LMFDB - Embedded newform 1792.2.b.l.897.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			897.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1792.897
Dual form			1792.2.b.l.897.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051i q^{3} -2.73205i q^{5} -1.00000 q^{7} +2.46410 q^{9} +O(q^{10})$	$q-0.732051i q^{3} -2.73205i q^{5} -1.00000 q^{7} +2.46410 q^{9} +1.46410i q^{11} +2.73205i q^{13} -2.00000 q^{15} +7.46410 q^{17} +6.19615i q^{19} +0.732051i q^{21} +8.92820 q^{23} -2.46410 q^{25} -4.00000i q^{27} +3.46410i q^{29} -2.53590 q^{31} +1.07180 q^{33} +2.73205i q^{35} -4.53590i q^{37} +2.00000 q^{39} +3.46410 q^{41} -2.53590i q^{43} -6.73205i q^{45} +1.46410 q^{47} +1.00000 q^{49} -5.46410i q^{51} -6.00000i q^{53} +4.00000 q^{55} +4.53590 q^{57} -6.19615i q^{59} -11.1244i q^{61} -2.46410 q^{63} +7.46410 q^{65} +14.9282i q^{67} -6.53590i q^{69} -13.8564 q^{71} -8.92820 q^{73} +1.80385i q^{75} -1.46410i q^{77} -8.00000 q^{79} +4.46410 q^{81} -7.26795i q^{83} -20.3923i q^{85} +2.53590 q^{87} -4.92820 q^{89} -2.73205i q^{91} +1.85641i q^{93} +16.9282 q^{95} +0.535898 q^{97} +3.60770i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 - 4 * q^9	$ 4 q - 4 q^{7} - 4 q^{9} - 8 q^{15} + 16 q^{17} + 8 q^{23} + 4 q^{25} - 24 q^{31} + 32 q^{33} + 8 q^{39} - 8 q^{47} + 4 q^{49} + 16 q^{55} + 32 q^{57} + 4 q^{63} + 16 q^{65} - 8 q^{73} - 32 q^{79} + 4 q^{81} + 24 q^{87} + 8 q^{89} + 40 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^9 - 8 * q^15 + 16 * q^17 + 8 * q^23 + 4 * q^25 - 24 * q^31 + 32 * q^33 + 8 * q^39 - 8 * q^47 + 4 * q^49 + 16 * q^55 + 32 * q^57 + 4 * q^63 + 16 * q^65 - 8 * q^73 - 32 * q^79 + 4 * q^81 + 24 * q^87 + 8 * q^89 + 40 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 0.732051i	− 0.422650i	−0.977416	−	0.211325i	$-0.932222\pi$
$3$	− 0.732051i	− 0.422650i	0.977416	−	0.211325i	$-0.0677778\pi$
$4$	0	0
$5$	− 2.73205i	− 1.22181i	−0.791704	−	0.610905i	$-0.790806\pi$
$5$	− 2.73205i	− 1.22181i	0.791704	−	0.610905i	$-0.209194\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	1.46410i	0.441443i	0.975337	+	0.220722i	$0.0708412\pi$
$11$	1.46410i	0.441443i	−0.975337	+	0.220722i	$0.929159\pi$
$12$	0	0
$13$	2.73205i	0.757735i	0.925451	+	0.378867i	$0.123686\pi$
$13$	2.73205i	0.757735i	−0.925451	+	0.378867i	$0.876314\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	7.46410	1.81031	0.905155	−	0.425081i	$-0.139754\pi$
$17$	7.46410	1.81031	0.905155	+	0.425081i	$0.139754\pi$
$18$	0	0
$19$	6.19615i	1.42149i	0.703447	+	0.710747i	$0.251643\pi$
$19$	6.19615i	1.42149i	−0.703447	+	0.710747i	$0.748357\pi$
$20$	0	0
$21$	0.732051i	0.159747i
$22$	0	0
$23$	8.92820	1.86166	0.930830	−	0.365454i	$-0.119086\pi$
$23$	8.92820	1.86166	0.930830	+	0.365454i	$0.119086\pi$
$24$	0	0
$25$	−2.46410	−0.492820
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	3.46410i	0.643268i	0.946864	+	0.321634i	$0.104232\pi$
$29$	3.46410i	0.643268i	−0.946864	+	0.321634i	$0.895768\pi$
$30$	0	0
$31$	−2.53590	−0.455461	−0.227730	−	0.973724i	$-0.573130\pi$
$31$	−2.53590	−0.455461	−0.227730	+	0.973724i	$0.573130\pi$
$32$	0	0
$33$	1.07180	0.186576
$34$	0	0
$35$	2.73205i	0.461801i
$36$	0	0
$37$	− 4.53590i	− 0.745697i	−0.927892	−	0.372849i	$-0.878381\pi$
$37$	− 4.53590i	− 0.745697i	0.927892	−	0.372849i	$-0.121619\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	− 2.53590i	− 0.386721i	−0.981128	−	0.193360i	$-0.938061\pi$
$43$	− 2.53590i	− 0.386721i	0.981128	−	0.193360i	$-0.0619387\pi$
$44$	0	0
$45$	− 6.73205i	− 1.00355i
$46$	0	0
$47$	1.46410	0.213561	0.106781	−	0.994283i	$-0.465946\pi$
$47$	1.46410	0.213561	0.106781	+	0.994283i	$0.465946\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 5.46410i	− 0.765127i
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	4.53590	0.600794
$58$	0	0
$59$	− 6.19615i	− 0.806670i	−0.915052	−	0.403335i	$-0.867851\pi$
$59$	− 6.19615i	− 0.806670i	0.915052	−	0.403335i	$-0.132149\pi$
$60$	0	0
$61$	− 11.1244i	− 1.42433i	−0.702013	−	0.712164i	$-0.747715\pi$
$61$	− 11.1244i	− 1.42433i	0.702013	−	0.712164i	$-0.252285\pi$
$62$	0	0
$63$	−2.46410	−0.310448
$64$	0	0
$65$	7.46410	0.925808
$66$	0	0
$67$	14.9282i	1.82377i	0.410445	+	0.911885i	$0.365373\pi$
$67$	14.9282i	1.82377i	−0.410445	+	0.911885i	$0.634627\pi$
$68$	0	0
$69$	− 6.53590i	− 0.786830i
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	−8.92820	−1.04497	−0.522484	−	0.852649i	$-0.674994\pi$
$73$	−8.92820	−1.04497	−0.522484	+	0.852649i	$0.674994\pi$
$74$	0	0
$75$	1.80385i	0.208290i
$76$	0	0
$77$	− 1.46410i	− 0.166850i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	− 7.26795i	− 0.797761i	−0.917003	−	0.398881i	$-0.869399\pi$
