LMFDB - Embedded newform 2268.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, 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("2268.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.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:	$18.1100711784$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	2268.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.94282 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.94282 q^{5} +1.00000 q^{7} +0.942820 q^{11} -1.00000 q^{13} +5.60301 q^{17} +1.28263 q^{19} -4.66019 q^{23} +10.5458 q^{25} -7.66019 q^{29} +7.82846 q^{31} -3.94282 q^{35} -9.82846 q^{37} +0.942820 q^{41} +9.26320 q^{43} -5.28263 q^{47} +1.00000 q^{49} -9.22545 q^{53} -3.71737 q^{55} -9.54583 q^{59} -10.5458 q^{61} +3.94282 q^{65} -1.71737 q^{67} +3.54583 q^{71} +2.71737 q^{73} +0.942820 q^{77} -16.3743 q^{79} -0.396990 q^{83} -22.0917 q^{85} +5.50808 q^{89} -1.00000 q^{91} -5.05718 q^{95} -13.5458 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7	$ 3 q - 3 q^{5} + 3 q^{7} - 6 q^{11} - 3 q^{13} + 3 q^{19} - 6 q^{23} + 6 q^{25} - 15 q^{29} - 3 q^{31} - 3 q^{35} - 3 q^{37} - 6 q^{41} + 3 q^{43} - 15 q^{47} + 3 q^{49} - 18 q^{53} - 12 q^{55} - 3 q^{59} - 6 q^{61} + 3 q^{65} - 6 q^{67} - 15 q^{71} + 9 q^{73} - 6 q^{77} + 3 q^{79} - 18 q^{83} - 15 q^{85} + 6 q^{89} - 3 q^{91} - 24 q^{95} - 15 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 - 6 * q^11 - 3 * q^13 + 3 * q^19 - 6 * q^23 + 6 * q^25 - 15 * q^29 - 3 * q^31 - 3 * q^35 - 3 * q^37 - 6 * q^41 + 3 * q^43 - 15 * q^47 + 3 * q^49 - 18 * q^53 - 12 * q^55 - 3 * q^59 - 6 * q^61 + 3 * q^65 - 6 * q^67 - 15 * q^71 + 9 * q^73 - 6 * q^77 + 3 * q^79 - 18 * q^83 - 15 * q^85 + 6 * q^89 - 3 * q^91 - 24 * q^95 - 15 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.94282	−1.76328	−0.881641	−	0.471920i	$-0.843561\pi$
$5$	−3.94282	−1.76328	−0.881641	+	0.471920i	$0.843561\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.942820	0.284271	0.142135	−	0.989847i	$-0.454603\pi$
$11$	0.942820	0.284271	0.142135	+	0.989847i	$0.454603\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.60301	1.35893	0.679465	−	0.733708i	$-0.262212\pi$
$17$	5.60301	1.35893	0.679465	+	0.733708i	$0.262212\pi$
$18$	0	0
$19$	1.28263	0.294256	0.147128	−	0.989117i	$-0.452997\pi$
$19$	1.28263	0.294256	0.147128	+	0.989117i	$0.452997\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.66019	−0.971717	−0.485858	−	0.874038i	$-0.661493\pi$
$23$	−4.66019	−0.971717	−0.485858	+	0.874038i	$0.661493\pi$
$24$	0	0
$25$	10.5458	2.10917
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.66019	−1.42246	−0.711231	−	0.702959i	$-0.751862\pi$
$29$	−7.66019	−1.42246	−0.711231	+	0.702959i	$0.751862\pi$
$30$	0	0
$31$	7.82846	1.40603	0.703016	−	0.711174i	$-0.251836\pi$
$31$	7.82846	1.40603	0.703016	+	0.711174i	$0.251836\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.94282	−0.666458
$36$	0	0
$37$	−9.82846	−1.61579	−0.807894	−	0.589327i	$-0.799393\pi$
$37$	−9.82846	−1.61579	−0.807894	+	0.589327i	$0.799393\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.942820	0.147244	0.0736219	−	0.997286i	$-0.476544\pi$
$41$	0.942820	0.147244	0.0736219	+	0.997286i	$0.476544\pi$
$42$	0	0
$43$	9.26320	1.41262	0.706312	−	0.707900i	$-0.250357\pi$
$43$	9.26320	1.41262	0.706312	+	0.707900i	$0.250357\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.28263	−0.770551	−0.385275	−	0.922802i	$-0.625894\pi$
$47$	−5.28263	−0.770551	−0.385275	+	0.922802i	$0.625894\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.22545	−1.26721	−0.633607	−	0.773655i	$-0.718426\pi$
$53$	−9.22545	−1.26721	−0.633607	+	0.773655i	$0.718426\pi$
$54$	0	0
$55$	−3.71737	−0.501250
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.54583	−1.24276	−0.621381	−	0.783509i	$-0.713428\pi$
$59$	−9.54583	−1.24276	−0.621381	+	0.783509i	$0.713428\pi$
$60$	0	0
$61$	−10.5458	−1.35026	−0.675128	−	0.737701i	$-0.735911\pi$
$61$	−10.5458	−1.35026	−0.675128	+	0.737701i	$0.735911\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.94282	0.489047
$66$	0	0
$67$	−1.71737	−0.209810	−0.104905	−	0.994482i	$-0.533454\pi$
$67$	−1.71737	−0.209810	−0.104905	+	0.994482i	$0.533454\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.54583	0.420813	0.210406	−	0.977614i	$-0.432521\pi$
$71$	3.54583	0.420813	0.210406	+	0.977614i	$0.432521\pi$
$72$	0	0
$73$	2.71737	0.318044	0.159022	−	0.987275i	$-0.449166\pi$
$73$	2.71737	0.318044	0.159022	+	0.987275i	$0.449166\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.942820	0.107444
$78$	0	0
$79$	−16.3743	−1.84225	−0.921126	−	0.389265i	$-0.872729\pi$
$79$	−16.3743	−1.84225	−0.921126	+	0.389265i	$0.872729\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.396990	−0.0435753	−0.0217877	−	0.999763i	$-0.506936\pi$
