LMFDB - Embedded newform 3344.2.a.j.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			1.2
Root			$-1.30278$ 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)

$f(q)$	$=$	$q+1.30278 q^{3} +1.30278 q^{5} -2.30278 q^{7} -1.30278 q^{9} +1.00000 q^{11} -1.69722 q^{13} +1.69722 q^{15} -2.60555 q^{17} +1.00000 q^{19} -3.00000 q^{21} +4.60555 q^{23} -3.30278 q^{25} -5.60555 q^{27} -9.30278 q^{29} -2.90833 q^{31} +1.30278 q^{33} -3.00000 q^{35} +8.00000 q^{37} -2.21110 q^{39} -3.69722 q^{41} -1.30278 q^{43} -1.69722 q^{45} -3.21110 q^{47} -1.69722 q^{49} -3.39445 q^{51} -0.605551 q^{53} +1.30278 q^{55} +1.30278 q^{57} +4.00000 q^{59} -2.00000 q^{61} +3.00000 q^{63} -2.21110 q^{65} +5.51388 q^{67} +6.00000 q^{69} +2.09167 q^{71} -16.6056 q^{73} -4.30278 q^{75} -2.30278 q^{77} -9.81665 q^{79} -3.39445 q^{81} -0.0916731 q^{83} -3.39445 q^{85} -12.1194 q^{87} -7.81665 q^{89} +3.90833 q^{91} -3.78890 q^{93} +1.30278 q^{95} -14.4222 q^{97} -1.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{11} - 7 q^{13} + 7 q^{15} + 2 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} - 3 q^{25} - 4 q^{27} - 15 q^{29} + 5 q^{31} - q^{33} - 6 q^{35} + 16 q^{37} + 10 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 - q^7 + q^9 + 2 * q^11 - 7 * q^13 + 7 * q^15 + 2 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 - 3 * q^25 - 4 * q^27 - 15 * q^29 + 5 * q^31 - q^33 - 6 * q^35 + 16 * q^37 + 10 * q^39 - 11 * q^41 + q^43 - 7 * q^45 + 8 * q^47 - 7 * q^49 - 14 * q^51 + 6 * q^53 - q^55 - q^57 + 8 * q^59 - 4 * q^61 + 6 * q^63 + 10 * q^65 - 7 * q^67 + 12 * q^69 + 15 * q^71 - 26 * q^73 - 5 * q^75 - q^77 + 2 * q^79 - 14 * q^81 - 11 * q^83 - 14 * q^85 + q^87 + 6 * q^89 - 3 * q^91 - 22 * q^93 - q^95 + q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.30278	0.752158	0.376079	−	0.926588i	$-0.377272\pi$
$3$	1.30278	0.752158	0.376079	+	0.926588i	$0.377272\pi$
$4$	0	0
$5$	1.30278	0.582619	0.291309	−	0.956629i	$-0.405909\pi$
$5$	1.30278	0.582619	0.291309	+	0.956629i	$0.405909\pi$
$6$	0	0
$7$	−2.30278	−0.870367	−0.435184	−	0.900342i	$-0.643317\pi$
$7$	−2.30278	−0.870367	−0.435184	+	0.900342i	$0.643317\pi$
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.69722	−0.470725	−0.235363	−	0.971908i	$-0.575628\pi$
$13$	−1.69722	−0.470725	−0.235363	+	0.971908i	$0.575628\pi$
$14$	0	0
$15$	1.69722	0.438221
$16$	0	0
$17$	−2.60555	−0.631939	−0.315970	−	0.948769i	$-0.602330\pi$
$17$	−2.60555	−0.631939	−0.315970	+	0.948769i	$0.602330\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	4.60555	0.960324	0.480162	−	0.877180i	$-0.340578\pi$
$23$	4.60555	0.960324	0.480162	+	0.877180i	$0.340578\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0	0
$27$	−5.60555	−1.07879
$28$	0	0
$29$	−9.30278	−1.72748	−0.863741	−	0.503936i	$-0.831885\pi$
$29$	−9.30278	−1.72748	−0.863741	+	0.503936i	$0.831885\pi$
$30$	0	0
$31$	−2.90833	−0.522351	−0.261175	−	0.965291i	$-0.584110\pi$
$31$	−2.90833	−0.522351	−0.261175	+	0.965291i	$0.584110\pi$
$32$	0	0
$33$	1.30278	0.226784
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−2.21110	−0.354060
$40$	0	0
$41$	−3.69722	−0.577409	−0.288705	−	0.957418i	$-0.593225\pi$
$41$	−3.69722	−0.577409	−0.288705	+	0.957418i	$0.593225\pi$
$42$	0	0
$43$	−1.30278	−0.198671	−0.0993357	−	0.995054i	$-0.531672\pi$
$43$	−1.30278	−0.198671	−0.0993357	+	0.995054i	$0.531672\pi$
$44$	0	0
$45$	−1.69722	−0.253007
$46$	0	0
$47$	−3.21110	−0.468387	−0.234194	−	0.972190i	$-0.575245\pi$
$47$	−3.21110	−0.468387	−0.234194	+	0.972190i	$0.575245\pi$
$48$	0	0
$49$	−1.69722	−0.242461
$50$	0	0
$51$	−3.39445	−0.475318
$52$	0	0
$53$	−0.605551	−0.0831789	−0.0415894	−	0.999135i	$-0.513242\pi$
$53$	−0.605551	−0.0831789	−0.0415894	+	0.999135i	$0.513242\pi$
$54$	0	0
$55$	1.30278	0.175666
$56$	0	0
$57$	1.30278	0.172557
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	−2.21110	−0.274254
$66$	0	0
$67$	5.51388	0.673627	0.336814	−	0.941571i	$-0.390651\pi$
$67$	5.51388	0.673627	0.336814	+	0.941571i	$0.390651\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	2.09167	0.248236	0.124118	−	0.992267i	$-0.460390\pi$
$71$	2.09167	0.248236	0.124118	+	0.992267i	$0.460390\pi$
$72$	0	0
$73$	−16.6056	−1.94353	−0.971766	−	0.235945i	$-0.924181\pi$
$73$	−16.6056	−1.94353	−0.971766	+	0.235945i	$0.924181\pi$
$74$	0	0
$75$	−4.30278	−0.496842
$76$	0	0
