LMFDB - Embedded newform 1792.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1,1792)]

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, 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("1792.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.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:	$14.3091920422$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 448)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1792.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -2.73205 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -2.73205 q^{5} -1.00000 q^{7} +4.46410 q^{9} +5.46410 q^{11} -6.73205 q^{13} +7.46410 q^{15} +2.00000 q^{17} +1.26795 q^{19} +2.73205 q^{21} +3.46410 q^{23} +2.46410 q^{25} -4.00000 q^{27} +1.46410 q^{29} +4.00000 q^{31} -14.9282 q^{33} +2.73205 q^{35} +1.46410 q^{37} +18.3923 q^{39} +2.00000 q^{41} +5.46410 q^{43} -12.1962 q^{45} +2.92820 q^{47} +1.00000 q^{49} -5.46410 q^{51} -12.0000 q^{53} -14.9282 q^{55} -3.46410 q^{57} -9.66025 q^{59} +11.1244 q^{61} -4.46410 q^{63} +18.3923 q^{65} -8.00000 q^{67} -9.46410 q^{69} -2.92820 q^{71} -12.9282 q^{73} -6.73205 q^{75} -5.46410 q^{77} -10.9282 q^{79} -2.46410 q^{81} +5.66025 q^{83} -5.46410 q^{85} -4.00000 q^{87} -11.8564 q^{89} +6.73205 q^{91} -10.9282 q^{93} -3.46410 q^{95} +8.92820 q^{97} +24.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 10 q^{13} + 8 q^{15} + 4 q^{17} + 6 q^{19} + 2 q^{21} - 2 q^{25} - 8 q^{27} - 4 q^{29} + 8 q^{31} - 16 q^{33} + 2 q^{35} - 4 q^{37} + 16 q^{39} + 4 q^{41} + 4 q^{43} - 14 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 24 q^{53} - 16 q^{55} - 2 q^{59} - 2 q^{61} - 2 q^{63} + 16 q^{65} - 16 q^{67} - 12 q^{69} + 8 q^{71} - 12 q^{73} - 10 q^{75} - 4 q^{77} - 8 q^{79} + 2 q^{81} - 6 q^{83} - 4 q^{85} - 8 q^{87} + 4 q^{89} + 10 q^{91} - 8 q^{93} + 4 q^{97} + 28 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 10 * q^13 + 8 * q^15 + 4 * q^17 + 6 * q^19 + 2 * q^21 - 2 * q^25 - 8 * q^27 - 4 * q^29 + 8 * q^31 - 16 * q^33 + 2 * q^35 - 4 * q^37 + 16 * q^39 + 4 * q^41 + 4 * q^43 - 14 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 - 24 * q^53 - 16 * q^55 - 2 * q^59 - 2 * q^61 - 2 * q^63 + 16 * q^65 - 16 * q^67 - 12 * q^69 + 8 * q^71 - 12 * q^73 - 10 * q^75 - 4 * q^77 - 8 * q^79 + 2 * q^81 - 6 * q^83 - 4 * q^85 - 8 * q^87 + 4 * q^89 + 10 * q^91 - 8 * q^93 + 4 * q^97 + 28 * 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
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	−6.73205	−1.86713	−0.933567	−	0.358402i	$-0.883322\pi$
$13$	−6.73205	−1.86713	−0.933567	+	0.358402i	$0.883322\pi$
$14$	0	0
$15$	7.46410	1.92722
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.26795	0.290887	0.145444	−	0.989367i	$-0.453539\pi$
$19$	1.26795	0.290887	0.145444	+	0.989367i	$0.453539\pi$
$20$	0	0
$21$	2.73205	0.596182
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	1.46410	0.271877	0.135938	−	0.990717i	$-0.456595\pi$
$29$	1.46410	0.271877	0.135938	+	0.990717i	$0.456595\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−14.9282	−2.59867
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	1.46410	0.240697	0.120348	−	0.992732i	$-0.461599\pi$
$37$	1.46410	0.240697	0.120348	+	0.992732i	$0.461599\pi$
$38$	0	0
$39$	18.3923	2.94513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	5.46410	0.833268	0.416634	−	0.909074i	$-0.363210\pi$
$43$	5.46410	0.833268	0.416634	+	0.909074i	$0.363210\pi$
$44$	0	0
$45$	−12.1962	−1.81810
$46$	0	0
$47$	2.92820	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$47$	2.92820	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.46410	−0.765127
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	−14.9282	−2.01292
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	−9.66025	−1.25766	−0.628829	−	0.777544i	$-0.716465\pi$
$59$	−9.66025	−1.25766	−0.628829	+	0.777544i	$0.716465\pi$
$60$	0	0
$61$	11.1244	1.42433	0.712164	−	0.702013i	$-0.247715\pi$
$61$	11.1244	1.42433	0.712164	+	0.702013i	$0.247715\pi$
$62$	0	0
$63$	−4.46410	−0.562424
$64$	0	0
$65$	18.3923	2.28128
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−9.46410	−1.13934
$70$	0	0
$71$	−2.92820	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$71$	−2.92820	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$72$	0	0
$73$	−12.9282	−1.51313	−0.756566	−	0.653917i	$-0.773124\pi$
$73$	−12.9282	−1.51313	−0.756566	+	0.653917i	$0.773124\pi$
$74$	0	0
$75$	−6.73205	−0.777350
$76$	0	0
$77$	−5.46410	−0.622692
$78$	0	0
$79$	−10.9282	−1.22952	−0.614759	−	0.788715i	$-0.710747\pi$
