LMFDB - Embedded newform 3344.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.71250$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.98063 q^{3} -3.49235 q^{5} -1.06736 q^{7} +5.88418 q^{9} +O(q^{10})$	$q+2.98063 q^{3} -3.49235 q^{5} -1.06736 q^{7} +5.88418 q^{9} -1.00000 q^{11} -0.0563258 q^{13} -10.4094 q^{15} -4.53628 q^{17} +1.00000 q^{19} -3.18141 q^{21} +1.07949 q^{23} +7.19651 q^{25} +8.59667 q^{27} +0.299905 q^{29} -9.18548 q^{31} -2.98063 q^{33} +3.72760 q^{35} +4.50448 q^{37} -0.167887 q^{39} +12.0009 q^{41} -10.7260 q^{43} -20.5496 q^{45} -2.89630 q^{47} -5.86074 q^{49} -13.5210 q^{51} -12.3213 q^{53} +3.49235 q^{55} +2.98063 q^{57} +1.14582 q^{59} -8.09599 q^{61} -6.28054 q^{63} +0.196709 q^{65} -11.2733 q^{67} +3.21755 q^{69} -13.4948 q^{71} -11.1470 q^{73} +21.4502 q^{75} +1.06736 q^{77} -11.4250 q^{79} +7.97100 q^{81} +13.9802 q^{83} +15.8423 q^{85} +0.893906 q^{87} -0.183185 q^{89} +0.0601200 q^{91} -27.3785 q^{93} -3.49235 q^{95} +5.66263 q^{97} -5.88418 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) $ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9	$ 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

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
$3$	2.98063	1.72087	0.860435	−	0.509561i	$-0.170192\pi$
$3$	2.98063	1.72087	0.860435	+	0.509561i	$0.170192\pi$
$4$	0	0
$5$	−3.49235	−1.56183	−0.780913	−	0.624639i	$-0.785246\pi$
$5$	−3.49235	−1.56183	−0.780913	+	0.624639i	$0.785246\pi$
$6$	0	0
$7$	−1.06736	−0.403424	−0.201712	−	0.979445i	$-0.564651\pi$
$7$	−1.06736	−0.403424	−0.201712	+	0.979445i	$0.564651\pi$
$8$	0	0
$9$	5.88418	1.96139
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.0563258	−0.0156220	−0.00781098	−	0.999969i	$-0.502486\pi$
$13$	−0.0563258	−0.0156220	−0.00781098	+	0.999969i	$0.502486\pi$
$14$	0	0
$15$	−10.4094	−2.68770
$16$	0	0
$17$	−4.53628	−1.10021	−0.550104	−	0.835096i	$-0.685412\pi$
$17$	−4.53628	−1.10021	−0.550104	+	0.835096i	$0.685412\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.18141	−0.694241
$22$	0	0
$23$	1.07949	0.225088	0.112544	−	0.993647i	$-0.464100\pi$
$23$	1.07949	0.225088	0.112544	+	0.993647i	$0.464100\pi$
$24$	0	0
$25$	7.19651	1.43930
$26$	0	0
$27$	8.59667	1.65443
$28$	0	0
$29$	0.299905	0.0556909	0.0278455	−	0.999612i	$-0.491135\pi$
$29$	0.299905	0.0556909	0.0278455	+	0.999612i	$0.491135\pi$
$30$	0	0
$31$	−9.18548	−1.64976	−0.824880	−	0.565307i	$-0.808758\pi$
$31$	−9.18548	−1.64976	−0.824880	+	0.565307i	$0.808758\pi$
$32$	0	0
$33$	−2.98063	−0.518862
$34$	0	0
$35$	3.72760	0.630079
$36$	0	0
$37$	4.50448	0.740531	0.370266	−	0.928926i	$-0.379267\pi$
$37$	4.50448	0.740531	0.370266	+	0.928926i	$0.379267\pi$
$38$	0	0
$39$	−0.167887	−0.0268834
$40$	0	0
$41$	12.0009	1.87423	0.937115	−	0.349020i	$-0.113486\pi$
$41$	12.0009	1.87423	0.937115	+	0.349020i	$0.113486\pi$
$42$	0	0
$43$	−10.7260	−1.63570	−0.817851	−	0.575430i	$-0.804835\pi$
$43$	−10.7260	−1.63570	−0.817851	+	0.575430i	$0.804835\pi$
$44$	0	0
$45$	−20.5496	−3.06335
$46$	0	0
$47$	−2.89630	−0.422469	−0.211235	−	0.977435i	$-0.567748\pi$
$47$	−2.89630	−0.422469	−0.211235	+	0.977435i	$0.567748\pi$
$48$	0	0
$49$	−5.86074	−0.837249
$50$	0	0
$51$	−13.5210	−1.89332
$52$	0	0
$53$	−12.3213	−1.69246	−0.846230	−	0.532818i	$-0.821133\pi$
$53$	−12.3213	−1.69246	−0.846230	+	0.532818i	$0.821133\pi$
$54$	0	0
$55$	3.49235	0.470908
$56$	0	0
$57$	2.98063	0.394795
$58$	0	0
$59$	1.14582	0.149173	0.0745863	−	0.997215i	$-0.476236\pi$
$59$	1.14582	0.149173	0.0745863	+	0.997215i	$0.476236\pi$
$60$	0	0
$61$	−8.09599	−1.03659	−0.518293	−	0.855203i	$-0.673432\pi$
$61$	−8.09599	−1.03659	−0.518293	+	0.855203i	$0.673432\pi$
$62$	0	0
$63$	−6.28054	−0.791273
$64$	0	0
$65$	0.196709	0.0243988
$66$	0	0
$67$	−11.2733	−1.37726	−0.688628	−	0.725115i	$-0.741787\pi$
$67$	−11.2733	−1.37726	−0.688628	+	0.725115i	$0.741787\pi$
$68$	0	0
$69$	3.21755	0.387348
$70$	0	0
$71$	−13.4948	−1.60154	−0.800771	−	0.598970i	$-0.795577\pi$
$71$	−13.4948	−1.60154	−0.800771	+	0.598970i	$0.795577\pi$
$72$	0	0
$73$	−11.1470	−1.30466	−0.652330	−	0.757935i	$-0.726208\pi$
$73$	−11.1470	−1.30466	−0.652330	+	0.757935i	$0.726208\pi$
$74$	0	0
$75$	21.4502	2.47685
$76$	0	0
$77$	1.06736	0.121637
$78$	0	0
$79$	−11.4250	−1.28541	−0.642706	−	0.766113i	$-0.722188\pi$
$79$	−11.4250	−1.28541	−0.642706	+	0.766113i	$0.722188\pi$
$80$	0	0
$81$	7.97100	0.885666
$82$	0	0
$83$	13.9802	1.53452	0.767261	−	0.641335i	$-0.221619\pi$
$83$	13.9802	1.53452	0.767261	+	0.641335i	$0.221619\pi$
