LMFDB - Embedded newform 1148.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, 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("1148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.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:	$9.16682615204$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.302776 q^{3} +0.302776 q^{5} +1.00000 q^{7} -2.90833 q^{9} +O(q^{10})$	$q+0.302776 q^{3} +0.302776 q^{5} +1.00000 q^{7} -2.90833 q^{9} +1.00000 q^{11} -5.00000 q^{13} +0.0916731 q^{15} -8.21110 q^{17} -6.90833 q^{19} +0.302776 q^{21} +0.302776 q^{23} -4.90833 q^{25} -1.78890 q^{27} +6.69722 q^{29} +10.9083 q^{31} +0.302776 q^{33} +0.302776 q^{35} +6.60555 q^{37} -1.51388 q^{39} -1.00000 q^{41} -4.39445 q^{43} -0.880571 q^{45} -12.6056 q^{47} +1.00000 q^{49} -2.48612 q^{51} -6.30278 q^{53} +0.302776 q^{55} -2.09167 q^{57} -4.90833 q^{59} +4.21110 q^{61} -2.90833 q^{63} -1.51388 q^{65} -8.90833 q^{67} +0.0916731 q^{69} +2.21110 q^{71} +7.00000 q^{73} -1.48612 q^{75} +1.00000 q^{77} -9.39445 q^{79} +8.18335 q^{81} +10.8167 q^{83} -2.48612 q^{85} +2.02776 q^{87} +9.51388 q^{89} -5.00000 q^{91} +3.30278 q^{93} -2.09167 q^{95} -16.1194 q^{97} -2.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	$ 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 10 q^{13} + 11 q^{15} - 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + q^{25} - 18 q^{27} + 17 q^{29} + 11 q^{31} - 3 q^{33} - 3 q^{35} + 6 q^{37} + 15 q^{39} - 2 q^{41} - 16 q^{43} - 27 q^{45} - 18 q^{47} + 2 q^{49} - 23 q^{51} - 9 q^{53} - 3 q^{55} - 15 q^{57} + q^{59} - 6 q^{61} + 5 q^{63} + 15 q^{65} - 7 q^{67} + 11 q^{69} - 10 q^{71} + 14 q^{73} - 21 q^{75} + 2 q^{77} - 26 q^{79} + 38 q^{81} - 23 q^{85} - 32 q^{87} + q^{89} - 10 q^{91} + 3 q^{93} - 15 q^{95} - 7 q^{97} + 5 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 + 2 * q^11 - 10 * q^13 + 11 * q^15 - 2 * q^17 - 3 * q^19 - 3 * q^21 - 3 * q^23 + q^25 - 18 * q^27 + 17 * q^29 + 11 * q^31 - 3 * q^33 - 3 * q^35 + 6 * q^37 + 15 * q^39 - 2 * q^41 - 16 * q^43 - 27 * q^45 - 18 * q^47 + 2 * q^49 - 23 * q^51 - 9 * q^53 - 3 * q^55 - 15 * q^57 + q^59 - 6 * q^61 + 5 * q^63 + 15 * q^65 - 7 * q^67 + 11 * q^69 - 10 * q^71 + 14 * q^73 - 21 * q^75 + 2 * q^77 - 26 * q^79 + 38 * q^81 - 23 * q^85 - 32 * q^87 + q^89 - 10 * q^91 + 3 * q^93 - 15 * q^95 - 7 * q^97 + 5 * q^99

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	0
$2$	0	0
$3$	0.302776	0.174808	0.0874038	−	0.996173i	$-0.472143\pi$
$3$	0.302776	0.174808	0.0874038	+	0.996173i	$0.472143\pi$
$4$	0	0
$5$	0.302776	0.135405	0.0677027	−	0.997706i	$-0.478433\pi$
$5$	0.302776	0.135405	0.0677027	+	0.997706i	$0.478433\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0.0916731	0.0236699
$16$	0	0
$17$	−8.21110	−1.99148	−0.995742	−	0.0921791i	$-0.970617\pi$
$17$	−8.21110	−1.99148	−0.995742	+	0.0921791i	$0.970617\pi$
$18$	0	0
$19$	−6.90833	−1.58488	−0.792439	−	0.609951i	$-0.791189\pi$
$19$	−6.90833	−1.58488	−0.792439	+	0.609951i	$0.791189\pi$
$20$	0	0
$21$	0.302776	0.0660711
$22$	0	0
$23$	0.302776	0.0631331	0.0315665	−	0.999502i	$-0.489950\pi$
$23$	0.302776	0.0631331	0.0315665	+	0.999502i	$0.489950\pi$
$24$	0	0
$25$	−4.90833	−0.981665
$26$	0	0
$27$	−1.78890	−0.344273
$28$	0	0
$29$	6.69722	1.24364	0.621822	−	0.783159i	$-0.286393\pi$
$29$	6.69722	1.24364	0.621822	+	0.783159i	$0.286393\pi$
$30$	0	0
$31$	10.9083	1.95919	0.979597	−	0.200974i	$-0.0644105\pi$
$31$	10.9083	1.95919	0.979597	+	0.200974i	$0.0644105\pi$
$32$	0	0
$33$	0.302776	0.0527065
$34$	0	0
$35$	0.302776	0.0511784
$36$	0	0
$37$	6.60555	1.08595	0.542973	−	0.839750i	$-0.317299\pi$
$37$	6.60555	1.08595	0.542973	+	0.839750i	$0.317299\pi$
$38$	0	0
$39$	−1.51388	−0.242415
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−4.39445	−0.670147	−0.335074	−	0.942192i	$-0.608761\pi$
$43$	−4.39445	−0.670147	−0.335074	+	0.942192i	$0.608761\pi$
$44$	0	0
$45$	−0.880571	−0.131268
$46$	0	0
$47$	−12.6056	−1.83871	−0.919354	−	0.393431i	$-0.871288\pi$
$47$	−12.6056	−1.83871	−0.919354	+	0.393431i	$0.871288\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.48612	−0.348127
$52$	0	0
$53$	−6.30278	−0.865753	−0.432876	−	0.901453i	$-0.642501\pi$
$53$	−6.30278	−0.865753	−0.432876	+	0.901453i	$0.642501\pi$
$54$	0	0
$55$	0.302776	0.0408263
$56$	0	0
$57$	−2.09167	−0.277049
$58$	0	0
$59$	−4.90833	−0.639010	−0.319505	−	0.947585i	$-0.603517\pi$
$59$	−4.90833	−0.639010	−0.319505	+	0.947585i	$0.603517\pi$
$60$	0	0
$61$	4.21110	0.539176	0.269588	−	0.962976i	$-0.413112\pi$
$61$	4.21110	0.539176	0.269588	+	0.962976i	$0.413112\pi$
$62$	0	0
$63$	−2.90833	−0.366415
$64$	0	0
$65$	−1.51388	−0.187773
$66$	0	0
