LMFDB - Embedded newform 4056.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +2.60555 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +2.60555 q^{7} +1.00000 q^{9} -2.60555 q^{11} +1.00000 q^{15} +1.60555 q^{17} +6.60555 q^{19} +2.60555 q^{21} +2.60555 q^{23} -4.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} -5.21110 q^{31} -2.60555 q^{33} +2.60555 q^{35} +2.39445 q^{37} +11.6056 q^{41} -2.60555 q^{43} +1.00000 q^{45} +1.39445 q^{47} -0.211103 q^{49} +1.60555 q^{51} +3.00000 q^{53} -2.60555 q^{55} +6.60555 q^{57} -9.21110 q^{59} -7.60555 q^{61} +2.60555 q^{63} +10.6056 q^{67} +2.60555 q^{69} +9.39445 q^{71} -7.00000 q^{73} -4.00000 q^{75} -6.78890 q^{77} +12.0000 q^{79} +1.00000 q^{81} +11.8167 q^{83} +1.60555 q^{85} +3.00000 q^{87} +16.4222 q^{89} -5.21110 q^{93} +6.60555 q^{95} -11.2111 q^{97} -2.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} - 2 q^{21} - 2 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} - 2 q^{35} + 12 q^{37} + 16 q^{41} + 2 q^{43} + 2 q^{45} + 10 q^{47} + 14 q^{49} - 4 q^{51} + 6 q^{53} + 2 q^{55} + 6 q^{57} - 4 q^{59} - 8 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 26 q^{71} - 14 q^{73} - 8 q^{75} - 28 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{83} - 4 q^{85} + 6 q^{87} + 4 q^{89} + 4 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 6 * q^19 - 2 * q^21 - 2 * q^23 - 8 * q^25 + 2 * q^27 + 6 * q^29 + 4 * q^31 + 2 * q^33 - 2 * q^35 + 12 * q^37 + 16 * q^41 + 2 * q^43 + 2 * q^45 + 10 * q^47 + 14 * q^49 - 4 * q^51 + 6 * q^53 + 2 * q^55 + 6 * q^57 - 4 * q^59 - 8 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 26 * q^71 - 14 * q^73 - 8 * q^75 - 28 * q^77 + 24 * q^79 + 2 * q^81 + 2 * q^83 - 4 * q^85 + 6 * q^87 + 4 * q^89 + 4 * q^93 + 6 * q^95 - 8 * q^97 + 2 * 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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	2.60555	0.984806	0.492403	−	0.870367i	$-0.336119\pi$
$7$	2.60555	0.984806	0.492403	+	0.870367i	$0.336119\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.60555	−0.785603	−0.392802	−	0.919623i	$-0.628494\pi$
$11$	−2.60555	−0.785603	−0.392802	+	0.919623i	$0.628494\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.60555	0.389403	0.194702	−	0.980863i	$-0.437626\pi$
$17$	1.60555	0.389403	0.194702	+	0.980863i	$0.437626\pi$
$18$	0	0
$19$	6.60555	1.51542	0.757709	−	0.652593i	$-0.226319\pi$
$19$	6.60555	1.51542	0.757709	+	0.652593i	$0.226319\pi$
$20$	0	0
$21$	2.60555	0.568578
$22$	0	0
$23$	2.60555	0.543295	0.271647	−	0.962397i	$-0.412432\pi$
$23$	2.60555	0.543295	0.271647	+	0.962397i	$0.412432\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−5.21110	−0.935942	−0.467971	−	0.883744i	$-0.655015\pi$
$31$	−5.21110	−0.935942	−0.467971	+	0.883744i	$0.655015\pi$
$32$	0	0
$33$	−2.60555	−0.453568
$34$	0	0
$35$	2.60555	0.440419
$36$	0	0
$37$	2.39445	0.393645	0.196822	−	0.980439i	$-0.436938\pi$
$37$	2.39445	0.393645	0.196822	+	0.980439i	$0.436938\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.6056	1.81248	0.906241	−	0.422761i	$-0.138939\pi$
$41$	11.6056	1.81248	0.906241	+	0.422761i	$0.138939\pi$
$42$	0	0
$43$	−2.60555	−0.397343	−0.198671	−	0.980066i	$-0.563663\pi$
$43$	−2.60555	−0.397343	−0.198671	+	0.980066i	$0.563663\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	1.39445	0.203401	0.101701	−	0.994815i	$-0.467572\pi$
$47$	1.39445	0.203401	0.101701	+	0.994815i	$0.467572\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	1.60555	0.224822
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−2.60555	−0.351332
$56$	0	0
$57$	6.60555	0.874927
$58$	0	0
$59$	−9.21110	−1.19918	−0.599592	−	0.800306i	$-0.704670\pi$
$59$	−9.21110	−1.19918	−0.599592	+	0.800306i	$0.704670\pi$
$60$	0	0
$61$	−7.60555	−0.973791	−0.486896	−	0.873460i	$-0.661871\pi$
$61$	−7.60555	−0.973791	−0.486896	+	0.873460i	$0.661871\pi$
$62$	0	0
$63$	2.60555	0.328269
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.6056	1.29567	0.647837	−	0.761779i	$-0.275674\pi$
$67$	10.6056	1.29567	0.647837	+	0.761779i	$0.275674\pi$
$68$	0	0
$69$	2.60555	0.313672
$70$	0	0
$71$	9.39445	1.11492	0.557458	−	0.830205i	$-0.311777\pi$
$71$	9.39445	1.11492	0.557458	+	0.830205i	$0.311777\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−6.78890	−0.773667
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.8167	1.29705	0.648523	−	0.761195i	$-0.275387\pi$
$83$	11.8167	1.29705	0.648523	+	0.761195i	$0.275387\pi$
