LMFDB - Embedded newform 5488.2.a.p.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5488,2,Mod(1,5488)] 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("5488.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5488, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5488 = 2^{4} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5488.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,5,0,-11,0,0,0,9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.8219006293$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1279733.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 $ x^6 - 2x^5 - 6x^4 + 10x^3 + 10x^2 - 11*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 343)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.33192$ of defining polynomial
Character	$\chi$	$=$	5488.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.95499 q^{3} -1.95286 q^{5} +5.73194 q^{9} -2.65101 q^{11} +1.50061 q^{13} -5.77069 q^{15} -2.09215 q^{17} -6.38881 q^{19} +2.12424 q^{23} -1.18632 q^{25} +8.07285 q^{27} -7.42740 q^{29} -0.0654672 q^{31} -7.83370 q^{33} +0.537796 q^{37} +4.43428 q^{39} -11.8724 q^{41} +0.588754 q^{43} -11.1937 q^{45} +3.02880 q^{47} -6.18227 q^{51} +1.04162 q^{53} +5.17707 q^{55} -18.8788 q^{57} -2.27140 q^{59} +10.2767 q^{61} -2.93048 q^{65} +0.221714 q^{67} +6.27711 q^{69} -10.7708 q^{71} -10.2207 q^{73} -3.50556 q^{75} +13.2859 q^{79} +6.65934 q^{81} -11.8543 q^{83} +4.08568 q^{85} -21.9479 q^{87} -3.51020 q^{89} -0.193455 q^{93} +12.4765 q^{95} -14.6497 q^{97} -15.1954 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} - 11 q^{5} + 9 q^{9} + q^{11} - 7 q^{13} + 5 q^{15} - 26 q^{17} - 3 q^{19} + 9 q^{23} + 15 q^{25} + 8 q^{27} - 2 q^{29} - 2 q^{31} - 18 q^{33} - 2 q^{37} - 7 q^{39} - 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 - 11 * q^5 + 9 * q^9 + q^11 - 7 * q^13 + 5 * q^15 - 26 * q^17 - 3 * q^19 + 9 * q^23 + 15 * q^25 + 8 * q^27 - 2 * q^29 - 2 * q^31 - 18 * q^33 - 2 * q^37 - 7 * q^39 - 28 * q^41 - 5 * q^43 - 3 * q^45 + 18 * q^47 - 13 * q^51 - q^53 - 22 * q^55 - 26 * q^57 + 5 * q^59 + 17 * q^61 - 14 * q^65 + 22 * q^67 + 20 * q^69 + 2 * q^71 - 12 * q^73 - 27 * q^75 + 2 * q^79 - 30 * q^81 + 7 * q^83 + 37 * q^85 - 25 * q^87 - 39 * q^89 - 22 * q^93 + 11 * q^95 - 35 * q^97 - 12 * q^99

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
$3$	2.95499	1.70606	0.853031	−	0.521860i	$-0.174762\pi$
$3$	2.95499	1.70606	0.853031	+	0.521860i	$0.174762\pi$
$4$	0	0
$5$	−1.95286	−0.873348	−0.436674	−	0.899620i	$-0.643844\pi$
$5$	−1.95286	−0.873348	−0.436674	+	0.899620i	$0.643844\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.73194	1.91065
$10$	0	0
$11$	−2.65101	−0.799310	−0.399655	−	0.916666i	$-0.630870\pi$
$11$	−2.65101	−0.799310	−0.399655	+	0.916666i	$0.630870\pi$
$12$	0	0
$13$	1.50061	0.416194	0.208097	−	0.978108i	$-0.433273\pi$
$13$	1.50061	0.416194	0.208097	+	0.978108i	$0.433273\pi$
$14$	0	0
$15$	−5.77069	−1.48999
$16$	0	0
$17$	−2.09215	−0.507421	−0.253710	−	0.967280i	$-0.581651\pi$
$17$	−2.09215	−0.507421	−0.253710	+	0.967280i	$0.581651\pi$
$18$	0	0
$19$	−6.38881	−1.46569	−0.732847	−	0.680394i	$-0.761809\pi$
$19$	−6.38881	−1.46569	−0.732847	+	0.680394i	$0.761809\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.12424	0.442936	0.221468	−	0.975168i	$-0.428915\pi$
$23$	2.12424	0.442936	0.221468	+	0.975168i	$0.428915\pi$
$24$	0	0
$25$	−1.18632	−0.237264
$26$	0	0
$27$	8.07285	1.55362
$28$	0	0
$29$	−7.42740	−1.37923	−0.689617	−	0.724174i	$-0.742221\pi$
$29$	−7.42740	−1.37923	−0.689617	+	0.724174i	$0.742221\pi$
$30$	0	0
$31$	−0.0654672	−0.0117582	−0.00587912	−	0.999983i	$-0.501871\pi$
$31$	−0.0654672	−0.0117582	−0.00587912	+	0.999983i	$0.501871\pi$
$32$	0	0
$33$	−7.83370	−1.36367
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.537796	0.0884130	0.0442065	−	0.999022i	$-0.485924\pi$
$37$	0.537796	0.0884130	0.0442065	+	0.999022i	$0.485924\pi$
$38$	0	0
$39$	4.43428	0.710053
$40$	0	0
$41$	−11.8724	−1.85416	−0.927082	−	0.374858i	$-0.877691\pi$
$41$	−11.8724	−1.85416	−0.927082	+	0.374858i	$0.877691\pi$
$42$	0	0
$43$	0.588754	0.0897842	0.0448921	−	0.998992i	$-0.485706\pi$
$43$	0.588754	0.0897842	0.0448921	+	0.998992i	$0.485706\pi$
$44$	0	0
$45$	−11.1937	−1.66866
$46$	0	0
$47$	3.02880	0.441796	0.220898	−	0.975297i	$-0.429101\pi$
$47$	3.02880	0.441796	0.220898	+	0.975297i	$0.429101\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.18227	−0.865691
$52$	0	0
$53$	1.04162	0.143078	0.0715390	−	0.997438i	$-0.477209\pi$
$53$	1.04162	0.143078	0.0715390	+	0.997438i	$0.477209\pi$
$54$	0	0
$55$	5.17707	0.698075
$56$	0	0
$57$	−18.8788	−2.50056
$58$	0	0
$59$	−2.27140	−0.295711	−0.147855	−	0.989009i	$-0.547237\pi$
$59$	−2.27140	−0.295711	−0.147855	+	0.989009i	$0.547237\pi$
