LMFDB - Embedded newform 1089.3.c.m.604.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.7
Root			$-1.43448 - 2.82504i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.m.604.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.982569i q^{2} +3.03456 q^{4} -0.397576 q^{5} +7.22681i q^{7} -6.91194i q^{8} +O(q^{10})$	$q-0.982569i q^{2} +3.03456 q^{4} -0.397576 q^{5} +7.22681i q^{7} -6.91194i q^{8} +0.390645i q^{10} -12.1597i q^{13} +7.10084 q^{14} +5.34678 q^{16} -20.9095i q^{17} +11.1128i q^{19} -1.20647 q^{20} +5.92990 q^{23} -24.8419 q^{25} -11.9477 q^{26} +21.9302i q^{28} -24.9546i q^{29} +59.6258 q^{31} -32.9013i q^{32} -20.5451 q^{34} -2.87320i q^{35} -5.96703 q^{37} +10.9191 q^{38} +2.74802i q^{40} -52.0396i q^{41} +17.6439i q^{43} -5.82653i q^{46} +55.3609 q^{47} -3.22680 q^{49} +24.4089i q^{50} -36.8993i q^{52} +94.6520 q^{53} +49.9513 q^{56} -24.5196 q^{58} +25.1052 q^{59} +33.4062i q^{61} -58.5865i q^{62} -10.9407 q^{64} +4.83439i q^{65} +94.6640 q^{67} -63.4513i q^{68} -2.82312 q^{70} -82.7241 q^{71} -46.5788i q^{73} +5.86302i q^{74} +33.7223i q^{76} -60.3191i q^{79} -2.12575 q^{80} -51.1325 q^{82} +150.673i q^{83} +8.31313i q^{85} +17.3363 q^{86} -134.190 q^{89} +87.8758 q^{91} +17.9946 q^{92} -54.3959i q^{94} -4.41816i q^{95} +37.6065 q^{97} +3.17056i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) $ 16 * q - 20 * q^4 + 4 * q^5	$ 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 0.982569i	− 0.491284i	−0.969361	−	0.245642i	$-0.921001\pi$
$2$	− 0.982569i	− 0.491284i	0.969361	−	0.245642i	$-0.0789988\pi$
$3$	0	0
$3$	0	0
$4$	3.03456	0.758640
$5$	−0.397576	−0.0795151	−0.0397576	−	0.999209i	$-0.512659\pi$
$5$	−0.397576	−0.0795151	−0.0397576	+	0.999209i	$0.512659\pi$
$6$	0	0
$7$	7.22681i	1.03240i	0.856468	+	0.516201i	$0.172654\pi$
$7$	7.22681i	1.03240i	−0.856468	+	0.516201i	$0.827346\pi$
$8$	− 6.91194i	− 0.863992i
$9$	0	0
$10$	0.390645i	0.0390645i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 12.1597i	− 0.935360i	−0.883898	−	0.467680i	$-0.845090\pi$
$13$	− 12.1597i	− 0.935360i	0.883898	−	0.467680i	$-0.154910\pi$
$14$	7.10084	0.507203
$15$	0	0
$16$	5.34678	0.334174
$17$	− 20.9095i	− 1.22997i	−0.788537	−	0.614987i	$-0.789161\pi$
$17$	− 20.9095i	− 1.22997i	0.788537	−	0.614987i	$-0.210839\pi$
$18$	0	0
$19$	11.1128i	0.584882i	0.956284	+	0.292441i	$0.0944676\pi$
$19$	11.1128i	0.584882i	−0.956284	+	0.292441i	$0.905532\pi$
$20$	−1.20647	−0.0603233
$21$	0	0
$22$	0	0
$23$	5.92990	0.257822	0.128911	−	0.991656i	$-0.458852\pi$
$23$	5.92990	0.257822	0.128911	+	0.991656i	$0.458852\pi$
$24$	0	0
$25$	−24.8419	−0.993677
$26$	−11.9477	−0.459528
$27$	0	0
$28$	21.9302i	0.783221i
$29$	− 24.9546i	− 0.860504i	−0.902709	−	0.430252i	$-0.858425\pi$
$29$	− 24.9546i	− 0.860504i	0.902709	−	0.430252i	$-0.141575\pi$
$30$	0	0
$31$	59.6258	1.92341	0.961707	−	0.274080i	$-0.0883732\pi$
$31$	59.6258	1.92341	0.961707	+	0.274080i	$0.0883732\pi$
$32$	− 32.9013i	− 1.02817i
$33$	0	0
$34$	−20.5451	−0.604267
$35$	− 2.87320i	− 0.0820915i
$36$	0	0
$37$	−5.96703	−0.161271	−0.0806355	−	0.996744i	$-0.525695\pi$
$37$	−5.96703	−0.161271	−0.0806355	+	0.996744i	$0.525695\pi$
$38$	10.9191	0.287344
$39$	0	0
$40$	2.74802i	0.0687004i
$41$	− 52.0396i	− 1.26926i	−0.772817	−	0.634629i	$-0.781153\pi$
$41$	− 52.0396i	− 1.26926i	0.772817	−	0.634629i	$-0.218847\pi$
$42$	0	0
$43$	17.6439i	0.410323i	0.978728	+	0.205161i	$0.0657719\pi$
$43$	17.6439i	0.410323i	−0.978728	+	0.205161i	$0.934228\pi$
$44$	0	0
$45$	0	0
$46$	− 5.82653i	− 0.126664i
$47$	55.3609	1.17789	0.588945	−	0.808173i	$-0.299543\pi$
$47$	55.3609	1.17789	0.588945	+	0.808173i	$0.299543\pi$
$48$	0	0
$49$	−3.22680	−0.0658531
$50$	24.4089i	0.488178i
$51$	0	0
$52$	− 36.8993i	− 0.709602i
$53$	94.6520	1.78589	0.892943	−	0.450170i	$-0.148636\pi$
$53$	94.6520	1.78589	0.892943	+	0.450170i	$0.148636\pi$
$54$	0	0
$55$	0	0
$56$	49.9513	0.891987
$57$	0	0
$58$	−24.5196	−0.422752
$59$	25.1052	0.425513	0.212756	−	0.977105i	$-0.431756\pi$
$59$	25.1052	0.425513	0.212756	+	0.977105i	$0.431756\pi$
$60$	0	0
$61$	33.4062i	0.547642i	0.961781	+	0.273821i	$0.0882876\pi$
$61$	33.4062i	0.547642i	−0.961781	+	0.273821i	$0.911712\pi$
$62$	− 58.5865i	− 0.944943i
$63$	0	0
$64$	−10.9407	−0.170948
$65$	4.83439i	0.0743753i
$66$	0	0
$67$	94.6640	1.41290	0.706448	−	0.707765i	$-0.250296\pi$
$67$	94.6640	1.41290	0.706448	+	0.707765i	$0.250296\pi$
$68$	− 63.4513i	− 0.933107i
$69$	0	0
$70$	−2.82312	−0.0403303
$71$	−82.7241	−1.16513	−0.582564	−	0.812785i	$-0.697951\pi$
$71$	−82.7241	−1.16513	−0.582564	+	0.812785i	$0.697951\pi$
$72$	0	0
$73$	− 46.5788i	− 0.638065i	−0.947744	−	0.319033i	$-0.896642\pi$
$73$	− 46.5788i	− 0.638065i	0.947744	−	0.319033i	$-0.103358\pi$
$74$	5.86302i	0.0792299i
$75$	0	0
$76$	33.7223i	0.443715i
$77$	0	0
$78$	0	0
$79$	− 60.3191i	− 0.763534i	−0.924259	−	0.381767i	$-0.875316\pi$
$79$	− 60.3191i	− 0.763534i	0.924259	−	0.381767i	$-0.124684\pi$
$80$	−2.12575	−0.0265719
$81$	0	0
$82$	−51.1325	−0.623567