$84$	0	0
$85$	1.60555	0.174146
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	16.4222	1.74075	0.870375	−	0.492389i	$-0.163876\pi$
$89$	16.4222	1.74075	0.870375	+	0.492389i	$0.163876\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.21110	−0.540366
$94$	0	0
$95$	6.60555	0.677715
$96$	0	0
$97$	−11.2111	−1.13831	−0.569157	−	0.822229i	$-0.692731\pi$
$97$	−11.2111	−1.13831	−0.569157	+	0.822229i	$0.692731\pi$
$98$	0	0
$99$	−2.60555	−0.261868
$100$	0	0
$101$	4.21110	0.419020	0.209510	−	0.977806i	$-0.432813\pi$
$101$	4.21110	0.419020	0.209510	+	0.977806i	$0.432813\pi$
$102$	0	0
$103$	−15.8167	−1.55846	−0.779231	−	0.626737i	$-0.784390\pi$
$103$	−15.8167	−1.55846	−0.779231	+	0.626737i	$0.784390\pi$
$104$	0	0
$105$	2.60555	0.254276
$106$	0	0
$107$	6.60555	0.638583	0.319291	−	0.947657i	$-0.396555\pi$
$107$	6.60555	0.638583	0.319291	+	0.947657i	$0.396555\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.39445	0.227271
$112$	0	0
$113$	−11.6056	−1.09176	−0.545879	−	0.837864i	$-0.683804\pi$
$113$	−11.6056	−1.09176	−0.545879	+	0.837864i	$0.683804\pi$
$114$	0	0
$115$	2.60555	0.242969
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.18335	0.383487
$120$	0	0
$121$	−4.21110	−0.382828
$122$	0	0
$123$	11.6056	1.04644
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	−2.60555	−0.229406
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	17.2111	1.49239
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	16.8167	1.43674	0.718372	−	0.695659i	$-0.244888\pi$
$137$	16.8167	1.43674	0.718372	+	0.695659i	$0.244888\pi$
$138$	0	0
$139$	−2.78890	−0.236551	−0.118276	−	0.992981i	$-0.537737\pi$
$139$	−2.78890	−0.236551	−0.118276	+	0.992981i	$0.537737\pi$
$140$	0	0
$141$	1.39445	0.117434
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	−0.211103	−0.0174114
$148$	0	0
$149$	10.2111	0.836526	0.418263	−	0.908326i	$-0.362639\pi$
$149$	10.2111	0.836526	0.418263	+	0.908326i	$0.362639\pi$
$150$	0	0
$151$	6.60555	0.537552	0.268776	−	0.963203i	$-0.413381\pi$
$151$	6.60555	0.537552	0.268776	+	0.963203i	$0.413381\pi$
$152$	0	0
$153$	1.60555	0.129801
$154$	0	0
$155$	−5.21110	−0.418566
$156$	0	0
$157$	8.39445	0.669950	0.334975	−	0.942227i	$-0.391272\pi$
$157$	8.39445	0.669950	0.334975	+	0.942227i	$0.391272\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	6.78890	0.535040
$162$	0	0
$163$	5.21110	0.408165	0.204083	−	0.978954i	$-0.434579\pi$
$163$	5.21110	0.408165	0.204083	+	0.978954i	$0.434579\pi$
$164$	0	0
$165$	−2.60555	−0.202842
$166$	0	0
$167$	−17.2111	−1.33184	−0.665918	−	0.746025i	$-0.731960\pi$
$167$	−17.2111	−1.33184	−0.665918	+	0.746025i	$0.731960\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	6.60555	0.505139
$172$	0	0
$173$	−21.6333	−1.64475	−0.822375	−	0.568946i	$-0.807351\pi$
$173$	−21.6333	−1.64475	−0.822375	+	0.568946i	$0.807351\pi$
$174$	0	0
$175$	−10.4222	−0.787845
$176$	0	0
$177$	−9.21110	−0.692349
$178$	0	0
$179$	−17.0278	−1.27271	−0.636357	−	0.771395i	$-0.719560\pi$
$179$	−17.0278	−1.27271	−0.636357	+	0.771395i	$0.719560\pi$
$180$	0	0
$181$	−24.8167	−1.84461	−0.922304	−	0.386466i	$-0.873696\pi$
$181$	−24.8167	−1.84461	−0.922304	+	0.386466i	$0.873696\pi$
$182$	0	0
$183$	−7.60555	−0.562219
$184$	0	0
$185$	2.39445	0.176043
$186$	0	0
$187$	−4.18335	−0.305917
$188$	0	0
$189$	2.60555	0.189526
$190$	0	0
$191$	−23.6333	−1.71005	−0.855023	−	0.518590i	$-0.826457\pi$
$191$	−23.6333	−1.71005	−0.855023	+	0.518590i	$0.826457\pi$
$192$	0	0
$193$	7.78890	0.560657	0.280329	−	0.959904i	$-0.409557\pi$
$193$	7.78890	0.560657	0.280329	+	0.959904i	$0.409557\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	19.8167	1.40477	0.702383	−	0.711800i	$-0.252120\pi$
$199$	19.8167	1.40477	0.702383	+	0.711800i	$0.252120\pi$
$200$	0	0
$201$	10.6056	0.748058
$202$	0	0
$203$	7.81665	0.548622
$204$	0	0
$205$	11.6056	0.810567
$206$	0	0
$207$	2.60555	0.181098
$208$	0	0
$209$	−17.2111	−1.19052
$210$	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
$213$	9.39445	0.643697
$214$	0	0
$215$	−2.60555	−0.177697
$216$	0	0
$217$	−13.5778	−0.921721
$218$	0	0
$219$	−7.00000	−0.473016
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	26.6056	1.76587	0.882936	−	0.469493i	$-0.155563\pi$
$227$	26.6056	1.76587	0.882936	+	0.469493i	$0.155563\pi$
$228$	0	0
$229$	−15.2111	−1.00518	−0.502589	−	0.864525i	$-0.667619\pi$
$229$	−15.2111	−1.00518	−0.502589	+	0.864525i	$0.667619\pi$
$230$	0	0
$231$	−6.78890	−0.446677
