LMFDB - Embedded newform 8112.2.a.cj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, 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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1014)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.15883 q^{5} +4.69202 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.15883 q^{5} +4.69202 q^{7} +1.00000 q^{9} -0.137063 q^{11} -3.15883 q^{15} -5.60388 q^{17} +4.98792 q^{19} +4.69202 q^{21} +6.09783 q^{23} +4.97823 q^{25} +1.00000 q^{27} -0.850855 q^{29} -6.23490 q^{31} -0.137063 q^{33} -14.8213 q^{35} +11.7017 q^{37} -4.27413 q^{41} +2.09783 q^{43} -3.15883 q^{45} -4.98792 q^{47} +15.0151 q^{49} -5.60388 q^{51} -1.82908 q^{53} +0.432960 q^{55} +4.98792 q^{57} +5.89977 q^{59} +4.39612 q^{61} +4.69202 q^{63} +4.71379 q^{67} +6.09783 q^{69} +0.0978347 q^{71} -2.32304 q^{73} +4.97823 q^{75} -0.643104 q^{77} -14.5157 q^{79} +1.00000 q^{81} +9.85623 q^{83} +17.7017 q^{85} -0.850855 q^{87} +17.0858 q^{89} -6.23490 q^{93} -15.7560 q^{95} +2.12737 q^{97} -0.137063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - q^{5} + 9 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - q^5 + 9 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - q^{5} + 9 q^{7} + 3 q^{9} + 5 q^{11} - q^{15} - 8 q^{17} - 4 q^{19} + 9 q^{21} + 18 q^{25} + 3 q^{27} + 11 q^{29} + 5 q^{31} + 5 q^{33} + 4 q^{35} + 8 q^{37} - 2 q^{41} - 12 q^{43} - q^{45} + 4 q^{47} + 20 q^{49} - 8 q^{51} + 5 q^{53} - 18 q^{55} - 4 q^{57} - 5 q^{59} + 22 q^{61} + 9 q^{63} + 6 q^{67} - 18 q^{71} + 13 q^{73} + 18 q^{75} - 6 q^{77} - 31 q^{79} + 3 q^{81} + 13 q^{83} + 26 q^{85} + 11 q^{87} + 14 q^{89} + 5 q^{93} - 8 q^{95} + 23 q^{97} + 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - q^5 + 9 * q^7 + 3 * q^9 + 5 * q^11 - q^15 - 8 * q^17 - 4 * q^19 + 9 * q^21 + 18 * q^25 + 3 * q^27 + 11 * q^29 + 5 * q^31 + 5 * q^33 + 4 * q^35 + 8 * q^37 - 2 * q^41 - 12 * q^43 - q^45 + 4 * q^47 + 20 * q^49 - 8 * q^51 + 5 * q^53 - 18 * q^55 - 4 * q^57 - 5 * q^59 + 22 * q^61 + 9 * q^63 + 6 * q^67 - 18 * q^71 + 13 * q^73 + 18 * q^75 - 6 * q^77 - 31 * q^79 + 3 * q^81 + 13 * q^83 + 26 * q^85 + 11 * q^87 + 14 * q^89 + 5 * q^93 - 8 * q^95 + 23 * q^97 + 5 * 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
$5$	−3.15883	−1.41267	−0.706337	−	0.707876i	$-0.749654\pi$
$5$	−3.15883	−1.41267	−0.706337	+	0.707876i	$0.749654\pi$
$6$	0	0
$7$	4.69202	1.77342	0.886709	−	0.462329i	$-0.152986\pi$
$7$	4.69202	1.77342	0.886709	+	0.462329i	$0.152986\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.137063	−0.0413262	−0.0206631	−	0.999786i	$-0.506578\pi$
$11$	−0.137063	−0.0413262	−0.0206631	+	0.999786i	$0.506578\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.15883	−0.815607
$16$	0	0
$17$	−5.60388	−1.35914	−0.679570	−	0.733611i	$-0.737833\pi$
$17$	−5.60388	−1.35914	−0.679570	+	0.733611i	$0.737833\pi$
$18$	0	0
$19$	4.98792	1.14431	0.572153	−	0.820147i	$-0.306108\pi$
$19$	4.98792	1.14431	0.572153	+	0.820147i	$0.306108\pi$
$20$	0	0
$21$	4.69202	1.02388
$22$	0	0
$23$	6.09783	1.27149	0.635743	−	0.771901i	$-0.280694\pi$
$23$	6.09783	1.27149	0.635743	+	0.771901i	$0.280694\pi$
$24$	0	0
$25$	4.97823	0.995646
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.850855	−0.158000	−0.0789999	−	0.996875i	$-0.525173\pi$
$29$	−0.850855	−0.158000	−0.0789999	+	0.996875i	$0.525173\pi$
$30$	0	0
$31$	−6.23490	−1.11982	−0.559910	−	0.828553i	$-0.689164\pi$
$31$	−6.23490	−1.11982	−0.559910	+	0.828553i	$0.689164\pi$
$32$	0	0
$33$	−0.137063	−0.0238597
$34$	0	0
$35$	−14.8213	−2.50526
$36$	0	0
$37$	11.7017	1.92375	0.961875	−	0.273491i	$-0.0881783\pi$
$37$	11.7017	1.92375	0.961875	+	0.273491i	$0.0881783\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.27413	−0.667506	−0.333753	−	0.942660i	$-0.608315\pi$
$41$	−4.27413	−0.667506	−0.333753	+	0.942660i	$0.608315\pi$
$42$	0	0
$43$	2.09783	0.319917	0.159958	−	0.987124i	$-0.448864\pi$
$43$	2.09783	0.319917	0.159958	+	0.987124i	$0.448864\pi$
$44$	0	0
$45$	−3.15883	−0.470891
$46$	0	0
$47$	−4.98792	−0.727563	−0.363781	−	0.931484i	$-0.618514\pi$
$47$	−4.98792	−0.727563	−0.363781	+	0.931484i	$0.618514\pi$
$48$	0	0
$49$	15.0151	2.14501
$50$	0	0
$51$	−5.60388	−0.784700
$52$	0	0
$53$	−1.82908	−0.251244	−0.125622	−	0.992078i	$-0.540093\pi$
$53$	−1.82908	−0.251244	−0.125622	+	0.992078i	$0.540093\pi$
$54$	0	0
$55$	0.432960	0.0583804
$56$	0	0
$57$	4.98792	0.660666
$58$	0	0
$59$	5.89977	0.768085	0.384042	−	0.923315i	$-0.374532\pi$
$59$	5.89977	0.768085	0.384042	+	0.923315i	$0.374532\pi$
$60$	0	0
$61$	4.39612	0.562866	0.281433	−	0.959581i	$-0.409190\pi$
$61$	4.39612	0.562866	0.281433	+	0.959581i	$0.409190\pi$
$62$	0	0
$63$	4.69202	0.591139
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.71379	0.575881	0.287941	−	0.957648i	$-0.407029\pi$
$67$	4.71379	0.575881	0.287941	+	0.957648i	$0.407029\pi$