$83$	150.673i	1.81534i	0.419686	+	0.907670i	$0.362140\pi$
$83$	150.673i	1.81534i	−0.419686	+	0.907670i	$0.637860\pi$
$84$	0	0
$85$	8.31313i	0.0978015i
$86$	17.3363	0.201585
$87$	0	0
$88$	0	0
$89$	−134.190	−1.50775	−0.753874	−	0.657019i	$-0.771817\pi$
$89$	−134.190	−1.50775	−0.753874	+	0.657019i	$0.771817\pi$
$90$	0	0
$91$	87.8758	0.965668
$92$	17.9946	0.195594
$93$	0	0
$94$	− 54.3959i	− 0.578679i
$95$	− 4.41816i	− 0.0465070i
$96$	0	0
$97$	37.6065	0.387696	0.193848	−	0.981032i	$-0.437903\pi$
$97$	37.6065	0.387696	0.193848	+	0.981032i	$0.437903\pi$
$98$	3.17056i	0.0323526i
$99$	0	0
$100$	−75.3843	−0.753843
$101$	− 80.8017i	− 0.800017i	−0.916512	−	0.400008i	$-0.869007\pi$
$101$	− 80.8017i	− 0.800017i	0.916512	−	0.400008i	$-0.130993\pi$
$102$	0	0
$103$	99.6632	0.967603	0.483802	−	0.875178i	$-0.339256\pi$
$103$	99.6632	0.967603	0.483802	+	0.875178i	$0.339256\pi$
$104$	−84.0470	−0.808144
$105$	0	0
$106$	− 93.0021i	− 0.877378i
$107$	108.254i	1.01172i	0.862617	+	0.505858i	$0.168824\pi$
$107$	108.254i	1.01172i	−0.862617	+	0.505858i	$0.831176\pi$
$108$	0	0
$109$	− 175.446i	− 1.60960i	−0.593546	−	0.804800i	$-0.702273\pi$
$109$	− 175.446i	− 1.60960i	0.593546	−	0.804800i	$-0.297727\pi$
$110$	0	0
$111$	0	0
$112$	38.6402i	0.345002i
$113$	−98.5234	−0.871888	−0.435944	−	0.899974i	$-0.643586\pi$
$113$	−98.5234	−0.871888	−0.435944	+	0.899974i	$0.643586\pi$
$114$	0	0
$115$	−2.35758	−0.0205007
$116$	− 75.7262i	− 0.652812i
$117$	0	0
$118$	− 24.6676i	− 0.209048i
$119$	151.109	1.26983
$120$	0	0
$121$	0	0
$122$	32.8238	0.269048
$123$	0	0
$124$	180.938	1.45918
$125$	19.8159	0.158527
$126$	0	0
$127$	− 21.8661i	− 0.172174i	−0.996288	−	0.0860872i	$-0.972564\pi$
$127$	− 21.8661i	− 0.172174i	0.996288	−	0.0860872i	$-0.0274364\pi$
$128$	− 120.855i	− 0.944182i
$129$	0	0
$130$	4.75013	0.0365394
$131$	− 16.3593i	− 0.124880i	−0.998049	−	0.0624402i	$-0.980112\pi$
$131$	− 16.3593i	− 0.124880i	0.998049	−	0.0624402i	$-0.0198883\pi$
$132$	0	0
$133$	−80.3099	−0.603834
$134$	− 93.0139i	− 0.694134i
$135$	0	0
$136$	−144.525	−1.06269
$137$	−30.5892	−0.223279	−0.111640	−	0.993749i	$-0.535610\pi$
$137$	−30.5892	−0.223279	−0.111640	+	0.993749i	$0.535610\pi$
$138$	0	0
$139$	− 27.7464i	− 0.199614i	−0.995007	−	0.0998072i	$-0.968177\pi$
$139$	− 27.7464i	− 0.199614i	0.995007	−	0.0998072i	$-0.0318226\pi$
$140$	− 8.71891i	− 0.0622779i
$141$	0	0
$142$	81.2821i	0.572409i
$143$	0	0
$144$	0	0
$145$	9.92134i	0.0684231i
$146$	−45.7668	−0.313471
$147$	0	0
$148$	−18.1073	−0.122347
$149$	198.806i	1.33427i	0.744936	+	0.667136i	$0.232480\pi$
$149$	198.806i	1.33427i	−0.744936	+	0.667136i	$0.767520\pi$
$150$	0	0
$151$	− 130.507i	− 0.864287i	−0.901805	−	0.432144i	$-0.857757\pi$
$151$	− 130.507i	− 0.864287i	0.901805	−	0.432144i	$-0.142243\pi$
$152$	76.8107	0.505334
$153$	0	0
$154$	0	0
$155$	−23.7058	−0.152940
$156$	0	0
$157$	−70.5872	−0.449600	−0.224800	−	0.974405i	$-0.572173\pi$
$157$	−70.5872	−0.449600	−0.224800	+	0.974405i	$0.572173\pi$
$158$	−59.2677	−0.375112
$159$	0	0
$160$	13.0808i	0.0817548i
$161$	42.8542i	0.266175i
$162$	0	0
$163$	−76.9098	−0.471840	−0.235920	−	0.971773i	$-0.575810\pi$
$163$	−76.9098	−0.471840	−0.235920	+	0.971773i	$0.575810\pi$
$164$	− 157.917i	− 0.962910i
$165$	0	0
$166$	148.047	0.891848
$167$	− 101.758i	− 0.609328i	−0.952460	−	0.304664i	$-0.901456\pi$
$167$	− 101.758i	− 0.609328i	0.952460	−	0.304664i	$-0.0985441\pi$
$168$	0	0
$169$	21.1420	0.125101
$170$	8.16822	0.0480483
$171$	0	0
$172$	53.5414i	0.311287i
$173$	− 124.902i	− 0.721979i	−0.932570	−	0.360989i	$-0.882439\pi$
$173$	− 124.902i	− 0.721979i	0.932570	−	0.360989i	$-0.117561\pi$
$174$	0	0
$175$	− 179.528i	− 1.02587i
$176$	0	0
$177$	0	0
$178$	131.850i	0.740733i
$179$	103.230	0.576706	0.288353	−	0.957524i	$-0.406892\pi$
$179$	103.230	0.576706	0.288353	+	0.957524i	$0.406892\pi$
$180$	0	0
$181$	−112.462	−0.621335	−0.310668	−	0.950519i	$-0.600553\pi$
$181$	−112.462	−0.621335	−0.310668	+	0.950519i	$0.600553\pi$
$182$	− 86.3440i	− 0.474417i
$183$	0	0
$184$	− 40.9871i	− 0.222756i
$185$	2.37235	0.0128235
$186$	0	0
$187$	0	0
$188$	167.996	0.893595
$189$	0	0
$190$	−4.34115	−0.0228482
$191$	−191.072	−1.00038	−0.500188	−	0.865917i	$-0.666736\pi$
$191$	−191.072	−1.00038	−0.500188	+	0.865917i	$0.666736\pi$
$192$	0	0
$193$	54.5578i	0.282683i	0.989961	+	0.141341i	$0.0451415\pi$
$193$	54.5578i	0.282683i	−0.989961	+	0.141341i	$0.954858\pi$
$194$	− 36.9510i	− 0.190469i
$195$	0	0
$196$	−9.79193	−0.0499588
$197$	− 75.2950i	− 0.382208i	−0.981570	−	0.191104i	$-0.938793\pi$
$197$	− 75.2950i	− 0.382208i	0.981570	−	0.191104i	$-0.0612068\pi$
$198$	0	0
$199$	−60.7193	−0.305122	−0.152561	−	0.988294i	$-0.548752\pi$
$199$	−60.7193	−0.305122	−0.152561	+	0.988294i	$0.548752\pi$
$200$	171.706i	0.858529i
$201$	0	0
$202$	−79.3932	−0.393036
$203$	180.342	0.888386
$204$	0	0
$205$	20.6897i	0.100925i
$206$	− 97.9259i	− 0.475368i
$207$	0	0
$208$	− 65.0152i	− 0.312573i
$209$	0	0
$210$	0	0
$211$	− 409.192i	− 1.93930i	−0.244501	−	0.969649i	$-0.578624\pi$
$211$	− 409.192i	− 1.93930i	0.244501	−	0.969649i	$-0.421376\pi$
