LMFDB - Embedded newform 160.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(1,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.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:	$9.44030560092$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.47214 q^{3} -5.00000 q^{5} -31.3050 q^{7} -7.00000 q^{9} +O(q^{10})$	$q+4.47214 q^{3} -5.00000 q^{5} -31.3050 q^{7} -7.00000 q^{9} -8.94427 q^{11} -62.0000 q^{13} -22.3607 q^{15} -46.0000 q^{17} +107.331 q^{19} -140.000 q^{21} +192.302 q^{23} +25.0000 q^{25} -152.053 q^{27} -90.0000 q^{29} -152.053 q^{31} -40.0000 q^{33} +156.525 q^{35} -214.000 q^{37} -277.272 q^{39} -10.0000 q^{41} -67.0820 q^{43} +35.0000 q^{45} +398.020 q^{47} +637.000 q^{49} -205.718 q^{51} -678.000 q^{53} +44.7214 q^{55} +480.000 q^{57} -411.437 q^{59} +250.000 q^{61} +219.135 q^{63} +310.000 q^{65} +49.1935 q^{67} +860.000 q^{69} -366.715 q^{71} +522.000 q^{73} +111.803 q^{75} +280.000 q^{77} +876.539 q^{79} -491.000 q^{81} +380.132 q^{83} +230.000 q^{85} -402.492 q^{87} +970.000 q^{89} +1940.91 q^{91} -680.000 q^{93} -536.656 q^{95} -934.000 q^{97} +62.6099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} - 14 q^{9}+O(q^{10}) $ 2 * q - 10 * q^5 - 14 * q^9	$ 2 q - 10 q^{5} - 14 q^{9} - 124 q^{13} - 92 q^{17} - 280 q^{21} + 50 q^{25} - 180 q^{29} - 80 q^{33} - 428 q^{37} - 20 q^{41} + 70 q^{45} + 1274 q^{49} - 1356 q^{53} + 960 q^{57} + 500 q^{61} + 620 q^{65} + 1720 q^{69} + 1044 q^{73} + 560 q^{77} - 982 q^{81} + 460 q^{85} + 1940 q^{89} - 1360 q^{93} - 1868 q^{97}+O(q^{100}) $ 2 * q - 10 * q^5 - 14 * q^9 - 124 * q^13 - 92 * q^17 - 280 * q^21 + 50 * q^25 - 180 * q^29 - 80 * q^33 - 428 * q^37 - 20 * q^41 + 70 * q^45 + 1274 * q^49 - 1356 * q^53 + 960 * q^57 + 500 * q^61 + 620 * q^65 + 1720 * q^69 + 1044 * q^73 + 560 * q^77 - 982 * q^81 + 460 * q^85 + 1940 * q^89 - 1360 * q^93 - 1868 * q^97

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
$3$	4.47214	0.860663	0.430331	−	0.902671i	$-0.358397\pi$
$3$	4.47214	0.860663	0.430331	+	0.902671i	$0.358397\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−31.3050	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−31.3050	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	0	0
$9$	−7.00000	−0.259259
$10$	0	0
$11$	−8.94427	−0.245164	−0.122582	−	0.992458i	$-0.539117\pi$
$11$	−8.94427	−0.245164	−0.122582	+	0.992458i	$0.539117\pi$
$12$	0	0
$13$	−62.0000	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$13$	−62.0000	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$14$	0	0
$15$	−22.3607	−0.384900
$16$	0	0
$17$	−46.0000	−0.656273	−0.328136	−	0.944630i	$-0.606421\pi$
$17$	−46.0000	−0.656273	−0.328136	+	0.944630i	$0.606421\pi$
$18$	0	0
$19$	107.331	1.29597	0.647986	−	0.761652i	$-0.275611\pi$
$19$	107.331	1.29597	0.647986	+	0.761652i	$0.275611\pi$
$20$	0	0
$21$	−140.000	−1.45479
$22$	0	0
$23$	192.302	1.74338	0.871689	−	0.490059i	$-0.163025\pi$
$23$	192.302	1.74338	0.871689	+	0.490059i	$0.163025\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−152.053	−1.08380
$28$	0	0
$29$	−90.0000	−0.576296	−0.288148	−	0.957586i	$-0.593039\pi$
$29$	−90.0000	−0.576296	−0.288148	+	0.957586i	$0.593039\pi$
$30$	0	0
$31$	−152.053	−0.880950	−0.440475	−	0.897765i	$-0.645190\pi$
$31$	−152.053	−0.880950	−0.440475	+	0.897765i	$0.645190\pi$
$32$	0	0
$33$	−40.0000	−0.211003
$34$	0	0
$35$	156.525	0.755929
$36$	0	0
$37$	−214.000	−0.950848	−0.475424	−	0.879757i	$-0.657705\pi$
$37$	−214.000	−0.950848	−0.475424	+	0.879757i	$0.657705\pi$
$38$	0	0
$39$	−277.272	−1.13844
$40$	0	0
$41$	−10.0000	−0.0380912	−0.0190456	−	0.999819i	$-0.506063\pi$
$41$	−10.0000	−0.0380912	−0.0190456	+	0.999819i	$0.506063\pi$
$42$	0	0
$43$	−67.0820	−0.237905	−0.118953	−	0.992900i	$-0.537954\pi$
$43$	−67.0820	−0.237905	−0.118953	+	0.992900i	$0.537954\pi$
$44$	0	0
$45$	35.0000	0.115944
$46$	0	0
$47$	398.020	1.23526	0.617630	−	0.786469i	$-0.288093\pi$
$47$	398.020	1.23526	0.617630	+	0.786469i	$0.288093\pi$
$48$	0	0
$49$	637.000	1.85714
$50$	0	0
$51$	−205.718	−0.564830
$52$	0	0
$53$	−678.000	−1.75718	−0.878589	−	0.477578i	$-0.841515\pi$
$53$	−678.000	−1.75718	−0.878589	+	0.477578i	$0.841515\pi$
$54$	0	0
$55$	44.7214	0.109640
$56$	0	0
$57$	480.000	1.11540
$58$	0	0
$59$	−411.437	−0.907872	−0.453936	−	0.891034i	$-0.649981\pi$
$59$	−411.437	−0.907872	−0.453936	+	0.891034i	$0.649981\pi$
$60$	0	0
$61$	250.000	0.524741	0.262371	−	0.964967i	$-0.415496\pi$
$61$	250.000	0.524741	0.262371	+	0.964967i	$0.415496\pi$
$62$	0	0
$63$	219.135	0.438228
$64$	0	0
$65$	310.000	0.591550
$66$	0	0
$67$	49.1935	0.0897006	0.0448503	−	0.998994i	$-0.485719\pi$
$67$	49.1935	0.0897006	0.0448503	+	0.998994i	$0.485719\pi$
$68$	0	0
$69$	860.000	1.50046
$70$	0	0
$71$	−366.715	−0.612973	−0.306486	−	0.951875i	$-0.599153\pi$
