LMFDB - Embedded newform 3192.2.a.s.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +5.46410 q^{13} -2.73205 q^{15} -6.73205 q^{17} +1.00000 q^{19} +1.00000 q^{21} +6.92820 q^{23} +2.46410 q^{25} +1.00000 q^{27} -6.73205 q^{29} -2.00000 q^{31} -4.00000 q^{33} -2.73205 q^{35} +8.92820 q^{37} +5.46410 q^{39} +4.92820 q^{41} -8.92820 q^{43} -2.73205 q^{45} -10.1962 q^{47} +1.00000 q^{49} -6.73205 q^{51} -9.66025 q^{53} +10.9282 q^{55} +1.00000 q^{57} +2.92820 q^{59} -12.9282 q^{61} +1.00000 q^{63} -14.9282 q^{65} -1.46410 q^{67} +6.92820 q^{69} -6.19615 q^{71} +10.3923 q^{73} +2.46410 q^{75} -4.00000 q^{77} -10.9282 q^{79} +1.00000 q^{81} -2.19615 q^{83} +18.3923 q^{85} -6.73205 q^{87} -6.00000 q^{89} +5.46410 q^{91} -2.00000 q^{93} -2.73205 q^{95} -15.8564 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−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
$9$	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
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	−2.73205	−0.705412
$16$	0	0
$17$	−6.73205	−1.63276	−0.816381	−	0.577514i	$-0.804023\pi$
$17$	−6.73205	−1.63276	−0.816381	+	0.577514i	$0.804023\pi$
$18$	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	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.73205	−1.25011	−0.625055	−	0.780581i	$-0.714924\pi$
$29$	−6.73205	−1.25011	−0.625055	+	0.780581i	$0.714924\pi$
$30$	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
$35$	−2.73205	−0.461801
$36$	0	0
$37$	8.92820	1.46779	0.733894	−	0.679264i	$-0.237701\pi$
$37$	8.92820	1.46779	0.733894	+	0.679264i	$0.237701\pi$
$38$	0	0
$39$	5.46410	0.874957
$40$	0	0
$41$	4.92820	0.769656	0.384828	−	0.922988i	$-0.374261\pi$
$41$	4.92820	0.769656	0.384828	+	0.922988i	$0.374261\pi$
$42$	0	0
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	0	0
$45$	−2.73205	−0.407270
$46$	0	0
$47$	−10.1962	−1.48726	−0.743631	−	0.668590i	$-0.766898\pi$
$47$	−10.1962	−1.48726	−0.743631	+	0.668590i	$0.766898\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.73205	−0.942676
$52$	0	0
$53$	−9.66025	−1.32694	−0.663469	−	0.748204i	$-0.730917\pi$
$53$	−9.66025	−1.32694	−0.663469	+	0.748204i	$0.730917\pi$
$54$	0	0
$55$	10.9282	1.47356
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	2.92820	0.381220	0.190610	−	0.981666i	$-0.438953\pi$
$59$	2.92820	0.381220	0.190610	+	0.981666i	$0.438953\pi$
$60$	0	0
$61$	−12.9282	−1.65529	−0.827643	−	0.561254i	$-0.810319\pi$
$61$	−12.9282	−1.65529	−0.827643	+	0.561254i	$0.810319\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−14.9282	−1.85162
$66$	0	0
$67$	−1.46410	−0.178868	−0.0894342	−	0.995993i	$-0.528506\pi$
$67$	−1.46410	−0.178868	−0.0894342	+	0.995993i	$0.528506\pi$
$68$	0	0
$69$	6.92820	0.834058
$70$	0	0
$71$	−6.19615	−0.735348	−0.367674	−	0.929955i	$-0.619846\pi$
$71$	−6.19615	−0.735348	−0.367674	+	0.929955i	$0.619846\pi$
$72$	0	0
$73$	10.3923	1.21633	0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923	1.21633	0.608164	+	0.793812i	$0.291906\pi$
$74$	0	0
$75$	2.46410	0.284530
$76$	0	0
$77$	−4.00000	−0.455842
$78$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.19615	−0.241059	−0.120530	−	0.992710i	$-0.538459\pi$
$83$	−2.19615	−0.241059	−0.120530	+	0.992710i	$0.538459\pi$
$84$	0	0
$85$	18.3923	1.99493
$86$	0	0
$87$	−6.73205	−0.721752
$88$	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
$91$	5.46410	0.572793
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−2.73205	−0.280302
$96$	0	0
$97$	−15.8564	−1.60997	−0.804987	−	0.593292i	$-0.797828\pi$
$97$	−15.8564	−1.60997	−0.804987	+	0.593292i	$0.797828\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−1.26795	−0.126166	−0.0630828	−	0.998008i	$-0.520093\pi$
$101$	−1.26795	−0.126166	−0.0630828	+	0.998008i	$0.520093\pi$
$102$	0	0
$103$	10.9282	1.07679	0.538394	−	0.842693i	$-0.319031\pi$
$103$	10.9282	1.07679	0.538394	+	0.842693i	$0.319031\pi$
$104$	0	0
$105$	−2.73205	−0.266621
$106$	0	0
$107$	−1.80385	−0.174385	−0.0871923	−	0.996192i	$-0.527789\pi$
$107$	−1.80385	−0.174385	−0.0871923	+	0.996192i	$0.527789\pi$
$108$	0	0
