LMFDB - Embedded newform 160.4.c.b.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(129,160)]

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, 1]))

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

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{29})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 15x^{2} + 49 $ x^4 + 15*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-3.19258i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.4.c.b.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{3} +(-3.00000 - 10.7703i) q^{5} -4.00000i q^{7} +11.0000 q^{9} -43.0813 q^{11} -21.5407i q^{13} +(-43.0813 + 12.0000i) q^{15} +43.0813i q^{17} -129.244 q^{19} -16.0000 q^{21} -52.0000i q^{23} +(-107.000 + 64.6220i) q^{25} -152.000i q^{27} +158.000 q^{29} -172.325 q^{31} +172.325i q^{33} +(-43.0813 + 12.0000i) q^{35} -280.029i q^{37} -86.1626 q^{39} -170.000 q^{41} +316.000i q^{43} +(-33.0000 - 118.474i) q^{45} -244.000i q^{47} +327.000 q^{49} +172.325 q^{51} -495.435i q^{53} +(129.244 + 464.000i) q^{55} +516.976i q^{57} +646.220 q^{59} +82.0000 q^{61} -44.0000i q^{63} +(-232.000 + 64.6220i) q^{65} -692.000i q^{67} -208.000 q^{69} +947.789 q^{71} +430.813i q^{73} +(258.488 + 428.000i) q^{75} +172.325i q^{77} +344.651 q^{79} -311.000 q^{81} +940.000i q^{83} +(464.000 - 129.244i) q^{85} -632.000i q^{87} -6.00000 q^{89} -86.1626 q^{91} +689.301i q^{93} +(387.732 + 1392.00i) q^{95} -1077.03i q^{97} -473.895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{5} + 44 q^{9} - 64 q^{21} - 428 q^{25} + 632 q^{29} - 680 q^{41} - 132 q^{45} + 1308 q^{49} + 328 q^{61} - 928 q^{65} - 832 q^{69} - 1244 q^{81} + 1856 q^{85} - 24 q^{89}+O(q^{100}) $ 4 * q - 12 * q^5 + 44 * q^9 - 64 * q^21 - 428 * q^25 + 632 * q^29 - 680 * q^41 - 132 * q^45 + 1308 * q^49 + 328 * q^61 - 928 * q^65 - 832 * q^69 - 1244 * q^81 + 1856 * q^85 - 24 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	4.00000i		−	0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$		−	4.00000i		−	0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$		0			0
$5$	−3.00000	−	10.7703i	−0.268328	−	0.963328i
$5$	−3.00000	−	10.7703i	−0.268328	−	0.963328i
$6$		0			0
$7$		−	4.00000i		−	0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$		−	4.00000i		−	0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$		0			0
$9$	11.0000			0.407407
$10$		0			0
$11$	−43.0813			−1.18086			−0.590432	−	0.807087i	$-0.701043\pi$
$11$	−43.0813			−1.18086			−0.590432	+	0.807087i	$0.701043\pi$
$12$		0			0
$13$		−	21.5407i		−	0.459562i	−0.973242	−	0.229781i	$-0.926199\pi$
$13$		−	21.5407i		−	0.459562i	0.973242	−	0.229781i	$-0.0738010\pi$
$14$		0			0
$15$	−43.0813	+	12.0000i	−0.741570	+	0.206559i
$16$		0			0
$17$			43.0813i			0.614633i	0.951607	+	0.307316i	$0.0994310\pi$
$17$			43.0813i			0.614633i	−0.951607	+	0.307316i	$0.900569\pi$
$18$		0			0
$19$	−129.244			−1.56056			−0.780279	−	0.625432i	$-0.784923\pi$
$19$	−129.244			−1.56056			−0.780279	+	0.625432i	$0.784923\pi$
$20$		0			0
$21$	−16.0000			−0.166261
$22$		0			0
$23$		−	52.0000i		−	0.471424i	−0.971823	−	0.235712i	$-0.924258\pi$
$23$		−	52.0000i		−	0.471424i	0.971823	−	0.235712i	$-0.0757422\pi$
$24$		0			0
$25$	−107.000	+	64.6220i	−0.856000	+	0.516976i
$26$		0			0
$27$		−	152.000i		−	1.08342i
$28$		0			0
$29$	158.000			1.01172			0.505860	−	0.862616i	$-0.331175\pi$
$29$	158.000			1.01172			0.505860	+	0.862616i	$0.331175\pi$
$30$		0			0
$31$	−172.325			−0.998404			−0.499202	−	0.866486i	$-0.666373\pi$
$31$	−172.325			−0.998404			−0.499202	+	0.866486i	$0.666373\pi$
$32$		0			0
$33$			172.325i			0.909030i
$34$		0			0
$35$	−43.0813	+	12.0000i	−0.208059	+	0.0579534i
$36$		0			0
$37$		−	280.029i		−	1.24423i	−0.782927	−	0.622114i	$-0.786274\pi$
$37$		−	280.029i		−	1.24423i	0.782927	−	0.622114i	$-0.213726\pi$
$38$		0			0
$39$	−86.1626			−0.353771
$40$		0			0
$41$	−170.000			−0.647550			−0.323775	−	0.946134i	$-0.604952\pi$
$41$	−170.000			−0.647550			−0.323775	+	0.946134i	$0.604952\pi$
$42$		0			0
$43$			316.000i			1.12069i	0.828260	+	0.560344i	$0.189331\pi$
$43$			316.000i			1.12069i	−0.828260	+	0.560344i	$0.810669\pi$
$44$		0			0
$45$	−33.0000	−	118.474i	−0.109319	−	0.392467i
$46$		0			0
$47$		−	244.000i		−	0.757257i	−0.925549	−	0.378628i	$-0.876396\pi$
$47$		−	244.000i		−	0.757257i	0.925549	−	0.378628i	$-0.123604\pi$
$48$		0			0
$49$	327.000			0.953353
$50$		0			0
$51$	172.325			0.473144
$52$		0			0
$53$		−	495.435i		−	1.28402i	−0.766695	−	0.642012i	$-0.778100\pi$
$53$		−	495.435i		−	1.28402i	0.766695	−	0.642012i	$-0.221900\pi$
$54$		0			0
$55$	129.244	+	464.000i	0.316859	+	1.13756i
$56$		0			0
$57$			516.976i			1.20132i
$58$		0			0
$59$	646.220			1.42594			0.712972	−	0.701193i	$-0.247349\pi$
$59$	646.220			1.42594			0.712972	+	0.701193i	$0.247349\pi$
$60$		0			0
$61$	82.0000			0.172115			0.0860576	−	0.996290i	$-0.472573\pi$
$61$	82.0000			0.172115			0.0860576	+	0.996290i	$0.472573\pi$
$62$		0			0
$63$		−	44.0000i		−	0.0879917i
$64$		0			0
$65$	−232.000	+	64.6220i	−0.442709	+	0.123313i
$66$		0			0
$67$		−	692.000i		−	1.26181i	−0.775860	−	0.630905i	$-0.782684\pi$
$67$		−	692.000i		−	1.26181i	0.775860	−	0.630905i	$-0.217316\pi$
$68$		0			0
$69$	−208.000			−0.362902
$70$		0			0
$71$	947.789			1.58425			0.792126	−	0.610358i	$-0.208974\pi$
$71$	947.789			1.58425			0.792126	+	0.610358i	$0.208974\pi$
$72$		0			0
$73$			430.813i			0.690724i	0.938470	+	0.345362i	$0.112244\pi$
$73$			430.813i			0.690724i	−0.938470	+	0.345362i	$0.887756\pi$
$74$		0			0
$75$	258.488	+	428.000i	0.397968	+	0.658949i
$76$		0			0
