$-0.659480\pi$
$313$	− 549.060i	− 1.75419i	0.480321	−	0.877093i	$-0.340520\pi$
$314$	48.4974i	0.154450i
$315$	0	0
$316$	−51.0000	−0.161392
$317$	−187.000	−0.589905	−0.294953	−	0.955512i	$-0.595304\pi$
$317$	−187.000	−0.589905	−0.294953	+	0.955512i	$0.595304\pi$
$318$	0	0
$319$	−275.000	−0.862069
$320$	− 67.5500i	− 0.211094i
$321$	0	0
$322$	0	0
$323$	84.0000	0.260062
$324$	0	0
$325$	− 13.8564i	− 0.0426351i
$326$	−212.000	−0.650307
$327$	0	0
$328$	− 24.2487i	− 0.0739290i
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.0483384	−0.0241692	−	0.999708i	$-0.507694\pi$
$331$	−16.0000	−0.0483384	−0.0241692	+	0.999708i	$0.507694\pi$
$332$	161.081i	0.485183i
$333$	0	0
$334$	96.9948i	0.290404i
$335$	270.200i	0.806567i
$336$	0	0
$337$	83.0000	0.246291	0.123145	−	0.992389i	$-0.460702\pi$
$337$	83.0000	0.246291	0.123145	+	0.992389i	$0.460702\pi$
$338$	−121.000	−0.357988
$339$	0	0
$340$	378.000	1.11176
$341$	− 361.999i	− 1.06158i
$342$	0	0
$343$	0	0
$344$	182.000	0.529070
$345$	0	0
$346$	− 214.774i	− 0.620735i
$347$	−358.000	−1.03170	−0.515850	−	0.856679i	$-0.672524\pi$
$347$	−358.000	−1.03170	−0.515850	+	0.856679i	$0.672524\pi$
$348$	0	0
$349$	678.964i	1.94546i	0.231950	+	0.972728i	$0.425489\pi$
$349$	678.964i	1.94546i	−0.231950	+	0.972728i	$0.574511\pi$
$350$	0	0
$351$	0	0
$352$	−363.000	−1.03125
$353$	644.323i	1.82528i	0.408767	+	0.912639i	$0.365959\pi$
$353$	644.323i	1.82528i	−0.408767	+	0.912639i	$0.634041\pi$
$354$	0	0
$355$	332.554i	0.936771i
$356$	239.023i	0.671413i
$357$	0	0
$358$	46.0000	0.128492
$359$	284.000	0.791086	0.395543	−	0.918447i	$-0.370556\pi$
$359$	284.000	0.791086	0.395543	+	0.918447i	$0.370556\pi$
$360$	0	0
$361$	349.000	0.966759
$362$	31.1769i	0.0861241i
$363$	0	0
$364$	0	0
$365$	36.0000	0.0986301
$366$	0	0
$367$	275.396i	0.750398i	0.926944	+	0.375199i	$0.122426\pi$
$367$	275.396i	0.750398i	−0.926944	+	0.375199i	$0.877574\pi$
$368$	−140.000	−0.380435
$369$	0	0
$370$	− 301.377i	− 0.814532i
$371$	0	0
$372$	0	0
$373$	50.0000	0.134048	0.0670241	−	0.997751i	$-0.478650\pi$
$373$	50.0000	0.134048	0.0670241	+	0.997751i	$0.478650\pi$
$374$	266.736i	0.713197i
$375$	0	0
$376$	− 533.472i	− 1.41881i
$377$	− 173.205i	− 0.459430i
$378$	0	0
$379$	458.000	1.20844	0.604222	−	0.796816i	$-0.293484\pi$
$379$	458.000	1.20844	0.604222	+	0.796816i	$0.293484\pi$
$380$	−54.0000	−0.142105
$381$	0	0
$382$	208.000	0.544503
$383$	405.300i	1.05822i	0.848552	+	0.529112i	$0.177475\pi$
$383$	405.300i	1.05822i	−0.848552	+	0.529112i	$0.822525\pi$
$384$	0	0
$385$	0	0
$386$	−239.000	−0.619171
$387$	0	0
$388$	− 275.396i	− 0.709784i
$389$	698.000	1.79434	0.897172	−	0.441681i	$-0.145618\pi$
$389$	698.000	1.79434	0.897172	+	0.441681i	$0.145618\pi$
$390$	0	0
$391$	678.964i	1.73648i
$392$	0	0
$393$	0	0
$394$	−26.0000	−0.0659898
$395$	− 88.3346i	− 0.223632i
$396$	0	0
$397$	575.041i	1.44847i	0.689556	+	0.724233i	$0.257806\pi$
$397$	575.041i	1.44847i	−0.689556	+	0.724233i	$0.742194\pi$
$398$	242.487i	0.609264i
$399$	0	0
$400$	−10.0000	−0.0250000
$401$	284.000	0.708229	0.354115	−	0.935202i	$-0.384782\pi$
$401$	284.000	0.708229	0.354115	+	0.935202i	$0.384782\pi$
$402$	0	0
$403$	228.000	0.565757
$404$	− 62.3538i	− 0.154341i
$405$	0	0
$406$	0	0
$407$	−638.000	−1.56757
$408$	0	0
$409$	− 209.578i	− 0.512416i	−0.966622	−	0.256208i	$-0.917527\pi$
$409$	− 209.578i	− 0.512416i	0.966622	−	0.256208i	$-0.0824732\pi$