$60$	0	0
$61$	10.2767	1.31580	0.657898	−	0.753107i	$-0.271446\pi$
$61$	10.2767	1.31580	0.657898	+	0.753107i	$0.271446\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.93048	−0.363482
$66$	0	0
$67$	0.221714	0.0270867	0.0135434	−	0.999908i	$-0.495689\pi$
$67$	0.221714	0.0270867	0.0135434	+	0.999908i	$0.495689\pi$
$68$	0	0
$69$	6.27711	0.755676
$70$	0	0
$71$	−10.7708	−1.27826	−0.639131	−	0.769098i	$-0.720706\pi$
$71$	−10.7708	−1.27826	−0.639131	+	0.769098i	$0.720706\pi$
$72$	0	0
$73$	−10.2207	−1.19624	−0.598122	−	0.801405i	$-0.704086\pi$
$73$	−10.2207	−1.19624	−0.598122	+	0.801405i	$0.704086\pi$
$74$	0	0
$75$	−3.50556	−0.404787
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.2859	1.49478	0.747392	−	0.664383i	$-0.231306\pi$
$79$	13.2859	1.49478	0.747392	+	0.664383i	$0.231306\pi$
$80$	0	0
$81$	6.65934	0.739927
$82$	0	0
$83$	−11.8543	−1.30118	−0.650592	−	0.759428i	$-0.725479\pi$
$83$	−11.8543	−1.30118	−0.650592	+	0.759428i	$0.725479\pi$
$84$	0	0
$85$	4.08568	0.443155
$86$	0	0
$87$	−21.9479	−2.35306
$88$	0	0
$89$	−3.51020	−0.372080	−0.186040	−	0.982542i	$-0.559565\pi$
$89$	−3.51020	−0.372080	−0.186040	+	0.982542i	$0.559565\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.193455	−0.0200603
$94$	0	0
$95$	12.4765	1.28006
$96$	0	0
$97$	−14.6497	−1.48746	−0.743728	−	0.668483i	$-0.766944\pi$
$97$	−14.6497	−1.48746	−0.743728	+	0.668483i	$0.766944\pi$
$98$	0	0
$99$	−15.1954	−1.52720
$100$	0	0
$101$	−0.324269	−0.0322659	−0.0161330	−	0.999870i	$-0.505136\pi$
$101$	−0.324269	−0.0322659	−0.0161330	+	0.999870i	$0.505136\pi$
$102$	0	0
$103$	10.1971	1.00475	0.502377	−	0.864648i	$-0.332459\pi$
$103$	10.1971	1.00475	0.502377	+	0.864648i	$0.332459\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.3684	1.48572	0.742861	−	0.669446i	$-0.233468\pi$
$107$	15.3684	1.48572	0.742861	+	0.669446i	$0.233468\pi$
$108$	0	0
$109$	−14.2387	−1.36382	−0.681911	−	0.731435i	$-0.738851\pi$
$109$	−14.2387	−1.36382	−0.681911	+	0.731435i	$0.738851\pi$
$110$	0	0
$111$	1.58918	0.150838
$112$	0	0
$113$	−8.22812	−0.774037	−0.387018	−	0.922072i	$-0.626495\pi$
$113$	−8.22812	−0.774037	−0.387018	+	0.922072i	$0.626495\pi$
$114$	0	0
$115$	−4.14836	−0.386837
$116$	0	0
$117$	8.60140	0.795200
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.97214	−0.361104
$122$	0	0
$123$	−35.0829	−3.16332
$124$	0	0
$125$	12.0810	1.08056
$126$	0	0
$127$	−15.7450	−1.39714	−0.698572	−	0.715540i	$-0.746181\pi$
$127$	−15.7450	−1.39714	−0.698572	+	0.715540i	$0.746181\pi$
$128$	0	0
$129$	1.73976	0.153177
$130$	0	0
$131$	8.25951	0.721636	0.360818	−	0.932636i	$-0.382497\pi$
$131$	8.25951	0.721636	0.360818	+	0.932636i	$0.382497\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−15.7652	−1.35685
$136$	0	0
$137$	0.0780530	0.00666851	0.00333426	−	0.999994i	$-0.498939\pi$
$137$	0.0780530	0.00666851	0.00333426	+	0.999994i	$0.498939\pi$
$138$	0	0
$139$	15.6375	1.32635	0.663177	−	0.748463i	$-0.269208\pi$
$139$	15.6375	1.32635	0.663177	+	0.748463i	$0.269208\pi$
$140$	0	0
$141$	8.95007	0.753732
$142$	0	0
$143$	−3.97813	−0.332668
$144$	0	0
$145$	14.5047	1.20455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.26566	−0.513303	−0.256652	−	0.966504i	$-0.582619\pi$
$149$	−6.26566	−0.513303	−0.256652	+	0.966504i	$0.582619\pi$
$150$	0	0
$151$	16.3222	1.32828	0.664142	−	0.747606i	$-0.268797\pi$
$151$	16.3222	1.32828	0.664142	+	0.747606i	$0.268797\pi$
$152$	0	0
$153$	−11.9921	−0.969502
$154$	0	0
$155$	0.127848	0.0102690
$156$	0	0
$157$	13.3664	1.06675	0.533376	−	0.845878i	$-0.320923\pi$
$157$	13.3664	1.06675	0.533376	+	0.845878i	$0.320923\pi$
$158$	0	0
$159$	3.07798	0.244100
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.1164	0.870700	0.435350	−	0.900261i	$-0.356625\pi$
$163$	11.1164	0.870700	0.435350	+	0.900261i	$0.356625\pi$
$164$	0	0
$165$	15.2982	1.19096
$166$	0	0
$167$	−8.91212	−0.689640	−0.344820	−	0.938669i	$-0.612060\pi$
$167$	−8.91212	−0.689640	−0.344820	+	0.938669i	$0.612060\pi$
$168$	0	0
$169$	−10.7482	−0.826783
$170$	0	0
$171$	−36.6203	−2.80042
$172$	0	0
$173$	−3.24509	−0.246720	−0.123360	−	0.992362i	$-0.539367\pi$
$173$	−3.24509	−0.246720	−0.123360	+	0.992362i	$0.539367\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.71194	−0.504501
$178$	0	0
$179$	4.04711	0.302495	0.151248	−	0.988496i	$-0.451671\pi$
$179$	4.04711	0.302495	0.151248	+	0.988496i	$0.451671\pi$
$180$	0	0
$181$	−8.66577	−0.644121	−0.322061	−	0.946719i	$-0.604376\pi$
$181$	−8.66577	−0.644121	−0.322061	+	0.946719i	$0.604376\pi$
$182$	0	0
$183$	30.3675	2.24483
$184$	0	0
$185$	−1.05024	−0.0772153
$186$	0	0
$187$	5.54631	0.405586
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.2165	−1.10102	−0.550512	−	0.834827i	$-0.685567\pi$
