LMFDB - Embedded newform 3192.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.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:	$25.4882483252$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.46410 q^{13} +0.732051 q^{15} -3.26795 q^{17} +1.00000 q^{19} +1.00000 q^{21} -6.92820 q^{23} -4.46410 q^{25} +1.00000 q^{27} -3.26795 q^{29} -2.00000 q^{31} -4.00000 q^{33} +0.732051 q^{35} -4.92820 q^{37} -1.46410 q^{39} -8.92820 q^{41} +4.92820 q^{43} +0.732051 q^{45} +0.196152 q^{47} +1.00000 q^{49} -3.26795 q^{51} +7.66025 q^{53} -2.92820 q^{55} +1.00000 q^{57} -10.9282 q^{59} +0.928203 q^{61} +1.00000 q^{63} -1.07180 q^{65} +5.46410 q^{67} -6.92820 q^{69} +4.19615 q^{71} -10.3923 q^{73} -4.46410 q^{75} -4.00000 q^{77} +2.92820 q^{79} +1.00000 q^{81} +8.19615 q^{83} -2.39230 q^{85} -3.26795 q^{87} -6.00000 q^{89} -1.46410 q^{91} -2.00000 q^{93} +0.732051 q^{95} +11.8564 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{11} + 4 q^{13} - 2 q^{15} - 10 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 4 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 10 q^{51} - 2 q^{53} + 8 q^{55} + 2 q^{57} - 8 q^{59} - 12 q^{61} + 2 q^{63} - 16 q^{65} + 4 q^{67} - 2 q^{71} - 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} + 6 q^{83} + 16 q^{85} - 10 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} - 2 q^{95} - 4 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 4 * q^13 - 2 * q^15 - 10 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 + 4 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 - 2 * q^45 - 10 * q^47 + 2 * q^49 - 10 * q^51 - 2 * q^53 + 8 * q^55 + 2 * q^57 - 8 * q^59 - 12 * q^61 + 2 * q^63 - 16 * q^65 + 4 * q^67 - 2 * q^71 - 2 * q^75 - 8 * q^77 - 8 * q^79 + 2 * q^81 + 6 * q^83 + 16 * q^85 - 10 * q^87 - 12 * q^89 + 4 * q^91 - 4 * q^93 - 2 * q^95 - 4 * q^97 - 8 * 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
$3$	1.00000	0.577350
$4$	0	0
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	−3.26795	−0.792594	−0.396297	−	0.918122i	$-0.629705\pi$
$17$	−3.26795	−0.792594	−0.396297	+	0.918122i	$0.629705\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.26795	−0.606843	−0.303421	−	0.952856i	$-0.598129\pi$
$29$	−3.26795	−0.606843	−0.303421	+	0.952856i	$0.598129\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0.732051	0.123739
$36$	0	0
$37$	−4.92820	−0.810192	−0.405096	−	0.914274i	$-0.632762\pi$
$37$	−4.92820	−0.810192	−0.405096	+	0.914274i	$0.632762\pi$
$38$	0	0
$39$	−1.46410	−0.234444
$40$	0	0
$41$	−8.92820	−1.39435	−0.697176	−	0.716900i	$-0.745560\pi$
$41$	−8.92820	−1.39435	−0.697176	+	0.716900i	$0.745560\pi$
$42$	0	0
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	0.732051	0.109128
$46$	0	0
$47$	0.196152	0.0286118	0.0143059	−	0.999898i	$-0.495446\pi$
$47$	0.196152	0.0286118	0.0143059	+	0.999898i	$0.495446\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.26795	−0.457604
$52$	0	0
$53$	7.66025	1.05222	0.526108	−	0.850418i	$-0.323651\pi$
$53$	7.66025	1.05222	0.526108	+	0.850418i	$0.323651\pi$
$54$	0	0
$55$	−2.92820	−0.394839
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−10.9282	−1.42273	−0.711365	−	0.702822i	$-0.751923\pi$
$59$	−10.9282	−1.42273	−0.711365	+	0.702822i	$0.751923\pi$
$60$	0	0
$61$	0.928203	0.118844	0.0594221	−	0.998233i	$-0.481074\pi$
$61$	0.928203	0.118844	0.0594221	+	0.998233i	$0.481074\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.07180	−0.132940
$66$	0	0
$67$	5.46410	0.667546	0.333773	−	0.942653i	$-0.391678\pi$
$67$	5.46410	0.667546	0.333773	+	0.942653i	$0.391678\pi$
$68$	0	0
$69$	−6.92820	−0.834058
$70$	0	0
$71$	4.19615	0.497992	0.248996	−	0.968505i	$-0.419899\pi$
$71$	4.19615	0.497992	0.248996	+	0.968505i	$0.419899\pi$
$72$	0	0
$73$	−10.3923	−1.21633	−0.608164	−	0.793812i	$-0.708094\pi$
$73$	−10.3923	−1.21633	−0.608164	+	0.793812i	$0.708094\pi$
$74$	0	0
$75$	−4.46410	−0.515470
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	2.92820	0.329449	0.164724	−	0.986340i	$-0.447327\pi$
$79$	2.92820	0.329449	0.164724	+	0.986340i	$0.447327\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.19615	0.899645	0.449822	−	0.893118i	$-0.351487\pi$
$83$	8.19615	0.899645	0.449822	+	0.893118i	$0.351487\pi$
$84$	0	0
$85$	−2.39230	−0.259482
$86$	0	0
$87$	−3.26795	−0.350361
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−1.46410	−0.153480
