LMFDB - Embedded newform 3344.2.a.q.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.523976$ 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)

$f(q)$	$=$	$q+0.523976 q^{3} +2.72545 q^{5} +4.67750 q^{7} -2.72545 q^{9} +1.00000 q^{11} -2.67750 q^{13} +1.42807 q^{15} +0.201472 q^{17} +1.00000 q^{19} +2.45090 q^{21} +1.79853 q^{23} +2.42807 q^{25} -3.00000 q^{27} -4.92692 q^{29} +2.57193 q^{31} +0.523976 q^{33} +12.7483 q^{35} +4.40294 q^{37} -1.40294 q^{39} +4.97487 q^{41} +11.7734 q^{43} -7.42807 q^{45} +12.4989 q^{47} +14.8790 q^{49} +0.105567 q^{51} -4.10557 q^{53} +2.72545 q^{55} +0.523976 q^{57} +5.24943 q^{59} +1.59706 q^{61} -12.7483 q^{63} -7.29738 q^{65} -9.08044 q^{67} +0.942386 q^{69} -12.8214 q^{71} +2.20147 q^{73} +1.27225 q^{75} +4.67750 q^{77} -11.8538 q^{79} +6.60442 q^{81} +13.6775 q^{83} +0.549103 q^{85} -2.58159 q^{87} -5.45090 q^{89} -12.5240 q^{91} +1.34763 q^{93} +2.72545 q^{95} +16.8059 q^{97} -2.72545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * 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
$3$	0.523976	0.302518	0.151259	−	0.988494i	$-0.451667\pi$
$3$	0.523976	0.302518	0.151259	+	0.988494i	$0.451667\pi$
$4$	0	0
$5$	2.72545	1.21886	0.609429	−	0.792841i	$-0.291399\pi$
$5$	2.72545	1.21886	0.609429	+	0.792841i	$0.291399\pi$
$6$	0	0
$7$	4.67750	1.76793	0.883964	−	0.467556i	$-0.154865\pi$
$7$	4.67750	1.76793	0.883964	+	0.467556i	$0.154865\pi$
$8$	0	0
$9$	−2.72545	−0.908483
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.67750	−0.742604	−0.371302	−	0.928512i	$-0.621089\pi$
$13$	−2.67750	−0.742604	−0.371302	+	0.928512i	$0.621089\pi$
$14$	0	0
$15$	1.42807	0.368726
$16$	0	0
$17$	0.201472	0.0488642	0.0244321	−	0.999701i	$-0.492222\pi$
$17$	0.201472	0.0488642	0.0244321	+	0.999701i	$0.492222\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.45090	0.534830
$22$	0	0
$23$	1.79853	0.375019	0.187509	−	0.982263i	$-0.439958\pi$
$23$	1.79853	0.375019	0.187509	+	0.982263i	$0.439958\pi$
$24$	0	0
$25$	2.42807	0.485614
$26$	0	0
$27$	−3.00000	−0.577350
$28$	0	0
$29$	−4.92692	−0.914906	−0.457453	−	0.889234i	$-0.651238\pi$
$29$	−4.92692	−0.914906	−0.457453	+	0.889234i	$0.651238\pi$
$30$	0	0
$31$	2.57193	0.461932	0.230966	−	0.972962i	$-0.425811\pi$
$31$	2.57193	0.461932	0.230966	+	0.972962i	$0.425811\pi$
$32$	0	0
$33$	0.523976	0.0912126
$34$	0	0
$35$	12.7483	2.15485
$36$	0	0
$37$	4.40294	0.723840	0.361920	−	0.932209i	$-0.382121\pi$
$37$	4.40294	0.723840	0.361920	+	0.932209i	$0.382121\pi$
$38$	0	0
$39$	−1.40294	−0.224651
$40$	0	0
$41$	4.97487	0.776945	0.388472	−	0.921460i	$-0.373003\pi$
$41$	4.97487	0.776945	0.388472	+	0.921460i	$0.373003\pi$
$42$	0	0
$43$	11.7734	1.79543	0.897713	−	0.440580i	$-0.145227\pi$
$43$	11.7734	1.79543	0.897713	+	0.440580i	$0.145227\pi$
$44$	0	0
$45$	−7.42807	−1.10731
$46$	0	0
$47$	12.4989	1.82314	0.911572	−	0.411140i	$-0.134869\pi$
$47$	12.4989	1.82314	0.911572	+	0.411140i	$0.134869\pi$
$48$	0	0
$49$	14.8790	2.12557
$50$	0	0
$51$	0.105567	0.0147823
$52$	0	0
$53$	−4.10557	−0.563943	−0.281971	−	0.959423i	$-0.590988\pi$
$53$	−4.10557	−0.563943	−0.281971	+	0.959423i	$0.590988\pi$
$54$	0	0
$55$	2.72545	0.367499
$56$	0	0
$57$	0.523976	0.0694024
$58$	0	0
$59$	5.24943	0.683417	0.341708	−	0.939806i	$-0.388994\pi$
$59$	5.24943	0.683417	0.341708	+	0.939806i	$0.388994\pi$
$60$	0	0
$61$	1.59706	0.204482	0.102241	−	0.994760i	$-0.467399\pi$
$61$	1.59706	0.204482	0.102241	+	0.994760i	$0.467399\pi$
$62$	0	0
$63$	−12.7483	−1.60613
$64$	0	0
$65$	−7.29738	−0.905128
$66$	0	0
$67$	−9.08044	−1.10935	−0.554676	−	0.832066i	$-0.687158\pi$
$67$	−9.08044	−1.10935	−0.554676	+	0.832066i	$0.687158\pi$
$68$	0	0
$69$	0.942386	0.113450
$70$	0	0
$71$	−12.8214	−1.52161	−0.760807	−	0.648978i	$-0.775197\pi$
$71$	−12.8214	−1.52161	−0.760807	+	0.648978i	$0.775197\pi$
$72$	0	0
$73$	2.20147	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$73$	2.20147	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$74$	0	0
$75$	1.27225	0.146907
$76$	0	0
$77$	4.67750	0.533050
$78$	0	0
$79$	−11.8538	−1.33366	−0.666831	−	0.745209i	$-0.732350\pi$
$79$	−11.8538	−1.33366	−0.666831	+	0.745209i	$0.732350\pi$
$80$	0	0
$81$	6.60442	0.733824
$82$	0	0
$83$	13.6775	1.50130	0.750650	−	0.660700i	$-0.229740\pi$
$83$	13.6775	1.50130	0.750650	+	0.660700i	$0.229740\pi$
$84$	0	0
$85$	0.549103	0.0595585
$86$	0	0
$87$	−2.58159	−0.276776
$88$	0	0
$89$	−5.45090	−0.577794	−0.288897	−	0.957360i	$-0.593289\pi$