$191$	−15.2165	−1.10102	−0.550512	+	0.834827i	$0.685567\pi$
$192$	0	0
$193$	10.2716	0.739364	0.369682	−	0.929158i	$-0.379467\pi$
$193$	10.2716	0.739364	0.369682	+	0.929158i	$0.379467\pi$
$194$	0	0
$195$	−8.65954	−0.620123
$196$	0	0
$197$	7.72522	0.550399	0.275200	−	0.961387i	$-0.411256\pi$
$197$	7.72522	0.550399	0.275200	+	0.961387i	$0.411256\pi$
$198$	0	0
$199$	−14.7692	−1.04696	−0.523482	−	0.852037i	$-0.675367\pi$
$199$	−14.7692	−1.04696	−0.523482	+	0.852037i	$0.675367\pi$
$200$	0	0
$201$	0.655163	0.0462116
$202$	0	0
$203$	0	0
$204$	0	0
$205$	23.1853	1.61933
$206$	0	0
$207$	12.1760	0.846294
$208$	0	0
$209$	16.9368	1.17154
$210$	0	0
$211$	4.88834	0.336527	0.168264	−	0.985742i	$-0.446184\pi$
$211$	4.88834	0.336527	0.168264	+	0.985742i	$0.446184\pi$
$212$	0	0
$213$	−31.8276	−2.18079
$214$	0	0
$215$	−1.14976	−0.0784128
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−30.2021	−2.04087
$220$	0	0
$221$	−3.13950	−0.211185
$222$	0	0
$223$	1.74696	0.116985	0.0584925	−	0.998288i	$-0.481371\pi$
$223$	1.74696	0.116985	0.0584925	+	0.998288i	$0.481371\pi$
$224$	0	0
$225$	−6.79992	−0.453328
$226$	0	0
$227$	23.6939	1.57262	0.786311	−	0.617831i	$-0.211988\pi$
$227$	23.6939	1.57262	0.786311	+	0.617831i	$0.211988\pi$
$228$	0	0
$229$	−13.1910	−0.871685	−0.435843	−	0.900023i	$-0.643550\pi$
$229$	−13.1910	−0.871685	−0.435843	+	0.900023i	$0.643550\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.2901	1.39476	0.697379	−	0.716703i	$-0.254349\pi$
$233$	21.2901	1.39476	0.697379	+	0.716703i	$0.254349\pi$
$234$	0	0
$235$	−5.91484	−0.385842
$236$	0	0
$237$	39.2598	2.55020
$238$	0	0
$239$	11.2612	0.728429	0.364215	−	0.931315i	$-0.381337\pi$
$239$	11.2612	0.728429	0.364215	+	0.931315i	$0.381337\pi$
$240$	0	0
$241$	−9.26049	−0.596521	−0.298260	−	0.954485i	$-0.596406\pi$
$241$	−9.26049	−0.596521	−0.298260	+	0.954485i	$0.596406\pi$
$242$	0	0
$243$	−4.54030	−0.291260
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.58710	−0.610013
$248$	0	0
$249$	−35.0294	−2.21990
$250$	0	0
$251$	−19.8599	−1.25355	−0.626773	−	0.779202i	$-0.715625\pi$
$251$	−19.8599	−1.25355	−0.626773	+	0.779202i	$0.715625\pi$
$252$	0	0
$253$	−5.63140	−0.354043
$254$	0	0
$255$	12.0731	0.756049
$256$	0	0
$257$	−25.1574	−1.56927	−0.784637	−	0.619955i	$-0.787151\pi$
$257$	−25.1574	−1.56927	−0.784637	+	0.619955i	$0.787151\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−42.5735	−2.63523
$262$	0	0
$263$	10.3286	0.636887	0.318444	−	0.947942i	$-0.396840\pi$
$263$	10.3286	0.636887	0.318444	+	0.947942i	$0.396840\pi$
$264$	0	0
$265$	−2.03415	−0.124957
$266$	0	0
$267$	−10.3726	−0.634792
$268$	0	0
$269$	0.448329	0.0273351	0.0136676	−	0.999907i	$-0.495649\pi$
$269$	0.448329	0.0273351	0.0136676	+	0.999907i	$0.495649\pi$
$270$	0	0
$271$	−23.9630	−1.45565	−0.727825	−	0.685763i	$-0.759469\pi$
$271$	−23.9630	−1.45565	−0.727825	+	0.685763i	$0.759469\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.14495	0.189648
$276$	0	0
$277$	15.1757	0.911822	0.455911	−	0.890025i	$-0.349314\pi$
$277$	15.1757	0.911822	0.455911	+	0.890025i	$0.349314\pi$
$278$	0	0
$279$	−0.375254	−0.0224659
$280$	0	0
$281$	10.0320	0.598460	0.299230	−	0.954181i	$-0.403270\pi$
$281$	10.0320	0.598460	0.299230	+	0.954181i	$0.403270\pi$
$282$	0	0
$283$	−2.10305	−0.125014	−0.0625068	−	0.998045i	$-0.519910\pi$
$283$	−2.10305	−0.125014	−0.0625068	+	0.998045i	$0.519910\pi$
$284$	0	0
$285$	36.8678	2.18386
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.6229	−0.742524
$290$	0	0
$291$	−43.2898	−2.53769
$292$	0	0
$293$	−18.6487	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$293$	−18.6487	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$294$	0	0
$295$	4.43573	0.258258
$296$	0	0
$297$	−21.4012	−1.24183
$298$	0	0
$299$	3.18766	0.184347
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.958209	−0.0550477
$304$	0	0
$305$	−20.0690	−1.14915
$306$	0	0
$307$	9.72652	0.555122	0.277561	−	0.960708i	$-0.410474\pi$
$307$	9.72652	0.555122	0.277561	+	0.960708i	$0.410474\pi$
$308$	0	0
$309$	30.1324	1.71417
$310$	0	0
$311$	−27.8427	−1.57881	−0.789406	−	0.613871i	$-0.789611\pi$
$311$	−27.8427	−1.57881	−0.789406	+	0.613871i	$0.789611\pi$
$312$	0	0
$313$	−20.2150	−1.14262	−0.571311	−	0.820734i	$-0.693565\pi$
$313$	−20.2150	−1.14262	−0.571311	+	0.820734i	$0.693565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.8535	0.665760	0.332880	−	0.942969i	$-0.391980\pi$
$317$	11.8535	0.665760	0.332880	+	0.942969i	$0.391980\pi$
$318$	0	0
$319$	19.6901	1.10244
$320$	0	0
$321$	45.4135	2.53473
$322$	0	0
$323$	13.3663	0.743723
$324$	0	0
$325$	−1.78020	−0.0987479
$326$	0	0
$327$	−42.0752	−2.32677
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.6856	−0.697261	−0.348631	−	0.937260i	$-0.613353\pi$