$71$	−366.715	−0.612973	−0.306486	+	0.951875i	$0.599153\pi$
$72$	0	0
$73$	522.000	0.836924	0.418462	−	0.908234i	$-0.362569\pi$
$73$	522.000	0.836924	0.418462	+	0.908234i	$0.362569\pi$
$74$	0	0
$75$	111.803	0.172133
$76$	0	0
$77$	280.000	0.414402
$78$	0	0
$79$	876.539	1.24833	0.624166	−	0.781291i	$-0.285439\pi$
$79$	876.539	1.24833	0.624166	+	0.781291i	$0.285439\pi$
$80$	0	0
$81$	−491.000	−0.673525
$82$	0	0
$83$	380.132	0.502709	0.251355	−	0.967895i	$-0.419124\pi$
$83$	380.132	0.502709	0.251355	+	0.967895i	$0.419124\pi$
$84$	0	0
$85$	230.000	0.293494
$86$	0	0
$87$	−402.492	−0.495997
$88$	0	0
$89$	970.000	1.15528	0.577639	−	0.816292i	$-0.303974\pi$
$89$	970.000	1.15528	0.577639	+	0.816292i	$0.303974\pi$
$90$	0	0
$91$	1940.91	2.23585
$92$	0	0
$93$	−680.000	−0.758201
$94$	0	0
$95$	−536.656	−0.579577
$96$	0	0
$97$	−934.000	−0.977663	−0.488832	−	0.872378i	$-0.662577\pi$
$97$	−934.000	−0.977663	−0.488832	+	0.872378i	$0.662577\pi$
$98$	0	0
$99$	62.6099	0.0635609
$100$	0	0
$101$	−602.000	−0.593082	−0.296541	−	0.955020i	$-0.595833\pi$
$101$	−602.000	−0.593082	−0.296541	+	0.955020i	$0.595833\pi$
$102$	0	0
$103$	−1829.10	−1.74978	−0.874888	−	0.484325i	$-0.839065\pi$
$103$	−1829.10	−1.74978	−0.874888	+	0.484325i	$0.839065\pi$
$104$	0	0
$105$	700.000	0.650600
$106$	0	0
$107$	−1525.00	−1.37782	−0.688912	−	0.724845i	$-0.741911\pi$
$107$	−1525.00	−1.37782	−0.688912	+	0.724845i	$0.741911\pi$
$108$	0	0
$109$	2154.00	1.89281	0.946403	−	0.322989i	$-0.104688\pi$
$109$	2154.00	1.89281	0.946403	+	0.322989i	$0.104688\pi$
$110$	0	0
$111$	−957.037	−0.818360
$112$	0	0
$113$	−2182.00	−1.81651	−0.908254	−	0.418420i	$-0.862584\pi$
$113$	−2182.00	−1.81651	−0.908254	+	0.418420i	$0.862584\pi$
$114$	0	0
$115$	−961.509	−0.779663
$116$	0	0
$117$	434.000	0.342934
$118$	0	0
$119$	1440.03	1.10930
$120$	0	0
$121$	−1251.00	−0.939895
$122$	0	0
$123$	−44.7214	−0.0327837
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1310.34	0.915539	0.457770	−	0.889071i	$-0.348648\pi$
$127$	1310.34	0.915539	0.457770	+	0.889071i	$0.348648\pi$
$128$	0	0
$129$	−300.000	−0.204756
$130$	0	0
$131$	−205.718	−0.137204	−0.0686019	−	0.997644i	$-0.521854\pi$
$131$	−205.718	−0.137204	−0.0686019	+	0.997644i	$0.521854\pi$
$132$	0	0
$133$	−3360.00	−2.19059
$134$	0	0
$135$	760.263	0.484689
$136$	0	0
$137$	−2094.00	−1.30586	−0.652929	−	0.757419i	$-0.726460\pi$
$137$	−2094.00	−1.30586	−0.652929	+	0.757419i	$0.726460\pi$
$138$	0	0
$139$	−1377.42	−0.840511	−0.420256	−	0.907406i	$-0.638060\pi$
$139$	−1377.42	−0.840511	−0.420256	+	0.907406i	$0.638060\pi$
$140$	0	0
$141$	1780.00	1.06314
$142$	0	0
$143$	554.545	0.324289
$144$	0	0
$145$	450.000	0.257727
$146$	0	0
$147$	2848.75	1.59837
$148$	0	0
$149$	−334.000	−0.183640	−0.0918200	−	0.995776i	$-0.529268\pi$
$149$	−334.000	−0.183640	−0.0918200	+	0.995776i	$0.529268\pi$
$150$	0	0
$151$	3139.44	1.69195	0.845973	−	0.533225i	$-0.179020\pi$
$151$	3139.44	1.69195	0.845973	+	0.533225i	$0.179020\pi$
$152$	0	0
$153$	322.000	0.170145
$154$	0	0
$155$	760.263	0.393973
$156$	0	0
$157$	834.000	0.423952	0.211976	−	0.977275i	$-0.432010\pi$
$157$	834.000	0.423952	0.211976	+	0.977275i	$0.432010\pi$
$158$	0	0
$159$	−3032.11	−1.51234
$160$	0	0
$161$	−6020.00	−2.94685
$162$	0	0
$163$	−3090.25	−1.48495	−0.742475	−	0.669874i	$-0.766348\pi$
$163$	−3090.25	−1.48495	−0.742475	+	0.669874i	$0.766348\pi$
$164$	0	0
$165$	200.000	0.0943635
$166$	0	0
$167$	4.47214	0.00207224	0.00103612	−	0.999999i	$-0.499670\pi$
$167$	4.47214	0.00207224	0.00103612	+	0.999999i	$0.499670\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	0	0
$171$	−751.319	−0.335993
$172$	0	0
$173$	−1838.00	−0.807749	−0.403874	−	0.914814i	$-0.632337\pi$
$173$	−1838.00	−0.807749	−0.403874	+	0.914814i	$0.632337\pi$
$174$	0	0
$175$	−782.624	−0.338062
$176$	0	0
$177$	−1840.00	−0.781372
$178$	0	0
$179$	1842.52	0.769365	0.384683	−	0.923049i	$-0.374311\pi$
$179$	1842.52	0.769365	0.384683	+	0.923049i	$0.374311\pi$
$180$	0	0
$181$	1862.00	0.764648	0.382324	−	0.924028i	$-0.375124\pi$
$181$	1862.00	0.764648	0.382324	+	0.924028i	$0.375124\pi$
$182$	0	0
$183$	1118.03	0.451625
$184$	0	0
$185$	1070.00	0.425232
$186$	0	0
$187$	411.437	0.160894
$188$	0	0
$189$	4760.00	1.83195
$190$	0	0
$191$	−2066.13	−0.782721	−0.391360	−	0.920237i	$-0.627995\pi$
$191$	−2066.13	−0.782721	−0.391360	+	0.920237i	$0.627995\pi$
$192$	0	0
$193$	3378.00	1.25986	0.629932	−	0.776650i	$-0.283083\pi$
$193$	3378.00	1.25986	0.629932	+	0.776650i	$0.283083\pi$
$194$	0	0
$195$	1386.36	0.509125
$196$	0	0
$197$	66.0000	0.0238696	0.0119348	−	0.999929i	$-0.496201\pi$
$197$	66.0000	0.0238696	0.0119348	+	0.999929i	$0.496201\pi$
$198$	0	0
$199$	−1216.42	−0.433316	−0.216658	−	0.976248i	$-0.569516\pi$
$199$	−1216.42	−0.433316	−0.216658	+	0.976248i	$0.569516\pi$
$200$	0	0
