LMFDB - Embedded newform 3344.2.a.y.1.3

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.57500224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 11x^2 - 18*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.12938$ 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)

$f(q)$	$=$	$q-0.129383 q^{3} -1.42012 q^{5} -5.10098 q^{7} -2.98326 q^{9} +O(q^{10})$	$q-0.129383 q^{3} -1.42012 q^{5} -5.10098 q^{7} -2.98326 q^{9} -1.00000 q^{11} -7.03336 q^{13} +0.183739 q^{15} -3.23187 q^{17} +1.00000 q^{19} +0.659980 q^{21} +6.34958 q^{23} -2.98326 q^{25} +0.774133 q^{27} +2.43101 q^{29} -8.51543 q^{31} +0.129383 q^{33} +7.24400 q^{35} -9.07874 q^{37} +0.909998 q^{39} -1.74376 q^{41} -2.02847 q^{43} +4.23659 q^{45} +5.67729 q^{47} +19.0200 q^{49} +0.418149 q^{51} -6.28923 q^{53} +1.42012 q^{55} -0.129383 q^{57} +1.86170 q^{59} +4.98792 q^{61} +15.2175 q^{63} +9.98821 q^{65} -4.01821 q^{67} -0.821529 q^{69} -4.62358 q^{71} -13.9755 q^{73} +0.385983 q^{75} +5.10098 q^{77} +7.19839 q^{79} +8.84962 q^{81} +9.72806 q^{83} +4.58964 q^{85} -0.314532 q^{87} +15.1572 q^{89} +35.8770 q^{91} +1.10175 q^{93} -1.42012 q^{95} -6.23850 q^{97} +2.98326 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * 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
$3$	−0.129383	−0.0746994	−0.0373497	−	0.999302i	$-0.511892\pi$
$3$	−0.129383	−0.0746994	−0.0373497	+	0.999302i	$0.511892\pi$
$4$	0	0
$5$	−1.42012	−0.635097	−0.317548	−	0.948242i	$-0.602860\pi$
$5$	−1.42012	−0.635097	−0.317548	+	0.948242i	$0.602860\pi$
$6$	0	0
$7$	−5.10098	−1.92799	−0.963994	−	0.265924i	$-0.914323\pi$
$7$	−5.10098	−1.92799	−0.963994	+	0.265924i	$0.914323\pi$
$8$	0	0
$9$	−2.98326	−0.994420
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−7.03336	−1.95070	−0.975352	−	0.220657i	$-0.929180\pi$
$13$	−7.03336	−1.95070	−0.975352	+	0.220657i	$0.929180\pi$
$14$	0	0
$15$	0.183739	0.0474413
$16$	0	0
$17$	−3.23187	−0.783843	−0.391922	−	0.919999i	$-0.628190\pi$
$17$	−3.23187	−0.783843	−0.391922	+	0.919999i	$0.628190\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.659980	0.144019
$22$	0	0
$23$	6.34958	1.32398	0.661990	−	0.749513i	$-0.269712\pi$
$23$	6.34958	1.32398	0.661990	+	0.749513i	$0.269712\pi$
$24$	0	0
$25$	−2.98326	−0.596652
$26$	0	0
$27$	0.774133	0.148982
$28$	0	0
$29$	2.43101	0.451427	0.225714	−	0.974194i	$-0.427529\pi$
$29$	2.43101	0.451427	0.225714	+	0.974194i	$0.427529\pi$
$30$	0	0
$31$	−8.51543	−1.52942	−0.764708	−	0.644377i	$-0.777117\pi$
$31$	−8.51543	−1.52942	−0.764708	+	0.644377i	$0.777117\pi$
$32$	0	0
$33$	0.129383	0.0225227
$34$	0	0
$35$	7.24400	1.22446
$36$	0	0
$37$	−9.07874	−1.49254	−0.746268	−	0.665646i	$-0.768156\pi$
$37$	−9.07874	−1.49254	−0.746268	+	0.665646i	$0.768156\pi$
$38$	0	0
$39$	0.909998	0.145716
$40$	0	0
$41$	−1.74376	−0.272329	−0.136164	−	0.990686i	$-0.543478\pi$
$41$	−1.74376	−0.272329	−0.136164	+	0.990686i	$0.543478\pi$
$42$	0	0
$43$	−2.02847	−0.309339	−0.154669	−	0.987966i	$-0.549431\pi$
$43$	−2.02847	−0.309339	−0.154669	+	0.987966i	$0.549431\pi$
$44$	0	0
$45$	4.23659	0.631553
$46$	0	0
$47$	5.67729	0.828118	0.414059	−	0.910250i	$-0.364111\pi$
$47$	5.67729	0.828118	0.414059	+	0.910250i	$0.364111\pi$
$48$	0	0
$49$	19.0200	2.71714
$50$	0	0
$51$	0.418149	0.0585526
$52$	0	0
$53$	−6.28923	−0.863892	−0.431946	−	0.901900i	$-0.642173\pi$
$53$	−6.28923	−0.863892	−0.431946	+	0.901900i	$0.642173\pi$
$54$	0	0
$55$	1.42012	0.191489
$56$	0	0
$57$	−0.129383	−0.0171372
$58$	0	0
$59$	1.86170	0.242372	0.121186	−	0.992630i	$-0.461330\pi$
$59$	1.86170	0.242372	0.121186	+	0.992630i	$0.461330\pi$
$60$	0	0
$61$	4.98792	0.638638	0.319319	−	0.947647i	$-0.396546\pi$
$61$	4.98792	0.638638	0.319319	+	0.947647i	$0.396546\pi$
$62$	0	0
$63$	15.2175	1.91723
$64$	0	0
$65$	9.98821	1.23889
$66$	0	0
$67$	−4.01821	−0.490903	−0.245451	−	0.969409i	$-0.578936\pi$
$67$	−4.01821	−0.490903	−0.245451	+	0.969409i	$0.578936\pi$
$68$	0	0
$69$	−0.821529	−0.0989005
$70$	0	0
$71$	−4.62358	−0.548718	−0.274359	−	0.961627i	$-0.588466\pi$
$71$	−4.62358	−0.548718	−0.274359	+	0.961627i	$0.588466\pi$
$72$	0	0
$73$	−13.9755	−1.63571	−0.817854	−	0.575425i	$-0.804837\pi$
$73$	−13.9755	−1.63571	−0.817854	+	0.575425i	$0.804837\pi$
$74$	0	0
$75$	0.385983	0.0445695
$76$	0	0
$77$	5.10098	0.581310
$78$	0	0
$79$	7.19839	0.809882	0.404941	−	0.914343i	$-0.367292\pi$
$79$	7.19839	0.809882	0.404941	+	0.914343i	$0.367292\pi$
$80$	0	0
$81$	8.84962	0.983291
$82$	0	0
$83$	9.72806	1.06779	0.533897	−	0.845550i	$-0.320727\pi$