$331$	−12.6856	−0.697261	−0.348631	+	0.937260i	$0.613353\pi$
$332$	0	0
$333$	3.08261	0.168926
$334$	0	0
$335$	−0.432978	−0.0236561
$336$	0	0
$337$	28.8310	1.57053	0.785264	−	0.619162i	$-0.212527\pi$
$337$	28.8310	1.57053	0.785264	+	0.619162i	$0.212527\pi$
$338$	0	0
$339$	−24.3140	−1.32055
$340$	0	0
$341$	0.173554	0.00939848
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−12.2583	−0.659967
$346$	0	0
$347$	−15.5749	−0.836103	−0.418052	−	0.908423i	$-0.637287\pi$
$347$	−15.5749	−0.836103	−0.418052	+	0.908423i	$0.637287\pi$
$348$	0	0
$349$	−9.08484	−0.486300	−0.243150	−	0.969989i	$-0.578181\pi$
$349$	−9.08484	−0.486300	−0.243150	+	0.969989i	$0.578181\pi$
$350$	0	0
$351$	12.1142	0.646608
$352$	0	0
$353$	−22.9922	−1.22375	−0.611876	−	0.790954i	$-0.709585\pi$
$353$	−22.9922	−1.22375	−0.611876	+	0.790954i	$0.709585\pi$
$354$	0	0
$355$	21.0340	1.11637
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.1163	0.745030	0.372515	−	0.928026i	$-0.378496\pi$
$359$	14.1163	0.745030	0.372515	+	0.928026i	$0.378496\pi$
$360$	0	0
$361$	21.8169	1.14826
$362$	0	0
$363$	−11.7376	−0.616065
$364$	0	0
$365$	19.9597	1.04474
$366$	0	0
$367$	13.1892	0.688469	0.344235	−	0.938884i	$-0.388138\pi$
$367$	13.1892	0.688469	0.344235	+	0.938884i	$0.388138\pi$
$368$	0	0
$369$	−68.0522	−3.54265
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9.15428	0.473991	0.236995	−	0.971511i	$-0.423837\pi$
$373$	9.15428	0.473991	0.236995	+	0.971511i	$0.423837\pi$
$374$	0	0
$375$	35.6993	1.84351
$376$	0	0
$377$	−11.1456	−0.574029
$378$	0	0
$379$	−3.07480	−0.157942	−0.0789710	−	0.996877i	$-0.525163\pi$
$379$	−3.07480	−0.157942	−0.0789710	+	0.996877i	$0.525163\pi$
$380$	0	0
$381$	−46.5263	−2.38361
$382$	0	0
$383$	1.09321	0.0558605	0.0279303	−	0.999610i	$-0.491108\pi$
$383$	1.09321	0.0558605	0.0279303	+	0.999610i	$0.491108\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.37471	0.171546
$388$	0	0
$389$	−11.8335	−0.599981	−0.299991	−	0.953942i	$-0.596984\pi$
$389$	−11.8335	−0.599981	−0.299991	+	0.953942i	$0.596984\pi$
$390$	0	0
$391$	−4.44424	−0.224755
$392$	0	0
$393$	24.4067	1.23116
$394$	0	0
$395$	−25.9456	−1.30547
$396$	0	0
$397$	28.2609	1.41838	0.709188	−	0.705020i	$-0.249062\pi$
$397$	28.2609	1.41838	0.709188	+	0.705020i	$0.249062\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.66256	−0.332713	−0.166356	−	0.986066i	$-0.553200\pi$
$401$	−6.66256	−0.332713	−0.166356	+	0.986066i	$0.553200\pi$
$402$	0	0
$403$	−0.0982406	−0.00489371
$404$	0	0
$405$	−13.0048	−0.646213
$406$	0	0
$407$	−1.42570	−0.0706694
$408$	0	0
$409$	−11.7556	−0.581276	−0.290638	−	0.956833i	$-0.593868\pi$
$409$	−11.7556	−0.581276	−0.290638	+	0.956833i	$0.593868\pi$
$410$	0	0
$411$	0.230645	0.0113769
$412$	0	0
$413$	0	0
$414$	0	0
$415$	23.1499	1.13639
$416$	0	0
$417$	46.2085	2.26284
$418$	0	0
$419$	6.77442	0.330952	0.165476	−	0.986214i	$-0.447084\pi$
$419$	6.77442	0.330952	0.165476	+	0.986214i	$0.447084\pi$
$420$	0	0
$421$	−1.52728	−0.0744351	−0.0372175	−	0.999307i	$-0.511849\pi$
$421$	−1.52728	−0.0744351	−0.0372175	+	0.999307i	$0.511849\pi$
$422$	0	0
$423$	17.3609	0.844117
$424$	0	0
$425$	2.48196	0.120393
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11.7553	−0.567552
$430$	0	0
$431$	−3.78461	−0.182298	−0.0911491	−	0.995837i	$-0.529054\pi$
$431$	−3.78461	−0.182298	−0.0911491	+	0.995837i	$0.529054\pi$
$432$	0	0
$433$	−3.53219	−0.169746	−0.0848730	−	0.996392i	$-0.527048\pi$
$433$	−3.53219	−0.169746	−0.0848730	+	0.996392i	$0.527048\pi$
$434$	0	0
$435$	42.8612	2.05504
$436$	0	0
$437$	−13.5714	−0.649208
$438$	0	0
$439$	11.4480	0.546381	0.273190	−	0.961960i	$-0.411921\pi$
$439$	11.4480	0.546381	0.273190	+	0.961960i	$0.411921\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.5209	0.642400	0.321200	−	0.947011i	$-0.395914\pi$
$443$	13.5209	0.642400	0.321200	+	0.947011i	$0.395914\pi$
$444$	0	0
$445$	6.85494	0.324955
$446$	0	0
$447$	−18.5150	−0.875727
$448$	0	0
$449$	−18.0667	−0.852621	−0.426310	−	0.904577i	$-0.640187\pi$
$449$	−18.0667	−0.852621	−0.426310	+	0.904577i	$0.640187\pi$
$450$	0	0
$451$	31.4740	1.48205
$452$	0	0
$453$	48.2320	2.26614
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.8916	1.30471	0.652357	−	0.757912i	$-0.273780\pi$
$457$	27.8916	1.30471	0.652357	+	0.757912i	$0.273780\pi$
$458$	0	0
$459$	−16.8896	−0.788340
$460$	0	0
$461$	−27.9948	−1.30385	−0.651925	−	0.758284i	$-0.726038\pi$
$461$	−27.9948	−1.30385	−0.651925	+	0.758284i	$0.726038\pi$
$462$	0	0
$463$	37.3527	1.73593	0.867964	−	0.496627i	$-0.165428\pi$
$463$	37.3527	1.73593	0.867964	+	0.496627i	$0.165428\pi$
$464$	0	0
$465$	0.377790	0.0175196
$466$	0	0
$467$	−28.8663	−1.33577	−0.667886	−	0.744264i	$-0.732801\pi$
$467$	−28.8663	−1.33577	−0.667886	+	0.744264i	$0.732801\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	39.4974	1.81995