$77$			172.325i			0.255043i
$78$		0			0
$79$	344.651			0.490838			0.245419	−	0.969417i	$-0.421074\pi$
$79$	344.651			0.490838			0.245419	+	0.969417i	$0.421074\pi$
$80$		0			0
$81$	−311.000			−0.426612
$82$		0			0
$83$			940.000i			1.24311i	0.783369	+	0.621557i	$0.213499\pi$
$83$			940.000i			1.24311i	−0.783369	+	0.621557i	$0.786501\pi$
$84$		0			0
$85$	464.000	−	129.244i	0.592093	−	0.164923i
$86$		0			0
$87$		−	632.000i		−	0.778822i
$88$		0			0
$89$	−6.00000			−0.00714605			−0.00357303	−	0.999994i	$-0.501137\pi$
$89$	−6.00000			−0.00714605			−0.00357303	+	0.999994i	$0.501137\pi$
$90$		0			0
$91$	−86.1626			−0.0992560
$92$		0			0
$93$			689.301i			0.768572i
$94$		0			0
$95$	387.732	+	1392.00i	0.418742	+	1.50333i
$96$		0			0
$97$		−	1077.03i		−	1.12738i	−0.825985	−	0.563691i	$-0.809381\pi$
$97$		−	1077.03i		−	1.12738i	0.825985	−	0.563691i	$-0.190619\pi$
$98$		0			0
$99$	−473.895			−0.481093
$100$		0			0
$101$	1014.00			0.998978			0.499489	−	0.866320i	$-0.333521\pi$
$101$	1014.00			0.998978			0.499489	+	0.866320i	$0.333521\pi$
$102$		0			0
$103$			44.0000i			0.0420917i	0.999779	+	0.0210459i	$0.00669960\pi$
$103$			44.0000i			0.0420917i	−0.999779	+	0.0210459i	$0.993300\pi$
$104$		0			0
$105$	48.0000	+	172.325i	0.0446126	+	0.160164i
$106$		0			0
$107$		−	1812.00i		−	1.63713i	−0.574416	−	0.818564i	$-0.694771\pi$
$107$		−	1812.00i		−	1.63713i	0.574416	−	0.818564i	$-0.305229\pi$
$108$		0			0
$109$	−2014.00			−1.76978			−0.884891	−	0.465798i	$-0.845767\pi$
$109$	−2014.00			−1.76978			−0.884891	+	0.465798i	$0.845767\pi$
$110$		0			0
$111$	−1120.11			−0.957807
$112$		0			0
$113$		−	1637.09i		−	1.36287i	−0.731878	−	0.681436i	$-0.761356\pi$
$113$		−	1637.09i		−	1.36287i	0.731878	−	0.681436i	$-0.238644\pi$
$114$		0			0
$115$	−560.057	+	156.000i	−0.454136	+	0.126496i
$116$		0			0
$117$		−	236.947i		−	0.187229i
$118$		0			0
$119$	172.325			0.132748
$120$		0			0
$121$	525.000			0.394440
$122$		0			0
$123$			680.000i			0.498484i
$124$		0			0
$125$	1017.00	+	958.559i	0.727706	+	0.685889i
$126$		0			0
$127$		−	2788.00i		−	1.94799i	−0.226565	−	0.973996i	$-0.572750\pi$
$127$		−	2788.00i		−	1.94799i	0.226565	−	0.973996i	$-0.427250\pi$
$128$		0			0
$129$	1264.00			0.862705
$130$		0			0
$131$	43.0813			0.0287331			0.0143665	−	0.999897i	$-0.495427\pi$
$131$	43.0813			0.0287331			0.0143665	+	0.999897i	$0.495427\pi$
$132$		0			0
$133$			516.976i			0.337049i
$134$		0			0
$135$	−1637.09	+	456.000i	−1.04369	+	0.290713i
$136$		0			0
$137$			473.895i			0.295529i	0.989023	+	0.147765i	$0.0472078\pi$
$137$			473.895i			0.295529i	−0.989023	+	0.147765i	$0.952792\pi$
$138$		0			0
$139$	−2110.98			−1.28814			−0.644070	−	0.764967i	$-0.722755\pi$
$139$	−2110.98			−1.28814			−0.644070	+	0.764967i	$0.722755\pi$
$140$		0			0
$141$	−976.000			−0.582936
$142$		0			0
$143$			928.000i			0.542680i
$144$		0			0
$145$	−474.000	−	1701.71i	−0.271473	−	0.974617i
$146$		0			0
$147$		−	1308.00i		−	0.733891i
$148$		0			0
$149$	−2550.00			−1.40204			−0.701021	−	0.713141i	$-0.747272\pi$
$149$	−2550.00			−1.40204			−0.701021	+	0.713141i	$0.747272\pi$
$150$		0			0
$151$	−2154.07			−1.16090			−0.580448	−	0.814297i	$-0.697123\pi$
$151$	−2154.07			−1.16090			−0.580448	+	0.814297i	$0.697123\pi$
$152$		0			0
$153$			473.895i			0.250406i
$154$		0			0
$155$	516.976	+	1856.00i	0.267900	+	0.961790i
$156$		0			0
$157$			3554.21i			1.80673i	0.428872	+	0.903365i	$0.358911\pi$
$157$			3554.21i			1.80673i	−0.428872	+	0.903365i	$0.641089\pi$
$158$		0			0
$159$	−1981.74			−0.988442
$160$		0			0
$161$	−208.000			−0.101818
$162$		0			0
$163$		−	228.000i		−	0.109560i	−0.998498	−	0.0547802i	$-0.982554\pi$
$163$		−	228.000i		−	0.109560i	0.998498	−	0.0547802i	$-0.0174458\pi$
$164$		0			0
$165$	1856.00	−	516.976i	0.875693	−	0.243918i
$166$		0			0
$167$		−	372.000i		−	0.172373i	−0.996279	−	0.0861863i	$-0.972532\pi$
$167$		−	372.000i		−	0.172373i	0.996279	−	0.0861863i	$-0.0274680\pi$
$168$		0			0
$169$	1733.00			0.788803
$170$		0			0
$171$	−1421.68			−0.635783
$172$		0			0
$173$			1917.12i			0.842519i	0.906940	+	0.421260i	$0.138412\pi$
$173$			1917.12i			0.842519i	−0.906940	+	0.421260i	$0.861588\pi$
$174$		0			0
$175$	258.488	+	428.000i	0.111656	+	0.184879i
$176$		0			0
$177$		−	2584.88i		−	1.09769i
$178$		0			0
$179$	2800.29			1.16929			0.584646	−	0.811289i	$-0.301234\pi$
$179$	2800.29			1.16929			0.584646	+	0.811289i	$0.301234\pi$
$180$		0			0
$181$	1542.00			0.633237			0.316619	−	0.948553i	$-0.397452\pi$
$181$	1542.00			0.633237			0.316619	+	0.948553i	$0.397452\pi$
$182$		0			0
$183$		−	328.000i		−	0.132494i
$184$		0			0
$185$	−3016.00	+	840.086i	−1.19860	+	0.333861i
$186$		0			0
$187$		−	1856.00i		−	0.725798i
$188$		0			0
$189$	−608.000			−0.233997
$190$		0			0
$191$	4480.46			1.69735			0.848677	−	0.528912i	$-0.177400\pi$
$191$	4480.46			1.69735			0.848677	+	0.528912i	$0.177400\pi$
$192$		0			0
$193$		−	2541.80i		−	0.947993i	−0.880527	−	0.473996i	$-0.842811\pi$
$193$		−	2541.80i		−	0.947993i	0.880527	−	0.473996i	$-0.157189\pi$
$194$		0			0
$195$	258.488	+	928.000i	0.0949267	+	0.340797i
$196$		0			0
$197$		−	2735.66i		−	0.989381i	−0.869069	−	0.494690i	$-0.835282\pi$
$197$		−	2735.66i		−	0.989381i	0.869069	−	0.494690i	$-0.164718\pi$
$198$		0			0
$199$	258.488			0.0920790			0.0460395	−	0.998940i	$-0.485340\pi$
$199$	258.488			0.0920790			0.0460395	+	0.998940i	$0.485340\pi$
$200$		0			0
$201$	−2768.00			−0.971342
$202$		0			0
$203$		−	632.000i		−	0.218511i
$204$		0			0
$205$	510.000	+	1830.96i	0.173756	+	0.623803i
$206$		0			0
$207$		−	572.000i		−	0.192062i
$208$		0			0
$209$	5568.00			1.84281