$83$	9.72806	1.06779	0.533897	+	0.845550i	$0.320727\pi$
$84$	0	0
$85$	4.58964	0.497816
$86$	0	0
$87$	−0.314532	−0.0337213
$88$	0	0
$89$	15.1572	1.60666	0.803328	−	0.595537i	$-0.203061\pi$
$89$	15.1572	1.60666	0.803328	+	0.595537i	$0.203061\pi$
$90$	0	0
$91$	35.8770	3.76093
$92$	0	0
$93$	1.10175	0.114246
$94$	0	0
$95$	−1.42012	−0.145701
$96$	0	0
$97$	−6.23850	−0.633424	−0.316712	−	0.948522i	$-0.602579\pi$
$97$	−6.23850	−0.633424	−0.316712	+	0.948522i	$0.602579\pi$
$98$	0	0
$99$	2.98326	0.299829
$100$	0	0
$101$	−11.1093	−1.10541	−0.552706	−	0.833376i	$-0.686405\pi$
$101$	−11.1093	−1.10541	−0.552706	+	0.833376i	$0.686405\pi$
$102$	0	0
$103$	−0.0393809	−0.00388031	−0.00194016	−	0.999998i	$-0.500618\pi$
$103$	−0.0393809	−0.00388031	−0.00194016	+	0.999998i	$0.500618\pi$
$104$	0	0
$105$	−0.937251	−0.0914663
$106$	0	0
$107$	13.9697	1.35050	0.675251	−	0.737588i	$-0.264035\pi$
$107$	13.9697	1.35050	0.675251	+	0.737588i	$0.264035\pi$
$108$	0	0
$109$	−13.4042	−1.28389	−0.641944	−	0.766752i	$-0.721872\pi$
$109$	−13.4042	−1.28389	−0.641944	+	0.766752i	$0.721872\pi$
$110$	0	0
$111$	1.17464	0.111491
$112$	0	0
$113$	−3.60293	−0.338935	−0.169468	−	0.985536i	$-0.554205\pi$
$113$	−3.60293	−0.338935	−0.169468	+	0.985536i	$0.554205\pi$
$114$	0	0
$115$	−9.01717	−0.840855
$116$	0	0
$117$	20.9823	1.93982
$118$	0	0
$119$	16.4857	1.51124
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.225612	0.0203428
$124$	0	0
$125$	11.3372	1.01403
$126$	0	0
$127$	−15.8979	−1.41071	−0.705355	−	0.708854i	$-0.749213\pi$
$127$	−15.8979	−1.41071	−0.705355	+	0.708854i	$0.749213\pi$
$128$	0	0
$129$	0.262450	0.0231074
$130$	0	0
$131$	12.1817	1.06432	0.532159	−	0.846645i	$-0.321381\pi$
$131$	12.1817	1.06432	0.532159	+	0.846645i	$0.321381\pi$
$132$	0	0
$133$	−5.10098	−0.442311
$134$	0	0
$135$	−1.09936	−0.0946179
$136$	0	0
$137$	−14.1220	−1.20652	−0.603261	−	0.797544i	$-0.706132\pi$
$137$	−14.1220	−1.20652	−0.603261	+	0.797544i	$0.706132\pi$
$138$	0	0
$139$	−7.86909	−0.667447	−0.333724	−	0.942671i	$-0.608305\pi$
$139$	−7.86909	−0.667447	−0.333724	+	0.942671i	$0.608305\pi$
$140$	0	0
$141$	−0.734546	−0.0618599
$142$	0	0
$143$	7.03336	0.588159
$144$	0	0
$145$	−3.45233	−0.286700
$146$	0	0
$147$	−2.46086	−0.202968
$148$	0	0
$149$	−18.6482	−1.52772	−0.763860	−	0.645382i	$-0.776698\pi$
$149$	−18.6482	−1.52772	−0.763860	+	0.645382i	$0.776698\pi$
$150$	0	0
$151$	−22.2749	−1.81270	−0.906352	−	0.422524i	$-0.861144\pi$
$151$	−22.2749	−1.81270	−0.906352	+	0.422524i	$0.861144\pi$
$152$	0	0
$153$	9.64150	0.779469
$154$	0	0
$155$	12.0929	0.971328
$156$	0	0
$157$	1.14571	0.0914379	0.0457189	−	0.998954i	$-0.485442\pi$
$157$	1.14571	0.0914379	0.0457189	+	0.998954i	$0.485442\pi$
$158$	0	0
$159$	0.813720	0.0645322
$160$	0	0
$161$	−32.3891	−2.55262
$162$	0	0
$163$	3.17176	0.248432	0.124216	−	0.992255i	$-0.460358\pi$
$163$	3.17176	0.248432	0.124216	+	0.992255i	$0.460358\pi$
$164$	0	0
$165$	−0.183739	−0.0143041
$166$	0	0
$167$	−13.7261	−1.06215	−0.531077	−	0.847323i	$-0.678213\pi$
$167$	−13.7261	−1.06215	−0.531077	+	0.847323i	$0.678213\pi$
$168$	0	0
$169$	36.4682	2.80524
$170$	0	0
$171$	−2.98326	−0.228136
$172$	0	0
$173$	−10.4861	−0.797243	−0.398622	−	0.917116i	$-0.630511\pi$
$173$	−10.4861	−0.797243	−0.398622	+	0.917116i	$0.630511\pi$
$174$	0	0
$175$	15.2175	1.15034
$176$	0	0
$177$	−0.240872	−0.0181051
$178$	0	0
$179$	15.9699	1.19365	0.596824	−	0.802372i	$-0.296429\pi$
$179$	15.9699	1.19365	0.596824	+	0.802372i	$0.296429\pi$
$180$	0	0
$181$	4.61072	0.342712	0.171356	−	0.985209i	$-0.445185\pi$
$181$	4.61072	0.342712	0.171356	+	0.985209i	$0.445185\pi$
$182$	0	0
$183$	−0.645353	−0.0477059
$184$	0	0
$185$	12.8929	0.947905
$186$	0	0
$187$	3.23187	0.236338
$188$	0	0
$189$	−3.94883	−0.287235
$190$	0	0
$191$	10.4763	0.758039	0.379019	−	0.925389i	$-0.376261\pi$
$191$	10.4763	0.758039	0.379019	+	0.925389i	$0.376261\pi$
$192$	0	0
$193$	20.5481	1.47908	0.739541	−	0.673112i	$-0.235043\pi$
$193$	20.5481	1.47908	0.739541	+	0.673112i	$0.235043\pi$
$194$	0	0
$195$	−1.29231	−0.0925439
$196$	0	0
$197$	17.1568	1.22237	0.611184	−	0.791488i	$-0.290693\pi$
$197$	17.1568	1.22237	0.611184	+	0.791488i	$0.290693\pi$
$198$	0	0
$199$	−9.84693	−0.698030	−0.349015	−	0.937117i	$-0.613484\pi$
$199$	−9.84693	−0.698030	−0.349015	+	0.937117i	$0.613484\pi$
$200$	0	0
$201$	0.519889	0.0366701
$202$	0	0
$203$	−12.4005	−0.870346
$204$	0	0
$205$	2.47634	0.172955
$206$	0	0
$207$	−18.9425	−1.31659