$472$	0	0
$473$	−1.56079	−0.0717654
$474$	0	0
$475$	7.57918	0.347757
$476$	0	0
$477$	5.97052	0.273372
$478$	0	0
$479$	25.0973	1.14672	0.573362	−	0.819302i	$-0.305639\pi$
$479$	25.0973	1.14672	0.573362	+	0.819302i	$0.305639\pi$
$480$	0	0
$481$	0.807021	0.0367970
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.6089	1.29907
$486$	0	0
$487$	41.0234	1.85895	0.929474	−	0.368888i	$-0.120262\pi$
$487$	41.0234	1.85895	0.929474	+	0.368888i	$0.120262\pi$
$488$	0	0
$489$	32.8487	1.48547
$490$	0	0
$491$	−0.985019	−0.0444533	−0.0222266	−	0.999753i	$-0.507076\pi$
$491$	−0.985019	−0.0444533	−0.0222266	+	0.999753i	$0.507076\pi$
$492$	0	0
$493$	15.5392	0.699852
$494$	0	0
$495$	29.6746	1.33378
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.68113	0.433387	0.216693	−	0.976240i	$-0.430473\pi$
$499$	9.68113	0.433387	0.216693	+	0.976240i	$0.430473\pi$
$500$	0	0
$501$	−26.3352	−1.17657
$502$	0	0
$503$	−12.4080	−0.553243	−0.276622	−	0.960979i	$-0.589215\pi$
$503$	−12.4080	−0.553243	−0.276622	+	0.960979i	$0.589215\pi$
$504$	0	0
$505$	0.633252	0.0281794
$506$	0	0
$507$	−31.7607	−1.41054
$508$	0	0
$509$	−13.7199	−0.608122	−0.304061	−	0.952653i	$-0.598343\pi$
$509$	−13.7199	−0.608122	−0.304061	+	0.952653i	$0.598343\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−51.5759	−2.27713
$514$	0	0
$515$	−19.9136	−0.877500
$516$	0	0
$517$	−8.02939	−0.353132
$518$	0	0
$519$	−9.58920	−0.420919
$520$	0	0
$521$	34.1034	1.49410	0.747049	−	0.664769i	$-0.231470\pi$
$521$	34.1034	1.49410	0.747049	+	0.664769i	$0.231470\pi$
$522$	0	0
$523$	5.24558	0.229373	0.114687	−	0.993402i	$-0.463414\pi$
$523$	5.24558	0.229373	0.114687	+	0.993402i	$0.463414\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.136967	0.00596638
$528$	0	0
$529$	−18.4876	−0.803808
$530$	0	0
$531$	−13.0195	−0.564999
$532$	0	0
$533$	−17.8159	−0.771692
$534$	0	0
$535$	−30.0125	−1.29755
$536$	0	0
$537$	11.9592	0.516075
$538$	0	0
$539$	0	0
$540$	0	0
$541$	43.6622	1.87719	0.938593	−	0.345027i	$-0.112130\pi$
$541$	43.6622	1.87719	0.938593	+	0.345027i	$0.112130\pi$
$542$	0	0
$543$	−25.6072	−1.09891
$544$	0	0
$545$	27.8063	1.19109
$546$	0	0
$547$	36.2818	1.55130	0.775648	−	0.631166i	$-0.217423\pi$
$547$	36.2818	1.55130	0.775648	+	0.631166i	$0.217423\pi$
$548$	0	0
$549$	58.9055	2.51402
$550$	0	0
$551$	47.4523	2.02153
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.10345	−0.131734
$556$	0	0
$557$	−7.19093	−0.304689	−0.152345	−	0.988327i	$-0.548682\pi$
$557$	−7.19093	−0.304689	−0.152345	+	0.988327i	$0.548682\pi$
$558$	0	0
$559$	0.883490	0.0373676
$560$	0	0
$561$	16.3893	0.691956
$562$	0	0
$563$	11.6990	0.493053	0.246527	−	0.969136i	$-0.420711\pi$
$563$	11.6990	0.493053	0.246527	+	0.969136i	$0.420711\pi$
$564$	0	0
$565$	16.0684	0.676003
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.2886	1.01823	0.509115	−	0.860698i	$-0.329973\pi$
$569$	24.2886	1.01823	0.509115	+	0.860698i	$0.329973\pi$
$570$	0	0
$571$	−1.58477	−0.0663206	−0.0331603	−	0.999450i	$-0.510557\pi$
$571$	−1.58477	−0.0663206	−0.0331603	+	0.999450i	$0.510557\pi$
$572$	0	0
$573$	−44.9644	−1.87842
$574$	0	0
$575$	−2.52004	−0.105093
$576$	0	0
$577$	17.9000	0.745188	0.372594	−	0.927994i	$-0.378468\pi$
$577$	17.9000	0.745188	0.372594	+	0.927994i	$0.378468\pi$
$578$	0	0
$579$	30.3523	1.26140
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.76135	−0.114364
$584$	0	0
$585$	−16.7974	−0.694486
$586$	0	0
$587$	−1.07134	−0.0442189	−0.0221094	−	0.999756i	$-0.507038\pi$
$587$	−1.07134	−0.0442189	−0.0221094	+	0.999756i	$0.507038\pi$
$588$	0	0
$589$	0.418257	0.0172340
$590$	0	0
$591$	22.8279	0.939015
$592$	0	0
$593$	5.21037	0.213964	0.106982	−	0.994261i	$-0.465881\pi$
$593$	5.21037	0.213964	0.106982	+	0.994261i	$0.465881\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−43.6429	−1.78619
$598$	0	0
$599$	−19.3597	−0.791015	−0.395508	−	0.918463i	$-0.629431\pi$
$599$	−19.3597	−0.791015	−0.395508	+	0.918463i	$0.629431\pi$
$600$	0	0
$601$	15.0693	0.614689	0.307345	−	0.951598i	$-0.400560\pi$
$601$	15.0693	0.614689	0.307345	+	0.951598i	$0.400560\pi$
$602$	0	0
$603$	1.27085	0.0517532
$604$	0	0
$605$	7.75705	0.315369
$606$	0	0
$607$	−30.9662	−1.25688	−0.628440	−	0.777858i	$-0.716306\pi$
$607$	−30.9662	−1.25688	−0.628440	+	0.777858i	$0.716306\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.54505	0.183873
$612$	0	0
$613$	3.75434	0.151636	0.0758182	−	0.997122i	$-0.475843\pi$
$613$	3.75434	0.151636	0.0758182	+	0.997122i	$0.475843\pi$
$614$	0	0
$615$	68.5122	2.76268
$616$	0	0
$617$	28.0758	1.13029	0.565145	−	0.824991i	$-0.308820\pi$
$617$	28.0758	1.13029	0.565145	+	0.824991i	$0.308820\pi$
$618$	0	0
$619$	−19.3674	−0.778441	−0.389221	−	0.921145i	$-0.627256\pi$
