LMFDB - Embedded newform 1143.2.a.k.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 $ x^16 - 26x^14 + 269x^12 - 1408x^10 + 3924x^8 - 5655x^6 + 3886x^4 - 1107*x^2 + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.518678$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.518678 q^{2} -1.73097 q^{4} -4.37543 q^{5} -5.00960 q^{7} -1.93517 q^{8} +O(q^{10})$	\(q+0.518678 q^{2} -1.73097 q^{4} -4.37543 q^{5} -5.00960 q^{7} -1.93517 q^{8} -2.26944 q^{10} -1.38644 q^{11} +3.17914 q^{13} -2.59837 q^{14} +2.45821 q^{16} +4.31283 q^{17} -1.85139 q^{19} +7.57376 q^{20} -0.719117 q^{22} -5.62281 q^{23} +14.1444 q^{25} +1.64895 q^{26} +8.67148 q^{28} -5.54058 q^{29} -6.48121 q^{31} +5.14537 q^{32} +2.23697 q^{34} +21.9192 q^{35} +7.44796 q^{37} -0.960275 q^{38} +8.46722 q^{40} -3.29646 q^{41} -1.77444 q^{43} +2.39989 q^{44} -2.91643 q^{46} -9.45154 q^{47} +18.0960 q^{49} +7.33640 q^{50} -5.50301 q^{52} +1.98709 q^{53} +6.06628 q^{55} +9.69444 q^{56} -2.87378 q^{58} +3.89711 q^{59} +3.80974 q^{61} -3.36166 q^{62} -2.24764 q^{64} -13.9101 q^{65} -6.46722 q^{67} -7.46539 q^{68} +11.3690 q^{70} -4.99965 q^{71} -0.636771 q^{73} +3.86309 q^{74} +3.20471 q^{76} +6.94551 q^{77} +6.86565 q^{79} -10.7558 q^{80} -1.70980 q^{82} -15.3781 q^{83} -18.8705 q^{85} -0.920363 q^{86} +2.68301 q^{88} +17.7972 q^{89} -15.9262 q^{91} +9.73294 q^{92} -4.90231 q^{94} +8.10063 q^{95} +8.19896 q^{97} +9.38602 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 10 * q^7	$ 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

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 $

$2$	0.518678	0.366761	0.183380	−	0.983042i	$-0.441296\pi$
$2$	0.518678	0.366761	0.183380	+	0.983042i	$0.441296\pi$
$3$	0	0
$3$	0	0
$4$	−1.73097	−0.865487
$5$	−4.37543	−1.95675	−0.978377	−	0.206831i	$-0.933685\pi$
$5$	−4.37543	−1.95675	−0.978377	+	0.206831i	$0.933685\pi$
$6$	0	0
$7$	−5.00960	−1.89345	−0.946725	−	0.322044i	$-0.895630\pi$
$7$	−5.00960	−1.89345	−0.946725	+	0.322044i	$0.895630\pi$
$8$	−1.93517	−0.684187
$9$	0	0
$10$	−2.26944	−0.717660
$11$	−1.38644	−0.418028	−0.209014	−	0.977913i	$-0.567025\pi$
$11$	−1.38644	−0.418028	−0.209014	+	0.977913i	$0.567025\pi$
$12$	0	0
$13$	3.17914	0.881736	0.440868	−	0.897572i	$-0.354671\pi$
$13$	3.17914	0.881736	0.440868	+	0.897572i	$0.354671\pi$
$14$	−2.59837	−0.694443
$15$	0	0
$16$	2.45821	0.614554
$17$	4.31283	1.04601	0.523007	−	0.852328i	$-0.324810\pi$
$17$	4.31283	1.04601	0.523007	+	0.852328i	$0.324810\pi$
$18$	0	0
$19$	−1.85139	−0.424738	−0.212369	−	0.977190i	$-0.568118\pi$
$19$	−1.85139	−0.424738	−0.212369	+	0.977190i	$0.568118\pi$
$20$	7.57376	1.69354
$21$	0	0
$22$	−0.719117	−0.153316
$23$	−5.62281	−1.17244	−0.586219	−	0.810153i	$-0.699384\pi$
$23$	−5.62281	−1.17244	−0.586219	+	0.810153i	$0.699384\pi$
$24$	0	0
$25$	14.1444	2.82888
$26$	1.64895	0.323386
$27$	0	0
$28$	8.67148	1.63875
$29$	−5.54058	−1.02886	−0.514430	−	0.857532i	$-0.671997\pi$
$29$	−5.54058	−1.02886	−0.514430	+	0.857532i	$0.671997\pi$
$30$	0	0
$31$	−6.48121	−1.16406	−0.582030	−	0.813167i	$-0.697741\pi$
$31$	−6.48121	−1.16406	−0.582030	+	0.813167i	$0.697741\pi$
$32$	5.14537	0.909581
$33$	0	0
$34$	2.23697	0.383637
$35$	21.9192	3.70501
$36$	0	0
$37$	7.44796	1.22444	0.612218	−	0.790689i	$-0.290277\pi$
$37$	7.44796	1.22444	0.612218	+	0.790689i	$0.290277\pi$
$38$	−0.960275	−0.155777
$39$	0	0
$40$	8.46722	1.33879
$41$	−3.29646	−0.514820	−0.257410	−	0.966302i	$-0.582869\pi$
$41$	−3.29646	−0.514820	−0.257410	+	0.966302i	$0.582869\pi$
$42$	0	0
$43$	−1.77444	−0.270600	−0.135300	−	0.990805i	$-0.543200\pi$
$43$	−1.77444	−0.270600	−0.135300	+	0.990805i	$0.543200\pi$
$44$	2.39989	0.361798
$45$	0	0
$46$	−2.91643	−0.430004
$47$	−9.45154	−1.37865	−0.689324	−	0.724453i	$-0.742093\pi$
$47$	−9.45154	−1.37865	−0.689324	+	0.724453i	$0.742093\pi$
$48$	0	0
$49$	18.0960	2.58515
$50$	7.33640	1.03752
$51$	0	0
$52$	−5.50301	−0.763131
$53$	1.98709	0.272948	0.136474	−	0.990644i	$-0.456423\pi$
$53$	1.98709	0.272948	0.136474	+	0.990644i	$0.456423\pi$
$54$	0	0
$55$	6.06628	0.817978
$56$	9.69444	1.29547
$57$	0	0
$58$	−2.87378	−0.377346
$59$	3.89711	0.507360	0.253680	−	0.967288i	$-0.418359\pi$
$59$	3.89711	0.507360	0.253680	+	0.967288i	$0.418359\pi$
$60$	0	0
$61$	3.80974	0.487787	0.243893	−	0.969802i	$-0.421575\pi$
$61$	3.80974	0.487787	0.243893	+	0.969802i	$0.421575\pi$
$62$	−3.36166	−0.426931
$63$	0	0
$64$	−2.24764	−0.280955
$65$	−13.9101	−1.72534
$66$	0	0
$67$	−6.46722	−0.790097	−0.395048	−	0.918660i	$-0.629272\pi$
$67$	−6.46722	−0.790097	−0.395048	+	0.918660i	$0.629272\pi$
$68$	−7.46539	−0.905311
$69$	0	0
$70$	11.3690	1.35885
$71$	−4.99965	−0.593349	−0.296675	−	0.954979i	$-0.595878\pi$
$71$	−4.99965	−0.593349	−0.296675	+	0.954979i	$0.595878\pi$
$72$	0	0
$73$	−0.636771	−0.0745284	−0.0372642	−	0.999305i	$-0.511864\pi$