$67$	−8.90833	−1.08833	−0.544163	−	0.838980i	$-0.683153\pi$
$67$	−8.90833	−1.08833	−0.544163	+	0.838980i	$0.683153\pi$
$68$	0	0
$69$	0.0916731	0.0110361
$70$	0	0
$71$	2.21110	0.262410	0.131205	−	0.991355i	$-0.458115\pi$
$71$	2.21110	0.262410	0.131205	+	0.991355i	$0.458115\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	−1.48612	−0.171603
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−9.39445	−1.05696	−0.528479	−	0.848946i	$-0.677237\pi$
$79$	−9.39445	−1.05696	−0.528479	+	0.848946i	$0.677237\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	10.8167	1.18728	0.593641	−	0.804730i	$-0.297690\pi$
$83$	10.8167	1.18728	0.593641	+	0.804730i	$0.297690\pi$
$84$	0	0
$85$	−2.48612	−0.269658
$86$	0	0
$87$	2.02776	0.217398
$88$	0	0
$89$	9.51388	1.00847	0.504235	−	0.863567i	$-0.331775\pi$
$89$	9.51388	1.00847	0.504235	+	0.863567i	$0.331775\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	0	0
$93$	3.30278	0.342482
$94$	0	0
$95$	−2.09167	−0.214601
$96$	0	0
$97$	−16.1194	−1.63668	−0.818340	−	0.574734i	$-0.805105\pi$
$97$	−16.1194	−1.63668	−0.818340	+	0.574734i	$0.805105\pi$
$98$	0	0
$99$	−2.90833	−0.292298
$100$	0	0
$101$	−1.60555	−0.159758	−0.0798792	−	0.996805i	$-0.525453\pi$
$101$	−1.60555	−0.159758	−0.0798792	+	0.996805i	$0.525453\pi$
$102$	0	0
$103$	−11.3028	−1.11370	−0.556848	−	0.830615i	$-0.687989\pi$
$103$	−11.3028	−1.11370	−0.556848	+	0.830615i	$0.687989\pi$
$104$	0	0
$105$	0.0916731	0.00894638
$106$	0	0
$107$	−5.81665	−0.562317	−0.281159	−	0.959661i	$-0.590719\pi$
$107$	−5.81665	−0.562317	−0.281159	+	0.959661i	$0.590719\pi$
$108$	0	0
$109$	−3.21110	−0.307568	−0.153784	−	0.988105i	$-0.549146\pi$
$109$	−3.21110	−0.307568	−0.153784	+	0.988105i	$0.549146\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	20.2111	1.90130	0.950650	−	0.310264i	$-0.100418\pi$
$113$	20.2111	1.90130	0.950650	+	0.310264i	$0.100418\pi$
$114$	0	0
$115$	0.0916731	0.00854856
$116$	0	0
$117$	14.5416	1.34437
$118$	0	0
$119$	−8.21110	−0.752711
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−0.302776	−0.0273004
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−1.33053	−0.117147
$130$	0	0
$131$	11.0917	0.969084	0.484542	−	0.874768i	$-0.338986\pi$
$131$	11.0917	0.969084	0.484542	+	0.874768i	$0.338986\pi$
$132$	0	0
$133$	−6.90833	−0.599028
$134$	0	0
$135$	−0.541635	−0.0466165
$136$	0	0
$137$	12.9083	1.10283	0.551416	−	0.834230i	$-0.314088\pi$
$137$	12.9083	1.10283	0.551416	+	0.834230i	$0.314088\pi$
$138$	0	0
$139$	−8.39445	−0.712008	−0.356004	−	0.934484i	$-0.615861\pi$
$139$	−8.39445	−0.712008	−0.356004	+	0.934484i	$0.615861\pi$
$140$	0	0
$141$	−3.81665	−0.321420
$142$	0	0
$143$	−5.00000	−0.418121
$144$	0	0
$145$	2.02776	0.168396
$146$	0	0
$147$	0.302776	0.0249725
$148$	0	0
$149$	−15.2111	−1.24614	−0.623071	−	0.782165i	$-0.714115\pi$
$149$	−15.2111	−1.24614	−0.623071	+	0.782165i	$0.714115\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	23.8806	1.93063
$154$	0	0
$155$	3.30278	0.265285
$156$	0	0
$157$	10.4222	0.831783	0.415891	−	0.909414i	$-0.363470\pi$
$157$	10.4222	0.831783	0.415891	+	0.909414i	$0.363470\pi$
$158$	0	0
$159$	−1.90833	−0.151340
$160$	0	0
$161$	0.302776	0.0238621
$162$	0	0
$163$	−2.78890	−0.218443	−0.109222	−	0.994017i	$-0.534836\pi$
$163$	−2.78890	−0.218443	−0.109222	+	0.994017i	$0.534836\pi$
$164$	0	0
$165$	0.0916731	0.00713674
$166$	0	0
$167$	−10.2111	−0.790159	−0.395079	−	0.918647i	$-0.629283\pi$
$167$	−10.2111	−0.790159	−0.395079	+	0.918647i	$0.629283\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	20.0917	1.53645
$172$	0	0
$173$	−10.8167	−0.822375	−0.411187	−	0.911551i	$-0.634886\pi$
$173$	−10.8167	−0.822375	−0.411187	+	0.911551i	$0.634886\pi$
$174$	0	0
$175$	−4.90833	−0.371035
$176$	0	0
$177$	−1.48612	−0.111704
$178$	0	0
$179$	16.1194	1.20482	0.602411	−	0.798186i	$-0.294207\pi$
$179$	16.1194	1.20482	0.602411	+	0.798186i	$0.294207\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	1.27502	0.0942521
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	−8.21110	−0.600455
$188$	0	0
$189$	−1.78890	−0.130123
$190$	0	0
$191$	9.21110	0.666492	0.333246	−	0.942840i	$-0.391856\pi$
$191$	9.21110	0.666492	0.333246	+	0.942840i	$0.391856\pi$
$192$	0	0
$193$	15.4222	1.11011	0.555057	−	0.831812i	$-0.312696\pi$
$193$	15.4222	1.11011	0.555057	+	0.831812i	$0.312696\pi$
$194$	0	0
$195$	−0.458365	−0.0328242
$196$	0	0
$197$	8.90833	0.634692	0.317346	−	0.948310i	$-0.397208\pi$
$197$	8.90833	0.634692	0.317346	+	0.948310i	$0.397208\pi$
$198$	0	0
$199$	14.2111	1.00740	0.503699	−	0.863879i	$-0.331972\pi$
$199$	14.2111	1.00740	0.503699	+	0.863879i	$0.331972\pi$
$200$	0	0
$201$	−2.69722	−0.190248
