LMFDB - Embedded newform 7744.2.a.dr.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7744,2,Mod(1,7744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7744.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,-1,0,-1,0,6,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.8361513253$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.88301$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04678 q^{3} -1.78180 q^{5} -0.353057 q^{7} +6.28284 q^{9} +5.54782 q^{13} -5.42874 q^{15} +3.81280 q^{17} +4.61803 q^{19} -1.07569 q^{21} +7.00209 q^{23} -1.82519 q^{25} +10.0021 q^{27} -2.11908 q^{29} -3.45427 q^{31} +0.629076 q^{35} +1.74050 q^{37} +16.9030 q^{39} -1.48318 q^{41} -3.92979 q^{43} -11.1948 q^{45} +1.21054 q^{47} -6.87535 q^{49} +11.6167 q^{51} -7.92641 q^{53} +14.0701 q^{57} -3.09017 q^{59} +7.60564 q^{61} -2.21820 q^{63} -9.88510 q^{65} +1.17352 q^{67} +21.3338 q^{69} +2.36957 q^{71} +3.90426 q^{73} -5.56095 q^{75} -4.10933 q^{79} +11.6256 q^{81} +0.818368 q^{83} -6.79364 q^{85} -6.45636 q^{87} -1.92979 q^{89} -1.95870 q^{91} -10.5244 q^{93} -8.22841 q^{95} -9.47761 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - q^{13} - 16 q^{15} - 12 q^{17} + 14 q^{19} + q^{21} + 2 q^{23} + 11 q^{25} + 14 q^{27} + 9 q^{29} - 11 q^{31} + 18 q^{35} - 13 q^{37} + 18 q^{39} - 8 q^{41}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - q^5 - q^7 + 6 * q^9 - q^13 - 16 * q^15 - 12 * q^17 + 14 * q^19 + q^21 + 2 * q^23 + 11 * q^25 + 14 * q^27 + 9 * q^29 - 11 * q^31 + 18 * q^35 - 13 * q^37 + 18 * q^39 - 8 * q^41 + 3 * q^43 - 22 * q^45 - 7 * q^47 - q^49 + 14 * q^51 - 11 * q^53 + 7 * q^57 + 10 * q^59 + 17 * q^61 - 15 * q^63 - 5 * q^65 - 5 * q^67 + 4 * q^69 + 5 * q^71 - 6 * q^73 + 7 * q^75 - 7 * q^79 - 8 * q^81 + 20 * q^83 + 13 * q^85 + 3 * q^87 + 11 * q^89 + 6 * q^91 + 10 * q^93 - 6 * q^95 + 4 * q^97

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
$3$	3.04678	1.75906	0.879528	−	0.475846i	$-0.157858\pi$
$3$	3.04678	1.75906	0.879528	+	0.475846i	$0.157858\pi$
$4$	0	0
$5$	−1.78180	−0.796845	−0.398422	−	0.917202i	$-0.630442\pi$
$5$	−1.78180	−0.796845	−0.398422	+	0.917202i	$0.630442\pi$
$6$	0	0
$7$	−0.353057	−0.133443	−0.0667215	−	0.997772i	$-0.521254\pi$
$7$	−0.353057	−0.133443	−0.0667215	+	0.997772i	$0.521254\pi$
$8$	0	0
$9$	6.28284	2.09428
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.54782	1.53869	0.769344	−	0.638834i	$-0.220583\pi$
$13$	5.54782	1.53869	0.769344	+	0.638834i	$0.220583\pi$
$14$	0	0
$15$	−5.42874	−1.40170
$16$	0	0
$17$	3.81280	0.924739	0.462370	−	0.886687i	$-0.346999\pi$
$17$	3.81280	0.924739	0.462370	+	0.886687i	$0.346999\pi$
$18$	0	0
$19$	4.61803	1.05945	0.529725	−	0.848170i	$-0.322295\pi$
$19$	4.61803	1.05945	0.529725	+	0.848170i	$0.322295\pi$
$20$	0	0
$21$	−1.07569	−0.234734
$22$	0	0
$23$	7.00209	1.46004	0.730018	−	0.683428i	$-0.239511\pi$
$23$	7.00209	1.46004	0.730018	+	0.683428i	$0.239511\pi$
$24$	0	0
$25$	−1.82519	−0.365039
$26$	0	0
$27$	10.0021	1.92490
$28$	0	0
$29$	−2.11908	−0.393503	−0.196752	−	0.980453i	$-0.563039\pi$
$29$	−2.11908	−0.393503	−0.196752	+	0.980453i	$0.563039\pi$
$30$	0	0
$31$	−3.45427	−0.620405	−0.310203	−	0.950670i	$-0.600397\pi$
$31$	−3.45427	−0.620405	−0.310203	+	0.950670i	$0.600397\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.629076	0.106333
$36$	0	0
$37$	1.74050	0.286136	0.143068	−	0.989713i	$-0.454303\pi$
$37$	1.74050	0.286136	0.143068	+	0.989713i	$0.454303\pi$
$38$	0	0
$39$	16.9030	2.70664
$40$	0	0
$41$	−1.48318	−0.231634	−0.115817	−	0.993271i	$-0.536949\pi$
$41$	−1.48318	−0.231634	−0.115817	+	0.993271i	$0.536949\pi$
$42$	0	0
$43$	−3.92979	−0.599287	−0.299643	−	0.954051i	$-0.596868\pi$
$43$	−3.92979	−0.599287	−0.299643	+	0.954051i	$0.596868\pi$
$44$	0	0
$45$	−11.1948	−1.66882
$46$	0	0
$47$	1.21054	0.176576	0.0882878	−	0.996095i	$-0.471860\pi$
$47$	1.21054	0.176576	0.0882878	+	0.996095i	$0.471860\pi$
$48$	0	0
$49$	−6.87535	−0.982193
$50$	0	0
$51$	11.6167	1.62667
$52$	0	0
$53$	−7.92641	−1.08878	−0.544388	−	0.838834i	$-0.683238\pi$
$53$	−7.92641	−1.08878	−0.544388	+	0.838834i	$0.683238\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.0701	1.86363
$58$	0	0
$59$	−3.09017	−0.402306	−0.201153	−	0.979560i	$-0.564469\pi$
$59$	−3.09017	−0.402306	−0.201153	+	0.979560i	$0.564469\pi$
$60$	0	0
$61$	7.60564	0.973802	0.486901	−	0.873457i	$-0.338127\pi$
$61$	7.60564	0.973802	0.486901	+	0.873457i	$0.338127\pi$
$62$	0	0
$63$	−2.21820	−0.279467
$64$	0	0
$65$	−9.88510	−1.22610
$66$	0	0
$67$	1.17352	0.143368	0.0716839	−	0.997427i	$-0.477163\pi$
$67$	1.17352	0.143368	0.0716839	+	0.997427i	$0.477163\pi$
$68$	0	0
$69$	21.3338	2.56829
$70$	0	0
$71$	2.36957	0.281216	0.140608	−	0.990065i	$-0.455094\pi$
$71$	2.36957	0.281216	0.140608	+	0.990065i	$0.455094\pi$
$72$	0	0