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	1.39445	0.0909638
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	−17.0278	−1.10143	−0.550717	−	0.834692i	$-0.685646\pi$
$239$	−17.0278	−1.10143	−0.550717	+	0.834692i	$0.685646\pi$
$240$	0	0
$241$	−14.6333	−0.942614	−0.471307	−	0.881969i	$-0.656218\pi$
$241$	−14.6333	−0.942614	−0.471307	+	0.881969i	$0.656218\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.211103	−0.0134868
$246$	0	0
$247$	0	0
$248$	0	0
$249$	11.8167	0.748850
$250$	0	0
$251$	29.2111	1.84379	0.921894	−	0.387442i	$-0.126641\pi$
$251$	29.2111	1.84379	0.921894	+	0.387442i	$0.126641\pi$
$252$	0	0
$253$	−6.78890	−0.426814
$254$	0	0
$255$	1.60555	0.100544
$256$	0	0
$257$	−23.6056	−1.47247	−0.736237	−	0.676724i	$-0.763399\pi$
$257$	−23.6056	−1.47247	−0.736237	+	0.676724i	$0.763399\pi$
$258$	0	0
$259$	6.23886	0.387664
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	18.6056	1.14727	0.573634	−	0.819112i	$-0.305533\pi$
$263$	18.6056	1.14727	0.573634	+	0.819112i	$0.305533\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	16.4222	1.00502
$268$	0	0
$269$	25.6333	1.56289	0.781445	−	0.623974i	$-0.214483\pi$
$269$	25.6333	1.56289	0.781445	+	0.623974i	$0.214483\pi$
$270$	0	0
$271$	1.21110	0.0735692	0.0367846	−	0.999323i	$-0.488288\pi$
$271$	1.21110	0.0735692	0.0367846	+	0.999323i	$0.488288\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.4222	0.628483
$276$	0	0
$277$	29.2389	1.75679	0.878396	−	0.477933i	$-0.158614\pi$
$277$	29.2389	1.75679	0.878396	+	0.477933i	$0.158614\pi$
$278$	0	0
$279$	−5.21110	−0.311981
$280$	0	0
$281$	−0.394449	−0.0235308	−0.0117654	−	0.999931i	$-0.503745\pi$
$281$	−0.394449	−0.0235308	−0.0117654	+	0.999931i	$0.503745\pi$
$282$	0	0
$283$	14.6056	0.868210	0.434105	−	0.900862i	$-0.357065\pi$
$283$	14.6056	0.868210	0.434105	+	0.900862i	$0.357065\pi$
$284$	0	0
$285$	6.60555	0.391279
$286$	0	0
$287$	30.2389	1.78494
$288$	0	0
$289$	−14.4222	−0.848365
$290$	0	0
$291$	−11.2111	−0.657206
$292$	0	0
$293$	−0.211103	−0.0123327	−0.00616637	−	0.999981i	$-0.501963\pi$
$293$	−0.211103	−0.0123327	−0.00616637	+	0.999981i	$0.501963\pi$
$294$	0	0
$295$	−9.21110	−0.536291
$296$	0	0
$297$	−2.60555	−0.151189
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−6.78890	−0.391306
$302$	0	0
$303$	4.21110	0.241922
$304$	0	0
$305$	−7.60555	−0.435493
$306$	0	0
$307$	−5.39445	−0.307877	−0.153939	−	0.988080i	$-0.549196\pi$
$307$	−5.39445	−0.307877	−0.153939	+	0.988080i	$0.549196\pi$
$308$	0	0
$309$	−15.8167	−0.899778
$310$	0	0
$311$	27.8167	1.57734	0.788669	−	0.614818i	$-0.210771\pi$
$311$	27.8167	1.57734	0.788669	+	0.614818i	$0.210771\pi$
$312$	0	0
$313$	3.21110	0.181502	0.0907511	−	0.995874i	$-0.471073\pi$
$313$	3.21110	0.181502	0.0907511	+	0.995874i	$0.471073\pi$
$314$	0	0
$315$	2.60555	0.146806
$316$	0	0
$317$	2.21110	0.124188	0.0620939	−	0.998070i	$-0.480222\pi$
$317$	2.21110	0.124188	0.0620939	+	0.998070i	$0.480222\pi$
$318$	0	0
$319$	−7.81665	−0.437649
$320$	0	0
$321$	6.60555	0.368686
$322$	0	0
$323$	10.6056	0.590109
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	3.63331	0.200311
$330$	0	0
$331$	15.6333	0.859284	0.429642	−	0.902999i	$-0.358640\pi$
$331$	15.6333	0.859284	0.429642	+	0.902999i	$0.358640\pi$
$332$	0	0
$333$	2.39445	0.131215
$334$	0	0
$335$	10.6056	0.579443
$336$	0	0
$337$	−2.57779	−0.140421	−0.0702107	−	0.997532i	$-0.522367\pi$
$337$	−2.57779	−0.140421	−0.0702107	+	0.997532i	$0.522367\pi$
$338$	0	0
$339$	−11.6056	−0.630327
$340$	0	0
$341$	13.5778	0.735279
$342$	0	0
$343$	−18.7889	−1.01451
$344$	0	0
$345$	2.60555	0.140278
$346$	0	0
$347$	17.0278	0.914098	0.457049	−	0.889442i	$-0.348906\pi$
$347$	17.0278	0.914098	0.457049	+	0.889442i	$0.348906\pi$
$348$	0	0
$349$	36.4222	1.94964	0.974818	−	0.223002i	$-0.0715857\pi$
$349$	36.4222	1.94964	0.974818	+	0.223002i	$0.0715857\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.0278	−0.853071	−0.426536	−	0.904471i	$-0.640266\pi$
$353$	−16.0278	−0.853071	−0.426536	+	0.904471i	$0.640266\pi$
$354$	0	0
$355$	9.39445	0.498606
$356$	0	0
$357$	4.18335	0.221406
$358$	0	0
$359$	−7.81665	−0.412547	−0.206274	−	0.978494i	$-0.566134\pi$
$359$	−7.81665	−0.412547	−0.206274	+	0.978494i	$0.566134\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	0	0
$363$	−4.21110	−0.221026
$364$	0	0
$365$	−7.00000	−0.366397
$366$	0	0
$367$	−1.39445	−0.0727896	−0.0363948	−	0.999337i	$-0.511587\pi$
$367$	−1.39445	−0.0727896	−0.0363948	+	0.999337i	$0.511587\pi$