$77$	−2.30278	−0.262426
$78$	0	0
$79$	−9.81665	−1.10446	−0.552230	−	0.833692i	$-0.686223\pi$
$79$	−9.81665	−1.10446	−0.552230	+	0.833692i	$0.686223\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−0.0916731	−0.0100624	−0.00503121	−	0.999987i	$-0.501601\pi$
$83$	−0.0916731	−0.0100624	−0.00503121	+	0.999987i	$0.501601\pi$
$84$	0	0
$85$	−3.39445	−0.368180
$86$	0	0
$87$	−12.1194	−1.29934
$88$	0	0
$89$	−7.81665	−0.828564	−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	0	0
$91$	3.90833	0.409704
$92$	0	0
$93$	−3.78890	−0.392890
$94$	0	0
$95$	1.30278	0.133662
$96$	0	0
$97$	−14.4222	−1.46435	−0.732177	−	0.681115i	$-0.761495\pi$
$97$	−14.4222	−1.46435	−0.732177	+	0.681115i	$0.761495\pi$
$98$	0	0
$99$	−1.30278	−0.130934
$100$	0	0
$101$	4.60555	0.458269	0.229135	−	0.973395i	$-0.426410\pi$
$101$	4.60555	0.458269	0.229135	+	0.973395i	$0.426410\pi$
$102$	0	0
$103$	4.51388	0.444766	0.222383	−	0.974959i	$-0.428617\pi$
$103$	4.51388	0.444766	0.222383	+	0.974959i	$0.428617\pi$
$104$	0	0
$105$	−3.90833	−0.381414
$106$	0	0
$107$	−7.21110	−0.697124	−0.348562	−	0.937286i	$-0.613330\pi$
$107$	−7.21110	−0.697124	−0.348562	+	0.937286i	$0.613330\pi$
$108$	0	0
$109$	3.21110	0.307568	0.153784	−	0.988105i	$-0.450854\pi$
$109$	3.21110	0.307568	0.153784	+	0.988105i	$0.450854\pi$
$110$	0	0
$111$	10.4222	0.989232
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	2.21110	0.204417
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.81665	−0.434303
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	−7.39445	−0.656151	−0.328076	−	0.944651i	$-0.606400\pi$
$127$	−7.39445	−0.656151	−0.328076	+	0.944651i	$0.606400\pi$
$128$	0	0
$129$	−1.69722	−0.149432
$130$	0	0
$131$	5.51388	0.481750	0.240875	−	0.970556i	$-0.422566\pi$
$131$	5.51388	0.481750	0.240875	+	0.970556i	$0.422566\pi$
$132$	0	0
$133$	−2.30278	−0.199676
$134$	0	0
$135$	−7.30278	−0.628523
$136$	0	0
$137$	−11.0917	−0.947626	−0.473813	−	0.880626i	$-0.657123\pi$
$137$	−11.0917	−0.947626	−0.473813	+	0.880626i	$0.657123\pi$
$138$	0	0
$139$	15.7250	1.33378	0.666888	−	0.745158i	$-0.267626\pi$
$139$	15.7250	1.33378	0.666888	+	0.745158i	$0.267626\pi$
$140$	0	0
$141$	−4.18335	−0.352301
$142$	0	0
$143$	−1.69722	−0.141929
$144$	0	0
$145$	−12.1194	−1.00646
$146$	0	0
$147$	−2.21110	−0.182369
$148$	0	0
$149$	−6.78890	−0.556168	−0.278084	−	0.960557i	$-0.589699\pi$
$149$	−6.78890	−0.556168	−0.278084	+	0.960557i	$0.589699\pi$
$150$	0	0
$151$	16.4222	1.33642	0.668210	−	0.743973i	$-0.267061\pi$
$151$	16.4222	1.33642	0.668210	+	0.743973i	$0.267061\pi$
$152$	0	0
$153$	3.39445	0.274425
$154$	0	0
$155$	−3.78890	−0.304332
$156$	0	0
$157$	14.6972	1.17297	0.586483	−	0.809961i	$-0.300512\pi$
$157$	14.6972	1.17297	0.586483	+	0.809961i	$0.300512\pi$
$158$	0	0
$159$	−0.788897	−0.0625636
$160$	0	0
$161$	−10.6056	−0.835835
$162$	0	0
$163$	−5.39445	−0.422526	−0.211263	−	0.977429i	$-0.567758\pi$
$163$	−5.39445	−0.422526	−0.211263	+	0.977429i	$0.567758\pi$
$164$	0	0
$165$	1.69722	0.132129
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−10.1194	−0.778418
$170$	0	0
$171$	−1.30278	−0.0996257
$172$	0	0
$173$	−9.11943	−0.693337	−0.346669	−	0.937988i	$-0.612687\pi$
$173$	−9.11943	−0.693337	−0.346669	+	0.937988i	$0.612687\pi$
$174$	0	0
$175$	7.60555	0.574926
$176$	0	0
$177$	5.21110	0.391690
$178$	0	0
$179$	−8.30278	−0.620579	−0.310289	−	0.950642i	$-0.600426\pi$
$179$	−8.30278	−0.620579	−0.310289	+	0.950642i	$0.600426\pi$
$180$	0	0
$181$	18.6056	1.38294	0.691470	−	0.722405i	$-0.256963\pi$
$181$	18.6056	1.38294	0.691470	+	0.722405i	$0.256963\pi$
$182$	0	0
$183$	−2.60555	−0.192608
$184$	0	0
$185$	10.4222	0.766256
$186$	0	0
$187$	−2.60555	−0.190537
$188$	0	0
$189$	12.9083	0.938943
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	19.1194	1.37625	0.688123	−	0.725594i	$-0.258435\pi$
$193$	19.1194	1.37625	0.688123	+	0.725594i	$0.258435\pi$
$194$	0	0
$195$	−2.88057	−0.206282
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	17.0278	1.20707	0.603533	−	0.797338i	$-0.293759\pi$
$199$	17.0278	1.20707	0.603533	+	0.797338i	$0.293759\pi$
$200$	0	0
$201$	7.18335	0.506674
$202$	0	0
$203$	21.4222	1.50354
$204$	0	0
$205$	−4.81665	−0.336410
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−10.6056	−0.730116	−0.365058	−	0.930985i	$-0.618951\pi$
$211$	−10.6056	−0.730116	−0.365058	+	0.930985i	$0.618951\pi$
$212$	0	0