$202$	0	0
$203$	6.69722	0.470053
$204$	0	0
$205$	−0.302776	−0.0211468
$206$	0	0
$207$	−0.880571	−0.0612039
$208$	0	0
$209$	−6.90833	−0.477859
$210$	0	0
$211$	9.11943	0.627807	0.313904	−	0.949455i	$-0.398363\pi$
$211$	9.11943	0.627807	0.313904	+	0.949455i	$0.398363\pi$
$212$	0	0
$213$	0.669468	0.0458712
$214$	0	0
$215$	−1.33053	−0.0907415
$216$	0	0
$217$	10.9083	0.740505
$218$	0	0
$219$	2.11943	0.143218
$220$	0	0
$221$	41.0555	2.76169
$222$	0	0
$223$	−8.30278	−0.555995	−0.277997	−	0.960582i	$-0.589671\pi$
$223$	−8.30278	−0.555995	−0.277997	+	0.960582i	$0.589671\pi$
$224$	0	0
$225$	14.2750	0.951668
$226$	0	0
$227$	−16.8167	−1.11616	−0.558080	−	0.829787i	$-0.688462\pi$
$227$	−16.8167	−1.11616	−0.558080	+	0.829787i	$0.688462\pi$
$228$	0	0
$229$	−21.3944	−1.41378	−0.706892	−	0.707321i	$-0.749904\pi$
$229$	−21.3944	−1.41378	−0.706892	+	0.707321i	$0.749904\pi$
$230$	0	0
$231$	0.302776	0.0199212
$232$	0	0
$233$	−10.0917	−0.661127	−0.330564	−	0.943784i	$-0.607239\pi$
$233$	−10.0917	−0.661127	−0.330564	+	0.943784i	$0.607239\pi$
$234$	0	0
$235$	−3.81665	−0.248971
$236$	0	0
$237$	−2.84441	−0.184764
$238$	0	0
$239$	−3.48612	−0.225498	−0.112749	−	0.993623i	$-0.535966\pi$
$239$	−3.48612	−0.225498	−0.112749	+	0.993623i	$0.535966\pi$
$240$	0	0
$241$	−9.81665	−0.632346	−0.316173	−	0.948702i	$-0.602398\pi$
$241$	−9.81665	−0.632346	−0.316173	+	0.948702i	$0.602398\pi$
$242$	0	0
$243$	7.84441	0.503219
$244$	0	0
$245$	0.302776	0.0193436
$246$	0	0
$247$	34.5416	2.19783
$248$	0	0
$249$	3.27502	0.207546
$250$	0	0
$251$	16.8167	1.06146	0.530729	−	0.847542i	$-0.321918\pi$
$251$	16.8167	1.06146	0.530729	+	0.847542i	$0.321918\pi$
$252$	0	0
$253$	0.302776	0.0190353
$254$	0	0
$255$	−0.752737	−0.0471382
$256$	0	0
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	0	0
$259$	6.60555	0.410449
$260$	0	0
$261$	−19.4777	−1.20564
$262$	0	0
$263$	15.6056	0.962280	0.481140	−	0.876644i	$-0.340223\pi$
$263$	15.6056	0.962280	0.481140	+	0.876644i	$0.340223\pi$
$264$	0	0
$265$	−1.90833	−0.117228
$266$	0	0
$267$	2.88057	0.176288
$268$	0	0
$269$	18.6972	1.13999	0.569995	−	0.821648i	$-0.306945\pi$
$269$	18.6972	1.13999	0.569995	+	0.821648i	$0.306945\pi$
$270$	0	0
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	0	0
$273$	−1.51388	−0.0916241
$274$	0	0
$275$	−4.90833	−0.295983
$276$	0	0
$277$	−4.39445	−0.264037	−0.132018	−	0.991247i	$-0.542146\pi$
$277$	−4.39445	−0.264037	−0.132018	+	0.991247i	$0.542146\pi$
$278$	0	0
$279$	−31.7250	−1.89932
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	0	0
$283$	−31.6333	−1.88040	−0.940202	−	0.340616i	$-0.889364\pi$
$283$	−31.6333	−1.88040	−0.940202	+	0.340616i	$0.889364\pi$
$284$	0	0
$285$	−0.633308	−0.0375139
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	50.4222	2.96601
$290$	0	0
$291$	−4.88057	−0.286104
$292$	0	0
$293$	29.8444	1.74353	0.871765	−	0.489925i	$-0.162976\pi$
$293$	29.8444	1.74353	0.871765	+	0.489925i	$0.162976\pi$
$294$	0	0
$295$	−1.48612	−0.0865254
$296$	0	0
$297$	−1.78890	−0.103802
$298$	0	0
$299$	−1.51388	−0.0875498
$300$	0	0
$301$	−4.39445	−0.253292
$302$	0	0
$303$	−0.486122	−0.0279270
$304$	0	0
$305$	1.27502	0.0730074
$306$	0	0
$307$	−11.2111	−0.639851	−0.319926	−	0.947443i	$-0.603658\pi$
$307$	−11.2111	−0.639851	−0.319926	+	0.947443i	$0.603658\pi$
$308$	0	0
$309$	−3.42221	−0.194682
$310$	0	0
$311$	5.90833	0.335030	0.167515	−	0.985869i	$-0.446426\pi$
$311$	5.90833	0.335030	0.167515	+	0.985869i	$0.446426\pi$
$312$	0	0
$313$	15.6056	0.882078	0.441039	−	0.897488i	$-0.354610\pi$
$313$	15.6056	0.882078	0.441039	+	0.897488i	$0.354610\pi$
$314$	0	0
$315$	−0.880571	−0.0496145
$316$	0	0
$317$	16.2111	0.910506	0.455253	−	0.890362i	$-0.349549\pi$
$317$	16.2111	0.910506	0.455253	+	0.890362i	$0.349549\pi$
$318$	0	0
$319$	6.69722	0.374973
$320$	0	0
$321$	−1.76114	−0.0982973
$322$	0	0
$323$	56.7250	3.15626
$324$	0	0
$325$	24.5416	1.36132
$326$	0	0
$327$	−0.972244	−0.0537652
$328$	0	0
$329$	−12.6056	−0.694967
$330$	0	0
$331$	−27.6333	−1.51886	−0.759432	−	0.650587i	$-0.774523\pi$
$331$	−27.6333	−1.51886	−0.759432	+	0.650587i	$0.774523\pi$
$332$	0	0
$333$	−19.2111	−1.05276
$334$	0	0
$335$	−2.69722	−0.147365
$336$	0	0
$337$	15.2111	0.828602	0.414301	−	0.910140i	$-0.364026\pi$
$337$	15.2111	0.828602	0.414301	+	0.910140i	$0.364026\pi$
$338$	0	0
$339$	6.11943	0.332362
$340$	0	0
$341$	10.9083	0.590719
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.0277564	0.00149435
$346$	0	0
$347$	−26.6056	−1.42826	−0.714130	−	0.700013i	$-0.753178\pi$
$347$	−26.6056	−1.42826	−0.714130	+	0.700013i	$0.753178\pi$
$348$	0	0
$349$	−12.0278	−0.643831	−0.321916	−	0.946768i	$-0.604327\pi$