$109$	−11.4641	−1.09806	−0.549031	−	0.835802i	$-0.685003\pi$
$109$	−11.4641	−1.09806	−0.549031	+	0.835802i	$0.685003\pi$
$110$	0	0
$111$	8.92820	0.847428
$112$	0	0
$113$	13.2679	1.24814	0.624072	−	0.781367i	$-0.285477\pi$
$113$	13.2679	1.24814	0.624072	+	0.781367i	$0.285477\pi$
$114$	0	0
$115$	−18.9282	−1.76506
$116$	0	0
$117$	5.46410	0.505156
$118$	0	0
$119$	−6.73205	−0.617126
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	4.92820	0.444361
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−13.4641	−1.19475	−0.597373	−	0.801964i	$-0.703789\pi$
$127$	−13.4641	−1.19475	−0.597373	+	0.801964i	$0.703789\pi$
$128$	0	0
$129$	−8.92820	−0.786084
$130$	0	0
$131$	7.26795	0.635004	0.317502	−	0.948258i	$-0.397156\pi$
$131$	7.26795	0.635004	0.317502	+	0.948258i	$0.397156\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−2.73205	−0.235137
$136$	0	0
$137$	−8.92820	−0.762788	−0.381394	−	0.924413i	$-0.624556\pi$
$137$	−8.92820	−0.762788	−0.381394	+	0.924413i	$0.624556\pi$
$138$	0	0
$139$	−15.3205	−1.29947	−0.649734	−	0.760161i	$-0.725120\pi$
$139$	−15.3205	−1.29947	−0.649734	+	0.760161i	$0.725120\pi$
$140$	0	0
$141$	−10.1962	−0.858671
$142$	0	0
$143$	−21.8564	−1.82772
$144$	0	0
$145$	18.3923	1.52740
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	3.46410	0.283790	0.141895	−	0.989882i	$-0.454680\pi$
$149$	3.46410	0.283790	0.141895	+	0.989882i	$0.454680\pi$
$150$	0	0
$151$	6.53590	0.531884	0.265942	−	0.963989i	$-0.414317\pi$
$151$	6.53590	0.531884	0.265942	+	0.963989i	$0.414317\pi$
$152$	0	0
$153$	−6.73205	−0.544254
$154$	0	0
$155$	5.46410	0.438887
$156$	0	0
$157$	−16.9282	−1.35102	−0.675509	−	0.737352i	$-0.736076\pi$
$157$	−16.9282	−1.35102	−0.675509	+	0.737352i	$0.736076\pi$
$158$	0	0
$159$	−9.66025	−0.766108
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	0.928203	0.0727025	0.0363512	−	0.999339i	$-0.488426\pi$
$163$	0.928203	0.0727025	0.0363512	+	0.999339i	$0.488426\pi$
$164$	0	0
$165$	10.9282	0.850759
$166$	0	0
$167$	2.53590	0.196234	0.0981169	−	0.995175i	$-0.468718\pi$
$167$	2.53590	0.196234	0.0981169	+	0.995175i	$0.468718\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	15.8564	1.20554	0.602770	−	0.797915i	$-0.294064\pi$
$173$	15.8564	1.20554	0.602770	+	0.797915i	$0.294064\pi$
$174$	0	0
$175$	2.46410	0.186269
$176$	0	0
$177$	2.92820	0.220097
$178$	0	0
$179$	−11.2679	−0.842206	−0.421103	−	0.907013i	$-0.638357\pi$
$179$	−11.2679	−0.842206	−0.421103	+	0.907013i	$0.638357\pi$
$180$	0	0
$181$	−20.9282	−1.55558	−0.777791	−	0.628524i	$-0.783660\pi$
$181$	−20.9282	−1.55558	−0.777791	+	0.628524i	$0.783660\pi$
$182$	0	0
$183$	−12.9282	−0.955680
$184$	0	0
$185$	−24.3923	−1.79336
$186$	0	0
$187$	26.9282	1.96919
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	0	0
$193$	−4.92820	−0.354740	−0.177370	−	0.984144i	$-0.556759\pi$
$193$	−4.92820	−0.354740	−0.177370	+	0.984144i	$0.556759\pi$
$194$	0	0
$195$	−14.9282	−1.06903
$196$	0	0
$197$	−12.9282	−0.921096	−0.460548	−	0.887635i	$-0.652347\pi$
$197$	−12.9282	−0.921096	−0.460548	+	0.887635i	$0.652347\pi$
$198$	0	0
$199$	−15.3205	−1.08604	−0.543021	−	0.839719i	$-0.682720\pi$
$199$	−15.3205	−1.08604	−0.543021	+	0.839719i	$0.682720\pi$
$200$	0	0
$201$	−1.46410	−0.103270
$202$	0	0
$203$	−6.73205	−0.472497
$204$	0	0
$205$	−13.4641	−0.940374
$206$	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
$213$	−6.19615	−0.424553
$214$	0	0
$215$	24.3923	1.66354
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	10.3923	0.702247
$220$	0	0
$221$	−36.7846	−2.47440
$222$	0	0
$223$	18.7846	1.25791	0.628955	−	0.777442i	$-0.283483\pi$
$223$	18.7846	1.25791	0.628955	+	0.777442i	$0.283483\pi$
$224$	0	0
$225$	2.46410	0.164273
$226$	0	0
$227$	27.3205	1.81333	0.906663	−	0.421856i	$-0.138621\pi$
$227$	27.3205	1.81333	0.906663	+	0.421856i	$0.138621\pi$
$228$	0	0
$229$	−18.3923	−1.21540	−0.607699	−	0.794168i	$-0.707907\pi$
$229$	−18.3923	−1.21540	−0.607699	+	0.794168i	$0.707907\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	17.3205	1.13470	0.567352	−	0.823475i	$-0.307968\pi$
$233$	17.3205	1.13470	0.567352	+	0.823475i	$0.307968\pi$