$84$	0	0
$85$	15.8423	1.71834
$86$	0	0
$87$	0.893906	0.0958368
$88$	0	0
$89$	−0.183185	−0.0194176	−0.00970878	−	0.999953i	$-0.503090\pi$
$89$	−0.183185	−0.0194176	−0.00970878	+	0.999953i	$0.503090\pi$
$90$	0	0
$91$	0.0601200	0.00630228
$92$	0	0
$93$	−27.3785	−2.83902
$94$	0	0
$95$	−3.49235	−0.358308
$96$	0	0
$97$	5.66263	0.574953	0.287477	−	0.957788i	$-0.407184\pi$
$97$	5.66263	0.574953	0.287477	+	0.957788i	$0.407184\pi$
$98$	0	0
$99$	−5.88418	−0.591382
$100$	0	0
$101$	8.00759	0.796785	0.398392	−	0.917215i	$-0.369568\pi$
$101$	8.00759	0.796785	0.398392	+	0.917215i	$0.369568\pi$
$102$	0	0
$103$	−6.18725	−0.609648	−0.304824	−	0.952409i	$-0.598598\pi$
$103$	−6.18725	−0.609648	−0.304824	+	0.952409i	$0.598598\pi$
$104$	0	0
$105$	11.1106	1.08428
$106$	0	0
$107$	7.29027	0.704777	0.352388	−	0.935854i	$-0.385370\pi$
$107$	7.29027	0.704777	0.352388	+	0.935854i	$0.385370\pi$
$108$	0	0
$109$	7.79895	0.747004	0.373502	−	0.927629i	$-0.378157\pi$
$109$	7.79895	0.747004	0.373502	+	0.927629i	$0.378157\pi$
$110$	0	0
$111$	13.4262	1.27436
$112$	0	0
$113$	−0.430558	−0.0405035	−0.0202517	−	0.999795i	$-0.506447\pi$
$113$	−0.430558	−0.0405035	−0.0202517	+	0.999795i	$0.506447\pi$
$114$	0	0
$115$	−3.76995	−0.351549
$116$	0	0
$117$	−0.331431	−0.0306408
$118$	0	0
$119$	4.84184	0.443851
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	35.7704	3.22531
$124$	0	0
$125$	−7.67100	−0.686115
$126$	0	0
$127$	3.13888	0.278531	0.139265	−	0.990255i	$-0.455526\pi$
$127$	3.13888	0.278531	0.139265	+	0.990255i	$0.455526\pi$
$128$	0	0
$129$	−31.9703	−2.81483
$130$	0	0
$131$	−12.4315	−1.08615	−0.543075	−	0.839684i	$-0.682740\pi$
$131$	−12.4315	−1.08615	−0.543075	+	0.839684i	$0.682740\pi$
$132$	0	0
$133$	−1.06736	−0.0925519
$134$	0	0
$135$	−30.0226	−2.58393
$136$	0	0
$137$	13.8301	1.18159	0.590794	−	0.806822i	$-0.298814\pi$
$137$	13.8301	1.18159	0.590794	+	0.806822i	$0.298814\pi$
$138$	0	0
$139$	−15.9968	−1.35683	−0.678417	−	0.734677i	$-0.737334\pi$
$139$	−15.9968	−1.35683	−0.678417	+	0.734677i	$0.737334\pi$
$140$	0	0
$141$	−8.63281	−0.727014
$142$	0	0
$143$	0.0563258	0.00471020
$144$	0	0
$145$	−1.04737	−0.0869796
$146$	0	0
$147$	−17.4687	−1.44080
$148$	0	0
$149$	11.3620	0.930812	0.465406	−	0.885097i	$-0.345908\pi$
$149$	11.3620	0.930812	0.465406	+	0.885097i	$0.345908\pi$
$150$	0	0
$151$	−4.66341	−0.379503	−0.189751	−	0.981832i	$-0.560768\pi$
$151$	−4.66341	−0.379503	−0.189751	+	0.981832i	$0.560768\pi$
$152$	0	0
$153$	−26.6923	−2.15794
$154$	0	0
$155$	32.0789	2.57664
$156$	0	0
$157$	−9.05206	−0.722433	−0.361217	−	0.932482i	$-0.617639\pi$
$157$	−9.05206	−0.722433	−0.361217	+	0.932482i	$0.617639\pi$
$158$	0	0
$159$	−36.7253	−2.91250
$160$	0	0
$161$	−1.15220	−0.0908062
$162$	0	0
$163$	−2.36761	−0.185446	−0.0927229	−	0.995692i	$-0.529557\pi$
$163$	−2.36761	−0.185446	−0.0927229	+	0.995692i	$0.529557\pi$
$164$	0	0
$165$	10.4094	0.810372
$166$	0	0
$167$	9.27361	0.717613	0.358807	−	0.933412i	$-0.383184\pi$
$167$	9.27361	0.717613	0.358807	+	0.933412i	$0.383184\pi$
$168$	0	0
$169$	−12.9968	−0.999756
$170$	0	0
$171$	5.88418	0.449974
$172$	0	0
$173$	10.7172	0.814812	0.407406	−	0.913247i	$-0.366433\pi$
$173$	10.7172	0.814812	0.407406	+	0.913247i	$0.366433\pi$
$174$	0	0
$175$	−7.68128	−0.580650
$176$	0	0
$177$	3.41526	0.256707
$178$	0	0
$179$	22.9070	1.71215	0.856073	−	0.516854i	$-0.172897\pi$
$179$	22.9070	1.71215	0.856073	+	0.516854i	$0.172897\pi$
$180$	0	0
$181$	5.90522	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$181$	5.90522	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$182$	0	0
$183$	−24.1312	−1.78383
$184$	0	0
$185$	−15.7312	−1.15658
$186$	0	0
$187$	4.53628	0.331725
$188$	0	0
$189$	−9.17575	−0.667438
$190$	0	0
$191$	−6.44628	−0.466437	−0.233218	−	0.972424i	$-0.574926\pi$
$191$	−6.44628	−0.466437	−0.233218	+	0.972424i	$0.574926\pi$
$192$	0	0
$193$	1.43606	0.103370	0.0516849	−	0.998663i	$-0.483541\pi$
$193$	1.43606	0.103370	0.0516849	+	0.998663i	$0.483541\pi$
$194$	0	0
$195$	0.586319	0.0419872
$196$	0	0
$197$	−4.75399	−0.338708	−0.169354	−	0.985555i	$-0.554168\pi$
$197$	−4.75399	−0.338708	−0.169354	+	0.985555i	$0.554168\pi$
$198$	0	0
$199$	2.36002	0.167298	0.0836489	−	0.996495i	$-0.473343\pi$
$199$	2.36002	0.167298	0.0836489	+	0.996495i	$0.473343\pi$
$200$	0	0
$201$	−33.6017	−2.37008
$202$	0	0
$203$	−0.320107	−0.0224671
$204$	0	0
$205$	−41.9115	−2.92722
$206$	0	0
$207$	6.35189	0.441487
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	24.5133	1.68756	0.843781	−	0.536687i	$-0.180324\pi$
