LMFDB - Embedded newform 315.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.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:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} +O(q^{10})$	\(q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} -0.414214 q^{10} +4.82843 q^{11} +0.828427 q^{13} +0.414214 q^{14} +3.00000 q^{16} +7.65685 q^{17} -2.82843 q^{19} -1.82843 q^{20} -2.00000 q^{22} -3.65685 q^{23} +1.00000 q^{25} -0.343146 q^{26} +1.82843 q^{28} +8.00000 q^{29} +8.48528 q^{31} -4.41421 q^{32} -3.17157 q^{34} -1.00000 q^{35} -6.00000 q^{37} +1.17157 q^{38} +1.58579 q^{40} -7.65685 q^{41} +1.65685 q^{43} -8.82843 q^{44} +1.51472 q^{46} -4.00000 q^{47} +1.00000 q^{49} -0.414214 q^{50} -1.51472 q^{52} +5.17157 q^{53} +4.82843 q^{55} -1.58579 q^{56} -3.31371 q^{58} -4.00000 q^{59} +6.00000 q^{61} -3.51472 q^{62} -4.17157 q^{64} +0.828427 q^{65} -15.3137 q^{67} -14.0000 q^{68} +0.414214 q^{70} +10.4853 q^{71} -12.1421 q^{73} +2.48528 q^{74} +5.17157 q^{76} -4.82843 q^{77} +5.65685 q^{79} +3.00000 q^{80} +3.17157 q^{82} -8.00000 q^{83} +7.65685 q^{85} -0.686292 q^{86} +7.65685 q^{88} -5.31371 q^{89} -0.828427 q^{91} +6.68629 q^{92} +1.65685 q^{94} -2.82843 q^{95} -3.17157 q^{97} -0.414214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 2 q^{10} + 4 q^{11} - 4 q^{13} - 2 q^{14} + 6 q^{16} + 4 q^{17} + 2 q^{20} - 4 q^{22} + 4 q^{23} + 2 q^{25} - 12 q^{26} - 2 q^{28} + 16 q^{29} - 6 q^{32} - 12 q^{34} - 2 q^{35} - 12 q^{37} + 8 q^{38} + 6 q^{40} - 4 q^{41} - 8 q^{43} - 12 q^{44} + 20 q^{46} - 8 q^{47} + 2 q^{49} + 2 q^{50} - 20 q^{52} + 16 q^{53} + 4 q^{55} - 6 q^{56} + 16 q^{58} - 8 q^{59} + 12 q^{61} - 24 q^{62} - 14 q^{64} - 4 q^{65} - 8 q^{67} - 28 q^{68} - 2 q^{70} + 4 q^{71} + 4 q^{73} - 12 q^{74} + 16 q^{76} - 4 q^{77} + 6 q^{80} + 12 q^{82} - 16 q^{83} + 4 q^{85} - 24 q^{86} + 4 q^{88} + 12 q^{89} + 4 q^{91} + 36 q^{92} - 8 q^{94} - 12 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 2 * q^10 + 4 * q^11 - 4 * q^13 - 2 * q^14 + 6 * q^16 + 4 * q^17 + 2 * q^20 - 4 * q^22 + 4 * q^23 + 2 * q^25 - 12 * q^26 - 2 * q^28 + 16 * q^29 - 6 * q^32 - 12 * q^34 - 2 * q^35 - 12 * q^37 + 8 * q^38 + 6 * q^40 - 4 * q^41 - 8 * q^43 - 12 * q^44 + 20 * q^46 - 8 * q^47 + 2 * q^49 + 2 * q^50 - 20 * q^52 + 16 * q^53 + 4 * q^55 - 6 * q^56 + 16 * q^58 - 8 * q^59 + 12 * q^61 - 24 * q^62 - 14 * q^64 - 4 * q^65 - 8 * q^67 - 28 * q^68 - 2 * q^70 + 4 * q^71 + 4 * q^73 - 12 * q^74 + 16 * q^76 - 4 * q^77 + 6 * q^80 + 12 * q^82 - 16 * q^83 + 4 * q^85 - 24 * q^86 + 4 * q^88 + 12 * q^89 + 4 * q^91 + 36 * q^92 - 8 * q^94 - 12 * q^97 + 2 * 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 $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.58579	0.560660
$9$	0	0
$10$	−0.414214	−0.130986
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	0.414214	0.110703
$15$	0	0
$16$	3.00000	0.750000
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	−1.82843	−0.408849
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.343146	−0.0672964
$27$	0	0
$28$	1.82843	0.345540
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	−3.17157	−0.543920
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	1.17157	0.190054
$39$	0	0
$40$	1.58579	0.250735
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	−8.82843	−1.33094
$45$	0	0
$46$	1.51472	0.223333
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.414214	−0.0585786
$51$	0	0
$52$	−1.51472	−0.210054
$53$	5.17157	0.710370	0.355185	−	0.934796i	$-0.384418\pi$
$53$	5.17157	0.710370	0.355185	+	0.934796i	$0.384418\pi$
$54$	0	0
$55$	4.82843	0.651065
$56$	−1.58579	−0.211910
$57$	0	0
$58$	−3.31371	−0.435111
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−3.51472	−0.446370
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0.828427	0.102754
$66$	0	0
$67$	−15.3137	−1.87087	−0.935434	−	0.353502i	$-0.884991\pi$
$67$	−15.3137	−1.87087	−0.935434	+	0.353502i	$0.884991\pi$
$68$	−14.0000	−1.69775
$69$	0	0
$70$	0.414214	0.0495080
$71$	10.4853	1.24437	0.622187	−	0.782869i	$-0.286244\pi$
$71$	10.4853	1.24437	0.622187	+	0.782869i	$0.286244\pi$
$72$	0	0
$73$	−12.1421	−1.42113	−0.710565	−	0.703632i	$-0.751560\pi$
$73$	−12.1421	−1.42113	−0.710565	+	0.703632i	$0.751560\pi$
$74$	2.48528	0.288908
$75$	0	0
$76$	5.17157	0.593220
$77$	−4.82843	−0.550250
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	3.17157	0.350242
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	7.65685	0.830502
$86$	−0.686292	−0.0740047
$87$	0	0
$88$	7.65685	0.816223
$89$	−5.31371	−0.563252	−0.281626	−	0.959524i	$-0.590874\pi$
$89$	−5.31371	−0.563252	−0.281626	+	0.959524i	$0.590874\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	6.68629	0.697094
$93$	0	0
$94$	1.65685	0.170891
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−3.17157	−0.322024	−0.161012	−	0.986952i	$-0.551476\pi$