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	3.34597	0.230346	0.115173	−	0.993345i	$-0.463258\pi$
$211$	3.34597	0.230346	0.115173	+	0.993345i	$0.463258\pi$
$212$	0	0
$213$	0.598213	0.0409889
$214$	0	0
$215$	2.88067	0.196460
$216$	0	0
$217$	43.4370	2.94870
$218$	0	0
$219$	1.80819	0.122186
$220$	0	0
$221$	22.7309	1.52905
$222$	0	0
$223$	−4.61466	−0.309021	−0.154510	−	0.987991i	$-0.549380\pi$
$223$	−4.61466	−0.309021	−0.154510	+	0.987991i	$0.549380\pi$
$224$	0	0
$225$	8.89984	0.593323
$226$	0	0
$227$	−25.5399	−1.69514	−0.847570	−	0.530684i	$-0.821935\pi$
$227$	−25.5399	−1.69514	−0.847570	+	0.530684i	$0.821935\pi$
$228$	0	0
$229$	−10.1376	−0.669913	−0.334956	−	0.942234i	$-0.608722\pi$
$229$	−10.1376	−0.669913	−0.334956	+	0.942234i	$0.608722\pi$
$230$	0	0
$231$	−0.659980	−0.0434235
$232$	0	0
$233$	−16.5488	−1.08415	−0.542074	−	0.840331i	$-0.682361\pi$
$233$	−16.5488	−1.08415	−0.542074	+	0.840331i	$0.682361\pi$
$234$	0	0
$235$	−8.06244	−0.525935
$236$	0	0
$237$	−0.931350	−0.0604977
$238$	0	0
$239$	−17.2743	−1.11738	−0.558692	−	0.829375i	$-0.688697\pi$
$239$	−17.2743	−1.11738	−0.558692	+	0.829375i	$0.688697\pi$
$240$	0	0
$241$	−8.20538	−0.528555	−0.264277	−	0.964447i	$-0.585133\pi$
$241$	−8.20538	−0.528555	−0.264277	+	0.964447i	$0.585133\pi$
$242$	0	0
$243$	−3.46739	−0.222433
$244$	0	0
$245$	−27.0106	−1.72564
$246$	0	0
$247$	−7.03336	−0.447522
$248$	0	0
$249$	−1.25865	−0.0797635
$250$	0	0
$251$	−14.1629	−0.893956	−0.446978	−	0.894545i	$-0.647500\pi$
$251$	−14.1629	−0.893956	−0.446978	+	0.894545i	$0.647500\pi$
$252$	0	0
$253$	−6.34958	−0.399195
$254$	0	0
$255$	−0.593822	−0.0371866
$256$	0	0
$257$	4.16251	0.259650	0.129825	−	0.991537i	$-0.458558\pi$
$257$	4.16251	0.259650	0.129825	+	0.991537i	$0.458558\pi$
$258$	0	0
$259$	46.3104	2.87759
$260$	0	0
$261$	−7.25234	−0.448908
$262$	0	0
$263$	21.0445	1.29766	0.648829	−	0.760934i	$-0.275259\pi$
$263$	21.0445	1.29766	0.648829	+	0.760934i	$0.275259\pi$
$264$	0	0
$265$	8.93146	0.548655
$266$	0	0
$267$	−1.96108	−0.120016
$268$	0	0
$269$	16.1828	0.986684	0.493342	−	0.869835i	$-0.335775\pi$
$269$	16.1828	0.986684	0.493342	+	0.869835i	$0.335775\pi$
$270$	0	0
$271$	−8.13639	−0.494250	−0.247125	−	0.968984i	$-0.579486\pi$
$271$	−8.13639	−0.494250	−0.247125	+	0.968984i	$0.579486\pi$
$272$	0	0
$273$	−4.64188	−0.280939
$274$	0	0
$275$	2.98326	0.179897
$276$	0	0
$277$	20.8547	1.25304	0.626519	−	0.779406i	$-0.284479\pi$
$277$	20.8547	1.25304	0.626519	+	0.779406i	$0.284479\pi$
$278$	0	0
$279$	25.4037	1.52088
$280$	0	0
$281$	−13.5480	−0.808207	−0.404104	−	0.914713i	$-0.632417\pi$
$281$	−13.5480	−0.808207	−0.404104	+	0.914713i	$0.632417\pi$
$282$	0	0
$283$	11.4618	0.681336	0.340668	−	0.940184i	$-0.389347\pi$
$283$	11.4618	0.681336	0.340668	+	0.940184i	$0.389347\pi$
$284$	0	0
$285$	0.183739	0.0108838
$286$	0	0
$287$	8.89485	0.525047
$288$	0	0
$289$	−6.55502	−0.385590
$290$	0	0
$291$	0.807157	0.0473164
$292$	0	0
$293$	−6.53367	−0.381701	−0.190850	−	0.981619i	$-0.561125\pi$
$293$	−6.53367	−0.381701	−0.190850	+	0.981619i	$0.561125\pi$
$294$	0	0
$295$	−2.64383	−0.153930
$296$	0	0
$297$	−0.774133	−0.0449197
$298$	0	0
$299$	−44.6589	−2.58269
$300$	0	0
$301$	10.3472	0.596402
$302$	0	0
$303$	1.43735	0.0825736
$304$	0	0
$305$	−7.08345	−0.405597
$306$	0	0
$307$	−14.6945	−0.838659	−0.419329	−	0.907834i	$-0.637735\pi$
$307$	−14.6945	−0.838659	−0.419329	+	0.907834i	$0.637735\pi$
$308$	0	0
$309$	0.00509522	0.000289857 0
$310$	0	0
$311$	−3.48959	−0.197877	−0.0989384	−	0.995094i	$-0.531545\pi$
$311$	−3.48959	−0.197877	−0.0989384	+	0.995094i	$0.531545\pi$
$312$	0	0
$313$	−33.6367	−1.90126	−0.950630	−	0.310327i	$-0.899561\pi$
$313$	−33.6367	−1.90126	−0.950630	+	0.310327i	$0.899561\pi$
$314$	0	0
$315$	−21.6107	−1.21763
$316$	0	0
$317$	19.5336	1.09712	0.548559	−	0.836112i	$-0.315177\pi$
$317$	19.5336	1.09712	0.548559	+	0.836112i	$0.315177\pi$
$318$	0	0
$319$	−2.43101	−0.136110
$320$	0	0
$321$	−1.80744	−0.100882
$322$	0	0
$323$	−3.23187	−0.179826
$324$	0	0
$325$	20.9823	1.16389
$326$	0	0
$327$	1.73427	0.0959056
$328$	0	0
$329$	−28.9597	−1.59660
$330$	0	0
$331$	7.76119	0.426594	0.213297	−	0.976987i	$-0.431580\pi$
$331$	7.76119	0.426594	0.213297	+	0.976987i	$0.431580\pi$
$332$	0	0
$333$	27.0842	1.48421
$334$	0	0
$335$	5.70634	0.311771
$336$	0	0
$337$	−3.11984	−0.169949	−0.0849743	−	0.996383i	$-0.527081\pi$
$337$	−3.11984	−0.169949	−0.0849743	+	0.996383i	$0.527081\pi$
$338$	0	0
$339$	0.466158	0.0253182
$340$	0	0
$341$	8.51543	0.461136
$342$	0	0
