LMFDB - Embedded newform 2960.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76156 q^{3} +1.00000 q^{5} -1.76156 q^{7} +0.103084 q^{9} +O(q^{10})$	$q+1.76156 q^{3} +1.00000 q^{5} -1.76156 q^{7} +0.103084 q^{9} -4.62620 q^{11} +1.76156 q^{15} +2.38776 q^{19} -3.10308 q^{21} -4.00000 q^{23} +1.00000 q^{25} -5.10308 q^{27} -7.25240 q^{29} -1.13536 q^{31} -8.14931 q^{33} -1.76156 q^{35} +1.00000 q^{37} +0.896916 q^{41} -9.25240 q^{43} +0.103084 q^{45} +8.74324 q^{47} -3.89692 q^{49} +8.35548 q^{53} -4.62620 q^{55} +4.20617 q^{57} +8.11704 q^{59} +5.04623 q^{61} -0.181588 q^{63} -16.1170 q^{67} -7.04623 q^{69} -12.1493 q^{71} -11.4017 q^{73} +1.76156 q^{75} +8.14931 q^{77} -11.6402 q^{79} -9.29862 q^{81} -18.0602 q^{83} -12.7755 q^{87} +1.45856 q^{89} -2.00000 q^{93} +2.38776 q^{95} +6.00000 q^{97} -0.476886 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 + 3 * q^5 + q^7 + 4 * q^9	$ 3 q - q^{3} + 3 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - q^{15} - 8 q^{19} - 13 q^{21} - 12 q^{23} + 3 q^{25} - 19 q^{27} - 4 q^{29} - 6 q^{31} - 3 q^{33} + q^{35} + 3 q^{37} - q^{41} - 10 q^{43} + 4 q^{45} - 3 q^{47} - 8 q^{49} + 11 q^{53} - 5 q^{55} + 20 q^{57} + 4 q^{59} - 10 q^{61} + 22 q^{63} - 28 q^{67} + 4 q^{69} - 15 q^{71} + 5 q^{73} - q^{75} + 3 q^{77} - 2 q^{79} + 15 q^{81} - 5 q^{83} - 8 q^{87} - 6 q^{89} - 6 q^{93} - 8 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - q^15 - 8 * q^19 - 13 * q^21 - 12 * q^23 + 3 * q^25 - 19 * q^27 - 4 * q^29 - 6 * q^31 - 3 * q^33 + q^35 + 3 * q^37 - q^41 - 10 * q^43 + 4 * q^45 - 3 * q^47 - 8 * q^49 + 11 * q^53 - 5 * q^55 + 20 * q^57 + 4 * q^59 - 10 * q^61 + 22 * q^63 - 28 * q^67 + 4 * q^69 - 15 * q^71 + 5 * q^73 - q^75 + 3 * q^77 - 2 * q^79 + 15 * q^81 - 5 * q^83 - 8 * q^87 - 6 * q^89 - 6 * q^93 - 8 * q^95 + 18 * q^97 - 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.76156	1.01704	0.508518	−	0.861052i	$-0.330194\pi$
$3$	1.76156	1.01704	0.508518	+	0.861052i	$0.330194\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.76156	−0.665806	−0.332903	−	0.942961i	$-0.608028\pi$
$7$	−1.76156	−0.665806	−0.332903	+	0.942961i	$0.608028\pi$
$8$	0	0
$9$	0.103084	0.0343612
$10$	0	0
$11$	−4.62620	−1.39485	−0.697426	−	0.716657i	$-0.745671\pi$
$11$	−4.62620	−1.39485	−0.697426	+	0.716657i	$0.745671\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.76156	0.454832
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.38776	0.547789	0.273894	−	0.961760i	$-0.411688\pi$
$19$	2.38776	0.547789	0.273894	+	0.961760i	$0.411688\pi$
$20$	0	0
$21$	−3.10308	−0.677148
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.10308	−0.982089
$28$	0	0
$29$	−7.25240	−1.34674	−0.673368	−	0.739307i	$-0.735153\pi$
$29$	−7.25240	−1.34674	−0.673368	+	0.739307i	$0.735153\pi$
$30$	0	0
$31$	−1.13536	−0.203917	−0.101958	−	0.994789i	$-0.532511\pi$
$31$	−1.13536	−0.203917	−0.101958	+	0.994789i	$0.532511\pi$
$32$	0	0
$33$	−8.14931	−1.41861
$34$	0	0
$35$	−1.76156	−0.297758
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.896916	0.140075	0.0700374	−	0.997544i	$-0.477688\pi$
$41$	0.896916	0.140075	0.0700374	+	0.997544i	$0.477688\pi$
$42$	0	0
$43$	−9.25240	−1.41098	−0.705489	−	0.708721i	$-0.749272\pi$
$43$	−9.25240	−1.41098	−0.705489	+	0.708721i	$0.749272\pi$
$44$	0	0
$45$	0.103084	0.0153668
$46$	0	0
$47$	8.74324	1.27533	0.637666	−	0.770313i	$-0.279900\pi$
$47$	8.74324	1.27533	0.637666	+	0.770313i	$0.279900\pi$
$48$	0	0
$49$	−3.89692	−0.556702
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.35548	1.14771	0.573857	−	0.818956i	$-0.305447\pi$
$53$	8.35548	1.14771	0.573857	+	0.818956i	$0.305447\pi$
$54$	0	0
$55$	−4.62620	−0.623796
$56$	0	0
$57$	4.20617	0.557120
$58$	0	0
$59$	8.11704	1.05675	0.528374	−	0.849012i	$-0.322802\pi$
$59$	8.11704	1.05675	0.528374	+	0.849012i	$0.322802\pi$
$60$	0	0
$61$	5.04623	0.646103	0.323052	−	0.946381i	$-0.395291\pi$
$61$	5.04623	0.646103	0.323052	+	0.946381i	$0.395291\pi$
$62$	0	0
$63$	−0.181588	−0.0228779
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−16.1170	−1.96901	−0.984505	−	0.175358i	$-0.943892\pi$
$67$	−16.1170	−1.96901	−0.984505	+	0.175358i	$0.943892\pi$
$68$	0	0
$69$	−7.04623	−0.848266
$70$	0	0
$71$	−12.1493	−1.44186	−0.720929	−	0.693009i	$-0.756285\pi$
$71$	−12.1493	−1.44186	−0.720929	+	0.693009i	$0.756285\pi$
$72$	0	0
$73$	−11.4017	−1.33447	−0.667235	−	0.744848i	$-0.732522\pi$
$73$	−11.4017	−1.33447	−0.667235	+	0.744848i	$0.732522\pi$
$74$	0	0
$75$	1.76156	0.203407
$76$	0	0
$77$	8.14931	0.928700
$78$	0	0