$201$	220.000	0.0772020
$202$	0	0
$203$	2817.45	0.974118
$204$	0	0
$205$	50.0000	0.0170349
$206$	0	0
$207$	−1346.11	−0.451987
$208$	0	0
$209$	−960.000	−0.317725
$210$	0	0
$211$	−5286.06	−1.72468	−0.862341	−	0.506329i	$-0.831002\pi$
$211$	−5286.06	−1.72468	−0.862341	+	0.506329i	$0.831002\pi$
$212$	0	0
$213$	−1640.00	−0.527563
$214$	0	0
$215$	335.410	0.106394
$216$	0	0
$217$	4760.00	1.48908
$218$	0	0
$219$	2334.45	0.720310
$220$	0	0
$221$	2852.00	0.868083
$222$	0	0
$223$	2965.03	0.890371	0.445186	−	0.895438i	$-0.353138\pi$
$223$	2965.03	0.890371	0.445186	+	0.895438i	$0.353138\pi$
$224$	0	0
$225$	−175.000	−0.0518519
$226$	0	0
$227$	−4369.28	−1.27753	−0.638765	−	0.769402i	$-0.720554\pi$
$227$	−4369.28	−1.27753	−0.638765	+	0.769402i	$0.720554\pi$
$228$	0	0
$229$	−3250.00	−0.937843	−0.468921	−	0.883240i	$-0.655357\pi$
$229$	−3250.00	−0.937843	−0.468921	+	0.883240i	$0.655357\pi$
$230$	0	0
$231$	1252.20	0.356661
$232$	0	0
$233$	3298.00	0.927293	0.463646	−	0.886020i	$-0.346541\pi$
$233$	3298.00	0.927293	0.463646	+	0.886020i	$0.346541\pi$
$234$	0	0
$235$	−1990.10	−0.552425
$236$	0	0
$237$	3920.00	1.07439
$238$	0	0
$239$	−554.545	−0.150086	−0.0750429	−	0.997180i	$-0.523909\pi$
$239$	−554.545	−0.150086	−0.0750429	+	0.997180i	$0.523909\pi$
$240$	0	0
$241$	5150.00	1.37652	0.688259	−	0.725465i	$-0.258375\pi$
$241$	5150.00	1.37652	0.688259	+	0.725465i	$0.258375\pi$
$242$	0	0
$243$	1909.60	0.504119
$244$	0	0
$245$	−3185.00	−0.830540
$246$	0	0
$247$	−6654.54	−1.71424
$248$	0	0
$249$	1700.00	0.432663
$250$	0	0
$251$	1386.36	0.348631	0.174316	−	0.984690i	$-0.444229\pi$
$251$	1386.36	0.348631	0.174316	+	0.984690i	$0.444229\pi$
$252$	0	0
$253$	−1720.00	−0.427413
$254$	0	0
$255$	1028.59	0.252600
$256$	0	0
$257$	−4166.00	−1.01116	−0.505580	−	0.862780i	$-0.668721\pi$
$257$	−4166.00	−1.01116	−0.505580	+	0.862780i	$0.668721\pi$
$258$	0	0
$259$	6699.26	1.60723
$260$	0	0
$261$	630.000	0.149410
$262$	0	0
$263$	961.509	0.225434	0.112717	−	0.993627i	$-0.464045\pi$
$263$	961.509	0.225434	0.112717	+	0.993627i	$0.464045\pi$
$264$	0	0
$265$	3390.00	0.785834
$266$	0	0
$267$	4337.97	0.994305
$268$	0	0
$269$	−1494.00	−0.338627	−0.169314	−	0.985562i	$-0.554155\pi$
$269$	−1494.00	−0.338627	−0.169314	+	0.985562i	$0.554155\pi$
$270$	0	0
$271$	5017.74	1.12474	0.562372	−	0.826884i	$-0.309889\pi$
$271$	5017.74	1.12474	0.562372	+	0.826884i	$0.309889\pi$
$272$	0	0
$273$	8680.00	1.92431
$274$	0	0
$275$	−223.607	−0.0490327
$276$	0	0
$277$	−1006.00	−0.218212	−0.109106	−	0.994030i	$-0.534799\pi$
$277$	−1006.00	−0.218212	−0.109106	+	0.994030i	$0.534799\pi$
$278$	0	0
$279$	1064.37	0.228395
$280$	0	0
$281$	−3210.00	−0.681468	−0.340734	−	0.940160i	$-0.610676\pi$
$281$	−3210.00	−0.681468	−0.340734	+	0.940160i	$0.610676\pi$
$282$	0	0
$283$	3635.85	0.763705	0.381853	−	0.924223i	$-0.375286\pi$
$283$	3635.85	0.763705	0.381853	+	0.924223i	$0.375286\pi$
$284$	0	0
$285$	−2400.00	−0.498820
$286$	0	0
$287$	313.050	0.0643858
$288$	0	0
$289$	−2797.00	−0.569306
$290$	0	0
$291$	−4176.97	−0.841439
$292$	0	0
$293$	−3622.00	−0.722183	−0.361091	−	0.932530i	$-0.617596\pi$
$293$	−3622.00	−0.722183	−0.361091	+	0.932530i	$0.617596\pi$
$294$	0	0
$295$	2057.18	0.406013
$296$	0	0
$297$	1360.00	0.265708
$298$	0	0
$299$	−11922.7	−2.30605
$300$	0	0
$301$	2100.00	0.402133
$302$	0	0
$303$	−2692.23	−0.510443
$304$	0	0
$305$	−1250.00	−0.234671
$306$	0	0
$307$	2088.49	0.388261	0.194131	−	0.980976i	$-0.437811\pi$
$307$	2088.49	0.388261	0.194131	+	0.980976i	$0.437811\pi$
$308$	0	0
$309$	−8180.00	−1.50597
$310$	0	0
$311$	−8899.55	−1.62266	−0.811330	−	0.584589i	$-0.801256\pi$
$311$	−8899.55	−1.62266	−0.811330	+	0.584589i	$0.801256\pi$
$312$	0	0
$313$	8778.00	1.58518	0.792591	−	0.609754i	$-0.208732\pi$
$313$	8778.00	1.58518	0.792591	+	0.609754i	$0.208732\pi$
$314$	0	0
$315$	−1095.67	−0.195982
$316$	0	0
$317$	−5046.00	−0.894043	−0.447021	−	0.894523i	$-0.647515\pi$
$317$	−5046.00	−0.894043	−0.447021	+	0.894523i	$0.647515\pi$
$318$	0	0
$319$	804.984	0.141287
$320$	0	0
$321$	−6820.00	−1.18584
$322$	0	0
$323$	−4937.24	−0.850512
$324$	0	0
$325$	−1550.00	−0.264549
$326$	0	0
$327$	9632.98	1.62907
$328$	0	0
$329$	−12460.0	−2.08797
$330$	0	0
$331$	−313.050	−0.0519842	−0.0259921	−	0.999662i	$-0.508274\pi$
$331$	−313.050	−0.0519842	−0.0259921	+	0.999662i	$0.508274\pi$
$332$	0	0
$333$	1498.00	0.246516
$334$	0	0
$335$	−245.967	−0.0401153
$336$	0	0
$337$	−2574.00	−0.416067	−0.208034	−	0.978122i	$-0.566706\pi$
$337$	−2574.00	−0.416067	−0.208034	+	0.978122i	$0.566706\pi$
$338$	0	0
$339$	−9758.20	−1.56340
$340$	0	0
$341$	1360.00	0.215977
$342$	0	0
$343$	−9203.66	−1.44884
$344$	0	0
$345$	−4300.00	−0.671027
$346$	0	0
$347$	2643.03	0.408892	0.204446	−	0.978878i	$-0.434461\pi$
$347$	2643.03	0.408892	0.204446	+	0.978878i	$0.434461\pi$
$348$	0	0
