LMFDB - Embedded newform 1336.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} + \cdots + 224 $ x^12 - 5x^11 - 15x^10 + 100x^9 + 36x^8 - 641x^7 + 129x^6 + 1804x^5 - 433x^4 - 2399x^3 + 148x^2 + 1224*x + 224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.22013$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.22013 q^{3} +3.37141 q^{5} +3.01986 q^{7} -1.51128 q^{9} +O(q^{10})$	$q-1.22013 q^{3} +3.37141 q^{5} +3.01986 q^{7} -1.51128 q^{9} +5.36789 q^{11} -1.17302 q^{13} -4.11357 q^{15} -0.433851 q^{17} +7.85722 q^{19} -3.68462 q^{21} -7.02037 q^{23} +6.36639 q^{25} +5.50436 q^{27} +6.74759 q^{29} -2.65090 q^{31} -6.54954 q^{33} +10.1812 q^{35} -6.36008 q^{37} +1.43124 q^{39} -10.3836 q^{41} -12.1954 q^{43} -5.09513 q^{45} +8.70775 q^{47} +2.11953 q^{49} +0.529356 q^{51} -9.67652 q^{53} +18.0974 q^{55} -9.58685 q^{57} +9.68896 q^{59} -0.450255 q^{61} -4.56384 q^{63} -3.95472 q^{65} +9.11189 q^{67} +8.56578 q^{69} +5.15668 q^{71} +10.9822 q^{73} -7.76784 q^{75} +16.2103 q^{77} +12.7337 q^{79} -2.18222 q^{81} -9.80856 q^{83} -1.46269 q^{85} -8.23295 q^{87} +1.00086 q^{89} -3.54234 q^{91} +3.23446 q^{93} +26.4899 q^{95} -2.76010 q^{97} -8.11237 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * 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
$3$	−1.22013	−0.704444	−0.352222	−	0.935916i	$-0.614574\pi$
$3$	−1.22013	−0.704444	−0.352222	+	0.935916i	$0.614574\pi$
$4$	0	0
$5$	3.37141	1.50774	0.753870	−	0.657024i	$-0.228185\pi$
$5$	3.37141	1.50774	0.753870	+	0.657024i	$0.228185\pi$
$6$	0	0
$7$	3.01986	1.14140	0.570699	−	0.821159i	$-0.306672\pi$
$7$	3.01986	1.14140	0.570699	+	0.821159i	$0.306672\pi$
$8$	0	0
$9$	−1.51128	−0.503759
$10$	0	0
$11$	5.36789	1.61848	0.809241	−	0.587477i	$-0.199879\pi$
$11$	5.36789	1.61848	0.809241	+	0.587477i	$0.199879\pi$
$12$	0	0
$13$	−1.17302	−0.325336	−0.162668	−	0.986681i	$-0.552010\pi$
$13$	−1.17302	−0.325336	−0.162668	+	0.986681i	$0.552010\pi$
$14$	0	0
$15$	−4.11357	−1.06212
$16$	0	0
$17$	−0.433851	−0.105224	−0.0526122	−	0.998615i	$-0.516755\pi$
$17$	−0.433851	−0.105224	−0.0526122	+	0.998615i	$0.516755\pi$
$18$	0	0
$19$	7.85722	1.80257	0.901285	−	0.433226i	$-0.142625\pi$
$19$	7.85722	1.80257	0.901285	+	0.433226i	$0.142625\pi$
$20$	0	0
$21$	−3.68462	−0.804051
$22$	0	0
$23$	−7.02037	−1.46385	−0.731924	−	0.681386i	$-0.761377\pi$
$23$	−7.02037	−1.46385	−0.731924	+	0.681386i	$0.761377\pi$
$24$	0	0
$25$	6.36639	1.27328
$26$	0	0
$27$	5.50436	1.05931
$28$	0	0
$29$	6.74759	1.25300	0.626498	−	0.779423i	$-0.284488\pi$
$29$	6.74759	1.25300	0.626498	+	0.779423i	$0.284488\pi$
$30$	0	0
$31$	−2.65090	−0.476117	−0.238058	−	0.971251i	$-0.576511\pi$
$31$	−2.65090	−0.476117	−0.238058	+	0.971251i	$0.576511\pi$
$32$	0	0
$33$	−6.54954	−1.14013
$34$	0	0
$35$	10.1812	1.72093
$36$	0	0
$37$	−6.36008	−1.04559	−0.522796	−	0.852458i	$-0.675111\pi$
$37$	−6.36008	−1.04559	−0.522796	+	0.852458i	$0.675111\pi$
$38$	0	0
$39$	1.43124	0.229181
$40$	0	0
$41$	−10.3836	−1.62165	−0.810827	−	0.585286i	$-0.800982\pi$
$41$	−10.3836	−1.62165	−0.810827	+	0.585286i	$0.800982\pi$
$42$	0	0
$43$	−12.1954	−1.85978	−0.929892	−	0.367834i	$-0.880100\pi$
$43$	−12.1954	−1.85978	−0.929892	+	0.367834i	$0.880100\pi$
$44$	0	0
$45$	−5.09513	−0.759537
$46$	0	0
$47$	8.70775	1.27016	0.635078	−	0.772448i	$-0.280968\pi$
$47$	8.70775	1.27016	0.635078	+	0.772448i	$0.280968\pi$
$48$	0	0
$49$	2.11953	0.302790
$50$	0	0
$51$	0.529356	0.0741247
$52$	0	0
$53$	−9.67652	−1.32917	−0.664586	−	0.747212i	$-0.731392\pi$
$53$	−9.67652	−1.32917	−0.664586	+	0.747212i	$0.731392\pi$
$54$	0	0
$55$	18.0974	2.44025
$56$	0	0
$57$	−9.58685	−1.26981
$58$	0	0
$59$	9.68896	1.26140	0.630698	−	0.776029i	$-0.282769\pi$
$59$	9.68896	1.26140	0.630698	+	0.776029i	$0.282769\pi$
$60$	0	0
$61$	−0.450255	−0.0576493	−0.0288247	−	0.999584i	$-0.509176\pi$
$61$	−0.450255	−0.0576493	−0.0288247	+	0.999584i	$0.509176\pi$
$62$	0	0
$63$	−4.56384	−0.574989
$64$	0	0
$65$	−3.95472	−0.490522
$66$	0	0
$67$	9.11189	1.11319	0.556597	−	0.830782i	$-0.312107\pi$
$67$	9.11189	1.11319	0.556597	+	0.830782i	$0.312107\pi$
$68$	0	0
$69$	8.56578	1.03120
$70$	0	0
$71$	5.15668	0.611986	0.305993	−	0.952034i	$-0.401012\pi$
$71$	5.15668	0.611986	0.305993	+	0.952034i	$0.401012\pi$
$72$	0	0
$73$	10.9822	1.28537	0.642687	−	0.766129i	$-0.277819\pi$
$73$	10.9822	1.28537	0.642687	+	0.766129i	$0.277819\pi$
$74$	0	0
$75$	−7.76784	−0.896953
$76$	0	0
$77$	16.2103	1.84733
$78$	0	0
$79$	12.7337	1.43265	0.716327	−	0.697765i	$-0.245822\pi$
