LMFDB - Embedded newform 7360.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -2.73205 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -2.73205 q^{7} -2.00000 q^{9} -2.73205 q^{11} +1.73205 q^{13} +1.00000 q^{15} +5.46410 q^{17} -2.73205 q^{19} -2.73205 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +5.92820 q^{29} +0.267949 q^{31} -2.73205 q^{33} -2.73205 q^{35} +2.73205 q^{37} +1.73205 q^{39} +2.46410 q^{41} -6.73205 q^{43} -2.00000 q^{45} +5.92820 q^{47} +0.464102 q^{49} +5.46410 q^{51} +0.535898 q^{53} -2.73205 q^{55} -2.73205 q^{57} +10.3923 q^{59} -7.66025 q^{61} +5.46410 q^{63} +1.73205 q^{65} -11.6603 q^{67} +1.00000 q^{69} -12.1244 q^{71} -12.2679 q^{73} +1.00000 q^{75} +7.46410 q^{77} +1.66025 q^{79} +1.00000 q^{81} -4.00000 q^{83} +5.46410 q^{85} +5.92820 q^{87} -17.6603 q^{89} -4.73205 q^{91} +0.267949 q^{93} -2.73205 q^{95} -7.66025 q^{97} +5.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9} - 2 q^{11} + 2 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{41} - 10 q^{43} - 4 q^{45} - 2 q^{47} - 6 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{55} - 2 q^{57} + 2 q^{61} + 4 q^{63} - 6 q^{67} + 2 q^{69} - 28 q^{73} + 2 q^{75} + 8 q^{77} - 14 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 2 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^9 - 2 * q^11 + 2 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 10 * q^27 - 2 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 + 2 * q^37 - 2 * q^41 - 10 * q^43 - 4 * q^45 - 2 * q^47 - 6 * q^49 + 4 * q^51 + 8 * q^53 - 2 * q^55 - 2 * q^57 + 2 * q^61 + 4 * q^63 - 6 * q^67 + 2 * q^69 - 28 * q^73 + 2 * q^75 + 8 * q^77 - 14 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 - 2 * q^87 - 18 * q^89 - 6 * q^91 + 4 * q^93 - 2 * q^95 + 2 * q^97 + 4 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−2.73205	−0.823744	−0.411872	−	0.911242i	$-0.635125\pi$
$11$	−2.73205	−0.823744	−0.411872	+	0.911242i	$0.635125\pi$
$12$	0	0
$13$	1.73205	0.480384	0.240192	−	0.970725i	$-0.422790\pi$
$13$	1.73205	0.480384	0.240192	+	0.970725i	$0.422790\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.46410	1.32524	0.662620	−	0.748956i	$-0.269445\pi$
$17$	5.46410	1.32524	0.662620	+	0.748956i	$0.269445\pi$
$18$	0	0
$19$	−2.73205	−0.626775	−0.313388	−	0.949625i	$-0.601464\pi$
$19$	−2.73205	−0.626775	−0.313388	+	0.949625i	$0.601464\pi$
$20$	0	0
$21$	−2.73205	−0.596182
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	5.92820	1.10084	0.550420	−	0.834888i	$-0.314468\pi$
$29$	5.92820	1.10084	0.550420	+	0.834888i	$0.314468\pi$
$30$	0	0
$31$	0.267949	0.0481251	0.0240625	−	0.999710i	$-0.492340\pi$
$31$	0.267949	0.0481251	0.0240625	+	0.999710i	$0.492340\pi$
$32$	0	0
$33$	−2.73205	−0.475589
$34$	0	0
$35$	−2.73205	−0.461801
$36$	0	0
$37$	2.73205	0.449146	0.224573	−	0.974457i	$-0.427901\pi$
$37$	2.73205	0.449146	0.224573	+	0.974457i	$0.427901\pi$
$38$	0	0
$39$	1.73205	0.277350
$40$	0	0
$41$	2.46410	0.384828	0.192414	−	0.981314i	$-0.438368\pi$
$41$	2.46410	0.384828	0.192414	+	0.981314i	$0.438368\pi$
$42$	0	0
$43$	−6.73205	−1.02663	−0.513314	−	0.858201i	$-0.671582\pi$
$43$	−6.73205	−1.02663	−0.513314	+	0.858201i	$0.671582\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	5.92820	0.864717	0.432359	−	0.901702i	$-0.357681\pi$
$47$	5.92820	0.864717	0.432359	+	0.901702i	$0.357681\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	5.46410	0.765127
$52$	0	0
$53$	0.535898	0.0736113	0.0368057	−	0.999322i	$-0.488282\pi$
$53$	0.535898	0.0736113	0.0368057	+	0.999322i	$0.488282\pi$
$54$	0	0
$55$	−2.73205	−0.368390
$56$	0	0
$57$	−2.73205	−0.361869
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	−7.66025	−0.980795	−0.490398	−	0.871499i	$-0.663148\pi$
$61$	−7.66025	−0.980795	−0.490398	+	0.871499i	$0.663148\pi$
$62$	0	0
$63$	5.46410	0.688412
$64$	0	0
$65$	1.73205	0.214834
$66$	0	0
$67$	−11.6603	−1.42453	−0.712263	−	0.701912i	$-0.752330\pi$
$67$	−11.6603	−1.42453	−0.712263	+	0.701912i	$0.752330\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−12.1244	−1.43890	−0.719448	−	0.694546i	$-0.755605\pi$
$71$	−12.1244	−1.43890	−0.719448	+	0.694546i	$0.755605\pi$
$72$	0	0
$73$	−12.2679	−1.43585	−0.717927	−	0.696118i	$-0.754909\pi$
$73$	−12.2679	−1.43585	−0.717927	+	0.696118i	$0.754909\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	7.46410	0.850613
$78$	0	0
$79$	1.66025	0.186793	0.0933966	−	0.995629i	$-0.470228\pi$
$79$	1.66025	0.186793	0.0933966	+	0.995629i	$0.470228\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	5.46410	0.592665
