LMFDB - Embedded newform 2205.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.a (trivial)

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:	yes
Analytic conductor:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} +2.53590 q^{8} +O(q^{10})$	$q-0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} +2.53590 q^{8} +0.732051 q^{10} +2.73205 q^{11} +5.73205 q^{13} +1.07180 q^{16} -6.73205 q^{17} -2.46410 q^{19} +1.46410 q^{20} -2.00000 q^{22} +1.26795 q^{23} +1.00000 q^{25} -4.19615 q^{26} -6.19615 q^{29} +6.46410 q^{31} -5.85641 q^{32} +4.92820 q^{34} +7.19615 q^{37} +1.80385 q^{38} -2.53590 q^{40} -2.73205 q^{41} -7.19615 q^{43} -4.00000 q^{44} -0.928203 q^{46} -2.00000 q^{47} -0.732051 q^{50} -8.39230 q^{52} +8.39230 q^{53} -2.73205 q^{55} +4.53590 q^{58} -10.1962 q^{59} +4.00000 q^{61} -4.73205 q^{62} +2.14359 q^{64} -5.73205 q^{65} +2.66025 q^{67} +9.85641 q^{68} +4.19615 q^{71} -4.66025 q^{73} -5.26795 q^{74} +3.60770 q^{76} +13.3923 q^{79} -1.07180 q^{80} +2.00000 q^{82} +9.12436 q^{83} +6.73205 q^{85} +5.26795 q^{86} +6.92820 q^{88} +9.12436 q^{89} -1.85641 q^{92} +1.46410 q^{94} +2.46410 q^{95} +1.07180 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 + 12 * q^8	$ 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8} - 2 q^{10} + 2 q^{11} + 8 q^{13} + 16 q^{16} - 10 q^{17} + 2 q^{19} - 4 q^{20} - 4 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{29} + 6 q^{31} + 16 q^{32} - 4 q^{34} + 4 q^{37} + 14 q^{38} - 12 q^{40} - 2 q^{41} - 4 q^{43} - 8 q^{44} + 12 q^{46} - 4 q^{47} + 2 q^{50} + 4 q^{52} - 4 q^{53} - 2 q^{55} + 16 q^{58} - 10 q^{59} + 8 q^{61} - 6 q^{62} + 32 q^{64} - 8 q^{65} - 12 q^{67} - 8 q^{68} - 2 q^{71} + 8 q^{73} - 14 q^{74} + 28 q^{76} + 6 q^{79} - 16 q^{80} + 4 q^{82} - 6 q^{83} + 10 q^{85} + 14 q^{86} - 6 q^{89} + 24 q^{92} - 4 q^{94} - 2 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 + 12 * q^8 - 2 * q^10 + 2 * q^11 + 8 * q^13 + 16 * q^16 - 10 * q^17 + 2 * q^19 - 4 * q^20 - 4 * q^22 + 6 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^29 + 6 * q^31 + 16 * q^32 - 4 * q^34 + 4 * q^37 + 14 * q^38 - 12 * q^40 - 2 * q^41 - 4 * q^43 - 8 * q^44 + 12 * q^46 - 4 * q^47 + 2 * q^50 + 4 * q^52 - 4 * q^53 - 2 * q^55 + 16 * q^58 - 10 * q^59 + 8 * q^61 - 6 * q^62 + 32 * q^64 - 8 * q^65 - 12 * q^67 - 8 * q^68 - 2 * q^71 + 8 * q^73 - 14 * q^74 + 28 * q^76 + 6 * q^79 - 16 * q^80 + 4 * q^82 - 6 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^89 + 24 * q^92 - 4 * q^94 - 2 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−0.732051	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	−0.732051	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.53590	0.896575
$9$	0	0
$10$	0.732051	0.231495
$11$	2.73205	0.823744	0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205	0.823744	0.411872	+	0.911242i	$0.364875\pi$
$12$	0	0
$13$	5.73205	1.58978	0.794892	−	0.606750i	$-0.207527\pi$
$13$	5.73205	1.58978	0.794892	+	0.606750i	$0.207527\pi$
$14$	0	0
$15$	0	0
$16$	1.07180	0.267949
$17$	−6.73205	−1.63276	−0.816381	−	0.577514i	$-0.804023\pi$
$17$	−6.73205	−1.63276	−0.816381	+	0.577514i	$0.804023\pi$
$18$	0	0
$19$	−2.46410	−0.565304	−0.282652	−	0.959223i	$-0.591214\pi$
$19$	−2.46410	−0.565304	−0.282652	+	0.959223i	$0.591214\pi$
$20$	1.46410	0.327383
$21$	0	0
$22$	−2.00000	−0.426401
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.19615	−0.822933
$27$	0	0
$28$	0	0
$29$	−6.19615	−1.15060	−0.575298	−	0.817944i	$-0.695114\pi$
$29$	−6.19615	−1.15060	−0.575298	+	0.817944i	$0.695114\pi$
$30$	0	0
$31$	6.46410	1.16099	0.580493	−	0.814265i	$-0.302860\pi$
$31$	6.46410	1.16099	0.580493	+	0.814265i	$0.302860\pi$
$32$	−5.85641	−1.03528
$33$	0	0
$34$	4.92820	0.845180
$35$	0	0
$36$	0	0
$37$	7.19615	1.18304	0.591520	−	0.806290i	$-0.298528\pi$
$37$	7.19615	1.18304	0.591520	+	0.806290i	$0.298528\pi$
$38$	1.80385	0.292623
$39$	0	0
$40$	−2.53590	−0.400961
$41$	−2.73205	−0.426675	−0.213337	−	0.976979i	$-0.568433\pi$
$41$	−2.73205	−0.426675	−0.213337	+	0.976979i	$0.568433\pi$
$42$	0	0
$43$	−7.19615	−1.09740	−0.548701	−	0.836018i	$-0.684878\pi$
$43$	−7.19615	−1.09740	−0.548701	+	0.836018i	$0.684878\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−0.928203	−0.136856
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	0	0
$50$	−0.732051	−0.103528
$51$	0	0
$52$	−8.39230	−1.16380
$53$	8.39230	1.15277	0.576386	−	0.817178i	$-0.304463\pi$
$53$	8.39230	1.15277	0.576386	+	0.817178i	$0.304463\pi$
$54$	0	0
$55$	−2.73205	−0.368390
$56$	0	0
$57$	0	0
$58$	4.53590	0.595593
$59$	−10.1962	−1.32743	−0.663713	−	0.747987i	$-0.731020\pi$
$59$	−10.1962	−1.32743	−0.663713	+	0.747987i	$0.731020\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−4.73205	−0.600971
$63$	0	0
$64$	2.14359	0.267949
$65$	−5.73205	−0.710973
$66$	0	0
$67$	2.66025	0.325002	0.162501	−	0.986708i	$-0.448044\pi$
$67$	2.66025	0.325002	0.162501	+	0.986708i	$0.448044\pi$
$68$	9.85641	1.19526
$69$	0	0
$70$	0	0
