LMFDB - Embedded newform 8112.2.a.by.1.2

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:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} -5.18598 q^{11} +1.44504 q^{15} -0.753020 q^{17} +7.96077 q^{19} -3.44504 q^{21} +2.82908 q^{23} -2.91185 q^{25} -1.00000 q^{27} -3.91185 q^{29} -4.89977 q^{31} +5.18598 q^{33} -4.97823 q^{35} -6.24698 q^{37} -1.80194 q^{41} +7.09783 q^{43} -1.44504 q^{45} +10.5526 q^{47} +4.86831 q^{49} +0.753020 q^{51} -3.08815 q^{53} +7.49396 q^{55} -7.96077 q^{57} +1.87800 q^{59} +3.34481 q^{61} +3.44504 q^{63} -4.54288 q^{67} -2.82908 q^{69} -9.11960 q^{71} -2.95108 q^{73} +2.91185 q^{75} -17.8659 q^{77} +9.43296 q^{79} +1.00000 q^{81} -6.46681 q^{83} +1.08815 q^{85} +3.91185 q^{87} -1.15883 q^{89} +4.89977 q^{93} -11.5036 q^{95} -8.65817 q^{97} -5.18598 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9} - q^{11} + 4 q^{15} - 7 q^{17} + 11 q^{19} - 10 q^{21} - 2 q^{23} - 5 q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + q^{33} - 18 q^{35} - 14 q^{37} - q^{41} + 3 q^{43} - 4 q^{45} - 9 q^{47} + 17 q^{49} + 7 q^{51} - 13 q^{53} + 13 q^{55} - 11 q^{57} - 14 q^{59} - 13 q^{61} + 10 q^{63} + 5 q^{67} + 2 q^{69} - 6 q^{71} - 18 q^{73} + 5 q^{75} - 15 q^{77} + 9 q^{79} + 3 q^{81} - 16 q^{83} + 7 q^{85} + 8 q^{87} + 5 q^{89} - 8 q^{93} - 3 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 - q^11 + 4 * q^15 - 7 * q^17 + 11 * q^19 - 10 * q^21 - 2 * q^23 - 5 * q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + q^33 - 18 * q^35 - 14 * q^37 - q^41 + 3 * q^43 - 4 * q^45 - 9 * q^47 + 17 * q^49 + 7 * q^51 - 13 * q^53 + 13 * q^55 - 11 * q^57 - 14 * q^59 - 13 * q^61 + 10 * q^63 + 5 * q^67 + 2 * q^69 - 6 * q^71 - 18 * q^73 + 5 * q^75 - 15 * q^77 + 9 * q^79 + 3 * q^81 - 16 * q^83 + 7 * q^85 + 8 * q^87 + 5 * q^89 - 8 * q^93 - 3 * q^95 - 5 * q^97 - 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
$5$	−1.44504	−0.646242	−0.323121	−	0.946358i	$-0.604732\pi$
$5$	−1.44504	−0.646242	−0.323121	+	0.946358i	$0.604732\pi$
$6$	0	0
$7$	3.44504	1.30210	0.651052	−	0.759033i	$-0.274328\pi$
$7$	3.44504	1.30210	0.651052	+	0.759033i	$0.274328\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.18598	−1.56363	−0.781816	−	0.623509i	$-0.785706\pi$
$11$	−5.18598	−1.56363	−0.781816	+	0.623509i	$0.785706\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.44504	0.373108
$16$	0	0
$17$	−0.753020	−0.182634	−0.0913171	−	0.995822i	$-0.529108\pi$
$17$	−0.753020	−0.182634	−0.0913171	+	0.995822i	$0.529108\pi$
$18$	0	0
$19$	7.96077	1.82633	0.913163	−	0.407594i	$-0.133632\pi$
$19$	7.96077	1.82633	0.913163	+	0.407594i	$0.133632\pi$
$20$	0	0
$21$	−3.44504	−0.751770
$22$	0	0
$23$	2.82908	0.589905	0.294952	−	0.955512i	$-0.404696\pi$
$23$	2.82908	0.589905	0.294952	+	0.955512i	$0.404696\pi$
$24$	0	0
$25$	−2.91185	−0.582371
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.91185	−0.726413	−0.363207	−	0.931709i	$-0.618318\pi$
$29$	−3.91185	−0.726413	−0.363207	+	0.931709i	$0.618318\pi$
$30$	0	0
$31$	−4.89977	−0.880025	−0.440013	−	0.897992i	$-0.645026\pi$
$31$	−4.89977	−0.880025	−0.440013	+	0.897992i	$0.645026\pi$
$32$	0	0
$33$	5.18598	0.902763
$34$	0	0
$35$	−4.97823	−0.841474
$36$	0	0
$37$	−6.24698	−1.02700	−0.513499	−	0.858090i	$-0.671651\pi$
$37$	−6.24698	−1.02700	−0.513499	+	0.858090i	$0.671651\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.80194	−0.281415	−0.140708	−	0.990051i	$-0.544938\pi$
$41$	−1.80194	−0.281415	−0.140708	+	0.990051i	$0.544938\pi$
$42$	0	0
$43$	7.09783	1.08241	0.541205	−	0.840891i	$-0.317968\pi$
$43$	7.09783	1.08241	0.541205	+	0.840891i	$0.317968\pi$
$44$	0	0
$45$	−1.44504	−0.215414
$46$	0	0
$47$	10.5526	1.53925	0.769625	−	0.638496i	$-0.220443\pi$
$47$	10.5526	1.53925	0.769625	+	0.638496i	$0.220443\pi$
$48$	0	0
$49$	4.86831	0.695473
$50$	0	0
$51$	0.753020	0.105444
$52$	0	0
$53$	−3.08815	−0.424189	−0.212095	−	0.977249i	$-0.568029\pi$
$53$	−3.08815	−0.424189	−0.212095	+	0.977249i	$0.568029\pi$
$54$	0	0
$55$	7.49396	1.01049
$56$	0	0
$57$	−7.96077	−1.05443
$58$	0	0
$59$	1.87800	0.244495	0.122248	−	0.992500i	$-0.460990\pi$
$59$	1.87800	0.244495	0.122248	+	0.992500i	$0.460990\pi$
$60$	0	0
$61$	3.34481	0.428260	0.214130	−	0.976805i	$-0.431308\pi$
$61$	3.34481	0.428260	0.214130	+	0.976805i	$0.431308\pi$
$62$	0	0
$63$	3.44504	0.434034
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.54288	−0.555001	−0.277500	−	0.960726i	$-0.589506\pi$
$67$	−4.54288	−0.555001	−0.277500	+	0.960726i	$0.589506\pi$
$68$	0	0
$69$	−2.82908	−0.340582
$70$	0	0
$71$	−9.11960	−1.08230	−0.541149	−	0.840927i	$-0.682011\pi$
$71$	−9.11960	−1.08230	−0.541149	+	0.840927i	$0.682011\pi$
$72$	0	0
$73$	−2.95108	−0.345398	−0.172699	−	0.984975i	$-0.555249\pi$
$73$	−2.95108	−0.345398	−0.172699	+	0.984975i	$0.555249\pi$
