LMFDB - Embedded newform 880.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,4,Mod(1,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("880.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	880.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:	$51.9216808051$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{3} +5.00000 q^{5} -8.00000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q+4.00000 q^{3} +5.00000 q^{5} -8.00000 q^{7} -11.0000 q^{9} -11.0000 q^{11} -58.0000 q^{13} +20.0000 q^{15} +114.000 q^{17} +4.00000 q^{19} -32.0000 q^{21} +152.000 q^{23} +25.0000 q^{25} -152.000 q^{27} -138.000 q^{29} -208.000 q^{31} -44.0000 q^{33} -40.0000 q^{35} -226.000 q^{37} -232.000 q^{39} -294.000 q^{41} -276.000 q^{43} -55.0000 q^{45} +240.000 q^{47} -279.000 q^{49} +456.000 q^{51} -370.000 q^{53} -55.0000 q^{55} +16.0000 q^{57} +716.000 q^{59} -650.000 q^{61} +88.0000 q^{63} -290.000 q^{65} -124.000 q^{67} +608.000 q^{69} -232.000 q^{71} -454.000 q^{73} +100.000 q^{75} +88.0000 q^{77} +144.000 q^{79} -311.000 q^{81} +692.000 q^{83} +570.000 q^{85} -552.000 q^{87} -1206.00 q^{89} +464.000 q^{91} -832.000 q^{93} +20.0000 q^{95} -1438.00 q^{97} +121.000 q^{99} +O(q^{100})$

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
$3$	4.00000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−58.0000	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$13$	−58.0000	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	114.000	1.62642	0.813208	−	0.581974i	$-0.197719\pi$
$17$	114.000	1.62642	0.813208	+	0.581974i	$0.197719\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	−32.0000	−0.332522
$22$	0	0
$23$	152.000	1.37801	0.689004	−	0.724757i	$-0.258048\pi$
$23$	152.000	1.37801	0.689004	+	0.724757i	$0.258048\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−152.000	−1.08342
$28$	0	0
$29$	−138.000	−0.883654	−0.441827	−	0.897100i	$-0.645669\pi$
$29$	−138.000	−0.883654	−0.441827	+	0.897100i	$0.645669\pi$
$30$	0	0
$31$	−208.000	−1.20509	−0.602547	−	0.798084i	$-0.705847\pi$
$31$	−208.000	−1.20509	−0.602547	+	0.798084i	$0.705847\pi$
$32$	0	0
$33$	−44.0000	−0.232104
$34$	0	0
$35$	−40.0000	−0.193178
$36$	0	0
$37$	−226.000	−1.00417	−0.502083	−	0.864819i	$-0.667433\pi$
$37$	−226.000	−1.00417	−0.502083	+	0.864819i	$0.667433\pi$
$38$	0	0
$39$	−232.000	−0.952557
$40$	0	0
$41$	−294.000	−1.11988	−0.559940	−	0.828533i	$-0.689176\pi$
$41$	−294.000	−1.11988	−0.559940	+	0.828533i	$0.689176\pi$
$42$	0	0
$43$	−276.000	−0.978828	−0.489414	−	0.872052i	$-0.662789\pi$
$43$	−276.000	−0.978828	−0.489414	+	0.872052i	$0.662789\pi$
$44$	0	0
$45$	−55.0000	−0.182198
$46$	0	0
$47$	240.000	0.744843	0.372421	−	0.928064i	$-0.378528\pi$
$47$	240.000	0.744843	0.372421	+	0.928064i	$0.378528\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	456.000	1.25202
$52$	0	0
$53$	−370.000	−0.958932	−0.479466	−	0.877560i	$-0.659170\pi$
$53$	−370.000	−0.958932	−0.479466	+	0.877560i	$0.659170\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	0	0
$57$	16.0000	0.0371799
$58$	0	0
$59$	716.000	1.57992	0.789960	−	0.613159i	$-0.210101\pi$
$59$	716.000	1.57992	0.789960	+	0.613159i	$0.210101\pi$
$60$	0	0
$61$	−650.000	−1.36433	−0.682164	−	0.731199i	$-0.738961\pi$
$61$	−650.000	−1.36433	−0.682164	+	0.731199i	$0.738961\pi$
$62$	0	0
$63$	88.0000	0.175983
$64$	0	0
$65$	−290.000	−0.553386
$66$	0	0
$67$	−124.000	−0.226105	−0.113052	−	0.993589i	$-0.536063\pi$
$67$	−124.000	−0.226105	−0.113052	+	0.993589i	$0.536063\pi$
$68$	0	0
$69$	608.000	1.06079
$70$	0	0
$71$	−232.000	−0.387793	−0.193897	−	0.981022i	$-0.562113\pi$
$71$	−232.000	−0.387793	−0.193897	+	0.981022i	$0.562113\pi$
$72$	0	0
$73$	−454.000	−0.727900	−0.363950	−	0.931419i	$-0.618572\pi$
$73$	−454.000	−0.727900	−0.363950	+	0.931419i	$0.618572\pi$
$74$	0	0
$75$	100.000	0.153960
$76$	0	0
$77$	88.0000	0.130241
$78$	0	0
$79$	144.000	0.205079	0.102540	−	0.994729i	$-0.467303\pi$
$79$	144.000	0.205079	0.102540	+	0.994729i	$0.467303\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	692.000	0.915143	0.457571	−	0.889173i	$-0.348719\pi$
$83$	692.000	0.915143	0.457571	+	0.889173i	$0.348719\pi$
$84$	0	0
$85$	570.000	0.727355
$86$	0	0
$87$	−552.000	−0.680237
$88$	0	0
$89$	−1206.00	−1.43636	−0.718178	−	0.695859i	$-0.755024\pi$
$89$	−1206.00	−1.43636	−0.718178	+	0.695859i	$0.755024\pi$