$86$	0	0
$87$	5.92820	0.635570
$88$	0	0
$89$	−17.6603	−1.87198	−0.935992	−	0.352022i	$-0.885494\pi$
$89$	−17.6603	−1.87198	−0.935992	+	0.352022i	$0.885494\pi$
$90$	0	0
$91$	−4.73205	−0.496054
$92$	0	0
$93$	0.267949	0.0277850
$94$	0	0
$95$	−2.73205	−0.280302
$96$	0	0
$97$	−7.66025	−0.777781	−0.388890	−	0.921284i	$-0.627142\pi$
$97$	−7.66025	−0.777781	−0.388890	+	0.921284i	$0.627142\pi$
$98$	0	0
$99$	5.46410	0.549163
$100$	0	0
$101$	17.8564	1.77678	0.888389	−	0.459091i	$-0.151825\pi$
$101$	17.8564	1.77678	0.888389	+	0.459091i	$0.151825\pi$
$102$	0	0
$103$	−5.46410	−0.538394	−0.269197	−	0.963085i	$-0.586758\pi$
$103$	−5.46410	−0.538394	−0.269197	+	0.963085i	$0.586758\pi$
$104$	0	0
$105$	−2.73205	−0.266621
$106$	0	0
$107$	−7.26795	−0.702619	−0.351310	−	0.936259i	$-0.614264\pi$
$107$	−7.26795	−0.702619	−0.351310	+	0.936259i	$0.614264\pi$
$108$	0	0
$109$	2.33975	0.224107	0.112054	−	0.993702i	$-0.464257\pi$
$109$	2.33975	0.224107	0.112054	+	0.993702i	$0.464257\pi$
$110$	0	0
$111$	2.73205	0.259315
$112$	0	0
$113$	−7.26795	−0.683711	−0.341856	−	0.939753i	$-0.611055\pi$
$113$	−7.26795	−0.683711	−0.341856	+	0.939753i	$0.611055\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−3.46410	−0.320256
$118$	0	0
$119$	−14.9282	−1.36847
$120$	0	0
$121$	−3.53590	−0.321445
$122$	0	0
$123$	2.46410	0.222181
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.8564	−0.963350	−0.481675	−	0.876350i	$-0.659971\pi$
$127$	−10.8564	−0.963350	−0.481675	+	0.876350i	$0.659971\pi$
$128$	0	0
$129$	−6.73205	−0.592724
$130$	0	0
$131$	12.6603	1.10613	0.553066	−	0.833138i	$-0.313458\pi$
$131$	12.6603	1.10613	0.553066	+	0.833138i	$0.313458\pi$
$132$	0	0
$133$	7.46410	0.647220
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	5.46410	0.466830	0.233415	−	0.972377i	$-0.425010\pi$
$137$	5.46410	0.466830	0.233415	+	0.972377i	$0.425010\pi$
$138$	0	0
$139$	−15.1962	−1.28892	−0.644460	−	0.764638i	$-0.722918\pi$
$139$	−15.1962	−1.28892	−0.644460	+	0.764638i	$0.722918\pi$
$140$	0	0
$141$	5.92820	0.499245
$142$	0	0
$143$	−4.73205	−0.395714
$144$	0	0
$145$	5.92820	0.492310
$146$	0	0
$147$	0.464102	0.0382785
$148$	0	0
$149$	−4.92820	−0.403734	−0.201867	−	0.979413i	$-0.564701\pi$
$149$	−4.92820	−0.403734	−0.201867	+	0.979413i	$0.564701\pi$
$150$	0	0
$151$	4.12436	0.335635	0.167818	−	0.985818i	$-0.446328\pi$
$151$	4.12436	0.335635	0.167818	+	0.985818i	$0.446328\pi$
$152$	0	0
$153$	−10.9282	−0.883493
$154$	0	0
$155$	0.267949	0.0215222
$156$	0	0
$157$	0.732051	0.0584240	0.0292120	−	0.999573i	$-0.490700\pi$
$157$	0.732051	0.0584240	0.0292120	+	0.999573i	$0.490700\pi$
$158$	0	0
$159$	0.535898	0.0424995
$160$	0	0
$161$	−2.73205	−0.215316
$162$	0	0
$163$	0.464102	0.0363512	0.0181756	−	0.999835i	$-0.494214\pi$
$163$	0.464102	0.0363512	0.0181756	+	0.999835i	$0.494214\pi$
$164$	0	0
$165$	−2.73205	−0.212690
$166$	0	0
$167$	−7.07180	−0.547232	−0.273616	−	0.961839i	$-0.588220\pi$
$167$	−7.07180	−0.547232	−0.273616	+	0.961839i	$0.588220\pi$
$168$	0	0
$169$	−10.0000	−0.769231
$170$	0	0
$171$	5.46410	0.417850
$172$	0	0
$173$	−19.8564	−1.50965	−0.754827	−	0.655924i	$-0.772279\pi$
$173$	−19.8564	−1.50965	−0.754827	+	0.655924i	$0.772279\pi$
$174$	0	0
$175$	−2.73205	−0.206524
$176$	0	0
$177$	10.3923	0.781133
$178$	0	0
$179$	−2.66025	−0.198837	−0.0994184	−	0.995046i	$-0.531698\pi$
$179$	−2.66025	−0.198837	−0.0994184	+	0.995046i	$0.531698\pi$
$180$	0	0
$181$	0.339746	0.0252531	0.0126266	−	0.999920i	$-0.495981\pi$
$181$	0.339746	0.0252531	0.0126266	+	0.999920i	$0.495981\pi$
$182$	0	0
$183$	−7.66025	−0.566262
$184$	0	0
$185$	2.73205	0.200864
$186$	0	0
$187$	−14.9282	−1.09166
$188$	0	0
$189$	13.6603	0.993637
$190$	0	0
$191$	2.39230	0.173101	0.0865506	−	0.996247i	$-0.472416\pi$
$191$	2.39230	0.173101	0.0865506	+	0.996247i	$0.472416\pi$
$192$	0	0
$193$	−7.33975	−0.528326	−0.264163	−	0.964478i	$-0.585096\pi$
$193$	−7.33975	−0.528326	−0.264163	+	0.964478i	$0.585096\pi$
$194$	0	0
$195$	1.73205	0.124035
$196$	0	0
$197$	−16.6603	−1.18699	−0.593497	−	0.804836i	$-0.702253\pi$
$197$	−16.6603	−1.18699	−0.593497	+	0.804836i	$0.702253\pi$
$198$	0	0
$199$	−11.8038	−0.836753	−0.418376	−	0.908274i	$-0.637401\pi$
$199$	−11.8038	−0.836753	−0.418376	+	0.908274i	$0.637401\pi$
$200$	0	0
$201$	−11.6603	−0.822451
$202$	0	0
$203$	−16.1962	−1.13675
$204$	0	0
$205$	2.46410	0.172100
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	7.46410	0.516303
$210$	0	0
$211$	8.39230	0.577750	0.288875	−	0.957367i	$-0.406719\pi$