$73$	−0.636771	−0.0745284	−0.0372642	+	0.999305i	$0.511864\pi$
$74$	3.86309	0.449075
$75$	0	0
$76$	3.20471	0.367605
$77$	6.94551	0.791515
$78$	0	0
$79$	6.86565	0.772446	0.386223	−	0.922405i	$-0.373779\pi$
$79$	6.86565	0.772446	0.386223	+	0.922405i	$0.373779\pi$
$80$	−10.7558	−1.20253
$81$	0	0
$82$	−1.70980	−0.188816
$83$	−15.3781	−1.68797	−0.843984	−	0.536369i	$-0.819796\pi$
$83$	−15.3781	−1.68797	−0.843984	+	0.536369i	$0.819796\pi$
$84$	0	0
$85$	−18.8705	−2.04679
$86$	−0.920363	−0.0992453
$87$	0	0
$88$	2.68301	0.286009
$89$	17.7972	1.88650	0.943252	−	0.332078i	$-0.107750\pi$
$89$	17.7972	1.88650	0.943252	+	0.332078i	$0.107750\pi$
$90$	0	0
$91$	−15.9262	−1.66952
$92$	9.73294	1.01473
$93$	0	0
$94$	−4.90231	−0.505634
$95$	8.10063	0.831108
$96$	0	0
$97$	8.19896	0.832478	0.416239	−	0.909255i	$-0.363348\pi$
$97$	8.19896	0.832478	0.416239	+	0.909255i	$0.363348\pi$
$98$	9.38602	0.948131
$99$	0	0
$100$	−24.4836	−2.44836
$101$	−3.11823	−0.310276	−0.155138	−	0.987893i	$-0.549582\pi$
$101$	−3.11823	−0.310276	−0.155138	+	0.987893i	$0.549582\pi$
$102$	0	0
$103$	14.4998	1.42871	0.714356	−	0.699783i	$-0.246720\pi$
$103$	14.4998	1.42871	0.714356	+	0.699783i	$0.246720\pi$
$104$	−6.15219	−0.603272
$105$	0	0
$106$	1.03066	0.100107
$107$	−6.81442	−0.658775	−0.329388	−	0.944195i	$-0.606842\pi$
$107$	−6.81442	−0.658775	−0.329388	+	0.944195i	$0.606842\pi$
$108$	0	0
$109$	9.82023	0.940608	0.470304	−	0.882505i	$-0.344144\pi$
$109$	9.82023	0.940608	0.470304	+	0.882505i	$0.344144\pi$
$110$	3.14645	0.300002
$111$	0	0
$112$	−12.3147	−1.16363
$113$	−15.6711	−1.47421	−0.737106	−	0.675777i	$-0.763808\pi$
$113$	−15.6711	−1.47421	−0.737106	+	0.675777i	$0.763808\pi$
$114$	0	0
$115$	24.6022	2.29417
$116$	9.59060	0.890465
$117$	0	0
$118$	2.02134	0.186080
$119$	−21.6055	−1.98057
$120$	0	0
$121$	−9.07778	−0.825253
$122$	1.97603	0.178901
$123$	0	0
$124$	11.2188	1.00748
$125$	−40.0108	−3.57867
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.4565	−1.01262
$129$	0	0
$130$	−7.21488	−0.632787
$131$	2.90282	0.253620	0.126810	−	0.991927i	$-0.459526\pi$
$131$	2.90282	0.253620	0.126810	+	0.991927i	$0.459526\pi$
$132$	0	0
$133$	9.27472	0.804220
$134$	−3.35441	−0.289777
$135$	0	0
$136$	−8.34607	−0.715670
$137$	18.0674	1.54360	0.771800	−	0.635865i	$-0.219357\pi$
$137$	18.0674	1.54360	0.771800	+	0.635865i	$0.219357\pi$
$138$	0	0
$139$	15.8789	1.34683	0.673413	−	0.739266i	$-0.264827\pi$
$139$	15.8789	1.34683	0.673413	+	0.739266i	$0.264827\pi$
$140$	−37.9415	−3.20664
$141$	0	0
$142$	−2.59321	−0.217617
$143$	−4.40770	−0.368590
$144$	0	0
$145$	24.2425	2.01323
$146$	−0.330279	−0.0273341
$147$	0	0
$148$	−12.8922	−1.05973
$149$	3.37867	0.276791	0.138396	−	0.990377i	$-0.455805\pi$
$149$	3.37867	0.276791	0.138396	+	0.990377i	$0.455805\pi$
$150$	0	0
$151$	−12.6521	−1.02962	−0.514808	−	0.857305i	$-0.672137\pi$
$151$	−12.6521	−1.02962	−0.514808	+	0.857305i	$0.672137\pi$
$152$	3.58276	0.290600
$153$	0	0
$154$	3.60248	0.290296
$155$	28.3581	2.27778
$156$	0	0
$157$	2.95253	0.235638	0.117819	−	0.993035i	$-0.462410\pi$
$157$	2.95253	0.235638	0.117819	+	0.993035i	$0.462410\pi$
$158$	3.56106	0.283303
$159$	0	0
$160$	−22.5132	−1.77983
$161$	28.1680	2.21995
$162$	0	0
$163$	11.7058	0.916867	0.458434	−	0.888729i	$-0.348411\pi$
$163$	11.7058	0.916867	0.458434	+	0.888729i	$0.348411\pi$
$164$	5.70608	0.445570
$165$	0	0
$166$	−7.97629	−0.619080
$167$	11.9329	0.923393	0.461696	−	0.887038i	$-0.347241\pi$
$167$	11.9329	0.923393	0.461696	+	0.887038i	$0.347241\pi$
$168$	0	0
$169$	−2.89304	−0.222542
$170$	−9.78771	−0.750683
$171$	0	0
$172$	3.07151	0.234200
$173$	6.91772	0.525945	0.262972	−	0.964803i	$-0.415297\pi$
$173$	6.91772	0.525945	0.262972	+	0.964803i	$0.415297\pi$
$174$	0	0
$175$	−70.8578	−5.35635
$176$	−3.40817	−0.256901
$177$	0	0
$178$	9.23104	0.691895
$179$	20.5064	1.53272	0.766361	−	0.642410i	$-0.222065\pi$
$179$	20.5064	1.53272	0.766361	+	0.642410i	$0.222065\pi$
$180$	0	0
$181$	−6.74447	−0.501313	−0.250656	−	0.968076i	$-0.580646\pi$
$181$	−6.74447	−0.501313	−0.250656	+	0.968076i	$0.580646\pi$
$182$	−8.26058	−0.612315
$183$	0	0
$184$	10.8811	0.802167
$185$	−32.5880	−2.39592
$186$	0	0
$187$	−5.97948	−0.437263
$188$	16.3604	1.19320
$189$	0	0
$190$	4.20162	0.304818
$191$	6.83932	0.494876	0.247438	−	0.968904i	$-0.420411\pi$
$191$	6.83932	0.494876	0.247438	+	0.968904i	$0.420411\pi$
$192$	0	0
$193$	−20.4255	−1.47026	−0.735131	−	0.677925i	$-0.762880\pi$
$193$	−20.4255	−1.47026	−0.735131	+	0.677925i	$0.762880\pi$
$194$	4.25262	0.305320
$195$	0	0
$196$	−31.3238	−2.23741
$197$	−7.19973	−0.512959	−0.256480	−	0.966550i	$-0.582563\pi$
$197$	−7.19973	−0.512959	−0.256480	+	0.966550i	$0.582563\pi$
$198$	0	0
$199$	8.72994	0.618849	0.309424	−	0.950924i	$-0.399864\pi$
$199$	8.72994	0.618849	0.309424	+	0.950924i	$0.399864\pi$
$200$	−27.3719	−1.93549
$201$	0	0
$202$	−1.61736	−0.113797