$349$	−12.0278	−0.643831	−0.321916	+	0.946768i	$0.604327\pi$
$350$	0	0
$351$	8.94449	0.477421
$352$	0	0
$353$	26.8444	1.42878	0.714392	−	0.699746i	$-0.246703\pi$
$353$	26.8444	1.42878	0.714392	+	0.699746i	$0.246703\pi$
$354$	0	0
$355$	0.669468	0.0355317
$356$	0	0
$357$	−2.48612	−0.131580
$358$	0	0
$359$	1.39445	0.0735962	0.0367981	−	0.999323i	$-0.488284\pi$
$359$	1.39445	0.0735962	0.0367981	+	0.999323i	$0.488284\pi$
$360$	0	0
$361$	28.7250	1.51184
$362$	0	0
$363$	−3.02776	−0.158916
$364$	0	0
$365$	2.11943	0.110936
$366$	0	0
$367$	−0.513878	−0.0268242	−0.0134121	−	0.999910i	$-0.504269\pi$
$367$	−0.513878	−0.0268242	−0.0134121	+	0.999910i	$0.504269\pi$
$368$	0	0
$369$	2.90833	0.151401
$370$	0	0
$371$	−6.30278	−0.327224
$372$	0	0
$373$	−18.3028	−0.947682	−0.473841	−	0.880610i	$-0.657133\pi$
$373$	−18.3028	−0.947682	−0.473841	+	0.880610i	$0.657133\pi$
$374$	0	0
$375$	−0.908327	−0.0469058
$376$	0	0
$377$	−33.4861	−1.72462
$378$	0	0
$379$	5.81665	0.298781	0.149391	−	0.988778i	$-0.452269\pi$
$379$	5.81665	0.298781	0.149391	+	0.988778i	$0.452269\pi$
$380$	0	0
$381$	2.11943	0.108582
$382$	0	0
$383$	6.39445	0.326741	0.163371	−	0.986565i	$-0.447763\pi$
$383$	6.39445	0.326741	0.163371	+	0.986565i	$0.447763\pi$
$384$	0	0
$385$	0.302776	0.0154309
$386$	0	0
$387$	12.7805	0.649669
$388$	0	0
$389$	−3.90833	−0.198160	−0.0990800	−	0.995079i	$-0.531590\pi$
$389$	−3.90833	−0.198160	−0.0990800	+	0.995079i	$0.531590\pi$
$390$	0	0
$391$	−2.48612	−0.125729
$392$	0	0
$393$	3.35829	0.169403
$394$	0	0
$395$	−2.84441	−0.143118
$396$	0	0
$397$	−18.7250	−0.939780	−0.469890	−	0.882725i	$-0.655706\pi$
$397$	−18.7250	−0.939780	−0.469890	+	0.882725i	$0.655706\pi$
$398$	0	0
$399$	−2.09167	−0.104715
$400$	0	0
$401$	−23.8167	−1.18935	−0.594673	−	0.803967i	$-0.702719\pi$
$401$	−23.8167	−1.18935	−0.594673	+	0.803967i	$0.702719\pi$
$402$	0	0
$403$	−54.5416	−2.71691
$404$	0	0
$405$	2.47772	0.123119
$406$	0	0
$407$	6.60555	0.327425
$408$	0	0
$409$	−3.60555	−0.178283	−0.0891415	−	0.996019i	$-0.528412\pi$
$409$	−3.60555	−0.178283	−0.0891415	+	0.996019i	$0.528412\pi$
$410$	0	0
$411$	3.90833	0.192784
$412$	0	0
$413$	−4.90833	−0.241523
$414$	0	0
$415$	3.27502	0.160764
$416$	0	0
$417$	−2.54163	−0.124464
$418$	0	0
$419$	11.7250	0.572803	0.286401	−	0.958110i	$-0.407541\pi$
$419$	11.7250	0.572803	0.286401	+	0.958110i	$0.407541\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	36.6611	1.78252
$424$	0	0
$425$	40.3028	1.95497
$426$	0	0
$427$	4.21110	0.203790
$428$	0	0
$429$	−1.51388	−0.0730907
$430$	0	0
$431$	−7.60555	−0.366347	−0.183173	−	0.983081i	$-0.558637\pi$
$431$	−7.60555	−0.366347	−0.183173	+	0.983081i	$0.558637\pi$
$432$	0	0
$433$	−22.9083	−1.10090	−0.550452	−	0.834867i	$-0.685545\pi$
$433$	−22.9083	−1.10090	−0.550452	+	0.834867i	$0.685545\pi$
$434$	0	0
$435$	0.613955	0.0294369
$436$	0	0
$437$	−2.09167	−0.100058
$438$	0	0
$439$	−22.6333	−1.08023	−0.540114	−	0.841592i	$-0.681619\pi$
$439$	−22.6333	−1.08023	−0.540114	+	0.841592i	$0.681619\pi$
$440$	0	0
$441$	−2.90833	−0.138492
$442$	0	0
$443$	−34.5416	−1.64112	−0.820561	−	0.571559i	$-0.806339\pi$
$443$	−34.5416	−1.64112	−0.820561	+	0.571559i	$0.806339\pi$
$444$	0	0
$445$	2.88057	0.136552
$446$	0	0
$447$	−4.60555	−0.217835
$448$	0	0
$449$	−8.78890	−0.414774	−0.207387	−	0.978259i	$-0.566496\pi$
$449$	−8.78890	−0.414774	−0.207387	+	0.978259i	$0.566496\pi$
$450$	0	0
$451$	−1.00000	−0.0470882
$452$	0	0
$453$	−1.51388	−0.0711282
$454$	0	0
$455$	−1.51388	−0.0709717
$456$	0	0
$457$	30.8167	1.44154	0.720771	−	0.693173i	$-0.243788\pi$
$457$	30.8167	1.44154	0.720771	+	0.693173i	$0.243788\pi$
$458$	0	0
$459$	14.6888	0.685615
$460$	0	0
$461$	18.8167	0.876379	0.438189	−	0.898883i	$-0.355620\pi$
$461$	18.8167	0.876379	0.438189	+	0.898883i	$0.355620\pi$
$462$	0	0
$463$	18.9361	0.880034	0.440017	−	0.897989i	$-0.354972\pi$
$463$	18.9361	0.880034	0.440017	+	0.897989i	$0.354972\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	8.21110	0.379965	0.189982	−	0.981788i	$-0.439157\pi$
$467$	8.21110	0.379965	0.189982	+	0.981788i	$0.439157\pi$
$468$	0	0
$469$	−8.90833	−0.411348
$470$	0	0
$471$	3.15559	0.145402
$472$	0	0
$473$	−4.39445	−0.202057
$474$	0	0
$475$	33.9083	1.55582
$476$	0	0
$477$	18.3305	0.839297
$478$	0	0
$479$	40.1194	1.83310	0.916552	−	0.399916i	$-0.130961\pi$
$479$	40.1194	1.83310	0.916552	+	0.399916i	$0.130961\pi$
$480$	0	0
$481$	−33.0278	−1.50594
$482$	0	0
$483$	0.0916731	0.00417127
$484$	0	0
$485$	−4.88057	−0.221615
$486$	0	0
$487$	−17.3944	−0.788218	−0.394109	−	0.919064i	$-0.628947\pi$
$487$	−17.3944	−0.788218	−0.394109	+	0.919064i	$0.628947\pi$
$488$	0	0
$489$	−0.844410	−0.0381855
$490$	0	0