$79$	−11.6402	−1.30962	−0.654810	−	0.755794i	$-0.727251\pi$
$79$	−11.6402	−1.30962	−0.654810	+	0.755794i	$0.727251\pi$
$80$	0	0
$81$	−9.29862	−1.03318
$82$	0	0
$83$	−18.0602	−1.98236	−0.991181	−	0.132513i	$-0.957695\pi$
$83$	−18.0602	−1.98236	−0.991181	+	0.132513i	$0.957695\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.7755	−1.36968
$88$	0	0
$89$	1.45856	0.154607	0.0773037	−	0.997008i	$-0.475369\pi$
$89$	1.45856	0.154607	0.0773037	+	0.997008i	$0.475369\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	2.38776	0.244979
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−0.476886	−0.0479288
$100$	0	0
$101$	4.69075	0.466747	0.233373	−	0.972387i	$-0.425024\pi$
$101$	4.69075	0.466747	0.233373	+	0.972387i	$0.425024\pi$
$102$	0	0
$103$	−14.5693	−1.43556	−0.717780	−	0.696270i	$-0.754842\pi$
$103$	−14.5693	−1.43556	−0.717780	+	0.696270i	$0.754842\pi$
$104$	0	0
$105$	−3.10308	−0.302830
$106$	0	0
$107$	5.84632	0.565185	0.282592	−	0.959240i	$-0.408806\pi$
$107$	5.84632	0.565185	0.282592	+	0.959240i	$0.408806\pi$
$108$	0	0
$109$	1.04623	0.100211	0.0501053	−	0.998744i	$-0.484044\pi$
$109$	1.04623	0.100211	0.0501053	+	0.998744i	$0.484044\pi$
$110$	0	0
$111$	1.76156	0.167200
$112$	0	0
$113$	8.29862	0.780669	0.390334	−	0.920673i	$-0.372359\pi$
$113$	8.29862	0.780669	0.390334	+	0.920673i	$0.372359\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.4017	0.945610
$122$	0	0
$123$	1.57997	0.142461
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.3126	1.71371	0.856857	−	0.515554i	$-0.172414\pi$
$127$	19.3126	1.71371	0.856857	+	0.515554i	$0.172414\pi$
$128$	0	0
$129$	−16.2986	−1.43501
$130$	0	0
$131$	1.13536	0.0991968	0.0495984	−	0.998769i	$-0.484206\pi$
$131$	1.13536	0.0991968	0.0495984	+	0.998769i	$0.484206\pi$
$132$	0	0
$133$	−4.20617	−0.364721
$134$	0	0
$135$	−5.10308	−0.439204
$136$	0	0
$137$	9.04623	0.772871	0.386436	−	0.922316i	$-0.373706\pi$
$137$	9.04623	0.772871	0.386436	+	0.922316i	$0.373706\pi$
$138$	0	0
$139$	7.45856	0.632627	0.316314	−	0.948655i	$-0.397555\pi$
$139$	7.45856	0.632627	0.316314	+	0.948655i	$0.397555\pi$
$140$	0	0
$141$	15.4017	1.29706
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.25240	−0.602279
$146$	0	0
$147$	−6.86464	−0.566186
$148$	0	0
$149$	1.10308	0.0903681	0.0451841	−	0.998979i	$-0.485613\pi$
$149$	1.10308	0.0903681	0.0451841	+	0.998979i	$0.485613\pi$
$150$	0	0
$151$	1.31695	0.107172	0.0535858	−	0.998563i	$-0.482935\pi$
$151$	1.31695	0.107172	0.0535858	+	0.998563i	$0.482935\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.13536	−0.0911942
$156$	0	0
$157$	−11.8140	−0.942863	−0.471432	−	0.881903i	$-0.656263\pi$
$157$	−11.8140	−0.942863	−0.471432	+	0.881903i	$0.656263\pi$
$158$	0	0
$159$	14.7187	1.16727
$160$	0	0
$161$	7.04623	0.555321
$162$	0	0
$163$	4.77551	0.374047	0.187023	−	0.982355i	$-0.440116\pi$
$163$	4.77551	0.374047	0.187023	+	0.982355i	$0.440116\pi$
$164$	0	0
$165$	−8.14931	−0.634423
$166$	0	0
$167$	−10.0279	−0.775983	−0.387991	−	0.921663i	$-0.626831\pi$
$167$	−10.0279	−0.775983	−0.387991	+	0.921663i	$0.626831\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0.246139	0.0188227
$172$	0	0
$173$	12.8969	0.980534	0.490267	−	0.871572i	$-0.336899\pi$
$173$	12.8969	0.980534	0.490267	+	0.871572i	$0.336899\pi$
$174$	0	0
$175$	−1.76156	−0.133161
$176$	0	0
$177$	14.2986	1.07475
$178$	0	0
$179$	−20.8925	−1.56158	−0.780791	−	0.624792i	$-0.785184\pi$
$179$	−20.8925	−1.56158	−0.780791	+	0.624792i	$0.785184\pi$
$180$	0	0
$181$	23.9065	1.77696	0.888478	−	0.458919i	$-0.151763\pi$
$181$	23.9065	1.77696	0.888478	+	0.458919i	$0.151763\pi$
$182$	0	0
$183$	8.88922	0.657110
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.98937	0.653881
$190$	0	0
$191$	9.43398	0.682619	0.341310	−	0.939951i	$-0.389130\pi$
$191$	9.43398	0.682619	0.341310	+	0.939951i	$0.389130\pi$
$192$	0	0
$193$	4.20617	0.302767	0.151383	−	0.988475i	$-0.451627\pi$
$193$	4.20617	0.302767	0.151383	+	0.988475i	$0.451627\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.14931	0.438120	0.219060	−	0.975711i	$-0.429701\pi$
$197$	6.14931	0.438120	0.219060	+	0.975711i	$0.429701\pi$
$198$	0	0
$199$	5.91087	0.419010	0.209505	−	0.977808i	$-0.432815\pi$
$199$	5.91087	0.419010	0.209505	+	0.977808i	$0.432815\pi$
$200$	0	0
$201$	−28.3911	−2.00255
$202$	0	0
$203$	12.7755	0.896665
$204$	0	0
$205$	0.896916	0.0626434
$206$	0	0
$207$	−0.412335	−0.0286593
$208$	0	0
$209$	−11.0462	−0.764084
$210$	0	0
$211$	27.6079	1.90060	0.950302	−	0.311329i	$-0.100774\pi$