$368$	0	0
$369$	11.6056	0.604161
$370$	0	0
$371$	7.81665	0.405820
$372$	0	0
$373$	−30.0278	−1.55478	−0.777389	−	0.629020i	$-0.783456\pi$
$373$	−30.0278	−1.55478	−0.777389	+	0.629020i	$0.783456\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.42221	−0.329887	−0.164943	−	0.986303i	$-0.552744\pi$
$379$	−6.42221	−0.329887	−0.164943	+	0.986303i	$0.552744\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	18.4222	0.941331	0.470665	−	0.882312i	$-0.344014\pi$
$383$	18.4222	0.941331	0.470665	+	0.882312i	$0.344014\pi$
$384$	0	0
$385$	−6.78890	−0.345994
$386$	0	0
$387$	−2.60555	−0.132448
$388$	0	0
$389$	−36.6333	−1.85738	−0.928691	−	0.370854i	$-0.879065\pi$
$389$	−36.6333	−1.85738	−0.928691	+	0.370854i	$0.879065\pi$
$390$	0	0
$391$	4.18335	0.211561
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	17.2111	0.861633
$400$	0	0
$401$	−26.4500	−1.32085	−0.660424	−	0.750893i	$-0.729623\pi$
$401$	−26.4500	−1.32085	−0.660424	+	0.750893i	$0.729623\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−6.23886	−0.309249
$408$	0	0
$409$	−4.57779	−0.226357	−0.113179	−	0.993575i	$-0.536103\pi$
$409$	−4.57779	−0.226357	−0.113179	+	0.993575i	$0.536103\pi$
$410$	0	0
$411$	16.8167	0.829504
$412$	0	0
$413$	−24.0000	−1.18096
$414$	0	0
$415$	11.8167	0.580057
$416$	0	0
$417$	−2.78890	−0.136573
$418$	0	0
$419$	−18.4222	−0.899984	−0.449992	−	0.893033i	$-0.648573\pi$
$419$	−18.4222	−0.899984	−0.449992	+	0.893033i	$0.648573\pi$
$420$	0	0
$421$	−24.3944	−1.18891	−0.594456	−	0.804128i	$-0.702633\pi$
$421$	−24.3944	−1.18891	−0.594456	+	0.804128i	$0.702633\pi$
$422$	0	0
$423$	1.39445	0.0678004
$424$	0	0
$425$	−6.42221	−0.311523
$426$	0	0
$427$	−19.8167	−0.958995
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8167	0.761861	0.380931	−	0.924604i	$-0.375604\pi$
$431$	15.8167	0.761861	0.380931	+	0.924604i	$0.375604\pi$
$432$	0	0
$433$	−35.4222	−1.70228	−0.851141	−	0.524937i	$-0.824089\pi$
$433$	−35.4222	−1.70228	−0.851141	+	0.524937i	$0.824089\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	17.2111	0.823319
$438$	0	0
$439$	35.8167	1.70944	0.854718	−	0.519093i	$-0.173730\pi$
$439$	35.8167	1.70944	0.854718	+	0.519093i	$0.173730\pi$
$440$	0	0
$441$	−0.211103	−0.0100525
$442$	0	0
$443$	−15.6333	−0.742761	−0.371380	−	0.928481i	$-0.621115\pi$
$443$	−15.6333	−0.742761	−0.371380	+	0.928481i	$0.621115\pi$
$444$	0	0
$445$	16.4222	0.778487
$446$	0	0
$447$	10.2111	0.482969
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−30.2389	−1.42389
$452$	0	0
$453$	6.60555	0.310356
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.6333	−1.62008	−0.810039	−	0.586376i	$-0.800554\pi$
$457$	−34.6333	−1.62008	−0.810039	+	0.586376i	$0.800554\pi$
$458$	0	0
$459$	1.60555	0.0749407
$460$	0	0
$461$	−18.6333	−0.867840	−0.433920	−	0.900951i	$-0.642870\pi$
$461$	−18.6333	−0.867840	−0.433920	+	0.900951i	$0.642870\pi$
$462$	0	0
$463$	4.18335	0.194417	0.0972083	−	0.995264i	$-0.469009\pi$
$463$	4.18335	0.194417	0.0972083	+	0.995264i	$0.469009\pi$
$464$	0	0
$465$	−5.21110	−0.241659
$466$	0	0
$467$	8.18335	0.378680	0.189340	−	0.981912i	$-0.439365\pi$
$467$	8.18335	0.378680	0.189340	+	0.981912i	$0.439365\pi$
$468$	0	0
$469$	27.6333	1.27599
$470$	0	0
$471$	8.39445	0.386796
$472$	0	0
$473$	6.78890	0.312154
$474$	0	0
$475$	−26.4222	−1.21233
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	3.63331	0.166010	0.0830050	−	0.996549i	$-0.473548\pi$
$479$	3.63331	0.166010	0.0830050	+	0.996549i	$0.473548\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	6.78890	0.308906
$484$	0	0
$485$	−11.2111	−0.509070
$486$	0	0
$487$	−30.6056	−1.38687	−0.693435	−	0.720519i	$-0.743904\pi$
$487$	−30.6056	−1.38687	−0.693435	+	0.720519i	$0.743904\pi$
$488$	0	0
$489$	5.21110	0.235654
$490$	0	0
$491$	19.8167	0.894313	0.447157	−	0.894456i	$-0.352437\pi$
$491$	19.8167	0.894313	0.447157	+	0.894456i	$0.352437\pi$
$492$	0	0
$493$	4.81665	0.216931
$494$	0	0
$495$	−2.60555	−0.117111
$496$	0	0
$497$	24.4777	1.09798
$498$	0	0
$499$	−33.2111	−1.48673	−0.743367	−	0.668884i	$-0.766772\pi$
$499$	−33.2111	−1.48673	−0.743367	+	0.668884i	$0.766772\pi$
$500$	0	0
$501$	−17.2111	−0.768935
$502$	0	0
$503$	15.8167	0.705230	0.352615	−	0.935769i	$-0.385293\pi$
$503$	15.8167	0.705230	0.352615	+	0.935769i	$0.385293\pi$
$504$	0	0
$505$	4.21110	0.187392
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.63331	0.205368	0.102684	−	0.994714i	$-0.467257\pi$