$74$	0	0
$75$	2.91185	0.336232
$76$	0	0
$77$	−17.8659	−2.03601
$78$	0	0
$79$	9.43296	1.06129	0.530645	−	0.847594i	$-0.321950\pi$
$79$	9.43296	1.06129	0.530645	+	0.847594i	$0.321950\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.46681	−0.709825	−0.354912	−	0.934900i	$-0.615489\pi$
$83$	−6.46681	−0.709825	−0.354912	+	0.934900i	$0.615489\pi$
$84$	0	0
$85$	1.08815	0.118026
$86$	0	0
$87$	3.91185	0.419395
$88$	0	0
$89$	−1.15883	−0.122836	−0.0614181	−	0.998112i	$-0.519562\pi$
$89$	−1.15883	−0.122836	−0.0614181	+	0.998112i	$0.519562\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.89977	0.508083
$94$	0	0
$95$	−11.5036	−1.18025
$96$	0	0
$97$	−8.65817	−0.879104	−0.439552	−	0.898217i	$-0.644863\pi$
$97$	−8.65817	−0.879104	−0.439552	+	0.898217i	$0.644863\pi$
$98$	0	0
$99$	−5.18598	−0.521211
$100$	0	0
$101$	−8.47650	−0.843443	−0.421722	−	0.906725i	$-0.638574\pi$
$101$	−8.47650	−0.843443	−0.421722	+	0.906725i	$0.638574\pi$
$102$	0	0
$103$	−5.64742	−0.556456	−0.278228	−	0.960515i	$-0.589747\pi$
$103$	−5.64742	−0.556456	−0.278228	+	0.960515i	$0.589747\pi$
$104$	0	0
$105$	4.97823	0.485825
$106$	0	0
$107$	6.73556	0.651151	0.325576	−	0.945516i	$-0.394442\pi$
$107$	6.73556	0.651151	0.325576	+	0.945516i	$0.394442\pi$
$108$	0	0
$109$	2.07606	0.198851	0.0994255	−	0.995045i	$-0.468300\pi$
$109$	2.07606	0.198851	0.0994255	+	0.995045i	$0.468300\pi$
$110$	0	0
$111$	6.24698	0.592937
$112$	0	0
$113$	6.16852	0.580286	0.290143	−	0.956983i	$-0.406297\pi$
$113$	6.16852	0.580286	0.290143	+	0.956983i	$0.406297\pi$
$114$	0	0
$115$	−4.08815	−0.381222
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.59419	−0.237809
$120$	0	0
$121$	15.8944	1.44495
$122$	0	0
$123$	1.80194	0.162475
$124$	0	0
$125$	11.4330	1.02260
$126$	0	0
$127$	14.2620	1.26555	0.632776	−	0.774335i	$-0.281915\pi$
$127$	14.2620	1.26555	0.632776	+	0.774335i	$0.281915\pi$
$128$	0	0
$129$	−7.09783	−0.624929
$130$	0	0
$131$	−22.6015	−1.97470	−0.987350	−	0.158554i	$-0.949317\pi$
$131$	−22.6015	−1.97470	−0.987350	+	0.158554i	$0.949317\pi$
$132$	0	0
$133$	27.4252	2.37807
$134$	0	0
$135$	1.44504	0.124369
$136$	0	0
$137$	13.6353	1.16495	0.582473	−	0.812850i	$-0.302085\pi$
$137$	13.6353	1.16495	0.582473	+	0.812850i	$0.302085\pi$
$138$	0	0
$139$	−17.6015	−1.49294	−0.746469	−	0.665420i	$-0.768252\pi$
$139$	−17.6015	−1.49294	−0.746469	+	0.665420i	$0.768252\pi$
$140$	0	0
$141$	−10.5526	−0.888686
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.65279	0.469439
$146$	0	0
$147$	−4.86831	−0.401532
$148$	0	0
$149$	−12.7385	−1.04358	−0.521791	−	0.853073i	$-0.674736\pi$
$149$	−12.7385	−1.04358	−0.521791	+	0.853073i	$0.674736\pi$
$150$	0	0
$151$	15.6407	1.27282	0.636412	−	0.771350i	$-0.280418\pi$
$151$	15.6407	1.27282	0.636412	+	0.771350i	$0.280418\pi$
$152$	0	0
$153$	−0.753020	−0.0608781
$154$	0	0
$155$	7.08038	0.568710
$156$	0	0
$157$	−0.823708	−0.0657391	−0.0328695	−	0.999460i	$-0.510465\pi$
$157$	−0.823708	−0.0657391	−0.0328695	+	0.999460i	$0.510465\pi$
$158$	0	0
$159$	3.08815	0.244906
$160$	0	0
$161$	9.74632	0.768117
$162$	0	0
$163$	6.26875	0.491006	0.245503	−	0.969396i	$-0.421047\pi$
$163$	6.26875	0.491006	0.245503	+	0.969396i	$0.421047\pi$
$164$	0	0
$165$	−7.49396	−0.583404
$166$	0	0
$167$	−7.45042	−0.576531	−0.288265	−	0.957551i	$-0.593079\pi$
$167$	−7.45042	−0.576531	−0.288265	+	0.957551i	$0.593079\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.96077	0.608775
$172$	0	0
$173$	−2.00969	−0.152794	−0.0763969	−	0.997077i	$-0.524342\pi$
$173$	−2.00969	−0.152794	−0.0763969	+	0.997077i	$0.524342\pi$
$174$	0	0
$175$	−10.0315	−0.758307
$176$	0	0
$177$	−1.87800	−0.141159
$178$	0	0
$179$	−20.0368	−1.49762	−0.748812	−	0.662783i	$-0.769375\pi$
$179$	−20.0368	−1.49762	−0.748812	+	0.662783i	$0.769375\pi$
$180$	0	0
$181$	24.1226	1.79302	0.896509	−	0.443026i	$-0.146095\pi$
$181$	24.1226	1.79302	0.896509	+	0.443026i	$0.146095\pi$
$182$	0	0
$183$	−3.34481	−0.247256
$184$	0	0
$185$	9.02715	0.663689
$186$	0	0
$187$	3.90515	0.285573
$188$	0	0
$189$	−3.44504	−0.250590
$190$	0	0
$191$	7.08038	0.512318	0.256159	−	0.966635i	$-0.417543\pi$
$191$	7.08038	0.512318	0.256159	+	0.966635i	$0.417543\pi$
$192$	0	0
$193$	−9.76809	−0.703122	−0.351561	−	0.936165i	$-0.614349\pi$
$193$	−9.76809	−0.703122	−0.351561	+	0.936165i	$0.614349\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.4112	−1.66798	−0.833989	−	0.551781i	$-0.813948\pi$
$197$	−23.4112	−1.66798	−0.833989	+	0.551781i	$0.813948\pi$
$198$	0	0
$199$	−4.02475	−0.285307	−0.142654	−	0.989773i	$-0.545563\pi$
$199$	−4.02475	−0.285307	−0.142654	+	0.989773i	$0.545563\pi$
$200$	0	0
$201$	4.54288	0.320430
$202$	0	0
$203$	−13.4765	−0.945865
$204$	0	0
$205$	2.60388	0.181863
$206$	0	0
$207$	2.82908	0.196635
