LMFDB - Embedded newform 3388.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3388,2,Mod(1,3388)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3388, 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("3388.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3388 = 2^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3388.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:	$27.0533162048$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	3388.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{3} +2.00000 q^{5} +1.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.44949 q^{3} +2.00000 q^{5} +1.00000 q^{7} +3.00000 q^{9} -4.44949 q^{13} -4.89898 q^{15} -4.44949 q^{17} -4.89898 q^{19} -2.44949 q^{21} +8.89898 q^{23} -1.00000 q^{25} +6.89898 q^{29} +1.55051 q^{31} +2.00000 q^{35} +4.00000 q^{37} +10.8990 q^{39} -5.34847 q^{41} +6.89898 q^{43} +6.00000 q^{45} -1.55051 q^{47} +1.00000 q^{49} +10.8990 q^{51} -9.79796 q^{53} +12.0000 q^{57} -7.34847 q^{59} +9.34847 q^{61} +3.00000 q^{63} -8.89898 q^{65} +14.6969 q^{67} -21.7980 q^{69} -3.10102 q^{71} -7.55051 q^{73} +2.44949 q^{75} +1.10102 q^{79} -9.00000 q^{81} +7.10102 q^{83} -8.89898 q^{85} -16.8990 q^{87} +6.00000 q^{89} -4.44949 q^{91} -3.79796 q^{93} -9.79796 q^{95} +14.8990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 4 * q^5 + 2 * q^7 + 6 * q^9	$ 2 q + 4 q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} + 8 q^{23} - 2 q^{25} + 4 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 4 q^{41} + 4 q^{43} + 12 q^{45} - 8 q^{47} + 2 q^{49} + 12 q^{51} + 24 q^{57} + 4 q^{61} + 6 q^{63} - 8 q^{65} - 24 q^{69} - 16 q^{71} - 20 q^{73} + 12 q^{79} - 18 q^{81} + 24 q^{83} - 8 q^{85} - 24 q^{87} + 12 q^{89} - 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 + 2 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 + 8 * q^23 - 2 * q^25 + 4 * q^29 + 8 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 + 4 * q^41 + 4 * q^43 + 12 * q^45 - 8 * q^47 + 2 * q^49 + 12 * q^51 + 24 * q^57 + 4 * q^61 + 6 * q^63 - 8 * q^65 - 24 * q^69 - 16 * q^71 - 20 * q^73 + 12 * q^79 - 18 * q^81 + 24 * q^83 - 8 * q^85 - 24 * q^87 + 12 * q^89 - 4 * q^91 + 12 * q^93 + 20 * 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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.44949	−1.23407	−0.617033	−	0.786937i	$-0.711666\pi$
$13$	−4.44949	−1.23407	−0.617033	+	0.786937i	$0.711666\pi$
$14$	0	0
$15$	−4.89898	−1.26491
$16$	0	0
$17$	−4.44949	−1.07916	−0.539580	−	0.841934i	$-0.681417\pi$
$17$	−4.44949	−1.07916	−0.539580	+	0.841934i	$0.681417\pi$
$18$	0	0
$19$	−4.89898	−1.12390	−0.561951	−	0.827170i	$-0.689949\pi$
$19$	−4.89898	−1.12390	−0.561951	+	0.827170i	$0.689949\pi$
$20$	0	0
$21$	−2.44949	−0.534522
$22$	0	0
$23$	8.89898	1.85557	0.927783	−	0.373121i	$-0.121712\pi$
$23$	8.89898	1.85557	0.927783	+	0.373121i	$0.121712\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.89898	1.28111	0.640554	−	0.767913i	$-0.278705\pi$
$29$	6.89898	1.28111	0.640554	+	0.767913i	$0.278705\pi$
$30$	0	0
$31$	1.55051	0.278480	0.139240	−	0.990259i	$-0.455534\pi$
$31$	1.55051	0.278480	0.139240	+	0.990259i	$0.455534\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	10.8990	1.74523
$40$	0	0
$41$	−5.34847	−0.835291	−0.417645	−	0.908610i	$-0.637145\pi$
$41$	−5.34847	−0.835291	−0.417645	+	0.908610i	$0.637145\pi$
$42$	0	0
$43$	6.89898	1.05208	0.526042	−	0.850458i	$-0.323675\pi$
$43$	6.89898	1.05208	0.526042	+	0.850458i	$0.323675\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	−1.55051	−0.226165	−0.113083	−	0.993586i	$-0.536072\pi$
$47$	−1.55051	−0.226165	−0.113083	+	0.993586i	$0.536072\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	10.8990	1.52616
$52$	0	0
$53$	−9.79796	−1.34585	−0.672927	−	0.739709i	$-0.734963\pi$
$53$	−9.79796	−1.34585	−0.672927	+	0.739709i	$0.734963\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.0000	1.58944
$58$	0	0
$59$	−7.34847	−0.956689	−0.478345	−	0.878172i	$-0.658763\pi$
$59$	−7.34847	−0.956689	−0.478345	+	0.878172i	$0.658763\pi$
$60$	0	0
$61$	9.34847	1.19695	0.598474	−	0.801142i	$-0.295774\pi$
$61$	9.34847	1.19695	0.598474	+	0.801142i	$0.295774\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	−8.89898	−1.10378
$66$	0	0
$67$	14.6969	1.79552	0.897758	−	0.440488i	$-0.145195\pi$
$67$	14.6969	1.79552	0.897758	+	0.440488i	$0.145195\pi$
$68$	0	0
$69$	−21.7980	−2.62417
$70$	0	0
$71$	−3.10102	−0.368023	−0.184012	−	0.982924i	$-0.558908\pi$
$71$	−3.10102	−0.368023	−0.184012	+	0.982924i	$0.558908\pi$