$73$	3.90426	0.456959	0.228480	−	0.973549i	$-0.426625\pi$
$73$	3.90426	0.456959	0.228480	+	0.973549i	$0.426625\pi$
$74$	0	0
$75$	−5.56095	−0.642124
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.10933	−0.462336	−0.231168	−	0.972914i	$-0.574255\pi$
$79$	−4.10933	−0.462336	−0.231168	+	0.972914i	$0.574255\pi$
$80$	0	0
$81$	11.6256	1.29173
$82$	0	0
$83$	0.818368	0.0898276	0.0449138	−	0.998991i	$-0.485699\pi$
$83$	0.818368	0.0898276	0.0449138	+	0.998991i	$0.485699\pi$
$84$	0	0
$85$	−6.79364	−0.736874
$86$	0	0
$87$	−6.45636	−0.692194
$88$	0	0
$89$	−1.92979	−0.204557	−0.102279	−	0.994756i	$-0.532613\pi$
$89$	−1.92979	−0.204557	−0.102279	+	0.994756i	$0.532613\pi$
$90$	0	0
$91$	−1.95870	−0.205327
$92$	0	0
$93$	−10.5244	−1.09133
$94$	0	0
$95$	−8.22841	−0.844217
$96$	0	0
$97$	−9.47761	−0.962305	−0.481153	−	0.876637i	$-0.659782\pi$
$97$	−9.47761	−0.962305	−0.481153	+	0.876637i	$0.659782\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.4053	1.43338	0.716691	−	0.697391i	$-0.245656\pi$
$101$	14.4053	1.43338	0.716691	+	0.697391i	$0.245656\pi$
$102$	0	0
$103$	−8.02553	−0.790779	−0.395389	−	0.918514i	$-0.629390\pi$
$103$	−8.02553	−0.790779	−0.395389	+	0.918514i	$0.629390\pi$
$104$	0	0
$105$	1.91665	0.187046
$106$	0	0
$107$	4.57464	0.442247	0.221124	−	0.975246i	$-0.429028\pi$
$107$	4.57464	0.442247	0.221124	+	0.975246i	$0.429028\pi$
$108$	0	0
$109$	20.6592	1.97880	0.989398	−	0.145228i	$-0.0463917\pi$
$109$	20.6592	1.97880	0.989398	+	0.145228i	$0.0463917\pi$
$110$	0	0
$111$	5.30290	0.503329
$112$	0	0
$113$	8.80514	0.828318	0.414159	−	0.910205i	$-0.364076\pi$
$113$	8.80514	0.828318	0.414159	+	0.910205i	$0.364076\pi$
$114$	0	0
$115$	−12.4763	−1.16342
$116$	0	0
$117$	34.8561	3.22245
$118$	0	0
$119$	−1.34613	−0.123400
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−4.51891	−0.407457
$124$	0	0
$125$	12.1611	1.08772
$126$	0	0
$127$	−17.7927	−1.57885	−0.789425	−	0.613847i	$-0.789621\pi$
$127$	−17.7927	−1.57885	−0.789425	+	0.613847i	$0.789621\pi$
$128$	0	0
$129$	−11.9732	−1.05418
$130$	0	0
$131$	18.3585	1.60399	0.801996	−	0.597329i	$-0.203771\pi$
$131$	18.3585	1.60399	0.801996	+	0.597329i	$0.203771\pi$
$132$	0	0
$133$	−1.63043	−0.141376
$134$	0	0
$135$	−17.8217	−1.53385
$136$	0	0
$137$	−13.4776	−1.15147	−0.575735	−	0.817636i	$-0.695284\pi$
$137$	−13.4776	−1.15147	−0.575735	+	0.817636i	$0.695284\pi$
$138$	0	0
$139$	14.2704	1.21040	0.605202	−	0.796072i	$-0.293092\pi$
$139$	14.2704	1.21040	0.605202	+	0.796072i	$0.293092\pi$
$140$	0	0
$141$	3.68825	0.310606
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.77577	0.313561
$146$	0	0
$147$	−20.9477	−1.72773
$148$	0	0
$149$	10.2029	0.835853	0.417926	−	0.908481i	$-0.362757\pi$
$149$	10.2029	0.835853	0.417926	+	0.908481i	$0.362757\pi$
$150$	0	0
$151$	−4.38879	−0.357155	−0.178577	−	0.983926i	$-0.557149\pi$
$151$	−4.38879	−0.357155	−0.178577	+	0.983926i	$0.557149\pi$
$152$	0	0
$153$	23.9552	1.93666
$154$	0	0
$155$	6.15481	0.494366
$156$	0	0
$157$	−22.1811	−1.77024	−0.885121	−	0.465360i	$-0.845925\pi$
$157$	−22.1811	−1.77024	−0.885121	+	0.465360i	$0.845925\pi$
$158$	0	0
$159$	−24.1500	−1.91522
$160$	0	0
$161$	−2.47214	−0.194832
$162$	0	0
$163$	19.7302	1.54539	0.772694	−	0.634779i	$-0.218909\pi$
$163$	19.7302	1.54539	0.772694	+	0.634779i	$0.218909\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.988897	0.0765232	0.0382616	−	0.999268i	$-0.487818\pi$
$167$	0.988897	0.0765232	0.0382616	+	0.999268i	$0.487818\pi$
$168$	0	0
$169$	17.7783	1.36756
$170$	0	0
$171$	29.0144	2.21879
$172$	0	0
$173$	17.4998	1.33048	0.665241	−	0.746629i	$-0.268329\pi$
$173$	17.4998	1.33048	0.665241	+	0.746629i	$0.268329\pi$
$174$	0	0
$175$	0.644397	0.0487118
$176$	0	0
$177$	−9.41506	−0.707679
$178$	0	0
$179$	−2.85201	−0.213169	−0.106585	−	0.994304i	$-0.533992\pi$
$179$	−2.85201	−0.213169	−0.106585	+	0.994304i	$0.533992\pi$
$180$	0	0
$181$	2.10256	0.156282	0.0781412	−	0.996942i	$-0.475102\pi$
$181$	2.10256	0.156282	0.0781412	+	0.996942i	$0.475102\pi$
$182$	0	0
$183$	23.1727	1.71297
$184$	0	0
$185$	−3.10121	−0.228006
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.53131	−0.256865
$190$	0	0
$191$	17.1212	1.23884	0.619422	−	0.785058i	$-0.287367\pi$
$191$	17.1212	1.23884	0.619422	+	0.785058i	$0.287367\pi$
$192$	0	0
$193$	20.1288	1.44890	0.724452	−	0.689325i	$-0.242093\pi$
$193$	20.1288	1.44890	0.724452	+	0.689325i	$0.242093\pi$
$194$	0	0
$195$	−30.1177	−2.15677
$196$	0	0
$197$	−12.5589	−0.894786	−0.447393	−	0.894337i	$-0.647648\pi$
$197$	−12.5589	−0.894786	−0.447393	+	0.894337i	$0.647648\pi$
$198$	0	0
$199$	3.32962	0.236031	0.118015	−	0.993012i	$-0.462347\pi$
$199$	3.32962	0.236031	0.118015	+	0.993012i	$0.462347\pi$
$200$	0	0
$201$	3.57544	0.252192
$202$	0	0
$203$	0.748155	0.0525102
$204$	0	0
$205$	2.64273	0.184576
$206$	0	0
