LMFDB - Embedded newform 5408.2.a.bt.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 25x^{10} + 225x^{8} - 885x^{6} + 1495x^{4} - 897x^{2} + 169 $ x^12 - 25x^10 + 225x^8 - 885x^6 + 1495x^4 - 897*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.62324$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.62324 q^{3} -3.72519 q^{5} -4.80058 q^{7} -0.365087 q^{9} -4.59139 q^{11} -6.04688 q^{15} -2.37985 q^{17} -0.974773 q^{19} -7.79251 q^{21} -6.51389 q^{23} +8.87701 q^{25} -5.46235 q^{27} -6.79705 q^{29} -3.35576 q^{31} -7.45294 q^{33} +17.8831 q^{35} +1.53476 q^{37} +4.85450 q^{41} +3.33270 q^{43} +1.36002 q^{45} -8.18495 q^{47} +16.0456 q^{49} -3.86307 q^{51} -9.39514 q^{53} +17.1038 q^{55} -1.58229 q^{57} -2.67262 q^{59} +12.4695 q^{61} +1.75263 q^{63} +5.60928 q^{67} -10.5736 q^{69} +15.3889 q^{71} -8.45473 q^{73} +14.4095 q^{75} +22.0414 q^{77} +1.33465 q^{79} -7.77145 q^{81} +0.817576 q^{83} +8.86537 q^{85} -11.0333 q^{87} -11.4245 q^{89} -5.44721 q^{93} +3.63121 q^{95} +5.81428 q^{97} +1.67626 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{5} + 14 q^{9} + 14 q^{17} - 28 q^{21} + 24 q^{25} + 18 q^{29} + 20 q^{33} - 8 q^{37} + 36 q^{41} - 66 q^{45} + 34 q^{49} + 38 q^{53} + 22 q^{57} + 36 q^{61} - 12 q^{73} + 64 q^{77} + 8 q^{81}+ \cdots - 8 q^{97}+O(q^{100}) $ 12 * q - 8 * q^5 + 14 * q^9 + 14 * q^17 - 28 * q^21 + 24 * q^25 + 18 * q^29 + 20 * q^33 - 8 * q^37 + 36 * q^41 - 66 * q^45 + 34 * q^49 + 38 * q^53 + 22 * q^57 + 36 * q^61 - 12 * q^73 + 64 * q^77 + 8 * q^81 - 18 * q^85 + 44 * q^89 - 4 * q^93 - 8 * 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
$3$	1.62324	0.937179	0.468589	−	0.883416i	$-0.344762\pi$
$3$	1.62324	0.937179	0.468589	+	0.883416i	$0.344762\pi$
$4$	0	0
$5$	−3.72519	−1.66595	−0.832977	−	0.553308i	$-0.813365\pi$
$5$	−3.72519	−1.66595	−0.832977	+	0.553308i	$0.813365\pi$
$6$	0	0
$7$	−4.80058	−1.81445	−0.907225	−	0.420645i	$-0.861804\pi$
$7$	−4.80058	−1.81445	−0.907225	+	0.420645i	$0.861804\pi$
$8$	0	0
$9$	−0.365087	−0.121696
$10$	0	0
$11$	−4.59139	−1.38436	−0.692179	−	0.721726i	$-0.743349\pi$
$11$	−4.59139	−1.38436	−0.692179	+	0.721726i	$0.743349\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−6.04688	−1.56130
$16$	0	0
$17$	−2.37985	−0.577197	−0.288599	−	0.957450i	$-0.593189\pi$
$17$	−2.37985	−0.577197	−0.288599	+	0.957450i	$0.593189\pi$
$18$	0	0
$19$	−0.974773	−0.223628	−0.111814	−	0.993729i	$-0.535666\pi$
$19$	−0.974773	−0.223628	−0.111814	+	0.993729i	$0.535666\pi$
$20$	0	0
$21$	−7.79251	−1.70046
$22$	0	0
$23$	−6.51389	−1.35824	−0.679120	−	0.734027i	$-0.737639\pi$
$23$	−6.51389	−1.35824	−0.679120	+	0.734027i	$0.737639\pi$
$24$	0	0
$25$	8.87701	1.77540
$26$	0	0
$27$	−5.46235	−1.05123
$28$	0	0
$29$	−6.79705	−1.26218	−0.631091	−	0.775709i	$-0.717392\pi$
$29$	−6.79705	−1.26218	−0.631091	+	0.775709i	$0.717392\pi$
$30$	0	0
$31$	−3.35576	−0.602713	−0.301356	−	0.953512i	$-0.597439\pi$
$31$	−3.35576	−0.602713	−0.301356	+	0.953512i	$0.597439\pi$
$32$	0	0
$33$	−7.45294	−1.29739
$34$	0	0
$35$	17.8831	3.02279
$36$	0	0
$37$	1.53476	0.252313	0.126157	−	0.992010i	$-0.459736\pi$
$37$	1.53476	0.252313	0.126157	+	0.992010i	$0.459736\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.85450	0.758146	0.379073	−	0.925367i	$-0.376243\pi$
$41$	4.85450	0.758146	0.379073	+	0.925367i	$0.376243\pi$
$42$	0	0
$43$	3.33270	0.508232	0.254116	−	0.967174i	$-0.418215\pi$
$43$	3.33270	0.508232	0.254116	+	0.967174i	$0.418215\pi$
$44$	0	0
$45$	1.36002	0.202739
$46$	0	0
$47$	−8.18495	−1.19390	−0.596949	−	0.802279i	$-0.703620\pi$
$47$	−8.18495	−1.19390	−0.596949	+	0.802279i	$0.703620\pi$
$48$	0	0
$49$	16.0456	2.29223
$50$	0	0
$51$	−3.86307	−0.540937
$52$	0	0
$53$	−9.39514	−1.29052	−0.645261	−	0.763962i	$-0.723251\pi$
$53$	−9.39514	−1.29052	−0.645261	+	0.763962i	$0.723251\pi$
$54$	0	0
$55$	17.1038	2.30627
$56$	0	0
$57$	−1.58229	−0.209580
$58$	0	0
$59$	−2.67262	−0.347945	−0.173973	−	0.984750i	$-0.555660\pi$
$59$	−2.67262	−0.347945	−0.173973	+	0.984750i	$0.555660\pi$
$60$	0	0
$61$	12.4695	1.59655	0.798277	−	0.602290i	$-0.205745\pi$
$61$	12.4695	1.59655	0.798277	+	0.602290i	$0.205745\pi$
$62$	0	0
$63$	1.75263	0.220811
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.60928	0.685283	0.342641	−	0.939466i	$-0.388678\pi$
$67$	5.60928	0.685283	0.342641	+	0.939466i	$0.388678\pi$
$68$	0	0
$69$	−10.5736	−1.27291
$70$	0	0
$71$	15.3889	1.82632	0.913161	−	0.407598i	$-0.133634\pi$
$71$	15.3889	1.82632	0.913161	+	0.407598i	$0.133634\pi$
$72$	0	0
$73$	−8.45473	−0.989551	−0.494776	−	0.869021i	$-0.664750\pi$