$72$	0	0
$73$	−7.55051	−0.883720	−0.441860	−	0.897084i	$-0.645681\pi$
$73$	−7.55051	−0.883720	−0.441860	+	0.897084i	$0.645681\pi$
$74$	0	0
$75$	2.44949	0.282843
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.10102	0.123874	0.0619372	−	0.998080i	$-0.480272\pi$
$79$	1.10102	0.123874	0.0619372	+	0.998080i	$0.480272\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	7.10102	0.779438	0.389719	−	0.920934i	$-0.372572\pi$
$83$	7.10102	0.779438	0.389719	+	0.920934i	$0.372572\pi$
$84$	0	0
$85$	−8.89898	−0.965230
$86$	0	0
$87$	−16.8990	−1.81176
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.44949	−0.466433
$92$	0	0
$93$	−3.79796	−0.393830
$94$	0	0
$95$	−9.79796	−1.00525
$96$	0	0
$97$	14.8990	1.51276	0.756381	−	0.654131i	$-0.226966\pi$
$97$	14.8990	1.51276	0.756381	+	0.654131i	$0.226966\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.55051	0.353289	0.176644	−	0.984275i	$-0.443476\pi$
$101$	3.55051	0.353289	0.176644	+	0.984275i	$0.443476\pi$
$102$	0	0
$103$	9.55051	0.941040	0.470520	−	0.882389i	$-0.344066\pi$
$103$	9.55051	0.941040	0.470520	+	0.882389i	$0.344066\pi$
$104$	0	0
$105$	−4.89898	−0.478091
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−5.10102	−0.488589	−0.244295	−	0.969701i	$-0.578556\pi$
$109$	−5.10102	−0.488589	−0.244295	+	0.969701i	$0.578556\pi$
$110$	0	0
$111$	−9.79796	−0.929981
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	17.7980	1.65967
$116$	0	0
$117$	−13.3485	−1.23407
$118$	0	0
$119$	−4.44949	−0.407884
$120$	0	0
$121$	0	0
$122$	0	0
$123$	13.1010	1.18128
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−17.7980	−1.57931	−0.789657	−	0.613549i	$-0.789741\pi$
$127$	−17.7980	−1.57931	−0.789657	+	0.613549i	$0.789741\pi$
$128$	0	0
$129$	−16.8990	−1.48787
$130$	0	0
$131$	9.79796	0.856052	0.428026	−	0.903767i	$-0.359209\pi$
$131$	9.79796	0.856052	0.428026	+	0.903767i	$0.359209\pi$
$132$	0	0
$133$	−4.89898	−0.424795
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.79796	−0.837096	−0.418548	−	0.908195i	$-0.637461\pi$
$137$	−9.79796	−0.837096	−0.418548	+	0.908195i	$0.637461\pi$
$138$	0	0
$139$	6.69694	0.568027	0.284013	−	0.958820i	$-0.408334\pi$
$139$	6.69694	0.568027	0.284013	+	0.958820i	$0.408334\pi$
$140$	0	0
$141$	3.79796	0.319846
$142$	0	0
$143$	0	0
$144$	0	0
$145$	13.7980	1.14586
$146$	0	0
$147$	−2.44949	−0.202031
$148$	0	0
$149$	7.79796	0.638834	0.319417	−	0.947614i	$-0.396513\pi$
$149$	7.79796	0.638834	0.319417	+	0.947614i	$0.396513\pi$
$150$	0	0
$151$	20.6969	1.68429	0.842146	−	0.539249i	$-0.181292\pi$
$151$	20.6969	1.68429	0.842146	+	0.539249i	$0.181292\pi$
$152$	0	0
$153$	−13.3485	−1.07916
$154$	0	0
$155$	3.10102	0.249080
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	24.0000	1.90332
$160$	0	0
$161$	8.89898	0.701338
$162$	0	0
$163$	−0.898979	−0.0704135	−0.0352068	−	0.999380i	$-0.511209\pi$
$163$	−0.898979	−0.0704135	−0.0352068	+	0.999380i	$0.511209\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.79796	0.448660	0.224330	−	0.974513i	$-0.427981\pi$
$167$	5.79796	0.448660	0.224330	+	0.974513i	$0.427981\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	−14.6969	−1.12390
$172$	0	0
$173$	14.2474	1.08321	0.541607	−	0.840632i	$-0.317816\pi$
$173$	14.2474	1.08321	0.541607	+	0.840632i	$0.317816\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	18.0000	1.35296
$178$	0	0
$179$	23.5959	1.76364	0.881821	−	0.471585i	$-0.156318\pi$
$179$	23.5959	1.76364	0.881821	+	0.471585i	$0.156318\pi$
$180$	0	0
$181$	18.8990	1.40475	0.702375	−	0.711807i	$-0.252123\pi$
$181$	18.8990	1.40475	0.702375	+	0.711807i	$0.252123\pi$
$182$	0	0
$183$	−22.8990	−1.69274
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.7980	1.28782	0.643908	−	0.765103i	$-0.277312\pi$
$191$	17.7980	1.28782	0.643908	+	0.765103i	$0.277312\pi$
$192$	0	0
$193$	17.5959	1.26658	0.633291	−	0.773914i	$-0.281704\pi$
$193$	17.5959	1.26658	0.633291	+	0.773914i	$0.281704\pi$
$194$	0	0
$195$	21.7980	1.56098
$196$	0	0
$197$	−13.5959	−0.968669	−0.484335	−	0.874883i	$-0.660938\pi$
$197$	−13.5959	−0.968669	−0.484335	+	0.874883i	$0.660938\pi$
$198$	0	0
$199$	−18.0454	−1.27921	−0.639603	−	0.768706i	$-0.720901\pi$
$199$	−18.0454	−1.27921	−0.639603	+	0.768706i	$0.720901\pi$
$200$	0	0
$201$	−36.0000	−2.53924
$202$	0	0
$203$	6.89898	0.484213
$204$	0	0
$205$	−10.6969	−0.747107
$206$	0	0
$207$	26.6969	1.85557
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	7.59592	0.520464
$214$	0	0
$215$	13.7980	0.941013
$216$	0	0
$217$	1.55051	0.105255