$79$	−10.9282	−1.22952	−0.614759	+	0.788715i	$0.710747\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	5.66025	0.621294	0.310647	−	0.950525i	$-0.399454\pi$
$83$	5.66025	0.621294	0.310647	+	0.950525i	$0.399454\pi$
$84$	0	0
$85$	−5.46410	−0.592665
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−11.8564	−1.25678	−0.628388	−	0.777900i	$-0.716285\pi$
$89$	−11.8564	−1.25678	−0.628388	+	0.777900i	$0.716285\pi$
$90$	0	0
$91$	6.73205	0.705711
$92$	0	0
$93$	−10.9282	−1.13320
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	8.92820	0.906522	0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820	0.906522	0.453261	+	0.891378i	$0.350261\pi$
$98$	0	0
$99$	24.3923	2.45152
$100$	0	0
$101$	1.66025	0.165201	0.0826007	−	0.996583i	$-0.473677\pi$
$101$	1.66025	0.165201	0.0826007	+	0.996583i	$0.473677\pi$
$102$	0	0
$103$	−9.85641	−0.971181	−0.485590	−	0.874187i	$-0.661395\pi$
$103$	−9.85641	−0.971181	−0.485590	+	0.874187i	$0.661395\pi$
$104$	0	0
$105$	−7.46410	−0.728422
$106$	0	0
$107$	5.07180	0.490309	0.245155	−	0.969484i	$-0.421161\pi$
$107$	5.07180	0.490309	0.245155	+	0.969484i	$0.421161\pi$
$108$	0	0
$109$	−12.3923	−1.18697	−0.593484	−	0.804846i	$-0.702248\pi$
$109$	−12.3923	−1.18697	−0.593484	+	0.804846i	$0.702248\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−6.53590	−0.614846	−0.307423	−	0.951573i	$-0.599467\pi$
$113$	−6.53590	−0.614846	−0.307423	+	0.951573i	$0.599467\pi$
$114$	0	0
$115$	−9.46410	−0.882532
$116$	0	0
$117$	−30.0526	−2.77836
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	−5.46410	−0.492681
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−11.4641	−1.01727	−0.508637	−	0.860981i	$-0.669851\pi$
$127$	−11.4641	−1.01727	−0.508637	+	0.860981i	$0.669851\pi$
$128$	0	0
$129$	−14.9282	−1.31436
$130$	0	0
$131$	10.7321	0.937664	0.468832	−	0.883287i	$-0.344675\pi$
$131$	10.7321	0.937664	0.468832	+	0.883287i	$0.344675\pi$
$132$	0	0
$133$	−1.26795	−0.109945
$134$	0	0
$135$	10.9282	0.940550
$136$	0	0
$137$	−19.8564	−1.69645	−0.848224	−	0.529638i	$-0.822328\pi$
$137$	−19.8564	−1.69645	−0.848224	+	0.529638i	$0.822328\pi$
$138$	0	0
$139$	−9.26795	−0.786097	−0.393049	−	0.919518i	$-0.628580\pi$
$139$	−9.26795	−0.786097	−0.393049	+	0.919518i	$0.628580\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−36.7846	−3.07608
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−2.73205	−0.225336
$148$	0	0
$149$	−1.07180	−0.0878050	−0.0439025	−	0.999036i	$-0.513979\pi$
$149$	−1.07180	−0.0878050	−0.0439025	+	0.999036i	$0.513979\pi$
$150$	0	0
$151$	2.39230	0.194683	0.0973415	−	0.995251i	$-0.468966\pi$
$151$	2.39230	0.194683	0.0973415	+	0.995251i	$0.468966\pi$
$152$	0	0
$153$	8.92820	0.721802
$154$	0	0
$155$	−10.9282	−0.877774
$156$	0	0
$157$	12.1962	0.973359	0.486679	−	0.873581i	$-0.338208\pi$
$157$	12.1962	0.973359	0.486679	+	0.873581i	$0.338208\pi$
$158$	0	0
$159$	32.7846	2.59999
$160$	0	0
$161$	−3.46410	−0.273009
$162$	0	0
$163$	13.4641	1.05459	0.527295	−	0.849682i	$-0.323206\pi$
$163$	13.4641	1.05459	0.527295	+	0.849682i	$0.323206\pi$
$164$	0	0
$165$	40.7846	3.17508
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	32.3205	2.48619
$170$	0	0
$171$	5.66025	0.432850
$172$	0	0
$173$	−3.80385	−0.289201	−0.144601	−	0.989490i	$-0.546190\pi$
$173$	−3.80385	−0.289201	−0.144601	+	0.989490i	$0.546190\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	26.3923	1.98377
$178$	0	0
$179$	2.92820	0.218864	0.109432	−	0.993994i	$-0.465097\pi$
$179$	2.92820	0.218864	0.109432	+	0.993994i	$0.465097\pi$
$180$	0	0
$181$	10.7321	0.797707	0.398854	−	0.917015i	$-0.369408\pi$
$181$	10.7321	0.797707	0.398854	+	0.917015i	$0.369408\pi$
$182$	0	0
$183$	−30.3923	−2.24666
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	10.9282	0.799149
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	6.53590	0.470464	0.235232	−	0.971939i	$-0.424415\pi$
$193$	6.53590	0.470464	0.235232	+	0.971939i	$0.424415\pi$
$194$	0	0
$195$	−50.2487	−3.59838
$196$	0	0
$197$	−25.8564	−1.84219	−0.921096	−	0.389335i	$-0.872705\pi$
$197$	−25.8564	−1.84219	−0.921096	+	0.389335i	$0.872705\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	21.8564	1.54163
$202$	0	0
$203$	−1.46410	−0.102760
$204$	0	0
$205$	−5.46410	−0.381629
$206$	0	0
$207$	15.4641	1.07483
$208$	0	0
$209$	6.92820	0.479234
$210$	0	0
$211$	−16.7846	−1.15550	−0.577750	−	0.816214i	$-0.696069\pi$
$211$	−16.7846	−1.15550	−0.577750	+	0.816214i	$0.696069\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	−14.9282	−1.01810
