LMFDB - Embedded newform 276.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [276,2,Mod(1,276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(276, 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("276.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 276 = 2^{2} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	276.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:	$2.20387109579$
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:	$\Z[a_1, \ldots, a_{5}]$
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	$\chi$	$=$	276.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.585786 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.585786 q^{5} -1.41421 q^{7} +1.00000 q^{9} +5.65685 q^{11} +5.65685 q^{13} +0.585786 q^{15} -2.24264 q^{17} -8.24264 q^{19} -1.41421 q^{21} +1.00000 q^{23} -4.65685 q^{25} +1.00000 q^{27} +8.82843 q^{29} -1.17157 q^{31} +5.65685 q^{33} -0.828427 q^{35} -3.17157 q^{37} +5.65685 q^{39} -2.00000 q^{41} -5.41421 q^{43} +0.585786 q^{45} -10.4853 q^{47} -5.00000 q^{49} -2.24264 q^{51} +4.58579 q^{53} +3.31371 q^{55} -8.24264 q^{57} -8.82843 q^{59} +3.17157 q^{61} -1.41421 q^{63} +3.31371 q^{65} -7.75736 q^{67} +1.00000 q^{69} +11.3137 q^{71} +0.343146 q^{73} -4.65685 q^{75} -8.00000 q^{77} -7.07107 q^{79} +1.00000 q^{81} -9.65685 q^{83} -1.31371 q^{85} +8.82843 q^{87} +13.0711 q^{89} -8.00000 q^{91} -1.17157 q^{93} -4.82843 q^{95} +16.1421 q^{97} +5.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{15} + 4 q^{17} - 8 q^{19} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} - 10 q^{49} + 4 q^{51} + 12 q^{53} - 16 q^{55} - 8 q^{57} - 12 q^{59} + 12 q^{61} - 16 q^{65} - 24 q^{67} + 2 q^{69} + 12 q^{73} + 2 q^{75} - 16 q^{77} + 2 q^{81} - 8 q^{83} + 20 q^{85} + 12 q^{87} + 12 q^{89} - 16 q^{91} - 8 q^{93} - 4 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^15 + 4 * q^17 - 8 * q^19 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 - 4 * q^41 - 8 * q^43 + 4 * q^45 - 4 * q^47 - 10 * q^49 + 4 * q^51 + 12 * q^53 - 16 * q^55 - 8 * q^57 - 12 * q^59 + 12 * q^61 - 16 * q^65 - 24 * q^67 + 2 * q^69 + 12 * q^73 + 2 * q^75 - 16 * q^77 + 2 * q^81 - 8 * q^83 + 20 * q^85 + 12 * q^87 + 12 * q^89 - 16 * q^91 - 8 * q^93 - 4 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	−2.24264	−0.543920	−0.271960	−	0.962309i	$-0.587672\pi$
$17$	−2.24264	−0.543920	−0.271960	+	0.962309i	$0.587672\pi$
$18$	0	0
$19$	−8.24264	−1.89099	−0.945496	−	0.325634i	$-0.894422\pi$
$19$	−8.24264	−1.89099	−0.945496	+	0.325634i	$0.894422\pi$
$20$	0	0
$21$	−1.41421	−0.308607
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.82843	1.63940	0.819699	−	0.572795i	$-0.194141\pi$
$29$	8.82843	1.63940	0.819699	+	0.572795i	$0.194141\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	5.65685	0.984732
$34$	0	0
$35$	−0.828427	−0.140030
$36$	0	0
$37$	−3.17157	−0.521403	−0.260702	−	0.965419i	$-0.583954\pi$
$37$	−3.17157	−0.521403	−0.260702	+	0.965419i	$0.583954\pi$
$38$	0	0
$39$	5.65685	0.905822
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−5.41421	−0.825660	−0.412830	−	0.910808i	$-0.635460\pi$
$43$	−5.41421	−0.825660	−0.412830	+	0.910808i	$0.635460\pi$
$44$	0	0
$45$	0.585786	0.0873239
$46$	0	0
$47$	−10.4853	−1.52944	−0.764718	−	0.644365i	$-0.777122\pi$
$47$	−10.4853	−1.52944	−0.764718	+	0.644365i	$0.777122\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	−2.24264	−0.314033
$52$	0	0
$53$	4.58579	0.629906	0.314953	−	0.949107i	$-0.398011\pi$
$53$	4.58579	0.629906	0.314953	+	0.949107i	$0.398011\pi$
$54$	0	0
$55$	3.31371	0.446820
$56$	0	0
$57$	−8.24264	−1.09176
$58$	0	0
$59$	−8.82843	−1.14936	−0.574682	−	0.818377i	$-0.694874\pi$
$59$	−8.82843	−1.14936	−0.574682	+	0.818377i	$0.694874\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	0	0
$63$	−1.41421	−0.178174
$64$	0	0
$65$	3.31371	0.411015
$66$	0	0
$67$	−7.75736	−0.947712	−0.473856	−	0.880602i	$-0.657138\pi$
$67$	−7.75736	−0.947712	−0.473856	+	0.880602i	$0.657138\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	0.343146	0.0401622	0.0200811	−	0.999798i	$-0.493608\pi$
$73$	0.343146	0.0401622	0.0200811	+	0.999798i	$0.493608\pi$
$74$	0	0
$75$	−4.65685	−0.537727
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−7.07107	−0.795557	−0.397779	−	0.917481i	$-0.630219\pi$
$79$	−7.07107	−0.795557	−0.397779	+	0.917481i	$0.630219\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.65685	−1.05998	−0.529989	−	0.848005i	$-0.677804\pi$
$83$	−9.65685	−1.05998	−0.529989	+	0.848005i	$0.677804\pi$
$84$	0	0
$85$	−1.31371	−0.142492
$86$	0	0