$89$	−5.45090	−0.577794	−0.288897	+	0.957360i	$0.593289\pi$
$90$	0	0
$91$	−12.5240	−1.31287
$92$	0	0
$93$	1.34763	0.139743
$94$	0	0
$95$	2.72545	0.279625
$96$	0	0
$97$	16.8059	1.70638	0.853190	−	0.521601i	$-0.174665\pi$
$97$	16.8059	1.70638	0.853190	+	0.521601i	$0.174665\pi$
$98$	0	0
$99$	−2.72545	−0.273918
$100$	0	0
$101$	7.45090	0.741392	0.370696	−	0.928754i	$-0.379119\pi$
$101$	7.45090	0.741392	0.370696	+	0.928754i	$0.379119\pi$
$102$	0	0
$103$	14.4834	1.42709	0.713545	−	0.700609i	$-0.247088\pi$
$103$	14.4834	1.42709	0.713545	+	0.700609i	$0.247088\pi$
$104$	0	0
$105$	6.67980	0.651881
$106$	0	0
$107$	−1.15352	−0.111515	−0.0557575	−	0.998444i	$-0.517757\pi$
$107$	−1.15352	−0.111515	−0.0557575	+	0.998444i	$0.517757\pi$
$108$	0	0
$109$	−9.55646	−0.915343	−0.457672	−	0.889121i	$-0.651316\pi$
$109$	−9.55646	−0.915343	−0.457672	+	0.889121i	$0.651316\pi$
$110$	0	0
$111$	2.30704	0.218974
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	4.90179	0.457095
$116$	0	0
$117$	7.29738	0.674643
$118$	0	0
$119$	0.942386	0.0863884
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.60672	0.235040
$124$	0	0
$125$	−7.00966	−0.626963
$126$	0	0
$127$	−5.75794	−0.510934	−0.255467	−	0.966818i	$-0.582229\pi$
$127$	−5.75794	−0.510934	−0.255467	+	0.966818i	$0.582229\pi$
$128$	0	0
$129$	6.16898	0.543149
$130$	0	0
$131$	−1.42807	−0.124771	−0.0623856	−	0.998052i	$-0.519871\pi$
$131$	−1.42807	−0.124771	−0.0623856	+	0.998052i	$0.519871\pi$
$132$	0	0
$133$	4.67750	0.405590
$134$	0	0
$135$	−8.17635	−0.703708
$136$	0	0
$137$	−14.5313	−1.24150	−0.620748	−	0.784010i	$-0.713171\pi$
$137$	−14.5313	−1.24150	−0.620748	+	0.784010i	$0.713171\pi$
$138$	0	0
$139$	−16.1763	−1.37206	−0.686030	−	0.727573i	$-0.740648\pi$
$139$	−16.1763	−1.37206	−0.686030	+	0.727573i	$0.740648\pi$
$140$	0	0
$141$	6.54910	0.551534
$142$	0	0
$143$	−2.67750	−0.223903
$144$	0	0
$145$	−13.4281	−1.11514
$146$	0	0
$147$	7.79623	0.643022
$148$	0	0
$149$	6.30704	0.516693	0.258346	−	0.966052i	$-0.416822\pi$
$149$	6.30704	0.516693	0.258346	+	0.966052i	$0.416822\pi$
$150$	0	0
$151$	−0.498850	−0.0405959	−0.0202979	−	0.999794i	$-0.506461\pi$
$151$	−0.498850	−0.0405959	−0.0202979	+	0.999794i	$0.506461\pi$
$152$	0	0
$153$	−0.549103	−0.0443923
$154$	0	0
$155$	7.00966	0.563030
$156$	0	0
$157$	−13.6272	−1.08757	−0.543786	−	0.839224i	$-0.683010\pi$
$157$	−13.6272	−1.08757	−0.543786	+	0.839224i	$0.683010\pi$
$158$	0	0
$159$	−2.15122	−0.170603
$160$	0	0
$161$	8.41261	0.663006
$162$	0	0
$163$	−15.9497	−1.24928	−0.624640	−	0.780913i	$-0.714754\pi$
$163$	−15.9497	−1.24928	−0.624640	+	0.780913i	$0.714754\pi$
$164$	0	0
$165$	1.42807	0.111175
$166$	0	0
$167$	−18.8059	−1.45524	−0.727622	−	0.685979i	$-0.759374\pi$
$167$	−18.8059	−1.45524	−0.727622	+	0.685979i	$0.759374\pi$
$168$	0	0
$169$	−5.83102	−0.448540
$170$	0	0
$171$	−2.72545	−0.208420
$172$	0	0
$173$	−3.22430	−0.245139	−0.122569	−	0.992460i	$-0.539113\pi$
$173$	−3.22430	−0.245139	−0.122569	+	0.992460i	$0.539113\pi$
$174$	0	0
$175$	11.3573	0.858531
$176$	0	0
$177$	2.75057	0.206746
$178$	0	0
$179$	−13.3778	−0.999905	−0.499953	−	0.866053i	$-0.666649\pi$
$179$	−13.3778	−0.999905	−0.499953	+	0.866053i	$0.666649\pi$
$180$	0	0
$181$	15.9497	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$181$	15.9497	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$182$	0	0
$183$	0.836819	0.0618595
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	0.201472	0.0147331
$188$	0	0
$189$	−14.0325	−1.02071
$190$	0	0
$191$	24.0553	1.74058	0.870291	−	0.492538i	$-0.163931\pi$
$191$	24.0553	1.74058	0.870291	+	0.492538i	$0.163931\pi$
$192$	0	0
$193$	13.0325	0.938099	0.469050	−	0.883172i	$-0.344597\pi$
$193$	13.0325	0.938099	0.469050	+	0.883172i	$0.344597\pi$
$194$	0	0
$195$	−3.82365	−0.273818
$196$	0	0
$197$	−5.59706	−0.398774	−0.199387	−	0.979921i	$-0.563895\pi$
$197$	−5.59706	−0.398774	−0.199387	+	0.979921i	$0.563895\pi$
$198$	0	0
$199$	7.39558	0.524259	0.262129	−	0.965033i	$-0.415575\pi$
$199$	7.39558	0.524259	0.262129	+	0.965033i	$0.415575\pi$
$200$	0	0
$201$	−4.75794	−0.335599
$202$	0	0
$203$	−23.0457	−1.61749
$204$	0	0
$205$	13.5588	0.946985
$206$	0	0
$207$	−4.90179	−0.340698
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	14.2974	0.984272	0.492136	−	0.870518i	$-0.336216\pi$
$211$	14.2974	0.984272	0.492136	+	0.870518i	$0.336216\pi$
$212$	0	0
$213$	−6.71809	−0.460316
$214$	0	0
$215$	32.0878	2.18837