$234$	0	0
$235$	27.8564	1.81715
$236$	0	0
$237$	−10.9282	−0.709863
$238$	0	0
$239$	−2.53590	−0.164034	−0.0820168	−	0.996631i	$-0.526136\pi$
$239$	−2.53590	−0.164034	−0.0820168	+	0.996631i	$0.526136\pi$
$240$	0	0
$241$	−30.7846	−1.98301	−0.991506	−	0.130065i	$-0.958482\pi$
$241$	−30.7846	−1.98301	−0.991506	+	0.130065i	$0.958482\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.73205	−0.174544
$246$	0	0
$247$	5.46410	0.347672
$248$	0	0
$249$	−2.19615	−0.139176
$250$	0	0
$251$	15.6603	0.988466	0.494233	−	0.869329i	$-0.335449\pi$
$251$	15.6603	0.988466	0.494233	+	0.869329i	$0.335449\pi$
$252$	0	0
$253$	−27.7128	−1.74229
$254$	0	0
$255$	18.3923	1.15177
$256$	0	0
$257$	−18.3923	−1.14728	−0.573640	−	0.819107i	$-0.694469\pi$
$257$	−18.3923	−1.14728	−0.573640	+	0.819107i	$0.694469\pi$
$258$	0	0
$259$	8.92820	0.554772
$260$	0	0
$261$	−6.73205	−0.416703
$262$	0	0
$263$	9.46410	0.583582	0.291791	−	0.956482i	$-0.405749\pi$
$263$	9.46410	0.583582	0.291791	+	0.956482i	$0.405749\pi$
$264$	0	0
$265$	26.3923	1.62127
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−20.5359	−1.25210	−0.626048	−	0.779785i	$-0.715329\pi$
$269$	−20.5359	−1.25210	−0.626048	+	0.779785i	$0.715329\pi$
$270$	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
$273$	5.46410	0.330702
$274$	0	0
$275$	−9.85641	−0.594364
$276$	0	0
$277$	−3.60770	−0.216765	−0.108383	−	0.994109i	$-0.534567\pi$
$277$	−3.60770	−0.216765	−0.108383	+	0.994109i	$0.534567\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−1.26795	−0.0756395	−0.0378198	−	0.999285i	$-0.512041\pi$
$281$	−1.26795	−0.0756395	−0.0378198	+	0.999285i	$0.512041\pi$
$282$	0	0
$283$	12.3923	0.736646	0.368323	−	0.929698i	$-0.379932\pi$
$283$	12.3923	0.736646	0.368323	+	0.929698i	$0.379932\pi$
$284$	0	0
$285$	−2.73205	−0.161833
$286$	0	0
$287$	4.92820	0.290903
$288$	0	0
$289$	28.3205	1.66591
$290$	0	0
$291$	−15.8564	−0.929519
$292$	0	0
$293$	−0.143594	−0.00838882	−0.00419441	−	0.999991i	$-0.501335\pi$
$293$	−0.143594	−0.00838882	−0.00419441	+	0.999991i	$0.501335\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	37.8564	2.18929
$300$	0	0
$301$	−8.92820	−0.514613
$302$	0	0
$303$	−1.26795	−0.0728418
$304$	0	0
$305$	35.3205	2.02245
$306$	0	0
$307$	3.07180	0.175317	0.0876584	−	0.996151i	$-0.472062\pi$
$307$	3.07180	0.175317	0.0876584	+	0.996151i	$0.472062\pi$
$308$	0	0
$309$	10.9282	0.621684
$310$	0	0
$311$	17.1244	0.971033	0.485517	−	0.874227i	$-0.338632\pi$
$311$	17.1244	0.971033	0.485517	+	0.874227i	$0.338632\pi$
$312$	0	0
$313$	32.9282	1.86121	0.930606	−	0.366022i	$-0.119281\pi$
$313$	32.9282	1.86121	0.930606	+	0.366022i	$0.119281\pi$
$314$	0	0
$315$	−2.73205	−0.153934
$316$	0	0
$317$	−22.7321	−1.27676	−0.638380	−	0.769722i	$-0.720395\pi$
$317$	−22.7321	−1.27676	−0.638380	+	0.769722i	$0.720395\pi$
$318$	0	0
$319$	26.9282	1.50769
$320$	0	0
$321$	−1.80385	−0.100681
$322$	0	0
$323$	−6.73205	−0.374581
$324$	0	0
$325$	13.4641	0.746854
$326$	0	0
$327$	−11.4641	−0.633966
$328$	0	0
$329$	−10.1962	−0.562132
$330$	0	0
$331$	−12.3923	−0.681143	−0.340571	−	0.940219i	$-0.610620\pi$
$331$	−12.3923	−0.681143	−0.340571	+	0.940219i	$0.610620\pi$
$332$	0	0
$333$	8.92820	0.489263
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	22.3923	1.21979	0.609893	−	0.792484i	$-0.291212\pi$
$337$	22.3923	1.21979	0.609893	+	0.792484i	$0.291212\pi$
$338$	0	0
$339$	13.2679	0.720616
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−18.9282	−1.01906
$346$	0	0
$347$	−15.3205	−0.822448	−0.411224	−	0.911534i	$-0.634899\pi$
$347$	−15.3205	−0.822448	−0.411224	+	0.911534i	$0.634899\pi$
$348$	0	0
$349$	18.7846	1.00552	0.502759	−	0.864427i	$-0.332318\pi$
$349$	18.7846	1.00552	0.502759	+	0.864427i	$0.332318\pi$
$350$	0	0
$351$	5.46410	0.291652
$352$	0	0
$353$	19.5167	1.03877	0.519384	−	0.854541i	$-0.326162\pi$
$353$	19.5167	1.03877	0.519384	+	0.854541i	$0.326162\pi$
$354$	0	0
$355$	16.9282	0.898456
$356$	0	0
$357$	−6.73205	−0.356298
$358$	0	0
$359$	2.14359	0.113135	0.0565673	−	0.998399i	$-0.481984\pi$