$213$	2.72498	0.186713
$214$	0	0
$215$	−1.69722	−0.115750
$216$	0	0
$217$	6.69722	0.454637
$218$	0	0
$219$	−21.6333	−1.46184
$220$	0	0
$221$	4.42221	0.297470
$222$	0	0
$223$	−1.21110	−0.0811014	−0.0405507	−	0.999177i	$-0.512911\pi$
$223$	−1.21110	−0.0811014	−0.0405507	+	0.999177i	$0.512911\pi$
$224$	0	0
$225$	4.30278	0.286852
$226$	0	0
$227$	9.39445	0.623531	0.311766	−	0.950159i	$-0.399080\pi$
$227$	9.39445	0.623531	0.311766	+	0.950159i	$0.399080\pi$
$228$	0	0
$229$	3.51388	0.232204	0.116102	−	0.993237i	$-0.462960\pi$
$229$	3.51388	0.232204	0.116102	+	0.993237i	$0.462960\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	−4.18335	−0.272891
$236$	0	0
$237$	−12.7889	−0.830728
$238$	0	0
$239$	−19.1194	−1.23673	−0.618367	−	0.785890i	$-0.712205\pi$
$239$	−19.1194	−1.23673	−0.618367	+	0.785890i	$0.712205\pi$
$240$	0	0
$241$	17.7250	1.14177	0.570883	−	0.821031i	$-0.306601\pi$
$241$	17.7250	1.14177	0.570883	+	0.821031i	$0.306601\pi$
$242$	0	0
$243$	12.3944	0.795104
$244$	0	0
$245$	−2.21110	−0.141262
$246$	0	0
$247$	−1.69722	−0.107992
$248$	0	0
$249$	−0.119429	−0.00756853
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	4.60555	0.289549
$254$	0	0
$255$	−4.42221	−0.276929
$256$	0	0
$257$	−13.3944	−0.835523	−0.417761	−	0.908557i	$-0.637185\pi$
$257$	−13.3944	−0.835523	−0.417761	+	0.908557i	$0.637185\pi$
$258$	0	0
$259$	−18.4222	−1.14470
$260$	0	0
$261$	12.1194	0.750174
$262$	0	0
$263$	4.51388	0.278338	0.139169	−	0.990269i	$-0.455557\pi$
$263$	4.51388	0.278338	0.139169	+	0.990269i	$0.455557\pi$
$264$	0	0
$265$	−0.788897	−0.0484616
$266$	0	0
$267$	−10.1833	−0.623211
$268$	0	0
$269$	27.0278	1.64791	0.823956	−	0.566654i	$-0.191762\pi$
$269$	27.0278	1.64791	0.823956	+	0.566654i	$0.191762\pi$
$270$	0	0
$271$	−13.6972	−0.832047	−0.416024	−	0.909354i	$-0.636577\pi$
$271$	−13.6972	−0.832047	−0.416024	+	0.909354i	$0.636577\pi$
$272$	0	0
$273$	5.09167	0.308162
$274$	0	0
$275$	−3.30278	−0.199165
$276$	0	0
$277$	−8.78890	−0.528074	−0.264037	−	0.964513i	$-0.585054\pi$
$277$	−8.78890	−0.528074	−0.264037	+	0.964513i	$0.585054\pi$
$278$	0	0
$279$	3.78890	0.226835
$280$	0	0
$281$	22.9361	1.36825	0.684126	−	0.729364i	$-0.260184\pi$
$281$	22.9361	1.36825	0.684126	+	0.729364i	$0.260184\pi$
$282$	0	0
$283$	7.09167	0.421556	0.210778	−	0.977534i	$-0.432400\pi$
$283$	7.09167	0.421556	0.210778	+	0.977534i	$0.432400\pi$
$284$	0	0
$285$	1.69722	0.100535
$286$	0	0
$287$	8.51388	0.502558
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0	0
$291$	−18.7889	−1.10142
$292$	0	0
$293$	−15.5139	−0.906330	−0.453165	−	0.891427i	$-0.649705\pi$
$293$	−15.5139	−0.906330	−0.453165	+	0.891427i	$0.649705\pi$
$294$	0	0
$295$	5.21110	0.303402
$296$	0	0
$297$	−5.60555	−0.325267
$298$	0	0
$299$	−7.81665	−0.452049
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−2.60555	−0.149193
$306$	0	0
$307$	−33.0278	−1.88499	−0.942497	−	0.334215i	$-0.891529\pi$
$307$	−33.0278	−1.88499	−0.942497	+	0.334215i	$0.891529\pi$
$308$	0	0
$309$	5.88057	0.334534
$310$	0	0
$311$	9.39445	0.532710	0.266355	−	0.963875i	$-0.414181\pi$
$311$	9.39445	0.532710	0.266355	+	0.963875i	$0.414181\pi$
$312$	0	0
$313$	34.9361	1.97470	0.987352	−	0.158543i	$-0.0506796\pi$
$313$	34.9361	1.97470	0.987352	+	0.158543i	$0.0506796\pi$
$314$	0	0
$315$	3.90833	0.220209
$316$	0	0
$317$	19.8167	1.11301	0.556507	−	0.830843i	$-0.312141\pi$
$317$	19.8167	1.11301	0.556507	+	0.830843i	$0.312141\pi$
$318$	0	0
$319$	−9.30278	−0.520855
$320$	0	0
$321$	−9.39445	−0.524347
$322$	0	0
$323$	−2.60555	−0.144977
$324$	0	0
$325$	5.60555	0.310940
$326$	0	0
$327$	4.18335	0.231340
$328$	0	0
$329$	7.39445	0.407669
$330$	0	0
$331$	−7.90833	−0.434681	−0.217340	−	0.976096i	$-0.569738\pi$
$331$	−7.90833	−0.434681	−0.217340	+	0.976096i	$0.569738\pi$
$332$	0	0
$333$	−10.4222	−0.571133
$334$	0	0
$335$	7.18335	0.392468
$336$	0	0
$337$	5.51388	0.300360	0.150180	−	0.988659i	$-0.452015\pi$
$337$	5.51388	0.300360	0.150180	+	0.988659i	$0.452015\pi$
$338$	0	0
$339$	−5.21110	−0.283028
$340$	0	0
$341$	−2.90833	−0.157495
$342$	0	0
$343$	20.0278	1.08140
$344$	0	0
$345$	7.81665	0.420834
$346$	0	0
$347$	14.7889	0.793910	0.396955	−	0.917838i	$-0.370067\pi$
$347$	14.7889	0.793910	0.396955	+	0.917838i	$0.370067\pi$
$348$	0	0
$349$	−19.2111	−1.02835	−0.514173	−	0.857686i	$-0.671901\pi$
$349$	−19.2111	−1.02835	−0.514173	+	0.857686i	$0.671901\pi$