$79$	12.7337	1.43265	0.716327	+	0.697765i	$0.245822\pi$
$80$	0	0
$81$	−2.18222	−0.242469
$82$	0	0
$83$	−9.80856	−1.07663	−0.538314	−	0.842744i	$-0.680939\pi$
$83$	−9.80856	−1.07663	−0.538314	+	0.842744i	$0.680939\pi$
$84$	0	0
$85$	−1.46269	−0.158651
$86$	0	0
$87$	−8.23295	−0.882665
$88$	0	0
$89$	1.00086	0.106091	0.0530454	−	0.998592i	$-0.483107\pi$
$89$	1.00086	0.106091	0.0530454	+	0.998592i	$0.483107\pi$
$90$	0	0
$91$	−3.54234	−0.371338
$92$	0	0
$93$	3.23446	0.335397
$94$	0	0
$95$	26.4899	2.71781
$96$	0	0
$97$	−2.76010	−0.280245	−0.140123	−	0.990134i	$-0.544750\pi$
$97$	−2.76010	−0.280245	−0.140123	+	0.990134i	$0.544750\pi$
$98$	0	0
$99$	−8.11237	−0.815324
$100$	0	0
$101$	1.99413	0.198423	0.0992116	−	0.995066i	$-0.468368\pi$
$101$	1.99413	0.198423	0.0992116	+	0.995066i	$0.468368\pi$
$102$	0	0
$103$	14.6133	1.43989	0.719944	−	0.694032i	$-0.244167\pi$
$103$	14.6133	1.43989	0.719944	+	0.694032i	$0.244167\pi$
$104$	0	0
$105$	−12.4224	−1.21230
$106$	0	0
$107$	1.32717	0.128303	0.0641513	−	0.997940i	$-0.479566\pi$
$107$	1.32717	0.128303	0.0641513	+	0.997940i	$0.479566\pi$
$108$	0	0
$109$	−3.18029	−0.304617	−0.152308	−	0.988333i	$-0.548671\pi$
$109$	−3.18029	−0.304617	−0.152308	+	0.988333i	$0.548671\pi$
$110$	0	0
$111$	7.76015	0.736561
$112$	0	0
$113$	7.29590	0.686341	0.343170	−	0.939273i	$-0.388499\pi$
$113$	7.29590	0.686341	0.343170	+	0.939273i	$0.388499\pi$
$114$	0	0
$115$	−23.6685	−2.20710
$116$	0	0
$117$	1.77275	0.163891
$118$	0	0
$119$	−1.31017	−0.120103
$120$	0	0
$121$	17.8143	1.61948
$122$	0	0
$123$	12.6694	1.14236
$124$	0	0
$125$	4.60667	0.412033
$126$	0	0
$127$	−14.8183	−1.31492	−0.657458	−	0.753492i	$-0.728368\pi$
$127$	−14.8183	−1.31492	−0.657458	+	0.753492i	$0.728368\pi$
$128$	0	0
$129$	14.8800	1.31011
$130$	0	0
$131$	−2.06955	−0.180817	−0.0904085	−	0.995905i	$-0.528817\pi$
$131$	−2.06955	−0.180817	−0.0904085	+	0.995905i	$0.528817\pi$
$132$	0	0
$133$	23.7277	2.05745
$134$	0	0
$135$	18.5574	1.59717
$136$	0	0
$137$	−13.3709	−1.14235	−0.571177	−	0.820827i	$-0.693513\pi$
$137$	−13.3709	−1.14235	−0.571177	+	0.820827i	$0.693513\pi$
$138$	0	0
$139$	−3.42500	−0.290505	−0.145252	−	0.989395i	$-0.546399\pi$
$139$	−3.42500	−0.290505	−0.145252	+	0.989395i	$0.546399\pi$
$140$	0	0
$141$	−10.6246	−0.894754
$142$	0	0
$143$	−6.29663	−0.526550
$144$	0	0
$145$	22.7489	1.88919
$146$	0	0
$147$	−2.58611	−0.213298
$148$	0	0
$149$	−6.79112	−0.556350	−0.278175	−	0.960530i	$-0.589730\pi$
$149$	−6.79112	−0.556350	−0.278175	+	0.960530i	$0.589730\pi$
$150$	0	0
$151$	3.86988	0.314927	0.157463	−	0.987525i	$-0.449668\pi$
$151$	3.86988	0.314927	0.157463	+	0.987525i	$0.449668\pi$
$152$	0	0
$153$	0.655669	0.0530077
$154$	0	0
$155$	−8.93728	−0.717860
$156$	0	0
$157$	−7.98752	−0.637473	−0.318737	−	0.947843i	$-0.603259\pi$
$157$	−7.98752	−0.637473	−0.318737	+	0.947843i	$0.603259\pi$
$158$	0	0
$159$	11.8066	0.936327
$160$	0	0
$161$	−21.2005	−1.67083
$162$	0	0
$163$	−12.4380	−0.974220	−0.487110	−	0.873341i	$-0.661949\pi$
$163$	−12.4380	−0.974220	−0.487110	+	0.873341i	$0.661949\pi$
$164$	0	0
$165$	−22.0812	−1.71902
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.6240	−0.894156
$170$	0	0
$171$	−11.8744	−0.908060
$172$	0	0
$173$	11.6617	0.886624	0.443312	−	0.896367i	$-0.353803\pi$
$173$	11.6617	0.886624	0.443312	+	0.896367i	$0.353803\pi$
$174$	0	0
$175$	19.2256	1.45332
$176$	0	0
$177$	−11.8218	−0.888583
$178$	0	0
$179$	2.71212	0.202713	0.101357	−	0.994850i	$-0.467682\pi$
$179$	2.71212	0.202713	0.101357	+	0.994850i	$0.467682\pi$
$180$	0	0
$181$	24.5380	1.82390	0.911948	−	0.410305i	$-0.134578\pi$
$181$	24.5380	1.82390	0.911948	+	0.410305i	$0.134578\pi$
$182$	0	0
$183$	0.549371	0.0406107
$184$	0	0
$185$	−21.4424	−1.57648
$186$	0	0
$187$	−2.32887	−0.170304
$188$	0	0
$189$	16.6224	1.20910
$190$	0	0
$191$	−17.0424	−1.23315	−0.616573	−	0.787298i	$-0.711479\pi$
$191$	−17.0424	−1.23315	−0.616573	+	0.787298i	$0.711479\pi$
$192$	0	0
$193$	23.2208	1.67147	0.835735	−	0.549133i	$-0.185042\pi$
$193$	23.2208	1.67147	0.835735	+	0.549133i	$0.185042\pi$
$194$	0	0
$195$	4.82528	0.345545
$196$	0	0
$197$	−3.75403	−0.267464	−0.133732	−	0.991018i	$-0.542696\pi$
$197$	−3.75403	−0.267464	−0.133732	+	0.991018i	$0.542696\pi$
$198$	0	0
$199$	−21.3356	−1.51244	−0.756220	−	0.654317i	$-0.772956\pi$
$199$	−21.3356	−1.51244	−0.756220	+	0.654317i	$0.772956\pi$
$200$	0	0
$201$	−11.1177	−0.784183
$202$	0	0
$203$	20.3767	1.43017
$204$	0	0
$205$	−35.0075	−2.44503