$73$	−8.45473	−0.989551	−0.494776	+	0.869021i	$0.664750\pi$
$74$	0	0
$75$	14.4095	1.66387
$76$	0	0
$77$	22.0414	2.51185
$78$	0	0
$79$	1.33465	0.150159	0.0750797	−	0.997178i	$-0.476079\pi$
$79$	1.33465	0.150159	0.0750797	+	0.997178i	$0.476079\pi$
$80$	0	0
$81$	−7.77145	−0.863495
$82$	0	0
$83$	0.817576	0.0897407	0.0448703	−	0.998993i	$-0.485713\pi$
$83$	0.817576	0.0897407	0.0448703	+	0.998993i	$0.485713\pi$
$84$	0	0
$85$	8.86537	0.961584
$86$	0	0
$87$	−11.0333	−1.18289
$88$	0	0
$89$	−11.4245	−1.21099	−0.605496	−	0.795848i	$-0.707025\pi$
$89$	−11.4245	−1.21099	−0.605496	+	0.795848i	$0.707025\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.44721	−0.564850
$94$	0	0
$95$	3.63121	0.372554
$96$	0	0
$97$	5.81428	0.590350	0.295175	−	0.955443i	$-0.404622\pi$
$97$	5.81428	0.590350	0.295175	+	0.955443i	$0.404622\pi$
$98$	0	0
$99$	1.67626	0.168470
$100$	0	0
$101$	−12.0588	−1.19989	−0.599947	−	0.800039i	$-0.704812\pi$
$101$	−12.0588	−1.19989	−0.599947	+	0.800039i	$0.704812\pi$
$102$	0	0
$103$	2.27273	0.223939	0.111969	−	0.993712i	$-0.464284\pi$
$103$	2.27273	0.223939	0.111969	+	0.993712i	$0.464284\pi$
$104$	0	0
$105$	29.0285	2.83290
$106$	0	0
$107$	−0.381695	−0.0368998	−0.0184499	−	0.999830i	$-0.505873\pi$
$107$	−0.381695	−0.0368998	−0.0184499	+	0.999830i	$0.505873\pi$
$108$	0	0
$109$	−8.24886	−0.790098	−0.395049	−	0.918660i	$-0.629272\pi$
$109$	−8.24886	−0.790098	−0.395049	+	0.918660i	$0.629272\pi$
$110$	0	0
$111$	2.49129	0.236463
$112$	0	0
$113$	−3.50293	−0.329528	−0.164764	−	0.986333i	$-0.552686\pi$
$113$	−3.50293	−0.329528	−0.164764	+	0.986333i	$0.552686\pi$
$114$	0	0
$115$	24.2655	2.26277
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.4247	1.04730
$120$	0	0
$121$	10.0809	0.916445
$122$	0	0
$123$	7.88003	0.710518
$124$	0	0
$125$	−14.4426	−1.29178
$126$	0	0
$127$	−12.4579	−1.10546	−0.552729	−	0.833361i	$-0.686413\pi$
$127$	−12.4579	−1.10546	−0.552729	+	0.833361i	$0.686413\pi$
$128$	0	0
$129$	5.40978	0.476304
$130$	0	0
$131$	−15.6343	−1.36597	−0.682987	−	0.730430i	$-0.739320\pi$
$131$	−15.6343	−1.36597	−0.682987	+	0.730430i	$0.739320\pi$
$132$	0	0
$133$	4.67948	0.405762
$134$	0	0
$135$	20.3483	1.75130
$136$	0	0
$137$	5.70457	0.487374	0.243687	−	0.969854i	$-0.421643\pi$
$137$	5.70457	0.487374	0.243687	+	0.969854i	$0.421643\pi$
$138$	0	0
$139$	−15.5844	−1.32185	−0.660927	−	0.750450i	$-0.729837\pi$
$139$	−15.5844	−1.32185	−0.660927	+	0.750450i	$0.729837\pi$
$140$	0	0
$141$	−13.2862	−1.11890
$142$	0	0
$143$	0	0
$144$	0	0
$145$	25.3203	2.10274
$146$	0	0
$147$	26.0459	2.14823
$148$	0	0
$149$	−9.58776	−0.785460	−0.392730	−	0.919654i	$-0.628469\pi$
$149$	−9.58776	−0.785460	−0.392730	+	0.919654i	$0.628469\pi$
$150$	0	0
$151$	−24.3979	−1.98547	−0.992735	−	0.120324i	$-0.961607\pi$
$151$	−24.3979	−1.98547	−0.992735	+	0.120324i	$0.961607\pi$
$152$	0	0
$153$	0.868850	0.0702424
$154$	0	0
$155$	12.5008	1.00409
$156$	0	0
$157$	7.31518	0.583815	0.291907	−	0.956447i	$-0.405710\pi$
$157$	7.31518	0.583815	0.291907	+	0.956447i	$0.405710\pi$
$158$	0	0
$159$	−15.2506	−1.20945
$160$	0	0
$161$	31.2705	2.46446
$162$	0	0
$163$	1.31921	0.103329	0.0516644	−	0.998665i	$-0.483547\pi$
$163$	1.31921	0.103329	0.0516644	+	0.998665i	$0.483547\pi$
$164$	0	0
$165$	27.7636	2.16139
$166$	0	0
$167$	2.38492	0.184550	0.0922752	−	0.995734i	$-0.470586\pi$
$167$	2.38492	0.184550	0.0922752	+	0.995734i	$0.470586\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.355877	0.0272146
$172$	0	0
$173$	10.6798	0.811971	0.405985	−	0.913880i	$-0.366928\pi$
$173$	10.6798	0.811971	0.405985	+	0.913880i	$0.366928\pi$
$174$	0	0
$175$	−42.6148	−3.22138
$176$	0	0
$177$	−4.33830	−0.326087
$178$	0	0
$179$	−1.33258	−0.0996016	−0.0498008	−	0.998759i	$-0.515859\pi$
$179$	−1.33258	−0.0996016	−0.0498008	+	0.998759i	$0.515859\pi$
$180$	0	0
$181$	−11.4536	−0.851342	−0.425671	−	0.904878i	$-0.639962\pi$
$181$	−11.4536	−0.851342	−0.425671	+	0.904878i	$0.639962\pi$
$182$	0	0
$183$	20.2410	1.49626
$184$	0	0
$185$	−5.71728	−0.420343
$186$	0	0
$187$	10.9268	0.799047
$188$	0	0
$189$	26.2225	1.90740
$190$	0	0
$191$	1.29421	0.0936456	0.0468228	−	0.998903i	$-0.485090\pi$
$191$	1.29421	0.0936456	0.0468228	+	0.998903i	$0.485090\pi$
$192$	0	0
$193$	−6.83758	−0.492180	−0.246090	−	0.969247i	$-0.579146\pi$
$193$	−6.83758	−0.492180	−0.246090	+	0.969247i	$0.579146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.59785	0.470077	0.235039	−	0.971986i	$-0.424478\pi$
$197$	6.59785	0.470077	0.235039	+	0.971986i	$0.424478\pi$
$198$	0	0
$199$	1.54389	0.109444	0.0547218	−	0.998502i	$-0.482573\pi$