$87$	8.82843	0.946507
$88$	0	0
$89$	13.0711	1.38553	0.692765	−	0.721163i	$-0.256392\pi$
$89$	13.0711	1.38553	0.692765	+	0.721163i	$0.256392\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−1.17157	−0.121486
$94$	0	0
$95$	−4.82843	−0.495386
$96$	0	0
$97$	16.1421	1.63899	0.819493	−	0.573090i	$-0.194255\pi$
$97$	16.1421	1.63899	0.819493	+	0.573090i	$0.194255\pi$
$98$	0	0
$99$	5.65685	0.568535
$100$	0	0
$101$	−5.31371	−0.528734	−0.264367	−	0.964422i	$-0.585163\pi$
$101$	−5.31371	−0.528734	−0.264367	+	0.964422i	$0.585163\pi$
$102$	0	0
$103$	−12.7279	−1.25412	−0.627060	−	0.778971i	$-0.715742\pi$
$103$	−12.7279	−1.25412	−0.627060	+	0.778971i	$0.715742\pi$
$104$	0	0
$105$	−0.828427	−0.0808462
$106$	0	0
$107$	5.65685	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$107$	5.65685	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$108$	0	0
$109$	3.65685	0.350263	0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685	0.350263	0.175132	+	0.984545i	$0.443965\pi$
$110$	0	0
$111$	−3.17157	−0.301032
$112$	0	0
$113$	−8.58579	−0.807683	−0.403841	−	0.914829i	$-0.632325\pi$
$113$	−8.58579	−0.807683	−0.403841	+	0.914829i	$0.632325\pi$
$114$	0	0
$115$	0.585786	0.0546249
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	3.17157	0.290738
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	16.4853	1.46283	0.731416	−	0.681931i	$-0.238860\pi$
$127$	16.4853	1.46283	0.731416	+	0.681931i	$0.238860\pi$
$128$	0	0
$129$	−5.41421	−0.476695
$130$	0	0
$131$	9.65685	0.843723	0.421862	−	0.906660i	$-0.361377\pi$
$131$	9.65685	0.843723	0.421862	+	0.906660i	$0.361377\pi$
$132$	0	0
$133$	11.6569	1.01078
$134$	0	0
$135$	0.585786	0.0504165
$136$	0	0
$137$	−13.0711	−1.11674	−0.558368	−	0.829593i	$-0.688572\pi$
$137$	−13.0711	−1.11674	−0.558368	+	0.829593i	$0.688572\pi$
$138$	0	0
$139$	9.65685	0.819084	0.409542	−	0.912291i	$-0.365689\pi$
$139$	9.65685	0.819084	0.409542	+	0.912291i	$0.365689\pi$
$140$	0	0
$141$	−10.4853	−0.883020
$142$	0	0
$143$	32.0000	2.67597
$144$	0	0
$145$	5.17157	0.429476
$146$	0	0
$147$	−5.00000	−0.412393
$148$	0	0
$149$	−6.24264	−0.511417	−0.255709	−	0.966754i	$-0.582309\pi$
$149$	−6.24264	−0.511417	−0.255709	+	0.966754i	$0.582309\pi$
$150$	0	0
$151$	2.34315	0.190682	0.0953412	−	0.995445i	$-0.469606\pi$
$151$	2.34315	0.190682	0.0953412	+	0.995445i	$0.469606\pi$
$152$	0	0
$153$	−2.24264	−0.181307
$154$	0	0
$155$	−0.686292	−0.0551243
$156$	0	0
$157$	−8.82843	−0.704585	−0.352293	−	0.935890i	$-0.614598\pi$
$157$	−8.82843	−0.704585	−0.352293	+	0.935890i	$0.614598\pi$
$158$	0	0
$159$	4.58579	0.363677
$160$	0	0
$161$	−1.41421	−0.111456
$162$	0	0
$163$	10.8284	0.848148	0.424074	−	0.905628i	$-0.360600\pi$
$163$	10.8284	0.848148	0.424074	+	0.905628i	$0.360600\pi$
$164$	0	0
$165$	3.31371	0.257972
$166$	0	0
$167$	2.34315	0.181318	0.0906590	−	0.995882i	$-0.471103\pi$
$167$	2.34315	0.181318	0.0906590	+	0.995882i	$0.471103\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	−8.24264	−0.630330
$172$	0	0
$173$	22.4853	1.70952	0.854762	−	0.519020i	$-0.173703\pi$
$173$	22.4853	1.70952	0.854762	+	0.519020i	$0.173703\pi$
$174$	0	0
$175$	6.58579	0.497839
$176$	0	0
$177$	−8.82843	−0.663585
$178$	0	0
$179$	−2.48528	−0.185759	−0.0928793	−	0.995677i	$-0.529607\pi$
$179$	−2.48528	−0.185759	−0.0928793	+	0.995677i	$0.529607\pi$
$180$	0	0
$181$	−17.3137	−1.28692	−0.643459	−	0.765481i	$-0.722501\pi$
$181$	−17.3137	−1.28692	−0.643459	+	0.765481i	$0.722501\pi$
$182$	0	0
$183$	3.17157	0.234449
$184$	0	0
$185$	−1.85786	−0.136593
$186$	0	0
$187$	−12.6863	−0.927714
$188$	0	0
$189$	−1.41421	−0.102869
$190$	0	0
$191$	−6.82843	−0.494088	−0.247044	−	0.969004i	$-0.579459\pi$
$191$	−6.82843	−0.494088	−0.247044	+	0.969004i	$0.579459\pi$
$192$	0	0
$193$	−11.3137	−0.814379	−0.407189	−	0.913344i	$-0.633491\pi$
$193$	−11.3137	−0.814379	−0.407189	+	0.913344i	$0.633491\pi$
$194$	0	0
$195$	3.31371	0.237300
$196$	0	0
$197$	−3.17157	−0.225965	−0.112983	−	0.993597i	$-0.536040\pi$
$197$	−3.17157	−0.225965	−0.112983	+	0.993597i	$0.536040\pi$
$198$	0	0
$199$	11.7574	0.833457	0.416729	−	0.909031i	$-0.363177\pi$
$199$	11.7574	0.833457	0.416729	+	0.909031i	$0.363177\pi$
$200$	0	0
$201$	−7.75736	−0.547162
$202$	0	0
$203$	−12.4853	−0.876295
$204$	0	0
$205$	−1.17157	−0.0818262
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−46.6274	−3.22529
$210$	0	0
$211$	16.4853	1.13489	0.567447	−	0.823410i	$-0.307931\pi$
$211$	16.4853	1.13489	0.567447	+	0.823410i	$0.307931\pi$
$212$	0	0
$213$	11.3137	0.775203
$214$	0	0
$215$	−3.17157	−0.216299
$216$	0	0
$217$	1.65685	0.112475
$218$	0	0