$71$	4.19615	0.497992	0.248996	−	0.968505i	$-0.419899\pi$
$71$	4.19615	0.497992	0.248996	+	0.968505i	$0.419899\pi$
$72$	0	0
$73$	−4.66025	−0.545441	−0.272721	−	0.962093i	$-0.587924\pi$
$73$	−4.66025	−0.545441	−0.272721	+	0.962093i	$0.587924\pi$
$74$	−5.26795	−0.612387
$75$	0	0
$76$	3.60770	0.413831
$77$	0	0
$78$	0	0
$79$	13.3923	1.50675	0.753376	−	0.657590i	$-0.228424\pi$
$79$	13.3923	1.50675	0.753376	+	0.657590i	$0.228424\pi$
$80$	−1.07180	−0.119831
$81$	0	0
$82$	2.00000	0.220863
$83$	9.12436	1.00153	0.500764	−	0.865584i	$-0.333052\pi$
$83$	9.12436	1.00153	0.500764	+	0.865584i	$0.333052\pi$
$84$	0	0
$85$	6.73205	0.730193
$86$	5.26795	0.568058
$87$	0	0
$88$	6.92820	0.738549
$89$	9.12436	0.967180	0.483590	−	0.875295i	$-0.339333\pi$
$89$	9.12436	0.967180	0.483590	+	0.875295i	$0.339333\pi$
$90$	0	0
$91$	0	0
$92$	−1.85641	−0.193544
$93$	0	0
$94$	1.46410	0.151011
$95$	2.46410	0.252811
$96$	0	0
$97$	1.07180	0.108824	0.0544122	−	0.998519i	$-0.482671\pi$
$97$	1.07180	0.108824	0.0544122	+	0.998519i	$0.482671\pi$
$98$	0	0
$99$	0	0
$100$	−1.46410	−0.146410
$101$	10.7321	1.06788	0.533939	−	0.845523i	$-0.320711\pi$
$101$	10.7321	1.06788	0.533939	+	0.845523i	$0.320711\pi$
$102$	0	0
$103$	1.19615	0.117860	0.0589302	−	0.998262i	$-0.481231\pi$
$103$	1.19615	0.117860	0.0589302	+	0.998262i	$0.481231\pi$
$104$	14.5359	1.42536
$105$	0	0
$106$	−6.14359	−0.596719
$107$	8.19615	0.792352	0.396176	−	0.918175i	$-0.370337\pi$
$107$	8.19615	0.792352	0.396176	+	0.918175i	$0.370337\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	0	0
$113$	4.92820	0.463606	0.231803	−	0.972763i	$-0.425537\pi$
$113$	4.92820	0.463606	0.231803	+	0.972763i	$0.425537\pi$
$114$	0	0
$115$	−1.26795	−0.118237
$116$	9.07180	0.842295
$117$	0	0
$118$	7.46410	0.687126
$119$	0	0
$120$	0	0
$121$	−3.53590	−0.321445
$122$	−2.92820	−0.265107
$123$	0	0
$124$	−9.46410	−0.849901
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.1962	1.34844	0.674220	−	0.738530i	$-0.264480\pi$
$127$	15.1962	1.34844	0.674220	+	0.738530i	$0.264480\pi$
$128$	10.1436	0.896575
$129$	0	0
$130$	4.19615	0.368027
$131$	−8.53590	−0.745785	−0.372892	−	0.927875i	$-0.621634\pi$
$131$	−8.53590	−0.745785	−0.372892	+	0.927875i	$0.621634\pi$
$132$	0	0
$133$	0	0
$134$	−1.94744	−0.168233
$135$	0	0
$136$	−17.0718	−1.46389
$137$	8.19615	0.700245	0.350122	−	0.936704i	$-0.386140\pi$
$137$	8.19615	0.700245	0.350122	+	0.936704i	$0.386140\pi$
$138$	0	0
$139$	−7.92820	−0.672461	−0.336231	−	0.941780i	$-0.609152\pi$
$139$	−7.92820	−0.672461	−0.336231	+	0.941780i	$0.609152\pi$
$140$	0	0
$141$	0	0
$142$	−3.07180	−0.257779
$143$	15.6603	1.30958
$144$	0	0
$145$	6.19615	0.514562
$146$	3.41154	0.282341
$147$	0	0
$148$	−10.5359	−0.866046
$149$	21.8564	1.79055	0.895273	−	0.445517i	$-0.146980\pi$
$149$	21.8564	1.79055	0.895273	+	0.445517i	$0.146980\pi$
$150$	0	0
$151$	4.92820	0.401051	0.200526	−	0.979688i	$-0.435735\pi$
$151$	4.92820	0.401051	0.200526	+	0.979688i	$0.435735\pi$
$152$	−6.24871	−0.506837
$153$	0	0
$154$	0	0
$155$	−6.46410	−0.519209
$156$	0	0
$157$	−14.3923	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$157$	−14.3923	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$158$	−9.80385	−0.779952
$159$	0	0
$160$	5.85641	0.462990
$161$	0	0
$162$	0	0
$163$	−5.85641	−0.458709	−0.229355	−	0.973343i	$-0.573662\pi$
$163$	−5.85641	−0.458709	−0.229355	+	0.973343i	$0.573662\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−6.67949	−0.518429
$167$	0.339746	0.0262903	0.0131452	−	0.999914i	$-0.495816\pi$
$167$	0.339746	0.0262903	0.0131452	+	0.999914i	$0.495816\pi$
$168$	0	0
$169$	19.8564	1.52742
$170$	−4.92820	−0.377976
$171$	0	0
$172$	10.5359	0.803355
$173$	−21.4641	−1.63189	−0.815943	−	0.578133i	$-0.803782\pi$
$173$	−21.4641	−1.63189	−0.815943	+	0.578133i	$0.803782\pi$
$174$	0	0
$175$	0	0
$176$	2.92820	0.220722
$177$	0	0
$178$	−6.67949	−0.500649
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	10.3205	0.767117	0.383559	−	0.923517i	$-0.374698\pi$
$181$	10.3205	0.767117	0.383559	+	0.923517i	$0.374698\pi$
$182$	0	0
$183$	0	0
$184$	3.21539	0.237042
$185$	−7.19615	−0.529072
$186$	0	0
$187$	−18.3923	−1.34498
$188$	2.92820	0.213561
$189$	0	0
$190$	−1.80385	−0.130865
$191$	−4.92820	−0.356592	−0.178296	−	0.983977i	$-0.557059\pi$
$191$	−4.92820	−0.356592	−0.178296	+	0.983977i	$0.557059\pi$
$192$	0	0
$193$	−9.19615	−0.661954	−0.330977	−	0.943639i	$-0.607378\pi$
$193$	−9.19615	−0.661954	−0.330977	+	0.943639i	$0.607378\pi$
$194$	−0.784610	−0.0563317
$195$	0	0
$196$	0	0
$197$	17.6603	1.25824	0.629121	−	0.777308i	$-0.283415\pi$
$197$	17.6603	1.25824	0.629121	+	0.777308i	$0.283415\pi$
$198$	0	0
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	2.53590	0.179315
$201$	0	0
$202$	−7.85641	−0.552775
$203$	0	0
$204$	0	0
$205$	2.73205	0.190815
$206$	−0.875644	−0.0610090
$207$	0	0
$208$	6.14359	0.425982
$209$	−6.73205	−0.465666
$210$	0	0
$211$	20.9282	1.44076	0.720378	−	0.693581i	$-0.243968\pi$