$216$	0	0
$217$	12.0302	0.816662
$218$	0	0
$219$	1.15352	0.0779476
$220$	0	0
$221$	−0.539441	−0.0362868
$222$	0	0
$223$	0.402945	0.0269832	0.0134916	−	0.999909i	$-0.495705\pi$
$223$	0.402945	0.0269832	0.0134916	+	0.999909i	$0.495705\pi$
$224$	0	0
$225$	−6.61758	−0.441172
$226$	0	0
$227$	21.0074	1.39431	0.697154	−	0.716922i	$-0.254449\pi$
$227$	21.0074	1.39431	0.697154	+	0.716922i	$0.254449\pi$
$228$	0	0
$229$	−14.3299	−0.946944	−0.473472	−	0.880809i	$-0.657000\pi$
$229$	−14.3299	−0.946944	−0.473472	+	0.880809i	$0.657000\pi$
$230$	0	0
$231$	2.45090	0.161257
$232$	0	0
$233$	−24.4029	−1.59869	−0.799345	−	0.600872i	$-0.794820\pi$
$233$	−24.4029	−1.59869	−0.799345	+	0.600872i	$0.794820\pi$
$234$	0	0
$235$	34.0650	2.22215
$236$	0	0
$237$	−6.21113	−0.403456
$238$	0	0
$239$	12.6849	0.820515	0.410258	−	0.911970i	$-0.365439\pi$
$239$	12.6849	0.820515	0.410258	+	0.911970i	$0.365439\pi$
$240$	0	0
$241$	−23.9343	−1.54174	−0.770871	−	0.636991i	$-0.780179\pi$
$241$	−23.9343	−1.54174	−0.770871	+	0.636991i	$0.780179\pi$
$242$	0	0
$243$	12.4606	0.799345
$244$	0	0
$245$	40.5519	2.59076
$246$	0	0
$247$	−2.67750	−0.170365
$248$	0	0
$249$	7.16669	0.454170
$250$	0	0
$251$	−18.4989	−1.16764	−0.583819	−	0.811884i	$-0.698442\pi$
$251$	−18.4989	−1.16764	−0.583819	+	0.811884i	$0.698442\pi$
$252$	0	0
$253$	1.79853	0.113072
$254$	0	0
$255$	0.287717	0.0180175
$256$	0	0
$257$	27.7579	1.73149	0.865746	−	0.500483i	$-0.166844\pi$
$257$	27.7579	1.73149	0.865746	+	0.500483i	$0.166844\pi$
$258$	0	0
$259$	20.5948	1.27970
$260$	0	0
$261$	13.4281	0.831177
$262$	0	0
$263$	−20.2723	−1.25004	−0.625020	−	0.780608i	$-0.714909\pi$
$263$	−20.2723	−1.25004	−0.625020	+	0.780608i	$0.714909\pi$
$264$	0	0
$265$	−11.1895	−0.687366
$266$	0	0
$267$	−2.85614	−0.174793
$268$	0	0
$269$	−15.6427	−0.953753	−0.476876	−	0.878970i	$-0.658231\pi$
$269$	−15.6427	−0.953753	−0.476876	+	0.878970i	$0.658231\pi$
$270$	0	0
$271$	12.3395	0.749573	0.374786	−	0.927111i	$-0.377716\pi$
$271$	12.3395	0.749573	0.374786	+	0.927111i	$0.377716\pi$
$272$	0	0
$273$	−6.56227	−0.397167
$274$	0	0
$275$	2.42807	0.146418
$276$	0	0
$277$	8.49885	0.510646	0.255323	−	0.966856i	$-0.417818\pi$
$277$	8.49885	0.510646	0.255323	+	0.966856i	$0.417818\pi$
$278$	0	0
$279$	−7.00966	−0.419657
$280$	0	0
$281$	−7.12839	−0.425244	−0.212622	−	0.977134i	$-0.568200\pi$
$281$	−7.12839	−0.425244	−0.212622	+	0.977134i	$0.568200\pi$
$282$	0	0
$283$	−11.5240	−0.685029	−0.342515	−	0.939512i	$-0.611279\pi$
$283$	−11.5240	−0.685029	−0.342515	+	0.939512i	$0.611279\pi$
$284$	0	0
$285$	1.42807	0.0845916
$286$	0	0
$287$	23.2700	1.37358
$288$	0	0
$289$	−16.9594	−0.997612
$290$	0	0
$291$	8.80589	0.516210
$292$	0	0
$293$	10.6775	0.623786	0.311893	−	0.950117i	$-0.399037\pi$
$293$	10.6775	0.623786	0.311893	+	0.950117i	$0.399037\pi$
$294$	0	0
$295$	14.3070	0.832988
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−4.81555	−0.278490
$300$	0	0
$301$	55.0700	3.17418
$302$	0	0
$303$	3.90409	0.224284
$304$	0	0
$305$	4.35269	0.249234
$306$	0	0
$307$	30.2568	1.72685	0.863423	−	0.504481i	$-0.168316\pi$
$307$	30.2568	1.72685	0.863423	+	0.504481i	$0.168316\pi$
$308$	0	0
$309$	7.58895	0.431720
$310$	0	0
$311$	−8.79623	−0.498788	−0.249394	−	0.968402i	$-0.580231\pi$
$311$	−8.79623	−0.498788	−0.249394	+	0.968402i	$0.580231\pi$
$312$	0	0
$313$	10.1860	0.575747	0.287874	−	0.957668i	$-0.407052\pi$
$313$	10.1860	0.575747	0.287874	+	0.957668i	$0.407052\pi$
$314$	0	0
$315$	−34.7448	−1.95765
$316$	0	0
$317$	−12.0097	−0.674530	−0.337265	−	0.941410i	$-0.609502\pi$
$317$	−12.0097	−0.674530	−0.337265	+	0.941410i	$0.609502\pi$
$318$	0	0
$319$	−4.92692	−0.275855
$320$	0	0
$321$	−0.604417	−0.0337353
$322$	0	0
$323$	0.201472	0.0112102
$324$	0	0
$325$	−6.50115	−0.360619
$326$	0	0
$327$	−5.00736	−0.276908
$328$	0	0
$329$	58.4633	3.22319
$330$	0	0
$331$	−26.1667	−1.43825	−0.719126	−	0.694880i	$-0.755457\pi$
$331$	−26.1667	−1.43825	−0.719126	+	0.694880i	$0.755457\pi$
$332$	0	0
$333$	−12.0000	−0.657596
$334$	0	0
$335$	−24.7483	−1.35214
$336$	0	0
$337$	17.3778	0.946630	0.473315	−	0.880893i	$-0.343057\pi$
$337$	17.3778	0.946630	0.473315	+	0.880893i	$0.343057\pi$
$338$	0	0
$339$	−4.19181	−0.227668
$340$	0	0
$341$	2.57193	0.139278
$342$	0	0
$343$	36.8538	1.98992
$344$	0	0
$345$	2.56842	0.138279
$346$	0	0
$347$	12.4029	0.665825	0.332912	−	0.942958i	$-0.391969\pi$
$347$	12.4029	0.665825	0.332912	+	0.942958i	$0.391969\pi$
$348$	0	0
