LMFDB - Embedded newform 160.6.f.a.49.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(49,160)]

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, 1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$160 = 2^{5} \cdot 5$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	160.f (of order $2$ , degree $1$ , not 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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.14
Character	$\chi$	$=$	160.49
Dual form			160.6.f.a.49.13

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29818 q^{3} +(51.3939 + 21.9924i) q^{5} -170.399i q^{7} -241.315 q^{9} -39.8166i q^{11} -537.234 q^{13} +(-66.7187 - 28.5502i) q^{15} +1355.83i q^{17} -1039.21i q^{19} +221.209i q^{21} -1610.62i q^{23} +(2157.67 + 2260.55i) q^{25} +628.730 q^{27} -5910.56i q^{29} -9611.53 q^{31} +51.6892i q^{33} +(3747.49 - 8757.46i) q^{35} -5295.85 q^{37} +697.429 q^{39} -12105.1 q^{41} -19338.2 q^{43} +(-12402.1 - 5307.10i) q^{45} +7894.36i q^{47} -12228.8 q^{49} -1760.11i q^{51} +18932.1 q^{53} +(875.664 - 2046.33i) q^{55} +1349.09i q^{57} -32322.2i q^{59} -48808.2i q^{61} +41119.8i q^{63} +(-27610.6 - 11815.1i) q^{65} +882.245 q^{67} +2090.88i q^{69} +8818.78 q^{71} -14380.4i q^{73} +(-2801.05 - 2934.62i) q^{75} -6784.70 q^{77} -49813.9 q^{79} +57823.3 q^{81} -13938.9 q^{83} +(-29817.9 + 69681.2i) q^{85} +7673.00i q^{87} +7581.49 q^{89} +91544.2i q^{91} +12477.5 q^{93} +(22854.8 - 53409.1i) q^{95} +66750.3i q^{97} +9608.33i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$28 q + 1940 q^{9} + 488 q^{15} + 1556 q^{25} - 4368 q^{31} - 23360 q^{39} - 2480 q^{41} - 38420 q^{49} + 48776 q^{55} + 37200 q^{65} + 69232 q^{71} + 35984 q^{79} + 122596 q^{81} - 178744 q^{89} - 89416 q^{95}+O(q^{100})$ 28 * q + 1940 * q^9 + 488 * q^15 + 1556 * q^25 - 4368 * q^31 - 23360 * q^39 - 2480 * q^41 - 38420 * q^49 + 48776 * q^55 + 37200 * q^65 + 69232 * q^71 + 35984 * q^79 + 122596 * q^81 - 178744 * q^89 - 89416 * q^95

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
$3$	−1.29818			−0.0832785			−0.0416393	−	0.999133i	$-0.513258\pi$
$3$	−1.29818			−0.0832785			−0.0416393	+	0.999133i	$0.513258\pi$
$4$		0			0
$5$	51.3939	+	21.9924i	0.919362	+	0.393413i
$5$	51.3939	+	21.9924i	0.919362	+	0.393413i
$6$		0			0
$7$		−	170.399i		−	1.31438i	−0.753724	−	0.657191i	$-0.771744\pi$
$7$		−	170.399i		−	1.31438i	0.753724	−	0.657191i	$-0.228256\pi$
$8$		0			0
$9$	−241.315			−0.993065
$10$		0			0
$11$		−	39.8166i		−	0.0992161i	−0.998769	−	0.0496081i	$-0.984203\pi$
$11$		−	39.8166i		−	0.0992161i	0.998769	−	0.0496081i	$-0.0157972\pi$
$12$		0			0
$13$	−537.234			−0.881669			−0.440834	−	0.897588i	$-0.645317\pi$
$13$	−537.234			−0.881669			−0.440834	+	0.897588i	$0.645317\pi$
$14$		0			0
$15$	−66.7187	−	28.5502i	−0.0765631	−	0.0327628i
$16$		0			0
$17$			1355.83i			1.13784i	0.822392	+	0.568921i	$0.192639\pi$
$17$			1355.83i			1.13784i	−0.822392	+	0.568921i	$0.807361\pi$
$18$		0			0
$19$		−	1039.21i		−	0.660419i	−0.943908	−	0.330209i	$-0.892881\pi$
$19$		−	1039.21i		−	0.660419i	0.943908	−	0.330209i	$-0.107119\pi$
$20$		0			0
