LMFDB - Embedded newform 441.3.d.d.244.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.46410i q^{5} -8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +3.46410i q^{5} -8.00000 q^{8} +6.92820i q^{10} -10.0000 q^{11} +12.1244i q^{13} -16.0000 q^{16} -6.92820i q^{17} +32.9090i q^{19} -20.0000 q^{22} -40.0000 q^{23} +13.0000 q^{25} +24.2487i q^{26} -16.0000 q^{29} -5.19615i q^{31} -13.8564i q^{34} +5.00000 q^{37} +65.8179i q^{38} -27.7128i q^{40} -24.2487i q^{41} -19.0000 q^{43} -80.0000 q^{46} +51.9615i q^{47} +26.0000 q^{50} +32.0000 q^{53} -34.6410i q^{55} -32.0000 q^{58} +41.5692i q^{59} +20.7846i q^{61} -10.3923i q^{62} +64.0000 q^{64} -42.0000 q^{65} +59.0000 q^{67} +26.0000 q^{71} +19.0526i q^{73} +10.0000 q^{74} +47.0000 q^{79} -55.4256i q^{80} -48.4974i q^{82} +24.2487i q^{83} +24.0000 q^{85} -38.0000 q^{86} +80.0000 q^{88} -117.779i q^{89} +103.923i q^{94} -114.000 q^{95} -48.4974i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 - 16 * q^8	$ 2 q + 4 q^{2} - 16 q^{8} - 20 q^{11} - 32 q^{16} - 40 q^{22} - 80 q^{23} + 26 q^{25} - 32 q^{29} + 10 q^{37} - 38 q^{43} - 160 q^{46} + 52 q^{50} + 64 q^{53} - 64 q^{58} + 128 q^{64} - 84 q^{65} + 118 q^{67} + 52 q^{71} + 20 q^{74} + 94 q^{79} + 48 q^{85} - 76 q^{86} + 160 q^{88} - 228 q^{95}+O(q^{100}) $ 2 * q + 4 * q^2 - 16 * q^8 - 20 * q^11 - 32 * q^16 - 40 * q^22 - 80 * q^23 + 26 * q^25 - 32 * q^29 + 10 * q^37 - 38 * q^43 - 160 * q^46 + 52 * q^50 + 64 * q^53 - 64 * q^58 + 128 * q^64 - 84 * q^65 + 118 * q^67 + 52 * q^71 + 20 * q^74 + 94 * q^79 + 48 * q^85 - 76 * q^86 + 160 * q^88 - 228 * 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 $

$2$	2.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	2.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.46410i	0.692820i	0.938083	+	0.346410i	$0.112599\pi$
$5$	3.46410i	0.692820i	−0.938083	+	0.346410i	$0.887401\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−1.00000
$9$	0	0
$10$	6.92820i	0.692820i
$11$	−10.0000	−0.909091	−0.454545	−	0.890724i	$-0.650198\pi$
$11$	−10.0000	−0.909091	−0.454545	+	0.890724i	$0.650198\pi$
$12$	0	0
$13$	12.1244i	0.932643i	0.884615	+	0.466321i	$0.154421\pi$
$13$	12.1244i	0.932643i	−0.884615	+	0.466321i	$0.845579\pi$
$14$	0	0
$15$	0	0
$16$	−16.0000	−1.00000
$17$	− 6.92820i	− 0.407541i	−0.979019	−	0.203771i	$-0.934680\pi$
$17$	− 6.92820i	− 0.407541i	0.979019	−	0.203771i	$-0.0653197\pi$
$18$	0	0
$19$	32.9090i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$19$	32.9090i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$20$	0	0
$21$	0	0
$22$	−20.0000	−0.909091
$23$	−40.0000	−1.73913	−0.869565	−	0.493818i	$-0.835601\pi$
$23$	−40.0000	−1.73913	−0.869565	+	0.493818i	$0.835601\pi$
$24$	0	0
$25$	13.0000	0.520000
$26$	24.2487i	0.932643i
$27$	0	0
$28$	0	0
$29$	−16.0000	−0.551724	−0.275862	−	0.961197i	$-0.588963\pi$
$29$	−16.0000	−0.551724	−0.275862	+	0.961197i	$0.588963\pi$
$30$	0	0
$31$	− 5.19615i	− 0.167618i	−0.996482	−	0.0838089i	$-0.973291\pi$
$31$	− 5.19615i	− 0.167618i	0.996482	−	0.0838089i	$-0.0267085\pi$
$32$	0	0
$33$	0	0
$34$	− 13.8564i	− 0.407541i
$35$	0	0
$36$	0	0
$37$	5.00000	0.135135	0.0675676	−	0.997715i	$-0.478476\pi$
$37$	5.00000	0.135135	0.0675676	+	0.997715i	$0.478476\pi$
$38$	65.8179i	1.73205i
$39$	0	0
$40$	− 27.7128i	− 0.692820i
$41$	− 24.2487i	− 0.591432i	−0.955276	−	0.295716i	$-0.904442\pi$
$41$	− 24.2487i	− 0.591432i	0.955276	−	0.295716i	$-0.0955582\pi$
$42$	0	0
$43$	−19.0000	−0.441860	−0.220930	−	0.975290i	$-0.570909\pi$
$43$	−19.0000	−0.441860	−0.220930	+	0.975290i	$0.570909\pi$
$44$	0	0
$45$	0	0
$46$	−80.0000	−1.73913
$47$	51.9615i	1.10556i	0.833326	+	0.552782i	$0.186434\pi$
$47$	51.9615i	1.10556i	−0.833326	+	0.552782i	$0.813566\pi$
$48$	0	0
$49$	0	0
$50$	26.0000	0.520000
$51$	0	0
$52$	0	0
$53$	32.0000	0.603774	0.301887	−	0.953344i	$-0.402384\pi$
$53$	32.0000	0.603774	0.301887	+	0.953344i	$0.402384\pi$
$54$	0	0
$55$	− 34.6410i	− 0.629837i
$56$	0	0
$57$	0	0
$58$	−32.0000	−0.551724
$59$	41.5692i	0.704563i	0.935894	+	0.352282i	$0.114594\pi$
$59$	41.5692i	0.704563i	−0.935894	+	0.352282i	$0.885406\pi$
$60$	0	0
$61$	20.7846i	0.340731i	0.985381	+	0.170366i	$0.0544949\pi$
$61$	20.7846i	0.340731i	−0.985381	+	0.170366i	$0.945505\pi$
$62$	− 10.3923i	− 0.167618i
$63$	0	0
$64$	64.0000	1.00000
$65$	−42.0000	−0.646154
$66$	0	0
$67$	59.0000	0.880597	0.440299	−	0.897851i	$-0.354873\pi$
$67$	59.0000	0.880597	0.440299	+	0.897851i	$0.354873\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	26.0000	0.366197	0.183099	−	0.983095i	$-0.441387\pi$
$71$	26.0000	0.366197	0.183099	+	0.983095i	$0.441387\pi$
$72$	0	0
$73$	19.0526i	0.260994i	0.991449	+	0.130497i	$0.0416573\pi$
$73$	19.0526i	0.260994i	−0.991449	+	0.130497i	$0.958343\pi$
$74$	10.0000	0.135135
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	47.0000	0.594937	0.297468	−	0.954732i	$-0.403858\pi$
$79$	47.0000	0.594937	0.297468	+	0.954732i	$0.403858\pi$