$212$	287.227	1.35484
$213$	0	0
$214$	106.367	0.497040
$215$	− 7.01478i	− 0.0326269i
$216$	0	0
$217$	430.905i	1.98574i
$218$	−172.388	−0.790771
$219$	0	0
$220$	0	0
$221$	−254.254	−1.15047
$222$	0	0
$223$	−147.312	−0.660590	−0.330295	−	0.943878i	$-0.607148\pi$
$223$	−147.312	−0.660590	−0.330295	+	0.943878i	$0.607148\pi$
$224$	237.772	1.06148
$225$	0	0
$226$	96.8060i	0.428345i
$227$	137.022i	0.603621i	0.953368	+	0.301810i	$0.0975910\pi$
$227$	137.022i	0.603621i	−0.953368	+	0.301810i	$0.902409\pi$
$228$	0	0
$229$	85.7061	0.374262	0.187131	−	0.982335i	$-0.440081\pi$
$229$	85.7061	0.374262	0.187131	+	0.982335i	$0.440081\pi$
$230$	2.31649i	0.0100717i
$231$	0	0
$232$	−172.485	−0.743469
$233$	216.431i	0.928887i	0.885603	+	0.464444i	$0.153746\pi$
$233$	216.431i	0.928887i	−0.885603	+	0.464444i	$0.846254\pi$
$234$	0	0
$235$	−22.0101	−0.0936601
$236$	76.1833	0.322811
$237$	0	0
$238$	− 148.475i	− 0.623846i
$239$	165.243i	0.691393i	0.938346	+	0.345697i	$0.112357\pi$
$239$	165.243i	0.691393i	−0.938346	+	0.345697i	$0.887643\pi$
$240$	0	0
$241$	233.818i	0.970199i	0.874459	+	0.485100i	$0.161217\pi$
$241$	233.818i	0.970199i	−0.874459	+	0.485100i	$0.838783\pi$
$242$	0	0
$243$	0	0
$244$	101.373i	0.415463i
$245$	1.28290	0.00523632
$246$	0	0
$247$	135.128	0.547076
$248$	− 412.130i	− 1.66181i
$249$	0	0
$250$	− 19.4705i	− 0.0778821i
$251$	−267.382	−1.06527	−0.532633	−	0.846346i	$-0.678798\pi$
$251$	−267.382	−1.06527	−0.532633	+	0.846346i	$0.678798\pi$
$252$	0	0
$253$	0	0
$254$	−21.4850	−0.0845866
$255$	0	0
$256$	−162.511	−0.634810
$257$	−49.5133	−0.192659	−0.0963293	−	0.995350i	$-0.530710\pi$
$257$	−49.5133	−0.192659	−0.0963293	+	0.995350i	$0.530710\pi$
$258$	0	0
$259$	− 43.1226i	− 0.166496i
$260$	14.6703i	0.0564241i
$261$	0	0
$262$	−16.0742	−0.0613518
$263$	198.763i	0.755752i	0.925856	+	0.377876i	$0.123345\pi$
$263$	198.763i	0.755752i	−0.925856	+	0.377876i	$0.876655\pi$
$264$	0	0
$265$	−37.6313	−0.142005
$266$	78.9100i	0.296654i
$267$	0	0
$268$	287.264	1.07188
$269$	−10.7729	−0.0400480	−0.0200240	−	0.999799i	$-0.506374\pi$
$269$	−10.7729	−0.0400480	−0.0200240	+	0.999799i	$0.506374\pi$
$270$	0	0
$271$	292.226i	1.07833i	0.842202	+	0.539163i	$0.181259\pi$
$271$	292.226i	1.07833i	−0.842202	+	0.539163i	$0.818741\pi$
$272$	− 111.799i	− 0.411025i
$273$	0	0
$274$	30.0560i	0.109693i
$275$	0	0
$276$	0	0
$277$	175.365i	0.633088i	0.948578	+	0.316544i	$0.102523\pi$
$277$	175.365i	0.633088i	−0.948578	+	0.316544i	$0.897477\pi$
$278$	−27.2628	−0.0980675
$279$	0	0
$280$	−19.8594	−0.0709265
$281$	− 227.381i	− 0.809184i	−0.914497	−	0.404592i	$-0.867414\pi$
$281$	− 227.381i	− 0.809184i	0.914497	−	0.404592i	$-0.132586\pi$
$282$	0	0
$283$	237.446i	0.839032i	0.907748	+	0.419516i	$0.137800\pi$
$283$	237.446i	0.839032i	−0.907748	+	0.419516i	$0.862200\pi$
$284$	−251.031	−0.883912
$285$	0	0
$286$	0	0
$287$	376.080	1.31038
$288$	0	0
$289$	−148.209	−0.512835
$290$	9.74840	0.0336152
$291$	0	0
$292$	− 141.346i	− 0.484062i
$293$	− 29.3337i	− 0.100115i	−0.998746	−	0.0500576i	$-0.984060\pi$
$293$	− 29.3337i	− 0.100115i	0.998746	−	0.0500576i	$-0.0159405\pi$
$294$	0	0
$295$	−9.98123	−0.0338347
$296$	41.2437i	0.139337i
$297$	0	0
$298$	195.341	0.655507
$299$	− 72.1057i	− 0.241156i
$300$	0	0
$301$	−127.509	−0.423618
$302$	−128.232	−0.424611
$303$	0	0
$304$	59.4175i	0.195452i
$305$	− 13.2815i	− 0.0435458i
$306$	0	0
$307$	347.331i	1.13137i	0.824621	+	0.565686i	$0.191389\pi$
$307$	347.331i	1.13137i	−0.824621	+	0.565686i	$0.808611\pi$
$308$	0	0
$309$	0	0
$310$	23.2926i	0.0751373i
$311$	−440.571	−1.41663	−0.708314	−	0.705898i	$-0.750544\pi$
$311$	−440.571	−1.41663	−0.708314	+	0.705898i	$0.750544\pi$
$312$	0	0
$313$	−25.4645	−0.0813562	−0.0406781	−	0.999172i	$-0.512952\pi$
$313$	−25.4645	−0.0813562	−0.0406781	+	0.999172i	$0.512952\pi$
$314$	69.3568i	0.220881i
$315$	0	0
$316$	− 183.042i	− 0.579247i
$317$	−91.3837	−0.288277	−0.144138	−	0.989558i	$-0.546041\pi$
$317$	−91.3837	−0.288277	−0.144138	+	0.989558i	$0.546041\pi$
$318$	0	0
$319$	0	0
$320$	4.34975	0.0135930
$321$	0	0
$322$	42.1072	0.130768
$323$	232.363	0.719390
$324$	0	0
$325$	302.070i	0.929446i
$326$	75.5692i	0.231807i
$327$	0	0
$328$	−359.695	−1.09663
$329$	400.083i	1.21606i
$330$	0	0
$331$	−318.761	−0.963024	−0.481512	−	0.876439i	$-0.659912\pi$
$331$	−318.761	−0.963024	−0.481512	+	0.876439i	$0.659912\pi$
$332$	457.226i	1.37719i
$333$	0	0
$334$	−99.9840	−0.299353
$335$	−37.6361	−0.112347
$336$	0	0
$337$	617.761i	1.83312i	0.399898	+	0.916560i	$0.369046\pi$
$337$	617.761i	1.83312i	−0.399898	+	0.916560i	$0.630954\pi$
$338$	− 20.7735i	− 0.0614601i
$339$	0	0
$340$	25.2267i	0.0741961i
$341$	0	0
$342$	0	0
$343$	330.794i	0.964415i
$344$	121.953	0.354516
$345$	0	0
$346$	−122.725	−0.354697
$347$	201.129i	0.579623i	0.957084	+	0.289811i	$0.0935925\pi$
$347$	201.129i	0.579623i	−0.957084	+	0.289811i	$0.906407\pi$
$348$	0	0
$349$	− 484.738i	− 1.38893i	−0.719525	−	0.694467i	$-0.755640\pi$
$349$	− 484.738i	− 1.38893i	0.719525	−	0.694467i	$-0.244360\pi$