$219$	0.343146	0.0231876
$220$	0	0
$221$	−12.6863	−0.853372
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	0	0
$227$	5.17157	0.343249	0.171625	−	0.985162i	$-0.445098\pi$
$227$	5.17157	0.343249	0.171625	+	0.985162i	$0.445098\pi$
$228$	0	0
$229$	6.97056	0.460628	0.230314	−	0.973116i	$-0.426025\pi$
$229$	6.97056	0.460628	0.230314	+	0.973116i	$0.426025\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−4.34315	−0.284529	−0.142264	−	0.989829i	$-0.545438\pi$
$233$	−4.34315	−0.284529	−0.142264	+	0.989829i	$0.545438\pi$
$234$	0	0
$235$	−6.14214	−0.400669
$236$	0	0
$237$	−7.07107	−0.459315
$238$	0	0
$239$	21.6569	1.40087	0.700433	−	0.713718i	$-0.252990\pi$
$239$	21.6569	1.40087	0.700433	+	0.713718i	$0.252990\pi$
$240$	0	0
$241$	17.7990	1.14653	0.573267	−	0.819369i	$-0.305676\pi$
$241$	17.7990	1.14653	0.573267	+	0.819369i	$0.305676\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.92893	−0.187123
$246$	0	0
$247$	−46.6274	−2.96683
$248$	0	0
$249$	−9.65685	−0.611978
$250$	0	0
$251$	−29.4558	−1.85924	−0.929618	−	0.368524i	$-0.879863\pi$
$251$	−29.4558	−1.85924	−0.929618	+	0.368524i	$0.879863\pi$
$252$	0	0
$253$	5.65685	0.355643
$254$	0	0
$255$	−1.31371	−0.0822676
$256$	0	0
$257$	−6.68629	−0.417079	−0.208540	−	0.978014i	$-0.566871\pi$
$257$	−6.68629	−0.417079	−0.208540	+	0.978014i	$0.566871\pi$
$258$	0	0
$259$	4.48528	0.278702
$260$	0	0
$261$	8.82843	0.546466
$262$	0	0
$263$	23.7990	1.46751	0.733754	−	0.679415i	$-0.237766\pi$
$263$	23.7990	1.46751	0.733754	+	0.679415i	$0.237766\pi$
$264$	0	0
$265$	2.68629	0.165018
$266$	0	0
$267$	13.0711	0.799936
$268$	0	0
$269$	28.8284	1.75770	0.878850	−	0.477098i	$-0.158311\pi$
$269$	28.8284	1.75770	0.878850	+	0.477098i	$0.158311\pi$
$270$	0	0
$271$	13.1716	0.800116	0.400058	−	0.916490i	$-0.368990\pi$
$271$	13.1716	0.800116	0.400058	+	0.916490i	$0.368990\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	−26.3431	−1.58855
$276$	0	0
$277$	−32.6274	−1.96039	−0.980196	−	0.198031i	$-0.936545\pi$
$277$	−32.6274	−1.96039	−0.980196	+	0.198031i	$0.936545\pi$
$278$	0	0
$279$	−1.17157	−0.0701402
$280$	0	0
$281$	10.9289	0.651965	0.325983	−	0.945376i	$-0.394305\pi$
$281$	10.9289	0.651965	0.325983	+	0.945376i	$0.394305\pi$
$282$	0	0
$283$	−2.10051	−0.124862	−0.0624310	−	0.998049i	$-0.519885\pi$
$283$	−2.10051	−0.124862	−0.0624310	+	0.998049i	$0.519885\pi$
$284$	0	0
$285$	−4.82843	−0.286011
$286$	0	0
$287$	2.82843	0.166957
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	16.1421	0.946269
$292$	0	0
$293$	−18.2426	−1.06575	−0.532873	−	0.846195i	$-0.678888\pi$
$293$	−18.2426	−1.06575	−0.532873	+	0.846195i	$0.678888\pi$
$294$	0	0
$295$	−5.17157	−0.301101
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	7.65685	0.441334
$302$	0	0
$303$	−5.31371	−0.305265
$304$	0	0
$305$	1.85786	0.106381
$306$	0	0
$307$	−29.4558	−1.68113	−0.840567	−	0.541708i	$-0.817778\pi$
$307$	−29.4558	−1.68113	−0.840567	+	0.541708i	$0.817778\pi$
$308$	0	0
$309$	−12.7279	−0.724066
$310$	0	0
$311$	3.17157	0.179843	0.0899217	−	0.995949i	$-0.471338\pi$
$311$	3.17157	0.179843	0.0899217	+	0.995949i	$0.471338\pi$
$312$	0	0
$313$	−15.6569	−0.884978	−0.442489	−	0.896774i	$-0.645904\pi$
$313$	−15.6569	−0.884978	−0.442489	+	0.896774i	$0.645904\pi$
$314$	0	0
$315$	−0.828427	−0.0466766
$316$	0	0
$317$	−3.17157	−0.178133	−0.0890666	−	0.996026i	$-0.528388\pi$
$317$	−3.17157	−0.178133	−0.0890666	+	0.996026i	$0.528388\pi$
$318$	0	0
$319$	49.9411	2.79617
$320$	0	0
$321$	5.65685	0.315735
$322$	0	0
$323$	18.4853	1.02855
$324$	0	0
$325$	−26.3431	−1.46125
$326$	0	0
$327$	3.65685	0.202225
$328$	0	0
$329$	14.8284	0.817518
$330$	0	0
$331$	−26.1421	−1.43690	−0.718451	−	0.695578i	$-0.755148\pi$
$331$	−26.1421	−1.43690	−0.718451	+	0.695578i	$0.755148\pi$
$332$	0	0
$333$	−3.17157	−0.173801
$334$	0	0
$335$	−4.54416	−0.248274
$336$	0	0
$337$	12.8284	0.698809	0.349404	−	0.936972i	$-0.386384\pi$
$337$	12.8284	0.698809	0.349404	+	0.936972i	$0.386384\pi$
$338$	0	0
$339$	−8.58579	−0.466316
$340$	0	0
$341$	−6.62742	−0.358895
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0.585786	0.0315377
$346$	0	0
$347$	−12.8284	−0.688666	−0.344333	−	0.938848i	$-0.611895\pi$
$347$	−12.8284	−0.688666	−0.344333	+	0.938848i	$0.611895\pi$
$348$	0	0
$349$	−19.3137	−1.03384	−0.516920	−	0.856034i	$-0.672921\pi$
$349$	−19.3137	−1.03384	−0.516920	+	0.856034i	$0.672921\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	−12.1421	−0.646261	−0.323130	−	0.946354i	$-0.604735\pi$
$353$	−12.1421	−0.646261	−0.323130	+	0.946354i	$0.604735\pi$
$354$	0	0
$355$	6.62742	0.351747
$356$	0	0