$83$	− 7.26795i	− 0.797761i	0.917003	−	0.398881i	$-0.130601\pi$
$84$	0	0
$85$	− 20.3923i	− 2.21186i
$86$	0	0
$87$	2.53590	0.271877
$88$	0	0
$89$	−4.92820	−0.522388	−0.261194	−	0.965286i	$-0.584116\pi$
$89$	−4.92820	−0.522388	−0.261194	+	0.965286i	$0.584116\pi$
$90$	0	0
$91$	− 2.73205i	− 0.286397i
$92$	0	0
$93$	1.85641i	0.192500i
$94$	0	0
$95$	16.9282	1.73680
$96$	0	0
$97$	0.535898	0.0544122	0.0272061	−	0.999630i	$-0.491339\pi$
$97$	0.535898	0.0544122	0.0272061	+	0.999630i	$0.491339\pi$
$98$	0	0
$99$	3.60770i	0.362587i
$100$	0	0
$101$	3.80385i	0.378497i	0.981929	+	0.189248i	$0.0606052\pi$
$101$	3.80385i	0.378497i	−0.981929	+	0.189248i	$0.939395\pi$
$102$	0	0
$103$	8.39230	0.826918	0.413459	−	0.910523i	$-0.364320\pi$
$103$	8.39230	0.826918	0.413459	+	0.910523i	$0.364320\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	0	0
$109$	− 3.46410i	− 0.331801i	−0.986143	−	0.165900i	$-0.946947\pi$
$109$	− 3.46410i	− 0.331801i	0.986143	−	0.165900i	$-0.0530530\pi$
$110$	0	0
$111$	−3.32051	−0.315169
$112$	0	0
$113$	0.392305	0.0369049	0.0184525	−	0.999830i	$-0.494126\pi$
$113$	0.392305	0.0369049	0.0184525	+	0.999830i	$0.494126\pi$
$114$	0	0
$115$	− 24.3923i	− 2.27459i
$116$	0	0
$117$	6.73205i	0.622378i
$118$	0	0
$119$	−7.46410	−0.684233
$120$	0	0
$121$	8.85641	0.805128
$122$	0	0
$123$	− 2.53590i	− 0.228654i
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−1.85641	−0.163447
$130$	0	0
$131$	− 2.19615i	− 0.191879i	−0.995387	−	0.0959394i	$-0.969415\pi$
$131$	− 2.19615i	− 0.191879i	0.995387	−	0.0959394i	$-0.0305855\pi$
$132$	0	0
$133$	− 6.19615i	− 0.537275i
$134$	0	0
$135$	−10.9282	−0.940550
$136$	0	0
$137$	20.9282	1.78802	0.894009	−	0.448050i	$-0.147881\pi$
$137$	20.9282	1.78802	0.894009	+	0.448050i	$0.147881\pi$
$138$	0	0
$139$	− 15.6603i	− 1.32829i	−0.747606	−	0.664143i	$-0.768797\pi$
$139$	− 15.6603i	− 1.32829i	0.747606	−	0.664143i	$-0.231203\pi$
$140$	0	0
$141$	− 1.07180i	− 0.0902616i
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	9.46410	0.785951
$146$	0	0
$147$	− 0.732051i	− 0.0603785i
$148$	0	0
$149$	19.8564i	1.62670i	0.581775	+	0.813350i	$0.302359\pi$
$149$	19.8564i	1.62670i	−0.581775	+	0.813350i	$0.697641\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	18.3923	1.48693
$154$	0	0
$155$	6.92820i	0.556487i
$156$	0	0
$157$	− 13.2679i	− 1.05890i	−0.848342	−	0.529449i	$-0.822399\pi$
$157$	− 13.2679i	− 1.05890i	0.848342	−	0.529449i	$-0.177601\pi$
$158$	0	0
$159$	−4.39230	−0.348332
$160$	0	0
$161$	−8.92820	−0.703641
$162$	0	0
$163$	20.3923i	1.59725i	0.601830	+	0.798624i	$0.294439\pi$
$163$	20.3923i	1.59725i	−0.601830	+	0.798624i	$0.705561\pi$
$164$	0	0
$165$	− 2.92820i	− 0.227960i
$166$	0	0
$167$	−6.53590	−0.505763	−0.252882	−	0.967497i	$-0.581378\pi$
$167$	−6.53590	−0.505763	−0.252882	+	0.967497i	$0.581378\pi$
$168$	0	0
$169$	5.53590	0.425838
$170$	0	0
$171$	15.2679i	1.16757i
$172$	0	0
$173$	5.66025i	0.430341i	0.976576	+	0.215171i	$0.0690307\pi$
$173$	5.66025i	0.430341i	−0.976576	+	0.215171i	$0.930969\pi$
$174$	0	0
$175$	2.46410	0.186269
$176$	0	0
$177$	−4.53590	−0.340939
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	1.66025i	0.123406i	0.998095	+	0.0617029i	$0.0196531\pi$
$181$	1.66025i	0.123406i	−0.998095	+	0.0617029i	$0.980347\pi$
$182$	0	0
$183$	−8.14359	−0.601992
$184$	0	0
$185$	−12.3923	−0.911100
$186$	0	0
$187$	10.9282i	0.799149i
$188$	0	0
$189$	4.00000i	0.290957i
$190$	0	0
$191$	−10.9282	−0.790737	−0.395369	−	0.918523i	$-0.629383\pi$
$191$	−10.9282	−0.790737	−0.395369	+	0.918523i	$0.629383\pi$
$192$	0	0
$193$	−12.3923	−0.892018	−0.446009	−	0.895029i	$-0.647155\pi$
$193$	−12.3923	−0.892018	−0.446009	+	0.895029i	$0.647155\pi$
$194$	0	0
$195$	− 5.46410i	− 0.391292i
$196$	0	0
$197$	23.8564i	1.69970i	0.527026	+	0.849849i	$0.323307\pi$
$197$	23.8564i	1.69970i	−0.527026	+	0.849849i	$0.676693\pi$
$198$	0	0
$199$	18.5359	1.31397	0.656987	−	0.753901i	$-0.271830\pi$
$199$	18.5359	1.31397	0.656987	+	0.753901i	$0.271830\pi$
$200$	0	0
$201$	10.9282	0.770816
$202$	0	0
$203$	− 3.46410i	− 0.243132i
$204$	0	0
$205$	− 9.46410i	− 0.661002i
$206$	0	0
$207$	22.0000	1.52911
$208$	0	0
$209$	−9.07180	−0.627509
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	10.1436i	0.695028i
$214$	0	0
$215$	−6.92820	−0.472500
$216$	0	0
$217$	2.53590	0.172148
$218$	0	0
$219$	6.53590i	0.441655i
$220$	0	0
$221$	20.3923i	1.37173i
$222$	0	0
$223$	25.8564	1.73147	0.865737	−	0.500500i	$-0.166850\pi$
$223$	25.8564	1.73147	0.865737	+	0.500500i	$0.166850\pi$
$224$	0	0
$225$	−6.07180	−0.404786
$226$	0	0
$227$	− 9.80385i	− 0.650704i	−0.945593	−	0.325352i	$-0.894517\pi$