$619$	−19.3674	−0.778441	−0.389221	+	0.921145i	$0.627256\pi$
$620$	0	0
$621$	17.1487	0.688154
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−17.6610	−0.706442
$626$	0	0
$627$	50.0480	1.99873
$628$	0	0
$629$	−1.12515	−0.0448626
$630$	0	0
$631$	−40.7248	−1.62123	−0.810615	−	0.585580i	$-0.800867\pi$
$631$	−40.7248	−1.62123	−0.810615	+	0.585580i	$0.800867\pi$
$632$	0	0
$633$	14.4450	0.574136
$634$	0	0
$635$	30.7479	1.22019
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−61.7377	−2.44231
$640$	0	0
$641$	0.706513	0.0279056	0.0139528	−	0.999903i	$-0.495559\pi$
$641$	0.706513	0.0279056	0.0139528	+	0.999903i	$0.495559\pi$
$642$	0	0
$643$	16.5495	0.652650	0.326325	−	0.945258i	$-0.394190\pi$
$643$	16.5495	0.652650	0.326325	+	0.945258i	$0.394190\pi$
$644$	0	0
$645$	−3.39752	−0.133777
$646$	0	0
$647$	−11.7596	−0.462317	−0.231159	−	0.972916i	$-0.574252\pi$
$647$	−11.7596	−0.462317	−0.231159	+	0.972916i	$0.574252\pi$
$648$	0	0
$649$	6.02150	0.236364
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.5877	0.883925	0.441962	−	0.897034i	$-0.354282\pi$
$653$	22.5877	0.883925	0.441962	+	0.897034i	$0.354282\pi$
$654$	0	0
$655$	−16.1297	−0.630239
$656$	0	0
$657$	−58.5846	−2.28560
$658$	0	0
$659$	−18.0135	−0.701705	−0.350852	−	0.936431i	$-0.614108\pi$
$659$	−18.0135	−0.701705	−0.350852	+	0.936431i	$0.614108\pi$
$660$	0	0
$661$	−37.6880	−1.46590	−0.732948	−	0.680285i	$-0.761856\pi$
$661$	−37.6880	−1.46590	−0.732948	+	0.680285i	$0.761856\pi$
$662$	0	0
$663$	−9.27717	−0.360295
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.7776	−0.610912
$668$	0	0
$669$	5.16223	0.199584
$670$	0	0
$671$	−27.2436	−1.05173
$672$	0	0
$673$	−18.7754	−0.723738	−0.361869	−	0.932229i	$-0.617861\pi$
$673$	−18.7754	−0.723738	−0.361869	+	0.932229i	$0.617861\pi$
$674$	0	0
$675$	−9.57699	−0.368619
$676$	0	0
$677$	3.85662	0.148222	0.0741110	−	0.997250i	$-0.476388\pi$
$677$	3.85662	0.148222	0.0741110	+	0.997250i	$0.476388\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	70.0153	2.68299
$682$	0	0
$683$	−40.0543	−1.53264	−0.766318	−	0.642461i	$-0.777913\pi$
$683$	−40.0543	−1.53264	−0.766318	+	0.642461i	$0.777913\pi$
$684$	0	0
$685$	−0.152427	−0.00582393
$686$	0	0
$687$	−38.9792	−1.48715
$688$	0	0
$689$	1.56307	0.0595482
$690$	0	0
$691$	−33.8236	−1.28671	−0.643356	−	0.765567i	$-0.722458\pi$
$691$	−33.8236	−1.28671	−0.643356	+	0.765567i	$0.722458\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−30.5379	−1.15837
$696$	0	0
$697$	24.8389	0.940842
$698$	0	0
$699$	62.9118	2.37954
$700$	0	0
$701$	41.0095	1.54891	0.774453	−	0.632631i	$-0.218025\pi$
$701$	41.0095	1.54891	0.774453	+	0.632631i	$0.218025\pi$
$702$	0	0
$703$	−3.43587	−0.129586
$704$	0	0
$705$	−17.4783	−0.658270
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.8640	−0.933787	−0.466893	−	0.884314i	$-0.654627\pi$
$709$	−24.8640	−0.933787	−0.466893	+	0.884314i	$0.654627\pi$
$710$	0	0
$711$	76.1542	2.85601
$712$	0	0
$713$	−0.139068	−0.00520815
$714$	0	0
$715$	7.76875	0.290535
$716$	0	0
$717$	33.2768	1.24275
$718$	0	0
$719$	20.1550	0.751654	0.375827	−	0.926690i	$-0.377359\pi$
$719$	20.1550	0.751654	0.375827	+	0.926690i	$0.377359\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−27.3646	−1.01770
$724$	0	0
$725$	8.81128	0.327243
$726$	0	0
$727$	39.6430	1.47028	0.735138	−	0.677918i	$-0.237117\pi$
$727$	39.6430	1.47028	0.735138	+	0.677918i	$0.237117\pi$
$728$	0	0
$729$	−33.3945	−1.23683
$730$	0	0
$731$	−1.23176	−0.0455584
$732$	0	0
$733$	4.69321	0.173348	0.0866738	−	0.996237i	$-0.472376\pi$
$733$	4.69321	0.173348	0.0866738	+	0.996237i	$0.472376\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.587767	−0.0216507
$738$	0	0
$739$	−24.4235	−0.898433	−0.449216	−	0.893423i	$-0.648297\pi$
$739$	−24.4235	−0.898433	−0.449216	+	0.893423i	$0.648297\pi$
$740$	0	0
$741$	−28.3298	−1.04072
$742$	0	0
$743$	−28.1516	−1.03278	−0.516391	−	0.856353i	$-0.672725\pi$
$743$	−28.1516	−1.03278	−0.516391	+	0.856353i	$0.672725\pi$
$744$	0	0
$745$	12.2360	0.448292
$746$	0	0
$747$	−67.9484	−2.48610
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.7070	1.04753	0.523766	−	0.851862i	$-0.324526\pi$
$751$	28.7070	1.04753	0.523766	+	0.851862i	$0.324526\pi$
$752$	0	0
$753$	−58.6858	−2.13863
$754$	0	0
$755$	−31.8751	−1.16005
$756$	0	0
$757$	−34.9823	−1.27145	−0.635726	−	0.771915i	$-0.719299\pi$
$757$	−34.9823	−1.27145	−0.635726	+	0.771915i	$0.719299\pi$
$758$	0	0
$759$	−16.6407	−0.604019
$760$	0	0
$761$	−47.2861	−1.71412	−0.857060	−	0.515216i	$-0.827712\pi$
$761$	−47.2861	−1.71412	−0.857060	+	0.515216i	$0.827712\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	23.4189	0.846712
$766$	0	0
$767$	−3.40848	−0.123073
$768$	0	0
$769$	41.8758	1.51008	0.755040	−	0.655678i	$-0.227617\pi$