$83$	−0.396990	−0.0435753	−0.0217877	+	0.999763i	$0.506936\pi$
$84$	0	0
$85$	−22.0917	−2.39618
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.50808	0.583855	0.291928	−	0.956440i	$-0.405703\pi$
$89$	5.50808	0.583855	0.291928	+	0.956440i	$0.405703\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.05718	−0.518856
$96$	0	0
$97$	−13.5458	−1.37537	−0.687685	−	0.726009i	$-0.741373\pi$
$97$	−13.5458	−1.37537	−0.687685	+	0.726009i	$0.741373\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.60301	−0.259009	−0.129505	−	0.991579i	$-0.541339\pi$
$101$	−2.60301	−0.259009	−0.129505	+	0.991579i	$0.541339\pi$
$102$	0	0
$103$	15.9806	1.57461	0.787306	−	0.616562i	$-0.211475\pi$
$103$	15.9806	1.57461	0.787306	+	0.616562i	$0.211475\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.8856	−1.05235	−0.526177	−	0.850375i	$-0.676375\pi$
$107$	−10.8856	−1.05235	−0.526177	+	0.850375i	$0.676375\pi$
$108$	0	0
$109$	−3.28263	−0.314419	−0.157209	−	0.987565i	$-0.550250\pi$
$109$	−3.28263	−0.314419	−0.157209	+	0.987565i	$0.550250\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.45090	−0.606850	−0.303425	−	0.952855i	$-0.598130\pi$
$113$	−6.45090	−0.606850	−0.303425	+	0.952855i	$0.598130\pi$
$114$	0	0
$115$	18.3743	1.71341
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.60301	0.513627
$120$	0	0
$121$	−10.1111	−0.919190
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−21.8662	−1.95577
$126$	0	0
$127$	−0.828460	−0.0735140	−0.0367570	−	0.999324i	$-0.511703\pi$
$127$	−0.828460	−0.0735140	−0.0367570	+	0.999324i	$0.511703\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.43147	0.474550	0.237275	−	0.971443i	$-0.423746\pi$
$131$	5.43147	0.474550	0.237275	+	0.971443i	$0.423746\pi$
$132$	0	0
$133$	1.28263	0.111218
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.1488	−1.80687	−0.903434	−	0.428728i	$-0.858962\pi$
$137$	−21.1488	−1.80687	−0.903434	+	0.428728i	$0.858962\pi$
$138$	0	0
$139$	−1.84789	−0.156736	−0.0783680	−	0.996924i	$-0.524971\pi$
$139$	−1.84789	−0.156736	−0.0783680	+	0.996924i	$0.524971\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.942820	−0.0788426
$144$	0	0
$145$	30.2028	2.50820
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.1488	0.995272	0.497636	−	0.867386i	$-0.334202\pi$
$149$	12.1488	0.995272	0.497636	+	0.867386i	$0.334202\pi$
$150$	0	0
$151$	−8.98057	−0.730828	−0.365414	−	0.930845i	$-0.619073\pi$
$151$	−8.98057	−0.730828	−0.365414	+	0.930845i	$0.619073\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−30.8662	−2.47923
$156$	0	0
$157$	−11.8090	−0.942463	−0.471232	−	0.882010i	$-0.656190\pi$
$157$	−11.8090	−0.942463	−0.471232	+	0.882010i	$0.656190\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.66019	−0.367274
$162$	0	0
$163$	−12.2826	−0.962050	−0.481025	−	0.876707i	$-0.659735\pi$
$163$	−12.2826	−0.962050	−0.481025	+	0.876707i	$0.659735\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.8856	−1.77094	−0.885472	−	0.464693i	$-0.846165\pi$
$167$	−22.8856	−1.77094	−0.885472	+	0.464693i	$0.846165\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.2632	1.46455	0.732277	−	0.681007i	$-0.238458\pi$
$173$	19.2632	1.46455	0.732277	+	0.681007i	$0.238458\pi$
$174$	0	0
$175$	10.5458	0.797190
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.83173	−0.361140	−0.180570	−	0.983562i	$-0.557794\pi$
$179$	−4.83173	−0.361140	−0.180570	+	0.983562i	$0.557794\pi$
$180$	0	0
$181$	3.26320	0.242552	0.121276	−	0.992619i	$-0.461301\pi$
$181$	3.26320	0.242552	0.121276	+	0.992619i	$0.461301\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	38.7518	2.84909
$186$	0	0
$187$	5.28263	0.386304
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.6979	1.06351	0.531753	−	0.846900i	$-0.321534\pi$
$191$	14.6979	1.06351	0.531753	+	0.846900i	$0.321534\pi$
$192$	0	0
$193$	10.1111	0.727812	0.363906	−	0.931436i	$-0.381443\pi$
$193$	10.1111	0.727812	0.363906	+	0.931436i	$0.381443\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.6375	1.18537	0.592686	−	0.805434i	$-0.298067\pi$
$197$	16.6375	1.18537	0.592686	+	0.805434i	$0.298067\pi$
$198$	0	0
$199$	13.1111	0.929421	0.464710	−	0.885463i	$-0.346158\pi$
$199$	13.1111	0.929421	0.464710	+	0.885463i	$0.346158\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.66019	−0.537640
$204$	0	0
$205$	−3.71737	−0.259632
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.20929	0.0836483
$210$	0	0
$211$	2.13052	0.146671	0.0733355	−	0.997307i	$-0.476636\pi$
$211$	2.13052	0.146671	0.0733355	+	0.997307i	$0.476636\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−36.5231	−2.49086
$216$	0	0
$217$	7.82846	0.531431
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.60301	−0.376899
$222$	0	0