$90$	0	0
$91$	464.000	0.534510
$92$	0	0
$93$	−832.000	−0.927682
$94$	0	0
$95$	20.0000	0.0215995
$96$	0	0
$97$	−1438.00	−1.50522	−0.752612	−	0.658464i	$-0.771206\pi$
$97$	−1438.00	−1.50522	−0.752612	+	0.658464i	$0.771206\pi$
$98$	0	0
$99$	121.000	0.122838
$100$	0	0
$101$	590.000	0.581259	0.290630	−	0.956836i	$-0.406135\pi$
$101$	590.000	0.581259	0.290630	+	0.956836i	$0.406135\pi$
$102$	0	0
$103$	104.000	0.0994896	0.0497448	−	0.998762i	$-0.484159\pi$
$103$	104.000	0.0994896	0.0497448	+	0.998762i	$0.484159\pi$
$104$	0	0
$105$	−160.000	−0.148709
$106$	0	0
$107$	−1364.00	−1.23236	−0.616182	−	0.787604i	$-0.711321\pi$
$107$	−1364.00	−1.23236	−0.616182	+	0.787604i	$0.711321\pi$
$108$	0	0
$109$	−826.000	−0.725839	−0.362920	−	0.931820i	$-0.618220\pi$
$109$	−826.000	−0.725839	−0.362920	+	0.931820i	$0.618220\pi$
$110$	0	0
$111$	−904.000	−0.773008
$112$	0	0
$113$	2322.00	1.93306	0.966528	−	0.256560i	$-0.0825892\pi$
$113$	2322.00	1.93306	0.966528	+	0.256560i	$0.0825892\pi$
$114$	0	0
$115$	760.000	0.616264
$116$	0	0
$117$	638.000	0.504129
$118$	0	0
$119$	−912.000	−0.702545
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1176.00	−0.862084
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1296.00	−0.905523	−0.452761	−	0.891632i	$-0.649561\pi$
$127$	−1296.00	−0.905523	−0.452761	+	0.891632i	$0.649561\pi$
$128$	0	0
$129$	−1104.00	−0.753502
$130$	0	0
$131$	2100.00	1.40059	0.700297	−	0.713851i	$-0.253051\pi$
$131$	2100.00	1.40059	0.700297	+	0.713851i	$0.253051\pi$
$132$	0	0
$133$	−32.0000	−0.0208628
$134$	0	0
$135$	−760.000	−0.484521
$136$	0	0
$137$	2394.00	1.49294	0.746472	−	0.665417i	$-0.231746\pi$
$137$	2394.00	1.49294	0.746472	+	0.665417i	$0.231746\pi$
$138$	0	0
$139$	−2340.00	−1.42789	−0.713943	−	0.700204i	$-0.753093\pi$
$139$	−2340.00	−1.42789	−0.713943	+	0.700204i	$0.753093\pi$
$140$	0	0
$141$	960.000	0.573380
$142$	0	0
$143$	638.000	0.373093
$144$	0	0
$145$	−690.000	−0.395182
$146$	0	0
$147$	−1116.00	−0.626164
$148$	0	0
$149$	−1826.00	−1.00397	−0.501986	−	0.864876i	$-0.667397\pi$
$149$	−1826.00	−1.00397	−0.501986	+	0.864876i	$0.667397\pi$
$150$	0	0
$151$	3128.00	1.68578	0.842891	−	0.538085i	$-0.180852\pi$
$151$	3128.00	1.68578	0.842891	+	0.538085i	$0.180852\pi$
$152$	0	0
$153$	−1254.00	−0.662614
$154$	0	0
$155$	−1040.00	−0.538934
$156$	0	0
$157$	−730.000	−0.371085	−0.185542	−	0.982636i	$-0.559404\pi$
$157$	−730.000	−0.371085	−0.185542	+	0.982636i	$0.559404\pi$
$158$	0	0
$159$	−1480.00	−0.738186
$160$	0	0
$161$	−1216.00	−0.595244
$162$	0	0
$163$	2500.00	1.20132	0.600660	−	0.799505i	$-0.294905\pi$
$163$	2500.00	1.20132	0.600660	+	0.799505i	$0.294905\pi$
$164$	0	0
$165$	−220.000	−0.103800
$166$	0	0
$167$	−3560.00	−1.64959	−0.824794	−	0.565434i	$-0.808709\pi$
$167$	−3560.00	−1.64959	−0.824794	+	0.565434i	$0.808709\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	−44.0000	−0.0196770
$172$	0	0
$173$	2406.00	1.05737	0.528684	−	0.848818i	$-0.322686\pi$
$173$	2406.00	1.05737	0.528684	+	0.848818i	$0.322686\pi$
$174$	0	0
$175$	−200.000	−0.0863919
$176$	0	0
$177$	2864.00	1.21622
$178$	0	0
$179$	−636.000	−0.265569	−0.132785	−	0.991145i	$-0.542392\pi$
$179$	−636.000	−0.265569	−0.132785	+	0.991145i	$0.542392\pi$
$180$	0	0
$181$	−1842.00	−0.756435	−0.378218	−	0.925717i	$-0.623463\pi$
$181$	−1842.00	−0.756435	−0.378218	+	0.925717i	$0.623463\pi$
$182$	0	0
$183$	−2600.00	−1.05026
$184$	0	0
$185$	−1130.00	−0.449077
$186$	0	0
$187$	−1254.00	−0.490383
$188$	0	0
$189$	1216.00	0.467995
$190$	0	0
$191$	−240.000	−0.0909204	−0.0454602	−	0.998966i	$-0.514475\pi$
$191$	−240.000	−0.0909204	−0.0454602	+	0.998966i	$0.514475\pi$
$192$	0	0
$193$	3778.00	1.40905	0.704524	−	0.709680i	$-0.251160\pi$
$193$	3778.00	1.40905	0.704524	+	0.709680i	$0.251160\pi$
$194$	0	0
$195$	−1160.00	−0.425997
$196$	0	0
$197$	−3378.00	−1.22169	−0.610844	−	0.791751i	$-0.709170\pi$
$197$	−3378.00	−1.22169	−0.610844	+	0.791751i	$0.709170\pi$
$198$	0	0
$199$	1784.00	0.635500	0.317750	−	0.948175i	$-0.397073\pi$
$199$	1784.00	0.635500	0.317750	+	0.948175i	$0.397073\pi$
$200$	0	0
$201$	−496.000	−0.174055
$202$	0	0
$203$	1104.00	0.381703
$204$	0	0
$205$	−1470.00	−0.500826
$206$	0	0
$207$	−1672.00	−0.561411
$208$	0	0
$209$	−44.0000	−0.0145624
$210$	0	0
$211$	−3804.00	−1.24113	−0.620564	−	0.784156i	$-0.713096\pi$