$359$	2.14359	0.113135	0.0565673	+	0.998399i	$0.481984\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−28.3923	−1.48612
$366$	0	0
$367$	19.3205	1.00852	0.504261	−	0.863551i	$-0.331765\pi$
$367$	19.3205	1.00852	0.504261	+	0.863551i	$0.331765\pi$
$368$	0	0
$369$	4.92820	0.256552
$370$	0	0
$371$	−9.66025	−0.501535
$372$	0	0
$373$	−10.7846	−0.558406	−0.279203	−	0.960232i	$-0.590070\pi$
$373$	−10.7846	−0.558406	−0.279203	+	0.960232i	$0.590070\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	−36.7846	−1.89450
$378$	0	0
$379$	−0.392305	−0.0201513	−0.0100757	−	0.999949i	$-0.503207\pi$
$379$	−0.392305	−0.0201513	−0.0100757	+	0.999949i	$0.503207\pi$
$380$	0	0
$381$	−13.4641	−0.689787
$382$	0	0
$383$	35.3205	1.80479	0.902397	−	0.430906i	$-0.141806\pi$
$383$	35.3205	1.80479	0.902397	+	0.430906i	$0.141806\pi$
$384$	0	0
$385$	10.9282	0.556953
$386$	0	0
$387$	−8.92820	−0.453846
$388$	0	0
$389$	−34.3923	−1.74376	−0.871880	−	0.489720i	$-0.837099\pi$
$389$	−34.3923	−1.74376	−0.871880	+	0.489720i	$0.837099\pi$
$390$	0	0
$391$	−46.6410	−2.35874
$392$	0	0
$393$	7.26795	0.366620
$394$	0	0
$395$	29.8564	1.50224
$396$	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
$401$	−24.5885	−1.22789	−0.613944	−	0.789349i	$-0.710418\pi$
$401$	−24.5885	−1.22789	−0.613944	+	0.789349i	$0.710418\pi$
$402$	0	0
$403$	−10.9282	−0.544373
$404$	0	0
$405$	−2.73205	−0.135757
$406$	0	0
$407$	−35.7128	−1.77022
$408$	0	0
$409$	−9.46410	−0.467970	−0.233985	−	0.972240i	$-0.575177\pi$
$409$	−9.46410	−0.467970	−0.233985	+	0.972240i	$0.575177\pi$
$410$	0	0
$411$	−8.92820	−0.440396
$412$	0	0
$413$	2.92820	0.144087
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	−15.3205	−0.750249
$418$	0	0
$419$	33.1244	1.61823	0.809115	−	0.587650i	$-0.199947\pi$
$419$	33.1244	1.61823	0.809115	+	0.587650i	$0.199947\pi$
$420$	0	0
$421$	−23.4641	−1.14357	−0.571785	−	0.820403i	$-0.693749\pi$
$421$	−23.4641	−1.14357	−0.571785	+	0.820403i	$0.693749\pi$
$422$	0	0
$423$	−10.1962	−0.495754
$424$	0	0
$425$	−16.5885	−0.804658
$426$	0	0
$427$	−12.9282	−0.625640
$428$	0	0
$429$	−21.8564	−1.05524
$430$	0	0
$431$	10.9808	0.528925	0.264462	−	0.964396i	$-0.414806\pi$
$431$	10.9808	0.528925	0.264462	+	0.964396i	$0.414806\pi$
$432$	0	0
$433$	−3.85641	−0.185327	−0.0926635	−	0.995697i	$-0.529538\pi$
$433$	−3.85641	−0.185327	−0.0926635	+	0.995697i	$0.529538\pi$
$434$	0	0
$435$	18.3923	0.881844
$436$	0	0
$437$	6.92820	0.331421
$438$	0	0
$439$	23.8564	1.13860	0.569302	−	0.822128i	$-0.307213\pi$
$439$	23.8564	1.13860	0.569302	+	0.822128i	$0.307213\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.39230	−0.208685	−0.104342	−	0.994541i	$-0.533274\pi$
$443$	−4.39230	−0.208685	−0.104342	+	0.994541i	$0.533274\pi$
$444$	0	0
$445$	16.3923	0.777070
$446$	0	0
$447$	3.46410	0.163846
$448$	0	0
$449$	2.33975	0.110419	0.0552097	−	0.998475i	$-0.482417\pi$
$449$	2.33975	0.110419	0.0552097	+	0.998475i	$0.482417\pi$
$450$	0	0
$451$	−19.7128	−0.928240
$452$	0	0
$453$	6.53590	0.307083
$454$	0	0
$455$	−14.9282	−0.699845
$456$	0	0
$457$	23.3205	1.09089	0.545444	−	0.838147i	$-0.316361\pi$
$457$	23.3205	1.09089	0.545444	+	0.838147i	$0.316361\pi$
$458$	0	0
$459$	−6.73205	−0.314225
$460$	0	0
$461$	−20.1962	−0.940629	−0.470314	−	0.882499i	$-0.655859\pi$
$461$	−20.1962	−0.940629	−0.470314	+	0.882499i	$0.655859\pi$
$462$	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
$465$	5.46410	0.253392
$466$	0	0
$467$	2.19615	0.101626	0.0508129	−	0.998708i	$-0.483819\pi$
$467$	2.19615	0.101626	0.0508129	+	0.998708i	$0.483819\pi$
$468$	0	0
$469$	−1.46410	−0.0676059
$470$	0	0
$471$	−16.9282	−0.780010
$472$	0	0
$473$	35.7128	1.64208
$474$	0	0
$475$	2.46410	0.113061
$476$	0	0
$477$	−9.66025	−0.442313
$478$	0	0
$479$	−25.5167	−1.16589	−0.582943	−	0.812513i	$-0.698099\pi$
$479$	−25.5167	−1.16589	−0.582943	+	0.812513i	$0.698099\pi$
$480$	0	0
$481$	48.7846	2.22439
$482$	0	0
$483$	6.92820	0.315244
$484$	0	0
$485$	43.3205	1.96708
$486$	0	0
$487$	−21.8564	−0.990408	−0.495204	−	0.868777i	$-0.664907\pi$
$487$	−21.8564	−0.990408	−0.495204	+	0.868777i	$0.664907\pi$
$488$	0	0
$489$	0.928203	0.0419748
$490$	0	0