$211$	8.39230	0.577750	0.288875	+	0.957367i	$0.406719\pi$
$212$	0	0
$213$	−12.1244	−0.830747
$214$	0	0
$215$	−6.73205	−0.459122
$216$	0	0
$217$	−0.732051	−0.0496948
$218$	0	0
$219$	−12.2679	−0.828991
$220$	0	0
$221$	9.46410	0.636624
$222$	0	0
$223$	16.9282	1.13360	0.566798	−	0.823857i	$-0.308182\pi$
$223$	16.9282	1.13360	0.566798	+	0.823857i	$0.308182\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	1.26795	0.0841567	0.0420784	−	0.999114i	$-0.486602\pi$
$227$	1.26795	0.0841567	0.0420784	+	0.999114i	$0.486602\pi$
$228$	0	0
$229$	−11.1244	−0.735118	−0.367559	−	0.930000i	$-0.619806\pi$
$229$	−11.1244	−0.735118	−0.367559	+	0.930000i	$0.619806\pi$
$230$	0	0
$231$	7.46410	0.491102
$232$	0	0
$233$	−23.0526	−1.51022	−0.755112	−	0.655596i	$-0.772417\pi$
$233$	−23.0526	−1.51022	−0.755112	+	0.655596i	$0.772417\pi$
$234$	0	0
$235$	5.92820	0.386713
$236$	0	0
$237$	1.66025	0.107845
$238$	0	0
$239$	−17.7321	−1.14699	−0.573496	−	0.819209i	$-0.694413\pi$
$239$	−17.7321	−1.14699	−0.573496	+	0.819209i	$0.694413\pi$
$240$	0	0
$241$	−24.2487	−1.56200	−0.780998	−	0.624533i	$-0.785289\pi$
$241$	−24.2487	−1.56200	−0.780998	+	0.624533i	$0.785289\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	0.464102	0.0296504
$246$	0	0
$247$	−4.73205	−0.301093
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−14.9282	−0.942260	−0.471130	−	0.882064i	$-0.656154\pi$
$251$	−14.9282	−0.942260	−0.471130	+	0.882064i	$0.656154\pi$
$252$	0	0
$253$	−2.73205	−0.171763
$254$	0	0
$255$	5.46410	0.342175
$256$	0	0
$257$	3.73205	0.232799	0.116399	−	0.993202i	$-0.462865\pi$
$257$	3.73205	0.232799	0.116399	+	0.993202i	$0.462865\pi$
$258$	0	0
$259$	−7.46410	−0.463797
$260$	0	0
$261$	−11.8564	−0.733893
$262$	0	0
$263$	−24.0526	−1.48314	−0.741572	−	0.670873i	$-0.765920\pi$
$263$	−24.0526	−1.48314	−0.741572	+	0.670873i	$0.765920\pi$
$264$	0	0
$265$	0.535898	0.0329200
$266$	0	0
$267$	−17.6603	−1.08079
$268$	0	0
$269$	21.9282	1.33699	0.668493	−	0.743718i	$-0.266940\pi$
$269$	21.9282	1.33699	0.668493	+	0.743718i	$0.266940\pi$
$270$	0	0
$271$	−22.2487	−1.35151	−0.675756	−	0.737125i	$-0.736183\pi$
$271$	−22.2487	−1.35151	−0.675756	+	0.737125i	$0.736183\pi$
$272$	0	0
$273$	−4.73205	−0.286397
$274$	0	0
$275$	−2.73205	−0.164749
$276$	0	0
$277$	29.0526	1.74560	0.872800	−	0.488079i	$-0.162302\pi$
$277$	29.0526	1.74560	0.872800	+	0.488079i	$0.162302\pi$
$278$	0	0
$279$	−0.535898	−0.0320834
$280$	0	0
$281$	11.1244	0.663623	0.331812	−	0.943346i	$-0.392340\pi$
$281$	11.1244	0.663623	0.331812	+	0.943346i	$0.392340\pi$
$282$	0	0
$283$	−13.3205	−0.791822	−0.395911	−	0.918289i	$-0.629571\pi$
$283$	−13.3205	−0.791822	−0.395911	+	0.918289i	$0.629571\pi$
$284$	0	0
$285$	−2.73205	−0.161833
$286$	0	0
$287$	−6.73205	−0.397380
$288$	0	0
$289$	12.8564	0.756259
$290$	0	0
$291$	−7.66025	−0.449052
$292$	0	0
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	0	0
$295$	10.3923	0.605063
$296$	0	0
$297$	13.6603	0.792648
$298$	0	0
$299$	1.73205	0.100167
$300$	0	0
$301$	18.3923	1.06011
$302$	0	0
$303$	17.8564	1.02582
$304$	0	0
$305$	−7.66025	−0.438625
$306$	0	0
$307$	11.4641	0.654291	0.327145	−	0.944974i	$-0.393913\pi$
$307$	11.4641	0.654291	0.327145	+	0.944974i	$0.393913\pi$
$308$	0	0
$309$	−5.46410	−0.310842
$310$	0	0
$311$	33.0526	1.87424	0.937119	−	0.349009i	$-0.113482\pi$
$311$	33.0526	1.87424	0.937119	+	0.349009i	$0.113482\pi$
$312$	0	0
$313$	−18.9282	−1.06989	−0.534943	−	0.844888i	$-0.679667\pi$
$313$	−18.9282	−1.06989	−0.534943	+	0.844888i	$0.679667\pi$
$314$	0	0
$315$	5.46410	0.307867
$316$	0	0
$317$	−20.9282	−1.17544	−0.587722	−	0.809063i	$-0.699975\pi$
$317$	−20.9282	−1.17544	−0.587722	+	0.809063i	$0.699975\pi$
$318$	0	0
$319$	−16.1962	−0.906810
$320$	0	0
$321$	−7.26795	−0.405657
$322$	0	0
$323$	−14.9282	−0.830627
$324$	0	0
$325$	1.73205	0.0960769
$326$	0	0
$327$	2.33975	0.129388
$328$	0	0
$329$	−16.1962	−0.892923
$330$	0	0
$331$	−0.267949	−0.0147278	−0.00736391	−	0.999973i	$-0.502344\pi$
$331$	−0.267949	−0.0147278	−0.00736391	+	0.999973i	$0.502344\pi$
$332$	0	0
$333$	−5.46410	−0.299431
$334$	0	0
$335$	−11.6603	−0.637068
$336$	0	0
$337$	34.5885	1.88415	0.942077	−	0.335398i	$-0.108871\pi$
$337$	34.5885	1.88415	0.942077	+	0.335398i	$0.108871\pi$
$338$	0	0
$339$	−7.26795	−0.394741
$340$	0	0
$341$	−0.732051	−0.0396428
$342$	0	0
$343$	17.8564	0.964155
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	7.85641	0.421754	0.210877	−	0.977513i	$-0.432368\pi$
$347$	7.85641	0.421754	0.210877	+	0.977513i	$0.432368\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	−8.66025	−0.462250