$80$	− 55.4256i	− 0.692820i
$81$	0	0
$82$	− 48.4974i	− 0.591432i
$83$	24.2487i	0.292153i	0.989273	+	0.146077i	$0.0466646\pi$
$83$	24.2487i	0.292153i	−0.989273	+	0.146077i	$0.953335\pi$
$84$	0	0
$85$	24.0000	0.282353
$86$	−38.0000	−0.441860
$87$	0	0
$88$	80.0000	0.909091
$89$	− 117.779i	− 1.32336i	−0.749784	−	0.661682i	$-0.769843\pi$
$89$	− 117.779i	− 1.32336i	0.749784	−	0.661682i	$-0.230157\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	103.923i	1.10556i
$95$	−114.000	−1.20000
$96$	0	0
$97$	− 48.4974i	− 0.499973i	−0.968249	−	0.249987i	$-0.919574\pi$
$97$	− 48.4974i	− 0.499973i	0.968249	−	0.249987i	$-0.0804263\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 128.172i	− 1.26903i	−0.772912	−	0.634514i	$-0.781200\pi$
$101$	− 128.172i	− 1.26903i	0.772912	−	0.634514i	$-0.218800\pi$
$102$	0	0
$103$	8.66025i	0.0840801i	0.999116	+	0.0420401i	$0.0133857\pi$
$103$	8.66025i	0.0840801i	−0.999116	+	0.0420401i	$0.986614\pi$
$104$	− 96.9948i	− 0.932643i
$105$	0	0
$106$	64.0000	0.603774
$107$	212.000	1.98131	0.990654	−	0.136397i	$-0.0435524\pi$
$107$	212.000	1.98131	0.990654	+	0.136397i	$0.0435524\pi$
$108$	0	0
$109$	17.0000	0.155963	0.0779817	−	0.996955i	$-0.475152\pi$
$109$	17.0000	0.155963	0.0779817	+	0.996955i	$0.475152\pi$
$110$	− 69.2820i	− 0.629837i
$111$	0	0
$112$	0	0
$113$	−142.000	−1.25664	−0.628319	−	0.777956i	$-0.716257\pi$
$113$	−142.000	−1.25664	−0.628319	+	0.777956i	$0.716257\pi$
$114$	0	0
$115$	− 138.564i	− 1.20490i
$116$	0	0
$117$	0	0
$118$	83.1384i	0.704563i
$119$	0	0
$120$	0	0
$121$	−21.0000	−0.173554
$122$	41.5692i	0.340731i
$123$	0	0
$124$	0	0
$125$	131.636i	1.05309i
$126$	0	0
$127$	−145.000	−1.14173	−0.570866	−	0.821043i	$-0.693392\pi$
$127$	−145.000	−1.14173	−0.570866	+	0.821043i	$0.693392\pi$
$128$	128.000	1.00000
$129$	0	0
$130$	−84.0000	−0.646154
$131$	148.956i	1.13707i	0.822659	+	0.568536i	$0.192490\pi$
$131$	148.956i	1.13707i	−0.822659	+	0.568536i	$0.807510\pi$
$132$	0	0
$133$	0	0
$134$	118.000	0.880597
$135$	0	0
$136$	55.4256i	0.407541i
$137$	116.000	0.846715	0.423358	−	0.905963i	$-0.360851\pi$
$137$	116.000	0.846715	0.423358	+	0.905963i	$0.360851\pi$
$138$	0	0
$139$	84.8705i	0.610579i	0.952260	+	0.305290i	$0.0987532\pi$
$139$	84.8705i	0.610579i	−0.952260	+	0.305290i	$0.901247\pi$
$140$	0	0
$141$	0	0
$142$	52.0000	0.366197
$143$	− 121.244i	− 0.847857i
$144$	0	0
$145$	− 55.4256i	− 0.382246i
$146$	38.1051i	0.260994i
$147$	0	0
$148$	0	0
$149$	−124.000	−0.832215	−0.416107	−	0.909315i	$-0.636606\pi$
$149$	−124.000	−0.832215	−0.416107	+	0.909315i	$0.636606\pi$
$150$	0	0
$151$	−46.0000	−0.304636	−0.152318	−	0.988332i	$-0.548674\pi$
$151$	−46.0000	−0.304636	−0.152318	+	0.988332i	$0.548674\pi$
$152$	− 263.272i	− 1.73205i
$153$	0	0
$154$	0	0
$155$	18.0000	0.116129
$156$	0	0
$157$	− 187.061i	− 1.19147i	−0.803179	−	0.595737i	$-0.796860\pi$
$157$	− 187.061i	− 1.19147i	0.803179	−	0.595737i	$-0.203140\pi$
$158$	94.0000	0.594937
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−58.0000	−0.355828	−0.177914	−	0.984046i	$-0.556935\pi$
$163$	−58.0000	−0.355828	−0.177914	+	0.984046i	$0.556935\pi$
$164$	0	0
$165$	0	0
$166$	48.4974i	0.292153i
$167$	266.736i	1.59722i	0.601849	+	0.798610i	$0.294431\pi$
$167$	266.736i	1.59722i	−0.601849	+	0.798610i	$0.705569\pi$
$168$	0	0
$169$	22.0000	0.130178
$170$	48.0000	0.282353
$171$	0	0
$172$	0	0
$173$	124.708i	0.720854i	0.932788	+	0.360427i	$0.117369\pi$
$173$	124.708i	0.720854i	−0.932788	+	0.360427i	$0.882631\pi$
$174$	0	0
$175$	0	0
$176$	160.000	0.909091
$177$	0	0
$178$	− 235.559i	− 1.32336i
$179$	−10.0000	−0.0558659	−0.0279330	−	0.999610i	$-0.508892\pi$
$179$	−10.0000	−0.0558659	−0.0279330	+	0.999610i	$0.508892\pi$
$180$	0	0
$181$	327.358i	1.80861i	0.426892	+	0.904303i	$0.359609\pi$
$181$	327.358i	1.80861i	−0.426892	+	0.904303i	$0.640391\pi$
$182$	0	0
$183$	0	0
$184$	320.000	1.73913
$185$	17.3205i	0.0936244i
$186$	0	0
$187$	69.2820i	0.370492i
$188$	0	0
$189$	0	0
$190$	−228.000	−1.20000
$191$	2.00000	0.0104712	0.00523560	−	0.999986i	$-0.498333\pi$
$191$	2.00000	0.0104712	0.00523560	+	0.999986i	$0.498333\pi$
$192$	0	0
$193$	−235.000	−1.21762	−0.608808	−	0.793317i	$-0.708352\pi$
$193$	−235.000	−1.21762	−0.608808	+	0.793317i	$0.708352\pi$
$194$	− 96.9948i	− 0.499973i
$195$	0	0
$196$	0	0
$197$	−100.000	−0.507614	−0.253807	−	0.967255i	$-0.581683\pi$
$197$	−100.000	−0.507614	−0.253807	+	0.967255i	$0.581683\pi$
$198$	0	0
$199$	200.918i	1.00964i	0.863225	+	0.504819i	$0.168441\pi$
$199$	200.918i	1.00964i	−0.863225	+	0.504819i	$0.831559\pi$
$200$	−104.000	−0.520000
$201$	0	0
$202$	− 256.344i	− 1.26903i
$203$	0	0
$204$	0	0
$205$	84.0000	0.409756
$206$	17.3205i	0.0840801i
$207$	0	0
$208$	− 193.990i	− 0.932643i
$209$	− 329.090i	− 1.57459i
$210$	0	0
$211$	2.00000	0.00947867	0.00473934	−	0.999989i	$-0.498491\pi$
$211$	2.00000	0.00947867	0.00473934	+	0.999989i	$0.498491\pi$
$212$	0	0
$213$	0	0