$208$	0	0
$209$	−41.2844	−2.85570
$210$	0	0
$211$	3.91185	0.269303	0.134652	−	0.990893i	$-0.457008\pi$
$211$	3.91185	0.269303	0.134652	+	0.990893i	$0.457008\pi$
$212$	0	0
$213$	9.11960	0.624865
$214$	0	0
$215$	−10.2567	−0.699499
$216$	0	0
$217$	−16.8799	−1.14588
$218$	0	0
$219$	2.95108	0.199416
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.44935	0.498846	0.249423	−	0.968395i	$-0.419759\pi$
$223$	7.44935	0.498846	0.249423	+	0.968395i	$0.419759\pi$
$224$	0	0
$225$	−2.91185	−0.194124
$226$	0	0
$227$	−21.2500	−1.41041	−0.705205	−	0.709004i	$-0.749145\pi$
$227$	−21.2500	−1.41041	−0.705205	+	0.709004i	$0.749145\pi$
$228$	0	0
$229$	−9.29590	−0.614290	−0.307145	−	0.951663i	$-0.599374\pi$
$229$	−9.29590	−0.614290	−0.307145	+	0.951663i	$0.599374\pi$
$230$	0	0
$231$	17.8659	1.17549
$232$	0	0
$233$	16.2107	1.06200	0.531000	−	0.847372i	$-0.321816\pi$
$233$	16.2107	1.06200	0.531000	+	0.847372i	$0.321816\pi$
$234$	0	0
$235$	−15.2489	−0.994728
$236$	0	0
$237$	−9.43296	−0.612737
$238$	0	0
$239$	−13.5090	−0.873826	−0.436913	−	0.899504i	$-0.643928\pi$
$239$	−13.5090	−0.873826	−0.436913	+	0.899504i	$0.643928\pi$
$240$	0	0
$241$	−6.26875	−0.403806	−0.201903	−	0.979406i	$-0.564713\pi$
$241$	−6.26875	−0.403806	−0.201903	+	0.979406i	$0.564713\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.03492	−0.449444
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.46681	0.409818
$250$	0	0
$251$	0.753020	0.0475302	0.0237651	−	0.999718i	$-0.492435\pi$
$251$	0.753020	0.0475302	0.0237651	+	0.999718i	$0.492435\pi$
$252$	0	0
$253$	−14.6716	−0.922394
$254$	0	0
$255$	−1.08815	−0.0681423
$256$	0	0
$257$	19.7265	1.23050	0.615252	−	0.788331i	$-0.289054\pi$
$257$	19.7265	1.23050	0.615252	+	0.788331i	$0.289054\pi$
$258$	0	0
$259$	−21.5211	−1.33726
$260$	0	0
$261$	−3.91185	−0.242138
$262$	0	0
$263$	17.6093	1.08583	0.542917	−	0.839787i	$-0.317320\pi$
$263$	17.6093	1.08583	0.542917	+	0.839787i	$0.317320\pi$
$264$	0	0
$265$	4.46250	0.274129
$266$	0	0
$267$	1.15883	0.0709195
$268$	0	0
$269$	−16.3870	−0.999135	−0.499567	−	0.866275i	$-0.666508\pi$
$269$	−16.3870	−0.999135	−0.499567	+	0.866275i	$0.666508\pi$
$270$	0	0
$271$	−0.795233	−0.0483070	−0.0241535	−	0.999708i	$-0.507689\pi$
$271$	−0.795233	−0.0483070	−0.0241535	+	0.999708i	$0.507689\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.1008	0.910614
$276$	0	0
$277$	−4.83340	−0.290411	−0.145205	−	0.989402i	$-0.546384\pi$
$277$	−4.83340	−0.290411	−0.145205	+	0.989402i	$0.546384\pi$
$278$	0	0
$279$	−4.89977	−0.293342
$280$	0	0
$281$	−18.7748	−1.12001	−0.560005	−	0.828489i	$-0.689201\pi$
$281$	−18.7748	−1.12001	−0.560005	+	0.828489i	$0.689201\pi$
$282$	0	0
$283$	7.91723	0.470631	0.235315	−	0.971919i	$-0.424388\pi$
$283$	7.91723	0.470631	0.235315	+	0.971919i	$0.424388\pi$
$284$	0	0
$285$	11.5036	0.681417
$286$	0	0
$287$	−6.20775	−0.366432
$288$	0	0
$289$	−16.4330	−0.966645
$290$	0	0
$291$	8.65817	0.507551
$292$	0	0
$293$	−6.57912	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$293$	−6.57912	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$294$	0	0
$295$	−2.71379	−0.158003
$296$	0	0
$297$	5.18598	0.300921
$298$	0	0
$299$	0	0
$300$	0	0
$301$	24.4523	1.40941
$302$	0	0
$303$	8.47650	0.486962
$304$	0	0
$305$	−4.83340	−0.276759
$306$	0	0
$307$	−24.8649	−1.41911	−0.709556	−	0.704649i	$-0.751105\pi$
$307$	−24.8649	−1.41911	−0.709556	+	0.704649i	$0.751105\pi$
$308$	0	0
$309$	5.64742	0.321270
$310$	0	0
$311$	17.0804	0.968539	0.484270	−	0.874919i	$-0.339085\pi$
$311$	17.0804	0.968539	0.484270	+	0.874919i	$0.339085\pi$
$312$	0	0
$313$	15.6974	0.887269	0.443635	−	0.896208i	$-0.353689\pi$
$313$	15.6974	0.887269	0.443635	+	0.896208i	$0.353689\pi$
$314$	0	0
$315$	−4.97823	−0.280491
$316$	0	0
$317$	−32.7821	−1.84123	−0.920613	−	0.390477i	$-0.872310\pi$
$317$	−32.7821	−1.84123	−0.920613	+	0.390477i	$0.872310\pi$
$318$	0	0
$319$	20.2868	1.13584
$320$	0	0
$321$	−6.73556	−0.375942
$322$	0	0
$323$	−5.99462	−0.333550
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.07606	−0.114807
$328$	0	0
$329$	36.3540	2.00426
$330$	0	0
$331$	−29.1618	−1.60288	−0.801439	−	0.598076i	$-0.795932\pi$
$331$	−29.1618	−1.60288	−0.801439	+	0.598076i	$0.795932\pi$
$332$	0	0
$333$	−6.24698	−0.342332
$334$	0	0
$335$	6.56465	0.358665
$336$	0	0
$337$	−33.2911	−1.81348	−0.906741	−	0.421688i	$-0.861438\pi$
$337$	−33.2911	−1.81348	−0.906741	+	0.421688i	$0.861438\pi$
$338$	0	0
$339$	−6.16852	−0.335028
$340$	0	0
$341$	25.4101	1.37604
$342$	0	0
$343$	−7.34375	−0.396525
$344$	0	0
$345$	4.08815	0.220098
$346$	0	0
$347$	0.873690	0.0469022	0.0234511	−	0.999725i	$-0.492535\pi$
$347$	0.873690	0.0469022	0.0234511	+	0.999725i	$0.492535\pi$
$348$	0	0
$349$	3.23191	0.173000	0.0865002	−	0.996252i	$-0.472432\pi$