$199$	1.54389	0.109444	0.0547218	+	0.998502i	$0.482573\pi$
$200$	0	0
$201$	9.10522	0.642233
$202$	0	0
$203$	32.6298	2.29016
$204$	0	0
$205$	−18.0839	−1.26304
$206$	0	0
$207$	2.37814	0.165292
$208$	0	0
$209$	4.47556	0.309581
$210$	0	0
$211$	−18.1929	−1.25245	−0.626227	−	0.779641i	$-0.715402\pi$
$211$	−18.1929	−1.25245	−0.626227	+	0.779641i	$0.715402\pi$
$212$	0	0
$213$	24.9799	1.71159
$214$	0	0
$215$	−12.4149	−0.846691
$216$	0	0
$217$	16.1096	1.09359
$218$	0	0
$219$	−13.7241	−0.927387
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.4030	1.36628	0.683142	−	0.730286i	$-0.260613\pi$
$223$	20.4030	1.36628	0.683142	+	0.730286i	$0.260613\pi$
$224$	0	0
$225$	−3.24088	−0.216059
$226$	0	0
$227$	3.95346	0.262400	0.131200	−	0.991356i	$-0.458117\pi$
$227$	3.95346	0.262400	0.131200	+	0.991356i	$0.458117\pi$
$228$	0	0
$229$	−2.92203	−0.193093	−0.0965467	−	0.995328i	$-0.530780\pi$
$229$	−2.92203	−0.193093	−0.0965467	+	0.995328i	$0.530780\pi$
$230$	0	0
$231$	35.7785	2.35405
$232$	0	0
$233$	18.8768	1.23666	0.618331	−	0.785918i	$-0.287809\pi$
$233$	18.8768	1.23666	0.618331	+	0.785918i	$0.287809\pi$
$234$	0	0
$235$	30.4905	1.98898
$236$	0	0
$237$	2.16645	0.140726
$238$	0	0
$239$	−15.0220	−0.971694	−0.485847	−	0.874044i	$-0.661489\pi$
$239$	−15.0220	−0.971694	−0.485847	+	0.874044i	$0.661489\pi$
$240$	0	0
$241$	24.7974	1.59734	0.798670	−	0.601769i	$-0.205537\pi$
$241$	24.7974	1.59734	0.798670	+	0.601769i	$0.205537\pi$
$242$	0	0
$243$	3.77210	0.241981
$244$	0	0
$245$	−59.7729	−3.81875
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.32712	0.0841031
$250$	0	0
$251$	−17.5753	−1.10934	−0.554672	−	0.832069i	$-0.687156\pi$
$251$	−17.5753	−1.10934	−0.554672	+	0.832069i	$0.687156\pi$
$252$	0	0
$253$	29.9078	1.88029
$254$	0	0
$255$	14.3906	0.901177
$256$	0	0
$257$	−6.35738	−0.396562	−0.198281	−	0.980145i	$-0.563536\pi$
$257$	−6.35738	−0.396562	−0.198281	+	0.980145i	$0.563536\pi$
$258$	0	0
$259$	−7.36776	−0.457810
$260$	0	0
$261$	2.48151	0.153602
$262$	0	0
$263$	0.724180	0.0446548	0.0223274	−	0.999751i	$-0.492892\pi$
$263$	0.724180	0.0446548	0.0223274	+	0.999751i	$0.492892\pi$
$264$	0	0
$265$	34.9987	2.14995
$266$	0	0
$267$	−18.5447	−1.13492
$268$	0	0
$269$	17.2850	1.05388	0.526941	−	0.849902i	$-0.323339\pi$
$269$	17.2850	1.05388	0.526941	+	0.849902i	$0.323339\pi$
$270$	0	0
$271$	4.94545	0.300415	0.150207	−	0.988655i	$-0.452006\pi$
$271$	4.94545	0.300415	0.150207	+	0.988655i	$0.452006\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−40.7578	−2.45779
$276$	0	0
$277$	−4.79606	−0.288167	−0.144084	−	0.989566i	$-0.546023\pi$
$277$	−4.79606	−0.288167	−0.144084	+	0.989566i	$0.546023\pi$
$278$	0	0
$279$	1.22514	0.0733475
$280$	0	0
$281$	20.8701	1.24501	0.622503	−	0.782618i	$-0.286116\pi$
$281$	20.8701	1.24501	0.622503	+	0.782618i	$0.286116\pi$
$282$	0	0
$283$	−27.0757	−1.60948	−0.804742	−	0.593624i	$-0.797697\pi$
$283$	−27.0757	−1.60948	−0.804742	+	0.593624i	$0.797697\pi$
$284$	0	0
$285$	5.89433	0.349150
$286$	0	0
$287$	−23.3044	−1.37562
$288$	0	0
$289$	−11.3363	−0.666843
$290$	0	0
$291$	9.43798	0.553264
$292$	0	0
$293$	7.14153	0.417213	0.208606	−	0.978000i	$-0.433107\pi$
$293$	7.14153	0.417213	0.208606	+	0.978000i	$0.433107\pi$
$294$	0	0
$295$	9.95600	0.579661
$296$	0	0
$297$	25.0798	1.45528
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−15.9989	−0.922162
$302$	0	0
$303$	−19.5743	−1.12452
$304$	0	0
$305$	−46.4512	−2.65979
$306$	0	0
$307$	−32.0190	−1.82742	−0.913711	−	0.406364i	$-0.866797\pi$
$307$	−32.0190	−1.82742	−0.913711	+	0.406364i	$0.866797\pi$
$308$	0	0
$309$	3.68919	0.209871
$310$	0	0
$311$	−33.9738	−1.92648	−0.963238	−	0.268648i	$-0.913423\pi$
$311$	−33.9738	−1.92648	−0.963238	+	0.268648i	$0.913423\pi$
$312$	0	0
$313$	32.2692	1.82396	0.911981	−	0.410232i	$-0.134552\pi$
$313$	32.2692	1.82396	0.911981	+	0.410232i	$0.134552\pi$
$314$	0	0
$315$	−6.52887	−0.367860
$316$	0	0
$317$	−26.7644	−1.50324	−0.751618	−	0.659599i	$-0.770726\pi$
$317$	−26.7644	−1.50324	−0.751618	+	0.659599i	$0.770726\pi$
$318$	0	0
$319$	31.2079	1.74731
$320$	0	0
$321$	−0.619583	−0.0345817
$322$	0	0
$323$	2.31981	0.129078
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.3899	−0.740463
$328$	0	0
$329$	39.2925	2.16627
$330$	0	0
$331$	11.0851	0.609290	0.304645	−	0.952466i	$-0.401462\pi$
$331$	11.0851	0.609290	0.304645	+	0.952466i	$0.401462\pi$
$332$	0	0
$333$	−0.560322	−0.0307054
$334$	0	0
$335$	−20.8956	−1.14165
$336$	0	0
$337$	15.3422	0.835741	0.417870	−	0.908507i	$-0.362777\pi$
$337$	15.3422	0.835741	0.417870	+	0.908507i	$0.362777\pi$