$491$	−7.72498	−0.348624	−0.174312	−	0.984691i	$-0.555770\pi$
$491$	−7.72498	−0.348624	−0.174312	+	0.984691i	$0.555770\pi$
$492$	0	0
$493$	−54.9916	−2.47670
$494$	0	0
$495$	−0.880571	−0.0395787
$496$	0	0
$497$	2.21110	0.0991815
$498$	0	0
$499$	−30.2389	−1.35368	−0.676839	−	0.736131i	$-0.736651\pi$
$499$	−30.2389	−1.35368	−0.676839	+	0.736131i	$0.736651\pi$
$500$	0	0
$501$	−3.09167	−0.138126
$502$	0	0
$503$	−12.6972	−0.566141	−0.283071	−	0.959099i	$-0.591353\pi$
$503$	−12.6972	−0.566141	−0.283071	+	0.959099i	$0.591353\pi$
$504$	0	0
$505$	−0.486122	−0.0216321
$506$	0	0
$507$	3.63331	0.161361
$508$	0	0
$509$	−24.0917	−1.06784	−0.533922	−	0.845534i	$-0.679283\pi$
$509$	−24.0917	−1.06784	−0.533922	+	0.845534i	$0.679283\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	12.3583	0.545632
$514$	0	0
$515$	−3.42221	−0.150800
$516$	0	0
$517$	−12.6056	−0.554392
$518$	0	0
$519$	−3.27502	−0.143757
$520$	0	0
$521$	10.3028	0.451373	0.225686	−	0.974200i	$-0.427538\pi$
$521$	10.3028	0.451373	0.225686	+	0.974200i	$0.427538\pi$
$522$	0	0
$523$	−29.6056	−1.29456	−0.647280	−	0.762252i	$-0.724094\pi$
$523$	−29.6056	−1.29456	−0.647280	+	0.762252i	$0.724094\pi$
$524$	0	0
$525$	−1.48612	−0.0648597
$526$	0	0
$527$	−89.5694	−3.90170
$528$	0	0
$529$	−22.9083	−0.996014
$530$	0	0
$531$	14.2750	0.619483
$532$	0	0
$533$	5.00000	0.216574
$534$	0	0
$535$	−1.76114	−0.0761408
$536$	0	0
$537$	4.88057	0.210612
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−40.3583	−1.73514	−0.867569	−	0.497317i	$-0.834319\pi$
$541$	−40.3583	−1.73514	−0.867569	+	0.497317i	$0.834319\pi$
$542$	0	0
$543$	−3.02776	−0.129933
$544$	0	0
$545$	−0.972244	−0.0416463
$546$	0	0
$547$	−2.57779	−0.110219	−0.0551093	−	0.998480i	$-0.517551\pi$
$547$	−2.57779	−0.110219	−0.0551093	+	0.998480i	$0.517551\pi$
$548$	0	0
$549$	−12.2473	−0.522700
$550$	0	0
$551$	−46.2666	−1.97102
$552$	0	0
$553$	−9.39445	−0.399493
$554$	0	0
$555$	0.605551	0.0257042
$556$	0	0
$557$	13.5416	0.573777	0.286889	−	0.957964i	$-0.407379\pi$
$557$	13.5416	0.573777	0.286889	+	0.957964i	$0.407379\pi$
$558$	0	0
$559$	21.9722	0.929327
$560$	0	0
$561$	−2.48612	−0.104964
$562$	0	0
$563$	−34.6333	−1.45962	−0.729810	−	0.683650i	$-0.760391\pi$
$563$	−34.6333	−1.45962	−0.729810	+	0.683650i	$0.760391\pi$
$564$	0	0
$565$	6.11943	0.257446
$566$	0	0
$567$	8.18335	0.343668
$568$	0	0
$569$	22.5416	0.944994	0.472497	−	0.881332i	$-0.343353\pi$
$569$	22.5416	0.944994	0.472497	+	0.881332i	$0.343353\pi$
$570$	0	0
$571$	45.6333	1.90969	0.954847	−	0.297097i	$-0.0960185\pi$
$571$	45.6333	1.90969	0.954847	+	0.297097i	$0.0960185\pi$
$572$	0	0
$573$	2.78890	0.116508
$574$	0	0
$575$	−1.48612	−0.0619756
$576$	0	0
$577$	7.42221	0.308990	0.154495	−	0.987994i	$-0.450625\pi$
$577$	7.42221	0.308990	0.154495	+	0.987994i	$0.450625\pi$
$578$	0	0
$579$	4.66947	0.194056
$580$	0	0
$581$	10.8167	0.448750
$582$	0	0
$583$	−6.30278	−0.261034
$584$	0	0
$585$	4.40285	0.182036
$586$	0	0
$587$	−33.2389	−1.37191	−0.685957	−	0.727642i	$-0.740616\pi$
$587$	−33.2389	−1.37191	−0.685957	+	0.727642i	$0.740616\pi$
$588$	0	0
$589$	−75.3583	−3.10508
$590$	0	0
$591$	2.69722	0.110949
$592$	0	0
$593$	−24.8444	−1.02024	−0.510119	−	0.860104i	$-0.670399\pi$
$593$	−24.8444	−1.02024	−0.510119	+	0.860104i	$0.670399\pi$
$594$	0	0
$595$	−2.48612	−0.101921
$596$	0	0
$597$	4.30278	0.176101
$598$	0	0
$599$	−1.81665	−0.0742265	−0.0371132	−	0.999311i	$-0.511816\pi$
$599$	−1.81665	−0.0742265	−0.0371132	+	0.999311i	$0.511816\pi$
$600$	0	0
$601$	9.30278	0.379468	0.189734	−	0.981836i	$-0.439237\pi$
$601$	9.30278	0.379468	0.189734	+	0.981836i	$0.439237\pi$
$602$	0	0
$603$	25.9083	1.05507
$604$	0	0
$605$	−3.02776	−0.123096
$606$	0	0
$607$	12.0278	0.488192	0.244096	−	0.969751i	$-0.421509\pi$
$607$	12.0278	0.488192	0.244096	+	0.969751i	$0.421509\pi$
$608$	0	0
$609$	2.02776	0.0821688
$610$	0	0
$611$	63.0278	2.54983
$612$	0	0
$613$	31.1472	1.25802	0.629011	−	0.777396i	$-0.283460\pi$
$613$	31.1472	1.25802	0.629011	+	0.777396i	$0.283460\pi$
$614$	0	0
$615$	−0.0916731	−0.00369662
$616$	0	0
$617$	7.97224	0.320950	0.160475	−	0.987040i	$-0.448697\pi$
$617$	7.97224	0.320950	0.160475	+	0.987040i	$0.448697\pi$
$618$	0	0
$619$	−32.5139	−1.30684	−0.653422	−	0.756994i	$-0.726667\pi$
$619$	−32.5139	−1.30684	−0.653422	+	0.756994i	$0.726667\pi$
$620$	0	0
$621$	−0.541635	−0.0217350
$622$	0	0
$623$	9.51388	0.381165
$624$	0	0
$625$	23.6333	0.945332
$626$	0	0
$627$	−2.09167	−0.0835334
$628$	0	0
$629$	−54.2389	−2.16264
$630$	0	0
$631$	−35.8444	−1.42694	−0.713472	−	0.700684i	$-0.752878\pi$
$631$	−35.8444	−1.42694	−0.713472	+	0.700684i	$0.752878\pi$
$632$	0	0
$633$	2.76114	0.109746