$211$	20.9282	1.44076	0.720378	+	0.693581i	$0.243968\pi$
$212$	−12.2872	−0.843887
$213$	0	0
$214$	−6.00000	−0.410152
$215$	7.19615	0.490774
$216$	0	0
$217$	0	0
$218$	−8.05256	−0.545388
$219$	0	0
$220$	4.00000	0.269680
$221$	−38.5885	−2.59574
$222$	0	0
$223$	0.392305	0.0262707	0.0131353	−	0.999914i	$-0.495819\pi$
$223$	0.392305	0.0262707	0.0131353	+	0.999914i	$0.495819\pi$
$224$	0	0
$225$	0	0
$226$	−3.60770	−0.239980
$227$	−15.6603	−1.03941	−0.519704	−	0.854347i	$-0.673958\pi$
$227$	−15.6603	−1.03941	−0.519704	+	0.854347i	$0.673958\pi$
$228$	0	0
$229$	−3.00000	−0.198246	−0.0991228	−	0.995075i	$-0.531604\pi$
$229$	−3.00000	−0.198246	−0.0991228	+	0.995075i	$0.531604\pi$
$230$	0.928203	0.0612039
$231$	0	0
$232$	−15.7128	−1.03160
$233$	17.3205	1.13470	0.567352	−	0.823475i	$-0.307968\pi$
$233$	17.3205	1.13470	0.567352	+	0.823475i	$0.307968\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	14.9282	0.971743
$237$	0	0
$238$	0	0
$239$	−20.9282	−1.35373	−0.676866	−	0.736106i	$-0.736663\pi$
$239$	−20.9282	−1.35373	−0.676866	+	0.736106i	$0.736663\pi$
$240$	0	0
$241$	6.53590	0.421014	0.210507	−	0.977592i	$-0.432488\pi$
$241$	6.53590	0.421014	0.210507	+	0.977592i	$0.432488\pi$
$242$	2.58846	0.166392
$243$	0	0
$244$	−5.85641	−0.374918
$245$	0	0
$246$	0	0
$247$	−14.1244	−0.898711
$248$	16.3923	1.04091
$249$	0	0
$250$	0.732051	0.0462990
$251$	−6.58846	−0.415860	−0.207930	−	0.978144i	$-0.566673\pi$
$251$	−6.58846	−0.415860	−0.207930	+	0.978144i	$0.566673\pi$
$252$	0	0
$253$	3.46410	0.217786
$254$	−11.1244	−0.698004
$255$	0	0
$256$	−11.7128	−0.732051
$257$	−11.6603	−0.727347	−0.363673	−	0.931527i	$-0.618478\pi$
$257$	−11.6603	−0.727347	−0.363673	+	0.931527i	$0.618478\pi$
$258$	0	0
$259$	0	0
$260$	8.39230	0.520469
$261$	0	0
$262$	6.24871	0.386047
$263$	12.3923	0.764142	0.382071	−	0.924133i	$-0.375211\pi$
$263$	12.3923	0.764142	0.382071	+	0.924133i	$0.375211\pi$
$264$	0	0
$265$	−8.39230	−0.515535
$266$	0	0
$267$	0	0
$268$	−3.89488	−0.237918
$269$	19.4641	1.18675	0.593374	−	0.804927i	$-0.297796\pi$
$269$	19.4641	1.18675	0.593374	+	0.804927i	$0.297796\pi$
$270$	0	0
$271$	16.9282	1.02832	0.514158	−	0.857696i	$-0.328105\pi$
$271$	16.9282	1.02832	0.514158	+	0.857696i	$0.328105\pi$
$272$	−7.21539	−0.437497
$273$	0	0
$274$	−6.00000	−0.362473
$275$	2.73205	0.164749
$276$	0	0
$277$	2.66025	0.159839	0.0799196	−	0.996801i	$-0.474534\pi$
$277$	2.66025	0.159839	0.0799196	+	0.996801i	$0.474534\pi$
$278$	5.80385	0.348092
$279$	0	0
$280$	0	0
$281$	13.8564	0.826604	0.413302	−	0.910594i	$-0.364375\pi$
$281$	13.8564	0.826604	0.413302	+	0.910594i	$0.364375\pi$
$282$	0	0
$283$	0.124356	0.00739218	0.00369609	−	0.999993i	$-0.498823\pi$
$283$	0.124356	0.00739218	0.00369609	+	0.999993i	$0.498823\pi$
$284$	−6.14359	−0.364555
$285$	0	0
$286$	−11.4641	−0.677887
$287$	0	0
$288$	0	0
$289$	28.3205	1.66591
$290$	−4.53590	−0.266357
$291$	0	0
$292$	6.82309	0.399291
$293$	−5.07180	−0.296298	−0.148149	−	0.988965i	$-0.547331\pi$
$293$	−5.07180	−0.296298	−0.148149	+	0.988965i	$0.547331\pi$
$294$	0	0
$295$	10.1962	0.593643
$296$	18.2487	1.06068
$297$	0	0
$298$	−16.0000	−0.926855
$299$	7.26795	0.420316
$300$	0	0
$301$	0	0
$302$	−3.60770	−0.207600
$303$	0	0
$304$	−2.64102	−0.151473
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−7.87564	−0.449487	−0.224743	−	0.974418i	$-0.572154\pi$
$307$	−7.87564	−0.449487	−0.224743	+	0.974418i	$0.572154\pi$
$308$	0	0
$309$	0	0
$310$	4.73205	0.268762
$311$	15.1244	0.857624	0.428812	−	0.903394i	$-0.358932\pi$
$311$	15.1244	0.857624	0.428812	+	0.903394i	$0.358932\pi$
$312$	0	0
$313$	4.66025	0.263413	0.131707	−	0.991289i	$-0.457954\pi$
$313$	4.66025	0.263413	0.131707	+	0.991289i	$0.457954\pi$
$314$	10.5359	0.594575
$315$	0	0
$316$	−19.6077	−1.10302
$317$	−30.4449	−1.70995	−0.854977	−	0.518666i	$-0.826429\pi$
$317$	−30.4449	−1.70995	−0.854977	+	0.518666i	$0.826429\pi$
$318$	0	0
$319$	−16.9282	−0.947797
$320$	−2.14359	−0.119831
$321$	0	0
$322$	0	0
$323$	16.5885	0.923006
$324$	0	0
$325$	5.73205	0.317957
$326$	4.28719	0.237445
$327$	0	0
$328$	−6.92820	−0.382546
$329$	0	0
$330$	0	0
$331$	21.9282	1.20528	0.602642	−	0.798012i	$-0.294115\pi$
$331$	21.9282	1.20528	0.602642	+	0.798012i	$0.294115\pi$
$332$	−13.3590	−0.733169
$333$	0	0
$334$	−0.248711	−0.0136089
$335$	−2.66025	−0.145345
$336$	0	0
$337$	−33.9808	−1.85105	−0.925525	−	0.378686i	$-0.876376\pi$
$337$	−33.9808	−1.85105	−0.925525	+	0.378686i	$0.876376\pi$
$338$	−14.5359	−0.790649
$339$	0	0
$340$	−9.85641	−0.534539
$341$	17.6603	0.956356
$342$	0	0
$343$	0	0
$344$	−18.2487	−0.983905
$345$	0	0
$346$	15.7128	0.844726
$347$	34.9282	1.87504	0.937522	−	0.347926i	$-0.113114\pi$
$347$	34.9282	1.87504	0.937522	+	0.347926i	$0.113114\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	0	0
$352$	−16.0000	−0.852803
$353$	−21.1244	−1.12434	−0.562168	−	0.827023i	$-0.690033\pi$
$353$	−21.1244	−1.12434	−0.562168	+	0.827023i	$0.690033\pi$
$354$	0	0
$355$	−4.19615	−0.222709
$356$	−13.3590	−0.708025
$357$	0	0