$218$	0	0
$219$	18.4949	1.24977
$220$	0	0
$221$	19.7980	1.33175
$222$	0	0
$223$	21.1464	1.41607	0.708035	−	0.706178i	$-0.249582\pi$
$223$	21.1464	1.41607	0.708035	+	0.706178i	$0.249582\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	17.7980	1.18129	0.590646	−	0.806931i	$-0.298873\pi$
$227$	17.7980	1.18129	0.590646	+	0.806931i	$0.298873\pi$
$228$	0	0
$229$	−14.8990	−0.984552	−0.492276	−	0.870439i	$-0.663835\pi$
$229$	−14.8990	−0.984552	−0.492276	+	0.870439i	$0.663835\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.8990	0.714016	0.357008	−	0.934101i	$-0.383797\pi$
$233$	10.8990	0.714016	0.357008	+	0.934101i	$0.383797\pi$
$234$	0	0
$235$	−3.10102	−0.202288
$236$	0	0
$237$	−2.69694	−0.175185
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−13.3485	−0.859850	−0.429925	−	0.902864i	$-0.641460\pi$
$241$	−13.3485	−0.859850	−0.429925	+	0.902864i	$0.641460\pi$
$242$	0	0
$243$	22.0454	1.41421
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	21.7980	1.38697
$248$	0	0
$249$	−17.3939	−1.10229
$250$	0	0
$251$	−2.44949	−0.154610	−0.0773052	−	0.997007i	$-0.524632\pi$
$251$	−2.44949	−0.154610	−0.0773052	+	0.997007i	$0.524632\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	21.7980	1.36504
$256$	0	0
$257$	15.7980	0.985450	0.492725	−	0.870185i	$-0.336001\pi$
$257$	15.7980	0.985450	0.492725	+	0.870185i	$0.336001\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	20.6969	1.28111
$262$	0	0
$263$	19.5959	1.20834	0.604168	−	0.796857i	$-0.293506\pi$
$263$	19.5959	1.20834	0.604168	+	0.796857i	$0.293506\pi$
$264$	0	0
$265$	−19.5959	−1.20377
$266$	0	0
$267$	−14.6969	−0.899438
$268$	0	0
$269$	−22.8990	−1.39618	−0.698088	−	0.716012i	$-0.745965\pi$
$269$	−22.8990	−1.39618	−0.698088	+	0.716012i	$0.745965\pi$
$270$	0	0
$271$	−30.6969	−1.86471	−0.932353	−	0.361549i	$-0.882248\pi$
$271$	−30.6969	−1.86471	−0.932353	+	0.361549i	$0.882248\pi$
$272$	0	0
$273$	10.8990	0.659636
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.79796	0.228197	0.114099	−	0.993469i	$-0.463602\pi$
$277$	3.79796	0.228197	0.114099	+	0.993469i	$0.463602\pi$
$278$	0	0
$279$	4.65153	0.278480
$280$	0	0
$281$	14.8990	0.888799	0.444399	−	0.895829i	$-0.353417\pi$
$281$	14.8990	0.888799	0.444399	+	0.895829i	$0.353417\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	0	0
$285$	24.0000	1.42164
$286$	0	0
$287$	−5.34847	−0.315710
$288$	0	0
$289$	2.79796	0.164586
$290$	0	0
$291$	−36.4949	−2.13937
$292$	0	0
$293$	25.3485	1.48087	0.740437	−	0.672126i	$-0.234619\pi$
$293$	25.3485	1.48087	0.740437	+	0.672126i	$0.234619\pi$
$294$	0	0
$295$	−14.6969	−0.855689
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−39.5959	−2.28989
$300$	0	0
$301$	6.89898	0.397651
$302$	0	0
$303$	−8.69694	−0.499626
$304$	0	0
$305$	18.6969	1.07058
$306$	0	0
$307$	−17.7980	−1.01578	−0.507892	−	0.861421i	$-0.669575\pi$
$307$	−17.7980	−1.01578	−0.507892	+	0.861421i	$0.669575\pi$
$308$	0	0
$309$	−23.3939	−1.33083
$310$	0	0
$311$	−6.44949	−0.365717	−0.182859	−	0.983139i	$-0.558535\pi$
$311$	−6.44949	−0.365717	−0.182859	+	0.983139i	$0.558535\pi$
$312$	0	0
$313$	−33.5959	−1.89895	−0.949477	−	0.313837i	$-0.898385\pi$
$313$	−33.5959	−1.89895	−0.949477	+	0.313837i	$0.898385\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	25.5959	1.43761	0.718805	−	0.695212i	$-0.244689\pi$
$317$	25.5959	1.43761	0.718805	+	0.695212i	$0.244689\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.79796	−0.546869
$322$	0	0
$323$	21.7980	1.21287
$324$	0	0
$325$	4.44949	0.246813
$326$	0	0
$327$	12.4949	0.690969
$328$	0	0
$329$	−1.55051	−0.0854824
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	0	0
$335$	29.3939	1.60596
$336$	0	0
$337$	8.69694	0.473752	0.236876	−	0.971540i	$-0.423876\pi$
$337$	8.69694	0.473752	0.236876	+	0.971540i	$0.423876\pi$
$338$	0	0
$339$	24.4949	1.33038
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−43.5959	−2.34713
$346$	0	0
$347$	22.4949	1.20759	0.603795	−	0.797140i	$-0.293655\pi$
$347$	22.4949	1.20759	0.603795	+	0.797140i	$0.293655\pi$
$348$	0	0
$349$	−26.2474	−1.40499	−0.702497	−	0.711687i	$-0.747932\pi$
$349$	−26.2474	−1.40499	−0.702497	+	0.711687i	$0.747932\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.5959	−0.723638	−0.361819	−	0.932248i	$-0.617844\pi$
$353$	−13.5959	−0.723638	−0.361819	+	0.932248i	$0.617844\pi$
$354$	0	0
$355$	−6.20204	−0.329170
$356$	0	0
$357$	10.8990	0.576835
$358$	0	0
$359$	−4.69694	−0.247895	−0.123947	−	0.992289i	$-0.539555\pi$