$203$	27.7561	1.94810
$204$	0	0
$205$	14.4234	1.00738
$206$	7.52074	0.523995
$207$	0	0
$208$	7.81502	0.541874
$209$	2.56685	0.177552
$210$	0	0
$211$	5.19771	0.357825	0.178912	−	0.983865i	$-0.442742\pi$
$211$	5.19771	0.357825	0.178912	+	0.983865i	$0.442742\pi$
$212$	−3.43960	−0.236233
$213$	0	0
$214$	−3.53449	−0.241613
$215$	7.76394	0.529496
$216$	0	0
$217$	32.4683	2.20409
$218$	5.09354	0.344978
$219$	0	0
$220$	−10.5006	−0.707949
$221$	13.7111	0.922308
$222$	0	0
$223$	5.00528	0.335178	0.167589	−	0.985857i	$-0.446402\pi$
$223$	5.00528	0.335178	0.167589	+	0.985857i	$0.446402\pi$
$224$	−25.7762	−1.72225
$225$	0	0
$226$	−8.12825	−0.540683
$227$	24.8220	1.64749	0.823746	−	0.566959i	$-0.191880\pi$
$227$	24.8220	1.64749	0.823746	+	0.566959i	$0.191880\pi$
$228$	0	0
$229$	−17.3219	−1.14466	−0.572331	−	0.820023i	$-0.693961\pi$
$229$	−17.3219	−1.14466	−0.572331	+	0.820023i	$0.693961\pi$
$230$	12.7606	0.841412
$231$	0	0
$232$	10.7220	0.703933
$233$	25.6319	1.67920	0.839602	−	0.543201i	$-0.182788\pi$
$233$	25.6319	1.67920	0.839602	+	0.543201i	$0.182788\pi$
$234$	0	0
$235$	41.3546	2.69768
$236$	−6.74579	−0.439113
$237$	0	0
$238$	−11.2063	−0.726397
$239$	11.6422	0.753069	0.376534	−	0.926403i	$-0.377116\pi$
$239$	11.6422	0.753069	0.376534	+	0.926403i	$0.377116\pi$
$240$	0	0
$241$	−16.9838	−1.09402	−0.547010	−	0.837126i	$-0.684234\pi$
$241$	−16.9838	−1.09402	−0.547010	+	0.837126i	$0.684234\pi$
$242$	−4.70844	−0.302670
$243$	0	0
$244$	−6.59455	−0.422173
$245$	−79.1780	−5.05850
$246$	0	0
$247$	−5.88584	−0.374507
$248$	12.5423	0.796435
$249$	0	0
$250$	−20.7527	−1.31252
$251$	−3.26492	−0.206080	−0.103040	−	0.994677i	$-0.532857\pi$
$251$	−3.26492	−0.206080	−0.103040	+	0.994677i	$0.532857\pi$
$252$	0	0
$253$	7.79570	0.490112
$254$	0.518678	0.0325447
$255$	0	0
$256$	−1.44697	−0.0904359
$257$	−1.08063	−0.0674078	−0.0337039	−	0.999432i	$-0.510730\pi$
$257$	−1.08063	−0.0674078	−0.0337039	+	0.999432i	$0.510730\pi$
$258$	0	0
$259$	−37.3113	−2.31841
$260$	24.0781	1.49326
$261$	0	0
$262$	1.50563	0.0930179
$263$	20.5571	1.26760	0.633801	−	0.773496i	$-0.281494\pi$
$263$	20.5571	1.26760	0.633801	+	0.773496i	$0.281494\pi$
$264$	0	0
$265$	−8.69438	−0.534092
$266$	4.81059	0.294956
$267$	0	0
$268$	11.1946	0.683818
$269$	−6.05815	−0.369372	−0.184686	−	0.982798i	$-0.559127\pi$
$269$	−6.05815	−0.369372	−0.184686	+	0.982798i	$0.559127\pi$
$270$	0	0
$271$	−22.5357	−1.36895	−0.684474	−	0.729037i	$-0.739968\pi$
$271$	−22.5357	−1.36895	−0.684474	+	0.729037i	$0.739968\pi$
$272$	10.6019	0.642832
$273$	0	0
$274$	9.37115	0.566132
$275$	−19.6104	−1.18255
$276$	0	0
$277$	2.36492	0.142094	0.0710472	−	0.997473i	$-0.477366\pi$
$277$	2.36492	0.142094	0.0710472	+	0.997473i	$0.477366\pi$
$278$	8.23601	0.493963
$279$	0	0
$280$	−42.4174	−2.53492
$281$	9.88523	0.589703	0.294852	−	0.955543i	$-0.404730\pi$
$281$	9.88523	0.589703	0.294852	+	0.955543i	$0.404730\pi$
$282$	0	0
$283$	10.7604	0.639642	0.319821	−	0.947478i	$-0.396377\pi$
$283$	10.7604	0.639642	0.319821	+	0.947478i	$0.396377\pi$
$284$	8.65426	0.513536
$285$	0	0
$286$	−2.28618	−0.135184
$287$	16.5139	0.974786
$288$	0	0
$289$	1.60048	0.0941458
$290$	12.5740	0.738372
$291$	0	0
$292$	1.10223	0.0645033
$293$	−20.0982	−1.17415	−0.587075	−	0.809533i	$-0.699721\pi$
$293$	−20.0982	−1.17415	−0.587075	+	0.809533i	$0.699721\pi$
$294$	0	0
$295$	−17.0515	−0.992778
$296$	−14.4131	−0.837744
$297$	0	0
$298$	1.75244	0.101516
$299$	−17.8757	−1.03378
$300$	0	0
$301$	8.88922	0.512366
$302$	−6.56238	−0.377623
$303$	0	0
$304$	−4.55111	−0.261024
$305$	−16.6692	−0.954478
$306$	0	0
$307$	−4.49261	−0.256407	−0.128203	−	0.991748i	$-0.540921\pi$
$307$	−4.49261	−0.256407	−0.128203	+	0.991748i	$0.540921\pi$
$308$	−12.0225	−0.685045
$309$	0	0
$310$	14.7087	0.835400
$311$	−12.1969	−0.691620	−0.345810	−	0.938304i	$-0.612396\pi$
$311$	−12.1969	−0.691620	−0.345810	+	0.938304i	$0.612396\pi$
$312$	0	0
$313$	29.6108	1.67370	0.836850	−	0.547432i	$-0.184394\pi$
$313$	29.6108	1.67370	0.836850	+	0.547432i	$0.184394\pi$
$314$	1.53141	0.0864227
$315$	0	0
$316$	−11.8843	−0.668542
$317$	−22.0932	−1.24088	−0.620439	−	0.784254i	$-0.713046\pi$
$317$	−22.0932	−1.24088	−0.620439	+	0.784254i	$0.713046\pi$
$318$	0	0
$319$	7.68170	0.430093
$320$	9.83440	0.549760
$321$	0	0
$322$	14.6101	0.814191
$323$	−7.98473	−0.444282
$324$	0	0
$325$	44.9671	2.49433
$326$	6.07153	0.336271
$327$	0	0
$328$	6.37922	0.352233
$329$	47.3484	2.61040
$330$	0	0
$331$	11.7978	0.648463	0.324232	−	0.945978i	$-0.394894\pi$
$331$	11.7978	0.648463	0.324232	+	0.945978i	$0.394894\pi$
$332$	26.6191	1.46091
$333$	0	0
$334$	6.18931	0.338664
$335$	28.2969	1.54602
$336$	0	0
$337$	−16.4030	−0.893528	−0.446764	−	0.894652i	$-0.647424\pi$
$337$	−16.4030	−0.893528	−0.446764	+	0.894652i	$0.647424\pi$
$338$	−1.50056	−0.0816196
$339$	0	0
$340$	32.6643	1.77147