$357$	3.17157	0.167857
$358$	0	0
$359$	25.1716	1.32850	0.664252	−	0.747508i	$-0.268750\pi$
$359$	25.1716	1.32850	0.664252	+	0.747508i	$0.268750\pi$
$360$	0	0
$361$	48.9411	2.57585
$362$	0	0
$363$	21.0000	1.10221
$364$	0	0
$365$	0.201010	0.0105214
$366$	0	0
$367$	−11.7574	−0.613729	−0.306865	−	0.951753i	$-0.599280\pi$
$367$	−11.7574	−0.613729	−0.306865	+	0.951753i	$0.599280\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−6.48528	−0.336699
$372$	0	0
$373$	−11.1716	−0.578442	−0.289221	−	0.957262i	$-0.593396\pi$
$373$	−11.1716	−0.578442	−0.289221	+	0.957262i	$0.593396\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	49.9411	2.57210
$378$	0	0
$379$	−8.72792	−0.448323	−0.224162	−	0.974552i	$-0.571964\pi$
$379$	−8.72792	−0.448323	−0.224162	+	0.974552i	$0.571964\pi$
$380$	0	0
$381$	16.4853	0.844567
$382$	0	0
$383$	−27.7990	−1.42046	−0.710231	−	0.703969i	$-0.751410\pi$
$383$	−27.7990	−1.42046	−0.710231	+	0.703969i	$0.751410\pi$
$384$	0	0
$385$	−4.68629	−0.238836
$386$	0	0
$387$	−5.41421	−0.275220
$388$	0	0
$389$	14.7279	0.746735	0.373368	−	0.927683i	$-0.378203\pi$
$389$	14.7279	0.746735	0.373368	+	0.927683i	$0.378203\pi$
$390$	0	0
$391$	−2.24264	−0.113415
$392$	0	0
$393$	9.65685	0.487124
$394$	0	0
$395$	−4.14214	−0.208413
$396$	0	0
$397$	−16.6274	−0.834506	−0.417253	−	0.908790i	$-0.637007\pi$
$397$	−16.6274	−0.834506	−0.417253	+	0.908790i	$0.637007\pi$
$398$	0	0
$399$	11.6569	0.583573
$400$	0	0
$401$	−15.8995	−0.793983	−0.396991	−	0.917822i	$-0.629946\pi$
$401$	−15.8995	−0.793983	−0.396991	+	0.917822i	$0.629946\pi$
$402$	0	0
$403$	−6.62742	−0.330135
$404$	0	0
$405$	0.585786	0.0291080
$406$	0	0
$407$	−17.9411	−0.889309
$408$	0	0
$409$	6.34315	0.313648	0.156824	−	0.987627i	$-0.449874\pi$
$409$	6.34315	0.313648	0.156824	+	0.987627i	$0.449874\pi$
$410$	0	0
$411$	−13.0711	−0.644748
$412$	0	0
$413$	12.4853	0.614361
$414$	0	0
$415$	−5.65685	−0.277684
$416$	0	0
$417$	9.65685	0.472898
$418$	0	0
$419$	31.3137	1.52977	0.764887	−	0.644164i	$-0.222795\pi$
$419$	31.3137	1.52977	0.764887	+	0.644164i	$0.222795\pi$
$420$	0	0
$421$	29.3137	1.42866	0.714331	−	0.699808i	$-0.246731\pi$
$421$	29.3137	1.42866	0.714331	+	0.699808i	$0.246731\pi$
$422$	0	0
$423$	−10.4853	−0.509812
$424$	0	0
$425$	10.4437	0.506591
$426$	0	0
$427$	−4.48528	−0.217058
$428$	0	0
$429$	32.0000	1.54497
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	2.97056	0.142756	0.0713781	−	0.997449i	$-0.477260\pi$
$433$	2.97056	0.142756	0.0713781	+	0.997449i	$0.477260\pi$
$434$	0	0
$435$	5.17157	0.247958
$436$	0	0
$437$	−8.24264	−0.394299
$438$	0	0
$439$	−2.34315	−0.111832	−0.0559161	−	0.998435i	$-0.517808\pi$
$439$	−2.34315	−0.111832	−0.0559161	+	0.998435i	$0.517808\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	40.2843	1.91396	0.956982	−	0.290148i	$-0.0937046\pi$
$443$	40.2843	1.91396	0.956982	+	0.290148i	$0.0937046\pi$
$444$	0	0
$445$	7.65685	0.362970
$446$	0	0
$447$	−6.24264	−0.295267
$448$	0	0
$449$	39.6569	1.87152	0.935761	−	0.352634i	$-0.114714\pi$
$449$	39.6569	1.87152	0.935761	+	0.352634i	$0.114714\pi$
$450$	0	0
$451$	−11.3137	−0.532742
$452$	0	0
$453$	2.34315	0.110091
$454$	0	0
$455$	−4.68629	−0.219697
$456$	0	0
$457$	−31.1716	−1.45814	−0.729072	−	0.684437i	$-0.760048\pi$
$457$	−31.1716	−1.45814	−0.729072	+	0.684437i	$0.760048\pi$
$458$	0	0
$459$	−2.24264	−0.104678
$460$	0	0
$461$	23.4558	1.09245	0.546224	−	0.837639i	$-0.316065\pi$
$461$	23.4558	1.09245	0.546224	+	0.837639i	$0.316065\pi$
$462$	0	0
$463$	−4.97056	−0.231002	−0.115501	−	0.993307i	$-0.536847\pi$
$463$	−4.97056	−0.231002	−0.115501	+	0.993307i	$0.536847\pi$
$464$	0	0
$465$	−0.686292	−0.0318260
$466$	0	0
$467$	35.1127	1.62482	0.812411	−	0.583085i	$-0.198155\pi$
$467$	35.1127	1.62482	0.812411	+	0.583085i	$0.198155\pi$
$468$	0	0
$469$	10.9706	0.506574
$470$	0	0
$471$	−8.82843	−0.406792
$472$	0	0
$473$	−30.6274	−1.40825
$474$	0	0
$475$	38.3848	1.76121
$476$	0	0
$477$	4.58579	0.209969
$478$	0	0
$479$	2.62742	0.120050	0.0600249	−	0.998197i	$-0.480882\pi$
$479$	2.62742	0.120050	0.0600249	+	0.998197i	$0.480882\pi$
$480$	0	0
$481$	−17.9411	−0.818045
$482$	0	0
$483$	−1.41421	−0.0643489
$484$	0	0
$485$	9.45584	0.429368
$486$	0	0
$487$	−14.3431	−0.649950	−0.324975	−	0.945723i	$-0.605356\pi$
$487$	−14.3431	−0.649950	−0.324975	+	0.945723i	$0.605356\pi$
$488$	0	0
$489$	10.8284	0.489678
$490$	0	0
$491$	0.142136	0.00641449	0.00320725	−	0.999995i	$-0.498979\pi$
$491$	0.142136	0.00641449	0.00320725	+	0.999995i	$0.498979\pi$
$492$	0	0
$493$	−19.7990	−0.891702
$494$	0	0
$495$	3.31371	0.148940