$352$	0	0
$353$	19.7321	1.05023	0.525116	−	0.851031i	$-0.324022\pi$
$353$	19.7321	1.05023	0.525116	+	0.851031i	$0.324022\pi$
$354$	0	0
$355$	−12.1244	−0.643494
$356$	0	0
$357$	−14.9282	−0.790084
$358$	0	0
$359$	−10.3397	−0.545711	−0.272855	−	0.962055i	$-0.587968\pi$
$359$	−10.3397	−0.545711	−0.272855	+	0.962055i	$0.587968\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	0	0
$363$	−3.53590	−0.185587
$364$	0	0
$365$	−12.2679	−0.642134
$366$	0	0
$367$	−14.5359	−0.758768	−0.379384	−	0.925239i	$-0.623864\pi$
$367$	−14.5359	−0.758768	−0.379384	+	0.925239i	$0.623864\pi$
$368$	0	0
$369$	−4.92820	−0.256552
$370$	0	0
$371$	−1.46410	−0.0760124
$372$	0	0
$373$	20.9282	1.08362	0.541811	−	0.840501i	$-0.317739\pi$
$373$	20.9282	1.08362	0.541811	+	0.840501i	$0.317739\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	10.2679	0.528826
$378$	0	0
$379$	−4.39230	−0.225618	−0.112809	−	0.993617i	$-0.535985\pi$
$379$	−4.39230	−0.225618	−0.112809	+	0.993617i	$0.535985\pi$
$380$	0	0
$381$	−10.8564	−0.556191
$382$	0	0
$383$	4.19615	0.214413	0.107207	−	0.994237i	$-0.465809\pi$
$383$	4.19615	0.214413	0.107207	+	0.994237i	$0.465809\pi$
$384$	0	0
$385$	7.46410	0.380406
$386$	0	0
$387$	13.4641	0.684419
$388$	0	0
$389$	11.6077	0.588534	0.294267	−	0.955723i	$-0.404925\pi$
$389$	11.6077	0.588534	0.294267	+	0.955723i	$0.404925\pi$
$390$	0	0
$391$	5.46410	0.276331
$392$	0	0
$393$	12.6603	0.638625
$394$	0	0
$395$	1.66025	0.0835364
$396$	0	0
$397$	5.73205	0.287683	0.143842	−	0.989601i	$-0.454054\pi$
$397$	5.73205	0.287683	0.143842	+	0.989601i	$0.454054\pi$
$398$	0	0
$399$	7.46410	0.373672
$400$	0	0
$401$	31.8564	1.59083	0.795417	−	0.606063i	$-0.207252\pi$
$401$	31.8564	1.59083	0.795417	+	0.606063i	$0.207252\pi$
$402$	0	0
$403$	0.464102	0.0231185
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−7.46410	−0.369982
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	5.46410	0.269524
$412$	0	0
$413$	−28.3923	−1.39709
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	−15.1962	−0.744159
$418$	0	0
$419$	4.73205	0.231176	0.115588	−	0.993297i	$-0.463125\pi$
$419$	4.73205	0.231176	0.115588	+	0.993297i	$0.463125\pi$
$420$	0	0
$421$	9.26795	0.451692	0.225846	−	0.974163i	$-0.427485\pi$
$421$	9.26795	0.451692	0.225846	+	0.974163i	$0.427485\pi$
$422$	0	0
$423$	−11.8564	−0.576478
$424$	0	0
$425$	5.46410	0.265048
$426$	0	0
$427$	20.9282	1.01279
$428$	0	0
$429$	−4.73205	−0.228466
$430$	0	0
$431$	5.80385	0.279562	0.139781	−	0.990182i	$-0.455360\pi$
$431$	5.80385	0.279562	0.139781	+	0.990182i	$0.455360\pi$
$432$	0	0
$433$	−4.92820	−0.236834	−0.118417	−	0.992964i	$-0.537782\pi$
$433$	−4.92820	−0.236834	−0.118417	+	0.992964i	$0.537782\pi$
$434$	0	0
$435$	5.92820	0.284236
$436$	0	0
$437$	−2.73205	−0.130692
$438$	0	0
$439$	−14.8038	−0.706549	−0.353275	−	0.935520i	$-0.614932\pi$
$439$	−14.8038	−0.706549	−0.353275	+	0.935520i	$0.614932\pi$
$440$	0	0
$441$	−0.928203	−0.0442002
$442$	0	0
$443$	33.2487	1.57969	0.789847	−	0.613304i	$-0.210160\pi$
$443$	33.2487	1.57969	0.789847	+	0.613304i	$0.210160\pi$
$444$	0	0
$445$	−17.6603	−0.837176
$446$	0	0
$447$	−4.92820	−0.233096
$448$	0	0
$449$	−9.85641	−0.465153	−0.232576	−	0.972578i	$-0.574716\pi$
$449$	−9.85641	−0.465153	−0.232576	+	0.972578i	$0.574716\pi$
$450$	0	0
$451$	−6.73205	−0.317000
$452$	0	0
$453$	4.12436	0.193779
$454$	0	0
$455$	−4.73205	−0.221842
$456$	0	0
$457$	17.0718	0.798585	0.399292	−	0.916824i	$-0.369256\pi$
$457$	17.0718	0.798585	0.399292	+	0.916824i	$0.369256\pi$
$458$	0	0
$459$	−27.3205	−1.27521
$460$	0	0
$461$	18.4641	0.859959	0.429979	−	0.902839i	$-0.358521\pi$
$461$	18.4641	0.859959	0.429979	+	0.902839i	$0.358521\pi$
$462$	0	0
$463$	−0.535898	−0.0249053	−0.0124527	−	0.999922i	$-0.503964\pi$
$463$	−0.535898	−0.0249053	−0.0124527	+	0.999922i	$0.503964\pi$
$464$	0	0
$465$	0.267949	0.0124258
$466$	0	0
$467$	31.8564	1.47414	0.737069	−	0.675817i	$-0.236209\pi$
$467$	31.8564	1.47414	0.737069	+	0.675817i	$0.236209\pi$
$468$	0	0
$469$	31.8564	1.47099
$470$	0	0
$471$	0.732051	0.0337311
$472$	0	0
$473$	18.3923	0.845679
$474$	0	0
$475$	−2.73205	−0.125355
$476$	0	0
$477$	−1.07180	−0.0490742
$478$	0	0
$479$	−25.1769	−1.15036	−0.575181	−	0.818026i	$-0.695068\pi$
$479$	−25.1769	−1.15036	−0.575181	+	0.818026i	$0.695068\pi$
$480$	0	0
$481$	4.73205	0.215763
$482$	0	0
$483$	−2.73205	−0.124313
$484$	0	0
$485$	−7.66025	−0.347834
$486$	0	0
$487$	23.3923	1.06001	0.530003	−	0.847996i	$-0.322191\pi$