$338$	0	0
$339$	−5.68611	−0.308827
$340$	0	0
$341$	15.4076	0.834370
$342$	0	0
$343$	−43.4242	−2.34469
$344$	0	0
$345$	39.3887	2.12062
$346$	0	0
$347$	−5.60958	−0.301138	−0.150569	−	0.988600i	$-0.548111\pi$
$347$	−5.60958	−0.301138	−0.150569	+	0.988600i	$0.548111\pi$
$348$	0	0
$349$	2.00631	0.107395	0.0536976	−	0.998557i	$-0.482899\pi$
$349$	2.00631	0.107395	0.0536976	+	0.998557i	$0.482899\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.1352	0.965239	0.482619	−	0.875830i	$-0.339685\pi$
$353$	18.1352	0.965239	0.482619	+	0.875830i	$0.339685\pi$
$354$	0	0
$355$	−57.3264	−3.04257
$356$	0	0
$357$	18.5450	0.981504
$358$	0	0
$359$	25.6223	1.35229	0.676147	−	0.736766i	$-0.263648\pi$
$359$	25.6223	1.35229	0.676147	+	0.736766i	$0.263648\pi$
$360$	0	0
$361$	−18.0498	−0.949990
$362$	0	0
$363$	16.3637	0.858873
$364$	0	0
$365$	31.4954	1.64855
$366$	0	0
$367$	14.0361	0.732676	0.366338	−	0.930482i	$-0.380611\pi$
$367$	14.0361	0.732676	0.366338	+	0.930482i	$0.380611\pi$
$368$	0	0
$369$	−1.77231	−0.0922630
$370$	0	0
$371$	45.1022	2.34159
$372$	0	0
$373$	20.7551	1.07466	0.537330	−	0.843372i	$-0.319433\pi$
$373$	20.7551	1.07466	0.537330	+	0.843372i	$0.319433\pi$
$374$	0	0
$375$	−23.4438	−1.21063
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.27500	0.0654921	0.0327461	−	0.999464i	$-0.489575\pi$
$379$	1.27500	0.0654921	0.0327461	+	0.999464i	$0.489575\pi$
$380$	0	0
$381$	−20.2221	−1.03601
$382$	0	0
$383$	6.41030	0.327551	0.163776	−	0.986498i	$-0.447633\pi$
$383$	6.41030	0.327551	0.163776	+	0.986498i	$0.447633\pi$
$384$	0	0
$385$	−82.1082	−4.18462
$386$	0	0
$387$	−1.21672	−0.0618496
$388$	0	0
$389$	17.3591	0.880144	0.440072	−	0.897963i	$-0.354953\pi$
$389$	17.3591	0.880144	0.440072	+	0.897963i	$0.354953\pi$
$390$	0	0
$391$	15.5021	0.783973
$392$	0	0
$393$	−25.3782	−1.28016
$394$	0	0
$395$	−4.97180	−0.250159
$396$	0	0
$397$	−37.3380	−1.87394	−0.936970	−	0.349409i	$-0.886383\pi$
$397$	−37.3380	−1.87394	−0.936970	+	0.349409i	$0.886383\pi$
$398$	0	0
$399$	7.59592	0.380272
$400$	0	0
$401$	−2.50016	−0.124852	−0.0624261	−	0.998050i	$-0.519884\pi$
$401$	−2.50016	−0.124852	−0.0624261	+	0.998050i	$0.519884\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	28.9501	1.43854
$406$	0	0
$407$	−7.04670	−0.349292
$408$	0	0
$409$	10.6213	0.525189	0.262594	−	0.964906i	$-0.415422\pi$
$409$	10.6213	0.525189	0.262594	+	0.964906i	$0.415422\pi$
$410$	0	0
$411$	9.25990	0.456757
$412$	0	0
$413$	12.8301	0.631329
$414$	0	0
$415$	−3.04562	−0.149504
$416$	0	0
$417$	−25.2973	−1.23881
$418$	0	0
$419$	8.12068	0.396721	0.198361	−	0.980129i	$-0.436438\pi$
$419$	8.12068	0.396721	0.198361	+	0.980129i	$0.436438\pi$
$420$	0	0
$421$	16.1985	0.789465	0.394733	−	0.918796i	$-0.370837\pi$
$421$	16.1985	0.789465	0.394733	+	0.918796i	$0.370837\pi$
$422$	0	0
$423$	2.98822	0.145292
$424$	0	0
$425$	−21.1259	−1.02476
$426$	0	0
$427$	−59.8608	−2.89687
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.6559	−1.09130	−0.545649	−	0.838014i	$-0.683717\pi$
$431$	−22.6559	−1.09130	−0.545649	+	0.838014i	$0.683717\pi$
$432$	0	0
$433$	−21.6077	−1.03840	−0.519201	−	0.854652i	$-0.673770\pi$
$433$	−21.6077	−1.03840	−0.519201	+	0.854652i	$0.673770\pi$
$434$	0	0
$435$	41.1009	1.97064
$436$	0	0
$437$	6.34957	0.303741
$438$	0	0
$439$	17.1517	0.818606	0.409303	−	0.912398i	$-0.365772\pi$
$439$	17.1517	0.818606	0.409303	+	0.912398i	$0.365772\pi$
$440$	0	0
$441$	−5.85804	−0.278954
$442$	0	0
$443$	−24.9778	−1.18673	−0.593365	−	0.804934i	$-0.702201\pi$
$443$	−24.9778	−1.18673	−0.593365	+	0.804934i	$0.702201\pi$
$444$	0	0
$445$	42.5583	2.01746
$446$	0	0
$447$	−15.5632	−0.736116
$448$	0	0
$449$	−32.7598	−1.54603	−0.773014	−	0.634389i	$-0.781252\pi$
$449$	−32.7598	−1.54603	−0.773014	+	0.634389i	$0.781252\pi$
$450$	0	0
$451$	−22.2889	−1.04954
$452$	0	0
$453$	−39.6036	−1.86074
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.41577	−0.159783	−0.0798915	−	0.996804i	$-0.525457\pi$
$457$	−3.41577	−0.159783	−0.0798915	+	0.996804i	$0.525457\pi$
$458$	0	0
$459$	12.9996	0.606767
$460$	0	0
$461$	−10.9229	−0.508730	−0.254365	−	0.967108i	$-0.581866\pi$
$461$	−10.9229	−0.508730	−0.254365	+	0.967108i	$0.581866\pi$
$462$	0	0
$463$	16.7908	0.780335	0.390168	−	0.920744i	$-0.372417\pi$
$463$	16.7908	0.780335	0.390168	+	0.920744i	$0.372417\pi$
$464$	0	0
$465$	20.2919	0.941014
$466$	0	0
$467$	21.3115	0.986179	0.493090	−	0.869979i	$-0.335867\pi$
$467$	21.3115	0.986179	0.493090	+	0.869979i	$0.335867\pi$
$468$	0	0
$469$	−26.9278	−1.24341
$470$	0	0
$471$	11.8743	0.547139
$472$	0	0
$473$	−15.3017	−0.703575