$349$	−10170.0	−1.55985	−0.779925	−	0.625873i	$-0.784743\pi$
$349$	−10170.0	−1.55985	−0.779925	+	0.625873i	$0.784743\pi$
$350$	0	0
$351$	9427.26	1.43359
$352$	0	0
$353$	−318.000	−0.0479474	−0.0239737	−	0.999713i	$-0.507632\pi$
$353$	−318.000	−0.0479474	−0.0239737	+	0.999713i	$0.507632\pi$
$354$	0	0
$355$	1833.58	0.274130
$356$	0	0
$357$	6440.00	0.954737
$358$	0	0
$359$	12378.9	1.81987	0.909933	−	0.414755i	$-0.136133\pi$
$359$	12378.9	1.81987	0.909933	+	0.414755i	$0.136133\pi$
$360$	0	0
$361$	4661.00	0.679545
$362$	0	0
$363$	−5594.64	−0.808933
$364$	0	0
$365$	−2610.00	−0.374284
$366$	0	0
$367$	−3072.36	−0.436991	−0.218496	−	0.975838i	$-0.570115\pi$
$367$	−3072.36	−0.436991	−0.218496	+	0.975838i	$0.570115\pi$
$368$	0	0
$369$	70.0000	0.00987549
$370$	0	0
$371$	21224.8	2.97017
$372$	0	0
$373$	−3278.00	−0.455036	−0.227518	−	0.973774i	$-0.573061\pi$
$373$	−3278.00	−0.455036	−0.227518	+	0.973774i	$0.573061\pi$
$374$	0	0
$375$	−559.017	−0.0769800
$376$	0	0
$377$	5580.00	0.762293
$378$	0	0
$379$	5116.12	0.693397	0.346699	−	0.937977i	$-0.387303\pi$
$379$	5116.12	0.693397	0.346699	+	0.937977i	$0.387303\pi$
$380$	0	0
$381$	5860.00	0.787971
$382$	0	0
$383$	−1149.34	−0.153338	−0.0766690	−	0.997057i	$-0.524428\pi$
$383$	−1149.34	−0.153338	−0.0766690	+	0.997057i	$0.524428\pi$
$384$	0	0
$385$	−1400.00	−0.185326
$386$	0	0
$387$	469.574	0.0616791
$388$	0	0
$389$	834.000	0.108703	0.0543515	−	0.998522i	$-0.482691\pi$
$389$	834.000	0.108703	0.0543515	+	0.998522i	$0.482691\pi$
$390$	0	0
$391$	−8845.88	−1.14413
$392$	0	0
$393$	−920.000	−0.118086
$394$	0	0
$395$	−4382.69	−0.558271
$396$	0	0
$397$	−8734.00	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−8734.00	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	−15026.4	−1.88536
$400$	0	0
$401$	242.000	0.0301369	0.0150685	−	0.999886i	$-0.495203\pi$
$401$	242.000	0.0301369	0.0150685	+	0.999886i	$0.495203\pi$
$402$	0	0
$403$	9427.26	1.16527
$404$	0	0
$405$	2455.00	0.301210
$406$	0	0
$407$	1914.07	0.233113
$408$	0	0
$409$	−6514.00	−0.787522	−0.393761	−	0.919213i	$-0.628826\pi$
$409$	−6514.00	−0.787522	−0.393761	+	0.919213i	$0.628826\pi$
$410$	0	0
$411$	−9364.65	−1.12390
$412$	0	0
$413$	12880.0	1.53458
$414$	0	0
$415$	−1900.66	−0.224818
$416$	0	0
$417$	−6160.00	−0.723397
$418$	0	0
$419$	16081.8	1.87505	0.937527	−	0.347913i	$-0.113110\pi$
$419$	16081.8	1.87505	0.937527	+	0.347913i	$0.113110\pi$
$420$	0	0
$421$	7250.00	0.839295	0.419648	−	0.907687i	$-0.362154\pi$
$421$	7250.00	0.839295	0.419648	+	0.907687i	$0.362154\pi$
$422$	0	0
$423$	−2786.14	−0.320252
$424$	0	0
$425$	−1150.00	−0.131255
$426$	0	0
$427$	−7826.24	−0.886975
$428$	0	0
$429$	2480.00	0.279104
$430$	0	0
$431$	4981.96	0.556781	0.278390	−	0.960468i	$-0.410199\pi$
$431$	4981.96	0.556781	0.278390	+	0.960468i	$0.410199\pi$
$432$	0	0
$433$	11482.0	1.27434	0.637171	−	0.770723i	$-0.280105\pi$
$433$	11482.0	1.27434	0.637171	+	0.770723i	$0.280105\pi$
$434$	0	0
$435$	2012.46	0.221816
$436$	0	0
$437$	20640.0	2.25937
$438$	0	0
$439$	3792.37	0.412301	0.206150	−	0.978520i	$-0.433906\pi$
$439$	3792.37	0.412301	0.206150	+	0.978520i	$0.433906\pi$
$440$	0	0
$441$	−4459.00	−0.481481
$442$	0	0
$443$	−746.847	−0.0800988	−0.0400494	−	0.999198i	$-0.512752\pi$
$443$	−746.847	−0.0800988	−0.0400494	+	0.999198i	$0.512752\pi$
$444$	0	0
$445$	−4850.00	−0.516656
$446$	0	0
$447$	−1493.69	−0.158052
$448$	0	0
$449$	−1306.00	−0.137269	−0.0686347	−	0.997642i	$-0.521864\pi$
$449$	−1306.00	−0.137269	−0.0686347	+	0.997642i	$0.521864\pi$
$450$	0	0
$451$	89.4427	0.00933857
$452$	0	0
$453$	14040.0	1.45620
$454$	0	0
$455$	−9704.54	−0.999902
$456$	0	0
$457$	−9526.00	−0.975071	−0.487536	−	0.873103i	$-0.662104\pi$
$457$	−9526.00	−0.975071	−0.487536	+	0.873103i	$0.662104\pi$
$458$	0	0
$459$	6994.42	0.711267
$460$	0	0
$461$	1518.00	0.153363	0.0766815	−	0.997056i	$-0.475568\pi$
$461$	1518.00	0.153363	0.0766815	+	0.997056i	$0.475568\pi$
$462$	0	0
$463$	17293.7	1.73587	0.867936	−	0.496676i	$-0.165446\pi$
$463$	17293.7	1.73587	0.867936	+	0.496676i	$0.165446\pi$
$464$	0	0
$465$	3400.00	0.339078
$466$	0	0
$467$	−16980.7	−1.68260	−0.841299	−	0.540570i	$-0.818208\pi$
$467$	−16980.7	−1.68260	−0.841299	+	0.540570i	$0.818208\pi$
$468$	0	0
$469$	−1540.00	−0.151622
$470$	0	0
$471$	3729.76	0.364880
$472$	0	0
$473$	600.000	0.0583256
$474$	0	0
$475$	2683.28	0.259195
$476$	0	0
$477$	4746.00	0.455565
$478$	0	0
$479$	3810.26	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$479$	3810.26	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$480$	0	0
$481$	13268.0	1.25773
$482$	0	0
$483$	−26922.3	−2.53624
$484$	0	0
$485$	4670.00	0.437224
$486$	0	0
$487$	1310.34	0.121924	0.0609620	−	0.998140i	$-0.480583\pi$
$487$	1310.34	0.121924	0.0609620	+	0.998140i	$0.480583\pi$
$488$	0	0
$489$	−13820.0	−1.27804
$490$	0	0