$634$	0	0
$635$	2.11943	0.0841070
$636$	0	0
$637$	−5.00000	−0.198107
$638$	0	0
$639$	−6.43061	−0.254391
$640$	0	0
$641$	42.6333	1.68391	0.841957	−	0.539544i	$-0.181403\pi$
$641$	42.6333	1.68391	0.841957	+	0.539544i	$0.181403\pi$
$642$	0	0
$643$	41.9638	1.65489	0.827446	−	0.561545i	$-0.189793\pi$
$643$	41.9638	1.65489	0.827446	+	0.561545i	$0.189793\pi$
$644$	0	0
$645$	−0.402853	−0.0158623
$646$	0	0
$647$	30.0278	1.18051	0.590256	−	0.807216i	$-0.299027\pi$
$647$	30.0278	1.18051	0.590256	+	0.807216i	$0.299027\pi$
$648$	0	0
$649$	−4.90833	−0.192669
$650$	0	0
$651$	3.30278	0.129446
$652$	0	0
$653$	2.72498	0.106637	0.0533184	−	0.998578i	$-0.483020\pi$
$653$	2.72498	0.106637	0.0533184	+	0.998578i	$0.483020\pi$
$654$	0	0
$655$	3.35829	0.131219
$656$	0	0
$657$	−20.3583	−0.794252
$658$	0	0
$659$	19.6056	0.763724	0.381862	−	0.924219i	$-0.375283\pi$
$659$	19.6056	0.763724	0.381862	+	0.924219i	$0.375283\pi$
$660$	0	0
$661$	21.3944	0.832148	0.416074	−	0.909331i	$-0.363406\pi$
$661$	21.3944	0.832148	0.416074	+	0.909331i	$0.363406\pi$
$662$	0	0
$663$	12.4306	0.482765
$664$	0	0
$665$	−2.09167	−0.0811116
$666$	0	0
$667$	2.02776	0.0785150
$668$	0	0
$669$	−2.51388	−0.0971921
$670$	0	0
$671$	4.21110	0.162568
$672$	0	0
$673$	17.1833	0.662369	0.331185	−	0.943566i	$-0.392552\pi$
$673$	17.1833	0.662369	0.331185	+	0.943566i	$0.392552\pi$
$674$	0	0
$675$	8.78049	0.337961
$676$	0	0
$677$	−30.3583	−1.16676	−0.583382	−	0.812198i	$-0.698271\pi$
$677$	−30.3583	−1.16676	−0.583382	+	0.812198i	$0.698271\pi$
$678$	0	0
$679$	−16.1194	−0.618607
$680$	0	0
$681$	−5.09167	−0.195113
$682$	0	0
$683$	−38.6333	−1.47826	−0.739131	−	0.673561i	$-0.764764\pi$
$683$	−38.6333	−1.47826	−0.739131	+	0.673561i	$0.764764\pi$
$684$	0	0
$685$	3.90833	0.149329
$686$	0	0
$687$	−6.47772	−0.247140
$688$	0	0
$689$	31.5139	1.20058
$690$	0	0
$691$	−25.3028	−0.962563	−0.481281	−	0.876566i	$-0.659829\pi$
$691$	−25.3028	−0.962563	−0.481281	+	0.876566i	$0.659829\pi$
$692$	0	0
$693$	−2.90833	−0.110478
$694$	0	0
$695$	−2.54163	−0.0964097
$696$	0	0
$697$	8.21110	0.311018
$698$	0	0
$699$	−3.05551	−0.115570
$700$	0	0
$701$	−22.9083	−0.865236	−0.432618	−	0.901577i	$-0.642410\pi$
$701$	−22.9083	−0.865236	−0.432618	+	0.901577i	$0.642410\pi$
$702$	0	0
$703$	−45.6333	−1.72109
$704$	0	0
$705$	−1.15559	−0.0435220
$706$	0	0
$707$	−1.60555	−0.0603830
$708$	0	0
$709$	−6.60555	−0.248077	−0.124038	−	0.992277i	$-0.539585\pi$
$709$	−6.60555	−0.248077	−0.124038	+	0.992277i	$0.539585\pi$
$710$	0	0
$711$	27.3221	1.02466
$712$	0	0
$713$	3.30278	0.123690
$714$	0	0
$715$	−1.51388	−0.0566158
$716$	0	0
$717$	−1.05551	−0.0394188
$718$	0	0
$719$	37.2389	1.38878	0.694388	−	0.719601i	$-0.255675\pi$
$719$	37.2389	1.38878	0.694388	+	0.719601i	$0.255675\pi$
$720$	0	0
$721$	−11.3028	−0.420937
$722$	0	0
$723$	−2.97224	−0.110539
$724$	0	0
$725$	−32.8722	−1.22084
$726$	0	0
$727$	5.66947	0.210269	0.105134	−	0.994458i	$-0.466473\pi$
$727$	5.66947	0.210269	0.105134	+	0.994458i	$0.466473\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	36.0833	1.33459
$732$	0	0
$733$	−37.4222	−1.38222	−0.691110	−	0.722749i	$-0.742878\pi$
$733$	−37.4222	−1.38222	−0.691110	+	0.722749i	$0.742878\pi$
$734$	0	0
$735$	0.0916731	0.00338141
$736$	0	0
$737$	−8.90833	−0.328142
$738$	0	0
$739$	18.5139	0.681044	0.340522	−	0.940237i	$-0.389396\pi$
$739$	18.5139	0.681044	0.340522	+	0.940237i	$0.389396\pi$
$740$	0	0
$741$	10.4584	0.384198
$742$	0	0
$743$	−38.5694	−1.41497	−0.707487	−	0.706726i	$-0.750171\pi$
$743$	−38.5694	−1.41497	−0.707487	+	0.706726i	$0.750171\pi$
$744$	0	0
$745$	−4.60555	−0.168734
$746$	0	0
$747$	−31.4584	−1.15100
$748$	0	0
$749$	−5.81665	−0.212536
$750$	0	0
$751$	−31.2389	−1.13992	−0.569961	−	0.821672i	$-0.693042\pi$
$751$	−31.2389	−1.13992	−0.569961	+	0.821672i	$0.693042\pi$
$752$	0	0
$753$	5.09167	0.185551
$754$	0	0
$755$	−1.51388	−0.0550957
$756$	0	0
$757$	29.0278	1.05503	0.527516	−	0.849545i	$-0.323124\pi$
$757$	29.0278	1.05503	0.527516	+	0.849545i	$0.323124\pi$
$758$	0	0
$759$	0.0916731	0.00332752
$760$	0	0
$761$	32.0000	1.16000	0.580000	−	0.814617i	$-0.303053\pi$
$761$	32.0000	1.16000	0.580000	+	0.814617i	$0.303053\pi$
$762$	0	0
$763$	−3.21110	−0.116250
$764$	0	0
$765$	7.23045	0.261418
$766$	0	0
$767$	24.5416	0.886147
$768$	0	0
$769$	−2.97224	−0.107182	−0.0535909	−	0.998563i	$-0.517067\pi$
$769$	−2.97224	−0.107182	−0.0535909	+	0.998563i	$0.517067\pi$
$770$	0	0
$771$	−5.75274	−0.207180
$772$	0	0
$773$	−21.8444	−0.785689	−0.392844	−	0.919605i	$-0.628509\pi$
$773$	−21.8444	−0.785689	−0.392844	+	0.919605i	$0.628509\pi$
$774$	0	0
$775$	−53.5416	−1.92327
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	6.90833	0.247516