$474$	0	0
$475$	−8.65307	−0.397030
$476$	0	0
$477$	3.43004	0.157051
$478$	0	0
$479$	−21.2213	−0.969627	−0.484814	−	0.874617i	$-0.661113\pi$
$479$	−21.2213	−0.969627	−0.484814	+	0.874617i	$0.661113\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	50.7596	2.30964
$484$	0	0
$485$	−21.6593	−0.983497
$486$	0	0
$487$	−9.64141	−0.436894	−0.218447	−	0.975849i	$-0.570099\pi$
$487$	−9.64141	−0.436894	−0.218447	+	0.975849i	$0.570099\pi$
$488$	0	0
$489$	2.14140	0.0968375
$490$	0	0
$491$	26.5711	1.19913	0.599567	−	0.800324i	$-0.295339\pi$
$491$	26.5711	1.19913	0.599567	+	0.800324i	$0.295339\pi$
$492$	0	0
$493$	16.1759	0.728528
$494$	0	0
$495$	−6.24437	−0.280663
$496$	0	0
$497$	−73.8756	−3.31377
$498$	0	0
$499$	−33.3474	−1.49283	−0.746417	−	0.665479i	$-0.768227\pi$
$499$	−33.3474	−1.49283	−0.746417	+	0.665479i	$0.768227\pi$
$500$	0	0
$501$	3.87130	0.172957
$502$	0	0
$503$	15.4191	0.687503	0.343752	−	0.939061i	$-0.388302\pi$
$503$	15.4191	0.687503	0.343752	+	0.939061i	$0.388302\pi$
$504$	0	0
$505$	44.9212	1.99897
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.6107	−0.913552	−0.456776	−	0.889582i	$-0.650996\pi$
$509$	−20.6107	−0.913552	−0.456776	+	0.889582i	$0.650996\pi$
$510$	0	0
$511$	40.5876	1.79549
$512$	0	0
$513$	5.32455	0.235085
$514$	0	0
$515$	−8.46634	−0.373072
$516$	0	0
$517$	37.5803	1.65278
$518$	0	0
$519$	17.3359	0.760962
$520$	0	0
$521$	−22.0921	−0.967874	−0.483937	−	0.875103i	$-0.660794\pi$
$521$	−22.0921	−0.967874	−0.483937	+	0.875103i	$0.660794\pi$
$522$	0	0
$523$	12.1869	0.532895	0.266448	−	0.963849i	$-0.414150\pi$
$523$	12.1869	0.532895	0.266448	+	0.963849i	$0.414150\pi$
$524$	0	0
$525$	−69.1742	−3.01901
$526$	0	0
$527$	7.98620	0.347884
$528$	0	0
$529$	19.4308	0.844817
$530$	0	0
$531$	0.975737	0.0423434
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1.42188	0.0614734
$536$	0	0
$537$	−2.16310	−0.0933445
$538$	0	0
$539$	−73.6717	−3.17326
$540$	0	0
$541$	1.72216	0.0740413	0.0370206	−	0.999315i	$-0.488213\pi$
$541$	1.72216	0.0740413	0.0370206	+	0.999315i	$0.488213\pi$
$542$	0	0
$543$	−18.5920	−0.797860
$544$	0	0
$545$	30.7285	1.31627
$546$	0	0
$547$	−14.5926	−0.623936	−0.311968	−	0.950093i	$-0.600988\pi$
$547$	−14.5926	−0.623936	−0.311968	+	0.950093i	$0.600988\pi$
$548$	0	0
$549$	−4.55245	−0.194294
$550$	0	0
$551$	6.62558	0.282259
$552$	0	0
$553$	−6.40708	−0.272457
$554$	0	0
$555$	−9.28052	−0.393936
$556$	0	0
$557$	−31.5611	−1.33729	−0.668644	−	0.743583i	$-0.733125\pi$
$557$	−31.5611	−1.33729	−0.668644	+	0.743583i	$0.733125\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	17.7369	0.748850
$562$	0	0
$563$	−14.9827	−0.631446	−0.315723	−	0.948851i	$-0.602247\pi$
$563$	−14.9827	−0.631446	−0.315723	+	0.948851i	$0.602247\pi$
$564$	0	0
$565$	13.0491	0.548979
$566$	0	0
$567$	37.3075	1.56677
$568$	0	0
$569$	−13.3208	−0.558438	−0.279219	−	0.960227i	$-0.590075\pi$
$569$	−13.3208	−0.558438	−0.279219	+	0.960227i	$0.590075\pi$
$570$	0	0
$571$	−37.8169	−1.58259	−0.791293	−	0.611437i	$-0.790592\pi$
$571$	−37.8169	−1.58259	−0.791293	+	0.611437i	$0.790592\pi$
$572$	0	0
$573$	2.10081	0.0877627
$574$	0	0
$575$	−57.8239	−2.41142
$576$	0	0
$577$	6.48477	0.269964	0.134982	−	0.990848i	$-0.456902\pi$
$577$	6.48477	0.269964	0.134982	+	0.990848i	$0.456902\pi$
$578$	0	0
$579$	−11.0990	−0.461260
$580$	0	0
$581$	−3.92484	−0.162830
$582$	0	0
$583$	43.1368	1.78654
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.5234	−0.434346	−0.217173	−	0.976133i	$-0.569684\pi$
$587$	−10.5234	−0.434346	−0.217173	+	0.976133i	$0.569684\pi$
$588$	0	0
$589$	3.27111	0.134784
$590$	0	0
$591$	10.7099	0.440546
$592$	0	0
$593$	27.5404	1.13095	0.565475	−	0.824766i	$-0.308693\pi$
$593$	27.5404	1.13095	0.565475	+	0.824766i	$0.308693\pi$
$594$	0	0
$595$	−42.5590	−1.74475
$596$	0	0
$597$	2.50611	0.102568
$598$	0	0
$599$	0.983892	0.0402007	0.0201004	−	0.999798i	$-0.493601\pi$
$599$	0.983892	0.0402007	0.0201004	+	0.999798i	$0.493601\pi$
$600$	0	0
$601$	−14.5903	−0.595151	−0.297575	−	0.954698i	$-0.596178\pi$
$601$	−14.5903	−0.595151	−0.297575	+	0.954698i	$0.596178\pi$
$602$	0	0
$603$	−2.04787	−0.0833959
$604$	0	0
$605$	−37.5532	−1.52675
$606$	0	0
$607$	−40.3088	−1.63609	−0.818043	−	0.575158i	$-0.804941\pi$
$607$	−40.3088	−1.63609	−0.818043	+	0.575158i	$0.804941\pi$
$608$	0	0
$609$	52.9661	2.14629
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−20.1566	−0.814118	−0.407059	−	0.913402i	$-0.633446\pi$
$613$	−20.1566	−0.814118	−0.407059	+	0.913402i	$0.633446\pi$
$614$	0	0
$615$	−29.3546	−1.18369
$616$	0	0