$223$	−14.0917	−0.943647	−0.471824	−	0.881693i	$-0.656404\pi$
$223$	−14.0917	−0.943647	−0.471824	+	0.881693i	$0.656404\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.79071	0.317971	0.158985	−	0.987281i	$-0.449178\pi$
$227$	4.79071	0.317971	0.158985	+	0.987281i	$0.449178\pi$
$228$	0	0
$229$	18.0917	1.19553	0.597765	−	0.801671i	$-0.296055\pi$
$229$	18.0917	1.19553	0.597765	+	0.801671i	$0.296055\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.14884	−0.206287	−0.103144	−	0.994666i	$-0.532890\pi$
$233$	−3.14884	−0.206287	−0.103144	+	0.994666i	$0.532890\pi$
$234$	0	0
$235$	20.8285	1.35870
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.3204	−0.796939	−0.398470	−	0.917182i	$-0.630459\pi$
$239$	−12.3204	−0.796939	−0.398470	+	0.917182i	$0.630459\pi$
$240$	0	0
$241$	14.5458	0.936979	0.468490	−	0.883469i	$-0.344798\pi$
$241$	14.5458	0.936979	0.468490	+	0.883469i	$0.344798\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.94282	−0.251898
$246$	0	0
$247$	−1.28263	−0.0816118
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.28263	−0.144078	−0.0720392	−	0.997402i	$-0.522951\pi$
$251$	−2.28263	−0.144078	−0.0720392	+	0.997402i	$0.522951\pi$
$252$	0	0
$253$	−4.39372	−0.276231
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.793980	0.0495271	0.0247636	−	0.999693i	$-0.492117\pi$
$257$	0.793980	0.0495271	0.0247636	+	0.999693i	$0.492117\pi$
$258$	0	0
$259$	−9.82846	−0.610711
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.4887	0.646758	0.323379	−	0.946270i	$-0.395181\pi$
$263$	10.4887	0.646758	0.323379	+	0.946270i	$0.395181\pi$
$264$	0	0
$265$	36.3743	2.23445
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.3398	0.630429	0.315215	−	0.949020i	$-0.397923\pi$
$269$	10.3398	0.630429	0.315215	+	0.949020i	$0.397923\pi$
$270$	0	0
$271$	−30.8285	−1.87270	−0.936348	−	0.351074i	$-0.885817\pi$
$271$	−30.8285	−1.87270	−0.936348	+	0.351074i	$0.885817\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.94282	0.599575
$276$	0	0
$277$	14.2438	0.855825	0.427913	−	0.903820i	$-0.359249\pi$
$277$	14.2438	0.855825	0.427913	+	0.903820i	$0.359249\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.92339	−0.532325	−0.266162	−	0.963928i	$-0.585756\pi$
$281$	−8.92339	−0.532325	−0.266162	+	0.963928i	$0.585756\pi$
$282$	0	0
$283$	−23.2632	−1.38285	−0.691427	−	0.722446i	$-0.743018\pi$
$283$	−23.2632	−1.38285	−0.691427	+	0.722446i	$0.743018\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.942820	0.0556529
$288$	0	0
$289$	14.3937	0.846689
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.8824	1.39522	0.697611	−	0.716476i	$-0.254246\pi$
$293$	23.8824	1.39522	0.697611	+	0.716476i	$0.254246\pi$
$294$	0	0
$295$	37.6375	2.19134
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.66019	0.269506
$300$	0	0
$301$	9.26320	0.533922
$302$	0	0
$303$	0	0
$304$	0	0
$305$	41.5803	2.38088
$306$	0	0
$307$	−7.37429	−0.420873	−0.210436	−	0.977608i	$-0.567489\pi$
$307$	−7.37429	−0.420873	−0.210436	+	0.977608i	$0.567489\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.17154	0.520070	0.260035	−	0.965599i	$-0.416266\pi$
$311$	9.17154	0.520070	0.260035	+	0.965599i	$0.416266\pi$
$312$	0	0
$313$	16.1111	0.910653	0.455326	−	0.890325i	$-0.349523\pi$
$313$	16.1111	0.910653	0.455326	+	0.890325i	$0.349523\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.26320	−0.239445	−0.119723	−	0.992807i	$-0.538201\pi$
$317$	−4.26320	−0.239445	−0.119723	+	0.992807i	$0.538201\pi$
$318$	0	0
$319$	−7.22218	−0.404364
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.18659	0.399873
$324$	0	0
$325$	−10.5458	−0.584977
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.28263	−0.291241
$330$	0	0
$331$	−12.2826	−0.675114	−0.337557	−	0.941305i	$-0.609601\pi$
$331$	−12.2826	−0.675114	−0.337557	+	0.941305i	$0.609601\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.77128	0.369955
$336$	0	0
$337$	−26.9806	−1.46973	−0.734863	−	0.678216i	$-0.762753\pi$
$337$	−26.9806	−1.46973	−0.734863	+	0.678216i	$0.762753\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.38083	0.399694
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.5048	−1.15444	−0.577219	−	0.816589i	$-0.695862\pi$
$347$	−21.5048	−1.15444	−0.577219	+	0.816589i	$0.695862\pi$
$348$	0	0
$349$	−5.09166	−0.272550	−0.136275	−	0.990671i	$-0.543513\pi$
$349$	−5.09166	−0.272550	−0.136275	+	0.990671i	$0.543513\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.56853	0.509281	0.254641	−	0.967036i	$-0.418043\pi$
$353$	9.56853	0.509281	0.254641	+	0.967036i	$0.418043\pi$
$354$	0	0
$355$	−13.9806	−0.742012
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.9623	0.578565	0.289283	−	0.957244i	$-0.406583\pi$
$359$	10.9623	0.578565	0.289283	+	0.957244i	$0.406583\pi$
$360$	0	0
$361$	−17.3549	−0.913414