$210$		0			0
$211$	904.708			0.295178			0.147589	−	0.989049i	$-0.452849\pi$
$211$	904.708			0.295178			0.147589	+	0.989049i	$0.452849\pi$
$212$		0			0
$213$		−	3791.16i		−	1.21956i
$214$		0			0
$215$	3403.42	−	948.000i	1.07959	−	0.300712i
$216$		0			0
$217$			689.301i			0.215635i
$218$		0			0
$219$	1723.25			0.531720
$220$		0			0
$221$	928.000			0.282462
$222$		0			0
$223$			4284.00i			1.28645i	0.765678	+	0.643224i	$0.222404\pi$
$223$			4284.00i			1.28645i	−0.765678	+	0.643224i	$0.777596\pi$
$224$		0			0
$225$	−1177.00	+	710.842i	−0.348741	+	0.210620i
$226$		0			0
$227$			4956.00i			1.44908i	0.689232	+	0.724540i	$0.257948\pi$
$227$			4956.00i			1.44908i	−0.689232	+	0.724540i	$0.742052\pi$
$228$		0			0
$229$	−1770.00			−0.510764			−0.255382	−	0.966840i	$-0.582201\pi$
$229$	−1770.00			−0.510764			−0.255382	+	0.966840i	$0.582201\pi$
$230$		0			0
$231$	689.301			0.196332
$232$		0			0
$233$			5428.25i			1.52625i	0.646251	+	0.763125i	$0.276336\pi$
$233$			5428.25i			1.52625i	−0.646251	+	0.763125i	$0.723664\pi$
$234$		0			0
$235$	−2627.96	+	732.000i	−0.729486	+	0.203193i
$236$		0			0
$237$		−	1378.60i		−	0.377847i
$238$		0			0
$239$	5514.41			1.49246			0.746229	−	0.665689i	$-0.231862\pi$
$239$	5514.41			1.49246			0.746229	+	0.665689i	$0.231862\pi$
$240$		0			0
$241$	−1618.00			−0.432467			−0.216233	−	0.976342i	$-0.569377\pi$
$241$	−1618.00			−0.432467			−0.216233	+	0.976342i	$0.569377\pi$
$242$		0			0
$243$		−	2860.00i		−	0.755017i
$244$		0			0
$245$	−981.000	−	3521.90i	−0.255811	−	0.918391i
$246$		0			0
$247$			2784.00i			0.717173i
$248$		0			0
$249$	3760.00			0.956949
$250$		0			0
$251$	4954.35			1.24588			0.622940	−	0.782270i	$-0.285938\pi$
$251$	4954.35			1.24588			0.622940	+	0.782270i	$0.285938\pi$
$252$		0			0
$253$			2240.23i			0.556688i
$254$		0			0
$255$	−516.976	−	1856.00i	−0.126958	−	0.455793i
$256$		0			0
$257$			4652.78i			1.12931i	0.825327	+	0.564655i	$0.190991\pi$
$257$			4652.78i			1.12931i	−0.825327	+	0.564655i	$0.809009\pi$
$258$		0			0
$259$	−1120.11			−0.268728
$260$		0			0
$261$	1738.00			0.412182
$262$		0			0
$263$			604.000i			0.141613i	0.997490	+	0.0708065i	$0.0225573\pi$
$263$			604.000i			0.141613i	−0.997490	+	0.0708065i	$0.977443\pi$
$264$		0			0
$265$	−5336.00	+	1486.31i	−1.23694	+	0.344540i
$266$		0			0
$267$			24.0000i			0.00550103i
$268$		0			0
$269$	−3262.00			−0.739359			−0.369680	−	0.929159i	$-0.620533\pi$
$269$	−3262.00			−0.739359			−0.369680	+	0.929159i	$0.620533\pi$
$270$		0			0
$271$	−2067.90			−0.463528			−0.231764	−	0.972772i	$-0.574450\pi$
$271$	−2067.90			−0.463528			−0.231764	+	0.972772i	$0.574450\pi$
$272$		0			0
$273$			344.651i			0.0764073i
$274$		0			0
$275$	4609.70	−	2784.00i	1.01082	−	0.610478i
$276$		0			0
$277$		−	1227.82i		−	0.266326i	−0.991094	−	0.133163i	$-0.957487\pi$
$277$		−	1227.82i		−	0.266326i	0.991094	−	0.133163i	$-0.0425134\pi$
$278$		0			0
$279$	−1895.58			−0.406757
$280$		0			0
$281$	−1290.00			−0.273861			−0.136931	−	0.990581i	$-0.543724\pi$
$281$	−1290.00			−0.273861			−0.136931	+	0.990581i	$0.543724\pi$
$282$		0			0
$283$		−	692.000i		−	0.145354i	−0.997356	−	0.0726769i	$-0.976846\pi$
$283$		−	692.000i		−	0.145354i	0.997356	−	0.0726769i	$-0.0231542\pi$
$284$		0			0
$285$	5568.00	−	1550.93i	1.15726	−	0.322347i
$286$		0			0
$287$			680.000i			0.139858i
$288$		0			0
$289$	3057.00			0.622227
$290$		0			0
$291$	−4308.13			−0.867860
$292$		0			0
$293$		−	5535.95i		−	1.10380i	−0.833910	−	0.551900i	$-0.813903\pi$
$293$		−	5535.95i		−	1.10380i	0.833910	−	0.551900i	$-0.186097\pi$
$294$		0			0
$295$	−1938.66	−	6960.00i	−0.382621	−	1.37365i
$296$		0			0
$297$			6548.36i			1.27938i
$298$		0			0
$299$	−1120.11			−0.216648
$300$		0			0
$301$	1264.00			0.242046
$302$		0			0
$303$		−	4056.00i		−	0.769014i
$304$		0			0
$305$	−246.000	−	883.167i	−0.0461833	−	0.165803i
$306$		0			0
$307$			7436.00i			1.38239i	0.722666	+	0.691197i	$0.242916\pi$
$307$			7436.00i			1.38239i	−0.722666	+	0.691197i	$0.757084\pi$
$308$		0			0
$309$	176.000			0.0324022
$310$		0			0
$311$	−2498.72			−0.455592			−0.227796	−	0.973709i	$-0.573152\pi$
$311$	−2498.72			−0.455592			−0.227796	+	0.973709i	$0.573152\pi$
$312$		0			0
$313$		−	2714.12i		−	0.490132i	−0.969506	−	0.245066i	$-0.921190\pi$
$313$		−	2714.12i		−	0.490132i	0.969506	−	0.245066i	$-0.0788096\pi$
$314$		0			0
$315$	−473.895	+	132.000i	−0.0847649	+	0.0236107i
$316$		0			0
$317$		−	3295.72i		−	0.583931i	−0.956429	−	0.291965i	$-0.905691\pi$
$317$		−	3295.72i		−	0.583931i	0.956429	−	0.291965i	$-0.0943092\pi$
$318$		0			0
$319$	−6806.85			−1.19470
$320$		0			0
$321$	−7248.00			−1.26026
$322$		0			0
$323$		−	5568.00i		−	0.959170i
$324$		0			0
$325$	1392.00	+	2304.85i	0.237582	+	0.393385i
$326$		0			0
$327$			8056.00i			1.36238i
$328$		0			0
$329$	−976.000			−0.163552
$330$		0			0
$331$	−2972.61			−0.493624			−0.246812	−	0.969063i	$-0.579383\pi$
$331$	−2972.61			−0.493624			−0.246812	+	0.969063i	$0.579383\pi$
$332$		0			0
$333$		−	3080.31i		−	0.506907i
$334$		0			0
$335$	−7453.07	+	2076.00i	−1.21554	+	0.338579i
$336$		0			0
$337$		−	9133.24i		−	1.47632i	−0.674627	−	0.738159i	$-0.735695\pi$
$337$		−	9133.24i		−	1.47632i	0.674627	−	0.738159i	$-0.264305\pi$
$338$		0			0
$339$	−6548.36			−1.04914
$340$		0			0
$341$	7424.00			1.17898
$342$		0			0
$343$		−	2680.00i		−	0.421885i
$344$		0			0
$345$	624.000	+	2240.23i	0.0973769	+	0.349594i
$346$		0			0
$347$			5916.00i			0.915238i	0.889148	+	0.457619i	$0.151298\pi$
$347$			5916.00i			0.915238i	−0.889148	+	0.457619i	$0.848702\pi$
$348$		0			0
$349$	−7522.00			−1.15371			−0.576853	−	0.816848i	$-0.695719\pi$