$21$			221.209i			0.109460i
$22$		0			0
$23$		−	1610.62i		−	0.634852i	−0.948283	−	0.317426i	$-0.897182\pi$
$23$		−	1610.62i		−	0.634852i	0.948283	−	0.317426i	$-0.102818\pi$
$24$		0			0
$25$	2157.67	+	2260.55i	0.690453	+	0.723377i
$26$		0			0
$27$	628.730			0.165980
$28$		0			0
$29$		−	5910.56i		−	1.30507i	−0.757759	−	0.652535i	$-0.773706\pi$
$29$		−	5910.56i		−	1.30507i	0.757759	−	0.652535i	$-0.226294\pi$
$30$		0			0
$31$	−9611.53			−1.79634			−0.898169	−	0.439650i	$-0.855102\pi$
$31$	−9611.53			−1.79634			−0.898169	+	0.439650i	$0.855102\pi$
$32$		0			0
$33$			51.6892i			0.00826257i
$34$		0			0
$35$	3747.49	−	8757.46i	0.517095	−	1.20839i
$36$		0			0
$37$	−5295.85			−0.635963			−0.317981	−	0.948097i	$-0.603005\pi$
$37$	−5295.85			−0.635963			−0.317981	+	0.948097i	$0.603005\pi$
$38$		0			0
$39$	697.429			0.0734241
$40$		0			0
$41$	−12105.1			−1.12463			−0.562315	−	0.826923i	$-0.690089\pi$
$41$	−12105.1			−1.12463			−0.562315	+	0.826923i	$0.690089\pi$
$42$		0			0
$43$	−19338.2			−1.59494			−0.797469	−	0.603360i	$-0.793828\pi$
$43$	−19338.2			−1.59494			−0.797469	+	0.603360i	$0.793828\pi$
$44$		0			0
$45$	−12402.1	−	5307.10i	−0.912986	−	0.390684i
$46$		0			0
$47$			7894.36i			0.521282i	0.965436	+	0.260641i	$0.0839338\pi$
$47$			7894.36i			0.521282i	−0.965436	+	0.260641i	$0.916066\pi$
$48$		0			0
$49$	−12228.8			−0.727601
$50$		0			0
$51$		−	1760.11i		−	0.0947578i
$52$		0			0
$53$	18932.1			0.925783			0.462891	−	0.886415i	$-0.346812\pi$
$53$	18932.1			0.925783			0.462891	+	0.886415i	$0.346812\pi$
$54$		0			0
$55$	875.664	−	2046.33i	0.0390329	−	0.0912155i
$56$		0			0
$57$			1349.09i			0.0549987i
$58$		0			0
$59$		−	32322.2i		−	1.20885i	−0.796663	−	0.604424i	$-0.793403\pi$
$59$		−	32322.2i		−	1.20885i	0.796663	−	0.604424i	$-0.206597\pi$
$60$		0			0
$61$		−	48808.2i		−	1.67945i	−0.543008	−	0.839727i	$-0.682715\pi$
$61$		−	48808.2i		−	1.67945i	0.543008	−	0.839727i	$-0.317285\pi$
$62$		0			0
$63$			41119.8i			1.30527i
$64$		0			0
$65$	−27610.6	−	11815.1i	−0.810573	−	0.346860i
$66$		0			0
$67$	882.245			0.0240106			0.0120053	−	0.999928i	$-0.496179\pi$
$67$	882.245			0.0240106			0.0120053	+	0.999928i	$0.496179\pi$
$68$		0			0
$69$			2090.88i			0.0528695i
$70$		0			0
$71$	8818.78			0.207617			0.103808	−	0.994597i	$-0.466897\pi$
$71$	8818.78			0.207617			0.103808	+	0.994597i	$0.466897\pi$
$72$		0			0
$73$		−	14380.4i		−	0.315837i	−0.987452	−	0.157919i	$-0.949522\pi$
$73$		−	14380.4i		−	0.315837i	0.987452	−	0.157919i	$-0.0504784\pi$
$74$		0			0
$75$	−2801.05	−	2934.62i	−0.0574999	−	0.0602418i
$76$		0			0
$77$	−6784.70			−0.130408
$78$		0			0
$79$	−49813.9			−0.898014			−0.449007	−	0.893528i	$-0.648222\pi$
$79$	−49813.9			−0.898014			−0.449007	+	0.893528i	$0.648222\pi$
$80$		0			0
$81$	57823.3			0.979242
$82$		0			0
$83$	−13938.9			−0.222092			−0.111046	−	0.993815i	$-0.535420\pi$
$83$	−13938.9			−0.222092			−0.111046	+	0.993815i	$0.535420\pi$
$84$		0			0
$85$	−29817.9	+	69681.2i	−0.447641	+	1.04609i
$86$		0			0
$87$			7673.00i			0.108684i
$88$		0			0
$89$	7581.49			0.101456			0.0507282	−	0.998712i	$-0.483846\pi$