$350$	0	0
$351$	9.51388	0.507813
$352$	0	0
$353$	23.2111	1.23540	0.617701	−	0.786413i	$-0.288064\pi$
$353$	23.2111	1.23540	0.617701	+	0.786413i	$0.288064\pi$
$354$	0	0
$355$	2.72498	0.144627
$356$	0	0
$357$	7.81665	0.413701
$358$	0	0
$359$	24.1194	1.27297	0.636487	−	0.771287i	$-0.280387\pi$
$359$	24.1194	1.27297	0.636487	+	0.771287i	$0.280387\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.30278	0.0683780
$364$	0	0
$365$	−21.6333	−1.13234
$366$	0	0
$367$	−4.42221	−0.230837	−0.115419	−	0.993317i	$-0.536821\pi$
$367$	−4.42221	−0.230837	−0.115419	+	0.993317i	$0.536821\pi$
$368$	0	0
$369$	4.81665	0.250745
$370$	0	0
$371$	1.39445	0.0723962
$372$	0	0
$373$	9.30278	0.481679	0.240840	−	0.970565i	$-0.422577\pi$
$373$	9.30278	0.481679	0.240840	+	0.970565i	$0.422577\pi$
$374$	0	0
$375$	−14.0917	−0.727691
$376$	0	0
$377$	15.7889	0.813170
$378$	0	0
$379$	−32.3028	−1.65928	−0.829641	−	0.558297i	$-0.811455\pi$
$379$	−32.3028	−1.65928	−0.829641	+	0.558297i	$0.811455\pi$
$380$	0	0
$381$	−9.63331	−0.493529
$382$	0	0
$383$	−22.9083	−1.17056	−0.585280	−	0.810831i	$-0.699016\pi$
$383$	−22.9083	−1.17056	−0.585280	+	0.810831i	$0.699016\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	1.69722	0.0862748
$388$	0	0
$389$	12.0917	0.613072	0.306536	−	0.951859i	$-0.400830\pi$
$389$	12.0917	0.613072	0.306536	+	0.951859i	$0.400830\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	7.18335	0.362352
$394$	0	0
$395$	−12.7889	−0.643479
$396$	0	0
$397$	−4.72498	−0.237140	−0.118570	−	0.992946i	$-0.537831\pi$
$397$	−4.72498	−0.237140	−0.118570	+	0.992946i	$0.537831\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−13.8167	−0.689971	−0.344985	−	0.938608i	$-0.612116\pi$
$401$	−13.8167	−0.689971	−0.344985	+	0.938608i	$0.612116\pi$
$402$	0	0
$403$	4.93608	0.245884
$404$	0	0
$405$	−4.42221	−0.219741
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−33.5416	−1.65853	−0.829264	−	0.558858i	$-0.811240\pi$
$409$	−33.5416	−1.65853	−0.829264	+	0.558858i	$0.811240\pi$
$410$	0	0
$411$	−14.4500	−0.712764
$412$	0	0
$413$	−9.21110	−0.453249
$414$	0	0
$415$	−0.119429	−0.00586256
$416$	0	0
$417$	20.4861	1.00321
$418$	0	0
$419$	−8.78890	−0.429366	−0.214683	−	0.976684i	$-0.568872\pi$
$419$	−8.78890	−0.429366	−0.214683	+	0.976684i	$0.568872\pi$
$420$	0	0
$421$	7.02776	0.342512	0.171256	−	0.985227i	$-0.445217\pi$
$421$	7.02776	0.342512	0.171256	+	0.985227i	$0.445217\pi$
$422$	0	0
$423$	4.18335	0.203401
$424$	0	0
$425$	8.60555	0.417431
$426$	0	0
$427$	4.60555	0.222878
$428$	0	0
$429$	−2.21110	−0.106753
$430$	0	0
$431$	2.78890	0.134336	0.0671682	−	0.997742i	$-0.478604\pi$
$431$	2.78890	0.134336	0.0671682	+	0.997742i	$0.478604\pi$
$432$	0	0
$433$	−15.8167	−0.760100	−0.380050	−	0.924966i	$-0.624093\pi$
$433$	−15.8167	−0.760100	−0.380050	+	0.924966i	$0.624093\pi$
$434$	0	0
$435$	−15.7889	−0.757020
$436$	0	0
$437$	4.60555	0.220313
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	2.21110	0.105291
$442$	0	0
$443$	17.2111	0.817724	0.408862	−	0.912596i	$-0.365926\pi$
$443$	17.2111	0.817724	0.408862	+	0.912596i	$0.365926\pi$
$444$	0	0
$445$	−10.1833	−0.482737
$446$	0	0
$447$	−8.84441	−0.418326
$448$	0	0
$449$	−13.8167	−0.652048	−0.326024	−	0.945361i	$-0.605709\pi$
$449$	−13.8167	−0.652048	−0.326024	+	0.945361i	$0.605709\pi$
$450$	0	0
$451$	−3.69722	−0.174095
$452$	0	0
$453$	21.3944	1.00520
$454$	0	0
$455$	5.09167	0.238701
$456$	0	0
$457$	−13.0278	−0.609413	−0.304706	−	0.952446i	$-0.598558\pi$
$457$	−13.0278	−0.609413	−0.304706	+	0.952446i	$0.598558\pi$
$458$	0	0
$459$	14.6056	0.681729
$460$	0	0
$461$	−9.21110	−0.429004	−0.214502	−	0.976724i	$-0.568813\pi$
$461$	−9.21110	−0.429004	−0.214502	+	0.976724i	$0.568813\pi$
$462$	0	0
$463$	34.4222	1.59974	0.799868	−	0.600176i	$-0.204903\pi$
$463$	34.4222	1.59974	0.799868	+	0.600176i	$0.204903\pi$
$464$	0	0
$465$	−4.93608	−0.228905
$466$	0	0
$467$	−7.02776	−0.325206	−0.162603	−	0.986692i	$-0.551989\pi$
$467$	−7.02776	−0.325206	−0.162603	+	0.986692i	$0.551989\pi$
$468$	0	0
$469$	−12.6972	−0.586303
$470$	0	0
$471$	19.1472	0.882256
$472$	0	0
$473$	−1.30278	−0.0599017
$474$	0	0
$475$	−3.30278	−0.151542
$476$	0	0
$477$	0.788897	0.0361211
$478$	0	0
$479$	−28.7250	−1.31248	−0.656239	−	0.754553i	$-0.727854\pi$
$479$	−28.7250	−1.31248	−0.656239	+	0.754553i	$0.727854\pi$
$480$	0	0
$481$	−13.5778	−0.619094