$97$	−3.17157	−0.322024	−0.161012	+	0.986952i	$0.551476\pi$
$98$	−0.414214	−0.0418419
$99$	0	0
$100$	−1.82843	−0.182843
$101$	4.34315	0.432159	0.216080	−	0.976376i	$-0.430673\pi$
$101$	4.34315	0.432159	0.216080	+	0.976376i	$0.430673\pi$
$102$	0	0
$103$	−15.3137	−1.50890	−0.754452	−	0.656355i	$-0.772097\pi$
$103$	−15.3137	−1.50890	−0.754452	+	0.656355i	$0.772097\pi$
$104$	1.31371	0.128820
$105$	0	0
$106$	−2.14214	−0.208063
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	−3.00000	−0.283473
$113$	12.4853	1.17452	0.587258	−	0.809400i	$-0.300207\pi$
$113$	12.4853	1.17452	0.587258	+	0.809400i	$0.300207\pi$
$114$	0	0
$115$	−3.65685	−0.341003
$116$	−14.6274	−1.35812
$117$	0	0
$118$	1.65685	0.152526
$119$	−7.65685	−0.701903
$120$	0	0
$121$	12.3137	1.11943
$122$	−2.48528	−0.225007
$123$	0	0
$124$	−15.5147	−1.39326
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.34315	0.207921	0.103960	−	0.994581i	$-0.466849\pi$
$127$	2.34315	0.207921	0.103960	+	0.994581i	$0.466849\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	−0.343146	−0.0300959
$131$	−13.6569	−1.19320	−0.596602	−	0.802537i	$-0.703483\pi$
$131$	−13.6569	−1.19320	−0.596602	+	0.802537i	$0.703483\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	6.34315	0.547964
$135$	0	0
$136$	12.1421	1.04118
$137$	22.1421	1.89173	0.945865	−	0.324560i	$-0.105216\pi$
$137$	22.1421	1.89173	0.945865	+	0.324560i	$0.105216\pi$
$138$	0	0
$139$	1.17157	0.0993715	0.0496858	−	0.998765i	$-0.484178\pi$
$139$	1.17157	0.0993715	0.0496858	+	0.998765i	$0.484178\pi$
$140$	1.82843	0.154530
$141$	0	0
$142$	−4.34315	−0.364469
$143$	4.00000	0.334497
$144$	0	0
$145$	8.00000	0.664364
$146$	5.02944	0.416239
$147$	0	0
$148$	10.9706	0.901775
$149$	9.65685	0.791120	0.395560	−	0.918440i	$-0.370550\pi$
$149$	9.65685	0.791120	0.395560	+	0.918440i	$0.370550\pi$
$150$	0	0
$151$	−5.65685	−0.460348	−0.230174	−	0.973149i	$-0.573930\pi$
$151$	−5.65685	−0.460348	−0.230174	+	0.973149i	$0.573930\pi$
$152$	−4.48528	−0.363804
$153$	0	0
$154$	2.00000	0.161165
$155$	8.48528	0.681554
$156$	0	0
$157$	−8.82843	−0.704585	−0.352293	−	0.935890i	$-0.614598\pi$
$157$	−8.82843	−0.704585	−0.352293	+	0.935890i	$0.614598\pi$
$158$	−2.34315	−0.186411
$159$	0	0
$160$	−4.41421	−0.348974
$161$	3.65685	0.288200
$162$	0	0
$163$	−6.34315	−0.496834	−0.248417	−	0.968653i	$-0.579910\pi$
$163$	−6.34315	−0.496834	−0.248417	+	0.968653i	$0.579910\pi$
$164$	14.0000	1.09322
$165$	0	0
$166$	3.31371	0.257194
$167$	−0.686292	−0.0531068	−0.0265534	−	0.999647i	$-0.508453\pi$
$167$	−0.686292	−0.0531068	−0.0265534	+	0.999647i	$0.508453\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	−3.17157	−0.243249
$171$	0	0
$172$	−3.02944	−0.230992
$173$	15.6569	1.19037	0.595184	−	0.803589i	$-0.297079\pi$
$173$	15.6569	1.19037	0.595184	+	0.803589i	$0.297079\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	14.4853	1.09187
$177$	0	0
$178$	2.20101	0.164973
$179$	5.51472	0.412189	0.206095	−	0.978532i	$-0.433925\pi$
$179$	5.51472	0.412189	0.206095	+	0.978532i	$0.433925\pi$
$180$	0	0
$181$	−11.6569	−0.866447	−0.433224	−	0.901286i	$-0.642624\pi$
$181$	−11.6569	−0.866447	−0.433224	+	0.901286i	$0.642624\pi$
$182$	0.343146	0.0254357
$183$	0	0
$184$	−5.79899	−0.427507
$185$	−6.00000	−0.441129
$186$	0	0
$187$	36.9706	2.70356
$188$	7.31371	0.533407
$189$	0	0
$190$	1.17157	0.0849948
$191$	4.82843	0.349373	0.174686	−	0.984624i	$-0.444109\pi$
$191$	4.82843	0.349373	0.174686	+	0.984624i	$0.444109\pi$
$192$	0	0
$193$	14.9706	1.07760	0.538802	−	0.842432i	$-0.318877\pi$
$193$	14.9706	1.07760	0.538802	+	0.842432i	$0.318877\pi$
$194$	1.31371	0.0943188
$195$	0	0
$196$	−1.82843	−0.130602
$197$	2.82843	0.201517	0.100759	−	0.994911i	$-0.467873\pi$
$197$	2.82843	0.201517	0.100759	+	0.994911i	$0.467873\pi$
$198$	0	0
$199$	17.1716	1.21726	0.608630	−	0.793454i	$-0.291719\pi$
$199$	17.1716	1.21726	0.608630	+	0.793454i	$0.291719\pi$
$200$	1.58579	0.112132
$201$	0	0
$202$	−1.79899	−0.126576
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−7.65685	−0.534778
$206$	6.34315	0.441948
$207$	0	0
$208$	2.48528	0.172323
$209$	−13.6569	−0.944664
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−9.45584	−0.649430
$213$	0	0
$214$	0.828427	0.0566301
$215$	1.65685	0.112997
$216$	0	0
$217$	−8.48528	−0.576018
$218$	−4.14214	−0.280541
$219$	0	0
$220$	−8.82843	−0.595212
$221$	6.34315	0.426686
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	4.41421	0.294937
$225$	0	0
$226$	−5.17157	−0.344008
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−17.3137	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$229$	−17.3137	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$230$	1.51472	0.0998776
$231$	0	0
$232$	12.6863	0.832896
$233$	−22.1421	−1.45058	−0.725290	−	0.688444i	$-0.758294\pi$
$233$	−22.1421	−1.45058	−0.725290	+	0.688444i	$0.758294\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	7.31371	0.476082