$206$	0	0
$207$	10.6097	0.737426
$208$	0	0
$209$	42.1767	2.91743
$210$	0	0
$211$	−8.59995	−0.592045	−0.296023	−	0.955181i	$-0.595660\pi$
$211$	−8.59995	−0.592045	−0.296023	+	0.955181i	$0.595660\pi$
$212$	0	0
$213$	−6.29184	−0.431110
$214$	0	0
$215$	−41.1157	−2.80407
$216$	0	0
$217$	−8.00535	−0.543439
$218$	0	0
$219$	−13.3998	−0.905475
$220$	0	0
$221$	0.508915	0.0342333
$222$	0	0
$223$	−12.1453	−0.813307	−0.406654	−	0.913582i	$-0.633304\pi$
$223$	−12.1453	−0.813307	−0.406654	+	0.913582i	$0.633304\pi$
$224$	0	0
$225$	−9.62138	−0.641425
$226$	0	0
$227$	−11.7768	−0.781657	−0.390828	−	0.920464i	$-0.627811\pi$
$227$	−11.7768	−0.781657	−0.390828	+	0.920464i	$0.627811\pi$
$228$	0	0
$229$	−14.9415	−0.987361	−0.493681	−	0.869643i	$-0.664349\pi$
$229$	−14.9415	−0.987361	−0.493681	+	0.869643i	$0.664349\pi$
$230$	0	0
$231$	−19.7787	−1.30134
$232$	0	0
$233$	10.4376	0.683791	0.341895	−	0.939738i	$-0.388931\pi$
$233$	10.4376	0.683791	0.341895	+	0.939738i	$0.388931\pi$
$234$	0	0
$235$	29.3574	1.91507
$236$	0	0
$237$	−15.5368	−1.00922
$238$	0	0
$239$	15.0527	0.973679	0.486840	−	0.873491i	$-0.338150\pi$
$239$	15.0527	0.973679	0.486840	+	0.873491i	$0.338150\pi$
$240$	0	0
$241$	−6.19153	−0.398831	−0.199416	−	0.979915i	$-0.563904\pi$
$241$	−6.19153	−0.398831	−0.199416	+	0.979915i	$0.563904\pi$
$242$	0	0
$243$	−13.8505	−0.888508
$244$	0	0
$245$	7.14579	0.456528
$246$	0	0
$247$	−9.21665	−0.586441
$248$	0	0
$249$	11.9677	0.758425
$250$	0	0
$251$	25.9970	1.64091	0.820457	−	0.571708i	$-0.193719\pi$
$251$	25.9970	1.64091	0.820457	+	0.571708i	$0.193719\pi$
$252$	0	0
$253$	−37.6846	−2.36921
$254$	0	0
$255$	1.78468	0.111761
$256$	0	0
$257$	14.2773	0.890591	0.445296	−	0.895384i	$-0.353099\pi$
$257$	14.2773	0.890591	0.445296	+	0.895384i	$0.353099\pi$
$258$	0	0
$259$	−19.2065	−1.19344
$260$	0	0
$261$	−10.1975	−0.631207
$262$	0	0
$263$	−9.75034	−0.601232	−0.300616	−	0.953745i	$-0.597192\pi$
$263$	−9.75034	−0.601232	−0.300616	+	0.953745i	$0.597192\pi$
$264$	0	0
$265$	−32.6235	−2.00405
$266$	0	0
$267$	−1.22118	−0.0747350
$268$	0	0
$269$	−16.1883	−0.987016	−0.493508	−	0.869741i	$-0.664286\pi$
$269$	−16.1883	−0.987016	−0.493508	+	0.869741i	$0.664286\pi$
$270$	0	0
$271$	−22.2060	−1.34892	−0.674458	−	0.738313i	$-0.735623\pi$
$271$	−22.2060	−1.34892	−0.674458	+	0.738313i	$0.735623\pi$
$272$	0	0
$273$	4.32212	0.261587
$274$	0	0
$275$	34.1741	2.06078
$276$	0	0
$277$	−15.4233	−0.926698	−0.463349	−	0.886176i	$-0.653352\pi$
$277$	−15.4233	−0.926698	−0.463349	+	0.886176i	$0.653352\pi$
$278$	0	0
$279$	4.00625	0.239848
$280$	0	0
$281$	−21.3606	−1.27427	−0.637135	−	0.770752i	$-0.719881\pi$
$281$	−21.3606	−1.27427	−0.637135	+	0.770752i	$0.719881\pi$
$282$	0	0
$283$	16.8971	1.00443	0.502215	−	0.864743i	$-0.332519\pi$
$283$	16.8971	1.00443	0.502215	+	0.864743i	$0.332519\pi$
$284$	0	0
$285$	−32.3212	−1.91454
$286$	0	0
$287$	−31.3571	−1.85095
$288$	0	0
$289$	−16.8118	−0.988928
$290$	0	0
$291$	3.36768	0.197417
$292$	0	0
$293$	15.1772	0.886661	0.443331	−	0.896358i	$-0.353797\pi$
$293$	15.1772	0.886661	0.443331	+	0.896358i	$0.353797\pi$
$294$	0	0
$295$	32.6655	1.90186
$296$	0	0
$297$	29.5468	1.71448
$298$	0	0
$299$	8.23500	0.476243
$300$	0	0
$301$	−36.8284	−2.12275
$302$	0	0
$303$	−2.43310	−0.139778
$304$	0	0
$305$	−1.51799	−0.0869201
$306$	0	0
$307$	−31.2377	−1.78283	−0.891415	−	0.453187i	$-0.850287\pi$
$307$	−31.2377	−1.78283	−0.891415	+	0.453187i	$0.850287\pi$
$308$	0	0
$309$	−17.8301	−1.01432
$310$	0	0
$311$	26.3617	1.49483	0.747417	−	0.664355i	$-0.231294\pi$
$311$	26.3617	1.49483	0.747417	+	0.664355i	$0.231294\pi$
$312$	0	0
$313$	−34.8574	−1.97026	−0.985128	−	0.171822i	$-0.945035\pi$
$313$	−34.8574	−1.97026	−0.985128	+	0.171822i	$0.945035\pi$
$314$	0	0
$315$	−15.3866	−0.866934
$316$	0	0
$317$	7.31466	0.410833	0.205416	−	0.978675i	$-0.434145\pi$
$317$	7.31466	0.410833	0.205416	+	0.978675i	$0.434145\pi$
$318$	0	0
$319$	36.2203	2.02795
$320$	0	0
$321$	−1.61933	−0.0903820
$322$	0	0
$323$	−3.40887	−0.189674
$324$	0	0
$325$	−7.46788	−0.414243
$326$	0	0
$327$	3.88038	0.214585
$328$	0	0
$329$	26.2962	1.44975
$330$	0	0
$331$	−4.90192	−0.269434	−0.134717	−	0.990884i	$-0.543013\pi$
$331$	−4.90192	−0.269434	−0.134717	+	0.990884i	$0.543013\pi$
$332$	0	0
$333$	9.61184	0.526726
$334$	0	0
$335$	30.7199	1.67841
$336$	0	0
$337$	21.8162	1.18840	0.594201	−	0.804316i	$-0.297468\pi$
$337$	21.8162	1.18840	0.594201	+	0.804316i	$0.297468\pi$
$338$	0	0
$339$	−8.90197	−0.483489
$340$	0	0