$349$	−23.8036	−1.27418	−0.637088	−	0.770791i	$-0.719861\pi$
$349$	−23.8036	−1.27418	−0.637088	+	0.770791i	$0.719861\pi$
$350$	0	0
$351$	8.03249	0.428742
$352$	0	0
$353$	−24.2664	−1.29157	−0.645786	−	0.763518i	$-0.723470\pi$
$353$	−24.2664	−1.29157	−0.645786	+	0.763518i	$0.723470\pi$
$354$	0	0
$355$	−34.9439	−1.85463
$356$	0	0
$357$	0.493788	0.0261340
$358$	0	0
$359$	−20.3395	−1.07348	−0.536740	−	0.843748i	$-0.680344\pi$
$359$	−20.3395	−1.07348	−0.536740	+	0.843748i	$0.680344\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.523976	0.0275016
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	23.0170	1.20148	0.600739	−	0.799445i	$-0.294873\pi$
$367$	23.0170	1.20148	0.600739	+	0.799445i	$0.294873\pi$
$368$	0	0
$369$	−13.5588	−0.705841
$370$	0	0
$371$	−19.2038	−0.997010
$372$	0	0
$373$	−18.5890	−0.962499	−0.481250	−	0.876584i	$-0.659817\pi$
$373$	−18.5890	−0.962499	−0.481250	+	0.876584i	$0.659817\pi$
$374$	0	0
$375$	−3.67290	−0.189668
$376$	0	0
$377$	13.1918	0.679413
$378$	0	0
$379$	10.5816	0.543540	0.271770	−	0.962362i	$-0.412391\pi$
$379$	10.5816	0.543540	0.271770	+	0.962362i	$0.412391\pi$
$380$	0	0
$381$	−3.01702	−0.154567
$382$	0	0
$383$	26.0878	1.33302	0.666512	−	0.745494i	$-0.267786\pi$
$383$	26.0878	1.33302	0.666512	+	0.745494i	$0.267786\pi$
$384$	0	0
$385$	12.7483	0.649712
$386$	0	0
$387$	−32.0878	−1.63111
$388$	0	0
$389$	−9.46636	−0.479964	−0.239982	−	0.970777i	$-0.577141\pi$
$389$	−9.46636	−0.479964	−0.239982	+	0.970777i	$0.577141\pi$
$390$	0	0
$391$	0.362354	0.0183250
$392$	0	0
$393$	−0.748275	−0.0377455
$394$	0	0
$395$	−32.3070	−1.62554
$396$	0	0
$397$	−25.1667	−1.26308	−0.631540	−	0.775343i	$-0.717577\pi$
$397$	−25.1667	−1.26308	−0.631540	+	0.775343i	$0.717577\pi$
$398$	0	0
$399$	2.45090	0.122698
$400$	0	0
$401$	−16.0650	−0.802247	−0.401123	−	0.916024i	$-0.631380\pi$
$401$	−16.0650	−0.802247	−0.401123	+	0.916024i	$0.631380\pi$
$402$	0	0
$403$	−6.88633	−0.343033
$404$	0	0
$405$	18.0000	0.894427
$406$	0	0
$407$	4.40294	0.218246
$408$	0	0
$409$	−19.5313	−0.965763	−0.482881	−	0.875686i	$-0.660410\pi$
$409$	−19.5313	−0.965763	−0.482881	+	0.875686i	$0.660410\pi$
$410$	0	0
$411$	−7.61408	−0.375575
$412$	0	0
$413$	24.5542	1.20823
$414$	0	0
$415$	37.2773	1.82987
$416$	0	0
$417$	−8.47602	−0.415073
$418$	0	0
$419$	−24.9018	−1.21653	−0.608266	−	0.793733i	$-0.708135\pi$
$419$	−24.9018	−1.21653	−0.608266	+	0.793733i	$0.708135\pi$
$420$	0	0
$421$	26.6044	1.29662	0.648310	−	0.761377i	$-0.275476\pi$
$421$	26.6044	1.29662	0.648310	+	0.761377i	$0.275476\pi$
$422$	0	0
$423$	−34.0650	−1.65630
$424$	0	0
$425$	0.489189	0.0237292
$426$	0	0
$427$	7.47022	0.361509
$428$	0	0
$429$	−1.40294	−0.0677348
$430$	0	0
$431$	32.5948	1.57003	0.785017	−	0.619474i	$-0.212654\pi$
$431$	32.5948	1.57003	0.785017	+	0.619474i	$0.212654\pi$
$432$	0	0
$433$	19.7386	0.948577	0.474289	−	0.880369i	$-0.342705\pi$
$433$	19.7386	0.948577	0.474289	+	0.880369i	$0.342705\pi$
$434$	0	0
$435$	−7.03599	−0.337350
$436$	0	0
$437$	1.79853	0.0860352
$438$	0	0
$439$	24.4029	1.16469	0.582345	−	0.812942i	$-0.302135\pi$
$439$	24.4029	1.16469	0.582345	+	0.812942i	$0.302135\pi$
$440$	0	0
$441$	−40.5519	−1.93104
$442$	0	0
$443$	16.5948	0.788441	0.394220	−	0.919016i	$-0.371015\pi$
$443$	16.5948	0.788441	0.394220	+	0.919016i	$0.371015\pi$
$444$	0	0
$445$	−14.8561	−0.704249
$446$	0	0
$447$	3.30474	0.156309
$448$	0	0
$449$	−20.9520	−0.988788	−0.494394	−	0.869238i	$-0.664610\pi$
$449$	−20.9520	−0.988788	−0.494394	+	0.869238i	$0.664610\pi$
$450$	0	0
$451$	4.97487	0.234258
$452$	0	0
$453$	−0.261386	−0.0122810
$454$	0	0
$455$	−34.1335	−1.60020
$456$	0	0
$457$	23.3144	1.09060	0.545301	−	0.838240i	$-0.316415\pi$
$457$	23.3144	1.09060	0.545301	+	0.838240i	$0.316415\pi$
$458$	0	0
$459$	−0.604417	−0.0282118
$460$	0	0
$461$	20.8059	0.969027	0.484513	−	0.874784i	$-0.338997\pi$
$461$	20.8059	0.969027	0.484513	+	0.874784i	$0.338997\pi$
$462$	0	0
$463$	9.29002	0.431744	0.215872	−	0.976422i	$-0.430741\pi$
$463$	9.29002	0.431744	0.215872	+	0.976422i	$0.430741\pi$
$464$	0	0
$465$	3.67290	0.170327
$466$	0	0
$467$	−12.0457	−0.557406	−0.278703	−	0.960377i	$-0.589905\pi$
$467$	−12.0457	−0.557406	−0.278703	+	0.960377i	$0.589905\pi$
$468$	0	0
$469$	−42.4737	−1.96125
$470$	0	0
$471$	−7.14035	−0.329010
$472$	0	0
$473$	11.7734	0.541342
$474$	0	0
$475$	2.42807	0.111408
$476$	0	0
$477$	11.1895	0.512333
$478$	0	0
$479$	−0.188307	−0.00860396	−0.00430198	−	0.999991i	$-0.501369\pi$