$487$	23.3923	1.06001	0.530003	+	0.847996i	$0.322191\pi$
$488$	0	0
$489$	0.464102	0.0209874
$490$	0	0
$491$	−16.6603	−0.751867	−0.375933	−	0.926647i	$-0.622678\pi$
$491$	−16.6603	−0.751867	−0.375933	+	0.926647i	$0.622678\pi$
$492$	0	0
$493$	32.3923	1.45888
$494$	0	0
$495$	5.46410	0.245593
$496$	0	0
$497$	33.1244	1.48583
$498$	0	0
$499$	−8.26795	−0.370124	−0.185062	−	0.982727i	$-0.559249\pi$
$499$	−8.26795	−0.370124	−0.185062	+	0.982727i	$0.559249\pi$
$500$	0	0
$501$	−7.07180	−0.315945
$502$	0	0
$503$	18.3923	0.820072	0.410036	−	0.912069i	$-0.365516\pi$
$503$	18.3923	0.820072	0.410036	+	0.912069i	$0.365516\pi$
$504$	0	0
$505$	17.8564	0.794600
$506$	0	0
$507$	−10.0000	−0.444116
$508$	0	0
$509$	−32.3205	−1.43258	−0.716291	−	0.697802i	$-0.754162\pi$
$509$	−32.3205	−1.43258	−0.716291	+	0.697802i	$0.754162\pi$
$510$	0	0
$511$	33.5167	1.48269
$512$	0	0
$513$	13.6603	0.603115
$514$	0	0
$515$	−5.46410	−0.240777
$516$	0	0
$517$	−16.1962	−0.712306
$518$	0	0
$519$	−19.8564	−0.871600
$520$	0	0
$521$	40.3923	1.76962	0.884810	−	0.465953i	$-0.154288\pi$
$521$	40.3923	1.76962	0.884810	+	0.465953i	$0.154288\pi$
$522$	0	0
$523$	−7.41154	−0.324084	−0.162042	−	0.986784i	$-0.551808\pi$
$523$	−7.41154	−0.324084	−0.162042	+	0.986784i	$0.551808\pi$
$524$	0	0
$525$	−2.73205	−0.119236
$526$	0	0
$527$	1.46410	0.0637773
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−20.7846	−0.901975
$532$	0	0
$533$	4.26795	0.184865
$534$	0	0
$535$	−7.26795	−0.314221
$536$	0	0
$537$	−2.66025	−0.114798
$538$	0	0
$539$	−1.26795	−0.0546144
$540$	0	0
$541$	−0.607695	−0.0261269	−0.0130634	−	0.999915i	$-0.504158\pi$
$541$	−0.607695	−0.0261269	−0.0130634	+	0.999915i	$0.504158\pi$
$542$	0	0
$543$	0.339746	0.0145799
$544$	0	0
$545$	2.33975	0.100224
$546$	0	0
$547$	27.7846	1.18798	0.593992	−	0.804471i	$-0.297551\pi$
$547$	27.7846	1.18798	0.593992	+	0.804471i	$0.297551\pi$
$548$	0	0
$549$	15.3205	0.653863
$550$	0	0
$551$	−16.1962	−0.689979
$552$	0	0
$553$	−4.53590	−0.192886
$554$	0	0
$555$	2.73205	0.115969
$556$	0	0
$557$	13.1244	0.556097	0.278048	−	0.960567i	$-0.410312\pi$
$557$	13.1244	0.556097	0.278048	+	0.960567i	$0.410312\pi$
$558$	0	0
$559$	−11.6603	−0.493176
$560$	0	0
$561$	−14.9282	−0.630269
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	−7.26795	−0.305765
$566$	0	0
$567$	−2.73205	−0.114735
$568$	0	0
$569$	19.8564	0.832424	0.416212	−	0.909268i	$-0.363357\pi$
$569$	19.8564	0.832424	0.416212	+	0.909268i	$0.363357\pi$
$570$	0	0
$571$	15.3205	0.641143	0.320572	−	0.947224i	$-0.396125\pi$
$571$	15.3205	0.641143	0.320572	+	0.947224i	$0.396125\pi$
$572$	0	0
$573$	2.39230	0.0999400
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−25.4449	−1.05928	−0.529642	−	0.848221i	$-0.677674\pi$
$577$	−25.4449	−1.05928	−0.529642	+	0.848221i	$0.677674\pi$
$578$	0	0
$579$	−7.33975	−0.305029
$580$	0	0
$581$	10.9282	0.453378
$582$	0	0
$583$	−1.46410	−0.0606369
$584$	0	0
$585$	−3.46410	−0.143223
$586$	0	0
$587$	−32.4641	−1.33994	−0.669968	−	0.742390i	$-0.733692\pi$
$587$	−32.4641	−1.33994	−0.669968	+	0.742390i	$0.733692\pi$
$588$	0	0
$589$	−0.732051	−0.0301636
$590$	0	0
$591$	−16.6603	−0.685311
$592$	0	0
$593$	15.7128	0.645248	0.322624	−	0.946527i	$-0.395435\pi$
$593$	15.7128	0.645248	0.322624	+	0.946527i	$0.395435\pi$
$594$	0	0
$595$	−14.9282	−0.611997
$596$	0	0
$597$	−11.8038	−0.483099
$598$	0	0
$599$	−41.3205	−1.68831	−0.844155	−	0.536099i	$-0.819897\pi$
$599$	−41.3205	−1.68831	−0.844155	+	0.536099i	$0.819897\pi$
$600$	0	0
$601$	1.92820	0.0786531	0.0393265	−	0.999226i	$-0.487479\pi$
$601$	1.92820	0.0786531	0.0393265	+	0.999226i	$0.487479\pi$
$602$	0	0
$603$	23.3205	0.949685
$604$	0	0
$605$	−3.53590	−0.143755
$606$	0	0
$607$	6.39230	0.259456	0.129728	−	0.991550i	$-0.458590\pi$
$607$	6.39230	0.259456	0.129728	+	0.991550i	$0.458590\pi$
$608$	0	0
$609$	−16.1962	−0.656301
$610$	0	0
$611$	10.2679	0.415397
$612$	0	0
$613$	−14.2487	−0.575500	−0.287750	−	0.957706i	$-0.592907\pi$
$613$	−14.2487	−0.575500	−0.287750	+	0.957706i	$0.592907\pi$
$614$	0	0
$615$	2.46410	0.0993622
$616$	0	0
$617$	25.2679	1.01725	0.508625	−	0.860988i	$-0.330154\pi$
$617$	25.2679	1.01725	0.508625	+	0.860988i	$0.330154\pi$
$618$	0	0
$619$	2.24871	0.0903833	0.0451917	−	0.998978i	$-0.485610\pi$
$619$	2.24871	0.0903833	0.0451917	+	0.998978i	$0.485610\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	48.2487	1.93304
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.46410	0.298088
$628$	0	0
$629$	14.9282	0.595226
$630$	0	0
$631$	−38.1962	−1.52056	−0.760282	−	0.649593i	$-0.774939\pi$
