LMFDB - Embedded newform 3344.2.a.s.1.4

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:	$4$
Coefficient field:	4.4.13676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 7x + 1 $ x^4 - x^3 - 6x^2 + 7x + 1
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			1.4
Root			$2.18363$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18363 q^{3} +0.493910 q^{5} -1.18363 q^{7} +1.76823 q^{9} -1.00000 q^{11} -1.80419 q^{13} +1.07852 q^{15} -6.21401 q^{17} -1.00000 q^{19} -2.58461 q^{21} -4.62940 q^{23} -4.75605 q^{25} -2.68972 q^{27} +4.44577 q^{29} -0.457953 q^{31} -2.18363 q^{33} -0.584606 q^{35} -3.58461 q^{37} -3.93968 q^{39} -1.33688 q^{41} +2.75605 q^{43} +0.873348 q^{45} -3.97564 q^{47} -5.59902 q^{49} -13.5691 q^{51} -6.98782 q^{53} -0.493910 q^{55} -2.18363 q^{57} -0.415394 q^{59} +7.75048 q^{61} -2.09293 q^{63} -0.891107 q^{65} -3.28874 q^{67} -10.1089 q^{69} +15.9911 q^{71} -0.391619 q^{73} -10.3855 q^{75} +1.18363 q^{77} +1.14107 q^{79} -11.1781 q^{81} +15.3617 q^{83} -3.06916 q^{85} +9.70792 q^{87} -12.4762 q^{89} +2.13549 q^{91} -1.00000 q^{93} -0.493910 q^{95} -3.06032 q^{97} -1.76823 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41}+ \cdots - q^{99}+O(q^{100}) $ 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - 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
$3$	2.18363	1.26072	0.630359	−	0.776304i	$-0.282908\pi$
$3$	2.18363	1.26072	0.630359	+	0.776304i	$0.282908\pi$
$4$	0	0
$5$	0.493910	0.220883	0.110442	−	0.993883i	$-0.464774\pi$
$5$	0.493910	0.220883	0.110442	+	0.993883i	$0.464774\pi$
$6$	0	0
$7$	−1.18363	−0.447370	−0.223685	−	0.974662i	$-0.571809\pi$
$7$	−1.18363	−0.447370	−0.223685	+	0.974662i	$0.571809\pi$
$8$	0	0
$9$	1.76823	0.589411
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.80419	−0.500392	−0.250196	−	0.968195i	$-0.580495\pi$
$13$	−1.80419	−0.500392	−0.250196	+	0.968195i	$0.580495\pi$
$14$	0	0
$15$	1.07852	0.278471
$16$	0	0
$17$	−6.21401	−1.50712	−0.753559	−	0.657380i	$-0.771665\pi$
$17$	−6.21401	−1.50712	−0.753559	+	0.657380i	$0.771665\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.58461	−0.564007
$22$	0	0
$23$	−4.62940	−0.965297	−0.482648	−	0.875814i	$-0.660325\pi$
$23$	−4.62940	−0.965297	−0.482648	+	0.875814i	$0.660325\pi$
$24$	0	0
$25$	−4.75605	−0.951211
$26$	0	0
$27$	−2.68972	−0.517637
$28$	0	0
$29$	4.44577	0.825559	0.412780	−	0.910831i	$-0.364558\pi$
$29$	4.44577	0.825559	0.412780	+	0.910831i	$0.364558\pi$
$30$	0	0
$31$	−0.457953	−0.0822508	−0.0411254	−	0.999154i	$-0.513094\pi$
$31$	−0.457953	−0.0822508	−0.0411254	+	0.999154i	$0.513094\pi$
$32$	0	0
$33$	−2.18363	−0.380121
$34$	0	0
$35$	−0.584606	−0.0988164
$36$	0	0
$37$	−3.58461	−0.589306	−0.294653	−	0.955604i	$-0.595204\pi$
$37$	−3.58461	−0.589306	−0.294653	+	0.955604i	$0.595204\pi$
$38$	0	0
$39$	−3.93968	−0.630854
$40$	0	0
$41$	−1.33688	−0.208785	−0.104393	−	0.994536i	$-0.533290\pi$
$41$	−1.33688	−0.208785	−0.104393	+	0.994536i	$0.533290\pi$
$42$	0	0
$43$	2.75605	0.420294	0.210147	−	0.977670i	$-0.432606\pi$
$43$	2.75605	0.420294	0.210147	+	0.977670i	$0.432606\pi$
$44$	0	0
$45$	0.873348	0.130191
$46$	0	0
$47$	−3.97564	−0.579906	−0.289953	−	0.957041i	$-0.593640\pi$
$47$	−3.97564	−0.579906	−0.289953	+	0.957041i	$0.593640\pi$
$48$	0	0
$49$	−5.59902	−0.799860
$50$	0	0
$51$	−13.5691	−1.90005
$52$	0	0
$53$	−6.98782	−0.959851	−0.479925	−	0.877309i	$-0.659336\pi$
$53$	−6.98782	−0.959851	−0.479925	+	0.877309i	$0.659336\pi$
$54$	0	0
$55$	−0.493910	−0.0665988
$56$	0	0
$57$	−2.18363	−0.289229
$58$	0	0
$59$	−0.415394	−0.0540798	−0.0270399	−	0.999634i	$-0.508608\pi$
$59$	−0.415394	−0.0540798	−0.0270399	+	0.999634i	$0.508608\pi$
$60$	0	0
$61$	7.75048	0.992347	0.496173	−	0.868223i	$-0.334738\pi$
$61$	7.75048	0.992347	0.496173	+	0.868223i	$0.334738\pi$
$62$	0	0
$63$	−2.09293	−0.263685
$64$	0	0
$65$	−0.891107	−0.110528
$66$	0	0
$67$	−3.28874	−0.401784	−0.200892	−	0.979613i	$-0.564384\pi$
$67$	−3.28874	−0.401784	−0.200892	+	0.979613i	$0.564384\pi$
$68$	0	0
$69$	−10.1089	−1.21697
$70$	0	0
$71$	15.9911	1.89779	0.948896	−	0.315589i	$-0.102202\pi$
$71$	15.9911	1.89779	0.948896	+	0.315589i	$0.102202\pi$
$72$	0	0
$73$	−0.391619	−0.0458355	−0.0229178	−	0.999737i	$-0.507296\pi$
$73$	−0.391619	−0.0458355	−0.0229178	+	0.999737i	$0.507296\pi$
$74$	0	0
$75$	−10.3855	−1.19921
$76$	0	0
$77$	1.18363	0.134887
$78$	0	0
$79$	1.14107	0.128380	0.0641902	−	0.997938i	$-0.479554\pi$
$79$	1.14107	0.128380	0.0641902	+	0.997938i	$0.479554\pi$