$211$	27.6079	1.90060	0.950302	+	0.311329i	$0.100774\pi$
$212$	0	0
$213$	−21.4017	−1.46642
$214$	0	0
$215$	−9.25240	−0.631008
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	−20.0848	−1.35720
$220$	0	0
$221$	0	0
$222$	0	0
$223$	21.2847	1.42533	0.712664	−	0.701506i	$-0.247489\pi$
$223$	21.2847	1.42533	0.712664	+	0.701506i	$0.247489\pi$
$224$	0	0
$225$	0.103084	0.00687225
$226$	0	0
$227$	−13.7293	−0.911244	−0.455622	−	0.890173i	$-0.650583\pi$
$227$	−13.7293	−0.911244	−0.455622	+	0.890173i	$0.650583\pi$
$228$	0	0
$229$	3.81404	0.252039	0.126020	−	0.992028i	$-0.459780\pi$
$229$	3.81404	0.252039	0.126020	+	0.992028i	$0.459780\pi$
$230$	0	0
$231$	14.3555	0.944521
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	8.74324	0.570346
$236$	0	0
$237$	−20.5048	−1.33193
$238$	0	0
$239$	4.72302	0.305507	0.152754	−	0.988264i	$-0.451186\pi$
$239$	4.72302	0.305507	0.152754	+	0.988264i	$0.451186\pi$
$240$	0	0
$241$	−26.7110	−1.72060	−0.860302	−	0.509785i	$-0.829725\pi$
$241$	−26.7110	−1.72060	−0.860302	+	0.509785i	$0.829725\pi$
$242$	0	0
$243$	−1.07081	−0.0686924
$244$	0	0
$245$	−3.89692	−0.248965
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−31.8140	−2.01613
$250$	0	0
$251$	−2.15368	−0.135939	−0.0679696	−	0.997687i	$-0.521652\pi$
$251$	−2.15368	−0.135939	−0.0679696	+	0.997687i	$0.521652\pi$
$252$	0	0
$253$	18.5048	1.16339
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.298625	−0.0186277	−0.00931385	−	0.999957i	$-0.502965\pi$
$257$	−0.298625	−0.0186277	−0.00931385	+	0.999957i	$0.502965\pi$
$258$	0	0
$259$	−1.76156	−0.109458
$260$	0	0
$261$	−0.747604	−0.0462755
$262$	0	0
$263$	11.0140	0.679149	0.339575	−	0.940579i	$-0.389717\pi$
$263$	11.0140	0.679149	0.339575	+	0.940579i	$0.389717\pi$
$264$	0	0
$265$	8.35548	0.513273
$266$	0	0
$267$	2.56934	0.157241
$268$	0	0
$269$	−16.7110	−1.01889	−0.509443	−	0.860505i	$-0.670148\pi$
$269$	−16.7110	−1.01889	−0.509443	+	0.860505i	$0.670148\pi$
$270$	0	0
$271$	7.43835	0.451848	0.225924	−	0.974145i	$-0.427460\pi$
$271$	7.43835	0.451848	0.225924	+	0.974145i	$0.427460\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.62620	−0.278970
$276$	0	0
$277$	12.7110	0.763728	0.381864	−	0.924219i	$-0.375282\pi$
$277$	12.7110	0.763728	0.381864	+	0.924219i	$0.375282\pi$
$278$	0	0
$279$	−0.117037	−0.00700682
$280$	0	0
$281$	−20.0925	−1.19862	−0.599308	−	0.800519i	$-0.704557\pi$
$281$	−20.0925	−1.19862	−0.599308	+	0.800519i	$0.704557\pi$
$282$	0	0
$283$	−16.2341	−0.965016	−0.482508	−	0.875892i	$-0.660274\pi$
$283$	−16.2341	−0.965016	−0.482508	+	0.875892i	$0.660274\pi$
$284$	0	0
$285$	4.20617	0.249152
$286$	0	0
$287$	−1.57997	−0.0932626
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	10.5693	0.619586
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	8.11704	0.472592
$296$	0	0
$297$	23.6079	1.36987
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.2986	0.939437
$302$	0	0
$303$	8.26302	0.474698
$304$	0	0
$305$	5.04623	0.288946
$306$	0	0
$307$	−4.03228	−0.230134	−0.115067	−	0.993358i	$-0.536708\pi$
$307$	−4.03228	−0.230134	−0.115067	+	0.993358i	$0.536708\pi$
$308$	0	0
$309$	−25.6647	−1.46002
$310$	0	0
$311$	9.07081	0.514358	0.257179	−	0.966364i	$-0.417207\pi$
$311$	9.07081	0.514358	0.257179	+	0.966364i	$0.417207\pi$
$312$	0	0
$313$	−24.0925	−1.36179	−0.680893	−	0.732383i	$-0.738408\pi$
$313$	−24.0925	−1.36179	−0.680893	+	0.732383i	$0.738408\pi$
$314$	0	0
$315$	−0.181588	−0.0102313
$316$	0	0
$317$	17.4586	0.980571	0.490285	−	0.871562i	$-0.336893\pi$
$317$	17.4586	0.980571	0.490285	+	0.871562i	$0.336893\pi$
$318$	0	0
$319$	33.5510	1.87850
$320$	0	0
$321$	10.2986	0.574813
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.84299	0.101918
$328$	0	0
$329$	−15.4017	−0.849124
$330$	0	0
$331$	1.07081	0.0588569	0.0294285	−	0.999567i	$-0.490631\pi$
$331$	1.07081	0.0588569	0.0294285	+	0.999567i	$0.490631\pi$
$332$	0	0
$333$	0.103084	0.00564895
$334$	0	0
$335$	−16.1170	−0.880568
$336$	0	0
$337$	−0.767815	−0.0418255	−0.0209128	−	0.999781i	$-0.506657\pi$
$337$	−0.767815	−0.0418255	−0.0209128	+	0.999781i	$0.506657\pi$
$338$	0	0
$339$	14.6185	0.793968
$340$	0	0
$341$	5.25240	0.284433
$342$	0	0
$343$	19.1955	1.03646
$344$	0	0
$345$	−7.04623	−0.379356
$346$	0	0
$347$	−15.0462	−0.807724	−0.403862	−	0.914820i	$-0.632332\pi$
$347$	−15.0462	−0.807724	−0.403862	+	0.914820i	$0.632332\pi$
$348$	0	0
$349$	−22.2062	−1.18867	−0.594334	−	0.804218i	$-0.702584\pi$
$349$	−22.2062	−1.18867	−0.594334	+	0.804218i	$0.702584\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.5048	1.51716	0.758579	−	0.651582i	$-0.225894\pi$