$358$	−7.32051	−0.386901
$359$	4.73205	0.249748	0.124874	−	0.992173i	$-0.460147\pi$
$359$	4.73205	0.249748	0.124874	+	0.992173i	$0.460147\pi$
$360$	0	0
$361$	−12.9282	−0.680432
$362$	−7.55514	−0.397089
$363$	0	0
$364$	0	0
$365$	4.66025	0.243929
$366$	0	0
$367$	0.803848	0.0419605	0.0209803	−	0.999780i	$-0.493321\pi$
$367$	0.803848	0.0419605	0.0209803	+	0.999780i	$0.493321\pi$
$368$	1.35898	0.0708419
$369$	0	0
$370$	5.26795	0.273868
$371$	0	0
$372$	0	0
$373$	−18.5167	−0.958756	−0.479378	−	0.877608i	$-0.659138\pi$
$373$	−18.5167	−0.958756	−0.479378	+	0.877608i	$0.659138\pi$
$374$	13.4641	0.696212
$375$	0	0
$376$	−5.07180	−0.261558
$377$	−35.5167	−1.82920
$378$	0	0
$379$	−28.3205	−1.45473	−0.727363	−	0.686253i	$-0.759255\pi$
$379$	−28.3205	−1.45473	−0.727363	+	0.686253i	$0.759255\pi$
$380$	−3.60770	−0.185071
$381$	0	0
$382$	3.60770	0.184586
$383$	−11.3205	−0.578451	−0.289225	−	0.957261i	$-0.593398\pi$
$383$	−11.3205	−0.578451	−0.289225	+	0.957261i	$0.593398\pi$
$384$	0	0
$385$	0	0
$386$	6.73205	0.342652
$387$	0	0
$388$	−1.56922	−0.0796650
$389$	−36.5885	−1.85511	−0.927554	−	0.373689i	$-0.878093\pi$
$389$	−36.5885	−1.85511	−0.927554	+	0.373689i	$0.878093\pi$
$390$	0	0
$391$	−8.53590	−0.431679
$392$	0	0
$393$	0	0
$394$	−12.9282	−0.651313
$395$	−13.3923	−0.673840
$396$	0	0
$397$	−20.8038	−1.04412	−0.522058	−	0.852910i	$-0.674835\pi$
$397$	−20.8038	−1.04412	−0.522058	+	0.852910i	$0.674835\pi$
$398$	−16.1051	−0.807277
$399$	0	0
$400$	1.07180	0.0535898
$401$	−4.39230	−0.219341	−0.109671	−	0.993968i	$-0.534980\pi$
$401$	−4.39230	−0.219341	−0.109671	+	0.993968i	$0.534980\pi$
$402$	0	0
$403$	37.0526	1.84572
$404$	−15.7128	−0.781742
$405$	0	0
$406$	0	0
$407$	19.6603	0.974523
$408$	0	0
$409$	30.8564	1.52575	0.762876	−	0.646545i	$-0.223787\pi$
$409$	30.8564	1.52575	0.762876	+	0.646545i	$0.223787\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−1.75129	−0.0862798
$413$	0	0
$414$	0	0
$415$	−9.12436	−0.447897
$416$	−33.5692	−1.64587
$417$	0	0
$418$	4.92820	0.241046
$419$	28.5359	1.39407	0.697035	−	0.717037i	$-0.254502\pi$
$419$	28.5359	1.39407	0.697035	+	0.717037i	$0.254502\pi$
$420$	0	0
$421$	13.9282	0.678819	0.339410	−	0.940639i	$-0.389773\pi$
$421$	13.9282	0.678819	0.339410	+	0.940639i	$0.389773\pi$
$422$	−15.3205	−0.745791
$423$	0	0
$424$	21.2820	1.03355
$425$	−6.73205	−0.326552
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−5.26795	−0.254043
$431$	17.3205	0.834300	0.417150	−	0.908838i	$-0.363029\pi$
$431$	17.3205	0.834300	0.417150	+	0.908838i	$0.363029\pi$
$432$	0	0
$433$	4.80385	0.230858	0.115429	−	0.993316i	$-0.463176\pi$
$433$	4.80385	0.230858	0.115429	+	0.993316i	$0.463176\pi$
$434$	0	0
$435$	0	0
$436$	−16.1051	−0.771295
$437$	−3.12436	−0.149458
$438$	0	0
$439$	7.46410	0.356242	0.178121	−	0.984009i	$-0.442998\pi$
$439$	7.46410	0.356242	0.178121	+	0.984009i	$0.442998\pi$
$440$	−6.92820	−0.330289
$441$	0	0
$442$	28.2487	1.34365
$443$	−2.53590	−0.120484	−0.0602421	−	0.998184i	$-0.519187\pi$
$443$	−2.53590	−0.120484	−0.0602421	+	0.998184i	$0.519187\pi$
$444$	0	0
$445$	−9.12436	−0.432536
$446$	−0.287187	−0.0135987
$447$	0	0
$448$	0	0
$449$	8.14359	0.384320	0.192160	−	0.981364i	$-0.438451\pi$
$449$	8.14359	0.384320	0.192160	+	0.981364i	$0.438451\pi$
$450$	0	0
$451$	−7.46410	−0.351471
$452$	−7.21539	−0.339383
$453$	0	0
$454$	11.4641	0.538037
$455$	0	0
$456$	0	0
$457$	0.660254	0.0308854	0.0154427	−	0.999881i	$-0.495084\pi$
$457$	0.660254	0.0308854	0.0154427	+	0.999881i	$0.495084\pi$
$458$	2.19615	0.102619
$459$	0	0
$460$	1.85641	0.0865554
$461$	34.9808	1.62922	0.814608	−	0.580012i	$-0.196952\pi$
$461$	34.9808	1.62922	0.814608	+	0.580012i	$0.196952\pi$
$462$	0	0
$463$	22.2679	1.03488	0.517440	−	0.855720i	$-0.326885\pi$
$463$	22.2679	1.03488	0.517440	+	0.855720i	$0.326885\pi$
$464$	−6.64102	−0.308301
$465$	0	0
$466$	−12.6795	−0.587366
$467$	27.8564	1.28904	0.644520	−	0.764587i	$-0.277057\pi$
$467$	27.8564	1.28904	0.644520	+	0.764587i	$0.277057\pi$
$468$	0	0
$469$	0	0
$470$	−1.46410	−0.0675340
$471$	0	0
$472$	−25.8564	−1.19014
$473$	−19.6603	−0.903979
$474$	0	0
$475$	−2.46410	−0.113061
$476$	0	0
$477$	0	0
$478$	15.3205	0.700744
$479$	32.7846	1.49797	0.748984	−	0.662589i	$-0.230542\pi$
$479$	32.7846	1.49797	0.748984	+	0.662589i	$0.230542\pi$
$480$	0	0
$481$	41.2487	1.88078
$482$	−4.78461	−0.217933
$483$	0	0
$484$	5.17691	0.235314
$485$	−1.07180	−0.0486678
$486$	0	0
$487$	−31.5885	−1.43141	−0.715705	−	0.698403i	$-0.753894\pi$
$487$	−31.5885	−1.43141	−0.715705	+	0.698403i	$0.753894\pi$
$488$	10.1436	0.459179
$489$	0	0
$490$	0	0
$491$	−10.2487	−0.462518	−0.231259	−	0.972892i	$-0.574284\pi$
$491$	−10.2487	−0.462518	−0.231259	+	0.972892i	$0.574284\pi$
$492$	0	0
$493$	41.7128	1.87865
$494$	10.3397	0.465207
$495$	0	0
$496$	6.92820	0.311086
$497$	0	0
$498$	0	0
$499$	−20.4641	−0.916099	−0.458050	−	0.888927i	$-0.651452\pi$
$499$	−20.4641	−0.916099	−0.458050	+	0.888927i	$0.651452\pi$
$500$	1.46410	0.0654766
$501$	0	0
