LMFDB - Embedded newform 2205.2.a.z.1.2

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			1.2
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)

$f(q)$	$=$	$q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} +9.46410 q^{8} +O(q^{10})$	$q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} +9.46410 q^{8} -2.73205 q^{10} -0.732051 q^{11} +2.26795 q^{13} +14.9282 q^{16} -3.26795 q^{17} +4.46410 q^{19} -5.46410 q^{20} -2.00000 q^{22} +4.73205 q^{23} +1.00000 q^{25} +6.19615 q^{26} +4.19615 q^{29} -0.464102 q^{31} +21.8564 q^{32} -8.92820 q^{34} -3.19615 q^{37} +12.1962 q^{38} -9.46410 q^{40} +0.732051 q^{41} +3.19615 q^{43} -4.00000 q^{44} +12.9282 q^{46} -2.00000 q^{47} +2.73205 q^{50} +12.3923 q^{52} -12.3923 q^{53} +0.732051 q^{55} +11.4641 q^{58} +0.196152 q^{59} +4.00000 q^{61} -1.26795 q^{62} +29.8564 q^{64} -2.26795 q^{65} -14.6603 q^{67} -17.8564 q^{68} -6.19615 q^{71} +12.6603 q^{73} -8.73205 q^{74} +24.3923 q^{76} -7.39230 q^{79} -14.9282 q^{80} +2.00000 q^{82} -15.1244 q^{83} +3.26795 q^{85} +8.73205 q^{86} -6.92820 q^{88} -15.1244 q^{89} +25.8564 q^{92} -5.46410 q^{94} -4.46410 q^{95} +14.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.73205	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$2$	2.73205	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$3$	0	0
$3$	0	0
$4$	5.46410	2.73205
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	9.46410	3.34607
$9$	0	0
$10$	−2.73205	−0.863950
$11$	−0.732051	−0.220722	−0.110361	−	0.993892i	$-0.535201\pi$
$11$	−0.732051	−0.220722	−0.110361	+	0.993892i	$0.535201\pi$
$12$	0	0
$13$	2.26795	0.629016	0.314508	−	0.949255i	$-0.398160\pi$
$13$	2.26795	0.629016	0.314508	+	0.949255i	$0.398160\pi$
$14$	0	0
$15$	0	0
$16$	14.9282	3.73205
$17$	−3.26795	−0.792594	−0.396297	−	0.918122i	$-0.629705\pi$
$17$	−3.26795	−0.792594	−0.396297	+	0.918122i	$0.629705\pi$
$18$	0	0
$19$	4.46410	1.02414	0.512068	−	0.858945i	$-0.328880\pi$
$19$	4.46410	1.02414	0.512068	+	0.858945i	$0.328880\pi$
$20$	−5.46410	−1.22181
$21$	0	0
$22$	−2.00000	−0.426401
$23$	4.73205	0.986701	0.493350	−	0.869831i	$-0.335772\pi$
$23$	4.73205	0.986701	0.493350	+	0.869831i	$0.335772\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.19615	1.21517
$27$	0	0
$28$	0	0
$29$	4.19615	0.779206	0.389603	−	0.920983i	$-0.372612\pi$
$29$	4.19615	0.779206	0.389603	+	0.920983i	$0.372612\pi$
$30$	0	0
$31$	−0.464102	−0.0833551	−0.0416776	−	0.999131i	$-0.513270\pi$
$31$	−0.464102	−0.0833551	−0.0416776	+	0.999131i	$0.513270\pi$
$32$	21.8564	3.86370
$33$	0	0
$34$	−8.92820	−1.53117
$35$	0	0
$36$	0	0
$37$	−3.19615	−0.525444	−0.262722	−	0.964872i	$-0.584620\pi$
$37$	−3.19615	−0.525444	−0.262722	+	0.964872i	$0.584620\pi$
$38$	12.1962	1.97848
$39$	0	0
$40$	−9.46410	−1.49641
$41$	0.732051	0.114327	0.0571636	−	0.998365i	$-0.481794\pi$
$41$	0.732051	0.114327	0.0571636	+	0.998365i	$0.481794\pi$
$42$	0	0
$43$	3.19615	0.487409	0.243704	−	0.969850i	$-0.421637\pi$
$43$	3.19615	0.487409	0.243704	+	0.969850i	$0.421637\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	12.9282	1.90616
$47$	−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
$50$	2.73205	0.386370
$51$	0	0
$52$	12.3923	1.71850
$53$	−12.3923	−1.70221	−0.851107	−	0.524992i	$-0.824068\pi$
$53$	−12.3923	−1.70221	−0.851107	+	0.524992i	$0.824068\pi$
$54$	0	0
$55$	0.732051	0.0987097
$56$	0	0
$57$	0	0
$58$	11.4641	1.50531
$59$	0.196152	0.0255369	0.0127684	−	0.999918i	$-0.495936\pi$
$59$	0.196152	0.0255369	0.0127684	+	0.999918i	$0.495936\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−1.26795	−0.161030
$63$	0	0
$64$	29.8564	3.73205
$65$	−2.26795	−0.281304
$66$	0	0
$67$	−14.6603	−1.79104	−0.895518	−	0.445026i	$-0.853194\pi$
$67$	−14.6603	−1.79104	−0.895518	+	0.445026i	$0.853194\pi$
$68$	−17.8564	−2.16541
$69$	0	0
$70$	0	0
$71$	−6.19615	−0.735348	−0.367674	−	0.929955i	$-0.619846\pi$
$71$	−6.19615	−0.735348	−0.367674	+	0.929955i	$0.619846\pi$
$72$	0	0
$73$	12.6603	1.48177	0.740885	−	0.671632i	$-0.234406\pi$
$73$	12.6603	1.48177	0.740885	+	0.671632i	$0.234406\pi$
$74$	−8.73205	−1.01508
$75$	0	0
$76$	24.3923	2.79799
$77$	0	0
$78$	0	0
$79$	−7.39230	−0.831699	−0.415850	−	0.909433i	$-0.636516\pi$
$79$	−7.39230	−0.831699	−0.415850	+	0.909433i	$0.636516\pi$
$80$	−14.9282	−1.66902
$81$	0	0
$82$	2.00000	0.220863
$83$	−15.1244	−1.66011	−0.830057	−	0.557679i	$-0.811692\pi$
$83$	−15.1244	−1.66011	−0.830057	+	0.557679i	$0.811692\pi$
$84$	0	0
$85$	3.26795	0.354459
$86$	8.73205	0.941601
$87$	0	0
$88$	−6.92820	−0.738549
$89$	−15.1244	−1.60318	−0.801589	−	0.597875i	$-0.796012\pi$
$89$	−15.1244	−1.60318	−0.801589	+	0.597875i	$0.796012\pi$
$90$	0	0
$91$	0	0
$92$	25.8564	2.69572