$214$	424.000	1.98131
$215$	− 65.8179i	− 0.306130i
$216$	0	0
$217$	0	0
$218$	34.0000	0.155963
$219$	0	0
$220$	0	0
$221$	84.0000	0.380090
$222$	0	0
$223$	− 339.482i	− 1.52234i	−0.648552	−	0.761170i	$-0.724625\pi$
$223$	− 339.482i	− 1.52234i	0.648552	−	0.761170i	$-0.275375\pi$
$224$	0	0
$225$	0	0
$226$	−284.000	−1.25664
$227$	162.813i	0.717237i	0.933484	+	0.358618i	$0.116752\pi$
$227$	162.813i	0.717237i	−0.933484	+	0.358618i	$0.883248\pi$
$228$	0	0
$229$	8.66025i	0.0378177i	0.999821	+	0.0189089i	$0.00601923\pi$
$229$	8.66025i	0.0378177i	−0.999821	+	0.0189089i	$0.993981\pi$
$230$	− 277.128i	− 1.20490i
$231$	0	0
$232$	128.000	0.551724
$233$	170.000	0.729614	0.364807	−	0.931083i	$-0.381135\pi$
$233$	170.000	0.729614	0.364807	+	0.931083i	$0.381135\pi$
$234$	0	0
$235$	−180.000	−0.765957
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−142.000	−0.594142	−0.297071	−	0.954855i	$-0.596010\pi$
$239$	−142.000	−0.594142	−0.297071	+	0.954855i	$0.596010\pi$
$240$	0	0
$241$	152.420i	0.632450i	0.948684	+	0.316225i	$0.102415\pi$
$241$	152.420i	0.632450i	−0.948684	+	0.316225i	$0.897585\pi$
$242$	−42.0000	−0.173554
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−399.000	−1.61538
$248$	41.5692i	0.167618i
$249$	0	0
$250$	263.272i	1.05309i
$251$	290.985i	1.15930i	0.814865	+	0.579650i	$0.196811\pi$
$251$	290.985i	1.15930i	−0.814865	+	0.579650i	$0.803189\pi$
$252$	0	0
$253$	400.000	1.58103
$254$	−290.000	−1.14173
$255$	0	0
$256$	0	0
$257$	439.941i	1.71183i	0.517115	+	0.855916i	$0.327006\pi$
$257$	439.941i	1.71183i	−0.517115	+	0.855916i	$0.672994\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	297.913i	1.13707i
$263$	−136.000	−0.517110	−0.258555	−	0.965997i	$-0.583246\pi$
$263$	−136.000	−0.517110	−0.258555	+	0.965997i	$0.583246\pi$
$264$	0	0
$265$	110.851i	0.418307i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 225.167i	− 0.837051i	−0.908205	−	0.418525i	$-0.862547\pi$
$269$	− 225.167i	− 0.837051i	0.908205	−	0.418525i	$-0.137453\pi$
$270$	0	0
$271$	− 367.195i	− 1.35496i	−0.735540	−	0.677481i	$-0.763072\pi$
$271$	− 367.195i	− 1.35496i	0.735540	−	0.677481i	$-0.236928\pi$
$272$	110.851i	0.407541i
$273$	0	0
$274$	232.000	0.846715
$275$	−130.000	−0.472727
$276$	0	0
$277$	395.000	1.42599	0.712996	−	0.701168i	$-0.247338\pi$
$277$	395.000	1.42599	0.712996	+	0.701168i	$0.247338\pi$
$278$	169.741i	0.610579i
$279$	0	0
$280$	0	0
$281$	−100.000	−0.355872	−0.177936	−	0.984042i	$-0.556942\pi$
$281$	−100.000	−0.355872	−0.177936	+	0.984042i	$0.556942\pi$
$282$	0	0
$283$	358.535i	1.26691i	0.773781	+	0.633453i	$0.218363\pi$
$283$	358.535i	1.26691i	−0.773781	+	0.633453i	$0.781637\pi$
$284$	0	0
$285$	0	0
$286$	− 242.487i	− 0.847857i
$287$	0	0
$288$	0	0
$289$	241.000	0.833910
$290$	− 110.851i	− 0.382246i
$291$	0	0
$292$	0	0
$293$	− 242.487i	− 0.827601i	−0.910368	−	0.413801i	$-0.864201\pi$
$293$	− 242.487i	− 0.827601i	0.910368	−	0.413801i	$-0.135799\pi$
$294$	0	0
$295$	−144.000	−0.488136
$296$	−40.0000	−0.135135
$297$	0	0
$298$	−248.000	−0.832215
$299$	− 484.974i	− 1.62199i
$300$	0	0
$301$	0	0
$302$	−92.0000	−0.304636
$303$	0	0
$304$	− 526.543i	− 1.73205i
$305$	−72.0000	−0.236066
$306$	0	0
$307$	181.865i	0.592395i	0.955127	+	0.296198i	$0.0957187\pi$
$307$	181.865i	0.592395i	−0.955127	+	0.296198i	$0.904281\pi$
$308$	0	0
$309$	0	0
$310$	36.0000	0.116129
$311$	550.792i	1.77104i	0.464605	+	0.885518i	$0.346196\pi$
$311$	550.792i	1.77104i	−0.464605	+	0.885518i	$0.653804\pi$
$312$	0	0
$313$	202.650i	0.647444i	0.946152	+	0.323722i	$0.104934\pi$
$313$	202.650i	0.647444i	−0.946152	+	0.323722i	$0.895066\pi$
$314$	− 374.123i	− 1.19147i
$315$	0	0
$316$	0	0
$317$	−292.000	−0.921136	−0.460568	−	0.887625i	$-0.652354\pi$
$317$	−292.000	−0.921136	−0.460568	+	0.887625i	$0.652354\pi$
$318$	0	0
$319$	160.000	0.501567
$320$	221.703i	0.692820i
$321$	0	0
$322$	0	0
$323$	228.000	0.705882
$324$	0	0
$325$	157.617i	0.484974i
$326$	−116.000	−0.355828
$327$	0	0
$328$	193.990i	0.591432i
$329$	0	0
$330$	0	0
$331$	5.00000	0.0151057	0.00755287	−	0.999971i	$-0.497596\pi$
$331$	5.00000	0.0151057	0.00755287	+	0.999971i	$0.497596\pi$
$332$	0	0
$333$	0	0
$334$	533.472i	1.59722i
$335$	204.382i	0.610096i
$336$	0	0
$337$	−439.000	−1.30267	−0.651335	−	0.758790i	$-0.725791\pi$
$337$	−439.000	−1.30267	−0.651335	+	0.758790i	$0.725791\pi$
$338$	44.0000	0.130178
$339$	0	0
$340$	0	0
$341$	51.9615i	0.152380i
$342$	0	0
$343$	0	0
$344$	152.000	0.441860
$345$	0	0
$346$	249.415i	0.720854i
$347$	−220.000	−0.634006	−0.317003	−	0.948425i	$-0.602676\pi$
$347$	−220.000	−0.634006	−0.317003	+	0.948425i	$0.602676\pi$
$348$	0	0
$349$	− 339.482i	− 0.972728i	−0.873756	−	0.486364i	$-0.838323\pi$
$349$	− 339.482i	− 0.972728i	0.873756	−	0.486364i	$-0.161677\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	308.305i	0.873385i	0.899611	+	0.436693i	$0.143850\pi$
$353$	308.305i	0.873385i	−0.899611	+	0.436693i	$0.856150\pi$
$354$	0	0