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	35.3205	2.38674
$220$	0	0
$221$	−13.4641	−0.905693
$222$	0	0
$223$	−6.92820	−0.463947	−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820	−0.463947	−0.231973	+	0.972722i	$0.574518\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	−3.80385	−0.252470	−0.126235	−	0.992000i	$-0.540289\pi$
$227$	−3.80385	−0.252470	−0.126235	+	0.992000i	$0.540289\pi$
$228$	0	0
$229$	17.2679	1.14110	0.570549	−	0.821263i	$-0.306730\pi$
$229$	17.2679	1.14110	0.570549	+	0.821263i	$0.306730\pi$
$230$	0	0
$231$	14.9282	0.982204
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	29.8564	1.93938
$238$	0	0
$239$	−12.5359	−0.810880	−0.405440	−	0.914122i	$-0.632882\pi$
$239$	−12.5359	−0.810880	−0.405440	+	0.914122i	$0.632882\pi$
$240$	0	0
$241$	−18.7846	−1.21002	−0.605012	−	0.796217i	$-0.706832\pi$
$241$	−18.7846	−1.21002	−0.605012	+	0.796217i	$0.706832\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−2.73205	−0.174544
$246$	0	0
$247$	−8.53590	−0.543126
$248$	0	0
$249$	−15.4641	−0.979998
$250$	0	0
$251$	−15.8038	−0.997530	−0.498765	−	0.866737i	$-0.666213\pi$
$251$	−15.8038	−0.997530	−0.498765	+	0.866737i	$0.666213\pi$
$252$	0	0
$253$	18.9282	1.19001
$254$	0	0
$255$	14.9282	0.934840
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−1.46410	−0.0909748
$260$	0	0
$261$	6.53590	0.404562
$262$	0	0
$263$	−10.9282	−0.673862	−0.336931	−	0.941529i	$-0.609389\pi$
$263$	−10.9282	−0.673862	−0.336931	+	0.941529i	$0.609389\pi$
$264$	0	0
$265$	32.7846	2.01394
$266$	0	0
$267$	32.3923	1.98238
$268$	0	0
$269$	−12.5885	−0.767532	−0.383766	−	0.923430i	$-0.625373\pi$
$269$	−12.5885	−0.767532	−0.383766	+	0.923430i	$0.625373\pi$
$270$	0	0
$271$	14.9282	0.906824	0.453412	−	0.891301i	$-0.350207\pi$
$271$	14.9282	0.906824	0.453412	+	0.891301i	$0.350207\pi$
$272$	0	0
$273$	−18.3923	−1.11315
$274$	0	0
$275$	13.4641	0.811916
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	17.8564	1.06904
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	16.5885	0.986081	0.493041	−	0.870006i	$-0.335885\pi$
$283$	16.5885	0.986081	0.493041	+	0.870006i	$0.335885\pi$
$284$	0	0
$285$	9.46410	0.560605
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−24.3923	−1.42990
$292$	0	0
$293$	−12.1962	−0.712507	−0.356253	−	0.934389i	$-0.615946\pi$
$293$	−12.1962	−0.712507	−0.356253	+	0.934389i	$0.615946\pi$
$294$	0	0
$295$	26.3923	1.53662
$296$	0	0
$297$	−21.8564	−1.26824
$298$	0	0
$299$	−23.3205	−1.34866
$300$	0	0
$301$	−5.46410	−0.314946
$302$	0	0
$303$	−4.53590	−0.260581
$304$	0	0
$305$	−30.3923	−1.74026
$306$	0	0
$307$	8.58846	0.490169	0.245085	−	0.969502i	$-0.421184\pi$
$307$	8.58846	0.490169	0.245085	+	0.969502i	$0.421184\pi$
$308$	0	0
$309$	26.9282	1.53189
$310$	0	0
$311$	5.85641	0.332086	0.166043	−	0.986118i	$-0.446901\pi$
$311$	5.85641	0.332086	0.166043	+	0.986118i	$0.446901\pi$
$312$	0	0
$313$	−19.8564	−1.12235	−0.561175	−	0.827697i	$-0.689651\pi$
$313$	−19.8564	−1.12235	−0.561175	+	0.827697i	$0.689651\pi$
$314$	0	0
$315$	12.1962	0.687175
$316$	0	0
$317$	−28.7846	−1.61670	−0.808352	−	0.588699i	$-0.799640\pi$
$317$	−28.7846	−1.61670	−0.808352	+	0.588699i	$0.799640\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−13.8564	−0.773389
$322$	0	0
$323$	2.53590	0.141101
$324$	0	0
$325$	−16.5885	−0.920162
$326$	0	0
$327$	33.8564	1.87226
$328$	0	0
$329$	−2.92820	−0.161437
$330$	0	0
$331$	−16.3923	−0.901003	−0.450501	−	0.892776i	$-0.648755\pi$
$331$	−16.3923	−0.901003	−0.450501	+	0.892776i	$0.648755\pi$
$332$	0	0
$333$	6.53590	0.358165
$334$	0	0
$335$	21.8564	1.19414
$336$	0	0
$337$	16.3923	0.892946	0.446473	−	0.894797i	$-0.352680\pi$
$337$	16.3923	0.892946	0.446473	+	0.894797i	$0.352680\pi$
$338$	0	0
$339$	17.8564	0.969827
$340$	0	0
$341$	21.8564	1.18359
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	25.8564	1.39206
$346$	0	0
$347$	−35.3205	−1.89610	−0.948052	−	0.318115i	$-0.896950\pi$
$347$	−35.3205	−1.89610	−0.948052	+	0.318115i	$0.896950\pi$
$348$	0	0
$349$	−14.0526	−0.752216	−0.376108	−	0.926576i	$-0.622738\pi$
$349$	−14.0526	−0.752216	−0.376108	+	0.926576i	$0.622738\pi$
$350$	0	0
$351$	26.9282	1.43732
$352$	0	0
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	5.46410	0.289191
$358$	0	0
$359$	27.4641	1.44950	0.724750	−	0.689012i	$-0.241955\pi$
$359$	27.4641	1.44950	0.724750	+	0.689012i	$0.241955\pi$
$360$	0	0
$361$	−17.3923	−0.915384
$362$	0	0
$363$	−51.5167	−2.70392
$364$	0	0
$365$	35.3205	1.84876
$366$	0	0
$367$	20.7846	1.08495	0.542474	−	0.840073i	$-0.317488\pi$
$367$	20.7846	1.08495	0.542474	+	0.840073i	$0.317488\pi$