$353$	28.5048	1.51716	0.758579	+	0.651582i	$0.225894\pi$
$354$	0	0
$355$	−12.1493	−0.644819
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.62620	0.455273	0.227637	−	0.973746i	$-0.426900\pi$
$359$	8.62620	0.455273	0.227637	+	0.973746i	$0.426900\pi$
$360$	0	0
$361$	−13.2986	−0.699928
$362$	0	0
$363$	18.3232	0.961719
$364$	0	0
$365$	−11.4017	−0.596793
$366$	0	0
$367$	−2.38776	−0.124640	−0.0623199	−	0.998056i	$-0.519850\pi$
$367$	−2.38776	−0.124640	−0.0623199	+	0.998056i	$0.519850\pi$
$368$	0	0
$369$	0.0924575	0.00481314
$370$	0	0
$371$	−14.7187	−0.764155
$372$	0	0
$373$	−25.6079	−1.32593	−0.662963	−	0.748652i	$-0.730701\pi$
$373$	−25.6079	−1.32593	−0.662963	+	0.748652i	$0.730701\pi$
$374$	0	0
$375$	1.76156	0.0909664
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−20.6262	−1.05950	−0.529748	−	0.848155i	$-0.677714\pi$
$379$	−20.6262	−1.05950	−0.529748	+	0.848155i	$0.677714\pi$
$380$	0	0
$381$	34.0202	1.74291
$382$	0	0
$383$	1.79383	0.0916606	0.0458303	−	0.998949i	$-0.485407\pi$
$383$	1.79383	0.0916606	0.0458303	+	0.998949i	$0.485407\pi$
$384$	0	0
$385$	8.14931	0.415327
$386$	0	0
$387$	−0.953771	−0.0484829
$388$	0	0
$389$	7.66473	0.388617	0.194309	−	0.980940i	$-0.437754\pi$
$389$	7.66473	0.388617	0.194309	+	0.980940i	$0.437754\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.00000	0.100887
$394$	0	0
$395$	−11.6402	−0.585680
$396$	0	0
$397$	14.1493	0.710134	0.355067	−	0.934841i	$-0.384458\pi$
$397$	14.1493	0.710134	0.355067	+	0.934841i	$0.384458\pi$
$398$	0	0
$399$	−7.40940	−0.370934
$400$	0	0
$401$	27.5510	1.37583	0.687916	−	0.725790i	$-0.258526\pi$
$401$	27.5510	1.37583	0.687916	+	0.725790i	$0.258526\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−9.29862	−0.462052
$406$	0	0
$407$	−4.62620	−0.229312
$408$	0	0
$409$	17.7572	0.878036	0.439018	−	0.898478i	$-0.355326\pi$
$409$	17.7572	0.878036	0.439018	+	0.898478i	$0.355326\pi$
$410$	0	0
$411$	15.9354	0.786038
$412$	0	0
$413$	−14.2986	−0.703589
$414$	0	0
$415$	−18.0602	−0.886539
$416$	0	0
$417$	13.1387	0.643404
$418$	0	0
$419$	2.29093	0.111919	0.0559596	−	0.998433i	$-0.482178\pi$
$419$	2.29093	0.111919	0.0559596	+	0.998433i	$0.482178\pi$
$420$	0	0
$421$	−17.7572	−0.865432	−0.432716	−	0.901530i	$-0.642445\pi$
$421$	−17.7572	−0.865432	−0.432716	+	0.901530i	$0.642445\pi$
$422$	0	0
$423$	0.901285	0.0438220
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.88922	−0.430180
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.2095	1.45514	0.727570	−	0.686033i	$-0.240650\pi$
$431$	30.2095	1.45514	0.727570	+	0.686033i	$0.240650\pi$
$432$	0	0
$433$	−17.6079	−0.846181	−0.423090	−	0.906088i	$-0.639055\pi$
$433$	−17.6079	−0.846181	−0.423090	+	0.906088i	$0.639055\pi$
$434$	0	0
$435$	−12.7755	−0.612539
$436$	0	0
$437$	−9.55102	−0.456887
$438$	0	0
$439$	−4.89255	−0.233509	−0.116754	−	0.993161i	$-0.537249\pi$
$439$	−4.89255	−0.233509	−0.116754	+	0.993161i	$0.537249\pi$
$440$	0	0
$441$	−0.401709	−0.0191290
$442$	0	0
$443$	7.61994	0.362034	0.181017	−	0.983480i	$-0.442061\pi$
$443$	7.61994	0.362034	0.181017	+	0.983480i	$0.442061\pi$
$444$	0	0
$445$	1.45856	0.0691425
$446$	0	0
$447$	1.94315	0.0919076
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−4.14931	−0.195383
$452$	0	0
$453$	2.31988	0.108997
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.5972	0.682831	0.341415	−	0.939913i	$-0.389094\pi$
$457$	14.5972	0.682831	0.341415	+	0.939913i	$0.389094\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19.4219	−0.904569	−0.452284	−	0.891874i	$-0.649391\pi$
$461$	−19.4219	−0.904569	−0.452284	+	0.891874i	$0.649391\pi$
$462$	0	0
$463$	14.3353	0.666216	0.333108	−	0.942889i	$-0.391903\pi$
$463$	14.3353	0.666216	0.333108	+	0.942889i	$0.391903\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	39.5789	1.83149	0.915747	−	0.401755i	$-0.131600\pi$
$467$	39.5789	1.83149	0.915747	+	0.401755i	$0.131600\pi$
$468$	0	0
$469$	28.3911	1.31098
$470$	0	0
$471$	−20.8111	−0.958925
$472$	0	0
$473$	42.8034	1.96810
$474$	0	0
$475$	2.38776	0.109558
$476$	0	0
$477$	0.861314	0.0394369
$478$	0	0
$479$	−24.1170	−1.10194	−0.550968	−	0.834527i	$-0.685741\pi$
$479$	−24.1170	−1.10194	−0.550968	+	0.834527i	$0.685741\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	12.4123	0.564781
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	−4.23407	−0.191864	−0.0959321	−	0.995388i	$-0.530583\pi$
$487$	−4.23407	−0.191864	−0.0959321	+	0.995388i	$0.530583\pi$
$488$	0	0
$489$	8.41233	0.380419
$490$	0	0