$479$	−0.188307	−0.00860396	−0.00430198	+	0.999991i	$0.501369\pi$
$480$	0	0
$481$	−11.7889	−0.537526
$482$	0	0
$483$	4.40801	0.200571
$484$	0	0
$485$	45.8036	2.07983
$486$	0	0
$487$	33.3852	1.51283	0.756413	−	0.654094i	$-0.226950\pi$
$487$	33.3852	1.51283	0.756413	+	0.654094i	$0.226950\pi$
$488$	0	0
$489$	−8.35729	−0.377930
$490$	0	0
$491$	−12.5217	−0.565095	−0.282548	−	0.959253i	$-0.591180\pi$
$491$	−12.5217	−0.565095	−0.282548	+	0.959253i	$0.591180\pi$
$492$	0	0
$493$	−0.992638	−0.0447062
$494$	0	0
$495$	−7.42807	−0.333867
$496$	0	0
$497$	−59.9718	−2.69010
$498$	0	0
$499$	−16.3070	−0.730003	−0.365002	−	0.931007i	$-0.618932\pi$
$499$	−16.3070	−0.730003	−0.365002	+	0.931007i	$0.618932\pi$
$500$	0	0
$501$	−9.85384	−0.440237
$502$	0	0
$503$	18.9726	0.845945	0.422973	−	0.906142i	$-0.360987\pi$
$503$	18.9726	0.845945	0.422973	+	0.906142i	$0.360987\pi$
$504$	0	0
$505$	20.3070	0.903651
$506$	0	0
$507$	−3.05531	−0.135691
$508$	0	0
$509$	−30.6907	−1.36034	−0.680170	−	0.733055i	$-0.738094\pi$
$509$	−30.6907	−1.36034	−0.680170	+	0.733055i	$0.738094\pi$
$510$	0	0
$511$	10.2974	0.455529
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	39.4737	1.73942
$516$	0	0
$517$	12.4989	0.549699
$518$	0	0
$519$	−1.68946	−0.0741589
$520$	0	0
$521$	−42.1106	−1.84490	−0.922450	−	0.386116i	$-0.873816\pi$
$521$	−42.1106	−1.84490	−0.922450	+	0.386116i	$0.873816\pi$
$522$	0	0
$523$	−37.8133	−1.65346	−0.826729	−	0.562600i	$-0.809801\pi$
$523$	−37.8133	−1.65346	−0.826729	+	0.562600i	$0.809801\pi$
$524$	0	0
$525$	5.95095	0.259721
$526$	0	0
$527$	0.518173	0.0225720
$528$	0	0
$529$	−19.7653	−0.859361
$530$	0	0
$531$	−14.3070	−0.620873
$532$	0	0
$533$	−13.3202	−0.576962
$534$	0	0
$535$	−3.14386	−0.135921
$536$	0	0
$537$	−7.00966	−0.302489
$538$	0	0
$539$	14.8790	0.640883
$540$	0	0
$541$	13.5661	0.583253	0.291627	−	0.956532i	$-0.405804\pi$
$541$	13.5661	0.583253	0.291627	+	0.956532i	$0.405804\pi$
$542$	0	0
$543$	8.35729	0.358646
$544$	0	0
$545$	−26.0457	−1.11567
$546$	0	0
$547$	−46.4177	−1.98468	−0.992338	−	0.123552i	$-0.960571\pi$
$547$	−46.4177	−1.98468	−0.992338	+	0.123552i	$0.960571\pi$
$548$	0	0
$549$	−4.35269	−0.185768
$550$	0	0
$551$	−4.92692	−0.209894
$552$	0	0
$553$	−55.4463	−2.35782
$554$	0	0
$555$	6.28772	0.266899
$556$	0	0
$557$	−38.2065	−1.61886	−0.809431	−	0.587214i	$-0.800225\pi$
$557$	−38.2065	−1.61886	−0.809431	+	0.587214i	$0.800225\pi$
$558$	0	0
$559$	−31.5232	−1.33329
$560$	0	0
$561$	0.105567	0.00445703
$562$	0	0
$563$	7.01702	0.295732	0.147866	−	0.989007i	$-0.452760\pi$
$563$	7.01702	0.295732	0.147866	+	0.989007i	$0.452760\pi$
$564$	0	0
$565$	−21.8036	−0.917284
$566$	0	0
$567$	30.8921	1.29735
$568$	0	0
$569$	−41.4235	−1.73656	−0.868281	−	0.496072i	$-0.834775\pi$
$569$	−41.4235	−1.73656	−0.868281	+	0.496072i	$0.834775\pi$
$570$	0	0
$571$	−12.1381	−0.507962	−0.253981	−	0.967209i	$-0.581740\pi$
$571$	−12.1381	−0.507962	−0.253981	+	0.967209i	$0.581740\pi$
$572$	0	0
$573$	12.6044	0.526557
$574$	0	0
$575$	4.36695	0.182115
$576$	0	0
$577$	12.8384	0.534469	0.267234	−	0.963632i	$-0.413890\pi$
$577$	12.8384	0.534469	0.267234	+	0.963632i	$0.413890\pi$
$578$	0	0
$579$	6.82872	0.283792
$580$	0	0
$581$	63.9764	2.65419
$582$	0	0
$583$	−4.10557	−0.170035
$584$	0	0
$585$	19.8886	0.822294
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	2.57193	0.105974
$590$	0	0
$591$	−2.93272	−0.120636
$592$	0	0
$593$	40.4177	1.65975	0.829877	−	0.557946i	$-0.188410\pi$
$593$	40.4177	1.65975	0.829877	+	0.557946i	$0.188410\pi$
$594$	0	0
$595$	2.56842	0.105295
$596$	0	0
$597$	3.87511	0.158598
$598$	0	0
$599$	−0.679795	−0.0277757	−0.0138878	−	0.999904i	$-0.504421\pi$
$599$	−0.679795	−0.0277757	−0.0138878	+	0.999904i	$0.504421\pi$
$600$	0	0
$601$	21.5240	0.877981	0.438991	−	0.898492i	$-0.355336\pi$
$601$	21.5240	0.877981	0.438991	+	0.898492i	$0.355336\pi$
$602$	0	0
$603$	24.7483	1.00783
$604$	0	0
$605$	2.72545	0.110805
$606$	0	0
$607$	−13.3550	−0.542062	−0.271031	−	0.962571i	$-0.587365\pi$
$607$	−13.3550	−0.542062	−0.271031	+	0.962571i	$0.587365\pi$
$608$	0	0
$609$	−12.0754	−0.489319
$610$	0	0
$611$	−33.4656	−1.35387
$612$	0	0
$613$	9.11293	0.368068	0.184034	−	0.982920i	$-0.441084\pi$
$613$	9.11293	0.368068	0.184034	+	0.982920i	$0.441084\pi$
$614$	0	0
$615$	7.10447	0.286480
$616$	0	0
$617$	0.329866	0.0132799	0.00663995	−	0.999978i	$-0.497886\pi$
$617$	0.329866	0.0132799	0.00663995	+	0.999978i	$0.497886\pi$