$349$	−7522.00			−1.15371			−0.576853	+	0.816848i	$0.695719\pi$
$350$		0			0
$351$	−3274.18			−0.497900
$352$		0			0
$353$			8185.45i			1.23419i	0.786890	+	0.617093i	$0.211690\pi$
$353$			8185.45i			1.23419i	−0.786890	+	0.617093i	$0.788310\pi$
$354$		0			0
$355$	−2843.37	−	10208.0i	−0.425099	−	1.52615i
$356$		0			0
$357$		−	689.301i		−	0.102190i
$358$		0			0
$359$	1464.76			0.215341			0.107670	−	0.994187i	$-0.465661\pi$
$359$	1464.76			0.215341			0.107670	+	0.994187i	$0.465661\pi$
$360$		0			0
$361$	9845.00			1.43534
$362$		0			0
$363$		−	2100.00i		−	0.303640i
$364$		0			0
$365$	4640.00	−	1292.44i	0.665394	−	0.185341i
$366$		0			0
$367$		−	516.000i		−	0.0733923i	−0.999326	−	0.0366962i	$-0.988317\pi$
$367$		−	516.000i		−	0.0733923i	0.999326	−	0.0366962i	$-0.0116834\pi$
$368$		0			0
$369$	−1870.00			−0.263817
$370$		0			0
$371$	−1981.74			−0.277323
$372$		0			0
$373$			8077.75i			1.12131i	0.828048	+	0.560657i	$0.189451\pi$
$373$			8077.75i			1.12131i	−0.828048	+	0.560657i	$0.810549\pi$
$374$		0			0
$375$	3834.24	−	4068.00i	0.527998	−	0.560188i
$376$		0			0
$377$		−	3403.42i		−	0.464948i
$378$		0			0
$379$	2197.15			0.297783			0.148892	−	0.988854i	$-0.452429\pi$
$379$	2197.15			0.297783			0.148892	+	0.988854i	$0.452429\pi$
$380$		0			0
$381$	−11152.0			−1.49957
$382$		0			0
$383$		−	5988.00i		−	0.798884i	−0.916759	−	0.399442i	$-0.869204\pi$
$383$		−	5988.00i		−	0.798884i	0.916759	−	0.399442i	$-0.130796\pi$
$384$		0			0
$385$	1856.00	−	516.976i	0.245690	−	0.0684351i
$386$		0			0
$387$			3476.00i			0.456576i
$388$		0			0
$389$	−7974.00			−1.03933			−0.519663	−	0.854371i	$-0.673943\pi$
$389$	−7974.00			−1.03933			−0.519663	+	0.854371i	$0.673943\pi$
$390$		0			0
$391$	2240.23			0.289753
$392$		0			0
$393$		−	172.325i		−	0.0221187i
$394$		0			0
$395$	−1033.95	−	3712.00i	−0.131706	−	0.472838i
$396$		0			0
$397$			7991.58i			1.01029i	0.863034	+	0.505146i	$0.168561\pi$
$397$			7991.58i			1.01029i	−0.863034	+	0.505146i	$0.831439\pi$
$398$		0			0
$399$	2067.90			0.259460
$400$		0			0
$401$	418.000			0.0520547			0.0260273	−	0.999661i	$-0.491714\pi$
$401$	418.000			0.0520547			0.0260273	+	0.999661i	$0.491714\pi$
$402$		0			0
$403$			3712.00i			0.458829i
$404$		0			0
$405$	933.000	+	3349.57i	0.114472	+	0.410967i
$406$		0			0
$407$			12064.0i			1.46926i
$408$		0			0
$409$	1414.00			0.170948			0.0854741	−	0.996340i	$-0.472760\pi$
$409$	1414.00			0.170948			0.0854741	+	0.996340i	$0.472760\pi$
$410$		0			0
$411$	1895.58			0.227499
$412$		0			0
$413$		−	2584.88i		−	0.307975i
$414$		0			0
$415$	10124.1	−	2820.00i	1.19753	−	0.333562i
$416$		0			0
$417$			8443.94i			0.991610i
$418$		0			0
$419$	3661.91			0.426960			0.213480	−	0.976947i	$-0.431520\pi$
$419$	3661.91			0.426960			0.213480	+	0.976947i	$0.431520\pi$
$420$		0			0
$421$	3290.00			0.380866			0.190433	−	0.981700i	$-0.439011\pi$
$421$	3290.00			0.380866			0.190433	+	0.981700i	$0.439011\pi$
$422$		0			0
$423$		−	2684.00i		−	0.308512i
$424$		0			0
$425$	−2784.00	−	4609.70i	−0.317750	−	0.526126i
$426$		0			0
$427$		−	328.000i		−	0.0371734i
$428$		0			0
$429$	3712.00			0.417755
$430$		0			0
$431$	1723.25			0.192590			0.0962949	−	0.995353i	$-0.469301\pi$
$431$	1723.25			0.192590			0.0962949	+	0.995353i	$0.469301\pi$
$432$		0			0
$433$		−	16414.0i		−	1.82172i	−0.412713	−	0.910861i	$-0.635419\pi$
$433$		−	16414.0i		−	1.82172i	0.412713	−	0.910861i	$-0.364581\pi$
$434$		0			0
$435$	−6806.85	+	1896.00i	−0.750261	+	0.208980i
$436$		0			0
$437$			6720.69i			0.735684i
$438$		0			0
$439$	3704.99			0.402801			0.201401	−	0.979509i	$-0.435451\pi$
$439$	3704.99			0.402801			0.201401	+	0.979509i	$0.435451\pi$
$440$		0			0
$441$	3597.00			0.388403
$442$		0			0
$443$			5484.00i			0.588155i	0.955782	+	0.294078i	$0.0950124\pi$
$443$			5484.00i			0.588155i	−0.955782	+	0.294078i	$0.904988\pi$
$444$		0			0
$445$	18.0000	+	64.6220i	0.00191749	+	0.00688399i
$446$		0			0
$447$			10200.0i			1.07929i
$448$		0			0
$449$	−3458.00			−0.363459			−0.181730	−	0.983349i	$-0.558170\pi$
$449$	−3458.00			−0.363459			−0.181730	+	0.983349i	$0.558170\pi$
$450$		0			0
$451$	7323.82			0.764668
$452$		0			0
$453$			8616.26i			0.893659i
$454$		0			0
$455$	258.488	+	928.000i	0.0266332	+	0.0956161i
$456$		0			0
$457$		−	10468.8i		−	1.07157i	−0.844354	−	0.535786i	$-0.820016\pi$
$457$		−	10468.8i		−	1.07157i	0.844354	−	0.535786i	$-0.179984\pi$
$458$		0			0
$459$	6548.36			0.665907
$460$		0			0
$461$	5806.00			0.586578			0.293289	−	0.956024i	$-0.405250\pi$
$461$	5806.00			0.586578			0.293289	+	0.956024i	$0.405250\pi$
$462$		0			0
$463$			5004.00i			0.502280i	0.967951	+	0.251140i	$0.0808054\pi$
$463$			5004.00i			0.502280i	−0.967951	+	0.251140i	$0.919195\pi$
$464$		0			0
$465$	7424.00	−	2067.90i	0.740387	−	0.206230i
$466$		0			0
$467$			5148.00i			0.510109i	0.966927	+	0.255055i	$0.0820935\pi$
$467$			5148.00i			0.510109i	−0.966927	+	0.255055i	$0.917907\pi$
$468$		0			0
$469$	−2768.00			−0.272525
$470$		0			0
$471$	14216.8			1.39082
$472$		0			0
$473$		−	13613.7i		−	1.32338i
$474$		0			0
$475$	13829.1	−	8352.00i	1.33584	−	0.806771i
$476$		0			0
$477$		−	5449.79i		−	0.523121i
$478$		0			0
$479$	−4480.46			−0.427385			−0.213692	−	0.976901i	$-0.568549\pi$
$479$	−4480.46			−0.427385			−0.213692	+	0.976901i	$0.568549\pi$
$480$		0			0
$481$	−6032.00			−0.571799
$482$		0			0
$483$			832.000i			0.0783795i
$484$		0			0
$485$	−11600.0	+	3231.10i	−1.08604	+	0.302509i
$486$		0			0
$487$		−	3044.00i		−	0.283238i	−0.989921	−	0.141619i	$-0.954769\pi$
$487$		−	3044.00i		−	0.283238i	0.989921	−	0.141619i	$-0.0452308\pi$
