LMFDB - Embedded newform 1521.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +4.56155 q^{4} +0.561553 q^{5} +3.56155 q^{7} +6.56155 q^{8} +O(q^{10})$	$q+2.56155 q^{2} +4.56155 q^{4} +0.561553 q^{5} +3.56155 q^{7} +6.56155 q^{8} +1.43845 q^{10} -2.00000 q^{11} +9.12311 q^{14} +7.68466 q^{16} -2.56155 q^{17} +1.12311 q^{19} +2.56155 q^{20} -5.12311 q^{22} -2.00000 q^{23} -4.68466 q^{25} +16.2462 q^{28} +5.68466 q^{29} +1.56155 q^{31} +6.56155 q^{32} -6.56155 q^{34} +2.00000 q^{35} -3.43845 q^{37} +2.87689 q^{38} +3.68466 q^{40} +2.56155 q^{41} +0.438447 q^{43} -9.12311 q^{44} -5.12311 q^{46} -8.24621 q^{47} +5.68466 q^{49} -12.0000 q^{50} -11.6847 q^{53} -1.12311 q^{55} +23.3693 q^{56} +14.5616 q^{58} -11.1231 q^{59} +12.1231 q^{61} +4.00000 q^{62} +1.43845 q^{64} -0.438447 q^{67} -11.6847 q^{68} +5.12311 q^{70} +14.0000 q^{71} +1.87689 q^{73} -8.80776 q^{74} +5.12311 q^{76} -7.12311 q^{77} +9.56155 q^{79} +4.31534 q^{80} +6.56155 q^{82} -9.12311 q^{83} -1.43845 q^{85} +1.12311 q^{86} -13.1231 q^{88} +13.1231 q^{89} -9.12311 q^{92} -21.1231 q^{94} +0.630683 q^{95} +4.43845 q^{97} +14.5616 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) $ 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8	$ 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8} + 7 q^{10} - 4 q^{11} + 10 q^{14} + 3 q^{16} - q^{17} - 6 q^{19} + q^{20} - 2 q^{22} - 4 q^{23} + 3 q^{25} + 16 q^{28} - q^{29} - q^{31} + 9 q^{32} - 9 q^{34} + 4 q^{35} - 11 q^{37} + 14 q^{38} - 5 q^{40} + q^{41} + 5 q^{43} - 10 q^{44} - 2 q^{46} - q^{49} - 24 q^{50} - 11 q^{53} + 6 q^{55} + 22 q^{56} + 25 q^{58} - 14 q^{59} + 16 q^{61} + 8 q^{62} + 7 q^{64} - 5 q^{67} - 11 q^{68} + 2 q^{70} + 28 q^{71} + 12 q^{73} + 3 q^{74} + 2 q^{76} - 6 q^{77} + 15 q^{79} + 21 q^{80} + 9 q^{82} - 10 q^{83} - 7 q^{85} - 6 q^{86} - 18 q^{88} + 18 q^{89} - 10 q^{92} - 34 q^{94} + 26 q^{95} + 13 q^{97} + 25 q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8 + 7 * q^10 - 4 * q^11 + 10 * q^14 + 3 * q^16 - q^17 - 6 * q^19 + q^20 - 2 * q^22 - 4 * q^23 + 3 * q^25 + 16 * q^28 - q^29 - q^31 + 9 * q^32 - 9 * q^34 + 4 * q^35 - 11 * q^37 + 14 * q^38 - 5 * q^40 + q^41 + 5 * q^43 - 10 * q^44 - 2 * q^46 - q^49 - 24 * q^50 - 11 * q^53 + 6 * q^55 + 22 * q^56 + 25 * q^58 - 14 * q^59 + 16 * q^61 + 8 * q^62 + 7 * q^64 - 5 * q^67 - 11 * q^68 + 2 * q^70 + 28 * q^71 + 12 * q^73 + 3 * q^74 + 2 * q^76 - 6 * q^77 + 15 * q^79 + 21 * q^80 + 9 * q^82 - 10 * q^83 - 7 * q^85 - 6 * q^86 - 18 * q^88 + 18 * q^89 - 10 * q^92 - 34 * q^94 + 26 * q^95 + 13 * q^97 + 25 * q^98

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 $

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	0	0
$3$	0	0
$4$	4.56155	2.28078
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	3.56155	1.34614	0.673070	−	0.739579i	$-0.264975\pi$
$7$	3.56155	1.34614	0.673070	+	0.739579i	$0.264975\pi$
$8$	6.56155	2.31986
$9$	0	0
$10$	1.43845	0.454877
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	9.12311	2.43825
$15$	0	0
$16$	7.68466	1.92116
$17$	−2.56155	−0.621268	−0.310634	−	0.950530i	$-0.600541\pi$
$17$	−2.56155	−0.621268	−0.310634	+	0.950530i	$0.600541\pi$
$18$	0	0
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	2.56155	0.572781
$21$	0	0
$22$	−5.12311	−1.09225
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	0	0
$28$	16.2462	3.07025
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	1.56155	0.280463	0.140232	−	0.990119i	$-0.455215\pi$
$31$	1.56155	0.280463	0.140232	+	0.990119i	$0.455215\pi$
$32$	6.56155	1.15993
$33$	0	0
$34$	−6.56155	−1.12530
$35$	2.00000	0.338062
$36$	0	0
$37$	−3.43845	−0.565277	−0.282639	−	0.959226i	$-0.591210\pi$
$37$	−3.43845	−0.565277	−0.282639	+	0.959226i	$0.591210\pi$
$38$	2.87689	0.466694
$39$	0	0
$40$	3.68466	0.582596
$41$	2.56155	0.400047	0.200024	−	0.979791i	$-0.435898\pi$
$41$	2.56155	0.400047	0.200024	+	0.979791i	$0.435898\pi$
$42$	0	0
$43$	0.438447	0.0668626	0.0334313	−	0.999441i	$-0.489357\pi$
$43$	0.438447	0.0668626	0.0334313	+	0.999441i	$0.489357\pi$
$44$	−9.12311	−1.37536
$45$	0	0
$46$	−5.12311	−0.755361
$47$	−8.24621	−1.20283	−0.601417	−	0.798935i	$-0.705397\pi$
$47$	−8.24621	−1.20283	−0.601417	+	0.798935i	$0.705397\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	−12.0000	−1.69706
$51$	0	0
$52$	0	0
$53$	−11.6847	−1.60501	−0.802506	−	0.596645i	$-0.796500\pi$
$53$	−11.6847	−1.60501	−0.802506	+	0.596645i	$0.796500\pi$
$54$	0	0
$55$	−1.12311	−0.151440
$56$	23.3693	3.12286
$57$	0	0
$58$	14.5616	1.91203
$59$	−11.1231	−1.44811	−0.724053	−	0.689745i	$-0.757723\pi$
$59$	−11.1231	−1.44811	−0.724053	+	0.689745i	$0.757723\pi$
$60$	0	0
$61$	12.1231	1.55220	0.776102	−	0.630607i	$-0.217194\pi$
$61$	12.1231	1.55220	0.776102	+	0.630607i	$0.217194\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.43845	0.179806
$65$	0	0
$66$	0	0
$67$	−0.438447	−0.0535648	−0.0267824	−	0.999641i	$-0.508526\pi$
$67$	−0.438447	−0.0535648	−0.0267824	+	0.999641i	$0.508526\pi$
$68$	−11.6847	−1.41697
$69$	0	0
$70$	5.12311	0.612328
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	1.87689	0.219674	0.109837	−	0.993950i	$-0.464967\pi$