$211$	−3804.00	−1.24113	−0.620564	+	0.784156i	$0.713096\pi$
$212$	0	0
$213$	−928.000	−0.298524
$214$	0	0
$215$	−1380.00	−0.437745
$216$	0	0
$217$	1664.00	0.520552
$218$	0	0
$219$	−1816.00	−0.560337
$220$	0	0
$221$	−6612.00	−2.01254
$222$	0	0
$223$	−3520.00	−1.05703	−0.528513	−	0.848925i	$-0.677250\pi$
$223$	−3520.00	−1.05703	−0.528513	+	0.848925i	$0.677250\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	4996.00	1.46078	0.730388	−	0.683032i	$-0.239339\pi$
$227$	4996.00	1.46078	0.730388	+	0.683032i	$0.239339\pi$
$228$	0	0
$229$	286.000	0.0825302	0.0412651	−	0.999148i	$-0.486861\pi$
$229$	286.000	0.0825302	0.0412651	+	0.999148i	$0.486861\pi$
$230$	0	0
$231$	352.000	0.100259
$232$	0	0
$233$	−678.000	−0.190632	−0.0953160	−	0.995447i	$-0.530386\pi$
$233$	−678.000	−0.190632	−0.0953160	+	0.995447i	$0.530386\pi$
$234$	0	0
$235$	1200.00	0.333104
$236$	0	0
$237$	576.000	0.157870
$238$	0	0
$239$	−4016.00	−1.08692	−0.543459	−	0.839436i	$-0.682886\pi$
$239$	−4016.00	−1.08692	−0.543459	+	0.839436i	$0.682886\pi$
$240$	0	0
$241$	2866.00	0.766039	0.383019	−	0.923740i	$-0.374884\pi$
$241$	2866.00	0.766039	0.383019	+	0.923740i	$0.374884\pi$
$242$	0	0
$243$	2860.00	0.755017
$244$	0	0
$245$	−1395.00	−0.363768
$246$	0	0
$247$	−232.000	−0.0597644
$248$	0	0
$249$	2768.00	0.704477
$250$	0	0
$251$	4428.00	1.11352	0.556759	−	0.830674i	$-0.312045\pi$
$251$	4428.00	1.11352	0.556759	+	0.830674i	$0.312045\pi$
$252$	0	0
$253$	−1672.00	−0.415485
$254$	0	0
$255$	2280.00	0.559918
$256$	0	0
$257$	7042.00	1.70921	0.854607	−	0.519276i	$-0.173798\pi$
$257$	7042.00	1.70921	0.854607	+	0.519276i	$0.173798\pi$
$258$	0	0
$259$	1808.00	0.433759
$260$	0	0
$261$	1518.00	0.360007
$262$	0	0
$263$	−1288.00	−0.301983	−0.150991	−	0.988535i	$-0.548247\pi$
$263$	−1288.00	−0.301983	−0.150991	+	0.988535i	$0.548247\pi$
$264$	0	0
$265$	−1850.00	−0.428848
$266$	0	0
$267$	−4824.00	−1.10571
$268$	0	0
$269$	7894.00	1.78924	0.894620	−	0.446827i	$-0.147446\pi$
$269$	7894.00	1.78924	0.894620	+	0.446827i	$0.147446\pi$
$270$	0	0
$271$	5104.00	1.14408	0.572040	−	0.820225i	$-0.306152\pi$
$271$	5104.00	1.14408	0.572040	+	0.820225i	$0.306152\pi$
$272$	0	0
$273$	1856.00	0.411466
$274$	0	0
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−4418.00	−0.958310	−0.479155	−	0.877730i	$-0.659057\pi$
$277$	−4418.00	−0.958310	−0.479155	+	0.877730i	$0.659057\pi$
$278$	0	0
$279$	2288.00	0.490964
$280$	0	0
$281$	−2358.00	−0.500592	−0.250296	−	0.968169i	$-0.580528\pi$
$281$	−2358.00	−0.500592	−0.250296	+	0.968169i	$0.580528\pi$
$282$	0	0
$283$	−644.000	−0.135271	−0.0676357	−	0.997710i	$-0.521546\pi$
$283$	−644.000	−0.135271	−0.0676357	+	0.997710i	$0.521546\pi$
$284$	0	0
$285$	80.0000	0.0166273
$286$	0	0
$287$	2352.00	0.483743
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	−5752.00	−1.15872
$292$	0	0
$293$	1678.00	0.334573	0.167286	−	0.985908i	$-0.446500\pi$
$293$	1678.00	0.334573	0.167286	+	0.985908i	$0.446500\pi$
$294$	0	0
$295$	3580.00	0.706562
$296$	0	0
$297$	1672.00	0.326664
$298$	0	0
$299$	−8816.00	−1.70516
$300$	0	0
$301$	2208.00	0.422814
$302$	0	0
$303$	2360.00	0.447454
$304$	0	0
$305$	−3250.00	−0.610146
$306$	0	0
$307$	−5676.00	−1.05520	−0.527600	−	0.849493i	$-0.676908\pi$
$307$	−5676.00	−1.05520	−0.527600	+	0.849493i	$0.676908\pi$
$308$	0	0
$309$	416.000	0.0765871
$310$	0	0
$311$	−4312.00	−0.786209	−0.393105	−	0.919494i	$-0.628599\pi$
$311$	−4312.00	−0.786209	−0.393105	+	0.919494i	$0.628599\pi$
$312$	0	0
$313$	−7190.00	−1.29841	−0.649206	−	0.760613i	$-0.724899\pi$
$313$	−7190.00	−1.29841	−0.649206	+	0.760613i	$0.724899\pi$
$314$	0	0
$315$	440.000	0.0787022
$316$	0	0
$317$	−4474.00	−0.792697	−0.396348	−	0.918100i	$-0.629723\pi$
$317$	−4474.00	−0.792697	−0.396348	+	0.918100i	$0.629723\pi$
$318$	0	0
$319$	1518.00	0.266432
$320$	0	0
$321$	−5456.00	−0.948674
$322$	0	0
$323$	456.000	0.0785527
$324$	0	0
$325$	−1450.00	−0.247482
$326$	0	0
$327$	−3304.00	−0.558751
$328$	0	0
$329$	−1920.00	−0.321742
$330$	0	0
$331$	7644.00	1.26934	0.634671	−	0.772782i	$-0.281136\pi$
$331$	7644.00	1.26934	0.634671	+	0.772782i	$0.281136\pi$
$332$	0	0
$333$	2486.00	0.409105
$334$	0	0
$335$	−620.000	−0.101117
$336$	0	0
$337$	−2926.00	−0.472966	−0.236483	−	0.971636i	$-0.575995\pi$
$337$	−2926.00	−0.472966	−0.236483	+	0.971636i	$0.575995\pi$
$338$	0	0
$339$	9288.00	1.48807
$340$	0	0