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	0.732051	0.0751068
$96$	0	0
$97$	11.8564	1.20384	0.601918	−	0.798558i	$-0.294403\pi$
$97$	11.8564	1.20384	0.601918	+	0.798558i	$0.294403\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−4.73205	−0.470857	−0.235428	−	0.971892i	$-0.575649\pi$
$101$	−4.73205	−0.470857	−0.235428	+	0.971892i	$0.575649\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	−12.1962	−1.17905	−0.589523	−	0.807751i	$-0.700684\pi$
$107$	−12.1962	−1.17905	−0.589523	+	0.807751i	$0.700684\pi$
$108$	0	0
$109$	−4.53590	−0.434460	−0.217230	−	0.976120i	$-0.569702\pi$
$109$	−4.53590	−0.434460	−0.217230	+	0.976120i	$0.569702\pi$
$110$	0	0
$111$	−4.92820	−0.467764
$112$	0	0
$113$	16.7321	1.57402	0.787009	−	0.616941i	$-0.211628\pi$
$113$	16.7321	1.57402	0.787009	+	0.616941i	$0.211628\pi$
$114$	0	0
$115$	−5.07180	−0.472947
$116$	0	0
$117$	−1.46410	−0.135356
$118$	0	0
$119$	−3.26795	−0.299572
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−8.92820	−0.805029
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−6.53590	−0.579967	−0.289984	−	0.957032i	$-0.593650\pi$
$127$	−6.53590	−0.579967	−0.289984	+	0.957032i	$0.593650\pi$
$128$	0	0
$129$	4.92820	0.433904
$130$	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
$133$	1.00000	0.0867110
$134$	0	0
$135$	0.732051	0.0630049
$136$	0	0
$137$	4.92820	0.421045	0.210522	−	0.977589i	$-0.432484\pi$
$137$	4.92820	0.421045	0.210522	+	0.977589i	$0.432484\pi$
$138$	0	0
$139$	19.3205	1.63874	0.819372	−	0.573262i	$-0.194322\pi$
$139$	19.3205	1.63874	0.819372	+	0.573262i	$0.194322\pi$
$140$	0	0
$141$	0.196152	0.0165190
$142$	0	0
$143$	5.85641	0.489737
$144$	0	0
$145$	−2.39230	−0.198670
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−3.46410	−0.283790	−0.141895	−	0.989882i	$-0.545320\pi$
$149$	−3.46410	−0.283790	−0.141895	+	0.989882i	$0.545320\pi$
$150$	0	0
$151$	13.4641	1.09569	0.547847	−	0.836579i	$-0.315448\pi$
$151$	13.4641	1.09569	0.547847	+	0.836579i	$0.315448\pi$
$152$	0	0
$153$	−3.26795	−0.264198
$154$	0	0
$155$	−1.46410	−0.117599
$156$	0	0
$157$	−3.07180	−0.245156	−0.122578	−	0.992459i	$-0.539116\pi$
$157$	−3.07180	−0.245156	−0.122578	+	0.992459i	$0.539116\pi$
$158$	0	0
$159$	7.66025	0.607498
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	−12.9282	−1.01262	−0.506308	−	0.862353i	$-0.668990\pi$
$163$	−12.9282	−1.01262	−0.506308	+	0.862353i	$0.668990\pi$
$164$	0	0
$165$	−2.92820	−0.227960
$166$	0	0
$167$	9.46410	0.732354	0.366177	−	0.930545i	$-0.380666\pi$
$167$	9.46410	0.732354	0.366177	+	0.930545i	$0.380666\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−11.8564	−0.901426	−0.450713	−	0.892669i	$-0.648830\pi$
$173$	−11.8564	−0.901426	−0.450713	+	0.892669i	$0.648830\pi$
$174$	0	0
$175$	−4.46410	−0.337454
$176$	0	0
$177$	−10.9282	−0.821414
$178$	0	0
$179$	−14.7321	−1.10113	−0.550563	−	0.834794i	$-0.685587\pi$
$179$	−14.7321	−1.10113	−0.550563	+	0.834794i	$0.685587\pi$
$180$	0	0
$181$	−7.07180	−0.525643	−0.262821	−	0.964845i	$-0.584653\pi$
$181$	−7.07180	−0.525643	−0.262821	+	0.964845i	$0.584653\pi$
$182$	0	0
$183$	0.928203	0.0686148
$184$	0	0
$185$	−3.60770	−0.265243
$186$	0	0
$187$	13.0718	0.955904
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	0	0
$193$	8.92820	0.642666	0.321333	−	0.946966i	$-0.395869\pi$
$193$	8.92820	0.642666	0.321333	+	0.946966i	$0.395869\pi$
$194$	0	0
$195$	−1.07180	−0.0767530
$196$	0	0
$197$	0.928203	0.0661317	0.0330659	−	0.999453i	$-0.489473\pi$
$197$	0.928203	0.0661317	0.0330659	+	0.999453i	$0.489473\pi$
$198$	0	0
$199$	19.3205	1.36959	0.684797	−	0.728734i	$-0.259891\pi$
$199$	19.3205	1.36959	0.684797	+	0.728734i	$0.259891\pi$
$200$	0	0
$201$	5.46410	0.385408
$202$	0	0
$203$	−3.26795	−0.229365
$204$	0	0
$205$	−6.53590	−0.456487
$206$	0	0
$207$	−6.92820	−0.481543
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	4.19615	0.287516
$214$	0	0
$215$	3.60770	0.246043
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	−10.3923	−0.702247
$220$	0	0
$221$	4.78461	0.321848
$222$	0	0
$223$	−22.7846	−1.52577	−0.762885	−	0.646534i	$-0.776218\pi$
$223$	−22.7846	−1.52577	−0.762885	+	0.646534i	$0.776218\pi$
$224$	0	0
$225$	−4.46410	−0.297607
$226$	0	0
$227$	−7.32051	−0.485879	−0.242940	−	0.970041i	$-0.578112\pi$
$227$	−7.32051	−0.485879	−0.242940	+	0.970041i	$0.578112\pi$