$68$	0	0
$69$	6.09783	0.734093
$70$	0	0
$71$	0.0978347	0.0116108	0.00580542	−	0.999983i	$-0.498152\pi$
$71$	0.0978347	0.0116108	0.00580542	+	0.999983i	$0.498152\pi$
$72$	0	0
$73$	−2.32304	−0.271892	−0.135946	−	0.990716i	$-0.543407\pi$
$73$	−2.32304	−0.271892	−0.135946	+	0.990716i	$0.543407\pi$
$74$	0	0
$75$	4.97823	0.574836
$76$	0	0
$77$	−0.643104	−0.0732885
$78$	0	0
$79$	−14.5157	−1.63315	−0.816574	−	0.577241i	$-0.804129\pi$
$79$	−14.5157	−1.63315	−0.816574	+	0.577241i	$0.804129\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.85623	1.08186	0.540931	−	0.841067i	$-0.318072\pi$
$83$	9.85623	1.08186	0.540931	+	0.841067i	$0.318072\pi$
$84$	0	0
$85$	17.7017	1.92002
$86$	0	0
$87$	−0.850855	−0.0912212
$88$	0	0
$89$	17.0858	1.81109	0.905543	−	0.424254i	$-0.139464\pi$
$89$	17.0858	1.81109	0.905543	+	0.424254i	$0.139464\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.23490	−0.646529
$94$	0	0
$95$	−15.7560	−1.61653
$96$	0	0
$97$	2.12737	0.216002	0.108001	−	0.994151i	$-0.465555\pi$
$97$	2.12737	0.216002	0.108001	+	0.994151i	$0.465555\pi$
$98$	0	0
$99$	−0.137063	−0.0137754
$100$	0	0
$101$	−9.18598	−0.914039	−0.457020	−	0.889457i	$-0.651083\pi$
$101$	−9.18598	−0.914039	−0.457020	+	0.889457i	$0.651083\pi$
$102$	0	0
$103$	−0.225209	−0.0221905	−0.0110953	−	0.999938i	$-0.503532\pi$
$103$	−0.225209	−0.0221905	−0.0110953	+	0.999938i	$0.503532\pi$
$104$	0	0
$105$	−14.8213	−1.44641
$106$	0	0
$107$	−11.2838	−1.09085	−0.545424	−	0.838160i	$-0.683631\pi$
$107$	−11.2838	−1.09085	−0.545424	+	0.838160i	$0.683631\pi$
$108$	0	0
$109$	−0.195669	−0.0187417	−0.00937086	−	0.999956i	$-0.502983\pi$
$109$	−0.195669	−0.0187417	−0.00937086	+	0.999956i	$0.502983\pi$
$110$	0	0
$111$	11.7017	1.11068
$112$	0	0
$113$	−0.439665	−0.0413602	−0.0206801	−	0.999786i	$-0.506583\pi$
$113$	−0.439665	−0.0413602	−0.0206801	+	0.999786i	$0.506583\pi$
$114$	0	0
$115$	−19.2620	−1.79619
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−26.2935	−2.41032
$120$	0	0
$121$	−10.9812	−0.998292
$122$	0	0
$123$	−4.27413	−0.385385
$124$	0	0
$125$	0.0687686	0.00615085
$126$	0	0
$127$	7.87263	0.698583	0.349291	−	0.937014i	$-0.386422\pi$
$127$	7.87263	0.698583	0.349291	+	0.937014i	$0.386422\pi$
$128$	0	0
$129$	2.09783	0.184704
$130$	0	0
$131$	0.621334	0.0542862	0.0271431	−	0.999632i	$-0.491359\pi$
$131$	0.621334	0.0542862	0.0271431	+	0.999632i	$0.491359\pi$
$132$	0	0
$133$	23.4034	2.02933
$134$	0	0
$135$	−3.15883	−0.271869
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	13.6582	1.15847	0.579235	−	0.815160i	$-0.303351\pi$
$139$	13.6582	1.15847	0.579235	+	0.815160i	$0.303351\pi$
$140$	0	0
$141$	−4.98792	−0.420059
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.68771	0.223202
$146$	0	0
$147$	15.0151	1.23842
$148$	0	0
$149$	16.0586	1.31557	0.657786	−	0.753205i	$-0.271493\pi$
$149$	16.0586	1.31557	0.657786	+	0.753205i	$0.271493\pi$
$150$	0	0
$151$	21.8823	1.78076	0.890379	−	0.455221i	$-0.150440\pi$
$151$	21.8823	1.78076	0.890379	+	0.455221i	$0.150440\pi$
$152$	0	0
$153$	−5.60388	−0.453046
$154$	0	0
$155$	19.6950	1.58194
$156$	0	0
$157$	−7.90217	−0.630661	−0.315331	−	0.948982i	$-0.602115\pi$
$157$	−7.90217	−0.630661	−0.315331	+	0.948982i	$0.602115\pi$
$158$	0	0
$159$	−1.82908	−0.145056
$160$	0	0
$161$	28.6112	2.25488
$162$	0	0
$163$	−8.01938	−0.628126	−0.314063	−	0.949402i	$-0.601690\pi$
$163$	−8.01938	−0.628126	−0.314063	+	0.949402i	$0.601690\pi$
$164$	0	0
$165$	0.432960	0.0337059
$166$	0	0
$167$	17.0858	1.32214	0.661068	−	0.750326i	$-0.270104\pi$
$167$	17.0858	1.32214	0.661068	+	0.750326i	$0.270104\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.98792	0.381436
$172$	0	0
$173$	−15.3448	−1.16664	−0.583322	−	0.812241i	$-0.698248\pi$
$173$	−15.3448	−1.16664	−0.583322	+	0.812241i	$0.698248\pi$
$174$	0	0
$175$	23.3580	1.76570
$176$	0	0
$177$	5.89977	0.443454
$178$	0	0
$179$	−0.523499	−0.0391282	−0.0195641	−	0.999809i	$-0.506228\pi$
$179$	−0.523499	−0.0391282	−0.0195641	+	0.999809i	$0.506228\pi$
$180$	0	0
$181$	8.89008	0.660795	0.330397	−	0.943842i	$-0.392817\pi$
$181$	8.89008	0.660795	0.330397	+	0.943842i	$0.392817\pi$
$182$	0	0
$183$	4.39612	0.324971
$184$	0	0
$185$	−36.9638	−2.71763
$186$	0	0
$187$	0.768086	0.0561680
$188$	0	0
$189$	4.69202	0.341294
$190$	0	0
$191$	7.03146	0.508779	0.254389	−	0.967102i	$-0.418126\pi$
$191$	7.03146	0.508779	0.254389	+	0.967102i	$0.418126\pi$
$192$	0	0
$193$	−17.7560	−1.27811	−0.639053	−	0.769163i	$-0.720673\pi$
$193$	−17.7560	−1.27811	−0.639053	+	0.769163i	$0.720673\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.6571	1.32926	0.664632	−	0.747171i	$-0.268588\pi$
$197$	18.6571	1.32926	0.664632	+	0.747171i	$0.268588\pi$
$198$	0	0
$199$	−7.66248	−0.543179	−0.271589	−	0.962413i	$-0.587549\pi$
$199$	−7.66248	−0.543179	−0.271589	+	0.962413i	$0.587549\pi$
$200$	0	0