$341$	2288.00	0.363349
$342$	0	0
$343$	4976.00	0.783320
$344$	0	0
$345$	3040.00	0.474400
$346$	0	0
$347$	2140.00	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$347$	2140.00	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$348$	0	0
$349$	−8522.00	−1.30708	−0.653542	−	0.756890i	$-0.726718\pi$
$349$	−8522.00	−1.30708	−0.653542	+	0.756890i	$0.726718\pi$
$350$	0	0
$351$	8816.00	1.34064
$352$	0	0
$353$	12834.0	1.93508	0.967542	−	0.252709i	$-0.0813215\pi$
$353$	12834.0	1.93508	0.967542	+	0.252709i	$0.0813215\pi$
$354$	0	0
$355$	−1160.00	−0.173427
$356$	0	0
$357$	−3648.00	−0.540820
$358$	0	0
$359$	4264.00	0.626867	0.313434	−	0.949610i	$-0.398521\pi$
$359$	4264.00	0.626867	0.313434	+	0.949610i	$0.398521\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	484.000	0.0699819
$364$	0	0
$365$	−2270.00	−0.325527
$366$	0	0
$367$	−9840.00	−1.39957	−0.699787	−	0.714351i	$-0.746722\pi$
$367$	−9840.00	−1.39957	−0.699787	+	0.714351i	$0.746722\pi$
$368$	0	0
$369$	3234.00	0.456247
$370$	0	0
$371$	2960.00	0.414220
$372$	0	0
$373$	1502.00	0.208500	0.104250	−	0.994551i	$-0.466756\pi$
$373$	1502.00	0.208500	0.104250	+	0.994551i	$0.466756\pi$
$374$	0	0
$375$	500.000	0.0688530
$376$	0	0
$377$	8004.00	1.09344
$378$	0	0
$379$	10700.0	1.45019	0.725095	−	0.688649i	$-0.241796\pi$
$379$	10700.0	1.45019	0.725095	+	0.688649i	$0.241796\pi$
$380$	0	0
$381$	−5184.00	−0.697072
$382$	0	0
$383$	−4000.00	−0.533657	−0.266828	−	0.963744i	$-0.585976\pi$
$383$	−4000.00	−0.533657	−0.266828	+	0.963744i	$0.585976\pi$
$384$	0	0
$385$	440.000	0.0582454
$386$	0	0
$387$	3036.00	0.398782
$388$	0	0
$389$	286.000	0.0372771	0.0186385	−	0.999826i	$-0.494067\pi$
$389$	286.000	0.0372771	0.0186385	+	0.999826i	$0.494067\pi$
$390$	0	0
$391$	17328.0	2.24121
$392$	0	0
$393$	8400.00	1.07818
$394$	0	0
$395$	720.000	0.0917143
$396$	0	0
$397$	2230.00	0.281916	0.140958	−	0.990016i	$-0.454982\pi$
$397$	2230.00	0.281916	0.140958	+	0.990016i	$0.454982\pi$
$398$	0	0
$399$	−128.000	−0.0160602
$400$	0	0
$401$	3186.00	0.396761	0.198381	−	0.980125i	$-0.436432\pi$
$401$	3186.00	0.396761	0.198381	+	0.980125i	$0.436432\pi$
$402$	0	0
$403$	12064.0	1.49119
$404$	0	0
$405$	−1555.00	−0.190787
$406$	0	0
$407$	2486.00	0.302768
$408$	0	0
$409$	5290.00	0.639544	0.319772	−	0.947494i	$-0.396394\pi$
$409$	5290.00	0.639544	0.319772	+	0.947494i	$0.396394\pi$
$410$	0	0
$411$	9576.00	1.14927
$412$	0	0
$413$	−5728.00	−0.682461
$414$	0	0
$415$	3460.00	0.409264
$416$	0	0
$417$	−9360.00	−1.09919
$418$	0	0
$419$	−3372.00	−0.393157	−0.196579	−	0.980488i	$-0.562983\pi$
$419$	−3372.00	−0.393157	−0.196579	+	0.980488i	$0.562983\pi$
$420$	0	0
$421$	−1410.00	−0.163228	−0.0816142	−	0.996664i	$-0.526008\pi$
$421$	−1410.00	−0.163228	−0.0816142	+	0.996664i	$0.526008\pi$
$422$	0	0
$423$	−2640.00	−0.303454
$424$	0	0
$425$	2850.00	0.325283
$426$	0	0
$427$	5200.00	0.589334
$428$	0	0
$429$	2552.00	0.287207
$430$	0	0
$431$	7728.00	0.863677	0.431838	−	0.901951i	$-0.357865\pi$
$431$	7728.00	0.863677	0.431838	+	0.901951i	$0.357865\pi$
$432$	0	0
$433$	466.000	0.0517195	0.0258597	−	0.999666i	$-0.491768\pi$
$433$	466.000	0.0517195	0.0258597	+	0.999666i	$0.491768\pi$
$434$	0	0
$435$	−2760.00	−0.304211
$436$	0	0
$437$	608.000	0.0665551
$438$	0	0
$439$	−8296.00	−0.901928	−0.450964	−	0.892542i	$-0.648920\pi$
$439$	−8296.00	−0.901928	−0.450964	+	0.892542i	$0.648920\pi$
$440$	0	0
$441$	3069.00	0.331390
$442$	0	0
$443$	5692.00	0.610463	0.305231	−	0.952278i	$-0.401266\pi$
$443$	5692.00	0.610463	0.305231	+	0.952278i	$0.401266\pi$
$444$	0	0
$445$	−6030.00	−0.642358
$446$	0	0
$447$	−7304.00	−0.772858
$448$	0	0
$449$	5698.00	0.598898	0.299449	−	0.954112i	$-0.403197\pi$
$449$	5698.00	0.598898	0.299449	+	0.954112i	$0.403197\pi$
$450$	0	0
$451$	3234.00	0.337657
$452$	0	0
$453$	12512.0	1.29772
$454$	0	0
$455$	2320.00	0.239040
$456$	0	0
$457$	−5990.00	−0.613130	−0.306565	−	0.951850i	$-0.599180\pi$
$457$	−5990.00	−0.613130	−0.306565	+	0.951850i	$0.599180\pi$
$458$	0	0
$459$	−17328.0	−1.76210
$460$	0	0
$461$	−8186.00	−0.827028	−0.413514	−	0.910498i	$-0.635699\pi$
$461$	−8186.00	−0.827028	−0.413514	+	0.910498i	$0.635699\pi$
$462$	0	0
$463$	8272.00	0.830308	0.415154	−	0.909751i	$-0.363728\pi$
$463$	8272.00	0.830308	0.415154	+	0.909751i	$0.363728\pi$
$464$	0	0
$465$	−4160.00	−0.414872
$466$	0	0