$93$	0	0
$94$	−5.46410	−0.563579
$95$	−4.46410	−0.458007
$96$	0	0
$97$	14.9282	1.51573	0.757865	−	0.652412i	$-0.226243\pi$
$97$	14.9282	1.51573	0.757865	+	0.652412i	$0.226243\pi$
$98$	0	0
$99$	0	0
$100$	5.46410	0.546410
$101$	7.26795	0.723188	0.361594	−	0.932336i	$-0.382233\pi$
$101$	7.26795	0.723188	0.361594	+	0.932336i	$0.382233\pi$
$102$	0	0
$103$	−9.19615	−0.906124	−0.453062	−	0.891479i	$-0.649668\pi$
$103$	−9.19615	−0.906124	−0.453062	+	0.891479i	$0.649668\pi$
$104$	21.4641	2.10473
$105$	0	0
$106$	−33.8564	−3.28842
$107$	−2.19615	−0.212310	−0.106155	−	0.994350i	$-0.533854\pi$
$107$	−2.19615	−0.212310	−0.106155	+	0.994350i	$0.533854\pi$
$108$	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
$113$	−8.92820	−0.839895	−0.419947	−	0.907548i	$-0.637951\pi$
$113$	−8.92820	−0.839895	−0.419947	+	0.907548i	$0.637951\pi$
$114$	0	0
$115$	−4.73205	−0.441266
$116$	22.9282	2.12883
$117$	0	0
$118$	0.535898	0.0493334
$119$	0	0
$120$	0	0
$121$	−10.4641	−0.951282
$122$	10.9282	0.989393
$123$	0	0
$124$	−2.53590	−0.227730
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.80385	0.426273	0.213136	−	0.977022i	$-0.431632\pi$
$127$	4.80385	0.426273	0.213136	+	0.977022i	$0.431632\pi$
$128$	37.8564	3.34607
$129$	0	0
$130$	−6.19615	−0.543439
$131$	−15.4641	−1.35110	−0.675552	−	0.737312i	$-0.736095\pi$
$131$	−15.4641	−1.35110	−0.675552	+	0.737312i	$0.736095\pi$
$132$	0	0
$133$	0	0
$134$	−40.0526	−3.46001
$135$	0	0
$136$	−30.9282	−2.65207
$137$	−2.19615	−0.187630	−0.0938150	−	0.995590i	$-0.529906\pi$
$137$	−2.19615	−0.187630	−0.0938150	+	0.995590i	$0.529906\pi$
$138$	0	0
$139$	5.92820	0.502824	0.251412	−	0.967880i	$-0.419105\pi$
$139$	5.92820	0.502824	0.251412	+	0.967880i	$0.419105\pi$
$140$	0	0
$141$	0	0
$142$	−16.9282	−1.42058
$143$	−1.66025	−0.138837
$144$	0	0
$145$	−4.19615	−0.348471
$146$	34.5885	2.86256
$147$	0	0
$148$	−17.4641	−1.43554
$149$	−5.85641	−0.479776	−0.239888	−	0.970801i	$-0.577111\pi$
$149$	−5.85641	−0.479776	−0.239888	+	0.970801i	$0.577111\pi$
$150$	0	0
$151$	−8.92820	−0.726567	−0.363283	−	0.931679i	$-0.618344\pi$
$151$	−8.92820	−0.726567	−0.363283	+	0.931679i	$0.618344\pi$
$152$	42.2487	3.42682
$153$	0	0
$154$	0	0
$155$	0.464102	0.0372775
$156$	0	0
$157$	6.39230	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$157$	6.39230	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$158$	−20.1962	−1.60672
$159$	0	0
$160$	−21.8564	−1.72790
$161$	0	0
$162$	0	0
$163$	21.8564	1.71193	0.855963	−	0.517037i	$-0.172965\pi$
$163$	21.8564	1.71193	0.855963	+	0.517037i	$0.172965\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−41.3205	−3.20709
$167$	17.6603	1.36659	0.683296	−	0.730142i	$-0.260546\pi$
$167$	17.6603	1.36659	0.683296	+	0.730142i	$0.260546\pi$
$168$	0	0
$169$	−7.85641	−0.604339
$170$	8.92820	0.684762
$171$	0	0
$172$	17.4641	1.33163
$173$	−14.5359	−1.10514	−0.552572	−	0.833465i	$-0.686354\pi$
$173$	−14.5359	−1.10514	−0.552572	+	0.833465i	$0.686354\pi$
$174$	0	0
$175$	0	0
$176$	−10.9282	−0.823744
$177$	0	0
$178$	−41.3205	−3.09710
$179$	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
$181$	−24.3205	−1.80773	−0.903865	−	0.427819i	$-0.859282\pi$
$181$	−24.3205	−1.80773	−0.903865	+	0.427819i	$0.859282\pi$
$182$	0	0
$183$	0	0
$184$	44.7846	3.30157
$185$	3.19615	0.234986
$186$	0	0
$187$	2.39230	0.174943
$188$	−10.9282	−0.797021
$189$	0	0
$190$	−12.1962	−0.884802
$191$	8.92820	0.646022	0.323011	−	0.946395i	$-0.395305\pi$
$191$	8.92820	0.646022	0.323011	+	0.946395i	$0.395305\pi$
$192$	0	0
$193$	1.19615	0.0861009	0.0430505	−	0.999073i	$-0.486292\pi$
$193$	1.19615	0.0861009	0.0430505	+	0.999073i	$0.486292\pi$
$194$	40.7846	2.92816
$195$	0	0
$196$	0	0
$197$	0.339746	0.0242059	0.0121029	−	0.999927i	$-0.496147\pi$
$197$	0.339746	0.0242059	0.0121029	+	0.999927i	$0.496147\pi$
$198$	0	0
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	9.46410	0.669213
$201$	0	0
$202$	19.8564	1.39709
$203$	0	0
$204$	0	0
$205$	−0.732051	−0.0511286
$206$	−25.1244	−1.75050
$207$	0	0
$208$	33.8564	2.34752
$209$	−3.26795	−0.226049
$210$	0	0
$211$	7.07180	0.486843	0.243421	−	0.969921i	$-0.421730\pi$
$211$	7.07180	0.486843	0.243421	+	0.969921i	$0.421730\pi$
$212$	−67.7128	−4.65054
$213$	0	0
$214$	−6.00000	−0.410152
$215$	−3.19615	−0.217976
$216$	0	0
$217$	0	0
$218$	30.0526	2.03542
$219$	0	0
$220$	4.00000	0.269680
$221$	−7.41154	−0.498554
$222$	0	0
$223$	−20.3923	−1.36557	−0.682785	−	0.730619i	$-0.739231\pi$
$223$	−20.3923	−1.36557	−0.682785	+	0.730619i	$0.739231\pi$
$224$	0	0
$225$	0	0
$226$	−24.3923	−1.62255
$227$	1.66025	0.110195	0.0550975	−	0.998481i	$-0.482453\pi$
$227$	1.66025	0.110195	0.0550975	+	0.998481i	$0.482453\pi$