$368$	0	0
$369$	8.92820	0.464784
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	30.9282	1.60140	0.800701	−	0.599064i	$-0.204461\pi$
$373$	30.9282	1.60140	0.800701	+	0.599064i	$0.204461\pi$
$374$	0	0
$375$	−18.9282	−0.977448
$376$	0	0
$377$	−9.85641	−0.507631
$378$	0	0
$379$	−14.2487	−0.731907	−0.365954	−	0.930633i	$-0.619257\pi$
$379$	−14.2487	−0.731907	−0.365954	+	0.930633i	$0.619257\pi$
$380$	0	0
$381$	31.3205	1.60460
$382$	0	0
$383$	−32.7846	−1.67522	−0.837608	−	0.546272i	$-0.816046\pi$
$383$	−32.7846	−1.67522	−0.837608	+	0.546272i	$0.816046\pi$
$384$	0	0
$385$	14.9282	0.760812
$386$	0	0
$387$	24.3923	1.23993
$388$	0	0
$389$	−23.3205	−1.18240	−0.591198	−	0.806526i	$-0.701345\pi$
$389$	−23.3205	−1.18240	−0.591198	+	0.806526i	$0.701345\pi$
$390$	0	0
$391$	6.92820	0.350374
$392$	0	0
$393$	−29.3205	−1.47902
$394$	0	0
$395$	29.8564	1.50224
$396$	0	0
$397$	4.87564	0.244702	0.122351	−	0.992487i	$-0.460957\pi$
$397$	4.87564	0.244702	0.122351	+	0.992487i	$0.460957\pi$
$398$	0	0
$399$	3.46410	0.173422
$400$	0	0
$401$	13.4641	0.672365	0.336183	−	0.941797i	$-0.390864\pi$
$401$	13.4641	0.672365	0.336183	+	0.941797i	$0.390864\pi$
$402$	0	0
$403$	−26.9282	−1.34139
$404$	0	0
$405$	6.73205	0.334518
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	3.07180	0.151891	0.0759453	−	0.997112i	$-0.475803\pi$
$409$	3.07180	0.151891	0.0759453	+	0.997112i	$0.475803\pi$
$410$	0	0
$411$	54.2487	2.67589
$412$	0	0
$413$	9.66025	0.475350
$414$	0	0
$415$	−15.4641	−0.759103
$416$	0	0
$417$	25.3205	1.23995
$418$	0	0
$419$	−23.8038	−1.16289	−0.581447	−	0.813584i	$-0.697513\pi$
$419$	−23.8038	−1.16289	−0.581447	+	0.813584i	$0.697513\pi$
$420$	0	0
$421$	25.8564	1.26016	0.630082	−	0.776529i	$-0.283021\pi$
$421$	25.8564	1.26016	0.630082	+	0.776529i	$0.283021\pi$
$422$	0	0
$423$	13.0718	0.635573
$424$	0	0
$425$	4.92820	0.239053
$426$	0	0
$427$	−11.1244	−0.538345
$428$	0	0
$429$	100.497	4.85206
$430$	0	0
$431$	−35.4641	−1.70825	−0.854123	−	0.520071i	$-0.825905\pi$
$431$	−35.4641	−1.70825	−0.854123	+	0.520071i	$0.825905\pi$
$432$	0	0
$433$	15.8564	0.762010	0.381005	−	0.924573i	$-0.375578\pi$
$433$	15.8564	0.762010	0.381005	+	0.924573i	$0.375578\pi$
$434$	0	0
$435$	10.9282	0.523967
$436$	0	0
$437$	4.39230	0.210112
$438$	0	0
$439$	−30.9282	−1.47612	−0.738061	−	0.674734i	$-0.764258\pi$
$439$	−30.9282	−1.47612	−0.738061	+	0.674734i	$0.764258\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	32.3923	1.53554
$446$	0	0
$447$	2.92820	0.138499
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	10.9282	0.514589
$452$	0	0
$453$	−6.53590	−0.307083
$454$	0	0
$455$	−18.3923	−0.862245
$456$	0	0
$457$	−15.3205	−0.716663	−0.358332	−	0.933594i	$-0.616654\pi$
$457$	−15.3205	−0.716663	−0.358332	+	0.933594i	$0.616654\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−24.1962	−1.12693	−0.563464	−	0.826141i	$-0.690531\pi$
$461$	−24.1962	−1.12693	−0.563464	+	0.826141i	$0.690531\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	29.8564	1.38456
$466$	0	0
$467$	−0.875644	−0.0405200	−0.0202600	−	0.999795i	$-0.506449\pi$
$467$	−0.875644	−0.0405200	−0.0202600	+	0.999795i	$0.506449\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−33.3205	−1.53533
$472$	0	0
$473$	29.8564	1.37280
$474$	0	0
$475$	3.12436	0.143355
$476$	0	0
$477$	−53.5692	−2.45277
$478$	0	0
$479$	−9.85641	−0.450351	−0.225175	−	0.974318i	$-0.572295\pi$
$479$	−9.85641	−0.450351	−0.225175	+	0.974318i	$0.572295\pi$
$480$	0	0
$481$	−9.85641	−0.449413
$482$	0	0
$483$	9.46410	0.430632
$484$	0	0
$485$	−24.3923	−1.10760
$486$	0	0
$487$	21.6077	0.979138	0.489569	−	0.871965i	$-0.337154\pi$
$487$	21.6077	0.979138	0.489569	+	0.871965i	$0.337154\pi$
$488$	0	0
$489$	−36.7846	−1.66346
$490$	0	0
$491$	32.7846	1.47955	0.739774	−	0.672855i	$-0.234932\pi$
$491$	32.7846	1.47955	0.739774	+	0.672855i	$0.234932\pi$
$492$	0	0
$493$	2.92820	0.131880
$494$	0	0
$495$	−66.6410	−2.99529
$496$	0	0
$497$	2.92820	0.131348
$498$	0	0
$499$	13.0718	0.585174	0.292587	−	0.956239i	$-0.405484\pi$
$499$	13.0718	0.585174	0.292587	+	0.956239i	$0.405484\pi$
$500$	0	0
$501$	−13.8564	−0.619059
$502$	0	0
$503$	−30.9282	−1.37902	−0.689510	−	0.724276i	$-0.742174\pi$
$503$	−30.9282	−1.37902	−0.689510	+	0.724276i	$0.742174\pi$
$504$	0	0
$505$	−4.53590	−0.201845
$506$	0	0
$507$	−88.3013	−3.92160
$508$	0	0
$509$	−3.12436	−0.138485	−0.0692423	−	0.997600i	$-0.522058\pi$
$509$	−3.12436	−0.138485	−0.0692423	+	0.997600i	$0.522058\pi$
$510$	0	0
$511$	12.9282	0.571910
$512$	0	0
$513$	−5.07180	−0.223925