$349$	3.23191	0.173000	0.0865002	+	0.996252i	$0.472432\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.14675	−0.433608	−0.216804	−	0.976215i	$-0.569563\pi$
$353$	−8.14675	−0.433608	−0.216804	+	0.976215i	$0.569563\pi$
$354$	0	0
$355$	13.1782	0.699427
$356$	0	0
$357$	2.59419	0.137299
$358$	0	0
$359$	2.64071	0.139371	0.0696857	−	0.997569i	$-0.477800\pi$
$359$	2.64071	0.139371	0.0696857	+	0.997569i	$0.477800\pi$
$360$	0	0
$361$	44.3739	2.33547
$362$	0	0
$363$	−15.8944	−0.834239
$364$	0	0
$365$	4.26444	0.223211
$366$	0	0
$367$	2.90408	0.151592	0.0757960	−	0.997123i	$-0.475850\pi$
$367$	2.90408	0.151592	0.0757960	+	0.997123i	$0.475850\pi$
$368$	0	0
$369$	−1.80194	−0.0938051
$370$	0	0
$371$	−10.6388	−0.552339
$372$	0	0
$373$	−8.39852	−0.434859	−0.217429	−	0.976076i	$-0.569767\pi$
$373$	−8.39852	−0.434859	−0.217429	+	0.976076i	$0.569767\pi$
$374$	0	0
$375$	−11.4330	−0.590396
$376$	0	0
$377$	0	0
$378$	0	0
$379$	15.7482	0.808932	0.404466	−	0.914553i	$-0.367457\pi$
$379$	15.7482	0.808932	0.404466	+	0.914553i	$0.367457\pi$
$380$	0	0
$381$	−14.2620	−0.730667
$382$	0	0
$383$	12.7385	0.650909	0.325455	−	0.945558i	$-0.394483\pi$
$383$	12.7385	0.650909	0.325455	+	0.945558i	$0.394483\pi$
$384$	0	0
$385$	25.8170	1.31576
$386$	0	0
$387$	7.09783	0.360803
$388$	0	0
$389$	−0.310371	−0.0157365	−0.00786823	−	0.999969i	$-0.502505\pi$
$389$	−0.310371	−0.0157365	−0.00786823	+	0.999969i	$0.502505\pi$
$390$	0	0
$391$	−2.13036	−0.107737
$392$	0	0
$393$	22.6015	1.14009
$394$	0	0
$395$	−13.6310	−0.685851
$396$	0	0
$397$	−1.49098	−0.0748299	−0.0374150	−	0.999300i	$-0.511912\pi$
$397$	−1.49098	−0.0748299	−0.0374150	+	0.999300i	$0.511912\pi$
$398$	0	0
$399$	−27.4252	−1.37298
$400$	0	0
$401$	23.8334	1.19018	0.595092	−	0.803658i	$-0.297116\pi$
$401$	23.8334	1.19018	0.595092	+	0.803658i	$0.297116\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.44504	−0.0718047
$406$	0	0
$407$	32.3967	1.60585
$408$	0	0
$409$	−4.26742	−0.211010	−0.105505	−	0.994419i	$-0.533646\pi$
$409$	−4.26742	−0.211010	−0.105505	+	0.994419i	$0.533646\pi$
$410$	0	0
$411$	−13.6353	−0.672581
$412$	0	0
$413$	6.46980	0.318358
$414$	0	0
$415$	9.34481	0.458719
$416$	0	0
$417$	17.6015	0.861948
$418$	0	0
$419$	−29.6896	−1.45043	−0.725217	−	0.688521i	$-0.758260\pi$
$419$	−29.6896	−1.45043	−0.725217	+	0.688521i	$0.758260\pi$
$420$	0	0
$421$	−29.3991	−1.43282	−0.716412	−	0.697677i	$-0.754217\pi$
$421$	−29.3991	−1.43282	−0.716412	+	0.697677i	$0.754217\pi$
$422$	0	0
$423$	10.5526	0.513083
$424$	0	0
$425$	2.19269	0.106361
$426$	0	0
$427$	11.5230	0.557638
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.0562	−1.59226	−0.796131	−	0.605124i	$-0.793123\pi$
$431$	−33.0562	−1.59226	−0.796131	+	0.605124i	$0.793123\pi$
$432$	0	0
$433$	−29.2664	−1.40645	−0.703226	−	0.710967i	$-0.748258\pi$
$433$	−29.2664	−1.40645	−0.703226	+	0.710967i	$0.748258\pi$
$434$	0	0
$435$	−5.65279	−0.271031
$436$	0	0
$437$	22.5217	1.07736
$438$	0	0
$439$	2.13169	0.101740	0.0508699	−	0.998705i	$-0.483801\pi$
$439$	2.13169	0.101740	0.0508699	+	0.998705i	$0.483801\pi$
$440$	0	0
$441$	4.86831	0.231824
$442$	0	0
$443$	22.9922	1.09239	0.546197	−	0.837657i	$-0.316075\pi$
$443$	22.9922	1.09239	0.546197	+	0.837657i	$0.316075\pi$
$444$	0	0
$445$	1.67456	0.0793819
$446$	0	0
$447$	12.7385	0.602513
$448$	0	0
$449$	12.9379	0.610579	0.305289	−	0.952260i	$-0.401247\pi$
$449$	12.9379	0.610579	0.305289	+	0.952260i	$0.401247\pi$
$450$	0	0
$451$	9.34481	0.440030
$452$	0	0
$453$	−15.6407	−0.734865
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.85325	0.227025	0.113513	−	0.993537i	$-0.463790\pi$
$457$	4.85325	0.227025	0.113513	+	0.993537i	$0.463790\pi$
$458$	0	0
$459$	0.753020	0.0351480
$460$	0	0
$461$	−18.8345	−0.877208	−0.438604	−	0.898680i	$-0.644527\pi$
$461$	−18.8345	−0.877208	−0.438604	+	0.898680i	$0.644527\pi$
$462$	0	0
$463$	−22.8767	−1.06317	−0.531585	−	0.847005i	$-0.678403\pi$
$463$	−22.8767	−1.06317	−0.531585	+	0.847005i	$0.678403\pi$
$464$	0	0
$465$	−7.08038	−0.328345
$466$	0	0
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	−15.6504	−0.722668
$470$	0	0
$471$	0.823708	0.0379545
$472$	0	0
$473$	−36.8092	−1.69249
$474$	0	0
$475$	−23.1806	−1.06360
$476$	0	0
$477$	−3.08815	−0.141396
$478$	0	0
$479$	−38.0901	−1.74038	−0.870190	−	0.492717i	$-0.836004\pi$
$479$	−38.0901	−1.74038	−0.870190	+	0.492717i	$0.836004\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−9.74632	−0.443473
$484$	0	0
$485$	12.5114	0.568114
$486$	0	0
$487$	−21.2500	−0.962928	−0.481464	−	0.876466i	$-0.659895\pi$
$487$	−21.2500	−0.962928	−0.481464	+	0.876466i	$0.659895\pi$
$488$	0	0
$489$	−6.26875	−0.283483
$490$	0	0
$491$	6.35019	0.286580	0.143290	−	0.989681i	$-0.454232\pi$
$491$	6.35019	0.286580	0.143290	+	0.989681i	$0.454232\pi$