$207$	43.9930	3.05773
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−24.0944	−1.65872	−0.829362	−	0.558712i	$-0.811296\pi$
$211$	−24.0944	−1.65872	−0.829362	+	0.558712i	$0.811296\pi$
$212$	0	0
$213$	7.21955	0.494676
$214$	0	0
$215$	7.00209	0.477539
$216$	0	0
$217$	1.21955	0.0827887
$218$	0	0
$219$	11.8954	0.803817
$220$	0	0
$221$	21.1527	1.42289
$222$	0	0
$223$	−0.529216	−0.0354389	−0.0177195	−	0.999843i	$-0.505641\pi$
$223$	−0.529216	−0.0354389	−0.0177195	+	0.999843i	$0.505641\pi$
$224$	0	0
$225$	−11.4674	−0.764493
$226$	0	0
$227$	−12.6800	−0.841603	−0.420802	−	0.907153i	$-0.638251\pi$
$227$	−12.6800	−0.841603	−0.420802	+	0.907153i	$0.638251\pi$
$228$	0	0
$229$	6.10256	0.403269	0.201634	−	0.979461i	$-0.435375\pi$
$229$	6.10256	0.403269	0.201634	+	0.979461i	$0.435375\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.49084	−0.163180	−0.0815901	−	0.996666i	$-0.526000\pi$
$233$	−2.49084	−0.163180	−0.0815901	+	0.996666i	$0.526000\pi$
$234$	0	0
$235$	−2.15694	−0.140703
$236$	0	0
$237$	−12.5202	−0.813275
$238$	0	0
$239$	−22.3682	−1.44688	−0.723439	−	0.690389i	$-0.757440\pi$
$239$	−22.3682	−1.44688	−0.723439	+	0.690389i	$0.757440\pi$
$240$	0	0
$241$	−24.1025	−1.55258	−0.776288	−	0.630378i	$-0.782900\pi$
$241$	−24.1025	−1.55258	−0.776288	+	0.630378i	$0.782900\pi$
$242$	0	0
$243$	5.41432	0.347329
$244$	0	0
$245$	12.2505	0.782655
$246$	0	0
$247$	25.6200	1.63016
$248$	0	0
$249$	2.49338	0.158012
$250$	0	0
$251$	−6.93108	−0.437486	−0.218743	−	0.975783i	$-0.570196\pi$
$251$	−6.93108	−0.437486	−0.218743	+	0.975783i	$0.570196\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−20.6987	−1.29620
$256$	0	0
$257$	−14.6529	−0.914021	−0.457011	−	0.889461i	$-0.651080\pi$
$257$	−14.6529	−0.914021	−0.457011	+	0.889461i	$0.651080\pi$
$258$	0	0
$259$	−0.614494	−0.0381828
$260$	0	0
$261$	−13.3138	−0.824106
$262$	0	0
$263$	22.9868	1.41742	0.708712	−	0.705497i	$-0.249276\pi$
$263$	22.9868	1.41742	0.708712	+	0.705497i	$0.249276\pi$
$264$	0	0
$265$	14.1233	0.867585
$266$	0	0
$267$	−5.87963	−0.359827
$268$	0	0
$269$	5.54782	0.338257	0.169128	−	0.985594i	$-0.445905\pi$
$269$	5.54782	0.338257	0.169128	+	0.985594i	$0.445905\pi$
$270$	0	0
$271$	13.5355	0.822222	0.411111	−	0.911585i	$-0.365141\pi$
$271$	13.5355	0.822222	0.411111	+	0.911585i	$0.365141\pi$
$272$	0	0
$273$	−5.96771	−0.361182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.96005	0.177852	0.0889260	−	0.996038i	$-0.471657\pi$
$277$	2.96005	0.177852	0.0889260	+	0.996038i	$0.471657\pi$
$278$	0	0
$279$	−21.7026	−1.29930
$280$	0	0
$281$	−8.96297	−0.534686	−0.267343	−	0.963601i	$-0.586146\pi$
$281$	−8.96297	−0.534686	−0.267343	+	0.963601i	$0.586146\pi$
$282$	0	0
$283$	18.0407	1.07241	0.536203	−	0.844089i	$-0.319858\pi$
$283$	18.0407	1.07241	0.536203	+	0.844089i	$0.319858\pi$
$284$	0	0
$285$	−25.0701	−1.48503
$286$	0	0
$287$	0.523646	0.0309099
$288$	0	0
$289$	−2.46257	−0.144857
$290$	0	0
$291$	−28.8761	−1.69275
$292$	0	0
$293$	−27.6400	−1.61475	−0.807373	−	0.590042i	$-0.799111\pi$
$293$	−27.6400	−1.61475	−0.807373	+	0.590042i	$0.799111\pi$
$294$	0	0
$295$	5.50606	0.320575
$296$	0	0
$297$	0	0
$298$	0	0
$299$	38.8463	2.24654
$300$	0	0
$301$	1.38744	0.0799706
$302$	0	0
$303$	43.8897	2.52140
$304$	0	0
$305$	−13.5517	−0.775969
$306$	0	0
$307$	0.662166	0.0377918	0.0188959	−	0.999821i	$-0.493985\pi$
$307$	0.662166	0.0377918	0.0188959	+	0.999821i	$0.493985\pi$
$308$	0	0
$309$	−24.4520	−1.39102
$310$	0	0
$311$	13.0701	0.741138	0.370569	−	0.928805i	$-0.379163\pi$
$311$	13.0701	0.741138	0.370569	+	0.928805i	$0.379163\pi$
$312$	0	0
$313$	19.8128	1.11989	0.559943	−	0.828531i	$-0.310823\pi$
$313$	19.8128	1.11989	0.559943	+	0.828531i	$0.310823\pi$
$314$	0	0
$315$	3.95239	0.222692
$316$	0	0
$317$	19.7391	1.10866	0.554329	−	0.832297i	$-0.312975\pi$
$317$	19.7391	1.10866	0.554329	+	0.832297i	$0.312975\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	13.9379	0.777938
$322$	0	0
$323$	17.6076	0.979715
$324$	0	0
$325$	−10.1258	−0.561681
$326$	0	0
$327$	62.9441	3.48082
$328$	0	0
$329$	−0.427390	−0.0235628
$330$	0	0
$331$	21.0860	1.15899	0.579494	−	0.814976i	$-0.303250\pi$
$331$	21.0860	1.15899	0.579494	+	0.814976i	$0.303250\pi$
$332$	0	0
$333$	10.9353	0.599249
$334$	0	0
$335$	−2.09097	−0.114242
$336$	0	0
$337$	−1.38615	−0.0755082	−0.0377541	−	0.999287i	$-0.512020\pi$
$337$	−1.38615	−0.0755082	−0.0377541	+	0.999287i	$0.512020\pi$
$338$	0	0
$339$	26.8273	1.45706
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.89879	0.264510
$344$	0	0
$345$	−38.0125	−2.04653
$346$	0	0
$347$	19.1162	1.02621	0.513104	−	0.858326i	$-0.328495\pi$
$347$	19.1162	1.02621	0.513104	+	0.858326i	$0.328495\pi$
$348$	0	0
$349$	−20.7163	−1.10892	−0.554460	−	0.832211i	$-0.687075\pi$
$349$	−20.7163	−1.10892	−0.554460	+	0.832211i	$0.687075\pi$
$350$	0	0