$502$	4.82309	0.215265
$503$	6.39230	0.285019	0.142509	−	0.989793i	$-0.454483\pi$
$503$	6.39230	0.285019	0.142509	+	0.989793i	$0.454483\pi$
$504$	0	0
$505$	−10.7321	−0.477570
$506$	−2.53590	−0.112734
$507$	0	0
$508$	−22.2487	−0.987127
$509$	−11.4641	−0.508137	−0.254069	−	0.967186i	$-0.581769\pi$
$509$	−11.4641	−0.508137	−0.254069	+	0.967186i	$0.581769\pi$
$510$	0	0
$511$	0	0
$512$	−11.7128	−0.517638
$513$	0	0
$514$	8.53590	0.376502
$515$	−1.19615	−0.0527088
$516$	0	0
$517$	−5.46410	−0.240311
$518$	0	0
$519$	0	0
$520$	−14.5359	−0.637441
$521$	−1.46410	−0.0641435	−0.0320717	−	0.999486i	$-0.510210\pi$
$521$	−1.46410	−0.0641435	−0.0320717	+	0.999486i	$0.510210\pi$
$522$	0	0
$523$	−24.2679	−1.06116	−0.530582	−	0.847634i	$-0.678026\pi$
$523$	−24.2679	−1.06116	−0.530582	+	0.847634i	$0.678026\pi$
$524$	12.4974	0.545952
$525$	0	0
$526$	−9.07180	−0.395549
$527$	−43.5167	−1.89562
$528$	0	0
$529$	−21.3923	−0.930100
$530$	6.14359	0.266861
$531$	0	0
$532$	0	0
$533$	−15.6603	−0.678321
$534$	0	0
$535$	−8.19615	−0.354351
$536$	6.74613	0.291389
$537$	0	0
$538$	−14.2487	−0.614306
$539$	0	0
$540$	0	0
$541$	−35.7846	−1.53850	−0.769250	−	0.638948i	$-0.779370\pi$
$541$	−35.7846	−1.53850	−0.769250	+	0.638948i	$0.779370\pi$
$542$	−12.3923	−0.532295
$543$	0	0
$544$	39.4256	1.69036
$545$	−11.0000	−0.471188
$546$	0	0
$547$	22.2487	0.951286	0.475643	−	0.879638i	$-0.342215\pi$
$547$	22.2487	0.951286	0.475643	+	0.879638i	$0.342215\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	15.2679	0.650437
$552$	0	0
$553$	0	0
$554$	−1.94744	−0.0827388
$555$	0	0
$556$	11.6077	0.492276
$557$	26.7846	1.13490	0.567450	−	0.823408i	$-0.307930\pi$
$557$	26.7846	1.13490	0.567450	+	0.823408i	$0.307930\pi$
$558$	0	0
$559$	−41.2487	−1.74463
$560$	0	0
$561$	0	0
$562$	−10.1436	−0.427882
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	−4.92820	−0.207331
$566$	−0.0910347	−0.00382647
$567$	0	0
$568$	10.6410	0.446487
$569$	26.4449	1.10863	0.554313	−	0.832308i	$-0.312981\pi$
$569$	26.4449	1.10863	0.554313	+	0.832308i	$0.312981\pi$
$570$	0	0
$571$	39.3923	1.64852	0.824258	−	0.566214i	$-0.191592\pi$
$571$	39.3923	1.64852	0.824258	+	0.566214i	$0.191592\pi$
$572$	−22.9282	−0.958676
$573$	0	0
$574$	0	0
$575$	1.26795	0.0528771
$576$	0	0
$577$	11.3397	0.472080	0.236040	−	0.971743i	$-0.424150\pi$
$577$	11.3397	0.472080	0.236040	+	0.971743i	$0.424150\pi$
$578$	−20.7321	−0.862340
$579$	0	0
$580$	−9.07180	−0.376686
$581$	0	0
$582$	0	0
$583$	22.9282	0.949589
$584$	−11.8179	−0.489029
$585$	0	0
$586$	3.71281	0.153375
$587$	−37.2679	−1.53821	−0.769106	−	0.639121i	$-0.779298\pi$
$587$	−37.2679	−1.53821	−0.769106	+	0.639121i	$0.779298\pi$
$588$	0	0
$589$	−15.9282	−0.656310
$590$	−7.46410	−0.307292
$591$	0	0
$592$	7.71281	0.316995
$593$	37.9090	1.55673	0.778367	−	0.627809i	$-0.216048\pi$
$593$	37.9090	1.55673	0.778367	+	0.627809i	$0.216048\pi$
$594$	0	0
$595$	0	0
$596$	−32.0000	−1.31077
$597$	0	0
$598$	−5.32051	−0.217572
$599$	−10.2487	−0.418751	−0.209375	−	0.977835i	$-0.567143\pi$
$599$	−10.2487	−0.418751	−0.209375	+	0.977835i	$0.567143\pi$
$600$	0	0
$601$	−13.9282	−0.568143	−0.284072	−	0.958803i	$-0.591685\pi$
$601$	−13.9282	−0.568143	−0.284072	+	0.958803i	$0.591685\pi$
$602$	0	0
$603$	0	0
$604$	−7.21539	−0.293590
$605$	3.53590	0.143755
$606$	0	0
$607$	−7.19615	−0.292083	−0.146041	−	0.989278i	$-0.546653\pi$
$607$	−7.19615	−0.292083	−0.146041	+	0.989278i	$0.546653\pi$
$608$	14.4308	0.585245
$609$	0	0
$610$	2.92820	0.118559
$611$	−11.4641	−0.463788
$612$	0	0
$613$	−13.0718	−0.527965	−0.263982	−	0.964527i	$-0.585036\pi$
$613$	−13.0718	−0.527965	−0.263982	+	0.964527i	$0.585036\pi$
$614$	5.76537	0.232671
$615$	0	0
$616$	0	0
$617$	−12.2487	−0.493115	−0.246557	−	0.969128i	$-0.579299\pi$
$617$	−12.2487	−0.493115	−0.246557	+	0.969128i	$0.579299\pi$
$618$	0	0
$619$	−43.9282	−1.76562	−0.882812	−	0.469727i	$-0.844352\pi$
$619$	−43.9282	−1.76562	−0.882812	+	0.469727i	$0.844352\pi$
$620$	9.46410	0.380087
$621$	0	0
$622$	−11.0718	−0.443939
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.41154	−0.136353
$627$	0	0
$628$	21.0718	0.840856
$629$	−48.4449	−1.93162
$630$	0	0
$631$	7.21539	0.287240	0.143620	−	0.989633i	$-0.454126\pi$
$631$	7.21539	0.287240	0.143620	+	0.989633i	$0.454126\pi$
$632$	33.9615	1.35092
$633$	0	0
$634$	22.2872	0.885137
$635$	−15.1962	−0.603041
$636$	0	0
$637$	0	0
$638$	12.3923	0.490616
$639$	0	0
$640$	−10.1436	−0.400961
$641$	−14.1962	−0.560714	−0.280357	−	0.959896i	$-0.590453\pi$
$641$	−14.1962	−0.560714	−0.280357	+	0.959896i	$0.590453\pi$
$642$	0	0
$643$	−40.5167	−1.59782	−0.798911	−	0.601450i	$-0.794590\pi$
$643$	−40.5167	−1.59782	−0.798911	+	0.601450i	$0.794590\pi$
$644$	0	0
$645$	0	0
$646$	−12.1436	−0.477783
$647$	−37.9090	−1.49036	−0.745178	−	0.666866i	$-0.767635\pi$
$647$	−37.9090	−1.49036	−0.745178	+	0.666866i	$0.767635\pi$
$648$	0	0
$649$	−27.8564	−1.09346
$650$	−4.19615	−0.164587
$651$	0	0
$652$	8.57437	0.335798