$343$	−61.3135	−3.31062
$344$	0	0
$345$	1.16667	0.0628114
$346$	0	0
$347$	12.5881	0.675764	0.337882	−	0.941188i	$-0.390290\pi$
$347$	12.5881	0.675764	0.337882	+	0.941188i	$0.390290\pi$
$348$	0	0
$349$	−0.822765	−0.0440416	−0.0220208	−	0.999758i	$-0.507010\pi$
$349$	−0.822765	−0.0440416	−0.0220208	+	0.999758i	$0.507010\pi$
$350$	0	0
$351$	−5.44475	−0.290619
$352$	0	0
$353$	−23.7990	−1.26670	−0.633348	−	0.773867i	$-0.718320\pi$
$353$	−23.7990	−1.26670	−0.633348	+	0.773867i	$0.718320\pi$
$354$	0	0
$355$	6.56604	0.348489
$356$	0	0
$357$	−2.13297	−0.112889
$358$	0	0
$359$	−9.93003	−0.524087	−0.262043	−	0.965056i	$-0.584396\pi$
$359$	−9.93003	−0.524087	−0.262043	+	0.965056i	$0.584396\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.129383	−0.00679085
$364$	0	0
$365$	19.8469	1.03883
$366$	0	0
$367$	14.2495	0.743819	0.371909	−	0.928269i	$-0.378703\pi$
$367$	14.2495	0.743819	0.371909	+	0.928269i	$0.378703\pi$
$368$	0	0
$369$	5.20207	0.270809
$370$	0	0
$371$	32.0812	1.66557
$372$	0	0
$373$	−15.8427	−0.820303	−0.410152	−	0.912017i	$-0.634524\pi$
$373$	−15.8427	−0.820303	−0.410152	+	0.912017i	$0.634524\pi$
$374$	0	0
$375$	−1.46684	−0.0757473
$376$	0	0
$377$	−17.0982	−0.880601
$378$	0	0
$379$	12.8261	0.658832	0.329416	−	0.944185i	$-0.393148\pi$
$379$	12.8261	0.658832	0.329416	+	0.944185i	$0.393148\pi$
$380$	0	0
$381$	2.05692	0.105379
$382$	0	0
$383$	−1.51579	−0.0774530	−0.0387265	−	0.999250i	$-0.512330\pi$
$383$	−1.51579	−0.0774530	−0.0387265	+	0.999250i	$0.512330\pi$
$384$	0	0
$385$	−7.24400	−0.369188
$386$	0	0
$387$	6.05146	0.307613
$388$	0	0
$389$	1.47555	0.0748132	0.0374066	−	0.999300i	$-0.488090\pi$
$389$	1.47555	0.0748132	0.0374066	+	0.999300i	$0.488090\pi$
$390$	0	0
$391$	−20.5210	−1.03779
$392$	0	0
$393$	−1.57610	−0.0795038
$394$	0	0
$395$	−10.2226	−0.514354
$396$	0	0
$397$	15.7378	0.789858	0.394929	−	0.918712i	$-0.370769\pi$
$397$	15.7378	0.789858	0.394929	+	0.918712i	$0.370769\pi$
$398$	0	0
$399$	0.659980	0.0330403
$400$	0	0
$401$	39.1154	1.95333	0.976665	−	0.214766i	$-0.0688990\pi$
$401$	39.1154	1.95333	0.976665	+	0.214766i	$0.0688990\pi$
$402$	0	0
$403$	59.8921	2.98344
$404$	0	0
$405$	−12.5675	−0.624485
$406$	0	0
$407$	9.07874	0.450017
$408$	0	0
$409$	32.9009	1.62684	0.813422	−	0.581674i	$-0.197602\pi$
$409$	32.9009	1.62684	0.813422	+	0.581674i	$0.197602\pi$
$410$	0	0
$411$	1.82715	0.0901264
$412$	0	0
$413$	−9.49647	−0.467291
$414$	0	0
$415$	−13.8150	−0.678152
$416$	0	0
$417$	1.01813	0.0498579
$418$	0	0
$419$	18.5406	0.905768	0.452884	−	0.891570i	$-0.350395\pi$
$419$	18.5406	0.905768	0.452884	+	0.891570i	$0.350395\pi$
$420$	0	0
$421$	6.49071	0.316338	0.158169	−	0.987412i	$-0.449441\pi$
$421$	6.49071	0.316338	0.158169	+	0.987412i	$0.449441\pi$
$422$	0	0
$423$	−16.9368	−0.823497
$424$	0	0
$425$	9.64150	0.467682
$426$	0	0
$427$	−25.4433	−1.23129
$428$	0	0
$429$	−0.909998	−0.0439351
$430$	0	0
$431$	31.5380	1.51913	0.759566	−	0.650431i	$-0.225412\pi$
$431$	31.5380	1.51913	0.759566	+	0.650431i	$0.225412\pi$
$432$	0	0
$433$	0.533263	0.0256270	0.0128135	−	0.999918i	$-0.495921\pi$
$433$	0.533263	0.0256270	0.0128135	+	0.999918i	$0.495921\pi$
$434$	0	0
$435$	0.446673	0.0214163
$436$	0	0
$437$	6.34958	0.303742
$438$	0	0
$439$	−22.2529	−1.06207	−0.531037	−	0.847349i	$-0.678198\pi$
$439$	−22.2529	−1.06207	−0.531037	+	0.847349i	$0.678198\pi$
$440$	0	0
$441$	−56.7415	−2.70198
$442$	0	0
$443$	−21.1549	−1.00510	−0.502550	−	0.864548i	$-0.667605\pi$
$443$	−21.1549	−1.00510	−0.502550	+	0.864548i	$0.667605\pi$
$444$	0	0
$445$	−21.5250	−1.02038
$446$	0	0
$447$	2.41276	0.114120
$448$	0	0
$449$	−19.9887	−0.943326	−0.471663	−	0.881779i	$-0.656346\pi$
$449$	−19.9887	−0.943326	−0.471663	+	0.881779i	$0.656346\pi$
$450$	0	0
$451$	1.74376	0.0821102
$452$	0	0
$453$	2.88199	0.135408
$454$	0	0
$455$	−50.9496	−2.38856
$456$	0	0
$457$	−19.0660	−0.891869	−0.445934	−	0.895066i	$-0.647129\pi$
$457$	−19.0660	−0.891869	−0.445934	+	0.895066i	$0.647129\pi$
$458$	0	0
$459$	−2.50190	−0.116778
$460$	0	0
$461$	7.00040	0.326041	0.163021	−	0.986623i	$-0.447876\pi$
$461$	7.00040	0.326041	0.163021	+	0.986623i	$0.447876\pi$
$462$	0	0
$463$	−21.8340	−1.01471	−0.507356	−	0.861736i	$-0.669377\pi$
$463$	−21.8340	−1.01471	−0.507356	+	0.861736i	$0.669377\pi$
$464$	0	0
$465$	−1.56462	−0.0725575
$466$	0	0
$467$	33.9124	1.56928	0.784640	−	0.619952i	$-0.212848\pi$
$467$	33.9124	1.56928	0.784640	+	0.619952i	$0.212848\pi$
$468$	0	0
$469$	20.4968	0.946455
$470$	0	0
$471$	−0.148236	−0.00683035