$359$	−4.69694	−0.247895	−0.123947	+	0.992289i	$0.539555\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.1010	−0.790424
$366$	0	0
$367$	−13.1464	−0.686238	−0.343119	−	0.939292i	$-0.611483\pi$
$367$	−13.1464	−0.686238	−0.343119	+	0.939292i	$0.611483\pi$
$368$	0	0
$369$	−16.0454	−0.835291
$370$	0	0
$371$	−9.79796	−0.508685
$372$	0	0
$373$	−23.7980	−1.23221	−0.616106	−	0.787663i	$-0.711291\pi$
$373$	−23.7980	−1.23221	−0.616106	+	0.787663i	$0.711291\pi$
$374$	0	0
$375$	29.3939	1.51789
$376$	0	0
$377$	−30.6969	−1.58097
$378$	0	0
$379$	−12.8990	−0.662576	−0.331288	−	0.943530i	$-0.607483\pi$
$379$	−12.8990	−0.662576	−0.331288	+	0.943530i	$0.607483\pi$
$380$	0	0
$381$	43.5959	2.23349
$382$	0	0
$383$	12.6515	0.646463	0.323232	−	0.946320i	$-0.395231\pi$
$383$	12.6515	0.646463	0.323232	+	0.946320i	$0.395231\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.6969	1.05208
$388$	0	0
$389$	25.5959	1.29776	0.648882	−	0.760889i	$-0.275237\pi$
$389$	25.5959	1.29776	0.648882	+	0.760889i	$0.275237\pi$
$390$	0	0
$391$	−39.5959	−2.00245
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	2.20204	0.110797
$396$	0	0
$397$	13.5959	0.682360	0.341180	−	0.939998i	$-0.389174\pi$
$397$	13.5959	0.682360	0.341180	+	0.939998i	$0.389174\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	−8.20204	−0.409590	−0.204795	−	0.978805i	$-0.565653\pi$
$401$	−8.20204	−0.409590	−0.204795	+	0.978805i	$0.565653\pi$
$402$	0	0
$403$	−6.89898	−0.343663
$404$	0	0
$405$	−18.0000	−0.894427
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16.4495	0.813375	0.406687	−	0.913567i	$-0.366684\pi$
$409$	16.4495	0.813375	0.406687	+	0.913567i	$0.366684\pi$
$410$	0	0
$411$	24.0000	1.18383
$412$	0	0
$413$	−7.34847	−0.361595
$414$	0	0
$415$	14.2020	0.697151
$416$	0	0
$417$	−16.4041	−0.803311
$418$	0	0
$419$	13.5505	0.661986	0.330993	−	0.943633i	$-0.392616\pi$
$419$	13.5505	0.661986	0.330993	+	0.943633i	$0.392616\pi$
$420$	0	0
$421$	13.7980	0.672471	0.336236	−	0.941778i	$-0.390846\pi$
$421$	13.7980	0.672471	0.336236	+	0.941778i	$0.390846\pi$
$422$	0	0
$423$	−4.65153	−0.226165
$424$	0	0
$425$	4.44949	0.215832
$426$	0	0
$427$	9.34847	0.452404
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.8990	0.910332	0.455166	−	0.890407i	$-0.349580\pi$
$431$	18.8990	0.910332	0.455166	+	0.890407i	$0.349580\pi$
$432$	0	0
$433$	30.8990	1.48491	0.742455	−	0.669896i	$-0.233661\pi$
$433$	30.8990	1.48491	0.742455	+	0.669896i	$0.233661\pi$
$434$	0	0
$435$	−33.7980	−1.62049
$436$	0	0
$437$	−43.5959	−2.08548
$438$	0	0
$439$	15.1010	0.720732	0.360366	−	0.932811i	$-0.382652\pi$
$439$	15.1010	0.720732	0.360366	+	0.932811i	$0.382652\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	−19.1010	−0.903447
$448$	0	0
$449$	−19.5959	−0.924789	−0.462394	−	0.886674i	$-0.653010\pi$
$449$	−19.5959	−0.924789	−0.462394	+	0.886674i	$0.653010\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−50.6969	−2.38195
$454$	0	0
$455$	−8.89898	−0.417191
$456$	0	0
$457$	39.3939	1.84277	0.921384	−	0.388654i	$-0.127060\pi$
$457$	39.3939	1.84277	0.921384	+	0.388654i	$0.127060\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.6515	1.05499	0.527493	−	0.849559i	$-0.323132\pi$
$461$	22.6515	1.05499	0.527493	+	0.849559i	$0.323132\pi$
$462$	0	0
$463$	−15.1010	−0.701804	−0.350902	−	0.936412i	$-0.614125\pi$
$463$	−15.1010	−0.701804	−0.350902	+	0.936412i	$0.614125\pi$
$464$	0	0
$465$	−7.59592	−0.352252
$466$	0	0
$467$	−7.34847	−0.340047	−0.170023	−	0.985440i	$-0.554384\pi$
$467$	−7.34847	−0.340047	−0.170023	+	0.985440i	$0.554384\pi$
$468$	0	0
$469$	14.6969	0.678642
$470$	0	0
$471$	−24.4949	−1.12867
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.89898	0.224781
$476$	0	0
$477$	−29.3939	−1.34585
$478$	0	0
$479$	8.89898	0.406605	0.203302	−	0.979116i	$-0.434833\pi$
$479$	8.89898	0.406605	0.203302	+	0.979116i	$0.434833\pi$
$480$	0	0
$481$	−17.7980	−0.811517
$482$	0	0
$483$	−21.7980	−0.991841
$484$	0	0
$485$	29.7980	1.35306
$486$	0	0
$487$	27.1010	1.22806	0.614032	−	0.789281i	$-0.289546\pi$
$487$	27.1010	1.22806	0.614032	+	0.789281i	$0.289546\pi$
$488$	0	0
$489$	2.20204	0.0995797
$490$	0	0
$491$	−3.30306	−0.149065	−0.0745325	−	0.997219i	$-0.523746\pi$
$491$	−3.30306	−0.149065	−0.0745325	+	0.997219i	$0.523746\pi$
$492$	0	0
$493$	−30.6969	−1.38252
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.10102	−0.139100
$498$	0	0
$499$	−13.3031	−0.595527	−0.297763	−	0.954640i	$-0.596241\pi$
$499$	−13.3031	−0.595527	−0.297763	+	0.954640i	$0.596241\pi$
$500$	0	0
$501$	−14.2020	−0.634500
$502$	0	0