$351$	55.4898	2.96183
$352$	0	0
$353$	5.09091	0.270962	0.135481	−	0.990780i	$-0.456742\pi$
$353$	5.09091	0.270962	0.135481	+	0.990780i	$0.456742\pi$
$354$	0	0
$355$	−4.22210	−0.224086
$356$	0	0
$357$	−4.10137	−0.217068
$358$	0	0
$359$	4.12437	0.217676	0.108838	−	0.994060i	$-0.465287\pi$
$359$	4.12437	0.217676	0.108838	+	0.994060i	$0.465287\pi$
$360$	0	0
$361$	2.32624	0.122434
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.95661	−0.364125
$366$	0	0
$367$	−36.5018	−1.90538	−0.952690	−	0.303943i	$-0.901697\pi$
$367$	−36.5018	−1.90538	−0.952690	+	0.303943i	$0.901697\pi$
$368$	0	0
$369$	−9.31858	−0.485106
$370$	0	0
$371$	2.79847	0.145289
$372$	0	0
$373$	5.01741	0.259792	0.129896	−	0.991528i	$-0.458536\pi$
$373$	5.01741	0.259792	0.129896	+	0.991528i	$0.458536\pi$
$374$	0	0
$375$	37.0522	1.91337
$376$	0	0
$377$	−11.7563	−0.605479
$378$	0	0
$379$	20.2483	1.04009	0.520043	−	0.854140i	$-0.325916\pi$
$379$	20.2483	1.04009	0.520043	+	0.854140i	$0.325916\pi$
$380$	0	0
$381$	−54.2105	−2.77729
$382$	0	0
$383$	9.40321	0.480482	0.240241	−	0.970713i	$-0.422774\pi$
$383$	9.40321	0.480482	0.240241	+	0.970713i	$0.422774\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.6902	−1.25508
$388$	0	0
$389$	−6.76467	−0.342982	−0.171491	−	0.985186i	$-0.554859\pi$
$389$	−6.76467	−0.342982	−0.171491	+	0.985186i	$0.554859\pi$
$390$	0	0
$391$	26.6976	1.35015
$392$	0	0
$393$	55.9343	2.82151
$394$	0	0
$395$	7.32200	0.368410
$396$	0	0
$397$	5.18919	0.260438	0.130219	−	0.991485i	$-0.458432\pi$
$397$	5.18919	0.260438	0.130219	+	0.991485i	$0.458432\pi$
$398$	0	0
$399$	−4.96755	−0.248689
$400$	0	0
$401$	5.88922	0.294094	0.147047	−	0.989130i	$-0.453023\pi$
$401$	5.88922	0.294094	0.147047	+	0.989130i	$0.453023\pi$
$402$	0	0
$403$	−19.1637	−0.954610
$404$	0	0
$405$	−20.7145	−1.02931
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−34.5388	−1.70783	−0.853916	−	0.520411i	$-0.825779\pi$
$409$	−34.5388	−1.70783	−0.853916	+	0.520411i	$0.825779\pi$
$410$	0	0
$411$	−41.0633	−2.02550
$412$	0	0
$413$	1.09101	0.0536849
$414$	0	0
$415$	−1.45817	−0.0715786
$416$	0	0
$417$	43.4789	2.12917
$418$	0	0
$419$	−7.32141	−0.357674	−0.178837	−	0.983879i	$-0.557233\pi$
$419$	−7.32141	−0.357674	−0.178837	+	0.983879i	$0.557233\pi$
$420$	0	0
$421$	−32.8786	−1.60241	−0.801203	−	0.598392i	$-0.795806\pi$
$421$	−32.8786	−1.60241	−0.801203	+	0.598392i	$0.795806\pi$
$422$	0	0
$423$	7.60564	0.369799
$424$	0	0
$425$	−6.95909	−0.337566
$426$	0	0
$427$	−2.68522	−0.129947
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.21611	0.251251	0.125626	−	0.992078i	$-0.459906\pi$
$431$	5.21611	0.251251	0.125626	+	0.992078i	$0.459906\pi$
$432$	0	0
$433$	11.0282	0.529980	0.264990	−	0.964251i	$-0.414631\pi$
$433$	11.0282	0.529980	0.264990	+	0.964251i	$0.414631\pi$
$434$	0	0
$435$	11.5039	0.551571
$436$	0	0
$437$	32.3359	1.54684
$438$	0	0
$439$	−22.5808	−1.07772	−0.538862	−	0.842394i	$-0.681145\pi$
$439$	−22.5808	−1.07772	−0.538862	+	0.842394i	$0.681145\pi$
$440$	0	0
$441$	−43.1968	−2.05699
$442$	0	0
$443$	−35.3181	−1.67801	−0.839006	−	0.544122i	$-0.816863\pi$
$443$	−35.3181	−1.67801	−0.839006	+	0.544122i	$0.816863\pi$
$444$	0	0
$445$	3.43849	0.163000
$446$	0	0
$447$	31.0859	1.47031
$448$	0	0
$449$	37.0598	1.74896	0.874480	−	0.485061i	$-0.161203\pi$
$449$	37.0598	1.74896	0.874480	+	0.485061i	$0.161203\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−13.3717	−0.628255
$454$	0	0
$455$	3.49000	0.163614
$456$	0	0
$457$	0.352996	0.0165125	0.00825624	−	0.999966i	$-0.497372\pi$
$457$	0.352996	0.0165125	0.00825624	+	0.999966i	$0.497372\pi$
$458$	0	0
$459$	38.1360	1.78003
$460$	0	0
$461$	−12.2449	−0.570303	−0.285151	−	0.958482i	$-0.592044\pi$
$461$	−12.2449	−0.570303	−0.285151	+	0.958482i	$0.592044\pi$
$462$	0	0
$463$	−20.9578	−0.973992	−0.486996	−	0.873404i	$-0.661907\pi$
$463$	−20.9578	−0.973992	−0.486996	+	0.873404i	$0.661907\pi$
$464$	0	0
$465$	18.7523	0.869619
$466$	0	0
$467$	29.5768	1.36865	0.684325	−	0.729177i	$-0.260097\pi$
$467$	29.5768	1.36865	0.684325	+	0.729177i	$0.260097\pi$
$468$	0	0
$469$	−0.414318	−0.0191314
$470$	0	0
$471$	−67.5808	−3.11396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−8.42880	−0.386740
$476$	0	0
$477$	−49.8004	−2.28020
$478$	0	0
$479$	5.82609	0.266201	0.133100	−	0.991103i	$-0.457507\pi$
$479$	5.82609	0.266201	0.133100	+	0.991103i	$0.457507\pi$
$480$	0	0
$481$	9.65596	0.440274
$482$	0	0
$483$	−7.53204	−0.342720
$484$	0	0
$485$	16.8872	0.766808
$486$	0	0
$487$	−30.9518	−1.40256	−0.701279	−	0.712887i	$-0.747387\pi$
$487$	−30.9518	−1.40256	−0.701279	+	0.712887i	$0.747387\pi$
$488$	0	0
$489$	60.1135	2.71843
$490$	0	0
$491$	10.6930	0.482567	0.241284	−	0.970455i	$-0.422432\pi$
$491$	10.6930	0.482567	0.241284	+	0.970455i	$0.422432\pi$
$492$	0	0
$493$	−8.07962	−0.363888