$472$	0	0
$473$	2.02847	0.0932692
$474$	0	0
$475$	−2.98326	−0.136881
$476$	0	0
$477$	18.7624	0.859071
$478$	0	0
$479$	−22.4815	−1.02720	−0.513602	−	0.858029i	$-0.671689\pi$
$479$	−22.4815	−1.02720	−0.513602	+	0.858029i	$0.671689\pi$
$480$	0	0
$481$	63.8541	2.91149
$482$	0	0
$483$	4.19060	0.190679
$484$	0	0
$485$	8.85942	0.402286
$486$	0	0
$487$	−2.01593	−0.0913506	−0.0456753	−	0.998956i	$-0.514544\pi$
$487$	−2.01593	−0.0913506	−0.0456753	+	0.998956i	$0.514544\pi$
$488$	0	0
$489$	−0.410373	−0.0185577
$490$	0	0
$491$	−13.6022	−0.613860	−0.306930	−	0.951732i	$-0.599302\pi$
$491$	−13.6022	−0.613860	−0.306930	+	0.951732i	$0.599302\pi$
$492$	0	0
$493$	−7.85671	−0.353848
$494$	0	0
$495$	−4.23659	−0.190420
$496$	0	0
$497$	23.5848	1.05792
$498$	0	0
$499$	−0.572450	−0.0256264	−0.0128132	−	0.999918i	$-0.504079\pi$
$499$	−0.572450	−0.0256264	−0.0128132	+	0.999918i	$0.504079\pi$
$500$	0	0
$501$	1.77592	0.0793423
$502$	0	0
$503$	−43.9411	−1.95924	−0.979619	−	0.200865i	$-0.935625\pi$
$503$	−43.9411	−1.95924	−0.979619	+	0.200865i	$0.935625\pi$
$504$	0	0
$505$	15.7765	0.702044
$506$	0	0
$507$	−4.71836	−0.209550
$508$	0	0
$509$	13.5696	0.601463	0.300731	−	0.953709i	$-0.402769\pi$
$509$	13.5696	0.601463	0.300731	+	0.953709i	$0.402769\pi$
$510$	0	0
$511$	71.2887	3.15363
$512$	0	0
$513$	0.774133	0.0341788
$514$	0	0
$515$	0.0559255	0.00246437
$516$	0	0
$517$	−5.67729	−0.249687
$518$	0	0
$519$	1.35672	0.0595535
$520$	0	0
$521$	3.64074	0.159504	0.0797519	−	0.996815i	$-0.474587\pi$
$521$	3.64074	0.159504	0.0797519	+	0.996815i	$0.474587\pi$
$522$	0	0
$523$	−13.5655	−0.593177	−0.296588	−	0.955005i	$-0.595849\pi$
$523$	−13.5655	−0.593177	−0.296588	+	0.955005i	$0.595849\pi$
$524$	0	0
$525$	−1.96889	−0.0859295
$526$	0	0
$527$	27.5208	1.19882
$528$	0	0
$529$	17.3172	0.752923
$530$	0	0
$531$	−5.55393	−0.241020
$532$	0	0
$533$	12.2645	0.531233
$534$	0	0
$535$	−19.8387	−0.857700
$536$	0	0
$537$	−2.06624	−0.0891647
$538$	0	0
$539$	−19.0200	−0.819248
$540$	0	0
$541$	9.76498	0.419829	0.209915	−	0.977720i	$-0.432681\pi$
$541$	9.76498	0.419829	0.209915	+	0.977720i	$0.432681\pi$
$542$	0	0
$543$	−0.596549	−0.0256004
$544$	0	0
$545$	19.0355	0.815393
$546$	0	0
$547$	16.0294	0.685367	0.342683	−	0.939451i	$-0.388664\pi$
$547$	16.0294	0.685367	0.342683	+	0.939451i	$0.388664\pi$
$548$	0	0
$549$	−14.8803	−0.635075
$550$	0	0
$551$	2.43101	0.103565
$552$	0	0
$553$	−36.7188	−1.56144
$554$	0	0
$555$	−1.66812	−0.0708079
$556$	0	0
$557$	40.9882	1.73673	0.868363	−	0.495929i	$-0.165173\pi$
$557$	40.9882	1.73673	0.868363	+	0.495929i	$0.165173\pi$
$558$	0	0
$559$	14.2670	0.603428
$560$	0	0
$561$	−0.418149	−0.0176543
$562$	0	0
$563$	−24.8857	−1.04881	−0.524403	−	0.851470i	$-0.675711\pi$
$563$	−24.8857	−1.04881	−0.524403	+	0.851470i	$0.675711\pi$
$564$	0	0
$565$	5.11659	0.215257
$566$	0	0
$567$	−45.1417	−1.89577
$568$	0	0
$569$	−30.3845	−1.27378	−0.636892	−	0.770953i	$-0.719780\pi$
$569$	−30.3845	−1.27378	−0.636892	+	0.770953i	$0.719780\pi$
$570$	0	0
$571$	46.6318	1.95148	0.975741	−	0.218929i	$-0.0702564\pi$
$571$	46.6318	1.95148	0.975741	+	0.218929i	$0.0702564\pi$
$572$	0	0
$573$	−1.35546	−0.0566250
$574$	0	0
$575$	−18.9425	−0.789955
$576$	0	0
$577$	−7.07057	−0.294352	−0.147176	−	0.989110i	$-0.547018\pi$
$577$	−7.07057	−0.294352	−0.147176	+	0.989110i	$0.547018\pi$
$578$	0	0
$579$	−2.65857	−0.110486
$580$	0	0
$581$	−49.6226	−2.05869
$582$	0	0
$583$	6.28923	0.260473
$584$	0	0
$585$	−29.7974	−1.23197
$586$	0	0
$587$	26.7052	1.10224	0.551120	−	0.834426i	$-0.314201\pi$
$587$	26.7052	1.10224	0.551120	+	0.834426i	$0.314201\pi$
$588$	0	0
$589$	−8.51543	−0.350872
$590$	0	0
$591$	−2.21979	−0.0913102
$592$	0	0
$593$	23.1728	0.951593	0.475797	−	0.879555i	$-0.342160\pi$
$593$	23.1728	0.951593	0.475797	+	0.879555i	$0.342160\pi$
$594$	0	0
$595$	−23.4116	−0.959784
$596$	0	0
$597$	1.27403	0.0521424
$598$	0	0
$599$	9.95574	0.406781	0.203390	−	0.979098i	$-0.434804\pi$
$599$	9.95574	0.406781	0.203390	+	0.979098i	$0.434804\pi$
$600$	0	0
$601$	10.3446	0.421967	0.210983	−	0.977490i	$-0.432333\pi$
$601$	10.3446	0.421967	0.210983	+	0.977490i	$0.432333\pi$
$602$	0	0
$603$	11.9874	0.488164
$604$	0	0
$605$	−1.42012	−0.0577361
$606$	0	0
$607$	−22.8505	−0.927473	−0.463736	−	0.885973i	$-0.653492\pi$
$607$	−22.8505	−0.927473	−0.463736	+	0.885973i	$0.653492\pi$
$608$	0	0
$609$	1.60442	0.0650143
$610$	0	0
$611$	−39.9304	−1.61541
$612$	0	0
$613$	28.4830	1.15042	0.575208	−	0.818007i	$-0.304921\pi$