$491$	2960.55	0.272114	0.136057	−	0.990701i	$-0.456557\pi$
$491$	2960.55	0.272114	0.136057	+	0.990701i	$0.456557\pi$
$492$	0	0
$493$	4140.00	0.378207
$494$	0	0
$495$	−313.050	−0.0284253
$496$	0	0
$497$	11480.0	1.03611
$498$	0	0
$499$	−19319.6	−1.73320	−0.866598	−	0.499006i	$-0.833699\pi$
$499$	−19319.6	−1.73320	−0.866598	+	0.499006i	$0.833699\pi$
$500$	0	0
$501$	20.0000	0.00178350
$502$	0	0
$503$	3072.36	0.272345	0.136173	−	0.990685i	$-0.456520\pi$
$503$	3072.36	0.272345	0.136173	+	0.990685i	$0.456520\pi$
$504$	0	0
$505$	3010.00	0.265234
$506$	0	0
$507$	7365.61	0.645203
$508$	0	0
$509$	18550.0	1.61535	0.807676	−	0.589626i	$-0.200725\pi$
$509$	18550.0	1.61535	0.807676	+	0.589626i	$0.200725\pi$
$510$	0	0
$511$	−16341.2	−1.41466
$512$	0	0
$513$	−16320.0	−1.40457
$514$	0	0
$515$	9145.52	0.782524
$516$	0	0
$517$	−3560.00	−0.302841
$518$	0	0
$519$	−8219.79	−0.695200
$520$	0	0
$521$	−2102.00	−0.176757	−0.0883784	−	0.996087i	$-0.528168\pi$
$521$	−2102.00	−0.176757	−0.0883784	+	0.996087i	$0.528168\pi$
$522$	0	0
$523$	17696.2	1.47955	0.739773	−	0.672856i	$-0.234933\pi$
$523$	17696.2	1.47955	0.739773	+	0.672856i	$0.234933\pi$
$524$	0	0
$525$	−3500.00	−0.290957
$526$	0	0
$527$	6994.42	0.578144
$528$	0	0
$529$	24813.0	2.03937
$530$	0	0
$531$	2880.06	0.235374
$532$	0	0
$533$	620.000	0.0503850
$534$	0	0
$535$	7624.99	0.616182
$536$	0	0
$537$	8240.00	0.662164
$538$	0	0
$539$	−5697.50	−0.455304
$540$	0	0
$541$	−9922.00	−0.788503	−0.394251	−	0.919003i	$-0.628996\pi$
$541$	−9922.00	−0.788503	−0.394251	+	0.919003i	$0.628996\pi$
$542$	0	0
$543$	8327.12	0.658105
$544$	0	0
$545$	−10770.0	−0.846488
$546$	0	0
$547$	3716.34	0.290493	0.145246	−	0.989396i	$-0.453603\pi$
$547$	3716.34	0.290493	0.145246	+	0.989396i	$0.453603\pi$
$548$	0	0
$549$	−1750.00	−0.136044
$550$	0	0
$551$	−9659.81	−0.746864
$552$	0	0
$553$	−27440.0	−2.11007
$554$	0	0
$555$	4785.19	0.365982
$556$	0	0
$557$	−15094.0	−1.14821	−0.574105	−	0.818781i	$-0.694650\pi$
$557$	−15094.0	−1.14821	−0.574105	+	0.818781i	$0.694650\pi$
$558$	0	0
$559$	4159.09	0.314688
$560$	0	0
$561$	1840.00	0.138476
$562$	0	0
$563$	5657.25	0.423490	0.211745	−	0.977325i	$-0.432085\pi$
$563$	5657.25	0.423490	0.211745	+	0.977325i	$0.432085\pi$
$564$	0	0
$565$	10910.0	0.812367
$566$	0	0
$567$	15370.7	1.13847
$568$	0	0
$569$	−5906.00	−0.435136	−0.217568	−	0.976045i	$-0.569812\pi$
$569$	−5906.00	−0.435136	−0.217568	+	0.976045i	$0.569812\pi$
$570$	0	0
$571$	−4892.52	−0.358573	−0.179287	−	0.983797i	$-0.557379\pi$
$571$	−4892.52	−0.358573	−0.179287	+	0.983797i	$0.557379\pi$
$572$	0	0
$573$	−9240.00	−0.673659
$574$	0	0
$575$	4807.55	0.348676
$576$	0	0
$577$	−13286.0	−0.958585	−0.479292	−	0.877655i	$-0.659107\pi$
$577$	−13286.0	−0.958585	−0.479292	+	0.877655i	$0.659107\pi$
$578$	0	0
$579$	15106.9	1.08432
$580$	0	0
$581$	−11900.0	−0.849734
$582$	0	0
$583$	6064.22	0.430796
$584$	0	0
$585$	−2170.00	−0.153365
$586$	0	0
$587$	9029.24	0.634884	0.317442	−	0.948278i	$-0.397176\pi$
$587$	9029.24	0.634884	0.317442	+	0.948278i	$0.397176\pi$
$588$	0	0
$589$	−16320.0	−1.14169
$590$	0	0
$591$	295.161	0.0205437
$592$	0	0
$593$	11442.0	0.792355	0.396178	−	0.918174i	$-0.370336\pi$
$593$	11442.0	0.792355	0.396178	+	0.918174i	$0.370336\pi$
$594$	0	0
$595$	−7200.14	−0.496096
$596$	0	0
$597$	−5440.00	−0.372939
$598$	0	0
$599$	−14149.8	−0.965187	−0.482593	−	0.875845i	$-0.660305\pi$
$599$	−14149.8	−0.965187	−0.482593	+	0.875845i	$0.660305\pi$
$600$	0	0
$601$	3110.00	0.211081	0.105540	−	0.994415i	$-0.466343\pi$
$601$	3110.00	0.211081	0.105540	+	0.994415i	$0.466343\pi$
$602$	0	0
$603$	−344.354	−0.0232557
$604$	0	0
$605$	6255.00	0.420334
$606$	0	0
$607$	−11193.8	−0.748502	−0.374251	−	0.927327i	$-0.622100\pi$
$607$	−11193.8	−0.748502	−0.374251	+	0.927327i	$0.622100\pi$
$608$	0	0
$609$	12600.0	0.838387
$610$	0	0
$611$	−24677.2	−1.63394
$612$	0	0
$613$	−5342.00	−0.351976	−0.175988	−	0.984392i	$-0.556312\pi$
$613$	−5342.00	−0.351976	−0.175988	+	0.984392i	$0.556312\pi$
$614$	0	0
$615$	223.607	0.0146613
$616$	0	0
$617$	19714.0	1.28631	0.643157	−	0.765734i	$-0.277624\pi$
$617$	19714.0	1.28631	0.643157	+	0.765734i	$0.277624\pi$
$618$	0	0
$619$	−13166.0	−0.854903	−0.427451	−	0.904038i	$-0.640589\pi$
$619$	−13166.0	−0.854903	−0.427451	+	0.904038i	$0.640589\pi$
$620$	0	0
$621$	−29240.0	−1.88947
$622$	0	0
$623$	−30365.8	−1.95278
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−4293.25	−0.273454
$628$	0	0
$629$	9844.00	0.624016
$630$	0	0
$631$	12262.6	0.773639	0.386820	−	0.922155i	$-0.373574\pi$
$631$	12262.6	0.773639	0.386820	+	0.922155i	$0.373574\pi$
$632$	0	0
$633$	−23640.0	−1.48437
$634$	0	0
$635$	−6551.68	−0.409442
$636$	0	0
$637$	−39494.0	−2.45653
$638$	0	0
$639$	2567.01	0.158919
$640$	0	0
$641$	−2690.00	−0.165754	−0.0828772	−	0.996560i	$-0.526411\pi$