$514$	0	0
$515$	26.9282	1.18660
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	10.3923	0.456172
$520$	0	0
$521$	14.7846	0.647726	0.323863	−	0.946104i	$-0.395018\pi$
$521$	14.7846	0.647726	0.323863	+	0.946104i	$0.395018\pi$
$522$	0	0
$523$	3.41154	0.149176	0.0745882	−	0.997214i	$-0.476236\pi$
$523$	3.41154	0.149176	0.0745882	+	0.997214i	$0.476236\pi$
$524$	0	0
$525$	6.73205	0.293811
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	−43.1244	−1.87144
$532$	0	0
$533$	−13.4641	−0.583195
$534$	0	0
$535$	−13.8564	−0.599065
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	5.46410	0.235356
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	0	0
$543$	−29.3205	−1.25826
$544$	0	0
$545$	33.8564	1.45025
$546$	0	0
$547$	46.2487	1.97745	0.988726	−	0.149736i	$-0.0478423\pi$
$547$	46.2487	1.97745	0.988726	+	0.149736i	$0.0478423\pi$
$548$	0	0
$549$	49.6603	2.11945
$550$	0	0
$551$	1.85641	0.0790856
$552$	0	0
$553$	10.9282	0.464714
$554$	0	0
$555$	10.9282	0.463876
$556$	0	0
$557$	23.7128	1.00474	0.502372	−	0.864652i	$-0.332461\pi$
$557$	23.7128	1.00474	0.502372	+	0.864652i	$0.332461\pi$
$558$	0	0
$559$	−36.7846	−1.55582
$560$	0	0
$561$	−29.8564	−1.26054
$562$	0	0
$563$	−34.0526	−1.43514	−0.717572	−	0.696484i	$-0.754747\pi$
$563$	−34.0526	−1.43514	−0.717572	+	0.696484i	$0.754747\pi$
$564$	0	0
$565$	17.8564	0.751225
$566$	0	0
$567$	2.46410	0.103483
$568$	0	0
$569$	5.46410	0.229067	0.114534	−	0.993419i	$-0.463463\pi$
$569$	5.46410	0.229067	0.114534	+	0.993419i	$0.463463\pi$
$570$	0	0
$571$	25.1769	1.05362	0.526811	−	0.849983i	$-0.323388\pi$
$571$	25.1769	1.05362	0.526811	+	0.849983i	$0.323388\pi$
$572$	0	0
$573$	−43.7128	−1.82613
$574$	0	0
$575$	8.53590	0.355972
$576$	0	0
$577$	−31.0718	−1.29354	−0.646768	−	0.762687i	$-0.723880\pi$
$577$	−31.0718	−1.29354	−0.646768	+	0.762687i	$0.723880\pi$
$578$	0	0
$579$	−17.8564	−0.742087
$580$	0	0
$581$	−5.66025	−0.234827
$582$	0	0
$583$	−65.5692	−2.71560
$584$	0	0
$585$	82.1051	3.39463
$586$	0	0
$587$	−33.2679	−1.37312	−0.686558	−	0.727075i	$-0.740879\pi$
$587$	−33.2679	−1.37312	−0.686558	+	0.727075i	$0.740879\pi$
$588$	0	0
$589$	5.07180	0.208980
$590$	0	0
$591$	70.6410	2.90578
$592$	0	0
$593$	12.1436	0.498678	0.249339	−	0.968416i	$-0.419787\pi$
$593$	12.1436	0.498678	0.249339	+	0.968416i	$0.419787\pi$
$594$	0	0
$595$	5.46410	0.224006
$596$	0	0
$597$	−10.9282	−0.447262
$598$	0	0
$599$	−27.7128	−1.13231	−0.566157	−	0.824297i	$-0.691571\pi$
$599$	−27.7128	−1.13231	−0.566157	+	0.824297i	$0.691571\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−35.7128	−1.45434
$604$	0	0
$605$	−51.5167	−2.09445
$606$	0	0
$607$	−27.7128	−1.12483	−0.562414	−	0.826856i	$-0.690127\pi$
$607$	−27.7128	−1.12483	−0.562414	+	0.826856i	$0.690127\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	−19.7128	−0.797495
$612$	0	0
$613$	−39.3205	−1.58814	−0.794070	−	0.607826i	$-0.792042\pi$
$613$	−39.3205	−1.58814	−0.794070	+	0.607826i	$0.792042\pi$
$614$	0	0
$615$	14.9282	0.601963
$616$	0	0
$617$	24.3923	0.981997	0.490999	−	0.871160i	$-0.336632\pi$
$617$	24.3923	0.981997	0.490999	+	0.871160i	$0.336632\pi$
$618$	0	0
$619$	−24.1962	−0.972525	−0.486263	−	0.873813i	$-0.661640\pi$
$619$	−24.1962	−0.972525	−0.486263	+	0.873813i	$0.661640\pi$
$620$	0	0
$621$	−13.8564	−0.556038
$622$	0	0
$623$	11.8564	0.475017
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	−18.9282	−0.755920
$628$	0	0
$629$	2.92820	0.116755
$630$	0	0
$631$	49.5692	1.97332	0.986660	−	0.162796i	$-0.0520513\pi$
$631$	49.5692	1.97332	0.986660	+	0.162796i	$0.0520513\pi$
$632$	0	0
$633$	45.8564	1.82263
$634$	0	0
$635$	31.3205	1.24292
$636$	0	0
$637$	−6.73205	−0.266734
$638$	0	0
$639$	−13.0718	−0.517112
$640$	0	0
$641$	−22.2487	−0.878771	−0.439386	−	0.898299i	$-0.644804\pi$
$641$	−22.2487	−0.878771	−0.439386	+	0.898299i	$0.644804\pi$
$642$	0	0
$643$	−14.3397	−0.565504	−0.282752	−	0.959193i	$-0.591247\pi$
$643$	−14.3397	−0.565504	−0.282752	+	0.959193i	$0.591247\pi$
$644$	0	0
$645$	40.7846	1.60589
$646$	0	0
$647$	1.85641	0.0729829	0.0364914	−	0.999334i	$-0.488382\pi$
$647$	1.85641	0.0729829	0.0364914	+	0.999334i	$0.488382\pi$
$648$	0	0
$649$	−52.7846	−2.07198
$650$	0	0
$651$	10.9282	0.428310
$652$	0	0
$653$	−38.5359	−1.50803	−0.754013	−	0.656859i	$-0.771885\pi$
$653$	−38.5359	−1.50803	−0.754013	+	0.656859i	$0.771885\pi$
$654$	0	0
$655$	−29.3205	−1.14565
$656$	0	0
$657$	−57.7128	−2.25159
$658$	0	0
$659$	−3.32051	−0.129349	−0.0646743	−	0.997906i	$-0.520601\pi$
$659$	−3.32051	−0.129349	−0.0646743	+	0.997906i	$0.520601\pi$