$73$	1.87689	0.219674	0.109837	+	0.993950i	$0.464967\pi$
$74$	−8.80776	−1.02388
$75$	0	0
$76$	5.12311	0.587661
$77$	−7.12311	−0.811753
$78$	0	0
$79$	9.56155	1.07576	0.537879	−	0.843022i	$-0.319226\pi$
$79$	9.56155	1.07576	0.537879	+	0.843022i	$0.319226\pi$
$80$	4.31534	0.482470
$81$	0	0
$82$	6.56155	0.724602
$83$	−9.12311	−1.00139	−0.500695	−	0.865624i	$-0.666922\pi$
$83$	−9.12311	−1.00139	−0.500695	+	0.865624i	$0.666922\pi$
$84$	0	0
$85$	−1.43845	−0.156022
$86$	1.12311	0.121108
$87$	0	0
$88$	−13.1231	−1.39893
$89$	13.1231	1.39105	0.695523	−	0.718504i	$-0.255173\pi$
$89$	13.1231	1.39105	0.695523	+	0.718504i	$0.255173\pi$
$90$	0	0
$91$	0	0
$92$	−9.12311	−0.951150
$93$	0	0
$94$	−21.1231	−2.17868
$95$	0.630683	0.0647067
$96$	0	0
$97$	4.43845	0.450656	0.225328	−	0.974283i	$-0.427655\pi$
$97$	4.43845	0.450656	0.225328	+	0.974283i	$0.427655\pi$
$98$	14.5616	1.47094
$99$	0	0
$100$	−21.3693	−2.13693
$101$	3.43845	0.342138	0.171069	−	0.985259i	$-0.445278\pi$
$101$	3.43845	0.342138	0.171069	+	0.985259i	$0.445278\pi$
$102$	0	0
$103$	−7.56155	−0.745062	−0.372531	−	0.928020i	$-0.621510\pi$
$103$	−7.56155	−0.745062	−0.372531	+	0.928020i	$0.621510\pi$
$104$	0	0
$105$	0	0
$106$	−29.9309	−2.90714
$107$	−8.24621	−0.797191	−0.398596	−	0.917127i	$-0.630502\pi$
$107$	−8.24621	−0.797191	−0.398596	+	0.917127i	$0.630502\pi$
$108$	0	0
$109$	−17.8078	−1.70567	−0.852837	−	0.522177i	$-0.825120\pi$
$109$	−17.8078	−1.70567	−0.852837	+	0.522177i	$0.825120\pi$
$110$	−2.87689	−0.274301
$111$	0	0
$112$	27.3693	2.58616
$113$	14.8078	1.39300	0.696499	−	0.717558i	$-0.254740\pi$
$113$	14.8078	1.39300	0.696499	+	0.717558i	$0.254740\pi$
$114$	0	0
$115$	−1.12311	−0.104730
$116$	25.9309	2.40762
$117$	0	0
$118$	−28.4924	−2.62294
$119$	−9.12311	−0.836314
$120$	0	0
$121$	−7.00000	−0.636364
$122$	31.0540	2.81149
$123$	0	0
$124$	7.12311	0.639674
$125$	−5.43845	−0.486430
$126$	0	0
$127$	9.56155	0.848451	0.424225	−	0.905557i	$-0.360546\pi$
$127$	9.56155	0.848451	0.424225	+	0.905557i	$0.360546\pi$
$128$	−9.43845	−0.834249
$129$	0	0
$130$	0	0
$131$	17.3693	1.51756	0.758782	−	0.651345i	$-0.225795\pi$
$131$	17.3693	1.51756	0.758782	+	0.651345i	$0.225795\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−1.12311	−0.0970215
$135$	0	0
$136$	−16.8078	−1.44125
$137$	−1.43845	−0.122895	−0.0614474	−	0.998110i	$-0.519572\pi$
$137$	−1.43845	−0.122895	−0.0614474	+	0.998110i	$0.519572\pi$
$138$	0	0
$139$	10.9309	0.927144	0.463572	−	0.886059i	$-0.346567\pi$
$139$	10.9309	0.927144	0.463572	+	0.886059i	$0.346567\pi$
$140$	9.12311	0.771043
$141$	0	0
$142$	35.8617	3.00945
$143$	0	0
$144$	0	0
$145$	3.19224	0.265101
$146$	4.80776	0.397893
$147$	0	0
$148$	−15.6847	−1.28927
$149$	−6.56155	−0.537543	−0.268772	−	0.963204i	$-0.586618\pi$
$149$	−6.56155	−0.537543	−0.268772	+	0.963204i	$0.586618\pi$
$150$	0	0
$151$	−15.3693	−1.25074	−0.625369	−	0.780329i	$-0.715051\pi$
$151$	−15.3693	−1.25074	−0.625369	+	0.780329i	$0.715051\pi$
$152$	7.36932	0.597731
$153$	0	0
$154$	−18.2462	−1.47032
$155$	0.876894	0.0704339
$156$	0	0
$157$	−4.36932	−0.348709	−0.174355	−	0.984683i	$-0.555784\pi$
$157$	−4.36932	−0.348709	−0.174355	+	0.984683i	$0.555784\pi$
$158$	24.4924	1.94851
$159$	0	0
$160$	3.68466	0.291298
$161$	−7.12311	−0.561379
$162$	0	0
$163$	−15.8078	−1.23816	−0.619080	−	0.785328i	$-0.712494\pi$
$163$	−15.8078	−1.23816	−0.619080	+	0.785328i	$0.712494\pi$
$164$	11.6847	0.912419
$165$	0	0
$166$	−23.3693	−1.81381
$167$	−6.24621	−0.483346	−0.241673	−	0.970358i	$-0.577696\pi$
$167$	−6.24621	−0.483346	−0.241673	+	0.970358i	$0.577696\pi$
$168$	0	0
$169$	0	0
$170$	−3.68466	−0.282600
$171$	0	0
$172$	2.00000	0.152499
$173$	−3.75379	−0.285395	−0.142698	−	0.989766i	$-0.545578\pi$
$173$	−3.75379	−0.285395	−0.142698	+	0.989766i	$0.545578\pi$
$174$	0	0
$175$	−16.6847	−1.26124
$176$	−15.3693	−1.15851
$177$	0	0
$178$	33.6155	2.51959
$179$	13.1231	0.980867	0.490433	−	0.871479i	$-0.336838\pi$
$179$	13.1231	0.980867	0.490433	+	0.871479i	$0.336838\pi$
$180$	0	0
$181$	9.68466	0.719855	0.359927	−	0.932980i	$-0.382801\pi$
$181$	9.68466	0.719855	0.359927	+	0.932980i	$0.382801\pi$
$182$	0	0
$183$	0	0
$184$	−13.1231	−0.967448
$185$	−1.93087	−0.141960
$186$	0	0
$187$	5.12311	0.374639
$188$	−37.6155	−2.74339
$189$	0	0
$190$	1.61553	0.117203
$191$	0.876894	0.0634499	0.0317249	−	0.999497i	$-0.489900\pi$
$191$	0.876894	0.0634499	0.0317249	+	0.999497i	$0.489900\pi$
$192$	0	0
$193$	−19.4924	−1.40310	−0.701548	−	0.712623i	$-0.747507\pi$
$193$	−19.4924	−1.40310	−0.701548	+	0.712623i	$0.747507\pi$
$194$	11.3693	0.816269
$195$	0	0
$196$	25.9309	1.85220
$197$	−11.3693	−0.810030	−0.405015	−	0.914310i	$-0.632734\pi$
$197$	−11.3693	−0.810030	−0.405015	+	0.914310i	$0.632734\pi$
$198$	0	0
$199$	−23.1771	−1.64298	−0.821490	−	0.570223i	$-0.806857\pi$
$199$	−23.1771	−1.64298	−0.821490	+	0.570223i	$0.806857\pi$
$200$	−30.7386	−2.17355
$201$	0	0
$202$	8.80776	0.619712
$203$	20.2462	1.42101
$204$	0	0
$205$	1.43845	0.100466
$206$	−19.3693	−1.34952
$207$	0	0
$208$	0	0
$209$	−2.24621	−0.155374
$210$	0	0
$211$	7.31534	0.503609	0.251804	−	0.967778i	$-0.418976\pi$