$410$	18.0000	0.0439024
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−279.000	−0.672289
$416$	− 228.631i	− 0.549593i
$417$	0	0
$418$	− 38.1051i	− 0.0911606i
$419$	131.636i	0.314167i	0.987585	+	0.157083i	$0.0502091\pi$
$419$	131.636i	0.314167i	−0.987585	+	0.157083i	$0.949791\pi$
$420$	0	0
$421$	−28.0000	−0.0665083	−0.0332542	−	0.999447i	$-0.510587\pi$
$421$	−28.0000	−0.0665083	−0.0332542	+	0.999447i	$0.510587\pi$
$422$	52.0000	0.123223
$423$	0	0
$424$	−217.000	−0.511792
$425$	48.4974i	0.114112i
$426$	0	0
$427$	0	0
$428$	93.0000	0.217290
$429$	0	0
$430$	135.100i	0.314186i
$431$	−118.000	−0.273782	−0.136891	−	0.990586i	$-0.543711\pi$
$431$	−118.000	−0.273782	−0.136891	+	0.990586i	$0.543711\pi$
$432$	0	0
$433$	− 561.184i	− 1.29604i	−0.761624	−	0.648019i	$-0.775598\pi$
$433$	− 561.184i	− 1.29604i	0.761624	−	0.648019i	$-0.224402\pi$
$434$	0	0
$435$	0	0
$436$	408.000	0.935780
$437$	− 96.9948i	− 0.221956i
$438$	0	0
$439$	739.586i	1.68471i	0.538927	+	0.842353i	$0.318830\pi$
$439$	739.586i	1.68471i	−0.538927	+	0.842353i	$0.681170\pi$
$440$	− 400.104i	− 0.909327i
$441$	0	0
$442$	−168.000	−0.380090
$443$	155.000	0.349887	0.174944	−	0.984578i	$-0.444026\pi$
$443$	155.000	0.349887	0.174944	+	0.984578i	$0.444026\pi$
$444$	0	0
$445$	−414.000	−0.930337
$446$	22.5167i	0.0504858i
$447$	0	0
$448$	0	0
$449$	368.000	0.819599	0.409800	−	0.912176i	$-0.365599\pi$
$449$	368.000	0.819599	0.409800	+	0.912176i	$0.365599\pi$
$450$	0	0
$451$	− 38.1051i	− 0.0844903i
$452$	−222.000	−0.491150
$453$	0	0
$454$	67.5500i	0.148789i
$455$	0	0
$456$	0	0
$457$	341.000	0.746171	0.373085	−	0.927797i	$-0.378300\pi$
$457$	341.000	0.746171	0.373085	+	0.927797i	$0.378300\pi$
$458$	− 31.1769i	− 0.0680719i
$459$	0	0
$460$	− 436.477i	− 0.948863i
$461$	55.4256i	0.120229i	0.998191	+	0.0601146i	$0.0191466\pi$
$461$	55.4256i	0.120229i	−0.998191	+	0.0601146i	$0.980853\pi$
$462$	0	0
$463$	−178.000	−0.384449	−0.192225	−	0.981351i	$-0.561570\pi$
$463$	−178.000	−0.384449	−0.192225	+	0.981351i	$0.561570\pi$
$464$	−125.000	−0.269397
$465$	0	0
$466$	262.000	0.562232
$467$	658.179i	1.40938i	0.709517	+	0.704689i	$0.248913\pi$
$467$	658.179i	1.40938i	−0.709517	+	0.704689i	$0.751087\pi$
$468$	0	0
$469$	0	0
$470$	396.000	0.842553
$471$	0	0
$472$	60.6218i	0.128436i
$473$	286.000	0.604651
$474$	0	0
$475$	− 6.92820i	− 0.0145857i
$476$	0	0
$477$	0	0
$478$	160.000	0.334728
$479$	− 509.223i	− 1.06310i	−0.847028	−	0.531548i	$-0.821611\pi$
$479$	− 509.223i	− 1.06310i	0.847028	−	0.531548i	$-0.178389\pi$
$480$	0	0
$481$	− 401.836i	− 0.835417i
$482$	− 472.850i	− 0.981016i
$483$	0	0
$484$	0	0
$485$	477.000	0.983505
$486$	0	0
$487$	−841.000	−1.72690	−0.863450	−	0.504435i	$-0.831701\pi$
$487$	−841.000	−1.72690	−0.863450	+	0.504435i	$0.831701\pi$
$488$	− 96.9948i	− 0.198760i
$489$	0	0
$490$	0	0
$491$	959.000	1.95316	0.976578	−	0.215162i	$-0.0690279\pi$
$491$	959.000	1.95316	0.976578	+	0.215162i	$0.0690279\pi$
$492$	0	0
$493$	606.218i	1.22965i
$494$	24.0000	0.0485830
$495$	0	0
$496$	− 164.545i	− 0.331744i
$497$	0	0
$498$	0	0
$499$	−118.000	−0.236473	−0.118236	−	0.992985i	$-0.537724\pi$
$499$	−118.000	−0.236473	−0.118236	+	0.992985i	$0.537724\pi$
$500$	358.535i	0.717069i
$501$	0	0
$502$	− 67.5500i	− 0.134562i
$503$	− 363.731i	− 0.723123i	−0.932348	−	0.361561i	$-0.882244\pi$