$503$	−24.4949	−1.09217	−0.546087	−	0.837729i	$-0.683883\pi$
$503$	−24.4949	−1.09217	−0.546087	+	0.837729i	$0.683883\pi$
$504$	0	0
$505$	7.10102	0.315991
$506$	0	0
$507$	−16.6515	−0.739520
$508$	0	0
$509$	4.69694	0.208188	0.104094	−	0.994567i	$-0.466806\pi$
$509$	4.69694	0.208188	0.104094	+	0.994567i	$0.466806\pi$
$510$	0	0
$511$	−7.55051	−0.334015
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.1010	0.841692
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−34.8990	−1.53190
$520$	0	0
$521$	−18.8990	−0.827979	−0.413990	−	0.910282i	$-0.635865\pi$
$521$	−18.8990	−0.827979	−0.413990	+	0.910282i	$0.635865\pi$
$522$	0	0
$523$	33.3939	1.46021	0.730106	−	0.683334i	$-0.239471\pi$
$523$	33.3939	1.46021	0.730106	+	0.683334i	$0.239471\pi$
$524$	0	0
$525$	2.44949	0.106904
$526$	0	0
$527$	−6.89898	−0.300524
$528$	0	0
$529$	56.1918	2.44312
$530$	0	0
$531$	−22.0454	−0.956689
$532$	0	0
$533$	23.7980	1.03080
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	−57.7980	−2.49417
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−21.5959	−0.928481	−0.464241	−	0.885709i	$-0.653673\pi$
$541$	−21.5959	−0.928481	−0.464241	+	0.885709i	$0.653673\pi$
$542$	0	0
$543$	−46.2929	−1.98662
$544$	0	0
$545$	−10.2020	−0.437007
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	28.0454	1.19695
$550$	0	0
$551$	−33.7980	−1.43984
$552$	0	0
$553$	1.10102	0.0468202
$554$	0	0
$555$	−19.5959	−0.831800
$556$	0	0
$557$	24.2020	1.02547	0.512737	−	0.858546i	$-0.328632\pi$
$557$	24.2020	1.02547	0.512737	+	0.858546i	$0.328632\pi$
$558$	0	0
$559$	−30.6969	−1.29834
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−42.2929	−1.78243	−0.891216	−	0.453580i	$-0.850147\pi$
$563$	−42.2929	−1.78243	−0.891216	+	0.453580i	$0.850147\pi$
$564$	0	0
$565$	−20.0000	−0.841406
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−20.6969	−0.867661	−0.433830	−	0.900995i	$-0.642838\pi$
$569$	−20.6969	−0.867661	−0.433830	+	0.900995i	$0.642838\pi$
$570$	0	0
$571$	29.7980	1.24701	0.623503	−	0.781821i	$-0.285709\pi$
$571$	29.7980	1.24701	0.623503	+	0.781821i	$0.285709\pi$
$572$	0	0
$573$	−43.5959	−1.82125
$574$	0	0
$575$	−8.89898	−0.371113
$576$	0	0
$577$	26.4949	1.10300	0.551499	−	0.834176i	$-0.314056\pi$
$577$	26.4949	1.10300	0.551499	+	0.834176i	$0.314056\pi$
$578$	0	0
$579$	−43.1010	−1.79122
$580$	0	0
$581$	7.10102	0.294600
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−26.6969	−1.10378
$586$	0	0
$587$	26.4495	1.09169	0.545844	−	0.837887i	$-0.316209\pi$
$587$	26.4495	1.09169	0.545844	+	0.837887i	$0.316209\pi$
$588$	0	0
$589$	−7.59592	−0.312984
$590$	0	0
$591$	33.3031	1.36990
$592$	0	0
$593$	35.1464	1.44329	0.721645	−	0.692263i	$-0.243386\pi$
$593$	35.1464	1.44329	0.721645	+	0.692263i	$0.243386\pi$
$594$	0	0
$595$	−8.89898	−0.364823
$596$	0	0
$597$	44.2020	1.80907
$598$	0	0
$599$	−16.8990	−0.690474	−0.345237	−	0.938516i	$-0.612201\pi$
$599$	−16.8990	−0.690474	−0.345237	+	0.938516i	$0.612201\pi$
$600$	0	0
$601$	−21.3485	−0.870822	−0.435411	−	0.900232i	$-0.643397\pi$
$601$	−21.3485	−0.870822	−0.435411	+	0.900232i	$0.643397\pi$
$602$	0	0
$603$	44.0908	1.79552
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.5959	0.957729	0.478864	−	0.877889i	$-0.341049\pi$
$607$	23.5959	0.957729	0.478864	+	0.877889i	$0.341049\pi$
$608$	0	0
$609$	−16.8990	−0.684781
$610$	0	0
$611$	6.89898	0.279103
$612$	0	0
$613$	−12.6969	−0.512825	−0.256412	−	0.966568i	$-0.582540\pi$
$613$	−12.6969	−0.512825	−0.256412	+	0.966568i	$0.582540\pi$
$614$	0	0
$615$	26.2020	1.05657
$616$	0	0
$617$	29.3939	1.18335	0.591676	−	0.806176i	$-0.298466\pi$
$617$	29.3939	1.18335	0.591676	+	0.806176i	$0.298466\pi$
$618$	0	0
$619$	10.4495	0.420000	0.210000	−	0.977701i	$-0.432654\pi$
$619$	10.4495	0.420000	0.210000	+	0.977701i	$0.432654\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17.7980	−0.709651
$630$	0	0
$631$	−21.7980	−0.867763	−0.433882	−	0.900970i	$-0.642856\pi$
$631$	−21.7980	−0.867763	−0.433882	+	0.900970i	$0.642856\pi$
$632$	0	0
$633$	−48.9898	−1.94717
$634$	0	0
$635$	−35.5959	−1.41258
$636$	0	0
$637$	−4.44949	−0.176295
$638$	0	0
$639$	−9.30306	−0.368023
$640$	0	0
$641$	31.5959	1.24796	0.623982	−	0.781439i	$-0.285514\pi$
$641$	31.5959	1.24796	0.623982	+	0.781439i	$0.285514\pi$
$642$	0	0
$643$	−20.2474	−0.798481	−0.399241	−	0.916846i	$-0.630726\pi$
$643$	−20.2474	−0.798481	−0.399241	+	0.916846i	$0.630726\pi$
$644$	0	0
$645$	−33.7980	−1.33079
$646$	0	0
$647$	−24.2474	−0.953266	−0.476633	−	0.879102i	$-0.658143\pi$