$350$	−176.399	−0.503996
$351$	0	0
$352$	0	0
$353$	−108.957	−0.308661	−0.154330	−	0.988019i	$-0.549322\pi$
$353$	−108.957	−0.308661	−0.154330	+	0.988019i	$0.549322\pi$
$354$	0	0
$355$	32.8891	0.0926453
$356$	−407.206	−1.14384
$357$	0	0
$358$	− 101.431i	− 0.283327i
$359$	428.512i	1.19363i	0.802380	+	0.596814i	$0.203567\pi$
$359$	428.512i	1.19363i	−0.802380	+	0.596814i	$0.796433\pi$
$360$	0	0
$361$	237.506	0.657913
$362$	110.501i	0.305252i
$363$	0	0
$364$	266.664	0.732594
$365$	18.5186i	0.0507358i
$366$	0	0
$367$	427.936	1.16604	0.583019	−	0.812459i	$-0.301871\pi$
$367$	427.936	1.16604	0.583019	+	0.812459i	$0.301871\pi$
$368$	31.7059	0.0861572
$369$	0	0
$370$	− 2.33099i	− 0.00629998i
$371$	684.032i	1.84375i
$372$	0	0
$373$	− 104.312i	− 0.279656i	−0.990176	−	0.139828i	$-0.955345\pi$
$373$	− 104.312i	− 0.279656i	0.990176	−	0.139828i	$-0.0446549\pi$
$374$	0	0
$375$	0	0
$376$	− 382.651i	− 1.01769i
$377$	−303.440	−0.804881
$378$	0	0
$379$	224.659	0.592767	0.296383	−	0.955069i	$-0.404219\pi$
$379$	224.659	0.592767	0.296383	+	0.955069i	$0.404219\pi$
$380$	− 13.4072i	− 0.0352820i
$381$	0	0
$382$	187.741i	0.491469i
$383$	−59.2675	−0.154745	−0.0773727	−	0.997002i	$-0.524653\pi$
$383$	−59.2675	−0.154745	−0.0773727	+	0.997002i	$0.524653\pi$
$384$	0	0
$385$	0	0
$386$	53.6068	0.138878
$387$	0	0
$388$	114.119	0.294122
$389$	412.174	1.05957	0.529786	−	0.848131i	$-0.322272\pi$
$389$	412.174	1.05957	0.529786	+	0.848131i	$0.322272\pi$
$390$	0	0
$391$	− 123.991i	− 0.317114i
$392$	22.3035i	0.0568966i
$393$	0	0
$394$	−73.9825	−0.187773
$395$	23.9814i	0.0607125i
$396$	0	0
$397$	619.925	1.56152	0.780762	−	0.624829i	$-0.214831\pi$
$397$	619.925	1.56152	0.780762	+	0.624829i	$0.214831\pi$
$398$	59.6609i	0.149902i
$399$	0	0
$400$	−132.824	−0.332061
$401$	672.081	1.67601	0.838006	−	0.545661i	$-0.183721\pi$
$401$	672.081	1.67601	0.838006	+	0.545661i	$0.183721\pi$
$402$	0	0
$403$	− 725.031i	− 1.79909i
$404$	− 245.197i	− 0.606924i
$405$	0	0
$406$	− 177.199i	− 0.436450i
$407$	0	0
$408$	0	0
$409$	614.345i	1.50207i	0.660264	+	0.751034i	$0.270444\pi$
$409$	614.345i	1.50207i	−0.660264	+	0.751034i	$0.729556\pi$
$410$	20.3290	0.0495830
$411$	0	0
$412$	302.434	0.734062
$413$	181.431i	0.439300i
$414$	0	0
$415$	− 59.9040i	− 0.144347i
$416$	−400.070	−0.961706
$417$	0	0
$418$	0	0
$419$	60.9488	0.145462	0.0727312	−	0.997352i	$-0.476828\pi$
$419$	60.9488	0.145462	0.0727312	+	0.997352i	$0.476828\pi$
$420$	0	0
$421$	208.608	0.495506	0.247753	−	0.968823i	$-0.420308\pi$
$421$	208.608	0.495506	0.247753	+	0.968823i	$0.420308\pi$
$422$	−402.059	−0.952747
$423$	0	0
$424$	− 654.229i	− 1.54299i
$425$	519.434i	1.22220i
$426$	0	0
$427$	−241.420	−0.565386
$428$	328.502i	0.767528i
$429$	0	0
$430$	−6.89250	−0.0160291
$431$	270.722i	0.628124i	0.949402	+	0.314062i	$0.101690\pi$
$431$	270.722i	0.628124i	−0.949402	+	0.314062i	$0.898310\pi$
$432$	0	0
$433$	−627.302	−1.44873	−0.724367	−	0.689414i	$-0.757868\pi$
$433$	−627.302	−1.44873	−0.724367	+	0.689414i	$0.757868\pi$
$434$	423.393	0.975561
$435$	0	0
$436$	− 532.402i	− 1.22111i
$437$	65.8976i	0.150795i
$438$	0	0
$439$	495.244i	1.12812i	0.825734	+	0.564060i	$0.190761\pi$
$439$	495.244i	1.12812i	−0.825734	+	0.564060i	$0.809239\pi$
$440$	0	0
$441$	0	0
$442$	249.822i	0.565207i
$443$	−742.004	−1.67495	−0.837477	−	0.546473i	$-0.815970\pi$
$443$	−742.004	−1.67495	−0.837477	+	0.546473i	$0.815970\pi$
$444$	0	0
$445$	53.3505	0.119889
$446$	144.744i	0.324537i
$447$	0	0
$448$	− 79.0664i	− 0.176487i
$449$	713.566	1.58923	0.794617	−	0.607111i	$-0.207672\pi$
$449$	713.566	1.58923	0.794617	+	0.607111i	$0.207672\pi$
$450$	0	0
$451$	0	0
$452$	−298.975	−0.661449
$453$	0	0
$454$	134.633	0.296549
$455$	−34.9373	−0.0767852
$456$	0	0
$457$	− 282.514i	− 0.618192i	−0.951031	−	0.309096i	$-0.899973\pi$
$457$	− 282.514i	− 0.618192i	0.951031	−	0.309096i	$-0.100027\pi$
$458$	− 84.2121i	− 0.183869i
$459$	0	0
$460$	−7.15422	−0.0155527
$461$	198.186i	0.429905i	0.976624	+	0.214953i	$0.0689597\pi$
$461$	198.186i	0.429905i	−0.976624	+	0.214953i	$0.931040\pi$
$462$	0	0
$463$	−226.915	−0.490097	−0.245049	−	0.969511i	$-0.578804\pi$
$463$	−226.915	−0.490097	−0.245049	+	0.969511i	$0.578804\pi$
$464$	− 133.427i	− 0.287558i
$465$	0	0
$466$	212.658	0.456348
$467$	−149.659	−0.320468	−0.160234	−	0.987079i	$-0.551225\pi$
$467$	−149.659	−0.320468	−0.160234	+	0.987079i	$0.551225\pi$
$468$	0	0
$469$	684.119i	1.45868i
$470$	21.6265i	0.0460138i
$471$	0	0
$472$	− 173.526i	− 0.367640i
$473$	0	0
$474$	0	0
$475$	− 276.063i	− 0.581184i
$476$	458.550	0.963341
$477$	0	0
$478$	162.363	0.339671
$479$	− 325.770i	− 0.680105i	−0.940407	−	0.340052i	$-0.889555\pi$
$479$	− 325.770i	− 0.680105i	0.940407	−	0.340052i	$-0.110445\pi$
$480$	0	0
$481$	72.5572i	0.150847i
$482$	229.742	0.476644
$483$	0	0
$484$	0	0
$485$	−14.9514	−0.0308277
$486$	0	0
$487$	−306.924	−0.630234	−0.315117	−	0.949053i	$-0.602044\pi$
$487$	−306.924	−0.630234	−0.315117	+	0.949053i	$0.602044\pi$
$488$	230.901	0.473158