$228$	0	0
$229$	−3.00000	−0.198246	−0.0991228	−	0.995075i	$-0.531604\pi$
$229$	−3.00000	−0.198246	−0.0991228	+	0.995075i	$0.531604\pi$
$230$	−12.9282	−0.852460
$231$	0	0
$232$	39.7128	2.60727
$233$	−17.3205	−1.13470	−0.567352	−	0.823475i	$-0.692032\pi$
$233$	−17.3205	−1.13470	−0.567352	+	0.823475i	$0.692032\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	1.07180	0.0697680
$237$	0	0
$238$	0	0
$239$	−7.07180	−0.457437	−0.228718	−	0.973493i	$-0.573453\pi$
$239$	−7.07180	−0.457437	−0.228718	+	0.973493i	$0.573453\pi$
$240$	0	0
$241$	13.4641	0.867299	0.433650	−	0.901082i	$-0.357226\pi$
$241$	13.4641	0.867299	0.433650	+	0.901082i	$0.357226\pi$
$242$	−28.5885	−1.83774
$243$	0	0
$244$	21.8564	1.39921
$245$	0	0
$246$	0	0
$247$	10.1244	0.644197
$248$	−4.39230	−0.278912
$249$	0	0
$250$	−2.73205	−0.172790
$251$	24.5885	1.55201	0.776005	−	0.630727i	$-0.217243\pi$
$251$	24.5885	1.55201	0.776005	+	0.630727i	$0.217243\pi$
$252$	0	0
$253$	−3.46410	−0.217786
$254$	13.1244	0.823495
$255$	0	0
$256$	43.7128	2.73205
$257$	5.66025	0.353077	0.176538	−	0.984294i	$-0.443510\pi$
$257$	5.66025	0.353077	0.176538	+	0.984294i	$0.443510\pi$
$258$	0	0
$259$	0	0
$260$	−12.3923	−0.768538
$261$	0	0
$262$	−42.2487	−2.61013
$263$	−8.39230	−0.517492	−0.258746	−	0.965945i	$-0.583309\pi$
$263$	−8.39230	−0.517492	−0.258746	+	0.965945i	$0.583309\pi$
$264$	0	0
$265$	12.3923	0.761253
$266$	0	0
$267$	0	0
$268$	−80.1051	−4.89320
$269$	12.5359	0.764327	0.382164	−	0.924095i	$-0.375179\pi$
$269$	12.5359	0.764327	0.382164	+	0.924095i	$0.375179\pi$
$270$	0	0
$271$	3.07180	0.186598	0.0932992	−	0.995638i	$-0.470259\pi$
$271$	3.07180	0.186598	0.0932992	+	0.995638i	$0.470259\pi$
$272$	−48.7846	−2.95800
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−0.732051	−0.0441443
$276$	0	0
$277$	−14.6603	−0.880849	−0.440425	−	0.897790i	$-0.645172\pi$
$277$	−14.6603	−0.880849	−0.440425	+	0.897790i	$0.645172\pi$
$278$	16.1962	0.971381
$279$	0	0
$280$	0	0
$281$	−13.8564	−0.826604	−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564	−0.826604	−0.413302	+	0.910594i	$0.635625\pi$
$282$	0	0
$283$	−24.1244	−1.43404	−0.717022	−	0.697050i	$-0.754495\pi$
$283$	−24.1244	−1.43404	−0.717022	+	0.697050i	$0.754495\pi$
$284$	−33.8564	−2.00901
$285$	0	0
$286$	−4.53590	−0.268213
$287$	0	0
$288$	0	0
$289$	−6.32051	−0.371795
$290$	−11.4641	−0.673195
$291$	0	0
$292$	69.1769	4.04827
$293$	−18.9282	−1.10580	−0.552899	−	0.833248i	$-0.686478\pi$
$293$	−18.9282	−1.10580	−0.552899	+	0.833248i	$0.686478\pi$
$294$	0	0
$295$	−0.196152	−0.0114204
$296$	−30.2487	−1.75817
$297$	0	0
$298$	−16.0000	−0.926855
$299$	10.7321	0.620651
$300$	0	0
$301$	0	0
$302$	−24.3923	−1.40362
$303$	0	0
$304$	66.6410	3.82212
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−32.1244	−1.83343	−0.916717	−	0.399537i	$-0.869171\pi$
$307$	−32.1244	−1.83343	−0.916717	+	0.399537i	$0.869171\pi$
$308$	0	0
$309$	0	0
$310$	1.26795	0.0720147
$311$	−9.12436	−0.517395	−0.258697	−	0.965958i	$-0.583293\pi$
$311$	−9.12436	−0.517395	−0.258697	+	0.965958i	$0.583293\pi$
$312$	0	0
$313$	−12.6603	−0.715600	−0.357800	−	0.933798i	$-0.616473\pi$
$313$	−12.6603	−0.715600	−0.357800	+	0.933798i	$0.616473\pi$
$314$	17.4641	0.985556
$315$	0	0
$316$	−40.3923	−2.27224
$317$	28.4449	1.59762	0.798811	−	0.601582i	$-0.205463\pi$
$317$	28.4449	1.59762	0.798811	+	0.601582i	$0.205463\pi$
$318$	0	0
$319$	−3.07180	−0.171988
$320$	−29.8564	−1.66902
$321$	0	0
$322$	0	0
$323$	−14.5885	−0.811723
$324$	0	0
$325$	2.26795	0.125803
$326$	59.7128	3.30719
$327$	0	0
$328$	6.92820	0.382546
$329$	0	0
$330$	0	0
$331$	8.07180	0.443666	0.221833	−	0.975085i	$-0.428796\pi$
$331$	8.07180	0.443666	0.221833	+	0.975085i	$0.428796\pi$
$332$	−82.6410	−4.53551
$333$	0	0
$334$	48.2487	2.64005
$335$	14.6603	0.800975
$336$	0	0
$337$	17.9808	0.979475	0.489737	−	0.871870i	$-0.337093\pi$
$337$	17.9808	0.979475	0.489737	+	0.871870i	$0.337093\pi$
$338$	−21.4641	−1.16749
$339$	0	0
$340$	17.8564	0.968400
$341$	0.339746	0.0183983
$342$	0	0
$343$	0	0
$344$	30.2487	1.63090
$345$	0	0
$346$	−39.7128	−2.13497
$347$	21.0718	1.13119	0.565597	−	0.824682i	$-0.308646\pi$
$347$	21.0718	1.13119	0.565597	+	0.824682i	$0.308646\pi$
$348$	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
$353$	3.12436	0.166293	0.0831463	−	0.996537i	$-0.473503\pi$
$353$	3.12436	0.166293	0.0831463	+	0.996537i	$0.473503\pi$
$354$	0	0
$355$	6.19615	0.328858
$356$	−82.6410	−4.37997
$357$	0	0
$358$	27.3205	1.44393
$359$	1.26795	0.0669198	0.0334599	−	0.999440i	$-0.489347\pi$
$359$	1.26795	0.0669198	0.0334599	+	0.999440i	$0.489347\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	−66.4449	−3.49226
$363$	0	0
$364$	0	0