$647$	−24.2474	−0.953266	−0.476633	+	0.879102i	$0.658143\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−3.79796	−0.148854
$652$	0	0
$653$	−43.3939	−1.69813	−0.849067	−	0.528285i	$-0.822835\pi$
$653$	−43.3939	−1.69813	−0.849067	+	0.528285i	$0.822835\pi$
$654$	0	0
$655$	19.5959	0.765676
$656$	0	0
$657$	−22.6515	−0.883720
$658$	0	0
$659$	−14.8990	−0.580382	−0.290191	−	0.956969i	$-0.593719\pi$
$659$	−14.8990	−0.580382	−0.290191	+	0.956969i	$0.593719\pi$
$660$	0	0
$661$	9.59592	0.373238	0.186619	−	0.982432i	$-0.440247\pi$
$661$	9.59592	0.373238	0.186619	+	0.982432i	$0.440247\pi$
$662$	0	0
$663$	−48.4949	−1.88339
$664$	0	0
$665$	−9.79796	−0.379949
$666$	0	0
$667$	61.3939	2.37718
$668$	0	0
$669$	−51.7980	−2.00262
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−36.6969	−1.41456	−0.707282	−	0.706932i	$-0.750079\pi$
$673$	−36.6969	−1.41456	−0.707282	+	0.706932i	$0.750079\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.9444	0.958691	0.479345	−	0.877626i	$-0.340874\pi$
$677$	24.9444	0.958691	0.479345	+	0.877626i	$0.340874\pi$
$678$	0	0
$679$	14.8990	0.571770
$680$	0	0
$681$	−43.5959	−1.67060
$682$	0	0
$683$	−45.7980	−1.75241	−0.876205	−	0.481938i	$-0.839933\pi$
$683$	−45.7980	−1.75241	−0.876205	+	0.481938i	$0.839933\pi$
$684$	0	0
$685$	−19.5959	−0.748722
$686$	0	0
$687$	36.4949	1.39237
$688$	0	0
$689$	43.5959	1.66087
$690$	0	0
$691$	−6.85357	−0.260722	−0.130361	−	0.991467i	$-0.541614\pi$
$691$	−6.85357	−0.260722	−0.130361	+	0.991467i	$0.541614\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.3939	0.508059
$696$	0	0
$697$	23.7980	0.901412
$698$	0	0
$699$	−26.6969	−1.00977
$700$	0	0
$701$	−8.69694	−0.328479	−0.164239	−	0.986421i	$-0.552517\pi$
$701$	−8.69694	−0.328479	−0.164239	+	0.986421i	$0.552517\pi$
$702$	0	0
$703$	−19.5959	−0.739074
$704$	0	0
$705$	7.59592	0.286079
$706$	0	0
$707$	3.55051	0.133531
$708$	0	0
$709$	−15.7980	−0.593305	−0.296652	−	0.954986i	$-0.595870\pi$
$709$	−15.7980	−0.593305	−0.296652	+	0.954986i	$0.595870\pi$
$710$	0	0
$711$	3.30306	0.123874
$712$	0	0
$713$	13.7980	0.516738
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.1918	1.46365
$718$	0	0
$719$	10.8536	0.404770	0.202385	−	0.979306i	$-0.435131\pi$
$719$	10.8536	0.404770	0.202385	+	0.979306i	$0.435131\pi$
$720$	0	0
$721$	9.55051	0.355680
$722$	0	0
$723$	32.6969	1.21601
$724$	0	0
$725$	−6.89898	−0.256222
$726$	0	0
$727$	−6.44949	−0.239198	−0.119599	−	0.992822i	$-0.538161\pi$
$727$	−6.44949	−0.239198	−0.119599	+	0.992822i	$0.538161\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−30.6969	−1.13537
$732$	0	0
$733$	−30.6515	−1.13214	−0.566070	−	0.824357i	$-0.691537\pi$
$733$	−30.6515	−1.13214	−0.566070	+	0.824357i	$0.691537\pi$
$734$	0	0
$735$	−4.89898	−0.180702
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−17.3939	−0.639844	−0.319922	−	0.947444i	$-0.603657\pi$
$739$	−17.3939	−0.639844	−0.319922	+	0.947444i	$0.603657\pi$
$740$	0	0
$741$	−53.3939	−1.96147
$742$	0	0
$743$	−19.5959	−0.718905	−0.359452	−	0.933163i	$-0.617036\pi$
$743$	−19.5959	−0.718905	−0.359452	+	0.933163i	$0.617036\pi$
$744$	0	0
$745$	15.5959	0.571390
$746$	0	0
$747$	21.3031	0.779438
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−3.10102	−0.113158	−0.0565789	−	0.998398i	$-0.518019\pi$
$751$	−3.10102	−0.113158	−0.0565789	+	0.998398i	$0.518019\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	41.3939	1.50648
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.8434	0.501822	0.250911	−	0.968010i	$-0.419270\pi$
$761$	13.8434	0.501822	0.250911	+	0.968010i	$0.419270\pi$
$762$	0	0
$763$	−5.10102	−0.184669
$764$	0	0
$765$	−26.6969	−0.965230
$766$	0	0
$767$	32.6969	1.18062
$768$	0	0
$769$	−4.85357	−0.175024	−0.0875121	−	0.996163i	$-0.527892\pi$
$769$	−4.85357	−0.175024	−0.0875121	+	0.996163i	$0.527892\pi$
$770$	0	0
$771$	−38.6969	−1.39364
$772$	0	0
$773$	32.6969	1.17603	0.588014	−	0.808851i	$-0.299910\pi$
$773$	32.6969	1.17603	0.588014	+	0.808851i	$0.299910\pi$
$774$	0	0
$775$	−1.55051	−0.0556960
$776$	0	0
$777$	−9.79796	−0.351500
$778$	0	0
$779$	26.2020	0.938786
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	18.6969	0.666474	0.333237	−	0.942843i	$-0.391859\pi$
$787$	18.6969	0.666474	0.333237	+	0.942843i	$0.391859\pi$
$788$	0	0
$789$	−48.0000	−1.70885
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	−41.5959	−1.47711
$794$	0	0
$795$	48.0000	1.70238
$796$	0	0
$797$	−12.2020	−0.432218	−0.216109	−	0.976369i	$-0.569337\pi$
$797$	−12.2020	−0.432218	−0.216109	+	0.976369i	$0.569337\pi$
$798$	0	0
$799$	6.89898	0.244068