$80$	0	0
$81$	−11.1781	−1.24201
$82$	0	0
$83$	15.3617	1.68616	0.843082	−	0.537786i	$-0.180739\pi$
$83$	15.3617	1.68616	0.843082	+	0.537786i	$0.180739\pi$
$84$	0	0
$85$	−3.06916	−0.332897
$86$	0	0
$87$	9.70792	1.04080
$88$	0	0
$89$	−12.4762	−1.32247	−0.661235	−	0.750179i	$-0.729967\pi$
$89$	−12.4762	−1.32247	−0.661235	+	0.750179i	$0.729967\pi$
$90$	0	0
$91$	2.13549	0.223860
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−0.493910	−0.0506741
$96$	0	0
$97$	−3.06032	−0.310728	−0.155364	−	0.987857i	$-0.549655\pi$
$97$	−3.06032	−0.310728	−0.155364	+	0.987857i	$0.549655\pi$
$98$	0	0
$99$	−1.76823	−0.177714
$100$	0	0
$101$	3.68972	0.367141	0.183570	−	0.983007i	$-0.441234\pi$
$101$	3.68972	0.367141	0.183570	+	0.983007i	$0.441234\pi$
$102$	0	0
$103$	3.19522	0.314835	0.157417	−	0.987532i	$-0.449683\pi$
$103$	3.19522	0.314835	0.157417	+	0.987532i	$0.449683\pi$
$104$	0	0
$105$	−1.27656	−0.124580
$106$	0	0
$107$	19.7831	1.91250	0.956252	−	0.292545i	$-0.0945021\pi$
$107$	19.7831	1.91250	0.956252	+	0.292545i	$0.0945021\pi$
$108$	0	0
$109$	−10.7345	−1.02818	−0.514090	−	0.857736i	$-0.671870\pi$
$109$	−10.7345	−1.02818	−0.514090	+	0.857736i	$0.671870\pi$
$110$	0	0
$111$	−7.82745	−0.742948
$112$	0	0
$113$	−3.89111	−0.366045	−0.183022	−	0.983109i	$-0.558588\pi$
$113$	−3.89111	−0.366045	−0.183022	+	0.983109i	$0.558588\pi$
$114$	0	0
$115$	−2.28651	−0.213218
$116$	0	0
$117$	−3.19023	−0.294937
$118$	0	0
$119$	7.35508	0.674239
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.91925	−0.263220
$124$	0	0
$125$	−4.81861	−0.430989
$126$	0	0
$127$	16.1177	1.43022	0.715109	−	0.699013i	$-0.246377\pi$
$127$	16.1177	1.43022	0.715109	+	0.699013i	$0.246377\pi$
$128$	0	0
$129$	6.01820	0.529873
$130$	0	0
$131$	−12.5425	−1.09584	−0.547921	−	0.836530i	$-0.684581\pi$
$131$	−12.5425	−1.09584	−0.547921	+	0.836530i	$0.684581\pi$
$132$	0	0
$133$	1.18363	0.102634
$134$	0	0
$135$	−1.32848	−0.114337
$136$	0	0
$137$	−17.0868	−1.45982	−0.729911	−	0.683543i	$-0.760438\pi$
$137$	−17.0868	−1.45982	−0.729911	+	0.683543i	$0.760438\pi$
$138$	0	0
$139$	−18.7411	−1.58960	−0.794800	−	0.606871i	$-0.792424\pi$
$139$	−18.7411	−1.58960	−0.794800	+	0.606871i	$0.792424\pi$
$140$	0	0
$141$	−8.68132	−0.731099
$142$	0	0
$143$	1.80419	0.150874
$144$	0	0
$145$	2.19581	0.182352
$146$	0	0
$147$	−12.2262	−1.00840
$148$	0	0
$149$	−5.30753	−0.434809	−0.217405	−	0.976082i	$-0.569759\pi$
$149$	−5.30753	−0.434809	−0.217405	+	0.976082i	$0.569759\pi$
$150$	0	0
$151$	6.11395	0.497546	0.248773	−	0.968562i	$-0.419973\pi$
$151$	6.11395	0.497546	0.248773	+	0.968562i	$0.419973\pi$
$152$	0	0
$153$	−10.9878	−0.888313
$154$	0	0
$155$	−0.226188	−0.0181678
$156$	0	0
$157$	12.6686	1.01107	0.505533	−	0.862807i	$-0.331296\pi$
$157$	12.6686	1.01107	0.505533	+	0.862807i	$0.331296\pi$
$158$	0	0
$159$	−15.2588	−1.21010
$160$	0	0
$161$	5.47949	0.431844
$162$	0	0
$163$	0.673759	0.0527729	0.0263864	−	0.999652i	$-0.491600\pi$
$163$	0.673759	0.0527729	0.0263864	+	0.999652i	$0.491600\pi$
$164$	0	0
$165$	−1.07852	−0.0839623
$166$	0	0
$167$	8.58126	0.664038	0.332019	−	0.943273i	$-0.392270\pi$
$167$	8.58126	0.664038	0.332019	+	0.943273i	$0.392270\pi$
$168$	0	0
$169$	−9.74490	−0.749607
$170$	0	0
$171$	−1.76823	−0.135220
$172$	0	0
$173$	2.78867	0.212019	0.106009	−	0.994365i	$-0.466193\pi$
$173$	2.78867	0.212019	0.106009	+	0.994365i	$0.466193\pi$
$174$	0	0
$175$	5.62940	0.425543
$176$	0	0
$177$	−0.907067	−0.0681794
$178$	0	0
$179$	−0.0906960	−0.00677893	−0.00338947	−	0.999994i	$-0.501079\pi$
$179$	−0.0906960	−0.00677893	−0.00338947	+	0.999994i	$0.501079\pi$
$180$	0	0
$181$	20.3799	1.51482	0.757412	−	0.652937i	$-0.226463\pi$
$181$	20.3799	1.51482	0.757412	+	0.652937i	$0.226463\pi$
$182$	0	0
$183$	16.9242	1.25107
$184$	0	0
$185$	−1.77047	−0.130168
$186$	0	0
$187$	6.21401	0.454413
$188$	0	0
$189$	3.18363	0.231575
$190$	0	0
$191$	−15.8594	−1.14754	−0.573772	−	0.819015i	$-0.694520\pi$
$191$	−15.8594	−1.14754	−0.573772	+	0.819015i	$0.694520\pi$
$192$	0	0
$193$	−16.8700	−1.21433	−0.607165	−	0.794576i	$-0.707693\pi$
$193$	−16.8700	−1.21433	−0.607165	+	0.794576i	$0.707693\pi$
$194$	0	0
$195$	−1.94585	−0.139345
$196$	0	0
$197$	11.1775	0.796361	0.398181	−	0.917307i	$-0.369642\pi$
$197$	11.1775	0.796361	0.398181	+	0.917307i	$0.369642\pi$
$198$	0	0
$199$	−14.6494	−1.03847	−0.519234	−	0.854632i	$-0.673783\pi$
$199$	−14.6494	−1.03847	−0.519234	+	0.854632i	$0.673783\pi$
$200$	0	0
$201$	−7.18139	−0.506536
$202$	0	0
$203$	−5.26214	−0.369330
$204$	0	0