$467$	−44.0000	−0.00435991	−0.00217995	−	0.999998i	$-0.500694\pi$
$467$	−44.0000	−0.00435991	−0.00217995	+	0.999998i	$0.500694\pi$
$468$	0	0
$469$	992.000	0.0976680
$470$	0	0
$471$	−2920.00	−0.285661
$472$	0	0
$473$	3036.00	0.295128
$474$	0	0
$475$	100.000	0.00965961
$476$	0	0
$477$	4070.00	0.390676
$478$	0	0
$479$	12864.0	1.22708	0.613540	−	0.789664i	$-0.289745\pi$
$479$	12864.0	1.22708	0.613540	+	0.789664i	$0.289745\pi$
$480$	0	0
$481$	13108.0	1.24256
$482$	0	0
$483$	−4864.00	−0.458219
$484$	0	0
$485$	−7190.00	−0.673157
$486$	0	0
$487$	5672.00	0.527768	0.263884	−	0.964554i	$-0.414996\pi$
$487$	5672.00	0.527768	0.263884	+	0.964554i	$0.414996\pi$
$488$	0	0
$489$	10000.0	0.924776
$490$	0	0
$491$	−11268.0	−1.03568	−0.517839	−	0.855478i	$-0.673263\pi$
$491$	−11268.0	−1.03568	−0.517839	+	0.855478i	$0.673263\pi$
$492$	0	0
$493$	−15732.0	−1.43719
$494$	0	0
$495$	605.000	0.0549348
$496$	0	0
$497$	1856.00	0.167511
$498$	0	0
$499$	13348.0	1.19747	0.598736	−	0.800946i	$-0.295670\pi$
$499$	13348.0	1.19747	0.598736	+	0.800946i	$0.295670\pi$
$500$	0	0
$501$	−14240.0	−1.26985
$502$	0	0
$503$	14504.0	1.28569	0.642844	−	0.765997i	$-0.277754\pi$
$503$	14504.0	1.28569	0.642844	+	0.765997i	$0.277754\pi$
$504$	0	0
$505$	2950.00	0.259947
$506$	0	0
$507$	4668.00	0.408902
$508$	0	0
$509$	4134.00	0.359993	0.179996	−	0.983667i	$-0.442391\pi$
$509$	4134.00	0.359993	0.179996	+	0.983667i	$0.442391\pi$
$510$	0	0
$511$	3632.00	0.314423
$512$	0	0
$513$	−608.000	−0.0523272
$514$	0	0
$515$	520.000	0.0444931
$516$	0	0
$517$	−2640.00	−0.224578
$518$	0	0
$519$	9624.00	0.813963
$520$	0	0
$521$	16410.0	1.37991	0.689957	−	0.723850i	$-0.257629\pi$
$521$	16410.0	1.37991	0.689957	+	0.723850i	$0.257629\pi$
$522$	0	0
$523$	4748.00	0.396970	0.198485	−	0.980104i	$-0.436398\pi$
$523$	4748.00	0.396970	0.198485	+	0.980104i	$0.436398\pi$
$524$	0	0
$525$	−800.000	−0.0665045
$526$	0	0
$527$	−23712.0	−1.95998
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	−7876.00	−0.643671
$532$	0	0
$533$	17052.0	1.38575
$534$	0	0
$535$	−6820.00	−0.551130
$536$	0	0
$537$	−2544.00	−0.204435
$538$	0	0
$539$	3069.00	0.245253
$540$	0	0
$541$	−5930.00	−0.471258	−0.235629	−	0.971843i	$-0.575715\pi$
$541$	−5930.00	−0.471258	−0.235629	+	0.971843i	$0.575715\pi$
$542$	0	0
$543$	−7368.00	−0.582304
$544$	0	0
$545$	−4130.00	−0.324605
$546$	0	0
$547$	−15836.0	−1.23784	−0.618920	−	0.785454i	$-0.712429\pi$
$547$	−15836.0	−1.23784	−0.618920	+	0.785454i	$0.712429\pi$
$548$	0	0
$549$	7150.00	0.555837
$550$	0	0
$551$	−552.000	−0.0426787
$552$	0	0
$553$	−1152.00	−0.0885859
$554$	0	0
$555$	−4520.00	−0.345700
$556$	0	0
$557$	−10554.0	−0.802850	−0.401425	−	0.915892i	$-0.631485\pi$
$557$	−10554.0	−0.802850	−0.401425	+	0.915892i	$0.631485\pi$
$558$	0	0
$559$	16008.0	1.21121
$560$	0	0
$561$	−5016.00	−0.377497
$562$	0	0
$563$	−24332.0	−1.82144	−0.910721	−	0.413023i	$-0.864473\pi$
$563$	−24332.0	−1.82144	−0.910721	+	0.413023i	$0.864473\pi$
$564$	0	0
$565$	11610.0	0.864489
$566$	0	0
$567$	2488.00	0.184279
$568$	0	0
$569$	18570.0	1.36818	0.684090	−	0.729397i	$-0.260199\pi$
$569$	18570.0	1.36818	0.684090	+	0.729397i	$0.260199\pi$
$570$	0	0
$571$	2092.00	0.153323	0.0766615	−	0.997057i	$-0.475574\pi$
$571$	2092.00	0.153323	0.0766615	+	0.997057i	$0.475574\pi$
$572$	0	0
$573$	−960.000	−0.0699905
$574$	0	0
$575$	3800.00	0.275602
$576$	0	0
$577$	16898.0	1.21919	0.609595	−	0.792713i	$-0.291332\pi$
$577$	16898.0	1.21919	0.609595	+	0.792713i	$0.291332\pi$
$578$	0	0
$579$	15112.0	1.08469
$580$	0	0
$581$	−5536.00	−0.395305
$582$	0	0
$583$	4070.00	0.289129
$584$	0	0
$585$	3190.00	0.225453
$586$	0	0
$587$	−26996.0	−1.89820	−0.949101	−	0.314973i	$-0.898005\pi$
$587$	−26996.0	−1.89820	−0.949101	+	0.314973i	$0.898005\pi$
$588$	0	0
$589$	−832.000	−0.0582037
$590$	0	0
$591$	−13512.0	−0.940456
$592$	0	0
$593$	−5934.00	−0.410928	−0.205464	−	0.978665i	$-0.565870\pi$
$593$	−5934.00	−0.410928	−0.205464	+	0.978665i	$0.565870\pi$
$594$	0	0
$595$	−4560.00	−0.314188
$596$	0	0
$597$	7136.00	0.489208
$598$	0	0
$599$	−24120.0	−1.64527	−0.822635	−	0.568570i	$-0.807497\pi$
$599$	−24120.0	−1.64527	−0.822635	+	0.568570i	$0.807497\pi$
$600$	0	0
$601$	−6070.00	−0.411981	−0.205990	−	0.978554i	$-0.566042\pi$
$601$	−6070.00	−0.411981	−0.205990	+	0.978554i	$0.566042\pi$
$602$	0	0
$603$	1364.00	0.0921167
$604$	0	0
$605$	605.000	0.0406558