$201$	4.71379	0.332485
$202$	0	0
$203$	−3.99223	−0.280200
$204$	0	0
$205$	13.5013	0.942969
$206$	0	0
$207$	6.09783	0.423829
$208$	0	0
$209$	−0.683661	−0.0472898
$210$	0	0
$211$	−11.1642	−0.768576	−0.384288	−	0.923213i	$-0.625553\pi$
$211$	−11.1642	−0.768576	−0.384288	+	0.923213i	$0.625553\pi$
$212$	0	0
$213$	0.0978347	0.00670352
$214$	0	0
$215$	−6.62671	−0.451938
$216$	0	0
$217$	−29.2543	−1.98591
$218$	0	0
$219$	−2.32304	−0.156977
$220$	0	0
$221$	0	0
$222$	0	0
$223$	24.6353	1.64970	0.824852	−	0.565349i	$-0.191258\pi$
$223$	24.6353	1.64970	0.824852	+	0.565349i	$0.191258\pi$
$224$	0	0
$225$	4.97823	0.331882
$226$	0	0
$227$	−7.47650	−0.496233	−0.248116	−	0.968730i	$-0.579812\pi$
$227$	−7.47650	−0.496233	−0.248116	+	0.968730i	$0.579812\pi$
$228$	0	0
$229$	19.2271	1.27056	0.635282	−	0.772280i	$-0.280884\pi$
$229$	19.2271	1.27056	0.635282	+	0.772280i	$0.280884\pi$
$230$	0	0
$231$	−0.643104	−0.0423131
$232$	0	0
$233$	3.70171	0.242507	0.121254	−	0.992622i	$-0.461309\pi$
$233$	3.70171	0.242507	0.121254	+	0.992622i	$0.461309\pi$
$234$	0	0
$235$	15.7560	1.02781
$236$	0	0
$237$	−14.5157	−0.942898
$238$	0	0
$239$	8.51334	0.550682	0.275341	−	0.961347i	$-0.411209\pi$
$239$	8.51334	0.550682	0.275341	+	0.961347i	$0.411209\pi$
$240$	0	0
$241$	−17.4330	−1.12296	−0.561478	−	0.827492i	$-0.689767\pi$
$241$	−17.4330	−1.12296	−0.561478	+	0.827492i	$0.689767\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−47.4301	−3.03020
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.85623	0.624613
$250$	0	0
$251$	3.48427	0.219925	0.109963	−	0.993936i	$-0.464927\pi$
$251$	3.48427	0.219925	0.109963	+	0.993936i	$0.464927\pi$
$252$	0	0
$253$	−0.835790	−0.0525456
$254$	0	0
$255$	17.7017	1.10852
$256$	0	0
$257$	−13.6039	−0.848586	−0.424293	−	0.905525i	$-0.639477\pi$
$257$	−13.6039	−0.848586	−0.424293	+	0.905525i	$0.639477\pi$
$258$	0	0
$259$	54.9047	3.41161
$260$	0	0
$261$	−0.850855	−0.0526666
$262$	0	0
$263$	−11.4577	−0.706513	−0.353256	−	0.935527i	$-0.614926\pi$
$263$	−11.4577	−0.706513	−0.353256	+	0.935527i	$0.614926\pi$
$264$	0	0
$265$	5.77777	0.354926
$266$	0	0
$267$	17.0858	1.04563
$268$	0	0
$269$	22.3666	1.36371	0.681857	−	0.731485i	$-0.261172\pi$
$269$	22.3666	1.36371	0.681857	+	0.731485i	$0.261172\pi$
$270$	0	0
$271$	−3.87263	−0.235245	−0.117623	−	0.993058i	$-0.537527\pi$
$271$	−3.87263	−0.235245	−0.117623	+	0.993058i	$0.537527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.682333	−0.0411462
$276$	0	0
$277$	28.7090	1.72496	0.862478	−	0.506094i	$-0.168911\pi$
$277$	28.7090	1.72496	0.862478	+	0.506094i	$0.168911\pi$
$278$	0	0
$279$	−6.23490	−0.373274
$280$	0	0
$281$	−29.0858	−1.73511	−0.867555	−	0.497341i	$-0.834310\pi$
$281$	−29.0858	−1.73511	−0.867555	+	0.497341i	$0.834310\pi$
$282$	0	0
$283$	−13.7560	−0.817710	−0.408855	−	0.912599i	$-0.634072\pi$
$283$	−13.7560	−0.817710	−0.408855	+	0.912599i	$0.634072\pi$
$284$	0	0
$285$	−15.7560	−0.933305
$286$	0	0
$287$	−20.0543	−1.18377
$288$	0	0
$289$	14.4034	0.847260
$290$	0	0
$291$	2.12737	0.124709
$292$	0	0
$293$	27.7362	1.62036	0.810182	−	0.586179i	$-0.199368\pi$
$293$	27.7362	1.62036	0.810182	+	0.586179i	$0.199368\pi$
$294$	0	0
$295$	−18.6364	−1.08505
$296$	0	0
$297$	−0.137063	−0.00795322
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.84309	0.567346
$302$	0	0
$303$	−9.18598	−0.527721
$304$	0	0
$305$	−13.8866	−0.795146
$306$	0	0
$307$	12.4590	0.711075	0.355538	−	0.934662i	$-0.384298\pi$
$307$	12.4590	0.711075	0.355538	+	0.934662i	$0.384298\pi$
$308$	0	0
$309$	−0.225209	−0.0128117
$310$	0	0
$311$	6.09783	0.345776	0.172888	−	0.984941i	$-0.444690\pi$
$311$	6.09783	0.345776	0.172888	+	0.984941i	$0.444690\pi$
$312$	0	0
$313$	−12.7385	−0.720025	−0.360013	−	0.932947i	$-0.617228\pi$
$313$	−12.7385	−0.720025	−0.360013	+	0.932947i	$0.617228\pi$
$314$	0	0
$315$	−14.8213	−0.835087
$316$	0	0
$317$	14.8140	0.832038	0.416019	−	0.909356i	$-0.363425\pi$
$317$	14.8140	0.832038	0.416019	+	0.909356i	$0.363425\pi$
$318$	0	0
$319$	0.116621	0.00652952
$320$	0	0
$321$	−11.2838	−0.629801
$322$	0	0
$323$	−27.9517	−1.55527
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.195669	−0.0108205
$328$	0	0
$329$	−23.4034	−1.29027
$330$	0	0
$331$	−7.70171	−0.423324	−0.211662	−	0.977343i	$-0.567888\pi$
$331$	−7.70171	−0.423324	−0.211662	+	0.977343i	$0.567888\pi$
$332$	0	0
$333$	11.7017	0.641250
$334$	0	0
$335$	−14.8901	−0.813532
$336$	0	0
$337$	26.5961	1.44878	0.724391	−	0.689389i	$-0.242121\pi$
$337$	26.5961	1.44878	0.724391	+	0.689389i	$0.242121\pi$
$338$	0	0
$339$	−0.439665	−0.0238793
$340$	0	0
$341$	0.854576	0.0462779
$342$	0	0
$343$	37.6069	2.03058
$344$	0	0
$345$	−19.2620	−1.03703
$346$	0	0
$347$	−0.911854	−0.0489509	−0.0244754	−	0.999700i	$-0.507792\pi$
$347$	−0.911854	−0.0489509	−0.0244754	+	0.999700i	$0.507792\pi$
$348$	0	0
$349$	−17.7211	−0.948588	−0.474294	−	0.880366i	$-0.657297\pi$