$491$	−3.87090	−0.174691	−0.0873456	−	0.996178i	$-0.527838\pi$
$491$	−3.87090	−0.174691	−0.0873456	+	0.996178i	$0.527838\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.476886	−0.0214344
$496$	0	0
$497$	21.4017	0.959998
$498$	0	0
$499$	−36.9571	−1.65443	−0.827213	−	0.561888i	$-0.810075\pi$
$499$	−36.9571	−1.65443	−0.827213	+	0.561888i	$0.810075\pi$
$500$	0	0
$501$	−17.6647	−0.789202
$502$	0	0
$503$	11.0462	0.492527	0.246263	−	0.969203i	$-0.420797\pi$
$503$	11.0462	0.492527	0.246263	+	0.969203i	$0.420797\pi$
$504$	0	0
$505$	4.69075	0.208736
$506$	0	0
$507$	−22.9002	−1.01704
$508$	0	0
$509$	−10.3921	−0.460623	−0.230311	−	0.973117i	$-0.573974\pi$
$509$	−10.3921	−0.460623	−0.230311	+	0.973117i	$0.573974\pi$
$510$	0	0
$511$	20.0848	0.888498
$512$	0	0
$513$	−12.1849	−0.537977
$514$	0	0
$515$	−14.5693	−0.642002
$516$	0	0
$517$	−40.4479	−1.77890
$518$	0	0
$519$	22.7187	0.997238
$520$	0	0
$521$	8.69075	0.380749	0.190374	−	0.981712i	$-0.439030\pi$
$521$	8.69075	0.380749	0.190374	+	0.981712i	$0.439030\pi$
$522$	0	0
$523$	17.2524	0.754395	0.377197	−	0.926133i	$-0.376888\pi$
$523$	17.2524	0.754395	0.377197	+	0.926133i	$0.376888\pi$
$524$	0	0
$525$	−3.10308	−0.135430
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0.836734	0.0363112
$532$	0	0
$533$	0	0
$534$	0	0
$535$	5.84632	0.252758
$536$	0	0
$537$	−36.8034	−1.58818
$538$	0	0
$539$	18.0279	0.776517
$540$	0	0
$541$	−20.0925	−0.863842	−0.431921	−	0.901911i	$-0.642164\pi$
$541$	−20.0925	−0.863842	−0.431921	+	0.901911i	$0.642164\pi$
$542$	0	0
$543$	42.1127	1.80723
$544$	0	0
$545$	1.04623	0.0448155
$546$	0	0
$547$	−44.1849	−1.88921	−0.944605	−	0.328209i	$-0.893555\pi$
$547$	−44.1849	−1.88921	−0.944605	+	0.328209i	$0.893555\pi$
$548$	0	0
$549$	0.520184	0.0222009
$550$	0	0
$551$	−17.3169	−0.737727
$552$	0	0
$553$	20.5048	0.871952
$554$	0	0
$555$	1.76156	0.0747739
$556$	0	0
$557$	−5.00958	−0.212263	−0.106131	−	0.994352i	$-0.533846\pi$
$557$	−5.00958	−0.212263	−0.106131	+	0.994352i	$0.533846\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.95377	−0.377356	−0.188678	−	0.982039i	$-0.560420\pi$
$563$	−8.95377	−0.377356	−0.188678	+	0.982039i	$0.560420\pi$
$564$	0	0
$565$	8.29862	0.349126
$566$	0	0
$567$	16.3801	0.687898
$568$	0	0
$569$	18.9538	0.794583	0.397292	−	0.917692i	$-0.369950\pi$
$569$	18.9538	0.794583	0.397292	+	0.917692i	$0.369950\pi$
$570$	0	0
$571$	−13.8140	−0.578100	−0.289050	−	0.957314i	$-0.593339\pi$
$571$	−13.8140	−0.578100	−0.289050	+	0.957314i	$0.593339\pi$
$572$	0	0
$573$	16.6185	0.694248
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−24.5972	−1.02400	−0.511998	−	0.858986i	$-0.671095\pi$
$577$	−24.5972	−1.02400	−0.511998	+	0.858986i	$0.671095\pi$
$578$	0	0
$579$	7.40940	0.307924
$580$	0	0
$581$	31.8140	1.31987
$582$	0	0
$583$	−38.6541	−1.60089
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.1695	−0.667388	−0.333694	−	0.942682i	$-0.608295\pi$
$587$	−16.1695	−0.667388	−0.333694	+	0.942682i	$0.608295\pi$
$588$	0	0
$589$	−2.71096	−0.111703
$590$	0	0
$591$	10.8324	0.445584
$592$	0	0
$593$	−36.4113	−1.49523	−0.747616	−	0.664131i	$-0.768802\pi$
$593$	−36.4113	−1.49523	−0.747616	+	0.664131i	$0.768802\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.4123	0.426148
$598$	0	0
$599$	2.53374	0.103526	0.0517629	−	0.998659i	$-0.483516\pi$
$599$	2.53374	0.103526	0.0517629	+	0.998659i	$0.483516\pi$
$600$	0	0
$601$	14.8034	0.603844	0.301922	−	0.953333i	$-0.402372\pi$
$601$	14.8034	0.603844	0.301922	+	0.953333i	$0.402372\pi$
$602$	0	0
$603$	−1.66140	−0.0676576
$604$	0	0
$605$	10.4017	0.422890
$606$	0	0
$607$	35.0462	1.42248	0.711241	−	0.702948i	$-0.248133\pi$
$607$	35.0462	1.42248	0.711241	+	0.702948i	$0.248133\pi$
$608$	0	0
$609$	22.5048	0.911940
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−3.94315	−0.159262	−0.0796311	−	0.996824i	$-0.525374\pi$
$613$	−3.94315	−0.159262	−0.0796311	+	0.996824i	$0.525374\pi$
$614$	0	0
$615$	1.57997	0.0637105
$616$	0	0
$617$	45.6233	1.83672	0.918362	−	0.395742i	$-0.129512\pi$
$617$	45.6233	1.83672	0.918362	+	0.395742i	$0.129512\pi$
$618$	0	0
$619$	−31.4942	−1.26586	−0.632929	−	0.774210i	$-0.718147\pi$
$619$	−31.4942	−1.26586	−0.632929	+	0.774210i	$0.718147\pi$
$620$	0	0
$621$	20.4123	0.819119
$622$	0	0
$623$	−2.56934	−0.102939
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−19.4586	−0.777100
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.36943	−0.0545163	−0.0272581	−	0.999628i	$-0.508678\pi$
$631$	−1.36943	−0.0545163	−0.0272581	+	0.999628i	$0.508678\pi$
$632$	0	0