$211$	7.31534	0.503609	0.251804	+	0.967778i	$0.418976\pi$
$212$	−53.3002	−3.66067
$213$	0	0
$214$	−21.1231	−1.44395
$215$	0.246211	0.0167915
$216$	0	0
$217$	5.56155	0.377543
$218$	−45.6155	−3.08947
$219$	0	0
$220$	−5.12311	−0.345400
$221$	0	0
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	23.3693	1.56143
$225$	0	0
$226$	37.9309	2.52312
$227$	−1.12311	−0.0745431	−0.0372716	−	0.999305i	$-0.511867\pi$
$227$	−1.12311	−0.0745431	−0.0372716	+	0.999305i	$0.511867\pi$
$228$	0	0
$229$	−0.246211	−0.0162701	−0.00813505	−	0.999967i	$-0.502589\pi$
$229$	−0.246211	−0.0162701	−0.00813505	+	0.999967i	$0.502589\pi$
$230$	−2.87689	−0.189697
$231$	0	0
$232$	37.3002	2.44888
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	−4.63068	−0.302072
$236$	−50.7386	−3.30280
$237$	0	0
$238$	−23.3693	−1.51481
$239$	−0.630683	−0.0407955	−0.0203977	−	0.999792i	$-0.506493\pi$
$239$	−0.630683	−0.0407955	−0.0203977	+	0.999792i	$0.506493\pi$
$240$	0	0
$241$	−2.80776	−0.180864	−0.0904320	−	0.995903i	$-0.528825\pi$
$241$	−2.80776	−0.180864	−0.0904320	+	0.995903i	$0.528825\pi$
$242$	−17.9309	−1.15264
$243$	0	0
$244$	55.3002	3.54023
$245$	3.19224	0.203944
$246$	0	0
$247$	0	0
$248$	10.2462	0.650635
$249$	0	0
$250$	−13.9309	−0.881066
$251$	30.7386	1.94021	0.970103	−	0.242695i	$-0.0780314\pi$
$251$	30.7386	1.94021	0.970103	+	0.242695i	$0.0780314\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	24.4924	1.53679
$255$	0	0
$256$	−27.0540	−1.69087
$257$	16.1771	1.00910	0.504549	−	0.863383i	$-0.331659\pi$
$257$	16.1771	1.00910	0.504549	+	0.863383i	$0.331659\pi$
$258$	0	0
$259$	−12.2462	−0.760943
$260$	0	0
$261$	0	0
$262$	44.4924	2.74875
$263$	15.3693	0.947713	0.473856	−	0.880602i	$-0.342862\pi$
$263$	15.3693	0.947713	0.473856	+	0.880602i	$0.342862\pi$
$264$	0	0
$265$	−6.56155	−0.403073
$266$	10.2462	0.628236
$267$	0	0
$268$	−2.00000	−0.122169
$269$	−3.36932	−0.205431	−0.102715	−	0.994711i	$-0.532753\pi$
$269$	−3.36932	−0.205431	−0.102715	+	0.994711i	$0.532753\pi$
$270$	0	0
$271$	1.06913	0.0649450	0.0324725	−	0.999473i	$-0.489662\pi$
$271$	1.06913	0.0649450	0.0324725	+	0.999473i	$0.489662\pi$
$272$	−19.6847	−1.19356
$273$	0	0
$274$	−3.68466	−0.222598
$275$	9.36932	0.564991
$276$	0	0
$277$	17.6847	1.06257	0.531284	−	0.847194i	$-0.321710\pi$
$277$	17.6847	1.06257	0.531284	+	0.847194i	$0.321710\pi$
$278$	28.0000	1.67933
$279$	0	0
$280$	13.1231	0.784256
$281$	−2.80776	−0.167497	−0.0837486	−	0.996487i	$-0.526689\pi$
$281$	−2.80776	−0.167497	−0.0837486	+	0.996487i	$0.526689\pi$
$282$	0	0
$283$	−1.31534	−0.0781889	−0.0390945	−	0.999236i	$-0.512447\pi$
$283$	−1.31534	−0.0781889	−0.0390945	+	0.999236i	$0.512447\pi$
$284$	63.8617	3.78950
$285$	0	0
$286$	0	0
$287$	9.12311	0.538520
$288$	0	0
$289$	−10.4384	−0.614026
$290$	8.17708	0.480175
$291$	0	0
$292$	8.56155	0.501027
$293$	24.5616	1.43490	0.717451	−	0.696609i	$-0.245309\pi$
$293$	24.5616	1.43490	0.717451	+	0.696609i	$0.245309\pi$
$294$	0	0
$295$	−6.24621	−0.363668
$296$	−22.5616	−1.31136
$297$	0	0
$298$	−16.8078	−0.973648
$299$	0	0
$300$	0	0
$301$	1.56155	0.0900064
$302$	−39.3693	−2.26545
$303$	0	0
$304$	8.63068	0.495004
$305$	6.80776	0.389811
$306$	0	0
$307$	−10.1922	−0.581702	−0.290851	−	0.956768i	$-0.593938\pi$
$307$	−10.1922	−0.581702	−0.290851	+	0.956768i	$0.593938\pi$
$308$	−32.4924	−1.85143
$309$	0	0
$310$	2.24621	0.127576
$311$	10.8769	0.616772	0.308386	−	0.951261i	$-0.400211\pi$
$311$	10.8769	0.616772	0.308386	+	0.951261i	$0.400211\pi$
$312$	0	0
$313$	−1.31534	−0.0743475	−0.0371738	−	0.999309i	$-0.511835\pi$
$313$	−1.31534	−0.0743475	−0.0371738	+	0.999309i	$0.511835\pi$
$314$	−11.1922	−0.631614
$315$	0	0
$316$	43.6155	2.45357
$317$	−23.0540	−1.29484	−0.647420	−	0.762133i	$-0.724152\pi$
$317$	−23.0540	−1.29484	−0.647420	+	0.762133i	$0.724152\pi$
$318$	0	0
$319$	−11.3693	−0.636560
$320$	0.807764	0.0451554
$321$	0	0
$322$	−18.2462	−1.01682
$323$	−2.87689	−0.160075
$324$	0	0
$325$	0	0
$326$	−40.4924	−2.24267
$327$	0	0
$328$	16.8078	0.928054
$329$	−29.3693	−1.61918
$330$	0	0
$331$	23.8078	1.30859	0.654297	−	0.756238i	$-0.272965\pi$
$331$	23.8078	1.30859	0.654297	+	0.756238i	$0.272965\pi$
$332$	−41.6155	−2.28395
$333$	0	0
$334$	−16.0000	−0.875481
$335$	−0.246211	−0.0134520
$336$	0	0
$337$	2.12311	0.115653	0.0578265	−	0.998327i	$-0.481583\pi$
$337$	2.12311	0.115653	0.0578265	+	0.998327i	$0.481583\pi$
$338$	0	0
$339$	0	0
$340$	−6.56155	−0.355850
$341$	−3.12311	−0.169126
$342$	0	0
$343$	−4.68466	−0.252948
$344$	2.87689	0.155112
$345$	0	0
$346$	−9.61553	−0.516934
$347$	13.6155	0.730920	0.365460	−	0.930827i	$-0.380912\pi$
$347$	13.6155	0.730920	0.365460	+	0.930827i	$0.380912\pi$
$348$	0	0
$349$	13.8078	0.739113	0.369556	−	0.929208i	$-0.379510\pi$
$349$	13.8078	0.739113	0.369556	+	0.929208i	$0.379510\pi$
$350$	−42.7386	−2.28448
$351$	0	0
$352$	−13.1231	−0.699464
$353$	17.6847	0.941259	0.470630	−	0.882331i	$-0.344027\pi$
$353$	17.6847	0.941259	0.470630	+	0.882331i	$0.344027\pi$
$354$	0	0
$355$	7.86174	0.417258
$356$	59.8617	3.17267
$357$	0	0
$358$	33.6155	1.77664
$359$	15.3693	0.811162	0.405581	−	0.914059i	$-0.367069\pi$