$509$	4.63331	0.205368	0.102684	+	0.994714i	$0.467257\pi$
$510$	0	0
$511$	−18.2389	−0.806840
$512$	0	0
$513$	6.60555	0.291642
$514$	0	0
$515$	−15.8167	−0.696965
$516$	0	0
$517$	−3.63331	−0.159793
$518$	0	0
$519$	−21.6333	−0.949597
$520$	0	0
$521$	−23.2389	−1.01811	−0.509056	−	0.860733i	$-0.670006\pi$
$521$	−23.2389	−1.01811	−0.509056	+	0.860733i	$0.670006\pi$
$522$	0	0
$523$	1.02776	0.0449406	0.0224703	−	0.999748i	$-0.492847\pi$
$523$	1.02776	0.0449406	0.0224703	+	0.999748i	$0.492847\pi$
$524$	0	0
$525$	−10.4222	−0.454862
$526$	0	0
$527$	−8.36669	−0.364459
$528$	0	0
$529$	−16.2111	−0.704831
$530$	0	0
$531$	−9.21110	−0.399728
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.60555	0.285583
$536$	0	0
$537$	−17.0278	−0.734802
$538$	0	0
$539$	0.550039	0.0236918
$540$	0	0
$541$	10.3944	0.446892	0.223446	−	0.974716i	$-0.428269\pi$
$541$	10.3944	0.446892	0.223446	+	0.974716i	$0.428269\pi$
$542$	0	0
$543$	−24.8167	−1.06498
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	17.0278	0.728054	0.364027	−	0.931388i	$-0.381402\pi$
$547$	17.0278	0.728054	0.364027	+	0.931388i	$0.381402\pi$
$548$	0	0
$549$	−7.60555	−0.324597
$550$	0	0
$551$	19.8167	0.844218
$552$	0	0
$553$	31.2666	1.32959
$554$	0	0
$555$	2.39445	0.101639
$556$	0	0
$557$	29.8444	1.26455	0.632274	−	0.774745i	$-0.282122\pi$
$557$	29.8444	1.26455	0.632274	+	0.774745i	$0.282122\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−4.18335	−0.176621
$562$	0	0
$563$	27.6333	1.16461	0.582303	−	0.812972i	$-0.302152\pi$
$563$	27.6333	1.16461	0.582303	+	0.812972i	$0.302152\pi$
$564$	0	0
$565$	−11.6056	−0.488249
$566$	0	0
$567$	2.60555	0.109423
$568$	0	0
$569$	−23.2111	−0.973060	−0.486530	−	0.873664i	$-0.661738\pi$
$569$	−23.2111	−0.973060	−0.486530	+	0.873664i	$0.661738\pi$
$570$	0	0
$571$	−19.4500	−0.813956	−0.406978	−	0.913438i	$-0.633417\pi$
$571$	−19.4500	−0.813956	−0.406978	+	0.913438i	$0.633417\pi$
$572$	0	0
$573$	−23.6333	−0.987296
$574$	0	0
$575$	−10.4222	−0.434636
$576$	0	0
$577$	26.2111	1.09118	0.545591	−	0.838051i	$-0.316305\pi$
$577$	26.2111	1.09118	0.545591	+	0.838051i	$0.316305\pi$
$578$	0	0
$579$	7.78890	0.323696
$580$	0	0
$581$	30.7889	1.27734
$582$	0	0
$583$	−7.81665	−0.323733
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.4222	−1.75095	−0.875476	−	0.483262i	$-0.839452\pi$
$587$	−42.4222	−1.75095	−0.875476	+	0.483262i	$0.839452\pi$
$588$	0	0
$589$	−34.4222	−1.41834
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	−16.0278	−0.658181	−0.329091	−	0.944298i	$-0.606742\pi$
$593$	−16.0278	−0.658181	−0.329091	+	0.944298i	$0.606742\pi$
$594$	0	0
$595$	4.18335	0.171500
$596$	0	0
$597$	19.8167	0.811042
$598$	0	0
$599$	−26.4222	−1.07958	−0.539791	−	0.841799i	$-0.681497\pi$
$599$	−26.4222	−1.07958	−0.539791	+	0.841799i	$0.681497\pi$
$600$	0	0
$601$	−39.4222	−1.60807	−0.804033	−	0.594585i	$-0.797316\pi$
$601$	−39.4222	−1.60807	−0.804033	+	0.594585i	$0.797316\pi$
$602$	0	0
$603$	10.6056	0.431891
$604$	0	0
$605$	−4.21110	−0.171206
$606$	0	0
$607$	30.0555	1.21992	0.609958	−	0.792434i	$-0.291186\pi$
$607$	30.0555	1.21992	0.609958	+	0.792434i	$0.291186\pi$
$608$	0	0
$609$	7.81665	0.316747
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−13.6056	−0.549523	−0.274762	−	0.961512i	$-0.588599\pi$
$613$	−13.6056	−0.549523	−0.274762	+	0.961512i	$0.588599\pi$
$614$	0	0
$615$	11.6056	0.467981
$616$	0	0
$617$	−12.3944	−0.498982	−0.249491	−	0.968377i	$-0.580263\pi$
$617$	−12.3944	−0.498982	−0.249491	+	0.968377i	$0.580263\pi$
$618$	0	0
$619$	−19.6333	−0.789129	−0.394565	−	0.918868i	$-0.629105\pi$
$619$	−19.6333	−0.789129	−0.394565	+	0.918868i	$0.629105\pi$
$620$	0	0
$621$	2.60555	0.104557
$622$	0	0
$623$	42.7889	1.71430
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−17.2111	−0.687345
$628$	0	0
$629$	3.84441	0.153287
$630$	0	0
$631$	26.7889	1.06645	0.533225	−	0.845974i	$-0.320980\pi$
$631$	26.7889	1.06645	0.533225	+	0.845974i	$0.320980\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.39445	0.371639
$640$	0	0
$641$	−26.3944	−1.04252	−0.521259	−	0.853399i	$-0.674537\pi$
$641$	−26.3944	−1.04252	−0.521259	+	0.853399i	$0.674537\pi$
$642$	0	0
$643$	−43.6333	−1.72073	−0.860365	−	0.509679i	$-0.829764\pi$
$643$	−43.6333	−1.72073	−0.860365	+	0.509679i	$0.829764\pi$
$644$	0	0
$645$	−2.60555	−0.102593
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	−13.5778	−0.532156
$652$	0	0
$653$	−21.6333	−0.846577	−0.423288	−	0.905995i	$-0.639124\pi$