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.7141	−0.560802
$366$	0	0
$367$	4.69794	0.245230	0.122615	−	0.992454i	$-0.460872\pi$
$367$	4.69794	0.245230	0.122615	+	0.992454i	$0.460872\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.22545	−0.478962
$372$	0	0
$373$	9.26320	0.479630	0.239815	−	0.970819i	$-0.422913\pi$
$373$	9.26320	0.479630	0.239815	+	0.970819i	$0.422913\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.66019	0.394520
$378$	0	0
$379$	−13.5458	−0.695803	−0.347901	−	0.937531i	$-0.613106\pi$
$379$	−13.5458	−0.695803	−0.347901	+	0.937531i	$0.613106\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.2794	0.627446	0.313723	−	0.949515i	$-0.398424\pi$
$383$	12.2794	0.627446	0.313723	+	0.949515i	$0.398424\pi$
$384$	0	0
$385$	−3.71737	−0.189455
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.51135	−0.380841	−0.190420	−	0.981703i	$-0.560985\pi$
$389$	−7.51135	−0.380841	−0.190420	+	0.981703i	$0.560985\pi$
$390$	0	0
$391$	−26.1111	−1.32049
$392$	0	0
$393$	0	0
$394$	0	0
$395$	64.5609	3.24841
$396$	0	0
$397$	−22.9201	−1.15033	−0.575164	−	0.818038i	$-0.695062\pi$
$397$	−22.9201	−1.15033	−0.575164	+	0.818038i	$0.695062\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.1261	−1.05499	−0.527495	−	0.849558i	$-0.676868\pi$
$401$	−21.1261	−1.05499	−0.527495	+	0.849558i	$0.676868\pi$
$402$	0	0
$403$	−7.82846	−0.389963
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.26647	−0.459322
$408$	0	0
$409$	3.43474	0.169837	0.0849185	−	0.996388i	$-0.472937\pi$
$409$	3.43474	0.169837	0.0849185	+	0.996388i	$0.472937\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.54583	−0.469720
$414$	0	0
$415$	1.56526	0.0768356
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.4315	−0.998143	−0.499071	−	0.866561i	$-0.666325\pi$
$419$	−20.4315	−0.998143	−0.499071	+	0.866561i	$0.666325\pi$
$420$	0	0
$421$	15.8090	0.770485	0.385242	−	0.922815i	$-0.374118\pi$
$421$	15.8090	0.770485	0.385242	+	0.922815i	$0.374118\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	59.0884	2.86621
$426$	0	0
$427$	−10.5458	−0.510348
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.7518	−1.28859	−0.644296	−	0.764777i	$-0.722849\pi$
$431$	−26.7518	−1.28859	−0.644296	+	0.764777i	$0.722849\pi$
$432$	0	0
$433$	−14.4347	−0.693689	−0.346845	−	0.937923i	$-0.612747\pi$
$433$	−14.4347	−0.693689	−0.346845	+	0.937923i	$0.612747\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.97730	−0.285933
$438$	0	0
$439$	−38.6375	−1.84407	−0.922033	−	0.387110i	$-0.873473\pi$
$439$	−38.6375	−1.84407	−0.922033	+	0.387110i	$0.873473\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.5653	0.501971	0.250985	−	0.967991i	$-0.419245\pi$
$443$	10.5653	0.501971	0.250985	+	0.967991i	$0.419245\pi$
$444$	0	0
$445$	−21.7174	−1.02950
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.37429	−0.442400	−0.221200	−	0.975228i	$-0.570997\pi$
$449$	−9.37429	−0.442400	−0.221200	+	0.975228i	$0.570997\pi$
$450$	0	0
$451$	0.888910	0.0418571
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.94282	0.184842
$456$	0	0
$457$	20.7174	0.969118	0.484559	−	0.874759i	$-0.338980\pi$
$457$	20.7174	0.969118	0.484559	+	0.874759i	$0.338980\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.2599	1.64222	0.821109	−	0.570771i	$-0.193356\pi$
$461$	35.2599	1.64222	0.821109	+	0.570771i	$0.193356\pi$
$462$	0	0
$463$	7.11109	0.330480	0.165240	−	0.986253i	$-0.447160\pi$
$463$	7.11109	0.330480	0.165240	+	0.986253i	$0.447160\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.03448	0.232968	0.116484	−	0.993193i	$-0.462838\pi$
$467$	5.03448	0.232968	0.116484	+	0.993193i	$0.462838\pi$
$468$	0	0
$469$	−1.71737	−0.0793008
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.73353	0.401568
$474$	0	0
$475$	13.5264	0.620634
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.26647	0.286322	0.143161	−	0.989699i	$-0.454273\pi$
$479$	6.26647	0.286322	0.143161	+	0.989699i	$0.454273\pi$
$480$	0	0
$481$	9.82846	0.448139
$482$	0	0
$483$	0	0
$484$	0	0
$485$	53.4088	2.42517
$486$	0	0
$487$	14.0722	0.637674	0.318837	−	0.947810i	$-0.396708\pi$
$487$	14.0722	0.637674	0.318837	+	0.947810i	$0.396708\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.94282	0.177937	0.0889685	−	0.996034i	$-0.471643\pi$
$491$	3.94282	0.177937	0.0889685	+	0.996034i	$0.471643\pi$
$492$	0	0
$493$	−42.9201	−1.93302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.54583	0.159052
$498$	0	0
$499$	19.6569	0.879965	0.439982	−	0.898006i	$-0.354985\pi$
$499$	19.6569	0.879965	0.439982	+	0.898006i	$0.354985\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.2632	−0.457613	−0.228807	−	0.973472i	$-0.573482\pi$
$503$	−10.2632	−0.457613	−0.228807	+	0.973472i	$0.573482\pi$
$504$	0	0
$505$	10.2632	0.456706