$653$	13.4115	0.524834	0.262417	−	0.964955i	$-0.415480\pi$
$653$	13.4115	0.524834	0.262417	+	0.964955i	$0.415480\pi$
$654$	0	0
$655$	8.53590	0.333525
$656$	−2.92820	−0.114327
$657$	0	0
$658$	0	0
$659$	10.9282	0.425702	0.212851	−	0.977085i	$-0.431725\pi$
$659$	10.9282	0.425702	0.212851	+	0.977085i	$0.431725\pi$
$660$	0	0
$661$	3.53590	0.137531	0.0687653	−	0.997633i	$-0.478094\pi$
$661$	3.53590	0.137531	0.0687653	+	0.997633i	$0.478094\pi$
$662$	−16.0526	−0.623900
$663$	0	0
$664$	23.1384	0.897946
$665$	0	0
$666$	0	0
$667$	−7.85641	−0.304201
$668$	−0.497423	−0.0192459
$669$	0	0
$670$	1.94744	0.0752362
$671$	10.9282	0.421879
$672$	0	0
$673$	−44.6603	−1.72153	−0.860763	−	0.509006i	$-0.830013\pi$
$673$	−44.6603	−1.72153	−0.860763	+	0.509006i	$0.830013\pi$
$674$	24.8756	0.958174
$675$	0	0
$676$	−29.0718	−1.11815
$677$	8.87564	0.341119	0.170559	−	0.985347i	$-0.445443\pi$
$677$	8.87564	0.341119	0.170559	+	0.985347i	$0.445443\pi$
$678$	0	0
$679$	0	0
$680$	17.0718	0.654674
$681$	0	0
$682$	−12.9282	−0.495046
$683$	−10.0526	−0.384650	−0.192325	−	0.981331i	$-0.561603\pi$
$683$	−10.0526	−0.384650	−0.192325	+	0.981331i	$0.561603\pi$
$684$	0	0
$685$	−8.19615	−0.313159
$686$	0	0
$687$	0	0
$688$	−7.71281	−0.294048
$689$	48.1051	1.83266
$690$	0	0
$691$	−18.8564	−0.717332	−0.358666	−	0.933466i	$-0.616768\pi$
$691$	−18.8564	−0.717332	−0.358666	+	0.933466i	$0.616768\pi$
$692$	31.4256	1.19462
$693$	0	0
$694$	−25.5692	−0.970594
$695$	7.92820	0.300734
$696$	0	0
$697$	18.3923	0.696658
$698$	−16.1051	−0.609588
$699$	0	0
$700$	0	0
$701$	−22.5885	−0.853154	−0.426577	−	0.904451i	$-0.640281\pi$
$701$	−22.5885	−0.853154	−0.426577	+	0.904451i	$0.640281\pi$
$702$	0	0
$703$	−17.7321	−0.668777
$704$	5.85641	0.220722
$705$	0	0
$706$	15.4641	0.581999
$707$	0	0
$708$	0	0
$709$	−14.9282	−0.560640	−0.280320	−	0.959907i	$-0.590441\pi$
$709$	−14.9282	−0.560640	−0.280320	+	0.959907i	$0.590441\pi$
$710$	3.07180	0.115282
$711$	0	0
$712$	23.1384	0.867150
$713$	8.19615	0.306948
$714$	0	0
$715$	−15.6603	−0.585660
$716$	−14.6410	−0.547160
$717$	0	0
$718$	−3.46410	−0.129279
$719$	27.4641	1.02424	0.512119	−	0.858914i	$-0.328861\pi$
$719$	27.4641	1.02424	0.512119	+	0.858914i	$0.328861\pi$
$720$	0	0
$721$	0	0
$722$	9.46410	0.352217
$723$	0	0
$724$	−15.1103	−0.561569
$725$	−6.19615	−0.230119
$726$	0	0
$727$	−30.6603	−1.13713	−0.568563	−	0.822640i	$-0.692500\pi$
$727$	−30.6603	−1.13713	−0.568563	+	0.822640i	$0.692500\pi$
$728$	0	0
$729$	0	0
$730$	−3.41154	−0.126267
$731$	48.4449	1.79180
$732$	0	0
$733$	−18.6603	−0.689232	−0.344616	−	0.938744i	$-0.611991\pi$
$733$	−18.6603	−0.689232	−0.344616	+	0.938744i	$0.611991\pi$
$734$	−0.588457	−0.0217204
$735$	0	0
$736$	−7.42563	−0.273712
$737$	7.26795	0.267718
$738$	0	0
$739$	−13.7846	−0.507075	−0.253538	−	0.967326i	$-0.581594\pi$
$739$	−13.7846	−0.507075	−0.253538	+	0.967326i	$0.581594\pi$
$740$	10.5359	0.387307
$741$	0	0
$742$	0	0
$743$	−49.9090	−1.83098	−0.915491	−	0.402338i	$-0.868198\pi$
$743$	−49.9090	−1.83098	−0.915491	+	0.402338i	$0.868198\pi$
$744$	0	0
$745$	−21.8564	−0.800757
$746$	13.5551	0.496289
$747$	0	0
$748$	26.9282	0.984593
$749$	0	0
$750$	0	0
$751$	31.9282	1.16508	0.582538	−	0.812803i	$-0.302060\pi$
$751$	31.9282	1.16508	0.582538	+	0.812803i	$0.302060\pi$
$752$	−2.14359	−0.0781688
$753$	0	0
$754$	26.0000	0.946864
$755$	−4.92820	−0.179356
$756$	0	0
$757$	−0.143594	−0.00521900	−0.00260950	−	0.999997i	$-0.500831\pi$
$757$	−0.143594	−0.00521900	−0.00260950	+	0.999997i	$0.500831\pi$
$758$	20.7321	0.753022
$759$	0	0
$760$	6.24871	0.226665
$761$	−43.2679	−1.56846	−0.784231	−	0.620469i	$-0.786942\pi$
$761$	−43.2679	−1.56846	−0.784231	+	0.620469i	$0.786942\pi$
$762$	0	0
$763$	0	0
$764$	7.21539	0.261044
$765$	0	0
$766$	8.28719	0.299428
$767$	−58.4449	−2.11032
$768$	0	0
$769$	17.6795	0.637539	0.318769	−	0.947832i	$-0.396730\pi$
$769$	17.6795	0.637539	0.318769	+	0.947832i	$0.396730\pi$
$770$	0	0
$771$	0	0
$772$	13.4641	0.484584
$773$	1.51666	0.0545505	0.0272752	−	0.999628i	$-0.491317\pi$
$773$	1.51666	0.0545505	0.0272752	+	0.999628i	$0.491317\pi$
$774$	0	0
$775$	6.46410	0.232197
$776$	2.71797	0.0975694
$777$	0	0
$778$	26.7846	0.960275
$779$	6.73205	0.241201
$780$	0	0
$781$	11.4641	0.410218
$782$	6.24871	0.223453
$783$	0	0
$784$	0	0
$785$	14.3923	0.513683
$786$	0	0
$787$	6.53590	0.232980	0.116490	−	0.993192i	$-0.462836\pi$
$787$	6.53590	0.232980	0.116490	+	0.993192i	$0.462836\pi$
$788$	−25.8564	−0.921096
$789$	0	0
$790$	9.80385	0.348805
$791$	0	0
$792$	0	0
$793$	22.9282	0.814204
$794$	15.2295	0.540474
$795$	0	0
$796$	−32.2102	−1.14166
$797$	42.0526	1.48958	0.744789	−	0.667300i	$-0.232550\pi$
$797$	42.0526	1.48958	0.744789	+	0.667300i	$0.232550\pi$
$798$	0	0
$799$	13.4641	0.476326
$800$	−5.85641	−0.207055
$801$	0	0
$802$	3.21539	0.113539
$803$	−12.7321	−0.449304
$804$	0	0
$805$	0	0
$806$	−27.1244	−0.955415
$807$	0	0
$808$	27.2154	0.957434
$809$	29.7128	1.04465	0.522323	−	0.852747i	$-0.325065\pi$
$809$	29.7128	1.04465	0.522323	+	0.852747i	$0.325065\pi$