$631$	−38.1962	−1.52056	−0.760282	+	0.649593i	$0.774939\pi$
$632$	0	0
$633$	8.39230	0.333564
$634$	0	0
$635$	−10.8564	−0.430823
$636$	0	0
$637$	0.803848	0.0318496
$638$	0	0
$639$	24.2487	0.959264
$640$	0	0
$641$	−19.5167	−0.770862	−0.385431	−	0.922737i	$-0.625947\pi$
$641$	−19.5167	−0.770862	−0.385431	+	0.922737i	$0.625947\pi$
$642$	0	0
$643$	27.4641	1.08308	0.541539	−	0.840675i	$-0.317842\pi$
$643$	27.4641	1.08308	0.541539	+	0.840675i	$0.317842\pi$
$644$	0	0
$645$	−6.73205	−0.265074
$646$	0	0
$647$	0.464102	0.0182457	0.00912286	−	0.999958i	$-0.497096\pi$
$647$	0.464102	0.0182457	0.00912286	+	0.999958i	$0.497096\pi$
$648$	0	0
$649$	−28.3923	−1.11450
$650$	0	0
$651$	−0.732051	−0.0286913
$652$	0	0
$653$	−23.1962	−0.907736	−0.453868	−	0.891069i	$-0.649956\pi$
$653$	−23.1962	−0.907736	−0.453868	+	0.891069i	$0.649956\pi$
$654$	0	0
$655$	12.6603	0.494677
$656$	0	0
$657$	24.5359	0.957237
$658$	0	0
$659$	−48.7846	−1.90038	−0.950189	−	0.311673i	$-0.899111\pi$
$659$	−48.7846	−1.90038	−0.950189	+	0.311673i	$0.899111\pi$
$660$	0	0
$661$	−25.7128	−1.00011	−0.500056	−	0.865993i	$-0.666687\pi$
$661$	−25.7128	−1.00011	−0.500056	+	0.865993i	$0.666687\pi$
$662$	0	0
$663$	9.46410	0.367555
$664$	0	0
$665$	7.46410	0.289445
$666$	0	0
$667$	5.92820	0.229541
$668$	0	0
$669$	16.9282	0.654482
$670$	0	0
$671$	20.9282	0.807924
$672$	0	0
$673$	−28.9090	−1.11436	−0.557179	−	0.830392i	$-0.688116\pi$
$673$	−28.9090	−1.11436	−0.557179	+	0.830392i	$0.688116\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−8.92820	−0.343139	−0.171569	−	0.985172i	$-0.554884\pi$
$677$	−8.92820	−0.343139	−0.171569	+	0.985172i	$0.554884\pi$
$678$	0	0
$679$	20.9282	0.803151
$680$	0	0
$681$	1.26795	0.0485879
$682$	0	0
$683$	−39.1051	−1.49632	−0.748158	−	0.663521i	$-0.769061\pi$
$683$	−39.1051	−1.49632	−0.748158	+	0.663521i	$0.769061\pi$
$684$	0	0
$685$	5.46410	0.208773
$686$	0	0
$687$	−11.1244	−0.424421
$688$	0	0
$689$	0.928203	0.0353617
$690$	0	0
$691$	−28.3923	−1.08009	−0.540047	−	0.841635i	$-0.681594\pi$
$691$	−28.3923	−1.08009	−0.540047	+	0.841635i	$0.681594\pi$
$692$	0	0
$693$	−14.9282	−0.567076
$694$	0	0
$695$	−15.1962	−0.576423
$696$	0	0
$697$	13.4641	0.509989
$698$	0	0
$699$	−23.0526	−0.871928
$700$	0	0
$701$	−10.7321	−0.405344	−0.202672	−	0.979247i	$-0.564963\pi$
$701$	−10.7321	−0.405344	−0.202672	+	0.979247i	$0.564963\pi$
$702$	0	0
$703$	−7.46410	−0.281514
$704$	0	0
$705$	5.92820	0.223269
$706$	0	0
$707$	−48.7846	−1.83473
$708$	0	0
$709$	4.58846	0.172323	0.0861616	−	0.996281i	$-0.472540\pi$
$709$	4.58846	0.172323	0.0861616	+	0.996281i	$0.472540\pi$
$710$	0	0
$711$	−3.32051	−0.124529
$712$	0	0
$713$	0.267949	0.0100348
$714$	0	0
$715$	−4.73205	−0.176969
$716$	0	0
$717$	−17.7321	−0.662216
$718$	0	0
$719$	−15.6077	−0.582069	−0.291034	−	0.956713i	$-0.593999\pi$
$719$	−15.6077	−0.582069	−0.291034	+	0.956713i	$0.593999\pi$
$720$	0	0
$721$	14.9282	0.555955
$722$	0	0
$723$	−24.2487	−0.901819
$724$	0	0
$725$	5.92820	0.220168
$726$	0	0
$727$	−35.1769	−1.30464	−0.652320	−	0.757944i	$-0.726204\pi$
$727$	−35.1769	−1.30464	−0.652320	+	0.757944i	$0.726204\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−36.7846	−1.36053
$732$	0	0
$733$	8.78461	0.324467	0.162233	−	0.986752i	$-0.448130\pi$
$733$	8.78461	0.324467	0.162233	+	0.986752i	$0.448130\pi$
$734$	0	0
$735$	0.464102	0.0171186
$736$	0	0
$737$	31.8564	1.17345
$738$	0	0
$739$	40.3731	1.48515	0.742574	−	0.669764i	$-0.233605\pi$
$739$	40.3731	1.48515	0.742574	+	0.669764i	$0.233605\pi$
$740$	0	0
$741$	−4.73205	−0.173836
$742$	0	0
$743$	13.1769	0.483414	0.241707	−	0.970349i	$-0.422293\pi$
$743$	13.1769	0.483414	0.241707	+	0.970349i	$0.422293\pi$
$744$	0	0
$745$	−4.92820	−0.180555
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	19.8564	0.725537
$750$	0	0
$751$	−26.9808	−0.984542	−0.492271	−	0.870442i	$-0.663833\pi$
$751$	−26.9808	−0.984542	−0.492271	+	0.870442i	$0.663833\pi$
$752$	0	0
$753$	−14.9282	−0.544014
$754$	0	0
$755$	4.12436	0.150101
$756$	0	0
$757$	−12.1962	−0.443277	−0.221638	−	0.975129i	$-0.571140\pi$
$757$	−12.1962	−0.443277	−0.221638	+	0.975129i	$0.571140\pi$
$758$	0	0
$759$	−2.73205	−0.0991672
$760$	0	0
$761$	39.7846	1.44219	0.721095	−	0.692836i	$-0.243639\pi$
$761$	39.7846	1.44219	0.721095	+	0.692836i	$0.243639\pi$
$762$	0	0
$763$	−6.39230	−0.231417
$764$	0	0
$765$	−10.9282	−0.395110
$766$	0	0
$767$	18.0000	0.649942
$768$	0	0
$769$	−12.9282	−0.466203	−0.233101	−	0.972452i	$-0.574887\pi$
$769$	−12.9282	−0.466203	−0.233101	+	0.972452i	$0.574887\pi$