$660$	0	0
$661$	−28.9808	−1.12722	−0.563611	−	0.826041i	$-0.690588\pi$
$661$	−28.9808	−1.12722	−0.563611	+	0.826041i	$0.690588\pi$
$662$	0	0
$663$	36.7846	1.42860
$664$	0	0
$665$	3.46410	0.134332
$666$	0	0
$667$	5.07180	0.196381
$668$	0	0
$669$	18.9282	0.731807
$670$	0	0
$671$	60.7846	2.34656
$672$	0	0
$673$	4.14359	0.159724	0.0798619	−	0.996806i	$-0.474552\pi$
$673$	4.14359	0.159724	0.0798619	+	0.996806i	$0.474552\pi$
$674$	0	0
$675$	−9.85641	−0.379373
$676$	0	0
$677$	14.4449	0.555161	0.277581	−	0.960702i	$-0.410467\pi$
$677$	14.4449	0.555161	0.277581	+	0.960702i	$0.410467\pi$
$678$	0	0
$679$	−8.92820	−0.342633
$680$	0	0
$681$	10.3923	0.398234
$682$	0	0
$683$	30.6410	1.17245	0.586223	−	0.810150i	$-0.300614\pi$
$683$	30.6410	1.17245	0.586223	+	0.810150i	$0.300614\pi$
$684$	0	0
$685$	54.2487	2.07274
$686$	0	0
$687$	−47.1769	−1.79991
$688$	0	0
$689$	80.7846	3.07765
$690$	0	0
$691$	−11.1244	−0.423190	−0.211595	−	0.977357i	$-0.567866\pi$
$691$	−11.1244	−0.423190	−0.211595	+	0.977357i	$0.567866\pi$
$692$	0	0
$693$	−24.3923	−0.926587
$694$	0	0
$695$	25.3205	0.960462
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	16.3923	0.620014
$700$	0	0
$701$	−19.6077	−0.740572	−0.370286	−	0.928918i	$-0.620740\pi$
$701$	−19.6077	−0.740572	−0.370286	+	0.928918i	$0.620740\pi$
$702$	0	0
$703$	1.85641	0.0700157
$704$	0	0
$705$	21.8564	0.823160
$706$	0	0
$707$	−1.66025	−0.0624403
$708$	0	0
$709$	−48.1051	−1.80663	−0.903313	−	0.428982i	$-0.858872\pi$
$709$	−48.1051	−1.80663	−0.903313	+	0.428982i	$0.858872\pi$
$710$	0	0
$711$	−48.7846	−1.82957
$712$	0	0
$713$	13.8564	0.518927
$714$	0	0
$715$	100.497	3.75839
$716$	0	0
$717$	34.2487	1.27904
$718$	0	0
$719$	21.0718	0.785845	0.392923	−	0.919572i	$-0.371464\pi$
$719$	21.0718	0.785845	0.392923	+	0.919572i	$0.371464\pi$
$720$	0	0
$721$	9.85641	0.367072
$722$	0	0
$723$	51.3205	1.90863
$724$	0	0
$725$	3.60770	0.133986
$726$	0	0
$727$	−10.9282	−0.405305	−0.202652	−	0.979251i	$-0.564956\pi$
$727$	−10.9282	−0.405305	−0.202652	+	0.979251i	$0.564956\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	10.9282	0.404194
$732$	0	0
$733$	−13.6603	−0.504553	−0.252276	−	0.967655i	$-0.581179\pi$
$733$	−13.6603	−0.504553	−0.252276	+	0.967655i	$0.581179\pi$
$734$	0	0
$735$	7.46410	0.275318
$736$	0	0
$737$	−43.7128	−1.61018
$738$	0	0
$739$	3.32051	0.122147	0.0610734	−	0.998133i	$-0.480548\pi$
$739$	3.32051	0.122147	0.0610734	+	0.998133i	$0.480548\pi$
$740$	0	0
$741$	23.3205	0.856700
$742$	0	0
$743$	−32.2487	−1.18309	−0.591545	−	0.806272i	$-0.701482\pi$
$743$	−32.2487	−1.18309	−0.591545	+	0.806272i	$0.701482\pi$
$744$	0	0
$745$	2.92820	0.107281
$746$	0	0
$747$	25.2679	0.924506
$748$	0	0
$749$	−5.07180	−0.185319
$750$	0	0
$751$	24.2487	0.884848	0.442424	−	0.896806i	$-0.354119\pi$
$751$	24.2487	0.884848	0.442424	+	0.896806i	$0.354119\pi$
$752$	0	0
$753$	43.1769	1.57345
$754$	0	0
$755$	−6.53590	−0.237866
$756$	0	0
$757$	10.2487	0.372496	0.186248	−	0.982503i	$-0.440367\pi$
$757$	10.2487	0.372496	0.186248	+	0.982503i	$0.440367\pi$
$758$	0	0
$759$	−51.7128	−1.87706
$760$	0	0
$761$	21.7128	0.787089	0.393544	−	0.919306i	$-0.371249\pi$
$761$	21.7128	0.787089	0.393544	+	0.919306i	$0.371249\pi$
$762$	0	0
$763$	12.3923	0.448632
$764$	0	0
$765$	−24.3923	−0.881906
$766$	0	0
$767$	65.0333	2.34822
$768$	0	0
$769$	−10.7846	−0.388903	−0.194451	−	0.980912i	$-0.562293\pi$
$769$	−10.7846	−0.388903	−0.194451	+	0.980912i	$0.562293\pi$
$770$	0	0
$771$	16.3923	0.590354
$772$	0	0
$773$	−20.8756	−0.750845	−0.375422	−	0.926854i	$-0.622502\pi$
$773$	−20.8756	−0.750845	−0.375422	+	0.926854i	$0.622502\pi$
$774$	0	0
$775$	9.85641	0.354053
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	2.53590	0.0908580
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	−5.85641	−0.209291
$784$	0	0
$785$	−33.3205	−1.18926
$786$	0	0
$787$	−20.5885	−0.733899	−0.366950	−	0.930241i	$-0.619598\pi$
$787$	−20.5885	−0.733899	−0.366950	+	0.930241i	$0.619598\pi$
$788$	0	0
$789$	29.8564	1.06292
$790$	0	0
$791$	6.53590	0.232390
$792$	0	0
$793$	−74.8897	−2.65941
$794$	0	0
$795$	−89.5692	−3.17669
$796$	0	0
$797$	12.9808	0.459802	0.229901	−	0.973214i	$-0.426160\pi$
$797$	12.9808	0.459802	0.229901	+	0.973214i	$0.426160\pi$
$798$	0	0
$799$	5.85641	0.207185
$800$	0	0
$801$	−52.9282	−1.87013
$802$	0	0
$803$	−70.6410	−2.49287
$804$	0	0
$805$	9.46410	0.333566
$806$	0	0
$807$	34.3923	1.21067
$808$	0	0
$809$	−23.3205	−0.819905	−0.409953	−	0.912107i	$-0.634455\pi$
$809$	−23.3205	−0.819905	−0.409953	+	0.912107i	$0.634455\pi$
$810$	0	0