$89$	7581.49			0.101456			0.0507282	+	0.998712i	$0.483846\pi$
$90$		0			0
$91$			91544.2i			1.15885i
$92$		0			0
$93$	12477.5			0.149596
$94$		0			0
$95$	22854.8	−	53409.1i	0.259817	−	0.607164i
$96$		0			0
$97$			66750.3i			0.720317i	0.932891	+	0.360159i	$0.117277\pi$
$97$			66750.3i			0.720317i	−0.932891	+	0.360159i	$0.882723\pi$
$98$		0			0
$99$			9608.33i			0.0985280i
$100$		0			0
$101$		−	66492.6i		−	0.648590i	−0.945956	−	0.324295i	$-0.894873\pi$
$101$		−	66492.6i		−	0.648590i	0.945956	−	0.324295i	$-0.105127\pi$
$102$		0			0
$103$		−	166310.i		−	1.54463i	−0.635237	−	0.772317i	$-0.719098\pi$
$103$		−	166310.i		−	1.54463i	0.635237	−	0.772317i	$-0.280902\pi$
$104$		0			0
$105$	−4864.93	+	11368.8i	−0.0430629	+	0.100633i
$106$		0			0
$107$	121647.			1.02717			0.513583	−	0.858040i	$-0.328318\pi$
$107$	121647.			1.02717			0.513583	+	0.858040i	$0.328318\pi$
$108$		0			0
$109$			170773.i			1.37674i	0.725359	+	0.688370i	$0.241674\pi$
$109$			170773.i			1.37674i	−0.725359	+	0.688370i	$0.758326\pi$
$110$		0			0
$111$	6874.99			0.0529621
$112$		0			0
$113$			3705.12i			0.0272964i	0.999907	+	0.0136482i	$0.00434449\pi$
$113$			3705.12i			0.0272964i	−0.999907	+	0.0136482i	$0.995656\pi$
$114$		0			0
$115$	35421.4	−	82775.8i	0.249759	−	0.583658i
$116$		0			0
$117$	129643.			0.875554
$118$		0			0
$119$	231031.			1.49556
$120$		0			0
$121$	159466.			0.990156
$122$		0			0
$123$	15714.7			0.0936576
$124$		0			0
$125$	61175.7	+	163631.i	0.350190	+	0.936679i
$126$		0			0
$127$			77574.6i			0.426786i	0.976966	+	0.213393i	$0.0684515\pi$
$127$			77574.6i			0.426786i	−0.976966	+	0.213393i	$0.931548\pi$
$128$		0			0
$129$	25104.5			0.132824
$130$		0			0
$131$			330023.i			1.68022i	0.542418	+	0.840109i	$0.317509\pi$
$131$			330023.i			1.68022i	−0.542418	+	0.840109i	$0.682491\pi$
$132$		0			0
$133$	−177080.			−0.868043
$134$		0			0
$135$	32312.9	+	13827.3i	0.152595	+	0.0652985i
$136$		0			0
$137$			135014.i			0.614578i	0.951616	+	0.307289i	$0.0994219\pi$
$137$			135014.i			0.614578i	−0.951616	+	0.307289i	$0.900578\pi$
$138$		0			0
$139$			130330.i			0.572146i	0.958208	+	0.286073i	$0.0923500\pi$
$139$			130330.i			0.572146i	−0.958208	+	0.286073i	$0.907650\pi$
$140$		0			0
$141$		−	10248.3i		−	0.0434116i
$142$		0			0
$143$			21390.8i			0.0874758i
$144$		0			0
$145$	129988.	−	303767.i	0.513431	−	1.19983i
$146$		0			0
$147$	15875.2			0.0605935
$148$		0			0
$149$		−	56728.8i		−	0.209333i	−0.994507	−	0.104667i	$-0.966622\pi$
$149$		−	56728.8i		−	0.209333i	0.994507	−	0.104667i	$-0.0333775\pi$
$150$		0			0
$151$	−272831.			−0.973757			−0.486879	−	0.873470i	$-0.661865\pi$
$151$	−272831.			−0.973757			−0.486879	+	0.873470i	$0.661865\pi$
$152$		0			0
$153$		−	327181.i		−	1.12995i
$154$		0			0
$155$	−493974.	−	211381.i	−1.65148	−	0.706702i
$156$		0			0
$157$	351466.			1.13798			0.568990	−	0.822345i	$-0.307334\pi$
$157$	351466.			1.13798			0.568990	+	0.822345i	$0.307334\pi$
$158$		0			0
$159$	−24577.4			−0.0770979
$160$		0			0
$161$	−274447.			−0.834438
$162$		0			0
$163$	434277.			1.28026			0.640129	−	0.768268i	$-0.278881\pi$
$163$	434277.			1.28026			0.640129	+	0.768268i	$0.278881\pi$