$617$	−11.7236	−0.471973	−0.235987	−	0.971756i	$-0.575832\pi$
$617$	−11.7236	−0.471973	−0.235987	+	0.971756i	$0.575832\pi$
$618$	0	0
$619$	−14.2432	−0.572481	−0.286240	−	0.958158i	$-0.592406\pi$
$619$	−14.2432	−0.572481	−0.286240	+	0.958158i	$0.592406\pi$
$620$	0	0
$621$	35.5812	1.42782
$622$	0	0
$623$	54.8442	2.19728
$624$	0	0
$625$	9.41624	0.376650
$626$	0	0
$627$	7.26492	0.290133
$628$	0	0
$629$	−3.65250	−0.145635
$630$	0	0
$631$	25.1165	0.999871	0.499936	−	0.866062i	$-0.333357\pi$
$631$	25.1165	0.999871	0.499936	+	0.866062i	$0.333357\pi$
$632$	0	0
$633$	−29.5315	−1.17377
$634$	0	0
$635$	46.4079	1.84164
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.61827	−0.222255
$640$	0	0
$641$	−9.80091	−0.387113	−0.193556	−	0.981089i	$-0.562002\pi$
$641$	−9.80091	−0.387113	−0.193556	+	0.981089i	$0.562002\pi$
$642$	0	0
$643$	−34.4216	−1.35746	−0.678728	−	0.734390i	$-0.737468\pi$
$643$	−34.4216	−1.35746	−0.678728	+	0.734390i	$0.737468\pi$
$644$	0	0
$645$	−20.1524	−0.793501
$646$	0	0
$647$	−1.80166	−0.0708307	−0.0354153	−	0.999373i	$-0.511275\pi$
$647$	−1.80166	−0.0708307	−0.0354153	+	0.999373i	$0.511275\pi$
$648$	0	0
$649$	12.2710	0.481680
$650$	0	0
$651$	26.1498	1.02489
$652$	0	0
$653$	−8.15092	−0.318970	−0.159485	−	0.987200i	$-0.550983\pi$
$653$	−8.15092	−0.318970	−0.159485	+	0.987200i	$0.550983\pi$
$654$	0	0
$655$	58.2407	2.27565
$656$	0	0
$657$	3.08671	0.120424
$658$	0	0
$659$	6.81340	0.265413	0.132706	−	0.991155i	$-0.457633\pi$
$659$	6.81340	0.265413	0.132706	+	0.991155i	$0.457633\pi$
$660$	0	0
$661$	19.7714	0.769018	0.384509	−	0.923121i	$-0.374371\pi$
$661$	19.7714	0.769018	0.384509	+	0.923121i	$0.374371\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−17.4319	−0.675981
$666$	0	0
$667$	44.2753	1.71435
$668$	0	0
$669$	33.1189	1.28045
$670$	0	0
$671$	−57.2523	−2.21020
$672$	0	0
$673$	30.9581	1.19335	0.596674	−	0.802484i	$-0.296489\pi$
$673$	30.9581	1.19335	0.596674	+	0.802484i	$0.296489\pi$
$674$	0	0
$675$	−48.4893	−1.86635
$676$	0	0
$677$	2.37832	0.0914062	0.0457031	−	0.998955i	$-0.485447\pi$
$677$	2.37832	0.0914062	0.0457031	+	0.998955i	$0.485447\pi$
$678$	0	0
$679$	−27.9119	−1.07116
$680$	0	0
$681$	6.41742	0.245916
$682$	0	0
$683$	−17.2262	−0.659143	−0.329571	−	0.944131i	$-0.606904\pi$
$683$	−17.2262	−0.659143	−0.329571	+	0.944131i	$0.606904\pi$
$684$	0	0
$685$	−21.2506	−0.811943
$686$	0	0
$687$	−4.74316	−0.180963
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−29.9969	−1.14113	−0.570567	−	0.821251i	$-0.693277\pi$
$691$	−29.9969	−1.14113	−0.570567	+	0.821251i	$0.693277\pi$
$692$	0	0
$693$	−8.04701	−0.305681
$694$	0	0
$695$	58.0549	2.20215
$696$	0	0
$697$	−11.5530	−0.437600
$698$	0	0
$699$	30.6417	1.15897
$700$	0	0
$701$	20.0628	0.757762	0.378881	−	0.925445i	$-0.376309\pi$
$701$	20.0628	0.757762	0.378881	+	0.925445i	$0.376309\pi$
$702$	0	0
$703$	−1.49605	−0.0564244
$704$	0	0
$705$	49.4934	1.86403
$706$	0	0
$707$	57.8892	2.17715
$708$	0	0
$709$	−23.2742	−0.874080	−0.437040	−	0.899442i	$-0.643973\pi$
$709$	−23.2742	−0.874080	−0.437040	+	0.899442i	$0.643973\pi$
$710$	0	0
$711$	−0.487262	−0.0182737
$712$	0	0
$713$	21.8591	0.818629
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.3844	−0.910651
$718$	0	0
$719$	26.4598	0.986786	0.493393	−	0.869807i	$-0.335757\pi$
$719$	26.4598	0.986786	0.493393	+	0.869807i	$0.335757\pi$
$720$	0	0
$721$	−10.9104	−0.406326
$722$	0	0
$723$	40.2521	1.49699
$724$	0	0
$725$	−60.3375	−2.24088
$726$	0	0
$727$	3.99866	0.148302	0.0741510	−	0.997247i	$-0.476375\pi$
$727$	3.99866	0.148302	0.0741510	+	0.997247i	$0.476375\pi$
$728$	0	0
$729$	29.4374	1.09027
$730$	0	0
$731$	−7.93131	−0.293350
$732$	0	0
$733$	−31.1240	−1.14959	−0.574796	−	0.818297i	$-0.694919\pi$
$733$	−31.1240	−1.14959	−0.574796	+	0.818297i	$0.694919\pi$
$734$	0	0
$735$	−97.0258	−3.57885
$736$	0	0
$737$	−25.7544	−0.948676
$738$	0	0
$739$	34.7596	1.27865	0.639327	−	0.768935i	$-0.279213\pi$
$739$	34.7596	1.27865	0.639327	+	0.768935i	$0.279213\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.9049	1.50066	0.750328	−	0.661065i	$-0.229895\pi$
$743$	40.9049	1.50066	0.750328	+	0.661065i	$0.229895\pi$
$744$	0	0
$745$	35.7162	1.30854
$746$	0	0
$747$	−0.298486	−0.0109210
$748$	0	0
$749$	1.83236	0.0669529
$750$	0	0
$751$	27.9558	1.02012	0.510061	−	0.860138i	$-0.329623\pi$
$751$	27.9558	1.02012	0.510061	+	0.860138i	$0.329623\pi$
$752$	0	0
$753$	−28.5290	−1.03965
$754$	0	0
$755$	90.8865	3.30770
$756$	0	0
$757$	47.0163	1.70884	0.854419	−	0.519585i	$-0.173913\pi$
$757$	47.0163	1.70884	0.854419	+	0.519585i	$0.173913\pi$
$758$	0	0
$759$	48.5477	1.76217