$503$	− 363.731i	− 0.723123i	0.932348	−	0.361561i	$-0.117756\pi$
$504$	0	0
$505$	108.000	0.213861
$506$	308.000	0.608696
$507$	0	0
$508$	3.00000	0.00590551
$509$	441.673i	0.867727i	0.900979	+	0.433863i	$0.142850\pi$
$509$	441.673i	0.867727i	−0.900979	+	0.433863i	$0.857150\pi$
$510$	0	0
$511$	0	0
$512$	−305.000	−0.595703
$513$	0	0
$514$	405.300i	0.788521i
$515$	0	0
$516$	0	0
$517$	− 838.313i	− 1.62149i
$518$	0	0
$519$	0	0
$520$	252.000	0.484615
$521$	973.413i	1.86835i	0.356810	+	0.934177i	$0.383864\pi$
$521$	973.413i	1.86835i	−0.356810	+	0.934177i	$0.616136\pi$
$522$	0	0
$523$	471.118i	0.900799i	0.892827	+	0.450399i	$0.148718\pi$
$523$	471.118i	0.900799i	−0.892827	+	0.450399i	$0.851282\pi$
$524$	− 545.596i	− 1.04121i
$525$	0	0
$526$	106.000	0.201521
$527$	−798.000	−1.51423
$528$	0	0
$529$	255.000	0.482042
$530$	− 161.081i	− 0.303926i
$531$	0	0
$532$	0	0
$533$	24.0000	0.0450281
$534$	0	0
$535$	161.081i	0.301085i
$536$	−364.000	−0.679104
$537$	0	0
$538$	434.745i	0.808076i
$539$	0	0
$540$	0	0
$541$	806.000	1.48983	0.744917	−	0.667157i	$-0.232489\pi$
$541$	806.000	1.48983	0.744917	+	0.667157i	$0.232489\pi$
$542$	126.440i	0.233284i
$543$	0	0
$544$	800.207i	1.47097i
$545$	706.677i	1.29665i
$546$	0	0
$547$	−154.000	−0.281536	−0.140768	−	0.990043i	$-0.544957\pi$
$547$	−154.000	−0.281536	−0.140768	+	0.990043i	$0.544957\pi$
$548$	264.000	0.481752
$549$	0	0
$550$	22.0000	0.0400000
$551$	− 86.6025i	− 0.157173i
$552$	0	0
$553$	0	0
$554$	−236.000	−0.425993
$555$	0	0
$556$	571.577i	1.02802i
$557$	851.000	1.52783	0.763914	−	0.645318i	$-0.223275\pi$
$557$	851.000	1.52783	0.763914	+	0.645318i	$0.223275\pi$
$558$	0	0
$559$	180.133i	0.322242i
$560$	0	0
$561$	0	0
$562$	−116.000	−0.206406
$563$	− 427.817i	− 0.759887i	−0.925010	−	0.379944i	$-0.875943\pi$
$563$	− 427.817i	− 0.759887i	0.925010	−	0.379944i	$-0.124057\pi$
$564$	0	0
$565$	− 384.515i	− 0.680558i
$566$	370.659i	0.654874i
$567$	0	0
$568$	−448.000	−0.788732
$569$	818.000	1.43761	0.718805	−	0.695212i	$-0.244689\pi$
$569$	818.000	1.43761	0.718805	+	0.695212i	$0.244689\pi$
$570$	0	0
$571$	284.000	0.497373	0.248687	−	0.968584i	$-0.420001\pi$
$571$	284.000	0.497373	0.248687	+	0.968584i	$0.420001\pi$
$572$	− 228.631i	− 0.399704i
$573$	0	0
$574$	0	0
$575$	56.0000	0.0973913
$576$	0	0
$577$	− 756.906i	− 1.31180i	−0.754850	−	0.655898i	$-0.772290\pi$
$577$	− 756.906i	− 1.31180i	0.754850	−	0.655898i	$-0.227710\pi$
$578$	299.000	0.517301
$579$	0	0
$580$	− 389.711i	− 0.671916i
$581$	0	0
$582$	0	0
$583$	−341.000	−0.584906
$584$	48.4974i	0.0830435i
$585$	0	0
$586$	19.0526i	0.0325129i
$587$	− 947.432i	− 1.61402i	−0.590535	−	0.807012i	$-0.701083\pi$
$587$	− 947.432i	− 1.61402i	0.590535	−	0.807012i	$-0.298917\pi$
$588$	0	0
$589$	114.000	0.193548
$590$	−45.0000	−0.0762712
$591$	0	0
$592$	−290.000	−0.489865
$593$	408.764i	0.689315i	0.938728	+	0.344658i	$0.112005\pi$
$593$	408.764i	0.689315i	−0.938728	+	0.344658i	$0.887995\pi$
$594$	0	0
$595$	0	0
$596$	−690.000	−1.15772
$597$	0	0
$598$	193.990i	0.324397i
$599$	−556.000	−0.928214	−0.464107	−	0.885779i	$-0.653625\pi$
$599$	−556.000	−0.928214	−0.464107	+	0.885779i	$0.653625\pi$
$600$	0	0
$601$	580.237i	0.965453i	0.875771	+	0.482726i	$0.160353\pi$
$601$	580.237i	0.965453i	−0.875771	+	0.482726i	$0.839647\pi$
$602$	0	0
$603$	0	0
$604$	−681.000	−1.12748
$605$	0	0
$606$	0	0
$607$	− 659.911i	− 1.08717i	−0.839355	−	0.543584i	$-0.817067\pi$