$228$	0	0
$229$	2.39230	0.158088	0.0790440	−	0.996871i	$-0.474813\pi$
$229$	2.39230	0.158088	0.0790440	+	0.996871i	$0.474813\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−17.3205	−1.13470	−0.567352	−	0.823475i	$-0.692032\pi$
$233$	−17.3205	−1.13470	−0.567352	+	0.823475i	$0.692032\pi$
$234$	0	0
$235$	0.143594	0.00936701
$236$	0	0
$237$	2.92820	0.190207
$238$	0	0
$239$	−9.46410	−0.612182	−0.306091	−	0.952002i	$-0.599021\pi$
$239$	−9.46410	−0.612182	−0.306091	+	0.952002i	$0.599021\pi$
$240$	0	0
$241$	10.7846	0.694698	0.347349	−	0.937736i	$-0.387082\pi$
$241$	10.7846	0.694698	0.347349	+	0.937736i	$0.387082\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.732051	0.0467690
$246$	0	0
$247$	−1.46410	−0.0931586
$248$	0	0
$249$	8.19615	0.519410
$250$	0	0
$251$	−1.66025	−0.104794	−0.0523972	−	0.998626i	$-0.516686\pi$
$251$	−1.66025	−0.104794	−0.0523972	+	0.998626i	$0.516686\pi$
$252$	0	0
$253$	27.7128	1.74229
$254$	0	0
$255$	−2.39230	−0.149812
$256$	0	0
$257$	2.39230	0.149228	0.0746139	−	0.997212i	$-0.476228\pi$
$257$	2.39230	0.149228	0.0746139	+	0.997212i	$0.476228\pi$
$258$	0	0
$259$	−4.92820	−0.306224
$260$	0	0
$261$	−3.26795	−0.202281
$262$	0	0
$263$	2.53590	0.156370	0.0781851	−	0.996939i	$-0.475087\pi$
$263$	2.53590	0.156370	0.0781851	+	0.996939i	$0.475087\pi$
$264$	0	0
$265$	5.60770	0.344478
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−27.4641	−1.67452	−0.837258	−	0.546808i	$-0.815843\pi$
$269$	−27.4641	−1.67452	−0.837258	+	0.546808i	$0.815843\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−1.46410	−0.0886115
$274$	0	0
$275$	17.8564	1.07678
$276$	0	0
$277$	−24.3923	−1.46559	−0.732796	−	0.680449i	$-0.761785\pi$
$277$	−24.3923	−1.46559	−0.732796	+	0.680449i	$0.761785\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−4.73205	−0.282290	−0.141145	−	0.989989i	$-0.545078\pi$
$281$	−4.73205	−0.282290	−0.141145	+	0.989989i	$0.545078\pi$
$282$	0	0
$283$	−8.39230	−0.498871	−0.249435	−	0.968391i	$-0.580245\pi$
$283$	−8.39230	−0.498871	−0.249435	+	0.968391i	$0.580245\pi$
$284$	0	0
$285$	0.732051	0.0433629
$286$	0	0
$287$	−8.92820	−0.527015
$288$	0	0
$289$	−6.32051	−0.371795
$290$	0	0
$291$	11.8564	0.695035
$292$	0	0
$293$	−27.8564	−1.62739	−0.813694	−	0.581293i	$-0.802547\pi$
$293$	−27.8564	−1.62739	−0.813694	+	0.581293i	$0.802547\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	10.1436	0.586619
$300$	0	0
$301$	4.92820	0.284057
$302$	0	0
$303$	−4.73205	−0.271849
$304$	0	0
$305$	0.679492	0.0389076
$306$	0	0
$307$	16.9282	0.966144	0.483072	−	0.875581i	$-0.339521\pi$
$307$	16.9282	0.966144	0.483072	+	0.875581i	$0.339521\pi$
$308$	0	0
$309$	−2.92820	−0.166580
$310$	0	0
$311$	−7.12436	−0.403985	−0.201993	−	0.979387i	$-0.564742\pi$
$311$	−7.12436	−0.403985	−0.201993	+	0.979387i	$0.564742\pi$
$312$	0	0
$313$	19.0718	1.07800	0.539001	−	0.842305i	$-0.318802\pi$
$313$	19.0718	1.07800	0.539001	+	0.842305i	$0.318802\pi$
$314$	0	0
$315$	0.732051	0.0412464
$316$	0	0
$317$	−19.2679	−1.08220	−0.541098	−	0.840960i	$-0.681991\pi$
$317$	−19.2679	−1.08220	−0.541098	+	0.840960i	$0.681991\pi$
$318$	0	0
$319$	13.0718	0.731880
$320$	0	0
$321$	−12.1962	−0.680723
$322$	0	0
$323$	−3.26795	−0.181834
$324$	0	0
$325$	6.53590	0.362546
$326$	0	0
$327$	−4.53590	−0.250836
$328$	0	0
$329$	0.196152	0.0108142
$330$	0	0
$331$	8.39230	0.461283	0.230641	−	0.973039i	$-0.425918\pi$
$331$	8.39230	0.461283	0.230641	+	0.973039i	$0.425918\pi$
$332$	0	0
$333$	−4.92820	−0.270064
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	1.60770	0.0875767	0.0437884	−	0.999041i	$-0.486057\pi$
$337$	1.60770	0.0875767	0.0437884	+	0.999041i	$0.486057\pi$
$338$	0	0
$339$	16.7321	0.908760
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.07180	−0.273056
$346$	0	0
$347$	19.3205	1.03718	0.518590	−	0.855023i	$-0.326457\pi$
$347$	19.3205	1.03718	0.518590	+	0.855023i	$0.326457\pi$
$348$	0	0
$349$	−22.7846	−1.21963	−0.609816	−	0.792543i	$-0.708757\pi$
$349$	−22.7846	−1.21963	−0.609816	+	0.792543i	$0.708757\pi$
$350$	0	0
$351$	−1.46410	−0.0781480
$352$	0	0
$353$	−25.5167	−1.35811	−0.679057	−	0.734085i	$-0.737611\pi$
$353$	−25.5167	−1.35811	−0.679057	+	0.734085i	$0.737611\pi$
$354$	0	0
$355$	3.07180	0.163034
$356$	0	0
$357$	−3.26795	−0.172958
$358$	0	0
$359$	29.8564	1.57576	0.787880	−	0.615828i	$-0.211178\pi$