$780$	0	0
$781$	2.21110	0.0791195
$782$	0	0
$783$	−11.9806	−0.428153
$784$	0	0
$785$	3.15559	0.112628
$786$	0	0
$787$	−10.6056	−0.378047	−0.189024	−	0.981973i	$-0.560532\pi$
$787$	−10.6056	−0.378047	−0.189024	+	0.981973i	$0.560532\pi$
$788$	0	0
$789$	4.72498	0.168214
$790$	0	0
$791$	20.2111	0.718624
$792$	0	0
$793$	−21.0555	−0.747703
$794$	0	0
$795$	−0.577795	−0.0204923
$796$	0	0
$797$	−28.6333	−1.01424	−0.507122	−	0.861874i	$-0.669291\pi$
$797$	−28.6333	−1.01424	−0.507122	+	0.861874i	$0.669291\pi$
$798$	0	0
$799$	103.505	3.66176
$800$	0	0
$801$	−27.6695	−0.977653
$802$	0	0
$803$	7.00000	0.247025
$804$	0	0
$805$	0.0916731	0.00323105
$806$	0	0
$807$	5.66106	0.199279
$808$	0	0
$809$	−45.3305	−1.59374	−0.796868	−	0.604153i	$-0.793512\pi$
$809$	−45.3305	−1.59374	−0.796868	+	0.604153i	$0.793512\pi$
$810$	0	0
$811$	−3.00000	−0.105344	−0.0526721	−	0.998612i	$-0.516774\pi$
$811$	−3.00000	−0.105344	−0.0526721	+	0.998612i	$0.516774\pi$
$812$	0	0
$813$	−5.75274	−0.201757
$814$	0	0
$815$	−0.844410	−0.0295784
$816$	0	0
$817$	30.3583	1.06210
$818$	0	0
$819$	14.5416	0.508126
$820$	0	0
$821$	39.8444	1.39058	0.695290	−	0.718730i	$-0.255276\pi$
$821$	39.8444	1.39058	0.695290	+	0.718730i	$0.255276\pi$
$822$	0	0
$823$	−31.4861	−1.09754	−0.548769	−	0.835974i	$-0.684903\pi$
$823$	−31.4861	−1.09754	−0.548769	+	0.835974i	$0.684903\pi$
$824$	0	0
$825$	−1.48612	−0.0517401
$826$	0	0
$827$	9.78890	0.340393	0.170197	−	0.985410i	$-0.445560\pi$
$827$	9.78890	0.340393	0.170197	+	0.985410i	$0.445560\pi$
$828$	0	0
$829$	−42.6333	−1.48072	−0.740358	−	0.672213i	$-0.765344\pi$
$829$	−42.6333	−1.48072	−0.740358	+	0.672213i	$0.765344\pi$
$830$	0	0
$831$	−1.33053	−0.0461556
$832$	0	0
$833$	−8.21110	−0.284498
$834$	0	0
$835$	−3.09167	−0.106992
$836$	0	0
$837$	−19.5139	−0.674498
$838$	0	0
$839$	−19.3944	−0.669571	−0.334785	−	0.942294i	$-0.608664\pi$
$839$	−19.3944	−0.669571	−0.334785	+	0.942294i	$0.608664\pi$
$840$	0	0
$841$	15.8528	0.546649
$842$	0	0
$843$	−1.51388	−0.0521407
$844$	0	0
$845$	3.63331	0.124990
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	−9.57779	−0.328709
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−53.6333	−1.83637	−0.918185	−	0.396152i	$-0.870345\pi$
$853$	−53.6333	−1.83637	−0.918185	+	0.396152i	$0.870345\pi$
$854$	0	0
$855$	6.08327	0.208043
$856$	0	0
$857$	−27.4861	−0.938908	−0.469454	−	0.882957i	$-0.655549\pi$
$857$	−27.4861	−0.938908	−0.469454	+	0.882957i	$0.655549\pi$
$858$	0	0
$859$	11.5139	0.392848	0.196424	−	0.980519i	$-0.437067\pi$
$859$	11.5139	0.392848	0.196424	+	0.980519i	$0.437067\pi$
$860$	0	0
$861$	−0.302776	−0.0103186
$862$	0	0
$863$	41.8444	1.42440	0.712200	−	0.701976i	$-0.247699\pi$
$863$	41.8444	1.42440	0.712200	+	0.701976i	$0.247699\pi$
$864$	0	0
$865$	−3.27502	−0.111354
$866$	0	0
$867$	15.2666	0.518481
$868$	0	0
$869$	−9.39445	−0.318685
$870$	0	0
$871$	44.5416	1.50924
$872$	0	0
$873$	46.8806	1.58667
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−27.9638	−0.944272	−0.472136	−	0.881526i	$-0.656517\pi$
$877$	−27.9638	−0.944272	−0.472136	+	0.881526i	$0.656517\pi$
$878$	0	0
$879$	9.03616	0.304782
$880$	0	0
$881$	−8.78890	−0.296105	−0.148053	−	0.988979i	$-0.547301\pi$
$881$	−8.78890	−0.296105	−0.148053	+	0.988979i	$0.547301\pi$
$882$	0	0
$883$	−43.5694	−1.46623	−0.733113	−	0.680106i	$-0.761934\pi$
$883$	−43.5694	−1.46623	−0.733113	+	0.680106i	$0.761934\pi$
$884$	0	0
$885$	−0.449961	−0.0151253
$886$	0	0
$887$	40.8444	1.37142	0.685711	−	0.727874i	$-0.259492\pi$
$887$	40.8444	1.37142	0.685711	+	0.727874i	$0.259492\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	0	0
$891$	8.18335	0.274152
$892$	0	0
$893$	87.0833	2.91413
$894$	0	0
$895$	4.88057	0.163139
$896$	0	0
$897$	−0.458365	−0.0153044
$898$	0	0
$899$	73.0555	2.43654
$900$	0	0
$901$	51.7527	1.72413
$902$	0	0
$903$	−1.33053	−0.0442773
$904$	0	0
$905$	−3.02776	−0.100646
$906$	0	0
$907$	4.88057	0.162057	0.0810283	−	0.996712i	$-0.474180\pi$
$907$	4.88057	0.162057	0.0810283	+	0.996712i	$0.474180\pi$
$908$	0	0
$909$	4.66947	0.154876
$910$	0	0
$911$	28.8444	0.955658	0.477829	−	0.878453i	$-0.341424\pi$
$911$	28.8444	0.955658	0.477829	+	0.878453i	$0.341424\pi$
$912$	0	0
$913$	10.8167	0.357979
$914$	0	0
$915$	0.386045	0.0127622
$916$	0	0
$917$	11.0917	0.366279
$918$	0	0
$919$	−16.5139	−0.544743	−0.272371	−	0.962192i	$-0.587808\pi$
$919$	−16.5139	−0.544743	−0.272371	+	0.962192i	$0.587808\pi$
$920$	0	0
$921$	−3.39445	−0.111851
$922$	0	0
$923$	−11.0555	−0.363897
$924$	0	0
$925$	−32.4222	−1.06604
$926$	0	0
$927$	32.8722	1.07966
$928$	0	0
$929$	21.5416	0.706758	0.353379	−	0.935480i	$-0.385033\pi$