$613$	28.4830	1.15042	0.575208	+	0.818007i	$0.304921\pi$
$614$	0	0
$615$	−0.320397	−0.0129196
$616$	0	0
$617$	−45.0965	−1.81552	−0.907759	−	0.419493i	$-0.862208\pi$
$617$	−45.0965	−1.81552	−0.907759	+	0.419493i	$0.862208\pi$
$618$	0	0
$619$	−11.5698	−0.465030	−0.232515	−	0.972593i	$-0.574696\pi$
$619$	−11.5698	−0.465030	−0.232515	+	0.972593i	$0.574696\pi$
$620$	0	0
$621$	4.91542	0.197249
$622$	0	0
$623$	−77.3163	−3.09761
$624$	0	0
$625$	−1.18386	−0.0473544
$626$	0	0
$627$	0.129383	0.00516706
$628$	0	0
$629$	29.3413	1.16991
$630$	0	0
$631$	31.8995	1.26990	0.634950	−	0.772554i	$-0.281021\pi$
$631$	31.8995	1.26990	0.634950	+	0.772554i	$0.281021\pi$
$632$	0	0
$633$	−0.432912	−0.0172067
$634$	0	0
$635$	22.5769	0.895938
$636$	0	0
$637$	−133.774	−5.30033
$638$	0	0
$639$	13.7933	0.545656
$640$	0	0
$641$	−34.7177	−1.37127	−0.685633	−	0.727947i	$-0.740474\pi$
$641$	−34.7177	−1.37127	−0.685633	+	0.727947i	$0.740474\pi$
$642$	0	0
$643$	−47.7147	−1.88168	−0.940842	−	0.338845i	$-0.889964\pi$
$643$	−47.7147	−1.88168	−0.940842	+	0.338845i	$0.889964\pi$
$644$	0	0
$645$	−0.372710	−0.0146754
$646$	0	0
$647$	−3.02131	−0.118780	−0.0593900	−	0.998235i	$-0.518916\pi$
$647$	−3.02131	−0.118780	−0.0593900	+	0.998235i	$0.518916\pi$
$648$	0	0
$649$	−1.86170	−0.0730780
$650$	0	0
$651$	−5.62001	−0.220266
$652$	0	0
$653$	−5.70288	−0.223171	−0.111585	−	0.993755i	$-0.535593\pi$
$653$	−5.70288	−0.223171	−0.111585	+	0.993755i	$0.535593\pi$
$654$	0	0
$655$	−17.2994	−0.675945
$656$	0	0
$657$	41.6926	1.62658
$658$	0	0
$659$	15.3068	0.596269	0.298135	−	0.954524i	$-0.403636\pi$
$659$	15.3068	0.596269	0.298135	+	0.954524i	$0.403636\pi$
$660$	0	0
$661$	−27.4937	−1.06938	−0.534691	−	0.845048i	$-0.679572\pi$
$661$	−27.4937	−1.06938	−0.534691	+	0.845048i	$0.679572\pi$
$662$	0	0
$663$	−2.94099	−0.114219
$664$	0	0
$665$	7.24400	0.280910
$666$	0	0
$667$	15.4359	0.597681
$668$	0	0
$669$	0.597059	0.0230836
$670$	0	0
$671$	−4.98792	−0.192557
$672$	0	0
$673$	−19.3829	−0.747156	−0.373578	−	0.927599i	$-0.621869\pi$
$673$	−19.3829	−0.747156	−0.373578	+	0.927599i	$0.621869\pi$
$674$	0	0
$675$	−2.30944	−0.0888903
$676$	0	0
$677$	−35.9454	−1.38150	−0.690748	−	0.723096i	$-0.742719\pi$
$677$	−35.9454	−1.38150	−0.690748	+	0.723096i	$0.742719\pi$
$678$	0	0
$679$	31.8225	1.22123
$680$	0	0
$681$	3.30443	0.126626
$682$	0	0
$683$	18.4639	0.706500	0.353250	−	0.935529i	$-0.385076\pi$
$683$	18.4639	0.706500	0.353250	+	0.935529i	$0.385076\pi$
$684$	0	0
$685$	20.0549	0.766259
$686$	0	0
$687$	1.31164	0.0500421
$688$	0	0
$689$	44.2344	1.68520
$690$	0	0
$691$	12.1021	0.460387	0.230194	−	0.973145i	$-0.426064\pi$
$691$	12.1021	0.460387	0.230194	+	0.973145i	$0.426064\pi$
$692$	0	0
$693$	−15.2175	−0.578066
$694$	0	0
$695$	11.1750	0.423894
$696$	0	0
$697$	5.63559	0.213463
$698$	0	0
$699$	2.14114	0.0809852
$700$	0	0
$701$	16.5740	0.625992	0.312996	−	0.949754i	$-0.398667\pi$
$701$	16.5740	0.625992	0.312996	+	0.949754i	$0.398667\pi$
$702$	0	0
$703$	−9.07874	−0.342411
$704$	0	0
$705$	1.04314	0.0392870
$706$	0	0
$707$	56.6681	2.13122
$708$	0	0
$709$	−16.7084	−0.627497	−0.313749	−	0.949506i	$-0.601585\pi$
$709$	−16.7084	−0.627497	−0.313749	+	0.949506i	$0.601585\pi$
$710$	0	0
$711$	−21.4747	−0.805363
$712$	0	0
$713$	−54.0694	−2.02492
$714$	0	0
$715$	−9.98821	−0.373538
$716$	0	0
$717$	2.23501	0.0834679
$718$	0	0
$719$	−4.86279	−0.181351	−0.0906756	−	0.995880i	$-0.528903\pi$
$719$	−4.86279	−0.181351	−0.0906756	+	0.995880i	$0.528903\pi$
$720$	0	0
$721$	0.200881	0.00748119
$722$	0	0
$723$	1.06164	0.0394827
$724$	0	0
$725$	−7.25234	−0.269345
$726$	0	0
$727$	27.7576	1.02947	0.514736	−	0.857349i	$-0.327890\pi$
$727$	27.7576	1.02947	0.514736	+	0.857349i	$0.327890\pi$
$728$	0	0
$729$	−26.1002	−0.966676
$730$	0	0
$731$	6.55575	0.242473
$732$	0	0
$733$	−14.0117	−0.517532	−0.258766	−	0.965940i	$-0.583316\pi$
$733$	−14.0117	−0.517532	−0.258766	+	0.965940i	$0.583316\pi$
$734$	0	0
$735$	3.49472	0.128905
$736$	0	0
$737$	4.01821	0.148013
$738$	0	0
$739$	4.27744	0.157348	0.0786740	−	0.996900i	$-0.474931\pi$
$739$	4.27744	0.157348	0.0786740	+	0.996900i	$0.474931\pi$
$740$	0	0
$741$	0.909998	0.0334296
$742$	0	0
$743$	10.2413	0.375717	0.187858	−	0.982196i	$-0.439845\pi$
$743$	10.2413	0.375717	0.187858	+	0.982196i	$0.439845\pi$
$744$	0	0
$745$	26.4827	0.970250
$746$	0	0
$747$	−29.0213	−1.06183
$748$	0	0
$749$	−71.2591	−2.60375
$750$	0	0
$751$	38.2639	1.39627	0.698134	−	0.715967i	$-0.254014\pi$
$751$	38.2639	1.39627	0.698134	+	0.715967i	$0.254014\pi$