$359$	29.8564	1.57576	0.787880	+	0.615828i	$0.211178\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−7.60770	−0.398205
$366$	0	0
$367$	−15.3205	−0.799724	−0.399862	−	0.916575i	$-0.630942\pi$
$367$	−15.3205	−0.799724	−0.399862	+	0.916575i	$0.630942\pi$
$368$	0	0
$369$	−8.92820	−0.464784
$370$	0	0
$371$	7.66025	0.397701
$372$	0	0
$373$	30.7846	1.59397	0.796983	−	0.604001i	$-0.206428\pi$
$373$	30.7846	1.59397	0.796983	+	0.604001i	$0.206428\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	4.78461	0.246420
$378$	0	0
$379$	20.3923	1.04748	0.523741	−	0.851877i	$-0.324536\pi$
$379$	20.3923	1.04748	0.523741	+	0.851877i	$0.324536\pi$
$380$	0	0
$381$	−6.53590	−0.334844
$382$	0	0
$383$	0.679492	0.0347204	0.0173602	−	0.999849i	$-0.494474\pi$
$383$	0.679492	0.0347204	0.0173602	+	0.999849i	$0.494474\pi$
$384$	0	0
$385$	−2.92820	−0.149235
$386$	0	0
$387$	4.92820	0.250515
$388$	0	0
$389$	−13.6077	−0.689938	−0.344969	−	0.938614i	$-0.612110\pi$
$389$	−13.6077	−0.689938	−0.344969	+	0.938614i	$0.612110\pi$
$390$	0	0
$391$	22.6410	1.14501
$392$	0	0
$393$	10.7321	0.541360
$394$	0	0
$395$	2.14359	0.107856
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	6.58846	0.329012	0.164506	−	0.986376i	$-0.447397\pi$
$401$	6.58846	0.329012	0.164506	+	0.986376i	$0.447397\pi$
$402$	0	0
$403$	2.92820	0.145864
$404$	0	0
$405$	0.732051	0.0363759
$406$	0	0
$407$	19.7128	0.977128
$408$	0	0
$409$	−2.53590	−0.125392	−0.0626961	−	0.998033i	$-0.519970\pi$
$409$	−2.53590	−0.125392	−0.0626961	+	0.998033i	$0.519970\pi$
$410$	0	0
$411$	4.92820	0.243090
$412$	0	0
$413$	−10.9282	−0.537742
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	19.3205	0.946129
$418$	0	0
$419$	8.87564	0.433604	0.216802	−	0.976216i	$-0.430437\pi$
$419$	8.87564	0.433604	0.216802	+	0.976216i	$0.430437\pi$
$420$	0	0
$421$	−16.5359	−0.805910	−0.402955	−	0.915220i	$-0.632017\pi$
$421$	−16.5359	−0.805910	−0.402955	+	0.915220i	$0.632017\pi$
$422$	0	0
$423$	0.196152	0.00953726
$424$	0	0
$425$	14.5885	0.707644
$426$	0	0
$427$	0.928203	0.0449189
$428$	0	0
$429$	5.85641	0.282750
$430$	0	0
$431$	−40.9808	−1.97397	−0.986987	−	0.160801i	$-0.948592\pi$
$431$	−40.9808	−1.97397	−0.986987	+	0.160801i	$0.948592\pi$
$432$	0	0
$433$	23.8564	1.14647	0.573233	−	0.819393i	$-0.305689\pi$
$433$	23.8564	1.14647	0.573233	+	0.819393i	$0.305689\pi$
$434$	0	0
$435$	−2.39230	−0.114702
$436$	0	0
$437$	−6.92820	−0.331421
$438$	0	0
$439$	−3.85641	−0.184056	−0.0920281	−	0.995756i	$-0.529335\pi$
$439$	−3.85641	−0.184056	−0.0920281	+	0.995756i	$0.529335\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	16.3923	0.778822	0.389411	−	0.921064i	$-0.372679\pi$
$443$	16.3923	0.778822	0.389411	+	0.921064i	$0.372679\pi$
$444$	0	0
$445$	−4.39230	−0.208215
$446$	0	0
$447$	−3.46410	−0.163846
$448$	0	0
$449$	19.6603	0.927825	0.463912	−	0.885881i	$-0.346445\pi$
$449$	19.6603	0.927825	0.463912	+	0.885881i	$0.346445\pi$
$450$	0	0
$451$	35.7128	1.68165
$452$	0	0
$453$	13.4641	0.632599
$454$	0	0
$455$	−1.07180	−0.0502466
$456$	0	0
$457$	−11.3205	−0.529551	−0.264776	−	0.964310i	$-0.585298\pi$
$457$	−11.3205	−0.529551	−0.264776	+	0.964310i	$0.585298\pi$
$458$	0	0
$459$	−3.26795	−0.152535
$460$	0	0
$461$	−9.80385	−0.456611	−0.228305	−	0.973590i	$-0.573318\pi$
$461$	−9.80385	−0.456611	−0.228305	+	0.973590i	$0.573318\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−1.46410	−0.0678961
$466$	0	0
$467$	−8.19615	−0.379273	−0.189636	−	0.981854i	$-0.560731\pi$
$467$	−8.19615	−0.379273	−0.189636	+	0.981854i	$0.560731\pi$
$468$	0	0
$469$	5.46410	0.252309
$470$	0	0
$471$	−3.07180	−0.141541
$472$	0	0
$473$	−19.7128	−0.906396
$474$	0	0
$475$	−4.46410	−0.204827
$476$	0	0
$477$	7.66025	0.350739
$478$	0	0
$479$	19.5167	0.891739	0.445869	−	0.895098i	$-0.352895\pi$
$479$	19.5167	0.891739	0.445869	+	0.895098i	$0.352895\pi$
$480$	0	0
$481$	7.21539	0.328993
$482$	0	0
$483$	−6.92820	−0.315244
$484$	0	0
$485$	8.67949	0.394115
$486$	0	0
$487$	5.85641	0.265379	0.132690	−	0.991158i	$-0.457639\pi$
$487$	5.85641	0.265379	0.132690	+	0.991158i	$0.457639\pi$
$488$	0	0
$489$	−12.9282	−0.584634
$490$	0	0
$491$	−6.24871	−0.282000	−0.141000	−	0.990010i	$-0.545032\pi$
$491$	−6.24871	−0.282000	−0.141000	+	0.990010i	$0.545032\pi$
$492$	0	0
$493$	10.6795	0.480980
$494$	0	0
$495$	−2.92820	−0.131613
$496$	0	0
$497$	4.19615	0.188223