$341$	−14.2298	−0.770586
$342$	0	0
$343$	−14.7383	−0.795795
$344$	0	0
$345$	28.8787	1.55478
$346$	0	0
$347$	−0.620461	−0.0333081	−0.0166540	−	0.999861i	$-0.505301\pi$
$347$	−0.620461	−0.0333081	−0.0166540	+	0.999861i	$0.505301\pi$
$348$	0	0
$349$	−5.01113	−0.268240	−0.134120	−	0.990965i	$-0.542821\pi$
$349$	−5.01113	−0.268240	−0.134120	+	0.990965i	$0.542821\pi$
$350$	0	0
$351$	−6.45670	−0.344633
$352$	0	0
$353$	25.6503	1.36523	0.682613	−	0.730780i	$-0.260843\pi$
$353$	25.6503	1.36523	0.682613	+	0.730780i	$0.260843\pi$
$354$	0	0
$355$	17.3853	0.922715
$356$	0	0
$357$	1.59858	0.0846058
$358$	0	0
$359$	16.9013	0.892019	0.446009	−	0.895028i	$-0.352845\pi$
$359$	16.9013	0.892019	0.446009	+	0.895028i	$0.352845\pi$
$360$	0	0
$361$	42.7359	2.24926
$362$	0	0
$363$	−21.7358	−1.14083
$364$	0	0
$365$	37.0256	1.93801
$366$	0	0
$367$	−35.3198	−1.84368	−0.921839	−	0.387573i	$-0.873313\pi$
$367$	−35.3198	−1.84368	−0.921839	+	0.387573i	$0.873313\pi$
$368$	0	0
$369$	15.6926	0.816922
$370$	0	0
$371$	−29.2217	−1.51711
$372$	0	0
$373$	−0.0515776	−0.00267059	−0.00133529	−	0.999999i	$-0.500425\pi$
$373$	−0.0515776	−0.00267059	−0.00133529	+	0.999999i	$0.500425\pi$
$374$	0	0
$375$	−5.62074	−0.290254
$376$	0	0
$377$	−7.91503	−0.407645
$378$	0	0
$379$	31.6188	1.62415	0.812075	−	0.583554i	$-0.198338\pi$
$379$	31.6188	1.62415	0.812075	+	0.583554i	$0.198338\pi$
$380$	0	0
$381$	18.0803	0.926284
$382$	0	0
$383$	31.5394	1.61159	0.805794	−	0.592196i	$-0.201739\pi$
$383$	31.5394	1.61159	0.805794	+	0.592196i	$0.201739\pi$
$384$	0	0
$385$	54.6514	2.78529
$386$	0	0
$387$	18.4306	0.936882
$388$	0	0
$389$	−11.0086	−0.558157	−0.279079	−	0.960268i	$-0.590029\pi$
$389$	−11.0086	−0.558157	−0.279079	+	0.960268i	$0.590029\pi$
$390$	0	0
$391$	3.04580	0.154033
$392$	0	0
$393$	2.52512	0.127376
$394$	0	0
$395$	42.9305	2.16007
$396$	0	0
$397$	−4.50509	−0.226104	−0.113052	−	0.993589i	$-0.536063\pi$
$397$	−4.50509	−0.226104	−0.113052	+	0.993589i	$0.536063\pi$
$398$	0	0
$399$	−28.9509	−1.44936
$400$	0	0
$401$	−20.2238	−1.00993	−0.504964	−	0.863140i	$-0.668494\pi$
$401$	−20.2238	−1.00993	−0.504964	+	0.863140i	$0.668494\pi$
$402$	0	0
$403$	3.10955	0.154898
$404$	0	0
$405$	−7.35714	−0.365579
$406$	0	0
$407$	−34.1403	−1.69227
$408$	0	0
$409$	−20.2311	−1.00036	−0.500182	−	0.865920i	$-0.666734\pi$
$409$	−20.2311	−1.00036	−0.500182	+	0.865920i	$0.666734\pi$
$410$	0	0
$411$	16.3143	0.804725
$412$	0	0
$413$	29.2593	1.43975
$414$	0	0
$415$	−33.0686	−1.62328
$416$	0	0
$417$	4.17896	0.204644
$418$	0	0
$419$	−30.7230	−1.50092	−0.750459	−	0.660917i	$-0.770168\pi$
$419$	−30.7230	−1.50092	−0.750459	+	0.660917i	$0.770168\pi$
$420$	0	0
$421$	−12.2297	−0.596038	−0.298019	−	0.954560i	$-0.596326\pi$
$421$	−12.2297	−0.596038	−0.298019	+	0.954560i	$0.596326\pi$
$422$	0	0
$423$	−13.1598	−0.639852
$424$	0	0
$425$	−2.76207	−0.133980
$426$	0	0
$427$	−1.35971	−0.0658008
$428$	0	0
$429$	7.68272	0.370925
$430$	0	0
$431$	−8.09356	−0.389853	−0.194927	−	0.980818i	$-0.562447\pi$
$431$	−8.09356	−0.389853	−0.194927	+	0.980818i	$0.562447\pi$
$432$	0	0
$433$	−15.7291	−0.755893	−0.377946	−	0.925827i	$-0.623370\pi$
$433$	−15.7291	−0.755893	−0.377946	+	0.925827i	$0.623370\pi$
$434$	0	0
$435$	−27.7566	−1.33083
$436$	0	0
$437$	−55.1606	−2.63869
$438$	0	0
$439$	−17.5036	−0.835402	−0.417701	−	0.908585i	$-0.637164\pi$
$439$	−17.5036	−0.835402	−0.417701	+	0.908585i	$0.637164\pi$
$440$	0	0
$441$	−3.20319	−0.152533
$442$	0	0
$443$	10.7528	0.510879	0.255440	−	0.966825i	$-0.417780\pi$
$443$	10.7528	0.510879	0.255440	+	0.966825i	$0.417780\pi$
$444$	0	0
$445$	3.37430	0.159957
$446$	0	0
$447$	8.28606	0.391917
$448$	0	0
$449$	24.8245	1.17154	0.585771	−	0.810476i	$-0.300792\pi$
$449$	24.8245	1.17154	0.585771	+	0.810476i	$0.300792\pi$
$450$	0	0
$451$	−55.7383	−2.62462
$452$	0	0
$453$	−4.72177	−0.221848
$454$	0	0
$455$	−11.9427	−0.559881
$456$	0	0
$457$	−18.2078	−0.851724	−0.425862	−	0.904788i	$-0.640029\pi$
$457$	−18.2078	−0.851724	−0.425862	+	0.904788i	$0.640029\pi$
$458$	0	0
$459$	−2.38807	−0.111466
$460$	0	0
$461$	7.64166	0.355908	0.177954	−	0.984039i	$-0.443052\pi$
$461$	7.64166	0.355908	0.177954	+	0.984039i	$0.443052\pi$
$462$	0	0
$463$	−3.76489	−0.174969	−0.0874847	−	0.996166i	$-0.527883\pi$
$463$	−3.76489	−0.174969	−0.0874847	+	0.996166i	$0.527883\pi$
$464$	0	0
$465$	10.9047	0.505692
$466$	0	0
$467$	−2.31606	−0.107174	−0.0535872	−	0.998563i	$-0.517066\pi$
$467$	−2.31606	−0.107174	−0.0535872	+	0.998563i	$0.517066\pi$
$468$	0	0