$227$	− 9.80385i	− 0.650704i	0.945593	−	0.325352i	$-0.105483\pi$
$228$	0	0
$229$	0.196152i	0.0129621i	0.999979	+	0.00648106i	$0.00206300\pi$
$229$	0.196152i	0.0129621i	−0.999979	+	0.00648106i	$0.997937\pi$
$230$	0	0
$231$	−1.07180	−0.0705191
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	− 4.00000i	− 0.260931i
$236$	0	0
$237$	5.85641i	0.380414i
$238$	0	0
$239$	12.9282	0.836256	0.418128	−	0.908388i	$-0.362686\pi$
$239$	12.9282	0.836256	0.418128	+	0.908388i	$0.362686\pi$
$240$	0	0
$241$	19.4641	1.25379	0.626897	−	0.779103i	$-0.284325\pi$
$241$	19.4641	1.25379	0.626897	+	0.779103i	$0.284325\pi$
$242$	0	0
$243$	− 15.2679i	− 0.979439i
$244$	0	0
$245$	− 2.73205i	− 0.174544i
$246$	0	0
$247$	−16.9282	−1.07712
$248$	0	0
$249$	−5.32051	−0.337173
$250$	0	0
$251$	− 26.1962i	− 1.65349i	−0.562579	−	0.826743i	$-0.690191\pi$
$251$	− 26.1962i	− 1.65349i	0.562579	−	0.826743i	$-0.309809\pi$
$252$	0	0
$253$	13.0718i	0.821817i
$254$	0	0
$255$	−14.9282	−0.934840
$256$	0	0
$257$	−24.9282	−1.55498	−0.777489	−	0.628896i	$-0.783507\pi$
$257$	−24.9282	−1.55498	−0.777489	+	0.628896i	$0.783507\pi$
$258$	0	0
$259$	4.53590i	0.281847i
$260$	0	0
$261$	8.53590i	0.528359i
$262$	0	0
$263$	13.8564	0.854423	0.427211	−	0.904152i	$-0.359496\pi$
$263$	13.8564	0.854423	0.427211	+	0.904152i	$0.359496\pi$
$264$	0	0
$265$	−16.3923	−1.00697
$266$	0	0
$267$	3.60770i	0.220787i
$268$	0	0
$269$	− 14.0526i	− 0.856800i	−0.903589	−	0.428400i	$-0.859077\pi$
$269$	− 14.0526i	− 0.856800i	0.903589	−	0.428400i	$-0.140923\pi$
$270$	0	0
$271$	−9.07180	−0.551072	−0.275536	−	0.961291i	$-0.588855\pi$
$271$	−9.07180	−0.551072	−0.275536	+	0.961291i	$0.588855\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	− 3.60770i	− 0.217552i
$276$	0	0
$277$	− 12.9282i	− 0.776780i	−0.921495	−	0.388390i	$-0.873031\pi$
$277$	− 12.9282i	− 0.776780i	0.921495	−	0.388390i	$-0.126969\pi$
$278$	0	0
$279$	−6.24871	−0.374101
$280$	0	0
$281$	7.85641	0.468674	0.234337	−	0.972155i	$-0.424708\pi$
$281$	7.85641	0.468674	0.234337	+	0.972155i	$0.424708\pi$
$282$	0	0
$283$	16.0526i	0.954226i	0.878842	+	0.477113i	$0.158317\pi$
$283$	16.0526i	0.954226i	−0.878842	+	0.477113i	$0.841683\pi$
$284$	0	0
$285$	− 12.3923i	− 0.734057i
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	38.7128	2.27722
$290$	0	0
$291$	− 0.392305i	− 0.0229973i
$292$	0	0
$293$	14.7321i	0.860656i	0.902673	+	0.430328i	$0.141602\pi$
$293$	14.7321i	0.860656i	−0.902673	+	0.430328i	$0.858398\pi$
$294$	0	0
$295$	−16.9282	−0.985598
$296$	0	0
$297$	5.85641	0.339823
$298$	0	0
$299$	24.3923i	1.41064i
$300$	0	0
$301$	2.53590i	0.146167i
$302$	0	0
$303$	2.78461	0.159972
$304$	0	0
$305$	−30.3923	−1.74026
$306$	0	0
$307$	− 26.1962i	− 1.49509i	−0.664209	−	0.747547i	$-0.731232\pi$
$307$	− 26.1962i	− 1.49509i	0.664209	−	0.747547i	$-0.268768\pi$
$308$	0	0
$309$	− 6.14359i	− 0.349497i
$310$	0	0
$311$	10.9282	0.619682	0.309841	−	0.950788i	$-0.399724\pi$
$311$	10.9282	0.619682	0.309841	+	0.950788i	$0.399724\pi$
$312$	0	0
$313$	−12.5359	−0.708571	−0.354285	−	0.935137i	$-0.615276\pi$
$313$	−12.5359	−0.708571	−0.354285	+	0.935137i	$0.615276\pi$
$314$	0	0
$315$	6.73205i	0.379308i
$316$	0	0
$317$	− 12.9282i	− 0.726120i	−0.931766	−	0.363060i	$-0.881732\pi$
$317$	− 12.9282i	− 0.726120i	0.931766	−	0.363060i	$-0.118268\pi$
$318$	0	0
$319$	−5.07180	−0.283966
$320$	0	0
$321$	11.7128	0.653745
$322$	0	0
$323$	46.2487i	2.57335i
$324$	0	0
$325$	− 6.73205i	− 0.373427i
$326$	0	0
$327$	−2.53590	−0.140236
$328$	0	0
$329$	−1.46410	−0.0807185
$330$	0	0
$331$	5.46410i	0.300334i	0.988661	+	0.150167i	$0.0479812\pi$
$331$	5.46410i	0.300334i	−0.988661	+	0.150167i	$0.952019\pi$
$332$	0	0
$333$	− 11.1769i	− 0.612491i
$334$	0	0
$335$	40.7846	2.22830
$336$	0	0
$337$	−26.2487	−1.42986	−0.714929	−	0.699197i	$-0.753541\pi$
$337$	−26.2487	−1.42986	−0.714929	+	0.699197i	$0.753541\pi$
$338$	0	0
$339$	− 0.287187i	− 0.0155979i
$340$	0	0
$341$	− 3.71281i	− 0.201060i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−17.8564	−0.961357
$346$	0	0
$347$	− 21.4641i	− 1.15225i	−0.817360	−	0.576127i	$-0.804563\pi$
$347$	− 21.4641i	− 1.15225i	0.817360	−	0.576127i	$-0.195437\pi$
$348$	0	0
$349$	23.1244i	1.23782i	0.785463	+	0.618909i	$0.212425\pi$
$349$	23.1244i	1.23782i	−0.785463	+	0.618909i	$0.787575\pi$
$350$	0	0
$351$	10.9282	0.583304
$352$	0	0
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	37.8564i	2.00921i
$356$	0	0
$357$	5.46410i	0.289191i
$358$	0	0
$359$	11.8564	0.625757	0.312879	−	0.949793i	$-0.398707\pi$
$359$	11.8564	0.625757	0.312879	+	0.949793i	$0.398707\pi$
$360$	0	0
$361$	−19.3923	−1.02065
$362$	0	0
$363$	− 6.48334i	− 0.340287i
$364$	0	0