$488$		0			0
$489$	−912.000			−0.0843396
$490$		0			0
$491$	−215.407			−0.0197987			−0.00989935	−	0.999951i	$-0.503151\pi$
$491$	−215.407			−0.0197987			−0.00989935	+	0.999951i	$0.503151\pi$
$492$		0			0
$493$			6806.85i			0.621836i
$494$		0			0
$495$	1421.68	+	5104.00i	0.129091	+	0.463450i
$496$		0			0
$497$		−	3791.16i		−	0.342166i
$498$		0			0
$499$	11416.5			1.02420			0.512099	−	0.858926i	$-0.328868\pi$
$499$	11416.5			1.02420			0.512099	+	0.858926i	$0.328868\pi$
$500$		0			0
$501$	−1488.00			−0.132692
$502$		0			0
$503$		−	15348.0i		−	1.36050i	−0.732978	−	0.680252i	$-0.761870\pi$
$503$		−	15348.0i		−	1.36050i	0.732978	−	0.680252i	$-0.238130\pi$
$504$		0			0
$505$	−3042.00	−	10921.1i	−0.268054	−	0.962343i
$506$		0			0
$507$		−	6932.00i		−	0.607221i
$508$		0			0
$509$	17502.0			1.52409			0.762046	−	0.647523i	$-0.224195\pi$
$509$	17502.0			1.52409			0.762046	+	0.647523i	$0.224195\pi$
$510$		0			0
$511$	1723.25			0.149182
$512$		0			0
$513$			19645.1i			1.69074i
$514$		0			0
$515$	473.895	−	132.000i	0.0405481	−	0.0112944i
$516$		0			0
$517$			10511.8i			0.894217i
$518$		0			0
$519$	7668.47			0.648572
$520$		0			0
$521$	2874.00			0.241674			0.120837	−	0.992672i	$-0.461442\pi$
$521$	2874.00			0.241674			0.120837	+	0.992672i	$0.461442\pi$
$522$		0			0
$523$			14604.0i			1.22101i	0.792012	+	0.610505i	$0.209034\pi$
$523$			14604.0i			1.22101i	−0.792012	+	0.610505i	$0.790966\pi$
$524$		0			0
$525$	1712.00	−	1033.95i	0.142320	−	0.0859530i
$526$		0			0
$527$		−	7424.00i		−	0.613652i
$528$		0			0
$529$	9463.00			0.777760
$530$		0			0
$531$	7108.42			0.580940
$532$		0			0
$533$			3661.91i			0.297589i
$534$		0			0
$535$	−19515.8	+	5436.00i	−1.57709	+	0.439287i
$536$		0			0
$537$		−	11201.1i		−	0.900121i
$538$		0			0
$539$	−14087.6			−1.12578
$540$		0			0
$541$	−19330.0			−1.53616			−0.768079	−	0.640355i	$-0.778787\pi$
$541$	−19330.0			−1.53616			−0.768079	+	0.640355i	$0.778787\pi$
$542$		0			0
$543$		−	6168.00i		−	0.487466i
$544$		0			0
$545$	6042.00	+	21691.4i	0.474882	+	1.70488i
$546$		0			0
$547$		−	708.000i		−	0.0553417i	−0.999617	−	0.0276708i	$-0.991191\pi$
$547$		−	708.000i		−	0.0553417i	0.999617	−	0.0276708i	$-0.00880902\pi$
$548$		0			0
$549$	902.000			0.0701210
$550$		0			0
$551$	−20420.5			−1.57885
$552$		0			0
$553$		−	1378.60i		−	0.106011i
$554$		0			0
$555$	3360.34	+	12064.0i	0.257007	+	0.922682i
$556$		0			0
$557$		−	9284.02i		−	0.706242i	−0.935578	−	0.353121i	$-0.885120\pi$
$557$		−	9284.02i		−	0.706242i	0.935578	−	0.353121i	$-0.114880\pi$
$558$		0			0
$559$	6806.85			0.515025
$560$		0			0
$561$	−7424.00			−0.558719
$562$		0			0
$563$		−	9220.00i		−	0.690189i	−0.938568	−	0.345095i	$-0.887847\pi$
$563$		−	9220.00i		−	0.690189i	0.938568	−	0.345095i	$-0.112153\pi$
$564$		0			0
$565$	−17632.0	+	4911.27i	−1.31289	+	0.365697i
$566$		0			0
$567$			1244.00i			0.0921395i
$568$		0			0
$569$	−10458.0			−0.770513			−0.385257	−	0.922809i	$-0.625887\pi$
$569$	−10458.0			−0.770513			−0.385257	+	0.922809i	$0.625887\pi$
$570$		0			0
$571$	−13312.1			−0.975648			−0.487824	−	0.872942i	$-0.662209\pi$
$571$	−13312.1			−0.975648			−0.487824	+	0.872942i	$0.662209\pi$
$572$		0			0
$573$		−	17921.8i		−	1.30662i
$574$		0			0
$575$	3360.34	+	5564.00i	0.243715	+	0.403539i
$576$		0			0
$577$			17491.0i			1.26198i	0.775792	+	0.630988i	$0.217350\pi$
$577$			17491.0i			1.26198i	−0.775792	+	0.630988i	$0.782650\pi$
$578$		0			0
$579$	−10167.2			−0.729765
$580$		0			0
$581$	3760.00			0.268487
$582$		0			0
$583$			21344.0i			1.51626i
$584$		0			0
$585$	−2552.00	+	710.842i	−0.180363	+	0.0502388i
$586$		0			0
$587$		−	11460.0i		−	0.805800i	−0.915244	−	0.402900i	$-0.868002\pi$
$587$		−	11460.0i		−	0.805800i	0.915244	−	0.402900i	$-0.131998\pi$
$588$		0			0
$589$	22272.0			1.55807
$590$		0			0
$591$	−10942.7			−0.761626
$592$		0			0
$593$			2326.39i			0.161102i	0.996750	+	0.0805510i	$0.0256680\pi$
$593$			2326.39i			0.161102i	−0.996750	+	0.0805510i	$0.974332\pi$
$594$		0			0
$595$	−516.976	−	1856.00i	−0.0356201	−	0.127880i
$596$		0			0
$597$		−	1033.95i		−	0.0708825i
$598$		0			0
$599$	−24384.0			−1.66328			−0.831640	−	0.555316i	$-0.812597\pi$
$599$	−24384.0			−1.66328			−0.831640	+	0.555316i	$0.812597\pi$
$600$		0			0
$601$	24054.0			1.63258			0.816292	−	0.577639i	$-0.196026\pi$
$601$	24054.0			1.63258			0.816292	+	0.577639i	$0.196026\pi$
$602$		0			0
$603$		−	7612.00i		−	0.514071i
$604$		0			0
$605$	−1575.00	−	5654.42i	−0.105839	−	0.379975i
$606$		0			0
$607$		−	17364.0i		−	1.16109i	−0.814227	−	0.580546i	$-0.802839\pi$
$607$		−	17364.0i		−	1.16109i	0.814227	−	0.580546i	$-0.197161\pi$
$608$		0			0
$609$	−2528.00			−0.168210
$610$		0			0
$611$	−5255.92			−0.348006
$612$		0			0
$613$			2434.09i			0.160379i	0.996780	+	0.0801894i	$0.0255525\pi$
$613$			2434.09i			0.160379i	−0.996780	+	0.0801894i	$0.974447\pi$
$614$		0			0
$615$	7323.82	−	2040.00i	0.480203	−	0.133757i
$616$		0			0
$617$			8616.26i			0.562201i	0.959678	+	0.281100i	$0.0906993\pi$
$617$			8616.26i			0.562201i	−0.959678	+	0.281100i	$0.909301\pi$
$618$		0			0
$619$	−9176.32			−0.595844			−0.297922	−	0.954590i	$-0.596294\pi$
$619$	−9176.32			−0.595844			−0.297922	+	0.954590i	$0.596294\pi$
$620$		0			0
$621$	−7904.00			−0.510751
$622$		0			0
$623$			24.0000i			0.00154340i
$624$		0			0
$625$	7273.00	−	13829.1i	0.465472	−	0.885063i
$626$		0			0
$627$		−	22272.0i		−	1.41859i
$628$		0			0
$629$	12064.0			0.764743
$630$		0			0
$631$	9219.40			0.581646			0.290823	−	0.956777i	$-0.406071\pi$
$631$	9219.40			0.581646			0.290823	+	0.956777i	$0.406071\pi$
$632$		0			0