$237$	0	0
$238$	3.17157	0.205583
$239$	−24.1421	−1.56162	−0.780812	−	0.624765i	$-0.785195\pi$
$239$	−24.1421	−1.56162	−0.780812	+	0.624765i	$0.785195\pi$
$240$	0	0
$241$	−7.65685	−0.493221	−0.246611	−	0.969115i	$-0.579317\pi$
$241$	−7.65685	−0.493221	−0.246611	+	0.969115i	$0.579317\pi$
$242$	−5.10051	−0.327873
$243$	0	0
$244$	−10.9706	−0.702318
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.34315	−0.149091
$248$	13.4558	0.854447
$249$	0	0
$250$	−0.414214	−0.0261972
$251$	−26.6274	−1.68071	−0.840354	−	0.542038i	$-0.817653\pi$
$251$	−26.6274	−1.68071	−0.840354	+	0.542038i	$0.817653\pi$
$252$	0	0
$253$	−17.6569	−1.11008
$254$	−0.970563	−0.0608985
$255$	0	0
$256$	3.97056	0.248160
$257$	−5.31371	−0.331460	−0.165730	−	0.986171i	$-0.552998\pi$
$257$	−5.31371	−0.331460	−0.165730	+	0.986171i	$0.552998\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	−1.51472	−0.0939389
$261$	0	0
$262$	5.65685	0.349482
$263$	−2.68629	−0.165644	−0.0828219	−	0.996564i	$-0.526393\pi$
$263$	−2.68629	−0.165644	−0.0828219	+	0.996564i	$0.526393\pi$
$264$	0	0
$265$	5.17157	0.317687
$266$	−1.17157	−0.0718337
$267$	0	0
$268$	28.0000	1.71037
$269$	14.9706	0.912771	0.456386	−	0.889782i	$-0.349144\pi$
$269$	14.9706	0.912771	0.456386	+	0.889782i	$0.349144\pi$
$270$	0	0
$271$	−24.4853	−1.48737	−0.743687	−	0.668527i	$-0.766925\pi$
$271$	−24.4853	−1.48737	−0.743687	+	0.668527i	$0.766925\pi$
$272$	22.9706	1.39279
$273$	0	0
$274$	−9.17157	−0.554075
$275$	4.82843	0.291165
$276$	0	0
$277$	−15.6569	−0.940729	−0.470365	−	0.882472i	$-0.655878\pi$
$277$	−15.6569	−0.940729	−0.470365	+	0.882472i	$0.655878\pi$
$278$	−0.485281	−0.0291052
$279$	0	0
$280$	−1.58579	−0.0947689
$281$	−25.6569	−1.53056	−0.765280	−	0.643698i	$-0.777399\pi$
$281$	−25.6569	−1.53056	−0.765280	+	0.643698i	$0.777399\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−19.1716	−1.13762
$285$	0	0
$286$	−1.65685	−0.0979718
$287$	7.65685	0.451970
$288$	0	0
$289$	41.6274	2.44867
$290$	−3.31371	−0.194588
$291$	0	0
$292$	22.2010	1.29922
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−9.51472	−0.553032
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−3.02944	−0.175197
$300$	0	0
$301$	−1.65685	−0.0954995
$302$	2.34315	0.134833
$303$	0	0
$304$	−8.48528	−0.486664
$305$	6.00000	0.343559
$306$	0	0
$307$	28.9706	1.65344	0.826719	−	0.562616i	$-0.190205\pi$
$307$	28.9706	1.65344	0.826719	+	0.562616i	$0.190205\pi$
$308$	8.82843	0.503046
$309$	0	0
$310$	−3.51472	−0.199623
$311$	−22.3431	−1.26696	−0.633482	−	0.773758i	$-0.718375\pi$
$311$	−22.3431	−1.26696	−0.633482	+	0.773758i	$0.718375\pi$
$312$	0	0
$313$	−15.1716	−0.857548	−0.428774	−	0.903412i	$-0.641054\pi$
$313$	−15.1716	−0.857548	−0.428774	+	0.903412i	$0.641054\pi$
$314$	3.65685	0.206368
$315$	0	0
$316$	−10.3431	−0.581847
$317$	−14.1421	−0.794301	−0.397151	−	0.917753i	$-0.630001\pi$
$317$	−14.1421	−0.794301	−0.397151	+	0.917753i	$0.630001\pi$
$318$	0	0
$319$	38.6274	2.16272
$320$	−4.17157	−0.233198
$321$	0	0
$322$	−1.51472	−0.0844120
$323$	−21.6569	−1.20502
$324$	0	0
$325$	0.828427	0.0459529
$326$	2.62742	0.145519
$327$	0	0
$328$	−12.1421	−0.670437
$329$	4.00000	0.220527
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	14.6274	0.802784
$333$	0	0
$334$	0.284271	0.0155546
$335$	−15.3137	−0.836677
$336$	0	0
$337$	34.2843	1.86758	0.933792	−	0.357817i	$-0.116479\pi$
$337$	34.2843	1.86758	0.933792	+	0.357817i	$0.116479\pi$
$338$	5.10051	0.277431
$339$	0	0
$340$	−14.0000	−0.759257
$341$	40.9706	2.21868
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.62742	0.141661
$345$	0	0
$346$	−6.48528	−0.348651
$347$	−11.6569	−0.625773	−0.312886	−	0.949791i	$-0.601296\pi$
$347$	−11.6569	−0.625773	−0.312886	+	0.949791i	$0.601296\pi$
$348$	0	0
$349$	−6.68629	−0.357909	−0.178954	−	0.983857i	$-0.557271\pi$
$349$	−6.68629	−0.357909	−0.178954	+	0.983857i	$0.557271\pi$
$350$	0.414214	0.0221406
$351$	0	0
$352$	−21.3137	−1.13602
$353$	−28.6274	−1.52368	−0.761842	−	0.647763i	$-0.775705\pi$
$353$	−28.6274	−1.52368	−0.761842	+	0.647763i	$0.775705\pi$
$354$	0	0
$355$	10.4853	0.556501
$356$	9.71573	0.514933
$357$	0	0
$358$	−2.28427	−0.120727
$359$	−6.48528	−0.342280	−0.171140	−	0.985247i	$-0.554745\pi$
$359$	−6.48528	−0.342280	−0.171140	+	0.985247i	$0.554745\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	4.82843	0.253776
$363$	0	0
$364$	1.51472	0.0793928
$365$	−12.1421	−0.635548
$366$	0	0
$367$	−8.97056	−0.468260	−0.234130	−	0.972205i	$-0.575224\pi$
$367$	−8.97056	−0.468260	−0.234130	+	0.972205i	$0.575224\pi$
$368$	−10.9706	−0.571880
$369$	0	0
$370$	2.48528	0.129204
$371$	−5.17157	−0.268495
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−15.3137	−0.791853
$375$	0	0
$376$	−6.34315	−0.327123
$377$	6.62742	0.341329
$378$	0	0
$379$	17.6569	0.906972	0.453486	−	0.891263i	$-0.350180\pi$
$379$	17.6569	0.906972	0.453486	+	0.891263i	$0.350180\pi$