$341$	8.98582	0.486610
$342$	0	0
$343$	−55.5867	−3.00140
$344$	3.43385	0.185141
$345$	0	0
$346$	3.58807	0.192896
$347$	−15.8358	−0.850109	−0.425054	−	0.905168i	$-0.639745\pi$
$347$	−15.8358	−0.850109	−0.425054	+	0.905168i	$0.639745\pi$
$348$	0	0
$349$	−29.3763	−1.57248	−0.786239	−	0.617922i	$-0.787975\pi$
$349$	−29.3763	−1.57248	−0.786239	+	0.617922i	$0.787975\pi$
$350$	−36.7524	−1.96450
$351$	0	0
$352$	−7.13376	−0.380230
$353$	1.89408	0.100812	0.0504058	−	0.998729i	$-0.483949\pi$
$353$	1.89408	0.100812	0.0504058	+	0.998729i	$0.483949\pi$
$354$	0	0
$355$	21.8756	1.16104
$356$	−30.8065	−1.63274
$357$	0	0
$358$	10.6362	0.562142
$359$	7.21134	0.380600	0.190300	−	0.981726i	$-0.439054\pi$
$359$	7.21134	0.380600	0.190300	+	0.981726i	$0.439054\pi$
$360$	0	0
$361$	−15.5724	−0.819598
$362$	−3.49821	−0.183862
$363$	0	0
$364$	27.5679	1.44495
$365$	2.78615	0.145834
$366$	0	0
$367$	9.31551	0.486266	0.243133	−	0.969993i	$-0.421825\pi$
$367$	9.31551	0.486266	0.243133	+	0.969993i	$0.421825\pi$
$368$	−13.8221	−0.720526
$369$	0	0
$370$	−16.9027	−0.878729
$371$	−9.95452	−0.516813
$372$	0	0
$373$	−19.4745	−1.00835	−0.504176	−	0.863601i	$-0.668204\pi$
$373$	−19.4745	−1.00835	−0.504176	+	0.863601i	$0.668204\pi$
$374$	−3.10143	−0.160371
$375$	0	0
$376$	18.2904	0.943254
$377$	−17.6143	−0.907183
$378$	0	0
$379$	−28.6039	−1.46929	−0.734643	−	0.678454i	$-0.762650\pi$
$379$	−28.6039	−1.46929	−0.734643	+	0.678454i	$0.762650\pi$
$380$	−14.0220	−0.719312
$381$	0	0
$382$	3.54741	0.181501
$383$	26.3936	1.34865	0.674324	−	0.738436i	$-0.264435\pi$
$383$	26.3936	1.34865	0.674324	+	0.738436i	$0.264435\pi$
$384$	0	0
$385$	−30.3896	−1.54880
$386$	−10.5943	−0.539235
$387$	0	0
$388$	−14.1922	−0.720499
$389$	−7.07951	−0.358946	−0.179473	−	0.983763i	$-0.557439\pi$
$389$	−7.07951	−0.358946	−0.179473	+	0.983763i	$0.557439\pi$
$390$	0	0
$391$	−24.2502	−1.22639
$392$	−35.0190	−1.76873
$393$	0	0
$394$	−3.73434	−0.188133
$395$	−30.0402	−1.51149
$396$	0	0
$397$	21.0189	1.05491	0.527455	−	0.849583i	$-0.323146\pi$
$397$	21.0189	1.05491	0.527455	+	0.849583i	$0.323146\pi$
$398$	4.52803	0.226969
$399$	0	0
$400$	34.7700	1.73850
$401$	5.11048	0.255205	0.127603	−	0.991825i	$-0.459272\pi$
$401$	5.11048	0.255205	0.127603	+	0.991825i	$0.459272\pi$
$402$	0	0
$403$	−20.6047	−1.02639
$404$	5.39758	0.268539
$405$	0	0
$406$	14.3965	0.714485
$407$	−10.3262	−0.511849
$408$	0	0
$409$	−1.55217	−0.0767498	−0.0383749	−	0.999263i	$-0.512218\pi$
$409$	−1.55217	−0.0767498	−0.0383749	+	0.999263i	$0.512218\pi$
$410$	7.48112	0.369466
$411$	0	0
$412$	−25.0988	−1.23653
$413$	−19.5229	−0.960660
$414$	0	0
$415$	67.2859	3.30294
$416$	16.3579	0.802010
$417$	0	0
$418$	1.33137	0.0651192
$419$	−29.4136	−1.43695	−0.718473	−	0.695555i	$-0.755159\pi$
$419$	−29.4136	−1.43695	−0.718473	+	0.695555i	$0.755159\pi$
$420$	0	0
$421$	28.1016	1.36959	0.684794	−	0.728737i	$-0.259892\pi$
$421$	28.1016	1.36959	0.684794	+	0.728737i	$0.259892\pi$
$422$	2.69594	0.131236
$423$	0	0
$424$	−3.84536	−0.186747
$425$	61.0024	2.95905
$426$	0	0
$427$	−19.0852	−0.923599
$428$	11.7956	0.570161
$429$	0	0
$430$	4.02699	0.194198
$431$	−3.88716	−0.187238	−0.0936190	−	0.995608i	$-0.529844\pi$
$431$	−3.88716	−0.187238	−0.0936190	+	0.995608i	$0.529844\pi$
$432$	0	0
$433$	8.90619	0.428004	0.214002	−	0.976833i	$-0.431350\pi$
$433$	8.90619	0.428004	0.214002	+	0.976833i	$0.431350\pi$
$434$	16.8406	0.808373
$435$	0	0
$436$	−16.9986	−0.814083
$437$	10.4100	0.497979
$438$	0	0
$439$	23.1429	1.10455	0.552276	−	0.833661i	$-0.313759\pi$
$439$	23.1429	1.10455	0.552276	+	0.833661i	$0.313759\pi$
$440$	−11.7393	−0.559650
$441$	0	0
$442$	7.11164	0.338266
$443$	−8.78994	−0.417622	−0.208811	−	0.977956i	$-0.566959\pi$
$443$	−8.78994	−0.417622	−0.208811	+	0.977956i	$0.566959\pi$
$444$	0	0
$445$	−77.8706	−3.69142
$446$	2.59613	0.122930
$447$	0	0
$448$	11.2598	0.531974
$449$	−13.3703	−0.630986	−0.315493	−	0.948928i	$-0.602170\pi$
$449$	−13.3703	−0.630986	−0.315493	+	0.948928i	$0.602170\pi$
$450$	0	0
$451$	4.57035	0.215209
$452$	27.1262	1.27591
$453$	0	0
$454$	12.8746	0.604235
$455$	69.6841	3.26684
$456$	0	0
$457$	24.3688	1.13992	0.569962	−	0.821671i	$-0.306958\pi$
$457$	24.3688	1.13992	0.569962	+	0.821671i	$0.306958\pi$
$458$	−8.98448	−0.419817
$459$	0	0
$460$	−42.5858	−1.98557
$461$	18.8215	0.876604	0.438302	−	0.898828i	$-0.355580\pi$
$461$	18.8215	0.876604	0.438302	+	0.898828i	$0.355580\pi$
$462$	0	0
$463$	8.35268	0.388182	0.194091	−	0.980984i	$-0.437824\pi$
$463$	8.35268	0.388182	0.194091	+	0.980984i	$0.437824\pi$
$464$	−13.6199	−0.632290
$465$	0	0
$466$	13.2947	0.615866
$467$	−1.29737	−0.0600354	−0.0300177	−	0.999549i	$-0.509556\pi$
$467$	−1.29737	−0.0600354	−0.0300177	+	0.999549i	$0.509556\pi$
$468$	0	0
$469$	32.3982	1.49601
$470$	21.4497	0.989401
$471$	0	0
$472$	−7.54158	−0.347129
$473$	2.46016	0.113118