$606$	0	0
$607$	1424.00	0.0952197	0.0476099	−	0.998866i	$-0.484840\pi$
$607$	1424.00	0.0952197	0.0476099	+	0.998866i	$0.484840\pi$
$608$	0	0
$609$	4416.00	0.293835
$610$	0	0
$611$	−13920.0	−0.921674
$612$	0	0
$613$	1742.00	0.114778	0.0573888	−	0.998352i	$-0.481723\pi$
$613$	1742.00	0.114778	0.0573888	+	0.998352i	$0.481723\pi$
$614$	0	0
$615$	−5880.00	−0.385536
$616$	0	0
$617$	−16518.0	−1.07778	−0.538889	−	0.842376i	$-0.681156\pi$
$617$	−16518.0	−1.07778	−0.538889	+	0.842376i	$0.681156\pi$
$618$	0	0
$619$	1084.00	0.0703871	0.0351936	−	0.999381i	$-0.488795\pi$
$619$	1084.00	0.0703871	0.0351936	+	0.999381i	$0.488795\pi$
$620$	0	0
$621$	−23104.0	−1.49297
$622$	0	0
$623$	9648.00	0.620448
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−176.000	−0.0112101
$628$	0	0
$629$	−25764.0	−1.63319
$630$	0	0
$631$	−248.000	−0.0156462	−0.00782308	−	0.999969i	$-0.502490\pi$
$631$	−248.000	−0.0156462	−0.00782308	+	0.999969i	$0.502490\pi$
$632$	0	0
$633$	−15216.0	−0.955421
$634$	0	0
$635$	−6480.00	−0.404962
$636$	0	0
$637$	16182.0	1.00652
$638$	0	0
$639$	2552.00	0.157990
$640$	0	0
$641$	21378.0	1.31729	0.658643	−	0.752456i	$-0.271131\pi$
$641$	21378.0	1.31729	0.658643	+	0.752456i	$0.271131\pi$
$642$	0	0
$643$	−18364.0	−1.12629	−0.563146	−	0.826358i	$-0.690409\pi$
$643$	−18364.0	−1.12629	−0.563146	+	0.826358i	$0.690409\pi$
$644$	0	0
$645$	−5520.00	−0.336976
$646$	0	0
$647$	−31064.0	−1.88756	−0.943780	−	0.330573i	$-0.892758\pi$
$647$	−31064.0	−1.88756	−0.943780	+	0.330573i	$0.892758\pi$
$648$	0	0
$649$	−7876.00	−0.476364
$650$	0	0
$651$	6656.00	0.400721
$652$	0	0
$653$	13398.0	0.802916	0.401458	−	0.915877i	$-0.368504\pi$
$653$	13398.0	0.802916	0.401458	+	0.915877i	$0.368504\pi$
$654$	0	0
$655$	10500.0	0.626365
$656$	0	0
$657$	4994.00	0.296552
$658$	0	0
$659$	14724.0	0.870358	0.435179	−	0.900344i	$-0.356685\pi$
$659$	14724.0	0.870358	0.435179	+	0.900344i	$0.356685\pi$
$660$	0	0
$661$	23182.0	1.36411	0.682054	−	0.731302i	$-0.261087\pi$
$661$	23182.0	1.36411	0.682054	+	0.731302i	$0.261087\pi$
$662$	0	0
$663$	−26448.0	−1.54925
$664$	0	0
$665$	−160.000	−0.00933013
$666$	0	0
$667$	−20976.0	−1.21768
$668$	0	0
$669$	−14080.0	−0.813698
$670$	0	0
$671$	7150.00	0.411360
$672$	0	0
$673$	−7902.00	−0.452600	−0.226300	−	0.974058i	$-0.572663\pi$
$673$	−7902.00	−0.452600	−0.226300	+	0.974058i	$0.572663\pi$
$674$	0	0
$675$	−3800.00	−0.216685
$676$	0	0
$677$	−27826.0	−1.57968	−0.789838	−	0.613316i	$-0.789835\pi$
$677$	−27826.0	−1.57968	−0.789838	+	0.613316i	$0.789835\pi$
$678$	0	0
$679$	11504.0	0.650196
$680$	0	0
$681$	19984.0	1.12451
$682$	0	0
$683$	−5780.00	−0.323815	−0.161907	−	0.986806i	$-0.551765\pi$
$683$	−5780.00	−0.323815	−0.161907	+	0.986806i	$0.551765\pi$
$684$	0	0
$685$	11970.0	0.667665
$686$	0	0
$687$	1144.00	0.0635318
$688$	0	0
$689$	21460.0	1.18659
$690$	0	0
$691$	−10044.0	−0.552955	−0.276477	−	0.961020i	$-0.589167\pi$
$691$	−10044.0	−0.552955	−0.276477	+	0.961020i	$0.589167\pi$
$692$	0	0
$693$	−968.000	−0.0530610
$694$	0	0
$695$	−11700.0	−0.638570
$696$	0	0
$697$	−33516.0	−1.82139
$698$	0	0
$699$	−2712.00	−0.146749
$700$	0	0
$701$	−5322.00	−0.286746	−0.143373	−	0.989669i	$-0.545795\pi$
$701$	−5322.00	−0.286746	−0.143373	+	0.989669i	$0.545795\pi$
$702$	0	0
$703$	−904.000	−0.0484993
$704$	0	0
$705$	4800.00	0.256423
$706$	0	0
$707$	−4720.00	−0.251080
$708$	0	0
$709$	19902.0	1.05421	0.527105	−	0.849800i	$-0.323277\pi$
$709$	19902.0	1.05421	0.527105	+	0.849800i	$0.323277\pi$
$710$	0	0
$711$	−1584.00	−0.0835508
$712$	0	0
$713$	−31616.0	−1.66063
$714$	0	0
$715$	3190.00	0.166852
$716$	0	0
$717$	−16064.0	−0.836710
$718$	0	0
$719$	18688.0	0.969325	0.484663	−	0.874701i	$-0.338942\pi$
$719$	18688.0	0.969325	0.484663	+	0.874701i	$0.338942\pi$
$720$	0	0
$721$	−832.000	−0.0429754
$722$	0	0
$723$	11464.0	0.589697
$724$	0	0
$725$	−3450.00	−0.176731
$726$	0	0
$727$	−6760.00	−0.344862	−0.172431	−	0.985022i	$-0.555162\pi$
$727$	−6760.00	−0.344862	−0.172431	+	0.985022i	$0.555162\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	−31464.0	−1.59198
$732$	0	0
$733$	−25162.0	−1.26791	−0.633956	−	0.773369i	$-0.718570\pi$
$733$	−25162.0	−1.26791	−0.633956	+	0.773369i	$0.718570\pi$
$734$	0	0
$735$	−5580.00	−0.280029
$736$	0	0
$737$	1364.00	0.0681731
$738$	0	0
$739$	14548.0	0.724164	0.362082	−	0.932146i	$-0.382066\pi$