$164$		0			0
$165$	−1136.77	+	2656.51i	−0.00325060	+	0.00759630i
$166$		0			0
$167$			580367.i			1.61032i	0.593059	+	0.805159i	$0.297920\pi$
$167$			580367.i			1.61032i	−0.593059	+	0.805159i	$0.702080\pi$
$168$		0			0
$169$	−82672.1			−0.222660
$170$		0			0
$171$			250777.i			0.655838i
$172$		0			0
$173$	−243764.			−0.619233			−0.309617	−	0.950861i	$-0.600201\pi$
$173$	−243764.			−0.619233			−0.309617	+	0.950861i	$0.600201\pi$
$174$		0			0
$175$	385196.	−	367664.i	0.950794	−	0.907519i
$176$		0			0
$177$			41960.2i			0.100671i
$178$		0			0
$179$		−	589838.i		−	1.37594i	−0.725738	−	0.687971i	$-0.758502\pi$
$179$		−	589838.i		−	1.37594i	0.725738	−	0.687971i	$-0.241498\pi$
$180$		0			0
$181$		−	306392.i		−	0.695154i	−0.937651	−	0.347577i	$-0.887005\pi$
$181$		−	306392.i		−	0.695154i	0.937651	−	0.347577i	$-0.112995\pi$
$182$		0			0
$183$			63362.0i			0.139863i
$184$		0			0
$185$	−272175.	−	116469.i	−0.584680	−	0.250196i
$186$		0			0
$187$	53984.4			0.112892
$188$		0			0
$189$		−	107135.i		−	0.218161i
$190$		0			0
$191$	123083.			0.244127			0.122064	−	0.992522i	$-0.461049\pi$
$191$	123083.			0.244127			0.122064	+	0.992522i	$0.461049\pi$
$192$		0			0
$193$			689437.i			1.33230i	0.745819	+	0.666149i	$0.232059\pi$
$193$			689437.i			1.33230i	−0.745819	+	0.666149i	$0.767941\pi$
$194$		0			0
$195$	35843.6	+	15338.2i	0.0675033	+	0.0288860i
$196$		0			0
$197$	70782.9			0.129946			0.0649730	−	0.997887i	$-0.479304\pi$
$197$	70782.9			0.129946			0.0649730	+	0.997887i	$0.479304\pi$
$198$		0			0
$199$	−314584.			−0.563123			−0.281561	−	0.959543i	$-0.590852\pi$
$199$	−314584.			−0.563123			−0.281561	+	0.959543i	$0.590852\pi$
$200$		0			0
$201$	−1145.32			−0.00199956
$202$		0			0
$203$	−1.00715e6			−1.71536
$204$		0			0
$205$	−622130.	−	266221.i	−1.03394	−	0.442444i
$206$		0			0
$207$			388665.i			0.630449i
$208$		0			0
$209$	−41377.8			−0.0655242
$210$		0			0
$211$		−	1.09357e6i		−	1.69098i	−0.533989	−	0.845491i	$-0.679308\pi$
$211$		−	1.09357e6i		−	1.69098i	0.533989	−	0.845491i	$-0.320692\pi$
$212$		0			0
$213$	−11448.4			−0.0172900
$214$		0			0
$215$	−993863.	−	425293.i	−1.46633	−	0.627469i
$216$		0			0
$217$			1.63779e6i			2.36107i
$218$		0			0
$219$			18668.4i			0.0263025i
$220$		0			0
$221$		−	728397.i		−	1.00320i
$222$		0			0
$223$		−	917927.i		−	1.23608i	−0.786147	−	0.618039i	$-0.787927\pi$
$223$		−	917927.i		−	1.23608i	0.786147	−	0.618039i	$-0.212073\pi$
$224$		0			0
$225$	−520676.	−	545505.i	−0.685664	−	0.718361i
$226$		0			0
$227$	1.20791e6			1.55585			0.777926	−	0.628356i	$-0.216272\pi$
$227$	1.20791e6			1.55585			0.777926	+	0.628356i	$0.216272\pi$
$228$		0			0
$229$			323648.i			0.407834i	0.978988	+	0.203917i	$0.0653673\pi$
$229$			323648.i			0.407834i	−0.978988	+	0.203917i	$0.934633\pi$
$230$		0			0
$231$	8807.79			0.0108602
$232$		0			0
$233$		−	1.34719e6i		−	1.62570i	−0.582476	−	0.812848i	$-0.697916\pi$
$233$		−	1.34719e6i		−	1.62570i	0.582476	−	0.812848i	$-0.302084\pi$
$234$		0			0
$235$	−173616.	+	405722.i	−0.205079	+	0.479247i
$236$		0			0
$237$	64667.7			0.0747853
$238$		0			0