$492$	0	0
$493$	2.94571	0.132668
$494$	0	0
$495$	7.49396	0.336828
$496$	0	0
$497$	−31.4174	−1.40926
$498$	0	0
$499$	−4.65087	−0.208202	−0.104101	−	0.994567i	$-0.533196\pi$
$499$	−4.65087	−0.208202	−0.104101	+	0.994567i	$0.533196\pi$
$500$	0	0
$501$	7.45042	0.332860
$502$	0	0
$503$	−15.4752	−0.690004	−0.345002	−	0.938602i	$-0.612122\pi$
$503$	−15.4752	−0.690004	−0.345002	+	0.938602i	$0.612122\pi$
$504$	0	0
$505$	12.2489	0.545069
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.5047	0.908855	0.454428	−	0.890784i	$-0.349844\pi$
$509$	20.5047	0.908855	0.454428	+	0.890784i	$0.349844\pi$
$510$	0	0
$511$	−10.1666	−0.449744
$512$	0	0
$513$	−7.96077	−0.351477
$514$	0	0
$515$	8.16075	0.359606
$516$	0	0
$517$	−54.7254	−2.40682
$518$	0	0
$519$	2.00969	0.0882155
$520$	0	0
$521$	−42.0267	−1.84122	−0.920611	−	0.390481i	$-0.872309\pi$
$521$	−42.0267	−1.84122	−0.920611	+	0.390481i	$0.872309\pi$
$522$	0	0
$523$	29.9885	1.31131	0.655653	−	0.755062i	$-0.272393\pi$
$523$	29.9885	1.31131	0.655653	+	0.755062i	$0.272393\pi$
$524$	0	0
$525$	10.0315	0.437809
$526$	0	0
$527$	3.68963	0.160723
$528$	0	0
$529$	−14.9963	−0.652012
$530$	0	0
$531$	1.87800	0.0814984
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−9.73317	−0.420802
$536$	0	0
$537$	20.0368	0.864653
$538$	0	0
$539$	−25.2470	−1.08746
$540$	0	0
$541$	−36.3803	−1.56411	−0.782056	−	0.623208i	$-0.785829\pi$
$541$	−36.3803	−1.56411	−0.782056	+	0.623208i	$0.785829\pi$
$542$	0	0
$543$	−24.1226	−1.03520
$544$	0	0
$545$	−3.00000	−0.128506
$546$	0	0
$547$	25.8159	1.10381	0.551905	−	0.833907i	$-0.313901\pi$
$547$	25.8159	1.10381	0.551905	+	0.833907i	$0.313901\pi$
$548$	0	0
$549$	3.34481	0.142753
$550$	0	0
$551$	−31.1414	−1.32667
$552$	0	0
$553$	32.4969	1.38191
$554$	0	0
$555$	−9.02715	−0.383181
$556$	0	0
$557$	−17.9903	−0.762274	−0.381137	−	0.924519i	$-0.624467\pi$
$557$	−17.9903	−0.762274	−0.381137	+	0.924519i	$0.624467\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.90515	−0.164876
$562$	0	0
$563$	39.1323	1.64923	0.824614	−	0.565695i	$-0.191392\pi$
$563$	39.1323	1.64923	0.824614	+	0.565695i	$0.191392\pi$
$564$	0	0
$565$	−8.91377	−0.375005
$566$	0	0
$567$	3.44504	0.144678
$568$	0	0
$569$	30.6002	1.28283	0.641413	−	0.767196i	$-0.278349\pi$
$569$	30.6002	1.28283	0.641413	+	0.767196i	$0.278349\pi$
$570$	0	0
$571$	2.96184	0.123949	0.0619745	−	0.998078i	$-0.480260\pi$
$571$	2.96184	0.123949	0.0619745	+	0.998078i	$0.480260\pi$
$572$	0	0
$573$	−7.08038	−0.295787
$574$	0	0
$575$	−8.23788	−0.343543
$576$	0	0
$577$	0.819396	0.0341119	0.0170560	−	0.999855i	$-0.494571\pi$
$577$	0.819396	0.0341119	0.0170560	+	0.999855i	$0.494571\pi$
$578$	0	0
$579$	9.76809	0.405948
$580$	0	0
$581$	−22.2784	−0.924265
$582$	0	0
$583$	16.0151	0.663276
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.7995	−1.31251	−0.656254	−	0.754540i	$-0.727860\pi$
$587$	−31.7995	−1.31251	−0.656254	+	0.754540i	$0.727860\pi$
$588$	0	0
$589$	−39.0060	−1.60721
$590$	0	0
$591$	23.4112	0.963008
$592$	0	0
$593$	4.26337	0.175076	0.0875379	−	0.996161i	$-0.472100\pi$
$593$	4.26337	0.175076	0.0875379	+	0.996161i	$0.472100\pi$
$594$	0	0
$595$	3.74871	0.153682
$596$	0	0
$597$	4.02475	0.164722
$598$	0	0
$599$	−24.7278	−1.01035	−0.505175	−	0.863017i	$-0.668572\pi$
$599$	−24.7278	−1.01035	−0.505175	+	0.863017i	$0.668572\pi$
$600$	0	0
$601$	6.82371	0.278345	0.139172	−	0.990268i	$-0.455556\pi$
$601$	6.82371	0.278345	0.139172	+	0.990268i	$0.455556\pi$
$602$	0	0
$603$	−4.54288	−0.185000
$604$	0	0
$605$	−22.9681	−0.933785
$606$	0	0
$607$	−31.9963	−1.29869	−0.649344	−	0.760494i	$-0.724957\pi$
$607$	−31.9963	−1.29869	−0.649344	+	0.760494i	$0.724957\pi$
$608$	0	0
$609$	13.4765	0.546095
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−33.5875	−1.35659	−0.678293	−	0.734792i	$-0.737280\pi$
$613$	−33.5875	−1.35659	−0.678293	+	0.734792i	$0.737280\pi$
$614$	0	0
$615$	−2.60388	−0.104998
$616$	0	0
$617$	26.5870	1.07035	0.535176	−	0.844740i	$-0.320245\pi$
$617$	26.5870	1.07035	0.535176	+	0.844740i	$0.320245\pi$
$618$	0	0
$619$	−9.17928	−0.368946	−0.184473	−	0.982838i	$-0.559058\pi$
$619$	−9.17928	−0.368946	−0.184473	+	0.982838i	$0.559058\pi$
$620$	0	0
$621$	−2.82908	−0.113527
$622$	0	0
$623$	−3.99223	−0.159945
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	0	0
$627$	41.2844	1.64874
$628$	0	0
$629$	4.70410	0.187565
$630$	0	0
$631$	17.0043	0.676931	0.338465	−	0.940979i	$-0.390092\pi$
$631$	17.0043	0.676931	0.338465	+	0.940979i	$0.390092\pi$
$632$	0	0
$633$	−3.91185	−0.155482
$634$	0	0
$635$	−20.6093	−0.817853
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.11960	−0.360766
$640$	0	0
$641$	−21.6649	−0.855711	−0.427856	−	0.903847i	$-0.640731\pi$
$641$	−21.6649	−0.855711	−0.427856	+	0.903847i	$0.640731\pi$