$474$	0	0
$475$	−26.1868	−1.20153
$476$	37.3986	1.71416
$477$	0	0
$478$	6.03853	0.276196
$479$	−14.3005	−0.653407	−0.326704	−	0.945127i	$-0.605938\pi$
$479$	−14.3005	−0.653407	−0.326704	+	0.945127i	$0.605938\pi$
$480$	0	0
$481$	23.6781	1.07963
$482$	−8.80911	−0.401244
$483$	0	0
$484$	15.7134	0.714245
$485$	−35.8740	−1.62895
$486$	0	0
$487$	−12.6930	−0.575174	−0.287587	−	0.957755i	$-0.592853\pi$
$487$	−12.6930	−0.575174	−0.287587	+	0.957755i	$0.592853\pi$
$488$	−7.37250	−0.333737
$489$	0	0
$490$	−41.0679	−1.85526
$491$	−33.0387	−1.49102	−0.745509	−	0.666496i	$-0.767793\pi$
$491$	−33.0387	−1.49102	−0.745509	+	0.666496i	$0.767793\pi$
$492$	0	0
$493$	−23.8956	−1.07620
$494$	−3.05285	−0.137354
$495$	0	0
$496$	−15.9322	−0.715377
$497$	25.0462	1.12348
$498$	0	0
$499$	11.0020	0.492516	0.246258	−	0.969204i	$-0.420799\pi$
$499$	11.0020	0.492516	0.246258	+	0.969204i	$0.420799\pi$
$500$	69.2576	3.09729
$501$	0	0
$502$	−1.69344	−0.0755819
$503$	13.1326	0.585555	0.292777	−	0.956181i	$-0.405421\pi$
$503$	13.1326	0.585555	0.292777	+	0.956181i	$0.405421\pi$
$504$	0	0
$505$	13.6436	0.607133
$506$	4.04346	0.179754
$507$	0	0
$508$	−1.73097	−0.0767995
$509$	26.3408	1.16754	0.583769	−	0.811920i	$-0.301578\pi$
$509$	26.3408	1.16754	0.583769	+	0.811920i	$0.301578\pi$
$510$	0	0
$511$	3.18996	0.141116
$512$	22.1626	0.979456
$513$	0	0
$514$	−0.560499	−0.0247225
$515$	−63.4431	−2.79564
$516$	0	0
$517$	13.1040	0.576314
$518$	−19.3525	−0.850301
$519$	0	0
$520$	26.9185	1.18046
$521$	−7.26112	−0.318115	−0.159058	−	0.987269i	$-0.550846\pi$
$521$	−7.26112	−0.318115	−0.159058	+	0.987269i	$0.550846\pi$
$522$	0	0
$523$	−23.2635	−1.01724	−0.508620	−	0.860991i	$-0.669844\pi$
$523$	−23.2635	−1.01724	−0.508620	+	0.860991i	$0.669844\pi$
$524$	−5.02470	−0.219505
$525$	0	0
$526$	10.6625	0.464907
$527$	−27.9523	−1.21762
$528$	0	0
$529$	8.61603	0.374610
$530$	−4.50958	−0.195884
$531$	0	0
$532$	−16.0543	−0.696042
$533$	−10.4799	−0.453935
$534$	0	0
$535$	29.8161	1.28906
$536$	12.5152	0.540574
$537$	0	0
$538$	−3.14223	−0.135471
$539$	−25.0891	−1.08066
$540$	0	0
$541$	0.291072	0.0125142	0.00625708	−	0.999980i	$-0.498008\pi$
$541$	0.291072	0.0125142	0.00625708	+	0.999980i	$0.498008\pi$
$542$	−11.6888	−0.502076
$543$	0	0
$544$	22.1911	0.951435
$545$	−42.9678	−1.84054
$546$	0	0
$547$	21.0576	0.900358	0.450179	−	0.892938i	$-0.351360\pi$
$547$	21.0576	0.900358	0.450179	+	0.892938i	$0.351360\pi$
$548$	−31.2741	−1.33596
$549$	0	0
$550$	−10.1715	−0.433714
$551$	10.2578	0.436996
$552$	0	0
$553$	−34.3942	−1.46259
$554$	1.22663	0.0521147
$555$	0	0
$556$	−27.4859	−1.16566
$557$	36.7313	1.55636	0.778178	−	0.628044i	$-0.216144\pi$
$557$	36.7313	1.55636	0.778178	+	0.628044i	$0.216144\pi$
$558$	0	0
$559$	−5.64120	−0.238597
$560$	53.8820	2.27693
$561$	0	0
$562$	5.12725	0.216280
$563$	39.0663	1.64645	0.823224	−	0.567717i	$-0.192173\pi$
$563$	39.0663	1.64645	0.823224	+	0.567717i	$0.192173\pi$
$564$	0	0
$565$	68.5678	2.88467
$566$	5.58120	0.234596
$567$	0	0
$568$	9.67519	0.405962
$569$	8.48803	0.355837	0.177918	−	0.984045i	$-0.443064\pi$
$569$	8.48803	0.355837	0.177918	+	0.984045i	$0.443064\pi$
$570$	0	0
$571$	−2.60653	−0.109080	−0.0545400	−	0.998512i	$-0.517369\pi$
$571$	−2.60653	−0.109080	−0.0545400	+	0.998512i	$0.517369\pi$
$572$	7.62961	0.319010
$573$	0	0
$574$	8.56541	0.357513
$575$	−79.5314	−3.31669
$576$	0	0
$577$	12.2218	0.508802	0.254401	−	0.967099i	$-0.418122\pi$
$577$	12.2218	0.508802	0.254401	+	0.967099i	$0.418122\pi$
$578$	0.830133	0.0345290
$579$	0	0
$580$	−41.9630	−1.74242
$581$	77.0381	3.19608
$582$	0	0
$583$	−2.75499	−0.114100
$584$	1.23226	0.0509914
$585$	0	0
$586$	−10.4245	−0.430632
$587$	0.523420	0.0216039	0.0108019	−	0.999942i	$-0.496562\pi$
$587$	0.523420	0.0216039	0.0108019	+	0.999942i	$0.496562\pi$
$588$	0	0
$589$	11.9993	0.494421
$590$	−8.84425	−0.364112
$591$	0	0
$592$	18.3087	0.752482
$593$	14.8809	0.611087	0.305543	−	0.952178i	$-0.401162\pi$
$593$	14.8809	0.611087	0.305543	+	0.952178i	$0.401162\pi$
$594$	0	0
$595$	94.5335	3.87550
$596$	−5.84838	−0.239559
$597$	0	0
$598$	−9.27175	−0.379150
$599$	26.1595	1.06885	0.534424	−	0.845217i	$-0.320529\pi$
$599$	26.1595	1.06885	0.534424	+	0.845217i	$0.320529\pi$
$600$	0	0
$601$	−11.5092	−0.469469	−0.234734	−	0.972060i	$-0.575422\pi$
$601$	−11.5092	−0.469469	−0.234734	+	0.972060i	$0.575422\pi$
$602$	4.61064	0.187916
$603$	0	0
$604$	21.9005	0.891119
$605$	39.7192	1.61482
$606$	0	0
$607$	19.9741	0.810722	0.405361	−	0.914157i	$-0.367146\pi$
$607$	19.9741	0.810722	0.405361	+	0.914157i	$0.367146\pi$
$608$	−9.52609	−0.386334
$609$	0	0
$610$	−8.64597	−0.350065
$611$	−30.0478	−1.21560
$612$	0	0
$613$	−24.5315	−0.990817	−0.495408	−	0.868660i	$-0.664982\pi$
$613$	−24.5315	−0.990817	−0.495408	+	0.868660i	$0.664982\pi$
$614$	−2.33022	−0.0940399
$615$	0	0
$616$	−13.4408	−0.541544