$653$	−21.6333	−0.846577	−0.423288	+	0.905995i	$0.639124\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.00000	−0.273096
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	20.8167	0.809674	0.404837	−	0.914389i	$-0.367328\pi$
$661$	20.8167	0.809674	0.404837	+	0.914389i	$0.367328\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17.2111	0.667418
$666$	0	0
$667$	7.81665	0.302662
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	19.8167	0.765013
$672$	0	0
$673$	15.0000	0.578208	0.289104	−	0.957298i	$-0.406643\pi$
$673$	15.0000	0.578208	0.289104	+	0.957298i	$0.406643\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	28.0555	1.07826	0.539130	−	0.842222i	$-0.318753\pi$
$677$	28.0555	1.07826	0.539130	+	0.842222i	$0.318753\pi$
$678$	0	0
$679$	−29.2111	−1.12102
$680$	0	0
$681$	26.6056	1.01953
$682$	0	0
$683$	−18.4222	−0.704906	−0.352453	−	0.935829i	$-0.614652\pi$
$683$	−18.4222	−0.704906	−0.352453	+	0.935829i	$0.614652\pi$
$684$	0	0
$685$	16.8167	0.642531
$686$	0	0
$687$	−15.2111	−0.580340
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.60555	0.251287	0.125644	−	0.992075i	$-0.459900\pi$
$691$	6.60555	0.251287	0.125644	+	0.992075i	$0.459900\pi$
$692$	0	0
$693$	−6.78890	−0.257889
$694$	0	0
$695$	−2.78890	−0.105789
$696$	0	0
$697$	18.6333	0.705787
$698$	0	0
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−6.36669	−0.240467	−0.120233	−	0.992746i	$-0.538364\pi$
$701$	−6.36669	−0.240467	−0.120233	+	0.992746i	$0.538364\pi$
$702$	0	0
$703$	15.8167	0.596536
$704$	0	0
$705$	1.39445	0.0525180
$706$	0	0
$707$	10.9722	0.412654
$708$	0	0
$709$	−32.3944	−1.21660	−0.608300	−	0.793708i	$-0.708148\pi$
$709$	−32.3944	−1.21660	−0.608300	+	0.793708i	$0.708148\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	−13.5778	−0.508492
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−17.0278	−0.635913
$718$	0	0
$719$	−14.7889	−0.551533	−0.275766	−	0.961225i	$-0.588932\pi$
$719$	−14.7889	−0.551533	−0.275766	+	0.961225i	$0.588932\pi$
$720$	0	0
$721$	−41.2111	−1.53478
$722$	0	0
$723$	−14.6333	−0.544219
$724$	0	0
$725$	−12.0000	−0.445669
$726$	0	0
$727$	−15.8167	−0.586607	−0.293304	−	0.956019i	$-0.594755\pi$
$727$	−15.8167	−0.586607	−0.293304	+	0.956019i	$0.594755\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.18335	−0.154727
$732$	0	0
$733$	−33.2389	−1.22771	−0.613853	−	0.789421i	$-0.710381\pi$
$733$	−33.2389	−1.22771	−0.613853	+	0.789421i	$0.710381\pi$
$734$	0	0
$735$	−0.211103	−0.00778663
$736$	0	0
$737$	−27.6333	−1.01789
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.7889	−0.395806	−0.197903	−	0.980222i	$-0.563413\pi$
$743$	−10.7889	−0.395806	−0.197903	+	0.980222i	$0.563413\pi$
$744$	0	0
$745$	10.2111	0.374106
$746$	0	0
$747$	11.8167	0.432349
$748$	0	0
$749$	17.2111	0.628880
$750$	0	0
$751$	−9.02776	−0.329428	−0.164714	−	0.986341i	$-0.552670\pi$
$751$	−9.02776	−0.329428	−0.164714	+	0.986341i	$0.552670\pi$
$752$	0	0
$753$	29.2111	1.06451
$754$	0	0
$755$	6.60555	0.240401
$756$	0	0
$757$	−28.4222	−1.03302	−0.516511	−	0.856280i	$-0.672770\pi$
$757$	−28.4222	−1.03302	−0.516511	+	0.856280i	$0.672770\pi$
$758$	0	0
$759$	−6.78890	−0.246421
$760$	0	0
$761$	11.5778	0.419695	0.209847	−	0.977734i	$-0.432703\pi$
$761$	11.5778	0.419695	0.209847	+	0.977734i	$0.432703\pi$
$762$	0	0
$763$	5.21110	0.188655
$764$	0	0
$765$	1.60555	0.0580488
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.36669	−0.229589	−0.114794	−	0.993389i	$-0.536621\pi$
$769$	−6.36669	−0.229589	−0.114794	+	0.993389i	$0.536621\pi$
$770$	0	0
$771$	−23.6056	−0.850133
$772$	0	0
$773$	21.6333	0.778096	0.389048	−	0.921217i	$-0.372804\pi$
$773$	21.6333	0.778096	0.389048	+	0.921217i	$0.372804\pi$
$774$	0	0
$775$	20.8444	0.748753
$776$	0	0
$777$	6.23886	0.223818
$778$	0	0
$779$	76.6611	2.74667
$780$	0	0
$781$	−24.4777	−0.875882
$782$	0	0
$783$	3.00000	0.107211
$784$	0	0
$785$	8.39445	0.299611
$786$	0	0
$787$	−21.2111	−0.756094	−0.378047	−	0.925786i	$-0.623404\pi$
$787$	−21.2111	−0.756094	−0.378047	+	0.925786i	$0.623404\pi$
$788$	0	0
$789$	18.6056	0.662375
$790$	0	0
$791$	−30.2389	−1.07517
$792$	0	0
$793$	0	0
$794$	0	0
$795$	3.00000	0.106399
$796$	0	0
$797$	2.84441	0.100754	0.0503771	−	0.998730i	$-0.483958\pi$
$797$	2.84441	0.100754	0.0503771	+	0.998730i	$0.483958\pi$
$798$	0	0
$799$	2.23886	0.0792051
$800$	0	0
$801$	16.4222	0.580250
$802$	0	0
$803$	18.2389	0.643635
$804$	0	0
$805$	6.78890	0.239277
$806$	0	0
$807$	25.6333	0.902335
$808$	0	0