$365$	−12.6603	−0.662668
$366$	0	0
$367$	11.1962	0.584434	0.292217	−	0.956352i	$-0.405607\pi$
$367$	11.1962	0.584434	0.292217	+	0.956352i	$0.405607\pi$
$368$	70.6410	3.68242
$369$	0	0
$370$	8.73205	0.453958
$371$	0	0
$372$	0	0
$373$	26.5167	1.37298	0.686490	−	0.727139i	$-0.259150\pi$
$373$	26.5167	1.37298	0.686490	+	0.727139i	$0.259150\pi$
$374$	6.53590	0.337963
$375$	0	0
$376$	−18.9282	−0.976148
$377$	9.51666	0.490133
$378$	0	0
$379$	6.32051	0.324663	0.162331	−	0.986736i	$-0.448099\pi$
$379$	6.32051	0.324663	0.162331	+	0.986736i	$0.448099\pi$
$380$	−24.3923	−1.25130
$381$	0	0
$382$	24.3923	1.24802
$383$	23.3205	1.19162	0.595811	−	0.803125i	$-0.296831\pi$
$383$	23.3205	1.19162	0.595811	+	0.803125i	$0.296831\pi$
$384$	0	0
$385$	0	0
$386$	3.26795	0.166334
$387$	0	0
$388$	81.5692	4.14105
$389$	−5.41154	−0.274376	−0.137188	−	0.990545i	$-0.543806\pi$
$389$	−5.41154	−0.274376	−0.137188	+	0.990545i	$0.543806\pi$
$390$	0	0
$391$	−15.4641	−0.782053
$392$	0	0
$393$	0	0
$394$	0.928203	0.0467622
$395$	7.39230	0.371947
$396$	0	0
$397$	−31.1962	−1.56569	−0.782845	−	0.622217i	$-0.786232\pi$
$397$	−31.1962	−1.56569	−0.782845	+	0.622217i	$0.786232\pi$
$398$	60.1051	3.01280
$399$	0	0
$400$	14.9282	0.746410
$401$	16.3923	0.818593	0.409296	−	0.912402i	$-0.365774\pi$
$401$	16.3923	0.818593	0.409296	+	0.912402i	$0.365774\pi$
$402$	0	0
$403$	−1.05256	−0.0524317
$404$	39.7128	1.97579
$405$	0	0
$406$	0	0
$407$	2.33975	0.115977
$408$	0	0
$409$	3.14359	0.155441	0.0777203	−	0.996975i	$-0.475236\pi$
$409$	3.14359	0.155441	0.0777203	+	0.996975i	$0.475236\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−50.2487	−2.47558
$413$	0	0
$414$	0	0
$415$	15.1244	0.742425
$416$	49.5692	2.43033
$417$	0	0
$418$	−8.92820	−0.436693
$419$	35.4641	1.73253	0.866267	−	0.499581i	$-0.166513\pi$
$419$	35.4641	1.73253	0.866267	+	0.499581i	$0.166513\pi$
$420$	0	0
$421$	0.0717968	0.00349916	0.00174958	−	0.999998i	$-0.499443\pi$
$421$	0.0717968	0.00349916	0.00174958	+	0.999998i	$0.499443\pi$
$422$	19.3205	0.940508
$423$	0	0
$424$	−117.282	−5.69572
$425$	−3.26795	−0.158519
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−8.73205	−0.421097
$431$	−17.3205	−0.834300	−0.417150	−	0.908838i	$-0.636971\pi$
$431$	−17.3205	−0.834300	−0.417150	+	0.908838i	$0.636971\pi$
$432$	0	0
$433$	15.1962	0.730280	0.365140	−	0.930953i	$-0.381021\pi$
$433$	15.1962	0.730280	0.365140	+	0.930953i	$0.381021\pi$
$434$	0	0
$435$	0	0
$436$	60.1051	2.87851
$437$	21.1244	1.01051
$438$	0	0
$439$	0.535898	0.0255770	0.0127885	−	0.999918i	$-0.495929\pi$
$439$	0.535898	0.0255770	0.0127885	+	0.999918i	$0.495929\pi$
$440$	6.92820	0.330289
$441$	0	0
$442$	−20.2487	−0.963133
$443$	−9.46410	−0.449653	−0.224827	−	0.974399i	$-0.572182\pi$
$443$	−9.46410	−0.449653	−0.224827	+	0.974399i	$0.572182\pi$
$444$	0	0
$445$	15.1244	0.716963
$446$	−55.7128	−2.63808
$447$	0	0
$448$	0	0
$449$	35.8564	1.69217	0.846084	−	0.533049i	$-0.178954\pi$
$449$	35.8564	1.69217	0.846084	+	0.533049i	$0.178954\pi$
$450$	0	0
$451$	−0.535898	−0.0252345
$452$	−48.7846	−2.29464
$453$	0	0
$454$	4.53590	0.212880
$455$	0	0
$456$	0	0
$457$	−16.6603	−0.779334	−0.389667	−	0.920956i	$-0.627410\pi$
$457$	−16.6603	−0.779334	−0.389667	+	0.920956i	$0.627410\pi$
$458$	−8.19615	−0.382981
$459$	0	0
$460$	−25.8564	−1.20556
$461$	−16.9808	−0.790873	−0.395436	−	0.918493i	$-0.629407\pi$
$461$	−16.9808	−0.790873	−0.395436	+	0.918493i	$0.629407\pi$
$462$	0	0
$463$	25.7321	1.19587	0.597935	−	0.801545i	$-0.295988\pi$
$463$	25.7321	1.19587	0.597935	+	0.801545i	$0.295988\pi$
$464$	62.6410	2.90804
$465$	0	0
$466$	−47.3205	−2.19208
$467$	0.143594	0.00664472	0.00332236	−	0.999994i	$-0.498942\pi$
$467$	0.143594	0.00664472	0.00332236	+	0.999994i	$0.498942\pi$
$468$	0	0
$469$	0	0
$470$	5.46410	0.252040
$471$	0	0
$472$	1.85641	0.0854480
$473$	−2.33975	−0.107582
$474$	0	0
$475$	4.46410	0.204827
$476$	0	0
$477$	0	0
$478$	−19.3205	−0.883699
$479$	−8.78461	−0.401379	−0.200690	−	0.979655i	$-0.564318\pi$
$479$	−8.78461	−0.401379	−0.200690	+	0.979655i	$0.564318\pi$
$480$	0	0
$481$	−7.24871	−0.330513
$482$	36.7846	1.67549
$483$	0	0
$484$	−57.1769	−2.59895
$485$	−14.9282	−0.677855
$486$	0	0
$487$	−0.411543	−0.0186488	−0.00932439	−	0.999957i	$-0.502968\pi$
$487$	−0.411543	−0.0186488	−0.00932439	+	0.999957i	$0.502968\pi$
$488$	37.8564	1.71368
$489$	0	0
$490$	0	0
$491$	38.2487	1.72614	0.863070	−	0.505084i	$-0.168539\pi$
$491$	38.2487	1.72614	0.863070	+	0.505084i	$0.168539\pi$
$492$	0	0
$493$	−13.7128	−0.617594
$494$	27.6603	1.24449
$495$	0	0
$496$	−6.92820	−0.311086
$497$	0	0
$498$	0	0
$499$	−13.5359	−0.605950	−0.302975	−	0.952998i	$-0.597980\pi$