$633$	48.6329	1.93298
$634$	0	0
$635$	19.3126	0.766396
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.25240	−0.0495440
$640$	0	0
$641$	36.2051	1.43002	0.715008	−	0.699116i	$-0.246423\pi$
$641$	36.2051	1.43002	0.715008	+	0.699116i	$0.246423\pi$
$642$	0	0
$643$	19.6926	0.776602	0.388301	−	0.921533i	$-0.373062\pi$
$643$	19.6926	0.776602	0.388301	+	0.921533i	$0.373062\pi$
$644$	0	0
$645$	−16.2986	−0.641758
$646$	0	0
$647$	5.12329	0.201417	0.100709	−	0.994916i	$-0.467889\pi$
$647$	5.12329	0.201417	0.100709	+	0.994916i	$0.467889\pi$
$648$	0	0
$649$	−37.5510	−1.47401
$650$	0	0
$651$	3.52311	0.138082
$652$	0	0
$653$	23.6801	0.926675	0.463337	−	0.886182i	$-0.346652\pi$
$653$	23.6801	0.926675	0.463337	+	0.886182i	$0.346652\pi$
$654$	0	0
$655$	1.13536	0.0443622
$656$	0	0
$657$	−1.17533	−0.0458540
$658$	0	0
$659$	−37.1589	−1.44751	−0.723753	−	0.690060i	$-0.757584\pi$
$659$	−37.1589	−1.44751	−0.723753	+	0.690060i	$0.757584\pi$
$660$	0	0
$661$	−20.3757	−0.792523	−0.396261	−	0.918138i	$-0.629693\pi$
$661$	−20.3757	−0.792523	−0.396261	+	0.918138i	$0.629693\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.20617	−0.163108
$666$	0	0
$667$	29.0096	1.12326
$668$	0	0
$669$	37.4942	1.44961
$670$	0	0
$671$	−23.3449	−0.901218
$672$	0	0
$673$	−2.44794	−0.0943610	−0.0471805	−	0.998886i	$-0.515024\pi$
$673$	−2.44794	−0.0943610	−0.0471805	+	0.998886i	$0.515024\pi$
$674$	0	0
$675$	−5.10308	−0.196418
$676$	0	0
$677$	−22.7466	−0.874221	−0.437111	−	0.899408i	$-0.643998\pi$
$677$	−22.7466	−0.874221	−0.437111	+	0.899408i	$0.643998\pi$
$678$	0	0
$679$	−10.5693	−0.405614
$680$	0	0
$681$	−24.1849	−0.926768
$682$	0	0
$683$	31.0462	1.18795	0.593975	−	0.804483i	$-0.297558\pi$
$683$	31.0462	1.18795	0.593975	+	0.804483i	$0.297558\pi$
$684$	0	0
$685$	9.04623	0.345639
$686$	0	0
$687$	6.71866	0.256333
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29.9142	1.13799	0.568995	−	0.822341i	$-0.307332\pi$
$691$	29.9142	1.13799	0.568995	+	0.822341i	$0.307332\pi$
$692$	0	0
$693$	0.840061	0.0319113
$694$	0	0
$695$	7.45856	0.282919
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−10.5693	−0.399769
$700$	0	0
$701$	−15.8496	−0.598633	−0.299316	−	0.954154i	$-0.596759\pi$
$701$	−15.8496	−0.598633	−0.299316	+	0.954154i	$0.596759\pi$
$702$	0	0
$703$	2.38776	0.0900559
$704$	0	0
$705$	15.4017	0.580062
$706$	0	0
$707$	−8.26302	−0.310763
$708$	0	0
$709$	−36.9325	−1.38703	−0.693515	−	0.720442i	$-0.743939\pi$
$709$	−36.9325	−1.38703	−0.693515	+	0.720442i	$0.743939\pi$
$710$	0	0
$711$	−1.19991	−0.0450001
$712$	0	0
$713$	4.54144	0.170078
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.31988	0.310712
$718$	0	0
$719$	27.4296	1.02295	0.511476	−	0.859298i	$-0.329099\pi$
$719$	27.4296	1.02295	0.511476	+	0.859298i	$0.329099\pi$
$720$	0	0
$721$	25.6647	0.955805
$722$	0	0
$723$	−47.0529	−1.74992
$724$	0	0
$725$	−7.25240	−0.269347
$726$	0	0
$727$	46.9325	1.74063	0.870315	−	0.492495i	$-0.163915\pi$
$727$	46.9325	1.74063	0.870315	+	0.492495i	$0.163915\pi$
$728$	0	0
$729$	26.0096	0.963318
$730$	0	0
$731$	0	0
$732$	0	0
$733$	11.1031	0.410102	0.205051	−	0.978751i	$-0.434264\pi$
$733$	11.1031	0.410102	0.205051	+	0.978751i	$0.434264\pi$
$734$	0	0
$735$	−6.86464	−0.253206
$736$	0	0
$737$	74.5606	2.74648
$738$	0	0
$739$	−4.26302	−0.156818	−0.0784089	−	0.996921i	$-0.524984\pi$
$739$	−4.26302	−0.156818	−0.0784089	+	0.996921i	$0.524984\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.6199	0.573040	0.286520	−	0.958074i	$-0.407502\pi$
$743$	15.6199	0.573040	0.286520	+	0.958074i	$0.407502\pi$
$744$	0	0
$745$	1.10308	0.0404139
$746$	0	0
$747$	−1.86171	−0.0681164
$748$	0	0
$749$	−10.2986	−0.376304
$750$	0	0
$751$	16.6262	0.606699	0.303349	−	0.952879i	$-0.401895\pi$
$751$	16.6262	0.606699	0.303349	+	0.952879i	$0.401895\pi$
$752$	0	0
$753$	−3.79383	−0.138255
$754$	0	0
$755$	1.31695	0.0479286
$756$	0	0
$757$	25.6281	0.931469	0.465734	−	0.884925i	$-0.345790\pi$
$757$	25.6281	0.931469	0.465734	+	0.884925i	$0.345790\pi$
$758$	0	0
$759$	32.5972	1.18321
$760$	0	0
$761$	1.60788	0.0582855	0.0291427	−	0.999575i	$-0.490722\pi$
$761$	1.60788	0.0582855	0.0291427	+	0.999575i	$0.490722\pi$
$762$	0	0
$763$	−1.84299	−0.0667208
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−25.7572	−0.928828	−0.464414	−	0.885618i	$-0.653735\pi$
$769$	−25.7572	−0.928828	−0.464414	+	0.885618i	$0.653735\pi$
$770$	0	0
$771$	−0.526045	−0.0189450
$772$	0	0
$773$	14.1493	0.508915	0.254458	−	0.967084i	$-0.418103\pi$
$773$	14.1493	0.508915	0.254458	+	0.967084i	$0.418103\pi$