$482$	0	0
$483$	−13.8167	−0.628680
$484$	0	0
$485$	−18.7889	−0.853160
$486$	0	0
$487$	−32.3305	−1.46504	−0.732518	−	0.680748i	$-0.761655\pi$
$487$	−32.3305	−1.46504	−0.732518	+	0.680748i	$0.761655\pi$
$488$	0	0
$489$	−7.02776	−0.317806
$490$	0	0
$491$	9.88057	0.445904	0.222952	−	0.974829i	$-0.428431\pi$
$491$	9.88057	0.445904	0.222952	+	0.974829i	$0.428431\pi$
$492$	0	0
$493$	24.2389	1.09166
$494$	0	0
$495$	−1.69722	−0.0762846
$496$	0	0
$497$	−4.81665	−0.216056
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	2.60555	0.116407
$502$	0	0
$503$	41.7250	1.86042	0.930212	−	0.367021i	$-0.119622\pi$
$503$	41.7250	1.86042	0.930212	+	0.367021i	$0.119622\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	−13.1833	−0.585493
$508$	0	0
$509$	25.2111	1.11746	0.558731	−	0.829349i	$-0.311288\pi$
$509$	25.2111	1.11746	0.558731	+	0.829349i	$0.311288\pi$
$510$	0	0
$511$	38.2389	1.69159
$512$	0	0
$513$	−5.60555	−0.247491
$514$	0	0
$515$	5.88057	0.259129
$516$	0	0
$517$	−3.21110	−0.141224
$518$	0	0
$519$	−11.8806	−0.521499
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	11.8167	0.516706	0.258353	−	0.966051i	$-0.416820\pi$
$523$	11.8167	0.516706	0.258353	+	0.966051i	$0.416820\pi$
$524$	0	0
$525$	9.90833	0.432435
$526$	0	0
$527$	7.57779	0.330094
$528$	0	0
$529$	−1.78890	−0.0777781
$530$	0	0
$531$	−5.21110	−0.226143
$532$	0	0
$533$	6.27502	0.271801
$534$	0	0
$535$	−9.39445	−0.406157
$536$	0	0
$537$	−10.8167	−0.466773
$538$	0	0
$539$	−1.69722	−0.0731046
$540$	0	0
$541$	12.2389	0.526190	0.263095	−	0.964770i	$-0.415257\pi$
$541$	12.2389	0.526190	0.263095	+	0.964770i	$0.415257\pi$
$542$	0	0
$543$	24.2389	1.04019
$544$	0	0
$545$	4.18335	0.179195
$546$	0	0
$547$	−10.4222	−0.445621	−0.222811	−	0.974862i	$-0.571523\pi$
$547$	−10.4222	−0.445621	−0.222811	+	0.974862i	$0.571523\pi$
$548$	0	0
$549$	2.60555	0.111202
$550$	0	0
$551$	−9.30278	−0.396312
$552$	0	0
$553$	22.6056	0.961286
$554$	0	0
$555$	13.5778	0.576345
$556$	0	0
$557$	−22.4222	−0.950059	−0.475030	−	0.879970i	$-0.657563\pi$
$557$	−22.4222	−0.950059	−0.475030	+	0.879970i	$0.657563\pi$
$558$	0	0
$559$	2.21110	0.0935197
$560$	0	0
$561$	−3.39445	−0.143314
$562$	0	0
$563$	0.788897	0.0332481	0.0166240	−	0.999862i	$-0.494708\pi$
$563$	0.788897	0.0332481	0.0166240	+	0.999862i	$0.494708\pi$
$564$	0	0
$565$	−5.21110	−0.219233
$566$	0	0
$567$	7.81665	0.328269
$568$	0	0
$569$	28.3028	1.18651	0.593257	−	0.805013i	$-0.297842\pi$
$569$	28.3028	1.18651	0.593257	+	0.805013i	$0.297842\pi$
$570$	0	0
$571$	−20.9083	−0.874986	−0.437493	−	0.899222i	$-0.644134\pi$
$571$	−20.9083	−0.874986	−0.437493	+	0.899222i	$0.644134\pi$
$572$	0	0
$573$	7.81665	0.326545
$574$	0	0
$575$	−15.2111	−0.634347
$576$	0	0
$577$	41.1472	1.71298	0.856490	−	0.516164i	$-0.172640\pi$
$577$	41.1472	1.71298	0.856490	+	0.516164i	$0.172640\pi$
$578$	0	0
$579$	24.9083	1.03515
$580$	0	0
$581$	0.211103	0.00875801
$582$	0	0
$583$	−0.605551	−0.0250794
$584$	0	0
$585$	2.88057	0.119097
$586$	0	0
$587$	44.8444	1.85093	0.925463	−	0.378837i	$-0.123676\pi$
$587$	44.8444	1.85093	0.925463	+	0.378837i	$0.123676\pi$
$588$	0	0
$589$	−2.90833	−0.119836
$590$	0	0
$591$	−7.81665	−0.321534
$592$	0	0
$593$	0.422205	0.0173379	0.00866894	−	0.999962i	$-0.497241\pi$
$593$	0.422205	0.0173379	0.00866894	+	0.999962i	$0.497241\pi$
$594$	0	0
$595$	7.81665	0.320452
$596$	0	0
$597$	22.1833	0.907904
$598$	0	0
$599$	47.9638	1.95975	0.979875	−	0.199614i	$-0.0639687\pi$
$599$	47.9638	1.95975	0.979875	+	0.199614i	$0.0639687\pi$
$600$	0	0
$601$	−27.0917	−1.10509	−0.552546	−	0.833482i	$-0.686344\pi$
$601$	−27.0917	−1.10509	−0.552546	+	0.833482i	$0.686344\pi$
$602$	0	0
$603$	−7.18335	−0.292528
$604$	0	0
$605$	1.30278	0.0529654
$606$	0	0
$607$	13.8167	0.560801	0.280400	−	0.959883i	$-0.409533\pi$
$607$	13.8167	0.560801	0.280400	+	0.959883i	$0.409533\pi$
$608$	0	0
$609$	27.9083	1.13090
$610$	0	0
$611$	5.44996	0.220482
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−6.27502	−0.253033
$616$	0	0
$617$	16.5416	0.665941	0.332971	−	0.942937i	$-0.391949\pi$
$617$	16.5416	0.665941	0.332971	+	0.942937i	$0.391949\pi$
$618$	0	0
$619$	−24.7889	−0.996350	−0.498175	−	0.867077i	$-0.665996\pi$
$619$	−24.7889	−0.996350	−0.498175	+	0.867077i	$0.665996\pi$
$620$	0	0
$621$	−25.8167	−1.03599
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	1.30278	0.0520278