$739$	14548.0	0.724164	0.362082	+	0.932146i	$0.382066\pi$
$740$	0	0
$741$	−928.000	−0.0460067
$742$	0	0
$743$	2168.00	0.107047	0.0535237	−	0.998567i	$-0.482955\pi$
$743$	2168.00	0.107047	0.0535237	+	0.998567i	$0.482955\pi$
$744$	0	0
$745$	−9130.00	−0.448990
$746$	0	0
$747$	−7612.00	−0.372836
$748$	0	0
$749$	10912.0	0.532331
$750$	0	0
$751$	8768.00	0.426030	0.213015	−	0.977049i	$-0.431672\pi$
$751$	8768.00	0.426030	0.213015	+	0.977049i	$0.431672\pi$
$752$	0	0
$753$	17712.0	0.857186
$754$	0	0
$755$	15640.0	0.753904
$756$	0	0
$757$	−19794.0	−0.950363	−0.475182	−	0.879888i	$-0.657618\pi$
$757$	−19794.0	−0.950363	−0.475182	+	0.879888i	$0.657618\pi$
$758$	0	0
$759$	−6688.00	−0.319841
$760$	0	0
$761$	−6262.00	−0.298288	−0.149144	−	0.988815i	$-0.547652\pi$
$761$	−6262.00	−0.298288	−0.149144	+	0.988815i	$0.547652\pi$
$762$	0	0
$763$	6608.00	0.313533
$764$	0	0
$765$	−6270.00	−0.296330
$766$	0	0
$767$	−41528.0	−1.95501
$768$	0	0
$769$	−39454.0	−1.85013	−0.925063	−	0.379813i	$-0.875988\pi$
$769$	−39454.0	−1.85013	−0.925063	+	0.379813i	$0.875988\pi$
$770$	0	0
$771$	28168.0	1.31575
$772$	0	0
$773$	−20034.0	−0.932177	−0.466089	−	0.884738i	$-0.654337\pi$
$773$	−20034.0	−0.932177	−0.466089	+	0.884738i	$0.654337\pi$
$774$	0	0
$775$	−5200.00	−0.241019
$776$	0	0
$777$	7232.00	0.333908
$778$	0	0
$779$	−1176.00	−0.0540880
$780$	0	0
$781$	2552.00	0.116924
$782$	0	0
$783$	20976.0	0.957370
$784$	0	0
$785$	−3650.00	−0.165954
$786$	0	0
$787$	35572.0	1.61119	0.805594	−	0.592468i	$-0.201846\pi$
$787$	35572.0	1.61119	0.805594	+	0.592468i	$0.201846\pi$
$788$	0	0
$789$	−5152.00	−0.232466
$790$	0	0
$791$	−18576.0	−0.835002
$792$	0	0
$793$	37700.0	1.68823
$794$	0	0
$795$	−7400.00	−0.330127
$796$	0	0
$797$	−35994.0	−1.59972	−0.799858	−	0.600190i	$-0.795092\pi$
$797$	−35994.0	−1.59972	−0.799858	+	0.600190i	$0.795092\pi$
$798$	0	0
$799$	27360.0	1.21142
$800$	0	0
$801$	13266.0	0.585182
$802$	0	0
$803$	4994.00	0.219470
$804$	0	0
$805$	−6080.00	−0.266201
$806$	0	0
$807$	31576.0	1.37736
$808$	0	0
$809$	−19110.0	−0.830497	−0.415248	−	0.909708i	$-0.636305\pi$
$809$	−19110.0	−0.830497	−0.415248	+	0.909708i	$0.636305\pi$
$810$	0	0
$811$	−3300.00	−0.142884	−0.0714418	−	0.997445i	$-0.522760\pi$
$811$	−3300.00	−0.142884	−0.0714418	+	0.997445i	$0.522760\pi$
$812$	0	0
$813$	20416.0	0.880714
$814$	0	0
$815$	12500.0	0.537247
$816$	0	0
$817$	−1104.00	−0.0472755
$818$	0	0
$819$	−5104.00	−0.217763
$820$	0	0
$821$	4862.00	0.206681	0.103340	−	0.994646i	$-0.467047\pi$
$821$	4862.00	0.206681	0.103340	+	0.994646i	$0.467047\pi$
$822$	0	0
$823$	−7176.00	−0.303936	−0.151968	−	0.988385i	$-0.548561\pi$
$823$	−7176.00	−0.303936	−0.151968	+	0.988385i	$0.548561\pi$
$824$	0	0
$825$	−1100.00	−0.0464207
$826$	0	0
$827$	−10276.0	−0.432082	−0.216041	−	0.976384i	$-0.569314\pi$
$827$	−10276.0	−0.432082	−0.216041	+	0.976384i	$0.569314\pi$
$828$	0	0
$829$	40646.0	1.70289	0.851444	−	0.524446i	$-0.175727\pi$
$829$	40646.0	1.70289	0.851444	+	0.524446i	$0.175727\pi$
$830$	0	0
$831$	−17672.0	−0.737707
$832$	0	0
$833$	−31806.0	−1.32294
$834$	0	0
$835$	−17800.0	−0.737718
$836$	0	0
$837$	31616.0	1.30563
$838$	0	0
$839$	−8808.00	−0.362439	−0.181219	−	0.983443i	$-0.558004\pi$
$839$	−8808.00	−0.362439	−0.181219	+	0.983443i	$0.558004\pi$
$840$	0	0
$841$	−5345.00	−0.219156
$842$	0	0
$843$	−9432.00	−0.385356
$844$	0	0
$845$	5835.00	0.237550
$846$	0	0
$847$	−968.000	−0.0392690
$848$	0	0
$849$	−2576.00	−0.104132
$850$	0	0
$851$	−34352.0	−1.38375
$852$	0	0
$853$	−18658.0	−0.748931	−0.374465	−	0.927241i	$-0.622174\pi$
$853$	−18658.0	−0.748931	−0.374465	+	0.927241i	$0.622174\pi$
$854$	0	0
$855$	−220.000	−0.00879981
$856$	0	0
$857$	22410.0	0.893245	0.446623	−	0.894722i	$-0.352627\pi$
$857$	22410.0	0.893245	0.446623	+	0.894722i	$0.352627\pi$
$858$	0	0
$859$	4780.00	0.189862	0.0949310	−	0.995484i	$-0.469737\pi$
$859$	4780.00	0.189862	0.0949310	+	0.995484i	$0.469737\pi$
$860$	0	0
$861$	9408.00	0.372385
$862$	0	0
$863$	−5088.00	−0.200692	−0.100346	−	0.994953i	$-0.531995\pi$
$863$	−5088.00	−0.200692	−0.100346	+	0.994953i	$0.531995\pi$
$864$	0	0
$865$	12030.0	0.472870
$866$	0	0
$867$	32332.0	1.26650
$868$	0	0
$869$	−1584.00	−0.0618337
$870$	0	0
$871$	7192.00	0.279784
$872$	0	0
$873$	15818.0	0.613240
$874$	0	0
$875$	−1000.00	−0.0386356
$876$	0	0
$877$	−22714.0	−0.874569	−0.437285	−	0.899323i	$-0.644060\pi$