$633$		−	3618.83i		−	0.227228i
$634$		0			0
$635$	−30027.7	+	8364.00i	−1.87655	+	0.522701i
$636$		0			0
$637$		−	7043.80i		−	0.438125i
$638$		0			0
$639$	10425.7			0.645436
$640$		0			0
$641$	−25538.0			−1.57362			−0.786810	−	0.617195i	$-0.788269\pi$
$641$	−25538.0			−1.57362			−0.786810	+	0.617195i	$0.788269\pi$
$642$		0			0
$643$			22060.0i			1.35297i	0.736455	+	0.676486i	$0.236498\pi$
$643$			22060.0i			1.35297i	−0.736455	+	0.676486i	$0.763502\pi$
$644$		0			0
$645$	−3792.00	−	13613.7i	−0.231488	−	0.831068i
$646$		0			0
$647$		−	29444.0i		−	1.78912i	−0.446944	−	0.894562i	$-0.647488\pi$
$647$		−	29444.0i		−	1.78912i	0.446944	−	0.894562i	$-0.352512\pi$
$648$		0			0
$649$	−27840.0			−1.68385
$650$		0			0
$651$	2757.20			0.165996
$652$		0			0
$653$		−	20786.7i		−	1.24571i	−0.782338	−	0.622854i	$-0.785973\pi$
$653$		−	20786.7i		−	1.24571i	0.782338	−	0.622854i	$-0.214027\pi$
$654$		0			0
$655$	−129.244	−	464.000i	−0.00770989	−	0.0276794i
$656$		0			0
$657$			4738.95i			0.281406i
$658$		0			0
$659$	−13742.9			−0.812366			−0.406183	−	0.913792i	$-0.633140\pi$
$659$	−13742.9			−0.812366			−0.406183	+	0.913792i	$0.633140\pi$
$660$		0			0
$661$	11530.0			0.678464			0.339232	−	0.940703i	$-0.389833\pi$
$661$	11530.0			0.678464			0.339232	+	0.940703i	$0.389833\pi$
$662$		0			0
$663$		−	3712.00i		−	0.217439i
$664$		0			0
$665$	5568.00	−	1550.93i	0.324688	−	0.0904397i
$666$		0			0
$667$		−	8216.00i		−	0.476949i
$668$		0			0
$669$	17136.0			0.990308
$670$		0			0
$671$	−3532.67			−0.203245
$672$		0			0
$673$			23910.1i			1.36949i	0.728782	+	0.684746i	$0.240087\pi$
$673$			23910.1i			1.36949i	−0.728782	+	0.684746i	$0.759913\pi$
$674$		0			0
$675$	9822.54	+	16264.0i	0.560103	+	0.927410i
$676$		0			0
$677$			2175.61i			0.123509i	0.998091	+	0.0617543i	$0.0196695\pi$
$677$			2175.61i			0.123509i	−0.998091	+	0.0617543i	$0.980330\pi$
$678$		0			0
$679$	−4308.13			−0.243492
$680$		0			0
$681$	19824.0			1.11550
$682$		0			0
$683$		−	20708.0i		−	1.16013i	−0.814570	−	0.580066i	$-0.803027\pi$
$683$		−	20708.0i		−	1.16013i	0.814570	−	0.580066i	$-0.196973\pi$
$684$		0			0
$685$	5104.00	−	1421.68i	0.284692	−	0.0792989i
$686$		0			0
$687$			7080.00i			0.393186i
$688$		0			0
$689$	−10672.0			−0.590088
$690$		0			0
$691$	−4609.70			−0.253779			−0.126890	−	0.991917i	$-0.540499\pi$
$691$	−4609.70			−0.253779			−0.126890	+	0.991917i	$0.540499\pi$
$692$		0			0
$693$			1895.58i			0.103906i
$694$		0			0
$695$	6332.95	+	22736.0i	0.345644	+	1.24090i
$696$		0			0
$697$		−	7323.82i		−	0.398005i
$698$		0			0
$699$	21713.0			1.17491
$700$		0			0
$701$	−8942.00			−0.481790			−0.240895	−	0.970551i	$-0.577441\pi$
$701$	−8942.00			−0.481790			−0.240895	+	0.970551i	$0.577441\pi$
$702$		0			0
$703$			36192.0i			1.94169i
$704$		0			0
$705$	2928.00	+	10511.8i	0.156418	+	0.561559i
$706$		0			0
$707$		−	4056.00i		−	0.215759i
$708$		0			0
$709$	3670.00			0.194400			0.0972001	−	0.995265i	$-0.469011\pi$
$709$	3670.00			0.194400			0.0972001	+	0.995265i	$0.469011\pi$
$710$		0			0
$711$	3791.16			0.199971
$712$		0			0
$713$			8960.91i			0.470672i
$714$		0			0
$715$	9994.87	−	2784.00i	0.522779	−	0.145616i
$716$		0			0
$717$		−	22057.6i		−	1.14889i
$718$		0			0
$719$	−8616.26			−0.446916			−0.223458	−	0.974714i	$-0.571735\pi$
$719$	−8616.26			−0.446916			−0.223458	+	0.974714i	$0.571735\pi$
$720$		0			0
$721$	176.000			0.00909096
$722$		0			0
$723$			6472.00i			0.332913i
$724$		0			0
$725$	−16906.0	+	10210.3i	−0.866032	+	0.523034i
$726$		0			0
$727$		−	32004.0i		−	1.63269i	−0.577567	−	0.816343i	$-0.695998\pi$
$727$		−	32004.0i		−	1.63269i	0.577567	−	0.816343i	$-0.304002\pi$
$728$		0			0
$729$	−19837.0			−1.00782
$730$		0			0
$731$	−13613.7			−0.688811
$732$		0			0
$733$			5363.62i			0.270273i	0.990827	+	0.135136i	$0.0431473\pi$
$733$			5363.62i			0.270273i	−0.990827	+	0.135136i	$0.956853\pi$
$734$		0			0
$735$	−14087.6	+	3924.00i	−0.706978	+	0.196924i
$736$		0			0
$737$			29812.3i			1.49003i
$738$		0			0
$739$	25374.9			1.26310			0.631550	−	0.775335i	$-0.282419\pi$
$739$	25374.9			1.26310			0.631550	+	0.775335i	$0.282419\pi$
$740$		0			0
$741$	11136.0			0.552080
$742$		0			0
$743$			17404.0i			0.859342i	0.902986	+	0.429671i	$0.141370\pi$
$743$			17404.0i			0.859342i	−0.902986	+	0.429671i	$0.858630\pi$
$744$		0			0
$745$	7650.00	+	27464.3i	0.376207	+	1.35062i
$746$		0			0
$747$			10340.0i			0.506454i
$748$		0			0
$749$	−7248.00			−0.353586
$750$		0			0
$751$	32569.5			1.58253			0.791263	−	0.611476i	$-0.209424\pi$
$751$	32569.5			1.58253			0.791263	+	0.611476i	$0.209424\pi$
$752$		0			0
$753$		−	19817.4i		−	0.959079i
$754$		0			0
$755$	6462.20	+	23200.0i	0.311501	+	1.11832i
$756$		0			0
$757$		−	6139.09i		−	0.294754i	−0.989080	−	0.147377i	$-0.952917\pi$
$757$		−	6139.09i		−	0.294754i	0.989080	−	0.147377i	$-0.0470831\pi$
$758$		0			0
$759$	8960.91			0.428538
$760$		0			0
$761$	27850.0			1.32663			0.663313	−	0.748342i	$-0.269150\pi$
$761$	27850.0			1.32663			0.663313	+	0.748342i	$0.269150\pi$
$762$		0			0
$763$			8056.00i			0.382237i
$764$		0			0
$765$	5104.00	−	1421.68i	0.241223	−	0.0671909i
$766$		0			0
$767$		−	13920.0i		−	0.655309i
$768$		0			0
$769$	−33550.0			−1.57327			−0.786635	−	0.617419i	$-0.788178\pi$
$769$	−33550.0			−1.57327			−0.786635	+	0.617419i	$0.788178\pi$
$770$		0			0
$771$	18611.1			0.869343
$772$		0			0
$773$			1658.63i			0.0771757i	0.999255	+	0.0385878i	$0.0122859\pi$
$773$			1658.63i			0.0771757i	−0.999255	+	0.0385878i	$0.987714\pi$
$774$		0			0
$775$	18438.8	−	11136.0i	0.854634	−	0.516151i
$776$		0			0
$777$			4480.46i			0.206867i