$489$	0	0
$490$	− 1.26054i	− 0.00257252i
$491$	613.874i	1.25025i	0.780524	+	0.625126i	$0.214953\pi$
$491$	613.874i	1.25025i	−0.780524	+	0.625126i	$0.785047\pi$
$492$	0	0
$493$	−521.790	−1.05840
$494$	− 132.772i	− 0.268770i
$495$	0	0
$496$	318.806	0.642754
$497$	− 597.832i	− 1.20288i
$498$	0	0
$499$	409.466	0.820573	0.410287	−	0.911957i	$-0.365429\pi$
$499$	409.466	0.820573	0.410287	+	0.911957i	$0.365429\pi$
$500$	60.1326	0.120265
$501$	0	0
$502$	262.721i	0.523349i
$503$	707.795i	1.40715i	0.710623	+	0.703573i	$0.248413\pi$
$503$	707.795i	1.40715i	−0.710623	+	0.703573i	$0.751587\pi$
$504$	0	0
$505$	32.1248i	0.0636134i
$506$	0	0
$507$	0	0
$508$	− 66.3541i	− 0.130618i
$509$	−205.948	−0.404613	−0.202306	−	0.979322i	$-0.564844\pi$
$509$	−205.948	−0.404613	−0.202306	+	0.979322i	$0.564844\pi$
$510$	0	0
$511$	336.616	0.658740
$512$	− 323.743i	− 0.632310i
$513$	0	0
$514$	48.6502i	0.0946502i
$515$	−39.6236	−0.0769391
$516$	0	0
$517$	0	0
$518$	−42.3709	−0.0817971
$519$	0	0
$520$	33.4150	0.0642597
$521$	332.454	0.638108	0.319054	−	0.947737i	$-0.396635\pi$
$521$	332.454	0.638108	0.319054	+	0.947737i	$0.396635\pi$
$522$	0	0
$523$	312.797i	0.598082i	0.954240	+	0.299041i	$0.0966668\pi$
$523$	312.797i	0.598082i	−0.954240	+	0.299041i	$0.903333\pi$
$524$	− 49.6434i	− 0.0947393i
$525$	0	0
$526$	195.298	0.371289
$527$	− 1246.75i	− 2.36575i
$528$	0	0
$529$	−493.836	−0.933528
$530$	36.9754i	0.0697648i
$531$	0	0
$532$	−243.705	−0.458092
$533$	−632.785	−1.18721
$534$	0	0
$535$	− 43.0390i	− 0.0804467i
$536$	− 654.312i	− 1.22073i
$537$	0	0
$538$	10.5851i	0.0196750i
$539$	0	0
$540$	0	0
$541$	− 61.0698i	− 0.112883i	−0.998406	−	0.0564416i	$-0.982025\pi$
$541$	− 61.0698i	− 0.112883i	0.998406	−	0.0564416i	$-0.0179755\pi$
$542$	287.132	0.529764
$543$	0	0
$544$	−687.952	−1.26462
$545$	69.7532i	0.127988i
$546$	0	0
$547$	− 571.085i	− 1.04403i	−0.852936	−	0.522016i	$-0.825180\pi$
$547$	− 571.085i	− 1.04403i	0.852936	−	0.522016i	$-0.174820\pi$
$548$	−92.8248	−0.169388
$549$	0	0
$550$	0	0
$551$	277.315	0.503294
$552$	0	0
$553$	435.915	0.788273
$554$	172.309	0.311026
$555$	0	0
$556$	− 84.1981i	− 0.151435i
$557$	− 704.122i	− 1.26413i	−0.774914	−	0.632066i	$-0.782207\pi$
$557$	− 704.122i	− 1.26413i	0.774914	−	0.632066i	$-0.217793\pi$
$558$	0	0
$559$	214.544	0.383800
$560$	− 15.3624i	− 0.0274328i
$561$	0	0
$562$	−223.417	−0.397539
$563$	− 505.018i	− 0.897013i	−0.893780	−	0.448506i	$-0.851956\pi$
$563$	− 505.018i	− 0.897013i	0.893780	−	0.448506i	$-0.148044\pi$
$564$	0	0
$565$	39.1705	0.0693283
$566$	233.307	0.412204
$567$	0	0
$568$	571.784i	1.00666i
$569$	− 673.489i	− 1.18364i	−0.806072	−	0.591818i	$-0.798410\pi$
$569$	− 673.489i	− 1.18364i	0.806072	−	0.591818i	$-0.201590\pi$
$570$	0	0
$571$	− 446.598i	− 0.782134i	−0.920362	−	0.391067i	$-0.872106\pi$
$571$	− 446.598i	− 0.782134i	0.920362	−	0.391067i	$-0.127894\pi$
$572$	0	0
$573$	0	0
$574$	− 369.525i	− 0.643772i
$575$	−147.310	−0.256191
$576$	0	0
$577$	−576.690	−0.999463	−0.499732	−	0.866180i	$-0.666568\pi$
$577$	−576.690	−0.999463	−0.499732	+	0.866180i	$0.666568\pi$
$578$	145.626i	0.251948i
$579$	0	0
$580$	30.1069i	0.0519085i
$581$	−1088.89	−1.87416
$582$	0	0
$583$	0	0
$584$	−321.949	−0.551283
$585$	0	0
$586$	−28.8224	−0.0491850
$587$	−906.327	−1.54400	−0.771999	−	0.635624i	$-0.780743\pi$
$587$	−906.327	−1.54400	−0.771999	+	0.635624i	$0.780743\pi$
$588$	0	0
$589$	662.608i	1.12497i
$590$	9.80725i	0.0166225i
$591$	0	0
$592$	−31.9044	−0.0538926
$593$	724.877i	1.22239i	0.791480	+	0.611195i	$0.209311\pi$
$593$	724.877i	1.22239i	−0.791480	+	0.611195i	$0.790689\pi$
$594$	0	0
$595$	−60.0774	−0.100970
$596$	603.290i	1.01223i
$597$	0	0
$598$	−70.8488	−0.118476
$599$	−398.301	−0.664944	−0.332472	−	0.943113i	$-0.607883\pi$
$599$	−398.301	−0.664944	−0.332472	+	0.943113i	$0.607883\pi$
$600$	0	0
$601$	− 985.468i	− 1.63971i	−0.572568	−	0.819857i	$-0.694053\pi$
$601$	− 985.468i	− 1.63971i	0.572568	−	0.819857i	$-0.305947\pi$
$602$	125.286i	0.208117i
$603$	0	0
$604$	− 396.032i	− 0.655682i
$605$	0	0
$606$	0	0
$607$	178.969i	0.294842i	0.989074	+	0.147421i	$0.0470972\pi$
$607$	178.969i	0.294842i	−0.989074	+	0.147421i	$0.952903\pi$
$608$	365.625	0.601357
$609$	0	0
$610$	−13.0500	−0.0213934
$611$	− 673.171i	− 1.10175i
$612$	0	0
$613$	1060.89i	1.73065i	0.501211	+	0.865325i	$0.332888\pi$
$613$	1060.89i	1.73065i	−0.501211	+	0.865325i	$0.667112\pi$
$614$	341.277	0.555825
$615$	0	0
$616$	0	0
$617$	−55.3570	−0.0897196	−0.0448598	−	0.998993i	$-0.514284\pi$
$617$	−55.3570	−0.0897196	−0.0448598	+	0.998993i	$0.514284\pi$
$618$	0	0
$619$	−752.536	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$619$	−752.536	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$620$	−71.9366	−0.116027
$621$	0	0
$622$	432.892i	0.695967i
$623$	− 969.762i	− 1.55660i
$624$	0	0
$625$	613.170	0.981072
$626$	25.0206i	0.0399690i
$627$	0	0
$628$	−214.201	−0.341084
$629$	124.768i	0.198359i
$630$	0	0
$631$	−335.720	−0.532045	−0.266022	−	0.963967i	$-0.585709\pi$