$365$	24.3923i	1.27675i
$366$	0	0
$367$	−28.7846	−1.50254	−0.751272	−	0.659992i	$-0.770559\pi$
$367$	−28.7846	−1.50254	−0.751272	+	0.659992i	$0.770559\pi$
$368$	0	0
$369$	8.53590	0.444361
$370$	0	0
$371$	6.00000i	0.311504i
$372$	0	0
$373$	11.8564i	0.613901i	0.951725	+	0.306951i	$0.0993087\pi$
$373$	11.8564i	0.613901i	−0.951725	+	0.306951i	$0.900691\pi$
$374$	0	0
$375$	−5.07180	−0.261906
$376$	0	0
$377$	−9.46410	−0.487426
$378$	0	0
$379$	− 38.2487i	− 1.96470i	−0.187041	−	0.982352i	$-0.559890\pi$
$379$	− 38.2487i	− 1.96470i	0.187041	−	0.982352i	$-0.440110\pi$
$380$	0	0
$381$	1.46410i	0.0750082i
$382$	0	0
$383$	−33.4641	−1.70994	−0.854968	−	0.518681i	$-0.826423\pi$
$383$	−33.4641	−1.70994	−0.854968	+	0.518681i	$0.826423\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	− 6.24871i	− 0.317640i
$388$	0	0
$389$	0.535898i	0.0271711i	0.999908	+	0.0135856i	$0.00432455\pi$
$389$	0.535898i	0.0271711i	−0.999908	+	0.0135856i	$0.995675\pi$
$390$	0	0
$391$	66.6410	3.37018
$392$	0	0
$393$	−1.60770	−0.0810975
$394$	0	0
$395$	21.8564i	1.09972i
$396$	0	0
$397$	28.9808i	1.45450i	0.686371	+	0.727251i	$0.259203\pi$
$397$	28.9808i	1.45450i	−0.686371	+	0.727251i	$0.740797\pi$
$398$	0	0
$399$	−4.53590	−0.227079
$400$	0	0
$401$	−30.2487	−1.51055	−0.755274	−	0.655409i	$-0.772496\pi$
$401$	−30.2487	−1.51055	−0.755274	+	0.655409i	$0.772496\pi$
$402$	0	0
$403$	− 6.92820i	− 0.345118i
$404$	0	0
$405$	− 12.1962i	− 0.606032i
$406$	0	0
$407$	6.64102	0.329183
$408$	0	0
$409$	−24.5359	−1.21322	−0.606611	−	0.794999i	$-0.707471\pi$
$409$	−24.5359	−1.21322	−0.606611	+	0.794999i	$0.707471\pi$
$410$	0	0
$411$	− 15.3205i	− 0.755705i
$412$	0	0
$413$	6.19615i	0.304893i
$414$	0	0
$415$	−19.8564	−0.974713
$416$	0	0
$417$	−11.4641	−0.561399
$418$	0	0
$419$	18.1962i	0.888940i	0.895794	+	0.444470i	$0.146608\pi$
$419$	18.1962i	0.888940i	−0.895794	+	0.444470i	$0.853392\pi$
$420$	0	0
$421$	− 20.9282i	− 1.01998i	−0.860181	−	0.509989i	$-0.829649\pi$
$421$	− 20.9282i	− 1.01998i	0.860181	−	0.509989i	$-0.170351\pi$
$422$	0	0
$423$	3.60770	0.175412
$424$	0	0
$425$	−18.3923	−0.892158
$426$	0	0
$427$	11.1244i	0.538345i
$428$	0	0
$429$	2.92820i	0.141375i
$430$	0	0
$431$	−10.7846	−0.519476	−0.259738	−	0.965679i	$-0.583636\pi$
$431$	−10.7846	−0.519476	−0.259738	+	0.965679i	$0.583636\pi$
$432$	0	0
$433$	−12.2487	−0.588636	−0.294318	−	0.955708i	$-0.595092\pi$
$433$	−12.2487	−0.588636	−0.294318	+	0.955708i	$0.595092\pi$
$434$	0	0
$435$	− 6.92820i	− 0.332182i
$436$	0	0
$437$	55.3205i	2.64634i
$438$	0	0
$439$	−15.7128	−0.749932	−0.374966	−	0.927039i	$-0.622346\pi$
$439$	−15.7128	−0.749932	−0.374966	+	0.927039i	$0.622346\pi$
$440$	0	0
$441$	2.46410	0.117338
$442$	0	0
$443$	6.92820i	0.329169i	0.986363	+	0.164584i	$0.0526283\pi$
$443$	6.92820i	0.329169i	−0.986363	+	0.164584i	$0.947372\pi$
$444$	0	0
$445$	13.4641i	0.638260i
$446$	0	0
$447$	14.5359	0.687524
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	5.07180i	0.238822i
$452$	0	0
$453$	− 1.46410i	− 0.0687895i
$454$	0	0
$455$	−7.46410	−0.349922
$456$	0	0
$457$	−9.46410	−0.442712	−0.221356	−	0.975193i	$-0.571048\pi$
$457$	−9.46410	−0.442712	−0.221356	+	0.975193i	$0.571048\pi$
$458$	0	0
$459$	− 29.8564i	− 1.39358i
$460$	0	0
$461$	14.3397i	0.667869i	0.942596	+	0.333934i	$0.108376\pi$
$461$	14.3397i	0.667869i	−0.942596	+	0.333934i	$0.891624\pi$
$462$	0	0
$463$	−35.7128	−1.65972	−0.829858	−	0.557975i	$-0.811578\pi$
$463$	−35.7128	−1.65972	−0.829858	+	0.557975i	$0.811578\pi$
$464$	0	0
$465$	5.07180	0.235199
$466$	0	0
$467$	− 24.4449i	− 1.13117i	−0.824689	−	0.565587i	$-0.808650\pi$
$467$	− 24.4449i	− 1.13117i	0.824689	−	0.565587i	$-0.191350\pi$
$468$	0	0
$469$	− 14.9282i	− 0.689320i
$470$	0	0
$471$	−9.71281	−0.447543
$472$	0	0
$473$	3.71281	0.170715
$474$	0	0
$475$	− 15.2679i	− 0.700542i
$476$	0	0
$477$	− 14.7846i	− 0.676941i
$478$	0	0
$479$	16.3923	0.748984	0.374492	−	0.927230i	$-0.377817\pi$
$479$	16.3923	0.748984	0.374492	+	0.927230i	$0.377817\pi$
$480$	0	0
$481$	12.3923	0.565040
$482$	0	0
$483$	6.53590i	0.297394i
$484$	0	0
$485$	− 1.46410i	− 0.0664814i
$486$	0	0
$487$	−32.6410	−1.47911	−0.739553	−	0.673099i	$-0.764963\pi$
$487$	−32.6410	−1.47911	−0.739553	+	0.673099i	$0.764963\pi$
$488$	0	0
$489$	14.9282	0.675077
$490$	0	0
$491$	20.0000i	0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$	20.0000i	0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$	0	0
$493$	25.8564i	1.16451i
$494$	0	0
$495$	9.85641	0.443013
$496$	0	0
$497$	13.8564	0.621545
$498$	0	0
$499$	16.0000i	0.716258i	0.933672	+	0.358129i	$0.116585\pi$
$499$	16.0000i	0.716258i	−0.933672	+	0.358129i	$0.883415\pi$
$500$	0	0