$760$	0	0
$761$	7.31631	0.265216	0.132608	−	0.991169i	$-0.457665\pi$
$761$	7.31631	0.265216	0.132608	+	0.991169i	$0.457665\pi$
$762$	0	0
$763$	39.5993	1.43359
$764$	0	0
$765$	−3.23663	−0.117021
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.6877	0.673896	0.336948	−	0.941523i	$-0.390605\pi$
$769$	18.6877	0.673896	0.336948	+	0.941523i	$0.390605\pi$
$770$	0	0
$771$	−10.3196	−0.371650
$772$	0	0
$773$	−39.3417	−1.41502	−0.707511	−	0.706702i	$-0.750182\pi$
$773$	−39.3417	−1.41502	−0.707511	+	0.706702i	$0.750182\pi$
$774$	0	0
$775$	−29.7891	−1.07006
$776$	0	0
$777$	−11.9597	−0.429050
$778$	0	0
$779$	−4.73203	−0.169543
$780$	0	0
$781$	−70.6564	−2.52828
$782$	0	0
$783$	37.1279	1.32684
$784$	0	0
$785$	−27.2504	−0.972608
$786$	0	0
$787$	−0.837653	−0.0298591	−0.0149296	−	0.999889i	$-0.504752\pi$
$787$	−0.837653	−0.0298591	−0.0149296	+	0.999889i	$0.504752\pi$
$788$	0	0
$789$	1.17552	0.0418496
$790$	0	0
$791$	16.8161	0.597913
$792$	0	0
$793$	0	0
$794$	0	0
$795$	56.8113	2.01489
$796$	0	0
$797$	1.54506	0.0547287	0.0273643	−	0.999626i	$-0.491289\pi$
$797$	1.54506	0.0547287	0.0273643	+	0.999626i	$0.491289\pi$
$798$	0	0
$799$	19.4789	0.689115
$800$	0	0
$801$	4.17093	0.147372
$802$	0	0
$803$	38.8190	1.36989
$804$	0	0
$805$	−116.488	−4.10568
$806$	0	0
$807$	28.0577	0.987677
$808$	0	0
$809$	24.9755	0.878090	0.439045	−	0.898465i	$-0.355317\pi$
$809$	24.9755	0.878090	0.439045	+	0.898465i	$0.355317\pi$
$810$	0	0
$811$	−30.3768	−1.06667	−0.533337	−	0.845903i	$-0.679062\pi$
$811$	−30.3768	−1.06667	−0.533337	+	0.845903i	$0.679062\pi$
$812$	0	0
$813$	8.02766	0.281542
$814$	0	0
$815$	−4.91431	−0.172141
$816$	0	0
$817$	−3.24863	−0.113655
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.7848	0.446192	0.223096	−	0.974796i	$-0.428384\pi$
$821$	12.7848	0.446192	0.223096	+	0.974796i	$0.428384\pi$
$822$	0	0
$823$	−38.3796	−1.33783	−0.668915	−	0.743339i	$-0.733241\pi$
$823$	−38.3796	−1.33783	−0.668915	+	0.743339i	$0.733241\pi$
$824$	0	0
$825$	−66.1598	−2.30339
$826$	0	0
$827$	−55.4794	−1.92921	−0.964604	−	0.263701i	$-0.915057\pi$
$827$	−55.4794	−1.92921	−0.964604	+	0.263701i	$0.915057\pi$
$828$	0	0
$829$	−10.3728	−0.360263	−0.180131	−	0.983643i	$-0.557652\pi$
$829$	−10.3728	−0.360263	−0.180131	+	0.983643i	$0.557652\pi$
$830$	0	0
$831$	−7.78516	−0.270064
$832$	0	0
$833$	−38.1861	−1.32307
$834$	0	0
$835$	−8.88426	−0.307452
$836$	0	0
$837$	18.3303	0.633590
$838$	0	0
$839$	29.1158	1.00519	0.502594	−	0.864522i	$-0.332379\pi$
$839$	29.1158	1.00519	0.502594	+	0.864522i	$0.332379\pi$
$840$	0	0
$841$	17.1999	0.593101
$842$	0	0
$843$	33.8772	1.16679
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−48.3942	−1.66284
$848$	0	0
$849$	−43.9504	−1.50838
$850$	0	0
$851$	−9.99728	−0.342702
$852$	0	0
$853$	45.2328	1.54874	0.774371	−	0.632732i	$-0.218067\pi$
$853$	45.2328	1.54874	0.774371	+	0.632732i	$0.218067\pi$
$854$	0	0
$855$	−1.32571	−0.0453382
$856$	0	0
$857$	−38.0791	−1.30076	−0.650379	−	0.759610i	$-0.725390\pi$
$857$	−38.0791	−1.30076	−0.650379	+	0.759610i	$0.725390\pi$
$858$	0	0
$859$	−42.5346	−1.45126	−0.725630	−	0.688085i	$-0.758452\pi$
$859$	−42.5346	−1.45126	−0.725630	+	0.688085i	$0.758452\pi$
$860$	0	0
$861$	−37.8287	−1.28920
$862$	0	0
$863$	33.1111	1.12712	0.563558	−	0.826077i	$-0.309432\pi$
$863$	33.1111	1.12712	0.563558	+	0.826077i	$0.309432\pi$
$864$	0	0
$865$	−39.7843	−1.35271
$866$	0	0
$867$	−18.4016	−0.624951
$868$	0	0
$869$	−6.12788	−0.207874
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.12272	−0.0718430
$874$	0	0
$875$	69.3328	2.34388
$876$	0	0
$877$	24.4117	0.824325	0.412162	−	0.911110i	$-0.364774\pi$
$877$	24.4117	0.824325	0.412162	+	0.911110i	$0.364774\pi$
$878$	0	0
$879$	11.5924	0.391003
$880$	0	0
$881$	33.4866	1.12819	0.564097	−	0.825709i	$-0.309225\pi$
$881$	33.4866	1.12819	0.564097	+	0.825709i	$0.309225\pi$
$882$	0	0
$883$	6.85897	0.230823	0.115411	−	0.993318i	$-0.463181\pi$
$883$	6.85897	0.230823	0.115411	+	0.993318i	$0.463181\pi$
$884$	0	0
$885$	16.1610	0.543246
$886$	0	0
$887$	−39.6035	−1.32975	−0.664877	−	0.746953i	$-0.731516\pi$
$887$	−39.6035	−1.32975	−0.664877	+	0.746953i	$0.731516\pi$
$888$	0	0
$889$	59.8051	2.00580
$890$	0	0
$891$	35.6818	1.19538
$892$	0	0
$893$	7.97847	0.266989
$894$	0	0
$895$	4.96410	0.165932
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22.8093	0.760733
$900$	0	0
$901$	22.3590	0.744886
$902$	0	0
$903$	−25.9701	−0.864231
$904$	0	0
$905$	42.6669	1.41830
$906$	0	0
$907$	−31.0417	−1.03072	−0.515362	−	0.856973i	$-0.672343\pi$
$907$	−31.0417	−1.03072	−0.515362	+	0.856973i	$0.672343\pi$