$631$	−335.720	−0.532045	−0.266022	+	0.963967i	$0.585709\pi$
$632$	−416.922	−0.659687
$633$	0	0
$634$	89.7908i	0.141626i
$635$	8.69344i	0.0136905i
$636$	0	0
$637$	39.2369i	0.0615964i
$638$	0	0
$639$	0	0
$640$	48.0491i	0.0750768i
$641$	552.092	0.861299	0.430649	−	0.902519i	$-0.358285\pi$
$641$	552.092	0.861299	0.430649	+	0.902519i	$0.358285\pi$
$642$	0	0
$643$	−138.358	−0.215176	−0.107588	−	0.994196i	$-0.534313\pi$
$643$	−138.358	−0.215176	−0.107588	+	0.994196i	$0.534313\pi$
$644$	130.044i	0.201931i
$645$	0	0
$646$	− 228.313i	− 0.353425i
$647$	1012.85	1.56546	0.782730	−	0.622362i	$-0.213827\pi$
$647$	1012.85	1.56546	0.782730	+	0.622362i	$0.213827\pi$
$648$	0	0
$649$	0	0
$650$	296.805	0.456623
$651$	0	0
$652$	−233.387	−0.357956
$653$	7.70732	0.0118029	0.00590147	−	0.999983i	$-0.498121\pi$
$653$	7.70732	0.0118029	0.00590147	+	0.999983i	$0.498121\pi$
$654$	0	0
$655$	6.50407i	0.00992988i
$656$	− 278.244i	− 0.424153i
$657$	0	0
$658$	393.109	0.597430
$659$	− 736.073i	− 1.11695i	−0.829520	−	0.558477i	$-0.811386\pi$
$659$	− 736.073i	− 1.11695i	0.829520	−	0.558477i	$-0.188614\pi$
$660$	0	0
$661$	−471.170	−0.712813	−0.356407	−	0.934331i	$-0.615998\pi$
$661$	−471.170	−0.712813	−0.356407	+	0.934331i	$0.615998\pi$
$662$	313.205i	0.473119i
$663$	0	0
$664$	1041.44	1.56844
$665$	31.9292	0.0480139
$666$	0	0
$667$	− 147.978i	− 0.221856i
$668$	− 308.790i	− 0.462260i
$669$	0	0
$670$	36.9801i	0.0551941i
$671$	0	0
$672$	0	0
$673$	− 1150.46i	− 1.70946i	−0.519077	−	0.854728i	$-0.673724\pi$
$673$	− 1150.46i	− 1.70946i	0.519077	−	0.854728i	$-0.326276\pi$
$674$	606.993	0.900583
$675$	0	0
$676$	64.1568	0.0949065
$677$	1172.40i	1.73176i	0.500252	+	0.865880i	$0.333241\pi$
$677$	1172.40i	1.73176i	−0.500252	+	0.865880i	$0.666759\pi$
$678$	0	0
$679$	271.775i	0.400258i
$680$	57.4598	0.0844997
$681$	0	0
$682$	0	0
$683$	329.083	0.481820	0.240910	−	0.970547i	$-0.422554\pi$
$683$	329.083	0.481820	0.240910	+	0.970547i	$0.422554\pi$
$684$	0	0
$685$	12.1615	0.0177541
$686$	325.028	0.473802
$687$	0	0
$688$	94.3380i	0.137119i
$689$	− 1150.94i	− 1.67045i
$690$	0	0
$691$	719.083	1.04064	0.520321	−	0.853971i	$-0.325812\pi$
$691$	719.083	1.04064	0.520321	+	0.853971i	$0.325812\pi$
$692$	− 379.023i	− 0.547722i
$693$	0	0
$694$	197.623	0.284760
$695$	11.0313i	0.0158724i
$696$	0	0
$697$	−1088.12	−1.56115
$698$	−476.288	−0.682361
$699$	0	0
$700$	− 544.788i	− 0.778269i
$701$	100.688i	0.143635i	0.997418	+	0.0718175i	$0.0228799\pi$
$701$	100.688i	0.143635i	−0.997418	+	0.0718175i	$0.977120\pi$
$702$	0	0
$703$	− 66.3102i	− 0.0943246i
$704$	0	0
$705$	0	0
$706$	107.058i	0.151640i
$707$	583.938	0.825938
$708$	0	0
$709$	275.579	0.388687	0.194343	−	0.980934i	$-0.437742\pi$
$709$	275.579	0.388687	0.194343	+	0.980934i	$0.437742\pi$
$710$	− 32.3158i	− 0.0455152i
$711$	0	0
$712$	927.510i	1.30268i
$713$	353.575	0.495898
$714$	0	0
$715$	0	0
$716$	313.258	0.437512
$717$	0	0
$718$	421.043	0.586411
$719$	−92.7087	−0.128941	−0.0644706	−	0.997920i	$-0.520536\pi$
$719$	−92.7087	−0.128941	−0.0644706	+	0.997920i	$0.520536\pi$
$720$	0	0
$721$	720.247i	0.998955i
$722$	− 233.366i	− 0.323222i
$723$	0	0
$724$	−341.272	−0.471370
$725$	619.921i	0.855063i
$726$	0	0
$727$	40.4150	0.0555915	0.0277958	−	0.999614i	$-0.491151\pi$
$727$	40.4150	0.0555915	0.0277958	+	0.999614i	$0.491151\pi$
$728$	− 607.392i	− 0.834329i
$729$	0	0
$730$	18.1958	0.0249257
$731$	368.926	0.504686
$732$	0	0
$733$	714.364i	0.974575i	0.873241	+	0.487288i	$0.162014\pi$
$733$	714.364i	0.974575i	−0.873241	+	0.487288i	$0.837986\pi$
$734$	− 420.476i	− 0.572856i
$735$	0	0
$736$	− 195.102i	− 0.265084i
$737$	0	0
$738$	0	0
$739$	51.5783i	0.0697947i	0.999391	+	0.0348974i	$0.0111104\pi$
$739$	51.5783i	0.0697947i	−0.999391	+	0.0348974i	$0.988890\pi$
$740$	7.19902	0.00972841
$741$	0	0
$742$	672.108	0.905807
$743$	898.577i	1.20939i	0.796457	+	0.604695i	$0.206705\pi$
$743$	898.577i	1.20939i	−0.796457	+	0.604695i	$0.793295\pi$
$744$	0	0
$745$	− 79.0406i	− 0.106095i
$746$	−102.493	−0.137391
$747$	0	0
$748$	0	0
$749$	−782.328	−1.04450
$750$	0	0
$751$	342.057	0.455469	0.227734	−	0.973723i	$-0.426868\pi$
$751$	342.057	0.455469	0.227734	+	0.973723i	$0.426868\pi$
$752$	296.002	0.393620
$753$	0	0
$754$	298.151i	0.395426i
$755$	51.8865i	0.0687239i
$756$	0	0
$757$	−807.960	−1.06732	−0.533659	−	0.845700i	$-0.679184\pi$
$757$	−807.960	−1.06732	−0.533659	+	0.845700i	$0.679184\pi$
$758$	− 220.743i	− 0.291217i
$759$	0	0
$760$	−30.5381	−0.0401817
$761$	− 782.358i	− 1.02807i	−0.857771	−	0.514033i	$-0.828151\pi$
$761$	− 782.358i	− 1.02807i	0.857771	−	0.514033i	$-0.171849\pi$
$762$	0	0
$763$	1267.92	1.66175
$764$	−579.818	−0.758925
$765$	0	0
$766$	58.2344i	0.0760240i
$767$	− 305.272i	− 0.398008i
$768$	0	0
$769$	652.678i	0.848736i	0.905490	+	0.424368i	$0.139504\pi$
$769$	652.678i	0.848736i	−0.905490	+	0.424368i	$0.860496\pi$
$770$	0	0
$771$	0	0
$772$	165.559i	0.214454i
$773$	864.726	1.11866	0.559331	−	0.828944i	$-0.311058\pi$
$773$	864.726	1.11866	0.559331	+	0.828944i	$0.311058\pi$
$774$	0	0
$775$	−1481.22	−1.91125