$211$	24.5133	1.68756	0.843781	+	0.536687i	$0.180324\pi$
$212$	0	0
$213$	−40.2232	−2.75605
$214$	0	0
$215$	37.4590	2.55468
$216$	0	0
$217$	9.80422	0.665554
$218$	0	0
$219$	−33.2252	−2.24515
$220$	0	0
$221$	0.255509	0.0171874
$222$	0	0
$223$	−8.47427	−0.567479	−0.283740	−	0.958901i	$-0.591575\pi$
$223$	−8.47427	−0.567479	−0.283740	+	0.958901i	$0.591575\pi$
$224$	0	0
$225$	42.3456	2.82304
$226$	0	0
$227$	−13.7491	−0.912558	−0.456279	−	0.889837i	$-0.650818\pi$
$227$	−13.7491	−0.912558	−0.456279	+	0.889837i	$0.650818\pi$
$228$	0	0
$229$	−1.99640	−0.131926	−0.0659629	−	0.997822i	$-0.521012\pi$
$229$	−1.99640	−0.131926	−0.0659629	+	0.997822i	$0.521012\pi$
$230$	0	0
$231$	3.18141	0.209321
$232$	0	0
$233$	11.3481	0.743437	0.371718	−	0.928346i	$-0.378769\pi$
$233$	11.3481	0.743437	0.371718	+	0.928346i	$0.378769\pi$
$234$	0	0
$235$	10.1149	0.659823
$236$	0	0
$237$	−34.0537	−2.21203
$238$	0	0
$239$	−5.21045	−0.337036	−0.168518	−	0.985699i	$-0.553898\pi$
$239$	−5.21045	−0.337036	−0.168518	+	0.985699i	$0.553898\pi$
$240$	0	0
$241$	6.60827	0.425676	0.212838	−	0.977087i	$-0.431729\pi$
$241$	6.60827	0.425676	0.212838	+	0.977087i	$0.431729\pi$
$242$	0	0
$243$	−2.03139	−0.130314
$244$	0	0
$245$	20.4678	1.30764
$246$	0	0
$247$	−0.0563258	−0.00358393
$248$	0	0
$249$	41.6697	2.64071
$250$	0	0
$251$	−24.0024	−1.51502	−0.757510	−	0.652823i	$-0.773585\pi$
$251$	−24.0024	−1.51502	−0.757510	+	0.652823i	$0.773585\pi$
$252$	0	0
$253$	−1.07949	−0.0678667
$254$	0	0
$255$	47.2200	2.95703
$256$	0	0
$257$	−5.68903	−0.354872	−0.177436	−	0.984132i	$-0.556780\pi$
$257$	−5.68903	−0.354872	−0.177436	+	0.984132i	$0.556780\pi$
$258$	0	0
$259$	−4.80790	−0.298748
$260$	0	0
$261$	1.76469	0.109232
$262$	0	0
$263$	−13.9857	−0.862395	−0.431197	−	0.902258i	$-0.641909\pi$
$263$	−13.9857	−0.862395	−0.431197	+	0.902258i	$0.641909\pi$
$264$	0	0
$265$	43.0303	2.64333
$266$	0	0
$267$	−0.546007	−0.0334151
$268$	0	0
$269$	15.4020	0.939075	0.469538	−	0.882912i	$-0.344421\pi$
$269$	15.4020	0.939075	0.469538	+	0.882912i	$0.344421\pi$
$270$	0	0
$271$	25.3911	1.54240	0.771199	−	0.636595i	$-0.219658\pi$
$271$	25.3911	1.54240	0.771199	+	0.636595i	$0.219658\pi$
$272$	0	0
$273$	0.179196	0.0108454
$274$	0	0
$275$	−7.19651	−0.433966
$276$	0	0
$277$	19.9798	1.20047	0.600235	−	0.799824i	$-0.295074\pi$
$277$	19.9798	1.20047	0.600235	+	0.799824i	$0.295074\pi$
$278$	0	0
$279$	−54.0490	−3.23583
$280$	0	0
$281$	6.18130	0.368745	0.184373	−	0.982856i	$-0.440975\pi$
$281$	6.18130	0.368745	0.184373	+	0.982856i	$0.440975\pi$
$282$	0	0
$283$	7.12127	0.423316	0.211658	−	0.977344i	$-0.432114\pi$
$283$	7.12127	0.423316	0.211658	+	0.977344i	$0.432114\pi$
$284$	0	0
$285$	−10.4094	−0.616601
$286$	0	0
$287$	−12.8093	−0.756110
$288$	0	0
$289$	3.57781	0.210459
$290$	0	0
$291$	16.8782	0.989420
$292$	0	0
$293$	19.1342	1.11783	0.558916	−	0.829224i	$-0.311217\pi$
$293$	19.1342	1.11783	0.558916	+	0.829224i	$0.311217\pi$
$294$	0	0
$295$	−4.00159	−0.232982
$296$	0	0
$297$	−8.59667	−0.498829
$298$	0	0
$299$	−0.0608030	−0.00351633
$300$	0	0
$301$	11.4485	0.659882
$302$	0	0
$303$	23.8677	1.37116
$304$	0	0
$305$	28.2740	1.61897
$306$	0	0
$307$	−6.82573	−0.389565	−0.194782	−	0.980846i	$-0.562400\pi$
$307$	−6.82573	−0.389565	−0.194782	+	0.980846i	$0.562400\pi$
$308$	0	0
$309$	−18.4419	−1.04912
$310$	0	0
$311$	12.8609	0.729276	0.364638	−	0.931149i	$-0.381193\pi$
$311$	12.8609	0.729276	0.364638	+	0.931149i	$0.381193\pi$
$312$	0	0
$313$	−3.01707	−0.170535	−0.0852673	−	0.996358i	$-0.527174\pi$
$313$	−3.01707	−0.170535	−0.0852673	+	0.996358i	$0.527174\pi$
$314$	0	0
$315$	21.9338	1.23583
$316$	0	0
$317$	17.7857	0.998943	0.499471	−	0.866330i	$-0.333528\pi$
$317$	17.7857	0.998943	0.499471	+	0.866330i	$0.333528\pi$
$318$	0	0
$319$	−0.299905	−0.0167914
$320$	0	0
$321$	21.7296	1.21283
$322$	0	0
$323$	−4.53628	−0.252405
$324$	0	0
$325$	−0.405349	−0.0224847
$326$	0	0
$327$	23.2458	1.28550
$328$	0	0
$329$	3.09140	0.170434
$330$	0	0
$331$	26.3860	1.45030	0.725152	−	0.688589i	$-0.241770\pi$
$331$	26.3860	1.45030	0.725152	+	0.688589i	$0.241770\pi$
$332$	0	0
$333$	26.5051	1.45247
$334$	0	0
$335$	39.3704	2.15104
$336$	0	0
$337$	−13.2024	−0.719181	−0.359590	−	0.933110i	$-0.617084\pi$
$337$	−13.2024	−0.719181	−0.359590	+	0.933110i	$0.617084\pi$
$338$	0	0
$339$	−1.28334	−0.0697012
$340$	0	0
$341$	9.18548	0.497422
$342$	0	0
$343$	13.7271	0.741191
$344$	0	0
$345$	−11.2368	−0.604970
$346$	0	0
$347$	−16.4809	−0.884740	−0.442370	−	0.896833i	$-0.645862\pi$