$355$	90.0666i	0.253709i
$356$	0	0
$357$	0	0
$358$	−20.0000	−0.0558659
$359$	−292.000	−0.813370	−0.406685	−	0.913568i	$-0.633315\pi$
$359$	−292.000	−0.813370	−0.406685	+	0.913568i	$0.633315\pi$
$360$	0	0
$361$	−722.000	−2.00000
$362$	654.715i	1.80861i
$363$	0	0
$364$	0	0
$365$	−66.0000	−0.180822
$366$	0	0
$367$	− 538.668i	− 1.46776i	−0.679279	−	0.733880i	$-0.737708\pi$
$367$	− 538.668i	− 1.46776i	0.679279	−	0.733880i	$-0.262292\pi$
$368$	640.000	1.73913
$369$	0	0
$370$	34.6410i	0.0936244i
$371$	0	0
$372$	0	0
$373$	−205.000	−0.549598	−0.274799	−	0.961502i	$-0.588611\pi$
$373$	−205.000	−0.549598	−0.274799	+	0.961502i	$0.588611\pi$
$374$	138.564i	0.370492i
$375$	0	0
$376$	− 415.692i	− 1.10556i
$377$	− 193.990i	− 0.514562i
$378$	0	0
$379$	−523.000	−1.37995	−0.689974	−	0.723835i	$-0.742378\pi$
$379$	−523.000	−1.37995	−0.689974	+	0.723835i	$0.742378\pi$
$380$	0	0
$381$	0	0
$382$	4.00000	0.0104712
$383$	76.2102i	0.198982i	0.995038	+	0.0994912i	$0.0317215\pi$
$383$	76.2102i	0.198982i	−0.995038	+	0.0994912i	$0.968278\pi$
$384$	0	0
$385$	0	0
$386$	−470.000	−1.21762
$387$	0	0
$388$	0	0
$389$	74.0000	0.190231	0.0951157	−	0.995466i	$-0.469678\pi$
$389$	74.0000	0.190231	0.0951157	+	0.995466i	$0.469678\pi$
$390$	0	0
$391$	277.128i	0.708768i
$392$	0	0
$393$	0	0
$394$	−200.000	−0.507614
$395$	162.813i	0.412184i
$396$	0	0
$397$	323.894i	0.815853i	0.913015	+	0.407926i	$0.133748\pi$
$397$	323.894i	0.815853i	−0.913015	+	0.407926i	$0.866252\pi$
$398$	401.836i	1.00964i
$399$	0	0
$400$	−208.000	−0.520000
$401$	128.000	0.319202	0.159601	−	0.987182i	$-0.448979\pi$
$401$	128.000	0.319202	0.159601	+	0.987182i	$0.448979\pi$
$402$	0	0
$403$	63.0000	0.156328
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−50.0000	−0.122850
$408$	0	0
$409$	− 296.181i	− 0.724158i	−0.932147	−	0.362079i	$-0.882067\pi$
$409$	− 296.181i	− 0.724158i	0.932147	−	0.362079i	$-0.117933\pi$
$410$	168.000	0.409756
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−84.0000	−0.202410
$416$	0	0
$417$	0	0
$418$	− 658.179i	− 1.57459i
$419$	412.228i	0.983838i	0.870641	+	0.491919i	$0.163704\pi$
$419$	412.228i	0.983838i	−0.870641	+	0.491919i	$0.836296\pi$
$420$	0	0
$421$	107.000	0.254157	0.127078	−	0.991893i	$-0.459440\pi$
$421$	107.000	0.254157	0.127078	+	0.991893i	$0.459440\pi$
$422$	4.00000	0.00947867
$423$	0	0
$424$	−256.000	−0.603774
$425$	− 90.0666i	− 0.211922i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	− 131.636i	− 0.306130i
$431$	−262.000	−0.607889	−0.303944	−	0.952690i	$-0.598304\pi$
$431$	−262.000	−0.607889	−0.303944	+	0.952690i	$0.598304\pi$
$432$	0	0
$433$	− 36.3731i	− 0.0840025i	−0.999118	−	0.0420012i	$-0.986627\pi$
$433$	− 36.3731i	− 0.0840025i	0.999118	−	0.0420012i	$-0.0133733\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1316.36i	− 3.01226i
$438$	0	0
$439$	311.769i	0.710180i	0.934832	+	0.355090i	$0.115550\pi$
$439$	311.769i	0.710180i	−0.934832	+	0.355090i	$0.884450\pi$
$440$	277.128i	0.629837i
$441$	0	0
$442$	168.000	0.380090
$443$	212.000	0.478555	0.239278	−	0.970951i	$-0.423089\pi$
$443$	212.000	0.478555	0.239278	+	0.970951i	$0.423089\pi$
$444$	0	0
$445$	408.000	0.916854
$446$	− 678.964i	− 1.52234i
$447$	0	0
$448$	0	0
$449$	782.000	1.74165	0.870824	−	0.491595i	$-0.163586\pi$
$449$	782.000	1.74165	0.870824	+	0.491595i	$0.163586\pi$
$450$	0	0
$451$	242.487i	0.537665i
$452$	0	0
$453$	0	0
$454$	325.626i	0.717237i
$455$	0	0
$456$	0	0
$457$	677.000	1.48140	0.740700	−	0.671836i	$-0.234494\pi$
$457$	677.000	1.48140	0.740700	+	0.671836i	$0.234494\pi$
$458$	17.3205i	0.0378177i
$459$	0	0
$460$	0	0
$461$	− 484.974i	− 1.05200i	−0.850483	−	0.526002i	$-0.823690\pi$
$461$	− 484.974i	− 1.05200i	0.850483	−	0.526002i	$-0.176310\pi$
$462$	0	0
$463$	443.000	0.956803	0.478402	−	0.878141i	$-0.341216\pi$
$463$	443.000	0.956803	0.478402	+	0.878141i	$0.341216\pi$
$464$	256.000	0.551724
$465$	0	0
$466$	340.000	0.729614
$467$	− 45.0333i	− 0.0964311i	−0.998837	−	0.0482155i	$-0.984647\pi$
$467$	− 45.0333i	− 0.0964311i	0.998837	−	0.0482155i	$-0.0153534\pi$
$468$	0	0
$469$	0	0
$470$	−360.000	−0.765957
$471$	0	0
$472$	− 332.554i	− 0.704563i
$473$	190.000	0.401691
$474$	0	0
$475$	427.817i	0.900666i
$476$	0	0
$477$	0	0
$478$	−284.000	−0.594142
$479$	− 55.4256i	− 0.115711i	−0.998325	−	0.0578556i	$-0.981574\pi$
$479$	− 55.4256i	− 0.115711i	0.998325	−	0.0578556i	$-0.0184263\pi$
$480$	0	0
$481$	60.6218i	0.126033i
$482$	304.841i	0.632450i
$483$	0	0
$484$	0	0
$485$	168.000	0.346392
$486$	0	0
$487$	−67.0000	−0.137577	−0.0687885	−	0.997631i	$-0.521913\pi$
$487$	−67.0000	−0.137577	−0.0687885	+	0.997631i	$0.521913\pi$
$488$	− 166.277i	− 0.340731i
$489$	0	0
$490$	0	0
$491$	68.0000	0.138493	0.0692464	−	0.997600i	$-0.477941\pi$
$491$	68.0000	0.138493	0.0692464	+	0.997600i	$0.477941\pi$
$492$	0	0
$493$	110.851i	0.224850i
$494$	−798.000	−1.61538
$495$	0	0
$496$	83.1384i	0.167618i
$497$	0	0
$498$	0	0