$642$	0	0
$643$	9.35557	0.368948	0.184474	−	0.982837i	$-0.440942\pi$
$643$	9.35557	0.368948	0.184474	+	0.982837i	$0.440942\pi$
$644$	0	0
$645$	10.2567	0.403856
$646$	0	0
$647$	0.702775	0.0276289	0.0138145	−	0.999905i	$-0.495603\pi$
$647$	0.702775	0.0276289	0.0138145	+	0.999905i	$0.495603\pi$
$648$	0	0
$649$	−9.73928	−0.382300
$650$	0	0
$651$	16.8799	0.661576
$652$	0	0
$653$	−37.3411	−1.46127	−0.730635	−	0.682768i	$-0.760776\pi$
$653$	−37.3411	−1.46127	−0.730635	+	0.682768i	$0.760776\pi$
$654$	0	0
$655$	32.6601	1.27614
$656$	0	0
$657$	−2.95108	−0.115133
$658$	0	0
$659$	0.735562	0.0286534	0.0143267	−	0.999897i	$-0.495440\pi$
$659$	0.735562	0.0286534	0.0143267	+	0.999897i	$0.495440\pi$
$660$	0	0
$661$	−13.8485	−0.538643	−0.269321	−	0.963050i	$-0.586799\pi$
$661$	−13.8485	−0.538643	−0.269321	+	0.963050i	$0.586799\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−39.6305	−1.53681
$666$	0	0
$667$	−11.0670	−0.428515
$668$	0	0
$669$	−7.44935	−0.288009
$670$	0	0
$671$	−17.3461	−0.669640
$672$	0	0
$673$	−6.35019	−0.244782	−0.122391	−	0.992482i	$-0.539056\pi$
$673$	−6.35019	−0.244782	−0.122391	+	0.992482i	$0.539056\pi$
$674$	0	0
$675$	2.91185	0.112077
$676$	0	0
$677$	33.7241	1.29612	0.648061	−	0.761589i	$-0.275580\pi$
$677$	33.7241	1.29612	0.648061	+	0.761589i	$0.275580\pi$
$678$	0	0
$679$	−29.8278	−1.14468
$680$	0	0
$681$	21.2500	0.814300
$682$	0	0
$683$	19.2687	0.737298	0.368649	−	0.929569i	$-0.379820\pi$
$683$	19.2687	0.737298	0.368649	+	0.929569i	$0.379820\pi$
$684$	0	0
$685$	−19.7036	−0.752837
$686$	0	0
$687$	9.29590	0.354661
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−39.4010	−1.49889	−0.749443	−	0.662069i	$-0.769679\pi$
$691$	−39.4010	−1.49889	−0.749443	+	0.662069i	$0.769679\pi$
$692$	0	0
$693$	−17.8659	−0.678670
$694$	0	0
$695$	25.4349	0.964800
$696$	0	0
$697$	1.35690	0.0513961
$698$	0	0
$699$	−16.2107	−0.613146
$700$	0	0
$701$	18.3985	0.694902	0.347451	−	0.937698i	$-0.387047\pi$
$701$	18.3985	0.694902	0.347451	+	0.937698i	$0.387047\pi$
$702$	0	0
$703$	−49.7308	−1.87563
$704$	0	0
$705$	15.2489	0.574307
$706$	0	0
$707$	−29.2019	−1.09825
$708$	0	0
$709$	38.4553	1.44422	0.722110	−	0.691778i	$-0.243172\pi$
$709$	38.4553	1.44422	0.722110	+	0.691778i	$0.243172\pi$
$710$	0	0
$711$	9.43296	0.353764
$712$	0	0
$713$	−13.8619	−0.519131
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.5090	0.504504
$718$	0	0
$719$	−48.4999	−1.80874	−0.904371	−	0.426747i	$-0.859659\pi$
$719$	−48.4999	−1.80874	−0.904371	+	0.426747i	$0.859659\pi$
$720$	0	0
$721$	−19.4556	−0.724564
$722$	0	0
$723$	6.26875	0.233137
$724$	0	0
$725$	11.3907	0.423042
$726$	0	0
$727$	−19.0344	−0.705948	−0.352974	−	0.935633i	$-0.614830\pi$
$727$	−19.0344	−0.705948	−0.352974	+	0.935633i	$0.614830\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.34481	−0.197685
$732$	0	0
$733$	−24.6213	−0.909410	−0.454705	−	0.890642i	$-0.650255\pi$
$733$	−24.6213	−0.909410	−0.454705	+	0.890642i	$0.650255\pi$
$734$	0	0
$735$	7.03492	0.259487
$736$	0	0
$737$	23.5593	0.867817
$738$	0	0
$739$	−44.5115	−1.63738	−0.818692	−	0.574233i	$-0.805300\pi$
$739$	−44.5115	−1.63738	−0.818692	+	0.574233i	$0.805300\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.4112	0.381950	0.190975	−	0.981595i	$-0.438835\pi$
$743$	10.4112	0.381950	0.190975	+	0.981595i	$0.438835\pi$
$744$	0	0
$745$	18.4077	0.674407
$746$	0	0
$747$	−6.46681	−0.236608
$748$	0	0
$749$	23.2043	0.847866
$750$	0	0
$751$	−1.69979	−0.0620263	−0.0310131	−	0.999519i	$-0.509873\pi$
$751$	−1.69979	−0.0620263	−0.0310131	+	0.999519i	$0.509873\pi$
$752$	0	0
$753$	−0.753020	−0.0274416
$754$	0	0
$755$	−22.6015	−0.822552
$756$	0	0
$757$	27.4252	0.996786	0.498393	−	0.866951i	$-0.333924\pi$
$757$	27.4252	0.996786	0.498393	+	0.866951i	$0.333924\pi$
$758$	0	0
$759$	14.6716	0.532545
$760$	0	0
$761$	5.02608	0.182195	0.0910977	−	0.995842i	$-0.470962\pi$
$761$	5.02608	0.182195	0.0910977	+	0.995842i	$0.470962\pi$
$762$	0	0
$763$	7.15213	0.258924
$764$	0	0
$765$	1.08815	0.0393420
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−42.4456	−1.53063	−0.765314	−	0.643657i	$-0.777416\pi$
$769$	−42.4456	−1.53063	−0.765314	+	0.643657i	$0.777416\pi$
$770$	0	0
$771$	−19.7265	−0.710431
$772$	0	0
$773$	−26.3593	−0.948078	−0.474039	−	0.880504i	$-0.657204\pi$
$773$	−26.3593	−0.948078	−0.474039	+	0.880504i	$0.657204\pi$
$774$	0	0
$775$	14.2674	0.512501
$776$	0	0
$777$	21.5211	0.772065
$778$	0	0
$779$	−14.3448	−0.513956
$780$	0	0
$781$	47.2941	1.69232
$782$	0	0
$783$	3.91185	0.139798
$784$	0	0
$785$	1.19029	0.0424834
$786$	0	0
$787$	17.1424	0.611062	0.305531	−	0.952182i	$-0.401166\pi$
$787$	17.1424	0.611062	0.305531	+	0.952182i	$0.401166\pi$
$788$	0	0
$789$	−17.6093	−0.626906
$790$	0	0
$791$	21.2508	0.755592