$347$	−16.4809	−0.884740	−0.442370	+	0.896833i	$0.645862\pi$
$348$	0	0
$349$	−19.0165	−1.01793	−0.508966	−	0.860787i	$-0.669972\pi$
$349$	−19.0165	−1.01793	−0.508966	+	0.860787i	$0.669972\pi$
$350$	0	0
$351$	−0.484214	−0.0258455
$352$	0	0
$353$	13.1955	0.702325	0.351162	−	0.936315i	$-0.385787\pi$
$353$	13.1955	0.702325	0.351162	+	0.936315i	$0.385787\pi$
$354$	0	0
$355$	47.1287	2.50133
$356$	0	0
$357$	14.4318	0.763810
$358$	0	0
$359$	2.88756	0.152400	0.0761998	−	0.997093i	$-0.475721\pi$
$359$	2.88756	0.152400	0.0761998	+	0.997093i	$0.475721\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.98063	0.156443
$364$	0	0
$365$	38.9293	2.03765
$366$	0	0
$367$	2.10833	0.110054	0.0550269	−	0.998485i	$-0.482476\pi$
$367$	2.10833	0.110054	0.0550269	+	0.998485i	$0.482476\pi$
$368$	0	0
$369$	70.6156	3.67610
$370$	0	0
$371$	13.1513	0.682780
$372$	0	0
$373$	8.07305	0.418007	0.209003	−	0.977915i	$-0.432978\pi$
$373$	8.07305	0.418007	0.209003	+	0.977915i	$0.432978\pi$
$374$	0	0
$375$	−22.8644	−1.18071
$376$	0	0
$377$	−0.0168924	−0.000870002 0
$378$	0	0
$379$	−14.3355	−0.736365	−0.368183	−	0.929754i	$-0.620020\pi$
$379$	−14.3355	−0.736365	−0.368183	+	0.929754i	$0.620020\pi$
$380$	0	0
$381$	9.35586	0.479315
$382$	0	0
$383$	9.79867	0.500689	0.250344	−	0.968157i	$-0.419456\pi$
$383$	9.79867	0.500689	0.250344	+	0.968157i	$0.419456\pi$
$384$	0	0
$385$	−3.72760	−0.189976
$386$	0	0
$387$	−63.1138	−3.20825
$388$	0	0
$389$	−19.9236	−1.01016	−0.505082	−	0.863071i	$-0.668538\pi$
$389$	−19.9236	−1.01016	−0.505082	+	0.863071i	$0.668538\pi$
$390$	0	0
$391$	−4.89685	−0.247644
$392$	0	0
$393$	−37.0539	−1.86912
$394$	0	0
$395$	39.9001	2.00759
$396$	0	0
$397$	13.9941	0.702342	0.351171	−	0.936311i	$-0.385784\pi$
$397$	13.9941	0.702342	0.351171	+	0.936311i	$0.385784\pi$
$398$	0	0
$399$	−3.18141	−0.159270
$400$	0	0
$401$	−18.6293	−0.930301	−0.465151	−	0.885232i	$-0.654000\pi$
$401$	−18.6293	−0.930301	−0.465151	+	0.885232i	$0.654000\pi$
$402$	0	0
$403$	0.517380	0.0257725
$404$	0	0
$405$	−27.8375	−1.38326
$406$	0	0
$407$	−4.50448	−0.223279
$408$	0	0
$409$	−30.8428	−1.52508	−0.762540	−	0.646941i	$-0.776048\pi$
$409$	−30.8428	−1.52508	−0.762540	+	0.646941i	$0.776048\pi$
$410$	0	0
$411$	41.2226	2.03336
$412$	0	0
$413$	−1.22300	−0.0601799
$414$	0	0
$415$	−48.8236	−2.39666
$416$	0	0
$417$	−47.6807	−2.33493
$418$	0	0
$419$	−37.9347	−1.85323	−0.926616	−	0.376009i	$-0.877296\pi$
$419$	−37.9347	−1.85323	−0.926616	+	0.376009i	$0.877296\pi$
$420$	0	0
$421$	−18.4642	−0.899891	−0.449945	−	0.893056i	$-0.648557\pi$
$421$	−18.4642	−0.899891	−0.449945	+	0.893056i	$0.648557\pi$
$422$	0	0
$423$	−17.0423	−0.828627
$424$	0	0
$425$	−32.6454	−1.58353
$426$	0	0
$427$	8.64134	0.418184
$428$	0	0
$429$	0.167887	0.00810564
$430$	0	0
$431$	−33.6742	−1.62203	−0.811013	−	0.585027i	$-0.801084\pi$
$431$	−33.6742	−1.62203	−0.811013	+	0.585027i	$0.801084\pi$
$432$	0	0
$433$	11.4041	0.548047	0.274024	−	0.961723i	$-0.411645\pi$
$433$	11.4041	0.548047	0.274024	+	0.961723i	$0.411645\pi$
$434$	0	0
$435$	−3.12183	−0.149680
$436$	0	0
$437$	1.07949	0.0516388
$438$	0	0
$439$	3.76890	0.179880	0.0899399	−	0.995947i	$-0.471333\pi$
$439$	3.76890	0.179880	0.0899399	+	0.995947i	$0.471333\pi$
$440$	0	0
$441$	−34.4856	−1.64217
$442$	0	0
$443$	−4.51841	−0.214676	−0.107338	−	0.994223i	$-0.534233\pi$
$443$	−4.51841	−0.214676	−0.107338	+	0.994223i	$0.534233\pi$
$444$	0	0
$445$	0.639746	0.0303269
$446$	0	0
$447$	33.8660	1.60181
$448$	0	0
$449$	−37.3611	−1.76318	−0.881590	−	0.472016i	$-0.843526\pi$
$449$	−37.3611	−1.76318	−0.881590	+	0.472016i	$0.843526\pi$
$450$	0	0
$451$	−12.0009	−0.565102
$452$	0	0
$453$	−13.8999	−0.653075
$454$	0	0
$455$	−0.209960	−0.00984308
$456$	0	0
$457$	9.37773	0.438672	0.219336	−	0.975649i	$-0.429611\pi$
$457$	9.37773	0.438672	0.219336	+	0.975649i	$0.429611\pi$
$458$	0	0
$459$	−38.9969	−1.82022
$460$	0	0
$461$	31.6785	1.47542	0.737708	−	0.675120i	$-0.235908\pi$
$461$	31.6785	1.47542	0.737708	+	0.675120i	$0.235908\pi$
$462$	0	0
$463$	37.1284	1.72550	0.862752	−	0.505628i	$-0.168739\pi$
$463$	37.1284	1.72550	0.862752	+	0.505628i	$0.168739\pi$
$464$	0	0
$465$	95.6155	4.43406
$466$	0	0
$467$	15.9678	0.738902	0.369451	−	0.929250i	$-0.379546\pi$
$467$	15.9678	0.738902	0.369451	+	0.929250i	$0.379546\pi$
$468$	0	0
$469$	12.0327	0.555619
$470$	0	0
$471$	−26.9809	−1.24321
$472$	0	0
$473$	10.7260	0.493183
$474$	0	0
$475$	7.19651	0.330199
$476$	0	0
$477$	−72.5006	−3.31958
$478$	0	0
$479$	−28.1729	−1.28725	−0.643627	−	0.765340i	$-0.722571\pi$