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.33981	0.0593860	0.0296930	−	0.999559i	$-0.490547\pi$
$509$	1.33981	0.0593860	0.0296930	+	0.999559i	$0.490547\pi$
$510$	0	0
$511$	2.71737	0.120209
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−63.0085	−2.77649
$516$	0	0
$517$	−4.98057	−0.219045
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.7940	0.691947	0.345973	−	0.938244i	$-0.387549\pi$
$521$	15.7940	0.691947	0.345973	+	0.938244i	$0.387549\pi$
$522$	0	0
$523$	−10.6764	−0.466844	−0.233422	−	0.972376i	$-0.574992\pi$
$523$	−10.6764	−0.466844	−0.233422	+	0.972376i	$0.574992\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	43.8629	1.91070
$528$	0	0
$529$	−1.28263	−0.0557665
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.942820	−0.0408381
$534$	0	0
$535$	42.9201	1.85560
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.942820	0.0406101
$540$	0	0
$541$	16.6569	0.716137	0.358068	−	0.933695i	$-0.383435\pi$
$541$	16.6569	0.716137	0.358068	+	0.933695i	$0.383435\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.9428	0.554409
$546$	0	0
$547$	15.6375	0.668611	0.334305	−	0.942465i	$-0.391498\pi$
$547$	15.6375	0.668611	0.334305	+	0.942465i	$0.391498\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.82519	−0.418567
$552$	0	0
$553$	−16.3743	−0.696306
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.7907	−0.711445	−0.355723	−	0.934592i	$-0.615765\pi$
$557$	−16.7907	−0.711445	−0.355723	+	0.934592i	$0.615765\pi$
$558$	0	0
$559$	−9.26320	−0.391792
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.1715	0.639404	0.319702	−	0.947518i	$-0.396417\pi$
$563$	15.1715	0.639404	0.319702	+	0.947518i	$0.396417\pi$
$564$	0	0
$565$	25.4347	1.07005
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.5264	1.48934	0.744672	−	0.667431i	$-0.232606\pi$
$569$	35.5264	1.48934	0.744672	+	0.667431i	$0.232606\pi$
$570$	0	0
$571$	−1.54583	−0.0646910	−0.0323455	−	0.999477i	$-0.510298\pi$
$571$	−1.54583	−0.0646910	−0.0323455	+	0.999477i	$0.510298\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−49.1456	−2.04951
$576$	0	0
$577$	−1.67635	−0.0697874	−0.0348937	−	0.999391i	$-0.511109\pi$
$577$	−1.67635	−0.0697874	−0.0348937	+	0.999391i	$0.511109\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.396990	−0.0164699
$582$	0	0
$583$	−8.69794	−0.360232
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.4692	−1.62907	−0.814535	−	0.580115i	$-0.803008\pi$
$587$	−39.4692	−1.62907	−0.814535	+	0.580115i	$0.803008\pi$
$588$	0	0
$589$	10.0410	0.413733
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.8317	0.937587	0.468793	−	0.883308i	$-0.344689\pi$
$593$	22.8317	0.937587	0.468793	+	0.883308i	$0.344689\pi$
$594$	0	0
$595$	−22.0917	−0.905670
$596$	0	0
$597$	0	0
$598$	0	0
$599$	45.2750	1.84989	0.924943	−	0.380106i	$-0.124113\pi$
$599$	45.2750	1.84989	0.924943	+	0.380106i	$0.124113\pi$
$600$	0	0
$601$	27.0506	1.10342	0.551709	−	0.834036i	$-0.313976\pi$
$601$	27.0506	1.10342	0.551709	+	0.834036i	$0.313976\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	39.8662	1.62079
$606$	0	0
$607$	−18.6569	−0.757261	−0.378631	−	0.925548i	$-0.623605\pi$
$607$	−18.6569	−0.757261	−0.378631	+	0.925548i	$0.623605\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.28263	0.213712
$612$	0	0
$613$	−28.9611	−1.16973	−0.584865	−	0.811131i	$-0.698852\pi$
$613$	−28.9611	−1.16973	−0.584865	+	0.811131i	$0.698852\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.6914	1.51740	0.758699	−	0.651441i	$-0.225835\pi$
$617$	37.6914	1.51740	0.758699	+	0.651441i	$0.225835\pi$
$618$	0	0
$619$	0.0194303	0.000780971 0	0.000390485	−	1.00000i	$-0.499876\pi$
$619$	0.0194303	0.000780971 0	0.000390485		1.00000i	$0.499876\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.50808	0.220677
$624$	0	0
$625$	33.4854	1.33942
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−55.0690	−2.19574
$630$	0	0
$631$	28.4854	1.13399	0.566993	−	0.823723i	$-0.308107\pi$
$631$	28.4854	1.13399	0.566993	+	0.823723i	$0.308107\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.26647	0.129626
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	44.5393	1.75920	0.879598	−	0.475718i	$-0.157812\pi$
$641$	44.5393	1.75920	0.879598	+	0.475718i	$0.157812\pi$
$642$	0	0
$643$	9.01943	0.355692	0.177846	−	0.984058i	$-0.443087\pi$
$643$	9.01943	0.355692	0.177846	+	0.984058i	$0.443087\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.8252	1.68363	0.841816	−	0.539765i	$-0.181487\pi$
$647$	42.8252	1.68363	0.841816	+	0.539765i	$0.181487\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.60301	0.219263	0.109631	−	0.993972i	$-0.465033\pi$
$653$	5.60301	0.219263	0.109631	+	0.993972i	$0.465033\pi$
$654$	0	0
$655$	−21.4153	−0.836765