$205$	−0.660298	−0.0461172
$206$	0	0
$207$	−8.18587	−0.568957
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	15.6830	1.07966	0.539832	−	0.841773i	$-0.318488\pi$
$211$	15.6830	1.07966	0.539832	+	0.841773i	$0.318488\pi$
$212$	0	0
$213$	34.9186	2.39258
$214$	0	0
$215$	1.36124	0.0928359
$216$	0	0
$217$	0.542047	0.0367965
$218$	0	0
$219$	−0.855151	−0.0577857
$220$	0	0
$221$	11.2113	0.754150
$222$	0	0
$223$	9.98070	0.668357	0.334178	−	0.942510i	$-0.391541\pi$
$223$	9.98070	0.668357	0.334178	+	0.942510i	$0.391541\pi$
$224$	0	0
$225$	−8.40982	−0.560654
$226$	0	0
$227$	2.62434	0.174184	0.0870918	−	0.996200i	$-0.472243\pi$
$227$	2.62434	0.174184	0.0870918	+	0.996200i	$0.472243\pi$
$228$	0	0
$229$	−11.5347	−0.762232	−0.381116	−	0.924527i	$-0.624460\pi$
$229$	−11.5347	−0.762232	−0.381116	+	0.924527i	$0.624460\pi$
$230$	0	0
$231$	2.58461	0.170055
$232$	0	0
$233$	1.53269	0.100410	0.0502049	−	0.998739i	$-0.484013\pi$
$233$	1.53269	0.100410	0.0502049	+	0.998739i	$0.484013\pi$
$234$	0	0
$235$	−1.96361	−0.128092
$236$	0	0
$237$	2.49167	0.161852
$238$	0	0
$239$	−11.2560	−0.728089	−0.364044	−	0.931382i	$-0.618604\pi$
$239$	−11.2560	−0.728089	−0.364044	+	0.931382i	$0.618604\pi$
$240$	0	0
$241$	−4.48507	−0.288909	−0.144454	−	0.989511i	$-0.546143\pi$
$241$	−4.48507	−0.288909	−0.144454	+	0.989511i	$0.546143\pi$
$242$	0	0
$243$	−16.3396	−1.04818
$244$	0	0
$245$	−2.76541	−0.176676
$246$	0	0
$247$	1.80419	0.114798
$248$	0	0
$249$	33.5442	2.12578
$250$	0	0
$251$	14.6902	0.927234	0.463617	−	0.886036i	$-0.346551\pi$
$251$	14.6902	0.927234	0.463617	+	0.886036i	$0.346551\pi$
$252$	0	0
$253$	4.62940	0.291048
$254$	0	0
$255$	−6.70190	−0.419689
$256$	0	0
$257$	14.3429	0.894685	0.447343	−	0.894363i	$-0.352370\pi$
$257$	14.3429	0.894685	0.447343	+	0.894363i	$0.352370\pi$
$258$	0	0
$259$	4.24284	0.263637
$260$	0	0
$261$	7.86117	0.486594
$262$	0	0
$263$	12.0747	0.744560	0.372280	−	0.928120i	$-0.378576\pi$
$263$	12.0747	0.744560	0.372280	+	0.928120i	$0.378576\pi$
$264$	0	0
$265$	−3.45135	−0.212015
$266$	0	0
$267$	−27.2433	−1.66726
$268$	0	0
$269$	−26.9568	−1.64358	−0.821792	−	0.569788i	$-0.807025\pi$
$269$	−26.9568	−1.64358	−0.821792	+	0.569788i	$0.807025\pi$
$270$	0	0
$271$	−18.2284	−1.10730	−0.553649	−	0.832750i	$-0.686765\pi$
$271$	−18.2284	−1.10730	−0.553649	+	0.832750i	$0.686765\pi$
$272$	0	0
$273$	4.66312	0.282225
$274$	0	0
$275$	4.75605	0.286801
$276$	0	0
$277$	17.1382	1.02973	0.514866	−	0.857270i	$-0.327842\pi$
$277$	17.1382	1.02973	0.514866	+	0.857270i	$0.327842\pi$
$278$	0	0
$279$	−0.809769	−0.0484796
$280$	0	0
$281$	−8.08229	−0.482149	−0.241075	−	0.970507i	$-0.577500\pi$
$281$	−8.08229	−0.482149	−0.241075	+	0.970507i	$0.577500\pi$
$282$	0	0
$283$	−20.1471	−1.19762	−0.598810	−	0.800891i	$-0.704360\pi$
$283$	−20.1471	−1.19762	−0.598810	+	0.800891i	$0.704360\pi$
$284$	0	0
$285$	−1.07852	−0.0638857
$286$	0	0
$287$	1.58237	0.0934043
$288$	0	0
$289$	21.6139	1.27140
$290$	0	0
$291$	−6.68260	−0.391741
$292$	0	0
$293$	2.84521	0.166219	0.0831094	−	0.996540i	$-0.473515\pi$
$293$	2.84521	0.166219	0.0831094	+	0.996540i	$0.473515\pi$
$294$	0	0
$295$	−0.205167	−0.0119453
$296$	0	0
$297$	2.68972	0.156073
$298$	0	0
$299$	8.35232	0.483027
$300$	0	0
$301$	−3.26214	−0.188027
$302$	0	0
$303$	8.05698	0.462861
$304$	0	0
$305$	3.82803	0.219193
$306$	0	0
$307$	8.97288	0.512109	0.256055	−	0.966662i	$-0.417577\pi$
$307$	8.97288	0.512109	0.256055	+	0.966662i	$0.417577\pi$
$308$	0	0
$309$	6.97718	0.396918
$310$	0	0
$311$	7.63274	0.432813	0.216407	−	0.976303i	$-0.430566\pi$
$311$	7.63274	0.432813	0.216407	+	0.976303i	$0.430566\pi$
$312$	0	0
$313$	−22.0956	−1.24892	−0.624459	−	0.781058i	$-0.714680\pi$
$313$	−22.0956	−1.24892	−0.624459	+	0.781058i	$0.714680\pi$
$314$	0	0
$315$	−1.03372	−0.0582435
$316$	0	0
$317$	22.8798	1.28506	0.642529	−	0.766261i	$-0.277885\pi$
$317$	22.8798	1.28506	0.642529	+	0.766261i	$0.277885\pi$
$318$	0	0
$319$	−4.44577	−0.248915
$320$	0	0
$321$	43.1989	2.41113
$322$	0	0
$323$	6.21401	0.345757
$324$	0	0
$325$	8.58083	0.475979
$326$	0	0
$327$	−23.4402	−1.29625
$328$	0	0
$329$	4.70568	0.259433
$330$	0	0
$331$	−29.6714	−1.63089	−0.815443	−	0.578837i	$-0.803507\pi$
$331$	−29.6714	−1.63089	−0.815443	+	0.578837i	$0.803507\pi$
$332$	0	0
$333$	−6.33842	−0.347343
$334$	0	0
$335$	−1.62434	−0.0887472
$336$	0	0
$337$	8.81637	0.480258	0.240129	−	0.970741i	$-0.422810\pi$
$337$	8.81637	0.480258	0.240129	+	0.970741i	$0.422810\pi$
$338$	0	0
$339$	−8.49673	−0.461479
$340$	0	0