$908$	0	0
$909$	4.40251	0.146022
$910$	0	0
$911$	−48.3759	−1.60277	−0.801383	−	0.598152i	$-0.795902\pi$
$911$	−48.3759	−1.60277	−0.801383	+	0.598152i	$0.795902\pi$
$912$	0	0
$913$	−3.75381	−0.124233
$914$	0	0
$915$	−75.4015	−2.49270
$916$	0	0
$917$	75.0537	2.47849
$918$	0	0
$919$	46.9584	1.54901	0.774507	−	0.632566i	$-0.217998\pi$
$919$	46.9584	1.54901	0.774507	+	0.632566i	$0.217998\pi$
$920$	0	0
$921$	−51.9746	−1.71262
$922$	0	0
$923$	0	0
$924$	0	0
$925$	13.6241	0.447958
$926$	0	0
$927$	−0.829744	−0.0272524
$928$	0	0
$929$	49.4181	1.62136	0.810678	−	0.585492i	$-0.199099\pi$
$929$	49.4181	1.62136	0.810678	+	0.585492i	$0.199099\pi$
$930$	0	0
$931$	−15.6408	−0.512607
$932$	0	0
$933$	−55.1477	−1.80545
$934$	0	0
$935$	−40.7044	−1.33118
$936$	0	0
$937$	35.6815	1.16566	0.582832	−	0.812593i	$-0.301945\pi$
$937$	35.6815	1.16566	0.582832	+	0.812593i	$0.301945\pi$
$938$	0	0
$939$	52.3807	1.70938
$940$	0	0
$941$	18.5877	0.605940	0.302970	−	0.953000i	$-0.402022\pi$
$941$	18.5877	0.605940	0.302970	+	0.953000i	$0.402022\pi$
$942$	0	0
$943$	−31.6217	−1.02974
$944$	0	0
$945$	−97.6836	−3.17765
$946$	0	0
$947$	12.7621	0.414713	0.207357	−	0.978265i	$-0.433514\pi$
$947$	12.7621	0.414713	0.207357	+	0.978265i	$0.433514\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−43.4450	−1.40880
$952$	0	0
$953$	28.9937	0.939199	0.469599	−	0.882880i	$-0.344398\pi$
$953$	28.9937	0.939199	0.469599	+	0.882880i	$0.344398\pi$
$954$	0	0
$955$	−4.82117	−0.156009
$956$	0	0
$957$	50.6580	1.63754
$958$	0	0
$959$	−27.3853	−0.884317
$960$	0	0
$961$	−19.7389	−0.636737
$962$	0	0
$963$	0.139352	0.00449054
$964$	0	0
$965$	25.4712	0.819948
$966$	0	0
$967$	14.7152	0.473209	0.236604	−	0.971606i	$-0.423965\pi$
$967$	14.7152	0.473209	0.236604	+	0.971606i	$0.423965\pi$
$968$	0	0
$969$	3.76561	0.120969
$970$	0	0
$971$	6.85405	0.219957	0.109979	−	0.993934i	$-0.464922\pi$
$971$	6.85405	0.219957	0.109979	+	0.993934i	$0.464922\pi$
$972$	0	0
$973$	74.8144	2.39844
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−61.7070	−1.97418	−0.987091	−	0.160160i	$-0.948799\pi$
$977$	−61.7070	−1.97418	−0.987091	+	0.160160i	$0.948799\pi$
$978$	0	0
$979$	52.4543	1.67645
$980$	0	0
$981$	3.01155	0.0961514
$982$	0	0
$983$	11.4681	0.365777	0.182888	−	0.983134i	$-0.441455\pi$
$983$	11.4681	0.365777	0.182888	+	0.983134i	$0.441455\pi$
$984$	0	0
$985$	−24.5782	−0.783127
$986$	0	0
$987$	63.7813	2.03018
$988$	0	0
$989$	−21.7089	−0.690301
$990$	0	0
$991$	45.3646	1.44105	0.720527	−	0.693427i	$-0.243900\pi$
$991$	45.3646	1.44105	0.720527	+	0.693427i	$0.243900\pi$
$992$	0	0
$993$	17.9937	0.571013
$994$	0	0
$995$	−5.75129	−0.182328
$996$	0	0
$997$	14.4474	0.457554	0.228777	−	0.973479i	$-0.426527\pi$
$997$	14.4474	0.457554	0.228777	+	0.973479i	$0.426527\pi$
$998$	0	0
$999$	−8.38341	−0.265239

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.bt.1.9	yes	12
4.3	odd	2	inner	5408.2.a.bt.1.4	✓	12
13.12	even	2		5408.2.a.bu.1.9	yes	12
52.51	odd	2		5408.2.a.bu.1.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5408.2.a.bt.1.4	✓	12	4.3	odd	2	inner
5408.2.a.bt.1.9	yes	12	1.1	even	1	trivial
5408.2.a.bu.1.4	yes	12	52.51	odd	2
5408.2.a.bu.1.9	yes	12	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q+1.62324 q^{3} -3.72519 q^{5} -4.80058 q^{7} -0.365087 q^{9} -4.59139 q^{11} -6.04688 q^{15} -2.37985 q^{17} -0.974773 q^{19} -7.79251 q^{21} -6.51389 q^{23} +8.87701 q^{25} -5.46235 q^{27} -6.79705 q^{29} -3.35576 q^{31} -7.45294 q^{33} +17.8831 q^{35} +1.53476 q^{37} +4.85450 q^{41} +3.33270 q^{43} +1.36002 q^{45} -8.18495 q^{47} +16.0456 q^{49} -3.86307 q^{51} -9.39514 q^{53} +17.1038 q^{55} -1.58229 q^{57} -2.67262 q^{59} +12.4695 q^{61} +1.75263 q^{63} +5.60928 q^{67} -10.5736 q^{69} +15.3889 q^{71} -8.45473 q^{73} +14.4095 q^{75} +22.0414 q^{77} +1.33465 q^{79} -7.77145 q^{81} +0.817576 q^{83} +8.86537 q^{85} -11.0333 q^{87} -11.4245 q^{89} -5.44721 q^{93} +3.63121 q^{95} +5.81428 q^{97} +1.67626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{5} + 14 q^{9} + 14 q^{17} - 28 q^{21} + 24 q^{25} + 18 q^{29} + 20 q^{33} - 8 q^{37} + 36 q^{41} - 66 q^{45} + 34 q^{49} + 38 q^{53} + 22 q^{57} + 36 q^{61} - 12 q^{73} + 64 q^{77} + 8 q^{81}+ \cdots - 8 q^{97}+O(q^{100}) \) 12 * q - 8 * q^5 + 14 * q^9 + 14 * q^17 - 28 * q^21 + 24 * q^25 + 18 * q^29 + 20 * q^33 - 8 * q^37 + 36 * q^41 - 66 * q^45 + 34 * q^49 + 38 * q^53 + 22 * q^57 + 36 * q^61 - 12 * q^73 + 64 * q^77 + 8 * q^81 - 18 * q^85 + 44 * q^89 - 4 * q^93 - 8 * q^97

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