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6.52751	−0.254276	−0.127138	−	0.991885i	$-0.540579\pi$
$659$	−6.52751	−0.254276	−0.127138	+	0.991885i	$0.540579\pi$
$660$	0	0
$661$	−11.0194	−0.428606	−0.214303	−	0.976767i	$-0.568748\pi$
$661$	−11.0194	−0.428606	−0.214303	+	0.976767i	$0.568748\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.05718	−0.196109
$666$	0	0
$667$	35.6979	1.38223
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.94282	−0.383838
$672$	0	0
$673$	−3.35486	−0.129320	−0.0646602	−	0.997907i	$-0.520596\pi$
$673$	−3.35486	−0.129320	−0.0646602	+	0.997907i	$0.520596\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−40.6375	−1.56183	−0.780913	−	0.624640i	$-0.785246\pi$
$677$	−40.6375	−1.56183	−0.780913	+	0.624640i	$0.785246\pi$
$678$	0	0
$679$	−13.5458	−0.519841
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14.0755	−0.538584	−0.269292	−	0.963059i	$-0.586790\pi$
$683$	−14.0755	−0.538584	−0.269292	+	0.963059i	$0.586790\pi$
$684$	0	0
$685$	83.3861	3.18602
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.22545	0.351462
$690$	0	0
$691$	−43.9201	−1.67080	−0.835400	−	0.549642i	$-0.814764\pi$
$691$	−43.9201	−1.67080	−0.835400	+	0.549642i	$0.814764\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.28590	0.276370
$696$	0	0
$697$	5.28263	0.200094
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.69578	0.290666	0.145333	−	0.989383i	$-0.453575\pi$
$701$	7.69578	0.290666	0.145333	+	0.989383i	$0.453575\pi$
$702$	0	0
$703$	−12.6063	−0.475455
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.60301	−0.0978963
$708$	0	0
$709$	0.222181	0.00834417	0.00417209	−	0.999991i	$-0.498672\pi$
$709$	0.222181	0.00834417	0.00417209	+	0.999991i	$0.498672\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36.4821	−1.36627
$714$	0	0
$715$	3.71737	0.139022
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.64403	−0.0986056	−0.0493028	−	0.998784i	$-0.515700\pi$
$719$	−2.64403	−0.0986056	−0.0493028	+	0.998784i	$0.515700\pi$
$720$	0	0
$721$	15.9806	0.595148
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−80.7831	−3.00021
$726$	0	0
$727$	25.4854	0.945200	0.472600	−	0.881277i	$-0.343315\pi$
$727$	25.4854	0.945200	0.472600	+	0.881277i	$0.343315\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	51.9018	1.91966
$732$	0	0
$733$	1.86948	0.0690508	0.0345254	−	0.999404i	$-0.489008\pi$
$733$	1.86948	0.0690508	0.0345254	+	0.999404i	$0.489008\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.61917	−0.0596429
$738$	0	0
$739$	51.1639	1.88209	0.941047	−	0.338276i	$-0.109844\pi$
$739$	51.1639	1.88209	0.941047	+	0.338276i	$0.109844\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.7335	−1.42100	−0.710498	−	0.703700i	$-0.751530\pi$
$743$	−38.7335	−1.42100	−0.710498	+	0.703700i	$0.751530\pi$
$744$	0	0
$745$	−47.9007	−1.75495
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.8856	−0.397753
$750$	0	0
$751$	6.19097	0.225912	0.112956	−	0.993600i	$-0.463968\pi$
$751$	6.19097	0.225912	0.112956	+	0.993600i	$0.463968\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	35.4088	1.28866
$756$	0	0
$757$	−13.7174	−0.498566	−0.249283	−	0.968431i	$-0.580195\pi$
$757$	−13.7174	−0.498566	−0.249283	+	0.968431i	$0.580195\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	39.3204	1.42536	0.712681	−	0.701488i	$-0.247481\pi$
$761$	39.3204	1.42536	0.712681	+	0.701488i	$0.247481\pi$
$762$	0	0
$763$	−3.28263	−0.118839
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.54583	0.344680
$768$	0	0
$769$	22.7292	0.819634	0.409817	−	0.912168i	$-0.365593\pi$
$769$	22.7292	0.819634	0.409817	+	0.912168i	$0.365593\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.7324	−1.03343	−0.516717	−	0.856156i	$-0.672846\pi$
$773$	−28.7324	−1.03343	−0.516717	+	0.856156i	$0.672846\pi$
$774$	0	0
$775$	82.5576	2.96556
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.20929	0.0433273
$780$	0	0
$781$	3.34308	0.119625
$782$	0	0
$783$	0	0
$784$	0	0
$785$	46.5609	1.66183
$786$	0	0
$787$	−54.0118	−1.92531	−0.962656	−	0.270728i	$-0.912736\pi$
$787$	−54.0118	−1.92531	−0.962656	+	0.270728i	$0.912736\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.45090	−0.229368
$792$	0	0
$793$	10.5458	0.374493
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.207131	−0.00733695	−0.00366848	−	0.999993i	$-0.501168\pi$
$797$	−0.207131	−0.00733695	−0.00366848	+	0.999993i	$0.501168\pi$
$798$	0	0
$799$	−29.5986	−1.04712
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.56199	0.0904107
$804$	0	0
$805$	18.3743	0.647609
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.0539	−1.26759	−0.633794	−	0.773502i	$-0.718503\pi$
$809$	−36.0539	−1.26759	−0.633794	+	0.773502i	$0.718503\pi$
$810$	0	0
$811$	−2.01943	−0.0709118	−0.0354559	−	0.999371i	$-0.511288\pi$