$491$	42.2487	1.90666	0.953329	−	0.301934i	$-0.0976322\pi$
$491$	42.2487	1.90666	0.953329	+	0.301934i	$0.0976322\pi$
$492$	0	0
$493$	45.3205	2.04113
$494$	0	0
$495$	10.9282	0.491186
$496$	0	0
$497$	−6.19615	−0.277935
$498$	0	0
$499$	25.8564	1.15749	0.578746	−	0.815508i	$-0.303542\pi$
$499$	25.8564	1.15749	0.578746	+	0.815508i	$0.303542\pi$
$500$	0	0
$501$	2.53590	0.113296
$502$	0	0
$503$	−28.7321	−1.28110	−0.640549	−	0.767917i	$-0.721293\pi$
$503$	−28.7321	−1.28110	−0.640549	+	0.767917i	$0.721293\pi$
$504$	0	0
$505$	3.46410	0.154150
$506$	0	0
$507$	16.8564	0.748619
$508$	0	0
$509$	−29.3205	−1.29961	−0.649804	−	0.760102i	$-0.725149\pi$
$509$	−29.3205	−1.29961	−0.649804	+	0.760102i	$0.725149\pi$
$510$	0	0
$511$	10.3923	0.459728
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−29.8564	−1.31563
$516$	0	0
$517$	40.7846	1.79371
$518$	0	0
$519$	15.8564	0.696019
$520$	0	0
$521$	12.2487	0.536626	0.268313	−	0.963332i	$-0.413534\pi$
$521$	12.2487	0.536626	0.268313	+	0.963332i	$0.413534\pi$
$522$	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
$525$	2.46410	0.107542
$526$	0	0
$527$	13.4641	0.586505
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	2.92820	0.127073
$532$	0	0
$533$	26.9282	1.16639
$534$	0	0
$535$	4.92820	0.213065
$536$	0	0
$537$	−11.2679	−0.486248
$538$	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
$543$	−20.9282	−0.898115
$544$	0	0
$545$	31.3205	1.34162
$546$	0	0
$547$	17.0718	0.729937	0.364969	−	0.931020i	$-0.381080\pi$
$547$	17.0718	0.729937	0.364969	+	0.931020i	$0.381080\pi$
$548$	0	0
$549$	−12.9282	−0.551762
$550$	0	0
$551$	−6.73205	−0.286795
$552$	0	0
$553$	−10.9282	−0.464714
$554$	0	0
$555$	−24.3923	−1.03540
$556$	0	0
$557$	38.1051	1.61457	0.807283	−	0.590165i	$-0.200937\pi$
$557$	38.1051	1.61457	0.807283	+	0.590165i	$0.200937\pi$
$558$	0	0
$559$	−48.7846	−2.06337
$560$	0	0
$561$	26.9282	1.13691
$562$	0	0
$563$	0.679492	0.0286372	0.0143186	−	0.999897i	$-0.495442\pi$
$563$	0.679492	0.0286372	0.0143186	+	0.999897i	$0.495442\pi$
$564$	0	0
$565$	−36.2487	−1.52499
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	15.5167	0.650492	0.325246	−	0.945629i	$-0.394553\pi$
$569$	15.5167	0.650492	0.325246	+	0.945629i	$0.394553\pi$
$570$	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
$573$	5.07180	0.211877
$574$	0	0
$575$	17.0718	0.711943
$576$	0	0
$577$	−26.7846	−1.11506	−0.557529	−	0.830157i	$-0.688250\pi$
$577$	−26.7846	−1.11506	−0.557529	+	0.830157i	$0.688250\pi$
$578$	0	0
$579$	−4.92820	−0.204809
$580$	0	0
$581$	−2.19615	−0.0911118
$582$	0	0
$583$	38.6410	1.60035
$584$	0	0
$585$	−14.9282	−0.617205
$586$	0	0
$587$	9.80385	0.404648	0.202324	−	0.979319i	$-0.435151\pi$
$587$	9.80385	0.404648	0.202324	+	0.979319i	$0.435151\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−12.9282	−0.531795
$592$	0	0
$593$	20.1962	0.829357	0.414678	−	0.909968i	$-0.363894\pi$
$593$	20.1962	0.829357	0.414678	+	0.909968i	$0.363894\pi$
$594$	0	0
$595$	18.3923	0.754011
$596$	0	0
$597$	−15.3205	−0.627027
$598$	0	0
$599$	−18.1962	−0.743475	−0.371737	−	0.928338i	$-0.621238\pi$
$599$	−18.1962	−0.743475	−0.371737	+	0.928338i	$0.621238\pi$
$600$	0	0
$601$	43.8564	1.78894	0.894470	−	0.447128i	$-0.147553\pi$
$601$	43.8564	1.78894	0.894470	+	0.447128i	$0.147553\pi$
$602$	0	0
$603$	−1.46410	−0.0596228
$604$	0	0
$605$	−13.6603	−0.555368
$606$	0	0
$607$	−24.7846	−1.00598	−0.502988	−	0.864293i	$-0.667766\pi$
$607$	−24.7846	−1.00598	−0.502988	+	0.864293i	$0.667766\pi$
$608$	0	0
$609$	−6.73205	−0.272796
$610$	0	0
$611$	−55.7128	−2.25390
$612$	0	0
$613$	−44.3923	−1.79299	−0.896494	−	0.443056i	$-0.853894\pi$
$613$	−44.3923	−1.79299	−0.896494	+	0.443056i	$0.853894\pi$
$614$	0	0
$615$	−13.4641	−0.542925
$616$	0	0
$617$	−18.7846	−0.756240	−0.378120	−	0.925757i	$-0.623429\pi$
$617$	−18.7846	−0.756240	−0.378120	+	0.925757i	$0.623429\pi$
$618$	0	0
$619$	−25.4641	−1.02349	−0.511744	−	0.859138i	$-0.671000\pi$
$619$	−25.4641	−1.02349	−0.511744	+	0.859138i	$0.671000\pi$
$620$	0	0
$621$	6.92820	0.278019
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	−60.1051	−2.39655