$239$	−997494.			−1.12958			−0.564788	−	0.825236i	$-0.691042\pi$
$239$	−997494.			−1.12958			−0.564788	+	0.825236i	$0.691042\pi$
$240$		0			0
$241$	610095.			0.676636			0.338318	−	0.941032i	$-0.390142\pi$
$241$	610095.			0.676636			0.338318	+	0.941032i	$0.390142\pi$
$242$		0			0
$243$	−227847.			−0.247529
$244$		0			0
$245$	−628485.	−	268941.i	−0.668928	−	0.286247i
$246$		0			0
$247$			558300.i			0.582271i
$248$		0			0
$249$	18095.2			0.0184955
$250$		0			0
$251$		−	691093.i		−	0.692392i	−0.938162	−	0.346196i	$-0.887473\pi$
$251$		−	691093.i		−	0.692392i	0.938162	−	0.346196i	$-0.112527\pi$
$252$		0			0
$253$	−64129.2			−0.0629875
$254$		0			0
$255$	38709.2	−	90459.0i	0.0372789	−	0.0871167i
$256$		0			0
$257$			755800.i			0.713796i	0.934143	+	0.356898i	$0.116166\pi$
$257$			755800.i			0.713796i	−0.934143	+	0.356898i	$0.883834\pi$
$258$		0			0
$259$			902408.i			0.835898i
$260$		0			0
$261$			1.42631e6i			1.29602i
$262$		0			0
$263$			316724.i			0.282353i	0.989984	+	0.141176i	$0.0450885\pi$
$263$			316724.i			0.282353i	−0.989984	+	0.141176i	$0.954912\pi$
$264$		0			0
$265$	972994.	+	416363.i	0.851130	+	0.364215i
$266$		0			0
$267$	−9842.17			−0.00844914
$268$		0			0
$269$		−	1.50082e6i		−	1.26458i	−0.774731	−	0.632291i	$-0.782115\pi$
$269$		−	1.50082e6i		−	1.26458i	0.774731	−	0.632291i	$-0.217885\pi$
$270$		0			0
$271$	660806.			0.546576			0.273288	−	0.961932i	$-0.411889\pi$
$271$	660806.			0.546576			0.273288	+	0.961932i	$0.411889\pi$
$272$		0			0
$273$		−	118841.i		−	0.0965073i
$274$		0			0
$275$	90007.5	−	85910.8i	0.0717707	−	0.0685040i
$276$		0			0
$277$	1.57567e6			1.23386			0.616931	−	0.787017i	$-0.288376\pi$
$277$	1.57567e6			1.23386			0.616931	+	0.787017i	$0.288376\pi$
$278$		0			0
$279$	2.31940e6			1.78388
$280$		0			0
$281$	−1.45729e6			−1.10098			−0.550491	−	0.834841i	$-0.685560\pi$
$281$	−1.45729e6			−1.10098			−0.550491	+	0.834841i	$0.685560\pi$
$282$		0			0
$283$	−1.38341e6			−1.02680			−0.513400	−	0.858149i	$-0.671614\pi$
$283$	−1.38341e6			−1.02680			−0.513400	+	0.858149i	$0.671614\pi$
$284$		0			0
$285$	−29669.7	+	69334.8i	−0.0216372	+	0.0505637i
$286$		0			0
$287$			2.06270e6i			1.47819i
$288$		0			0
$289$	−418409.			−0.294684
$290$		0			0
$291$		−	86654.2i		−	0.0599870i
$292$		0			0
$293$	544523.			0.370551			0.185275	−	0.982687i	$-0.440682\pi$
$293$	544523.			0.370551			0.185275	+	0.982687i	$0.440682\pi$
$294$		0			0
$295$	710845.	−	1.66117e6i	0.475576	−	1.11137i
$296$		0			0
$297$		−	25033.9i		−	0.0164678i
$298$		0			0
$299$			865278.i			0.559729i
$300$		0			0
$301$			3.29520e6i			2.09636i
$302$		0			0
$303$			86319.7i			0.0540136i
$304$		0			0
$305$	1.07341e6	−	2.50844e6i	0.660719	−	1.54403i
$306$		0			0
$307$	2.70567e6			1.63843			0.819216	−	0.573485i	$-0.194409\pi$
$307$	2.70567e6			1.63843			0.819216	+	0.573485i	$0.194409\pi$
$308$		0			0
$309$			215901.i			0.128635i
$310$		0			0
$311$	309689.			0.181562			0.0907810	−	0.995871i	$-0.471064\pi$
$311$	309689.			0.181562			0.0907810	+	0.995871i	$0.471064\pi$
$312$		0			0
$313$		−	1.89714e6i		−	1.09456i	−0.836950	−	0.547279i	$-0.815664\pi$