$607$	− 659.911i	− 1.08717i	0.839355	−	0.543584i	$-0.182933\pi$
$608$	− 114.315i	− 0.188019i
$609$	0	0
$610$	72.0000	0.118033
$611$	528.000	0.864157
$612$	0	0
$613$	320.000	0.522023	0.261011	−	0.965336i	$-0.415944\pi$
$613$	320.000	0.522023	0.261011	+	0.965336i	$0.415944\pi$
$614$	457.261i	0.744725i
$615$	0	0
$616$	0	0
$617$	−652.000	−1.05673	−0.528363	−	0.849019i	$-0.677194\pi$
$617$	−652.000	−1.05673	−0.528363	+	0.849019i	$0.677194\pi$
$618$	0	0
$619$	− 644.323i	− 1.04091i	−0.853889	−	0.520455i	$-0.825762\pi$
$619$	− 644.323i	− 1.04091i	0.853889	−	0.520455i	$-0.174238\pi$
$620$	513.000	0.827419
$621$	0	0
$622$	329.090i	0.529083i
$623$	0	0
$624$	0	0
$625$	−671.000	−1.07360
$626$	549.060i	0.877093i
$627$	0	0
$628$	145.492i	0.231676i
$629$	1406.43i	2.23597i
$630$	0	0
$631$	−97.0000	−0.153724	−0.0768621	−	0.997042i	$-0.524490\pi$
$631$	−97.0000	−0.153724	−0.0768621	+	0.997042i	$0.524490\pi$
$632$	119.000	0.188291
$633$	0	0
$634$	187.000	0.294953
$635$	5.19615i	0.00818292i
$636$	0	0
$637$	0	0
$638$	275.000	0.431034
$639$	0	0
$640$	− 618.342i	− 0.966160i
$641$	−700.000	−1.09204	−0.546022	−	0.837771i	$-0.683858\pi$
$641$	−700.000	−1.09204	−0.546022	+	0.837771i	$0.683858\pi$
$642$	0	0
$643$	− 325.626i	− 0.506416i	−0.967412	−	0.253208i	$-0.918514\pi$
$643$	− 325.626i	− 0.506416i	0.967412	−	0.253208i	$-0.0814857\pi$
$644$	0	0
$645$	0	0
$646$	−84.0000	−0.130031
$647$	360.267i	0.556826i	0.960461	+	0.278413i	$0.0898084\pi$
$647$	360.267i	0.556826i	−0.960461	+	0.278413i	$0.910192\pi$
$648$	0	0
$649$	95.2628i	0.146784i
$650$	13.8564i	0.0213175i
$651$	0	0
$652$	−636.000	−0.975460
$653$	−373.000	−0.571210	−0.285605	−	0.958347i	$-0.592194\pi$
$653$	−373.000	−0.571210	−0.285605	+	0.958347i	$0.592194\pi$
$654$	0	0
$655$	945.000	1.44275
$656$	− 17.3205i	− 0.0264032i
$657$	0	0
$658$	0	0
$659$	818.000	1.24127	0.620637	−	0.784098i	$-0.286874\pi$
$659$	818.000	1.24127	0.620637	+	0.784098i	$0.286874\pi$
$660$	0	0
$661$	377.587i	0.571236i	0.958344	+	0.285618i	$0.0921989\pi$
$661$	377.587i	0.571236i	−0.958344	+	0.285618i	$0.907801\pi$
$662$	16.0000	0.0241692
$663$	0	0
$664$	− 375.855i	− 0.566047i
$665$	0	0
$666$	0	0
$667$	700.000	1.04948
$668$	290.985i	0.435606i
$669$	0	0
$670$	− 270.200i	− 0.403283i
$671$	− 152.420i	− 0.227154i
$672$	0	0
$673$	1205.00	1.79049	0.895245	−	0.445574i	$-0.147000\pi$
$673$	1205.00	1.79049	0.895245	+	0.445574i	$0.147000\pi$
$674$	−83.0000	−0.123145
$675$	0	0
$676$	−363.000	−0.536982
$677$	− 538.668i	− 0.795669i	−0.917457	−	0.397834i	$-0.869762\pi$
$677$	− 538.668i	− 0.795669i	0.917457	−	0.397834i	$-0.130238\pi$
$678$	0	0
$679$	0	0
$680$	−882.000	−1.29706
$681$	0	0
$682$	361.999i	0.530790i
$683$	−949.000	−1.38946	−0.694729	−	0.719271i	$-0.744476\pi$
$683$	−949.000	−1.38946	−0.694729	+	0.719271i	$0.744476\pi$
$684$	0	0
$685$	457.261i	0.667535i
$686$	0	0
$687$	0	0
$688$	130.000	0.188953
$689$	− 214.774i	− 0.311719i
$690$	0	0
$691$	308.305i	0.446172i	0.974799	+	0.223086i	$0.0716131\pi$
$691$	308.305i	0.446172i	−0.974799	+	0.223086i	$0.928387\pi$
$692$	− 644.323i	− 0.931102i
$693$	0	0
$694$	358.000	0.515850
$695$	−990.000	−1.42446
$696$	0	0
$697$	−84.0000	−0.120516
$698$	− 678.964i	− 0.972728i
$699$	0	0
$700$	0	0
$701$	413.000	0.589158	0.294579	−	0.955627i	$-0.404821\pi$
$701$	413.000	0.589158	0.294579	+	0.955627i	$0.404821\pi$