$341$	0.457953	0.0247996
$342$	0	0
$343$	14.9126	0.805203
$344$	0	0
$345$	−4.99288	−0.268808
$346$	0	0
$347$	−9.96070	−0.534719	−0.267359	−	0.963597i	$-0.586151\pi$
$347$	−9.96070	−0.534719	−0.267359	+	0.963597i	$0.586151\pi$
$348$	0	0
$349$	−1.31578	−0.0704320	−0.0352160	−	0.999380i	$-0.511212\pi$
$349$	−1.31578	−0.0704320	−0.0352160	+	0.999380i	$0.511212\pi$
$350$	0	0
$351$	4.85277	0.259021
$352$	0	0
$353$	−2.76324	−0.147073	−0.0735363	−	0.997293i	$-0.523428\pi$
$353$	−2.76324	−0.147073	−0.0735363	+	0.997293i	$0.523428\pi$
$354$	0	0
$355$	7.89815	0.419190
$356$	0	0
$357$	16.0608	0.850025
$358$	0	0
$359$	−16.2892	−0.859710	−0.429855	−	0.902898i	$-0.641435\pi$
$359$	−16.2892	−0.859710	−0.429855	+	0.902898i	$0.641435\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.18363	0.114611
$364$	0	0
$365$	−0.193424	−0.0101243
$366$	0	0
$367$	−32.0873	−1.67494	−0.837471	−	0.546482i	$-0.815967\pi$
$367$	−32.0873	−1.67494	−0.837471	+	0.546482i	$0.815967\pi$
$368$	0	0
$369$	−2.36392	−0.123061
$370$	0	0
$371$	8.27098	0.429408
$372$	0	0
$373$	−30.8345	−1.59655	−0.798275	−	0.602294i	$-0.794254\pi$
$373$	−30.8345	−1.59655	−0.798275	+	0.602294i	$0.794254\pi$
$374$	0	0
$375$	−10.5221	−0.543356
$376$	0	0
$377$	−8.02102	−0.413104
$378$	0	0
$379$	−19.2112	−0.986812	−0.493406	−	0.869799i	$-0.664248\pi$
$379$	−19.2112	−0.986812	−0.493406	+	0.869799i	$0.664248\pi$
$380$	0	0
$381$	35.1951	1.80310
$382$	0	0
$383$	36.1803	1.84873	0.924363	−	0.381514i	$-0.124597\pi$
$383$	36.1803	1.84873	0.924363	+	0.381514i	$0.124597\pi$
$384$	0	0
$385$	0.584606	0.0297943
$386$	0	0
$387$	4.87335	0.247726
$388$	0	0
$389$	−16.0481	−0.813669	−0.406834	−	0.913502i	$-0.633367\pi$
$389$	−16.0481	−0.813669	−0.406834	+	0.913502i	$0.633367\pi$
$390$	0	0
$391$	28.7671	1.45482
$392$	0	0
$393$	−27.3881	−1.38155
$394$	0	0
$395$	0.563585	0.0283571
$396$	0	0
$397$	31.4584	1.57885	0.789426	−	0.613846i	$-0.210378\pi$
$397$	31.4584	1.57885	0.789426	+	0.613846i	$0.210378\pi$
$398$	0	0
$399$	2.58461	0.129392
$400$	0	0
$401$	−19.8262	−0.990072	−0.495036	−	0.868873i	$-0.664845\pi$
$401$	−19.8262	−0.990072	−0.495036	+	0.868873i	$0.664845\pi$
$402$	0	0
$403$	0.826235	0.0411577
$404$	0	0
$405$	−5.52095	−0.274338
$406$	0	0
$407$	3.58461	0.177682
$408$	0	0
$409$	34.3540	1.69869	0.849347	−	0.527834i	$-0.176996\pi$
$409$	34.3540	1.69869	0.849347	+	0.527834i	$0.176996\pi$
$410$	0	0
$411$	−37.3112	−1.84042
$412$	0	0
$413$	0.491673	0.0241936
$414$	0	0
$415$	7.58728	0.372445
$416$	0	0
$417$	−40.9236	−2.00404
$418$	0	0
$419$	−6.78324	−0.331383	−0.165691	−	0.986178i	$-0.552986\pi$
$419$	−6.78324	−0.331383	−0.165691	+	0.986178i	$0.552986\pi$
$420$	0	0
$421$	−7.08409	−0.345258	−0.172629	−	0.984987i	$-0.555226\pi$
$421$	−7.08409	−0.345258	−0.172629	+	0.984987i	$0.555226\pi$
$422$	0	0
$423$	−7.02986	−0.341803
$424$	0	0
$425$	29.5541	1.43359
$426$	0	0
$427$	−9.17368	−0.443946
$428$	0	0
$429$	3.93968	0.190210
$430$	0	0
$431$	−18.6139	−0.896599	−0.448299	−	0.893883i	$-0.647970\pi$
$431$	−18.6139	−0.896599	−0.448299	+	0.893883i	$0.647970\pi$
$432$	0	0
$433$	9.88751	0.475163	0.237582	−	0.971368i	$-0.423645\pi$
$433$	9.88751	0.475163	0.237582	+	0.971368i	$0.423645\pi$
$434$	0	0
$435$	4.79483	0.229895
$436$	0	0
$437$	4.62940	0.221454
$438$	0	0
$439$	−1.49270	−0.0712425	−0.0356213	−	0.999365i	$-0.511341\pi$
$439$	−1.49270	−0.0712425	−0.0356213	+	0.999365i	$0.511341\pi$
$440$	0	0
$441$	−9.90038	−0.471447
$442$	0	0
$443$	16.0276	0.761492	0.380746	−	0.924680i	$-0.375667\pi$
$443$	16.0276	0.761492	0.380746	+	0.924680i	$0.375667\pi$
$444$	0	0
$445$	−6.16209	−0.292111
$446$	0	0
$447$	−11.5897	−0.548172
$448$	0	0
$449$	6.40975	0.302495	0.151247	−	0.988496i	$-0.451671\pi$
$449$	6.40975	0.302495	0.151247	+	0.988496i	$0.451671\pi$
$450$	0	0
$451$	1.33688	0.0629512
$452$	0	0
$453$	13.3506	0.627266
$454$	0	0
$455$	1.05474	0.0494470
$456$	0	0
$457$	24.9906	1.16901	0.584505	−	0.811390i	$-0.301289\pi$
$457$	24.9906	1.16901	0.584505	+	0.811390i	$0.301289\pi$
$458$	0	0
$459$	16.7139	0.780140
$460$	0	0
$461$	−25.8318	−1.20311	−0.601554	−	0.798832i	$-0.705451\pi$
$461$	−25.8318	−1.20311	−0.601554	+	0.798832i	$0.705451\pi$
$462$	0	0
$463$	14.5762	0.677414	0.338707	−	0.940892i	$-0.390011\pi$
$463$	14.5762	0.677414	0.338707	+	0.940892i	$0.390011\pi$
$464$	0	0
$465$	−0.493910	−0.0229045
$466$	0	0
$467$	17.9445	0.830372	0.415186	−	0.909737i	$-0.363717\pi$
$467$	17.9445	0.830372	0.415186	+	0.909737i	$0.363717\pi$
$468$	0	0
$469$	3.89265	0.179746