$752$	0	0
$753$	1.83244	0.0667779
$754$	0	0
$755$	31.6330	1.15124
$756$	0	0
$757$	−34.3797	−1.24955	−0.624776	−	0.780804i	$-0.714810\pi$
$757$	−34.3797	−1.24955	−0.624776	+	0.780804i	$0.714810\pi$
$758$	0	0
$759$	0.821529	0.0298196
$760$	0	0
$761$	−27.3794	−0.992501	−0.496250	−	0.868179i	$-0.665290\pi$
$761$	−27.3794	−0.992501	−0.496250	+	0.868179i	$0.665290\pi$
$762$	0	0
$763$	68.3744	2.47532
$764$	0	0
$765$	−13.6921	−0.495039
$766$	0	0
$767$	−13.0940	−0.472796
$768$	0	0
$769$	16.1105	0.580959	0.290479	−	0.956881i	$-0.406185\pi$
$769$	16.1105	0.580959	0.290479	+	0.956881i	$0.406185\pi$
$770$	0	0
$771$	−0.538558	−0.0193957
$772$	0	0
$773$	1.40536	0.0505473	0.0252737	−	0.999681i	$-0.491954\pi$
$773$	1.40536	0.0505473	0.0252737	+	0.999681i	$0.491954\pi$
$774$	0	0
$775$	25.4037	0.912529
$776$	0	0
$777$	−5.99179	−0.214954
$778$	0	0
$779$	−1.74376	−0.0624765
$780$	0	0
$781$	4.62358	0.165445
$782$	0	0
$783$	1.88192	0.0672545
$784$	0	0
$785$	−1.62705	−0.0580719
$786$	0	0
$787$	7.58377	0.270332	0.135166	−	0.990823i	$-0.456843\pi$
$787$	7.58377	0.270332	0.135166	+	0.990823i	$0.456843\pi$
$788$	0	0
$789$	−2.72280	−0.0969343
$790$	0	0
$791$	18.3785	0.653463
$792$	0	0
$793$	−35.0819	−1.24579
$794$	0	0
$795$	−1.15558	−0.0409842
$796$	0	0
$797$	−23.4257	−0.829782	−0.414891	−	0.909871i	$-0.636180\pi$
$797$	−23.4257	−0.829782	−0.414891	+	0.909871i	$0.636180\pi$
$798$	0	0
$799$	−18.3483	−0.649115
$800$	0	0
$801$	−45.2178	−1.59769
$802$	0	0
$803$	13.9755	0.493185
$804$	0	0
$805$	45.9964	1.62116
$806$	0	0
$807$	−2.09378	−0.0737047
$808$	0	0
$809$	−11.5024	−0.404404	−0.202202	−	0.979344i	$-0.564810\pi$
$809$	−11.5024	−0.404404	−0.202202	+	0.979344i	$0.564810\pi$
$810$	0	0
$811$	−46.7550	−1.64179	−0.820895	−	0.571079i	$-0.806525\pi$
$811$	−46.7550	−1.64179	−0.820895	+	0.571079i	$0.806525\pi$
$812$	0	0
$813$	1.05271	0.0369202
$814$	0	0
$815$	−4.50428	−0.157778
$816$	0	0
$817$	−2.02847	−0.0709672
$818$	0	0
$819$	−107.030	−3.73995
$820$	0	0
$821$	12.4924	0.435986	0.217993	−	0.975950i	$-0.430049\pi$
$821$	12.4924	0.435986	0.217993	+	0.975950i	$0.430049\pi$
$822$	0	0
$823$	0.669106	0.0233236	0.0116618	−	0.999932i	$-0.496288\pi$
$823$	0.669106	0.0233236	0.0116618	+	0.999932i	$0.496288\pi$
$824$	0	0
$825$	−0.385983	−0.0134382
$826$	0	0
$827$	13.4167	0.466543	0.233272	−	0.972412i	$-0.425057\pi$
$827$	13.4167	0.466543	0.233272	+	0.972412i	$0.425057\pi$
$828$	0	0
$829$	19.8478	0.689343	0.344671	−	0.938723i	$-0.387990\pi$
$829$	19.8478	0.689343	0.344671	+	0.938723i	$0.387990\pi$
$830$	0	0
$831$	−2.69825	−0.0936012
$832$	0	0
$833$	−61.4700	−2.12981
$834$	0	0
$835$	19.4927	0.674571
$836$	0	0
$837$	−6.59207	−0.227855
$838$	0	0
$839$	−11.5515	−0.398802	−0.199401	−	0.979918i	$-0.563900\pi$
$839$	−11.5515	−0.398802	−0.199401	+	0.979918i	$0.563900\pi$
$840$	0	0
$841$	−23.0902	−0.796213
$842$	0	0
$843$	1.75289	0.0603726
$844$	0	0
$845$	−51.7891	−1.78160
$846$	0	0
$847$	−5.10098	−0.175272
$848$	0	0
$849$	−1.48297	−0.0508953
$850$	0	0
$851$	−57.6462	−1.97609
$852$	0	0
$853$	−18.0519	−0.618085	−0.309043	−	0.951048i	$-0.600009\pi$
$853$	−18.0519	−0.618085	−0.309043	+	0.951048i	$0.600009\pi$
$854$	0	0
$855$	4.23659	0.144888
$856$	0	0
$857$	4.15199	0.141829	0.0709146	−	0.997482i	$-0.477408\pi$
$857$	4.15199	0.141829	0.0709146	+	0.997482i	$0.477408\pi$
$858$	0	0
$859$	5.08887	0.173630	0.0868149	−	0.996224i	$-0.472331\pi$
$859$	5.08887	0.173630	0.0868149	+	0.996224i	$0.472331\pi$
$860$	0	0
$861$	−1.15084	−0.0392206
$862$	0	0
$863$	−11.9500	−0.406782	−0.203391	−	0.979098i	$-0.565196\pi$
$863$	−11.9500	−0.406782	−0.203391	+	0.979098i	$0.565196\pi$
$864$	0	0
$865$	14.8915	0.506327
$866$	0	0
$867$	0.848109	0.0288033
$868$	0	0
$869$	−7.19839	−0.244189
$870$	0	0
$871$	28.2615	0.957606
$872$	0	0
$873$	18.6111	0.629889
$874$	0	0
$875$	−57.8307	−1.95503
$876$	0	0
$877$	33.5552	1.13308	0.566538	−	0.824035i	$-0.308282\pi$
$877$	33.5552	1.13308	0.566538	+	0.824035i	$0.308282\pi$
$878$	0	0
$879$	0.845346	0.0285128
$880$	0	0
$881$	−31.5508	−1.06297	−0.531486	−	0.847067i	$-0.678366\pi$
$881$	−31.5508	−1.06297	−0.531486	+	0.847067i	$0.678366\pi$
$882$	0	0
$883$	58.9397	1.98348	0.991740	−	0.128263i	$-0.0409401\pi$
$883$	58.9397	1.98348	0.991740	+	0.128263i	$0.0409401\pi$
$884$	0	0
$885$	0.342067	0.0114985
$886$	0	0
$887$	−18.3167	−0.615014	−0.307507	−	0.951546i	$-0.599495\pi$
$887$	−18.3167	−0.615014	−0.307507	+	0.951546i	$0.599495\pi$
$888$	0	0
$889$	81.0948	2.71983
$890$	0	0