$499$	509.000	1.02004	0.510020	−	0.860163i	$-0.329638\pi$
$499$	509.000	1.02004	0.510020	+	0.860163i	$0.329638\pi$
$500$	0	0
$501$	0	0
$502$	581.969i	1.15930i
$503$	− 654.715i	− 1.30162i	−0.759240	−	0.650810i	$-0.774429\pi$
$503$	− 654.715i	− 1.30162i	0.759240	−	0.650810i	$-0.225571\pi$
$504$	0	0
$505$	444.000	0.879208
$506$	800.000	1.58103
$507$	0	0
$508$	0	0
$509$	− 869.490i	− 1.70823i	−0.520083	−	0.854115i	$-0.674099\pi$
$509$	− 869.490i	− 1.70823i	0.520083	−	0.854115i	$-0.325901\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−1.00000
$513$	0	0
$514$	879.882i	1.71183i
$515$	−30.0000	−0.0582524
$516$	0	0
$517$	− 519.615i	− 1.00506i
$518$	0	0
$519$	0	0
$520$	336.000	0.646154
$521$	429.549i	0.824469i	0.911078	+	0.412235i	$0.135252\pi$
$521$	429.549i	0.824469i	−0.911078	+	0.412235i	$0.864748\pi$
$522$	0	0
$523$	− 985.537i	− 1.88439i	−0.335063	−	0.942196i	$-0.608757\pi$
$523$	− 985.537i	− 1.88439i	0.335063	−	0.942196i	$-0.391243\pi$
$524$	0	0
$525$	0	0
$526$	−272.000	−0.517110
$527$	−36.0000	−0.0683112
$528$	0	0
$529$	1071.00	2.02457
$530$	221.703i	0.418307i
$531$	0	0
$532$	0	0
$533$	294.000	0.551595
$534$	0	0
$535$	734.390i	1.37269i
$536$	−472.000	−0.880597
$537$	0	0
$538$	− 450.333i	− 0.837051i
$539$	0	0
$540$	0	0
$541$	−121.000	−0.223660	−0.111830	−	0.993727i	$-0.535671\pi$
$541$	−121.000	−0.223660	−0.111830	+	0.993727i	$0.535671\pi$
$542$	− 734.390i	− 1.35496i
$543$	0	0
$544$	0	0
$545$	58.8897i	0.108055i
$546$	0	0
$547$	926.000	1.69287	0.846435	−	0.532492i	$-0.178744\pi$
$547$	926.000	1.69287	0.846435	+	0.532492i	$0.178744\pi$
$548$	0	0
$549$	0	0
$550$	−260.000	−0.472727
$551$	− 526.543i	− 0.955614i
$552$	0	0
$553$	0	0
$554$	790.000	1.42599
$555$	0	0
$556$	0	0
$557$	662.000	1.18851	0.594255	−	0.804277i	$-0.297447\pi$
$557$	662.000	1.18851	0.594255	+	0.804277i	$0.297447\pi$
$558$	0	0
$559$	− 230.363i	− 0.412098i
$560$	0	0
$561$	0	0
$562$	−200.000	−0.355872
$563$	− 322.161i	− 0.572223i	−0.958196	−	0.286111i	$-0.907637\pi$
$563$	− 322.161i	− 0.572223i	0.958196	−	0.286111i	$-0.0923627\pi$
$564$	0	0
$565$	− 491.902i	− 0.870624i
$566$	717.069i	1.26691i
$567$	0	0
$568$	−208.000	−0.366197
$569$	758.000	1.33216	0.666081	−	0.745880i	$-0.267971\pi$
$569$	758.000	1.33216	0.666081	+	0.745880i	$0.267971\pi$
$570$	0	0
$571$	−865.000	−1.51489	−0.757443	−	0.652901i	$-0.773552\pi$
$571$	−865.000	−1.51489	−0.757443	+	0.652901i	$0.773552\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−520.000	−0.904348
$576$	0	0
$577$	− 1072.14i	− 1.85813i	−0.369920	−	0.929064i	$-0.620615\pi$
$577$	− 1072.14i	− 1.85813i	0.369920	−	0.929064i	$-0.379385\pi$
$578$	482.000	0.833910
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−320.000	−0.548885
$584$	− 152.420i	− 0.260994i
$585$	0	0
$586$	− 484.974i	− 0.827601i
$587$	− 339.482i	− 0.578334i	−0.957279	−	0.289167i	$-0.906622\pi$
$587$	− 339.482i	− 0.578334i	0.957279	−	0.289167i	$-0.0933783\pi$
$588$	0	0
$589$	171.000	0.290323
$590$	−288.000	−0.488136
$591$	0	0
$592$	−80.0000	−0.135135
$593$	245.951i	0.414758i	0.978261	+	0.207379i	$0.0664933\pi$
$593$	245.951i	0.414758i	−0.978261	+	0.207379i	$0.933507\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	− 969.948i	− 1.62199i
$599$	284.000	0.474124	0.237062	−	0.971495i	$-0.423816\pi$
$599$	284.000	0.474124	0.237062	+	0.971495i	$0.423816\pi$
$600$	0	0
$601$	− 594.093i	− 0.988508i	−0.869317	−	0.494254i	$-0.835441\pi$
$601$	− 594.093i	− 0.988508i	0.869317	−	0.494254i	$-0.164559\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 72.7461i	− 0.120242i
$606$	0	0
$607$	8.66025i	0.0142673i	0.999975	+	0.00713365i	$0.00227073\pi$
$607$	8.66025i	0.0142673i	−0.999975	+	0.00713365i	$0.997729\pi$
$608$	0	0
$609$	0	0
$610$	−144.000	−0.236066
$611$	−630.000	−1.03110
$612$	0	0
$613$	878.000	1.43230	0.716150	−	0.697946i	$-0.245903\pi$
$613$	878.000	1.43230	0.716150	+	0.697946i	$0.245903\pi$
$614$	363.731i	0.592395i
$615$	0	0
$616$	0	0
$617$	194.000	0.314425	0.157212	−	0.987565i	$-0.449749\pi$
$617$	194.000	0.314425	0.157212	+	0.987565i	$0.449749\pi$
$618$	0	0
$619$	− 611.414i	− 0.987745i	−0.869534	−	0.493872i	$-0.835581\pi$
$619$	− 611.414i	− 0.987745i	0.869534	−	0.493872i	$-0.164419\pi$
$620$	0	0
$621$	0	0
$622$	1101.58i	1.77104i
$623$	0	0
$624$	0	0
$625$	−131.000	−0.209600
$626$	405.300i	0.647444i
$627$	0	0
$628$	0	0
$629$	− 34.6410i	− 0.0550732i
$630$	0	0
$631$	−250.000	−0.396197	−0.198098	−	0.980182i	$-0.563477\pi$
$631$	−250.000	−0.396197	−0.198098	+	0.980182i	$0.563477\pi$
$632$	−376.000	−0.594937
$633$	0	0
$634$	−584.000	−0.921136
$635$	− 502.295i	− 0.791015i
$636$	0	0
$637$	0	0
$638$	320.000	0.501567
$639$	0	0
$640$	443.405i	0.692820i
$641$	1124.00	1.75351	0.876755	−	0.480937i	$-0.159703\pi$
$641$	1124.00	1.75351	0.876755	+	0.480937i	$0.159703\pi$
$642$	0	0
$643$	− 569.845i	− 0.886228i	−0.896465	−	0.443114i	$-0.853874\pi$
$643$	− 569.845i	− 0.886228i	0.896465	−	0.443114i	$-0.146126\pi$
$644$	0	0
$645$	0	0