$501$	4.78461i	0.213761i
$502$	0	0
$503$	9.07180	0.404491	0.202246	−	0.979335i	$-0.435176\pi$
$503$	9.07180	0.404491	0.202246	+	0.979335i	$0.435176\pi$
$504$	0	0
$505$	10.3923	0.462451
$506$	0	0
$507$	− 4.05256i	− 0.179980i
$508$	0	0
$509$	9.26795i	0.410795i	0.978679	+	0.205397i	$0.0658487\pi$
$509$	9.26795i	0.410795i	−0.978679	+	0.205397i	$0.934151\pi$
$510$	0	0
$511$	8.92820	0.394960
$512$	0	0
$513$	24.7846	1.09427
$514$	0	0
$515$	− 22.9282i	− 1.01034i
$516$	0	0
$517$	2.14359i	0.0942751i
$518$	0	0
$519$	4.14359	0.181884
$520$	0	0
$521$	−30.3923	−1.33151	−0.665756	−	0.746170i	$-0.731891\pi$
$521$	−30.3923	−1.33151	−0.665756	+	0.746170i	$0.731891\pi$
$522$	0	0
$523$	2.58846i	0.113185i	0.998397	+	0.0565927i	$0.0180236\pi$
$523$	2.58846i	0.113185i	−0.998397	+	0.0565927i	$0.981976\pi$
$524$	0	0
$525$	− 1.80385i	− 0.0787264i
$526$	0	0
$527$	−18.9282	−0.824525
$528$	0	0
$529$	56.7128	2.46577
$530$	0	0
$531$	− 15.2679i	− 0.662573i
$532$	0	0
$533$	9.46410i	0.409936i
$534$	0	0
$535$	43.7128	1.88987
$536$	0	0
$537$	−8.78461	−0.379084
$538$	0	0
$539$	1.46410i	0.0630633i
$540$	0	0
$541$	34.7846i	1.49551i	0.663976	+	0.747754i	$0.268868\pi$
$541$	34.7846i	1.49551i	−0.663976	+	0.747754i	$0.731132\pi$
$542$	0	0
$543$	1.21539	0.0521574
$544$	0	0
$545$	−9.46410	−0.405398
$546$	0	0
$547$	3.60770i	0.154254i	0.997021	+	0.0771270i	$0.0245747\pi$
$547$	3.60770i	0.154254i	−0.997021	+	0.0771270i	$0.975425\pi$
$548$	0	0
$549$	− 27.4115i	− 1.16990i
$550$	0	0
$551$	−21.4641	−0.914401
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	9.07180i	0.385076i
$556$	0	0
$557$	25.7128i	1.08949i	0.838603	+	0.544743i	$0.183373\pi$
$557$	25.7128i	1.08949i	−0.838603	+	0.544743i	$0.816627\pi$
$558$	0	0
$559$	6.92820	0.293032
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 9.12436i	− 0.384546i	−0.981342	−	0.192273i	$-0.938414\pi$
$563$	− 9.12436i	− 0.384546i	0.981342	−	0.192273i	$-0.0615859\pi$
$564$	0	0
$565$	− 1.07180i	− 0.0450908i
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	22.2487	0.932714	0.466357	−	0.884596i	$-0.345566\pi$
$569$	22.2487	0.932714	0.466357	+	0.884596i	$0.345566\pi$
$570$	0	0
$571$	− 32.3923i	− 1.35558i	−0.735257	−	0.677788i	$-0.762939\pi$
$571$	− 32.3923i	− 1.35558i	0.735257	−	0.677788i	$-0.237061\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	−22.0000	−0.917463
$576$	0	0
$577$	8.92820	0.371686	0.185843	−	0.982579i	$-0.440498\pi$
$577$	8.92820	0.371686	0.185843	+	0.982579i	$0.440498\pi$
$578$	0	0
$579$	9.07180i	0.377011i
$580$	0	0
$581$	7.26795i	0.301525i
$582$	0	0
$583$	8.78461	0.363821
$584$	0	0
$585$	18.3923	0.760428
$586$	0	0
$587$	− 10.5885i	− 0.437032i	−0.975833	−	0.218516i	$-0.929878\pi$
$587$	− 10.5885i	− 0.437032i	0.975833	−	0.218516i	$-0.0701217\pi$
$588$	0	0
$589$	− 15.7128i	− 0.647435i
$590$	0	0
$591$	17.4641	0.718377
$592$	0	0
$593$	7.85641	0.322624	0.161312	−	0.986903i	$-0.448427\pi$
$593$	7.85641	0.322624	0.161312	+	0.986903i	$0.448427\pi$
$594$	0	0
$595$	20.3923i	0.836003i
$596$	0	0
$597$	− 13.5692i	− 0.555351i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	0.928203	0.0378622	0.0189311	−	0.999821i	$-0.493974\pi$
$601$	0.928203	0.0378622	0.0189311	+	0.999821i	$0.493974\pi$
$602$	0	0
$603$	36.7846i	1.49799i
$604$	0	0
$605$	− 24.1962i	− 0.983713i
$606$	0	0
$607$	26.9282	1.09298	0.546491	−	0.837465i	$-0.315963\pi$
$607$	26.9282	1.09298	0.546491	+	0.837465i	$0.315963\pi$
$608$	0	0
$609$	−2.53590	−0.102760
$610$	0	0
$611$	4.00000i	0.161823i
$612$	0	0
$613$	21.3205i	0.861127i	0.902560	+	0.430564i	$0.141685\pi$
$613$	21.3205i	0.861127i	−0.902560	+	0.430564i	$0.858315\pi$
$614$	0	0
$615$	−6.92820	−0.279372
$616$	0	0
$617$	−15.3205	−0.616780	−0.308390	−	0.951260i	$-0.599790\pi$
$617$	−15.3205	−0.616780	−0.308390	+	0.951260i	$0.599790\pi$
$618$	0	0
$619$	− 47.3731i	− 1.90408i	−0.305967	−	0.952042i	$-0.598980\pi$
$619$	− 47.3731i	− 1.90408i	0.305967	−	0.952042i	$-0.401020\pi$
$620$	0	0
$621$	− 35.7128i	− 1.43311i
$622$	0	0
$623$	4.92820	0.197444
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	6.64102i	0.265217i
$628$	0	0
$629$	− 33.8564i	− 1.34994i
$630$	0	0
$631$	−21.0718	−0.838855	−0.419427	−	0.907789i	$-0.637769\pi$
$631$	−21.0718	−0.838855	−0.419427	+	0.907789i	$0.637769\pi$
$632$	0	0
$633$	2.92820	0.116386
$634$	0	0
$635$	5.46410i	0.216836i
$636$	0	0
$637$	2.73205i	0.108248i
$638$	0	0
$639$	−34.1436	−1.35070
$640$	0	0
$641$	−12.6795	−0.500810	−0.250405	−	0.968141i	$-0.580564\pi$
$641$	−12.6795	−0.500810	−0.250405	+	0.968141i	$0.580564\pi$
$642$	0	0
$643$	− 19.2679i	− 0.759854i	−0.925017	−	0.379927i	$-0.875949\pi$
$643$	− 19.2679i	− 0.759854i	0.925017	−	0.379927i	$-0.124051\pi$