$499$	−13.5359	−0.605950	−0.302975	+	0.952998i	$0.597980\pi$
$500$	−5.46410	−0.244362
$501$	0	0
$502$	67.1769	2.99825
$503$	−14.3923	−0.641721	−0.320861	−	0.947126i	$-0.603972\pi$
$503$	−14.3923	−0.641721	−0.320861	+	0.947126i	$0.603972\pi$
$504$	0	0
$505$	−7.26795	−0.323419
$506$	−9.46410	−0.420731
$507$	0	0
$508$	26.2487	1.16460
$509$	−4.53590	−0.201050	−0.100525	−	0.994935i	$-0.532052\pi$
$509$	−4.53590	−0.201050	−0.100525	+	0.994935i	$0.532052\pi$
$510$	0	0
$511$	0	0
$512$	43.7128	1.93185
$513$	0	0
$514$	15.4641	0.682092
$515$	9.19615	0.405231
$516$	0	0
$517$	1.46410	0.0643911
$518$	0	0
$519$	0	0
$520$	−21.4641	−0.941263
$521$	5.46410	0.239387	0.119693	−	0.992811i	$-0.461809\pi$
$521$	5.46410	0.239387	0.119693	+	0.992811i	$0.461809\pi$
$522$	0	0
$523$	−27.7321	−1.21264	−0.606319	−	0.795222i	$-0.707355\pi$
$523$	−27.7321	−1.21264	−0.606319	+	0.795222i	$0.707355\pi$
$524$	−84.4974	−3.69129
$525$	0	0
$526$	−22.9282	−0.999717
$527$	1.51666	0.0660668
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	33.8564	1.47063
$531$	0	0
$532$	0	0
$533$	1.66025	0.0719136
$534$	0	0
$535$	2.19615	0.0949479
$536$	−138.746	−5.99292
$537$	0	0
$538$	34.2487	1.47657
$539$	0	0
$540$	0	0
$541$	5.78461	0.248700	0.124350	−	0.992238i	$-0.460315\pi$
$541$	5.78461	0.248700	0.124350	+	0.992238i	$0.460315\pi$
$542$	8.39230	0.360480
$543$	0	0
$544$	−71.4256	−3.06235
$545$	−11.0000	−0.471188
$546$	0	0
$547$	−26.2487	−1.12231	−0.561157	−	0.827709i	$-0.689644\pi$
$547$	−26.2487	−1.12231	−0.561157	+	0.827709i	$0.689644\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	18.7321	0.798012
$552$	0	0
$553$	0	0
$554$	−40.0526	−1.70167
$555$	0	0
$556$	32.3923	1.37374
$557$	−14.7846	−0.626444	−0.313222	−	0.949680i	$-0.601408\pi$
$557$	−14.7846	−0.626444	−0.313222	+	0.949680i	$0.601408\pi$
$558$	0	0
$559$	7.24871	0.306588
$560$	0	0
$561$	0	0
$562$	−37.8564	−1.59688
$563$	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
$565$	8.92820	0.375612
$566$	−65.9090	−2.77036
$567$	0	0
$568$	−58.6410	−2.46052
$569$	−32.4449	−1.36016	−0.680080	−	0.733138i	$-0.738055\pi$
$569$	−32.4449	−1.36016	−0.680080	+	0.733138i	$0.738055\pi$
$570$	0	0
$571$	18.6077	0.778708	0.389354	−	0.921088i	$-0.372698\pi$
$571$	18.6077	0.778708	0.389354	+	0.921088i	$0.372698\pi$
$572$	−9.07180	−0.379311
$573$	0	0
$574$	0	0
$575$	4.73205	0.197340
$576$	0	0
$577$	28.6603	1.19314	0.596571	−	0.802560i	$-0.296529\pi$
$577$	28.6603	1.19314	0.596571	+	0.802560i	$0.296529\pi$
$578$	−17.2679	−0.718252
$579$	0	0
$580$	−22.9282	−0.952042
$581$	0	0
$582$	0	0
$583$	9.07180	0.375715
$584$	119.818	4.95810
$585$	0	0
$586$	−51.7128	−2.13624
$587$	−40.7321	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$587$	−40.7321	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$588$	0	0
$589$	−2.07180	−0.0853669
$590$	−0.535898	−0.0220626
$591$	0	0
$592$	−47.7128	−1.96098
$593$	−27.9090	−1.14608	−0.573042	−	0.819526i	$-0.694237\pi$
$593$	−27.9090	−1.14608	−0.573042	+	0.819526i	$0.694237\pi$
$594$	0	0
$595$	0	0
$596$	−32.0000	−1.31077
$597$	0	0
$598$	29.3205	1.19900
$599$	38.2487	1.56280	0.781400	−	0.624030i	$-0.214506\pi$
$599$	38.2487	1.56280	0.781400	+	0.624030i	$0.214506\pi$
$600$	0	0
$601$	−0.0717968	−0.00292865	−0.00146433	−	0.999999i	$-0.500466\pi$
$601$	−0.0717968	−0.00292865	−0.00146433	+	0.999999i	$0.500466\pi$
$602$	0	0
$603$	0	0
$604$	−48.7846	−1.98502
$605$	10.4641	0.425426
$606$	0	0
$607$	3.19615	0.129728	0.0648639	−	0.997894i	$-0.479339\pi$
$607$	3.19615	0.129728	0.0648639	+	0.997894i	$0.479339\pi$
$608$	97.5692	3.95695
$609$	0	0
$610$	−10.9282	−0.442470
$611$	−4.53590	−0.183503
$612$	0	0
$613$	−26.9282	−1.08762	−0.543810	−	0.839208i	$-0.683019\pi$
$613$	−26.9282	−1.08762	−0.543810	+	0.839208i	$0.683019\pi$
$614$	−87.7654	−3.54192
$615$	0	0
$616$	0	0
$617$	36.2487	1.45932	0.729659	−	0.683811i	$-0.239679\pi$
$617$	36.2487	1.45932	0.729659	+	0.683811i	$0.239679\pi$
$618$	0	0
$619$	−30.0718	−1.20869	−0.604344	−	0.796724i	$-0.706565\pi$
$619$	−30.0718	−1.20869	−0.604344	+	0.796724i	$0.706565\pi$
$620$	2.53590	0.101844
$621$	0	0
$622$	−24.9282	−0.999530
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−34.5885	−1.38243
$627$	0	0
$628$	34.9282	1.39379
$629$	10.4449	0.416464
$630$	0	0
$631$	48.7846	1.94208	0.971042	−	0.238908i	$-0.0767893\pi$
$631$	48.7846	1.94208	0.971042	+	0.238908i	$0.0767893\pi$
$632$	−69.9615	−2.78292
$633$	0	0
$634$	77.7128	3.08637
$635$	−4.80385	−0.190635
$636$	0	0
$637$	0	0
$638$	−8.39230	−0.332255
$639$	0	0
$640$	−37.8564	−1.49641
$641$	−3.80385	−0.150243	−0.0751215	−	0.997174i	$-0.523934\pi$
$641$	−3.80385	−0.150243	−0.0751215	+	0.997174i	$0.523934\pi$
$642$	0	0