$349$	−17.7211	−0.948588	−0.474294	+	0.880366i	$0.657297\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.4349	−1.40699	−0.703493	−	0.710702i	$-0.748378\pi$
$353$	−26.4349	−1.40699	−0.703493	+	0.710702i	$0.748378\pi$
$354$	0	0
$355$	−0.309043	−0.0164023
$356$	0	0
$357$	−26.2935	−1.39160
$358$	0	0
$359$	−7.76941	−0.410054	−0.205027	−	0.978756i	$-0.565728\pi$
$359$	−7.76941	−0.410054	−0.205027	+	0.978756i	$0.565728\pi$
$360$	0	0
$361$	5.87933	0.309438
$362$	0	0
$363$	−10.9812	−0.576364
$364$	0	0
$365$	7.33811	0.384094
$366$	0	0
$367$	13.3274	0.695682	0.347841	−	0.937553i	$-0.386915\pi$
$367$	13.3274	0.695682	0.347841	+	0.937553i	$0.386915\pi$
$368$	0	0
$369$	−4.27413	−0.222502
$370$	0	0
$371$	−8.58211	−0.445561
$372$	0	0
$373$	6.70304	0.347070	0.173535	−	0.984828i	$-0.444481\pi$
$373$	6.70304	0.347070	0.173535	+	0.984828i	$0.444481\pi$
$374$	0	0
$375$	0.0687686	0.00355120
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.41550	−0.124076	−0.0620380	−	0.998074i	$-0.519760\pi$
$379$	−2.41550	−0.124076	−0.0620380	+	0.998074i	$0.519760\pi$
$380$	0	0
$381$	7.87263	0.403327
$382$	0	0
$383$	−10.0978	−0.515975	−0.257988	−	0.966148i	$-0.583059\pi$
$383$	−10.0978	−0.515975	−0.257988	+	0.966148i	$0.583059\pi$
$384$	0	0
$385$	2.03146	0.103533
$386$	0	0
$387$	2.09783	0.106639
$388$	0	0
$389$	25.1336	1.27432	0.637162	−	0.770730i	$-0.280108\pi$
$389$	25.1336	1.27432	0.637162	+	0.770730i	$0.280108\pi$
$390$	0	0
$391$	−34.1715	−1.72813
$392$	0	0
$393$	0.621334	0.0313421
$394$	0	0
$395$	45.8528	2.30710
$396$	0	0
$397$	20.8358	1.04572	0.522859	−	0.852419i	$-0.324865\pi$
$397$	20.8358	1.04572	0.522859	+	0.852419i	$0.324865\pi$
$398$	0	0
$399$	23.4034	1.17164
$400$	0	0
$401$	−5.95646	−0.297451	−0.148726	−	0.988878i	$-0.547517\pi$
$401$	−5.95646	−0.297451	−0.148726	+	0.988878i	$0.547517\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.15883	−0.156964
$406$	0	0
$407$	−1.60388	−0.0795012
$408$	0	0
$409$	−1.80194	−0.0891001	−0.0445500	−	0.999007i	$-0.514185\pi$
$409$	−1.80194	−0.0891001	−0.0445500	+	0.999007i	$0.514185\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	27.6819	1.36214
$414$	0	0
$415$	−31.1342	−1.52832
$416$	0	0
$417$	13.6582	0.668843
$418$	0	0
$419$	28.4499	1.38987	0.694935	−	0.719072i	$-0.255433\pi$
$419$	28.4499	1.38987	0.694935	+	0.719072i	$0.255433\pi$
$420$	0	0
$421$	13.9323	0.679019	0.339509	−	0.940603i	$-0.389739\pi$
$421$	13.9323	0.679019	0.339509	+	0.940603i	$0.389739\pi$
$422$	0	0
$423$	−4.98792	−0.242521
$424$	0	0
$425$	−27.8974	−1.35322
$426$	0	0
$427$	20.6267	0.998196
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.9022	0.765980	0.382990	−	0.923752i	$-0.374894\pi$
$431$	15.9022	0.765980	0.382990	+	0.923752i	$0.374894\pi$
$432$	0	0
$433$	4.77718	0.229577	0.114788	−	0.993390i	$-0.463381\pi$
$433$	4.77718	0.229577	0.114788	+	0.993390i	$0.463381\pi$
$434$	0	0
$435$	2.68771	0.128866
$436$	0	0
$437$	30.4155	1.45497
$438$	0	0
$439$	33.6316	1.60515	0.802575	−	0.596552i	$-0.203463\pi$
$439$	33.6316	1.60515	0.802575	+	0.596552i	$0.203463\pi$
$440$	0	0
$441$	15.0151	0.715003
$442$	0	0
$443$	35.3749	1.68071	0.840357	−	0.542033i	$-0.182345\pi$
$443$	35.3749	1.68071	0.840357	+	0.542033i	$0.182345\pi$
$444$	0	0
$445$	−53.9711	−2.55847
$446$	0	0
$447$	16.0586	0.759546
$448$	0	0
$449$	18.0629	0.852442	0.426221	−	0.904619i	$-0.359845\pi$
$449$	18.0629	0.852442	0.426221	+	0.904619i	$0.359845\pi$
$450$	0	0
$451$	0.585826	0.0275855
$452$	0	0
$453$	21.8823	1.02812
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.4668	0.723507	0.361753	−	0.932274i	$-0.382178\pi$
$457$	15.4668	0.723507	0.361753	+	0.932274i	$0.382178\pi$
$458$	0	0
$459$	−5.60388	−0.261567
$460$	0	0
$461$	18.8092	0.876033	0.438017	−	0.898967i	$-0.355681\pi$
$461$	18.8092	0.876033	0.438017	+	0.898967i	$0.355681\pi$
$462$	0	0
$463$	−15.8431	−0.736291	−0.368145	−	0.929768i	$-0.620007\pi$
$463$	−15.8431	−0.736291	−0.368145	+	0.929768i	$0.620007\pi$
$464$	0	0
$465$	19.6950	0.913334
$466$	0	0
$467$	−22.0006	−1.01807	−0.509033	−	0.860747i	$-0.669997\pi$
$467$	−22.0006	−1.01807	−0.509033	+	0.860747i	$0.669997\pi$
$468$	0	0
$469$	22.1172	1.02128
$470$	0	0
$471$	−7.90217	−0.364113
$472$	0	0
$473$	−0.287536	−0.0132209
$474$	0	0
$475$	24.8310	1.13932
$476$	0	0
$477$	−1.82908	−0.0837480
$478$	0	0
$479$	−21.3491	−0.975466	−0.487733	−	0.872993i	$-0.662176\pi$
$479$	−21.3491	−0.975466	−0.487733	+	0.872993i	$0.662176\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	28.6112	1.30185
$484$	0	0
$485$	−6.72002	−0.305141
$486$	0	0
$487$	−31.6394	−1.43372	−0.716859	−	0.697219i	$-0.754421\pi$
$487$	−31.6394	−1.43372	−0.716859	+	0.697219i	$0.754421\pi$
$488$	0	0
$489$	−8.01938	−0.362649
$490$	0	0
$491$	1.39911	0.0631409	0.0315704	−	0.999502i	$-0.489949\pi$
$491$	1.39911	0.0631409	0.0315704	+	0.999502i	$0.489949\pi$