$479$	−28.1729	−1.28725	−0.643627	+	0.765340i	$0.722571\pi$
$480$	0	0
$481$	−0.253718	−0.0115686
$482$	0	0
$483$	−3.43429	−0.156266
$484$	0	0
$485$	−19.7759	−0.897977
$486$	0	0
$487$	−24.0613	−1.09032	−0.545161	−	0.838331i	$-0.683532\pi$
$487$	−24.0613	−1.09032	−0.545161	+	0.838331i	$0.683532\pi$
$488$	0	0
$489$	−7.05699	−0.319128
$490$	0	0
$491$	29.0197	1.30964	0.654821	−	0.755784i	$-0.272744\pi$
$491$	29.0197	1.30964	0.654821	+	0.755784i	$0.272744\pi$
$492$	0	0
$493$	−1.36045	−0.0612716
$494$	0	0
$495$	20.5496	0.923636
$496$	0	0
$497$	14.4039	0.646102
$498$	0	0
$499$	15.0257	0.672642	0.336321	−	0.941747i	$-0.390817\pi$
$499$	15.0257	0.672642	0.336321	+	0.941747i	$0.390817\pi$
$500$	0	0
$501$	27.6412	1.23492
$502$	0	0
$503$	28.7785	1.28317	0.641585	−	0.767052i	$-0.278277\pi$
$503$	28.7785	1.28317	0.641585	+	0.767052i	$0.278277\pi$
$504$	0	0
$505$	−27.9653	−1.24444
$506$	0	0
$507$	−38.7388	−1.72045
$508$	0	0
$509$	−10.4772	−0.464395	−0.232197	−	0.972669i	$-0.574592\pi$
$509$	−10.4772	−0.464395	−0.232197	+	0.972669i	$0.574592\pi$
$510$	0	0
$511$	11.8979	0.526332
$512$	0	0
$513$	8.59667	0.379552
$514$	0	0
$515$	21.6081	0.952165
$516$	0	0
$517$	2.89630	0.127379
$518$	0	0
$519$	31.9440	1.40219
$520$	0	0
$521$	−24.7590	−1.08471	−0.542357	−	0.840148i	$-0.682468\pi$
$521$	−24.7590	−1.08471	−0.542357	+	0.840148i	$0.682468\pi$
$522$	0	0
$523$	−14.8566	−0.649635	−0.324818	−	0.945777i	$-0.605303\pi$
$523$	−14.8566	−0.649635	−0.324818	+	0.945777i	$0.605303\pi$
$524$	0	0
$525$	−22.8951	−0.999223
$526$	0	0
$527$	41.6679	1.81508
$528$	0	0
$529$	−21.8347	−0.949335
$530$	0	0
$531$	6.74219	0.292586
$532$	0	0
$533$	−0.675962	−0.0292792
$534$	0	0
$535$	−25.4602	−1.10074
$536$	0	0
$537$	68.2773	2.94638
$538$	0	0
$539$	5.86074	0.252440
$540$	0	0
$541$	3.88960	0.167227	0.0836134	−	0.996498i	$-0.473354\pi$
$541$	3.88960	0.167227	0.0836134	+	0.996498i	$0.473354\pi$
$542$	0	0
$543$	17.6013	0.755343
$544$	0	0
$545$	−27.2367	−1.16669
$546$	0	0
$547$	−17.5180	−0.749015	−0.374507	−	0.927224i	$-0.622188\pi$
$547$	−17.5180	−0.749015	−0.374507	+	0.927224i	$0.622188\pi$
$548$	0	0
$549$	−47.6382	−2.03315
$550$	0	0
$551$	0.299905	0.0127764
$552$	0	0
$553$	12.1946	0.518567
$554$	0	0
$555$	−46.8890	−1.99033
$556$	0	0
$557$	28.9860	1.22818	0.614088	−	0.789237i	$-0.289524\pi$
$557$	28.9860	1.22818	0.614088	+	0.789237i	$0.289524\pi$
$558$	0	0
$559$	0.604152	0.0255529
$560$	0	0
$561$	13.5210	0.570856
$562$	0	0
$563$	−29.6112	−1.24796	−0.623982	−	0.781438i	$-0.714486\pi$
$563$	−29.6112	−1.24796	−0.623982	+	0.781438i	$0.714486\pi$
$564$	0	0
$565$	1.50366	0.0632594
$566$	0	0
$567$	−8.50793	−0.357299
$568$	0	0
$569$	−4.05818	−0.170128	−0.0850640	−	0.996375i	$-0.527109\pi$
$569$	−4.05818	−0.170128	−0.0850640	+	0.996375i	$0.527109\pi$
$570$	0	0
$571$	−20.3380	−0.851117	−0.425559	−	0.904931i	$-0.639922\pi$
$571$	−20.3380	−0.851117	−0.425559	+	0.904931i	$0.639922\pi$
$572$	0	0
$573$	−19.2140	−0.802677
$574$	0	0
$575$	7.76854	0.323971
$576$	0	0
$577$	24.4768	1.01898	0.509492	−	0.860476i	$-0.329833\pi$
$577$	24.4768	1.01898	0.509492	+	0.860476i	$0.329833\pi$
$578$	0	0
$579$	4.28037	0.177886
$580$	0	0
$581$	−14.9219	−0.619064
$582$	0	0
$583$	12.3213	0.510296
$584$	0	0
$585$	1.15747	0.0478556
$586$	0	0
$587$	3.34628	0.138116	0.0690579	−	0.997613i	$-0.478001\pi$
$587$	3.34628	0.138116	0.0690579	+	0.997613i	$0.478001\pi$
$588$	0	0
$589$	−9.18548	−0.378481
$590$	0	0
$591$	−14.1699	−0.582872
$592$	0	0
$593$	−39.2063	−1.61001	−0.805006	−	0.593267i	$-0.797838\pi$
$593$	−39.2063	−1.61001	−0.805006	+	0.593267i	$0.797838\pi$
$594$	0	0
$595$	−16.9094	−0.693219
$596$	0	0
$597$	7.03437	0.287898
$598$	0	0
$599$	−5.72987	−0.234116	−0.117058	−	0.993125i	$-0.537346\pi$
$599$	−5.72987	−0.234116	−0.117058	+	0.993125i	$0.537346\pi$
$600$	0	0
$601$	13.9163	0.567656	0.283828	−	0.958875i	$-0.408395\pi$
$601$	13.9163	0.567656	0.283828	+	0.958875i	$0.408395\pi$
$602$	0	0
$603$	−66.3343	−2.70134
$604$	0	0
$605$	−3.49235	−0.141984
$606$	0	0
$607$	−0.156175	−0.00633897	−0.00316948	−	0.999995i	$-0.501009\pi$
$607$	−0.156175	−0.00633897	−0.00316948	+	0.999995i	$0.501009\pi$
$608$	0	0
$609$	−0.954120	−0.0386629
$610$	0	0
$611$	0.163137	0.00659980
$612$	0	0
$613$	40.1902	1.62327	0.811634	−	0.584166i	$-0.198578\pi$
$613$	40.1902	1.62327	0.811634	+	0.584166i	$0.198578\pi$
$614$	0	0
$615$	−124.923	−5.03737
$616$	0	0
$617$	12.8606	0.517747	0.258874	−	0.965911i	$-0.416649\pi$
$617$	12.8606	0.517747	0.258874	+	0.965911i	$0.416649\pi$