$643$	4.51666	0.178120	0.0890599	−	0.996026i	$-0.471614\pi$
$643$	4.51666	0.178120	0.0890599	+	0.996026i	$0.471614\pi$
$644$	0	0
$645$	0	0
$646$	−39.8564	−1.56813
$647$	27.9090	1.09721	0.548607	−	0.836080i	$-0.315158\pi$
$647$	27.9090	1.09721	0.548607	+	0.836080i	$0.315158\pi$
$648$	0	0
$649$	−0.143594	−0.00563654
$650$	6.19615	0.243033
$651$	0	0
$652$	119.426	4.67707
$653$	44.5885	1.74488	0.872441	−	0.488720i	$-0.162536\pi$
$653$	44.5885	1.74488	0.872441	+	0.488720i	$0.162536\pi$
$654$	0	0
$655$	15.4641	0.604232
$656$	10.9282	0.426675
$657$	0	0
$658$	0	0
$659$	−2.92820	−0.114067	−0.0570333	−	0.998372i	$-0.518164\pi$
$659$	−2.92820	−0.114067	−0.0570333	+	0.998372i	$0.518164\pi$
$660$	0	0
$661$	10.4641	0.407006	0.203503	−	0.979074i	$-0.434767\pi$
$661$	10.4641	0.407006	0.203503	+	0.979074i	$0.434767\pi$
$662$	22.0526	0.857097
$663$	0	0
$664$	−143.138	−5.55485
$665$	0	0
$666$	0	0
$667$	19.8564	0.768843
$668$	96.4974	3.73360
$669$	0	0
$670$	40.0526	1.54737
$671$	−2.92820	−0.113042
$672$	0	0
$673$	−27.3397	−1.05387	−0.526935	−	0.849906i	$-0.676659\pi$
$673$	−27.3397	−1.05387	−0.526935	+	0.849906i	$0.676659\pi$
$674$	49.1244	1.89220
$675$	0	0
$676$	−42.9282	−1.65108
$677$	33.1244	1.27307	0.636536	−	0.771247i	$-0.280367\pi$
$677$	33.1244	1.27307	0.636536	+	0.771247i	$0.280367\pi$
$678$	0	0
$679$	0	0
$680$	30.9282	1.18604
$681$	0	0
$682$	0.928203	0.0355427
$683$	28.0526	1.07340	0.536701	−	0.843773i	$-0.319670\pi$
$683$	28.0526	1.07340	0.536701	+	0.843773i	$0.319670\pi$
$684$	0	0
$685$	2.19615	0.0839107
$686$	0	0
$687$	0	0
$688$	47.7128	1.81903
$689$	−28.1051	−1.07072
$690$	0	0
$691$	8.85641	0.336914	0.168457	−	0.985709i	$-0.446122\pi$
$691$	8.85641	0.336914	0.168457	+	0.985709i	$0.446122\pi$
$692$	−79.4256	−3.01931
$693$	0	0
$694$	57.5692	2.18530
$695$	−5.92820	−0.224870
$696$	0	0
$697$	−2.39230	−0.0906150
$698$	60.1051	2.27501
$699$	0	0
$700$	0	0
$701$	8.58846	0.324382	0.162191	−	0.986759i	$-0.448144\pi$
$701$	8.58846	0.324382	0.162191	+	0.986759i	$0.448144\pi$
$702$	0	0
$703$	−14.2679	−0.538126
$704$	−21.8564	−0.823744
$705$	0	0
$706$	8.53590	0.321253
$707$	0	0
$708$	0	0
$709$	−1.07180	−0.0402522	−0.0201261	−	0.999797i	$-0.506407\pi$
$709$	−1.07180	−0.0402522	−0.0201261	+	0.999797i	$0.506407\pi$
$710$	16.9282	0.635304
$711$	0	0
$712$	−143.138	−5.36434
$713$	−2.19615	−0.0822466
$714$	0	0
$715$	1.66025	0.0620900
$716$	54.6410	2.04203
$717$	0	0
$718$	3.46410	0.129279
$719$	20.5359	0.765860	0.382930	−	0.923777i	$-0.374915\pi$
$719$	20.5359	0.765860	0.382930	+	0.923777i	$0.374915\pi$
$720$	0	0
$721$	0	0
$722$	2.53590	0.0943764
$723$	0	0
$724$	−132.890	−4.93881
$725$	4.19615	0.155841
$726$	0	0
$727$	−13.3397	−0.494744	−0.247372	−	0.968921i	$-0.579567\pi$
$727$	−13.3397	−0.494744	−0.247372	+	0.968921i	$0.579567\pi$
$728$	0	0
$729$	0	0
$730$	−34.5885	−1.28018
$731$	−10.4449	−0.386317
$732$	0	0
$733$	−1.33975	−0.0494846	−0.0247423	−	0.999694i	$-0.507877\pi$
$733$	−1.33975	−0.0494846	−0.0247423	+	0.999694i	$0.507877\pi$
$734$	30.5885	1.12904
$735$	0	0
$736$	103.426	3.81232
$737$	10.7321	0.395320
$738$	0	0
$739$	27.7846	1.02207	0.511037	−	0.859559i	$-0.329262\pi$
$739$	27.7846	1.02207	0.511037	+	0.859559i	$0.329262\pi$
$740$	17.4641	0.641993
$741$	0	0
$742$	0	0
$743$	15.9090	0.583643	0.291822	−	0.956473i	$-0.405739\pi$
$743$	15.9090	0.583643	0.291822	+	0.956473i	$0.405739\pi$
$744$	0	0
$745$	5.85641	0.214562
$746$	72.4449	2.65239
$747$	0	0
$748$	13.0718	0.477952
$749$	0	0
$750$	0	0
$751$	18.0718	0.659449	0.329725	−	0.944077i	$-0.393044\pi$
$751$	18.0718	0.659449	0.329725	+	0.944077i	$0.393044\pi$
$752$	−29.8564	−1.08875
$753$	0	0
$754$	26.0000	0.946864
$755$	8.92820	0.324931
$756$	0	0
$757$	−27.8564	−1.01246	−0.506229	−	0.862399i	$-0.668961\pi$
$757$	−27.8564	−1.01246	−0.506229	+	0.862399i	$0.668961\pi$
$758$	17.2679	0.627200
$759$	0	0
$760$	−42.2487	−1.53252
$761$	−46.7321	−1.69404	−0.847018	−	0.531565i	$-0.821604\pi$
$761$	−46.7321	−1.69404	−0.847018	+	0.531565i	$0.821604\pi$
$762$	0	0
$763$	0	0
$764$	48.7846	1.76497
$765$	0	0
$766$	63.7128	2.30204
$767$	0.444864	0.0160631
$768$	0	0
$769$	52.3205	1.88673	0.943363	−	0.331763i	$-0.107643\pi$
$769$	52.3205	1.88673	0.943363	+	0.331763i	$0.107643\pi$
$770$	0	0
$771$	0	0
$772$	6.53590	0.235232
$773$	−43.5167	−1.56519	−0.782593	−	0.622534i	$-0.786103\pi$
$773$	−43.5167	−1.56519	−0.782593	+	0.622534i	$0.786103\pi$
$774$	0	0
$775$	−0.464102	−0.0166710
$776$	141.282	5.07173
$777$	0	0
$778$	−14.7846	−0.530054
$779$	3.26795	0.117086
$780$	0	0
$781$	4.53590	0.162307
$782$	−42.2487	−1.51081
$783$	0	0
$784$	0	0
$785$	−6.39230	−0.228151
$786$	0	0
$787$	13.4641	0.479943	0.239972	−	0.970780i	$-0.422862\pi$