$492$	0	0
$493$	4.76809	0.214744
$494$	0	0
$495$	0.432960	0.0194601
$496$	0	0
$497$	0.459042	0.0205909
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	17.0858	0.763335
$502$	0	0
$503$	18.3827	0.819645	0.409822	−	0.912165i	$-0.365591\pi$
$503$	18.3827	0.819645	0.409822	+	0.912165i	$0.365591\pi$
$504$	0	0
$505$	29.0170	1.29124
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.132751	−0.00588411	−0.00294205	−	0.999996i	$-0.500936\pi$
$509$	−0.132751	−0.00588411	−0.00294205	+	0.999996i	$0.500936\pi$
$510$	0	0
$511$	−10.8998	−0.482178
$512$	0	0
$513$	4.98792	0.220222
$514$	0	0
$515$	0.711399	0.0313480
$516$	0	0
$517$	0.683661	0.0300674
$518$	0	0
$519$	−15.3448	−0.673563
$520$	0	0
$521$	37.0508	1.62323	0.811613	−	0.584195i	$-0.198590\pi$
$521$	37.0508	1.62323	0.811613	+	0.584195i	$0.198590\pi$
$522$	0	0
$523$	3.15346	0.137891	0.0689455	−	0.997620i	$-0.478037\pi$
$523$	3.15346	0.137891	0.0689455	+	0.997620i	$0.478037\pi$
$524$	0	0
$525$	23.3580	1.01942
$526$	0	0
$527$	34.9396	1.52199
$528$	0	0
$529$	14.1836	0.616678
$530$	0	0
$531$	5.89977	0.256028
$532$	0	0
$533$	0	0
$534$	0	0
$535$	35.6437	1.54101
$536$	0	0
$537$	−0.523499	−0.0225907
$538$	0	0
$539$	−2.05802	−0.0886450
$540$	0	0
$541$	4.07846	0.175347	0.0876733	−	0.996149i	$-0.472057\pi$
$541$	4.07846	0.175347	0.0876733	+	0.996149i	$0.472057\pi$
$542$	0	0
$543$	8.89008	0.381510
$544$	0	0
$545$	0.618087	0.0264759
$546$	0	0
$547$	23.0508	0.985583	0.492791	−	0.870148i	$-0.335977\pi$
$547$	23.0508	0.985583	0.492791	+	0.870148i	$0.335977\pi$
$548$	0	0
$549$	4.39612	0.187622
$550$	0	0
$551$	−4.24400	−0.180800
$552$	0	0
$553$	−68.1081	−2.89625
$554$	0	0
$555$	−36.9638	−1.56902
$556$	0	0
$557$	20.4155	0.865033	0.432516	−	0.901626i	$-0.357626\pi$
$557$	20.4155	0.865033	0.432516	+	0.901626i	$0.357626\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.768086	0.0324286
$562$	0	0
$563$	−21.5609	−0.908685	−0.454342	−	0.890827i	$-0.650126\pi$
$563$	−21.5609	−0.908685	−0.454342	+	0.890827i	$0.650126\pi$
$564$	0	0
$565$	1.38883	0.0584285
$566$	0	0
$567$	4.69202	0.197046
$568$	0	0
$569$	8.98792	0.376793	0.188397	−	0.982093i	$-0.439671\pi$
$569$	8.98792	0.376793	0.188397	+	0.982093i	$0.439671\pi$
$570$	0	0
$571$	−13.5603	−0.567482	−0.283741	−	0.958901i	$-0.591576\pi$
$571$	−13.5603	−0.567482	−0.283741	+	0.958901i	$0.591576\pi$
$572$	0	0
$573$	7.03146	0.293743
$574$	0	0
$575$	30.3564	1.26595
$576$	0	0
$577$	16.2825	0.677849	0.338924	−	0.940814i	$-0.389937\pi$
$577$	16.2825	0.677849	0.338924	+	0.940814i	$0.389937\pi$
$578$	0	0
$579$	−17.7560	−0.737914
$580$	0	0
$581$	46.2457	1.91859
$582$	0	0
$583$	0.250700	0.0103830
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−47.5706	−1.96345	−0.981725	−	0.190307i	$-0.939052\pi$
$587$	−47.5706	−1.96345	−0.981725	+	0.190307i	$0.939052\pi$
$588$	0	0
$589$	−31.0992	−1.28142
$590$	0	0
$591$	18.6571	0.767451
$592$	0	0
$593$	31.0267	1.27411	0.637056	−	0.770817i	$-0.280152\pi$
$593$	31.0267	1.27411	0.637056	+	0.770817i	$0.280152\pi$
$594$	0	0
$595$	83.0568	3.40500
$596$	0	0
$597$	−7.66248	−0.313604
$598$	0	0
$599$	−22.3263	−0.912228	−0.456114	−	0.889921i	$-0.650759\pi$
$599$	−22.3263	−0.912228	−0.456114	+	0.889921i	$0.650759\pi$
$600$	0	0
$601$	−8.18060	−0.333694	−0.166847	−	0.985983i	$-0.553359\pi$
$601$	−8.18060	−0.333694	−0.166847	+	0.985983i	$0.553359\pi$
$602$	0	0
$603$	4.71379	0.191960
$604$	0	0
$605$	34.6878	1.41026
$606$	0	0
$607$	−3.30798	−0.134267	−0.0671334	−	0.997744i	$-0.521385\pi$
$607$	−3.30798	−0.134267	−0.0671334	+	0.997744i	$0.521385\pi$
$608$	0	0
$609$	−3.99223	−0.161773
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.8853	−0.439653	−0.219827	−	0.975539i	$-0.570549\pi$
$613$	−10.8853	−0.439653	−0.219827	+	0.975539i	$0.570549\pi$
$614$	0	0
$615$	13.5013	0.544423
$616$	0	0
$617$	−34.5676	−1.39164	−0.695820	−	0.718216i	$-0.744959\pi$
$617$	−34.5676	−1.39164	−0.695820	+	0.718216i	$0.744959\pi$
$618$	0	0
$619$	−2.86592	−0.115191	−0.0575955	−	0.998340i	$-0.518343\pi$
$619$	−2.86592	−0.115191	−0.0575955	+	0.998340i	$0.518343\pi$
$620$	0	0
$621$	6.09783	0.244698
$622$	0	0
$623$	80.1667	3.21181
$624$	0	0
$625$	−25.1084	−1.00434
$626$	0	0
$627$	−0.683661	−0.0273028
$628$	0	0
$629$	−65.5749	−2.61464
$630$	0	0
$631$	42.6631	1.69839	0.849195	−	0.528079i	$-0.177088\pi$
$631$	42.6631	1.69839	0.849195	+	0.528079i	$0.177088\pi$
$632$	0	0
$633$	−11.1642	−0.443738
$634$	0	0
$635$	−24.8683	−0.986869
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.0978347	0.00387028
$640$	0	0
$641$	41.3927	1.63491	0.817456	−	0.575991i	$-0.195384\pi$
$641$	41.3927	1.63491	0.817456	+	0.575991i	$0.195384\pi$
$642$	0	0
$643$	−13.7125	−0.540767	−0.270383	−	0.962753i	$-0.587150\pi$
$643$	−13.7125	−0.540767	−0.270383	+	0.962753i	$0.587150\pi$
$644$	0	0
$645$	−6.62671	−0.260926
$646$	0	0