$646$	456.000	0.705882
$647$	1084.26i	1.67583i	0.545799	+	0.837916i	$0.316226\pi$
$647$	1084.26i	1.67583i	−0.545799	+	0.837916i	$0.683774\pi$
$648$	0	0
$649$	− 415.692i	− 0.640512i
$650$	315.233i	0.484974i
$651$	0	0
$652$	0	0
$653$	1010.00	1.54671	0.773354	−	0.633975i	$-0.218578\pi$
$653$	1010.00	1.54671	0.773354	+	0.633975i	$0.218578\pi$
$654$	0	0
$655$	−516.000	−0.787786
$656$	387.979i	0.591432i
$657$	0	0
$658$	0	0
$659$	908.000	1.37785	0.688923	−	0.724835i	$-0.258084\pi$
$659$	908.000	1.37785	0.688923	+	0.724835i	$0.258084\pi$
$660$	0	0
$661$	722.265i	1.09269i	0.837562	+	0.546343i	$0.183980\pi$
$661$	722.265i	1.09269i	−0.837562	+	0.546343i	$0.816020\pi$
$662$	10.0000	0.0151057
$663$	0	0
$664$	− 193.990i	− 0.292153i
$665$	0	0
$666$	0	0
$667$	640.000	0.959520
$668$	0	0
$669$	0	0
$670$	408.764i	0.610096i
$671$	− 207.846i	− 0.309756i
$672$	0	0
$673$	−1027.00	−1.52600	−0.763001	−	0.646397i	$-0.776275\pi$
$673$	−1027.00	−1.52600	−0.763001	+	0.646397i	$0.776275\pi$
$674$	−878.000	−1.30267
$675$	0	0
$676$	0	0
$677$	561.184i	0.828928i	0.910066	+	0.414464i	$0.136031\pi$
$677$	561.184i	0.828928i	−0.910066	+	0.414464i	$0.863969\pi$
$678$	0	0
$679$	0	0
$680$	−192.000	−0.282353
$681$	0	0
$682$	103.923i	0.152380i
$683$	−976.000	−1.42899	−0.714495	−	0.699641i	$-0.753343\pi$
$683$	−976.000	−1.42899	−0.714495	+	0.699641i	$0.753343\pi$
$684$	0	0
$685$	401.836i	0.586622i
$686$	0	0
$687$	0	0
$688$	304.000	0.441860
$689$	387.979i	0.563105i
$690$	0	0
$691$	566.381i	0.819654i	0.912163	+	0.409827i	$0.134411\pi$
$691$	566.381i	0.819654i	−0.912163	+	0.409827i	$0.865589\pi$
$692$	0	0
$693$	0	0
$694$	−440.000	−0.634006
$695$	−294.000	−0.423022
$696$	0	0
$697$	−168.000	−0.241033
$698$	− 678.964i	− 0.972728i
$699$	0	0
$700$	0	0
$701$	−352.000	−0.502140	−0.251070	−	0.967969i	$-0.580782\pi$
$701$	−352.000	−0.502140	−0.251070	+	0.967969i	$0.580782\pi$
$702$	0	0
$703$	164.545i	0.234061i
$704$	−640.000	−0.909091
$705$	0	0
$706$	616.610i	0.873385i
$707$	0	0
$708$	0	0
$709$	−1150.00	−1.62200	−0.811001	−	0.585044i	$-0.801077\pi$
$709$	−1150.00	−1.62200	−0.811001	+	0.585044i	$0.801077\pi$
$710$	180.133i	0.253709i
$711$	0	0
$712$	942.236i	1.32336i
$713$	207.846i	0.291509i
$714$	0	0
$715$	420.000	0.587413
$716$	0	0
$717$	0	0
$718$	−584.000	−0.813370
$719$	973.413i	1.35384i	0.736056	+	0.676921i	$0.236686\pi$
$719$	973.413i	1.35384i	−0.736056	+	0.676921i	$0.763314\pi$
$720$	0	0
$721$	0	0
$722$	−1444.00	−2.00000
$723$	0	0
$724$	0	0
$725$	−208.000	−0.286897
$726$	0	0
$727$	206.114i	0.283513i	0.989902	+	0.141757i	$0.0452750\pi$
$727$	206.114i	0.283513i	−0.989902	+	0.141757i	$0.954725\pi$
$728$	0	0
$729$	0	0
$730$	−132.000	−0.180822
$731$	131.636i	0.180076i
$732$	0	0
$733$	1245.34i	1.69897i	0.527613	+	0.849485i	$0.323087\pi$
$733$	1245.34i	1.69897i	−0.527613	+	0.849485i	$0.676913\pi$
$734$	− 1077.34i	− 1.46776i
$735$	0	0
$736$	0	0
$737$	−590.000	−0.800543
$738$	0	0
$739$	311.000	0.420839	0.210419	−	0.977611i	$-0.432517\pi$
$739$	311.000	0.420839	0.210419	+	0.977611i	$0.432517\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−394.000	−0.530283	−0.265141	−	0.964210i	$-0.585419\pi$
$743$	−394.000	−0.530283	−0.265141	+	0.964210i	$0.585419\pi$
$744$	0	0
$745$	− 429.549i	− 0.576575i
$746$	−410.000	−0.549598
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−79.0000	−0.105193	−0.0525965	−	0.998616i	$-0.516750\pi$
$751$	−79.0000	−0.105193	−0.0525965	+	0.998616i	$0.516750\pi$
$752$	− 831.384i	− 1.10556i
$753$	0	0
$754$	− 387.979i	− 0.514562i
$755$	− 159.349i	− 0.211058i
$756$	0	0
$757$	−250.000	−0.330251	−0.165125	−	0.986273i	$-0.552803\pi$
$757$	−250.000	−0.330251	−0.165125	+	0.986273i	$0.552803\pi$
$758$	−1046.00	−1.37995
$759$	0	0
$760$	912.000	1.20000
$761$	949.164i	1.24726i	0.781720	+	0.623629i	$0.214343\pi$
$761$	949.164i	1.24726i	−0.781720	+	0.623629i	$0.785657\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	152.420i	0.198982i
$767$	−504.000	−0.657106
$768$	0	0
$769$	− 860.829i	− 1.11941i	−0.828691	−	0.559707i	$-0.810914\pi$
$769$	− 860.829i	− 1.11941i	0.828691	−	0.559707i	$-0.189086\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 225.167i	− 0.291289i	−0.989337	−	0.145645i	$-0.953474\pi$
$773$	− 225.167i	− 0.291289i	0.989337	−	0.145645i	$-0.0465256\pi$
$774$	0	0
$775$	− 67.5500i	− 0.0871613i
$776$	387.979i	0.499973i
$777$	0	0
$778$	148.000	0.190231
$779$	798.000	1.02439
$780$	0	0
$781$	−260.000	−0.332907
$782$	554.256i	0.708768i
$783$	0	0
$784$	0	0
$785$	648.000	0.825478
$786$	0	0
$787$	249.415i	0.316919i	0.987365	+	0.158460i	$0.0506527\pi$
$787$	249.415i	0.316919i	−0.987365	+	0.158460i	$0.949347\pi$
$788$	0	0
$789$	0	0
$790$	325.626i	0.412184i
$791$	0	0
$792$	0	0
$793$	−252.000	−0.317781
$794$	647.787i	0.815853i
$795$	0	0
$796$	0	0
$797$	1357.93i	1.70380i	0.523705	+	0.851900i	$0.324549\pi$
$797$	1357.93i	1.70380i	−0.523705	+	0.851900i	$0.675451\pi$
$798$	0	0