$469$	27.5166	1.27060
$470$	0	0
$471$	9.74583	0.449064
$472$	0	0
$473$	−65.4637	−3.01002
$474$	0	0
$475$	50.0222	2.29517
$476$	0	0
$477$	14.6239	0.669582
$478$	0	0
$479$	−6.70237	−0.306239	−0.153120	−	0.988208i	$-0.548932\pi$
$479$	−6.70237	−0.306239	−0.153120	+	0.988208i	$0.548932\pi$
$480$	0	0
$481$	7.46048	0.340169
$482$	0	0
$483$	25.8674	1.17701
$484$	0	0
$485$	−9.30541	−0.422537
$486$	0	0
$487$	18.4076	0.834127	0.417064	−	0.908877i	$-0.363059\pi$
$487$	18.4076	0.834127	0.417064	+	0.908877i	$0.363059\pi$
$488$	0	0
$489$	15.1760	0.686283
$490$	0	0
$491$	−15.4313	−0.696405	−0.348202	−	0.937419i	$-0.613208\pi$
$491$	−15.4313	−0.696405	−0.348202	+	0.937419i	$0.613208\pi$
$492$	0	0
$493$	−2.92745	−0.131846
$494$	0	0
$495$	−27.3501	−1.22930
$496$	0	0
$497$	15.5724	0.698519
$498$	0	0
$499$	−7.80105	−0.349223	−0.174611	−	0.984637i	$-0.555867\pi$
$499$	−7.80105	−0.349223	−0.174611	+	0.984637i	$0.555867\pi$
$500$	0	0
$501$	−1.22013	−0.0545115
$502$	0	0
$503$	−14.3823	−0.641273	−0.320636	−	0.947202i	$-0.603897\pi$
$503$	−14.3823	−0.641273	−0.320636	+	0.947202i	$0.603897\pi$
$504$	0	0
$505$	6.72302	0.299171
$506$	0	0
$507$	14.1829	0.629883
$508$	0	0
$509$	−1.22032	−0.0540897	−0.0270449	−	0.999634i	$-0.508610\pi$
$509$	−1.22032	−0.0540897	−0.0270449	+	0.999634i	$0.508610\pi$
$510$	0	0
$511$	33.1648	1.46712
$512$	0	0
$513$	43.2489	1.90949
$514$	0	0
$515$	49.2673	2.17098
$516$	0	0
$517$	46.7423	2.05572
$518$	0	0
$519$	−14.2288	−0.624577
$520$	0	0
$521$	30.9701	1.35683	0.678413	−	0.734681i	$-0.262668\pi$
$521$	30.9701	1.35683	0.678413	+	0.734681i	$0.262668\pi$
$522$	0	0
$523$	5.00811	0.218989	0.109495	−	0.993987i	$-0.465077\pi$
$523$	5.00811	0.218989	0.109495	+	0.993987i	$0.465077\pi$
$524$	0	0
$525$	−23.4578	−1.02378
$526$	0	0
$527$	1.15010	0.0500991
$528$	0	0
$529$	26.2856	1.14285
$530$	0	0
$531$	−14.6427	−0.635439
$532$	0	0
$533$	12.1802	0.527582
$534$	0	0
$535$	4.47444	0.193447
$536$	0	0
$537$	−3.30914	−0.142800
$538$	0	0
$539$	11.3774	0.490059
$540$	0	0
$541$	−35.7939	−1.53890	−0.769450	−	0.638707i	$-0.779470\pi$
$541$	−35.7939	−1.53890	−0.769450	+	0.638707i	$0.779470\pi$
$542$	0	0
$543$	−29.9396	−1.28483
$544$	0	0
$545$	−10.7221	−0.459282
$546$	0	0
$547$	13.2758	0.567631	0.283816	−	0.958879i	$-0.408400\pi$
$547$	13.2758	0.567631	0.283816	+	0.958879i	$0.408400\pi$
$548$	0	0
$549$	0.680460	0.0290413
$550$	0	0
$551$	53.0173	2.25861
$552$	0	0
$553$	38.4540	1.63523
$554$	0	0
$555$	26.1626	1.11054
$556$	0	0
$557$	−38.2998	−1.62281	−0.811407	−	0.584482i	$-0.801298\pi$
$557$	−38.2998	−1.62281	−0.811407	+	0.584482i	$0.801298\pi$
$558$	0	0
$559$	14.3054	0.605055
$560$	0	0
$561$	2.84153	0.119969
$562$	0	0
$563$	−14.3925	−0.606570	−0.303285	−	0.952900i	$-0.598083\pi$
$563$	−14.3925	−0.606570	−0.303285	+	0.952900i	$0.598083\pi$
$564$	0	0
$565$	24.5975	1.03482
$566$	0	0
$567$	−6.58998	−0.276753
$568$	0	0
$569$	6.60055	0.276709	0.138355	−	0.990383i	$-0.455819\pi$
$569$	6.60055	0.276709	0.138355	+	0.990383i	$0.455819\pi$
$570$	0	0
$571$	−13.0074	−0.544345	−0.272172	−	0.962249i	$-0.587742\pi$
$571$	−13.0074	−0.544345	−0.272172	+	0.962249i	$0.587742\pi$
$572$	0	0
$573$	20.7940	0.868682
$574$	0	0
$575$	−44.6944	−1.86389
$576$	0	0
$577$	−8.25627	−0.343713	−0.171856	−	0.985122i	$-0.554977\pi$
$577$	−8.25627	−0.343713	−0.171856	+	0.985122i	$0.554977\pi$
$578$	0	0
$579$	−28.3325	−1.17746
$580$	0	0
$581$	−29.6204	−1.22886
$582$	0	0
$583$	−51.9425	−2.15124
$584$	0	0
$585$	5.97667	0.247105
$586$	0	0
$587$	31.0312	1.28079	0.640397	−	0.768044i	$-0.278770\pi$
$587$	31.0312	1.28079	0.640397	+	0.768044i	$0.278770\pi$
$588$	0	0
$589$	−20.8287	−0.858234
$590$	0	0
$591$	4.58042	0.188413
$592$	0	0
$593$	−8.65997	−0.355623	−0.177811	−	0.984065i	$-0.556902\pi$
$593$	−8.65997	−0.355623	−0.177811	+	0.984065i	$0.556902\pi$
$594$	0	0
$595$	−4.41711	−0.181084
$596$	0	0
$597$	26.0323	1.06543
$598$	0	0
$599$	−12.0228	−0.491238	−0.245619	−	0.969366i	$-0.578991\pi$
$599$	−12.0228	−0.491238	−0.245619	+	0.969366i	$0.578991\pi$
$600$	0	0
$601$	−18.0040	−0.734400	−0.367200	−	0.930142i	$-0.619683\pi$
$601$	−18.0040	−0.734400	−0.367200	+	0.930142i	$0.619683\pi$
$602$	0	0
$603$	−13.7706	−0.560782
$604$	0	0
$605$	60.0593	2.44176
$606$	0	0
$607$	21.8854	0.888300	0.444150	−	0.895953i	$-0.353506\pi$
$607$	21.8854	0.888300	0.444150	+	0.895953i	$0.353506\pi$
$608$	0	0
$609$	−24.8623	−1.00747
$610$	0	0
$611$	−10.2143	−0.413228
$612$	0	0
$613$	−18.7365	−0.756762	−0.378381	−	0.925650i	$-0.623519\pi$