$618$	0	0
$619$	2.03398	0.0817526	0.0408763	−	0.999164i	$-0.486985\pi$
$619$	2.03398	0.0817526	0.0408763	+	0.999164i	$0.486985\pi$
$620$	0	0
$621$	9.27999	0.372393
$622$	0	0
$623$	0.195524	0.00783352
$624$	0	0
$625$	−9.19275	−0.367710
$626$	0	0
$627$	−2.98063	−0.119035
$628$	0	0
$629$	−20.4336	−0.814739
$630$	0	0
$631$	−37.6984	−1.50075	−0.750375	−	0.661012i	$-0.770127\pi$
$631$	−37.6984	−1.50075	−0.750375	+	0.661012i	$0.770127\pi$
$632$	0	0
$633$	73.0651	2.90408
$634$	0	0
$635$	−10.9621	−0.435017
$636$	0	0
$637$	0.330111	0.0130795
$638$	0	0
$639$	−79.4060	−3.14125
$640$	0	0
$641$	2.84360	0.112315	0.0561576	−	0.998422i	$-0.482115\pi$
$641$	2.84360	0.112315	0.0561576	+	0.998422i	$0.482115\pi$
$642$	0	0
$643$	−30.6389	−1.20828	−0.604140	−	0.796878i	$-0.706483\pi$
$643$	−30.6389	−1.20828	−0.604140	+	0.796878i	$0.706483\pi$
$644$	0	0
$645$	111.652	4.39628
$646$	0	0
$647$	6.76617	0.266006	0.133003	−	0.991116i	$-0.457538\pi$
$647$	6.76617	0.266006	0.133003	+	0.991116i	$0.457538\pi$
$648$	0	0
$649$	−1.14582	−0.0449772
$650$	0	0
$651$	29.2228	1.14533
$652$	0	0
$653$	11.9015	0.465743	0.232872	−	0.972508i	$-0.425188\pi$
$653$	11.9015	0.465743	0.232872	+	0.972508i	$0.425188\pi$
$654$	0	0
$655$	43.4153	1.69638
$656$	0	0
$657$	−65.5910	−2.55895
$658$	0	0
$659$	11.2353	0.437664	0.218832	−	0.975763i	$-0.429775\pi$
$659$	11.2353	0.437664	0.218832	+	0.975763i	$0.429775\pi$
$660$	0	0
$661$	−22.9273	−0.891768	−0.445884	−	0.895091i	$-0.647111\pi$
$661$	−22.9273	−0.891768	−0.445884	+	0.895091i	$0.647111\pi$
$662$	0	0
$663$	0.761580	0.0295773
$664$	0	0
$665$	3.72760	0.144550
$666$	0	0
$667$	0.323743	0.0125354
$668$	0	0
$669$	−25.2587	−0.976558
$670$	0	0
$671$	8.09599	0.312542
$672$	0	0
$673$	−3.09828	−0.119430	−0.0597150	−	0.998215i	$-0.519019\pi$
$673$	−3.09828	−0.119430	−0.0597150	+	0.998215i	$0.519019\pi$
$674$	0	0
$675$	61.8661	2.38123
$676$	0	0
$677$	33.3985	1.28361	0.641804	−	0.766868i	$-0.278186\pi$
$677$	33.3985	1.28361	0.641804	+	0.766868i	$0.278186\pi$
$678$	0	0
$679$	−6.04407	−0.231950
$680$	0	0
$681$	−40.9810	−1.57039
$682$	0	0
$683$	−6.01634	−0.230209	−0.115104	−	0.993353i	$-0.536720\pi$
$683$	−6.01634	−0.230209	−0.115104	+	0.993353i	$0.536720\pi$
$684$	0	0
$685$	−48.2997	−1.84544
$686$	0	0
$687$	−5.95054	−0.227027
$688$	0	0
$689$	0.694007	0.0264396
$690$	0	0
$691$	15.6730	0.596227	0.298114	−	0.954530i	$-0.403643\pi$
$691$	15.6730	0.596227	0.298114	+	0.954530i	$0.403643\pi$
$692$	0	0
$693$	6.28054	0.238578
$694$	0	0
$695$	55.8665	2.11914
$696$	0	0
$697$	−54.4395	−2.06204
$698$	0	0
$699$	33.8244	1.27936
$700$	0	0
$701$	18.0567	0.681992	0.340996	−	0.940065i	$-0.389236\pi$
$701$	18.0567	0.681992	0.340996	+	0.940065i	$0.389236\pi$
$702$	0	0
$703$	4.50448	0.169890
$704$	0	0
$705$	30.1488	1.13547
$706$	0	0
$707$	−8.54699	−0.321442
$708$	0	0
$709$	16.2048	0.608586	0.304293	−	0.952579i	$-0.401580\pi$
$709$	16.2048	0.608586	0.304293	+	0.952579i	$0.401580\pi$
$710$	0	0
$711$	−67.2266	−2.52120
$712$	0	0
$713$	−9.91560	−0.371342
$714$	0	0
$715$	−0.196709	−0.00735652
$716$	0	0
$717$	−15.5304	−0.579995
$718$	0	0
$719$	27.9403	1.04200	0.520998	−	0.853558i	$-0.325560\pi$
$719$	27.9403	1.04200	0.520998	+	0.853558i	$0.325560\pi$
$720$	0	0
$721$	6.60403	0.245947
$722$	0	0
$723$	19.6968	0.732533
$724$	0	0
$725$	2.15827	0.0801561
$726$	0	0
$727$	30.7020	1.13867	0.569337	−	0.822104i	$-0.307200\pi$
$727$	30.7020	1.13867	0.569337	+	0.822104i	$0.307200\pi$
$728$	0	0
$729$	−29.9678	−1.10992
$730$	0	0
$731$	48.6562	1.79961
$732$	0	0
$733$	44.9333	1.65965	0.829826	−	0.558023i	$-0.188440\pi$
$733$	44.9333	1.65965	0.829826	+	0.558023i	$0.188440\pi$
$734$	0	0
$735$	61.0069	2.25027
$736$	0	0
$737$	11.2733	0.415258
$738$	0	0
$739$	3.19172	0.117409	0.0587047	−	0.998275i	$-0.481303\pi$
$739$	3.19172	0.117409	0.0587047	+	0.998275i	$0.481303\pi$
$740$	0	0
$741$	−0.167887	−0.00616747
$742$	0	0
$743$	−17.4348	−0.639619	−0.319810	−	0.947482i	$-0.603619\pi$
$743$	−17.4348	−0.639619	−0.319810	+	0.947482i	$0.603619\pi$
$744$	0	0
$745$	−39.6801	−1.45377
$746$	0	0
$747$	82.2617	3.00980
$748$	0	0
$749$	−7.78135	−0.284324
$750$	0	0
$751$	−9.55633	−0.348715	−0.174358	−	0.984682i	$-0.555785\pi$
$751$	−9.55633	−0.348715	−0.174358	+	0.984682i	$0.555785\pi$
$752$	0	0
$753$	−71.5425	−2.60715
$754$	0	0
$755$	16.2863	0.592718
$756$	0	0
$757$	−48.5259	−1.76370	−0.881852	−	0.471527i	$-0.843703\pi$
$757$	−48.5259	−1.76370	−0.881852	+	0.471527i	$0.843703\pi$
$758$	0	0
$759$	−3.21755	−0.116790