$799$	360.000	0.450563
$800$	0	0
$801$	0	0
$802$	256.000	0.319202
$803$	− 190.526i	− 0.237267i
$804$	0	0
$805$	0	0
$806$	126.000	0.156328
$807$	0	0
$808$	1025.37i	1.26903i
$809$	1418.00	1.75278	0.876391	−	0.481601i	$-0.159945\pi$
$809$	1418.00	1.75278	0.876391	+	0.481601i	$0.159945\pi$
$810$	0	0
$811$	872.954i	1.07639i	0.842820	+	0.538196i	$0.180894\pi$
$811$	872.954i	1.07639i	−0.842820	+	0.538196i	$0.819106\pi$
$812$	0	0
$813$	0	0
$814$	−100.000	−0.122850
$815$	− 200.918i	− 0.246525i
$816$	0	0
$817$	− 625.270i	− 0.765325i
$818$	− 592.361i	− 0.724158i
$819$	0	0
$820$	0	0
$821$	−250.000	−0.304507	−0.152253	−	0.988341i	$-0.548653\pi$
$821$	−250.000	−0.304507	−0.152253	+	0.988341i	$0.548653\pi$
$822$	0	0
$823$	206.000	0.250304	0.125152	−	0.992138i	$-0.460058\pi$
$823$	206.000	0.250304	0.125152	+	0.992138i	$0.460058\pi$
$824$	− 69.2820i	− 0.0840801i
$825$	0	0
$826$	0	0
$827$	−1234.00	−1.49214	−0.746070	−	0.665867i	$-0.768062\pi$
$827$	−1234.00	−1.49214	−0.746070	+	0.665867i	$0.768062\pi$
$828$	0	0
$829$	− 344.678i	− 0.415776i	−0.978153	−	0.207888i	$-0.933341\pi$
$829$	− 344.678i	− 0.415776i	0.978153	−	0.207888i	$-0.0666589\pi$
$830$	−168.000	−0.202410
$831$	0	0
$832$	775.959i	0.932643i
$833$	0	0
$834$	0	0
$835$	−924.000	−1.10659
$836$	0	0
$837$	0	0
$838$	824.456i	0.983838i
$839$	− 484.974i	− 0.578038i	−0.957323	−	0.289019i	$-0.906671\pi$
$839$	− 484.974i	− 0.578038i	0.957323	−	0.289019i	$-0.0933291\pi$
$840$	0	0
$841$	−585.000	−0.695600
$842$	214.000	0.254157
$843$	0	0
$844$	0	0
$845$	76.2102i	0.0901896i
$846$	0	0
$847$	0	0
$848$	−512.000	−0.603774
$849$	0	0
$850$	− 180.133i	− 0.211922i
$851$	−200.000	−0.235018
$852$	0	0
$853$	278.860i	0.326917i	0.986550	+	0.163458i	$0.0522650\pi$
$853$	278.860i	0.326917i	−0.986550	+	0.163458i	$0.947735\pi$
$854$	0	0
$855$	0	0
$856$	−1696.00	−1.98131
$857$	− 637.395i	− 0.743751i	−0.928283	−	0.371876i	$-0.878715\pi$
$857$	− 637.395i	− 0.743751i	0.928283	−	0.371876i	$-0.121285\pi$
$858$	0	0
$859$	− 609.682i	− 0.709758i	−0.934912	−	0.354879i	$-0.884522\pi$
$859$	− 609.682i	− 0.709758i	0.934912	−	0.354879i	$-0.115478\pi$
$860$	0	0
$861$	0	0
$862$	−524.000	−0.607889
$863$	−670.000	−0.776362	−0.388181	−	0.921583i	$-0.626896\pi$
$863$	−670.000	−0.776362	−0.388181	+	0.921583i	$0.626896\pi$
$864$	0	0
$865$	−432.000	−0.499422
$866$	− 72.7461i	− 0.0840025i
$867$	0	0
$868$	0	0
$869$	−470.000	−0.540852
$870$	0	0
$871$	715.337i	0.821282i
$872$	−136.000	−0.155963
$873$	0	0
$874$	− 2632.72i	− 3.01226i
$875$	0	0
$876$	0	0
$877$	−394.000	−0.449259	−0.224629	−	0.974444i	$-0.572117\pi$
$877$	−394.000	−0.449259	−0.224629	+	0.974444i	$0.572117\pi$
$878$	623.538i	0.710180i
$879$	0	0
$880$	554.256i	0.629837i
$881$	− 1163.94i	− 1.32116i	−0.750758	−	0.660578i	$-0.770311\pi$
$881$	− 1163.94i	− 1.32116i	0.750758	−	0.660578i	$-0.229689\pi$
$882$	0	0
$883$	737.000	0.834655	0.417327	−	0.908756i	$-0.362967\pi$
$883$	737.000	0.834655	0.417327	+	0.908756i	$0.362967\pi$
$884$	0	0
$885$	0	0
$886$	424.000	0.478555
$887$	730.925i	0.824042i	0.911174	+	0.412021i	$0.135177\pi$
$887$	730.925i	0.824042i	−0.911174	+	0.412021i	$0.864823\pi$
$888$	0	0
$889$	0	0
$890$	816.000	0.916854
$891$	0	0
$892$	0	0
$893$	−1710.00	−1.91489
$894$	0	0
$895$	− 34.6410i	− 0.0387050i
$896$	0	0
$897$	0	0
$898$	1564.00	1.74165
$899$	83.1384i	0.0924788i
$900$	0	0
$901$	− 221.703i	− 0.246063i
$902$	484.974i	0.537665i
$903$	0	0
$904$	1136.00	1.25664
$905$	−1134.00	−1.25304
$906$	0	0
$907$	−235.000	−0.259096	−0.129548	−	0.991573i	$-0.541353\pi$
$907$	−235.000	−0.259096	−0.129548	+	0.991573i	$0.541353\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	740.000	0.812294	0.406147	−	0.913808i	$-0.366872\pi$
$911$	740.000	0.812294	0.406147	+	0.913808i	$0.366872\pi$
$912$	0	0
$913$	− 242.487i	− 0.265594i
$914$	1354.00	1.48140
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1517.00	1.65071	0.825354	−	0.564616i	$-0.190976\pi$
$919$	1517.00	1.65071	0.825354	+	0.564616i	$0.190976\pi$
$920$	1108.51i	1.20490i
$921$	0	0
$922$	− 969.948i	− 1.05200i
$923$	315.233i	0.341531i
$924$	0	0
$925$	65.0000	0.0702703
$926$	886.000	0.956803
$927$	0	0
$928$	0	0
$929$	− 1111.98i	− 1.19696i	−0.801137	−	0.598480i	$-0.795771\pi$
$929$	− 1111.98i	− 1.19696i	0.801137	−	0.598480i	$-0.204229\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	− 90.0666i	− 0.0964311i
$935$	−240.000	−0.256684
$936$	0	0
$937$	− 836.581i	− 0.892829i	−0.894826	−	0.446414i	$-0.852701\pi$
$937$	− 836.581i	− 0.892829i	0.894826	−	0.446414i	$-0.147299\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 394.908i	− 0.419668i	−0.977737	−	0.209834i	$-0.932708\pi$
$941$	− 394.908i	− 0.419668i	0.977737	−	0.209834i	$-0.0672924\pi$
$942$	0	0
$943$	969.948i	1.02858i
$944$	− 665.108i	− 0.704563i
$945$	0	0
$946$	380.000	0.401691
$947$	338.000	0.356917	0.178458	−	0.983947i	$-0.442889\pi$
$947$	338.000	0.356917	0.178458	+	0.983947i	$0.442889\pi$
$948$	0	0