$770$	0	0
$771$	3.73205	0.134407
$772$	0	0
$773$	−15.6603	−0.563260	−0.281630	−	0.959523i	$-0.590875\pi$
$773$	−15.6603	−0.563260	−0.281630	+	0.959523i	$0.590875\pi$
$774$	0	0
$775$	0.267949	0.00962502
$776$	0	0
$777$	−7.46410	−0.267773
$778$	0	0
$779$	−6.73205	−0.241201
$780$	0	0
$781$	33.1244	1.18528
$782$	0	0
$783$	−29.6410	−1.05928
$784$	0	0
$785$	0.732051	0.0261280
$786$	0	0
$787$	−23.7128	−0.845270	−0.422635	−	0.906300i	$-0.638895\pi$
$787$	−23.7128	−0.845270	−0.422635	+	0.906300i	$0.638895\pi$
$788$	0	0
$789$	−24.0526	−0.856294
$790$	0	0
$791$	19.8564	0.706013
$792$	0	0
$793$	−13.2679	−0.471159
$794$	0	0
$795$	0.535898	0.0190064
$796$	0	0
$797$	−13.9474	−0.494044	−0.247022	−	0.969010i	$-0.579452\pi$
$797$	−13.9474	−0.494044	−0.247022	+	0.969010i	$0.579452\pi$
$798$	0	0
$799$	32.3923	1.14596
$800$	0	0
$801$	35.3205	1.24799
$802$	0	0
$803$	33.5167	1.18278
$804$	0	0
$805$	−2.73205	−0.0962921
$806$	0	0
$807$	21.9282	0.771909
$808$	0	0
$809$	−30.9282	−1.08738	−0.543689	−	0.839287i	$-0.682973\pi$
$809$	−30.9282	−1.08738	−0.543689	+	0.839287i	$0.682973\pi$
$810$	0	0
$811$	−27.0526	−0.949944	−0.474972	−	0.880001i	$-0.657542\pi$
$811$	−27.0526	−0.949944	−0.474972	+	0.880001i	$0.657542\pi$
$812$	0	0
$813$	−22.2487	−0.780296
$814$	0	0
$815$	0.464102	0.0162568
$816$	0	0
$817$	18.3923	0.643465
$818$	0	0
$819$	9.46410	0.330702
$820$	0	0
$821$	45.8564	1.60040	0.800200	−	0.599733i	$-0.204727\pi$
$821$	45.8564	1.60040	0.800200	+	0.599733i	$0.204727\pi$
$822$	0	0
$823$	−5.78461	−0.201639	−0.100819	−	0.994905i	$-0.532146\pi$
$823$	−5.78461	−0.201639	−0.100819	+	0.994905i	$0.532146\pi$
$824$	0	0
$825$	−2.73205	−0.0951178
$826$	0	0
$827$	−20.3923	−0.709110	−0.354555	−	0.935035i	$-0.615368\pi$
$827$	−20.3923	−0.709110	−0.354555	+	0.935035i	$0.615368\pi$
$828$	0	0
$829$	4.67949	0.162525	0.0812627	−	0.996693i	$-0.474105\pi$
$829$	4.67949	0.162525	0.0812627	+	0.996693i	$0.474105\pi$
$830$	0	0
$831$	29.0526	1.00782
$832$	0	0
$833$	2.53590	0.0878637
$834$	0	0
$835$	−7.07180	−0.244730
$836$	0	0
$837$	−1.33975	−0.0463084
$838$	0	0
$839$	−35.6603	−1.23113	−0.615564	−	0.788087i	$-0.711072\pi$
$839$	−35.6603	−1.23113	−0.615564	+	0.788087i	$0.711072\pi$
$840$	0	0
$841$	6.14359	0.211848
$842$	0	0
$843$	11.1244	0.383143
$844$	0	0
$845$	−10.0000	−0.344010
$846$	0	0
$847$	9.66025	0.331930
$848$	0	0
$849$	−13.3205	−0.457159
$850$	0	0
$851$	2.73205	0.0936535
$852$	0	0
$853$	−18.7846	−0.643173	−0.321586	−	0.946880i	$-0.604216\pi$
$853$	−18.7846	−0.643173	−0.321586	+	0.946880i	$0.604216\pi$
$854$	0	0
$855$	5.46410	0.186868
$856$	0	0
$857$	13.0526	0.445867	0.222933	−	0.974834i	$-0.428437\pi$
$857$	13.0526	0.445867	0.222933	+	0.974834i	$0.428437\pi$
$858$	0	0
$859$	−6.66025	−0.227245	−0.113622	−	0.993524i	$-0.536245\pi$
$859$	−6.66025	−0.227245	−0.113622	+	0.993524i	$0.536245\pi$
$860$	0	0
$861$	−6.73205	−0.229428
$862$	0	0
$863$	−5.24871	−0.178668	−0.0893341	−	0.996002i	$-0.528474\pi$
$863$	−5.24871	−0.178668	−0.0893341	+	0.996002i	$0.528474\pi$
$864$	0	0
$865$	−19.8564	−0.675138
$866$	0	0
$867$	12.8564	0.436626
$868$	0	0
$869$	−4.53590	−0.153870
$870$	0	0
$871$	−20.1962	−0.684321
$872$	0	0
$873$	15.3205	0.518521
$874$	0	0
$875$	−2.73205	−0.0923602
$876$	0	0
$877$	−26.3923	−0.891205	−0.445602	−	0.895231i	$-0.647010\pi$
$877$	−26.3923	−0.891205	−0.445602	+	0.895231i	$0.647010\pi$
$878$	0	0
$879$	27.7128	0.934730
$880$	0	0
$881$	−8.39230	−0.282744	−0.141372	−	0.989957i	$-0.545151\pi$
$881$	−8.39230	−0.282744	−0.141372	+	0.989957i	$0.545151\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	10.3923	0.349334
$886$	0	0
$887$	37.1051	1.24587	0.622934	−	0.782274i	$-0.285941\pi$
$887$	37.1051	1.24587	0.622934	+	0.782274i	$0.285941\pi$
$888$	0	0
$889$	29.6603	0.994773
$890$	0	0
$891$	−2.73205	−0.0915271
$892$	0	0
$893$	−16.1962	−0.541984
$894$	0	0
$895$	−2.66025	−0.0889225
$896$	0	0
$897$	1.73205	0.0578315
$898$	0	0
$899$	1.58846	0.0529780
$900$	0	0
$901$	2.92820	0.0975526
$902$	0	0
$903$	18.3923	0.612058
$904$	0	0
$905$	0.339746	0.0112935
$906$	0	0
$907$	24.7321	0.821214	0.410607	−	0.911812i	$-0.365317\pi$
$907$	24.7321	0.821214	0.410607	+	0.911812i	$0.365317\pi$
$908$	0	0
$909$	−35.7128	−1.18452
$910$	0	0
$911$	−7.41154	−0.245555	−0.122778	−	0.992434i	$-0.539180\pi$
$911$	−7.41154	−0.245555	−0.122778	+	0.992434i	$0.539180\pi$
$912$	0	0
$913$	10.9282	0.361671
$914$	0	0
$915$	−7.66025	−0.253240
$916$	0	0
$917$	−34.5885	−1.14221
$918$	0	0
$919$	−17.8038	−0.587295	−0.293647	−	0.955914i	$-0.594869\pi$