$702$	0	0
$703$	− 200.918i	− 0.285801i
$704$	143.000	0.203125
$705$	0	0
$706$	− 644.323i	− 0.912639i
$707$	0	0
$708$	0	0
$709$	890.000	1.25529	0.627645	−	0.778500i	$-0.284019\pi$
$709$	890.000	1.25529	0.627645	+	0.778500i	$0.284019\pi$
$710$	− 332.554i	− 0.468386i
$711$	0	0
$712$	− 557.720i	− 0.783315i
$713$	921.451i	1.29236i
$714$	0	0
$715$	396.000	0.553846
$716$	138.000	0.192737
$717$	0	0
$718$	−284.000	−0.395543
$719$	668.572i	0.929863i	0.885347	+	0.464932i	$0.153921\pi$
$719$	668.572i	0.929863i	−0.885347	+	0.464932i	$0.846079\pi$
$720$	0	0
$721$	0	0
$722$	−349.000	−0.483380
$723$	0	0
$724$	93.5307i	0.129186i
$725$	50.0000	0.0689655
$726$	0	0
$727$	− 417.424i	− 0.574174i	−0.957905	−	0.287087i	$-0.907313\pi$
$727$	− 417.424i	− 0.574174i	0.957905	−	0.287087i	$-0.0926868\pi$
$728$	0	0
$729$	0	0
$730$	−36.0000	−0.0493151
$731$	− 630.466i	− 0.862471i
$732$	0	0
$733$	− 197.454i	− 0.269378i	−0.990888	−	0.134689i	$-0.956997\pi$
$733$	− 197.454i	− 0.269378i	0.990888	−	0.134689i	$-0.0430035\pi$
$734$	− 275.396i	− 0.375199i
$735$	0	0
$736$	924.000	1.25543
$737$	−572.000	−0.776119
$738$	0	0
$739$	−310.000	−0.419486	−0.209743	−	0.977757i	$-0.567263\pi$
$739$	−310.000	−0.419486	−0.209743	+	0.977757i	$0.567263\pi$
$740$	− 904.131i	− 1.22180i
$741$	0	0
$742$	0	0
$743$	812.000	1.09287	0.546433	−	0.837503i	$-0.315985\pi$
$743$	812.000	1.09287	0.546433	+	0.837503i	$0.315985\pi$
$744$	0	0
$745$	− 1195.12i	− 1.60418i
$746$	−50.0000	−0.0670241
$747$	0	0
$748$	800.207i	1.06980i
$749$	0	0
$750$	0	0
$751$	−1075.00	−1.43142	−0.715712	−	0.698395i	$-0.753898\pi$
$751$	−1075.00	−1.43142	−0.715712	+	0.698395i	$0.753898\pi$
$752$	− 381.051i	− 0.506717i
$753$	0	0
$754$	173.205i	0.229715i
$755$	− 1179.53i	− 1.56229i
$756$	0	0
$757$	−484.000	−0.639366	−0.319683	−	0.947525i	$-0.603576\pi$
$757$	−484.000	−0.639366	−0.319683	+	0.947525i	$0.603576\pi$
$758$	−458.000	−0.604222
$759$	0	0
$760$	126.000	0.165789
$761$	166.277i	0.218498i	0.994014	+	0.109249i	$0.0348446\pi$
$761$	166.277i	0.218498i	−0.994014	+	0.109249i	$0.965155\pi$
$762$	0	0
$763$	0	0
$764$	624.000	0.816754
$765$	0	0
$766$	− 405.300i	− 0.529112i
$767$	−60.0000	−0.0782269
$768$	0	0
$769$	594.093i	0.772553i	0.922383	+	0.386277i	$0.126239\pi$
$769$	594.093i	0.772553i	−0.922383	+	0.386277i	$0.873761\pi$
$770$	0	0
$771$	0	0
$772$	−717.000	−0.928756
$773$	− 1032.30i	− 1.33545i	−0.744408	−	0.667725i	$-0.767268\pi$
$773$	− 1032.30i	− 1.33545i	0.744408	−	0.667725i	$-0.232732\pi$
$774$	0	0
$775$	65.8179i	0.0849264i
$776$	642.591i	0.828081i
$777$	0	0
$778$	−698.000	−0.897172
$779$	12.0000	0.0154044
$780$	0	0
$781$	−704.000	−0.901408
$782$	− 678.964i	− 0.868240i
$783$	0	0
$784$	0	0
$785$	−252.000	−0.321019
$786$	0	0
$787$	408.764i	0.519395i	0.965690	+	0.259698i	$0.0836229\pi$
$787$	408.764i	0.519395i	−0.965690	+	0.259698i	$0.916377\pi$
$788$	−78.0000	−0.0989848
$789$	0	0
$790$	88.3346i	0.111816i
$791$	0	0
$792$	0	0
$793$	96.0000	0.121059
$794$	− 575.041i	− 0.724233i
$795$	0	0
$796$	727.461i	0.913896i
$797$	625.270i	0.784530i	0.919852	+	0.392265i	$0.128308\pi$
$797$	625.270i	0.784530i	−0.919852	+	0.392265i	$0.871692\pi$
$798$	0	0
$799$	−1848.00	−2.31289
$800$	66.0000	0.0825000
$801$	0	0
$802$	−284.000	−0.354115
$803$	76.2102i	0.0949069i
$804$	0	0
$805$	0	0
$806$	−228.000	−0.282878
$807$	0	0
$808$	145.492i	0.180065i