$949$	−231.000	−0.243414
$950$	855.633i	0.900666i
$951$	0	0
$952$	0	0
$953$	1244.00	1.30535	0.652676	−	0.757637i	$-0.273646\pi$
$953$	1244.00	1.30535	0.652676	+	0.757637i	$0.273646\pi$
$954$	0	0
$955$	6.92820i	0.00725466i
$956$	0	0
$957$	0	0
$958$	− 110.851i	− 0.115711i
$959$	0	0
$960$	0	0
$961$	934.000	0.971904
$962$	121.244i	0.126033i
$963$	0	0
$964$	0	0
$965$	− 814.064i	− 0.843590i
$966$	0	0
$967$	−1741.00	−1.80041	−0.900207	−	0.435463i	$-0.856585\pi$
$967$	−1741.00	−1.80041	−0.900207	+	0.435463i	$0.856585\pi$
$968$	168.000	0.173554
$969$	0	0
$970$	336.000	0.346392
$971$	− 1281.72i	− 1.32000i	−0.751267	−	0.659999i	$-0.770557\pi$
$971$	− 1281.72i	− 1.32000i	0.751267	−	0.659999i	$-0.229443\pi$
$972$	0	0
$973$	0	0
$974$	−134.000	−0.137577
$975$	0	0
$976$	− 332.554i	− 0.340731i
$977$	−262.000	−0.268168	−0.134084	−	0.990970i	$-0.542809\pi$
$977$	−262.000	−0.268168	−0.134084	+	0.990970i	$0.542809\pi$
$978$	0	0
$979$	1177.79i	1.20306i
$980$	0	0
$981$	0	0
$982$	136.000	0.138493
$983$	1108.51i	1.12768i	0.825883	+	0.563842i	$0.190677\pi$
$983$	1108.51i	1.12768i	−0.825883	+	0.563842i	$0.809323\pi$
$984$	0	0
$985$	− 346.410i	− 0.351685i
$986$	221.703i	0.224850i
$987$	0	0
$988$	0	0
$989$	760.000	0.768453
$990$	0	0
$991$	−67.0000	−0.0676085	−0.0338042	−	0.999428i	$-0.510762\pi$
$991$	−67.0000	−0.0676085	−0.0338042	+	0.999428i	$0.510762\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−696.000	−0.699497
$996$	0	0
$997$	989.001i	0.991977i	0.868329	+	0.495988i	$0.165194\pi$
$997$	989.001i	0.991977i	−0.868329	+	0.495988i	$0.834806\pi$
$998$	1018.00	1.02004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.d.d.244.2		2
3.2	odd	2		147.3.d.a.97.2		2
7.2	even	3		63.3.m.a.10.1		2
7.3	odd	6		63.3.m.a.19.1		2
7.4	even	3		441.3.m.b.19.1		2
7.5	odd	6		441.3.m.b.325.1		2
7.6	odd	2	inner	441.3.d.d.244.1		2
12.11	even	2		2352.3.f.b.97.1		2
21.2	odd	6		21.3.f.c.10.1	✓	2
21.5	even	6		147.3.f.e.31.1		2
21.11	odd	6		147.3.f.e.19.1		2
21.17	even	6		21.3.f.c.19.1	yes	2
21.20	even	2		147.3.d.a.97.1		2
28.3	even	6		1008.3.cg.f.145.1		2
28.23	odd	6		1008.3.cg.f.577.1		2
84.23	even	6		336.3.bh.c.241.1		2
84.59	odd	6		336.3.bh.c.145.1		2
84.83	odd	2		2352.3.f.b.97.2		2
105.2	even	12		525.3.s.d.199.1		4
105.17	odd	12		525.3.s.d.124.2		4
105.23	even	12		525.3.s.d.199.2		4
105.38	odd	12		525.3.s.d.124.1		4
105.44	odd	6		525.3.o.b.451.1		2
105.59	even	6		525.3.o.b.376.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.3.f.c.10.1	✓	2	21.2	odd	6
21.3.f.c.19.1	yes	2	21.17	even	6
63.3.m.a.10.1		2	7.2	even	3
63.3.m.a.19.1		2	7.3	odd	6
147.3.d.a.97.1		2	21.20	even	2
147.3.d.a.97.2		2	3.2	odd	2
147.3.f.e.19.1		2	21.11	odd	6
147.3.f.e.31.1		2	21.5	even	6
336.3.bh.c.145.1		2	84.59	odd	6
336.3.bh.c.241.1		2	84.23	even	6
441.3.d.d.244.1		2	7.6	odd	2	inner
441.3.d.d.244.2		2	1.1	even	1	trivial
441.3.m.b.19.1		2	7.4	even	3
441.3.m.b.325.1		2	7.5	odd	6
525.3.o.b.376.1		2	105.59	even	6
525.3.o.b.451.1		2	105.44	odd	6
525.3.s.d.124.1		4	105.38	odd	12
525.3.s.d.124.2		4	105.17	odd	12
525.3.s.d.199.1		4	105.2	even	12
525.3.s.d.199.2		4	105.23	even	12
1008.3.cg.f.145.1		2	28.3	even	6
1008.3.cg.f.577.1		2	28.23	odd	6
2352.3.f.b.97.1		2	12.11	even	2
2352.3.f.b.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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.46410i q^{5} -8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +3.46410i q^{5} -8.00000 q^{8} +6.92820i q^{10} -10.0000 q^{11} +12.1244i q^{13} -16.0000 q^{16} -6.92820i q^{17} +32.9090i q^{19} -20.0000 q^{22} -40.0000 q^{23} +13.0000 q^{25} +24.2487i q^{26} -16.0000 q^{29} -5.19615i q^{31} -13.8564i q^{34} +5.00000 q^{37} +65.8179i q^{38} -27.7128i q^{40} -24.2487i q^{41} -19.0000 q^{43} -80.0000 q^{46} +51.9615i q^{47} +26.0000 q^{50} +32.0000 q^{53} -34.6410i q^{55} -32.0000 q^{58} +41.5692i q^{59} +20.7846i q^{61} -10.3923i q^{62} +64.0000 q^{64} -42.0000 q^{65} +59.0000 q^{67} +26.0000 q^{71} +19.0526i q^{73} +10.0000 q^{74} +47.0000 q^{79} -55.4256i q^{80} -48.4974i q^{82} +24.2487i q^{83} +24.0000 q^{85} -38.0000 q^{86} +80.0000 q^{88} -117.779i q^{89} +103.923i q^{94} -114.000 q^{95} -48.4974i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 - 16 * q^8	\( 2 q + 4 q^{2} - 16 q^{8} - 20 q^{11} - 32 q^{16} - 40 q^{22} - 80 q^{23} + 26 q^{25} - 32 q^{29} + 10 q^{37} - 38 q^{43} - 160 q^{46} + 52 q^{50} + 64 q^{53} - 64 q^{58} + 128 q^{64} - 84 q^{65} + 118 q^{67} + 52 q^{71} + 20 q^{74} + 94 q^{79} + 48 q^{85} - 76 q^{86} + 160 q^{88} - 228 q^{95}+O(q^{100}) \) 2 * q + 4 * q^2 - 16 * q^8 - 20 * q^11 - 32 * q^16 - 40 * q^22 - 80 * q^23 + 26 * q^25 - 32 * q^29 + 10 * q^37 - 38 * q^43 - 160 * q^46 + 52 * q^50 + 64 * q^53 - 64 * q^58 + 128 * q^64 - 84 * q^65 + 118 * q^67 + 52 * q^71 + 20 * q^74 + 94 * q^79 + 48 * q^85 - 76 * q^86 + 160 * q^88 - 228 * q^95

Introduction

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