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−24.4853	−1.09611	−0.548056	−	0.836442i	$-0.684632\pi$
$499$	−24.4853	−1.09611	−0.548056	+	0.836442i	$0.684632\pi$
$500$	0	0
$501$	2.34315	0.104684
$502$	0	0
$503$	0.686292	0.0306002	0.0153001	−	0.999883i	$-0.495130\pi$
$503$	0.686292	0.0306002	0.0153001	+	0.999883i	$0.495130\pi$
$504$	0	0
$505$	−3.11270	−0.138513
$506$	0	0
$507$	19.0000	0.843820
$508$	0	0
$509$	−22.4853	−0.996643	−0.498321	−	0.866992i	$-0.666050\pi$
$509$	−22.4853	−0.996643	−0.498321	+	0.866992i	$0.666050\pi$
$510$	0	0
$511$	−0.485281	−0.0214676
$512$	0	0
$513$	−8.24264	−0.363921
$514$	0	0
$515$	−7.45584	−0.328544
$516$	0	0
$517$	−59.3137	−2.60861
$518$	0	0
$519$	22.4853	0.986994
$520$	0	0
$521$	−6.24264	−0.273495	−0.136748	−	0.990606i	$-0.543665\pi$
$521$	−6.24264	−0.273495	−0.136748	+	0.990606i	$0.543665\pi$
$522$	0	0
$523$	1.21320	0.0530497	0.0265248	−	0.999648i	$-0.491556\pi$
$523$	1.21320	0.0530497	0.0265248	+	0.999648i	$0.491556\pi$
$524$	0	0
$525$	6.58579	0.287427
$526$	0	0
$527$	2.62742	0.114452
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.82843	−0.383121
$532$	0	0
$533$	−11.3137	−0.490051
$534$	0	0
$535$	3.31371	0.143264
$536$	0	0
$537$	−2.48528	−0.107248
$538$	0	0
$539$	−28.2843	−1.21829
$540$	0	0
$541$	24.2843	1.04406	0.522031	−	0.852926i	$-0.325174\pi$
$541$	24.2843	1.04406	0.522031	+	0.852926i	$0.325174\pi$
$542$	0	0
$543$	−17.3137	−0.743002
$544$	0	0
$545$	2.14214	0.0917590
$546$	0	0
$547$	30.1421	1.28878	0.644392	−	0.764695i	$-0.277110\pi$
$547$	30.1421	1.28878	0.644392	+	0.764695i	$0.277110\pi$
$548$	0	0
$549$	3.17157	0.135359
$550$	0	0
$551$	−72.7696	−3.10009
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	−1.85786	−0.0788620
$556$	0	0
$557$	21.7574	0.921889	0.460944	−	0.887429i	$-0.347511\pi$
$557$	21.7574	0.921889	0.460944	+	0.887429i	$0.347511\pi$
$558$	0	0
$559$	−30.6274	−1.29540
$560$	0	0
$561$	−12.6863	−0.535616
$562$	0	0
$563$	26.8284	1.13068	0.565342	−	0.824857i	$-0.308744\pi$
$563$	26.8284	1.13068	0.565342	+	0.824857i	$0.308744\pi$
$564$	0	0
$565$	−5.02944	−0.211590
$566$	0	0
$567$	−1.41421	−0.0593914
$568$	0	0
$569$	18.2426	0.764771	0.382386	−	0.924003i	$-0.375103\pi$
$569$	18.2426	0.764771	0.382386	+	0.924003i	$0.375103\pi$
$570$	0	0
$571$	0.727922	0.0304626	0.0152313	−	0.999884i	$-0.495152\pi$
$571$	0.727922	0.0304626	0.0152313	+	0.999884i	$0.495152\pi$
$572$	0	0
$573$	−6.82843	−0.285262
$574$	0	0
$575$	−4.65685	−0.194204
$576$	0	0
$577$	−1.37258	−0.0571414	−0.0285707	−	0.999592i	$-0.509096\pi$
$577$	−1.37258	−0.0571414	−0.0285707	+	0.999592i	$0.509096\pi$
$578$	0	0
$579$	−11.3137	−0.470182
$580$	0	0
$581$	13.6569	0.566582
$582$	0	0
$583$	25.9411	1.07437
$584$	0	0
$585$	3.31371	0.137005
$586$	0	0
$587$	−22.3431	−0.922200	−0.461100	−	0.887348i	$-0.652545\pi$
$587$	−22.3431	−0.922200	−0.461100	+	0.887348i	$0.652545\pi$
$588$	0	0
$589$	9.65685	0.397904
$590$	0	0
$591$	−3.17157	−0.130461
$592$	0	0
$593$	−43.6569	−1.79277	−0.896386	−	0.443274i	$-0.853817\pi$
$593$	−43.6569	−1.79277	−0.896386	+	0.443274i	$0.853817\pi$
$594$	0	0
$595$	1.85786	0.0761650
$596$	0	0
$597$	11.7574	0.481197
$598$	0	0
$599$	−12.2843	−0.501922	−0.250961	−	0.967997i	$-0.580747\pi$
$599$	−12.2843	−0.501922	−0.250961	+	0.967997i	$0.580747\pi$
$600$	0	0
$601$	12.6863	0.517485	0.258742	−	0.965946i	$-0.416692\pi$
$601$	12.6863	0.517485	0.258742	+	0.965946i	$0.416692\pi$
$602$	0	0
$603$	−7.75736	−0.315904
$604$	0	0
$605$	12.3015	0.500128
$606$	0	0
$607$	−10.3431	−0.419815	−0.209908	−	0.977721i	$-0.567316\pi$
$607$	−10.3431	−0.419815	−0.209908	+	0.977721i	$0.567316\pi$
$608$	0	0
$609$	−12.4853	−0.505929
$610$	0	0
$611$	−59.3137	−2.39958
$612$	0	0
$613$	18.4853	0.746613	0.373307	−	0.927708i	$-0.378224\pi$
$613$	18.4853	0.746613	0.373307	+	0.927708i	$0.378224\pi$
$614$	0	0
$615$	−1.17157	−0.0472424
$616$	0	0
$617$	−11.8995	−0.479056	−0.239528	−	0.970890i	$-0.576993\pi$
$617$	−11.8995	−0.479056	−0.239528	+	0.970890i	$0.576993\pi$
$618$	0	0
$619$	−0.242641	−0.00975255	−0.00487628	−	0.999988i	$-0.501552\pi$
$619$	−0.242641	−0.00975255	−0.00487628	+	0.999988i	$0.501552\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−18.4853	−0.740597
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	−46.6274	−1.86212
$628$	0	0
$629$	7.11270	0.283602
$630$	0	0
$631$	−4.24264	−0.168897	−0.0844484	−	0.996428i	$-0.526913\pi$
$631$	−4.24264	−0.168897	−0.0844484	+	0.996428i	$0.526913\pi$
$632$	0	0
$633$	16.4853	0.655231
$634$	0	0
$635$	9.65685	0.383221
$636$	0	0
$637$	−28.2843	−1.12066
$638$	0	0
$639$	11.3137	0.447563
$640$	0	0