$809$	752.000	0.929543	0.464771	−	0.885431i	$-0.346137\pi$
$809$	752.000	0.929543	0.464771	+	0.885431i	$0.346137\pi$
$810$	0	0
$811$	270.200i	0.333169i	0.986027	+	0.166584i	$0.0532738\pi$
$811$	270.200i	0.333169i	−0.986027	+	0.166584i	$0.946726\pi$
$812$	0	0
$813$	0	0
$814$	638.000	0.783784
$815$	− 1101.58i	− 1.35164i
$816$	0	0
$817$	90.0666i	0.110241i
$818$	209.578i	0.256208i
$819$	0	0
$820$	54.0000	0.0658537
$821$	−883.000	−1.07552	−0.537759	−	0.843099i	$-0.680729\pi$
$821$	−883.000	−1.07552	−0.537759	+	0.843099i	$0.680729\pi$
$822$	0	0
$823$	−490.000	−0.595383	−0.297691	−	0.954662i	$-0.596217\pi$
$823$	−490.000	−0.595383	−0.297691	+	0.954662i	$0.596217\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1279.00	−1.54655	−0.773277	−	0.634068i	$-0.781384\pi$
$827$	−1279.00	−1.54655	−0.773277	+	0.634068i	$0.781384\pi$
$828$	0	0
$829$	1513.81i	1.82607i	0.407881	+	0.913035i	$0.366268\pi$
$829$	1513.81i	1.82607i	−0.407881	+	0.913035i	$0.633732\pi$
$830$	279.000	0.336145
$831$	0	0
$832$	90.0666i	0.108253i
$833$	0	0
$834$	0	0
$835$	−504.000	−0.603593
$836$	− 114.315i	− 0.136741i
$837$	0	0
$838$	− 131.636i	− 0.157083i
$839$	523.079i	0.623456i	0.950171	+	0.311728i	$0.100908\pi$
$839$	523.079i	0.623456i	−0.950171	+	0.311728i	$0.899092\pi$
$840$	0	0
$841$	−216.000	−0.256837
$842$	28.0000	0.0332542
$843$	0	0
$844$	156.000	0.184834
$845$	− 628.734i	− 0.744064i
$846$	0	0
$847$	0	0
$848$	−155.000	−0.182783
$849$	0	0
$850$	− 48.4974i	− 0.0570558i
$851$	1624.00	1.90834
$852$	0	0
$853$	387.979i	0.454841i	0.973797	+	0.227421i	$0.0730292\pi$
$853$	387.979i	0.454841i	−0.973797	+	0.227421i	$0.926971\pi$
$854$	0	0
$855$	0	0
$856$	−217.000	−0.253505
$857$	831.384i	0.970110i	0.874484	+	0.485055i	$0.161200\pi$
$857$	831.384i	0.970110i	−0.874484	+	0.485055i	$0.838800\pi$
$858$	0	0
$859$	− 1132.76i	− 1.31870i	−0.751837	−	0.659349i	$-0.770832\pi$
$859$	− 1132.76i	− 1.31870i	0.751837	−	0.659349i	$-0.229168\pi$
$860$	405.300i	0.471279i
$861$	0	0
$862$	118.000	0.136891
$863$	−22.0000	−0.0254925	−0.0127462	−	0.999919i	$-0.504057\pi$
$863$	−22.0000	−0.0254925	−0.0127462	+	0.999919i	$0.504057\pi$
$864$	0	0
$865$	1116.00	1.29017
$866$	561.184i	0.648019i
$867$	0	0
$868$	0	0
$869$	187.000	0.215190
$870$	0	0
$871$	− 360.267i	− 0.413624i
$872$	−952.000	−1.09174
$873$	0	0
$874$	96.9948i	0.110978i
$875$	0	0
$876$	0	0
$877$	−40.0000	−0.0456100	−0.0228050	−	0.999740i	$-0.507260\pi$
$877$	−40.0000	−0.0456100	−0.0228050	+	0.999740i	$0.507260\pi$
$878$	− 739.586i	− 0.842353i
$879$	0	0
$880$	− 285.788i	− 0.324760i
$881$	− 20.7846i	− 0.0235921i	−0.999930	−	0.0117960i	$-0.996245\pi$
$881$	− 20.7846i	− 0.0235921i	0.999930	−	0.0117960i	$-0.00375488\pi$
$882$	0	0
$883$	386.000	0.437146	0.218573	−	0.975821i	$-0.429860\pi$
$883$	386.000	0.437146	0.218573	+	0.975821i	$0.429860\pi$
$884$	−504.000	−0.570136
$885$	0	0
$886$	−155.000	−0.174944
$887$	− 1725.12i	− 1.94490i	−0.233120	−	0.972448i	$-0.574894\pi$
$887$	− 1725.12i	− 1.94490i	0.233120	−	0.972448i	$-0.425106\pi$
$888$	0	0
$889$	0	0
$890$	414.000	0.465169
$891$	0	0
$892$	67.5500i	0.0757287i
$893$	264.000	0.295633
$894$	0	0
$895$	239.023i	0.267065i
$896$	0	0
$897$	0	0
$898$	−368.000	−0.409800
$899$	822.724i	0.915155i
$900$	0	0
$901$	751.710i	0.834306i
$902$	38.1051i	0.0422451i
$903$	0	0
$904$	518.000	0.573009
$905$	−162.000	−0.179006
$906$	0	0