$760$	0	0
$761$	38.6035	1.39937	0.699687	−	0.714449i	$-0.253323\pi$
$761$	38.6035	1.39937	0.699687	+	0.714449i	$0.253323\pi$
$762$	0	0
$763$	−8.32429	−0.301360
$764$	0	0
$765$	93.2187	3.37033
$766$	0	0
$767$	−0.0645391	−0.00233037
$768$	0	0
$769$	53.2658	1.92081	0.960406	−	0.278603i	$-0.0898712\pi$
$769$	53.2658	1.92081	0.960406	+	0.278603i	$0.0898712\pi$
$770$	0	0
$771$	−16.9569	−0.610688
$772$	0	0
$773$	9.31748	0.335126	0.167563	−	0.985861i	$-0.446410\pi$
$773$	9.31748	0.335126	0.167563	+	0.985861i	$0.446410\pi$
$774$	0	0
$775$	−66.1034	−2.37451
$776$	0	0
$777$	−14.3306	−0.514107
$778$	0	0
$779$	12.0009	0.429978
$780$	0	0
$781$	13.4948	0.482883
$782$	0	0
$783$	2.57818	0.0921367
$784$	0	0
$785$	31.6130	1.12832
$786$	0	0
$787$	15.0616	0.536889	0.268444	−	0.963295i	$-0.413490\pi$
$787$	15.0616	0.536889	0.268444	+	0.963295i	$0.413490\pi$
$788$	0	0
$789$	−41.6862	−1.48407
$790$	0	0
$791$	0.459561	0.0163401
$792$	0	0
$793$	0.456013	0.0161935
$794$	0	0
$795$	128.257	4.54882
$796$	0	0
$797$	−19.0593	−0.675114	−0.337557	−	0.941305i	$-0.609601\pi$
$797$	−19.0593	−0.675114	−0.337557	+	0.941305i	$0.609601\pi$
$798$	0	0
$799$	13.1384	0.464804
$800$	0	0
$801$	−1.07789	−0.0380855
$802$	0	0
$803$	11.1470	0.393370
$804$	0	0
$805$	4.02389	0.141824
$806$	0	0
$807$	45.9077	1.61603
$808$	0	0
$809$	−15.2273	−0.535363	−0.267681	−	0.963507i	$-0.586257\pi$
$809$	−15.2273	−0.535363	−0.267681	+	0.963507i	$0.586257\pi$
$810$	0	0
$811$	43.3354	1.52171	0.760857	−	0.648920i	$-0.224779\pi$
$811$	43.3354	1.52171	0.760857	+	0.648920i	$0.224779\pi$
$812$	0	0
$813$	75.6814	2.65426
$814$	0	0
$815$	8.26854	0.289634
$816$	0	0
$817$	−10.7260	−0.375256
$818$	0	0
$819$	0.353756	0.0123612
$820$	0	0
$821$	19.4279	0.678040	0.339020	−	0.940779i	$-0.389905\pi$
$821$	19.4279	0.678040	0.339020	+	0.940779i	$0.389905\pi$
$822$	0	0
$823$	−2.35683	−0.0821539	−0.0410769	−	0.999156i	$-0.513079\pi$
$823$	−2.35683	−0.0821539	−0.0410769	+	0.999156i	$0.513079\pi$
$824$	0	0
$825$	−21.4502	−0.746799
$826$	0	0
$827$	29.8127	1.03669	0.518344	−	0.855172i	$-0.326549\pi$
$827$	29.8127	1.03669	0.518344	+	0.855172i	$0.326549\pi$
$828$	0	0
$829$	52.6510	1.82864	0.914322	−	0.404988i	$-0.132724\pi$
$829$	52.6510	1.82864	0.914322	+	0.404988i	$0.132724\pi$
$830$	0	0
$831$	59.5524	2.06585
$832$	0	0
$833$	26.5859	0.921148
$834$	0	0
$835$	−32.3867	−1.12079
$836$	0	0
$837$	−78.9645	−2.72941
$838$	0	0
$839$	−29.7892	−1.02844	−0.514218	−	0.857660i	$-0.671918\pi$
$839$	−29.7892	−1.02844	−0.514218	+	0.857660i	$0.671918\pi$
$840$	0	0
$841$	−28.9101	−0.996899
$842$	0	0
$843$	18.4242	0.634563
$844$	0	0
$845$	45.3895	1.56145
$846$	0	0
$847$	−1.06736	−0.0366749
$848$	0	0
$849$	21.2259	0.728471
$850$	0	0
$851$	4.86252	0.166685
$852$	0	0
$853$	−35.2393	−1.20657	−0.603285	−	0.797526i	$-0.706142\pi$
$853$	−35.2393	−1.20657	−0.603285	+	0.797526i	$0.706142\pi$
$854$	0	0
$855$	−20.5496	−0.702782
$856$	0	0
$857$	−21.5877	−0.737421	−0.368711	−	0.929544i	$-0.620201\pi$
$857$	−21.5877	−0.737421	−0.368711	+	0.929544i	$0.620201\pi$
$858$	0	0
$859$	−27.7669	−0.947396	−0.473698	−	0.880687i	$-0.657081\pi$
$859$	−27.7669	−0.947396	−0.473698	+	0.880687i	$0.657081\pi$
$860$	0	0
$861$	−38.1799	−1.30117
$862$	0	0
$863$	−7.11774	−0.242291	−0.121145	−	0.992635i	$-0.538657\pi$
$863$	−7.11774	−0.242291	−0.121145	+	0.992635i	$0.538657\pi$
$864$	0	0
$865$	−37.4282	−1.27260
$866$	0	0
$867$	10.6641	0.362173
$868$	0	0
$869$	11.4250	0.387566
$870$	0	0
$871$	0.634980	0.0215155
$872$	0	0
$873$	33.3199	1.12771
$874$	0	0
$875$	8.18772	0.276796
$876$	0	0
$877$	−44.1031	−1.48925	−0.744627	−	0.667481i	$-0.767373\pi$
$877$	−44.1031	−1.48925	−0.744627	+	0.667481i	$0.767373\pi$
$878$	0	0
$879$	57.0321	1.92364
$880$	0	0
$881$	−37.9204	−1.27757	−0.638785	−	0.769385i	$-0.720563\pi$
$881$	−37.9204	−1.27757	−0.638785	+	0.769385i	$0.720563\pi$
$882$	0	0
$883$	−4.85252	−0.163300	−0.0816502	−	0.996661i	$-0.526019\pi$
$883$	−4.85252	−0.163300	−0.0816502	+	0.996661i	$0.526019\pi$
$884$	0	0
$885$	−11.9273	−0.400931
$886$	0	0
$887$	−11.2754	−0.378591	−0.189295	−	0.981920i	$-0.560620\pi$
$887$	−11.2754	−0.378591	−0.189295	+	0.981920i	$0.560620\pi$
$888$	0	0
$889$	−3.35032	−0.112366
$890$	0	0
$891$	−7.97100	−0.267038
$892$	0	0
$893$	−2.89630	−0.0969210
$894$	0	0
$895$	−79.9991	−2.67408
$896$	0	0
$897$	−0.181231	−0.00605114
$898$	0	0
$899$	−2.75477	−0.0918767
$900$	0	0
$901$	55.8928	1.86206
$902$	0	0
$903$	34.1239	1.13557
$904$	0	0
$905$	−20.6231	−0.685534
$906$	0	0