$647$	−18.0086	−0.707992	−0.353996	−	0.935247i	$-0.615177\pi$
$647$	−18.0086	−0.707992	−0.353996	+	0.935247i	$0.615177\pi$
$648$	0	0
$649$	−0.808643	−0.0317420
$650$	0	0
$651$	−29.2543	−1.14657
$652$	0	0
$653$	46.5652	1.82224	0.911119	−	0.412143i	$-0.135220\pi$
$653$	46.5652	1.82224	0.911119	+	0.412143i	$0.135220\pi$
$654$	0	0
$655$	−1.96269	−0.0766887
$656$	0	0
$657$	−2.32304	−0.0906306
$658$	0	0
$659$	−13.8562	−0.539762	−0.269881	−	0.962894i	$-0.586984\pi$
$659$	−13.8562	−0.539762	−0.269881	+	0.962894i	$0.586984\pi$
$660$	0	0
$661$	−43.1051	−1.67660	−0.838298	−	0.545213i	$-0.816449\pi$
$661$	−43.1051	−1.67660	−0.838298	+	0.545213i	$0.816449\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−73.9275	−2.86679
$666$	0	0
$667$	−5.18837	−0.200895
$668$	0	0
$669$	24.6353	0.952457
$670$	0	0
$671$	−0.602548	−0.0232611
$672$	0	0
$673$	30.7415	1.18500	0.592499	−	0.805571i	$-0.298141\pi$
$673$	30.7415	1.18500	0.592499	+	0.805571i	$0.298141\pi$
$674$	0	0
$675$	4.97823	0.191612
$676$	0	0
$677$	16.5894	0.637582	0.318791	−	0.947825i	$-0.396723\pi$
$677$	16.5894	0.637582	0.318791	+	0.947825i	$0.396723\pi$
$678$	0	0
$679$	9.98169	0.383062
$680$	0	0
$681$	−7.47650	−0.286500
$682$	0	0
$683$	34.9885	1.33880	0.669399	−	0.742903i	$-0.266552\pi$
$683$	34.9885	1.33880	0.669399	+	0.742903i	$0.266552\pi$
$684$	0	0
$685$	−12.6353	−0.482771
$686$	0	0
$687$	19.2271	0.733561
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−14.0871	−0.535898	−0.267949	−	0.963433i	$-0.586346\pi$
$691$	−14.0871	−0.535898	−0.267949	+	0.963433i	$0.586346\pi$
$692$	0	0
$693$	−0.643104	−0.0244295
$694$	0	0
$695$	−43.1439	−1.63654
$696$	0	0
$697$	23.9517	0.907234
$698$	0	0
$699$	3.70171	0.140012
$700$	0	0
$701$	48.6112	1.83602	0.918009	−	0.396559i	$-0.129796\pi$
$701$	48.6112	1.83602	0.918009	+	0.396559i	$0.129796\pi$
$702$	0	0
$703$	58.3672	2.20136
$704$	0	0
$705$	15.7560	0.593405
$706$	0	0
$707$	−43.1008	−1.62097
$708$	0	0
$709$	17.2862	0.649197	0.324599	−	0.945852i	$-0.394771\pi$
$709$	17.2862	0.649197	0.324599	+	0.945852i	$0.394771\pi$
$710$	0	0
$711$	−14.5157	−0.544382
$712$	0	0
$713$	−38.0194	−1.42384
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.51334	0.317936
$718$	0	0
$719$	−29.1207	−1.08602	−0.543009	−	0.839727i	$-0.682715\pi$
$719$	−29.1207	−1.08602	−0.543009	+	0.839727i	$0.682715\pi$
$720$	0	0
$721$	−1.05669	−0.0393531
$722$	0	0
$723$	−17.4330	−0.648339
$724$	0	0
$725$	−4.23575	−0.157312
$726$	0	0
$727$	−45.5666	−1.68997	−0.844985	−	0.534790i	$-0.820391\pi$
$727$	−45.5666	−1.68997	−0.844985	+	0.534790i	$0.820391\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.7560	−0.434812
$732$	0	0
$733$	−21.7995	−0.805185	−0.402592	−	0.915379i	$-0.631891\pi$
$733$	−21.7995	−0.805185	−0.402592	+	0.915379i	$0.631891\pi$
$734$	0	0
$735$	−47.4301	−1.74949
$736$	0	0
$737$	−0.646088	−0.0237990
$738$	0	0
$739$	41.5663	1.52904	0.764521	−	0.644599i	$-0.222976\pi$
$739$	41.5663	1.52904	0.764521	+	0.644599i	$0.222976\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.8224	−1.05739	−0.528695	−	0.848812i	$-0.677319\pi$
$743$	−28.8224	−1.05739	−0.528695	+	0.848812i	$0.677319\pi$
$744$	0	0
$745$	−50.7265	−1.85847
$746$	0	0
$747$	9.85623	0.360621
$748$	0	0
$749$	−52.9439	−1.93453
$750$	0	0
$751$	−16.6203	−0.606482	−0.303241	−	0.952914i	$-0.598069\pi$
$751$	−16.6203	−0.606482	−0.303241	+	0.952914i	$0.598069\pi$
$752$	0	0
$753$	3.48427	0.126974
$754$	0	0
$755$	−69.1226	−2.51563
$756$	0	0
$757$	−40.3913	−1.46805	−0.734024	−	0.679123i	$-0.762360\pi$
$757$	−40.3913	−1.46805	−0.734024	+	0.679123i	$0.762360\pi$
$758$	0	0
$759$	−0.835790	−0.0303372
$760$	0	0
$761$	−3.29483	−0.119438	−0.0597188	−	0.998215i	$-0.519020\pi$
$761$	−3.29483	−0.119438	−0.0597188	+	0.998215i	$0.519020\pi$
$762$	0	0
$763$	−0.918085	−0.0332369
$764$	0	0
$765$	17.7017	0.640007
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.35258	−0.0848363	−0.0424182	−	0.999100i	$-0.513506\pi$
$769$	−2.35258	−0.0848363	−0.0424182	+	0.999100i	$0.513506\pi$
$770$	0	0
$771$	−13.6039	−0.489932
$772$	0	0
$773$	−20.3937	−0.733512	−0.366756	−	0.930317i	$-0.619532\pi$
$773$	−20.3937	−0.733512	−0.366756	+	0.930317i	$0.619532\pi$
$774$	0	0
$775$	−31.0388	−1.11494
$776$	0	0
$777$	54.9047	1.96969
$778$	0	0
$779$	−21.3190	−0.763832
$780$	0	0
$781$	−0.0134095	−0.000479831 0
$782$	0	0
$783$	−0.850855	−0.0304071
$784$	0	0
$785$	24.9616	0.890919
$786$	0	0
$787$	−19.6775	−0.701429	−0.350714	−	0.936482i	$-0.614061\pi$
$787$	−19.6775	−0.701429	−0.350714	+	0.936482i	$0.614061\pi$
$788$	0	0
$789$	−11.4577	−0.407905
$790$	0	0
$791$	−2.06292	−0.0733489
$792$	0	0
$793$	0	0
$794$	0	0
$795$	5.77777	0.204917
$796$	0	0
$797$	−45.8689	−1.62476	−0.812380	−	0.583128i	$-0.801828\pi$
$797$	−45.8689	−1.62476	−0.812380	+	0.583128i	$0.801828\pi$
$798$	0	0
$799$	27.9517	0.988859
$800$	0	0