$811$	−27.1244	−0.952465	−0.476232	−	0.879319i	$-0.657998\pi$
$811$	−27.1244	−0.952465	−0.476232	+	0.879319i	$0.657998\pi$
$812$	0	0
$813$	−40.7846	−1.43038
$814$	0	0
$815$	−36.7846	−1.28851
$816$	0	0
$817$	6.92820	0.242387
$818$	0	0
$819$	30.0526	1.05012
$820$	0	0
$821$	6.92820	0.241796	0.120898	−	0.992665i	$-0.461423\pi$
$821$	6.92820	0.241796	0.120898	+	0.992665i	$0.461423\pi$
$822$	0	0
$823$	2.92820	0.102071	0.0510354	−	0.998697i	$-0.483748\pi$
$823$	2.92820	0.102071	0.0510354	+	0.998697i	$0.483748\pi$
$824$	0	0
$825$	−36.7846	−1.28068
$826$	0	0
$827$	−13.8564	−0.481834	−0.240917	−	0.970546i	$-0.577448\pi$
$827$	−13.8564	−0.481834	−0.240917	+	0.970546i	$0.577448\pi$
$828$	0	0
$829$	40.9808	1.42332	0.711660	−	0.702524i	$-0.247944\pi$
$829$	40.9808	1.42332	0.711660	+	0.702524i	$0.247944\pi$
$830$	0	0
$831$	−10.9282	−0.379095
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−13.8564	−0.479521
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	39.7128	1.37104	0.685519	−	0.728054i	$-0.259575\pi$
$839$	39.7128	1.37104	0.685519	+	0.728054i	$0.259575\pi$
$840$	0	0
$841$	−26.8564	−0.926083
$842$	0	0
$843$	5.46410	0.188194
$844$	0	0
$845$	−88.3013	−3.03766
$846$	0	0
$847$	−18.8564	−0.647914
$848$	0	0
$849$	−45.3205	−1.55540
$850$	0	0
$851$	5.07180	0.173859
$852$	0	0
$853$	4.98076	0.170538	0.0852690	−	0.996358i	$-0.472825\pi$
$853$	4.98076	0.170538	0.0852690	+	0.996358i	$0.472825\pi$
$854$	0	0
$855$	−15.4641	−0.528861
$856$	0	0
$857$	30.7846	1.05158	0.525791	−	0.850614i	$-0.323769\pi$
$857$	30.7846	1.05158	0.525791	+	0.850614i	$0.323769\pi$
$858$	0	0
$859$	−30.4449	−1.03877	−0.519383	−	0.854542i	$-0.673838\pi$
$859$	−30.4449	−1.03877	−0.519383	+	0.854542i	$0.673838\pi$
$860$	0	0
$861$	5.46410	0.186216
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	10.3923	0.353349
$866$	0	0
$867$	35.5167	1.20621
$868$	0	0
$869$	−59.7128	−2.02562
$870$	0	0
$871$	53.8564	1.82485
$872$	0	0
$873$	39.8564	1.34893
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	20.3923	0.688599	0.344300	−	0.938860i	$-0.388116\pi$
$877$	20.3923	0.688599	0.344300	+	0.938860i	$0.388116\pi$
$878$	0	0
$879$	33.3205	1.12387
$880$	0	0
$881$	−50.7846	−1.71098	−0.855488	−	0.517822i	$-0.826743\pi$
$881$	−50.7846	−1.71098	−0.855488	+	0.517822i	$0.826743\pi$
$882$	0	0
$883$	−5.07180	−0.170680	−0.0853398	−	0.996352i	$-0.527198\pi$
$883$	−5.07180	−0.170680	−0.0853398	+	0.996352i	$0.527198\pi$
$884$	0	0
$885$	−72.1051	−2.42379
$886$	0	0
$887$	46.6410	1.56605	0.783026	−	0.621989i	$-0.213675\pi$
$887$	46.6410	1.56605	0.783026	+	0.621989i	$0.213675\pi$
$888$	0	0
$889$	11.4641	0.384494
$890$	0	0
$891$	−13.4641	−0.451064
$892$	0	0
$893$	3.71281	0.124245
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	63.7128	2.12731
$898$	0	0
$899$	5.85641	0.195322
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	14.9282	0.496779
$904$	0	0
$905$	−29.3205	−0.974647
$906$	0	0
$907$	−43.7128	−1.45146	−0.725730	−	0.687980i	$-0.758498\pi$
$907$	−43.7128	−1.45146	−0.725730	+	0.687980i	$0.758498\pi$
$908$	0	0
$909$	7.41154	0.245825
$910$	0	0
$911$	−4.53590	−0.150281	−0.0751405	−	0.997173i	$-0.523941\pi$
$911$	−4.53590	−0.150281	−0.0751405	+	0.997173i	$0.523941\pi$
$912$	0	0
$913$	30.9282	1.02357
$914$	0	0
$915$	83.0333	2.74500
$916$	0	0
$917$	−10.7321	−0.354404
$918$	0	0
$919$	7.21539	0.238014	0.119007	−	0.992893i	$-0.462029\pi$
$919$	7.21539	0.238014	0.119007	+	0.992893i	$0.462029\pi$
$920$	0	0
$921$	−23.4641	−0.773168
$922$	0	0
$923$	19.7128	0.648855
$924$	0	0
$925$	3.60770	0.118620
$926$	0	0
$927$	−44.0000	−1.44515
$928$	0	0
$929$	37.7128	1.23732	0.618659	−	0.785660i	$-0.287676\pi$
$929$	37.7128	1.23732	0.618659	+	0.785660i	$0.287676\pi$
$930$	0	0
$931$	1.26795	0.0415554
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	−29.8564	−0.976409
$936$	0	0
$937$	−8.64102	−0.282290	−0.141145	−	0.989989i	$-0.545078\pi$
$937$	−8.64102	−0.282290	−0.141145	+	0.989989i	$0.545078\pi$
$938$	0	0
$939$	54.2487	1.77034
$940$	0	0
$941$	−44.9808	−1.46633	−0.733165	−	0.680050i	$-0.761958\pi$
$941$	−44.9808	−1.46633	−0.733165	+	0.680050i	$0.761958\pi$
$942$	0	0
$943$	6.92820	0.225613
$944$	0	0
$945$	−10.9282	−0.355494
$946$	0	0
$947$	24.3923	0.792643	0.396322	−	0.918112i	$-0.370286\pi$
$947$	24.3923	0.792643	0.396322	+	0.918112i	$0.370286\pi$
$948$	0	0
$949$	87.0333	2.82522
$950$	0	0
$951$	78.6410	2.55011
$952$	0	0
$953$	19.8564	0.643212	0.321606	−	0.946874i	$-0.395777\pi$
$953$	19.8564	0.643212	0.321606	+	0.946874i	$0.395777\pi$
$954$	0	0
$955$	−43.7128	−1.41451
$956$	0	0