$644$	0	0
$645$	5.07180i	0.199702i
$646$	0	0
$647$	−11.3205	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$647$	−11.3205	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$648$	0	0
$649$	9.07180	0.356099
$650$	0	0
$651$	− 1.85641i	− 0.0727583i
$652$	0	0
$653$	27.4641i	1.07475i	0.843342	+	0.537377i	$0.180585\pi$
$653$	27.4641i	1.07475i	−0.843342	+	0.537377i	$0.819415\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	23.3205i	0.908438i	0.890890	+	0.454219i	$0.150082\pi$
$659$	23.3205i	0.908438i	−0.890890	+	0.454219i	$0.849918\pi$
$660$	0	0
$661$	28.5885i	1.11196i	0.831195	+	0.555981i	$0.187657\pi$
$661$	28.5885i	1.11196i	−0.831195	+	0.555981i	$0.812343\pi$
$662$	0	0
$663$	14.9282	0.579763
$664$	0	0
$665$	−16.9282	−0.656448
$666$	0	0
$667$	30.9282i	1.19754i
$668$	0	0
$669$	− 18.9282i	− 0.731807i
$670$	0	0
$671$	16.2872	0.628760
$672$	0	0
$673$	3.07180	0.118409	0.0592045	−	0.998246i	$-0.481144\pi$
$673$	3.07180	0.118409	0.0592045	+	0.998246i	$0.481144\pi$
$674$	0	0
$675$	9.85641i	0.379373i
$676$	0	0
$677$	39.5167i	1.51875i	0.650654	+	0.759374i	$0.274495\pi$
$677$	39.5167i	1.51875i	−0.650654	+	0.759374i	$0.725505\pi$
$678$	0	0
$679$	−0.535898	−0.0205659
$680$	0	0
$681$	−7.17691	−0.275020
$682$	0	0
$683$	− 13.8564i	− 0.530201i	−0.964221	−	0.265100i	$-0.914595\pi$
$683$	− 13.8564i	− 0.530201i	0.964221	−	0.265100i	$-0.0854051\pi$
$684$	0	0
$685$	− 57.1769i	− 2.18462i
$686$	0	0
$687$	0.143594	0.00547844
$688$	0	0
$689$	16.3923	0.624497
$690$	0	0
$691$	− 2.19615i	− 0.0835456i	−0.999127	−	0.0417728i	$-0.986699\pi$
$691$	− 2.19615i	− 0.0835456i	0.999127	−	0.0417728i	$-0.0133006\pi$
$692$	0	0
$693$	− 3.60770i	− 0.137045i
$694$	0	0
$695$	−42.7846	−1.62291
$696$	0	0
$697$	25.8564	0.979381
$698$	0	0
$699$	13.1769i	0.498397i
$700$	0	0
$701$	0.535898i	0.0202406i	0.999949	+	0.0101203i	$0.00322145\pi$
$701$	0.535898i	0.0202406i	−0.999949	+	0.0101203i	$0.996779\pi$
$702$	0	0
$703$	28.1051	1.06000
$704$	0	0
$705$	−2.92820	−0.110283
$706$	0	0
$707$	− 3.80385i	− 0.143058i
$708$	0	0
$709$	− 36.5359i	− 1.37213i	−0.727538	−	0.686067i	$-0.759336\pi$
$709$	− 36.5359i	− 1.37213i	0.727538	−	0.686067i	$-0.240664\pi$
$710$	0	0
$711$	−19.7128	−0.739288
$712$	0	0
$713$	−22.6410	−0.847913
$714$	0	0
$715$	10.9282i	0.408692i
$716$	0	0
$717$	− 9.46410i	− 0.353443i
$718$	0	0
$719$	−42.2487	−1.57561	−0.787806	−	0.615924i	$-0.788783\pi$
$719$	−42.2487	−1.57561	−0.787806	+	0.615924i	$0.788783\pi$
$720$	0	0
$721$	−8.39230	−0.312546
$722$	0	0
$723$	− 14.2487i	− 0.529915i
$724$	0	0
$725$	− 8.53590i	− 0.317015i
$726$	0	0
$727$	38.5359	1.42922	0.714609	−	0.699524i	$-0.246605\pi$
$727$	38.5359	1.42922	0.714609	+	0.699524i	$0.246605\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	− 18.9282i	− 0.700085i
$732$	0	0
$733$	0.588457i	0.0217352i	0.999941	+	0.0108676i	$0.00345933\pi$
$733$	0.588457i	0.0217352i	−0.999941	+	0.0108676i	$0.996541\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−21.8564	−0.805091
$738$	0	0
$739$	− 23.3205i	− 0.857859i	−0.903338	−	0.428929i	$-0.858891\pi$
$739$	− 23.3205i	− 0.857859i	0.903338	−	0.428929i	$-0.141109\pi$
$740$	0	0
$741$	12.3923i	0.455243i
$742$	0	0
$743$	−29.7128	−1.09006	−0.545029	−	0.838417i	$-0.683481\pi$
$743$	−29.7128	−1.09006	−0.545029	+	0.838417i	$0.683481\pi$
$744$	0	0
$745$	54.2487	1.98752
$746$	0	0
$747$	− 17.9090i	− 0.655255i
$748$	0	0
$749$	− 16.0000i	− 0.584627i
$750$	0	0
$751$	−4.92820	−0.179833	−0.0899163	−	0.995949i	$-0.528660\pi$
$751$	−4.92820	−0.179833	−0.0899163	+	0.995949i	$0.528660\pi$
$752$	0	0
$753$	−19.1769	−0.698846
$754$	0	0
$755$	− 5.46410i	− 0.198859i
$756$	0	0
$757$	− 0.535898i	− 0.0194776i	−0.999953	−	0.00973878i	$-0.996900\pi$
$757$	− 0.535898i	− 0.0194776i	0.999953	−	0.00973878i	$-0.00310000\pi$
$758$	0	0
$759$	9.56922	0.347341
$760$	0	0
$761$	1.32051	0.0478684	0.0239342	−	0.999714i	$-0.492381\pi$
$761$	1.32051	0.0478684	0.0239342	+	0.999714i	$0.492381\pi$
$762$	0	0
$763$	3.46410i	0.125409i
$764$	0	0
$765$	− 50.2487i	− 1.81675i
$766$	0	0
$767$	16.9282	0.611242
$768$	0	0
$769$	40.5359	1.46176	0.730881	−	0.682505i	$-0.239109\pi$
$769$	40.5359	1.46176	0.730881	+	0.682505i	$0.239109\pi$
$770$	0	0
$771$	18.2487i	0.657211i
$772$	0	0
$773$	− 43.5167i	− 1.56519i	−0.622534	−	0.782593i	$-0.713897\pi$
$773$	− 43.5167i	− 1.56519i	0.622534	−	0.782593i	$-0.286103\pi$
$774$	0	0
$775$	6.24871	0.224460
$776$	0	0
$777$	3.32051	0.119123
$778$	0	0
$779$	21.4641i	0.769031i
$780$	0	0
$781$	− 20.2872i	− 0.725933i
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	−36.2487	−1.29377
$786$	0	0
$787$	− 12.7321i	− 0.453849i	−0.973912	−	0.226924i	$-0.927133\pi$