$470$	0	0
$471$	27.6636	1.27467
$472$	0	0
$473$	−2.75605	−0.126723
$474$	0	0
$475$	4.75605	0.218223
$476$	0	0
$477$	−12.3561	−0.565747
$478$	0	0
$479$	−1.82187	−0.0832433	−0.0416217	−	0.999133i	$-0.513252\pi$
$479$	−1.82187	−0.0832433	−0.0416217	+	0.999133i	$0.513252\pi$
$480$	0	0
$481$	6.46731	0.294884
$482$	0	0
$483$	11.9652	0.544434
$484$	0	0
$485$	−1.51152	−0.0686346
$486$	0	0
$487$	24.2489	1.09882	0.549410	−	0.835553i	$-0.314852\pi$
$487$	24.2489	1.09882	0.549410	+	0.835553i	$0.314852\pi$
$488$	0	0
$489$	1.47124	0.0665317
$490$	0	0
$491$	−5.86495	−0.264681	−0.132341	−	0.991204i	$-0.542249\pi$
$491$	−5.86495	−0.264681	−0.132341	+	0.991204i	$0.542249\pi$
$492$	0	0
$493$	−27.6261	−1.24422
$494$	0	0
$495$	−0.873348	−0.0392541
$496$	0	0
$497$	−18.9275	−0.849014
$498$	0	0
$499$	−31.5830	−1.41385	−0.706924	−	0.707289i	$-0.749918\pi$
$499$	−31.5830	−1.41385	−0.706924	+	0.707289i	$0.749918\pi$
$500$	0	0
$501$	18.7383	0.837165
$502$	0	0
$503$	−24.5160	−1.09311	−0.546556	−	0.837422i	$-0.684062\pi$
$503$	−24.5160	−1.09311	−0.546556	+	0.837422i	$0.684062\pi$
$504$	0	0
$505$	1.82239	0.0810952
$506$	0	0
$507$	−21.2792	−0.945044
$508$	0	0
$509$	−22.3423	−0.990305	−0.495153	−	0.868806i	$-0.664888\pi$
$509$	−22.3423	−0.990305	−0.495153	+	0.868806i	$0.664888\pi$
$510$	0	0
$511$	0.463532	0.0205054
$512$	0	0
$513$	2.68972	0.118754
$514$	0	0
$515$	1.57815	0.0695417
$516$	0	0
$517$	3.97564	0.174848
$518$	0	0
$519$	6.08942	0.267296
$520$	0	0
$521$	27.5779	1.20821	0.604105	−	0.796904i	$-0.293531\pi$
$521$	27.5779	1.20821	0.604105	+	0.796904i	$0.293531\pi$
$522$	0	0
$523$	−20.7289	−0.906410	−0.453205	−	0.891406i	$-0.649719\pi$
$523$	−20.7289	−0.906410	−0.453205	+	0.891406i	$0.649719\pi$
$524$	0	0
$525$	12.2925	0.536490
$526$	0	0
$527$	2.84572	0.123962
$528$	0	0
$529$	−1.56865	−0.0682020
$530$	0	0
$531$	−0.734515	−0.0318752
$532$	0	0
$533$	2.41199	0.104475
$534$	0	0
$535$	9.77106	0.422440
$536$	0	0
$537$	−0.198046	−0.00854633
$538$	0	0
$539$	5.59902	0.241167
$540$	0	0
$541$	−5.19735	−0.223452	−0.111726	−	0.993739i	$-0.535638\pi$
$541$	−5.19735	−0.223452	−0.111726	+	0.993739i	$0.535638\pi$
$542$	0	0
$543$	44.5021	1.90977
$544$	0	0
$545$	−5.30188	−0.227108
$546$	0	0
$547$	−45.0887	−1.92786	−0.963928	−	0.266164i	$-0.914244\pi$
$547$	−45.0887	−1.92786	−0.963928	+	0.266164i	$0.914244\pi$
$548$	0	0
$549$	13.7047	0.584900
$550$	0	0
$551$	−4.44577	−0.189396
$552$	0	0
$553$	−1.35060	−0.0574335
$554$	0	0
$555$	−3.86605	−0.164105
$556$	0	0
$557$	−44.0905	−1.86817	−0.934087	−	0.357047i	$-0.883784\pi$
$557$	−44.0905	−1.86817	−0.934087	+	0.357047i	$0.883784\pi$
$558$	0	0
$559$	−4.97245	−0.210312
$560$	0	0
$561$	13.5691	0.572887
$562$	0	0
$563$	25.2372	1.06362	0.531810	−	0.846864i	$-0.321512\pi$
$563$	25.2372	1.06362	0.531810	+	0.846864i	$0.321512\pi$
$564$	0	0
$565$	−1.92185	−0.0808530
$566$	0	0
$567$	13.2307	0.555636
$568$	0	0
$569$	−3.44258	−0.144320	−0.0721602	−	0.997393i	$-0.522989\pi$
$569$	−3.44258	−0.144320	−0.0721602	+	0.997393i	$0.522989\pi$
$570$	0	0
$571$	−2.16389	−0.0905559	−0.0452780	−	0.998974i	$-0.514417\pi$
$571$	−2.16389	−0.0905559	−0.0452780	+	0.998974i	$0.514417\pi$
$572$	0	0
$573$	−34.6310	−1.44673
$574$	0	0
$575$	22.0177	0.918201
$576$	0	0
$577$	−19.1000	−0.795142	−0.397571	−	0.917571i	$-0.630147\pi$
$577$	−19.1000	−0.795142	−0.397571	+	0.917571i	$0.630147\pi$
$578$	0	0
$579$	−36.8378	−1.53093
$580$	0	0
$581$	−18.1825	−0.754338
$582$	0	0
$583$	6.98782	0.289406
$584$	0	0
$585$	−1.57569	−0.0651466
$586$	0	0
$587$	−7.78221	−0.321206	−0.160603	−	0.987019i	$-0.551344\pi$
$587$	−7.78221	−0.321206	−0.160603	+	0.987019i	$0.551344\pi$
$588$	0	0
$589$	0.457953	0.0188696
$590$	0	0
$591$	24.4074	1.00399
$592$	0	0
$593$	−40.6214	−1.66812	−0.834061	−	0.551672i	$-0.813990\pi$
$593$	−40.6214	−1.66812	−0.834061	+	0.551672i	$0.813990\pi$
$594$	0	0
$595$	3.63274	0.148928
$596$	0	0
$597$	−31.9888	−1.30922
$598$	0	0
$599$	−17.6299	−0.720339	−0.360169	−	0.932887i	$-0.617281\pi$
$599$	−17.6299	−0.720339	−0.360169	+	0.932887i	$0.617281\pi$
$600$	0	0
$601$	−12.5751	−0.512949	−0.256475	−	0.966551i	$-0.582561\pi$
$601$	−12.5751	−0.512949	−0.256475	+	0.966551i	$0.582561\pi$
$602$	0	0
$603$	−5.81527	−0.236816
$604$	0	0
$605$	0.493910	0.0200803
$606$	0	0
$607$	−23.4661	−0.952461	−0.476230	−	0.879321i	$-0.657997\pi$
$607$	−23.4661	−0.952461	−0.476230	+	0.879321i	$0.657997\pi$
$608$	0	0
$609$	−11.4906	−0.465621
$610$	0	0
$611$	7.17281	0.290181
$612$	0	0