$628$	0	0
$629$	−20.8444	−0.831121
$630$	0	0
$631$	39.3944	1.56827	0.784134	−	0.620591i	$-0.213107\pi$
$631$	39.3944	1.56827	0.784134	+	0.620591i	$0.213107\pi$
$632$	0	0
$633$	−13.8167	−0.549163
$634$	0	0
$635$	−9.63331	−0.382286
$636$	0	0
$637$	2.88057	0.114132
$638$	0	0
$639$	−2.72498	−0.107799
$640$	0	0
$641$	−4.18335	−0.165232	−0.0826161	−	0.996581i	$-0.526328\pi$
$641$	−4.18335	−0.165232	−0.0826161	+	0.996581i	$0.526328\pi$
$642$	0	0
$643$	1.21110	0.0477612	0.0238806	−	0.999715i	$-0.492398\pi$
$643$	1.21110	0.0477612	0.0238806	+	0.999715i	$0.492398\pi$
$644$	0	0
$645$	−2.21110	−0.0870621
$646$	0	0
$647$	−2.60555	−0.102435	−0.0512174	−	0.998688i	$-0.516310\pi$
$647$	−2.60555	−0.102435	−0.0512174	+	0.998688i	$0.516310\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	8.72498	0.341959
$652$	0	0
$653$	−43.7527	−1.71218	−0.856088	−	0.516829i	$-0.827112\pi$
$653$	−43.7527	−1.71218	−0.856088	+	0.516829i	$0.827112\pi$
$654$	0	0
$655$	7.18335	0.280677
$656$	0	0
$657$	21.6333	0.843996
$658$	0	0
$659$	44.2389	1.72330	0.861651	−	0.507502i	$-0.169431\pi$
$659$	44.2389	1.72330	0.861651	+	0.507502i	$0.169431\pi$
$660$	0	0
$661$	35.6333	1.38598	0.692988	−	0.720950i	$-0.256294\pi$
$661$	35.6333	1.38598	0.692988	+	0.720950i	$0.256294\pi$
$662$	0	0
$663$	5.76114	0.223744
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	−42.8444	−1.65894
$668$	0	0
$669$	−1.57779	−0.0610011
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−51.5694	−1.98785	−0.993927	−	0.110042i	$-0.964902\pi$
$673$	−51.5694	−1.98785	−0.993927	+	0.110042i	$0.964902\pi$
$674$	0	0
$675$	18.5139	0.712600
$676$	0	0
$677$	14.3305	0.550767	0.275384	−	0.961334i	$-0.411195\pi$
$677$	14.3305	0.550767	0.275384	+	0.961334i	$0.411195\pi$
$678$	0	0
$679$	33.2111	1.27453
$680$	0	0
$681$	12.2389	0.468994
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	−14.4500	−0.552105
$686$	0	0
$687$	4.57779	0.174654
$688$	0	0
$689$	1.02776	0.0391544
$690$	0	0
$691$	−39.2111	−1.49166	−0.745830	−	0.666136i	$-0.767947\pi$
$691$	−39.2111	−1.49166	−0.745830	+	0.666136i	$0.767947\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	20.4861	0.777083
$696$	0	0
$697$	9.63331	0.364888
$698$	0	0
$699$	−20.8444	−0.788408
$700$	0	0
$701$	−33.2111	−1.25437	−0.627183	−	0.778872i	$-0.715792\pi$
$701$	−33.2111	−1.25437	−0.627183	+	0.778872i	$0.715792\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	−5.44996	−0.205257
$706$	0	0
$707$	−10.6056	−0.398863
$708$	0	0
$709$	5.93608	0.222934	0.111467	−	0.993768i	$-0.464445\pi$
$709$	5.93608	0.222934	0.111467	+	0.993768i	$0.464445\pi$
$710$	0	0
$711$	12.7889	0.479621
$712$	0	0
$713$	−13.3944	−0.501626
$714$	0	0
$715$	−2.21110	−0.0826905
$716$	0	0
$717$	−24.9083	−0.930219
$718$	0	0
$719$	−10.2389	−0.381845	−0.190923	−	0.981605i	$-0.561148\pi$
$719$	−10.2389	−0.381845	−0.190923	+	0.981605i	$0.561148\pi$
$720$	0	0
$721$	−10.3944	−0.387109
$722$	0	0
$723$	23.0917	0.858789
$724$	0	0
$725$	30.7250	1.14110
$726$	0	0
$727$	23.6333	0.876511	0.438255	−	0.898850i	$-0.355597\pi$
$727$	23.6333	0.876511	0.438255	+	0.898850i	$0.355597\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	3.39445	0.125548
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	−2.88057	−0.106251
$736$	0	0
$737$	5.51388	0.203106
$738$	0	0
$739$	−35.5416	−1.30742	−0.653710	−	0.756745i	$-0.726788\pi$
$739$	−35.5416	−1.30742	−0.653710	+	0.756745i	$0.726788\pi$
$740$	0	0
$741$	−2.21110	−0.0812269
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	−8.84441	−0.324034
$746$	0	0
$747$	0.119429	0.00436970
$748$	0	0
$749$	16.6056	0.606754
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−26.0555	−0.949516
$754$	0	0
$755$	21.3944	0.778624
$756$	0	0
$757$	−5.11943	−0.186069	−0.0930344	−	0.995663i	$-0.529657\pi$
$757$	−5.11943	−0.186069	−0.0930344	+	0.995663i	$0.529657\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	−2.60555	−0.0944512	−0.0472256	−	0.998884i	$-0.515038\pi$
$761$	−2.60555	−0.0944512	−0.0472256	+	0.998884i	$0.515038\pi$
$762$	0	0
$763$	−7.39445	−0.267697
$764$	0	0
$765$	4.42221	0.159885
$766$	0	0
$767$	−6.78890	−0.245133
$768$	0	0
$769$	48.0555	1.73293	0.866463	−	0.499241i	$-0.166388\pi$
$769$	48.0555	1.73293	0.866463	+	0.499241i	$0.166388\pi$
$770$	0	0
$771$	−17.4500	−0.628445
$772$	0	0
$773$	44.0555	1.58457	0.792283	−	0.610153i	$-0.208892\pi$