$380$	5.17157	0.265296
$381$	0	0
$382$	−2.00000	−0.102329
$383$	4.68629	0.239458	0.119729	−	0.992807i	$-0.461797\pi$
$383$	4.68629	0.239458	0.119729	+	0.992807i	$0.461797\pi$
$384$	0	0
$385$	−4.82843	−0.246079
$386$	−6.20101	−0.315623
$387$	0	0
$388$	5.79899	0.294399
$389$	18.3431	0.930034	0.465017	−	0.885302i	$-0.346048\pi$
$389$	18.3431	0.930034	0.465017	+	0.885302i	$0.346048\pi$
$390$	0	0
$391$	−28.0000	−1.41602
$392$	1.58579	0.0800943
$393$	0	0
$394$	−1.17157	−0.0590230
$395$	5.65685	0.284627
$396$	0	0
$397$	32.1421	1.61317	0.806584	−	0.591120i	$-0.201314\pi$
$397$	32.1421	1.61317	0.806584	+	0.591120i	$0.201314\pi$
$398$	−7.11270	−0.356527
$399$	0	0
$400$	3.00000	0.150000
$401$	26.6274	1.32971	0.664855	−	0.746973i	$-0.268493\pi$
$401$	26.6274	1.32971	0.664855	+	0.746973i	$0.268493\pi$
$402$	0	0
$403$	7.02944	0.350161
$404$	−7.94113	−0.395086
$405$	0	0
$406$	3.31371	0.164457
$407$	−28.9706	−1.43602
$408$	0	0
$409$	25.3137	1.25168	0.625841	−	0.779951i	$-0.284756\pi$
$409$	25.3137	1.25168	0.625841	+	0.779951i	$0.284756\pi$
$410$	3.17157	0.156633
$411$	0	0
$412$	28.0000	1.37946
$413$	4.00000	0.196827
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−3.65685	−0.179292
$417$	0	0
$418$	5.65685	0.276686
$419$	34.6274	1.69166	0.845830	−	0.533453i	$-0.179106\pi$
$419$	34.6274	1.69166	0.845830	+	0.533453i	$0.179106\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	4.97056	0.241963
$423$	0	0
$424$	8.20101	0.398276
$425$	7.65685	0.371412
$426$	0	0
$427$	−6.00000	−0.290360
$428$	3.65685	0.176761
$429$	0	0
$430$	−0.686292	−0.0330959
$431$	7.17157	0.345443	0.172721	−	0.984971i	$-0.444744\pi$
$431$	7.17157	0.345443	0.172721	+	0.984971i	$0.444744\pi$
$432$	0	0
$433$	−13.7990	−0.663137	−0.331569	−	0.943431i	$-0.607578\pi$
$433$	−13.7990	−0.663137	−0.331569	+	0.943431i	$0.607578\pi$
$434$	3.51472	0.168712
$435$	0	0
$436$	−18.2843	−0.875658
$437$	10.3431	0.494780
$438$	0	0
$439$	−37.4558	−1.78767	−0.893835	−	0.448396i	$-0.851995\pi$
$439$	−37.4558	−1.78767	−0.893835	+	0.448396i	$0.851995\pi$
$440$	7.65685	0.365026
$441$	0	0
$442$	−2.62742	−0.124973
$443$	34.9706	1.66150	0.830751	−	0.556645i	$-0.187911\pi$
$443$	34.9706	1.66150	0.830751	+	0.556645i	$0.187911\pi$
$444$	0	0
$445$	−5.31371	−0.251894
$446$	−2.34315	−0.110951
$447$	0	0
$448$	4.17157	0.197088
$449$	−19.3137	−0.911470	−0.455735	−	0.890115i	$-0.650624\pi$
$449$	−19.3137	−0.911470	−0.455735	+	0.890115i	$0.650624\pi$
$450$	0	0
$451$	−36.9706	−1.74088
$452$	−22.8284	−1.07376
$453$	0	0
$454$	8.28427	0.388800
$455$	−0.828427	−0.0388373
$456$	0	0
$457$	−21.3137	−0.997013	−0.498507	−	0.866886i	$-0.666118\pi$
$457$	−21.3137	−0.997013	−0.498507	+	0.866886i	$0.666118\pi$
$458$	7.17157	0.335106
$459$	0	0
$460$	6.68629	0.311750
$461$	16.6274	0.774416	0.387208	−	0.921992i	$-0.373440\pi$
$461$	16.6274	0.774416	0.387208	+	0.921992i	$0.373440\pi$
$462$	0	0
$463$	−21.6569	−1.00648	−0.503240	−	0.864147i	$-0.667859\pi$
$463$	−21.6569	−1.00648	−0.503240	+	0.864147i	$0.667859\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	9.17157	0.424865
$467$	31.3137	1.44903	0.724513	−	0.689261i	$-0.242065\pi$
$467$	31.3137	1.44903	0.724513	+	0.689261i	$0.242065\pi$
$468$	0	0
$469$	15.3137	0.707121
$470$	1.65685	0.0764250
$471$	0	0
$472$	−6.34315	−0.291967
$473$	8.00000	0.367840
$474$	0	0
$475$	−2.82843	−0.129777
$476$	14.0000	0.641689
$477$	0	0
$478$	10.0000	0.457389
$479$	−30.3431	−1.38641	−0.693207	−	0.720739i	$-0.743803\pi$
$479$	−30.3431	−1.38641	−0.693207	+	0.720739i	$0.743803\pi$
$480$	0	0
$481$	−4.97056	−0.226638
$482$	3.17157	0.144461
$483$	0	0
$484$	−22.5147	−1.02340
$485$	−3.17157	−0.144014
$486$	0	0
$487$	−2.34315	−0.106178	−0.0530890	−	0.998590i	$-0.516907\pi$
$487$	−2.34315	−0.106178	−0.0530890	+	0.998590i	$0.516907\pi$
$488$	9.51472	0.430711
$489$	0	0
$490$	−0.414214	−0.0187123
$491$	12.1421	0.547967	0.273983	−	0.961734i	$-0.411659\pi$
$491$	12.1421	0.547967	0.273983	+	0.961734i	$0.411659\pi$
$492$	0	0
$493$	61.2548	2.75878
$494$	0.970563	0.0436677
$495$	0	0
$496$	25.4558	1.14300
$497$	−10.4853	−0.470329
$498$	0	0
$499$	14.3431	0.642087	0.321044	−	0.947064i	$-0.395966\pi$
$499$	14.3431	0.642087	0.321044	+	0.947064i	$0.395966\pi$
$500$	−1.82843	−0.0817697
$501$	0	0
$502$	11.0294	0.492268
$503$	30.6274	1.36561	0.682805	−	0.730601i	$-0.260760\pi$
$503$	30.6274	1.36561	0.682805	+	0.730601i	$0.260760\pi$
$504$	0	0
$505$	4.34315	0.193267
$506$	7.31371	0.325134
$507$	0	0
$508$	−4.28427	−0.190084
$509$	27.6569	1.22587	0.612934	−	0.790134i	$-0.289989\pi$
$509$	27.6569	1.22587	0.612934	+	0.790134i	$0.289989\pi$
$510$	0	0
$511$	12.1421	0.537136
$512$	−22.7574	−1.00574
$513$	0	0
$514$	2.20101	0.0970824
$515$	−15.3137	−0.674803
$516$	0	0
$517$	−19.3137	−0.849416
$518$	−2.48528	−0.109197
$519$	0	0
$520$	1.31371	0.0576099
$521$	−6.97056	−0.305386	−0.152693	−	0.988274i	$-0.548795\pi$