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−4.46250	−0.158269
$796$	0	0
$797$	30.1629	1.06842	0.534212	−	0.845351i	$-0.320608\pi$
$797$	30.1629	1.06842	0.534212	+	0.845351i	$0.320608\pi$
$798$	0	0
$799$	−7.94630	−0.281120
$800$	0	0
$801$	−1.15883	−0.0409454
$802$	0	0
$803$	15.3043	0.540076
$804$	0	0
$805$	−14.0838	−0.496390
$806$	0	0
$807$	16.3870	0.576851
$808$	0	0
$809$	−29.8504	−1.04948	−0.524742	−	0.851261i	$-0.675838\pi$
$809$	−29.8504	−1.04948	−0.524742	+	0.851261i	$0.675838\pi$
$810$	0	0
$811$	47.7362	1.67624	0.838122	−	0.545484i	$-0.183654\pi$
$811$	47.7362	1.67624	0.838122	+	0.545484i	$0.183654\pi$
$812$	0	0
$813$	0.795233	0.0278900
$814$	0	0
$815$	−9.05861	−0.317309
$816$	0	0
$817$	56.5042	1.97683
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.9299	0.625758	0.312879	−	0.949793i	$-0.398707\pi$
$821$	17.9299	0.625758	0.312879	+	0.949793i	$0.398707\pi$
$822$	0	0
$823$	−54.3196	−1.89346	−0.946731	−	0.322026i	$-0.895636\pi$
$823$	−54.3196	−1.89346	−0.946731	+	0.322026i	$0.895636\pi$
$824$	0	0
$825$	−15.1008	−0.525743
$826$	0	0
$827$	49.1041	1.70752	0.853758	−	0.520670i	$-0.174318\pi$
$827$	49.1041	1.70752	0.853758	+	0.520670i	$0.174318\pi$
$828$	0	0
$829$	−7.35796	−0.255553	−0.127776	−	0.991803i	$-0.540784\pi$
$829$	−7.35796	−0.255553	−0.127776	+	0.991803i	$0.540784\pi$
$830$	0	0
$831$	4.83340	0.167669
$832$	0	0
$833$	−3.66594	−0.127017
$834$	0	0
$835$	10.7662	0.372579
$836$	0	0
$837$	4.89977	0.169361
$838$	0	0
$839$	−36.5013	−1.26016	−0.630082	−	0.776529i	$-0.716979\pi$
$839$	−36.5013	−1.26016	−0.630082	+	0.776529i	$0.716979\pi$
$840$	0	0
$841$	−13.6974	−0.472324
$842$	0	0
$843$	18.7748	0.646638
$844$	0	0
$845$	0	0
$846$	0	0
$847$	54.7569	1.88147
$848$	0	0
$849$	−7.91723	−0.271719
$850$	0	0
$851$	−17.6732	−0.605831
$852$	0	0
$853$	9.73855	0.333441	0.166721	−	0.986004i	$-0.446682\pi$
$853$	9.73855	0.333441	0.166721	+	0.986004i	$0.446682\pi$
$854$	0	0
$855$	−11.5036	−0.393416
$856$	0	0
$857$	−15.2030	−0.519323	−0.259662	−	0.965700i	$-0.583611\pi$
$857$	−15.2030	−0.519323	−0.259662	+	0.965700i	$0.583611\pi$
$858$	0	0
$859$	31.9885	1.09143	0.545717	−	0.837970i	$-0.316257\pi$
$859$	31.9885	1.09143	0.545717	+	0.837970i	$0.316257\pi$
$860$	0	0
$861$	6.20775	0.211560
$862$	0	0
$863$	−35.2905	−1.20130	−0.600652	−	0.799511i	$-0.705092\pi$
$863$	−35.2905	−1.20130	−0.600652	+	0.799511i	$0.705092\pi$
$864$	0	0
$865$	2.90408	0.0987418
$866$	0	0
$867$	16.4330	0.558093
$868$	0	0
$869$	−48.9191	−1.65947
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−8.65817	−0.293035
$874$	0	0
$875$	39.3870	1.33152
$876$	0	0
$877$	15.3263	0.517532	0.258766	−	0.965940i	$-0.416684\pi$
$877$	15.3263	0.517532	0.258766	+	0.965940i	$0.416684\pi$
$878$	0	0
$879$	6.57912	0.221908
$880$	0	0
$881$	−36.4306	−1.22738	−0.613689	−	0.789548i	$-0.710315\pi$
$881$	−36.4306	−1.22738	−0.613689	+	0.789548i	$0.710315\pi$
$882$	0	0
$883$	37.6819	1.26810	0.634048	−	0.773294i	$-0.281392\pi$
$883$	37.6819	1.26810	0.634048	+	0.773294i	$0.281392\pi$
$884$	0	0
$885$	2.71379	0.0912231
$886$	0	0
$887$	−4.24890	−0.142664	−0.0713320	−	0.997453i	$-0.522725\pi$
$887$	−4.24890	−0.142664	−0.0713320	+	0.997453i	$0.522725\pi$
$888$	0	0
$889$	49.1333	1.64788
$890$	0	0
$891$	−5.18598	−0.173737
$892$	0	0
$893$	84.0066	2.81117
$894$	0	0
$895$	28.9541	0.967828
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.1672	0.639262
$900$	0	0
$901$	2.32544	0.0774715
$902$	0	0
$903$	−24.4523	−0.813723
$904$	0	0
$905$	−34.8582	−1.15872
$906$	0	0
$907$	12.6183	0.418985	0.209493	−	0.977810i	$-0.432819\pi$
$907$	12.6183	0.418985	0.209493	+	0.977810i	$0.432819\pi$
$908$	0	0
$909$	−8.47650	−0.281148
$910$	0	0
$911$	−6.77777	−0.224558	−0.112279	−	0.993677i	$-0.535815\pi$
$911$	−6.77777	−0.224558	−0.112279	+	0.993677i	$0.535815\pi$
$912$	0	0
$913$	33.5368	1.10990
$914$	0	0
$915$	4.83340	0.159787
$916$	0	0
$917$	−77.8631	−2.57126
$918$	0	0
$919$	20.4674	0.675157	0.337579	−	0.941297i	$-0.390392\pi$
$919$	20.4674	0.675157	0.337579	+	0.941297i	$0.390392\pi$
$920$	0	0
$921$	24.8649	0.819325
$922$	0	0
$923$	0	0
$924$	0	0
$925$	18.1903	0.598093
$926$	0	0
$927$	−5.64742	−0.185485
$928$	0	0
$929$	4.65220	0.152634	0.0763169	−	0.997084i	$-0.475684\pi$
$929$	4.65220	0.152634	0.0763169	+	0.997084i	$0.475684\pi$
$930$	0	0
$931$	38.7555	1.27016
$932$	0	0
$933$	−17.0804	−0.559186
$934$	0	0
$935$	−5.64310	−0.184549
$936$	0	0
$937$	−41.8544	−1.36732	−0.683662	−	0.729798i	$-0.739614\pi$
$937$	−41.8544	−1.36732	−0.683662	+	0.729798i	$0.739614\pi$
$938$	0	0
$939$	−15.6974	−0.512265
$940$	0	0
$941$	−30.3454	−0.989232	−0.494616	−	0.869112i	$-0.664691\pi$
$941$	−30.3454	−0.989232	−0.494616	+	0.869112i	$0.664691\pi$
$942$	0	0
$943$	−5.09783	−0.166008
$944$	0	0
$945$	4.97823	0.161942