$613$	−18.8159	−0.759965	−0.379983	−	0.924994i	$-0.624070\pi$
$613$	−18.8159	−0.759965	−0.379983	+	0.924994i	$0.624070\pi$
$614$	0	0
$615$	−1.44184	−0.0581408
$616$	0	0
$617$	1.65094	0.0664643	0.0332322	−	0.999448i	$-0.489420\pi$
$617$	1.65094	0.0664643	0.0332322	+	0.999448i	$0.489420\pi$
$618$	0	0
$619$	20.4799	0.823158	0.411579	−	0.911374i	$-0.364977\pi$
$619$	20.4799	0.823158	0.411579	+	0.911374i	$0.364977\pi$
$620$	0	0
$621$	12.4518	0.499673
$622$	0	0
$623$	14.7671	0.591633
$624$	0	0
$625$	21.4003	0.856012
$626$	0	0
$627$	2.18363	0.0872057
$628$	0	0
$629$	22.2748	0.888153
$630$	0	0
$631$	26.8892	1.07044	0.535222	−	0.844712i	$-0.320228\pi$
$631$	26.8892	1.07044	0.535222	+	0.844712i	$0.320228\pi$
$632$	0	0
$633$	34.2459	1.36115
$634$	0	0
$635$	7.96070	0.315911
$636$	0	0
$637$	10.1017	0.400244
$638$	0	0
$639$	28.2760	1.11858
$640$	0	0
$641$	12.0171	0.474647	0.237323	−	0.971431i	$-0.423730\pi$
$641$	12.0171	0.474647	0.237323	+	0.971431i	$0.423730\pi$
$642$	0	0
$643$	2.84527	0.112207	0.0561033	−	0.998425i	$-0.482132\pi$
$643$	2.84527	0.112207	0.0561033	+	0.998425i	$0.482132\pi$
$644$	0	0
$645$	2.97245	0.117040
$646$	0	0
$647$	19.2975	0.758664	0.379332	−	0.925261i	$-0.376154\pi$
$647$	19.2975	0.758664	0.379332	+	0.925261i	$0.376154\pi$
$648$	0	0
$649$	0.415394	0.0163057
$650$	0	0
$651$	1.18363	0.0463901
$652$	0	0
$653$	34.7017	1.35798	0.678991	−	0.734147i	$-0.262418\pi$
$653$	34.7017	1.35798	0.678991	+	0.734147i	$0.262418\pi$
$654$	0	0
$655$	−6.19485	−0.242053
$656$	0	0
$657$	−0.692474	−0.0270160
$658$	0	0
$659$	−4.36057	−0.169864	−0.0849319	−	0.996387i	$-0.527067\pi$
$659$	−4.36057	−0.169864	−0.0849319	+	0.996387i	$0.527067\pi$
$660$	0	0
$661$	4.43307	0.172427	0.0862133	−	0.996277i	$-0.472523\pi$
$661$	4.43307	0.172427	0.0862133	+	0.996277i	$0.472523\pi$
$662$	0	0
$663$	24.4812	0.950771
$664$	0	0
$665$	0.584606	0.0226700
$666$	0	0
$667$	−20.5813	−0.796910
$668$	0	0
$669$	21.7941	0.842610
$670$	0	0
$671$	−7.75048	−0.299204
$672$	0	0
$673$	21.9264	0.845200	0.422600	−	0.906316i	$-0.361117\pi$
$673$	21.9264	0.845200	0.422600	+	0.906316i	$0.361117\pi$
$674$	0	0
$675$	12.7924	0.492382
$676$	0	0
$677$	35.2924	1.35640	0.678198	−	0.734879i	$-0.262761\pi$
$677$	35.2924	1.35640	0.678198	+	0.734879i	$0.262761\pi$
$678$	0	0
$679$	3.62228	0.139010
$680$	0	0
$681$	5.73059	0.219597
$682$	0	0
$683$	41.1138	1.57318	0.786588	−	0.617478i	$-0.211846\pi$
$683$	41.1138	1.57318	0.786588	+	0.617478i	$0.211846\pi$
$684$	0	0
$685$	−8.43932	−0.322450
$686$	0	0
$687$	−25.1874	−0.960961
$688$	0	0
$689$	12.6074	0.480302
$690$	0	0
$691$	−21.6912	−0.825171	−0.412586	−	0.910919i	$-0.635374\pi$
$691$	−21.6912	−0.825171	−0.412586	+	0.910919i	$0.635374\pi$
$692$	0	0
$693$	2.09293	0.0795039
$694$	0	0
$695$	−9.25642	−0.351116
$696$	0	0
$697$	8.30738	0.314664
$698$	0	0
$699$	3.34682	0.126588
$700$	0	0
$701$	−16.6458	−0.628703	−0.314352	−	0.949307i	$-0.601787\pi$
$701$	−16.6458	−0.628703	−0.314352	+	0.949307i	$0.601787\pi$
$702$	0	0
$703$	3.58461	0.135196
$704$	0	0
$705$	−4.28779	−0.161487
$706$	0	0
$707$	−4.36726	−0.164248
$708$	0	0
$709$	−23.1731	−0.870282	−0.435141	−	0.900362i	$-0.643302\pi$
$709$	−23.1731	−0.870282	−0.435141	+	0.900362i	$0.643302\pi$
$710$	0	0
$711$	2.01768	0.0756689
$712$	0	0
$713$	2.12005	0.0793965
$714$	0	0
$715$	0.891107	0.0333255
$716$	0	0
$717$	−24.5789	−0.917915
$718$	0	0
$719$	27.3892	1.02144	0.510722	−	0.859746i	$-0.329378\pi$
$719$	27.3892	1.02144	0.510722	+	0.859746i	$0.329378\pi$
$720$	0	0
$721$	−3.78196	−0.140847
$722$	0	0
$723$	−9.79373	−0.364233
$724$	0	0
$725$	−21.1443	−0.785281
$726$	0	0
$727$	2.97898	0.110484	0.0552421	−	0.998473i	$-0.482407\pi$
$727$	2.97898	0.110484	0.0552421	+	0.998473i	$0.482407\pi$
$728$	0	0
$729$	−2.14537	−0.0794581
$730$	0	0
$731$	−17.1261	−0.633433
$732$	0	0
$733$	12.6851	0.468535	0.234267	−	0.972172i	$-0.424731\pi$
$733$	12.6851	0.468535	0.234267	+	0.972172i	$0.424731\pi$
$734$	0	0
$735$	−6.03863	−0.222738
$736$	0	0
$737$	3.28874	0.121142
$738$	0	0
$739$	−18.6272	−0.685211	−0.342606	−	0.939479i	$-0.611309\pi$
$739$	−18.6272	−0.685211	−0.342606	+	0.939479i	$0.611309\pi$
$740$	0	0
$741$	3.93968	0.144728
$742$	0	0
$743$	40.8878	1.50003	0.750013	−	0.661423i	$-0.230047\pi$
$743$	40.8878	1.50003	0.750013	+	0.661423i	$0.230047\pi$
$744$	0	0
$745$	−2.62144	−0.0960420
$746$	0	0
$747$	27.1630	0.993844
$748$	0	0
$749$	−23.4158	−0.855596
$750$	0	0
$751$	−15.2482	−0.556416	−0.278208	−	0.960521i	$-0.589740\pi$