$787$	13.4641	0.479943	0.239972	+	0.970780i	$0.422862\pi$
$788$	1.85641	0.0661317
$789$	0	0
$790$	20.1962	0.718547
$791$	0	0
$792$	0	0
$793$	9.07180	0.322149
$794$	−85.2295	−3.02468
$795$	0	0
$796$	120.210	4.26074
$797$	3.94744	0.139826	0.0699128	−	0.997553i	$-0.477728\pi$
$797$	3.94744	0.139826	0.0699128	+	0.997553i	$0.477728\pi$
$798$	0	0
$799$	6.53590	0.231223
$800$	21.8564	0.772741
$801$	0	0
$802$	44.7846	1.58140
$803$	−9.26795	−0.327059
$804$	0	0
$805$	0	0
$806$	−2.87564	−0.101290
$807$	0	0
$808$	68.7846	2.41983
$809$	−25.7128	−0.904014	−0.452007	−	0.892014i	$-0.649292\pi$
$809$	−25.7128	−0.904014	−0.452007	+	0.892014i	$0.649292\pi$
$810$	0	0
$811$	−3.46410	−0.121641	−0.0608205	−	0.998149i	$-0.519372\pi$
$811$	−3.46410	−0.121641	−0.0608205	+	0.998149i	$0.519372\pi$
$812$	0	0
$813$	0	0
$814$	6.39230	0.224050
$815$	−21.8564	−0.765597
$816$	0	0
$817$	14.2679	0.499172
$818$	8.58846	0.300288
$819$	0	0
$820$	−4.00000	−0.139686
$821$	25.5167	0.890538	0.445269	−	0.895397i	$-0.353108\pi$
$821$	25.5167	0.890538	0.445269	+	0.895397i	$0.353108\pi$
$822$	0	0
$823$	39.1769	1.36562	0.682811	−	0.730595i	$-0.260757\pi$
$823$	39.1769	1.36562	0.682811	+	0.730595i	$0.260757\pi$
$824$	−87.0333	−3.03195
$825$	0	0
$826$	0	0
$827$	3.75129	0.130445	0.0652225	−	0.997871i	$-0.479224\pi$
$827$	3.75129	0.130445	0.0652225	+	0.997871i	$0.479224\pi$
$828$	0	0
$829$	−4.60770	−0.160032	−0.0800159	−	0.996794i	$-0.525497\pi$
$829$	−4.60770	−0.160032	−0.0800159	+	0.996794i	$0.525497\pi$
$830$	41.3205	1.43426
$831$	0	0
$832$	67.7128	2.34752
$833$	0	0
$834$	0	0
$835$	−17.6603	−0.611158
$836$	−17.8564	−0.617577
$837$	0	0
$838$	96.8897	3.34700
$839$	−18.4449	−0.636787	−0.318394	−	0.947959i	$-0.603143\pi$
$839$	−18.4449	−0.636787	−0.318394	+	0.947959i	$0.603143\pi$
$840$	0	0
$841$	−11.3923	−0.392838
$842$	0.196152	0.00675986
$843$	0	0
$844$	38.6410	1.33008
$845$	7.85641	0.270269
$846$	0	0
$847$	0	0
$848$	−184.995	−6.35275
$849$	0	0
$850$	−8.92820	−0.306235
$851$	−15.1244	−0.518456
$852$	0	0
$853$	−31.9808	−1.09500	−0.547500	−	0.836806i	$-0.684420\pi$
$853$	−31.9808	−1.09500	−0.547500	+	0.836806i	$0.684420\pi$
$854$	0	0
$855$	0	0
$856$	−20.7846	−0.710403
$857$	−29.1244	−0.994869	−0.497435	−	0.867502i	$-0.665725\pi$
$857$	−29.1244	−0.994869	−0.497435	+	0.867502i	$0.665725\pi$
$858$	0	0
$859$	7.46410	0.254672	0.127336	−	0.991860i	$-0.459357\pi$
$859$	7.46410	0.254672	0.127336	+	0.991860i	$0.459357\pi$
$860$	−17.4641	−0.595521
$861$	0	0
$862$	−47.3205	−1.61174
$863$	−14.3923	−0.489920	−0.244960	−	0.969533i	$-0.578775\pi$
$863$	−14.3923	−0.489920	−0.244960	+	0.969533i	$0.578775\pi$
$864$	0	0
$865$	14.5359	0.494235
$866$	41.5167	1.41079
$867$	0	0
$868$	0	0
$869$	5.41154	0.183574
$870$	0	0
$871$	−33.2487	−1.12659
$872$	104.105	3.52544
$873$	0	0
$874$	57.7128	1.95217
$875$	0	0
$876$	0	0
$877$	−4.14359	−0.139919	−0.0699596	−	0.997550i	$-0.522287\pi$
$877$	−4.14359	−0.139919	−0.0699596	+	0.997550i	$0.522287\pi$
$878$	1.46410	0.0494110
$879$	0	0
$880$	10.9282	0.368390
$881$	−9.85641	−0.332071	−0.166035	−	0.986120i	$-0.553097\pi$
$881$	−9.85641	−0.332071	−0.166035	+	0.986120i	$0.553097\pi$
$882$	0	0
$883$	53.5885	1.80340	0.901698	−	0.432367i	$-0.142322\pi$
$883$	53.5885	1.80340	0.901698	+	0.432367i	$0.142322\pi$
$884$	−40.4974	−1.36208
$885$	0	0
$886$	−25.8564	−0.868663
$887$	25.2679	0.848415	0.424207	−	0.905565i	$-0.360553\pi$
$887$	25.2679	0.848415	0.424207	+	0.905565i	$0.360553\pi$
$888$	0	0
$889$	0	0
$890$	41.3205	1.38507
$891$	0	0
$892$	−111.426	−3.73081
$893$	−8.92820	−0.298771
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0	0
$898$	97.9615	3.26902
$899$	−1.94744	−0.0649508
$900$	0	0
$901$	40.4974	1.34916
$902$	−1.46410	−0.0487493
$903$	0	0
$904$	−84.4974	−2.81034
$905$	24.3205	0.808441
$906$	0	0
$907$	33.5885	1.11529	0.557643	−	0.830081i	$-0.311706\pi$
$907$	33.5885	1.11529	0.557643	+	0.830081i	$0.311706\pi$
$908$	9.07180	0.301058
$909$	0	0
$910$	0	0
$911$	14.7321	0.488095	0.244047	−	0.969763i	$-0.421525\pi$
$911$	14.7321	0.488095	0.244047	+	0.969763i	$0.421525\pi$
$912$	0	0
$913$	11.0718	0.366423
$914$	−45.5167	−1.50556
$915$	0	0
$916$	−16.3923	−0.541617
$917$	0	0
$918$	0	0
$919$	−30.8564	−1.01786	−0.508929	−	0.860808i	$-0.669959\pi$
$919$	−30.8564	−1.01786	−0.508929	+	0.860808i	$0.669959\pi$
$920$	−44.7846	−1.47650
$921$	0	0
$922$	−46.3923	−1.52785
$923$	−14.0526	−0.462546
$924$	0	0
$925$	−3.19615	−0.105089
$926$	70.3013	2.31024
$927$	0	0
$928$	91.7128	3.01062
$929$	52.4449	1.72066	0.860330	−	0.509737i	$-0.170257\pi$
$929$	52.4449	1.72066	0.860330	+	0.509737i	$0.170257\pi$
$930$	0	0
$931$	0	0
$932$	−94.6410	−3.10007
$933$	0	0
$934$	0.392305	0.0128366