$521$	−6.97056	−0.305386	−0.152693	+	0.988274i	$0.548795\pi$
$522$	0	0
$523$	−29.6569	−1.29680	−0.648402	−	0.761298i	$-0.724562\pi$
$523$	−29.6569	−1.29680	−0.648402	+	0.761298i	$0.724562\pi$
$524$	24.9706	1.09084
$525$	0	0
$526$	1.11270	0.0485160
$527$	64.9706	2.83016
$528$	0	0
$529$	−9.62742	−0.418583
$530$	−2.14214	−0.0930484
$531$	0	0
$532$	−5.17157	−0.224216
$533$	−6.34315	−0.274752
$534$	0	0
$535$	−2.00000	−0.0864675
$536$	−24.2843	−1.04892
$537$	0	0
$538$	−6.20101	−0.267345
$539$	4.82843	0.207975
$540$	0	0
$541$	10.6863	0.459440	0.229720	−	0.973257i	$-0.426219\pi$
$541$	10.6863	0.459440	0.229720	+	0.973257i	$0.426219\pi$
$542$	10.1421	0.435642
$543$	0	0
$544$	−33.7990	−1.44912
$545$	10.0000	0.428353
$546$	0	0
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	−40.4853	−1.72945
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	−22.6274	−0.963960
$552$	0	0
$553$	−5.65685	−0.240554
$554$	6.48528	0.275533
$555$	0	0
$556$	−2.14214	−0.0908468
$557$	21.1716	0.897068	0.448534	−	0.893766i	$-0.351946\pi$
$557$	21.1716	0.897068	0.448534	+	0.893766i	$0.351946\pi$
$558$	0	0
$559$	1.37258	0.0580541
$560$	−3.00000	−0.126773
$561$	0	0
$562$	10.6274	0.448291
$563$	−37.9411	−1.59903	−0.799514	−	0.600648i	$-0.794909\pi$
$563$	−37.9411	−1.59903	−0.799514	+	0.600648i	$0.794909\pi$
$564$	0	0
$565$	12.4853	0.525260
$566$	1.65685	0.0696428
$567$	0	0
$568$	16.6274	0.697671
$569$	−16.6863	−0.699526	−0.349763	−	0.936838i	$-0.613738\pi$
$569$	−16.6863	−0.699526	−0.349763	+	0.936838i	$0.613738\pi$
$570$	0	0
$571$	−0.686292	−0.0287204	−0.0143602	−	0.999897i	$-0.504571\pi$
$571$	−0.686292	−0.0287204	−0.0143602	+	0.999897i	$0.504571\pi$
$572$	−7.31371	−0.305802
$573$	0	0
$574$	−3.17157	−0.132379
$575$	−3.65685	−0.152501
$576$	0	0
$577$	−42.4853	−1.76869	−0.884343	−	0.466838i	$-0.845393\pi$
$577$	−42.4853	−1.76869	−0.884343	+	0.466838i	$0.845393\pi$
$578$	−17.2426	−0.717199
$579$	0	0
$580$	−14.6274	−0.607370
$581$	8.00000	0.331896
$582$	0	0
$583$	24.9706	1.03418
$584$	−19.2548	−0.796771
$585$	0	0
$586$	−0.828427	−0.0342220
$587$	−10.6274	−0.438640	−0.219320	−	0.975653i	$-0.570384\pi$
$587$	−10.6274	−0.438640	−0.219320	+	0.975653i	$0.570384\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	1.65685	0.0682116
$591$	0	0
$592$	−18.0000	−0.739795
$593$	5.31371	0.218208	0.109104	−	0.994030i	$-0.465202\pi$
$593$	5.31371	0.218208	0.109104	+	0.994030i	$0.465202\pi$
$594$	0	0
$595$	−7.65685	−0.313900
$596$	−17.6569	−0.723253
$597$	0	0
$598$	1.25483	0.0513140
$599$	8.14214	0.332679	0.166339	−	0.986069i	$-0.446805\pi$
$599$	8.14214	0.332679	0.166339	+	0.986069i	$0.446805\pi$
$600$	0	0
$601$	30.2843	1.23532	0.617661	−	0.786445i	$-0.288081\pi$
$601$	30.2843	1.23532	0.617661	+	0.786445i	$0.288081\pi$
$602$	0.686292	0.0279712
$603$	0	0
$604$	10.3431	0.420857
$605$	12.3137	0.500623
$606$	0	0
$607$	48.9706	1.98765	0.993827	−	0.110942i	$-0.0353867\pi$
$607$	48.9706	1.98765	0.993827	+	0.110942i	$0.0353867\pi$
$608$	12.4853	0.506345
$609$	0	0
$610$	−2.48528	−0.100626
$611$	−3.31371	−0.134058
$612$	0	0
$613$	−18.9706	−0.766214	−0.383107	−	0.923704i	$-0.625146\pi$
$613$	−18.9706	−0.766214	−0.383107	+	0.923704i	$0.625146\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	−7.65685	−0.308503
$617$	−26.8284	−1.08007	−0.540036	−	0.841642i	$-0.681589\pi$
$617$	−26.8284	−1.08007	−0.540036	+	0.841642i	$0.681589\pi$
$618$	0	0
$619$	−11.5147	−0.462816	−0.231408	−	0.972857i	$-0.574333\pi$
$619$	−11.5147	−0.462816	−0.231408	+	0.972857i	$0.574333\pi$
$620$	−15.5147	−0.623086
$621$	0	0
$622$	9.25483	0.371085
$623$	5.31371	0.212889
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.28427	0.251170
$627$	0	0
$628$	16.1421	0.644141
$629$	−45.9411	−1.83179
$630$	0	0
$631$	5.65685	0.225196	0.112598	−	0.993641i	$-0.464083\pi$
$631$	5.65685	0.225196	0.112598	+	0.993641i	$0.464083\pi$
$632$	8.97056	0.356830
$633$	0	0
$634$	5.85786	0.232646
$635$	2.34315	0.0929849
$636$	0	0
$637$	0.828427	0.0328235
$638$	−16.0000	−0.633446
$639$	0	0
$640$	10.5563	0.417276
$641$	−20.9706	−0.828287	−0.414144	−	0.910212i	$-0.635919\pi$
$641$	−20.9706	−0.828287	−0.414144	+	0.910212i	$0.635919\pi$
$642$	0	0
$643$	33.6569	1.32730	0.663648	−	0.748045i	$-0.269007\pi$
$643$	33.6569	1.32730	0.663648	+	0.748045i	$0.269007\pi$
$644$	−6.68629	−0.263477
$645$	0	0
$646$	8.97056	0.352942
$647$	−26.6274	−1.04683	−0.523416	−	0.852077i	$-0.675343\pi$
$647$	−26.6274	−1.04683	−0.523416	+	0.852077i	$0.675343\pi$
$648$	0	0
$649$	−19.3137	−0.758129
$650$	−0.343146	−0.0134593
$651$	0	0
$652$	11.5980	0.454212
$653$	34.1421	1.33609	0.668043	−	0.744123i	$-0.267132\pi$
$653$	34.1421	1.33609	0.668043	+	0.744123i	$0.267132\pi$
$654$	0	0
$655$	−13.6569	−0.533617
$656$	−22.9706	−0.896850
$657$	0	0
$658$	−1.65685	−0.0645909
$659$	3.85786	0.150281	0.0751405	−	0.997173i	$-0.476059\pi$
$659$	3.85786	0.150281	0.0751405	+	0.997173i	$0.476059\pi$