$498$	0	0
$499$	−1.85641	−0.0831042	−0.0415521	−	0.999136i	$-0.513230\pi$
$499$	−1.85641	−0.0831042	−0.0415521	+	0.999136i	$0.513230\pi$
$500$	0	0
$501$	9.46410	0.422825
$502$	0	0
$503$	−25.2679	−1.12664	−0.563321	−	0.826238i	$-0.690477\pi$
$503$	−25.2679	−1.12664	−0.563321	+	0.826238i	$0.690477\pi$
$504$	0	0
$505$	−3.46410	−0.154150
$506$	0	0
$507$	−10.8564	−0.482150
$508$	0	0
$509$	5.32051	0.235827	0.117914	−	0.993024i	$-0.462379\pi$
$509$	5.32051	0.235827	0.117914	+	0.993024i	$0.462379\pi$
$510$	0	0
$511$	−10.3923	−0.459728
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−2.14359	−0.0944580
$516$	0	0
$517$	−0.784610	−0.0345071
$518$	0	0
$519$	−11.8564	−0.520438
$520$	0	0
$521$	−36.2487	−1.58808	−0.794042	−	0.607862i	$-0.792027\pi$
$521$	−36.2487	−1.58808	−0.794042	+	0.607862i	$0.792027\pi$
$522$	0	0
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	0	0
$525$	−4.46410	−0.194829
$526$	0	0
$527$	6.53590	0.284708
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	−10.9282	−0.474244
$532$	0	0
$533$	13.0718	0.566202
$534$	0	0
$535$	−8.92820	−0.386000
$536$	0	0
$537$	−14.7321	−0.635735
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	−7.07180	−0.303480
$544$	0	0
$545$	−3.32051	−0.142235
$546$	0	0
$547$	30.9282	1.32239	0.661197	−	0.750212i	$-0.270049\pi$
$547$	30.9282	1.32239	0.661197	+	0.750212i	$0.270049\pi$
$548$	0	0
$549$	0.928203	0.0396147
$550$	0	0
$551$	−3.26795	−0.139219
$552$	0	0
$553$	2.92820	0.124520
$554$	0	0
$555$	−3.60770	−0.153138
$556$	0	0
$557$	−38.1051	−1.61457	−0.807283	−	0.590165i	$-0.799063\pi$
$557$	−38.1051	−1.61457	−0.807283	+	0.590165i	$0.799063\pi$
$558$	0	0
$559$	−7.21539	−0.305178
$560$	0	0
$561$	13.0718	0.551892
$562$	0	0
$563$	35.3205	1.48858	0.744291	−	0.667855i	$-0.232788\pi$
$563$	35.3205	1.48858	0.744291	+	0.667855i	$0.232788\pi$
$564$	0	0
$565$	12.2487	0.515307
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−29.5167	−1.23740	−0.618701	−	0.785626i	$-0.712341\pi$
$569$	−29.5167	−1.23740	−0.618701	+	0.785626i	$0.712341\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	18.9282	0.790737
$574$	0	0
$575$	30.9282	1.28980
$576$	0	0
$577$	14.7846	0.615491	0.307746	−	0.951469i	$-0.400425\pi$
$577$	14.7846	0.615491	0.307746	+	0.951469i	$0.400425\pi$
$578$	0	0
$579$	8.92820	0.371043
$580$	0	0
$581$	8.19615	0.340034
$582$	0	0
$583$	−30.6410	−1.26902
$584$	0	0
$585$	−1.07180	−0.0443133
$586$	0	0
$587$	20.1962	0.833584	0.416792	−	0.909002i	$-0.363154\pi$
$587$	20.1962	0.833584	0.416792	+	0.909002i	$0.363154\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0.928203	0.0381812
$592$	0	0
$593$	9.80385	0.402596	0.201298	−	0.979530i	$-0.435484\pi$
$593$	9.80385	0.402596	0.201298	+	0.979530i	$0.435484\pi$
$594$	0	0
$595$	−2.39230	−0.0980749
$596$	0	0
$597$	19.3205	0.790736
$598$	0	0
$599$	−7.80385	−0.318857	−0.159428	−	0.987210i	$-0.550965\pi$
$599$	−7.80385	−0.318857	−0.159428	+	0.987210i	$0.550965\pi$
$600$	0	0
$601$	16.1436	0.658511	0.329255	−	0.944241i	$-0.393202\pi$
$601$	16.1436	0.658511	0.329255	+	0.944241i	$0.393202\pi$
$602$	0	0
$603$	5.46410	0.222515
$604$	0	0
$605$	3.66025	0.148810
$606$	0	0
$607$	16.7846	0.681266	0.340633	−	0.940196i	$-0.389359\pi$
$607$	16.7846	0.681266	0.340633	+	0.940196i	$0.389359\pi$
$608$	0	0
$609$	−3.26795	−0.132424
$610$	0	0
$611$	−0.287187	−0.0116183
$612$	0	0
$613$	−23.6077	−0.953506	−0.476753	−	0.879037i	$-0.658186\pi$
$613$	−23.6077	−0.953506	−0.476753	+	0.879037i	$0.658186\pi$
$614$	0	0
$615$	−6.53590	−0.263553
$616$	0	0
$617$	22.7846	0.917274	0.458637	−	0.888624i	$-0.348338\pi$
$617$	22.7846	0.917274	0.458637	+	0.888624i	$0.348338\pi$
$618$	0	0
$619$	−18.5359	−0.745021	−0.372510	−	0.928028i	$-0.621503\pi$
$619$	−18.5359	−0.745021	−0.372510	+	0.928028i	$0.621503\pi$
$620$	0	0
$621$	−6.92820	−0.278019
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	16.1051	0.642153
$630$	0	0
$631$	−34.7846	−1.38475	−0.692377	−	0.721536i	$-0.743436\pi$
$631$	−34.7846	−1.38475	−0.692377	+	0.721536i	$0.743436\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	−4.78461	−0.189871
$636$	0	0
$637$	−1.46410	−0.0580098
$638$	0	0
$639$	4.19615	0.165997
$640$	0	0
$641$	3.26795	0.129076	0.0645381	−	0.997915i	$-0.479443\pi$
$641$	3.26795	0.129076	0.0645381	+	0.997915i	$0.479443\pi$
$642$	0	0
$643$	0.392305	0.0154710	0.00773550	−	0.999970i	$-0.497538\pi$