$359$	15.3693	0.811162	0.405581	+	0.914059i	$0.367069\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	24.8078	1.30387
$363$	0	0
$364$	0	0
$365$	1.05398	0.0551676
$366$	0	0
$367$	20.0540	1.04681	0.523404	−	0.852084i	$-0.324662\pi$
$367$	20.0540	1.04681	0.523404	+	0.852084i	$0.324662\pi$
$368$	−15.3693	−0.801181
$369$	0	0
$370$	−4.94602	−0.257132
$371$	−41.6155	−2.16057
$372$	0	0
$373$	3.63068	0.187990	0.0939948	−	0.995573i	$-0.470036\pi$
$373$	3.63068	0.187990	0.0939948	+	0.995573i	$0.470036\pi$
$374$	13.1231	0.678580
$375$	0	0
$376$	−54.1080	−2.79040
$377$	0	0
$378$	0	0
$379$	−11.3153	−0.581230	−0.290615	−	0.956840i	$-0.593860\pi$
$379$	−11.3153	−0.581230	−0.290615	+	0.956840i	$0.593860\pi$
$380$	2.87689	0.147582
$381$	0	0
$382$	2.24621	0.114926
$383$	26.7386	1.36628	0.683140	−	0.730287i	$-0.260614\pi$
$383$	26.7386	1.36628	0.683140	+	0.730287i	$0.260614\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−49.9309	−2.54141
$387$	0	0
$388$	20.2462	1.02785
$389$	3.05398	0.154843	0.0774213	−	0.996998i	$-0.475331\pi$
$389$	3.05398	0.154843	0.0774213	+	0.996998i	$0.475331\pi$
$390$	0	0
$391$	5.12311	0.259087
$392$	37.3002	1.88394
$393$	0	0
$394$	−29.1231	−1.46720
$395$	5.36932	0.270160
$396$	0	0
$397$	12.0540	0.604972	0.302486	−	0.953154i	$-0.402184\pi$
$397$	12.0540	0.604972	0.302486	+	0.953154i	$0.402184\pi$
$398$	−59.3693	−2.97591
$399$	0	0
$400$	−36.0000	−1.80000
$401$	18.5616	0.926920	0.463460	−	0.886118i	$-0.346608\pi$
$401$	18.5616	0.926920	0.463460	+	0.886118i	$0.346608\pi$
$402$	0	0
$403$	0	0
$404$	15.6847	0.780341
$405$	0	0
$406$	51.8617	2.57385
$407$	6.87689	0.340875
$408$	0	0
$409$	18.3693	0.908304	0.454152	−	0.890924i	$-0.349942\pi$
$409$	18.3693	0.908304	0.454152	+	0.890924i	$0.349942\pi$
$410$	3.68466	0.181972
$411$	0	0
$412$	−34.4924	−1.69932
$413$	−39.6155	−1.94935
$414$	0	0
$415$	−5.12311	−0.251483
$416$	0	0
$417$	0	0
$418$	−5.75379	−0.281427
$419$	17.7538	0.867329	0.433665	−	0.901074i	$-0.357220\pi$
$419$	17.7538	0.867329	0.433665	+	0.901074i	$0.357220\pi$
$420$	0	0
$421$	−14.7538	−0.719056	−0.359528	−	0.933134i	$-0.617062\pi$
$421$	−14.7538	−0.719056	−0.359528	+	0.933134i	$0.617062\pi$
$422$	18.7386	0.912182
$423$	0	0
$424$	−76.6695	−3.72340
$425$	12.0000	0.582086
$426$	0	0
$427$	43.1771	2.08949
$428$	−37.6155	−1.81822
$429$	0	0
$430$	0.630683	0.0304142
$431$	−2.87689	−0.138575	−0.0692876	−	0.997597i	$-0.522073\pi$
$431$	−2.87689	−0.138575	−0.0692876	+	0.997597i	$0.522073\pi$
$432$	0	0
$433$	25.2462	1.21326	0.606628	−	0.794986i	$-0.292522\pi$
$433$	25.2462	1.21326	0.606628	+	0.794986i	$0.292522\pi$
$434$	14.2462	0.683840
$435$	0	0
$436$	−81.2311	−3.89026
$437$	−2.24621	−0.107451
$438$	0	0
$439$	−1.31534	−0.0627778	−0.0313889	−	0.999507i	$-0.509993\pi$
$439$	−1.31534	−0.0627778	−0.0313889	+	0.999507i	$0.509993\pi$
$440$	−7.36932	−0.351318
$441$	0	0
$442$	0	0
$443$	−14.7386	−0.700254	−0.350127	−	0.936702i	$-0.613862\pi$
$443$	−14.7386	−0.700254	−0.350127	+	0.936702i	$0.613862\pi$
$444$	0	0
$445$	7.36932	0.349339
$446$	−20.4924	−0.970344
$447$	0	0
$448$	5.12311	0.242044
$449$	8.24621	0.389163	0.194581	−	0.980886i	$-0.437665\pi$
$449$	8.24621	0.389163	0.194581	+	0.980886i	$0.437665\pi$
$450$	0	0
$451$	−5.12311	−0.241238
$452$	67.5464	3.17712
$453$	0	0
$454$	−2.87689	−0.135019
$455$	0	0
$456$	0	0
$457$	28.6155	1.33858	0.669289	−	0.743002i	$-0.266599\pi$
$457$	28.6155	1.33858	0.669289	+	0.743002i	$0.266599\pi$
$458$	−0.630683	−0.0294699
$459$	0	0
$460$	−5.12311	−0.238866
$461$	−36.8078	−1.71431	−0.857154	−	0.515060i	$-0.827770\pi$
$461$	−36.8078	−1.71431	−0.857154	+	0.515060i	$0.827770\pi$
$462$	0	0
$463$	26.6847	1.24014	0.620071	−	0.784546i	$-0.287104\pi$
$463$	26.6847	1.24014	0.620071	+	0.784546i	$0.287104\pi$
$464$	43.6847	2.02801
$465$	0	0
$466$	−66.6004	−3.08520
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	0	0
$469$	−1.56155	−0.0721058
$470$	−11.8617	−0.547141
$471$	0	0
$472$	−72.9848	−3.35940
$473$	−0.876894	−0.0403196
$474$	0	0
$475$	−5.26137	−0.241408
$476$	−41.6155	−1.90744
$477$	0	0
$478$	−1.61553	−0.0738925
$479$	−6.24621	−0.285397	−0.142698	−	0.989766i	$-0.545578\pi$
$479$	−6.24621	−0.285397	−0.142698	+	0.989766i	$0.545578\pi$
$480$	0	0
$481$	0	0
$482$	−7.19224	−0.327597
$483$	0	0
$484$	−31.9309	−1.45140
$485$	2.49242	0.113175
$486$	0	0
$487$	1.12311	0.0508928	0.0254464	−	0.999676i	$-0.491899\pi$
$487$	1.12311	0.0508928	0.0254464	+	0.999676i	$0.491899\pi$
$488$	79.5464	3.60090
$489$	0	0
$490$	8.17708	0.369403
$491$	19.7538	0.891476	0.445738	−	0.895163i	$-0.352941\pi$
$491$	19.7538	0.891476	0.445738	+	0.895163i	$0.352941\pi$
$492$	0	0
$493$	−14.5616	−0.655819
$494$	0	0
$495$	0	0
$496$	12.0000	0.538816
$497$	49.8617	2.23660
$498$	0	0
$499$	28.4924	1.27550	0.637748	−	0.770245i	$-0.279866\pi$
$499$	28.4924	1.27550	0.637748	+	0.770245i	$0.279866\pi$
$500$	−24.8078	−1.10944
$501$	0	0
$502$	78.7386	3.51428
$503$	11.7538	0.524076	0.262038	−	0.965058i	$-0.415605\pi$
$503$	11.7538	0.524076	0.262038	+	0.965058i	$0.415605\pi$
$504$	0	0
$505$	1.93087	0.0859226
$506$	10.2462	0.455500
$507$	0	0
$508$	43.6155	1.93513