$809$	−31.6056	−1.11119	−0.555596	−	0.831452i	$-0.687510\pi$
$809$	−31.6056	−1.11119	−0.555596	+	0.831452i	$0.687510\pi$
$810$	0	0
$811$	−23.6333	−0.829878	−0.414939	−	0.909849i	$-0.636197\pi$
$811$	−23.6333	−0.829878	−0.414939	+	0.909849i	$0.636197\pi$
$812$	0	0
$813$	1.21110	0.0424752
$814$	0	0
$815$	5.21110	0.182537
$816$	0	0
$817$	−17.2111	−0.602140
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.21110	−0.251669	−0.125835	−	0.992051i	$-0.540161\pi$
$821$	−7.21110	−0.251669	−0.125835	+	0.992051i	$0.540161\pi$
$822$	0	0
$823$	19.6333	0.684374	0.342187	−	0.939632i	$-0.388832\pi$
$823$	19.6333	0.684374	0.342187	+	0.939632i	$0.388832\pi$
$824$	0	0
$825$	10.4222	0.362855
$826$	0	0
$827$	1.21110	0.0421142	0.0210571	−	0.999778i	$-0.493297\pi$
$827$	1.21110	0.0421142	0.0210571	+	0.999778i	$0.493297\pi$
$828$	0	0
$829$	46.8167	1.62601	0.813005	−	0.582257i	$-0.197830\pi$
$829$	46.8167	1.62601	0.813005	+	0.582257i	$0.197830\pi$
$830$	0	0
$831$	29.2389	1.01428
$832$	0	0
$833$	−0.338936	−0.0117434
$834$	0	0
$835$	−17.2111	−0.595615
$836$	0	0
$837$	−5.21110	−0.180122
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−0.394449	−0.0135855
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.9722	−0.377011
$848$	0	0
$849$	14.6056	0.501261
$850$	0	0
$851$	6.23886	0.213865
$852$	0	0
$853$	36.8167	1.26058	0.630289	−	0.776361i	$-0.282936\pi$
$853$	36.8167	1.26058	0.630289	+	0.776361i	$0.282936\pi$
$854$	0	0
$855$	6.60555	0.225905
$856$	0	0
$857$	4.39445	0.150112	0.0750558	−	0.997179i	$-0.476087\pi$
$857$	4.39445	0.150112	0.0750558	+	0.997179i	$0.476087\pi$
$858$	0	0
$859$	−47.8167	−1.63148	−0.815742	−	0.578417i	$-0.803671\pi$
$859$	−47.8167	−1.63148	−0.815742	+	0.578417i	$0.803671\pi$
$860$	0	0
$861$	30.2389	1.03054
$862$	0	0
$863$	33.0278	1.12428	0.562139	−	0.827043i	$-0.309979\pi$
$863$	33.0278	1.12428	0.562139	+	0.827043i	$0.309979\pi$
$864$	0	0
$865$	−21.6333	−0.735555
$866$	0	0
$867$	−14.4222	−0.489804
$868$	0	0
$869$	−31.2666	−1.06065
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−11.2111	−0.379438
$874$	0	0
$875$	−23.4500	−0.792753
$876$	0	0
$877$	−18.8167	−0.635393	−0.317697	−	0.948192i	$-0.602909\pi$
$877$	−18.8167	−0.635393	−0.317697	+	0.948192i	$0.602909\pi$
$878$	0	0
$879$	−0.211103	−0.00712031
$880$	0	0
$881$	−28.8167	−0.970858	−0.485429	−	0.874276i	$-0.661337\pi$
$881$	−28.8167	−0.970858	−0.485429	+	0.874276i	$0.661337\pi$
$882$	0	0
$883$	9.21110	0.309978	0.154989	−	0.987916i	$-0.450466\pi$
$883$	9.21110	0.309978	0.154989	+	0.987916i	$0.450466\pi$
$884$	0	0
$885$	−9.21110	−0.309628
$886$	0	0
$887$	3.63331	0.121995	0.0609973	−	0.998138i	$-0.480572\pi$
$887$	3.63331	0.121995	0.0609973	+	0.998138i	$0.480572\pi$
$888$	0	0
$889$	10.4222	0.349550
$890$	0	0
$891$	−2.60555	−0.0872893
$892$	0	0
$893$	9.21110	0.308238
$894$	0	0
$895$	−17.0278	−0.569175
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.6333	−0.521400
$900$	0	0
$901$	4.81665	0.160466
$902$	0	0
$903$	−6.78890	−0.225920
$904$	0	0
$905$	−24.8167	−0.824933
$906$	0	0
$907$	−10.7889	−0.358239	−0.179120	−	0.983827i	$-0.557325\pi$
$907$	−10.7889	−0.358239	−0.179120	+	0.983827i	$0.557325\pi$
$908$	0	0
$909$	4.21110	0.139673
$910$	0	0
$911$	−19.6333	−0.650481	−0.325240	−	0.945631i	$-0.605445\pi$
$911$	−19.6333	−0.650481	−0.325240	+	0.945631i	$0.605445\pi$
$912$	0	0
$913$	−30.7889	−1.01896
$914$	0	0
$915$	−7.60555	−0.251432
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.63331	−0.119852	−0.0599259	−	0.998203i	$-0.519086\pi$
$919$	−3.63331	−0.119852	−0.0599259	+	0.998203i	$0.519086\pi$
$920$	0	0
$921$	−5.39445	−0.177753
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−9.57779	−0.314916
$926$	0	0
$927$	−15.8167	−0.519487
$928$	0	0
$929$	−20.3944	−0.669120	−0.334560	−	0.942374i	$-0.608588\pi$
$929$	−20.3944	−0.669120	−0.334560	+	0.942374i	$0.608588\pi$
$930$	0	0
$931$	−1.39445	−0.0457012
$932$	0	0
$933$	27.8167	0.910676
$934$	0	0
$935$	−4.18335	−0.136810
$936$	0	0
$937$	−27.0555	−0.883865	−0.441933	−	0.897048i	$-0.645707\pi$
$937$	−27.0555	−0.883865	−0.441933	+	0.897048i	$0.645707\pi$
$938$	0	0
$939$	3.21110	0.104790
$940$	0	0
$941$	−47.2111	−1.53904	−0.769519	−	0.638624i	$-0.779504\pi$
$941$	−47.2111	−1.53904	−0.769519	+	0.638624i	$0.779504\pi$
$942$	0	0
$943$	30.2389	0.984713
$944$	0	0
$945$	2.60555	0.0847586
$946$	0	0
$947$	3.63331	0.118067	0.0590333	−	0.998256i	$-0.481198\pi$