$907$	−24.0191	−0.797540	−0.398770	−	0.917051i	$-0.630563\pi$
$907$	−24.0191	−0.797540	−0.398770	+	0.917051i	$0.630563\pi$
$908$	0	0
$909$	47.1181	1.56281
$910$	0	0
$911$	−7.48540	−0.248002	−0.124001	−	0.992282i	$-0.539573\pi$
$911$	−7.48540	−0.248002	−0.124001	+	0.992282i	$0.539573\pi$
$912$	0	0
$913$	−13.9802	−0.462676
$914$	0	0
$915$	84.2745	2.78603
$916$	0	0
$917$	13.2689	0.438179
$918$	0	0
$919$	−40.0187	−1.32009	−0.660047	−	0.751224i	$-0.729464\pi$
$919$	−40.0187	−1.32009	−0.660047	+	0.751224i	$0.729464\pi$
$920$	0	0
$921$	−20.3450	−0.670390
$922$	0	0
$923$	0.760108	0.0250193
$924$	0	0
$925$	32.4165	1.06585
$926$	0	0
$927$	−36.4069	−1.19576
$928$	0	0
$929$	−32.3099	−1.06005	−0.530027	−	0.847981i	$-0.677818\pi$
$929$	−32.3099	−1.06005	−0.530027	+	0.847981i	$0.677818\pi$
$930$	0	0
$931$	−5.86074	−0.192078
$932$	0	0
$933$	38.3337	1.25499
$934$	0	0
$935$	−15.8423	−0.518098
$936$	0	0
$937$	−54.1000	−1.76737	−0.883684	−	0.468083i	$-0.844945\pi$
$937$	−54.1000	−1.76737	−0.883684	+	0.468083i	$0.844945\pi$
$938$	0	0
$939$	−8.99277	−0.293468
$940$	0	0
$941$	26.6955	0.870248	0.435124	−	0.900371i	$-0.356705\pi$
$941$	26.6955	0.870248	0.435124	+	0.900371i	$0.356705\pi$
$942$	0	0
$943$	12.9548	0.421868
$944$	0	0
$945$	32.0449	1.04242
$946$	0	0
$947$	39.4463	1.28183	0.640916	−	0.767611i	$-0.278555\pi$
$947$	39.4463	1.28183	0.640916	+	0.767611i	$0.278555\pi$
$948$	0	0
$949$	0.627865	0.0203814
$950$	0	0
$951$	53.0126	1.71905
$952$	0	0
$953$	−48.7523	−1.57924	−0.789621	−	0.613595i	$-0.789723\pi$
$953$	−48.7523	−1.57924	−0.789621	+	0.613595i	$0.789723\pi$
$954$	0	0
$955$	22.5127	0.728493
$956$	0	0
$957$	−0.893906	−0.0288959
$958$	0	0
$959$	−14.7618	−0.476682
$960$	0	0
$961$	53.3730	1.72171
$962$	0	0
$963$	42.8972	1.38234
$964$	0	0
$965$	−5.01522	−0.161446
$966$	0	0
$967$	1.39478	0.0448532	0.0224266	−	0.999748i	$-0.492861\pi$
$967$	1.39478	0.0448532	0.0224266	+	0.999748i	$0.492861\pi$
$968$	0	0
$969$	−13.5210	−0.434356
$970$	0	0
$971$	3.11733	0.100040	0.0500200	−	0.998748i	$-0.484071\pi$
$971$	3.11733	0.100040	0.0500200	+	0.998748i	$0.484071\pi$
$972$	0	0
$973$	17.0744	0.547380
$974$	0	0
$975$	−1.20820	−0.0386933
$976$	0	0
$977$	−22.3690	−0.715647	−0.357823	−	0.933789i	$-0.616481\pi$
$977$	−22.3690	−0.715647	−0.357823	+	0.933789i	$0.616481\pi$
$978$	0	0
$979$	0.183185	0.00585462
$980$	0	0
$981$	45.8904	1.46517
$982$	0	0
$983$	−42.2421	−1.34731	−0.673657	−	0.739044i	$-0.735277\pi$
$983$	−42.2421	−1.34731	−0.673657	+	0.739044i	$0.735277\pi$
$984$	0	0
$985$	16.6026	0.529003
$986$	0	0
$987$	9.21433	0.293295
$988$	0	0
$989$	−11.5786	−0.368178
$990$	0	0
$991$	1.69828	0.0539477	0.0269738	−	0.999636i	$-0.491413\pi$
$991$	1.69828	0.0539477	0.0269738	+	0.999636i	$0.491413\pi$
$992$	0	0
$993$	78.6469	2.49578
$994$	0	0
$995$	−8.24203	−0.261290
$996$	0	0
$997$	18.9376	0.599759	0.299879	−	0.953977i	$-0.403054\pi$
$997$	18.9376	0.599759	0.299879	+	0.953977i	$0.403054\pi$
$998$	0	0
$999$	38.7235	1.22516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.t.1.5		5
4.3	odd	2		209.2.a.c.1.2	✓	5
12.11	even	2		1881.2.a.k.1.4		5
20.19	odd	2		5225.2.a.h.1.4		5
44.43	even	2		2299.2.a.n.1.4		5
76.75	even	2		3971.2.a.h.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.2	✓	5	4.3	odd	2
1881.2.a.k.1.4		5	12.11	even	2
2299.2.a.n.1.4		5	44.43	even	2
3344.2.a.t.1.5		5	1.1	even	1	trivial
3971.2.a.h.1.4		5	76.75	even	2
5225.2.a.h.1.4		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+2.98063 q^{3} -3.49235 q^{5} -1.06736 q^{7} +5.88418 q^{9} +O(q^{10})\)	\(q+2.98063 q^{3} -3.49235 q^{5} -1.06736 q^{7} +5.88418 q^{9} -1.00000 q^{11} -0.0563258 q^{13} -10.4094 q^{15} -4.53628 q^{17} +1.00000 q^{19} -3.18141 q^{21} +1.07949 q^{23} +7.19651 q^{25} +8.59667 q^{27} +0.299905 q^{29} -9.18548 q^{31} -2.98063 q^{33} +3.72760 q^{35} +4.50448 q^{37} -0.167887 q^{39} +12.0009 q^{41} -10.7260 q^{43} -20.5496 q^{45} -2.89630 q^{47} -5.86074 q^{49} -13.5210 q^{51} -12.3213 q^{53} +3.49235 q^{55} +2.98063 q^{57} +1.14582 q^{59} -8.09599 q^{61} -6.28054 q^{63} +0.196709 q^{65} -11.2733 q^{67} +3.21755 q^{69} -13.4948 q^{71} -11.1470 q^{73} +21.4502 q^{75} +1.06736 q^{77} -11.4250 q^{79} +7.97100 q^{81} +13.9802 q^{83} +15.8423 q^{85} +0.893906 q^{87} -0.183185 q^{89} +0.0601200 q^{91} -27.3785 q^{93} -3.49235 q^{95} +5.66263 q^{97} -5.88418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9	\( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

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