$641$	−2690.00	−0.165754	−0.0828772	+	0.996560i	$0.526411\pi$
$642$	0	0
$643$	12240.2	0.750712	0.375356	−	0.926881i	$-0.377520\pi$
$643$	12240.2	0.750712	0.375356	+	0.926881i	$0.377520\pi$
$644$	0	0
$645$	1500.00	0.0915697
$646$	0	0
$647$	−17973.5	−1.09214	−0.546068	−	0.837741i	$-0.683876\pi$
$647$	−17973.5	−1.09214	−0.546068	+	0.837741i	$0.683876\pi$
$648$	0	0
$649$	3680.00	0.222577
$650$	0	0
$651$	21287.4	1.28159
$652$	0	0
$653$	−3478.00	−0.208430	−0.104215	−	0.994555i	$-0.533233\pi$
$653$	−3478.00	−0.208430	−0.104215	+	0.994555i	$0.533233\pi$
$654$	0	0
$655$	1028.59	0.0613594
$656$	0	0
$657$	−3654.00	−0.216980
$658$	0	0
$659$	−10572.1	−0.624934	−0.312467	−	0.949929i	$-0.601155\pi$
$659$	−10572.1	−0.624934	−0.312467	+	0.949929i	$0.601155\pi$
$660$	0	0
$661$	−110.000	−0.00647277	−0.00323639	−	0.999995i	$-0.501030\pi$
$661$	−110.000	−0.00647277	−0.00323639	+	0.999995i	$0.501030\pi$
$662$	0	0
$663$	12754.5	0.747127
$664$	0	0
$665$	16800.0	0.979663
$666$	0	0
$667$	−17307.2	−1.00470
$668$	0	0
$669$	13260.0	0.766310
$670$	0	0
$671$	−2236.07	−0.128647
$672$	0	0
$673$	−14278.0	−0.817796	−0.408898	−	0.912580i	$-0.634087\pi$
$673$	−14278.0	−0.817796	−0.408898	+	0.912580i	$0.634087\pi$
$674$	0	0
$675$	−3801.32	−0.216760
$676$	0	0
$677$	18386.0	1.04377	0.521884	−	0.853016i	$-0.325229\pi$
$677$	18386.0	1.04377	0.521884	+	0.853016i	$0.325229\pi$
$678$	0	0
$679$	29238.8	1.65255
$680$	0	0
$681$	−19540.0	−1.09952
$682$	0	0
$683$	15317.1	0.858113	0.429057	−	0.903278i	$-0.358846\pi$
$683$	15317.1	0.858113	0.429057	+	0.903278i	$0.358846\pi$
$684$	0	0
$685$	10470.0	0.583997
$686$	0	0
$687$	−14534.4	−0.807167
$688$	0	0
$689$	42036.0	2.32430
$690$	0	0
$691$	9507.76	0.523433	0.261717	−	0.965145i	$-0.415711\pi$
$691$	9507.76	0.523433	0.261717	+	0.965145i	$0.415711\pi$
$692$	0	0
$693$	−1960.00	−0.107438
$694$	0	0
$695$	6887.09	0.375888
$696$	0	0
$697$	460.000	0.0249982
$698$	0	0
$699$	14749.1	0.798086
$700$	0	0
$701$	−15830.0	−0.852911	−0.426456	−	0.904508i	$-0.640238\pi$
$701$	−15830.0	−0.852911	−0.426456	+	0.904508i	$0.640238\pi$
$702$	0	0
$703$	−22968.9	−1.23227
$704$	0	0
$705$	−8900.00	−0.475452
$706$	0	0
$707$	18845.6	1.00249
$708$	0	0
$709$	−20050.0	−1.06205	−0.531025	−	0.847356i	$-0.678193\pi$
$709$	−20050.0	−1.06205	−0.531025	+	0.847356i	$0.678193\pi$
$710$	0	0
$711$	−6135.77	−0.323642
$712$	0	0
$713$	−29240.0	−1.53583
$714$	0	0
$715$	−2772.72	−0.145027
$716$	0	0
$717$	−2480.00	−0.129173
$718$	0	0
$719$	21126.4	1.09580	0.547900	−	0.836544i	$-0.315427\pi$
$719$	21126.4	1.09580	0.547900	+	0.836544i	$0.315427\pi$
$720$	0	0
$721$	57260.0	2.95766
$722$	0	0
$723$	23031.5	1.18472
$724$	0	0
$725$	−2250.00	−0.115259
$726$	0	0
$727$	−11336.9	−0.578351	−0.289175	−	0.957276i	$-0.593381\pi$
$727$	−11336.9	−0.578351	−0.289175	+	0.957276i	$0.593381\pi$
$728$	0	0
$729$	21797.0	1.10740
$730$	0	0
$731$	3085.77	0.156131
$732$	0	0
$733$	−17198.0	−0.866607	−0.433303	−	0.901248i	$-0.642652\pi$
$733$	−17198.0	−0.866607	−0.433303	+	0.901248i	$0.642652\pi$
$734$	0	0
$735$	−14243.8	−0.714815
$736$	0	0
$737$	−440.000	−0.0219913
$738$	0	0
$739$	4597.36	0.228845	0.114423	−	0.993432i	$-0.463498\pi$
$739$	4597.36	0.228845	0.114423	+	0.993432i	$0.463498\pi$
$740$	0	0
$741$	−29760.0	−1.47539
$742$	0	0
$743$	−2419.43	−0.119462	−0.0597309	−	0.998215i	$-0.519024\pi$
$743$	−2419.43	−0.119462	−0.0597309	+	0.998215i	$0.519024\pi$
$744$	0	0
$745$	1670.00	0.0821263
$746$	0	0
$747$	−2660.92	−0.130332
$748$	0	0
$749$	47740.0	2.32895
$750$	0	0
$751$	7432.69	0.361149	0.180574	−	0.983561i	$-0.442204\pi$
$751$	7432.69	0.361149	0.180574	+	0.983561i	$0.442204\pi$
$752$	0	0
$753$	6200.00	0.300054
$754$	0	0
$755$	−15697.2	−0.756662
$756$	0	0
$757$	11474.0	0.550898	0.275449	−	0.961316i	$-0.411174\pi$
$757$	11474.0	0.550898	0.275449	+	0.961316i	$0.411174\pi$
$758$	0	0
$759$	−7692.07	−0.367858
$760$	0	0
$761$	31802.0	1.51488	0.757439	−	0.652906i	$-0.226450\pi$
$761$	31802.0	1.51488	0.757439	+	0.652906i	$0.226450\pi$
$762$	0	0
$763$	−67430.9	−3.19942
$764$	0	0
$765$	−1610.00	−0.0760911
$766$	0	0
$767$	25509.1	1.20089
$768$	0	0
$769$	−5310.00	−0.249003	−0.124502	−	0.992219i	$-0.539733\pi$
$769$	−5310.00	−0.249003	−0.124502	+	0.992219i	$0.539733\pi$
$770$	0	0
$771$	−18630.9	−0.870267
$772$	0	0
$773$	37938.0	1.76525	0.882623	−	0.470082i	$-0.155776\pi$
$773$	37938.0	1.76525	0.882623	+	0.470082i	$0.155776\pi$
$774$	0	0
$775$	−3801.32	−0.176190
$776$	0	0
$777$	29960.0	1.38328
$778$	0	0
$779$	−1073.31	−0.0493651
$780$	0	0
$781$	3280.00	0.150279
$782$	0	0
$783$	13684.7	0.624588
$784$	0	0
$785$	−4170.00	−0.189597
$786$	0	0
$787$	37633.0	1.70454	0.852270	−	0.523103i	$-0.175226\pi$
$787$	37633.0	1.70454	0.852270	+	0.523103i	$0.175226\pi$
$788$	0	0
$789$	4300.00	0.194023
$790$	0	0
$791$	68307.4	3.07046