$769$	41.8758	1.51008	0.755040	+	0.655678i	$0.227617\pi$
$770$	0	0
$771$	−74.3397	−2.67728
$772$	0	0
$773$	21.0942	0.758706	0.379353	−	0.925252i	$-0.376147\pi$
$773$	21.0942	0.758706	0.379353	+	0.925252i	$0.376147\pi$
$774$	0	0
$775$	0.0776650	0.00278981
$776$	0	0
$777$	0	0
$778$	0	0
$779$	75.8508	2.71764
$780$	0	0
$781$	28.5536	1.02173
$782$	0	0
$783$	−59.9603	−2.14281
$784$	0	0
$785$	−26.1027	−0.931646
$786$	0	0
$787$	40.2501	1.43476	0.717380	−	0.696682i	$-0.245341\pi$
$787$	40.2501	1.43476	0.717380	+	0.696682i	$0.245341\pi$
$788$	0	0
$789$	30.5208	1.08657
$790$	0	0
$791$	0	0
$792$	0	0
$793$	15.4213	0.547627
$794$	0	0
$795$	−6.01088	−0.213184
$796$	0	0
$797$	5.32667	0.188680	0.0943401	−	0.995540i	$-0.469926\pi$
$797$	5.32667	0.188680	0.0943401	+	0.995540i	$0.469926\pi$
$798$	0	0
$799$	−6.33671	−0.224177
$800$	0	0
$801$	−20.1202	−0.710914
$802$	0	0
$803$	27.0952	0.956170
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.32481	0.0466354
$808$	0	0
$809$	38.3657	1.34887	0.674434	−	0.738335i	$-0.264388\pi$
$809$	38.3657	1.34887	0.674434	+	0.738335i	$0.264388\pi$
$810$	0	0
$811$	33.5120	1.17676	0.588382	−	0.808583i	$-0.299765\pi$
$811$	33.5120	1.17676	0.588382	+	0.808583i	$0.299765\pi$
$812$	0	0
$813$	−70.8104	−2.48343
$814$	0	0
$815$	−21.7087	−0.760424
$816$	0	0
$817$	−3.76144	−0.131596
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.9875	−1.29087	−0.645437	−	0.763814i	$-0.723325\pi$
$821$	−36.9875	−1.29087	−0.645437	+	0.763814i	$0.723325\pi$
$822$	0	0
$823$	−8.10944	−0.282677	−0.141339	−	0.989961i	$-0.545141\pi$
$823$	−8.10944	−0.282677	−0.141339	+	0.989961i	$0.545141\pi$
$824$	0	0
$825$	9.29328	0.323551
$826$	0	0
$827$	47.9413	1.66708	0.833541	−	0.552458i	$-0.186310\pi$
$827$	47.9413	1.66708	0.833541	+	0.552458i	$0.186310\pi$
$828$	0	0
$829$	−29.4695	−1.02352	−0.511758	−	0.859129i	$-0.671006\pi$
$829$	−29.4695	−1.02352	−0.511758	+	0.859129i	$0.671006\pi$
$830$	0	0
$831$	44.8441	1.55562
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17.4042	0.602296
$836$	0	0
$837$	−0.528507	−0.0182679
$838$	0	0
$839$	42.6216	1.47146	0.735731	−	0.677274i	$-0.236839\pi$
$839$	42.6216	1.47146	0.735731	+	0.677274i	$0.236839\pi$
$840$	0	0
$841$	26.1663	0.902287
$842$	0	0
$843$	29.6445	1.02101
$844$	0	0
$845$	20.9897	0.722069
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6.21449	−0.213281
$850$	0	0
$851$	1.14241	0.0391613
$852$	0	0
$853$	44.8312	1.53499	0.767496	−	0.641054i	$-0.221502\pi$
$853$	44.8312	1.53499	0.767496	+	0.641054i	$0.221502\pi$
$854$	0	0
$855$	71.5145	2.44574
$856$	0	0
$857$	−32.5489	−1.11185	−0.555924	−	0.831233i	$-0.687636\pi$
$857$	−32.5489	−1.11185	−0.555924	+	0.831233i	$0.687636\pi$
$858$	0	0
$859$	−34.8831	−1.19020	−0.595099	−	0.803653i	$-0.702887\pi$
$859$	−34.8831	−1.19020	−0.595099	+	0.803653i	$0.702887\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.66021	0.0565141	0.0282571	−	0.999601i	$-0.491004\pi$
$863$	1.66021	0.0565141	0.0282571	+	0.999601i	$0.491004\pi$
$864$	0	0
$865$	6.33722	0.215472
$866$	0	0
$867$	−37.3005	−1.26679
$868$	0	0
$869$	−35.2212	−1.19480
$870$	0	0
$871$	0.332706	0.0112733
$872$	0	0
$873$	−83.9714	−2.84200
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.5981	1.03323	0.516613	−	0.856219i	$-0.327193\pi$
$877$	30.5981	1.03323	0.516613	+	0.856219i	$0.327193\pi$
$878$	0	0
$879$	−55.1067	−1.85870
$880$	0	0
$881$	−51.9678	−1.75084	−0.875419	−	0.483365i	$-0.839414\pi$
$881$	−51.9678	−1.75084	−0.875419	+	0.483365i	$0.839414\pi$
$882$	0	0
$883$	−22.8041	−0.767420	−0.383710	−	0.923454i	$-0.625354\pi$
$883$	−22.8041	−0.767420	−0.383710	+	0.923454i	$0.625354\pi$
$884$	0	0
$885$	13.1075	0.440604
$886$	0	0
$887$	−14.9208	−0.500993	−0.250496	−	0.968118i	$-0.580594\pi$
$887$	−14.9208	−0.500993	−0.250496	+	0.968118i	$0.580594\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−17.6540	−0.591431
$892$	0	0
$893$	−19.3505	−0.647538
$894$	0	0
$895$	−7.90346	−0.264183
$896$	0	0
$897$	9.41949	0.314508
$898$	0	0
$899$	0.486251	0.0162174
$900$	0	0
$901$	−2.17923	−0.0726007
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.9231	0.562542
$906$	0	0
$907$	41.6207	1.38199	0.690996	−	0.722858i	$-0.257172\pi$
$907$	41.6207	1.38199	0.690996	+	0.722858i	$0.257172\pi$
$908$	0	0
$909$	−1.85869	−0.0616488
$910$	0	0
$911$	−21.8850	−0.725083	−0.362541	−	0.931968i	$-0.618091\pi$
$911$	−21.8850	−0.725083	−0.362541	+	0.931968i	$0.618091\pi$
$912$	0	0
$913$	31.4260	1.04005
$914$	0	0
$915$	−59.3036	−1.96052
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.45638	−0.147002	−0.0735012	−	0.997295i	$-0.523417\pi$
$919$	−4.45638	−0.147002	−0.0735012	+	0.997295i	$0.523417\pi$
$920$	0	0
$921$	28.7417	0.947072
$922$	0	0
$923$	−16.1628	−0.532005
$924$	0	0