$787$	− 12.7321i	− 0.453849i	0.973912	−	0.226924i	$-0.0728670\pi$
$788$	0	0
$789$	− 10.1436i	− 0.361121i
$790$	0	0
$791$	−0.392305	−0.0139488
$792$	0	0
$793$	30.3923	1.07926
$794$	0	0
$795$	12.0000i	0.425596i
$796$	0	0
$797$	5.66025i	0.200496i	0.994962	+	0.100248i	$0.0319637\pi$
$797$	5.66025i	0.200496i	−0.994962	+	0.100248i	$0.968036\pi$
$798$	0	0
$799$	10.9282	0.386612
$800$	0	0
$801$	−12.1436	−0.429073
$802$	0	0
$803$	− 13.0718i	− 0.461294i
$804$	0	0
$805$	24.3923i	0.859716i
$806$	0	0
$807$	−10.2872	−0.362126
$808$	0	0
$809$	−9.46410	−0.332740	−0.166370	−	0.986063i	$-0.553205\pi$
$809$	−9.46410	−0.332740	−0.166370	+	0.986063i	$0.553205\pi$
$810$	0	0
$811$	34.9808i	1.22834i	0.789173	+	0.614170i	$0.210509\pi$
$811$	34.9808i	1.22834i	−0.789173	+	0.614170i	$0.789491\pi$
$812$	0	0
$813$	6.64102i	0.232911i
$814$	0	0
$815$	55.7128	1.95153
$816$	0	0
$817$	15.7128	0.549722
$818$	0	0
$819$	− 6.73205i	− 0.235237i
$820$	0	0
$821$	− 5.71281i	− 0.199379i	−0.995019	−	0.0996893i	$-0.968215\pi$
$821$	− 5.71281i	− 0.199379i	0.995019	−	0.0996893i	$-0.0317849\pi$
$822$	0	0
$823$	0.784610	0.0273498	0.0136749	−	0.999906i	$-0.495647\pi$
$823$	0.784610	0.0273498	0.0136749	+	0.999906i	$0.495647\pi$
$824$	0	0
$825$	−2.64102	−0.0919484
$826$	0	0
$827$	34.9282i	1.21457i	0.794483	+	0.607286i	$0.207742\pi$
$827$	34.9282i	1.21457i	−0.794483	+	0.607286i	$0.792258\pi$
$828$	0	0
$829$	− 29.2679i	− 1.01652i	−0.861204	−	0.508259i	$-0.830289\pi$
$829$	− 29.2679i	− 1.01652i	0.861204	−	0.508259i	$-0.169711\pi$
$830$	0	0
$831$	−9.46410	−0.328306
$832$	0	0
$833$	7.46410	0.258616
$834$	0	0
$835$	17.8564i	0.617946i
$836$	0	0
$837$	10.1436i	0.350614i
$838$	0	0
$839$	−2.53590	−0.0875489	−0.0437745	−	0.999041i	$-0.513938\pi$
$839$	−2.53590	−0.0875489	−0.0437745	+	0.999041i	$0.513938\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	− 5.75129i	− 0.198085i
$844$	0	0
$845$	− 15.1244i	− 0.520294i
$846$	0	0
$847$	−8.85641	−0.304310
$848$	0	0
$849$	11.7513	0.403303
$850$	0	0
$851$	− 40.4974i	− 1.38823i
$852$	0	0
$853$	− 11.5167i	− 0.394323i	−0.980371	−	0.197161i	$-0.936828\pi$
$853$	− 11.5167i	− 0.394323i	0.980371	−	0.197161i	$-0.0631723\pi$
$854$	0	0
$855$	41.7128	1.42655
$856$	0	0
$857$	20.5359	0.701493	0.350746	−	0.936470i	$-0.385928\pi$
$857$	20.5359	0.701493	0.350746	+	0.936470i	$0.385928\pi$
$858$	0	0
$859$	3.66025i	0.124886i	0.998049	+	0.0624431i	$0.0198892\pi$
$859$	3.66025i	0.124886i	−0.998049	+	0.0624431i	$0.980111\pi$
$860$	0	0
$861$	2.53590i	0.0864232i
$862$	0	0
$863$	21.0718	0.717292	0.358646	−	0.933474i	$-0.383238\pi$
$863$	21.0718	0.717292	0.358646	+	0.933474i	$0.383238\pi$
$864$	0	0
$865$	15.4641	0.525795
$866$	0	0
$867$	− 28.3397i	− 0.962468i
$868$	0	0
$869$	− 11.7128i	− 0.397330i
$870$	0	0
$871$	−40.7846	−1.38193
$872$	0	0
$873$	1.32051	0.0446924
$874$	0	0
$875$	6.92820i	0.234216i
$876$	0	0
$877$	− 34.1051i	− 1.15165i	−0.817574	−	0.575824i	$-0.804681\pi$
$877$	− 34.1051i	− 1.15165i	0.817574	−	0.575824i	$-0.195319\pi$
$878$	0	0
$879$	10.7846	0.363756
$880$	0	0
$881$	−13.7128	−0.461996	−0.230998	−	0.972954i	$-0.574199\pi$
$881$	−13.7128	−0.461996	−0.230998	+	0.972954i	$0.574199\pi$
$882$	0	0
$883$	8.00000i	0.269221i	0.990899	+	0.134611i	$0.0429784\pi$
$883$	8.00000i	0.269221i	−0.990899	+	0.134611i	$0.957022\pi$
$884$	0	0
$885$	12.3923i	0.416563i
$886$	0	0
$887$	−33.4641	−1.12361	−0.561807	−	0.827268i	$-0.689894\pi$
$887$	−33.4641	−1.12361	−0.561807	+	0.827268i	$0.689894\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	6.53590i	0.218961i
$892$	0	0
$893$	9.07180i	0.303576i
$894$	0	0
$895$	−32.7846	−1.09587
$896$	0	0
$897$	17.8564	0.596208
$898$	0	0
$899$	− 8.78461i	− 0.292983i
$900$	0	0
$901$	− 44.7846i	− 1.49199i
$902$	0	0
$903$	1.85641	0.0617773
$904$	0	0
$905$	4.53590	0.150778
$906$	0	0
$907$	42.6410i	1.41587i	0.706277	+	0.707936i	$0.250373\pi$
$907$	42.6410i	1.41587i	−0.706277	+	0.707936i	$0.749627\pi$
$908$	0	0
$909$	9.37307i	0.310885i
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	10.6410	0.352166
$914$	0	0
$915$	22.2487i	0.735520i
$916$	0	0
$917$	2.19615i	0.0725233i
$918$	0	0
$919$	8.78461	0.289778	0.144889	−	0.989448i	$-0.453718\pi$
$919$	8.78461	0.289778	0.144889	+	0.989448i	$0.453718\pi$
$920$	0	0
$921$	−19.1769	−0.631901
$922$	0	0
$923$	− 37.8564i	− 1.24606i
$924$	0	0
$925$	11.1769i	0.367495i
$926$	0	0
$927$	20.6795	0.679204
$928$	0	0
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	6.19615i	0.203071i
$932$	0	0
$933$	− 8.00000i	− 0.261908i
$934$	0	0
$935$	29.8564	0.976409
$936$	0	0
$937$	43.5692	1.42334	0.711672	−	0.702512i	$-0.247938\pi$