$792$	0	0
$793$	−15500.0	−0.694100
$794$	0	0
$795$	15160.5	0.676338
$796$	0	0
$797$	−17526.0	−0.778924	−0.389462	−	0.921042i	$-0.627339\pi$
$797$	−17526.0	−0.778924	−0.389462	+	0.921042i	$0.627339\pi$
$798$	0	0
$799$	−18308.9	−0.810667
$800$	0	0
$801$	−6790.00	−0.299517
$802$	0	0
$803$	−4668.91	−0.205183
$804$	0	0
$805$	30100.0	1.31787
$806$	0	0
$807$	−6681.37	−0.291444
$808$	0	0
$809$	8970.00	0.389825	0.194912	−	0.980821i	$-0.437558\pi$
$809$	8970.00	0.389825	0.194912	+	0.980821i	$0.437558\pi$
$810$	0	0
$811$	−3550.88	−0.153746	−0.0768731	−	0.997041i	$-0.524494\pi$
$811$	−3550.88	−0.153746	−0.0768731	+	0.997041i	$0.524494\pi$
$812$	0	0
$813$	22440.0	0.968026
$814$	0	0
$815$	15451.2	0.664090
$816$	0	0
$817$	−7200.00	−0.308318
$818$	0	0
$819$	−13586.3	−0.579665
$820$	0	0
$821$	−15550.0	−0.661022	−0.330511	−	0.943802i	$-0.607221\pi$
$821$	−15550.0	−0.661022	−0.330511	+	0.943802i	$0.607221\pi$
$822$	0	0
$823$	−26712.1	−1.13138	−0.565689	−	0.824619i	$-0.691390\pi$
$823$	−26712.1	−1.13138	−0.565689	+	0.824619i	$0.691390\pi$
$824$	0	0
$825$	−1000.00	−0.0422006
$826$	0	0
$827$	−863.122	−0.0362923	−0.0181461	−	0.999835i	$-0.505776\pi$
$827$	−863.122	−0.0362923	−0.0181461	+	0.999835i	$0.505776\pi$
$828$	0	0
$829$	19066.0	0.798781	0.399391	−	0.916781i	$-0.369222\pi$
$829$	19066.0	0.798781	0.399391	+	0.916781i	$0.369222\pi$
$830$	0	0
$831$	−4498.97	−0.187807
$832$	0	0
$833$	−29302.0	−1.21879
$834$	0	0
$835$	−22.3607	−0.000926734 0
$836$	0	0
$837$	23120.0	0.954772
$838$	0	0
$839$	−47744.5	−1.96463	−0.982315	−	0.187238i	$-0.940047\pi$
$839$	−47744.5	−1.96463	−0.982315	+	0.187238i	$0.940047\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	0	0
$843$	−14355.6	−0.586514
$844$	0	0
$845$	−8235.00	−0.335258
$846$	0	0
$847$	39162.5	1.58871
$848$	0	0
$849$	16260.0	0.657293
$850$	0	0
$851$	−41152.6	−1.65769
$852$	0	0
$853$	−14462.0	−0.580503	−0.290252	−	0.956950i	$-0.593739\pi$
$853$	−14462.0	−0.580503	−0.290252	+	0.956950i	$0.593739\pi$
$854$	0	0
$855$	3756.59	0.150261
$856$	0	0
$857$	29346.0	1.16971	0.584854	−	0.811138i	$-0.301152\pi$
$857$	29346.0	1.16971	0.584854	+	0.811138i	$0.301152\pi$
$858$	0	0
$859$	22807.9	0.905932	0.452966	−	0.891528i	$-0.350366\pi$
$859$	22807.9	0.905932	0.452966	+	0.891528i	$0.350366\pi$
$860$	0	0
$861$	1400.00	0.0554145
$862$	0	0
$863$	24753.3	0.976375	0.488187	−	0.872739i	$-0.337658\pi$
$863$	24753.3	0.976375	0.488187	+	0.872739i	$0.337658\pi$
$864$	0	0
$865$	9190.00	0.361236
$866$	0	0
$867$	−12508.6	−0.489981
$868$	0	0
$869$	−7840.00	−0.306046
$870$	0	0
$871$	−3050.00	−0.118651
$872$	0	0
$873$	6538.00	0.253468
$874$	0	0
$875$	3913.12	0.151186
$876$	0	0
$877$	−32126.0	−1.23696	−0.618482	−	0.785799i	$-0.712252\pi$
$877$	−32126.0	−1.23696	−0.618482	+	0.785799i	$0.712252\pi$
$878$	0	0
$879$	−16198.1	−0.621556
$880$	0	0
$881$	−33570.0	−1.28377	−0.641885	−	0.766801i	$-0.721848\pi$
$881$	−33570.0	−1.28377	−0.641885	+	0.766801i	$0.721848\pi$
$882$	0	0
$883$	−6435.40	−0.245265	−0.122632	−	0.992452i	$-0.539134\pi$
$883$	−6435.40	−0.245265	−0.122632	+	0.992452i	$0.539134\pi$
$884$	0	0
$885$	9200.00	0.349440
$886$	0	0
$887$	−46827.7	−1.77263	−0.886314	−	0.463084i	$-0.846743\pi$
$887$	−46827.7	−1.77263	−0.886314	+	0.463084i	$0.846743\pi$
$888$	0	0
$889$	−41020.0	−1.54754
$890$	0	0
$891$	4391.64	0.165124
$892$	0	0
$893$	42720.0	1.60086
$894$	0	0
$895$	−9212.60	−0.344071
$896$	0	0
$897$	−53320.0	−1.98473
$898$	0	0
$899$	13684.7	0.507688
$900$	0	0
$901$	31188.0	1.15319
$902$	0	0
$903$	9391.49	0.346101
$904$	0	0
$905$	−9310.00	−0.341961
$906$	0	0
$907$	11980.9	0.438608	0.219304	−	0.975657i	$-0.429621\pi$
$907$	11980.9	0.438608	0.219304	+	0.975657i	$0.429621\pi$
$908$	0	0
$909$	4214.00	0.153762
$910$	0	0
$911$	−24194.3	−0.879903	−0.439951	−	0.898022i	$-0.645004\pi$
$911$	−24194.3	−0.879903	−0.439951	+	0.898022i	$0.645004\pi$
$912$	0	0
$913$	−3400.00	−0.123246
$914$	0	0
$915$	−5590.17	−0.201973
$916$	0	0
$917$	6440.00	0.231917
$918$	0	0
$919$	−37512.3	−1.34648	−0.673240	−	0.739424i	$-0.735098\pi$
$919$	−37512.3	−1.34648	−0.673240	+	0.739424i	$0.735098\pi$
$920$	0	0
$921$	9340.00	0.334162
$922$	0	0
$923$	22736.3	0.810808
$924$	0	0
$925$	−5350.00	−0.190170
$926$	0	0
$927$	12803.7	0.453646
$928$	0	0
$929$	−21994.0	−0.776749	−0.388374	−	0.921502i	$-0.626963\pi$
$929$	−21994.0	−0.776749	−0.388374	+	0.921502i	$0.626963\pi$
$930$	0	0
$931$	68370.0	2.40681
$932$	0	0
$933$	−39800.0	−1.39656
$934$	0	0
$935$	−2057.18	−0.0719541
$936$	0	0
$937$	−16286.0	−0.567813	−0.283906	−	0.958852i	$-0.591630\pi$
$937$	−16286.0	−0.567813	−0.283906	+	0.958852i	$0.591630\pi$
$938$	0	0
$939$	39256.4	1.36431
$940$	0	0
$941$	24302.0	0.841894	0.420947	−	0.907085i	$-0.361698\pi$
$941$	24302.0	0.841894	0.420947	+	0.907085i	$0.361698\pi$
$942$	0	0