$617$	38.8697	1.56484	0.782418	−	0.622754i	$-0.213986\pi$
$617$	38.8697	1.56484	0.782418	+	0.622754i	$0.213986\pi$
$618$	0	0
$619$	21.3462	0.857977	0.428988	−	0.903310i	$-0.358870\pi$
$619$	21.3462	0.857977	0.428988	+	0.903310i	$0.358870\pi$
$620$	−49.0871	−1.97139
$621$	0	0
$622$	−6.32624	−0.253659
$623$	−89.1570	−3.57200
$624$	0	0
$625$	104.342	4.17370
$626$	15.3585	0.613848
$627$	0	0
$628$	−5.11075	−0.203941
$629$	32.1218	1.28078
$630$	0	0
$631$	−9.86303	−0.392641	−0.196321	−	0.980540i	$-0.562899\pi$
$631$	−9.86303	−0.392641	−0.196321	+	0.980540i	$0.562899\pi$
$632$	−13.2862	−0.528498
$633$	0	0
$634$	−11.4593	−0.455106
$635$	−4.37543	−0.173634
$636$	0	0
$637$	57.5299	2.27942
$638$	3.98433	0.157741
$639$	0	0
$640$	50.1273	1.98146
$641$	−42.5134	−1.67918	−0.839589	−	0.543222i	$-0.817204\pi$
$641$	−42.5134	−1.67918	−0.839589	+	0.543222i	$0.817204\pi$
$642$	0	0
$643$	33.8845	1.33627	0.668137	−	0.744038i	$-0.267092\pi$
$643$	33.8845	1.33627	0.668137	+	0.744038i	$0.267092\pi$
$644$	−48.7581	−1.92134
$645$	0	0
$646$	−4.14150	−0.162945
$647$	23.1058	0.908382	0.454191	−	0.890904i	$-0.349928\pi$
$647$	23.1058	0.908382	0.454191	+	0.890904i	$0.349928\pi$
$648$	0	0
$649$	−5.40311	−0.212091
$650$	23.3235	0.914821
$651$	0	0
$652$	−20.2624	−0.793536
$653$	−22.7425	−0.889982	−0.444991	−	0.895535i	$-0.646793\pi$
$653$	−22.7425	−0.889982	−0.444991	+	0.895535i	$0.646793\pi$
$654$	0	0
$655$	−12.7011	−0.496272
$656$	−8.10340	−0.316385
$657$	0	0
$658$	24.5586	0.957393
$659$	−10.0402	−0.391109	−0.195554	−	0.980693i	$-0.562651\pi$
$659$	−10.0402	−0.391109	−0.195554	+	0.980693i	$0.562651\pi$
$660$	0	0
$661$	20.8078	0.809331	0.404666	−	0.914465i	$-0.367388\pi$
$661$	20.8078	0.809331	0.404666	+	0.914465i	$0.367388\pi$
$662$	6.11923	0.237831
$663$	0	0
$664$	29.7593	1.15489
$665$	−40.5809	−1.57366
$666$	0	0
$667$	31.1537	1.20627
$668$	−20.6555	−0.799184
$669$	0	0
$670$	14.6770	0.567021
$671$	−5.28198	−0.203909
$672$	0	0
$673$	−1.35696	−0.0523068	−0.0261534	−	0.999658i	$-0.508326\pi$
$673$	−1.35696	−0.0523068	−0.0261534	+	0.999658i	$0.508326\pi$
$674$	−8.50787	−0.327711
$675$	0	0
$676$	5.00778	0.192607
$677$	−20.6490	−0.793607	−0.396804	−	0.917904i	$-0.629881\pi$
$677$	−20.6490	−0.793607	−0.396804	+	0.917904i	$0.629881\pi$
$678$	0	0
$679$	−41.0735	−1.57625
$680$	36.5177	1.40039
$681$	0	0
$682$	4.66075	0.178469
$683$	−4.38268	−0.167699	−0.0838494	−	0.996478i	$-0.526721\pi$
$683$	−4.38268	−0.167699	−0.0838494	+	0.996478i	$0.526721\pi$
$684$	0	0
$685$	−79.0526	−3.02044
$686$	−28.8316	−1.10080
$687$	0	0
$688$	−4.36195	−0.166298
$689$	6.31725	0.240668
$690$	0	0
$691$	−17.6485	−0.671382	−0.335691	−	0.941972i	$-0.608970\pi$
$691$	−17.6485	−0.671382	−0.335691	+	0.941972i	$0.608970\pi$
$692$	−11.9744	−0.455198
$693$	0	0
$694$	−8.21367	−0.311787
$695$	−69.4769	−2.63541
$696$	0	0
$697$	−14.2171	−0.538509
$698$	−15.2368	−0.576723
$699$	0	0
$700$	122.653	4.63585
$701$	−1.87490	−0.0708141	−0.0354070	−	0.999373i	$-0.511273\pi$
$701$	−1.87490	−0.0708141	−0.0354070	+	0.999373i	$0.511273\pi$
$702$	0	0
$703$	−13.7891	−0.520065
$704$	3.11622	0.117447
$705$	0	0
$706$	0.982417	0.0369737
$707$	15.6211	0.587491
$708$	0	0
$709$	−13.2986	−0.499440	−0.249720	−	0.968318i	$-0.580339\pi$
$709$	−13.2986	−0.499440	−0.249720	+	0.968318i	$0.580339\pi$
$710$	11.3464	0.425823
$711$	0	0
$712$	−34.4408	−1.29072
$713$	36.4426	1.36479
$714$	0	0
$715$	19.2856	0.721240
$716$	−35.4961	−1.32655
$717$	0	0
$718$	3.74036	0.139589
$719$	44.4726	1.65855	0.829274	−	0.558843i	$-0.188754\pi$
$719$	44.4726	1.65855	0.829274	+	0.558843i	$0.188754\pi$
$720$	0	0
$721$	−72.6383	−2.70519
$722$	−8.07704	−0.300596
$723$	0	0
$724$	11.6745	0.433880
$725$	−78.3683	−2.91053
$726$	0	0
$727$	38.9553	1.44477	0.722386	−	0.691490i	$-0.243045\pi$
$727$	38.9553	1.44477	0.722386	+	0.691490i	$0.243045\pi$
$728$	30.8200	1.14227
$729$	0	0
$730$	1.44511	0.0534861
$731$	−7.65285	−0.283051
$732$	0	0
$733$	36.3745	1.34352	0.671762	−	0.740767i	$-0.265538\pi$
$733$	36.3745	1.34352	0.671762	+	0.740767i	$0.265538\pi$
$734$	4.83175	0.178343
$735$	0	0
$736$	−28.9314	−1.06643
$737$	8.96643	0.330283
$738$	0	0
$739$	15.3558	0.564871	0.282435	−	0.959286i	$-0.408858\pi$
$739$	15.3558	0.564871	0.282435	+	0.959286i	$0.408858\pi$
$740$	56.4090	2.07364
$741$	0	0
$742$	−5.16319	−0.189547
$743$	45.8367	1.68159	0.840793	−	0.541357i	$-0.182089\pi$
$743$	45.8367	1.68159	0.840793	+	0.541357i	$0.182089\pi$
$744$	0	0
$745$	−14.7831	−0.541612
$746$	−10.1010	−0.369824
$747$	0	0
$748$	10.3503	0.378445
$749$	34.1375	1.24736
$750$	0	0
$751$	9.11412	0.332579	0.166290	−	0.986077i	$-0.446821\pi$
$751$	9.11412	0.332579	0.166290	+	0.986077i	$0.446821\pi$
$752$	−23.2339	−0.847254
$753$	0	0
$754$	−9.13616	−0.332719
$755$	55.3586	2.01470
$756$	0	0
$757$	−1.09942	−0.0399590	−0.0199795	−	0.999800i	$-0.506360\pi$