$643$	0.392305	0.0154710	0.00773550	+	0.999970i	$0.497538\pi$
$644$	0	0
$645$	3.60770	0.142053
$646$	0	0
$647$	10.3397	0.406497	0.203249	−	0.979127i	$-0.434850\pi$
$647$	10.3397	0.406497	0.203249	+	0.979127i	$0.434850\pi$
$648$	0	0
$649$	43.7128	1.71588
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	−45.7128	−1.78888	−0.894440	−	0.447187i	$-0.852426\pi$
$653$	−45.7128	−1.78888	−0.894440	+	0.447187i	$0.852426\pi$
$654$	0	0
$655$	7.85641	0.306975
$656$	0	0
$657$	−10.3923	−0.405442
$658$	0	0
$659$	−11.1244	−0.433343	−0.216672	−	0.976245i	$-0.569520\pi$
$659$	−11.1244	−0.433343	−0.216672	+	0.976245i	$0.569520\pi$
$660$	0	0
$661$	5.46410	0.212529	0.106264	−	0.994338i	$-0.466111\pi$
$661$	5.46410	0.212529	0.106264	+	0.994338i	$0.466111\pi$
$662$	0	0
$663$	4.78461	0.185819
$664$	0	0
$665$	0.732051	0.0283877
$666$	0	0
$667$	22.6410	0.876664
$668$	0	0
$669$	−22.7846	−0.880904
$670$	0	0
$671$	−3.71281	−0.143332
$672$	0	0
$673$	43.8564	1.69054	0.845270	−	0.534339i	$-0.179440\pi$
$673$	43.8564	1.69054	0.845270	+	0.534339i	$0.179440\pi$
$674$	0	0
$675$	−4.46410	−0.171823
$676$	0	0
$677$	0.535898	0.0205962	0.0102981	−	0.999947i	$-0.496722\pi$
$677$	0.535898	0.0205962	0.0102981	+	0.999947i	$0.496722\pi$
$678$	0	0
$679$	11.8564	0.455007
$680$	0	0
$681$	−7.32051	−0.280522
$682$	0	0
$683$	−27.8038	−1.06388	−0.531942	−	0.846781i	$-0.678538\pi$
$683$	−27.8038	−1.06388	−0.531942	+	0.846781i	$0.678538\pi$
$684$	0	0
$685$	3.60770	0.137843
$686$	0	0
$687$	2.39230	0.0912721
$688$	0	0
$689$	−11.2154	−0.427272
$690$	0	0
$691$	29.8564	1.13579	0.567896	−	0.823101i	$-0.307758\pi$
$691$	29.8564	1.13579	0.567896	+	0.823101i	$0.307758\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	14.1436	0.536497
$696$	0	0
$697$	29.1769	1.10515
$698$	0	0
$699$	−17.3205	−0.655122
$700$	0	0
$701$	−5.21539	−0.196983	−0.0984913	−	0.995138i	$-0.531402\pi$
$701$	−5.21539	−0.196983	−0.0984913	+	0.995138i	$0.531402\pi$
$702$	0	0
$703$	−4.92820	−0.185871
$704$	0	0
$705$	0.143594	0.00540805
$706$	0	0
$707$	−4.73205	−0.177967
$708$	0	0
$709$	39.3205	1.47671	0.738356	−	0.674411i	$-0.235602\pi$
$709$	39.3205	1.47671	0.738356	+	0.674411i	$0.235602\pi$
$710$	0	0
$711$	2.92820	0.109816
$712$	0	0
$713$	13.8564	0.518927
$714$	0	0
$715$	4.28719	0.160332
$716$	0	0
$717$	−9.46410	−0.353443
$718$	0	0
$719$	11.4115	0.425579	0.212789	−	0.977098i	$-0.431745\pi$
$719$	11.4115	0.425579	0.212789	+	0.977098i	$0.431745\pi$
$720$	0	0
$721$	−2.92820	−0.109052
$722$	0	0
$723$	10.7846	0.401084
$724$	0	0
$725$	14.5885	0.541802
$726$	0	0
$727$	−35.7128	−1.32451	−0.662257	−	0.749276i	$-0.730401\pi$
$727$	−35.7128	−1.32451	−0.662257	+	0.749276i	$0.730401\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.1051	−0.595669
$732$	0	0
$733$	37.3205	1.37846	0.689232	−	0.724541i	$-0.257948\pi$
$733$	37.3205	1.37846	0.689232	+	0.724541i	$0.257948\pi$
$734$	0	0
$735$	0.732051	0.0270021
$736$	0	0
$737$	−21.8564	−0.805091
$738$	0	0
$739$	−47.8564	−1.76043	−0.880213	−	0.474578i	$-0.842601\pi$
$739$	−47.8564	−1.76043	−0.880213	+	0.474578i	$0.842601\pi$
$740$	0	0
$741$	−1.46410	−0.0537851
$742$	0	0
$743$	5.66025	0.207655	0.103827	−	0.994595i	$-0.466891\pi$
$743$	5.66025	0.207655	0.103827	+	0.994595i	$0.466891\pi$
$744$	0	0
$745$	−2.53590	−0.0929081
$746$	0	0
$747$	8.19615	0.299882
$748$	0	0
$749$	−12.1962	−0.445638
$750$	0	0
$751$	24.3923	0.890088	0.445044	−	0.895509i	$-0.353188\pi$
$751$	24.3923	0.890088	0.445044	+	0.895509i	$0.353188\pi$
$752$	0	0
$753$	−1.66025	−0.0605030
$754$	0	0
$755$	9.85641	0.358711
$756$	0	0
$757$	−14.7846	−0.537356	−0.268678	−	0.963230i	$-0.586587\pi$
$757$	−14.7846	−0.537356	−0.268678	+	0.963230i	$0.586587\pi$
$758$	0	0
$759$	27.7128	1.00591
$760$	0	0
$761$	−28.0526	−1.01690	−0.508452	−	0.861090i	$-0.669782\pi$
$761$	−28.0526	−1.01690	−0.508452	+	0.861090i	$0.669782\pi$
$762$	0	0
$763$	−4.53590	−0.164211
$764$	0	0
$765$	−2.39230	−0.0864940
$766$	0	0
$767$	16.0000	0.577727
$768$	0	0
$769$	8.92820	0.321959	0.160980	−	0.986958i	$-0.448535\pi$
$769$	8.92820	0.321959	0.160980	+	0.986958i	$0.448535\pi$
$770$	0	0
$771$	2.39230	0.0861568
$772$	0	0
$773$	42.7846	1.53886	0.769428	−	0.638734i	$-0.220542\pi$
$773$	42.7846	1.53886	0.769428	+	0.638734i	$0.220542\pi$
$774$	0	0
$775$	8.92820	0.320711
$776$	0	0
$777$	−4.92820	−0.176798
$778$	0	0