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.836593	−0.0375263
$498$	0	0
$499$	25.6798	1.14958	0.574792	−	0.818300i	$-0.305083\pi$
$499$	25.6798	1.14958	0.574792	+	0.818300i	$0.305083\pi$
$500$	0	0
$501$	3.01295	0.134609
$502$	0	0
$503$	17.2572	0.769461	0.384731	−	0.923029i	$-0.374294\pi$
$503$	17.2572	0.769461	0.384731	+	0.923029i	$0.374294\pi$
$504$	0	0
$505$	−25.6674	−1.14218
$506$	0	0
$507$	54.1666	2.40562
$508$	0	0
$509$	−34.9229	−1.54793	−0.773966	−	0.633227i	$-0.781730\pi$
$509$	−34.9229	−1.54793	−0.773966	+	0.633227i	$0.781730\pi$
$510$	0	0
$511$	−1.37843	−0.0609780
$512$	0	0
$513$	46.1900	2.03934
$514$	0	0
$515$	14.2999	0.630128
$516$	0	0
$517$	0	0
$518$	0	0
$519$	53.3178	2.34039
$520$	0	0
$521$	−7.96986	−0.349166	−0.174583	−	0.984642i	$-0.555858\pi$
$521$	−7.96986	−0.349166	−0.174583	+	0.984642i	$0.555858\pi$
$522$	0	0
$523$	22.8353	0.998518	0.499259	−	0.866453i	$-0.333606\pi$
$523$	22.8353	0.998518	0.499259	+	0.866453i	$0.333606\pi$
$524$	0	0
$525$	1.96333	0.0856869
$526$	0	0
$527$	−13.1704	−0.573713
$528$	0	0
$529$	26.0293	1.13171
$530$	0	0
$531$	−19.4151	−0.842542
$532$	0	0
$533$	−8.22841	−0.356412
$534$	0	0
$535$	−8.15109	−0.352402
$536$	0	0
$537$	−8.68944	−0.374977
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−39.9578	−1.71792	−0.858959	−	0.512044i	$-0.828888\pi$
$541$	−39.9578	−1.71792	−0.858959	+	0.512044i	$0.828888\pi$
$542$	0	0
$543$	6.40604	0.274910
$544$	0	0
$545$	−36.8106	−1.57679
$546$	0	0
$547$	−16.6384	−0.711409	−0.355704	−	0.934599i	$-0.615759\pi$
$547$	−16.6384	−0.711409	−0.355704	+	0.934599i	$0.615759\pi$
$548$	0	0
$549$	47.7850	2.03942
$550$	0	0
$551$	−9.78598	−0.416897
$552$	0	0
$553$	1.45083	0.0616954
$554$	0	0
$555$	−9.44870	−0.401075
$556$	0	0
$557$	19.7600	0.837258	0.418629	−	0.908157i	$-0.362511\pi$
$557$	19.7600	0.837258	0.418629	+	0.908157i	$0.362511\pi$
$558$	0	0
$559$	−21.8018	−0.922116
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.2663	1.02270	0.511352	−	0.859371i	$-0.329145\pi$
$563$	24.2663	1.02270	0.511352	+	0.859371i	$0.329145\pi$
$564$	0	0
$565$	−15.6890	−0.660041
$566$	0	0
$567$	−4.10450	−0.172373
$568$	0	0
$569$	−28.4513	−1.19274	−0.596371	−	0.802709i	$-0.703391\pi$
$569$	−28.4513	−1.19274	−0.596371	+	0.802709i	$0.703391\pi$
$570$	0	0
$571$	−26.9485	−1.12776	−0.563879	−	0.825858i	$-0.690692\pi$
$571$	−26.9485	−1.12776	−0.563879	+	0.825858i	$0.690692\pi$
$572$	0	0
$573$	52.1644	2.17920
$574$	0	0
$575$	−12.7802	−0.532970
$576$	0	0
$577$	10.8780	0.452857	0.226428	−	0.974028i	$-0.427295\pi$
$577$	10.8780	0.452857	0.226428	+	0.974028i	$0.427295\pi$
$578$	0	0
$579$	61.3280	2.54871
$580$	0	0
$581$	−0.288931	−0.0119869
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−62.1065	−2.56779
$586$	0	0
$587$	−27.0501	−1.11648	−0.558238	−	0.829681i	$-0.688522\pi$
$587$	−27.0501	−1.11648	−0.558238	+	0.829681i	$0.688522\pi$
$588$	0	0
$589$	−15.9519	−0.657288
$590$	0	0
$591$	−38.2642	−1.57398
$592$	0	0
$593$	6.72088	0.275993	0.137997	−	0.990433i	$-0.455934\pi$
$593$	6.72088	0.275993	0.137997	+	0.990433i	$0.455934\pi$
$594$	0	0
$595$	2.39854	0.0983306
$596$	0	0
$597$	10.1446	0.415191
$598$	0	0
$599$	−28.0336	−1.14542	−0.572711	−	0.819757i	$-0.694108\pi$
$599$	−28.0336	−1.14542	−0.572711	+	0.819757i	$0.694108\pi$
$600$	0	0
$601$	−17.1250	−0.698544	−0.349272	−	0.937021i	$-0.613571\pi$
$601$	−17.1250	−0.698544	−0.349272	+	0.937021i	$0.613571\pi$
$602$	0	0
$603$	7.37301	0.300252
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−37.5051	−1.52229	−0.761143	−	0.648584i	$-0.775362\pi$
$607$	−37.5051	−1.52229	−0.761143	+	0.648584i	$0.775362\pi$
$608$	0	0
$609$	2.27946	0.0923685
$610$	0	0
$611$	6.71586	0.271695
$612$	0	0
$613$	−26.6202	−1.07518	−0.537590	−	0.843207i	$-0.680665\pi$
$613$	−26.6202	−1.07518	−0.537590	+	0.843207i	$0.680665\pi$
$614$	0	0
$615$	8.05179	0.324680
$616$	0	0
$617$	13.8616	0.558047	0.279024	−	0.960284i	$-0.409989\pi$
$617$	13.8616	0.558047	0.279024	+	0.960284i	$0.409989\pi$
$618$	0	0
$619$	8.21318	0.330116	0.165058	−	0.986284i	$-0.447219\pi$
$619$	8.21318	0.330116	0.165058	+	0.986284i	$0.447219\pi$
$620$	0	0
$621$	70.0355	2.81043
$622$	0	0
$623$	0.681325	0.0272967
$624$	0	0
$625$	−12.5427	−0.501708
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.63616	0.264601
$630$	0	0
$631$	−39.4638	−1.57103	−0.785515	−	0.618843i	$-0.787602\pi$
$631$	−39.4638	−1.57103	−0.785515	+	0.618843i	$0.787602\pi$
$632$	0	0
$633$	−73.4101	−2.91779
$634$	0	0
$635$	31.7031	1.25810
$636$	0	0
$637$	−38.1432	−1.51129
$638$	0	0
$639$	14.8876	0.588946
$640$	0	0
$641$	−36.1424	−1.42754	−0.713770	−	0.700380i	$-0.753014\pi$
$641$	−36.1424	−1.42754	−0.713770	+	0.700380i	$0.753014\pi$
$642$	0	0
$643$	−23.9855	−0.945895	−0.472947	−	0.881091i	$-0.656810\pi$
$643$	−23.9855	−0.945895	−0.472947	+	0.881091i	$0.656810\pi$