$613$	−18.7365	−0.756762	−0.378381	+	0.925650i	$0.623519\pi$
$614$	0	0
$615$	42.7138	1.72239
$616$	0	0
$617$	22.0482	0.887627	0.443813	−	0.896119i	$-0.353625\pi$
$617$	22.0482	0.887627	0.443813	+	0.896119i	$0.353625\pi$
$618$	0	0
$619$	17.6145	0.707987	0.353994	−	0.935248i	$-0.384823\pi$
$619$	17.6145	0.707987	0.353994	+	0.935248i	$0.384823\pi$
$620$	0	0
$621$	−38.6426	−1.55067
$622$	0	0
$623$	3.02245	0.121092
$624$	0	0
$625$	−16.3010	−0.652040
$626$	0	0
$627$	−51.4612	−2.05516
$628$	0	0
$629$	2.75933	0.110022
$630$	0	0
$631$	20.1905	0.803772	0.401886	−	0.915690i	$-0.368355\pi$
$631$	20.1905	0.803772	0.401886	+	0.915690i	$0.368355\pi$
$632$	0	0
$633$	10.4931	0.417063
$634$	0	0
$635$	−49.9587	−1.98255
$636$	0	0
$637$	−2.48624	−0.0985084
$638$	0	0
$639$	−7.79317	−0.308293
$640$	0	0
$641$	24.4865	0.967159	0.483580	−	0.875300i	$-0.339336\pi$
$641$	24.4865	0.967159	0.483580	+	0.875300i	$0.339336\pi$
$642$	0	0
$643$	19.9847	0.788121	0.394061	−	0.919084i	$-0.371070\pi$
$643$	19.9847	0.788121	0.394061	+	0.919084i	$0.371070\pi$
$644$	0	0
$645$	50.1666	1.97531
$646$	0	0
$647$	19.1292	0.752046	0.376023	−	0.926610i	$-0.377291\pi$
$647$	19.1292	0.752046	0.376023	+	0.926610i	$0.377291\pi$
$648$	0	0
$649$	52.0093	2.04155
$650$	0	0
$651$	9.76759	0.382822
$652$	0	0
$653$	−24.2424	−0.948679	−0.474339	−	0.880342i	$-0.657313\pi$
$653$	−24.2424	−0.948679	−0.474339	+	0.880342i	$0.657313\pi$
$654$	0	0
$655$	−6.97728	−0.272625
$656$	0	0
$657$	−16.5972	−0.647519
$658$	0	0
$659$	13.0334	0.507710	0.253855	−	0.967242i	$-0.418301\pi$
$659$	13.0334	0.507710	0.253855	+	0.967242i	$0.418301\pi$
$660$	0	0
$661$	1.42817	0.0555492	0.0277746	−	0.999614i	$-0.491158\pi$
$661$	1.42817	0.0555492	0.0277746	+	0.999614i	$0.491158\pi$
$662$	0	0
$663$	−0.620943	−0.0241154
$664$	0	0
$665$	79.9957	3.10210
$666$	0	0
$667$	−47.3705	−1.83419
$668$	0	0
$669$	14.8188	0.572929
$670$	0	0
$671$	−2.41692	−0.0933043
$672$	0	0
$673$	−23.1117	−0.890889	−0.445444	−	0.895310i	$-0.646954\pi$
$673$	−23.1117	−0.890889	−0.445444	+	0.895310i	$0.646954\pi$
$674$	0	0
$675$	35.0429	1.34880
$676$	0	0
$677$	−20.8404	−0.800961	−0.400480	−	0.916305i	$-0.631157\pi$
$677$	−20.8404	−0.800961	−0.400480	+	0.916305i	$0.631157\pi$
$678$	0	0
$679$	−8.33509	−0.319872
$680$	0	0
$681$	14.3693	0.550633
$682$	0	0
$683$	−32.5785	−1.24658	−0.623291	−	0.781990i	$-0.714205\pi$
$683$	−32.5785	−1.24658	−0.623291	+	0.781990i	$0.714205\pi$
$684$	0	0
$685$	−45.0788	−1.72237
$686$	0	0
$687$	18.2306	0.695541
$688$	0	0
$689$	11.3507	0.432428
$690$	0	0
$691$	−34.9907	−1.33111	−0.665555	−	0.746349i	$-0.731805\pi$
$691$	−34.9907	−1.33111	−0.665555	+	0.746349i	$0.731805\pi$
$692$	0	0
$693$	−24.4982	−0.930609
$694$	0	0
$695$	−11.5471	−0.438006
$696$	0	0
$697$	4.50496	0.170638
$698$	0	0
$699$	−12.7353	−0.481692
$700$	0	0
$701$	45.6480	1.72410	0.862050	−	0.506824i	$-0.169180\pi$
$701$	45.6480	1.72410	0.862050	+	0.506824i	$0.169180\pi$
$702$	0	0
$703$	−49.9726	−1.88475
$704$	0	0
$705$	−35.8199	−1.34906
$706$	0	0
$707$	6.02198	0.226480
$708$	0	0
$709$	−20.6054	−0.773852	−0.386926	−	0.922111i	$-0.626463\pi$
$709$	−20.6054	−0.773852	−0.386926	+	0.922111i	$0.626463\pi$
$710$	0	0
$711$	−19.2441	−0.721712
$712$	0	0
$713$	18.6103	0.696962
$714$	0	0
$715$	−21.2285	−0.793901
$716$	0	0
$717$	−18.3663	−0.685903
$718$	0	0
$719$	20.9113	0.779860	0.389930	−	0.920845i	$-0.372499\pi$
$719$	20.9113	0.779860	0.389930	+	0.920845i	$0.372499\pi$
$720$	0	0
$721$	44.1300	1.64349
$722$	0	0
$723$	7.55448	0.280954
$724$	0	0
$725$	42.9578	1.59541
$726$	0	0
$727$	16.4675	0.610744	0.305372	−	0.952233i	$-0.401219\pi$
$727$	16.4675	0.610744	0.305372	+	0.952233i	$0.401219\pi$
$728$	0	0
$729$	23.4461	0.868373
$730$	0	0
$731$	5.29100	0.195695
$732$	0	0
$733$	−41.8380	−1.54532	−0.772660	−	0.634820i	$-0.781074\pi$
$733$	−41.8380	−1.54532	−0.772660	+	0.634820i	$0.781074\pi$
$734$	0	0
$735$	−8.71882	−0.321598
$736$	0	0
$737$	48.9117	1.80168
$738$	0	0
$739$	8.91182	0.327826	0.163913	−	0.986475i	$-0.447588\pi$
$739$	8.91182	0.327826	0.163913	+	0.986475i	$0.447588\pi$
$740$	0	0
$741$	11.2455	0.413115
$742$	0	0
$743$	−2.36093	−0.0866142	−0.0433071	−	0.999062i	$-0.513789\pi$
$743$	−2.36093	−0.0866142	−0.0433071	+	0.999062i	$0.513789\pi$
$744$	0	0
$745$	−22.8956	−0.838831
$746$	0	0
$747$	14.8234	0.542361
$748$	0	0
$749$	4.00787	0.146444
$750$	0	0
$751$	38.9903	1.42278	0.711389	−	0.702799i	$-0.248066\pi$
$751$	38.9903	1.42278	0.711389	+	0.702799i	$0.248066\pi$