$935$	−2.39230	−0.0782367
$936$	0	0
$937$	31.7321	1.03664	0.518320	−	0.855186i	$-0.326557\pi$
$937$	31.7321	1.03664	0.518320	+	0.855186i	$0.326557\pi$
$938$	0	0
$939$	0	0
$940$	10.9282	0.356439
$941$	30.0526	0.979685	0.489843	−	0.871811i	$-0.337054\pi$
$941$	30.0526	0.979685	0.489843	+	0.871811i	$0.337054\pi$
$942$	0	0
$943$	3.46410	0.112807
$944$	2.92820	0.0953049
$945$	0	0
$946$	−6.39230	−0.207832
$947$	5.66025	0.183934	0.0919668	−	0.995762i	$-0.470685\pi$
$947$	5.66025	0.183934	0.0919668	+	0.995762i	$0.470685\pi$
$948$	0	0
$949$	28.7128	0.932057
$950$	12.1962	0.395695
$951$	0	0
$952$	0	0
$953$	36.1051	1.16956	0.584780	−	0.811192i	$-0.301181\pi$
$953$	36.1051	1.16956	0.584780	+	0.811192i	$0.301181\pi$
$954$	0	0
$955$	−8.92820	−0.288910
$956$	−38.6410	−1.24974
$957$	0	0
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	−30.7846	−0.993052
$962$	−19.8038	−0.638502
$963$	0	0
$964$	73.5692	2.36951
$965$	−1.19615	−0.0385055
$966$	0	0
$967$	−10.1244	−0.325577	−0.162789	−	0.986661i	$-0.552049\pi$
$967$	−10.1244	−0.325577	−0.162789	+	0.986661i	$0.552049\pi$
$968$	−99.0333	−3.18305
$969$	0	0
$970$	−40.7846	−1.30951
$971$	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
$974$	−1.12436	−0.0360267
$975$	0	0
$976$	59.7128	1.91136
$977$	16.5885	0.530712	0.265356	−	0.964151i	$-0.414511\pi$
$977$	16.5885	0.530712	0.265356	+	0.964151i	$0.414511\pi$
$978$	0	0
$979$	11.0718	0.353856
$980$	0	0
$981$	0	0
$982$	104.497	3.33465
$983$	−9.80385	−0.312694	−0.156347	−	0.987702i	$-0.549972\pi$
$983$	−9.80385	−0.312694	−0.156347	+	0.987702i	$0.549972\pi$
$984$	0	0
$985$	−0.339746	−0.0108252
$986$	−37.4641	−1.19310
$987$	0	0
$988$	55.3205	1.75998
$989$	15.1244	0.480927
$990$	0	0
$991$	−21.1051	−0.670426	−0.335213	−	0.942142i	$-0.608808\pi$
$991$	−21.1051	−0.670426	−0.335213	+	0.942142i	$0.608808\pi$
$992$	−10.1436	−0.322059
$993$	0	0
$994$	0	0
$995$	−22.0000	−0.697447
$996$	0	0
$997$	−55.9808	−1.77293	−0.886464	−	0.462797i	$-0.846846\pi$
$997$	−55.9808	−1.77293	−0.886464	+	0.462797i	$0.846846\pi$
$998$	−36.9808	−1.17061
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.z.1.2		2
3.2	odd	2		735.2.a.g.1.1		2
7.2	even	3		315.2.j.c.46.1		4
7.4	even	3		315.2.j.c.226.1		4
7.6	odd	2		2205.2.a.ba.1.2		2
15.14	odd	2		3675.2.a.bg.1.2		2
21.2	odd	6		105.2.i.d.46.2	yes	4
21.5	even	6		735.2.i.l.361.2		4
21.11	odd	6		105.2.i.d.16.2	✓	4
21.17	even	6		735.2.i.l.226.2		4
21.20	even	2		735.2.a.h.1.1		2
84.11	even	6		1680.2.bg.o.961.1		4
84.23	even	6		1680.2.bg.o.1201.1		4
105.2	even	12		525.2.r.a.424.1		4
105.23	even	12		525.2.r.f.424.2		4
105.32	even	12		525.2.r.f.499.2		4
105.44	odd	6		525.2.i.f.151.1		4
105.53	even	12		525.2.r.a.499.1		4
105.74	odd	6		525.2.i.f.226.1		4
105.104	even	2		3675.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.i.d.16.2	✓	4	21.11	odd	6
105.2.i.d.46.2	yes	4	21.2	odd	6
315.2.j.c.46.1		4	7.2	even	3
315.2.j.c.226.1		4	7.4	even	3
525.2.i.f.151.1		4	105.44	odd	6
525.2.i.f.226.1		4	105.74	odd	6
525.2.r.a.424.1		4	105.2	even	12
525.2.r.a.499.1		4	105.53	even	12
525.2.r.f.424.2		4	105.23	even	12
525.2.r.f.499.2		4	105.32	even	12
735.2.a.g.1.1		2	3.2	odd	2
735.2.a.h.1.1		2	21.20	even	2
735.2.i.l.226.2		4	21.17	even	6
735.2.i.l.361.2		4	21.5	even	6
1680.2.bg.o.961.1		4	84.11	even	6
1680.2.bg.o.1201.1		4	84.23	even	6
2205.2.a.z.1.2		2	1.1	even	1	trivial
2205.2.a.ba.1.2		2	7.6	odd	2
3675.2.a.be.1.2		2	105.104	even	2
3675.2.a.bg.1.2		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)

\(f(q)\)	\(=\)	\(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} +9.46410 q^{8} +O(q^{10})\)	\(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} +9.46410 q^{8} -2.73205 q^{10} -0.732051 q^{11} +2.26795 q^{13} +14.9282 q^{16} -3.26795 q^{17} +4.46410 q^{19} -5.46410 q^{20} -2.00000 q^{22} +4.73205 q^{23} +1.00000 q^{25} +6.19615 q^{26} +4.19615 q^{29} -0.464102 q^{31} +21.8564 q^{32} -8.92820 q^{34} -3.19615 q^{37} +12.1962 q^{38} -9.46410 q^{40} +0.732051 q^{41} +3.19615 q^{43} -4.00000 q^{44} +12.9282 q^{46} -2.00000 q^{47} +2.73205 q^{50} +12.3923 q^{52} -12.3923 q^{53} +0.732051 q^{55} +11.4641 q^{58} +0.196152 q^{59} +4.00000 q^{61} -1.26795 q^{62} +29.8564 q^{64} -2.26795 q^{65} -14.6603 q^{67} -17.8564 q^{68} -6.19615 q^{71} +12.6603 q^{73} -8.73205 q^{74} +24.3923 q^{76} -7.39230 q^{79} -14.9282 q^{80} +2.00000 q^{82} -15.1244 q^{83} +3.26795 q^{85} +8.73205 q^{86} -6.92820 q^{88} -15.1244 q^{89} +25.8564 q^{92} -5.46410 q^{94} -4.46410 q^{95} +14.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8}+O(q^{10}) \) 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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