$644$	0	0
$645$	21.3338	0.840018
$646$	0	0
$647$	41.5698	1.63428	0.817139	−	0.576440i	$-0.195559\pi$
$647$	41.5698	1.63428	0.817139	+	0.576440i	$0.195559\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	3.71571	0.145630
$652$	0	0
$653$	−30.0501	−1.17595	−0.587976	−	0.808878i	$-0.700075\pi$
$653$	−30.0501	−1.17595	−0.587976	+	0.808878i	$0.700075\pi$
$654$	0	0
$655$	−32.7112	−1.27813
$656$	0	0
$657$	24.5299	0.957001
$658$	0	0
$659$	25.2450	0.983405	0.491702	−	0.870763i	$-0.336375\pi$
$659$	25.2450	0.983405	0.491702	+	0.870763i	$0.336375\pi$
$660$	0	0
$661$	−9.51852	−0.370227	−0.185114	−	0.982717i	$-0.559265\pi$
$661$	−9.51852	−0.370227	−0.185114	+	0.982717i	$0.559265\pi$
$662$	0	0
$663$	64.4476	2.50294
$664$	0	0
$665$	2.90510	0.112655
$666$	0	0
$667$	−14.8380	−0.574529
$668$	0	0
$669$	−1.61240	−0.0623391
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11.2923	0.435287	0.217643	−	0.976028i	$-0.430163\pi$
$673$	11.2923	0.435287	0.217643	+	0.976028i	$0.430163\pi$
$674$	0	0
$675$	−18.2557	−0.702664
$676$	0	0
$677$	35.4830	1.36372	0.681862	−	0.731481i	$-0.261170\pi$
$677$	35.4830	1.36372	0.681862	+	0.731481i	$0.261170\pi$
$678$	0	0
$679$	3.34613	0.128413
$680$	0	0
$681$	−38.6332	−1.48043
$682$	0	0
$683$	24.8947	0.952569	0.476284	−	0.879291i	$-0.341983\pi$
$683$	24.8947	0.952569	0.476284	+	0.879291i	$0.341983\pi$
$684$	0	0
$685$	24.0144	0.917543
$686$	0	0
$687$	18.5931	0.709373
$688$	0	0
$689$	−43.9743	−1.67529
$690$	0	0
$691$	−0.634549	−0.0241394	−0.0120697	−	0.999927i	$-0.503842\pi$
$691$	−0.634549	−0.0241394	−0.0120697	+	0.999927i	$0.503842\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.4271	−0.964504
$696$	0	0
$697$	−5.65506	−0.214201
$698$	0	0
$699$	−7.58903	−0.287043
$700$	0	0
$701$	−17.4622	−0.659539	−0.329770	−	0.944061i	$-0.606971\pi$
$701$	−17.4622	−0.659539	−0.329770	+	0.944061i	$0.606971\pi$
$702$	0	0
$703$	8.03767	0.303146
$704$	0	0
$705$	−6.57171	−0.247505
$706$	0	0
$707$	−5.08589	−0.191275
$708$	0	0
$709$	−32.6051	−1.22451	−0.612256	−	0.790660i	$-0.709738\pi$
$709$	−32.6051	−1.22451	−0.612256	+	0.790660i	$0.709738\pi$
$710$	0	0
$711$	−25.8183	−0.968261
$712$	0	0
$713$	−24.1871	−0.905814
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−68.1508	−2.54514
$718$	0	0
$719$	34.9849	1.30472	0.652359	−	0.757910i	$-0.273780\pi$
$719$	34.9849	1.30472	0.652359	+	0.757910i	$0.273780\pi$
$720$	0	0
$721$	2.83347	0.105524
$722$	0	0
$723$	−73.4348	−2.73107
$724$	0	0
$725$	3.86773	0.143644
$726$	0	0
$727$	16.2644	0.603214	0.301607	−	0.953432i	$-0.402477\pi$
$727$	16.2644	0.603214	0.301607	+	0.953432i	$0.402477\pi$
$728$	0	0
$729$	−18.3806	−0.680762
$730$	0	0
$731$	−14.9835	−0.554184
$732$	0	0
$733$	−45.0811	−1.66511	−0.832553	−	0.553945i	$-0.813122\pi$
$733$	−45.0811	−1.66511	−0.832553	+	0.553945i	$0.813122\pi$
$734$	0	0
$735$	37.3245	1.37674
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−14.6482	−0.538843	−0.269421	−	0.963022i	$-0.586832\pi$
$739$	−14.6482	−0.538843	−0.269421	+	0.963022i	$0.586832\pi$
$740$	0	0
$741$	78.0585	2.86755
$742$	0	0
$743$	−4.23352	−0.155313	−0.0776564	−	0.996980i	$-0.524744\pi$
$743$	−4.23352	−0.155313	−0.0776564	+	0.996980i	$0.524744\pi$
$744$	0	0
$745$	−18.1795	−0.666045
$746$	0	0
$747$	5.14168	0.188124
$748$	0	0
$749$	−1.61511	−0.0590148
$750$	0	0
$751$	22.8387	0.833397	0.416698	−	0.909045i	$-0.363187\pi$
$751$	22.8387	0.833397	0.416698	+	0.909045i	$0.363187\pi$
$752$	0	0
$753$	−21.1174	−0.769562
$754$	0	0
$755$	7.81994	0.284597
$756$	0	0
$757$	30.9973	1.12661	0.563307	−	0.826248i	$-0.309529\pi$
$757$	30.9973	1.12661	0.563307	+	0.826248i	$0.309529\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.9137	−1.04812	−0.524061	−	0.851681i	$-0.675584\pi$
$761$	−28.9137	−1.04812	−0.524061	+	0.851681i	$0.675584\pi$
$762$	0	0
$763$	−7.29389	−0.264056
$764$	0	0
$765$	−42.6834	−1.54322
$766$	0	0
$767$	−17.1437	−0.619023
$768$	0	0
$769$	5.37741	0.193914	0.0969571	−	0.995289i	$-0.469089\pi$
$769$	5.37741	0.193914	0.0969571	+	0.995289i	$0.469089\pi$
$770$	0	0
$771$	−44.6440	−1.60782
$772$	0	0
$773$	−44.5996	−1.60414	−0.802068	−	0.597233i	$-0.796267\pi$
$773$	−44.5996	−1.60414	−0.802068	+	0.597233i	$0.796267\pi$
$774$	0	0
$775$	6.30471	0.226472
$776$	0	0
$777$	−1.87222	−0.0671657
$778$	0	0
$779$	−6.84937	−0.245404
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−21.1952	−0.757455
$784$	0	0
$785$	39.5222	1.41061
$786$	0	0
$787$	−41.6282	−1.48389	−0.741943	−	0.670463i	$-0.766095\pi$
$787$	−41.6282	−1.48389	−0.741943	+	0.670463i	$0.766095\pi$
$788$	0	0
$789$	70.0355	2.49333
$790$	0	0
$791$	−3.10871	−0.110533
$792$	0	0
$793$	42.1947	1.49838
$794$	0	0
$795$	43.0304	1.52613
$796$	0	0
$797$	5.04266	0.178620	0.0893100	−	0.996004i	$-0.471534\pi$