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	17.7980	0.627296
$806$	0	0
$807$	56.0908	1.97449
$808$	0	0
$809$	−13.5959	−0.478007	−0.239004	−	0.971019i	$-0.576821\pi$
$809$	−13.5959	−0.478007	−0.239004	+	0.971019i	$0.576821\pi$
$810$	0	0
$811$	−15.5959	−0.547647	−0.273823	−	0.961780i	$-0.588288\pi$
$811$	−15.5959	−0.547647	−0.273823	+	0.961780i	$0.588288\pi$
$812$	0	0
$813$	75.1918	2.63709
$814$	0	0
$815$	−1.79796	−0.0629798
$816$	0	0
$817$	−33.7980	−1.18244
$818$	0	0
$819$	−13.3485	−0.466433
$820$	0	0
$821$	−3.79796	−0.132550	−0.0662748	−	0.997801i	$-0.521111\pi$
$821$	−3.79796	−0.132550	−0.0662748	+	0.997801i	$0.521111\pi$
$822$	0	0
$823$	−43.1918	−1.50557	−0.752786	−	0.658265i	$-0.771291\pi$
$823$	−43.1918	−1.50557	−0.752786	+	0.658265i	$0.771291\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.3031	−0.810327	−0.405163	−	0.914244i	$-0.632785\pi$
$827$	−23.3031	−0.810327	−0.405163	+	0.914244i	$0.632785\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−9.30306	−0.322720
$832$	0	0
$833$	−4.44949	−0.154166
$834$	0	0
$835$	11.5959	0.401293
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14.4495	0.498852	0.249426	−	0.968394i	$-0.419758\pi$
$839$	14.4495	0.498852	0.249426	+	0.968394i	$0.419758\pi$
$840$	0	0
$841$	18.5959	0.641239
$842$	0	0
$843$	−36.4949	−1.25695
$844$	0	0
$845$	13.5959	0.467714
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−58.7878	−2.01759
$850$	0	0
$851$	35.5959	1.22021
$852$	0	0
$853$	−8.85357	−0.303141	−0.151570	−	0.988446i	$-0.548433\pi$
$853$	−8.85357	−0.303141	−0.151570	+	0.988446i	$0.548433\pi$
$854$	0	0
$855$	−29.3939	−1.00525
$856$	0	0
$857$	−43.1464	−1.47385	−0.736927	−	0.675972i	$-0.763724\pi$
$857$	−43.1464	−1.47385	−0.736927	+	0.675972i	$0.763724\pi$
$858$	0	0
$859$	−39.8434	−1.35944	−0.679719	−	0.733473i	$-0.737898\pi$
$859$	−39.8434	−1.35944	−0.679719	+	0.733473i	$0.737898\pi$
$860$	0	0
$861$	13.1010	0.446482
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	28.4949	0.968856
$866$	0	0
$867$	−6.85357	−0.232760
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−65.3939	−2.21579
$872$	0	0
$873$	44.6969	1.51276
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	−7.30306	−0.246607	−0.123303	−	0.992369i	$-0.539349\pi$
$877$	−7.30306	−0.246607	−0.123303	+	0.992369i	$0.539349\pi$
$878$	0	0
$879$	−62.0908	−2.09427
$880$	0	0
$881$	8.69694	0.293007	0.146504	−	0.989210i	$-0.453198\pi$
$881$	8.69694	0.293007	0.146504	+	0.989210i	$0.453198\pi$
$882$	0	0
$883$	−10.2020	−0.343326	−0.171663	−	0.985156i	$-0.554914\pi$
$883$	−10.2020	−0.343326	−0.171663	+	0.985156i	$0.554914\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	0	0
$887$	−38.2929	−1.28575	−0.642874	−	0.765972i	$-0.722258\pi$
$887$	−38.2929	−1.28575	−0.642874	+	0.765972i	$0.722258\pi$
$888$	0	0
$889$	−17.7980	−0.596924
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.59592	0.254188
$894$	0	0
$895$	47.1918	1.57745
$896$	0	0
$897$	96.9898	3.23839
$898$	0	0
$899$	10.6969	0.356763
$900$	0	0
$901$	43.5959	1.45239
$902$	0	0
$903$	−16.8990	−0.562363
$904$	0	0
$905$	37.7980	1.25645
$906$	0	0
$907$	12.8990	0.428304	0.214152	−	0.976800i	$-0.431301\pi$
$907$	12.8990	0.428304	0.214152	+	0.976800i	$0.431301\pi$
$908$	0	0
$909$	10.6515	0.353289
$910$	0	0
$911$	−12.4949	−0.413974	−0.206987	−	0.978344i	$-0.566366\pi$
$911$	−12.4949	−0.413974	−0.206987	+	0.978344i	$0.566366\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−45.7980	−1.51403
$916$	0	0
$917$	9.79796	0.323557
$918$	0	0
$919$	−43.5959	−1.43810	−0.719048	−	0.694960i	$-0.755422\pi$
$919$	−43.5959	−1.43810	−0.719048	+	0.694960i	$0.755422\pi$
$920$	0	0
$921$	43.5959	1.43653
$922$	0	0
$923$	13.7980	0.454165
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	28.6515	0.941040
$928$	0	0
$929$	16.2929	0.534551	0.267276	−	0.963620i	$-0.413877\pi$
$929$	16.2929	0.534551	0.267276	+	0.963620i	$0.413877\pi$
$930$	0	0
$931$	−4.89898	−0.160558
$932$	0	0
$933$	15.7980	0.517202
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.9444	0.684223	0.342112	−	0.939659i	$-0.388858\pi$
$937$	20.9444	0.684223	0.342112	+	0.939659i	$0.388858\pi$
$938$	0	0
$939$	82.2929	2.68553
$940$	0	0
$941$	−48.4495	−1.57941	−0.789704	−	0.613488i	$-0.789766\pi$
$941$	−48.4495	−1.57941	−0.789704	+	0.613488i	$0.789766\pi$
$942$	0	0
$943$	−47.5959	−1.54994
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.5959	1.28669	0.643347	−	0.765575i	$-0.277545\pi$
$947$	39.5959	1.28669	0.643347	+	0.765575i	$0.277545\pi$
$948$	0	0
$949$	33.5959	1.09057