$757$	−1.09942	−0.0399590	−0.0199795	+	0.999800i	$0.506360\pi$
$758$	−14.8362	−0.538876
$759$	0	0
$760$	−15.6761	−0.568633
$761$	2.31666	0.0839787	0.0419894	−	0.999118i	$-0.486630\pi$
$761$	2.31666	0.0839787	0.0419894	+	0.999118i	$0.486630\pi$
$762$	0	0
$763$	−49.1954	−1.78099
$764$	−11.8387	−0.428309
$765$	0	0
$766$	13.6898	0.494631
$767$	12.3895	0.447358
$768$	0	0
$769$	25.6417	0.924664	0.462332	−	0.886707i	$-0.347013\pi$
$769$	25.6417	0.924664	0.462332	+	0.886707i	$0.347013\pi$
$770$	−15.7624	−0.568039
$771$	0	0
$772$	35.3561	1.27249
$773$	−33.9995	−1.22288	−0.611439	−	0.791292i	$-0.709409\pi$
$773$	−33.9995	−1.22288	−0.611439	+	0.791292i	$0.709409\pi$
$774$	0	0
$775$	−91.6730	−3.29299
$776$	−15.8664	−0.569571
$777$	0	0
$778$	−3.67199	−0.131647
$779$	6.10303	0.218664
$780$	0	0
$781$	6.93173	0.248037
$782$	−12.5781	−0.449790
$783$	0	0
$784$	44.4840	1.58871
$785$	−12.9186	−0.461085
$786$	0	0
$787$	13.4836	0.480638	0.240319	−	0.970694i	$-0.422748\pi$
$787$	13.4836	0.480638	0.240319	+	0.970694i	$0.422748\pi$
$788$	12.4625	0.443959
$789$	0	0
$790$	−15.5812	−0.554354
$791$	78.5058	2.79135
$792$	0	0
$793$	12.1117	0.430099
$794$	10.9020	0.386899
$795$	0	0
$796$	−15.1113	−0.535605
$797$	−18.8515	−0.667753	−0.333877	−	0.942617i	$-0.608357\pi$
$797$	−18.8515	−0.667753	−0.333877	+	0.942617i	$0.608357\pi$
$798$	0	0
$799$	−40.7629	−1.44209
$800$	72.7782	2.57310
$801$	0	0
$802$	2.65069	0.0935992
$803$	0.882846	0.0311549
$804$	0	0
$805$	−123.247	−4.34390
$806$	−10.6872	−0.376441
$807$	0	0
$808$	6.03432	0.212287
$809$	23.3041	0.819330	0.409665	−	0.912236i	$-0.365646\pi$
$809$	23.3041	0.819330	0.409665	+	0.912236i	$0.365646\pi$
$810$	0	0
$811$	−15.3436	−0.538786	−0.269393	−	0.963030i	$-0.586823\pi$
$811$	−15.3436	−0.538786	−0.269393	+	0.963030i	$0.586823\pi$
$812$	−48.0450	−1.68605
$813$	0	0
$814$	−5.35595	−0.187726
$815$	−51.2179	−1.79408
$816$	0	0
$817$	3.28518	0.114934
$818$	−0.805076	−0.0281488
$819$	0	0
$820$	−24.9666	−0.871871
$821$	−31.8535	−1.11170	−0.555848	−	0.831284i	$-0.687606\pi$
$821$	−31.8535	−1.11170	−0.555848	+	0.831284i	$0.687606\pi$
$822$	0	0
$823$	−18.5931	−0.648116	−0.324058	−	0.946037i	$-0.605047\pi$
$823$	−18.5931	−0.648116	−0.324058	+	0.946037i	$0.605047\pi$
$824$	−28.0597	−0.977506
$825$	0	0
$826$	−10.1261	−0.352332
$827$	26.7858	0.931433	0.465717	−	0.884934i	$-0.345797\pi$
$827$	26.7858	0.931433	0.465717	+	0.884934i	$0.345797\pi$
$828$	0	0
$829$	−14.6572	−0.509066	−0.254533	−	0.967064i	$-0.581922\pi$
$829$	−14.6572	−0.509066	−0.254533	+	0.967064i	$0.581922\pi$
$830$	34.8997	1.21139
$831$	0	0
$832$	−7.14557	−0.247728
$833$	78.0451	2.70410
$834$	0	0
$835$	−52.2114	−1.80685
$836$	−4.44314	−0.153669
$837$	0	0
$838$	−15.2562	−0.527016
$839$	−28.2922	−0.976755	−0.488378	−	0.872632i	$-0.662411\pi$
$839$	−28.2922	−0.976755	−0.488378	+	0.872632i	$0.662411\pi$
$840$	0	0
$841$	1.69807	0.0585543
$842$	14.5757	0.502311
$843$	0	0
$844$	−8.99709	−0.309693
$845$	12.6583	0.435460
$846$	0	0
$847$	45.4760	1.56257
$848$	4.88469	0.167741
$849$	0	0
$850$	31.6406	1.08526
$851$	−41.8785	−1.43558
$852$	0	0
$853$	33.4782	1.14627	0.573137	−	0.819460i	$-0.305726\pi$
$853$	33.4782	1.14627	0.573137	+	0.819460i	$0.305726\pi$
$854$	−9.89909	−0.338740
$855$	0	0
$856$	13.1871	0.450726
$857$	−18.0325	−0.615979	−0.307989	−	0.951390i	$-0.599656\pi$
$857$	−18.0325	−0.615979	−0.307989	+	0.951390i	$0.599656\pi$
$858$	0	0
$859$	−19.5316	−0.666411	−0.333205	−	0.942854i	$-0.608130\pi$
$859$	−19.5316	−0.666411	−0.333205	+	0.942854i	$0.608130\pi$
$860$	−13.4392	−0.458272
$861$	0	0
$862$	−2.01618	−0.0686715
$863$	−18.1328	−0.617248	−0.308624	−	0.951184i	$-0.599868\pi$
$863$	−18.1328	−0.617248	−0.308624	+	0.951184i	$0.599868\pi$
$864$	0	0
$865$	−30.2680	−1.02914
$866$	4.61944	0.156975
$867$	0	0
$868$	−56.2017	−1.90761
$869$	−9.51883	−0.322904
$870$	0	0
$871$	−20.5602	−0.696657
$872$	−19.0039	−0.643552
$873$	0	0
$874$	5.39945	0.182639
$875$	200.438	6.77604
$876$	0	0
$877$	−33.4041	−1.12798	−0.563988	−	0.825783i	$-0.690734\pi$
$877$	−33.4041	−1.12798	−0.563988	+	0.825783i	$0.690734\pi$
$878$	12.0037	0.405106
$879$	0	0
$880$	14.9122	0.502691
$881$	28.3637	0.955596	0.477798	−	0.878470i	$-0.341435\pi$
$881$	28.3637	0.955596	0.477798	+	0.878470i	$0.341435\pi$
$882$	0	0
$883$	−21.8544	−0.735459	−0.367729	−	0.929933i	$-0.619865\pi$
$883$	−21.8544	−0.735459	−0.367729	+	0.929933i	$0.619865\pi$
$884$	−23.7335	−0.798245
$885$	0	0
$886$	−4.55915	−0.153167
$887$	20.3567	0.683511	0.341755	−	0.939789i	$-0.388979\pi$
$887$	20.3567	0.683511	0.341755	+	0.939789i	$0.388979\pi$
$888$	0	0
$889$	−5.00960	−0.168016
$890$	−40.3898	−1.35387
$891$	0	0
$892$	−8.66400	−0.290092
$893$	17.4985	0.585565
$894$	0	0
$895$	−89.7245	−2.99916
$896$	57.3926	1.91735
$897$	0	0
$898$	−6.93490	−0.231421
$899$	35.9097	1.19766
$900$	0	0