$774$	0	0
$775$	−1.13536	−0.0407833
$776$	0	0
$777$	−3.10308	−0.111323
$778$	0	0
$779$	2.14162	0.0767314
$780$	0	0
$781$	56.2051	2.01118
$782$	0	0
$783$	37.0096	1.32261
$784$	0	0
$785$	−11.8140	−0.421661
$786$	0	0
$787$	−0.210536	−0.00750480	−0.00375240	−	0.999993i	$-0.501194\pi$
$787$	−0.210536	−0.00750480	−0.00375240	+	0.999993i	$0.501194\pi$
$788$	0	0
$789$	19.4017	0.690719
$790$	0	0
$791$	−14.6185	−0.519774
$792$	0	0
$793$	0	0
$794$	0	0
$795$	14.7187	0.522017
$796$	0	0
$797$	−14.0925	−0.499180	−0.249590	−	0.968352i	$-0.580296\pi$
$797$	−14.0925	−0.499180	−0.249590	+	0.968352i	$0.580296\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.150354	0.00531250
$802$	0	0
$803$	52.7466	1.86139
$804$	0	0
$805$	7.04623	0.248347
$806$	0	0
$807$	−29.4373	−1.03624
$808$	0	0
$809$	−14.2986	−0.502713	−0.251356	−	0.967895i	$-0.580877\pi$
$809$	−14.2986	−0.502713	−0.251356	+	0.967895i	$0.580877\pi$
$810$	0	0
$811$	0.0202108	0.000709697 0	0.000354849	−	1.00000i	$-0.499887\pi$
$811$	0.0202108	0.000709697 0	0.000354849		1.00000i	$0.499887\pi$
$812$	0	0
$813$	13.1031	0.459545
$814$	0	0
$815$	4.77551	0.167279
$816$	0	0
$817$	−22.0925	−0.772917
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45.4942	−1.58776	−0.793879	−	0.608076i	$-0.791942\pi$
$821$	−45.4942	−1.58776	−0.793879	+	0.608076i	$0.791942\pi$
$822$	0	0
$823$	30.0804	1.04854	0.524268	−	0.851553i	$-0.324339\pi$
$823$	30.0804	1.04854	0.524268	+	0.851553i	$0.324339\pi$
$824$	0	0
$825$	−8.14931	−0.283723
$826$	0	0
$827$	11.8217	0.411082	0.205541	−	0.978648i	$-0.434105\pi$
$827$	11.8217	0.411082	0.205541	+	0.978648i	$0.434105\pi$
$828$	0	0
$829$	12.6185	0.438259	0.219129	−	0.975696i	$-0.429678\pi$
$829$	12.6185	0.438259	0.219129	+	0.975696i	$0.429678\pi$
$830$	0	0
$831$	22.3911	0.776738
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−10.0279	−0.347030
$836$	0	0
$837$	5.79383	0.200264
$838$	0	0
$839$	−27.4586	−0.947975	−0.473987	−	0.880532i	$-0.657186\pi$
$839$	−27.4586	−0.947975	−0.473987	+	0.880532i	$0.657186\pi$
$840$	0	0
$841$	23.5972	0.813698
$842$	0	0
$843$	−35.3940	−1.21903
$844$	0	0
$845$	−13.0000	−0.447214
$846$	0	0
$847$	−18.3232	−0.629593
$848$	0	0
$849$	−28.5972	−0.981455
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−55.7205	−1.90784	−0.953918	−	0.300069i	$-0.902990\pi$
$853$	−55.7205	−1.90784	−0.953918	+	0.300069i	$0.902990\pi$
$854$	0	0
$855$	0.246139	0.00841776
$856$	0	0
$857$	−4.50479	−0.153881	−0.0769404	−	0.997036i	$-0.524515\pi$
$857$	−4.50479	−0.153881	−0.0769404	+	0.997036i	$0.524515\pi$
$858$	0	0
$859$	−19.3415	−0.659924	−0.329962	−	0.943994i	$-0.607036\pi$
$859$	−19.3415	−0.659924	−0.329962	+	0.943994i	$0.607036\pi$
$860$	0	0
$861$	−2.78321	−0.0948514
$862$	0	0
$863$	−28.1729	−0.959015	−0.479507	−	0.877538i	$-0.659185\pi$
$863$	−28.1729	−0.959015	−0.479507	+	0.877538i	$0.659185\pi$
$864$	0	0
$865$	12.8969	0.438508
$866$	0	0
$867$	−29.9465	−1.01704
$868$	0	0
$869$	53.8496	1.82672
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0.618502	0.0209331
$874$	0	0
$875$	−1.76156	−0.0595515
$876$	0	0
$877$	−24.9171	−0.841392	−0.420696	−	0.907202i	$-0.638214\pi$
$877$	−24.9171	−0.841392	−0.420696	+	0.907202i	$0.638214\pi$
$878$	0	0
$879$	−17.6156	−0.594158
$880$	0	0
$881$	0.784248	0.0264220	0.0132110	−	0.999913i	$-0.495795\pi$
$881$	0.784248	0.0264220	0.0132110	+	0.999913i	$0.495795\pi$
$882$	0	0
$883$	−19.2245	−0.646956	−0.323478	−	0.946236i	$-0.604852\pi$
$883$	−19.2245	−0.646956	−0.323478	+	0.946236i	$0.604852\pi$
$884$	0	0
$885$	14.2986	0.480643
$886$	0	0
$887$	13.4783	0.452558	0.226279	−	0.974063i	$-0.427344\pi$
$887$	13.4783	0.452558	0.226279	+	0.974063i	$0.427344\pi$
$888$	0	0
$889$	−34.0202	−1.14100
$890$	0	0
$891$	43.0173	1.44113
$892$	0	0
$893$	20.8767	0.698612
$894$	0	0
$895$	−20.8925	−0.698361
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.23407	0.274622
$900$	0	0
$901$	0	0
$902$	0	0
$903$	28.7110	0.955441
$904$	0	0
$905$	23.9065	0.794679
$906$	0	0
$907$	−12.3478	−0.410001	−0.205001	−	0.978762i	$-0.565720\pi$
$907$	−12.3478	−0.410001	−0.205001	+	0.978762i	$0.565720\pi$
$908$	0	0
$909$	0.483540	0.0160380
$910$	0	0
$911$	−36.8280	−1.22017	−0.610083	−	0.792338i	$-0.708864\pi$
$911$	−36.8280	−1.22017	−0.610083	+	0.792338i	$0.708864\pi$
$912$	0	0
$913$	83.5500	2.76510
$914$	0	0
$915$	8.88922	0.293869
$916$	0	0
$917$	−2.00000	−0.0660458
$918$	0	0
$919$	8.35111	0.275478	0.137739	−	0.990469i	$-0.456017\pi$
$919$	8.35111	0.275478	0.137739	+	0.990469i	$0.456017\pi$
$920$	0	0