$313$		−	1.89714e6i		−	1.09456i	0.836950	−	0.547279i	$-0.184336\pi$
$314$		0			0
$315$	−904324.	+	2.11330e6i	−0.513508	+	1.20001i
$316$		0			0
$317$	546064.			0.305207			0.152604	−	0.988287i	$-0.451234\pi$
$317$	546064.			0.305207			0.152604	+	0.988287i	$0.451234\pi$
$318$		0			0
$319$	−235338.			−0.129484
$320$		0			0
$321$	−157920.			−0.0855409
$322$		0			0
$323$	1.40899e6			0.751452
$324$		0			0
$325$	−1.15917e6	−	1.21445e6i	−0.608751	−	0.637779i
$326$		0			0
$327$		−	221694.i		−	0.114653i
$328$		0			0
$329$	1.34519e6			0.685163
$330$		0			0
$331$			1.57476e6i			0.790029i	0.918675	+	0.395015i	$0.129260\pi$
$331$			1.57476e6i			0.790029i	−0.918675	+	0.395015i	$0.870740\pi$
$332$		0			0
$333$	1.27797e6			0.631552
$334$		0			0
$335$	45342.0	+	19402.7i	0.0220744	+	0.00944606i
$336$		0			0
$337$		−	77460.4i		−	0.0371540i	−0.999827	−	0.0185770i	$-0.994086\pi$
$337$		−	77460.4i		−	0.0371540i	0.999827	−	0.0185770i	$-0.00591358\pi$
$338$		0			0
$339$		−	4809.92i		−	0.00227321i
$340$		0			0
$341$			382698.i			0.178226i
$342$		0			0
$343$		−	780123.i		−	0.358037i
$344$		0			0
$345$	−45983.4	+	107458.i	−0.0207995	+	0.0486062i
$346$		0			0
$347$	−2.74014e6			−1.22165			−0.610827	−	0.791764i	$-0.709163\pi$
$347$	−2.74014e6			−1.22165			−0.610827	+	0.791764i	$0.709163\pi$
$348$		0			0
$349$			2.65302e6i			1.16594i	0.812493	+	0.582972i	$0.198110\pi$
$349$			2.65302e6i			1.16594i	−0.812493	+	0.582972i	$0.801890\pi$
$350$		0			0
$351$	−337775.			−0.146339
$352$		0			0
$353$			2.88565e6i			1.23256i	0.787528	+	0.616279i	$0.211360\pi$
$353$			2.88565e6i			1.23256i	−0.787528	+	0.616279i	$0.788640\pi$
$354$		0			0
$355$	453232.	+	193947.i	0.190875	+	0.0816792i
$356$		0			0
$357$	−299921.			−0.124548
$358$		0			0
$359$	−1.66246e6			−0.680794			−0.340397	−	0.940282i	$-0.610561\pi$
$359$	−1.66246e6			−0.680794			−0.340397	+	0.940282i	$0.610561\pi$
$360$		0			0
$361$	1.39614e6			0.563847
$362$		0			0
$363$	−207016.			−0.0824588
$364$		0			0
$365$	316260.	−	739064.i	0.124254	−	0.290369i
$366$		0			0
$367$		−	183532.i		−	0.0711291i	−0.999367	−	0.0355645i	$-0.988677\pi$
$367$		−	183532.i		−	0.0711291i	0.999367	−	0.0355645i	$-0.0113229\pi$
$368$		0			0
$369$	2.92115e6			1.11683
$370$		0			0
$371$		−	3.22601e6i		−	1.21683i
$372$		0			0
$373$	−2.57701e6			−0.959056			−0.479528	−	0.877527i	$-0.659192\pi$
$373$	−2.57701e6			−0.959056			−0.479528	+	0.877527i	$0.659192\pi$
$374$		0			0
$375$	−79417.3	−	212423.i	−0.0291633	−	0.0780052i
$376$		0			0
$377$			3.17536e6i			1.15064i
$378$		0			0
$379$			1.87753e6i			0.671412i	0.941967	+	0.335706i	$0.108975\pi$
$379$			1.87753e6i			0.671412i	−0.941967	+	0.335706i	$0.891025\pi$
$380$		0			0
$381$		−	100706.i		−	0.0355421i
$382$		0			0
$383$		−	1.41226e6i		−	0.491945i	−0.969277	−	0.245973i	$-0.920893\pi$
$383$		−	1.41226e6i		−	0.491945i	0.969277	−	0.245973i	$-0.0791073\pi$
$384$		0			0
$385$	−348692.	−	149212.i	−0.119892	−	0.0513041i
$386$		0			0
$387$	4.66658e6			1.58388
$388$		0			0
$389$		−	2.90646e6i		−	0.973846i	−0.873445	−	0.486923i	$-0.838119\pi$
$389$		−	2.90646e6i		−	0.973846i	0.873445	−	0.486923i	$-0.161881\pi$