$929$	21.5416	0.706758	0.353379	+	0.935480i	$0.385033\pi$
$930$	0	0
$931$	−6.90833	−0.226411
$932$	0	0
$933$	1.78890	0.0585659
$934$	0	0
$935$	−2.48612	−0.0813049
$936$	0	0
$937$	−1.97224	−0.0644304	−0.0322152	−	0.999481i	$-0.510256\pi$
$937$	−1.97224	−0.0644304	−0.0322152	+	0.999481i	$0.510256\pi$
$938$	0	0
$939$	4.72498	0.154194
$940$	0	0
$941$	−1.69722	−0.0553279	−0.0276640	−	0.999617i	$-0.508807\pi$
$941$	−1.69722	−0.0553279	−0.0276640	+	0.999617i	$0.508807\pi$
$942$	0	0
$943$	−0.302776	−0.00985973
$944$	0	0
$945$	−0.541635	−0.0176194
$946$	0	0
$947$	44.0917	1.43279	0.716393	−	0.697697i	$-0.245792\pi$
$947$	44.0917	1.43279	0.716393	+	0.697697i	$0.245792\pi$
$948$	0	0
$949$	−35.0000	−1.13615
$950$	0	0
$951$	4.90833	0.159163
$952$	0	0
$953$	−10.1472	−0.328700	−0.164350	−	0.986402i	$-0.552553\pi$
$953$	−10.1472	−0.328700	−0.164350	+	0.986402i	$0.552553\pi$
$954$	0	0
$955$	2.78890	0.0902466
$956$	0	0
$957$	2.02776	0.0655481
$958$	0	0
$959$	12.9083	0.416832
$960$	0	0
$961$	87.9916	2.83844
$962$	0	0
$963$	16.9167	0.545134
$964$	0	0
$965$	4.66947	0.150315
$966$	0	0
$967$	−9.57779	−0.308001	−0.154001	−	0.988071i	$-0.549216\pi$
$967$	−9.57779	−0.308001	−0.154001	+	0.988071i	$0.549216\pi$
$968$	0	0
$969$	17.1749	0.551739
$970$	0	0
$971$	−15.9722	−0.512574	−0.256287	−	0.966601i	$-0.582499\pi$
$971$	−15.9722	−0.512574	−0.256287	+	0.966601i	$0.582499\pi$
$972$	0	0
$973$	−8.39445	−0.269114
$974$	0	0
$975$	7.43061	0.237970
$976$	0	0
$977$	−23.3028	−0.745522	−0.372761	−	0.927927i	$-0.621589\pi$
$977$	−23.3028	−0.745522	−0.372761	+	0.927927i	$0.621589\pi$
$978$	0	0
$979$	9.51388	0.304065
$980$	0	0
$981$	9.33894	0.298169
$982$	0	0
$983$	32.0278	1.02153	0.510763	−	0.859721i	$-0.329363\pi$
$983$	32.0278	1.02153	0.510763	+	0.859721i	$0.329363\pi$
$984$	0	0
$985$	2.69722	0.0859407
$986$	0	0
$987$	−3.81665	−0.121485
$988$	0	0
$989$	−1.33053	−0.0423085
$990$	0	0
$991$	15.1833	0.482315	0.241157	−	0.970486i	$-0.422473\pi$
$991$	15.1833	0.482315	0.241157	+	0.970486i	$0.422473\pi$
$992$	0	0
$993$	−8.36669	−0.265509
$994$	0	0
$995$	4.30278	0.136407
$996$	0	0
$997$	−37.7250	−1.19476	−0.597381	−	0.801958i	$-0.703792\pi$
$997$	−37.7250	−1.19476	−0.597381	+	0.801958i	$0.703792\pi$
$998$	0	0
$999$	−11.8167	−0.373862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.a.a.1.2	✓	2
4.3	odd	2		4592.2.a.o.1.1		2
7.6	odd	2		8036.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.a.1.2	✓	2	1.1	even	1	trivial
4592.2.a.o.1.1		2	4.3	odd	2
8036.2.a.g.1.1		2	7.6	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+0.302776 q^{3} +0.302776 q^{5} +1.00000 q^{7} -2.90833 q^{9} +O(q^{10})\)	\(q+0.302776 q^{3} +0.302776 q^{5} +1.00000 q^{7} -2.90833 q^{9} +1.00000 q^{11} -5.00000 q^{13} +0.0916731 q^{15} -8.21110 q^{17} -6.90833 q^{19} +0.302776 q^{21} +0.302776 q^{23} -4.90833 q^{25} -1.78890 q^{27} +6.69722 q^{29} +10.9083 q^{31} +0.302776 q^{33} +0.302776 q^{35} +6.60555 q^{37} -1.51388 q^{39} -1.00000 q^{41} -4.39445 q^{43} -0.880571 q^{45} -12.6056 q^{47} +1.00000 q^{49} -2.48612 q^{51} -6.30278 q^{53} +0.302776 q^{55} -2.09167 q^{57} -4.90833 q^{59} +4.21110 q^{61} -2.90833 q^{63} -1.51388 q^{65} -8.90833 q^{67} +0.0916731 q^{69} +2.21110 q^{71} +7.00000 q^{73} -1.48612 q^{75} +1.00000 q^{77} -9.39445 q^{79} +8.18335 q^{81} +10.8167 q^{83} -2.48612 q^{85} +2.02776 q^{87} +9.51388 q^{89} -5.00000 q^{91} +3.30278 q^{93} -2.09167 q^{95} -16.1194 q^{97} -2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	\( 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 10 q^{13} + 11 q^{15} - 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + q^{25} - 18 q^{27} + 17 q^{29} + 11 q^{31} - 3 q^{33} - 3 q^{35} + 6 q^{37} + 15 q^{39} - 2 q^{41} - 16 q^{43} - 27 q^{45} - 18 q^{47} + 2 q^{49} - 23 q^{51} - 9 q^{53} - 3 q^{55} - 15 q^{57} + q^{59} - 6 q^{61} + 5 q^{63} + 15 q^{65} - 7 q^{67} + 11 q^{69} - 10 q^{71} + 14 q^{73} - 21 q^{75} + 2 q^{77} - 26 q^{79} + 38 q^{81} - 23 q^{85} - 32 q^{87} + q^{89} - 10 q^{91} + 3 q^{93} - 15 q^{95} - 7 q^{97} + 5 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 + 2 * q^11 - 10 * q^13 + 11 * q^15 - 2 * q^17 - 3 * q^19 - 3 * q^21 - 3 * q^23 + q^25 - 18 * q^27 + 17 * q^29 + 11 * q^31 - 3 * q^33 - 3 * q^35 + 6 * q^37 + 15 * q^39 - 2 * q^41 - 16 * q^43 - 27 * q^45 - 18 * q^47 + 2 * q^49 - 23 * q^51 - 9 * q^53 - 3 * q^55 - 15 * q^57 + q^59 - 6 * q^61 + 5 * q^63 + 15 * q^65 - 7 * q^67 + 11 * q^69 - 10 * q^71 + 14 * q^73 - 21 * q^75 + 2 * q^77 - 26 * q^79 + 38 * q^81 - 23 * q^85 - 32 * q^87 + q^89 - 10 * q^91 + 3 * q^93 - 15 * q^95 - 7 * q^97 + 5 * q^99

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