$618$	0	0
$619$	−8.92112	−0.358570	−0.179285	−	0.983797i	$-0.557378\pi$
$619$	−8.92112	−0.358570	−0.179285	+	0.983797i	$0.557378\pi$
$620$	0	0
$621$	−5.39558	−0.216517
$622$	0	0
$623$	−25.4966	−1.02150
$624$	0	0
$625$	−31.2448	−1.24979
$626$	0	0
$627$	0.523976	0.0209256
$628$	0	0
$629$	0.887072	0.0353699
$630$	0	0
$631$	−44.6597	−1.77788	−0.888938	−	0.458028i	$-0.848556\pi$
$631$	−44.6597	−1.77788	−0.888938	+	0.458028i	$0.848556\pi$
$632$	0	0
$633$	7.49149	0.297760
$634$	0	0
$635$	−15.6930	−0.622756
$636$	0	0
$637$	−39.8384	−1.57845
$638$	0	0
$639$	34.9439	1.38236
$640$	0	0
$641$	−6.93272	−0.273826	−0.136913	−	0.990583i	$-0.543718\pi$
$641$	−6.93272	−0.273826	−0.136913	+	0.990583i	$0.543718\pi$
$642$	0	0
$643$	−23.4966	−0.926614	−0.463307	−	0.886198i	$-0.653337\pi$
$643$	−23.4966	−0.926614	−0.463307	+	0.886198i	$0.653337\pi$
$644$	0	0
$645$	16.8133	0.662021
$646$	0	0
$647$	−20.9880	−0.825125	−0.412562	−	0.910929i	$-0.635366\pi$
$647$	−20.9880	−0.825125	−0.412562	+	0.910929i	$0.635366\pi$
$648$	0	0
$649$	5.24943	0.206058
$650$	0	0
$651$	6.30353	0.247055
$652$	0	0
$653$	−44.1335	−1.72708	−0.863538	−	0.504284i	$-0.831756\pi$
$653$	−44.1335	−1.72708	−0.863538	+	0.504284i	$0.831756\pi$
$654$	0	0
$655$	−3.89213	−0.152078
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−24.0862	−0.938267	−0.469133	−	0.883127i	$-0.655434\pi$
$659$	−24.0862	−0.938267	−0.469133	+	0.883127i	$0.655434\pi$
$660$	0	0
$661$	−9.65237	−0.375434	−0.187717	−	0.982223i	$-0.560109\pi$
$661$	−9.65237	−0.375434	−0.187717	+	0.982223i	$0.560109\pi$
$662$	0	0
$663$	−0.282655	−0.0109774
$664$	0	0
$665$	12.7483	0.494357
$666$	0	0
$667$	−8.86120	−0.343107
$668$	0	0
$669$	0.211133	0.00816289
$670$	0	0
$671$	1.59706	0.0616536
$672$	0	0
$673$	24.6176	0.948938	0.474469	−	0.880272i	$-0.342640\pi$
$673$	24.6176	0.948938	0.474469	+	0.880272i	$0.342640\pi$
$674$	0	0
$675$	−7.28421	−0.280369
$676$	0	0
$677$	21.5719	0.829077	0.414538	−	0.910032i	$-0.363943\pi$
$677$	21.5719	0.829077	0.414538	+	0.910032i	$0.363943\pi$
$678$	0	0
$679$	78.6095	3.01675
$680$	0	0
$681$	11.0074	0.421803
$682$	0	0
$683$	0.997701	0.0381759	0.0190880	−	0.999818i	$-0.493924\pi$
$683$	0.997701	0.0381759	0.0190880	+	0.999818i	$0.493924\pi$
$684$	0	0
$685$	−39.6044	−1.51321
$686$	0	0
$687$	−7.50851	−0.286468
$688$	0	0
$689$	10.9926	0.418786
$690$	0	0
$691$	9.00230	0.342464	0.171232	−	0.985231i	$-0.445225\pi$
$691$	9.00230	0.342464	0.171232	+	0.985231i	$0.445225\pi$
$692$	0	0
$693$	−12.7483	−0.484267
$694$	0	0
$695$	−44.0878	−1.67235
$696$	0	0
$697$	1.00230	0.0379648
$698$	0	0
$699$	−12.7866	−0.483632
$700$	0	0
$701$	−15.0936	−0.570078	−0.285039	−	0.958516i	$-0.592006\pi$
$701$	−15.0936	−0.570078	−0.285039	+	0.958516i	$0.592006\pi$
$702$	0	0
$703$	4.40294	0.166060
$704$	0	0
$705$	17.8492	0.672241
$706$	0	0
$707$	34.8515	1.31073
$708$	0	0
$709$	−22.9246	−0.860952	−0.430476	−	0.902602i	$-0.641654\pi$
$709$	−22.9246	−0.860952	−0.430476	+	0.902602i	$0.641654\pi$
$710$	0	0
$711$	32.3070	1.21161
$712$	0	0
$713$	4.62569	0.173233
$714$	0	0
$715$	−7.29738	−0.272906
$716$	0	0
$717$	6.64657	0.248221
$718$	0	0
$719$	−6.70032	−0.249880	−0.124940	−	0.992164i	$-0.539874\pi$
$719$	−6.70032	−0.249880	−0.124940	+	0.992164i	$0.539874\pi$
$720$	0	0
$721$	67.7460	2.52299
$722$	0	0
$723$	−12.5410	−0.466405
$724$	0	0
$725$	−11.9629	−0.444291
$726$	0	0
$727$	18.4583	0.684579	0.342289	−	0.939595i	$-0.388798\pi$
$727$	18.4583	0.684579	0.342289	+	0.939595i	$0.388798\pi$
$728$	0	0
$729$	−13.2842	−0.492008
$730$	0	0
$731$	2.37201	0.0877321
$732$	0	0
$733$	41.8995	1.54759	0.773797	−	0.633434i	$-0.218355\pi$
$733$	41.8995	1.54759	0.773797	+	0.633434i	$0.218355\pi$
$734$	0	0
$735$	21.2482	0.783752
$736$	0	0
$737$	−9.08044	−0.334482
$738$	0	0
$739$	34.8361	1.28147	0.640733	−	0.767764i	$-0.278631\pi$
$739$	34.8361	1.28147	0.640733	+	0.767764i	$0.278631\pi$
$740$	0	0
$741$	−1.40294	−0.0515385
$742$	0	0
$743$	25.7889	0.946102	0.473051	−	0.881035i	$-0.343153\pi$
$743$	25.7889	0.946102	0.473051	+	0.881035i	$0.343153\pi$
$744$	0	0
$745$	17.1895	0.629775
$746$	0	0
$747$	−37.2773	−1.36391
$748$	0	0
$749$	−5.39558	−0.197150
$750$	0	0
$751$	33.5011	1.22247	0.611237	−	0.791447i	$-0.290672\pi$
$751$	33.5011	1.22247	0.611237	+	0.791447i	$0.290672\pi$
$752$	0	0
$753$	−9.69296	−0.353231
$754$	0	0
$755$	−1.35959	−0.0494806
$756$	0	0
$757$	7.53134	0.273731	0.136866	−	0.990590i	$-0.456297\pi$