$779$	−8.92820	−0.319886
$780$	0	0
$781$	−16.7846	−0.600601
$782$	0	0
$783$	−3.26795	−0.116787
$784$	0	0
$785$	−2.24871	−0.0802599
$786$	0	0
$787$	4.14359	0.147703	0.0738516	−	0.997269i	$-0.476471\pi$
$787$	4.14359	0.147703	0.0738516	+	0.997269i	$0.476471\pi$
$788$	0	0
$789$	2.53590	0.0902804
$790$	0	0
$791$	16.7321	0.594923
$792$	0	0
$793$	−1.35898	−0.0482589
$794$	0	0
$795$	5.60770	0.198884
$796$	0	0
$797$	14.7846	0.523698	0.261849	−	0.965109i	$-0.415668\pi$
$797$	14.7846	0.523698	0.261849	+	0.965109i	$0.415668\pi$
$798$	0	0
$799$	−0.641016	−0.0226775
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	41.5692	1.46695
$804$	0	0
$805$	−5.07180	−0.178757
$806$	0	0
$807$	−27.4641	−0.966782
$808$	0	0
$809$	49.7128	1.74781	0.873905	−	0.486097i	$-0.161580\pi$
$809$	49.7128	1.74781	0.873905	+	0.486097i	$0.161580\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	−9.46410	−0.331513
$816$	0	0
$817$	4.92820	0.172416
$818$	0	0
$819$	−1.46410	−0.0511599
$820$	0	0
$821$	−41.3205	−1.44210	−0.721048	−	0.692885i	$-0.756339\pi$
$821$	−41.3205	−1.44210	−0.721048	+	0.692885i	$0.756339\pi$
$822$	0	0
$823$	22.7846	0.794222	0.397111	−	0.917771i	$-0.370013\pi$
$823$	22.7846	0.794222	0.397111	+	0.917771i	$0.370013\pi$
$824$	0	0
$825$	17.8564	0.621680
$826$	0	0
$827$	16.9808	0.590479	0.295239	−	0.955423i	$-0.404601\pi$
$827$	16.9808	0.590479	0.295239	+	0.955423i	$0.404601\pi$
$828$	0	0
$829$	−9.71281	−0.337340	−0.168670	−	0.985673i	$-0.553947\pi$
$829$	−9.71281	−0.337340	−0.168670	+	0.985673i	$0.553947\pi$
$830$	0	0
$831$	−24.3923	−0.846160
$832$	0	0
$833$	−3.26795	−0.113228
$834$	0	0
$835$	6.92820	0.239760
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	42.6410	1.47213	0.736066	−	0.676910i	$-0.236681\pi$
$839$	42.6410	1.47213	0.736066	+	0.676910i	$0.236681\pi$
$840$	0	0
$841$	−18.3205	−0.631742
$842$	0	0
$843$	−4.73205	−0.162980
$844$	0	0
$845$	−7.94744	−0.273400
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	−8.39230	−0.288023
$850$	0	0
$851$	34.1436	1.17043
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	0	0
$855$	0.732051	0.0250356
$856$	0	0
$857$	−7.85641	−0.268370	−0.134185	−	0.990956i	$-0.542842\pi$
$857$	−7.85641	−0.268370	−0.134185	+	0.990956i	$0.542842\pi$
$858$	0	0
$859$	37.8564	1.29164	0.645822	−	0.763488i	$-0.276515\pi$
$859$	37.8564	1.29164	0.645822	+	0.763488i	$0.276515\pi$
$860$	0	0
$861$	−8.92820	−0.304272
$862$	0	0
$863$	−41.3731	−1.40836	−0.704178	−	0.710024i	$-0.748684\pi$
$863$	−41.3731	−1.40836	−0.704178	+	0.710024i	$0.748684\pi$
$864$	0	0
$865$	−8.67949	−0.295112
$866$	0	0
$867$	−6.32051	−0.214656
$868$	0	0
$869$	−11.7128	−0.397330
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	11.8564	0.401279
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	−35.5692	−1.20109	−0.600544	−	0.799592i	$-0.705049\pi$
$877$	−35.5692	−1.20109	−0.600544	+	0.799592i	$0.705049\pi$
$878$	0	0
$879$	−27.8564	−0.939573
$880$	0	0
$881$	23.2679	0.783917	0.391959	−	0.919983i	$-0.371798\pi$
$881$	23.2679	0.783917	0.391959	+	0.919983i	$0.371798\pi$
$882$	0	0
$883$	−28.7846	−0.968679	−0.484340	−	0.874880i	$-0.660940\pi$
$883$	−28.7846	−0.968679	−0.484340	+	0.874880i	$0.660940\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	41.1769	1.38259	0.691293	−	0.722575i	$-0.257042\pi$
$887$	41.1769	1.38259	0.691293	+	0.722575i	$0.257042\pi$
$888$	0	0
$889$	−6.53590	−0.219207
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	0.196152	0.00656399
$894$	0	0
$895$	−10.7846	−0.360490
$896$	0	0
$897$	10.1436	0.338685
$898$	0	0
$899$	6.53590	0.217984
$900$	0	0
$901$	−25.0333	−0.833981
$902$	0	0
$903$	4.92820	0.164000
$904$	0	0
$905$	−5.17691	−0.172086
$906$	0	0
$907$	−34.6410	−1.15024	−0.575118	−	0.818070i	$-0.695044\pi$
$907$	−34.6410	−1.15024	−0.575118	+	0.818070i	$0.695044\pi$
$908$	0	0
$909$	−4.73205	−0.156952
$910$	0	0
$911$	47.5167	1.57430	0.787149	−	0.616763i	$-0.211556\pi$
$911$	47.5167	1.57430	0.787149	+	0.616763i	$0.211556\pi$
$912$	0	0
$913$	−32.7846	−1.08501
$914$	0	0
$915$	0.679492	0.0224633
$916$	0	0
$917$	10.7321	0.354404
$918$	0	0
$919$	−3.71281	−0.122474	−0.0612372	−	0.998123i	$-0.519505\pi$
$919$	−3.71281	−0.122474	−0.0612372	+	0.998123i	$0.519505\pi$
$920$	0	0
$921$	16.9282	0.557803
$922$	0	0
$923$	−6.14359	−0.202219
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	−2.92820	−0.0961748