$921$	−7.10308	−0.234055
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−1.50186	−0.0493276
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−9.30488	−0.304955
$932$	0	0
$933$	15.9787	0.523121
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.0202	−1.24207	−0.621033	−	0.783784i	$-0.713287\pi$
$937$	−38.0202	−1.24207	−0.621033	+	0.783784i	$0.713287\pi$
$938$	0	0
$939$	−42.4402	−1.38498
$940$	0	0
$941$	31.2158	1.01760	0.508802	−	0.860883i	$-0.330088\pi$
$941$	31.2158	1.01760	0.508802	+	0.860883i	$0.330088\pi$
$942$	0	0
$943$	−3.58767	−0.116830
$944$	0	0
$945$	8.98937	0.292424
$946$	0	0
$947$	2.14162	0.0695932	0.0347966	−	0.999394i	$-0.488922\pi$
$947$	2.14162	0.0695932	0.0347966	+	0.999394i	$0.488922\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	30.7543	0.997275
$952$	0	0
$953$	−15.5154	−0.502594	−0.251297	−	0.967910i	$-0.580857\pi$
$953$	−15.5154	−0.502594	−0.251297	+	0.967910i	$0.580857\pi$
$954$	0	0
$955$	9.43398	0.305277
$956$	0	0
$957$	59.1020	1.91050
$958$	0	0
$959$	−15.9354	−0.514582
$960$	0	0
$961$	−29.7110	−0.958418
$962$	0	0
$963$	0.602660	0.0194205
$964$	0	0
$965$	4.20617	0.135401
$966$	0	0
$967$	35.4498	1.13999	0.569995	−	0.821648i	$-0.306945\pi$
$967$	35.4498	1.13999	0.569995	+	0.821648i	$0.306945\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−13.5510	−0.434873	−0.217436	−	0.976074i	$-0.569769\pi$
$971$	−13.5510	−0.434873	−0.217436	+	0.976074i	$0.569769\pi$
$972$	0	0
$973$	−13.1387	−0.421207
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.2062	0.518481	0.259241	−	0.965813i	$-0.416528\pi$
$977$	16.2062	0.518481	0.259241	+	0.965813i	$0.416528\pi$
$978$	0	0
$979$	−6.74760	−0.215654
$980$	0	0
$981$	0.107849	0.00344336
$982$	0	0
$983$	6.17389	0.196917	0.0984583	−	0.995141i	$-0.468609\pi$
$983$	6.17389	0.196917	0.0984583	+	0.995141i	$0.468609\pi$
$984$	0	0
$985$	6.14931	0.195933
$986$	0	0
$987$	−27.1310	−0.863589
$988$	0	0
$989$	37.0096	1.17684
$990$	0	0
$991$	−13.9109	−0.441893	−0.220947	−	0.975286i	$-0.570915\pi$
$991$	−13.9109	−0.441893	−0.220947	+	0.975286i	$0.570915\pi$
$992$	0	0
$993$	1.88629	0.0598596
$994$	0	0
$995$	5.91087	0.187387
$996$	0	0
$997$	−8.87671	−0.281128	−0.140564	−	0.990072i	$-0.544892\pi$
$997$	−8.87671	−0.281128	−0.140564	+	0.990072i	$0.544892\pi$
$998$	0	0
$999$	−5.10308	−0.161454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.s.1.3		3
4.3	odd	2		1480.2.a.f.1.1	✓	3
20.19	odd	2		7400.2.a.m.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.f.1.1	✓	3	4.3	odd	2
2960.2.a.s.1.3		3	1.1	even	1	trivial
7400.2.a.m.1.3		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q+1.76156 q^{3} +1.00000 q^{5} -1.76156 q^{7} +0.103084 q^{9} +O(q^{10})\)	\(q+1.76156 q^{3} +1.00000 q^{5} -1.76156 q^{7} +0.103084 q^{9} -4.62620 q^{11} +1.76156 q^{15} +2.38776 q^{19} -3.10308 q^{21} -4.00000 q^{23} +1.00000 q^{25} -5.10308 q^{27} -7.25240 q^{29} -1.13536 q^{31} -8.14931 q^{33} -1.76156 q^{35} +1.00000 q^{37} +0.896916 q^{41} -9.25240 q^{43} +0.103084 q^{45} +8.74324 q^{47} -3.89692 q^{49} +8.35548 q^{53} -4.62620 q^{55} +4.20617 q^{57} +8.11704 q^{59} +5.04623 q^{61} -0.181588 q^{63} -16.1170 q^{67} -7.04623 q^{69} -12.1493 q^{71} -11.4017 q^{73} +1.76156 q^{75} +8.14931 q^{77} -11.6402 q^{79} -9.29862 q^{81} -18.0602 q^{83} -12.7755 q^{87} +1.45856 q^{89} -2.00000 q^{93} +2.38776 q^{95} +6.00000 q^{97} -0.476886 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 + 3 * q^5 + q^7 + 4 * q^9	\( 3 q - q^{3} + 3 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - q^{15} - 8 q^{19} - 13 q^{21} - 12 q^{23} + 3 q^{25} - 19 q^{27} - 4 q^{29} - 6 q^{31} - 3 q^{33} + q^{35} + 3 q^{37} - q^{41} - 10 q^{43} + 4 q^{45} - 3 q^{47} - 8 q^{49} + 11 q^{53} - 5 q^{55} + 20 q^{57} + 4 q^{59} - 10 q^{61} + 22 q^{63} - 28 q^{67} + 4 q^{69} - 15 q^{71} + 5 q^{73} - q^{75} + 3 q^{77} - 2 q^{79} + 15 q^{81} - 5 q^{83} - 8 q^{87} - 6 q^{89} - 6 q^{93} - 8 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - q^15 - 8 * q^19 - 13 * q^21 - 12 * q^23 + 3 * q^25 - 19 * q^27 - 4 * q^29 - 6 * q^31 - 3 * q^33 + q^35 + 3 * q^37 - q^41 - 10 * q^43 + 4 * q^45 - 3 * q^47 - 8 * q^49 + 11 * q^53 - 5 * q^55 + 20 * q^57 + 4 * q^59 - 10 * q^61 + 22 * q^63 - 28 * q^67 + 4 * q^69 - 15 * q^71 + 5 * q^73 - q^75 + 3 * q^77 - 2 * q^79 + 15 * q^81 - 5 * q^83 - 8 * q^87 - 6 * q^89 - 6 * q^93 - 8 * q^95 + 18 * q^97 - 14 * q^99

Introduction

L-functions

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Database

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