$811$	−2.01943	−0.0709118	−0.0354559	+	0.999371i	$0.511288\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	48.4282	1.69637
$816$	0	0
$817$	11.8813	0.415673
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.3743	−1.06007	−0.530035	−	0.847976i	$-0.677821\pi$
$821$	−30.3743	−1.06007	−0.530035	+	0.847976i	$0.677821\pi$
$822$	0	0
$823$	−13.1305	−0.457701	−0.228851	−	0.973462i	$-0.573497\pi$
$823$	−13.1305	−0.457701	−0.228851	+	0.973462i	$0.573497\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.3236	−0.393762	−0.196881	−	0.980427i	$-0.563081\pi$
$827$	−11.3236	−0.393762	−0.196881	+	0.980427i	$0.563081\pi$
$828$	0	0
$829$	−11.0604	−0.384145	−0.192073	−	0.981381i	$-0.561521\pi$
$829$	−11.0604	−0.384145	−0.192073	+	0.981381i	$0.561521\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.60301	0.194133
$834$	0	0
$835$	90.2340	3.12267
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.5426	0.363970	0.181985	−	0.983301i	$-0.441748\pi$
$839$	10.5426	0.363970	0.181985	+	0.983301i	$0.441748\pi$
$840$	0	0
$841$	29.6785	1.02340
$842$	0	0
$843$	0	0
$844$	0	0
$845$	47.3138	1.62765
$846$	0	0
$847$	−10.1111	−0.347421
$848$	0	0
$849$	0	0
$850$	0	0
$851$	45.8025	1.57009
$852$	0	0
$853$	−27.2826	−0.934139	−0.467070	−	0.884220i	$-0.654690\pi$
$853$	−27.2826	−0.934139	−0.467070	+	0.884220i	$0.654690\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.43147	−0.0830574	−0.0415287	−	0.999137i	$-0.513223\pi$
$857$	−2.43147	−0.0830574	−0.0415287	+	0.999137i	$0.513223\pi$
$858$	0	0
$859$	−31.7486	−1.08325	−0.541624	−	0.840621i	$-0.682190\pi$
$859$	−31.7486	−1.08325	−0.541624	+	0.840621i	$0.682190\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.8252	−0.947181	−0.473590	−	0.880745i	$-0.657042\pi$
$863$	−27.8252	−0.947181	−0.473590	+	0.880745i	$0.657042\pi$
$864$	0	0
$865$	−75.9513	−2.58242
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−15.4380	−0.523699
$870$	0	0
$871$	1.71737	0.0581909
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−21.8662	−0.739213
$876$	0	0
$877$	−4.58685	−0.154887	−0.0774434	−	0.996997i	$-0.524676\pi$
$877$	−4.58685	−0.154887	−0.0774434	+	0.996997i	$0.524676\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	57.4465	1.93542	0.967711	−	0.252062i	$-0.0811085\pi$
$881$	57.4465	1.93542	0.967711	+	0.252062i	$0.0811085\pi$
$882$	0	0
$883$	36.3937	1.22475	0.612373	−	0.790569i	$-0.290215\pi$
$883$	36.3937	1.22475	0.612373	+	0.790569i	$0.290215\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.4476	−1.45883	−0.729414	−	0.684072i	$-0.760207\pi$
$887$	−43.4476	−1.45883	−0.729414	+	0.684072i	$0.760207\pi$
$888$	0	0
$889$	−0.828460	−0.0277857
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.77566	−0.226739
$894$	0	0
$895$	19.0506	0.636793
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−59.9675	−2.00003
$900$	0	0
$901$	−51.6903	−1.72205
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.8662	−0.427687
$906$	0	0
$907$	22.3138	0.740919	0.370459	−	0.928849i	$-0.379200\pi$
$907$	22.3138	0.740919	0.370459	+	0.928849i	$0.379200\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15.3560	−0.508766	−0.254383	−	0.967104i	$-0.581872\pi$
$911$	−15.3560	−0.508766	−0.254383	+	0.967104i	$0.581872\pi$
$912$	0	0
$913$	−0.374290	−0.0123872
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.43147	0.179363
$918$	0	0
$919$	−36.8285	−1.21486	−0.607429	−	0.794374i	$-0.707799\pi$
$919$	−36.8285	−1.21486	−0.607429	+	0.794374i	$0.707799\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.54583	−0.116712
$924$	0	0
$925$	−103.649	−3.40797
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−41.7292	−1.36909	−0.684545	−	0.728971i	$-0.739999\pi$
$929$	−41.7292	−1.36909	−0.684545	+	0.728971i	$0.739999\pi$
$930$	0	0
$931$	1.28263	0.0420365
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20.8285	−0.681163
$936$	0	0
$937$	−25.3743	−0.828942	−0.414471	−	0.910063i	$-0.636033\pi$
$937$	−25.3743	−0.828942	−0.414471	+	0.910063i	$0.636033\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47.1423	1.53680	0.768398	−	0.639973i	$-0.221054\pi$
$941$	47.1423	1.53680	0.768398	+	0.639973i	$0.221054\pi$
$942$	0	0
$943$	−4.39372	−0.143079
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.5070	−1.12133	−0.560663	−	0.828044i	$-0.689454\pi$
$947$	−34.5070	−1.12133	−0.560663	+	0.828044i	$0.689454\pi$
$948$	0	0
$949$	−2.71737	−0.0882096
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44.1833	−1.43124	−0.715619	−	0.698491i	$-0.753855\pi$
$953$	−44.1833	−1.43124	−0.715619	+	0.698491i	$0.753855\pi$
$954$	0	0
$955$	−57.9513	−1.87526
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−21.1488	−0.682932
$960$	0	0
$961$	30.2848	0.976929
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−39.8662	−1.28334