$660$	0	0
$661$	26.6863	1.03798	0.518988	−	0.854781i	$-0.326309\pi$
$661$	26.6863	1.03798	0.518988	+	0.854781i	$0.326309\pi$
$662$	4.97056	0.193186
$663$	0	0
$664$	−12.6863	−0.492324
$665$	2.82843	0.109682
$666$	0	0
$667$	−29.2548	−1.13275
$668$	1.25483	0.0485510
$669$	0	0
$670$	6.34315	0.245057
$671$	28.9706	1.11840
$672$	0	0
$673$	−40.6274	−1.56607	−0.783036	−	0.621976i	$-0.786330\pi$
$673$	−40.6274	−1.56607	−0.783036	+	0.621976i	$0.786330\pi$
$674$	−14.2010	−0.547002
$675$	0	0
$676$	22.5147	0.865951
$677$	−1.31371	−0.0504899	−0.0252450	−	0.999681i	$-0.508037\pi$
$677$	−1.31371	−0.0504899	−0.0252450	+	0.999681i	$0.508037\pi$
$678$	0	0
$679$	3.17157	0.121714
$680$	12.1421	0.465630
$681$	0	0
$682$	−16.9706	−0.649836
$683$	34.9706	1.33811	0.669056	−	0.743212i	$-0.266699\pi$
$683$	34.9706	1.33811	0.669056	+	0.743212i	$0.266699\pi$
$684$	0	0
$685$	22.1421	0.846008
$686$	0.414214	0.0158147
$687$	0	0
$688$	4.97056	0.189501
$689$	4.28427	0.163218
$690$	0	0
$691$	18.1421	0.690159	0.345080	−	0.938573i	$-0.387852\pi$
$691$	18.1421	0.690159	0.345080	+	0.938573i	$0.387852\pi$
$692$	−28.6274	−1.08825
$693$	0	0
$694$	4.82843	0.183285
$695$	1.17157	0.0444403
$696$	0	0
$697$	−58.6274	−2.22067
$698$	2.76955	0.104829
$699$	0	0
$700$	1.82843	0.0691080
$701$	−19.3137	−0.729469	−0.364734	−	0.931112i	$-0.618840\pi$
$701$	−19.3137	−0.729469	−0.364734	+	0.931112i	$0.618840\pi$
$702$	0	0
$703$	16.9706	0.640057
$704$	−20.1421	−0.759135
$705$	0	0
$706$	11.8579	0.446277
$707$	−4.34315	−0.163341
$708$	0	0
$709$	−28.6274	−1.07513	−0.537563	−	0.843224i	$-0.680655\pi$
$709$	−28.6274	−1.07513	−0.537563	+	0.843224i	$0.680655\pi$
$710$	−4.34315	−0.162995
$711$	0	0
$712$	−8.42641	−0.315793
$713$	−31.0294	−1.16206
$714$	0	0
$715$	4.00000	0.149592
$716$	−10.0833	−0.376829
$717$	0	0
$718$	2.68629	0.100252
$719$	43.3137	1.61533	0.807664	−	0.589642i	$-0.200731\pi$
$719$	43.3137	1.61533	0.807664	+	0.589642i	$0.200731\pi$
$720$	0	0
$721$	15.3137	0.570312
$722$	4.55635	0.169570
$723$	0	0
$724$	21.3137	0.792118
$725$	8.00000	0.297113
$726$	0	0
$727$	−17.6569	−0.654856	−0.327428	−	0.944876i	$-0.606182\pi$
$727$	−17.6569	−0.654856	−0.327428	+	0.944876i	$0.606182\pi$
$728$	−1.31371	−0.0486893
$729$	0	0
$730$	5.02944	0.186148
$731$	12.6863	0.469219
$732$	0	0
$733$	−18.4853	−0.682769	−0.341385	−	0.939924i	$-0.610896\pi$
$733$	−18.4853	−0.682769	−0.341385	+	0.939924i	$0.610896\pi$
$734$	3.71573	0.137150
$735$	0	0
$736$	16.1421	0.595007
$737$	−73.9411	−2.72366
$738$	0	0
$739$	26.6274	0.979505	0.489753	−	0.871861i	$-0.337087\pi$
$739$	26.6274	0.979505	0.489753	+	0.871861i	$0.337087\pi$
$740$	10.9706	0.403286
$741$	0	0
$742$	2.14214	0.0786403
$743$	−6.68629	−0.245296	−0.122648	−	0.992450i	$-0.539139\pi$
$743$	−6.68629	−0.245296	−0.122648	+	0.992450i	$0.539139\pi$
$744$	0	0
$745$	9.65685	0.353800
$746$	4.14214	0.151654
$747$	0	0
$748$	−67.5980	−2.47163
$749$	2.00000	0.0730784
$750$	0	0
$751$	1.37258	0.0500863	0.0250431	−	0.999686i	$-0.492028\pi$
$751$	1.37258	0.0500863	0.0250431	+	0.999686i	$0.492028\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−2.74517	−0.0999730
$755$	−5.65685	−0.205874
$756$	0	0
$757$	1.02944	0.0374155	0.0187078	−	0.999825i	$-0.494045\pi$
$757$	1.02944	0.0374155	0.0187078	+	0.999825i	$0.494045\pi$
$758$	−7.31371	−0.265646
$759$	0	0
$760$	−4.48528	−0.162698
$761$	20.6274	0.747743	0.373872	−	0.927480i	$-0.378030\pi$
$761$	20.6274	0.747743	0.373872	+	0.927480i	$0.378030\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	−8.82843	−0.319401
$765$	0	0
$766$	−1.94113	−0.0701357
$767$	−3.31371	−0.119651
$768$	0	0
$769$	−8.34315	−0.300862	−0.150431	−	0.988621i	$-0.548066\pi$
$769$	−8.34315	−0.300862	−0.150431	+	0.988621i	$0.548066\pi$
$770$	2.00000	0.0720750
$771$	0	0
$772$	−27.3726	−0.985161
$773$	47.6569	1.71410	0.857049	−	0.515235i	$-0.172295\pi$
$773$	47.6569	1.71410	0.857049	+	0.515235i	$0.172295\pi$
$774$	0	0
$775$	8.48528	0.304800
$776$	−5.02944	−0.180546
$777$	0	0
$778$	−7.59798	−0.272401
$779$	21.6569	0.775937
$780$	0	0
$781$	50.6274	1.81159
$782$	11.5980	0.414743
$783$	0	0
$784$	3.00000	0.107143
$785$	−8.82843	−0.315100
$786$	0	0
$787$	13.6569	0.486814	0.243407	−	0.969924i	$-0.421735\pi$
$787$	13.6569	0.486814	0.243407	+	0.969924i	$0.421735\pi$
$788$	−5.17157	−0.184230
$789$	0	0
$790$	−2.34315	−0.0833654
$791$	−12.4853	−0.443925
$792$	0	0
$793$	4.97056	0.176510
$794$	−13.3137	−0.472486
$795$	0	0
$796$	−31.3970	−1.11284
$797$	17.0294	0.603214	0.301607	−	0.953432i	$-0.402477\pi$
$797$	17.0294	0.603214	0.301607	+	0.953432i	$0.402477\pi$
$798$	0	0
$799$	−30.6274	−1.08352
$800$	−4.41421	−0.156066
$801$	0	0
$802$	−11.0294	−0.389463
$803$	−58.6274	−2.06892
$804$	0	0
$805$	3.65685	0.128887
$806$	−2.91169	−0.102560
$807$	0	0
$808$	6.88730	0.242294
$809$	−13.6569	−0.480149	−0.240075	−	0.970754i	$-0.577172\pi$