$509$	6.80776	0.301749	0.150874	−	0.988553i	$-0.451791\pi$
$509$	6.80776	0.301749	0.150874	+	0.988553i	$0.451791\pi$
$510$	0	0
$511$	6.68466	0.295712
$512$	−50.4233	−2.22842
$513$	0	0
$514$	41.4384	1.82777
$515$	−4.24621	−0.187110
$516$	0	0
$517$	16.4924	0.725336
$518$	−31.3693	−1.37829
$519$	0	0
$520$	0	0
$521$	37.9309	1.66178	0.830891	−	0.556436i	$-0.187831\pi$
$521$	37.9309	1.66178	0.830891	+	0.556436i	$0.187831\pi$
$522$	0	0
$523$	−23.8617	−1.04340	−0.521701	−	0.853129i	$-0.674702\pi$
$523$	−23.8617	−1.04340	−0.521701	+	0.853129i	$0.674702\pi$
$524$	79.2311	3.46122
$525$	0	0
$526$	39.3693	1.71658
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−16.8078	−0.730083
$531$	0	0
$532$	18.2462	0.791074
$533$	0	0
$534$	0	0
$535$	−4.63068	−0.200202
$536$	−2.87689	−0.124263
$537$	0	0
$538$	−8.63068	−0.372095
$539$	−11.3693	−0.489711
$540$	0	0
$541$	29.7386	1.27856	0.639282	−	0.768972i	$-0.279232\pi$
$541$	29.7386	1.27856	0.639282	+	0.768972i	$0.279232\pi$
$542$	2.73863	0.117634
$543$	0	0
$544$	−16.8078	−0.720627
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−24.9309	−1.06597	−0.532984	−	0.846126i	$-0.678929\pi$
$547$	−24.9309	−1.06597	−0.532984	+	0.846126i	$0.678929\pi$
$548$	−6.56155	−0.280296
$549$	0	0
$550$	24.0000	1.02336
$551$	6.38447	0.271988
$552$	0	0
$553$	34.0540	1.44812
$554$	45.3002	1.92462
$555$	0	0
$556$	49.8617	2.11461
$557$	−14.0691	−0.596128	−0.298064	−	0.954546i	$-0.596341\pi$
$557$	−14.0691	−0.596128	−0.298064	+	0.954546i	$0.596341\pi$
$558$	0	0
$559$	0	0
$560$	15.3693	0.649472
$561$	0	0
$562$	−7.19224	−0.303386
$563$	−1.36932	−0.0577098	−0.0288549	−	0.999584i	$-0.509186\pi$
$563$	−1.36932	−0.0577098	−0.0288549	+	0.999584i	$0.509186\pi$
$564$	0	0
$565$	8.31534	0.349829
$566$	−3.36932	−0.141623
$567$	0	0
$568$	91.8617	3.85443
$569$	−40.7386	−1.70785	−0.853926	−	0.520394i	$-0.825785\pi$
$569$	−40.7386	−1.70785	−0.853926	+	0.520394i	$0.825785\pi$
$570$	0	0
$571$	19.3693	0.810581	0.405290	−	0.914188i	$-0.367170\pi$
$571$	19.3693	0.810581	0.405290	+	0.914188i	$0.367170\pi$
$572$	0	0
$573$	0	0
$574$	23.3693	0.975416
$575$	9.36932	0.390728
$576$	0	0
$577$	29.6847	1.23579	0.617894	−	0.786261i	$-0.287986\pi$
$577$	29.6847	1.23579	0.617894	+	0.786261i	$0.287986\pi$
$578$	−26.7386	−1.11218
$579$	0	0
$580$	14.5616	0.604636
$581$	−32.4924	−1.34801
$582$	0	0
$583$	23.3693	0.967858
$584$	12.3153	0.509612
$585$	0	0
$586$	62.9157	2.59902
$587$	14.6307	0.603873	0.301936	−	0.953328i	$-0.402367\pi$
$587$	14.6307	0.603873	0.301936	+	0.953328i	$0.402367\pi$
$588$	0	0
$589$	1.75379	0.0722636
$590$	−16.0000	−0.658710
$591$	0	0
$592$	−26.4233	−1.08599
$593$	44.4233	1.82425	0.912123	−	0.409917i	$-0.134442\pi$
$593$	44.4233	1.82425	0.912123	+	0.409917i	$0.134442\pi$
$594$	0	0
$595$	−5.12311	−0.210027
$596$	−29.9309	−1.22602
$597$	0	0
$598$	0	0
$599$	0.384472	0.0157091	0.00785455	−	0.999969i	$-0.497500\pi$
$599$	0.384472	0.0157091	0.00785455	+	0.999969i	$0.497500\pi$
$600$	0	0
$601$	−35.9309	−1.46565	−0.732825	−	0.680417i	$-0.761799\pi$
$601$	−35.9309	−1.46565	−0.732825	+	0.680417i	$0.761799\pi$
$602$	4.00000	0.163028
$603$	0	0
$604$	−70.1080	−2.85265
$605$	−3.93087	−0.159813
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	7.36932	0.298865
$609$	0	0
$610$	17.4384	0.706062
$611$	0	0
$612$	0	0
$613$	−22.8617	−0.923377	−0.461688	−	0.887042i	$-0.652756\pi$
$613$	−22.8617	−0.923377	−0.461688	+	0.887042i	$0.652756\pi$
$614$	−26.1080	−1.05363
$615$	0	0
$616$	−46.7386	−1.88315
$617$	−10.8078	−0.435104	−0.217552	−	0.976049i	$-0.569807\pi$
$617$	−10.8078	−0.435104	−0.217552	+	0.976049i	$0.569807\pi$
$618$	0	0
$619$	24.3002	0.976707	0.488353	−	0.872646i	$-0.337598\pi$
$619$	24.3002	0.976707	0.488353	+	0.872646i	$0.337598\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	27.8617	1.11715
$623$	46.7386	1.87254
$624$	0	0
$625$	20.3693	0.814773
$626$	−3.36932	−0.134665
$627$	0	0
$628$	−19.9309	−0.795328
$629$	8.80776	0.351189
$630$	0	0
$631$	14.4384	0.574786	0.287393	−	0.957813i	$-0.407212\pi$
$631$	14.4384	0.574786	0.287393	+	0.957813i	$0.407212\pi$
$632$	62.7386	2.49561
$633$	0	0
$634$	−59.0540	−2.34533
$635$	5.36932	0.213075
$636$	0	0
$637$	0	0
$638$	−29.1231	−1.15299
$639$	0	0
$640$	−5.30019	−0.209508
$641$	−26.1771	−1.03393	−0.516966	−	0.856006i	$-0.672939\pi$
$641$	−26.1771	−1.03393	−0.516966	+	0.856006i	$0.672939\pi$
$642$	0	0
$643$	−38.5464	−1.52012	−0.760061	−	0.649852i	$-0.774831\pi$
$643$	−38.5464	−1.52012	−0.760061	+	0.649852i	$0.774831\pi$
$644$	−32.4924	−1.28038
$645$	0	0
$646$	−7.36932	−0.289942
$647$	47.6155	1.87196	0.935980	−	0.352054i	$-0.114517\pi$
$647$	47.6155	1.87196	0.935980	+	0.352054i	$0.114517\pi$
$648$	0	0
$649$	22.2462	0.873240
$650$	0	0
$651$	0	0
$652$	−72.1080	−2.82397
$653$	−14.8769	−0.582178	−0.291089	−	0.956696i	$-0.594018\pi$
$653$	−14.8769	−0.582178	−0.291089	+	0.956696i	$0.594018\pi$
$654$	0	0
$655$	9.75379	0.381112
$656$	19.6847	0.768557
$657$	0	0
$658$	−75.2311	−2.93281
$659$	−14.2462	−0.554954	−0.277477	−	0.960732i	$-0.589498\pi$
$659$	−14.2462	−0.554954	−0.277477	+	0.960732i	$0.589498\pi$
$660$	0	0