$641$	39.6985	1.56800	0.783998	−	0.620763i	$-0.213177\pi$
$641$	39.6985	1.56800	0.783998	+	0.620763i	$0.213177\pi$
$642$	0	0
$643$	−8.72792	−0.344196	−0.172098	−	0.985080i	$-0.555055\pi$
$643$	−8.72792	−0.344196	−0.172098	+	0.985080i	$0.555055\pi$
$644$	0	0
$645$	−3.17157	−0.124881
$646$	0	0
$647$	29.7990	1.17152	0.585760	−	0.810485i	$-0.300796\pi$
$647$	29.7990	1.17152	0.585760	+	0.810485i	$0.300796\pi$
$648$	0	0
$649$	−49.9411	−1.96036
$650$	0	0
$651$	1.65685	0.0649372
$652$	0	0
$653$	−14.9706	−0.585843	−0.292922	−	0.956136i	$-0.594628\pi$
$653$	−14.9706	−0.585843	−0.292922	+	0.956136i	$0.594628\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	0	0
$657$	0.343146	0.0133874
$658$	0	0
$659$	−39.7990	−1.55035	−0.775174	−	0.631747i	$-0.782338\pi$
$659$	−39.7990	−1.55035	−0.775174	+	0.631747i	$0.782338\pi$
$660$	0	0
$661$	−20.1421	−0.783438	−0.391719	−	0.920085i	$-0.628120\pi$
$661$	−20.1421	−0.783438	−0.391719	+	0.920085i	$0.628120\pi$
$662$	0	0
$663$	−12.6863	−0.492695
$664$	0	0
$665$	6.82843	0.264795
$666$	0	0
$667$	8.82843	0.341838
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	17.9411	0.692609
$672$	0	0
$673$	0.686292	0.0264546	0.0132273	−	0.999913i	$-0.495789\pi$
$673$	0.686292	0.0264546	0.0132273	+	0.999913i	$0.495789\pi$
$674$	0	0
$675$	−4.65685	−0.179242
$676$	0	0
$677$	43.6985	1.67947	0.839735	−	0.542997i	$-0.182711\pi$
$677$	43.6985	1.67947	0.839735	+	0.542997i	$0.182711\pi$
$678$	0	0
$679$	−22.8284	−0.876075
$680$	0	0
$681$	5.17157	0.198175
$682$	0	0
$683$	26.6274	1.01887	0.509435	−	0.860509i	$-0.329854\pi$
$683$	26.6274	1.01887	0.509435	+	0.860509i	$0.329854\pi$
$684$	0	0
$685$	−7.65685	−0.292553
$686$	0	0
$687$	6.97056	0.265944
$688$	0	0
$689$	25.9411	0.988278
$690$	0	0
$691$	25.4558	0.968386	0.484193	−	0.874961i	$-0.339113\pi$
$691$	25.4558	0.968386	0.484193	+	0.874961i	$0.339113\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	4.48528	0.169892
$698$	0	0
$699$	−4.34315	−0.164273
$700$	0	0
$701$	47.2132	1.78322	0.891609	−	0.452806i	$-0.149577\pi$
$701$	47.2132	1.78322	0.891609	+	0.452806i	$0.149577\pi$
$702$	0	0
$703$	26.1421	0.985969
$704$	0	0
$705$	−6.14214	−0.231326
$706$	0	0
$707$	7.51472	0.282620
$708$	0	0
$709$	−5.31371	−0.199561	−0.0997803	−	0.995009i	$-0.531814\pi$
$709$	−5.31371	−0.199561	−0.0997803	+	0.995009i	$0.531814\pi$
$710$	0	0
$711$	−7.07107	−0.265186
$712$	0	0
$713$	−1.17157	−0.0438757
$714$	0	0
$715$	18.7452	0.701029
$716$	0	0
$717$	21.6569	0.808790
$718$	0	0
$719$	−20.1421	−0.751175	−0.375587	−	0.926787i	$-0.622559\pi$
$719$	−20.1421	−0.751175	−0.375587	+	0.926787i	$0.622559\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	17.7990	0.661952
$724$	0	0
$725$	−41.1127	−1.52689
$726$	0	0
$727$	24.0416	0.891655	0.445827	−	0.895119i	$-0.352910\pi$
$727$	24.0416	0.891655	0.445827	+	0.895119i	$0.352910\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.1421	0.449093
$732$	0	0
$733$	10.9706	0.405207	0.202603	−	0.979261i	$-0.435060\pi$
$733$	10.9706	0.405207	0.202603	+	0.979261i	$0.435060\pi$
$734$	0	0
$735$	−2.92893	−0.108035
$736$	0	0
$737$	−43.8823	−1.61642
$738$	0	0
$739$	24.6863	0.908100	0.454050	−	0.890976i	$-0.349979\pi$
$739$	24.6863	0.908100	0.454050	+	0.890976i	$0.349979\pi$
$740$	0	0
$741$	−46.6274	−1.71290
$742$	0	0
$743$	−17.4558	−0.640393	−0.320196	−	0.947351i	$-0.603749\pi$
$743$	−17.4558	−0.640393	−0.320196	+	0.947351i	$0.603749\pi$
$744$	0	0
$745$	−3.65685	−0.133977
$746$	0	0
$747$	−9.65685	−0.353326
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−5.69848	−0.207941	−0.103970	−	0.994580i	$-0.533155\pi$
$751$	−5.69848	−0.207941	−0.103970	+	0.994580i	$0.533155\pi$
$752$	0	0
$753$	−29.4558	−1.07343
$754$	0	0
$755$	1.37258	0.0499534
$756$	0	0
$757$	21.3137	0.774660	0.387330	−	0.921941i	$-0.373397\pi$
$757$	21.3137	0.774660	0.387330	+	0.921941i	$0.373397\pi$
$758$	0	0
$759$	5.65685	0.205331
$760$	0	0
$761$	−35.1716	−1.27497	−0.637484	−	0.770463i	$-0.720025\pi$
$761$	−35.1716	−1.27497	−0.637484	+	0.770463i	$0.720025\pi$
$762$	0	0
$763$	−5.17157	−0.187224
$764$	0	0
$765$	−1.31371	−0.0474972
$766$	0	0
$767$	−49.9411	−1.80327
$768$	0	0
$769$	19.6569	0.708844	0.354422	−	0.935086i	$-0.384678\pi$
$769$	19.6569	0.708844	0.354422	+	0.935086i	$0.384678\pi$
$770$	0	0
$771$	−6.68629	−0.240801
$772$	0	0
$773$	−44.8701	−1.61386	−0.806932	−	0.590644i	$-0.798874\pi$
$773$	−44.8701	−1.61386	−0.806932	+	0.590644i	$0.798874\pi$
$774$	0	0
$775$	5.45584	0.195980
$776$	0	0
$777$	4.48528	0.160909
$778$	0	0
$779$	16.4853	0.590647
$780$	0	0
$781$	64.0000	2.29010
$782$	0	0