$810$	0	0
$811$	3.46410	0.121641	0.0608205	−	0.998149i	$-0.480628\pi$
$811$	3.46410	0.121641	0.0608205	+	0.998149i	$0.480628\pi$
$812$	0	0
$813$	0	0
$814$	−14.3923	−0.504450
$815$	5.85641	0.205141
$816$	0	0
$817$	17.7321	0.620366
$818$	−22.5885	−0.789787
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−19.5167	−0.681136	−0.340568	−	0.940220i	$-0.610619\pi$
$821$	−19.5167	−0.681136	−0.340568	+	0.940220i	$0.610619\pi$
$822$	0	0
$823$	−23.1769	−0.807896	−0.403948	−	0.914782i	$-0.632362\pi$
$823$	−23.1769	−0.807896	−0.403948	+	0.914782i	$0.632362\pi$
$824$	3.03332	0.105671
$825$	0	0
$826$	0	0
$827$	52.2487	1.81687	0.908433	−	0.418031i	$-0.137280\pi$
$827$	52.2487	1.81687	0.908433	+	0.418031i	$0.137280\pi$
$828$	0	0
$829$	−25.3923	−0.881911	−0.440956	−	0.897529i	$-0.645360\pi$
$829$	−25.3923	−0.881911	−0.440956	+	0.897529i	$0.645360\pi$
$830$	6.67949	0.231849
$831$	0	0
$832$	12.2872	0.425982
$833$	0	0
$834$	0	0
$835$	−0.339746	−0.0117574
$836$	9.85641	0.340891
$837$	0	0
$838$	−20.8897	−0.721624
$839$	40.4449	1.39631	0.698156	−	0.715946i	$-0.254004\pi$
$839$	40.4449	1.39631	0.698156	+	0.715946i	$0.254004\pi$
$840$	0	0
$841$	9.39230	0.323873
$842$	−10.1962	−0.351383
$843$	0	0
$844$	−30.6410	−1.05471
$845$	−19.8564	−0.683081
$846$	0	0
$847$	0	0
$848$	8.99485	0.308884
$849$	0	0
$850$	4.92820	0.169036
$851$	9.12436	0.312779
$852$	0	0
$853$	19.9808	0.684128	0.342064	−	0.939677i	$-0.388874\pi$
$853$	19.9808	0.684128	0.342064	+	0.939677i	$0.388874\pi$
$854$	0	0
$855$	0	0
$856$	20.7846	0.710403
$857$	−4.87564	−0.166549	−0.0832744	−	0.996527i	$-0.526538\pi$
$857$	−4.87564	−0.166549	−0.0832744	+	0.996527i	$0.526538\pi$
$858$	0	0
$859$	0.535898	0.0182846	0.00914231	−	0.999958i	$-0.497090\pi$
$859$	0.535898	0.0182846	0.00914231	+	0.999958i	$0.497090\pi$
$860$	−10.5359	−0.359271
$861$	0	0
$862$	−12.6795	−0.431865
$863$	6.39230	0.217597	0.108798	−	0.994064i	$-0.465300\pi$
$863$	6.39230	0.217597	0.108798	+	0.994064i	$0.465300\pi$
$864$	0	0
$865$	21.4641	0.729801
$866$	−3.51666	−0.119501
$867$	0	0
$868$	0	0
$869$	36.5885	1.24118
$870$	0	0
$871$	15.2487	0.516683
$872$	27.8949	0.944640
$873$	0	0
$874$	2.28719	0.0773653
$875$	0	0
$876$	0	0
$877$	−31.8564	−1.07571	−0.537857	−	0.843036i	$-0.680766\pi$
$877$	−31.8564	−1.07571	−0.537857	+	0.843036i	$0.680766\pi$
$878$	−5.46410	−0.184404
$879$	0	0
$880$	−2.92820	−0.0987097
$881$	17.8564	0.601598	0.300799	−	0.953688i	$-0.402747\pi$
$881$	17.8564	0.601598	0.300799	+	0.953688i	$0.402747\pi$
$882$	0	0
$883$	22.4115	0.754208	0.377104	−	0.926171i	$-0.376920\pi$
$883$	22.4115	0.754208	0.377104	+	0.926171i	$0.376920\pi$
$884$	56.4974	1.90021
$885$	0	0
$886$	1.85641	0.0623672
$887$	28.7321	0.964728	0.482364	−	0.875971i	$-0.339778\pi$
$887$	28.7321	0.964728	0.482364	+	0.875971i	$0.339778\pi$
$888$	0	0
$889$	0	0
$890$	6.67949	0.223897
$891$	0	0
$892$	−0.574374	−0.0192315
$893$	4.92820	0.164916
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0	0
$898$	−5.96152	−0.198939
$899$	−40.0526	−1.33583
$900$	0	0
$901$	−56.4974	−1.88220
$902$	5.46410	0.181935
$903$	0	0
$904$	12.4974	0.415658
$905$	−10.3205	−0.343065
$906$	0	0
$907$	2.41154	0.0800740	0.0400370	−	0.999198i	$-0.487252\pi$
$907$	2.41154	0.0800740	0.0400370	+	0.999198i	$0.487252\pi$
$908$	22.9282	0.760899
$909$	0	0
$910$	0	0
$911$	11.2679	0.373324	0.186662	−	0.982424i	$-0.440233\pi$
$911$	11.2679	0.373324	0.186662	+	0.982424i	$0.440233\pi$
$912$	0	0
$913$	24.9282	0.825003
$914$	−0.483340	−0.0159874
$915$	0	0
$916$	4.39230	0.145126
$917$	0	0
$918$	0	0
$919$	−3.14359	−0.103698	−0.0518488	−	0.998655i	$-0.516511\pi$
$919$	−3.14359	−0.103698	−0.0518488	+	0.998655i	$0.516511\pi$
$920$	−3.21539	−0.106008
$921$	0	0
$922$	−25.6077	−0.843345
$923$	24.0526	0.791700
$924$	0	0
$925$	7.19615	0.236608
$926$	−16.3013	−0.535693
$927$	0	0
$928$	36.2872	1.19119
$929$	−6.44486	−0.211449	−0.105725	−	0.994395i	$-0.533716\pi$
$929$	−6.44486	−0.211449	−0.105725	+	0.994395i	$0.533716\pi$
$930$	0	0
$931$	0	0
$932$	−25.3590	−0.830661
$933$	0	0
$934$	−20.3923	−0.667257
$935$	18.3923	0.601493
$936$	0	0
$937$	28.2679	0.923474	0.461737	−	0.887017i	$-0.347226\pi$
$937$	28.2679	0.923474	0.461737	+	0.887017i	$0.347226\pi$
$938$	0	0
$939$	0	0
$940$	−2.92820	−0.0955075
$941$	−8.05256	−0.262506	−0.131253	−	0.991349i	$-0.541900\pi$
$941$	−8.05256	−0.262506	−0.131253	+	0.991349i	$0.541900\pi$
$942$	0	0
$943$	−3.46410	−0.112807
$944$	−10.9282	−0.355683
$945$	0	0
$946$	14.3923	0.467934
$947$	−11.6603	−0.378907	−0.189454	−	0.981890i	$-0.560672\pi$
$947$	−11.6603	−0.378907	−0.189454	+	0.981890i	$0.560672\pi$
$948$	0	0
$949$	−26.7128	−0.867135
$950$	1.80385	0.0585245
$951$	0	0
$952$	0	0
$953$	−40.1051	−1.29913	−0.649566	−	0.760305i	$-0.725049\pi$
$953$	−40.1051	−1.29913	−0.649566	+	0.760305i	$0.725049\pi$
$954$	0	0
$955$	4.92820	0.159473
$956$	30.6410	0.991001
$957$	0	0
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	10.7846	0.347891