$661$	−30.3693	−1.18123	−0.590615	−	0.806954i	$-0.701115\pi$
$661$	−30.3693	−1.18123	−0.590615	+	0.806954i	$0.701115\pi$
$662$	60.9848	2.37024
$663$	0	0
$664$	−59.8617	−2.32309
$665$	2.24621	0.0871043
$666$	0	0
$667$	−11.3693	−0.440222
$668$	−28.4924	−1.10240
$669$	0	0
$670$	−0.630683	−0.0243654
$671$	−24.2462	−0.936015
$672$	0	0
$673$	−6.75379	−0.260339	−0.130170	−	0.991492i	$-0.541552\pi$
$673$	−6.75379	−0.260339	−0.130170	+	0.991492i	$0.541552\pi$
$674$	5.43845	0.209481
$675$	0	0
$676$	0	0
$677$	−25.6155	−0.984485	−0.492242	−	0.870458i	$-0.663823\pi$
$677$	−25.6155	−0.984485	−0.492242	+	0.870458i	$0.663823\pi$
$678$	0	0
$679$	15.8078	0.606646
$680$	−9.43845	−0.361948
$681$	0	0
$682$	−8.00000	−0.306336
$683$	36.1080	1.38163	0.690816	−	0.723030i	$-0.257251\pi$
$683$	36.1080	1.38163	0.690816	+	0.723030i	$0.257251\pi$
$684$	0	0
$685$	−0.807764	−0.0308631
$686$	−12.0000	−0.458162
$687$	0	0
$688$	3.36932	0.128454
$689$	0	0
$690$	0	0
$691$	−2.30019	−0.0875032	−0.0437516	−	0.999042i	$-0.513931\pi$
$691$	−2.30019	−0.0875032	−0.0437516	+	0.999042i	$0.513931\pi$
$692$	−17.1231	−0.650923
$693$	0	0
$694$	34.8769	1.32391
$695$	6.13826	0.232837
$696$	0	0
$697$	−6.56155	−0.248537
$698$	35.3693	1.33875
$699$	0	0
$700$	−76.1080	−2.87661
$701$	−19.3693	−0.731569	−0.365785	−	0.930700i	$-0.619199\pi$
$701$	−19.3693	−0.731569	−0.365785	+	0.930700i	$0.619199\pi$
$702$	0	0
$703$	−3.86174	−0.145648
$704$	−2.87689	−0.108427
$705$	0	0
$706$	45.3002	1.70490
$707$	12.2462	0.460566
$708$	0	0
$709$	−25.4924	−0.957388	−0.478694	−	0.877982i	$-0.658890\pi$
$709$	−25.4924	−0.957388	−0.478694	+	0.877982i	$0.658890\pi$
$710$	20.1383	0.755775
$711$	0	0
$712$	86.1080	3.22703
$713$	−3.12311	−0.116961
$714$	0	0
$715$	0	0
$716$	59.8617	2.23714
$717$	0	0
$718$	39.3693	1.46925
$719$	−1.36932	−0.0510669	−0.0255335	−	0.999674i	$-0.508128\pi$
$719$	−1.36932	−0.0510669	−0.0255335	+	0.999674i	$0.508128\pi$
$720$	0	0
$721$	−26.9309	−1.00296
$722$	−45.4384	−1.69104
$723$	0	0
$724$	44.1771	1.64183
$725$	−26.6307	−0.989039
$726$	0	0
$727$	39.6695	1.47126	0.735630	−	0.677383i	$-0.236886\pi$
$727$	39.6695	1.47126	0.735630	+	0.677383i	$0.236886\pi$
$728$	0	0
$729$	0	0
$730$	2.69981	0.0999246
$731$	−1.12311	−0.0415396
$732$	0	0
$733$	−53.4924	−1.97579	−0.987894	−	0.155131i	$-0.950420\pi$
$733$	−53.4924	−1.97579	−0.987894	+	0.155131i	$0.950420\pi$
$734$	51.3693	1.89608
$735$	0	0
$736$	−13.1231	−0.483724
$737$	0.876894	0.0323008
$738$	0	0
$739$	6.24621	0.229771	0.114885	−	0.993379i	$-0.463350\pi$
$739$	6.24621	0.229771	0.114885	+	0.993379i	$0.463350\pi$
$740$	−8.80776	−0.323780
$741$	0	0
$742$	−106.600	−3.91342
$743$	−37.3693	−1.37095	−0.685474	−	0.728097i	$-0.740405\pi$
$743$	−37.3693	−1.37095	−0.685474	+	0.728097i	$0.740405\pi$
$744$	0	0
$745$	−3.68466	−0.134995
$746$	9.30019	0.340504
$747$	0	0
$748$	23.3693	0.854467
$749$	−29.3693	−1.07313
$750$	0	0
$751$	−30.1080	−1.09865	−0.549327	−	0.835607i	$-0.685116\pi$
$751$	−30.1080	−1.09865	−0.549327	+	0.835607i	$0.685116\pi$
$752$	−63.3693	−2.31084
$753$	0	0
$754$	0	0
$755$	−8.63068	−0.314103
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−28.9848	−1.05278
$759$	0	0
$760$	4.13826	0.150110
$761$	−15.3693	−0.557137	−0.278569	−	0.960416i	$-0.589860\pi$
$761$	−15.3693	−0.557137	−0.278569	+	0.960416i	$0.589860\pi$
$762$	0	0
$763$	−63.4233	−2.29608
$764$	4.00000	0.144715
$765$	0	0
$766$	68.4924	2.47473
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	−10.2462	−0.369248
$771$	0	0
$772$	−88.9157	−3.20015
$773$	−7.75379	−0.278884	−0.139442	−	0.990230i	$-0.544531\pi$
$773$	−7.75379	−0.278884	−0.139442	+	0.990230i	$0.544531\pi$
$774$	0	0
$775$	−7.31534	−0.262775
$776$	29.1231	1.04546
$777$	0	0
$778$	7.82292	0.280465
$779$	2.87689	0.103075
$780$	0	0
$781$	−28.0000	−1.00192
$782$	13.1231	0.469281
$783$	0	0
$784$	43.6847	1.56017
$785$	−2.45360	−0.0875728
$786$	0	0
$787$	1.17708	0.0419584	0.0209792	−	0.999780i	$-0.493322\pi$
$787$	1.17708	0.0419584	0.0209792	+	0.999780i	$0.493322\pi$
$788$	−51.8617	−1.84750
$789$	0	0
$790$	13.7538	0.489338
$791$	52.7386	1.87517
$792$	0	0
$793$	0	0
$794$	30.8769	1.09578
$795$	0	0
$796$	−105.723	−3.74727
$797$	41.6155	1.47410	0.737049	−	0.675840i	$-0.236219\pi$
$797$	41.6155	1.47410	0.737049	+	0.675840i	$0.236219\pi$
$798$	0	0
$799$	21.1231	0.747282
$800$	−30.7386	−1.08677
$801$	0	0
$802$	47.5464	1.67892
$803$	−3.75379	−0.132468
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	22.5616	0.793713
$809$	−37.3002	−1.31140	−0.655702	−	0.755019i	$-0.727627\pi$
$809$	−37.3002	−1.31140	−0.655702	+	0.755019i	$0.727627\pi$
$810$	0	0
$811$	1.56155	0.0548335	0.0274168	−	0.999624i	$-0.491272\pi$
$811$	1.56155	0.0548335	0.0274168	+	0.999624i	$0.491272\pi$
$812$	92.3542	3.24100
$813$	0	0
$814$	17.6155	0.617424
$815$	−8.87689	−0.310944
$816$	0	0
$817$	0.492423	0.0172277
$818$	47.0540	1.64520
$819$	0	0
$820$	6.56155	0.229139
$821$	−26.4924	−0.924592	−0.462296	−	0.886726i	$-0.652974\pi$
$821$	−26.4924	−0.924592	−0.462296	+	0.886726i	$0.652974\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−49.6155	−1.72844
$825$	0	0
$826$	−101.477	−3.53085