$801$	17.0858	0.603695
$802$	0	0
$803$	0.318404	0.0112362
$804$	0	0
$805$	−90.3779	−3.18540
$806$	0	0
$807$	22.3666	0.787341
$808$	0	0
$809$	38.1414	1.34098	0.670490	−	0.741919i	$-0.266084\pi$
$809$	38.1414	1.34098	0.670490	+	0.741919i	$0.266084\pi$
$810$	0	0
$811$	−46.6983	−1.63980	−0.819899	−	0.572509i	$-0.805970\pi$
$811$	−46.6983	−1.63980	−0.819899	+	0.572509i	$0.805970\pi$
$812$	0	0
$813$	−3.87263	−0.135819
$814$	0	0
$815$	25.3319	0.887337
$816$	0	0
$817$	10.4638	0.366083
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.6950	1.14106	0.570532	−	0.821276i	$-0.306737\pi$
$821$	32.6950	1.14106	0.570532	+	0.821276i	$0.306737\pi$
$822$	0	0
$823$	21.6799	0.755715	0.377858	−	0.925864i	$-0.376661\pi$
$823$	21.6799	0.755715	0.377858	+	0.925864i	$0.376661\pi$
$824$	0	0
$825$	−0.682333	−0.0237558
$826$	0	0
$827$	13.9172	0.483950	0.241975	−	0.970283i	$-0.422205\pi$
$827$	13.9172	0.483950	0.241975	+	0.970283i	$0.422205\pi$
$828$	0	0
$829$	−16.4047	−0.569760	−0.284880	−	0.958563i	$-0.591954\pi$
$829$	−16.4047	−0.569760	−0.284880	+	0.958563i	$0.591954\pi$
$830$	0	0
$831$	28.7090	0.995904
$832$	0	0
$833$	−84.1426	−2.91537
$834$	0	0
$835$	−53.9711	−1.86775
$836$	0	0
$837$	−6.23490	−0.215510
$838$	0	0
$839$	−11.6146	−0.400982	−0.200491	−	0.979696i	$-0.564254\pi$
$839$	−11.6146	−0.400982	−0.200491	+	0.979696i	$0.564254\pi$
$840$	0	0
$841$	−28.2760	−0.975036
$842$	0	0
$843$	−29.0858	−1.00177
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−51.5241	−1.77039
$848$	0	0
$849$	−13.7560	−0.472105
$850$	0	0
$851$	71.3551	2.44602
$852$	0	0
$853$	26.2983	0.900436	0.450218	−	0.892919i	$-0.351346\pi$
$853$	26.2983	0.900436	0.450218	+	0.892919i	$0.351346\pi$
$854$	0	0
$855$	−15.7560	−0.538844
$856$	0	0
$857$	−48.6305	−1.66119	−0.830594	−	0.556879i	$-0.811999\pi$
$857$	−48.6305	−1.66119	−0.830594	+	0.556879i	$0.811999\pi$
$858$	0	0
$859$	−33.6185	−1.14705	−0.573524	−	0.819189i	$-0.694424\pi$
$859$	−33.6185	−1.14705	−0.573524	+	0.819189i	$0.694424\pi$
$860$	0	0
$861$	−20.0543	−0.683449
$862$	0	0
$863$	5.78879	0.197053	0.0985264	−	0.995134i	$-0.468587\pi$
$863$	5.78879	0.197053	0.0985264	+	0.995134i	$0.468587\pi$
$864$	0	0
$865$	48.4717	1.64809
$866$	0	0
$867$	14.4034	0.489166
$868$	0	0
$869$	1.98957	0.0674917
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.12737	0.0720007
$874$	0	0
$875$	0.322664	0.0109080
$876$	0	0
$877$	9.50604	0.320996	0.160498	−	0.987036i	$-0.448690\pi$
$877$	9.50604	0.320996	0.160498	+	0.987036i	$0.448690\pi$
$878$	0	0
$879$	27.7362	0.935517
$880$	0	0
$881$	−46.7875	−1.57631	−0.788155	−	0.615477i	$-0.788963\pi$
$881$	−46.7875	−1.57631	−0.788155	+	0.615477i	$0.788963\pi$
$882$	0	0
$883$	−3.03146	−0.102017	−0.0510084	−	0.998698i	$-0.516244\pi$
$883$	−3.03146	−0.102017	−0.0510084	+	0.998698i	$0.516244\pi$
$884$	0	0
$885$	−18.6364	−0.626456
$886$	0	0
$887$	−37.0180	−1.24294	−0.621472	−	0.783436i	$-0.713465\pi$
$887$	−37.0180	−1.24294	−0.621472	+	0.783436i	$0.713465\pi$
$888$	0	0
$889$	36.9385	1.23888
$890$	0	0
$891$	−0.137063	−0.00459179
$892$	0	0
$893$	−24.8793	−0.832555
$894$	0	0
$895$	1.65365	0.0552753
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.30499	0.176931
$900$	0	0
$901$	10.2500	0.341476
$902$	0	0
$903$	9.84309	0.327557
$904$	0	0
$905$	−28.0823	−0.933487
$906$	0	0
$907$	−19.0965	−0.634089	−0.317045	−	0.948411i	$-0.602690\pi$
$907$	−19.0965	−0.634089	−0.317045	+	0.948411i	$0.602690\pi$
$908$	0	0
$909$	−9.18598	−0.304680
$910$	0	0
$911$	31.3142	1.03749	0.518743	−	0.854930i	$-0.326400\pi$
$911$	31.3142	1.03749	0.518743	+	0.854930i	$0.326400\pi$
$912$	0	0
$913$	−1.35093	−0.0447092
$914$	0	0
$915$	−13.8866	−0.459078
$916$	0	0
$917$	2.91531	0.0962721
$918$	0	0
$919$	−30.3967	−1.00270	−0.501348	−	0.865246i	$-0.667162\pi$
$919$	−30.3967	−1.00270	−0.501348	+	0.865246i	$0.667162\pi$
$920$	0	0
$921$	12.4590	0.410539
$922$	0	0
$923$	0	0
$924$	0	0
$925$	58.2538	1.91537
$926$	0	0
$927$	−0.225209	−0.00739685
$928$	0	0
$929$	−40.5810	−1.33142	−0.665710	−	0.746210i	$-0.731871\pi$
$929$	−40.5810	−1.33142	−0.665710	+	0.746210i	$0.731871\pi$
$930$	0	0
$931$	74.8939	2.45455
$932$	0	0
$933$	6.09783	0.199634
$934$	0	0
$935$	−2.42626	−0.0793470
$936$	0	0
$937$	−18.7047	−0.611056	−0.305528	−	0.952183i	$-0.598833\pi$
$937$	−18.7047	−0.611056	−0.305528	+	0.952183i	$0.598833\pi$
$938$	0	0
$939$	−12.7385	−0.415707
$940$	0	0
$941$	−4.04998	−0.132026	−0.0660128	−	0.997819i	$-0.521028\pi$
$941$	−4.04998	−0.132026	−0.0660128	+	0.997819i	$0.521028\pi$
$942$	0	0
$943$	−26.0629	−0.848725
$944$	0	0
$945$	−14.8213	−0.482137
$946$	0	0
$947$	−11.5356	−0.374856	−0.187428	−	0.982278i	$-0.560015\pi$
$947$	−11.5356	−0.374856	−0.187428	+	0.982278i	$0.560015\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	14.8140	0.480377
$952$	0	0
$953$	−9.57109	−0.310038	−0.155019	−	0.987911i	$-0.549544\pi$