$946$	0	0
$947$	12.0325	0.391004	0.195502	−	0.980703i	$-0.437366\pi$
$947$	12.0325	0.391004	0.195502	+	0.980703i	$0.437366\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	32.7821	1.06303
$952$	0	0
$953$	22.9825	0.744478	0.372239	−	0.928137i	$-0.378590\pi$
$953$	22.9825	0.744478	0.372239	+	0.928137i	$0.378590\pi$
$954$	0	0
$955$	−10.2314	−0.331082
$956$	0	0
$957$	−20.2868	−0.655779
$958$	0	0
$959$	46.9743	1.51688
$960$	0	0
$961$	−6.99223	−0.225556
$962$	0	0
$963$	6.73556	0.217050
$964$	0	0
$965$	14.1153	0.454387
$966$	0	0
$967$	−38.8883	−1.25056	−0.625281	−	0.780399i	$-0.715016\pi$
$967$	−38.8883	−1.25056	−0.625281	+	0.780399i	$0.715016\pi$
$968$	0	0
$969$	5.99462	0.192575
$970$	0	0
$971$	57.5133	1.84569	0.922845	−	0.385171i	$-0.125857\pi$
$971$	57.5133	1.84569	0.922845	+	0.385171i	$0.125857\pi$
$972$	0	0
$973$	−60.6378	−1.94396
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.3690	−0.523690	−0.261845	−	0.965110i	$-0.584331\pi$
$977$	−16.3690	−0.523690	−0.261845	+	0.965110i	$0.584331\pi$
$978$	0	0
$979$	6.00969	0.192070
$980$	0	0
$981$	2.07606	0.0662836
$982$	0	0
$983$	−15.6963	−0.500635	−0.250318	−	0.968164i	$-0.580535\pi$
$983$	−15.6963	−0.500635	−0.250318	+	0.968164i	$0.580535\pi$
$984$	0	0
$985$	33.8301	1.07792
$986$	0	0
$987$	−36.3540	−1.15716
$988$	0	0
$989$	20.0804	0.638519
$990$	0	0
$991$	11.2644	0.357827	0.178913	−	0.983865i	$-0.442742\pi$
$991$	11.2644	0.357827	0.178913	+	0.983865i	$0.442742\pi$
$992$	0	0
$993$	29.1618	0.925422
$994$	0	0
$995$	5.81594	0.184378
$996$	0	0
$997$	−7.70112	−0.243897	−0.121948	−	0.992536i	$-0.538914\pi$
$997$	−7.70112	−0.243897	−0.121948	+	0.992536i	$0.538914\pi$
$998$	0	0
$999$	6.24698	0.197646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.by.1.2		3
4.3	odd	2		507.2.a.j.1.1	✓	3
12.11	even	2		1521.2.a.q.1.3		3
13.12	even	2		8112.2.a.cf.1.2		3
52.3	odd	6		507.2.e.k.22.3		6
52.7	even	12		507.2.j.h.361.1		12
52.11	even	12		507.2.j.h.316.6		12
52.15	even	12		507.2.j.h.316.1		12
52.19	even	12		507.2.j.h.361.6		12
52.23	odd	6		507.2.e.j.22.1		6
52.31	even	4		507.2.b.g.337.6		6
52.35	odd	6		507.2.e.k.484.3		6
52.43	odd	6		507.2.e.j.484.1		6
52.47	even	4		507.2.b.g.337.1		6
52.51	odd	2		507.2.a.k.1.3	yes	3
156.47	odd	4		1521.2.b.m.1351.6		6
156.83	odd	4		1521.2.b.m.1351.1		6
156.155	even	2		1521.2.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.1	✓	3	4.3	odd	2
507.2.a.k.1.3	yes	3	52.51	odd	2
507.2.b.g.337.1		6	52.47	even	4
507.2.b.g.337.6		6	52.31	even	4
507.2.e.j.22.1		6	52.23	odd	6
507.2.e.j.484.1		6	52.43	odd	6
507.2.e.k.22.3		6	52.3	odd	6
507.2.e.k.484.3		6	52.35	odd	6
507.2.j.h.316.1		12	52.15	even	12
507.2.j.h.316.6		12	52.11	even	12
507.2.j.h.361.1		12	52.7	even	12
507.2.j.h.361.6		12	52.19	even	12
1521.2.a.p.1.1		3	156.155	even	2
1521.2.a.q.1.3		3	12.11	even	2
1521.2.b.m.1351.1		6	156.83	odd	4
1521.2.b.m.1351.6		6	156.47	odd	4
8112.2.a.by.1.2		3	1.1	even	1	trivial
8112.2.a.cf.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} -5.18598 q^{11} +1.44504 q^{15} -0.753020 q^{17} +7.96077 q^{19} -3.44504 q^{21} +2.82908 q^{23} -2.91185 q^{25} -1.00000 q^{27} -3.91185 q^{29} -4.89977 q^{31} +5.18598 q^{33} -4.97823 q^{35} -6.24698 q^{37} -1.80194 q^{41} +7.09783 q^{43} -1.44504 q^{45} +10.5526 q^{47} +4.86831 q^{49} +0.753020 q^{51} -3.08815 q^{53} +7.49396 q^{55} -7.96077 q^{57} +1.87800 q^{59} +3.34481 q^{61} +3.44504 q^{63} -4.54288 q^{67} -2.82908 q^{69} -9.11960 q^{71} -2.95108 q^{73} +2.91185 q^{75} -17.8659 q^{77} +9.43296 q^{79} +1.00000 q^{81} -6.46681 q^{83} +1.08815 q^{85} +3.91185 q^{87} -1.15883 q^{89} +4.89977 q^{93} -11.5036 q^{95} -8.65817 q^{97} -5.18598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9} - q^{11} + 4 q^{15} - 7 q^{17} + 11 q^{19} - 10 q^{21} - 2 q^{23} - 5 q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + q^{33} - 18 q^{35} - 14 q^{37} - q^{41} + 3 q^{43} - 4 q^{45} - 9 q^{47} + 17 q^{49} + 7 q^{51} - 13 q^{53} + 13 q^{55} - 11 q^{57} - 14 q^{59} - 13 q^{61} + 10 q^{63} + 5 q^{67} + 2 q^{69} - 6 q^{71} - 18 q^{73} + 5 q^{75} - 15 q^{77} + 9 q^{79} + 3 q^{81} - 16 q^{83} + 7 q^{85} + 8 q^{87} + 5 q^{89} - 8 q^{93} - 3 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 - q^11 + 4 * q^15 - 7 * q^17 + 11 * q^19 - 10 * q^21 - 2 * q^23 - 5 * q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + q^33 - 18 * q^35 - 14 * q^37 - q^41 + 3 * q^43 - 4 * q^45 - 9 * q^47 + 17 * q^49 + 7 * q^51 - 13 * q^53 + 13 * q^55 - 11 * q^57 - 14 * q^59 - 13 * q^61 + 10 * q^63 + 5 * q^67 + 2 * q^69 - 6 * q^71 - 18 * q^73 + 5 * q^75 - 15 * q^77 + 9 * q^79 + 3 * q^81 - 16 * q^83 + 7 * q^85 + 8 * q^87 + 5 * q^89 - 8 * q^93 - 3 * q^95 - 5 * q^97 - q^99

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