$827$	34.7386	1.20798	0.603990	−	0.796992i	$-0.293577\pi$
$827$	34.7386	1.20798	0.603990	+	0.796992i	$0.293577\pi$
$828$	0	0
$829$	−19.4924	−0.677000	−0.338500	−	0.940966i	$-0.609919\pi$
$829$	−19.4924	−0.677000	−0.338500	+	0.940966i	$0.609919\pi$
$830$	−13.1231	−0.455510
$831$	0	0
$832$	0	0
$833$	−14.5616	−0.504528
$834$	0	0
$835$	−3.50758	−0.121385
$836$	−10.2462	−0.354373
$837$	0	0
$838$	45.4773	1.57099
$839$	−19.6155	−0.677203	−0.338602	−	0.940930i	$-0.609954\pi$
$839$	−19.6155	−0.677203	−0.338602	+	0.940930i	$0.609954\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	−37.7926	−1.30242
$843$	0	0
$844$	33.3693	1.14862
$845$	0	0
$846$	0	0
$847$	−24.9309	−0.856635
$848$	−89.7926	−3.08349
$849$	0	0
$850$	30.7386	1.05433
$851$	6.87689	0.235737
$852$	0	0
$853$	6.12311	0.209651	0.104826	−	0.994491i	$-0.466572\pi$
$853$	6.12311	0.209651	0.104826	+	0.994491i	$0.466572\pi$
$854$	110.600	3.78467
$855$	0	0
$856$	−54.1080	−1.84937
$857$	−31.4384	−1.07392	−0.536958	−	0.843609i	$-0.680427\pi$
$857$	−31.4384	−1.07392	−0.536958	+	0.843609i	$0.680427\pi$
$858$	0	0
$859$	20.4384	0.697351	0.348675	−	0.937244i	$-0.386632\pi$
$859$	20.4384	0.697351	0.348675	+	0.937244i	$0.386632\pi$
$860$	1.12311	0.0382976
$861$	0	0
$862$	−7.36932	−0.251000
$863$	−2.49242	−0.0848430	−0.0424215	−	0.999100i	$-0.513507\pi$
$863$	−2.49242	−0.0848430	−0.0424215	+	0.999100i	$0.513507\pi$
$864$	0	0
$865$	−2.10795	−0.0716725
$866$	64.6695	2.19756
$867$	0	0
$868$	25.3693	0.861091
$869$	−19.1231	−0.648707
$870$	0	0
$871$	0	0
$872$	−116.847	−3.95692
$873$	0	0
$874$	−5.75379	−0.194625
$875$	−19.3693	−0.654802
$876$	0	0
$877$	−19.4384	−0.656390	−0.328195	−	0.944610i	$-0.606440\pi$
$877$	−19.4384	−0.656390	−0.328195	+	0.944610i	$0.606440\pi$
$878$	−3.36932	−0.113709
$879$	0	0
$880$	−8.63068	−0.290940
$881$	37.9309	1.27792	0.638962	−	0.769239i	$-0.279364\pi$
$881$	37.9309	1.27792	0.638962	+	0.769239i	$0.279364\pi$
$882$	0	0
$883$	−11.8078	−0.397363	−0.198681	−	0.980064i	$-0.563666\pi$
$883$	−11.8078	−0.397363	−0.198681	+	0.980064i	$0.563666\pi$
$884$	0	0
$885$	0	0
$886$	−37.7538	−1.26836
$887$	−49.3693	−1.65766	−0.828830	−	0.559501i	$-0.810993\pi$
$887$	−49.3693	−1.65766	−0.828830	+	0.559501i	$0.810993\pi$
$888$	0	0
$889$	34.0540	1.14213
$890$	18.8769	0.632755
$891$	0	0
$892$	−36.4924	−1.22186
$893$	−9.26137	−0.309920
$894$	0	0
$895$	7.36932	0.246329
$896$	−33.6155	−1.12302
$897$	0	0
$898$	21.1231	0.704887
$899$	8.87689	0.296061
$900$	0	0
$901$	29.9309	0.997142
$902$	−13.1231	−0.436952
$903$	0	0
$904$	97.1619	3.23156
$905$	5.43845	0.180780
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−5.12311	−0.170016
$909$	0	0
$910$	0	0
$911$	10.7386	0.355787	0.177893	−	0.984050i	$-0.443072\pi$
$911$	10.7386	0.355787	0.177893	+	0.984050i	$0.443072\pi$
$912$	0	0
$913$	18.2462	0.603861
$914$	73.3002	2.42455
$915$	0	0
$916$	−1.12311	−0.0371085
$917$	61.8617	2.04285
$918$	0	0
$919$	44.4924	1.46767	0.733835	−	0.679328i	$-0.237729\pi$
$919$	44.4924	1.46767	0.733835	+	0.679328i	$0.237729\pi$
$920$	−7.36932	−0.242959
$921$	0	0
$922$	−94.2850	−3.10511
$923$	0	0
$924$	0	0
$925$	16.1080	0.529626
$926$	68.3542	2.24626
$927$	0	0
$928$	37.3002	1.22444
$929$	12.8078	0.420209	0.210105	−	0.977679i	$-0.432620\pi$
$929$	12.8078	0.420209	0.210105	+	0.977679i	$0.432620\pi$
$930$	0	0
$931$	6.38447	0.209243
$932$	−118.600	−3.88488
$933$	0	0
$934$	66.6004	2.17923
$935$	2.87689	0.0940845
$936$	0	0
$937$	−3.43845	−0.112329	−0.0561646	−	0.998422i	$-0.517887\pi$
$937$	−3.43845	−0.112329	−0.0561646	+	0.998422i	$0.517887\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−21.1231	−0.688960
$941$	−2.49242	−0.0812507	−0.0406253	−	0.999174i	$-0.512935\pi$
$941$	−2.49242	−0.0812507	−0.0406253	+	0.999174i	$0.512935\pi$
$942$	0	0
$943$	−5.12311	−0.166831
$944$	−85.4773	−2.78205
$945$	0	0
$946$	−2.24621	−0.0730306
$947$	10.7386	0.348959	0.174479	−	0.984661i	$-0.444176\pi$
$947$	10.7386	0.348959	0.174479	+	0.984661i	$0.444176\pi$
$948$	0	0
$949$	0	0
$950$	−13.4773	−0.437260
$951$	0	0
$952$	−59.8617	−1.94013
$953$	34.9848	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$953$	34.9848	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$954$	0	0
$955$	0.492423	0.0159344
$956$	−2.87689	−0.0930454
$957$	0	0
$958$	−16.0000	−0.516937
$959$	−5.12311	−0.165434
$960$	0	0
$961$	−28.5616	−0.921340
$962$	0	0
$963$	0	0
$964$	−12.8078	−0.412510
$965$	−10.9460	−0.352365
$966$	0	0
$967$	9.12311	0.293379	0.146690	−	0.989183i	$-0.453138\pi$
$967$	9.12311	0.293379	0.146690	+	0.989183i	$0.453138\pi$
$968$	−45.9309	−1.47627
$969$	0	0
$970$	6.38447	0.204993
$971$	−52.9848	−1.70036	−0.850182	−	0.526488i	$-0.823508\pi$
$971$	−52.9848	−1.70036	−0.850182	+	0.526488i	$0.823508\pi$
$972$	0	0
$973$	38.9309	1.24807
$974$	2.87689	0.0921816
$975$	0	0
$976$	93.1619	2.98204
$977$	15.8229	0.506220	0.253110	−	0.967438i	$-0.418547\pi$
$977$	15.8229	0.506220	0.253110	+	0.967438i	$0.418547\pi$
$978$	0	0
$979$	−26.2462	−0.838833
$980$	14.5616	0.465152
$981$	0	0
$982$	50.6004	1.61472
$983$	−27.6155	−0.880799	−0.440399	−	0.897802i	$-0.645163\pi$