$751$	−15.2482	−0.556416	−0.278208	+	0.960521i	$0.589740\pi$
$752$	0	0
$753$	32.0778	1.16898
$754$	0	0
$755$	3.01974	0.109900
$756$	0	0
$757$	1.09565	0.0398220	0.0199110	−	0.999802i	$-0.493662\pi$
$757$	1.09565	0.0398220	0.0199110	+	0.999802i	$0.493662\pi$
$758$	0	0
$759$	10.1089	0.366930
$760$	0	0
$761$	47.6858	1.72861	0.864304	−	0.502970i	$-0.167759\pi$
$761$	47.6858	1.72861	0.864304	+	0.502970i	$0.167759\pi$
$762$	0	0
$763$	12.7057	0.459976
$764$	0	0
$765$	−5.42699	−0.196213
$766$	0	0
$767$	0.749451	0.0270611
$768$	0	0
$769$	−7.70913	−0.277998	−0.138999	−	0.990293i	$-0.544389\pi$
$769$	−7.70913	−0.277998	−0.138999	+	0.990293i	$0.544389\pi$
$770$	0	0
$771$	31.3196	1.12795
$772$	0	0
$773$	17.5085	0.629737	0.314869	−	0.949135i	$-0.398040\pi$
$773$	17.5085	0.629737	0.314869	+	0.949135i	$0.398040\pi$
$774$	0	0
$775$	2.17805	0.0782379
$776$	0	0
$777$	9.26479	0.332373
$778$	0	0
$779$	1.33688	0.0478987
$780$	0	0
$781$	−15.9911	−0.572206
$782$	0	0
$783$	−11.9579	−0.427340
$784$	0	0
$785$	6.25715	0.223327
$786$	0	0
$787$	−5.47468	−0.195151	−0.0975756	−	0.995228i	$-0.531109\pi$
$787$	−5.47468	−0.195151	−0.0975756	+	0.995228i	$0.531109\pi$
$788$	0	0
$789$	26.3667	0.938681
$790$	0	0
$791$	4.60563	0.163757
$792$	0	0
$793$	−13.9833	−0.496563
$794$	0	0
$795$	−7.53647	−0.267291
$796$	0	0
$797$	−33.0404	−1.17035	−0.585176	−	0.810906i	$-0.698975\pi$
$797$	−33.0404	−1.17035	−0.585176	+	0.810906i	$0.698975\pi$
$798$	0	0
$799$	24.7046	0.873987
$800$	0	0
$801$	−22.0608	−0.779478
$802$	0	0
$803$	0.391619	0.0138199
$804$	0	0
$805$	2.70637	0.0953871
$806$	0	0
$807$	−58.8636	−2.07210
$808$	0	0
$809$	−14.1534	−0.497608	−0.248804	−	0.968554i	$-0.580038\pi$
$809$	−14.1534	−0.497608	−0.248804	+	0.968554i	$0.580038\pi$
$810$	0	0
$811$	−43.2514	−1.51876	−0.759381	−	0.650646i	$-0.774498\pi$
$811$	−43.2514	−1.51876	−0.759381	+	0.650646i	$0.774498\pi$
$812$	0	0
$813$	−39.8041	−1.39599
$814$	0	0
$815$	0.332776	0.0116566
$816$	0	0
$817$	−2.75605	−0.0964221
$818$	0	0
$819$	3.77605	0.131946
$820$	0	0
$821$	47.1419	1.64527	0.822633	−	0.568573i	$-0.192504\pi$
$821$	47.1419	1.64527	0.822633	+	0.568573i	$0.192504\pi$
$822$	0	0
$823$	29.3098	1.02168	0.510838	−	0.859677i	$-0.329335\pi$
$823$	29.3098	1.02168	0.510838	+	0.859677i	$0.329335\pi$
$824$	0	0
$825$	10.3855	0.361575
$826$	0	0
$827$	−11.8963	−0.413677	−0.206838	−	0.978375i	$-0.566317\pi$
$827$	−11.8963	−0.413677	−0.206838	+	0.978375i	$0.566317\pi$
$828$	0	0
$829$	8.91532	0.309642	0.154821	−	0.987943i	$-0.450520\pi$
$829$	8.91532	0.309642	0.154821	+	0.987943i	$0.450520\pi$
$830$	0	0
$831$	37.4234	1.29820
$832$	0	0
$833$	34.7924	1.20548
$834$	0	0
$835$	4.23837	0.146675
$836$	0	0
$837$	1.23177	0.0425761
$838$	0	0
$839$	47.2566	1.63148	0.815739	−	0.578420i	$-0.196331\pi$
$839$	47.2566	1.63148	0.815739	+	0.578420i	$0.196331\pi$
$840$	0	0
$841$	−9.23511	−0.318452
$842$	0	0
$843$	−17.6487	−0.607855
$844$	0	0
$845$	−4.81310	−0.165576
$846$	0	0
$847$	−1.18363	−0.0406700
$848$	0	0
$849$	−43.9938	−1.50986
$850$	0	0
$851$	16.5946	0.568855
$852$	0	0
$853$	−7.04189	−0.241110	−0.120555	−	0.992707i	$-0.538467\pi$
$853$	−7.04189	−0.241110	−0.120555	+	0.992707i	$0.538467\pi$
$854$	0	0
$855$	−0.873348	−0.0298679
$856$	0	0
$857$	51.1303	1.74658	0.873288	−	0.487204i	$-0.161983\pi$
$857$	51.1303	1.74658	0.873288	+	0.487204i	$0.161983\pi$
$858$	0	0
$859$	−9.52646	−0.325038	−0.162519	−	0.986705i	$-0.551962\pi$
$859$	−9.52646	−0.325038	−0.162519	+	0.986705i	$0.551962\pi$
$860$	0	0
$861$	3.45531	0.117757
$862$	0	0
$863$	−35.2935	−1.20140	−0.600702	−	0.799473i	$-0.705112\pi$
$863$	−35.2935	−1.20140	−0.600702	+	0.799473i	$0.705112\pi$
$864$	0	0
$865$	1.37735	0.0468313
$866$	0	0
$867$	47.1967	1.60288
$868$	0	0
$869$	−1.14107	−0.0387081
$870$	0	0
$871$	5.93352	0.201050
$872$	0	0
$873$	−5.41136	−0.183147
$874$	0	0
$875$	5.70344	0.192812
$876$	0	0
$877$	−22.5103	−0.760119	−0.380060	−	0.924962i	$-0.624097\pi$
$877$	−22.5103	−0.760119	−0.380060	+	0.924962i	$0.624097\pi$
$878$	0	0
$879$	6.21287	0.209555
$880$	0	0
$881$	−34.4317	−1.16003	−0.580017	−	0.814604i	$-0.696954\pi$
$881$	−34.4317	−1.16003	−0.580017	+	0.814604i	$0.696954\pi$
$882$	0	0
$883$	−14.3992	−0.484571	−0.242285	−	0.970205i	$-0.577897\pi$
$883$	−14.3992	−0.484571	−0.242285	+	0.970205i	$0.577897\pi$
$884$	0	0
$885$	−0.448009	−0.0150597
$886$	0	0
$887$	43.3560	1.45575	0.727875	−	0.685710i	$-0.240508\pi$
$887$	43.3560	1.45575	0.727875	+	0.685710i	$0.240508\pi$
$888$	0	0
$889$	−19.0774	−0.639836
$890$	0	0