$947$	3.63331	0.118067	0.0590333	+	0.998256i	$0.481198\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	2.21110	0.0716999
$952$	0	0
$953$	8.78890	0.284700	0.142350	−	0.989816i	$-0.454534\pi$
$953$	8.78890	0.284700	0.142350	+	0.989816i	$0.454534\pi$
$954$	0	0
$955$	−23.6333	−0.764756
$956$	0	0
$957$	−7.81665	−0.252677
$958$	0	0
$959$	43.8167	1.41491
$960$	0	0
$961$	−3.84441	−0.124013
$962$	0	0
$963$	6.60555	0.212861
$964$	0	0
$965$	7.78890	0.250733
$966$	0	0
$967$	−2.97224	−0.0955809	−0.0477905	−	0.998857i	$-0.515218\pi$
$967$	−2.97224	−0.0955809	−0.0477905	+	0.998857i	$0.515218\pi$
$968$	0	0
$969$	10.6056	0.340699
$970$	0	0
$971$	−43.2666	−1.38849	−0.694246	−	0.719738i	$-0.744262\pi$
$971$	−43.2666	−1.38849	−0.694246	+	0.719738i	$0.744262\pi$
$972$	0	0
$973$	−7.26662	−0.232957
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.4500	−1.61404	−0.807019	−	0.590526i	$-0.798920\pi$
$977$	−50.4500	−1.61404	−0.807019	+	0.590526i	$0.798920\pi$
$978$	0	0
$979$	−42.7889	−1.36754
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	37.2111	1.18685	0.593425	−	0.804889i	$-0.297775\pi$
$983$	37.2111	1.18685	0.593425	+	0.804889i	$0.297775\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	3.63331	0.115649
$988$	0	0
$989$	−6.78890	−0.215874
$990$	0	0
$991$	−21.3944	−0.679617	−0.339809	−	0.940495i	$-0.610362\pi$
$991$	−21.3944	−0.679617	−0.339809	+	0.940495i	$0.610362\pi$
$992$	0	0
$993$	15.6333	0.496108
$994$	0	0
$995$	19.8167	0.628230
$996$	0	0
$997$	58.4500	1.85113	0.925564	−	0.378590i	$-0.123591\pi$
$997$	58.4500	1.85113	0.925564	+	0.378590i	$0.123591\pi$
$998$	0	0
$999$	2.39445	0.0757570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.w.1.2		2
4.3	odd	2		8112.2.a.bn.1.1		2
13.4	even	6		312.2.q.d.289.2	yes	4
13.5	odd	4		4056.2.c.l.337.2		4
13.8	odd	4		4056.2.c.l.337.3		4
13.10	even	6		312.2.q.d.217.2	✓	4
13.12	even	2		4056.2.a.v.1.1		2
39.17	odd	6		936.2.t.e.289.2		4
39.23	odd	6		936.2.t.e.217.2		4
52.23	odd	6		624.2.q.i.529.1		4
52.43	odd	6		624.2.q.i.289.1		4
52.51	odd	2		8112.2.a.bl.1.2		2
156.23	even	6		1872.2.t.q.1153.1		4
156.95	even	6		1872.2.t.q.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.d.217.2	✓	4	13.10	even	6
312.2.q.d.289.2	yes	4	13.4	even	6
624.2.q.i.289.1		4	52.43	odd	6
624.2.q.i.529.1		4	52.23	odd	6
936.2.t.e.217.2		4	39.23	odd	6
936.2.t.e.289.2		4	39.17	odd	6
1872.2.t.q.289.1		4	156.95	even	6
1872.2.t.q.1153.1		4	156.23	even	6
4056.2.a.v.1.1		2	13.12	even	2
4056.2.a.w.1.2		2	1.1	even	1	trivial
4056.2.c.l.337.2		4	13.5	odd	4
4056.2.c.l.337.3		4	13.8	odd	4
8112.2.a.bl.1.2		2	52.51	odd	2
8112.2.a.bn.1.1		2	4.3	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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.60555 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.60555 q^{7} +1.00000 q^{9} -2.60555 q^{11} +1.00000 q^{15} +1.60555 q^{17} +6.60555 q^{19} +2.60555 q^{21} +2.60555 q^{23} -4.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} -5.21110 q^{31} -2.60555 q^{33} +2.60555 q^{35} +2.39445 q^{37} +11.6056 q^{41} -2.60555 q^{43} +1.00000 q^{45} +1.39445 q^{47} -0.211103 q^{49} +1.60555 q^{51} +3.00000 q^{53} -2.60555 q^{55} +6.60555 q^{57} -9.21110 q^{59} -7.60555 q^{61} +2.60555 q^{63} +10.6056 q^{67} +2.60555 q^{69} +9.39445 q^{71} -7.00000 q^{73} -4.00000 q^{75} -6.78890 q^{77} +12.0000 q^{79} +1.00000 q^{81} +11.8167 q^{83} +1.60555 q^{85} +3.00000 q^{87} +16.4222 q^{89} -5.21110 q^{93} +6.60555 q^{95} -11.2111 q^{97} -2.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} - 2 q^{21} - 2 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} - 2 q^{35} + 12 q^{37} + 16 q^{41} + 2 q^{43} + 2 q^{45} + 10 q^{47} + 14 q^{49} - 4 q^{51} + 6 q^{53} + 2 q^{55} + 6 q^{57} - 4 q^{59} - 8 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 26 q^{71} - 14 q^{73} - 8 q^{75} - 28 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{83} - 4 q^{85} + 6 q^{87} + 4 q^{89} + 4 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 6 * q^19 - 2 * q^21 - 2 * q^23 - 8 * q^25 + 2 * q^27 + 6 * q^29 + 4 * q^31 + 2 * q^33 - 2 * q^35 + 12 * q^37 + 16 * q^41 + 2 * q^43 + 2 * q^45 + 10 * q^47 + 14 * q^49 - 4 * q^51 + 6 * q^53 + 2 * q^55 + 6 * q^57 - 4 * q^59 - 8 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 26 * q^71 - 14 * q^73 - 8 * q^75 - 28 * q^77 + 24 * q^79 + 2 * q^81 + 2 * q^83 - 4 * q^85 + 6 * q^87 + 4 * q^89 + 4 * q^93 + 6 * q^95 - 8 * q^97 + 2 * q^99

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