$757$	7.53134	0.273731	0.136866	+	0.990590i	$0.456297\pi$
$758$	0	0
$759$	0.942386	0.0342064
$760$	0	0
$761$	38.1969	1.38464	0.692318	−	0.721593i	$-0.256590\pi$
$761$	38.1969	1.38464	0.692318	+	0.721593i	$0.256590\pi$
$762$	0	0
$763$	−44.7003	−1.61826
$764$	0	0
$765$	−1.49655	−0.0541079
$766$	0	0
$767$	−14.0553	−0.507508
$768$	0	0
$769$	16.6501	0.600417	0.300208	−	0.953874i	$-0.402944\pi$
$769$	16.6501	0.600417	0.300208	+	0.953874i	$0.402944\pi$
$770$	0	0
$771$	14.5445	0.523808
$772$	0	0
$773$	−23.6330	−0.850022	−0.425011	−	0.905188i	$-0.639730\pi$
$773$	−23.6330	−0.850022	−0.425011	+	0.905188i	$0.639730\pi$
$774$	0	0
$775$	6.24483	0.224321
$776$	0	0
$777$	10.7912	0.387131
$778$	0	0
$779$	4.97487	0.178243
$780$	0	0
$781$	−12.8214	−0.458784
$782$	0	0
$783$	14.7808	0.528221
$784$	0	0
$785$	−37.1404	−1.32560
$786$	0	0
$787$	20.3933	0.726942	0.363471	−	0.931606i	$-0.381592\pi$
$787$	20.3933	0.726942	0.363471	+	0.931606i	$0.381592\pi$
$788$	0	0
$789$	−10.6222	−0.378160
$790$	0	0
$791$	−37.4200	−1.33050
$792$	0	0
$793$	−4.27611	−0.151849
$794$	0	0
$795$	−5.86304	−0.207941
$796$	0	0
$797$	−9.70262	−0.343685	−0.171842	−	0.985124i	$-0.554972\pi$
$797$	−9.70262	−0.343685	−0.171842	+	0.985124i	$0.554972\pi$
$798$	0	0
$799$	2.51817	0.0890865
$800$	0	0
$801$	14.8561	0.524916
$802$	0	0
$803$	2.20147	0.0776883
$804$	0	0
$805$	22.9281	0.808110
$806$	0	0
$807$	−8.19641	−0.288527
$808$	0	0
$809$	−27.0723	−0.951813	−0.475906	−	0.879496i	$-0.657880\pi$
$809$	−27.0723	−0.951813	−0.475906	+	0.879496i	$0.657880\pi$
$810$	0	0
$811$	−51.9548	−1.82438	−0.912190	−	0.409767i	$-0.865610\pi$
$811$	−51.9548	−1.82438	−0.912190	+	0.409767i	$0.865610\pi$
$812$	0	0
$813$	6.46562	0.226759
$814$	0	0
$815$	−43.4702	−1.52270
$816$	0	0
$817$	11.7734	0.411899
$818$	0	0
$819$	34.1335	1.19272
$820$	0	0
$821$	−25.6574	−0.895451	−0.447725	−	0.894171i	$-0.647766\pi$
$821$	−25.6574	−0.895451	−0.447725	+	0.894171i	$0.647766\pi$
$822$	0	0
$823$	41.8635	1.45927	0.729635	−	0.683837i	$-0.239690\pi$
$823$	41.8635	1.45927	0.729635	+	0.683837i	$0.239690\pi$
$824$	0	0
$825$	1.27225	0.0442941
$826$	0	0
$827$	−6.15582	−0.214059	−0.107029	−	0.994256i	$-0.534134\pi$
$827$	−6.15582	−0.214059	−0.107029	+	0.994256i	$0.534134\pi$
$828$	0	0
$829$	−0.247126	−0.00858303	−0.00429151	−	0.999991i	$-0.501366\pi$
$829$	−0.247126	−0.00858303	−0.00429151	+	0.999991i	$0.501366\pi$
$830$	0	0
$831$	4.45320	0.154480
$832$	0	0
$833$	2.99770	0.103864
$834$	0	0
$835$	−51.2545	−1.77373
$836$	0	0
$837$	−7.71579	−0.266697
$838$	0	0
$839$	−16.9320	−0.584557	−0.292278	−	0.956333i	$-0.594413\pi$
$839$	−16.9320	−0.584557	−0.292278	+	0.956333i	$0.594413\pi$
$840$	0	0
$841$	−4.72545	−0.162947
$842$	0	0
$843$	−3.73511	−0.128644
$844$	0	0
$845$	−15.8921	−0.546706
$846$	0	0
$847$	4.67750	0.160721
$848$	0	0
$849$	−6.03829	−0.207234
$850$	0	0
$851$	7.91882	0.271454
$852$	0	0
$853$	21.6930	0.742753	0.371376	−	0.928482i	$-0.378886\pi$
$853$	21.6930	0.742753	0.371376	+	0.928482i	$0.378886\pi$
$854$	0	0
$855$	−7.42807	−0.254035
$856$	0	0
$857$	34.7328	1.18645	0.593225	−	0.805037i	$-0.297854\pi$
$857$	34.7328	1.18645	0.593225	+	0.805037i	$0.297854\pi$
$858$	0	0
$859$	15.3047	0.522191	0.261095	−	0.965313i	$-0.415916\pi$
$859$	15.3047	0.522191	0.261095	+	0.965313i	$0.415916\pi$
$860$	0	0
$861$	12.1929	0.415533
$862$	0	0
$863$	−35.8457	−1.22020	−0.610102	−	0.792323i	$-0.708871\pi$
$863$	−35.8457	−1.22020	−0.610102	+	0.792323i	$0.708871\pi$
$864$	0	0
$865$	−8.78766	−0.298789
$866$	0	0
$867$	−8.88633	−0.301796
$868$	0	0
$869$	−11.8538	−0.402114
$870$	0	0
$871$	24.3128	0.823809
$872$	0	0
$873$	−45.8036	−1.55022
$874$	0	0
$875$	−32.7877	−1.10843
$876$	0	0
$877$	−45.4907	−1.53611	−0.768057	−	0.640382i	$-0.778776\pi$
$877$	−45.4907	−1.53611	−0.768057	+	0.640382i	$0.778776\pi$
$878$	0	0
$879$	5.59476	0.188706
$880$	0	0
$881$	31.3322	1.05561	0.527804	−	0.849366i	$-0.323016\pi$
$881$	31.3322	1.05561	0.527804	+	0.849366i	$0.323016\pi$
$882$	0	0
$883$	−35.3550	−1.18979	−0.594895	−	0.803803i	$-0.702806\pi$
$883$	−35.3550	−1.18979	−0.594895	+	0.803803i	$0.702806\pi$
$884$	0	0
$885$	7.49655	0.251994
$886$	0	0
$887$	16.6141	0.557846	0.278923	−	0.960313i	$-0.410023\pi$
$887$	16.6141	0.557846	0.278923	+	0.960313i	$0.410023\pi$
$888$	0	0
$889$	−26.9327	−0.903295
$890$	0	0
$891$	6.60442	0.221256
$892$	0	0
$893$	12.4989	0.418258
$894$	0	0
$895$	−36.4606	−1.21874
$896$	0	0
$897$	−2.52323	−0.0842484
$898$	0	0