$891$	11.1781	0.374479
$892$	0	0
$893$	3.97564	0.133040
$894$	0	0
$895$	−0.0447956	−0.00149735
$896$	0	0
$897$	18.2384	0.608961
$898$	0	0
$899$	−2.03596	−0.0679029
$900$	0	0
$901$	43.4224	1.44661
$902$	0	0
$903$	−7.12331	−0.237049
$904$	0	0
$905$	10.0658	0.334599
$906$	0	0
$907$	−0.306501	−0.0101772	−0.00508861	−	0.999987i	$-0.501620\pi$
$907$	−0.306501	−0.0101772	−0.00508861	+	0.999987i	$0.501620\pi$
$908$	0	0
$909$	6.52429	0.216397
$910$	0	0
$911$	12.0563	0.399442	0.199721	−	0.979853i	$-0.435996\pi$
$911$	12.0563	0.399442	0.199721	+	0.979853i	$0.435996\pi$
$912$	0	0
$913$	−15.3617	−0.508397
$914$	0	0
$915$	8.35900	0.276340
$916$	0	0
$917$	14.8456	0.490246
$918$	0	0
$919$	10.0917	0.332895	0.166448	−	0.986050i	$-0.446770\pi$
$919$	10.0917	0.332895	0.166448	+	0.986050i	$0.446770\pi$
$920$	0	0
$921$	19.5934	0.645626
$922$	0	0
$923$	−28.8510	−0.949641
$924$	0	0
$925$	17.0486	0.560554
$926$	0	0
$927$	5.64990	0.185567
$928$	0	0
$929$	48.6178	1.59510	0.797550	−	0.603253i	$-0.206129\pi$
$929$	48.6178	1.59510	0.797550	+	0.603253i	$0.206129\pi$
$930$	0	0
$931$	5.59902	0.183501
$932$	0	0
$933$	16.6671	0.545656
$934$	0	0
$935$	3.06916	0.100372
$936$	0	0
$937$	13.8047	0.450980	0.225490	−	0.974245i	$-0.427602\pi$
$937$	13.8047	0.450980	0.225490	+	0.974245i	$0.427602\pi$
$938$	0	0
$939$	−48.2486	−1.57453
$940$	0	0
$941$	33.4773	1.09133	0.545664	−	0.838004i	$-0.316277\pi$
$941$	33.4773	1.09133	0.545664	+	0.838004i	$0.316277\pi$
$942$	0	0
$943$	6.18895	0.201540
$944$	0	0
$945$	1.57242	0.0511510
$946$	0	0
$947$	25.9114	0.842007	0.421004	−	0.907059i	$-0.361678\pi$
$947$	25.9114	0.842007	0.421004	+	0.907059i	$0.361678\pi$
$948$	0	0
$949$	0.706555	0.0229358
$950$	0	0
$951$	49.9610	1.62010
$952$	0	0
$953$	25.8963	0.838865	0.419433	−	0.907787i	$-0.362229\pi$
$953$	25.8963	0.838865	0.419433	+	0.907787i	$0.362229\pi$
$954$	0	0
$955$	−7.83309	−0.253473
$956$	0	0
$957$	−9.70792	−0.313812
$958$	0	0
$959$	20.2244	0.653080
$960$	0	0
$961$	−30.7903	−0.993235
$962$	0	0
$963$	34.9811	1.12725
$964$	0	0
$965$	−8.33226	−0.268225
$966$	0	0
$967$	1.73280	0.0557230	0.0278615	−	0.999612i	$-0.491130\pi$
$967$	1.73280	0.0557230	0.0278615	+	0.999612i	$0.491130\pi$
$968$	0	0
$969$	13.5691	0.435902
$970$	0	0
$971$	15.0088	0.481654	0.240827	−	0.970568i	$-0.422581\pi$
$971$	15.0088	0.481654	0.240827	+	0.970568i	$0.422581\pi$
$972$	0	0
$973$	22.1825	0.711139
$974$	0	0
$975$	18.7373	0.600075
$976$	0	0
$977$	−9.33234	−0.298568	−0.149284	−	0.988794i	$-0.547697\pi$
$977$	−9.33234	−0.298568	−0.149284	+	0.988794i	$0.547697\pi$
$978$	0	0
$979$	12.4762	0.398739
$980$	0	0
$981$	−18.9811	−0.606021
$982$	0	0
$983$	4.83086	0.154080	0.0770402	−	0.997028i	$-0.475453\pi$
$983$	4.83086	0.154080	0.0770402	+	0.997028i	$0.475453\pi$
$984$	0	0
$985$	5.52066	0.175903
$986$	0	0
$987$	10.2755	0.327071
$988$	0	0
$989$	−12.7589	−0.405709
$990$	0	0
$991$	−35.0069	−1.11203	−0.556015	−	0.831172i	$-0.687670\pi$
$991$	−35.0069	−1.11203	−0.556015	+	0.831172i	$0.687670\pi$
$992$	0	0
$993$	−64.7913	−2.05609
$994$	0	0
$995$	−7.23548	−0.229380
$996$	0	0
$997$	−22.1287	−0.700824	−0.350412	−	0.936596i	$-0.613958\pi$
$997$	−22.1287	−0.700824	−0.350412	+	0.936596i	$0.613958\pi$
$998$	0	0
$999$	9.64158	0.305046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.s.1.4		4
4.3	odd	2		1672.2.a.f.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.f.1.1	✓	4	4.3	odd	2
3344.2.a.s.1.4		4	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)

\(f(q)\)	\(=\)	\(q+2.18363 q^{3} +0.493910 q^{5} -1.18363 q^{7} +1.76823 q^{9} -1.00000 q^{11} -1.80419 q^{13} +1.07852 q^{15} -6.21401 q^{17} -1.00000 q^{19} -2.58461 q^{21} -4.62940 q^{23} -4.75605 q^{25} -2.68972 q^{27} +4.44577 q^{29} -0.457953 q^{31} -2.18363 q^{33} -0.584606 q^{35} -3.58461 q^{37} -3.93968 q^{39} -1.33688 q^{41} +2.75605 q^{43} +0.873348 q^{45} -3.97564 q^{47} -5.59902 q^{49} -13.5691 q^{51} -6.98782 q^{53} -0.493910 q^{55} -2.18363 q^{57} -0.415394 q^{59} +7.75048 q^{61} -2.09293 q^{63} -0.891107 q^{65} -3.28874 q^{67} -10.1089 q^{69} +15.9911 q^{71} -0.391619 q^{73} -10.3855 q^{75} +1.18363 q^{77} +1.14107 q^{79} -11.1781 q^{81} +15.3617 q^{83} -3.06916 q^{85} +9.70792 q^{87} -12.4762 q^{89} +2.13549 q^{91} -1.00000 q^{93} -0.493910 q^{95} -3.06032 q^{97} -1.76823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41}+ \cdots - q^{99}+O(q^{100}) \) 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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