$966$	0	0
$967$	42.4660	1.36561	0.682806	−	0.730599i	$-0.260759\pi$
$967$	42.4660	1.36561	0.682806	+	0.730599i	$0.260759\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.05391	0.194279	0.0971396	−	0.995271i	$-0.469031\pi$
$971$	6.05391	0.194279	0.0971396	+	0.995271i	$0.469031\pi$
$972$	0	0
$973$	−1.84789	−0.0592406
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44.0528	−1.40937	−0.704687	−	0.709518i	$-0.748913\pi$
$977$	−44.0528	−1.40937	−0.704687	+	0.709518i	$0.748913\pi$
$978$	0	0
$979$	5.19313	0.165973
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.47033	−0.142581	−0.0712907	−	0.997456i	$-0.522712\pi$
$983$	−4.47033	−0.142581	−0.0712907	+	0.997456i	$0.522712\pi$
$984$	0	0
$985$	−65.5986	−2.09015
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43.1683	−1.37267
$990$	0	0
$991$	−31.5048	−1.00078	−0.500392	−	0.865799i	$-0.666811\pi$
$991$	−31.5048	−1.00078	−0.500392	+	0.865799i	$0.666811\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−51.6947	−1.63883
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.a.g.1.1		3
3.2	odd	2		2268.2.a.j.1.3		3
4.3	odd	2		9072.2.a.bt.1.1		3
9.2	odd	6		756.2.j.a.253.1		6
9.4	even	3		252.2.j.b.169.1	yes	6
9.5	odd	6		756.2.j.a.505.1		6
9.7	even	3		252.2.j.b.85.1	✓	6
12.11	even	2		9072.2.a.bz.1.3		3
36.7	odd	6		1008.2.r.g.337.3		6
36.11	even	6		3024.2.r.i.1009.1		6
36.23	even	6		3024.2.r.i.2017.1		6
36.31	odd	6		1008.2.r.g.673.3		6
63.2	odd	6		5292.2.l.g.361.3		6
63.4	even	3		1764.2.l.d.961.2		6
63.5	even	6		5292.2.i.g.2125.3		6
63.11	odd	6		5292.2.i.d.1549.1		6
63.13	odd	6		1764.2.j.d.1177.3		6
63.16	even	3		1764.2.l.d.949.2		6
63.20	even	6		5292.2.j.e.1765.3		6
63.23	odd	6		5292.2.i.d.2125.1		6
63.25	even	3		1764.2.i.f.373.3		6
63.31	odd	6		1764.2.l.g.961.2		6
63.32	odd	6		5292.2.l.g.3313.3		6
63.34	odd	6		1764.2.j.d.589.3		6
63.38	even	6		5292.2.i.g.1549.3		6
63.40	odd	6		1764.2.i.e.1537.1		6
63.41	even	6		5292.2.j.e.3529.3		6
63.47	even	6		5292.2.l.d.361.1		6
63.52	odd	6		1764.2.i.e.373.1		6
63.58	even	3		1764.2.i.f.1537.3		6
63.59	even	6		5292.2.l.d.3313.1		6
63.61	odd	6		1764.2.l.g.949.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.b.85.1	✓	6	9.7	even	3
252.2.j.b.169.1	yes	6	9.4	even	3
756.2.j.a.253.1		6	9.2	odd	6
756.2.j.a.505.1		6	9.5	odd	6
1008.2.r.g.337.3		6	36.7	odd	6
1008.2.r.g.673.3		6	36.31	odd	6
1764.2.i.e.373.1		6	63.52	odd	6
1764.2.i.e.1537.1		6	63.40	odd	6
1764.2.i.f.373.3		6	63.25	even	3
1764.2.i.f.1537.3		6	63.58	even	3
1764.2.j.d.589.3		6	63.34	odd	6
1764.2.j.d.1177.3		6	63.13	odd	6
1764.2.l.d.949.2		6	63.16	even	3
1764.2.l.d.961.2		6	63.4	even	3
1764.2.l.g.949.2		6	63.61	odd	6
1764.2.l.g.961.2		6	63.31	odd	6
2268.2.a.g.1.1		3	1.1	even	1	trivial
2268.2.a.j.1.3		3	3.2	odd	2
3024.2.r.i.1009.1		6	36.11	even	6
3024.2.r.i.2017.1		6	36.23	even	6
5292.2.i.d.1549.1		6	63.11	odd	6
5292.2.i.d.2125.1		6	63.23	odd	6
5292.2.i.g.1549.3		6	63.38	even	6
5292.2.i.g.2125.3		6	63.5	even	6
5292.2.j.e.1765.3		6	63.20	even	6
5292.2.j.e.3529.3		6	63.41	even	6
5292.2.l.d.361.1		6	63.47	even	6
5292.2.l.d.3313.1		6	63.59	even	6
5292.2.l.g.361.3		6	63.2	odd	6
5292.2.l.g.3313.3		6	63.32	odd	6
9072.2.a.bt.1.1		3	4.3	odd	2
9072.2.a.bz.1.3		3	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.a (trivial)

\(f(q)\)	\(=\)	\(q-3.94282 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.94282 q^{5} +1.00000 q^{7} +0.942820 q^{11} -1.00000 q^{13} +5.60301 q^{17} +1.28263 q^{19} -4.66019 q^{23} +10.5458 q^{25} -7.66019 q^{29} +7.82846 q^{31} -3.94282 q^{35} -9.82846 q^{37} +0.942820 q^{41} +9.26320 q^{43} -5.28263 q^{47} +1.00000 q^{49} -9.22545 q^{53} -3.71737 q^{55} -9.54583 q^{59} -10.5458 q^{61} +3.94282 q^{65} -1.71737 q^{67} +3.54583 q^{71} +2.71737 q^{73} +0.942820 q^{77} -16.3743 q^{79} -0.396990 q^{83} -22.0917 q^{85} +5.50808 q^{89} -1.00000 q^{91} -5.05718 q^{95} -13.5458 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7	\( 3 q - 3 q^{5} + 3 q^{7} - 6 q^{11} - 3 q^{13} + 3 q^{19} - 6 q^{23} + 6 q^{25} - 15 q^{29} - 3 q^{31} - 3 q^{35} - 3 q^{37} - 6 q^{41} + 3 q^{43} - 15 q^{47} + 3 q^{49} - 18 q^{53} - 12 q^{55} - 3 q^{59} - 6 q^{61} + 3 q^{65} - 6 q^{67} - 15 q^{71} + 9 q^{73} - 6 q^{77} + 3 q^{79} - 18 q^{83} - 15 q^{85} + 6 q^{89} - 3 q^{91} - 24 q^{95} - 15 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 - 6 * q^11 - 3 * q^13 + 3 * q^19 - 6 * q^23 + 6 * q^25 - 15 * q^29 - 3 * q^31 - 3 * q^35 - 3 * q^37 - 6 * q^41 + 3 * q^43 - 15 * q^47 + 3 * q^49 - 18 * q^53 - 12 * q^55 - 3 * q^59 - 6 * q^61 + 3 * q^65 - 6 * q^67 - 15 * q^71 + 9 * q^73 - 6 * q^77 + 3 * q^79 - 18 * q^83 - 15 * q^85 + 6 * q^89 - 3 * q^91 - 24 * q^95 - 15 * q^97

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