$797$	5.04266	0.178620	0.0893100	+	0.996004i	$0.471534\pi$
$798$	0	0
$799$	4.61555	0.163286
$800$	0	0
$801$	−12.1246	−0.428400
$802$	0	0
$803$	0	0
$804$	0	0
$805$	4.40485	0.155251
$806$	0	0
$807$	16.9030	0.595013
$808$	0	0
$809$	28.0083	0.984718	0.492359	−	0.870392i	$-0.336135\pi$
$809$	28.0083	0.984718	0.492359	+	0.870392i	$0.336135\pi$
$810$	0	0
$811$	−33.3989	−1.17279	−0.586397	−	0.810024i	$-0.699454\pi$
$811$	−33.3989	−1.17279	−0.586397	+	0.810024i	$0.699454\pi$
$812$	0	0
$813$	41.2396	1.44634
$814$	0	0
$815$	−35.1552	−1.23143
$816$	0	0
$817$	−18.1479	−0.634914
$818$	0	0
$819$	−12.3062	−0.430013
$820$	0	0
$821$	−30.3529	−1.05932	−0.529661	−	0.848209i	$-0.677681\pi$
$821$	−30.3529	−1.05932	−0.529661	+	0.848209i	$0.677681\pi$
$822$	0	0
$823$	−6.75910	−0.235607	−0.117804	−	0.993037i	$-0.537585\pi$
$823$	−6.75910	−0.235607	−0.117804	+	0.993037i	$0.537585\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.8311	−1.14165	−0.570825	−	0.821072i	$-0.693377\pi$
$827$	−32.8311	−1.14165	−0.570825	+	0.821072i	$0.693377\pi$
$828$	0	0
$829$	−15.4529	−0.536701	−0.268350	−	0.963321i	$-0.586478\pi$
$829$	−15.4529	−0.536701	−0.268350	+	0.963321i	$0.586478\pi$
$830$	0	0
$831$	9.01860	0.312852
$832$	0	0
$833$	−26.2143	−0.908273
$834$	0	0
$835$	−1.76202	−0.0609771
$836$	0	0
$837$	−34.5499	−1.19422
$838$	0	0
$839$	−27.9580	−0.965219	−0.482609	−	0.875836i	$-0.660311\pi$
$839$	−27.9580	−0.965219	−0.482609	+	0.875836i	$0.660311\pi$
$840$	0	0
$841$	−24.5095	−0.845155
$842$	0	0
$843$	−27.3082	−0.940544
$844$	0	0
$845$	−31.6774	−1.08974
$846$	0	0
$847$	0	0
$848$	0	0
$849$	54.9658	1.88642
$850$	0	0
$851$	12.1871	0.417769
$852$	0	0
$853$	−14.0787	−0.482045	−0.241023	−	0.970520i	$-0.577483\pi$
$853$	−14.0787	−0.482045	−0.241023	+	0.970520i	$0.577483\pi$
$854$	0	0
$855$	−51.6978	−1.76803
$856$	0	0
$857$	26.4776	0.904457	0.452228	−	0.891902i	$-0.350629\pi$
$857$	26.4776	0.904457	0.452228	+	0.891902i	$0.350629\pi$
$858$	0	0
$859$	−47.5614	−1.62277	−0.811387	−	0.584510i	$-0.801287\pi$
$859$	−47.5614	−1.62277	−0.811387	+	0.584510i	$0.801287\pi$
$860$	0	0
$861$	1.59543	0.0543722
$862$	0	0
$863$	−43.4169	−1.47793	−0.738964	−	0.673745i	$-0.764684\pi$
$863$	−43.4169	−1.47793	−0.738964	+	0.673745i	$0.764684\pi$
$864$	0	0
$865$	−31.1810	−1.06019
$866$	0	0
$867$	−7.50290	−0.254812
$868$	0	0
$869$	0	0
$870$	0	0
$871$	6.51045	0.220598
$872$	0	0
$873$	−59.5463	−2.01534
$874$	0	0
$875$	−4.29357	−0.145149
$876$	0	0
$877$	−13.1347	−0.443527	−0.221764	−	0.975100i	$-0.571181\pi$
$877$	−13.1347	−0.443527	−0.221764	+	0.975100i	$0.571181\pi$
$878$	0	0
$879$	−84.2128	−2.84043
$880$	0	0
$881$	−36.3388	−1.22429	−0.612143	−	0.790747i	$-0.709692\pi$
$881$	−36.3388	−1.22429	−0.612143	+	0.790747i	$0.709692\pi$
$882$	0	0
$883$	33.6371	1.13198	0.565989	−	0.824413i	$-0.308495\pi$
$883$	33.6371	1.13198	0.565989	+	0.824413i	$0.308495\pi$
$884$	0	0
$885$	16.7757	0.563910
$886$	0	0
$887$	35.0138	1.17565	0.587824	−	0.808989i	$-0.299985\pi$
$887$	35.0138	1.17565	0.587824	+	0.808989i	$0.299985\pi$
$888$	0	0
$889$	6.28185	0.210686
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.59032	0.187073
$894$	0	0
$895$	5.08171	0.169863
$896$	0	0
$897$	118.356	3.95180
$898$	0	0
$899$	7.31987	0.244131
$900$	0	0
$901$	−30.2218	−1.00683
$902$	0	0
$903$	4.22721	0.140673
$904$	0	0
$905$	−3.74635	−0.124533
$906$	0	0
$907$	51.8416	1.72137	0.860686	−	0.509137i	$-0.170035\pi$
$907$	51.8416	1.72137	0.860686	+	0.509137i	$0.170035\pi$
$908$	0	0
$909$	90.5063	3.00190
$910$	0	0
$911$	29.6304	0.981700	0.490850	−	0.871244i	$-0.336686\pi$
$911$	29.6304	0.981700	0.490850	+	0.871244i	$0.336686\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−41.2891	−1.36497
$916$	0	0
$917$	−6.48160	−0.214041
$918$	0	0
$919$	30.1065	0.993123	0.496562	−	0.868001i	$-0.334596\pi$
$919$	30.1065	0.993123	0.496562	+	0.868001i	$0.334596\pi$
$920$	0	0
$921$	2.01747	0.0664779
$922$	0	0
$923$	13.1460	0.432704
$924$	0	0
$925$	−3.17674	−0.104451
$926$	0	0
$927$	−50.4231	−1.65611
$928$	0	0
$929$	−51.0725	−1.67564	−0.837818	−	0.545950i	$-0.816169\pi$
$929$	−51.0725	−1.67564	−0.837818	+	0.545950i	$0.816169\pi$
$930$	0	0
$931$	−31.7506	−1.04058
$932$	0	0
$933$	39.8217	1.30370
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−8.35131	−0.272825	−0.136413	−	0.990652i	$-0.543557\pi$
$937$	−8.35131	−0.272825	−0.136413	+	0.990652i	$0.543557\pi$
$938$	0	0
$939$	60.3652	1.96994
$940$	0	0
$941$	−53.3937	−1.74058	−0.870292	−	0.492536i	$-0.836070\pi$
$941$	−53.3937	−1.74058	−0.870292	+	0.492536i	$0.836070\pi$
$942$	0	0
$943$	−10.3853	−0.338193
$944$	0	0
$945$	6.29208	0.204681
$946$	0	0
$947$	−19.2532	−0.625646	−0.312823	−	0.949811i	$-0.601275\pi$
$947$	−19.2532	−0.625646	−0.312823	+	0.949811i	$0.601275\pi$
$948$	0	0
$949$	21.6601	0.703118
$950$	0	0
$951$	60.1406	1.95019
$952$	0	0