$877$	−22714.0	−0.874569	−0.437285	+	0.899323i	$0.644060\pi$
$878$	0	0
$879$	6712.00	0.257554
$880$	0	0
$881$	38354.0	1.46672	0.733359	−	0.679841i	$-0.237951\pi$
$881$	38354.0	1.46672	0.733359	+	0.679841i	$0.237951\pi$
$882$	0	0
$883$	−10892.0	−0.415113	−0.207557	−	0.978223i	$-0.566551\pi$
$883$	−10892.0	−0.415113	−0.207557	+	0.978223i	$0.566551\pi$
$884$	0	0
$885$	14320.0	0.543911
$886$	0	0
$887$	32104.0	1.21527	0.607636	−	0.794215i	$-0.292118\pi$
$887$	32104.0	1.21527	0.607636	+	0.794215i	$0.292118\pi$
$888$	0	0
$889$	10368.0	0.391149
$890$	0	0
$891$	3421.00	0.128628
$892$	0	0
$893$	960.000	0.0359744
$894$	0	0
$895$	−3180.00	−0.118766
$896$	0	0
$897$	−35264.0	−1.31263
$898$	0	0
$899$	28704.0	1.06489
$900$	0	0
$901$	−42180.0	−1.55962
$902$	0	0
$903$	8832.00	0.325482
$904$	0	0
$905$	−9210.00	−0.338288
$906$	0	0
$907$	20396.0	0.746679	0.373340	−	0.927695i	$-0.378213\pi$
$907$	20396.0	0.746679	0.373340	+	0.927695i	$0.378213\pi$
$908$	0	0
$909$	−6490.00	−0.236809
$910$	0	0
$911$	10432.0	0.379394	0.189697	−	0.981843i	$-0.439249\pi$
$911$	10432.0	0.379394	0.189697	+	0.981843i	$0.439249\pi$
$912$	0	0
$913$	−7612.00	−0.275926
$914$	0	0
$915$	−13000.0	−0.469690
$916$	0	0
$917$	−16800.0	−0.605000
$918$	0	0
$919$	−3080.00	−0.110555	−0.0552774	−	0.998471i	$-0.517604\pi$
$919$	−3080.00	−0.110555	−0.0552774	+	0.998471i	$0.517604\pi$
$920$	0	0
$921$	−22704.0	−0.812293
$922$	0	0
$923$	13456.0	0.479859
$924$	0	0
$925$	−5650.00	−0.200833
$926$	0	0
$927$	−1144.00	−0.0405328
$928$	0	0
$929$	−15582.0	−0.550300	−0.275150	−	0.961401i	$-0.588728\pi$
$929$	−15582.0	−0.550300	−0.275150	+	0.961401i	$0.588728\pi$
$930$	0	0
$931$	−1116.00	−0.0392862
$932$	0	0
$933$	−17248.0	−0.605224
$934$	0	0
$935$	−6270.00	−0.219306
$936$	0	0
$937$	−21862.0	−0.762220	−0.381110	−	0.924530i	$-0.624458\pi$
$937$	−21862.0	−0.762220	−0.381110	+	0.924530i	$0.624458\pi$
$938$	0	0
$939$	−28760.0	−0.999518
$940$	0	0
$941$	−33850.0	−1.17267	−0.586333	−	0.810070i	$-0.699429\pi$
$941$	−33850.0	−1.17267	−0.586333	+	0.810070i	$0.699429\pi$
$942$	0	0
$943$	−44688.0	−1.54320
$944$	0	0
$945$	6080.00	0.209294
$946$	0	0
$947$	26836.0	0.920858	0.460429	−	0.887696i	$-0.347696\pi$
$947$	26836.0	0.920858	0.460429	+	0.887696i	$0.347696\pi$
$948$	0	0
$949$	26332.0	0.900709
$950$	0	0
$951$	−17896.0	−0.610218
$952$	0	0
$953$	25194.0	0.856363	0.428181	−	0.903693i	$-0.359154\pi$
$953$	25194.0	0.856363	0.428181	+	0.903693i	$0.359154\pi$
$954$	0	0
$955$	−1200.00	−0.0406608
$956$	0	0
$957$	6072.00	0.205099
$958$	0	0
$959$	−19152.0	−0.644891
$960$	0	0
$961$	13473.0	0.452251
$962$	0	0
$963$	15004.0	0.502074
$964$	0	0
$965$	18890.0	0.630146
$966$	0	0
$967$	10296.0	0.342396	0.171198	−	0.985237i	$-0.445236\pi$
$967$	10296.0	0.342396	0.171198	+	0.985237i	$0.445236\pi$
$968$	0	0
$969$	1824.00	0.0604699
$970$	0	0
$971$	−46116.0	−1.52413	−0.762066	−	0.647499i	$-0.775815\pi$
$971$	−46116.0	−1.52413	−0.762066	+	0.647499i	$0.775815\pi$
$972$	0	0
$973$	18720.0	0.616789
$974$	0	0
$975$	−5800.00	−0.190511
$976$	0	0
$977$	−37070.0	−1.21389	−0.606947	−	0.794742i	$-0.707606\pi$
$977$	−37070.0	−1.21389	−0.606947	+	0.794742i	$0.707606\pi$
$978$	0	0
$979$	13266.0	0.433078
$980$	0	0
$981$	9086.00	0.295712
$982$	0	0
$983$	−13896.0	−0.450879	−0.225439	−	0.974257i	$-0.572382\pi$
$983$	−13896.0	−0.450879	−0.225439	+	0.974257i	$0.572382\pi$
$984$	0	0
$985$	−16890.0	−0.546355
$986$	0	0
$987$	−7680.00	−0.247677
$988$	0	0
$989$	−41952.0	−1.34883
$990$	0	0
$991$	39856.0	1.27757	0.638783	−	0.769387i	$-0.279438\pi$
$991$	39856.0	1.27757	0.638783	+	0.769387i	$0.279438\pi$
$992$	0	0
$993$	30576.0	0.977140
$994$	0	0
$995$	8920.00	0.284204
$996$	0	0
$997$	−13138.0	−0.417337	−0.208668	−	0.977986i	$-0.566913\pi$
$997$	−13138.0	−0.417337	−0.208668	+	0.977986i	$0.566913\pi$
$998$	0	0
$999$	34352.0	1.08794

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.4.a.l.1.1		1
4.3	odd	2		440.4.a.b.1.1	✓	1
20.19	odd	2		2200.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.4.a.b.1.1	✓	1	4.3	odd	2
880.4.a.l.1.1		1	1.1	even	1	trivial
2200.4.a.h.1.1		1	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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	880.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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