$630$	0	0
$631$	6.78461	0.270091	0.135046	−	0.990839i	$-0.456882\pi$
$631$	6.78461	0.270091	0.135046	+	0.990839i	$0.456882\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	36.7846	1.45975
$636$	0	0
$637$	5.46410	0.216496
$638$	0	0
$639$	−6.19615	−0.245116
$640$	0	0
$641$	6.73205	0.265900	0.132950	−	0.991123i	$-0.457555\pi$
$641$	6.73205	0.265900	0.132950	+	0.991123i	$0.457555\pi$
$642$	0	0
$643$	−20.3923	−0.804194	−0.402097	−	0.915597i	$-0.631719\pi$
$643$	−20.3923	−0.804194	−0.402097	+	0.915597i	$0.631719\pi$
$644$	0	0
$645$	24.3923	0.960446
$646$	0	0
$647$	27.6603	1.08744	0.543718	−	0.839268i	$-0.317016\pi$
$647$	27.6603	1.08744	0.543718	+	0.839268i	$0.317016\pi$
$648$	0	0
$649$	−11.7128	−0.459768
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	9.71281	0.380092	0.190046	−	0.981775i	$-0.439136\pi$
$653$	9.71281	0.380092	0.190046	+	0.981775i	$0.439136\pi$
$654$	0	0
$655$	−19.8564	−0.775854
$656$	0	0
$657$	10.3923	0.405442
$658$	0	0
$659$	13.1244	0.511252	0.255626	−	0.966776i	$-0.417718\pi$
$659$	13.1244	0.511252	0.255626	+	0.966776i	$0.417718\pi$
$660$	0	0
$661$	−1.46410	−0.0569470	−0.0284735	−	0.999595i	$-0.509065\pi$
$661$	−1.46410	−0.0569470	−0.0284735	+	0.999595i	$0.509065\pi$
$662$	0	0
$663$	−36.7846	−1.42860
$664$	0	0
$665$	−2.73205	−0.105944
$666$	0	0
$667$	−46.6410	−1.80595
$668$	0	0
$669$	18.7846	0.726255
$670$	0	0
$671$	51.7128	1.99635
$672$	0	0
$673$	16.1436	0.622290	0.311145	−	0.950362i	$-0.399288\pi$
$673$	16.1436	0.622290	0.311145	+	0.950362i	$0.399288\pi$
$674$	0	0
$675$	2.46410	0.0948433
$676$	0	0
$677$	7.46410	0.286869	0.143434	−	0.989660i	$-0.454185\pi$
$677$	7.46410	0.286869	0.143434	+	0.989660i	$0.454185\pi$
$678$	0	0
$679$	−15.8564	−0.608513
$680$	0	0
$681$	27.3205	1.04692
$682$	0	0
$683$	−38.1962	−1.46154	−0.730768	−	0.682626i	$-0.760838\pi$
$683$	−38.1962	−1.46154	−0.730768	+	0.682626i	$0.760838\pi$
$684$	0	0
$685$	24.3923	0.931982
$686$	0	0
$687$	−18.3923	−0.701710
$688$	0	0
$689$	−52.7846	−2.01093
$690$	0	0
$691$	2.14359	0.0815461	0.0407731	−	0.999168i	$-0.487018\pi$
$691$	2.14359	0.0815461	0.0407731	+	0.999168i	$0.487018\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	41.8564	1.58770
$696$	0	0
$697$	−33.1769	−1.25667
$698$	0	0
$699$	17.3205	0.655122
$700$	0	0
$701$	−46.7846	−1.76703	−0.883515	−	0.468403i	$-0.844830\pi$
$701$	−46.7846	−1.76703	−0.883515	+	0.468403i	$0.844830\pi$
$702$	0	0
$703$	8.92820	0.336734
$704$	0	0
$705$	27.8564	1.04913
$706$	0	0
$707$	−1.26795	−0.0476861
$708$	0	0
$709$	4.67949	0.175742	0.0878710	−	0.996132i	$-0.471994\pi$
$709$	4.67949	0.175742	0.0878710	+	0.996132i	$0.471994\pi$
$710$	0	0
$711$	−10.9282	−0.409840
$712$	0	0
$713$	−13.8564	−0.518927
$714$	0	0
$715$	59.7128	2.23313
$716$	0	0
$717$	−2.53590	−0.0947049
$718$	0	0
$719$	42.5885	1.58828	0.794141	−	0.607734i	$-0.207921\pi$
$719$	42.5885	1.58828	0.794141	+	0.607734i	$0.207921\pi$
$720$	0	0
$721$	10.9282	0.406988
$722$	0	0
$723$	−30.7846	−1.14489
$724$	0	0
$725$	−16.5885	−0.616080
$726$	0	0
$727$	19.7128	0.731108	0.365554	−	0.930790i	$-0.380880\pi$
$727$	19.7128	0.731108	0.365554	+	0.930790i	$0.380880\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	60.1051	2.22307
$732$	0	0
$733$	2.67949	0.0989693	0.0494846	−	0.998775i	$-0.484242\pi$
$733$	2.67949	0.0989693	0.0494846	+	0.998775i	$0.484242\pi$
$734$	0	0
$735$	−2.73205	−0.100773
$736$	0	0
$737$	5.85641	0.215724
$738$	0	0
$739$	−20.1436	−0.740994	−0.370497	−	0.928834i	$-0.620813\pi$
$739$	−20.1436	−0.740994	−0.370497	+	0.928834i	$0.620813\pi$
$740$	0	0
$741$	5.46410	0.200729
$742$	0	0
$743$	−11.6603	−0.427773	−0.213887	−	0.976858i	$-0.568612\pi$
$743$	−11.6603	−0.427773	−0.213887	+	0.976858i	$0.568612\pi$
$744$	0	0
$745$	−9.46410	−0.346738
$746$	0	0
$747$	−2.19615	−0.0803530
$748$	0	0
$749$	−1.80385	−0.0659112
$750$	0	0
$751$	3.60770	0.131647	0.0658233	−	0.997831i	$-0.479033\pi$
$751$	3.60770	0.131647	0.0658233	+	0.997831i	$0.479033\pi$
$752$	0	0
$753$	15.6603	0.570691
$754$	0	0
$755$	−17.8564	−0.649861
$756$	0	0
$757$	26.7846	0.973503	0.486752	−	0.873540i	$-0.338182\pi$