$950$	0	0
$951$	−62.6969	−2.03309
$952$	0	0
$953$	24.2020	0.783981	0.391991	−	0.919969i	$-0.371787\pi$
$953$	24.2020	0.783981	0.391991	+	0.919969i	$0.371787\pi$
$954$	0	0
$955$	35.5959	1.15186
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.79796	−0.316393
$960$	0	0
$961$	−28.5959	−0.922449
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	35.1918	1.13287
$966$	0	0
$967$	29.1010	0.935826	0.467913	−	0.883775i	$-0.345006\pi$
$967$	29.1010	0.935826	0.467913	+	0.883775i	$0.345006\pi$
$968$	0	0
$969$	−53.3939	−1.71526
$970$	0	0
$971$	6.04541	0.194006	0.0970032	−	0.995284i	$-0.469074\pi$
$971$	6.04541	0.194006	0.0970032	+	0.995284i	$0.469074\pi$
$972$	0	0
$973$	6.69694	0.214694
$974$	0	0
$975$	−10.8990	−0.349047
$976$	0	0
$977$	23.7980	0.761364	0.380682	−	0.924706i	$-0.375689\pi$
$977$	23.7980	0.761364	0.380682	+	0.924706i	$0.375689\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−15.3031	−0.488589
$982$	0	0
$983$	4.65153	0.148361	0.0741804	−	0.997245i	$-0.476366\pi$
$983$	4.65153	0.148361	0.0741804	+	0.997245i	$0.476366\pi$
$984$	0	0
$985$	−27.1918	−0.866404
$986$	0	0
$987$	3.79796	0.120890
$988$	0	0
$989$	61.3939	1.95221
$990$	0	0
$991$	−28.0908	−0.892334	−0.446167	−	0.894950i	$-0.647211\pi$
$991$	−28.0908	−0.892334	−0.446167	+	0.894950i	$0.647211\pi$
$992$	0	0
$993$	78.3837	2.48743
$994$	0	0
$995$	−36.0908	−1.14416
$996$	0	0
$997$	−32.9444	−1.04336	−0.521680	−	0.853141i	$-0.674694\pi$
$997$	−32.9444	−1.04336	−0.521680	+	0.853141i	$0.674694\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3388.2.a.h.1.1		2
11.10	odd	2		308.2.a.b.1.1	✓	2
33.32	even	2		2772.2.a.n.1.2		2
44.43	even	2		1232.2.a.n.1.2		2
55.32	even	4		7700.2.e.j.1849.3		4
55.43	even	4		7700.2.e.j.1849.2		4
55.54	odd	2		7700.2.a.s.1.2		2
77.10	even	6		2156.2.i.i.177.1		4
77.32	odd	6		2156.2.i.e.177.2		4
77.54	even	6		2156.2.i.i.1145.1		4
77.65	odd	6		2156.2.i.e.1145.2		4
77.76	even	2		2156.2.a.c.1.2		2
88.21	odd	2		4928.2.a.bp.1.2		2
88.43	even	2		4928.2.a.bq.1.1		2
308.307	odd	2		8624.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.a.b.1.1	✓	2	11.10	odd	2
1232.2.a.n.1.2		2	44.43	even	2
2156.2.a.c.1.2		2	77.76	even	2
2156.2.i.e.177.2		4	77.32	odd	6
2156.2.i.e.1145.2		4	77.65	odd	6
2156.2.i.i.177.1		4	77.10	even	6
2156.2.i.i.1145.1		4	77.54	even	6
2772.2.a.n.1.2		2	33.32	even	2
3388.2.a.h.1.1		2	1.1	even	1	trivial
4928.2.a.bp.1.2		2	88.21	odd	2
4928.2.a.bq.1.1		2	88.43	even	2
7700.2.a.s.1.2		2	55.54	odd	2
7700.2.e.j.1849.2		4	55.43	even	4
7700.2.e.j.1849.3		4	55.32	even	4
8624.2.a.bj.1.1		2	308.307	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3388 = 2^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3388.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44949 q^{3} +2.00000 q^{5} +1.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.44949 q^{3} +2.00000 q^{5} +1.00000 q^{7} +3.00000 q^{9} -4.44949 q^{13} -4.89898 q^{15} -4.44949 q^{17} -4.89898 q^{19} -2.44949 q^{21} +8.89898 q^{23} -1.00000 q^{25} +6.89898 q^{29} +1.55051 q^{31} +2.00000 q^{35} +4.00000 q^{37} +10.8990 q^{39} -5.34847 q^{41} +6.89898 q^{43} +6.00000 q^{45} -1.55051 q^{47} +1.00000 q^{49} +10.8990 q^{51} -9.79796 q^{53} +12.0000 q^{57} -7.34847 q^{59} +9.34847 q^{61} +3.00000 q^{63} -8.89898 q^{65} +14.6969 q^{67} -21.7980 q^{69} -3.10102 q^{71} -7.55051 q^{73} +2.44949 q^{75} +1.10102 q^{79} -9.00000 q^{81} +7.10102 q^{83} -8.89898 q^{85} -16.8990 q^{87} +6.00000 q^{89} -4.44949 q^{91} -3.79796 q^{93} -9.79796 q^{95} +14.8990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 4 * q^5 + 2 * q^7 + 6 * q^9	\( 2 q + 4 q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} + 8 q^{23} - 2 q^{25} + 4 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 4 q^{41} + 4 q^{43} + 12 q^{45} - 8 q^{47} + 2 q^{49} + 12 q^{51} + 24 q^{57} + 4 q^{61} + 6 q^{63} - 8 q^{65} - 24 q^{69} - 16 q^{71} - 20 q^{73} + 12 q^{79} - 18 q^{81} + 24 q^{83} - 8 q^{85} - 24 q^{87} + 12 q^{89} - 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 + 2 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 + 8 * q^23 - 2 * q^25 + 4 * q^29 + 8 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 + 4 * q^41 + 4 * q^43 + 12 * q^45 - 8 * q^47 + 2 * q^49 + 12 * q^51 + 24 * q^57 + 4 * q^61 + 6 * q^63 - 8 * q^65 - 24 * q^69 - 16 * q^71 - 20 * q^73 + 12 * q^79 - 18 * q^81 + 24 * q^83 - 8 * q^85 - 24 * q^87 + 12 * q^89 - 4 * q^91 + 12 * q^93 + 20 * q^97

Introduction

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