$943$	−1923.02	−0.0664073
$944$	0	0
$945$	−23800.0	−0.819274
$946$	0	0
$947$	19869.7	0.681815	0.340907	−	0.940097i	$-0.389266\pi$
$947$	19869.7	0.681815	0.340907	+	0.940097i	$0.389266\pi$
$948$	0	0
$949$	−32364.0	−1.10704
$950$	0	0
$951$	−22566.4	−0.769470
$952$	0	0
$953$	−22422.0	−0.762140	−0.381070	−	0.924546i	$-0.624444\pi$
$953$	−22422.0	−0.762140	−0.381070	+	0.924546i	$0.624444\pi$
$954$	0	0
$955$	10330.6	0.350043
$956$	0	0
$957$	3600.00	0.121600
$958$	0	0
$959$	65552.6	2.20730
$960$	0	0
$961$	−6671.00	−0.223927
$962$	0	0
$963$	10675.0	0.357214
$964$	0	0
$965$	−16890.0	−0.563428
$966$	0	0
$967$	43777.7	1.45584	0.727920	−	0.685662i	$-0.240487\pi$
$967$	43777.7	1.45584	0.727920	+	0.685662i	$0.240487\pi$
$968$	0	0
$969$	−22080.0	−0.732004
$970$	0	0
$971$	−25714.8	−0.849873	−0.424936	−	0.905223i	$-0.639704\pi$
$971$	−25714.8	−0.849873	−0.424936	+	0.905223i	$0.639704\pi$
$972$	0	0
$973$	43120.0	1.42072
$974$	0	0
$975$	−6931.81	−0.227688
$976$	0	0
$977$	28986.0	0.949175	0.474588	−	0.880208i	$-0.342597\pi$
$977$	28986.0	0.949175	0.474588	+	0.880208i	$0.342597\pi$
$978$	0	0
$979$	−8675.94	−0.283232
$980$	0	0
$981$	−15078.0	−0.490727
$982$	0	0
$983$	32123.4	1.04229	0.521147	−	0.853467i	$-0.325504\pi$
$983$	32123.4	1.04229	0.521147	+	0.853467i	$0.325504\pi$
$984$	0	0
$985$	−330.000	−0.0106748
$986$	0	0
$987$	−55722.8	−1.79704
$988$	0	0
$989$	−12900.0	−0.414758
$990$	0	0
$991$	−11994.3	−0.384471	−0.192235	−	0.981349i	$-0.561574\pi$
$991$	−11994.3	−0.384471	−0.192235	+	0.981349i	$0.561574\pi$
$992$	0	0
$993$	−1400.00	−0.0447408
$994$	0	0
$995$	6082.10	0.193785
$996$	0	0
$997$	−406.000	−0.0128968	−0.00644842	−	0.999979i	$-0.502053\pi$
$997$	−406.000	−0.0128968	−0.00644842	+	0.999979i	$0.502053\pi$
$998$	0	0
$999$	32539.3	1.03053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.a.d.1.2	yes	2
3.2	odd	2		1440.4.a.bb.1.1		2
4.3	odd	2	inner	160.4.a.d.1.1	✓	2
5.2	odd	4		800.4.c.l.449.1		4
5.3	odd	4		800.4.c.l.449.3		4
5.4	even	2		800.4.a.o.1.1		2
8.3	odd	2		320.4.a.q.1.2		2
8.5	even	2		320.4.a.q.1.1		2
12.11	even	2		1440.4.a.bb.1.2		2
16.3	odd	4		1280.4.d.v.641.2		4
16.5	even	4		1280.4.d.v.641.1		4
16.11	odd	4		1280.4.d.v.641.3		4
16.13	even	4		1280.4.d.v.641.4		4
20.3	even	4		800.4.c.l.449.2		4
20.7	even	4		800.4.c.l.449.4		4
20.19	odd	2		800.4.a.o.1.2		2
40.19	odd	2		1600.4.a.cg.1.1		2
40.29	even	2		1600.4.a.cg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.d.1.1	✓	2	4.3	odd	2	inner
160.4.a.d.1.2	yes	2	1.1	even	1	trivial
320.4.a.q.1.1		2	8.5	even	2
320.4.a.q.1.2		2	8.3	odd	2
800.4.a.o.1.1		2	5.4	even	2
800.4.a.o.1.2		2	20.19	odd	2
800.4.c.l.449.1		4	5.2	odd	4
800.4.c.l.449.2		4	20.3	even	4
800.4.c.l.449.3		4	5.3	odd	4
800.4.c.l.449.4		4	20.7	even	4
1280.4.d.v.641.1		4	16.5	even	4
1280.4.d.v.641.2		4	16.3	odd	4
1280.4.d.v.641.3		4	16.11	odd	4
1280.4.d.v.641.4		4	16.13	even	4
1440.4.a.bb.1.1		2	3.2	odd	2
1440.4.a.bb.1.2		2	12.11	even	2
1600.4.a.cg.1.1		2	40.19	odd	2
1600.4.a.cg.1.2		2	40.29	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)

\(f(q)\)	\(=\)	\(q+4.47214 q^{3} -5.00000 q^{5} -31.3050 q^{7} -7.00000 q^{9} +O(q^{10})\)	\(q+4.47214 q^{3} -5.00000 q^{5} -31.3050 q^{7} -7.00000 q^{9} -8.94427 q^{11} -62.0000 q^{13} -22.3607 q^{15} -46.0000 q^{17} +107.331 q^{19} -140.000 q^{21} +192.302 q^{23} +25.0000 q^{25} -152.053 q^{27} -90.0000 q^{29} -152.053 q^{31} -40.0000 q^{33} +156.525 q^{35} -214.000 q^{37} -277.272 q^{39} -10.0000 q^{41} -67.0820 q^{43} +35.0000 q^{45} +398.020 q^{47} +637.000 q^{49} -205.718 q^{51} -678.000 q^{53} +44.7214 q^{55} +480.000 q^{57} -411.437 q^{59} +250.000 q^{61} +219.135 q^{63} +310.000 q^{65} +49.1935 q^{67} +860.000 q^{69} -366.715 q^{71} +522.000 q^{73} +111.803 q^{75} +280.000 q^{77} +876.539 q^{79} -491.000 q^{81} +380.132 q^{83} +230.000 q^{85} -402.492 q^{87} +970.000 q^{89} +1940.91 q^{91} -680.000 q^{93} -536.656 q^{95} -934.000 q^{97} +62.6099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} - 14 q^{9}+O(q^{10}) \) 2 * q - 10 * q^5 - 14 * q^9	\( 2 q - 10 q^{5} - 14 q^{9} - 124 q^{13} - 92 q^{17} - 280 q^{21} + 50 q^{25} - 180 q^{29} - 80 q^{33} - 428 q^{37} - 20 q^{41} + 70 q^{45} + 1274 q^{49} - 1356 q^{53} + 960 q^{57} + 500 q^{61} + 620 q^{65} + 1720 q^{69} + 1044 q^{73} + 560 q^{77} - 982 q^{81} + 460 q^{85} + 1940 q^{89} - 1360 q^{93} - 1868 q^{97}+O(q^{100}) \) 2 * q - 10 * q^5 - 14 * q^9 - 124 * q^13 - 92 * q^17 - 280 * q^21 + 50 * q^25 - 180 * q^29 - 80 * q^33 - 428 * q^37 - 20 * q^41 + 70 * q^45 + 1274 * q^49 - 1356 * q^53 + 960 * q^57 + 500 * q^61 + 620 * q^65 + 1720 * q^69 + 1044 * q^73 + 560 * q^77 - 982 * q^81 + 460 * q^85 + 1940 * q^89 - 1360 * q^93 - 1868 * q^97

Introduction

L-functions

Modular forms

Varieties

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