$907$	−592.000	−0.652701	−0.326351	−	0.945249i	$-0.605819\pi$
$907$	−592.000	−0.652701	−0.326351	+	0.945249i	$0.605819\pi$
$908$	202.650i	0.223183i
$909$	0	0
$910$	0	0
$911$	416.000	0.456641	0.228321	−	0.973586i	$-0.426677\pi$
$911$	416.000	0.456641	0.228321	+	0.973586i	$0.426677\pi$
$912$	0	0
$913$	− 590.629i	− 0.646911i
$914$	−341.000	−0.373085
$915$	0	0
$916$	− 93.5307i	− 0.102108i
$917$	0	0
$918$	0	0
$919$	50.0000	0.0544070	0.0272035	−	0.999630i	$-0.491340\pi$
$919$	50.0000	0.0544070	0.0272035	+	0.999630i	$0.491340\pi$
$920$	1018.45i	1.10701i
$921$	0	0
$922$	− 55.4256i	− 0.0601146i
$923$	− 443.405i	− 0.480395i
$924$	0	0
$925$	116.000	0.125405
$926$	178.000	0.192225
$927$	0	0
$928$	825.000	0.889009
$929$	412.228i	0.443733i	0.975077	+	0.221867i	$0.0712149\pi$
$929$	412.228i	0.443733i	−0.975077	+	0.221867i	$0.928785\pi$
$930$	0	0
$931$	0	0
$932$	786.000	0.843348
$933$	0	0
$934$	− 658.179i	− 0.704689i
$935$	−1386.00	−1.48235
$936$	0	0
$937$	− 1605.61i	− 1.71357i	−0.515677	−	0.856783i	$-0.672460\pi$
$937$	− 1605.61i	− 1.71357i	0.515677	−	0.856783i	$-0.327540\pi$
$938$	0	0
$939$	0	0
$940$	1188.00	1.26383
$941$	1328.48i	1.41178i	0.708323	+	0.705889i	$0.249452\pi$
$941$	1328.48i	1.41178i	−0.708323	+	0.705889i	$0.750548\pi$
$942$	0	0
$943$	96.9948i	0.102858i
$944$	43.3013i	0.0458700i
$945$	0	0
$946$	−286.000	−0.302326
$947$	1370.00	1.44667	0.723337	−	0.690495i	$-0.242607\pi$
$947$	1370.00	1.44667	0.723337	+	0.690495i	$0.242607\pi$
$948$	0	0
$949$	−48.0000	−0.0505796
$950$	6.92820i	0.00729285i
$951$	0	0
$952$	0	0
$953$	−1150.00	−1.20672	−0.603358	−	0.797471i	$-0.706171\pi$
$953$	−1150.00	−1.20672	−0.603358	+	0.797471i	$0.706171\pi$
$954$	0	0
$955$	1080.80i	1.13173i
$956$	480.000	0.502092
$957$	0	0
$958$	509.223i	0.531548i
$959$	0	0
$960$	0	0
$961$	−122.000	−0.126951
$962$	401.836i	0.417709i
$963$	0	0
$964$	− 1418.55i	− 1.47152i
$965$	− 1241.88i	− 1.28692i
$966$	0	0
$967$	5.00000	0.00517063	0.00258532	−	0.999997i	$-0.499177\pi$
$967$	5.00000	0.00517063	0.00258532	+	0.999997i	$0.499177\pi$
$968$	0	0
$969$	0	0
$970$	−477.000	−0.491753
$971$	445.137i	0.458432i	0.973376	+	0.229216i	$0.0736161\pi$
$971$	445.137i	0.458432i	−0.973376	+	0.229216i	$0.926384\pi$
$972$	0	0
$973$	0	0
$974$	841.000	0.863450
$975$	0	0
$976$	− 69.2820i	− 0.0709857i
$977$	−958.000	−0.980553	−0.490276	−	0.871567i	$-0.663104\pi$
$977$	−958.000	−0.980553	−0.490276	+	0.871567i	$0.663104\pi$
$978$	0	0
$979$	− 876.418i	− 0.895217i
$980$	0	0
$981$	0	0
$982$	−959.000	−0.976578
$983$	− 280.592i	− 0.285445i	−0.989763	−	0.142722i	$-0.954414\pi$
$983$	− 280.592i	− 0.285445i	0.989763	−	0.142722i	$-0.0455856\pi$
$984$	0	0
$985$	− 135.100i	− 0.137157i
$986$	− 606.218i	− 0.614825i
$987$	0	0
$988$	72.0000	0.0728745
$989$	−728.000	−0.736097
$990$	0	0
$991$	923.000	0.931382	0.465691	−	0.884947i	$-0.345806\pi$
$991$	923.000	0.931382	0.465691	+	0.884947i	$0.345806\pi$
$992$	1086.00i	1.09475i
$993$	0	0
$994$	0	0
$995$	−1260.00	−1.26633
$996$	0	0
$997$	20.7846i	0.0208472i	0.999946	+	0.0104236i	$0.00331799\pi$
$997$	20.7846i	0.0208472i	−0.999946	+	0.0104236i	$0.996682\pi$
$998$	118.000	0.118236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.d.b.244.1		2
3.2	odd	2		147.3.d.b.97.2		2
7.2	even	3		441.3.m.e.325.1		2
7.3	odd	6		441.3.m.e.19.1		2
7.4	even	3		63.3.m.c.19.1		2
7.5	odd	6		63.3.m.c.10.1		2