$937$	43.5692	1.42334	0.711672	+	0.702512i	$0.247938\pi$
$938$	0	0
$939$	9.17691i	0.299477i
$940$	0	0
$941$	55.9090i	1.82258i	0.411765	+	0.911290i	$0.364912\pi$
$941$	55.9090i	1.82258i	−0.411765	+	0.911290i	$0.635088\pi$
$942$	0	0
$943$	30.9282	1.00716
$944$	0	0
$945$	10.9282	0.355494
$946$	0	0
$947$	7.32051i	0.237885i	0.992901	+	0.118942i	$0.0379503\pi$
$947$	7.32051i	0.237885i	−0.992901	+	0.118942i	$0.962050\pi$
$948$	0	0
$949$	− 24.3923i	− 0.791808i
$950$	0	0
$951$	−9.46410	−0.306895
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	29.8564i	0.966131i
$956$	0	0
$957$	3.71281i	0.120018i
$958$	0	0
$959$	−20.9282	−0.675807
$960$	0	0
$961$	−24.5692	−0.792555
$962$	0	0
$963$	39.4256i	1.27047i
$964$	0	0
$965$	33.8564i	1.08988i
$966$	0	0
$967$	−32.6410	−1.04966	−0.524832	−	0.851206i	$-0.675872\pi$
$967$	−32.6410	−1.04966	−0.524832	+	0.851206i	$0.675872\pi$
$968$	0	0
$969$	33.8564	1.08762
$970$	0	0
$971$	17.8038i	0.571353i	0.958326	+	0.285676i	$0.0922182\pi$
$971$	17.8038i	0.571353i	−0.958326	+	0.285676i	$0.907782\pi$
$972$	0	0
$973$	15.6603i	0.502045i
$974$	0	0
$975$	−4.92820	−0.157829
$976$	0	0
$977$	16.6410	0.532393	0.266197	−	0.963919i	$-0.414233\pi$
$977$	16.6410	0.532393	0.266197	+	0.963919i	$0.414233\pi$
$978$	0	0
$979$	− 7.21539i	− 0.230605i
$980$	0	0
$981$	− 8.53590i	− 0.272530i
$982$	0	0
$983$	−0.392305	−0.0125126	−0.00625629	−	0.999980i	$-0.501991\pi$
$983$	−0.392305	−0.0125126	−0.00625629	+	0.999980i	$0.501991\pi$
$984$	0	0
$985$	65.1769	2.07671
$986$	0	0
$987$	1.07180i	0.0341157i
$988$	0	0
$989$	− 22.6410i	− 0.719942i
$990$	0	0
$991$	10.9282	0.347146	0.173573	−	0.984821i	$-0.444469\pi$
$991$	10.9282	0.347146	0.173573	+	0.984821i	$0.444469\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	− 50.6410i	− 1.60543i
$996$	0	0
$997$	− 53.6603i	− 1.69944i	−0.527236	−	0.849719i	$-0.676772\pi$
$997$	− 53.6603i	− 1.69944i	0.527236	−	0.849719i	$-0.323228\pi$
$998$	0	0
$999$	−18.1436	−0.574038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.b.l.897.2		4
4.3	odd	2		1792.2.b.n.897.3		4
8.3	odd	2		1792.2.b.n.897.2		4
8.5	even	2	inner	1792.2.b.l.897.3		4
16.3	odd	4		896.2.a.g.1.1	yes	2
16.5	even	4		896.2.a.h.1.1	yes	2
16.11	odd	4		896.2.a.f.1.2	yes	2
16.13	even	4		896.2.a.e.1.2	✓	2
48.5	odd	4		8064.2.a.bf.1.1		2
48.11	even	4		8064.2.a.be.1.1		2
48.29	odd	4		8064.2.a.br.1.2		2
48.35	even	4		8064.2.a.bm.1.2		2
112.13	odd	4		6272.2.a.t.1.1		2
112.27	even	4		6272.2.a.s.1.1		2
112.69	odd	4		6272.2.a.i.1.2		2
112.83	even	4		6272.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.e.1.2	✓	2	16.13	even	4
896.2.a.f.1.2	yes	2	16.11	odd	4
896.2.a.g.1.1	yes	2	16.3	odd	4
896.2.a.h.1.1	yes	2	16.5	even	4
1792.2.b.l.897.2		4	1.1	even	1	trivial
1792.2.b.l.897.3		4	8.5	even	2	inner
1792.2.b.n.897.2		4	8.3	odd	2
1792.2.b.n.897.3		4	4.3	odd	2
6272.2.a.i.1.2		2	112.69	odd	4
6272.2.a.j.1.2		2	112.83	even	4
6272.2.a.s.1.1		2	112.27	even	4
6272.2.a.t.1.1		2	112.13	odd	4
8064.2.a.be.1.1		2	48.11	even	4
8064.2.a.bf.1.1		2	48.5	odd	4
8064.2.a.bm.1.2		2	48.35	even	4
8064.2.a.br.1.2		2	48.29	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.732051i q^{3} -2.73205i q^{5} -1.00000 q^{7} +2.46410 q^{9} +O(q^{10})\)	\(q-0.732051i q^{3} -2.73205i q^{5} -1.00000 q^{7} +2.46410 q^{9} +1.46410i q^{11} +2.73205i q^{13} -2.00000 q^{15} +7.46410 q^{17} +6.19615i q^{19} +0.732051i q^{21} +8.92820 q^{23} -2.46410 q^{25} -4.00000i q^{27} +3.46410i q^{29} -2.53590 q^{31} +1.07180 q^{33} +2.73205i q^{35} -4.53590i q^{37} +2.00000 q^{39} +3.46410 q^{41} -2.53590i q^{43} -6.73205i q^{45} +1.46410 q^{47} +1.00000 q^{49} -5.46410i q^{51} -6.00000i q^{53} +4.00000 q^{55} +4.53590 q^{57} -6.19615i q^{59} -11.1244i q^{61} -2.46410 q^{63} +7.46410 q^{65} +14.9282i q^{67} -6.53590i q^{69} -13.8564 q^{71} -8.92820 q^{73} +1.80385i q^{75} -1.46410i q^{77} -8.00000 q^{79} +4.46410 q^{81} -7.26795i q^{83} -20.3923i q^{85} +2.53590 q^{87} -4.92820 q^{89} -2.73205i q^{91} +1.85641i q^{93} +16.9282 q^{95} +0.535898 q^{97} +3.60770i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 - 4 * q^9	\( 4 q - 4 q^{7} - 4 q^{9} - 8 q^{15} + 16 q^{17} + 8 q^{23} + 4 q^{25} - 24 q^{31} + 32 q^{33} + 8 q^{39} - 8 q^{47} + 4 q^{49} + 16 q^{55} + 32 q^{57} + 4 q^{63} + 16 q^{65} - 8 q^{73} - 32 q^{79} + 4 q^{81} + 24 q^{87} + 8 q^{89} + 40 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^9 - 8 * q^15 + 16 * q^17 + 8 * q^23 + 4 * q^25 - 24 * q^31 + 32 * q^33 + 8 * q^39 - 8 * q^47 + 4 * q^49 + 16 * q^55 + 32 * q^57 + 4 * q^63 + 16 * q^65 - 8 * q^73 - 32 * q^79 + 4 * q^81 + 24 * q^87 + 8 * q^89 + 40 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more