$776$	− 259.934i	− 0.334967i
$777$	0	0
$778$	− 404.989i	− 0.520551i
$779$	578.304	0.742367
$780$	0	0
$781$	0	0
$782$	−121.830	−0.155793
$783$	0	0
$784$	−17.2530	−0.0220064
$785$	28.0637	0.0357500
$786$	0	0
$787$	− 436.626i	− 0.554798i	−0.960755	−	0.277399i	$-0.910528\pi$
$787$	− 436.626i	− 0.554798i	0.960755	−	0.277399i	$-0.0894724\pi$
$788$	− 228.487i	− 0.289958i
$789$	0	0
$790$	23.5634	0.0298271
$791$	− 712.010i	− 0.900139i
$792$	0	0
$793$	406.208	0.512243
$794$	− 609.119i	− 0.767152i
$795$	0	0
$796$	−184.256	−0.231478
$797$	−453.109	−0.568519	−0.284259	−	0.958747i	$-0.591748\pi$
$797$	−453.109	−0.568519	−0.284259	+	0.958747i	$0.591748\pi$
$798$	0	0
$799$	− 1157.57i	− 1.44877i
$800$	817.333i	1.02167i
$801$	0	0
$802$	− 660.366i	− 0.823399i
$803$	0	0
$804$	0	0
$805$	− 17.0378i	− 0.0211650i
$806$	−712.393	−0.883863
$807$	0	0
$808$	−558.496	−0.691208
$809$	− 115.243i	− 0.142451i	−0.997460	−	0.0712256i	$-0.977309\pi$
$809$	− 115.243i	− 0.142451i	0.997460	−	0.0712256i	$-0.0226910\pi$
$810$	0	0
$811$	− 33.2681i	− 0.0410211i	−0.999790	−	0.0205105i	$-0.993471\pi$
$811$	− 33.2681i	− 0.0410211i	0.999790	−	0.0205105i	$-0.00652917\pi$
$812$	547.259	0.673965
$813$	0	0
$814$	0	0
$815$	30.5775	0.0375184
$816$	0	0
$817$	−196.072	−0.239991
$818$	603.637	0.737942
$819$	0	0
$820$	62.7840i	0.0765659i
$821$	715.058i	0.870960i	0.900198	+	0.435480i	$0.143421\pi$
$821$	715.058i	0.870960i	−0.900198	+	0.435480i	$0.856579\pi$
$822$	0	0
$823$	−157.039	−0.190813	−0.0954066	−	0.995438i	$-0.530415\pi$
$823$	−157.039	−0.190813	−0.0954066	+	0.995438i	$0.530415\pi$
$824$	− 688.866i	− 0.836002i
$825$	0	0
$826$	178.268	0.215821
$827$	− 854.310i	− 1.03302i	−0.856280	−	0.516511i	$-0.827230\pi$
$827$	− 854.310i	− 1.03302i	0.856280	−	0.516511i	$-0.172770\pi$
$828$	0	0
$829$	−1374.17	−1.65763	−0.828813	−	0.559526i	$-0.810983\pi$
$829$	−1374.17	−1.65763	−0.828813	+	0.559526i	$0.810983\pi$
$830$	−58.8598	−0.0709154
$831$	0	0
$832$	133.035i	0.159898i
$833$	67.4710i	0.0809976i
$834$	0	0
$835$	40.4564i	0.0484508i
$836$	0	0
$837$	0	0
$838$	− 59.8864i	− 0.0714634i
$839$	−652.536	−0.777755	−0.388877	−	0.921290i	$-0.627137\pi$
$839$	−652.536	−0.777755	−0.388877	+	0.921290i	$0.627137\pi$
$840$	0	0
$841$	218.267	0.259533
$842$	− 204.972i	− 0.243435i
$843$	0	0
$844$	− 1241.72i	− 1.47123i
$845$	−8.40556	−0.00994741
$846$	0	0
$847$	0	0
$848$	506.083	0.596796
$849$	0	0
$850$	510.379	0.600446
$851$	−35.3839	−0.0415792
$852$	0	0
$853$	− 1323.71i	− 1.55183i	−0.630835	−	0.775917i	$-0.717288\pi$
$853$	− 1323.71i	− 1.55183i	0.630835	−	0.775917i	$-0.282712\pi$
$854$	237.212i	0.277765i
$855$	0	0
$856$	748.242	0.874114
$857$	− 38.8452i	− 0.0453269i	−0.999743	−	0.0226634i	$-0.992785\pi$
$857$	− 38.8452i	− 0.0453269i	0.999743	−	0.0226634i	$-0.00721462\pi$
$858$	0	0
$859$	227.261	0.264564	0.132282	−	0.991212i	$-0.457770\pi$
$859$	227.261	0.264564	0.132282	+	0.991212i	$0.457770\pi$
$860$	− 21.2868i	− 0.0247520i
$861$	0	0
$862$	266.003	0.308588
$863$	−957.225	−1.10918	−0.554592	−	0.832123i	$-0.687126\pi$
$863$	−957.225	−1.10918	−0.554592	+	0.832123i	$0.687126\pi$
$864$	0	0
$865$	49.6581i	0.0574082i
$866$	616.367i	0.711741i
$867$	0	0
$868$	1307.61i	1.50646i
$869$	0	0
$870$	0	0
$871$	− 1151.08i	− 1.32157i
$872$	−1212.67	−1.39068
$873$	0	0
$874$	64.7489	0.0740834
$875$	143.206i	0.163664i
$876$	0	0
$877$	− 1203.94i	− 1.37280i	−0.727225	−	0.686399i	$-0.759190\pi$
$877$	− 1203.94i	− 1.37280i	0.727225	−	0.686399i	$-0.240810\pi$
$878$	486.612	0.554228
$879$	0	0
$880$	0	0
$881$	−1170.77	−1.32892	−0.664458	−	0.747326i	$-0.731337\pi$
$881$	−1170.77	−1.32892	−0.664458	+	0.747326i	$0.731337\pi$
$882$	0	0
$883$	364.218	0.412478	0.206239	−	0.978502i	$-0.433878\pi$
$883$	364.218	0.412478	0.206239	+	0.978502i	$0.433878\pi$
$884$	−771.547	−0.872791
$885$	0	0
$886$	729.070i	0.822878i
$887$	284.728i	0.321001i	0.987036	+	0.160501i	$0.0513108\pi$
$887$	284.728i	0.321001i	−0.987036	+	0.160501i	$0.948689\pi$
$888$	0	0
$889$	158.022	0.177753
$890$	− 52.4205i	− 0.0588995i
$891$	0	0
$892$	−447.025	−0.501150
$893$	615.212i	0.688928i
$894$	0	0
$895$	−41.0419	−0.0458568
$896$	873.399	0.974775
$897$	0	0
$898$	− 701.127i	− 0.780766i
$899$	− 1487.94i	− 1.65511i
$900$	0	0
$901$	− 1979.13i	− 2.19659i
$902$	0	0
$903$	0	0
$904$	680.988i	0.753305i
$905$	44.7120	0.0494056
$906$	0	0
$907$	756.730	0.834322	0.417161	−	0.908833i	$-0.363025\pi$
$907$	756.730	0.834322	0.417161	+	0.908833i	$0.363025\pi$
$908$	415.801i	0.457931i
$909$	0	0
$910$	34.3283i	0.0377234i
$911$	659.343	0.723757	0.361879	−	0.932225i	$-0.382135\pi$
$911$	659.343	0.723757	0.361879	+	0.932225i	$0.382135\pi$
$912$	0	0
$913$	0	0
$914$	−277.589	−0.303708
$915$	0	0
$916$	260.080	0.283930
$917$	118.226	0.128927
$918$	0	0
$919$	− 746.936i	− 0.812771i	−0.913702	−	0.406385i	$-0.866789\pi$
$919$	− 746.936i	− 0.812771i	0.913702	−	0.406385i	$-0.133211\pi$
$920$	16.2955i	0.0177125i
$921$	0	0
$922$	194.732	0.211206
$923$	1005.90i	1.08981i
$924$	0	0
$925$	148.233	0.160251