$390$		0			0
$391$	2.18372e6			0.722361
$392$		0			0
$393$		−	428430.i		−	0.139926i
$394$		0			0
$395$	−2.56013e6	−	1.09553e6i	−0.825600	−	0.353290i
$396$		0			0
$397$	2.30169e6			0.732943			0.366472	−	0.930429i	$-0.380566\pi$
$397$	2.30169e6			0.732943			0.366472	+	0.930429i	$0.380566\pi$
$398$		0			0
$399$	229883.			0.0722893
$400$		0			0
$401$	−2.58744e6			−0.803544			−0.401772	−	0.915740i	$-0.631606\pi$
$401$	−2.58744e6			−0.803544			−0.401772	+	0.915740i	$0.631606\pi$
$402$		0			0
$403$	5.16364e6			1.58378
$404$		0			0
$405$	2.97176e6	+	1.27167e6i	0.900278	+	0.385246i
$406$		0			0
$407$			210863.i			0.0630978i
$408$		0			0
$409$	−3.17301e6			−0.937916			−0.468958	−	0.883221i	$-0.655370\pi$
$409$	−3.17301e6			−0.937916			−0.468958	+	0.883221i	$0.655370\pi$
$410$		0			0
$411$		−	175273.i		−	0.0511812i
$412$		0			0
$413$	−5.50768e6			−1.58889
$414$		0			0
$415$	−716374.	−	306550.i	−0.204183	−	0.0873738i
$416$		0			0
$417$		−	169192.i		−	0.0476475i
$418$		0			0
$419$		−	6.16028e6i		−	1.71422i	−0.515137	−	0.857108i	$-0.672259\pi$
$419$		−	6.16028e6i		−	1.71422i	0.515137	−	0.857108i	$-0.327741\pi$
$420$		0			0
$421$		−	2.93941e6i		−	0.808268i	−0.914700	−	0.404134i	$-0.867573\pi$
$421$		−	2.93941e6i		−	0.808268i	0.914700	−	0.404134i	$-0.132427\pi$
$422$		0			0
$423$		−	1.90503e6i		−	0.517666i
$424$		0			0
$425$	−3.06492e6	+	2.92542e6i	−0.823089	+	0.785626i
$426$		0			0
$427$	−8.31687e6			−2.20745
$428$		0			0
$429$		−	27769.2i		−	0.00728485i
$430$		0			0
$431$	−2.60556e6			−0.675629			−0.337815	−	0.941213i	$-0.609688\pi$
$431$	−2.60556e6			−0.675629			−0.337815	+	0.941213i	$0.609688\pi$
$432$		0			0
$433$		−	525118.i		−	0.134597i	−0.997733	−	0.0672987i	$-0.978562\pi$
$433$		−	525118.i		−	0.134597i	0.997733	−	0.0672987i	$-0.0214381\pi$
$434$		0			0
$435$	−168748.	+	394345.i	−0.0427578	+	0.0999202i
$436$		0			0
$437$	−1.67377e6			−0.419268
$438$		0			0
$439$	−3.48971e6			−0.864228			−0.432114	−	0.901819i	$-0.642232\pi$
$439$	−3.48971e6			−0.864228			−0.432114	+	0.901819i	$0.642232\pi$
$440$		0			0
$441$	2.95099e6			0.722555
$442$		0			0
$443$	−5.36353e6			−1.29850			−0.649250	−	0.760575i	$-0.724917\pi$
$443$	−5.36353e6			−1.29850			−0.649250	+	0.760575i	$0.724917\pi$
$444$		0			0
$445$	389642.	+	166735.i	0.0932752	+	0.0399143i
$446$		0			0
$447$			73644.4i			0.0174330i
$448$		0			0
$449$	−5.64299e6			−1.32097			−0.660486	−	0.750838i	$-0.729650\pi$
$449$	−5.64299e6			−1.32097			−0.660486	+	0.750838i	$0.729650\pi$
$450$		0			0
$451$			481985.i			0.111581i
$452$		0			0
$453$	354184.			0.0810931
$454$		0			0
$455$	−2.01328e6	+	4.70481e6i	−0.455906	+	1.06540i
$456$		0			0
$457$		−	6.54027e6i		−	1.46489i	−0.680825	−	0.732446i	$-0.738379\pi$
$457$		−	6.54027e6i		−	1.46489i	0.680825	−	0.732446i	$-0.261621\pi$
$458$		0			0
$459$			852448.i			0.188858i
$460$		0			0
$461$		−	7.92791e6i		−	1.73743i	−0.495315	−	0.868713i	$-0.664947\pi$
$461$		−	7.92791e6i		−	1.73743i	0.495315	−	0.868713i	$-0.335053\pi$
$462$		0			0
$463$			5.43590e6i			1.17847i	0.807961	+	0.589235i	$0.200571\pi$