$757$	26.7846	0.973503	0.486752	+	0.873540i	$0.338182\pi$
$758$	0	0
$759$	−27.7128	−1.00591
$760$	0	0
$761$	10.0526	0.364405	0.182202	−	0.983261i	$-0.441677\pi$
$761$	10.0526	0.364405	0.182202	+	0.983261i	$0.441677\pi$
$762$	0	0
$763$	−11.4641	−0.415028
$764$	0	0
$765$	18.3923	0.664975
$766$	0	0
$767$	16.0000	0.577727
$768$	0	0
$769$	−4.92820	−0.177716	−0.0888578	−	0.996044i	$-0.528322\pi$
$769$	−4.92820	−0.177716	−0.0888578	+	0.996044i	$0.528322\pi$
$770$	0	0
$771$	−18.3923	−0.662383
$772$	0	0
$773$	1.21539	0.0437146	0.0218573	−	0.999761i	$-0.493042\pi$
$773$	1.21539	0.0437146	0.0218573	+	0.999761i	$0.493042\pi$
$774$	0	0
$775$	−4.92820	−0.177026
$776$	0	0
$777$	8.92820	0.320298
$778$	0	0
$779$	4.92820	0.176571
$780$	0	0
$781$	24.7846	0.886863
$782$	0	0
$783$	−6.73205	−0.240584
$784$	0	0
$785$	46.2487	1.65069
$786$	0	0
$787$	31.8564	1.13556	0.567779	−	0.823181i	$-0.307803\pi$
$787$	31.8564	1.13556	0.567779	+	0.823181i	$0.307803\pi$
$788$	0	0
$789$	9.46410	0.336931
$790$	0	0
$791$	13.2679	0.471754
$792$	0	0
$793$	−70.6410	−2.50854
$794$	0	0
$795$	26.3923	0.936039
$796$	0	0
$797$	−26.7846	−0.948760	−0.474380	−	0.880320i	$-0.657328\pi$
$797$	−26.7846	−0.948760	−0.474380	+	0.880320i	$0.657328\pi$
$798$	0	0
$799$	68.6410	2.42834
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−41.5692	−1.46695
$804$	0	0
$805$	−18.9282	−0.667132
$806$	0	0
$807$	−20.5359	−0.722898
$808$	0	0
$809$	−5.71281	−0.200852	−0.100426	−	0.994945i	$-0.532021\pi$
$809$	−5.71281	−0.200852	−0.100426	+	0.994945i	$0.532021\pi$
$810$	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
$815$	−2.53590	−0.0888286
$816$	0	0
$817$	−8.92820	−0.312358
$818$	0	0
$819$	5.46410	0.190931
$820$	0	0
$821$	−6.67949	−0.233116	−0.116558	−	0.993184i	$-0.537186\pi$
$821$	−6.67949	−0.233116	−0.116558	+	0.993184i	$0.537186\pi$
$822$	0	0
$823$	−18.7846	−0.654790	−0.327395	−	0.944888i	$-0.606171\pi$
$823$	−18.7846	−0.654790	−0.327395	+	0.944888i	$0.606171\pi$
$824$	0	0
$825$	−9.85641	−0.343156
$826$	0	0
$827$	−34.9808	−1.21640	−0.608200	−	0.793784i	$-0.708108\pi$
$827$	−34.9808	−1.21640	−0.608200	+	0.793784i	$0.708108\pi$
$828$	0	0
$829$	45.7128	1.58767	0.793836	−	0.608132i	$-0.208081\pi$
$829$	45.7128	1.58767	0.793836	+	0.608132i	$0.208081\pi$
$830$	0	0
$831$	−3.60770	−0.125150
$832$	0	0
$833$	−6.73205	−0.233252
$834$	0	0
$835$	−6.92820	−0.239760
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−26.6410	−0.919750	−0.459875	−	0.887984i	$-0.652106\pi$
$839$	−26.6410	−0.919750	−0.459875	+	0.887984i	$0.652106\pi$
$840$	0	0
$841$	16.3205	0.562776
$842$	0	0
$843$	−1.26795	−0.0436705
$844$	0	0
$845$	−46.0526	−1.58426
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	12.3923	0.425303
$850$	0	0
$851$	61.8564	2.12041
$852$	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
$855$	−2.73205	−0.0934342
$856$	0	0
$857$	19.8564	0.678282	0.339141	−	0.940736i	$-0.389864\pi$
$857$	19.8564	0.678282	0.339141	+	0.940736i	$0.389864\pi$
$858$	0	0
$859$	10.1436	0.346095	0.173047	−	0.984913i	$-0.444639\pi$
$859$	10.1436	0.346095	0.173047	+	0.984913i	$0.444639\pi$
$860$	0	0
$861$	4.92820	0.167953
$862$	0	0
$863$	31.3731	1.06795	0.533976	−	0.845500i	$-0.320697\pi$
$863$	31.3731	1.06795	0.533976	+	0.845500i	$0.320697\pi$
$864$	0	0
$865$	−43.3205	−1.47294
$866$	0	0
$867$	28.3205	0.961815
$868$	0	0
$869$	43.7128	1.48286
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−15.8564	−0.536658
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	47.5692	1.60630	0.803149	−	0.595778i	$-0.203156\pi$
$877$	47.5692	1.60630	0.803149	+	0.595778i	$0.203156\pi$
$878$	0	0
$879$	−0.143594	−0.00484329
$880$	0	0
$881$	26.7321	0.900626	0.450313	−	0.892871i	$-0.351312\pi$
$881$	26.7321	0.900626	0.450313	+	0.892871i	$0.351312\pi$
$882$	0	0
$883$	12.7846	0.430236	0.215118	−	0.976588i	$-0.430986\pi$
$883$	12.7846	0.430236	0.215118	+	0.976588i	$0.430986\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	−21.1769	−0.711051	−0.355526	−	0.934667i	$-0.615698\pi$
$887$	−21.1769	−0.711051	−0.355526	+	0.934667i	$0.615698\pi$