$926$	222.960i	0.240777i
$927$	0	0
$928$	−821.040	−0.884741
$929$	1212.62	1.30529	0.652647	−	0.757662i	$-0.273658\pi$
$929$	1212.62	1.30529	0.652647	+	0.757662i	$0.273658\pi$
$930$	0	0
$931$	− 35.8587i	− 0.0385163i
$932$	656.772i	0.704691i
$933$	0	0
$934$	147.050i	0.157441i
$935$	0	0
$936$	0	0
$937$	1591.84i	1.69887i	0.527697	+	0.849433i	$0.323056\pi$
$937$	1591.84i	1.69887i	−0.527697	+	0.849433i	$0.676944\pi$
$938$	672.194	0.716625
$939$	0	0
$940$	−66.7910	−0.0710543
$941$	− 596.807i	− 0.634226i	−0.948388	−	0.317113i	$-0.897287\pi$
$941$	− 596.807i	− 0.634226i	0.948388	−	0.317113i	$-0.102713\pi$
$942$	0	0
$943$	− 308.590i	− 0.327242i
$944$	134.232	0.142195
$945$	0	0
$946$	0	0
$947$	883.411	0.932852	0.466426	−	0.884560i	$-0.345541\pi$
$947$	883.411	0.932852	0.466426	+	0.884560i	$0.345541\pi$
$948$	0	0
$949$	−566.383	−0.596821
$950$	−271.250	−0.285527
$951$	0	0
$952$	− 1044.46i	− 1.09712i
$953$	282.867i	0.296817i	0.988926	+	0.148409i	$0.0474151\pi$
$953$	282.867i	0.296817i	−0.988926	+	0.148409i	$0.952585\pi$
$954$	0	0
$955$	75.9655	0.0795450
$956$	501.440i	0.524518i
$957$	0	0
$958$	−320.091	−0.334125
$959$	− 221.063i	− 0.230514i
$960$	0	0
$961$	2594.24	2.69952
$962$	71.2924	0.0741086
$963$	0	0
$964$	709.535i	0.736032i
$965$	− 21.6908i	− 0.0224776i
$966$	0	0
$967$	− 335.731i	− 0.347188i	−0.984817	−	0.173594i	$-0.944462\pi$
$967$	− 335.731i	− 0.347188i	0.984817	−	0.173594i	$-0.0555381\pi$
$968$	0	0
$969$	0	0
$970$	14.6908i	0.0151452i
$971$	−1395.34	−1.43701	−0.718505	−	0.695522i	$-0.755173\pi$
$971$	−1395.34	−1.43701	−0.718505	+	0.695522i	$0.755173\pi$
$972$	0	0
$973$	200.518	0.206082
$974$	301.574i	0.309624i
$975$	0	0
$976$	178.615i	0.183008i
$977$	1163.83	1.19123	0.595614	−	0.803271i	$-0.296909\pi$
$977$	1163.83	1.19123	0.595614	+	0.803271i	$0.296909\pi$
$978$	0	0
$979$	0	0
$980$	3.89303	0.00397248
$981$	0	0
$982$	603.173	0.614230
$983$	409.788	0.416875	0.208438	−	0.978036i	$-0.433162\pi$
$983$	409.788	0.416875	0.208438	+	0.978036i	$0.433162\pi$
$984$	0	0
$985$	29.9354i	0.0303913i
$986$	512.694i	0.519974i
$987$	0	0
$988$	410.053	0.415033
$989$	104.626i	0.105790i
$990$	0	0
$991$	1604.17	1.61874	0.809372	−	0.587297i	$-0.199808\pi$
$991$	1604.17	1.61874	0.809372	+	0.587297i	$0.199808\pi$
$992$	− 1961.77i	− 1.97759i
$993$	0	0
$994$	−587.411	−0.590956
$995$	24.1405	0.0242618
$996$	0	0
$997$	72.3843i	0.0726021i	0.999341	+	0.0363010i	$0.0115575\pi$
$997$	72.3843i	0.0726021i	−0.999341	+	0.0363010i	$0.988442\pi$
$998$	− 402.329i	− 0.403135i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.m.604.7		16
3.2	odd	2		363.3.c.e.241.10		16
11.2	odd	10		99.3.k.c.73.4		16
11.5	even	5		99.3.k.c.19.4		16
11.10	odd	2	inner	1089.3.c.m.604.10		16
33.2	even	10		33.3.g.a.7.1	✓	16
33.5	odd	10		33.3.g.a.19.1	yes	16
33.8	even	10		363.3.g.g.112.1		16
33.14	odd	10		363.3.g.a.112.4		16
33.17	even	10		363.3.g.f.118.4		16
33.20	odd	10		363.3.g.f.40.4		16
33.26	odd	10		363.3.g.g.94.1		16
33.29	even	10		363.3.g.a.94.4		16
33.32	even	2		363.3.c.e.241.7		16
132.35	odd	10		528.3.bf.b.337.3		16
132.71	even	10		528.3.bf.b.481.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.1	✓	16	33.2	even	10
33.3.g.a.19.1	yes	16	33.5	odd	10
99.3.k.c.19.4		16	11.5	even	5
99.3.k.c.73.4		16	11.2	odd	10
363.3.c.e.241.7		16	33.32	even	2
363.3.c.e.241.10		16	3.2	odd	2
363.3.g.a.94.4		16	33.29	even	10
363.3.g.a.112.4		16	33.14	odd	10
363.3.g.f.40.4		16	33.20	odd	10
363.3.g.f.118.4		16	33.17	even	10
363.3.g.g.94.1		16	33.26	odd	10
363.3.g.g.112.1		16	33.8	even	10
528.3.bf.b.337.3		16	132.35	odd	10
528.3.bf.b.481.3		16	132.71	even	10
1089.3.c.m.604.7		16	1.1	even	1	trivial
1089.3.c.m.604.10		16	11.10	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.982569i q^{2} +3.03456 q^{4} -0.397576 q^{5} +7.22681i q^{7} -6.91194i q^{8} +O(q^{10})\)	\(q-0.982569i q^{2} +3.03456 q^{4} -0.397576 q^{5} +7.22681i q^{7} -6.91194i q^{8} +0.390645i q^{10} -12.1597i q^{13} +7.10084 q^{14} +5.34678 q^{16} -20.9095i q^{17} +11.1128i q^{19} -1.20647 q^{20} +5.92990 q^{23} -24.8419 q^{25} -11.9477 q^{26} +21.9302i q^{28} -24.9546i q^{29} +59.6258 q^{31} -32.9013i q^{32} -20.5451 q^{34} -2.87320i q^{35} -5.96703 q^{37} +10.9191 q^{38} +2.74802i q^{40} -52.0396i q^{41} +17.6439i q^{43} -5.82653i q^{46} +55.3609 q^{47} -3.22680 q^{49} +24.4089i q^{50} -36.8993i q^{52} +94.6520 q^{53} +49.9513 q^{56} -24.5196 q^{58} +25.1052 q^{59} +33.4062i q^{61} -58.5865i q^{62} -10.9407 q^{64} +4.83439i q^{65} +94.6640 q^{67} -63.4513i q^{68} -2.82312 q^{70} -82.7241 q^{71} -46.5788i q^{73} +5.86302i q^{74} +33.7223i q^{76} -60.3191i q^{79} -2.12575 q^{80} -51.1325 q^{82} +150.673i q^{83} +8.31313i q^{85} +17.3363 q^{86} -134.190 q^{89} +87.8758 q^{91} +17.9946 q^{92} -54.3959i q^{94} -4.41816i q^{95} +37.6065 q^{97} +3.17056i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) 16 * q - 20 * q^4 + 4 * q^5	\( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Introduction

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