$953$	−36.6494	−1.18719	−0.593595	−	0.804764i	$-0.702292\pi$
$953$	−36.6494	−1.18719	−0.593595	+	0.804764i	$0.702292\pi$
$954$	0	0
$955$	−30.5065	−0.987167
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.75836	0.153656
$960$	0	0
$961$	−19.0680	−0.615098
$962$	0	0
$963$	28.7417	0.926190
$964$	0	0
$965$	−35.8655	−1.15455
$966$	0	0
$967$	45.6501	1.46801	0.734004	−	0.679146i	$-0.237649\pi$
$967$	45.6501	1.46801	0.734004	+	0.679146i	$0.237649\pi$
$968$	0	0
$969$	53.6465	1.72337
$970$	0	0
$971$	46.1836	1.48210	0.741051	−	0.671449i	$-0.234328\pi$
$971$	46.1836	1.48210	0.741051	+	0.671449i	$0.234328\pi$
$972$	0	0
$973$	−5.03828	−0.161520
$974$	0	0
$975$	−30.8512	−0.988028
$976$	0	0
$977$	53.3452	1.70666	0.853332	−	0.521368i	$-0.174578\pi$
$977$	53.3452	1.70666	0.853332	+	0.521368i	$0.174578\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	129.799	4.14416
$982$	0	0
$983$	−12.0857	−0.385475	−0.192738	−	0.981250i	$-0.561737\pi$
$983$	−12.0857	−0.385475	−0.192738	+	0.981250i	$0.561737\pi$
$984$	0	0
$985$	22.3775	0.713006
$986$	0	0
$987$	−1.30216	−0.0414482
$988$	0	0
$989$	−27.5167	−0.874981
$990$	0	0
$991$	36.0620	1.14555	0.572774	−	0.819714i	$-0.305867\pi$
$991$	36.0620	1.14555	0.572774	+	0.819714i	$0.305867\pi$
$992$	0	0
$993$	64.2442	2.03873
$994$	0	0
$995$	−5.93271	−0.188080
$996$	0	0
$997$	9.32212	0.295234	0.147617	−	0.989045i	$-0.452840\pi$
$997$	9.32212	0.295234	0.147617	+	0.989045i	$0.452840\pi$
$998$	0	0
$999$	17.4086	0.550783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.dr.1.4		4
4.3	odd	2		7744.2.a.di.1.1		4
8.3	odd	2		968.2.a.n.1.4		4
8.5	even	2		1936.2.a.bb.1.1		4
11.5	even	5		704.2.m.i.641.2		8
11.9	even	5		704.2.m.i.257.2		8
11.10	odd	2		7744.2.a.ds.1.4		4
24.11	even	2		8712.2.a.ce.1.2		4
44.27	odd	10		704.2.m.l.641.1		8
44.31	odd	10		704.2.m.l.257.1		8
44.43	even	2		7744.2.a.dh.1.1		4
88.3	odd	10		968.2.i.s.9.1		8
88.5	even	10		176.2.m.d.113.1		8
88.19	even	10		968.2.i.t.9.1		8
88.21	odd	2		1936.2.a.bc.1.1		4
88.27	odd	10		88.2.i.b.25.2	✓	8
88.35	even	10		968.2.i.p.81.2		8
88.43	even	2		968.2.a.m.1.4		4
88.51	even	10		968.2.i.t.753.1		8
88.53	even	10		176.2.m.d.81.1		8
88.59	odd	10		968.2.i.s.753.1		8
88.75	odd	10		88.2.i.b.81.2	yes	8
88.83	even	10		968.2.i.p.729.2		8
264.131	odd	2		8712.2.a.cd.1.2		4
264.203	even	10		792.2.r.g.289.2		8
264.251	even	10		792.2.r.g.433.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.i.b.25.2	✓	8	88.27	odd	10
88.2.i.b.81.2	yes	8	88.75	odd	10
176.2.m.d.81.1		8	88.53	even	10
176.2.m.d.113.1		8	88.5	even	10
704.2.m.i.257.2		8	11.9	even	5
704.2.m.i.641.2		8	11.5	even	5
704.2.m.l.257.1		8	44.31	odd	10
704.2.m.l.641.1		8	44.27	odd	10
792.2.r.g.289.2		8	264.203	even	10
792.2.r.g.433.2		8	264.251	even	10
968.2.a.m.1.4		4	88.43	even	2
968.2.a.n.1.4		4	8.3	odd	2
968.2.i.p.81.2		8	88.35	even	10
968.2.i.p.729.2		8	88.83	even	10
968.2.i.s.9.1		8	88.3	odd	10
968.2.i.s.753.1		8	88.59	odd	10
968.2.i.t.9.1		8	88.19	even	10
968.2.i.t.753.1		8	88.51	even	10
1936.2.a.bb.1.1		4	8.5	even	2
1936.2.a.bc.1.1		4	88.21	odd	2
7744.2.a.dh.1.1		4	44.43	even	2
7744.2.a.di.1.1		4	4.3	odd	2
7744.2.a.dr.1.4		4	1.1	even	1	trivial
7744.2.a.ds.1.4		4	11.10	odd	2
8712.2.a.cd.1.2		4	264.131	odd	2
8712.2.a.ce.1.2		4	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04678 q^{3} -1.78180 q^{5} -0.353057 q^{7} +6.28284 q^{9} +5.54782 q^{13} -5.42874 q^{15} +3.81280 q^{17} +4.61803 q^{19} -1.07569 q^{21} +7.00209 q^{23} -1.82519 q^{25} +10.0021 q^{27} -2.11908 q^{29} -3.45427 q^{31} +0.629076 q^{35} +1.74050 q^{37} +16.9030 q^{39} -1.48318 q^{41} -3.92979 q^{43} -11.1948 q^{45} +1.21054 q^{47} -6.87535 q^{49} +11.6167 q^{51} -7.92641 q^{53} +14.0701 q^{57} -3.09017 q^{59} +7.60564 q^{61} -2.21820 q^{63} -9.88510 q^{65} +1.17352 q^{67} +21.3338 q^{69} +2.36957 q^{71} +3.90426 q^{73} -5.56095 q^{75} -4.10933 q^{79} +11.6256 q^{81} +0.818368 q^{83} -6.79364 q^{85} -6.45636 q^{87} -1.92979 q^{89} -1.95870 q^{91} -10.5244 q^{93} -8.22841 q^{95} -9.47761 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - q^{13} - 16 q^{15} - 12 q^{17} + 14 q^{19} + q^{21} + 2 q^{23} + 11 q^{25} + 14 q^{27} + 9 q^{29} - 11 q^{31} + 18 q^{35} - 13 q^{37} + 18 q^{39} - 8 q^{41}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - q^5 - q^7 + 6 * q^9 - q^13 - 16 * q^15 - 12 * q^17 + 14 * q^19 + q^21 + 2 * q^23 + 11 * q^25 + 14 * q^27 + 9 * q^29 - 11 * q^31 + 18 * q^35 - 13 * q^37 + 18 * q^39 - 8 * q^41 + 3 * q^43 - 22 * q^45 - 7 * q^47 - q^49 + 14 * q^51 - 11 * q^53 + 7 * q^57 + 10 * q^59 + 17 * q^61 - 15 * q^63 - 5 * q^65 - 5 * q^67 + 4 * q^69 + 5 * q^71 - 6 * q^73 + 7 * q^75 - 7 * q^79 - 8 * q^81 + 20 * q^83 + 13 * q^85 + 3 * q^87 + 11 * q^89 + 6 * q^91 + 10 * q^93 - 6 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more