$809$	−13.6569	−0.480149	−0.240075	+	0.970754i	$0.577172\pi$
$810$	0	0
$811$	−30.8284	−1.08253	−0.541266	−	0.840851i	$-0.682055\pi$
$811$	−30.8284	−1.08253	−0.541266	+	0.840851i	$0.682055\pi$
$812$	14.6274	0.513322
$813$	0	0
$814$	12.0000	0.420600
$815$	−6.34315	−0.222191
$816$	0	0
$817$	−4.68629	−0.163953
$818$	−10.4853	−0.366609
$819$	0	0
$820$	14.0000	0.488901
$821$	−37.9411	−1.32415	−0.662077	−	0.749436i	$-0.730325\pi$
$821$	−37.9411	−1.32415	−0.662077	+	0.749436i	$0.730325\pi$
$822$	0	0
$823$	21.6569	0.754910	0.377455	−	0.926028i	$-0.376799\pi$
$823$	21.6569	0.754910	0.377455	+	0.926028i	$0.376799\pi$
$824$	−24.2843	−0.845983
$825$	0	0
$826$	−1.65685	−0.0576493
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	0	0
$829$	−0.343146	−0.0119179	−0.00595897	−	0.999982i	$-0.501897\pi$
$829$	−0.343146	−0.0119179	−0.00595897	+	0.999982i	$0.501897\pi$
$830$	3.31371	0.115021
$831$	0	0
$832$	−3.45584	−0.119810
$833$	7.65685	0.265294
$834$	0	0
$835$	−0.686292	−0.0237501
$836$	24.9706	0.863625
$837$	0	0
$838$	−14.3431	−0.495476
$839$	6.34315	0.218990	0.109495	−	0.993987i	$-0.465077\pi$
$839$	6.34315	0.218990	0.109495	+	0.993987i	$0.465077\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−5.79899	−0.199846
$843$	0	0
$844$	21.9411	0.755245
$845$	−12.3137	−0.423604
$846$	0	0
$847$	−12.3137	−0.423104
$848$	15.5147	0.532778
$849$	0	0
$850$	−3.17157	−0.108784
$851$	21.9411	0.752132
$852$	0	0
$853$	25.5147	0.873607	0.436804	−	0.899557i	$-0.356110\pi$
$853$	25.5147	0.873607	0.436804	+	0.899557i	$0.356110\pi$
$854$	2.48528	0.0850446
$855$	0	0
$856$	−3.17157	−0.108402
$857$	11.6569	0.398191	0.199095	−	0.979980i	$-0.436200\pi$
$857$	11.6569	0.398191	0.199095	+	0.979980i	$0.436200\pi$
$858$	0	0
$859$	−9.85786	−0.336346	−0.168173	−	0.985757i	$-0.553787\pi$
$859$	−9.85786	−0.336346	−0.168173	+	0.985757i	$0.553787\pi$
$860$	−3.02944	−0.103303
$861$	0	0
$862$	−2.97056	−0.101178
$863$	−2.28427	−0.0777575	−0.0388787	−	0.999244i	$-0.512379\pi$
$863$	−2.28427	−0.0777575	−0.0388787	+	0.999244i	$0.512379\pi$
$864$	0	0
$865$	15.6569	0.532349
$866$	5.71573	0.194228
$867$	0	0
$868$	15.5147	0.526604
$869$	27.3137	0.926554
$870$	0	0
$871$	−12.6863	−0.429859
$872$	15.8579	0.537015
$873$	0	0
$874$	−4.28427	−0.144918
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−31.6569	−1.06898	−0.534488	−	0.845176i	$-0.679496\pi$
$877$	−31.6569	−1.06898	−0.534488	+	0.845176i	$0.679496\pi$
$878$	15.5147	0.523596
$879$	0	0
$880$	14.4853	0.488299
$881$	−33.3137	−1.12237	−0.561184	−	0.827691i	$-0.689654\pi$
$881$	−33.3137	−1.12237	−0.561184	+	0.827691i	$0.689654\pi$
$882$	0	0
$883$	−15.3137	−0.515347	−0.257674	−	0.966232i	$-0.582956\pi$
$883$	−15.3137	−0.515347	−0.257674	+	0.966232i	$0.582956\pi$
$884$	−11.5980	−0.390082
$885$	0	0
$886$	−14.4853	−0.486643
$887$	−8.68629	−0.291657	−0.145829	−	0.989310i	$-0.546585\pi$
$887$	−8.68629	−0.291657	−0.145829	+	0.989310i	$0.546585\pi$
$888$	0	0
$889$	−2.34315	−0.0785866
$890$	2.20101	0.0737780
$891$	0	0
$892$	−10.3431	−0.346314
$893$	11.3137	0.378599
$894$	0	0
$895$	5.51472	0.184337
$896$	−10.5563	−0.352663
$897$	0	0
$898$	8.00000	0.266963
$899$	67.8823	2.26400
$900$	0	0
$901$	39.5980	1.31920
$902$	15.3137	0.509891
$903$	0	0
$904$	19.7990	0.658505
$905$	−11.6569	−0.387487
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	36.5685	1.21357
$909$	0	0
$910$	0.343146	0.0113752
$911$	32.1421	1.06492	0.532458	−	0.846456i	$-0.321268\pi$
$911$	32.1421	1.06492	0.532458	+	0.846456i	$0.321268\pi$
$912$	0	0
$913$	−38.6274	−1.27838
$914$	8.82843	0.292018
$915$	0	0
$916$	31.6569	1.04597
$917$	13.6569	0.450989
$918$	0	0
$919$	22.6274	0.746410	0.373205	−	0.927749i	$-0.378259\pi$
$919$	22.6274	0.746410	0.373205	+	0.927749i	$0.378259\pi$
$920$	−5.79899	−0.191187
$921$	0	0
$922$	−6.88730	−0.226821
$923$	8.68629	0.285913
$924$	0	0
$925$	−6.00000	−0.197279
$926$	8.97056	0.294791
$927$	0	0
$928$	−35.3137	−1.15923
$929$	15.9411	0.523011	0.261506	−	0.965202i	$-0.415781\pi$
$929$	15.9411	0.523011	0.261506	+	0.965202i	$0.415781\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	40.4853	1.32614
$933$	0	0
$934$	−12.9706	−0.424410
$935$	36.9706	1.20907
$936$	0	0
$937$	25.7990	0.842816	0.421408	−	0.906871i	$-0.361536\pi$
$937$	25.7990	0.842816	0.421408	+	0.906871i	$0.361536\pi$
$938$	−6.34315	−0.207111
$939$	0	0
$940$	7.31371	0.238547
$941$	−40.6274	−1.32442	−0.662208	−	0.749320i	$-0.730380\pi$
$941$	−40.6274	−1.32442	−0.662208	+	0.749320i	$0.730380\pi$
$942$	0	0
$943$	28.0000	0.911805
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−3.31371	−0.107738
$947$	14.9706	0.486478	0.243239	−	0.969966i	$-0.421790\pi$
$947$	14.9706	0.486478	0.243239	+	0.969966i	$0.421790\pi$
$948$	0	0
$949$	−10.0589	−0.326525
$950$	1.17157	0.0380108
$951$	0	0
$952$	−12.1421	−0.393529
$953$	11.1127	0.359976	0.179988	−	0.983669i	$-0.442394\pi$