$783$	8.82843	0.315502
$784$	0	0
$785$	−5.17157	−0.184581
$786$	0	0
$787$	44.5269	1.58721	0.793606	−	0.608431i	$-0.208201\pi$
$787$	44.5269	1.58721	0.793606	+	0.608431i	$0.208201\pi$
$788$	0	0
$789$	23.7990	0.847266
$790$	0	0
$791$	12.1421	0.431725
$792$	0	0
$793$	17.9411	0.637108
$794$	0	0
$795$	2.68629	0.0952729
$796$	0	0
$797$	−0.100505	−0.00356007	−0.00178004	−	0.999998i	$-0.500567\pi$
$797$	−0.100505	−0.00356007	−0.00178004	+	0.999998i	$0.500567\pi$
$798$	0	0
$799$	23.5147	0.831891
$800$	0	0
$801$	13.0711	0.461843
$802$	0	0
$803$	1.94113	0.0685008
$804$	0	0
$805$	−0.828427	−0.0291982
$806$	0	0
$807$	28.8284	1.01481
$808$	0	0
$809$	48.8284	1.71672	0.858358	−	0.513051i	$-0.171485\pi$
$809$	48.8284	1.71672	0.858358	+	0.513051i	$0.171485\pi$
$810$	0	0
$811$	−10.8284	−0.380238	−0.190119	−	0.981761i	$-0.560887\pi$
$811$	−10.8284	−0.380238	−0.190119	+	0.981761i	$0.560887\pi$
$812$	0	0
$813$	13.1716	0.461947
$814$	0	0
$815$	6.34315	0.222191
$816$	0	0
$817$	44.6274	1.56132
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	25.3137	0.883455	0.441727	−	0.897149i	$-0.354366\pi$
$821$	25.3137	0.883455	0.441727	+	0.897149i	$0.354366\pi$
$822$	0	0
$823$	4.20101	0.146438	0.0732190	−	0.997316i	$-0.476673\pi$
$823$	4.20101	0.146438	0.0732190	+	0.997316i	$0.476673\pi$
$824$	0	0
$825$	−26.3431	−0.917151
$826$	0	0
$827$	24.7696	0.861322	0.430661	−	0.902514i	$-0.358281\pi$
$827$	24.7696	0.861322	0.430661	+	0.902514i	$0.358281\pi$
$828$	0	0
$829$	−19.0294	−0.660920	−0.330460	−	0.943820i	$-0.607204\pi$
$829$	−19.0294	−0.660920	−0.330460	+	0.943820i	$0.607204\pi$
$830$	0	0
$831$	−32.6274	−1.13183
$832$	0	0
$833$	11.2132	0.388514
$834$	0	0
$835$	1.37258	0.0475002
$836$	0	0
$837$	−1.17157	−0.0404955
$838$	0	0
$839$	−1.17157	−0.0404472	−0.0202236	−	0.999795i	$-0.506438\pi$
$839$	−1.17157	−0.0404472	−0.0202236	+	0.999795i	$0.506438\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	0	0
$843$	10.9289	0.376412
$844$	0	0
$845$	11.1299	0.382882
$846$	0	0
$847$	−29.6985	−1.02045
$848$	0	0
$849$	−2.10051	−0.0720891
$850$	0	0
$851$	−3.17157	−0.108720
$852$	0	0
$853$	−47.9411	−1.64147	−0.820736	−	0.571307i	$-0.806437\pi$
$853$	−47.9411	−1.64147	−0.820736	+	0.571307i	$0.806437\pi$
$854$	0	0
$855$	−4.82843	−0.165129
$856$	0	0
$857$	6.97056	0.238110	0.119055	−	0.992888i	$-0.462014\pi$
$857$	6.97056	0.238110	0.119055	+	0.992888i	$0.462014\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	2.82843	0.0963925
$862$	0	0
$863$	−26.4853	−0.901569	−0.450785	−	0.892633i	$-0.648856\pi$
$863$	−26.4853	−0.901569	−0.450785	+	0.892633i	$0.648856\pi$
$864$	0	0
$865$	13.1716	0.447847
$866$	0	0
$867$	−11.9706	−0.406542
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	−43.8823	−1.48689
$872$	0	0
$873$	16.1421	0.546329
$874$	0	0
$875$	8.00000	0.270449
$876$	0	0
$877$	2.97056	0.100309	0.0501544	−	0.998741i	$-0.484029\pi$
$877$	2.97056	0.100309	0.0501544	+	0.998741i	$0.484029\pi$
$878$	0	0
$879$	−18.2426	−0.615309
$880$	0	0
$881$	−41.7574	−1.40684	−0.703421	−	0.710774i	$-0.748345\pi$
$881$	−41.7574	−1.40684	−0.703421	+	0.710774i	$0.748345\pi$
$882$	0	0
$883$	31.7990	1.07012	0.535061	−	0.844814i	$-0.320289\pi$
$883$	31.7990	1.07012	0.535061	+	0.844814i	$0.320289\pi$
$884$	0	0
$885$	−5.17157	−0.173841
$886$	0	0
$887$	−9.79899	−0.329018	−0.164509	−	0.986376i	$-0.552604\pi$
$887$	−9.79899	−0.329018	−0.164509	+	0.986376i	$0.552604\pi$
$888$	0	0
$889$	−23.3137	−0.781917
$890$	0	0
$891$	5.65685	0.189512
$892$	0	0
$893$	86.4264	2.89215
$894$	0	0
$895$	−1.45584	−0.0486635
$896$	0	0
$897$	5.65685	0.188877
$898$	0	0
$899$	−10.3431	−0.344963
$900$	0	0
$901$	−10.2843	−0.342619
$902$	0	0
$903$	7.65685	0.254804
$904$	0	0
$905$	−10.1421	−0.337136
$906$	0	0
$907$	−11.0711	−0.367609	−0.183804	−	0.982963i	$-0.558841\pi$
$907$	−11.0711	−0.367609	−0.183804	+	0.982963i	$0.558841\pi$
$908$	0	0
$909$	−5.31371	−0.176245
$910$	0	0
$911$	45.9411	1.52210	0.761049	−	0.648695i	$-0.224685\pi$
$911$	45.9411	1.52210	0.761049	+	0.648695i	$0.224685\pi$
$912$	0	0
$913$	−54.6274	−1.80790
$914$	0	0
$915$	1.85786	0.0614191
$916$	0	0
$917$	−13.6569	−0.450989
$918$	0	0
$919$	24.0416	0.793060	0.396530	−	0.918022i	$-0.370214\pi$
$919$	24.0416	0.793060	0.396530	+	0.918022i	$0.370214\pi$
$920$	0	0
$921$	−29.4558	−0.970603
$922$	0	0
$923$	64.0000	2.10659
$924$	0	0
$925$	14.7696	0.485620
$926$	0	0
$927$	−12.7279	−0.418040
$928$	0	0
$929$	−2.97056	−0.0974610	−0.0487305	−	0.998812i	$-0.515518\pi$
$929$	−2.97056	−0.0974610	−0.0487305	+	0.998812i	$0.515518\pi$
$930$	0	0
$931$	41.2132	1.35071
$932$	0	0