$888$	0	0
$889$	−13.4641	−0.451571
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−10.1962	−0.341201
$894$	0	0
$895$	30.7846	1.02902
$896$	0	0
$897$	37.8564	1.26399
$898$	0	0
$899$	13.4641	0.449053
$900$	0	0
$901$	65.0333	2.16657
$902$	0	0
$903$	−8.92820	−0.297112
$904$	0	0
$905$	57.1769	1.90062
$906$	0	0
$907$	34.6410	1.15024	0.575118	−	0.818070i	$-0.304956\pi$
$907$	34.6410	1.15024	0.575118	+	0.818070i	$0.304956\pi$
$908$	0	0
$909$	−1.26795	−0.0420552
$910$	0	0
$911$	2.48334	0.0822767	0.0411384	−	0.999153i	$-0.486902\pi$
$911$	2.48334	0.0822767	0.0411384	+	0.999153i	$0.486902\pi$
$912$	0	0
$913$	8.78461	0.290728
$914$	0	0
$915$	35.3205	1.16766
$916$	0	0
$917$	7.26795	0.240009
$918$	0	0
$919$	51.7128	1.70585	0.852924	−	0.522035i	$-0.174827\pi$
$919$	51.7128	1.70585	0.852924	+	0.522035i	$0.174827\pi$
$920$	0	0
$921$	3.07180	0.101219
$922$	0	0
$923$	−33.8564	−1.11440
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	10.9282	0.358929
$928$	0	0
$929$	−4.58846	−0.150542	−0.0752712	−	0.997163i	$-0.523982\pi$
$929$	−4.58846	−0.150542	−0.0752712	+	0.997163i	$0.523982\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	17.1244	0.560626
$934$	0	0
$935$	−73.5692	−2.40597
$936$	0	0
$937$	39.0718	1.27642	0.638210	−	0.769862i	$-0.279675\pi$
$937$	39.0718	1.27642	0.638210	+	0.769862i	$0.279675\pi$
$938$	0	0
$939$	32.9282	1.07457
$940$	0	0
$941$	9.60770	0.313202	0.156601	−	0.987662i	$-0.449946\pi$
$941$	9.60770	0.313202	0.156601	+	0.987662i	$0.449946\pi$
$942$	0	0
$943$	34.1436	1.11187
$944$	0	0
$945$	−2.73205	−0.0888736
$946$	0	0
$947$	37.8564	1.23017	0.615084	−	0.788462i	$-0.289122\pi$
$947$	37.8564	1.23017	0.615084	+	0.788462i	$0.289122\pi$
$948$	0	0
$949$	56.7846	1.84331
$950$	0	0
$951$	−22.7321	−0.737137
$952$	0	0
$953$	−32.5885	−1.05564	−0.527822	−	0.849355i	$-0.676991\pi$
$953$	−32.5885	−1.05564	−0.527822	+	0.849355i	$0.676991\pi$
$954$	0	0
$955$	−13.8564	−0.448383
$956$	0	0
$957$	26.9282	0.870465
$958$	0	0
$959$	−8.92820	−0.288307
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−1.80385	−0.0581282
$964$	0	0
$965$	13.4641	0.433425
$966$	0	0
$967$	10.7846	0.346810	0.173405	−	0.984851i	$-0.444523\pi$
$967$	10.7846	0.346810	0.173405	+	0.984851i	$0.444523\pi$
$968$	0	0
$969$	−6.73205	−0.216265
$970$	0	0
$971$	−56.7846	−1.82230	−0.911152	−	0.412069i	$-0.864806\pi$
$971$	−56.7846	−1.82230	−0.911152	+	0.412069i	$0.864806\pi$
$972$	0	0
$973$	−15.3205	−0.491153
$974$	0	0
$975$	13.4641	0.431196
$976$	0	0
$977$	−19.5167	−0.624393	−0.312197	−	0.950018i	$-0.601065\pi$
$977$	−19.5167	−0.624393	−0.312197	+	0.950018i	$0.601065\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−11.4641	−0.366021
$982$	0	0
$983$	−32.7846	−1.04567	−0.522833	−	0.852435i	$-0.675125\pi$
$983$	−32.7846	−1.04567	−0.522833	+	0.852435i	$0.675125\pi$
$984$	0	0
$985$	35.3205	1.12540
$986$	0	0
$987$	−10.1962	−0.324547
$988$	0	0
$989$	−61.8564	−1.96692
$990$	0	0
$991$	38.6410	1.22747	0.613736	−	0.789511i	$-0.289666\pi$
$991$	38.6410	1.22747	0.613736	+	0.789511i	$0.289666\pi$
$992$	0	0
$993$	−12.3923	−0.393258
$994$	0	0
$995$	41.8564	1.32694
$996$	0	0
$997$	−56.9282	−1.80293	−0.901467	−	0.432848i	$-0.857509\pi$
$997$	−56.9282	−1.80293	−0.901467	+	0.432848i	$0.857509\pi$
$998$	0	0
$999$	8.92820	0.282476

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +5.46410 q^{13} -2.73205 q^{15} -6.73205 q^{17} +1.00000 q^{19} +1.00000 q^{21} +6.92820 q^{23} +2.46410 q^{25} +1.00000 q^{27} -6.73205 q^{29} -2.00000 q^{31} -4.00000 q^{33} -2.73205 q^{35} +8.92820 q^{37} +5.46410 q^{39} +4.92820 q^{41} -8.92820 q^{43} -2.73205 q^{45} -10.1962 q^{47} +1.00000 q^{49} -6.73205 q^{51} -9.66025 q^{53} +10.9282 q^{55} +1.00000 q^{57} +2.92820 q^{59} -12.9282 q^{61} +1.00000 q^{63} -14.9282 q^{65} -1.46410 q^{67} +6.92820 q^{69} -6.19615 q^{71} +10.3923 q^{73} +2.46410 q^{75} -4.00000 q^{77} -10.9282 q^{79} +1.00000 q^{81} -2.19615 q^{83} +18.3923 q^{85} -6.73205 q^{87} -6.00000 q^{89} +5.46410 q^{91} -2.00000 q^{93} -2.73205 q^{95} -15.8564 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more