$953$	−9.57109	−0.310038	−0.155019	+	0.987911i	$0.549544\pi$
$954$	0	0
$955$	−22.2112	−0.718738
$956$	0	0
$957$	0.116621	0.00376982
$958$	0	0
$959$	18.7681	0.606053
$960$	0	0
$961$	7.87395	0.253998
$962$	0	0
$963$	−11.2838	−0.363616
$964$	0	0
$965$	56.0883	1.80555
$966$	0	0
$967$	61.2073	1.96829	0.984147	−	0.177357i	$-0.0567546\pi$
$967$	61.2073	1.96829	0.984147	+	0.177357i	$0.0567546\pi$
$968$	0	0
$969$	−27.9517	−0.897937
$970$	0	0
$971$	28.8595	0.926145	0.463072	−	0.886320i	$-0.346747\pi$
$971$	28.8595	0.926145	0.463072	+	0.886320i	$0.346747\pi$
$972$	0	0
$973$	64.0844	2.05445
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46.6305	−1.49184	−0.745922	−	0.666034i	$-0.767991\pi$
$977$	−46.6305	−1.49184	−0.745922	+	0.666034i	$0.767991\pi$
$978$	0	0
$979$	−2.34183	−0.0748452
$980$	0	0
$981$	−0.195669	−0.00624724
$982$	0	0
$983$	55.6883	1.77618	0.888090	−	0.459669i	$-0.152032\pi$
$983$	55.6883	1.77618	0.888090	+	0.459669i	$0.152032\pi$
$984$	0	0
$985$	−58.9347	−1.87782
$986$	0	0
$987$	−23.4034	−0.744939
$988$	0	0
$989$	12.7922	0.406770
$990$	0	0
$991$	9.32172	0.296114	0.148057	−	0.988979i	$-0.452698\pi$
$991$	9.32172	0.296114	0.148057	+	0.988979i	$0.452698\pi$
$992$	0	0
$993$	−7.70171	−0.244406
$994$	0	0
$995$	24.2045	0.767334
$996$	0	0
$997$	46.0253	1.45764	0.728819	−	0.684707i	$-0.240070\pi$
$997$	46.0253	1.45764	0.728819	+	0.684707i	$0.240070\pi$
$998$	0	0
$999$	11.7017	0.370226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cj.1.1		3
4.3	odd	2		1014.2.a.l.1.1	✓	3
12.11	even	2		3042.2.a.bh.1.3		3
13.12	even	2		8112.2.a.cm.1.3		3
52.3	odd	6		1014.2.e.n.529.1		6
52.7	even	12		1014.2.i.h.361.1		12
52.11	even	12		1014.2.i.h.823.4		12
52.15	even	12		1014.2.i.h.823.3		12
52.19	even	12		1014.2.i.h.361.6		12
52.23	odd	6		1014.2.e.l.529.3		6
52.31	even	4		1014.2.b.f.337.6		6
52.35	odd	6		1014.2.e.n.991.1		6
52.43	odd	6		1014.2.e.l.991.3		6
52.47	even	4		1014.2.b.f.337.1		6
52.51	odd	2		1014.2.a.n.1.3	yes	3
156.47	odd	4		3042.2.b.o.1351.6		6
156.83	odd	4		3042.2.b.o.1351.1		6
156.155	even	2		3042.2.a.ba.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.l.1.1	✓	3	4.3	odd	2
1014.2.a.n.1.3	yes	3	52.51	odd	2
1014.2.b.f.337.1		6	52.47	even	4
1014.2.b.f.337.6		6	52.31	even	4
1014.2.e.l.529.3		6	52.23	odd	6
1014.2.e.l.991.3		6	52.43	odd	6
1014.2.e.n.529.1		6	52.3	odd	6
1014.2.e.n.991.1		6	52.35	odd	6
1014.2.i.h.361.1		12	52.7	even	12
1014.2.i.h.361.6		12	52.19	even	12
1014.2.i.h.823.3		12	52.15	even	12
1014.2.i.h.823.4		12	52.11	even	12
3042.2.a.ba.1.1		3	156.155	even	2
3042.2.a.bh.1.3		3	12.11	even	2
3042.2.b.o.1351.1		6	156.83	odd	4
3042.2.b.o.1351.6		6	156.47	odd	4
8112.2.a.cj.1.1		3	1.1	even	1	trivial
8112.2.a.cm.1.3		3	13.12	even	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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.15883 q^{5} +4.69202 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.15883 q^{5} +4.69202 q^{7} +1.00000 q^{9} -0.137063 q^{11} -3.15883 q^{15} -5.60388 q^{17} +4.98792 q^{19} +4.69202 q^{21} +6.09783 q^{23} +4.97823 q^{25} +1.00000 q^{27} -0.850855 q^{29} -6.23490 q^{31} -0.137063 q^{33} -14.8213 q^{35} +11.7017 q^{37} -4.27413 q^{41} +2.09783 q^{43} -3.15883 q^{45} -4.98792 q^{47} +15.0151 q^{49} -5.60388 q^{51} -1.82908 q^{53} +0.432960 q^{55} +4.98792 q^{57} +5.89977 q^{59} +4.39612 q^{61} +4.69202 q^{63} +4.71379 q^{67} +6.09783 q^{69} +0.0978347 q^{71} -2.32304 q^{73} +4.97823 q^{75} -0.643104 q^{77} -14.5157 q^{79} +1.00000 q^{81} +9.85623 q^{83} +17.7017 q^{85} -0.850855 q^{87} +17.0858 q^{89} -6.23490 q^{93} -15.7560 q^{95} +2.12737 q^{97} -0.137063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - q^{5} + 9 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - q^5 + 9 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - q^{5} + 9 q^{7} + 3 q^{9} + 5 q^{11} - q^{15} - 8 q^{17} - 4 q^{19} + 9 q^{21} + 18 q^{25} + 3 q^{27} + 11 q^{29} + 5 q^{31} + 5 q^{33} + 4 q^{35} + 8 q^{37} - 2 q^{41} - 12 q^{43} - q^{45} + 4 q^{47} + 20 q^{49} - 8 q^{51} + 5 q^{53} - 18 q^{55} - 4 q^{57} - 5 q^{59} + 22 q^{61} + 9 q^{63} + 6 q^{67} - 18 q^{71} + 13 q^{73} + 18 q^{75} - 6 q^{77} - 31 q^{79} + 3 q^{81} + 13 q^{83} + 26 q^{85} + 11 q^{87} + 14 q^{89} + 5 q^{93} - 8 q^{95} + 23 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - q^5 + 9 * q^7 + 3 * q^9 + 5 * q^11 - q^15 - 8 * q^17 - 4 * q^19 + 9 * q^21 + 18 * q^25 + 3 * q^27 + 11 * q^29 + 5 * q^31 + 5 * q^33 + 4 * q^35 + 8 * q^37 - 2 * q^41 - 12 * q^43 - q^45 + 4 * q^47 + 20 * q^49 - 8 * q^51 + 5 * q^53 - 18 * q^55 - 4 * q^57 - 5 * q^59 + 22 * q^61 + 9 * q^63 + 6 * q^67 - 18 * q^71 + 13 * q^73 + 18 * q^75 - 6 * q^77 - 31 * q^79 + 3 * q^81 + 13 * q^83 + 26 * q^85 + 11 * q^87 + 14 * q^89 + 5 * q^93 - 8 * q^95 + 23 * q^97 + 5 * q^99

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