$928$	0	0
$929$	26.5885	0.872339	0.436169	−	0.899865i	$-0.356335\pi$
$929$	26.5885	0.872339	0.436169	+	0.899865i	$0.356335\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−7.12436	−0.233241
$934$	0	0
$935$	9.56922	0.312947
$936$	0	0
$937$	52.9282	1.72909	0.864545	−	0.502556i	$-0.167607\pi$
$937$	52.9282	1.72909	0.864545	+	0.502556i	$0.167607\pi$
$938$	0	0
$939$	19.0718	0.622385
$940$	0	0
$941$	30.3923	0.990761	0.495380	−	0.868676i	$-0.335029\pi$
$941$	30.3923	0.990761	0.495380	+	0.868676i	$0.335029\pi$
$942$	0	0
$943$	61.8564	2.01432
$944$	0	0
$945$	0.732051	0.0238136
$946$	0	0
$947$	10.1436	0.329622	0.164811	−	0.986325i	$-0.447299\pi$
$947$	10.1436	0.329622	0.164811	+	0.986325i	$0.447299\pi$
$948$	0	0
$949$	15.2154	0.493912
$950$	0	0
$951$	−19.2679	−0.624806
$952$	0	0
$953$	−1.41154	−0.0457244	−0.0228622	−	0.999739i	$-0.507278\pi$
$953$	−1.41154	−0.0457244	−0.0228622	+	0.999739i	$0.507278\pi$
$954$	0	0
$955$	13.8564	0.448383
$956$	0	0
$957$	13.0718	0.422551
$958$	0	0
$959$	4.92820	0.159140
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−12.1962	−0.393016
$964$	0	0
$965$	6.53590	0.210398
$966$	0	0
$967$	−30.7846	−0.989966	−0.494983	−	0.868903i	$-0.664826\pi$
$967$	−30.7846	−0.989966	−0.494983	+	0.868903i	$0.664826\pi$
$968$	0	0
$969$	−3.26795	−0.104982
$970$	0	0
$971$	−15.2154	−0.488285	−0.244143	−	0.969739i	$-0.578506\pi$
$971$	−15.2154	−0.488285	−0.244143	+	0.969739i	$0.578506\pi$
$972$	0	0
$973$	19.3205	0.619387
$974$	0	0
$975$	6.53590	0.209316
$976$	0	0
$977$	25.5167	0.816350	0.408175	−	0.912904i	$-0.366165\pi$
$977$	25.5167	0.816350	0.408175	+	0.912904i	$0.366165\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−4.53590	−0.144820
$982$	0	0
$983$	8.78461	0.280186	0.140093	−	0.990138i	$-0.455260\pi$
$983$	8.78461	0.280186	0.140093	+	0.990138i	$0.455260\pi$
$984$	0	0
$985$	0.679492	0.0216504
$986$	0	0
$987$	0.196152	0.00624360
$988$	0	0
$989$	−34.1436	−1.08570
$990$	0	0
$991$	−30.6410	−0.973344	−0.486672	−	0.873585i	$-0.661789\pi$
$991$	−30.6410	−0.973344	−0.486672	+	0.873585i	$0.661789\pi$
$992$	0	0
$993$	8.39230	0.266322
$994$	0	0
$995$	14.1436	0.448382
$996$	0	0
$997$	−43.0718	−1.36410	−0.682049	−	0.731307i	$-0.738911\pi$
$997$	−43.0718	−1.36410	−0.682049	+	0.731307i	$0.738911\pi$
$998$	0	0
$999$	−4.92820	−0.155921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.s.1.2	✓	2
3.2	odd	2		9576.2.a.bw.1.1		2
4.3	odd	2		6384.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.s.1.2	✓	2	1.1	even	1	trivial
6384.2.a.bi.1.2		2	4.3	odd	2
9576.2.a.bw.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.46410 q^{13} +0.732051 q^{15} -3.26795 q^{17} +1.00000 q^{19} +1.00000 q^{21} -6.92820 q^{23} -4.46410 q^{25} +1.00000 q^{27} -3.26795 q^{29} -2.00000 q^{31} -4.00000 q^{33} +0.732051 q^{35} -4.92820 q^{37} -1.46410 q^{39} -8.92820 q^{41} +4.92820 q^{43} +0.732051 q^{45} +0.196152 q^{47} +1.00000 q^{49} -3.26795 q^{51} +7.66025 q^{53} -2.92820 q^{55} +1.00000 q^{57} -10.9282 q^{59} +0.928203 q^{61} +1.00000 q^{63} -1.07180 q^{65} +5.46410 q^{67} -6.92820 q^{69} +4.19615 q^{71} -10.3923 q^{73} -4.46410 q^{75} -4.00000 q^{77} +2.92820 q^{79} +1.00000 q^{81} +8.19615 q^{83} -2.39230 q^{85} -3.26795 q^{87} -6.00000 q^{89} -1.46410 q^{91} -2.00000 q^{93} +0.732051 q^{95} +11.8564 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 4 q^{13} - 2 q^{15} - 10 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 4 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 10 q^{51} - 2 q^{53} + 8 q^{55} + 2 q^{57} - 8 q^{59} - 12 q^{61} + 2 q^{63} - 16 q^{65} + 4 q^{67} - 2 q^{71} - 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} + 6 q^{83} + 16 q^{85} - 10 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} - 2 q^{95} - 4 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 4 * q^13 - 2 * q^15 - 10 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 + 4 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 - 2 * q^45 - 10 * q^47 + 2 * q^49 - 10 * q^51 - 2 * q^53 + 8 * q^55 + 2 * q^57 - 8 * q^59 - 12 * q^61 + 2 * q^63 - 16 * q^65 + 4 * q^67 - 2 * q^71 - 2 * q^75 - 8 * q^77 - 8 * q^79 + 2 * q^81 + 6 * q^83 + 16 * q^85 - 10 * q^87 - 12 * q^89 + 4 * q^91 - 4 * q^93 - 2 * q^95 - 4 * q^97 - 8 * q^99

Introduction

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