$957$	−21.8564	−0.706517
$958$	0	0
$959$	19.8564	0.641197
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	22.6410	0.729597
$964$	0	0
$965$	−17.8564	−0.574818
$966$	0	0
$967$	15.1769	0.488057	0.244028	−	0.969768i	$-0.421531\pi$
$967$	15.1769	0.488057	0.244028	+	0.969768i	$0.421531\pi$
$968$	0	0
$969$	−6.92820	−0.222566
$970$	0	0
$971$	−55.5167	−1.78161	−0.890807	−	0.454381i	$-0.849860\pi$
$971$	−55.5167	−1.78161	−0.890807	+	0.454381i	$0.849860\pi$
$972$	0	0
$973$	9.26795	0.297117
$974$	0	0
$975$	45.3205	1.45142
$976$	0	0
$977$	−28.1436	−0.900393	−0.450197	−	0.892929i	$-0.648646\pi$
$977$	−28.1436	−0.900393	−0.450197	+	0.892929i	$0.648646\pi$
$978$	0	0
$979$	−64.7846	−2.07053
$980$	0	0
$981$	−55.3205	−1.76625
$982$	0	0
$983$	47.7128	1.52180	0.760901	−	0.648868i	$-0.224757\pi$
$983$	47.7128	1.52180	0.760901	+	0.648868i	$0.224757\pi$
$984$	0	0
$985$	70.6410	2.25081
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	18.9282	0.601882
$990$	0	0
$991$	24.7846	0.787309	0.393655	−	0.919258i	$-0.371211\pi$
$991$	24.7846	0.787309	0.393655	+	0.919258i	$0.371211\pi$
$992$	0	0
$993$	44.7846	1.42120
$994$	0	0
$995$	−10.9282	−0.346447
$996$	0	0
$997$	5.26795	0.166838	0.0834188	−	0.996515i	$-0.473416\pi$
$997$	5.26795	0.166838	0.0834188	+	0.996515i	$0.473416\pi$
$998$	0	0
$999$	−5.85641	−0.185289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.a.i.1.1		2
4.3	odd	2		1792.2.a.q.1.2		2
8.3	odd	2		1792.2.a.k.1.1		2
8.5	even	2		1792.2.a.s.1.2		2
16.3	odd	4		448.2.b.c.225.4	yes	4
16.5	even	4		448.2.b.d.225.4	yes	4
16.11	odd	4		448.2.b.c.225.1	✓	4
16.13	even	4		448.2.b.d.225.1	yes	4
48.5	odd	4		4032.2.c.n.2017.4		4
48.11	even	4		4032.2.c.k.2017.4		4
48.29	odd	4		4032.2.c.n.2017.1		4
48.35	even	4		4032.2.c.k.2017.1		4
112.13	odd	4		3136.2.b.h.1569.4		4
112.27	even	4		3136.2.b.g.1569.4		4
112.69	odd	4		3136.2.b.h.1569.1		4
112.83	even	4		3136.2.b.g.1569.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.b.c.225.1	✓	4	16.11	odd	4
448.2.b.c.225.4	yes	4	16.3	odd	4
448.2.b.d.225.1	yes	4	16.13	even	4
448.2.b.d.225.4	yes	4	16.5	even	4
1792.2.a.i.1.1		2	1.1	even	1	trivial
1792.2.a.k.1.1		2	8.3	odd	2
1792.2.a.q.1.2		2	4.3	odd	2
1792.2.a.s.1.2		2	8.5	even	2
3136.2.b.g.1569.1		4	112.83	even	4
3136.2.b.g.1569.4		4	112.27	even	4
3136.2.b.h.1569.1		4	112.69	odd	4
3136.2.b.h.1569.4		4	112.13	odd	4
4032.2.c.k.2017.1		4	48.35	even	4
4032.2.c.k.2017.4		4	48.11	even	4
4032.2.c.n.2017.1		4	48.29	odd	4
4032.2.c.n.2017.4		4	48.5	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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -2.73205 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -2.73205 q^{5} -1.00000 q^{7} +4.46410 q^{9} +5.46410 q^{11} -6.73205 q^{13} +7.46410 q^{15} +2.00000 q^{17} +1.26795 q^{19} +2.73205 q^{21} +3.46410 q^{23} +2.46410 q^{25} -4.00000 q^{27} +1.46410 q^{29} +4.00000 q^{31} -14.9282 q^{33} +2.73205 q^{35} +1.46410 q^{37} +18.3923 q^{39} +2.00000 q^{41} +5.46410 q^{43} -12.1962 q^{45} +2.92820 q^{47} +1.00000 q^{49} -5.46410 q^{51} -12.0000 q^{53} -14.9282 q^{55} -3.46410 q^{57} -9.66025 q^{59} +11.1244 q^{61} -4.46410 q^{63} +18.3923 q^{65} -8.00000 q^{67} -9.46410 q^{69} -2.92820 q^{71} -12.9282 q^{73} -6.73205 q^{75} -5.46410 q^{77} -10.9282 q^{79} -2.46410 q^{81} +5.66025 q^{83} -5.46410 q^{85} -4.00000 q^{87} -11.8564 q^{89} +6.73205 q^{91} -10.9282 q^{93} -3.46410 q^{95} +8.92820 q^{97} +24.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 10 q^{13} + 8 q^{15} + 4 q^{17} + 6 q^{19} + 2 q^{21} - 2 q^{25} - 8 q^{27} - 4 q^{29} + 8 q^{31} - 16 q^{33} + 2 q^{35} - 4 q^{37} + 16 q^{39} + 4 q^{41} + 4 q^{43} - 14 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 24 q^{53} - 16 q^{55} - 2 q^{59} - 2 q^{61} - 2 q^{63} + 16 q^{65} - 16 q^{67} - 12 q^{69} + 8 q^{71} - 12 q^{73} - 10 q^{75} - 4 q^{77} - 8 q^{79} + 2 q^{81} - 6 q^{83} - 4 q^{85} - 8 q^{87} + 4 q^{89} + 10 q^{91} - 8 q^{93} + 4 q^{97} + 28 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 10 * q^13 + 8 * q^15 + 4 * q^17 + 6 * q^19 + 2 * q^21 - 2 * q^25 - 8 * q^27 - 4 * q^29 + 8 * q^31 - 16 * q^33 + 2 * q^35 - 4 * q^37 + 16 * q^39 + 4 * q^41 + 4 * q^43 - 14 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 - 24 * q^53 - 16 * q^55 - 2 * q^59 - 2 * q^61 - 2 * q^63 + 16 * q^65 - 16 * q^67 - 12 * q^69 + 8 * q^71 - 12 * q^73 - 10 * q^75 - 4 * q^77 - 8 * q^79 + 2 * q^81 - 6 * q^83 - 4 * q^85 - 8 * q^87 + 4 * q^89 + 10 * q^91 - 8 * q^93 + 4 * q^97 + 28 * q^99

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