$752$	0	0
$753$	−31.7198	−1.15593
$754$	0	0
$755$	13.0470	0.474827
$756$	0	0
$757$	−0.620984	−0.0225701	−0.0112850	−	0.999936i	$-0.503592\pi$
$757$	−0.620984	−0.0225701	−0.0112850	+	0.999936i	$0.503592\pi$
$758$	0	0
$759$	45.9802	1.66898
$760$	0	0
$761$	−13.5348	−0.490635	−0.245318	−	0.969443i	$-0.578892\pi$
$761$	−13.5348	−0.490635	−0.245318	+	0.969443i	$0.578892\pi$
$762$	0	0
$763$	−9.60402	−0.347689
$764$	0	0
$765$	2.21053	0.0799218
$766$	0	0
$767$	−11.3653	−0.410378
$768$	0	0
$769$	33.7996	1.21884	0.609422	−	0.792846i	$-0.291402\pi$
$769$	33.7996	1.21884	0.609422	+	0.792846i	$0.291402\pi$
$770$	0	0
$771$	−17.4202	−0.627372
$772$	0	0
$773$	22.9993	0.827229	0.413614	−	0.910452i	$-0.364266\pi$
$773$	22.9993	0.827229	0.413614	+	0.910452i	$0.364266\pi$
$774$	0	0
$775$	−16.8767	−0.606229
$776$	0	0
$777$	23.4345	0.840709
$778$	0	0
$779$	−81.5866	−2.92314
$780$	0	0
$781$	27.6805	0.990487
$782$	0	0
$783$	37.1411	1.32732
$784$	0	0
$785$	−26.9292	−0.961144
$786$	0	0
$787$	11.6216	0.414267	0.207133	−	0.978313i	$-0.433587\pi$
$787$	11.6216	0.414267	0.207133	+	0.978313i	$0.433587\pi$
$788$	0	0
$789$	11.8967	0.423534
$790$	0	0
$791$	22.0326	0.783388
$792$	0	0
$793$	0.528157	0.0187554
$794$	0	0
$795$	39.8050	1.41174
$796$	0	0
$797$	1.24625	0.0441444	0.0220722	−	0.999756i	$-0.492974\pi$
$797$	1.24625	0.0441444	0.0220722	+	0.999756i	$0.492974\pi$
$798$	0	0
$799$	−3.77787	−0.133651
$800$	0	0
$801$	−1.51257	−0.0534441
$802$	0	0
$803$	58.9515	2.08035
$804$	0	0
$805$	−71.4755	−2.51918
$806$	0	0
$807$	19.7518	0.695297
$808$	0	0
$809$	−6.40863	−0.225315	−0.112658	−	0.993634i	$-0.535936\pi$
$809$	−6.40863	−0.225315	−0.112658	+	0.993634i	$0.535936\pi$
$810$	0	0
$811$	5.30520	0.186291	0.0931454	−	0.995653i	$-0.470308\pi$
$811$	5.30520	0.186291	0.0931454	+	0.995653i	$0.470308\pi$
$812$	0	0
$813$	27.0942	0.950236
$814$	0	0
$815$	−41.9336	−1.46887
$816$	0	0
$817$	−95.8221	−3.35239
$818$	0	0
$819$	5.35345	0.187065
$820$	0	0
$821$	36.9282	1.28880	0.644402	−	0.764687i	$-0.277106\pi$
$821$	36.9282	1.28880	0.644402	+	0.764687i	$0.277106\pi$
$822$	0	0
$823$	−46.7332	−1.62902	−0.814509	−	0.580151i	$-0.802994\pi$
$823$	−46.7332	−1.62902	−0.814509	+	0.580151i	$0.802994\pi$
$824$	0	0
$825$	−41.6970	−1.45170
$826$	0	0
$827$	15.5845	0.541924	0.270962	−	0.962590i	$-0.412658\pi$
$827$	15.5845	0.541924	0.270962	+	0.962590i	$0.412658\pi$
$828$	0	0
$829$	9.76531	0.339163	0.169582	−	0.985516i	$-0.445758\pi$
$829$	9.76531	0.339163	0.169582	+	0.985516i	$0.445758\pi$
$830$	0	0
$831$	18.8185	0.652807
$832$	0	0
$833$	−0.919560	−0.0318609
$834$	0	0
$835$	3.37141	0.116672
$836$	0	0
$837$	−14.5915	−0.504357
$838$	0	0
$839$	14.5004	0.500609	0.250304	−	0.968167i	$-0.419469\pi$
$839$	14.5004	0.500609	0.250304	+	0.968167i	$0.419469\pi$
$840$	0	0
$841$	16.5299	0.569997
$842$	0	0
$843$	26.0628	0.897651
$844$	0	0
$845$	−39.1894	−1.34815
$846$	0	0
$847$	53.7966	1.84847
$848$	0	0
$849$	−20.6167	−0.707564
$850$	0	0
$851$	44.6501	1.53059
$852$	0	0
$853$	18.9275	0.648066	0.324033	−	0.946046i	$-0.394961\pi$
$853$	18.9275	0.648066	0.324033	+	0.946046i	$0.394961\pi$
$854$	0	0
$855$	−40.0336	−1.36912
$856$	0	0
$857$	1.43471	0.0490088	0.0245044	−	0.999700i	$-0.492199\pi$
$857$	1.43471	0.0490088	0.0245044	+	0.999700i	$0.492199\pi$
$858$	0	0
$859$	21.1641	0.722110	0.361055	−	0.932545i	$-0.382417\pi$
$859$	21.1641	0.722110	0.361055	+	0.932545i	$0.382417\pi$
$860$	0	0
$861$	38.2598	1.30389
$862$	0	0
$863$	4.63916	0.157919	0.0789594	−	0.996878i	$-0.474840\pi$
$863$	4.63916	0.157919	0.0789594	+	0.996878i	$0.474840\pi$
$864$	0	0
$865$	39.3164	1.33680
$866$	0	0
$867$	20.5126	0.696644
$868$	0	0
$869$	68.3532	2.31872
$870$	0	0
$871$	−10.6884	−0.362162
$872$	0	0
$873$	4.17127	0.141176
$874$	0	0
$875$	13.9115	0.470293
$876$	0	0
$877$	−35.1486	−1.18688	−0.593442	−	0.804877i	$-0.702231\pi$
$877$	−35.1486	−1.18688	−0.593442	+	0.804877i	$0.702231\pi$
$878$	0	0
$879$	−18.5182	−0.624603
$880$	0	0
$881$	−30.2199	−1.01814	−0.509068	−	0.860726i	$-0.670010\pi$
$881$	−30.2199	−1.01814	−0.509068	+	0.860726i	$0.670010\pi$
$882$	0	0
$883$	31.0584	1.04520	0.522599	−	0.852579i	$-0.324963\pi$
$883$	31.0584	1.04520	0.522599	+	0.852579i	$0.324963\pi$
$884$	0	0
$885$	−39.8562	−1.33975
$886$	0	0
$887$	−26.5968	−0.893035	−0.446517	−	0.894775i	$-0.647336\pi$
$887$	−26.5968	−0.893035	−0.446517	+	0.894775i	$0.647336\pi$
$888$	0	0
$889$	−44.7492	−1.50084
$890$	0	0
$891$	−11.7139	−0.392431
$892$	0	0