$899$	−12.6717	−0.422625
$900$	0	0
$901$	−0.827158	−0.0275566
$902$	0	0
$903$	28.8554	0.960248
$904$	0	0
$905$	43.4702	1.44500
$906$	0	0
$907$	−54.8612	−1.82164	−0.910818	−	0.412808i	$-0.864548\pi$
$907$	−54.8612	−1.82164	−0.910818	+	0.412808i	$0.864548\pi$
$908$	0	0
$909$	−20.3070	−0.673542
$910$	0	0
$911$	−2.98298	−0.0988304	−0.0494152	−	0.998778i	$-0.515736\pi$
$911$	−2.98298	−0.0988304	−0.0494152	+	0.998778i	$0.515736\pi$
$912$	0	0
$913$	13.6775	0.452659
$914$	0	0
$915$	2.28071	0.0753979
$916$	0	0
$917$	−6.67980	−0.220586
$918$	0	0
$919$	26.6701	0.879767	0.439883	−	0.898055i	$-0.355020\pi$
$919$	26.6701	0.879767	0.439883	+	0.898055i	$0.355020\pi$
$920$	0	0
$921$	15.8538	0.522402
$922$	0	0
$923$	34.3291	1.12996
$924$	0	0
$925$	10.6907	0.351507
$926$	0	0
$927$	−39.4737	−1.29649
$928$	0	0
$929$	43.7595	1.43570	0.717851	−	0.696197i	$-0.245126\pi$
$929$	43.7595	1.43570	0.717851	+	0.696197i	$0.245126\pi$
$930$	0	0
$931$	14.8790	0.487638
$932$	0	0
$933$	−4.60902	−0.150892
$934$	0	0
$935$	0.549103	0.0179576
$936$	0	0
$937$	47.7823	1.56098	0.780490	−	0.625168i	$-0.214970\pi$
$937$	47.7823	1.56098	0.780490	+	0.625168i	$0.214970\pi$
$938$	0	0
$939$	5.33723	0.174174
$940$	0	0
$941$	36.2664	1.18225	0.591126	−	0.806579i	$-0.298684\pi$
$941$	36.2664	1.18225	0.591126	+	0.806579i	$0.298684\pi$
$942$	0	0
$943$	8.94745	0.291369
$944$	0	0
$945$	−38.2448	−1.24410
$946$	0	0
$947$	−35.5468	−1.15512	−0.577558	−	0.816350i	$-0.695994\pi$
$947$	−35.5468	−1.15512	−0.577558	+	0.816350i	$0.695994\pi$
$948$	0	0
$949$	−5.89443	−0.191341
$950$	0	0
$951$	−6.29278	−0.204057
$952$	0	0
$953$	23.2088	0.751808	0.375904	−	0.926659i	$-0.377332\pi$
$953$	23.2088	0.751808	0.375904	+	0.926659i	$0.377332\pi$
$954$	0	0
$955$	65.5615	2.12152
$956$	0	0
$957$	−2.58159	−0.0834510
$958$	0	0
$959$	−67.9703	−2.19487
$960$	0	0
$961$	−24.3852	−0.786619
$962$	0	0
$963$	3.14386	0.101309
$964$	0	0
$965$	35.5194	1.14341
$966$	0	0
$967$	6.09591	0.196031	0.0980156	−	0.995185i	$-0.468751\pi$
$967$	6.09591	0.196031	0.0980156	+	0.995185i	$0.468751\pi$
$968$	0	0
$969$	0.105567	0.00339129
$970$	0	0
$971$	−34.9822	−1.12263	−0.561317	−	0.827601i	$-0.689705\pi$
$971$	−34.9822	−1.12263	−0.561317	+	0.827601i	$0.689705\pi$
$972$	0	0
$973$	−75.6648	−2.42570
$974$	0	0
$975$	−3.40645	−0.109094
$976$	0	0
$977$	−10.9018	−0.348779	−0.174390	−	0.984677i	$-0.555795\pi$
$977$	−10.9018	−0.348779	−0.174390	+	0.984677i	$0.555795\pi$
$978$	0	0
$979$	−5.45090	−0.174211
$980$	0	0
$981$	26.0457	0.831574
$982$	0	0
$983$	25.4281	0.811030	0.405515	−	0.914089i	$-0.367092\pi$
$983$	25.4281	0.811030	0.405515	+	0.914089i	$0.367092\pi$
$984$	0	0
$985$	−15.2545	−0.486048
$986$	0	0
$987$	30.6334	0.975072
$988$	0	0
$989$	21.1748	0.673319
$990$	0	0
$991$	−49.3925	−1.56901	−0.784503	−	0.620125i	$-0.787082\pi$
$991$	−49.3925	−1.56901	−0.784503	+	0.620125i	$0.787082\pi$
$992$	0	0
$993$	−13.7107	−0.435097
$994$	0	0
$995$	20.1563	0.638997
$996$	0	0
$997$	20.7409	0.656871	0.328436	−	0.944526i	$-0.393479\pi$
$997$	20.7409	0.656871	0.328436	+	0.944526i	$0.393479\pi$
$998$	0	0
$999$	−13.2088	−0.417909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.q.1.2		3
4.3	odd	2		418.2.a.g.1.2	✓	3
12.11	even	2		3762.2.a.bg.1.1		3
44.43	even	2		4598.2.a.bo.1.2		3
76.75	even	2		7942.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.2	✓	3	4.3	odd	2
3344.2.a.q.1.2		3	1.1	even	1	trivial
3762.2.a.bg.1.1		3	12.11	even	2
4598.2.a.bo.1.2		3	44.43	even	2
7942.2.a.bi.1.2		3	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.523976 q^{3} +2.72545 q^{5} +4.67750 q^{7} -2.72545 q^{9} +1.00000 q^{11} -2.67750 q^{13} +1.42807 q^{15} +0.201472 q^{17} +1.00000 q^{19} +2.45090 q^{21} +1.79853 q^{23} +2.42807 q^{25} -3.00000 q^{27} -4.92692 q^{29} +2.57193 q^{31} +0.523976 q^{33} +12.7483 q^{35} +4.40294 q^{37} -1.40294 q^{39} +4.97487 q^{41} +11.7734 q^{43} -7.42807 q^{45} +12.4989 q^{47} +14.8790 q^{49} +0.105567 q^{51} -4.10557 q^{53} +2.72545 q^{55} +0.523976 q^{57} +5.24943 q^{59} +1.59706 q^{61} -12.7483 q^{63} -7.29738 q^{65} -9.08044 q^{67} +0.942386 q^{69} -12.8214 q^{71} +2.20147 q^{73} +1.27225 q^{75} +4.67750 q^{77} -11.8538 q^{79} +6.60442 q^{81} +13.6775 q^{83} +0.549103 q^{85} -2.58159 q^{87} -5.45090 q^{89} -12.5240 q^{91} +1.34763 q^{93} +2.72545 q^{95} +16.8059 q^{97} -2.72545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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