$773$	44.0555	1.58457	0.792283	+	0.610153i	$0.208892\pi$
$774$	0	0
$775$	9.60555	0.345042
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	−3.69722	−0.132467
$780$	0	0
$781$	2.09167	0.0748459
$782$	0	0
$783$	52.1472	1.86359
$784$	0	0
$785$	19.1472	0.683392
$786$	0	0
$787$	−1.02776	−0.0366356	−0.0183178	−	0.999832i	$-0.505831\pi$
$787$	−1.02776	−0.0366356	−0.0183178	+	0.999832i	$0.505831\pi$
$788$	0	0
$789$	5.88057	0.209354
$790$	0	0
$791$	9.21110	0.327509
$792$	0	0
$793$	3.39445	0.120540
$794$	0	0
$795$	−1.02776	−0.0364508
$796$	0	0
$797$	−2.23886	−0.0793045	−0.0396522	−	0.999214i	$-0.512625\pi$
$797$	−2.23886	−0.0793045	−0.0396522	+	0.999214i	$0.512625\pi$
$798$	0	0
$799$	8.36669	0.295992
$800$	0	0
$801$	10.1833	0.359811
$802$	0	0
$803$	−16.6056	−0.585997
$804$	0	0
$805$	−13.8167	−0.486973
$806$	0	0
$807$	35.2111	1.23949
$808$	0	0
$809$	−51.6333	−1.81533	−0.907665	−	0.419695i	$-0.862137\pi$
$809$	−51.6333	−1.81533	−0.907665	+	0.419695i	$0.862137\pi$
$810$	0	0
$811$	−6.84441	−0.240340	−0.120170	−	0.992753i	$-0.538344\pi$
$811$	−6.84441	−0.240340	−0.120170	+	0.992753i	$0.538344\pi$
$812$	0	0
$813$	−17.8444	−0.625831
$814$	0	0
$815$	−7.02776	−0.246172
$816$	0	0
$817$	−1.30278	−0.0455784
$818$	0	0
$819$	−5.09167	−0.177917
$820$	0	0
$821$	3.02776	0.105669	0.0528347	−	0.998603i	$-0.483174\pi$
$821$	3.02776	0.105669	0.0528347	+	0.998603i	$0.483174\pi$
$822$	0	0
$823$	−21.5778	−0.752155	−0.376077	−	0.926588i	$-0.622727\pi$
$823$	−21.5778	−0.752155	−0.376077	+	0.926588i	$0.622727\pi$
$824$	0	0
$825$	−4.30278	−0.149803
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	3.21110	0.111526	0.0557631	−	0.998444i	$-0.482241\pi$
$829$	3.21110	0.111526	0.0557631	+	0.998444i	$0.482241\pi$
$830$	0	0
$831$	−11.4500	−0.397195
$832$	0	0
$833$	4.42221	0.153220
$834$	0	0
$835$	2.60555	0.0901688
$836$	0	0
$837$	16.3028	0.563506
$838$	0	0
$839$	17.7250	0.611934	0.305967	−	0.952042i	$-0.401020\pi$
$839$	17.7250	0.611934	0.305967	+	0.952042i	$0.401020\pi$
$840$	0	0
$841$	57.5416	1.98419
$842$	0	0
$843$	29.8806	1.02914
$844$	0	0
$845$	−13.1833	−0.453521
$846$	0	0
$847$	−2.30278	−0.0791243
$848$	0	0
$849$	9.23886	0.317077
$850$	0	0
$851$	36.8444	1.26301
$852$	0	0
$853$	−42.4222	−1.45251	−0.726254	−	0.687426i	$-0.758741\pi$
$853$	−42.4222	−1.45251	−0.726254	+	0.687426i	$0.758741\pi$
$854$	0	0
$855$	−1.69722	−0.0580438
$856$	0	0
$857$	23.3305	0.796956	0.398478	−	0.917178i	$-0.369539\pi$
$857$	23.3305	0.796956	0.398478	+	0.917178i	$0.369539\pi$
$858$	0	0
$859$	−35.6333	−1.21579	−0.607896	−	0.794016i	$-0.707986\pi$
$859$	−35.6333	−1.21579	−0.607896	+	0.794016i	$0.707986\pi$
$860$	0	0
$861$	11.0917	0.378003
$862$	0	0
$863$	53.9361	1.83601	0.918003	−	0.396574i	$-0.129801\pi$
$863$	53.9361	1.83601	0.918003	+	0.396574i	$0.129801\pi$
$864$	0	0
$865$	−11.8806	−0.403952
$866$	0	0
$867$	−13.3028	−0.451786
$868$	0	0
$869$	−9.81665	−0.333007
$870$	0	0
$871$	−9.35829	−0.317094
$872$	0	0
$873$	18.7889	0.635908
$874$	0	0
$875$	24.9083	0.842055
$876$	0	0
$877$	0.936083	0.0316093	0.0158046	−	0.999875i	$-0.494969\pi$
$877$	0.936083	0.0316093	0.0158046	+	0.999875i	$0.494969\pi$
$878$	0	0
$879$	−20.2111	−0.681704
$880$	0	0
$881$	−1.51388	−0.0510039	−0.0255019	−	0.999675i	$-0.508118\pi$
$881$	−1.51388	−0.0510039	−0.0255019	+	0.999675i	$0.508118\pi$
$882$	0	0
$883$	23.8167	0.801494	0.400747	−	0.916189i	$-0.368751\pi$
$883$	23.8167	0.801494	0.400747	+	0.916189i	$0.368751\pi$
$884$	0	0
$885$	6.78890	0.228206
$886$	0	0
$887$	−4.36669	−0.146619	−0.0733096	−	0.997309i	$-0.523356\pi$
$887$	−4.36669	−0.146619	−0.0733096	+	0.997309i	$0.523356\pi$
$888$	0	0
$889$	17.0278	0.571093
$890$	0	0
$891$	−3.39445	−0.113718
$892$	0	0
$893$	−3.21110	−0.107455
$894$	0	0
$895$	−10.8167	−0.361561
$896$	0	0
$897$	−10.1833	−0.340012
$898$	0	0
$899$	27.0555	0.902352
$900$	0	0
$901$	1.57779	0.0525640
$902$	0	0
$903$	3.90833	0.130061
$904$	0	0
$905$	24.2389	0.805727
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−14.0555	−0.465680	−0.232840	−	0.972515i	$-0.574802\pi$
$911$	−14.0555	−0.465680	−0.232840	+	0.972515i	$0.574802\pi$
$912$	0	0
$913$	−0.0916731	−0.00303394
$914$	0	0
$915$	−3.39445	−0.112217
$916$	0	0
$917$	−12.6972	−0.419299
$918$	0	0
$919$	−37.9083	−1.25048	−0.625240	−	0.780432i	$-0.714999\pi$
$919$	−37.9083	−1.25048	−0.625240	+	0.780432i	$0.714999\pi$
$920$	0	0