7.6	odd	2	inner	441.3.d.b.244.2		2
12.11	even	2		2352.3.f.d.97.1		2
21.2	odd	6		147.3.f.c.31.1		2
21.5	even	6		21.3.f.b.10.1	✓	2
21.11	odd	6		21.3.f.b.19.1	yes	2
21.17	even	6		147.3.f.c.19.1		2
21.20	even	2		147.3.d.b.97.1		2
28.11	odd	6		1008.3.cg.g.145.1		2
28.19	even	6		1008.3.cg.g.577.1		2
84.11	even	6		336.3.bh.a.145.1		2
84.47	odd	6		336.3.bh.a.241.1		2
84.83	odd	2		2352.3.f.d.97.2		2
105.32	even	12		525.3.s.c.124.1		4
105.47	odd	12		525.3.s.c.199.2		4
105.53	even	12		525.3.s.c.124.2		4
105.68	odd	12		525.3.s.c.199.1		4
105.74	odd	6		525.3.o.g.376.1		2
105.89	even	6		525.3.o.g.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.3.f.b.10.1	✓	2	21.5	even	6
21.3.f.b.19.1	yes	2	21.11	odd	6
63.3.m.c.10.1		2	7.5	odd	6
63.3.m.c.19.1		2	7.4	even	3
147.3.d.b.97.1		2	21.20	even	2
147.3.d.b.97.2		2	3.2	odd	2
147.3.f.c.19.1		2	21.17	even	6
147.3.f.c.31.1		2	21.2	odd	6
336.3.bh.a.145.1		2	84.11	even	6
336.3.bh.a.241.1		2	84.47	odd	6
441.3.d.b.244.1		2	1.1	even	1	trivial
441.3.d.b.244.2		2	7.6	odd	2	inner
441.3.m.e.19.1		2	7.3	odd	6
441.3.m.e.325.1		2	7.2	even	3
525.3.o.g.376.1		2	105.74	odd	6
525.3.o.g.451.1		2	105.89	even	6
525.3.s.c.124.1		4	105.32	even	12
525.3.s.c.124.2		4	105.53	even	12
525.3.s.c.199.1		4	105.68	odd	12
525.3.s.c.199.2		4	105.47	odd	12
1008.3.cg.g.145.1		2	28.11	odd	6
1008.3.cg.g.577.1		2	28.19	even	6
2352.3.f.d.97.1		2	12.11	even	2
2352.3.f.d.97.2		2	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.00000 q^{4} -5.19615i q^{5} +7.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} -3.00000 q^{4} -5.19615i q^{5} +7.00000 q^{8} +5.19615i q^{10} +11.0000 q^{11} +6.92820i q^{13} +5.00000 q^{16} -24.2487i q^{17} +3.46410i q^{19} +15.5885i q^{20} -11.0000 q^{22} -28.0000 q^{23} -2.00000 q^{25} -6.92820i q^{26} -25.0000 q^{29} -32.9090i q^{31} -33.0000 q^{32} +24.2487i q^{34} -58.0000 q^{37} -3.46410i q^{38} -36.3731i q^{40} -3.46410i q^{41} +26.0000 q^{43} -33.0000 q^{44} +28.0000 q^{46} -76.2102i q^{47} +2.00000 q^{50} -20.7846i q^{52} -31.0000 q^{53} -57.1577i q^{55} +25.0000 q^{58} +8.66025i q^{59} -13.8564i q^{61} +32.9090i q^{62} +13.0000 q^{64} +36.0000 q^{65} -52.0000 q^{67} +72.7461i q^{68} -64.0000 q^{71} +6.92820i q^{73} +58.0000 q^{74} -10.3923i q^{76} +17.0000 q^{79} -25.9808i q^{80} +3.46410i q^{82} -53.6936i q^{83} -126.000 q^{85} -26.0000 q^{86} +77.0000 q^{88} -79.6743i q^{89} +84.0000 q^{92} +76.2102i q^{94} +18.0000 q^{95} +91.7987i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 6 q^{4} + 14 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 6 * q^4 + 14 * q^8	\( 2 q - 2 q^{2} - 6 q^{4} + 14 q^{8} + 22 q^{11} + 10 q^{16} - 22 q^{22} - 56 q^{23} - 4 q^{25} - 50 q^{29} - 66 q^{32} - 116 q^{37} + 52 q^{43} - 66 q^{44} + 56 q^{46} + 4 q^{50} - 62 q^{53} + 50 q^{58} + 26 q^{64} + 72 q^{65} - 104 q^{67} - 128 q^{71} + 116 q^{74} + 34 q^{79} - 252 q^{85} - 52 q^{86} + 154 q^{88} + 168 q^{92} + 36 q^{95}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^4 + 14 * q^8 + 22 * q^11 + 10 * q^16 - 22 * q^22 - 56 * q^23 - 4 * q^25 - 50 * q^29 - 66 * q^32 - 116 * q^37 + 52 * q^43 - 66 * q^44 + 56 * q^46 + 4 * q^50 - 62 * q^53 + 50 * q^58 + 26 * q^64 + 72 * q^65 - 104 * q^67 - 128 * q^71 + 116 * q^74 + 34 * q^79 - 252 * q^85 - 52 * q^86 + 154 * q^88 + 168 * q^92 + 36 * q^95

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