$901$	8.56998	0.285507
$902$	2.37054	0.0789303
$903$	0	0
$904$	30.3263	1.00864
$905$	29.5100	0.980945
$906$	0	0
$907$	28.4450	0.944502	0.472251	−	0.881464i	$-0.343442\pi$
$907$	28.4450	0.944502	0.472251	+	0.881464i	$0.343442\pi$
$908$	−42.9662	−1.42588
$909$	0	0
$910$	36.1436	1.19815
$911$	9.14532	0.302998	0.151499	−	0.988457i	$-0.451590\pi$
$911$	9.14532	0.302998	0.151499	+	0.988457i	$0.451590\pi$
$912$	0	0
$913$	21.3209	0.705617
$914$	12.6396	0.418079
$915$	0	0
$916$	29.9837	0.990690
$917$	−14.5419	−0.480217
$918$	0	0
$919$	4.66335	0.153830	0.0769148	−	0.997038i	$-0.475493\pi$
$919$	4.66335	0.153830	0.0769148	+	0.997038i	$0.475493\pi$
$920$	−47.6096	−1.56964
$921$	0	0
$922$	9.76229	0.321504
$923$	−15.8946	−0.523177
$924$	0	0
$925$	105.347	3.46379
$926$	4.33235	0.142370
$927$	0	0
$928$	−28.5083	−0.935832
$929$	52.0030	1.70616	0.853081	−	0.521778i	$-0.174731\pi$
$929$	52.0030	1.70616	0.853081	+	0.521778i	$0.174731\pi$
$930$	0	0
$931$	−33.5028	−1.09801
$932$	−44.3682	−1.45333
$933$	0	0
$934$	−0.672920	−0.0220186
$935$	26.1628	0.855616
$936$	0	0
$937$	20.2510	0.661570	0.330785	−	0.943706i	$-0.392687\pi$
$937$	20.2510	0.661570	0.330785	+	0.943706i	$0.392687\pi$
$938$	16.8042	0.548677
$939$	0	0
$940$	−71.5837	−2.33480
$941$	−18.0520	−0.588478	−0.294239	−	0.955732i	$-0.595066\pi$
$941$	−18.0520	−0.588478	−0.294239	+	0.955732i	$0.595066\pi$
$942$	0	0
$943$	18.5354	0.603595
$944$	9.57992	0.311800
$945$	0	0
$946$	1.27603	0.0414873
$947$	−5.50730	−0.178963	−0.0894815	−	0.995988i	$-0.528521\pi$
$947$	−5.50730	−0.178963	−0.0894815	+	0.995988i	$0.528521\pi$
$948$	0	0
$949$	−2.02439	−0.0657143
$950$	−13.5825	−0.440676
$951$	0	0
$952$	41.8104	1.35508
$953$	−44.3563	−1.43684	−0.718420	−	0.695610i	$-0.755134\pi$
$953$	−44.3563	−1.43684	−0.718420	+	0.695610i	$0.755134\pi$
$954$	0	0
$955$	−29.9250	−0.968350
$956$	−20.1523	−0.651771
$957$	0	0
$958$	−7.41736	−0.239644
$959$	−90.5102	−2.92273
$960$	0	0
$961$	11.0061	0.355036
$962$	12.2813	0.395966
$963$	0	0
$964$	29.3984	0.946860
$965$	89.3706	2.87694
$966$	0	0
$967$	24.0839	0.774485	0.387242	−	0.921978i	$-0.373428\pi$
$967$	24.0839	0.774485	0.387242	+	0.921978i	$0.373428\pi$
$968$	17.5671	0.564627
$969$	0	0
$970$	−18.6070	−0.597436
$971$	31.6938	1.01710	0.508551	−	0.861032i	$-0.330181\pi$
$971$	31.6938	1.01710	0.508551	+	0.861032i	$0.330181\pi$
$972$	0	0
$973$	−79.5466	−2.55015
$974$	−6.58357	−0.210951
$975$	0	0
$976$	9.36515	0.299771
$977$	47.3546	1.51501	0.757504	−	0.652831i	$-0.226419\pi$
$977$	47.3546	1.51501	0.757504	+	0.652831i	$0.226419\pi$
$978$	0	0
$979$	−24.6748	−0.788611
$980$	137.055	4.37806
$981$	0	0
$982$	−17.1365	−0.546847
$983$	−11.9369	−0.380727	−0.190363	−	0.981714i	$-0.560967\pi$
$983$	−11.9369	−0.380727	−0.190363	+	0.981714i	$0.560967\pi$
$984$	0	0
$985$	31.5019	1.00373
$986$	−12.3941	−0.394709
$987$	0	0
$988$	10.1882	0.324131
$989$	9.97734	0.317261
$990$	0	0
$991$	0.199970	0.00635224	0.00317612	−	0.999995i	$-0.498989\pi$
$991$	0.199970	0.00635224	0.00317612	+	0.999995i	$0.498989\pi$
$992$	−33.3482	−1.05881
$993$	0	0
$994$	12.9909	0.412047
$995$	−38.1973	−1.21093
$996$	0	0
$997$	51.7428	1.63871	0.819355	−	0.573287i	$-0.194332\pi$
$997$	51.7428	1.63871	0.819355	+	0.573287i	$0.194332\pi$
$998$	5.70649	0.180636
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.k.1.10	yes	16
3.2	odd	2	inner	1143.2.a.k.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.2.a.k.1.7	✓	16	3.2	odd	2	inner
1143.2.a.k.1.10	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+0.518678 q^{2} -1.73097 q^{4} -4.37543 q^{5} -5.00960 q^{7} -1.93517 q^{8} +O(q^{10})\)	\(q+0.518678 q^{2} -1.73097 q^{4} -4.37543 q^{5} -5.00960 q^{7} -1.93517 q^{8} -2.26944 q^{10} -1.38644 q^{11} +3.17914 q^{13} -2.59837 q^{14} +2.45821 q^{16} +4.31283 q^{17} -1.85139 q^{19} +7.57376 q^{20} -0.719117 q^{22} -5.62281 q^{23} +14.1444 q^{25} +1.64895 q^{26} +8.67148 q^{28} -5.54058 q^{29} -6.48121 q^{31} +5.14537 q^{32} +2.23697 q^{34} +21.9192 q^{35} +7.44796 q^{37} -0.960275 q^{38} +8.46722 q^{40} -3.29646 q^{41} -1.77444 q^{43} +2.39989 q^{44} -2.91643 q^{46} -9.45154 q^{47} +18.0960 q^{49} +7.33640 q^{50} -5.50301 q^{52} +1.98709 q^{53} +6.06628 q^{55} +9.69444 q^{56} -2.87378 q^{58} +3.89711 q^{59} +3.80974 q^{61} -3.36166 q^{62} -2.24764 q^{64} -13.9101 q^{65} -6.46722 q^{67} -7.46539 q^{68} +11.3690 q^{70} -4.99965 q^{71} -0.636771 q^{73} +3.86309 q^{74} +3.20471 q^{76} +6.94551 q^{77} +6.86565 q^{79} -10.7558 q^{80} -1.70980 q^{82} -15.3781 q^{83} -18.8705 q^{85} -0.920363 q^{86} +2.68301 q^{88} +17.7972 q^{89} -15.9262 q^{91} +9.73294 q^{92} -4.90231 q^{94} +8.10063 q^{95} +8.19896 q^{97} +9.38602 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 10 * q^7	\( 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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