$962$	−30.1962	−0.973563
$963$	0	0
$964$	−9.56922	−0.308204
$965$	9.19615	0.296035
$966$	0	0
$967$	14.1244	0.454209	0.227104	−	0.973870i	$-0.427074\pi$
$967$	14.1244	0.454209	0.227104	+	0.973870i	$0.427074\pi$
$968$	−8.96668	−0.288200
$969$	0	0
$970$	0.784610	0.0251923
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	0	0
$974$	23.1244	0.740952
$975$	0	0
$976$	4.28719	0.137230
$977$	−14.5885	−0.466726	−0.233363	−	0.972390i	$-0.574973\pi$
$977$	−14.5885	−0.466726	−0.233363	+	0.972390i	$0.574973\pi$
$978$	0	0
$979$	24.9282	0.796709
$980$	0	0
$981$	0	0
$982$	7.50258	0.239417
$983$	−20.1962	−0.644157	−0.322079	−	0.946713i	$-0.604382\pi$
$983$	−20.1962	−0.644157	−0.322079	+	0.946713i	$0.604382\pi$
$984$	0	0
$985$	−17.6603	−0.562702
$986$	−30.5359	−0.972461
$987$	0	0
$988$	20.6795	0.657902
$989$	−9.12436	−0.290138
$990$	0	0
$991$	55.1051	1.75047	0.875236	−	0.483696i	$-0.160706\pi$
$991$	55.1051	1.75047	0.875236	+	0.483696i	$0.160706\pi$
$992$	−37.8564	−1.20194
$993$	0	0
$994$	0	0
$995$	−22.0000	−0.697447
$996$	0	0
$997$	−4.01924	−0.127291	−0.0636453	−	0.997973i	$-0.520273\pi$
$997$	−4.01924	−0.127291	−0.0636453	+	0.997973i	$0.520273\pi$
$998$	14.9808	0.474208
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.z.1.1		2
3.2	odd	2		735.2.a.g.1.2		2
7.2	even	3		315.2.j.c.46.2		4
7.4	even	3		315.2.j.c.226.2		4
7.6	odd	2		2205.2.a.ba.1.1		2
15.14	odd	2		3675.2.a.bg.1.1		2
21.2	odd	6		105.2.i.d.46.1	yes	4
21.5	even	6		735.2.i.l.361.1		4
21.11	odd	6		105.2.i.d.16.1	✓	4
21.17	even	6		735.2.i.l.226.1		4
21.20	even	2		735.2.a.h.1.2		2
84.11	even	6		1680.2.bg.o.961.2		4
84.23	even	6		1680.2.bg.o.1201.2		4
105.2	even	12		525.2.r.f.424.1		4
105.23	even	12		525.2.r.a.424.2		4
105.32	even	12		525.2.r.a.499.2		4
105.44	odd	6		525.2.i.f.151.2		4
105.53	even	12		525.2.r.f.499.1		4
105.74	odd	6		525.2.i.f.226.2		4
105.104	even	2		3675.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.i.d.16.1	✓	4	21.11	odd	6
105.2.i.d.46.1	yes	4	21.2	odd	6
315.2.j.c.46.2		4	7.2	even	3
315.2.j.c.226.2		4	7.4	even	3
525.2.i.f.151.2		4	105.44	odd	6
525.2.i.f.226.2		4	105.74	odd	6
525.2.r.a.424.2		4	105.23	even	12
525.2.r.a.499.2		4	105.32	even	12
525.2.r.f.424.1		4	105.2	even	12
525.2.r.f.499.1		4	105.53	even	12
735.2.a.g.1.2		2	3.2	odd	2
735.2.a.h.1.2		2	21.20	even	2
735.2.i.l.226.1		4	21.17	even	6
735.2.i.l.361.1		4	21.5	even	6
1680.2.bg.o.961.2		4	84.11	even	6
1680.2.bg.o.1201.2		4	84.23	even	6
2205.2.a.z.1.1		2	1.1	even	1	trivial
2205.2.a.ba.1.1		2	7.6	odd	2
3675.2.a.be.1.1		2	105.104	even	2
3675.2.a.bg.1.1		2	15.14	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} +2.53590 q^{8} +O(q^{10})\)	\(q-0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} +2.53590 q^{8} +0.732051 q^{10} +2.73205 q^{11} +5.73205 q^{13} +1.07180 q^{16} -6.73205 q^{17} -2.46410 q^{19} +1.46410 q^{20} -2.00000 q^{22} +1.26795 q^{23} +1.00000 q^{25} -4.19615 q^{26} -6.19615 q^{29} +6.46410 q^{31} -5.85641 q^{32} +4.92820 q^{34} +7.19615 q^{37} +1.80385 q^{38} -2.53590 q^{40} -2.73205 q^{41} -7.19615 q^{43} -4.00000 q^{44} -0.928203 q^{46} -2.00000 q^{47} -0.732051 q^{50} -8.39230 q^{52} +8.39230 q^{53} -2.73205 q^{55} +4.53590 q^{58} -10.1962 q^{59} +4.00000 q^{61} -4.73205 q^{62} +2.14359 q^{64} -5.73205 q^{65} +2.66025 q^{67} +9.85641 q^{68} +4.19615 q^{71} -4.66025 q^{73} -5.26795 q^{74} +3.60770 q^{76} +13.3923 q^{79} -1.07180 q^{80} +2.00000 q^{82} +9.12436 q^{83} +6.73205 q^{85} +5.26795 q^{86} +6.92820 q^{88} +9.12436 q^{89} -1.85641 q^{92} +1.46410 q^{94} +2.46410 q^{95} +1.07180 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 + 12 * q^8	\( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8} - 2 q^{10} + 2 q^{11} + 8 q^{13} + 16 q^{16} - 10 q^{17} + 2 q^{19} - 4 q^{20} - 4 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{29} + 6 q^{31} + 16 q^{32} - 4 q^{34} + 4 q^{37} + 14 q^{38} - 12 q^{40} - 2 q^{41} - 4 q^{43} - 8 q^{44} + 12 q^{46} - 4 q^{47} + 2 q^{50} + 4 q^{52} - 4 q^{53} - 2 q^{55} + 16 q^{58} - 10 q^{59} + 8 q^{61} - 6 q^{62} + 32 q^{64} - 8 q^{65} - 12 q^{67} - 8 q^{68} - 2 q^{71} + 8 q^{73} - 14 q^{74} + 28 q^{76} + 6 q^{79} - 16 q^{80} + 4 q^{82} - 6 q^{83} + 10 q^{85} + 14 q^{86} - 6 q^{89} + 24 q^{92} - 4 q^{94} - 2 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 + 12 * q^8 - 2 * q^10 + 2 * q^11 + 8 * q^13 + 16 * q^16 - 10 * q^17 + 2 * q^19 - 4 * q^20 - 4 * q^22 + 6 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^29 + 6 * q^31 + 16 * q^32 - 4 * q^34 + 4 * q^37 + 14 * q^38 - 12 * q^40 - 2 * q^41 - 4 * q^43 - 8 * q^44 + 12 * q^46 - 4 * q^47 + 2 * q^50 + 4 * q^52 - 4 * q^53 - 2 * q^55 + 16 * q^58 - 10 * q^59 + 8 * q^61 - 6 * q^62 + 32 * q^64 - 8 * q^65 - 12 * q^67 - 8 * q^68 - 2 * q^71 + 8 * q^73 - 14 * q^74 + 28 * q^76 + 6 * q^79 - 16 * q^80 + 4 * q^82 - 6 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^89 + 24 * q^92 - 4 * q^94 - 2 * q^95 + 16 * q^97

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