$933$	3.17157	0.103833
$934$	0	0
$935$	−7.43146	−0.243035
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	0	0
$939$	−15.6569	−0.510942
$940$	0	0
$941$	−30.0416	−0.979329	−0.489665	−	0.871911i	$-0.662881\pi$
$941$	−30.0416	−0.979329	−0.489665	+	0.871911i	$0.662881\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	−0.828427	−0.0269487
$946$	0	0
$947$	−29.7990	−0.968337	−0.484169	−	0.874975i	$-0.660878\pi$
$947$	−29.7990	−0.968337	−0.484169	+	0.874975i	$0.660878\pi$
$948$	0	0
$949$	1.94113	0.0630116
$950$	0	0
$951$	−3.17157	−0.102845
$952$	0	0
$953$	15.8995	0.515035	0.257518	−	0.966274i	$-0.417095\pi$
$953$	15.8995	0.515035	0.257518	+	0.966274i	$0.417095\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	0	0
$957$	49.9411	1.61437
$958$	0	0
$959$	18.4853	0.596921
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	5.65685	0.182290
$964$	0	0
$965$	−6.62742	−0.213344
$966$	0	0
$967$	45.4558	1.46176	0.730881	−	0.682505i	$-0.239110\pi$
$967$	45.4558	1.46176	0.730881	+	0.682505i	$0.239110\pi$
$968$	0	0
$969$	18.4853	0.593833
$970$	0	0
$971$	−9.45584	−0.303452	−0.151726	−	0.988423i	$-0.548483\pi$
$971$	−9.45584	−0.303452	−0.151726	+	0.988423i	$0.548483\pi$
$972$	0	0
$973$	−13.6569	−0.437819
$974$	0	0
$975$	−26.3431	−0.843656
$976$	0	0
$977$	0.786797	0.0251719	0.0125859	−	0.999921i	$-0.495994\pi$
$977$	0.786797	0.0251719	0.0125859	+	0.999921i	$0.495994\pi$
$978$	0	0
$979$	73.9411	2.36317
$980$	0	0
$981$	3.65685	0.116754
$982$	0	0
$983$	20.0000	0.637901	0.318950	−	0.947771i	$-0.396670\pi$
$983$	20.0000	0.637901	0.318950	+	0.947771i	$0.396670\pi$
$984$	0	0
$985$	−1.85786	−0.0591965
$986$	0	0
$987$	14.8284	0.471994
$988$	0	0
$989$	−5.41421	−0.172162
$990$	0	0
$991$	60.7696	1.93041	0.965204	−	0.261497i	$-0.0842162\pi$
$991$	60.7696	1.93041	0.965204	+	0.261497i	$0.0842162\pi$
$992$	0	0
$993$	−26.1421	−0.829596
$994$	0	0
$995$	6.88730	0.218342
$996$	0	0
$997$	−27.3137	−0.865034	−0.432517	−	0.901626i	$-0.642374\pi$
$997$	−27.3137	−0.865034	−0.432517	+	0.901626i	$0.642374\pi$
$998$	0	0
$999$	−3.17157	−0.100344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	276.2.a.b.1.1	✓	2
3.2	odd	2		828.2.a.e.1.2		2
4.3	odd	2		1104.2.a.l.1.1		2
5.2	odd	4		6900.2.f.l.6349.1		4
5.3	odd	4		6900.2.f.l.6349.4		4
5.4	even	2		6900.2.a.m.1.2		2
8.3	odd	2		4416.2.a.bi.1.2		2
8.5	even	2		4416.2.a.bc.1.2		2
12.11	even	2		3312.2.a.s.1.2		2
23.22	odd	2		6348.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.a.b.1.1	✓	2	1.1	even	1	trivial
828.2.a.e.1.2		2	3.2	odd	2
1104.2.a.l.1.1		2	4.3	odd	2
3312.2.a.s.1.2		2	12.11	even	2
4416.2.a.bc.1.2		2	8.5	even	2
4416.2.a.bi.1.2		2	8.3	odd	2
6348.2.a.h.1.2		2	23.22	odd	2
6900.2.a.m.1.2		2	5.4	even	2
6900.2.f.l.6349.1		4	5.2	odd	4
6900.2.f.l.6349.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	276.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.585786 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.585786 q^{5} -1.41421 q^{7} +1.00000 q^{9} +5.65685 q^{11} +5.65685 q^{13} +0.585786 q^{15} -2.24264 q^{17} -8.24264 q^{19} -1.41421 q^{21} +1.00000 q^{23} -4.65685 q^{25} +1.00000 q^{27} +8.82843 q^{29} -1.17157 q^{31} +5.65685 q^{33} -0.828427 q^{35} -3.17157 q^{37} +5.65685 q^{39} -2.00000 q^{41} -5.41421 q^{43} +0.585786 q^{45} -10.4853 q^{47} -5.00000 q^{49} -2.24264 q^{51} +4.58579 q^{53} +3.31371 q^{55} -8.24264 q^{57} -8.82843 q^{59} +3.17157 q^{61} -1.41421 q^{63} +3.31371 q^{65} -7.75736 q^{67} +1.00000 q^{69} +11.3137 q^{71} +0.343146 q^{73} -4.65685 q^{75} -8.00000 q^{77} -7.07107 q^{79} +1.00000 q^{81} -9.65685 q^{83} -1.31371 q^{85} +8.82843 q^{87} +13.0711 q^{89} -8.00000 q^{91} -1.17157 q^{93} -4.82843 q^{95} +16.1421 q^{97} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{15} + 4 q^{17} - 8 q^{19} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} - 10 q^{49} + 4 q^{51} + 12 q^{53} - 16 q^{55} - 8 q^{57} - 12 q^{59} + 12 q^{61} - 16 q^{65} - 24 q^{67} + 2 q^{69} + 12 q^{73} + 2 q^{75} - 16 q^{77} + 2 q^{81} - 8 q^{83} + 20 q^{85} + 12 q^{87} + 12 q^{89} - 16 q^{91} - 8 q^{93} - 4 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 2 * q^9 + 4 * q^15 + 4 * q^17 - 8 * q^19 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 - 4 * q^41 - 8 * q^43 + 4 * q^45 - 4 * q^47 - 10 * q^49 + 4 * q^51 + 12 * q^53 - 16 * q^55 - 8 * q^57 - 12 * q^59 + 12 * q^61 - 16 * q^65 - 24 * q^67 + 2 * q^69 + 12 * q^73 + 2 * q^75 - 16 * q^77 + 2 * q^81 - 8 * q^83 + 20 * q^85 + 12 * q^87 + 12 * q^89 - 16 * q^91 - 8 * q^93 - 4 * q^95 + 4 * q^97

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