$925$	−0.637998	−0.0209772
$926$	0	0
$927$	58.4495	1.91973
$928$	0	0
$929$	−18.0749	−0.593019	−0.296509	−	0.955030i	$-0.595823\pi$
$929$	−18.0749	−0.593019	−0.296509	+	0.955030i	$0.595823\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−82.2747	−2.69355
$934$	0	0
$935$	−10.8312	−0.354218
$936$	0	0
$937$	23.0218	0.752088	0.376044	−	0.926602i	$-0.377284\pi$
$937$	23.0218	0.752088	0.376044	+	0.926602i	$0.377284\pi$
$938$	0	0
$939$	−59.7352	−1.94938
$940$	0	0
$941$	11.2388	0.366373	0.183186	−	0.983078i	$-0.441359\pi$
$941$	11.2388	0.366373	0.183186	+	0.983078i	$0.441359\pi$
$942$	0	0
$943$	−25.2200	−0.821275
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.60672	−0.279681	−0.139840	−	0.990174i	$-0.544659\pi$
$947$	−8.60672	−0.279681	−0.139840	+	0.990174i	$0.544659\pi$
$948$	0	0
$949$	−15.3373	−0.497870
$950$	0	0
$951$	35.0270	1.13583
$952$	0	0
$953$	−16.0513	−0.519951	−0.259976	−	0.965615i	$-0.583715\pi$
$953$	−16.0513	−0.519951	−0.259976	+	0.965615i	$0.583715\pi$
$954$	0	0
$955$	29.7157	0.961577
$956$	0	0
$957$	58.1841	1.88082
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9957	−0.999862
$962$	0	0
$963$	88.0909	2.83869
$964$	0	0
$965$	−20.0590	−0.645721
$966$	0	0
$967$	−15.6295	−0.502610	−0.251305	−	0.967908i	$-0.580860\pi$
$967$	−15.6295	−0.502610	−0.251305	+	0.967908i	$0.580860\pi$
$968$	0	0
$969$	39.4974	1.26884
$970$	0	0
$971$	5.03310	0.161520	0.0807600	−	0.996734i	$-0.474265\pi$
$971$	5.03310	0.161520	0.0807600	+	0.996734i	$0.474265\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5.26047	−0.168470
$976$	0	0
$977$	17.0593	0.545776	0.272888	−	0.962046i	$-0.412021\pi$
$977$	17.0593	0.545776	0.272888	+	0.962046i	$0.412021\pi$
$978$	0	0
$979$	9.30557	0.297407
$980$	0	0
$981$	−81.6156	−2.60578
$982$	0	0
$983$	26.8870	0.857561	0.428781	−	0.903409i	$-0.358943\pi$
$983$	26.8870	0.857561	0.428781	+	0.903409i	$0.358943\pi$
$984$	0	0
$985$	−15.0863	−0.480690
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.25066	0.0397686
$990$	0	0
$991$	−30.0379	−0.954184	−0.477092	−	0.878853i	$-0.658309\pi$
$991$	−30.0379	−0.954184	−0.477092	+	0.878853i	$0.658309\pi$
$992$	0	0
$993$	−37.4857	−1.18957
$994$	0	0
$995$	28.8423	0.914364
$996$	0	0
$997$	42.6067	1.34937	0.674684	−	0.738107i	$-0.264280\pi$
$997$	42.6067	1.34937	0.674684	+	0.738107i	$0.264280\pi$
$998$	0	0
$999$	4.34155	0.137360

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.p.1.6		6
4.3	odd	2		343.2.a.c.1.1	✓	6
7.6	odd	2		5488.2.a.h.1.1		6
12.11	even	2		3087.2.a.k.1.6		6
20.19	odd	2		8575.2.a.o.1.6		6
28.3	even	6		343.2.c.d.324.6		12
28.11	odd	6		343.2.c.e.324.6		12
28.19	even	6		343.2.c.d.18.6		12
28.23	odd	6		343.2.c.e.18.6		12
28.27	even	2		343.2.a.d.1.1	yes	6
84.83	odd	2		3087.2.a.j.1.6		6
140.139	even	2		8575.2.a.n.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.c.1.1	✓	6	4.3	odd	2
343.2.a.d.1.1	yes	6	28.27	even	2
343.2.c.d.18.6		12	28.19	even	6
343.2.c.d.324.6		12	28.3	even	6
343.2.c.e.18.6		12	28.23	odd	6
343.2.c.e.324.6		12	28.11	odd	6
3087.2.a.j.1.6		6	84.83	odd	2
3087.2.a.k.1.6		6	12.11	even	2
5488.2.a.h.1.1		6	7.6	odd	2
5488.2.a.p.1.6		6	1.1	even	1	trivial
8575.2.a.n.1.6		6	140.139	even	2
8575.2.a.o.1.6		6	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)

\(f(q)\)	\(=\)	\(q+2.95499 q^{3} -1.95286 q^{5} +5.73194 q^{9} -2.65101 q^{11} +1.50061 q^{13} -5.77069 q^{15} -2.09215 q^{17} -6.38881 q^{19} +2.12424 q^{23} -1.18632 q^{25} +8.07285 q^{27} -7.42740 q^{29} -0.0654672 q^{31} -7.83370 q^{33} +0.537796 q^{37} +4.43428 q^{39} -11.8724 q^{41} +0.588754 q^{43} -11.1937 q^{45} +3.02880 q^{47} -6.18227 q^{51} +1.04162 q^{53} +5.17707 q^{55} -18.8788 q^{57} -2.27140 q^{59} +10.2767 q^{61} -2.93048 q^{65} +0.221714 q^{67} +6.27711 q^{69} -10.7708 q^{71} -10.2207 q^{73} -3.50556 q^{75} +13.2859 q^{79} +6.65934 q^{81} -11.8543 q^{83} +4.08568 q^{85} -21.9479 q^{87} -3.51020 q^{89} -0.193455 q^{93} +12.4765 q^{95} -14.6497 q^{97} -15.1954 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 11 q^{5} + 9 q^{9} + q^{11} - 7 q^{13} + 5 q^{15} - 26 q^{17} - 3 q^{19} + 9 q^{23} + 15 q^{25} + 8 q^{27} - 2 q^{29} - 2 q^{31} - 18 q^{33} - 2 q^{37} - 7 q^{39} - 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 - 11 * q^5 + 9 * q^9 + q^11 - 7 * q^13 + 5 * q^15 - 26 * q^17 - 3 * q^19 + 9 * q^23 + 15 * q^25 + 8 * q^27 - 2 * q^29 - 2 * q^31 - 18 * q^33 - 2 * q^37 - 7 * q^39 - 28 * q^41 - 5 * q^43 - 3 * q^45 + 18 * q^47 - 13 * q^51 - q^53 - 22 * q^55 - 26 * q^57 + 5 * q^59 + 17 * q^61 - 14 * q^65 + 22 * q^67 + 20 * q^69 + 2 * q^71 - 12 * q^73 - 27 * q^75 + 2 * q^79 - 30 * q^81 + 7 * q^83 + 37 * q^85 - 25 * q^87 - 39 * q^89 - 22 * q^93 + 11 * q^95 - 35 * q^97 - 12 * q^99

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