$778$		0			0
$779$	21971.5			1.01054
$780$		0			0
$781$	−40832.0			−1.87079
$782$		0			0
$783$		−	24016.0i		−	1.09612i
$784$		0			0
$785$	38280.0	−	10662.6i	1.74047	−	0.484797i
$786$		0			0
$787$		−	27028.0i		−	1.22420i	−0.790781	−	0.612099i	$-0.790325\pi$
$787$		−	27028.0i		−	1.22420i	0.790781	−	0.612099i	$-0.209675\pi$
$788$		0			0
$789$	2416.00			0.109014
$790$		0			0
$791$	−6548.36			−0.294353
$792$		0			0
$793$		−	1766.33i		−	0.0790976i
$794$		0			0
$795$	5945.22	+	21344.0i	0.265227	+	0.952193i
$796$		0			0
$797$		−	2692.58i		−	0.119669i	−0.998208	−	0.0598345i	$-0.980943\pi$
$797$		−	2692.58i		−	0.119669i	0.998208	−	0.0598345i	$-0.0190573\pi$
$798$		0			0
$799$	10511.8			0.465435
$800$		0			0
$801$	−66.0000			−0.00291135
$802$		0			0
$803$		−	18560.0i		−	0.815652i
$804$		0			0
$805$	624.000	+	2240.23i	0.0273206	+	0.0980841i
$806$		0			0
$807$			13048.0i			0.569159i
$808$		0			0
$809$	1434.00			0.0623198			0.0311599	−	0.999514i	$-0.490080\pi$
$809$	1434.00			0.0623198			0.0311599	+	0.999514i	$0.490080\pi$
$810$		0			0
$811$	42004.3			1.81871			0.909353	−	0.416026i	$-0.136578\pi$
$811$	42004.3			1.81871			0.909353	+	0.416026i	$0.136578\pi$
$812$		0			0
$813$			8271.61i			0.356824i
$814$		0			0
$815$	−2455.64	+	684.000i	−0.105543	+	0.0293981i
$816$		0			0
$817$		−	40841.1i		−	1.74890i
$818$		0			0
$819$	−947.789			−0.0404376
$820$		0			0
$821$	21322.0			0.906386			0.453193	−	0.891412i	$-0.350285\pi$
$821$	21322.0			0.906386			0.453193	+	0.891412i	$0.350285\pi$
$822$		0			0
$823$			21548.0i			0.912656i	0.889812	+	0.456328i	$0.150836\pi$
$823$			21548.0i			0.912656i	−0.889812	+	0.456328i	$0.849164\pi$
$824$		0			0
$825$	−11136.0	−	18438.8i	−0.469946	−	0.778129i
$826$		0			0
$827$			9660.00i			0.406180i	0.979160	+	0.203090i	$0.0650985\pi$
$827$			9660.00i			0.406180i	−0.979160	+	0.203090i	$0.934902\pi$
$828$		0			0
$829$	24082.0			1.00893			0.504465	−	0.863432i	$-0.331690\pi$
$829$	24082.0			1.00893			0.504465	+	0.863432i	$0.331690\pi$
$830$		0			0
$831$	−4911.27			−0.205018
$832$		0			0
$833$			14087.6i			0.585962i
$834$		0			0
$835$	−4006.56	+	1116.00i	−0.166051	+	0.0462524i
$836$		0			0
$837$			26193.4i			1.08169i
$838$		0			0
$839$	14906.1			0.613369			0.306685	−	0.951811i	$-0.400780\pi$
$839$	14906.1			0.613369			0.306685	+	0.951811i	$0.400780\pi$
$840$		0			0
$841$	575.000			0.0235762
$842$		0			0
$843$			5160.00i			0.210818i
$844$		0			0
$845$	−5199.00	−	18665.0i	−0.211658	−	0.759876i
$846$		0			0
$847$		−	2100.00i		−	0.0851911i
$848$		0			0
$849$	−2768.00			−0.111893
$850$		0			0
$851$	−14561.5			−0.586559
$852$		0			0
$853$		−	21002.1i		−	0.843024i	−0.906823	−	0.421512i	$-0.861499\pi$
$853$		−	21002.1i		−	0.843024i	0.906823	−	0.421512i	$-0.138501\pi$
$854$		0			0
$855$	4265.05	+	15312.0i	0.170598	+	0.612467i
$856$		0			0
$857$		−	19860.5i		−	0.791624i	−0.918332	−	0.395812i	$-0.870463\pi$
$857$		−	19860.5i		−	0.791624i	0.918332	−	0.395812i	$-0.129537\pi$
$858$		0			0
$859$	−34852.8			−1.38436			−0.692178	−	0.721727i	$-0.743349\pi$
$859$	−34852.8			−1.38436			−0.692178	+	0.721727i	$0.743349\pi$
$860$		0			0
$861$	2720.00			0.107662
$862$		0			0
$863$			11468.0i			0.452347i	0.974087	+	0.226173i	$0.0726216\pi$
$863$			11468.0i			0.452347i	−0.974087	+	0.226173i	$0.927378\pi$
$864$		0			0
$865$	20648.0	−	5751.36i	0.811622	−	0.226072i
$866$		0			0
$867$		−	12228.0i		−	0.478990i
$868$		0			0
$869$	−14848.0			−0.579613
$870$		0			0
$871$	−14906.1			−0.579880
$872$		0			0
$873$		−	11847.4i		−	0.459304i
$874$		0			0
$875$	3834.24	−	4068.00i	0.148138	−	0.157170i
$876$		0			0
$877$			34572.8i			1.33117i	0.746321	+	0.665587i	$0.231819\pi$
$877$			34572.8i			1.33117i	−0.746321	+	0.665587i	$0.768181\pi$
$878$		0			0
$879$	−22143.8			−0.849706
$880$		0			0
$881$	−33186.0			−1.26909			−0.634543	−	0.772888i	$-0.718812\pi$
$881$	−33186.0			−1.26909			−0.634543	+	0.772888i	$0.718812\pi$
$882$		0			0
$883$			15196.0i			0.579146i	0.957156	+	0.289573i	$0.0935133\pi$
$883$			15196.0i			0.579146i	−0.957156	+	0.289573i	$0.906487\pi$
$884$		0			0
$885$	−27840.0	+	7754.64i	−1.05744	+	0.294542i
$886$		0			0
$887$			3036.00i			0.114925i	0.998348	+	0.0574627i	$0.0183010\pi$
$887$			3036.00i			0.114925i	−0.998348	+	0.0574627i	$0.981699\pi$
$888$		0			0
$889$	−11152.0			−0.420727
$890$		0			0
$891$	13398.3			0.503771
$892$		0			0
$893$			31535.5i			1.18174i
$894$		0			0
$895$	−8400.86	−	30160.0i	−0.313754	−	1.12641i
$896$		0			0
$897$			4480.46i			0.166776i
$898$		0			0
$899$	−27227.4			−1.01011
$900$		0			0
$901$	21344.0			0.789203
$902$		0			0
$903$		−	5056.00i		−	0.186327i
$904$		0			0
$905$	−4626.00	−	16607.8i	−0.169915	−	0.610015i
$906$		0			0
$907$			7404.00i			0.271054i	0.990774	+	0.135527i	$0.0432727\pi$
$907$			7404.00i			0.271054i	−0.990774	+	0.135527i	$0.956727\pi$
$908$		0			0
$909$	11154.0			0.406991
$910$		0			0
$911$	−6548.36			−0.238152			−0.119076	−	0.992885i	$-0.537993\pi$
$911$	−6548.36			−0.238152			−0.119076	+	0.992885i	$0.537993\pi$
$912$		0			0
$913$		−	40496.4i		−	1.46795i
$914$		0			0
$915$	−3532.67	+	984.000i	−0.127635	+	0.0355520i
$916$		0			0
$917$		−	172.325i		−	0.00620576i
$918$		0			0
$919$	34206.6			1.22782			0.613912	−	0.789374i	$-0.289595\pi$
$919$	34206.6			1.22782			0.613912	+	0.789374i	$0.289595\pi$
$920$		0			0
$921$	29744.0			1.06417
$922$		0			0
$923$		−	20416.0i		−	0.728062i
$924$		0			0
$925$	18096.0	+	29963.1i	0.643235	+	1.06506i
$926$		0			0
$927$			484.000i			0.0171485i
$928$		0			0
$929$	24286.0			0.857694			0.428847	−	0.903377i	$-0.358920\pi$