$953$	11.1127	0.359976	0.179988	+	0.983669i	$0.442394\pi$
$954$	0	0
$955$	4.82843	0.156244
$956$	44.1421	1.42766
$957$	0	0
$958$	12.5685	0.406071
$959$	−22.1421	−0.715007
$960$	0	0
$961$	41.0000	1.32258
$962$	2.05887	0.0663808
$963$	0	0
$964$	14.0000	0.450910
$965$	14.9706	0.481919
$966$	0	0
$967$	6.62742	0.213123	0.106562	−	0.994306i	$-0.466016\pi$
$967$	6.62742	0.213123	0.106562	+	0.994306i	$0.466016\pi$
$968$	19.5269	0.627619
$969$	0	0
$970$	1.31371	0.0421806
$971$	16.9706	0.544611	0.272306	−	0.962211i	$-0.412214\pi$
$971$	16.9706	0.544611	0.272306	+	0.962211i	$0.412214\pi$
$972$	0	0
$973$	−1.17157	−0.0375589
$974$	0.970563	0.0310988
$975$	0	0
$976$	18.0000	0.576166
$977$	−22.1421	−0.708390	−0.354195	−	0.935172i	$-0.615245\pi$
$977$	−22.1421	−0.708390	−0.354195	+	0.935172i	$0.615245\pi$
$978$	0	0
$979$	−25.6569	−0.819997
$980$	−1.82843	−0.0584070
$981$	0	0
$982$	−5.02944	−0.160496
$983$	25.9411	0.827393	0.413697	−	0.910415i	$-0.364237\pi$
$983$	25.9411	0.827393	0.413697	+	0.910415i	$0.364237\pi$
$984$	0	0
$985$	2.82843	0.0901212
$986$	−25.3726	−0.808028
$987$	0	0
$988$	4.28427	0.136301
$989$	−6.05887	−0.192661
$990$	0	0
$991$	−28.6863	−0.911250	−0.455625	−	0.890172i	$-0.650584\pi$
$991$	−28.6863	−0.911250	−0.455625	+	0.890172i	$0.650584\pi$
$992$	−37.4558	−1.18922
$993$	0	0
$994$	4.34315	0.137756
$995$	17.1716	0.544375
$996$	0	0
$997$	−54.4853	−1.72557	−0.862783	−	0.505574i	$-0.831281\pi$
$997$	−54.4853	−1.72557	−0.862783	+	0.505574i	$0.831281\pi$
$998$	−5.94113	−0.188063
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.a.f.1.1	yes	2
3.2	odd	2		315.2.a.c.1.2	✓	2
4.3	odd	2		5040.2.a.bx.1.1		2
5.2	odd	4		1575.2.d.j.1324.2		4
5.3	odd	4		1575.2.d.j.1324.3		4
5.4	even	2		1575.2.a.m.1.2		2
7.6	odd	2		2205.2.a.y.1.1		2
12.11	even	2		5040.2.a.bu.1.2		2
15.2	even	4		1575.2.d.h.1324.3		4
15.8	even	4		1575.2.d.h.1324.2		4
15.14	odd	2		1575.2.a.u.1.1		2
21.20	even	2		2205.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.a.c.1.2	✓	2	3.2	odd	2
315.2.a.f.1.1	yes	2	1.1	even	1	trivial
1575.2.a.m.1.2		2	5.4	even	2
1575.2.a.u.1.1		2	15.14	odd	2
1575.2.d.h.1324.2		4	15.8	even	4
1575.2.d.h.1324.3		4	15.2	even	4
1575.2.d.j.1324.2		4	5.2	odd	4
1575.2.d.j.1324.3		4	5.3	odd	4
2205.2.a.p.1.2		2	21.20	even	2
2205.2.a.y.1.1		2	7.6	odd	2
5040.2.a.bu.1.2		2	12.11	even	2
5040.2.a.bx.1.1		2	4.3	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} +O(q^{10})\)	\(q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} -0.414214 q^{10} +4.82843 q^{11} +0.828427 q^{13} +0.414214 q^{14} +3.00000 q^{16} +7.65685 q^{17} -2.82843 q^{19} -1.82843 q^{20} -2.00000 q^{22} -3.65685 q^{23} +1.00000 q^{25} -0.343146 q^{26} +1.82843 q^{28} +8.00000 q^{29} +8.48528 q^{31} -4.41421 q^{32} -3.17157 q^{34} -1.00000 q^{35} -6.00000 q^{37} +1.17157 q^{38} +1.58579 q^{40} -7.65685 q^{41} +1.65685 q^{43} -8.82843 q^{44} +1.51472 q^{46} -4.00000 q^{47} +1.00000 q^{49} -0.414214 q^{50} -1.51472 q^{52} +5.17157 q^{53} +4.82843 q^{55} -1.58579 q^{56} -3.31371 q^{58} -4.00000 q^{59} +6.00000 q^{61} -3.51472 q^{62} -4.17157 q^{64} +0.828427 q^{65} -15.3137 q^{67} -14.0000 q^{68} +0.414214 q^{70} +10.4853 q^{71} -12.1421 q^{73} +2.48528 q^{74} +5.17157 q^{76} -4.82843 q^{77} +5.65685 q^{79} +3.00000 q^{80} +3.17157 q^{82} -8.00000 q^{83} +7.65685 q^{85} -0.686292 q^{86} +7.65685 q^{88} -5.31371 q^{89} -0.828427 q^{91} +6.68629 q^{92} +1.65685 q^{94} -2.82843 q^{95} -3.17157 q^{97} -0.414214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 2 q^{10} + 4 q^{11} - 4 q^{13} - 2 q^{14} + 6 q^{16} + 4 q^{17} + 2 q^{20} - 4 q^{22} + 4 q^{23} + 2 q^{25} - 12 q^{26} - 2 q^{28} + 16 q^{29} - 6 q^{32} - 12 q^{34} - 2 q^{35} - 12 q^{37} + 8 q^{38} + 6 q^{40} - 4 q^{41} - 8 q^{43} - 12 q^{44} + 20 q^{46} - 8 q^{47} + 2 q^{49} + 2 q^{50} - 20 q^{52} + 16 q^{53} + 4 q^{55} - 6 q^{56} + 16 q^{58} - 8 q^{59} + 12 q^{61} - 24 q^{62} - 14 q^{64} - 4 q^{65} - 8 q^{67} - 28 q^{68} - 2 q^{70} + 4 q^{71} + 4 q^{73} - 12 q^{74} + 16 q^{76} - 4 q^{77} + 6 q^{80} + 12 q^{82} - 16 q^{83} + 4 q^{85} - 24 q^{86} + 4 q^{88} + 12 q^{89} + 4 q^{91} + 36 q^{92} - 8 q^{94} - 12 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 2 * q^10 + 4 * q^11 - 4 * q^13 - 2 * q^14 + 6 * q^16 + 4 * q^17 + 2 * q^20 - 4 * q^22 + 4 * q^23 + 2 * q^25 - 12 * q^26 - 2 * q^28 + 16 * q^29 - 6 * q^32 - 12 * q^34 - 2 * q^35 - 12 * q^37 + 8 * q^38 + 6 * q^40 - 4 * q^41 - 8 * q^43 - 12 * q^44 + 20 * q^46 - 8 * q^47 + 2 * q^49 + 2 * q^50 - 20 * q^52 + 16 * q^53 + 4 * q^55 - 6 * q^56 + 16 * q^58 - 8 * q^59 + 12 * q^61 - 24 * q^62 - 14 * q^64 - 4 * q^65 - 8 * q^67 - 28 * q^68 - 2 * q^70 + 4 * q^71 + 4 * q^73 - 12 * q^74 + 16 * q^76 - 4 * q^77 + 6 * q^80 + 12 * q^82 - 16 * q^83 + 4 * q^85 - 24 * q^86 + 4 * q^88 + 12 * q^89 + 4 * q^91 + 36 * q^92 - 8 * q^94 - 12 * q^97 + 2 * q^98

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