$891$	−8.84962	−0.296473
$892$	0	0
$893$	5.67729	0.189983
$894$	0	0
$895$	−22.6792	−0.758082
$896$	0	0
$897$	5.77811	0.192925
$898$	0	0
$899$	−20.7011	−0.690420
$900$	0	0
$901$	20.3260	0.677156
$902$	0	0
$903$	−1.33875	−0.0445508
$904$	0	0
$905$	−6.54777	−0.217655
$906$	0	0
$907$	−11.7843	−0.391292	−0.195646	−	0.980675i	$-0.562680\pi$
$907$	−11.7843	−0.391292	−0.195646	+	0.980675i	$0.562680\pi$
$908$	0	0
$909$	33.1418	1.09924
$910$	0	0
$911$	−6.83787	−0.226549	−0.113274	−	0.993564i	$-0.536134\pi$
$911$	−6.83787	−0.226549	−0.113274	+	0.993564i	$0.536134\pi$
$912$	0	0
$913$	−9.72806	−0.321952
$914$	0	0
$915$	0.916478	0.0302978
$916$	0	0
$917$	−62.1384	−2.05199
$918$	0	0
$919$	44.7520	1.47623	0.738116	−	0.674673i	$-0.235715\pi$
$919$	44.7520	1.47623	0.738116	+	0.674673i	$0.235715\pi$
$920$	0	0
$921$	1.90122	0.0626473
$922$	0	0
$923$	32.5193	1.07039
$924$	0	0
$925$	27.0842	0.890525
$926$	0	0
$927$	0.117483	0.00385866
$928$	0	0
$929$	−6.54839	−0.214846	−0.107423	−	0.994213i	$-0.534260\pi$
$929$	−6.54839	−0.214846	−0.107423	+	0.994213i	$0.534260\pi$
$930$	0	0
$931$	19.0200	0.623354
$932$	0	0
$933$	0.451495	0.0147813
$934$	0	0
$935$	−4.58964	−0.150097
$936$	0	0
$937$	16.9825	0.554793	0.277396	−	0.960756i	$-0.410529\pi$
$937$	16.9825	0.554793	0.277396	+	0.960756i	$0.410529\pi$
$938$	0	0
$939$	4.35202	0.142023
$940$	0	0
$941$	−44.4202	−1.44806	−0.724029	−	0.689770i	$-0.757712\pi$
$941$	−44.4202	−1.44806	−0.724029	+	0.689770i	$0.757712\pi$
$942$	0	0
$943$	−11.0721	−0.360558
$944$	0	0
$945$	5.60781	0.182422
$946$	0	0
$947$	−36.6101	−1.18967	−0.594834	−	0.803849i	$-0.702782\pi$
$947$	−36.6101	−1.18967	−0.594834	+	0.803849i	$0.702782\pi$
$948$	0	0
$949$	98.2947	3.19078
$950$	0	0
$951$	−2.52732	−0.0819540
$952$	0	0
$953$	18.9783	0.614768	0.307384	−	0.951586i	$-0.400546\pi$
$953$	18.9783	0.614768	0.307384	+	0.951586i	$0.400546\pi$
$954$	0	0
$955$	−14.8776	−0.481428
$956$	0	0
$957$	0.314532	0.0101674
$958$	0	0
$959$	72.0359	2.32616
$960$	0	0
$961$	41.5126	1.33911
$962$	0	0
$963$	−41.6753	−1.34297
$964$	0	0
$965$	−29.1807	−0.939360
$966$	0	0
$967$	−4.43383	−0.142582	−0.0712912	−	0.997456i	$-0.522712\pi$
$967$	−4.43383	−0.142582	−0.0712912	+	0.997456i	$0.522712\pi$
$968$	0	0
$969$	0.418149	0.0134329
$970$	0	0
$971$	59.2517	1.90148	0.950738	−	0.309995i	$-0.100327\pi$
$971$	59.2517	1.90148	0.950738	+	0.309995i	$0.100327\pi$
$972$	0	0
$973$	40.1400	1.28683
$974$	0	0
$975$	−2.71476	−0.0869419
$976$	0	0
$977$	18.2522	0.583939	0.291970	−	0.956428i	$-0.405689\pi$
$977$	18.2522	0.583939	0.291970	+	0.956428i	$0.405689\pi$
$978$	0	0
$979$	−15.1572	−0.484425
$980$	0	0
$981$	39.9881	1.27672
$982$	0	0
$983$	18.0170	0.574654	0.287327	−	0.957833i	$-0.407233\pi$
$983$	18.0170	0.574654	0.287327	+	0.957833i	$0.407233\pi$
$984$	0	0
$985$	−24.3647	−0.776323
$986$	0	0
$987$	3.74690	0.119265
$988$	0	0
$989$	−12.8799	−0.409558
$990$	0	0
$991$	−34.6175	−1.09966	−0.549831	−	0.835276i	$-0.685308\pi$
$991$	−34.6175	−1.09966	−0.549831	+	0.835276i	$0.685308\pi$
$992$	0	0
$993$	−1.00417	−0.0318663
$994$	0	0
$995$	13.9838	0.443317
$996$	0	0
$997$	6.73230	0.213214	0.106607	−	0.994301i	$-0.466001\pi$
$997$	6.73230	0.213214	0.106607	+	0.994301i	$0.466001\pi$
$998$	0	0
$999$	−7.02815	−0.222361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.y.1.3		6
4.3	odd	2		1672.2.a.h.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.h.1.4	✓	6	4.3	odd	2
3344.2.a.y.1.3		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-0.129383 q^{3} -1.42012 q^{5} -5.10098 q^{7} -2.98326 q^{9} +O(q^{10})\)	\(q-0.129383 q^{3} -1.42012 q^{5} -5.10098 q^{7} -2.98326 q^{9} -1.00000 q^{11} -7.03336 q^{13} +0.183739 q^{15} -3.23187 q^{17} +1.00000 q^{19} +0.659980 q^{21} +6.34958 q^{23} -2.98326 q^{25} +0.774133 q^{27} +2.43101 q^{29} -8.51543 q^{31} +0.129383 q^{33} +7.24400 q^{35} -9.07874 q^{37} +0.909998 q^{39} -1.74376 q^{41} -2.02847 q^{43} +4.23659 q^{45} +5.67729 q^{47} +19.0200 q^{49} +0.418149 q^{51} -6.28923 q^{53} +1.42012 q^{55} -0.129383 q^{57} +1.86170 q^{59} +4.98792 q^{61} +15.2175 q^{63} +9.98821 q^{65} -4.01821 q^{67} -0.821529 q^{69} -4.62358 q^{71} -13.9755 q^{73} +0.385983 q^{75} +5.10098 q^{77} +7.19839 q^{79} +8.84962 q^{81} +9.72806 q^{83} +4.58964 q^{85} -0.314532 q^{87} +15.1572 q^{89} +35.8770 q^{91} +1.10175 q^{93} -1.42012 q^{95} -6.23850 q^{97} +2.98326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Introduction

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