$929$	24286.0			0.857694			0.428847	+	0.903377i	$0.358920\pi$
$930$		0			0
$931$	−42262.8			−1.48776
$932$		0			0
$933$			9994.87i			0.350715i
$934$		0			0
$935$	−19989.7	+	5568.00i	−0.699181	+	0.194752i
$936$		0			0
$937$			37997.7i			1.32479i	0.749153	+	0.662397i	$0.230461\pi$
$937$			37997.7i			1.32479i	−0.749153	+	0.662397i	$0.769539\pi$
$938$		0			0
$939$	−10856.5			−0.377304
$940$		0			0
$941$	−36722.0			−1.27216			−0.636080	−	0.771623i	$-0.719445\pi$
$941$	−36722.0			−1.27216			−0.636080	+	0.771623i	$0.719445\pi$
$942$		0			0
$943$			8840.00i			0.305270i
$944$		0			0
$945$	1824.00	+	6548.36i	0.0627881	+	0.225416i
$946$		0			0
$947$			16364.0i			0.561519i	0.959778	+	0.280760i	$0.0905863\pi$
$947$			16364.0i			0.561519i	−0.959778	+	0.280760i	$0.909414\pi$
$948$		0			0
$949$	9280.00			0.317431
$950$		0			0
$951$	−13182.9			−0.449510
$952$		0			0
$953$		−	7797.72i		−	0.265050i	−0.991180	−	0.132525i	$-0.957691\pi$
$953$		−	7797.72i		−	0.265050i	0.991180	−	0.132525i	$-0.0423085\pi$
$954$		0			0
$955$	−13441.4	−	48256.0i	−0.455448	−	1.63511i
$956$		0			0
$957$			27227.4i			0.919683i
$958$		0			0
$959$	1895.58			0.0638284
$960$		0			0
$961$	−95.0000			−0.00318888
$962$		0			0
$963$		−	19932.0i		−	0.666978i
$964$		0			0
$965$	−27376.0	+	7625.39i	−0.913228	+	0.254373i
$966$		0			0
$967$		−	16756.0i		−	0.557225i	−0.960404	−	0.278613i	$-0.910125\pi$
$967$		−	16756.0i		−	0.557225i	0.960404	−	0.278613i	$-0.0898745\pi$
$968$		0			0
$969$	−22272.0			−0.738369
$970$		0			0
$971$	−16414.0			−0.542482			−0.271241	−	0.962512i	$-0.587434\pi$
$971$	−16414.0			−0.542482			−0.271241	+	0.962512i	$0.587434\pi$
$972$		0			0
$973$			8443.94i			0.278212i
$974$		0			0
$975$	9219.40	−	5568.00i	0.302828	−	0.182891i
$976$		0			0
$977$		−	50362.1i		−	1.64916i	−0.565749	−	0.824578i	$-0.691413\pi$
$977$		−	50362.1i		−	1.64916i	0.565749	−	0.824578i	$-0.308587\pi$
$978$		0			0
$979$	258.488			0.00843852
$980$		0			0
$981$	−22154.0			−0.721022
$982$		0			0
$983$		−	60612.0i		−	1.96666i	−0.181841	−	0.983328i	$-0.558206\pi$
$983$		−	60612.0i		−	1.96666i	0.181841	−	0.983328i	$-0.441794\pi$
$984$		0			0
$985$	−29464.0	+	8206.99i	−0.953098	+	0.265479i
$986$		0			0
$987$			3904.00i			0.125902i
$988$		0			0
$989$	16432.0			0.528319
$990$		0			0
$991$	−16198.6			−0.519238			−0.259619	−	0.965711i	$-0.583597\pi$
$991$	−16198.6			−0.519238			−0.259619	+	0.965711i	$0.583597\pi$
$992$		0			0
$993$			11890.4i			0.379992i
$994$		0			0
$995$	−775.464	−	2784.00i	−0.0247074	−	0.0887022i
$996$		0			0
$997$		−	54002.4i		−	1.71542i	−0.514133	−	0.857710i	$-0.671886\pi$
$997$		−	54002.4i		−	1.71542i	0.514133	−	0.857710i	$-0.328114\pi$
$998$		0			0
$999$	−42564.3			−1.34802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.c.b.129.1	✓	4
3.2	odd	2		1440.4.f.h.289.3		4
4.3	odd	2	inner	160.4.c.b.129.3	yes	4
5.2	odd	4		800.4.a.n.1.1		2
5.3	odd	4		800.4.a.r.1.1		2
5.4	even	2	inner	160.4.c.b.129.4	yes	4
8.3	odd	2		320.4.c.i.129.2		4
8.5	even	2		320.4.c.i.129.4		4
12.11	even	2		1440.4.f.h.289.4		4
15.14	odd	2		1440.4.f.h.289.2		4
20.3	even	4		800.4.a.n.1.2		2
20.7	even	4		800.4.a.r.1.2		2
20.19	odd	2	inner	160.4.c.b.129.2	yes	4
40.3	even	4		1600.4.a.co.1.1		2
40.13	odd	4		1600.4.a.cc.1.2		2
40.19	odd	2		320.4.c.i.129.3		4
40.27	even	4		1600.4.a.cc.1.1		2
40.29	even	2		320.4.c.i.129.1		4
40.37	odd	4		1600.4.a.co.1.2		2
60.59	even	2		1440.4.f.h.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.b.129.1	✓	4	1.1	even	1	trivial
160.4.c.b.129.2	yes	4	20.19	odd	2	inner
160.4.c.b.129.3	yes	4	4.3	odd	2	inner
160.4.c.b.129.4	yes	4	5.4	even	2	inner
320.4.c.i.129.1		4	40.29	even	2
320.4.c.i.129.2		4	8.3	odd	2
320.4.c.i.129.3		4	40.19	odd	2
320.4.c.i.129.4		4	8.5	even	2
800.4.a.n.1.1		2	5.2	odd	4
800.4.a.n.1.2		2	20.3	even	4
800.4.a.r.1.1		2	5.3	odd	4
800.4.a.r.1.2		2	20.7	even	4
1440.4.f.h.289.1		4	60.59	even	2
1440.4.f.h.289.2		4	15.14	odd	2
1440.4.f.h.289.3		4	3.2	odd	2
1440.4.f.h.289.4		4	12.11	even	2
1600.4.a.cc.1.1		2	40.27	even	4
1600.4.a.cc.1.2		2	40.13	odd	4
1600.4.a.co.1.1		2	40.3	even	4
1600.4.a.co.1.2		2	40.37	odd	4

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{3} +(-3.00000 - 10.7703i) q^{5} -4.00000i q^{7} +11.0000 q^{9} -43.0813 q^{11} -21.5407i q^{13} +(-43.0813 + 12.0000i) q^{15} +43.0813i q^{17} -129.244 q^{19} -16.0000 q^{21} -52.0000i q^{23} +(-107.000 + 64.6220i) q^{25} -152.000i q^{27} +158.000 q^{29} -172.325 q^{31} +172.325i q^{33} +(-43.0813 + 12.0000i) q^{35} -280.029i q^{37} -86.1626 q^{39} -170.000 q^{41} +316.000i q^{43} +(-33.0000 - 118.474i) q^{45} -244.000i q^{47} +327.000 q^{49} +172.325 q^{51} -495.435i q^{53} +(129.244 + 464.000i) q^{55} +516.976i q^{57} +646.220 q^{59} +82.0000 q^{61} -44.0000i q^{63} +(-232.000 + 64.6220i) q^{65} -692.000i q^{67} -208.000 q^{69} +947.789 q^{71} +430.813i q^{73} +(258.488 + 428.000i) q^{75} +172.325i q^{77} +344.651 q^{79} -311.000 q^{81} +940.000i q^{83} +(464.000 - 129.244i) q^{85} -632.000i q^{87} -6.00000 q^{89} -86.1626 q^{91} +689.301i q^{93} +(387.732 + 1392.00i) q^{95} -1077.03i q^{97} -473.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{5} + 44 q^{9} - 64 q^{21} - 428 q^{25} + 632 q^{29} - 680 q^{41} - 132 q^{45} + 1308 q^{49} + 328 q^{61} - 928 q^{65} - 832 q^{69} - 1244 q^{81} + 1856 q^{85} - 24 q^{89}+O(q^{100}) \) 4 * q - 12 * q^5 + 44 * q^9 - 64 * q^21 - 428 * q^25 + 632 * q^29 - 680 * q^41 - 132 * q^45 + 1308 * q^49 + 328 * q^61 - 928 * q^65 - 832 * q^69 - 1244 * q^81 + 1856 * q^85 - 24 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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