$919$	−17.8038	−0.587295	−0.293647	+	0.955914i	$0.594869\pi$
$920$	0	0
$921$	11.4641	0.377755
$922$	0	0
$923$	−21.0000	−0.691223
$924$	0	0
$925$	2.73205	0.0898293
$926$	0	0
$927$	10.9282	0.358929
$928$	0	0
$929$	−19.3923	−0.636241	−0.318120	−	0.948050i	$-0.603052\pi$
$929$	−19.3923	−0.636241	−0.318120	+	0.948050i	$0.603052\pi$
$930$	0	0
$931$	−1.26795	−0.0415554
$932$	0	0
$933$	33.0526	1.08209
$934$	0	0
$935$	−14.9282	−0.488204
$936$	0	0
$937$	45.6603	1.49166	0.745828	−	0.666139i	$-0.232054\pi$
$937$	45.6603	1.49166	0.745828	+	0.666139i	$0.232054\pi$
$938$	0	0
$939$	−18.9282	−0.617699
$940$	0	0
$941$	−37.5692	−1.22472	−0.612361	−	0.790578i	$-0.709780\pi$
$941$	−37.5692	−1.22472	−0.612361	+	0.790578i	$0.709780\pi$
$942$	0	0
$943$	2.46410	0.0802422
$944$	0	0
$945$	13.6603	0.444368
$946$	0	0
$947$	40.4641	1.31491	0.657453	−	0.753495i	$-0.271634\pi$
$947$	40.4641	1.31491	0.657453	+	0.753495i	$0.271634\pi$
$948$	0	0
$949$	−21.2487	−0.689762
$950$	0	0
$951$	−20.9282	−0.678643
$952$	0	0
$953$	−39.1769	−1.26906	−0.634532	−	0.772896i	$-0.718807\pi$
$953$	−39.1769	−1.26906	−0.634532	+	0.772896i	$0.718807\pi$
$954$	0	0
$955$	2.39230	0.0774132
$956$	0	0
$957$	−16.1962	−0.523547
$958$	0	0
$959$	−14.9282	−0.482057
$960$	0	0
$961$	−30.9282	−0.997684
$962$	0	0
$963$	14.5359	0.468413
$964$	0	0
$965$	−7.33975	−0.236275
$966$	0	0
$967$	−3.53590	−0.113707	−0.0568534	−	0.998383i	$-0.518107\pi$
$967$	−3.53590	−0.113707	−0.0568534	+	0.998383i	$0.518107\pi$
$968$	0	0
$969$	−14.9282	−0.479563
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	41.5167	1.33096
$974$	0	0
$975$	1.73205	0.0554700
$976$	0	0
$977$	−2.44486	−0.0782181	−0.0391091	−	0.999235i	$-0.512452\pi$
$977$	−2.44486	−0.0782181	−0.0391091	+	0.999235i	$0.512452\pi$
$978$	0	0
$979$	48.2487	1.54204
$980$	0	0
$981$	−4.67949	−0.149405
$982$	0	0
$983$	−39.1769	−1.24955	−0.624775	−	0.780805i	$-0.714809\pi$
$983$	−39.1769	−1.24955	−0.624775	+	0.780805i	$0.714809\pi$
$984$	0	0
$985$	−16.6603	−0.530840
$986$	0	0
$987$	−16.1962	−0.515529
$988$	0	0
$989$	−6.73205	−0.214067
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	0	0
$993$	−0.267949	−0.00850311
$994$	0	0
$995$	−11.8038	−0.374207
$996$	0	0
$997$	6.67949	0.211542	0.105771	−	0.994391i	$-0.466269\pi$
$997$	6.67949	0.211542	0.105771	+	0.994391i	$0.466269\pi$
$998$	0	0
$999$	−13.6603	−0.432191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.bs.1.1		2
4.3	odd	2		7360.2.a.bg.1.2		2
8.3	odd	2		3680.2.a.n.1.2	yes	2
8.5	even	2		3680.2.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.l.1.1	✓	2	8.5	even	2
3680.2.a.n.1.2	yes	2	8.3	odd	2
7360.2.a.bg.1.2		2	4.3	odd	2
7360.2.a.bs.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.73205 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.73205 q^{7} -2.00000 q^{9} -2.73205 q^{11} +1.73205 q^{13} +1.00000 q^{15} +5.46410 q^{17} -2.73205 q^{19} -2.73205 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +5.92820 q^{29} +0.267949 q^{31} -2.73205 q^{33} -2.73205 q^{35} +2.73205 q^{37} +1.73205 q^{39} +2.46410 q^{41} -6.73205 q^{43} -2.00000 q^{45} +5.92820 q^{47} +0.464102 q^{49} +5.46410 q^{51} +0.535898 q^{53} -2.73205 q^{55} -2.73205 q^{57} +10.3923 q^{59} -7.66025 q^{61} +5.46410 q^{63} +1.73205 q^{65} -11.6603 q^{67} +1.00000 q^{69} -12.1244 q^{71} -12.2679 q^{73} +1.00000 q^{75} +7.46410 q^{77} +1.66025 q^{79} +1.00000 q^{81} -4.00000 q^{83} +5.46410 q^{85} +5.92820 q^{87} -17.6603 q^{89} -4.73205 q^{91} +0.267949 q^{93} -2.73205 q^{95} -7.66025 q^{97} +5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9} - 2 q^{11} + 2 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{41} - 10 q^{43} - 4 q^{45} - 2 q^{47} - 6 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{55} - 2 q^{57} + 2 q^{61} + 4 q^{63} - 6 q^{67} + 2 q^{69} - 28 q^{73} + 2 q^{75} + 8 q^{77} - 14 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 2 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^9 - 2 * q^11 + 2 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 10 * q^27 - 2 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 + 2 * q^37 - 2 * q^41 - 10 * q^43 - 4 * q^45 - 2 * q^47 - 6 * q^49 + 4 * q^51 + 8 * q^53 - 2 * q^55 - 2 * q^57 + 2 * q^61 + 4 * q^63 - 6 * q^67 + 2 * q^69 - 28 * q^73 + 2 * q^75 + 8 * q^77 - 14 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 - 2 * q^87 - 18 * q^89 - 6 * q^91 + 4 * q^93 - 2 * q^95 + 2 * q^97 + 4 * q^99

Introduction

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