$893$	68.4188	2.28955
$894$	0	0
$895$	9.14365	0.305638
$896$	0	0
$897$	−10.0478	−0.335486
$898$	0	0
$899$	−17.8872	−0.596572
$900$	0	0
$901$	4.19817	0.139861
$902$	0	0
$903$	44.9355	1.49536
$904$	0	0
$905$	82.7277	2.74996
$906$	0	0
$907$	−2.47688	−0.0822434	−0.0411217	−	0.999154i	$-0.513093\pi$
$907$	−2.47688	−0.0822434	−0.0411217	+	0.999154i	$0.513093\pi$
$908$	0	0
$909$	−3.01368	−0.0999574
$910$	0	0
$911$	44.2810	1.46710	0.733548	−	0.679637i	$-0.237863\pi$
$911$	44.2810	1.46710	0.733548	+	0.679637i	$0.237863\pi$
$912$	0	0
$913$	−52.6513	−1.74250
$914$	0	0
$915$	1.85216	0.0612304
$916$	0	0
$917$	−6.24973	−0.206384
$918$	0	0
$919$	53.3990	1.76147	0.880734	−	0.473610i	$-0.157050\pi$
$919$	53.3990	1.76147	0.880734	+	0.473610i	$0.157050\pi$
$920$	0	0
$921$	38.1142	1.25590
$922$	0	0
$923$	−6.04887	−0.199101
$924$	0	0
$925$	−40.4908	−1.33133
$926$	0	0
$927$	−22.0847	−0.725356
$928$	0	0
$929$	27.4457	0.900465	0.450233	−	0.892911i	$-0.351341\pi$
$929$	27.4457	0.900465	0.450233	+	0.892911i	$0.351341\pi$
$930$	0	0
$931$	16.6536	0.545800
$932$	0	0
$933$	−32.1648	−1.05303
$934$	0	0
$935$	−7.85157	−0.256774
$936$	0	0
$937$	−30.0530	−0.981789	−0.490894	−	0.871219i	$-0.663330\pi$
$937$	−30.0530	−0.981789	−0.490894	+	0.871219i	$0.663330\pi$
$938$	0	0
$939$	42.5306	1.38794
$940$	0	0
$941$	−19.8946	−0.648546	−0.324273	−	0.945964i	$-0.605120\pi$
$941$	−19.8946	−0.648546	−0.324273	+	0.945964i	$0.605120\pi$
$942$	0	0
$943$	72.8970	2.37385
$944$	0	0
$945$	56.0408	1.82301
$946$	0	0
$947$	47.2656	1.53593	0.767963	−	0.640495i	$-0.221271\pi$
$947$	47.2656	1.53593	0.767963	+	0.640495i	$0.221271\pi$
$948$	0	0
$949$	−12.8824	−0.418179
$950$	0	0
$951$	−8.92486	−0.289408
$952$	0	0
$953$	−42.6479	−1.38150	−0.690750	−	0.723094i	$-0.742719\pi$
$953$	−42.6479	−1.38150	−0.690750	+	0.723094i	$0.742719\pi$
$954$	0	0
$955$	−57.4569	−1.85926
$956$	0	0
$957$	−44.1936	−1.42858
$958$	0	0
$959$	−40.3782	−1.30388
$960$	0	0
$961$	−23.9727	−0.773313
$962$	0	0
$963$	−2.00572	−0.0646336
$964$	0	0
$965$	78.2868	2.52014
$966$	0	0
$967$	55.4826	1.78420	0.892101	−	0.451837i	$-0.149231\pi$
$967$	55.4826	1.78420	0.892101	+	0.451837i	$0.149231\pi$
$968$	0	0
$969$	4.15927	0.133615
$970$	0	0
$971$	33.0831	1.06169	0.530843	−	0.847470i	$-0.321875\pi$
$971$	33.0831	1.06169	0.530843	+	0.847470i	$0.321875\pi$
$972$	0	0
$973$	−10.3430	−0.331582
$974$	0	0
$975$	9.11181	0.291811
$976$	0	0
$977$	57.4911	1.83930	0.919651	−	0.392736i	$-0.128471\pi$
$977$	57.4911	1.83930	0.919651	+	0.392736i	$0.128471\pi$
$978$	0	0
$979$	5.37250	0.171706
$980$	0	0
$981$	4.80630	0.153453
$982$	0	0
$983$	−35.4313	−1.13008	−0.565042	−	0.825062i	$-0.691140\pi$
$983$	−35.4313	−1.13008	−0.565042	+	0.825062i	$0.691140\pi$
$984$	0	0
$985$	−12.6564	−0.403265
$986$	0	0
$987$	−32.0848	−1.02127
$988$	0	0
$989$	85.6163	2.72244
$990$	0	0
$991$	−62.2447	−1.97727	−0.988634	−	0.150342i	$-0.951962\pi$
$991$	−62.2447	−1.97727	−0.988634	+	0.150342i	$0.951962\pi$
$992$	0	0
$993$	5.98099	0.189801
$994$	0	0
$995$	−71.9310	−2.28037
$996$	0	0
$997$	−34.8632	−1.10413	−0.552064	−	0.833802i	$-0.686159\pi$
$997$	−34.8632	−1.10413	−0.552064	+	0.833802i	$0.686159\pi$
$998$	0	0
$999$	−35.0082	−1.10761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.e.1.4	✓	12
4.3	odd	2		2672.2.a.n.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.e.1.4	✓	12	1.1	even	1	trivial
2672.2.a.n.1.9		12	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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-1.22013 q^{3} +3.37141 q^{5} +3.01986 q^{7} -1.51128 q^{9} +O(q^{10})\)	\(q-1.22013 q^{3} +3.37141 q^{5} +3.01986 q^{7} -1.51128 q^{9} +5.36789 q^{11} -1.17302 q^{13} -4.11357 q^{15} -0.433851 q^{17} +7.85722 q^{19} -3.68462 q^{21} -7.02037 q^{23} +6.36639 q^{25} +5.50436 q^{27} +6.74759 q^{29} -2.65090 q^{31} -6.54954 q^{33} +10.1812 q^{35} -6.36008 q^{37} +1.43124 q^{39} -10.3836 q^{41} -12.1954 q^{43} -5.09513 q^{45} +8.70775 q^{47} +2.11953 q^{49} +0.529356 q^{51} -9.67652 q^{53} +18.0974 q^{55} -9.58685 q^{57} +9.68896 q^{59} -0.450255 q^{61} -4.56384 q^{63} -3.95472 q^{65} +9.11189 q^{67} +8.56578 q^{69} +5.15668 q^{71} +10.9822 q^{73} -7.76784 q^{75} +16.2103 q^{77} +12.7337 q^{79} -2.18222 q^{81} -9.80856 q^{83} -1.46269 q^{85} -8.23295 q^{87} +1.00086 q^{89} -3.54234 q^{91} +3.23446 q^{93} +26.4899 q^{95} -2.76010 q^{97} -8.11237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * q^99

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