$983$	−27.6155	−0.880799	−0.440399	+	0.897802i	$0.645163\pi$
$984$	0	0
$985$	−6.38447	−0.203426
$986$	−37.3002	−1.18788
$987$	0	0
$988$	0	0
$989$	−0.876894	−0.0278836
$990$	0	0
$991$	40.3542	1.28189	0.640946	−	0.767586i	$-0.278542\pi$
$991$	40.3542	1.28189	0.640946	+	0.767586i	$0.278542\pi$
$992$	10.2462	0.325318
$993$	0	0
$994$	127.723	4.05114
$995$	−13.0152	−0.412608
$996$	0	0
$997$	20.6155	0.652900	0.326450	−	0.945214i	$-0.394147\pi$
$997$	20.6155	0.652900	0.326450	+	0.945214i	$0.394147\pi$
$998$	72.9848	2.31029
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.m.1.2		2
3.2	odd	2		507.2.a.d.1.1		2
12.11	even	2		8112.2.a.bo.1.1		2
13.4	even	6		117.2.g.c.55.2		4
13.5	odd	4		1521.2.b.h.1351.1		4
13.8	odd	4		1521.2.b.h.1351.4		4
13.10	even	6		117.2.g.c.100.2		4
13.12	even	2		1521.2.a.g.1.1		2
39.2	even	12		507.2.j.g.316.1		8
39.5	even	4		507.2.b.d.337.4		4
39.8	even	4		507.2.b.d.337.1		4
39.11	even	12		507.2.j.g.316.4		8
39.17	odd	6		39.2.e.b.16.1	✓	4
39.20	even	12		507.2.j.g.361.1		8
39.23	odd	6		39.2.e.b.22.1	yes	4
39.29	odd	6		507.2.e.g.22.2		4
39.32	even	12		507.2.j.g.361.4		8
39.35	odd	6		507.2.e.g.484.2		4
39.38	odd	2		507.2.a.g.1.2		2
52.23	odd	6		1872.2.t.r.1153.1		4
52.43	odd	6		1872.2.t.r.289.1		4
156.23	even	6		624.2.q.h.529.2		4
156.95	even	6		624.2.q.h.289.2		4
156.155	even	2		8112.2.a.bk.1.2		2
195.17	even	12		975.2.bb.i.874.4		8
195.23	even	12		975.2.bb.i.724.4		8
195.62	even	12		975.2.bb.i.724.1		8
195.134	odd	6		975.2.i.k.601.2		4
195.173	even	12		975.2.bb.i.874.1		8
195.179	odd	6		975.2.i.k.451.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.1	✓	4	39.17	odd	6
39.2.e.b.22.1	yes	4	39.23	odd	6
117.2.g.c.55.2		4	13.4	even	6
117.2.g.c.100.2		4	13.10	even	6
507.2.a.d.1.1		2	3.2	odd	2
507.2.a.g.1.2		2	39.38	odd	2
507.2.b.d.337.1		4	39.8	even	4
507.2.b.d.337.4		4	39.5	even	4
507.2.e.g.22.2		4	39.29	odd	6
507.2.e.g.484.2		4	39.35	odd	6
507.2.j.g.316.1		8	39.2	even	12
507.2.j.g.316.4		8	39.11	even	12
507.2.j.g.361.1		8	39.20	even	12
507.2.j.g.361.4		8	39.32	even	12
624.2.q.h.289.2		4	156.95	even	6
624.2.q.h.529.2		4	156.23	even	6
975.2.i.k.451.2		4	195.179	odd	6
975.2.i.k.601.2		4	195.134	odd	6
975.2.bb.i.724.1		8	195.62	even	12
975.2.bb.i.724.4		8	195.23	even	12
975.2.bb.i.874.1		8	195.173	even	12
975.2.bb.i.874.4		8	195.17	even	12
1521.2.a.g.1.1		2	13.12	even	2
1521.2.a.m.1.2		2	1.1	even	1	trivial
1521.2.b.h.1351.1		4	13.5	odd	4
1521.2.b.h.1351.4		4	13.8	odd	4
1872.2.t.r.289.1		4	52.43	odd	6
1872.2.t.r.1153.1		4	52.23	odd	6
8112.2.a.bk.1.2		2	156.155	even	2
8112.2.a.bo.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} +4.56155 q^{4} +0.561553 q^{5} +3.56155 q^{7} +6.56155 q^{8} +O(q^{10})\)	\(q+2.56155 q^{2} +4.56155 q^{4} +0.561553 q^{5} +3.56155 q^{7} +6.56155 q^{8} +1.43845 q^{10} -2.00000 q^{11} +9.12311 q^{14} +7.68466 q^{16} -2.56155 q^{17} +1.12311 q^{19} +2.56155 q^{20} -5.12311 q^{22} -2.00000 q^{23} -4.68466 q^{25} +16.2462 q^{28} +5.68466 q^{29} +1.56155 q^{31} +6.56155 q^{32} -6.56155 q^{34} +2.00000 q^{35} -3.43845 q^{37} +2.87689 q^{38} +3.68466 q^{40} +2.56155 q^{41} +0.438447 q^{43} -9.12311 q^{44} -5.12311 q^{46} -8.24621 q^{47} +5.68466 q^{49} -12.0000 q^{50} -11.6847 q^{53} -1.12311 q^{55} +23.3693 q^{56} +14.5616 q^{58} -11.1231 q^{59} +12.1231 q^{61} +4.00000 q^{62} +1.43845 q^{64} -0.438447 q^{67} -11.6847 q^{68} +5.12311 q^{70} +14.0000 q^{71} +1.87689 q^{73} -8.80776 q^{74} +5.12311 q^{76} -7.12311 q^{77} +9.56155 q^{79} +4.31534 q^{80} +6.56155 q^{82} -9.12311 q^{83} -1.43845 q^{85} +1.12311 q^{86} -13.1231 q^{88} +13.1231 q^{89} -9.12311 q^{92} -21.1231 q^{94} +0.630683 q^{95} +4.43845 q^{97} +14.5616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8	\( 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8} + 7 q^{10} - 4 q^{11} + 10 q^{14} + 3 q^{16} - q^{17} - 6 q^{19} + q^{20} - 2 q^{22} - 4 q^{23} + 3 q^{25} + 16 q^{28} - q^{29} - q^{31} + 9 q^{32} - 9 q^{34} + 4 q^{35} - 11 q^{37} + 14 q^{38} - 5 q^{40} + q^{41} + 5 q^{43} - 10 q^{44} - 2 q^{46} - q^{49} - 24 q^{50} - 11 q^{53} + 6 q^{55} + 22 q^{56} + 25 q^{58} - 14 q^{59} + 16 q^{61} + 8 q^{62} + 7 q^{64} - 5 q^{67} - 11 q^{68} + 2 q^{70} + 28 q^{71} + 12 q^{73} + 3 q^{74} + 2 q^{76} - 6 q^{77} + 15 q^{79} + 21 q^{80} + 9 q^{82} - 10 q^{83} - 7 q^{85} - 6 q^{86} - 18 q^{88} + 18 q^{89} - 10 q^{92} - 34 q^{94} + 26 q^{95} + 13 q^{97} + 25 q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8 + 7 * q^10 - 4 * q^11 + 10 * q^14 + 3 * q^16 - q^17 - 6 * q^19 + q^20 - 2 * q^22 - 4 * q^23 + 3 * q^25 + 16 * q^28 - q^29 - q^31 + 9 * q^32 - 9 * q^34 + 4 * q^35 - 11 * q^37 + 14 * q^38 - 5 * q^40 + q^41 + 5 * q^43 - 10 * q^44 - 2 * q^46 - q^49 - 24 * q^50 - 11 * q^53 + 6 * q^55 + 22 * q^56 + 25 * q^58 - 14 * q^59 + 16 * q^61 + 8 * q^62 + 7 * q^64 - 5 * q^67 - 11 * q^68 + 2 * q^70 + 28 * q^71 + 12 * q^73 + 3 * q^74 + 2 * q^76 - 6 * q^77 + 15 * q^79 + 21 * q^80 + 9 * q^82 - 10 * q^83 - 7 * q^85 - 6 * q^86 - 18 * q^88 + 18 * q^89 - 10 * q^92 - 34 * q^94 + 26 * q^95 + 13 * q^97 + 25 * q^98

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