$921$	−43.0278	−1.41781
$922$	0	0
$923$	−3.55004	−0.116851
$924$	0	0
$925$	−26.4222	−0.868757
$926$	0	0
$927$	−5.88057	−0.193143
$928$	0	0
$929$	−6.48612	−0.212803	−0.106401	−	0.994323i	$-0.533933\pi$
$929$	−6.48612	−0.212803	−0.106401	+	0.994323i	$0.533933\pi$
$930$	0	0
$931$	−1.69722	−0.0556243
$932$	0	0
$933$	12.2389	0.400682
$934$	0	0
$935$	−3.39445	−0.111010
$936$	0	0
$937$	16.7889	0.548469	0.274235	−	0.961663i	$-0.411575\pi$
$937$	16.7889	0.548469	0.274235	+	0.961663i	$0.411575\pi$
$938$	0	0
$939$	45.5139	1.48529
$940$	0	0
$941$	1.15559	0.0376712	0.0188356	−	0.999823i	$-0.494004\pi$
$941$	1.15559	0.0376712	0.0188356	+	0.999823i	$0.494004\pi$
$942$	0	0
$943$	−17.0278	−0.554500
$944$	0	0
$945$	16.8167	0.547046
$946$	0	0
$947$	20.1833	0.655871	0.327935	−	0.944700i	$-0.393647\pi$
$947$	20.1833	0.655871	0.327935	+	0.944700i	$0.393647\pi$
$948$	0	0
$949$	28.1833	0.914870
$950$	0	0
$951$	25.8167	0.837162
$952$	0	0
$953$	22.8444	0.740003	0.370002	−	0.929031i	$-0.379357\pi$
$953$	22.8444	0.740003	0.370002	+	0.929031i	$0.379357\pi$
$954$	0	0
$955$	7.81665	0.252941
$956$	0	0
$957$	−12.1194	−0.391766
$958$	0	0
$959$	25.5416	0.824782
$960$	0	0
$961$	−22.5416	−0.727150
$962$	0	0
$963$	9.39445	0.302732
$964$	0	0
$965$	24.9083	0.801827
$966$	0	0
$967$	−9.21110	−0.296209	−0.148105	−	0.988972i	$-0.547317\pi$
$967$	−9.21110	−0.296209	−0.148105	+	0.988972i	$0.547317\pi$
$968$	0	0
$969$	−3.39445	−0.109045
$970$	0	0
$971$	−31.7250	−1.01810	−0.509052	−	0.860736i	$-0.670004\pi$
$971$	−31.7250	−1.01810	−0.509052	+	0.860736i	$0.670004\pi$
$972$	0	0
$973$	−36.2111	−1.16087
$974$	0	0
$975$	7.30278	0.233876
$976$	0	0
$977$	−35.6333	−1.14001	−0.570005	−	0.821641i	$-0.693059\pi$
$977$	−35.6333	−1.14001	−0.570005	+	0.821641i	$0.693059\pi$
$978$	0	0
$979$	−7.81665	−0.249821
$980$	0	0
$981$	−4.18335	−0.133564
$982$	0	0
$983$	−23.1472	−0.738281	−0.369140	−	0.929374i	$-0.620348\pi$
$983$	−23.1472	−0.738281	−0.369140	+	0.929374i	$0.620348\pi$
$984$	0	0
$985$	−7.81665	−0.249059
$986$	0	0
$987$	9.63331	0.306632
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−27.8806	−0.885656	−0.442828	−	0.896607i	$-0.646025\pi$
$991$	−27.8806	−0.885656	−0.442828	+	0.896607i	$0.646025\pi$
$992$	0	0
$993$	−10.3028	−0.326949
$994$	0	0
$995$	22.1833	0.703259
$996$	0	0
$997$	−24.6056	−0.779266	−0.389633	−	0.920970i	$-0.627398\pi$
$997$	−24.6056	−0.779266	−0.389633	+	0.920970i	$0.627398\pi$
$998$	0	0
$999$	−44.8444	−1.41881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.j.1.2		2
4.3	odd	2		1672.2.a.d.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.d.1.1	✓	2	4.3	odd	2
3344.2.a.j.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.30278 q^{3} +1.30278 q^{5} -2.30278 q^{7} -1.30278 q^{9} +1.00000 q^{11} -1.69722 q^{13} +1.69722 q^{15} -2.60555 q^{17} +1.00000 q^{19} -3.00000 q^{21} +4.60555 q^{23} -3.30278 q^{25} -5.60555 q^{27} -9.30278 q^{29} -2.90833 q^{31} +1.30278 q^{33} -3.00000 q^{35} +8.00000 q^{37} -2.21110 q^{39} -3.69722 q^{41} -1.30278 q^{43} -1.69722 q^{45} -3.21110 q^{47} -1.69722 q^{49} -3.39445 q^{51} -0.605551 q^{53} +1.30278 q^{55} +1.30278 q^{57} +4.00000 q^{59} -2.00000 q^{61} +3.00000 q^{63} -2.21110 q^{65} +5.51388 q^{67} +6.00000 q^{69} +2.09167 q^{71} -16.6056 q^{73} -4.30278 q^{75} -2.30278 q^{77} -9.81665 q^{79} -3.39445 q^{81} -0.0916731 q^{83} -3.39445 q^{85} -12.1194 q^{87} -7.81665 q^{89} +3.90833 q^{91} -3.78890 q^{93} +1.30278 q^{95} -14.4222 q^{97} -1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{11} - 7 q^{13} + 7 q^{15} + 2 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} - 3 q^{25} - 4 q^{27} - 15 q^{29} + 5 q^{31} - q^{33} - 6 q^{35} + 16 q^{37} + 10 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 - q^7 + q^9 + 2 * q^11 - 7 * q^13 + 7 * q^15 + 2 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 - 3 * q^25 - 4 * q^27 - 15 * q^29 + 5 * q^31 - q^33 - 6 * q^35 + 16 * q^37 + 10 * q^39 - 11 * q^41 + q^43 - 7 * q^45 + 8 * q^47 - 7 * q^49 - 14 * q^51 + 6 * q^53 - q^55 - q^57 + 8 * q^59 - 4 * q^61 + 6 * q^63 + 10 * q^65 - 7 * q^67 + 12 * q^69 + 15 * q^71 - 26 * q^73 - 5 * q^75 - q^77 + 2 * q^79 - 14 * q^81 - 11 * q^83 - 14 * q^85 + q^87 + 6 * q^89 - 3 * q^91 - 22 * q^93 - q^95 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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