$463$			5.43590e6i			1.17847i	−0.807961	+	0.589235i	$0.799429\pi$
$464$		0			0
$465$	641269.	+	274411.i	0.137533	+	0.0588531i
$466$		0			0
$467$	1.47407e6			0.312771			0.156386	−	0.987696i	$-0.450016\pi$
$467$	1.47407e6			0.312771			0.156386	+	0.987696i	$0.450016\pi$
$468$		0			0
$469$		−	150334.i		−	0.0315590i
$470$		0			0
$471$	−456268.			−0.0947692
$472$		0			0
$473$			769979.i			0.158244i
$474$		0			0
$475$	2.34919e6	−	2.24227e6i	0.477732	−	0.455988i
$476$		0			0
$477$	−4.56859e6			−0.919362
$478$		0			0
$479$	4.13083e6			0.822618			0.411309	−	0.911496i	$-0.365072\pi$
$479$	4.13083e6			0.822618			0.411309	+	0.911496i	$0.365072\pi$
$480$		0			0
$481$	2.84512e6			0.560709
$482$		0			0
$483$	356283.			0.0694908
$484$		0			0
$485$	−1.46800e6	+	3.43056e6i	−0.283382	+	0.662232i
$486$		0			0
$487$			2.24605e6i			0.429138i	0.976709	+	0.214569i	$0.0688347\pi$
$487$			2.24605e6i			0.429138i	−0.976709	+	0.214569i	$0.931165\pi$
$488$		0			0
$489$	−563771.			−0.106618
$490$		0			0
$491$			578316.i			0.108258i	0.998534	+	0.0541292i	$0.0172383\pi$
$491$			578316.i			0.108258i	−0.998534	+	0.0541292i	$0.982762\pi$
$492$		0			0
$493$	8.01370e6			1.48496
$494$		0			0
$495$	−211311.	+	493809.i	−0.0387622	+	0.0905829i
$496$		0			0
$497$		−	1.50271e6i		−	0.272888i
$498$		0			0
$499$			307235.i			0.0552357i	0.999619	+	0.0276178i	$0.00879215\pi$
$499$			307235.i			0.0552357i	−0.999619	+	0.0276178i	$0.991208\pi$
$500$		0			0
$501$		−	753424.i		−	0.134105i
$502$		0			0
$503$			3.87678e6i			0.683206i	0.939844	+	0.341603i	$0.110970\pi$
$503$			3.87678e6i			0.683206i	−0.939844	+	0.341603i	$0.889030\pi$
$504$		0			0
$505$	1.46234e6	−	3.41732e6i	0.255164	−	0.596289i
$506$		0			0
$507$	107324.			0.0185428
$508$		0			0
$509$		−	4.90268e6i		−	0.838762i	−0.907810	−	0.419381i	$-0.862247\pi$
$509$		−	4.90268e6i		−	0.838762i	0.907810	−	0.419381i	$-0.137753\pi$
$510$		0			0
$511$	−2.45040e6			−0.415131
$512$		0			0
$513$		−	653382.i		−	0.109616i
$514$		0			0
$515$	3.65757e6	−	8.54733e6i	0.607679	−	1.42008i
$516$		0			0
$517$	314327.			0.0517195
$518$		0			0
$519$	316451.			0.0515689
$520$		0			0
$521$	−5.57462e6			−0.899749			−0.449874	−	0.893092i	$-0.648531\pi$
$521$	−5.57462e6			−0.899749			−0.449874	+	0.893092i	$0.648531\pi$
$522$		0			0
$523$	−1.23492e6			−0.197417			−0.0987087	−	0.995116i	$-0.531471\pi$
$523$	−1.23492e6			−0.197417			−0.0987087	+	0.995116i	$0.531471\pi$
$524$		0			0
$525$	−500055.	+	477295.i	−0.0791808	+	0.0755769i
$526$		0			0
$527$		−	1.30316e7i		−	2.04395i
$528$		0			0
$529$	3.84226e6			0.596963
$530$		0			0
$531$			7.79983e6i			1.20046i
$532$		0			0
$533$	6.50329e6			0.991552
$534$		0			0
$535$	6.25189e6	+	2.67531e6i	0.944337	+	0.404100i
$536$		0			0
$537$			765718.i			0.114586i
$538$		0			0
$539$			486908.i			0.0721897i
$540$		0			0
$541$			2.07857e6i			0.305332i	0.988278	+	0.152666i	$0.0487858\pi$
$541$			2.07857e6i			0.305332i	−0.988278	+	0.152666i	$0.951214\pi$
$542$		0			0
$543$			397753.i			0.0578914i
$544$		0			0
$545$	−3.75571e6	+	8.77668e6i	−0.541627	+	1.26572i
$546$		0			0
$547$	−6.52487e6			−0.932403			−0.466201	−	0.884679i	$-0.654378\pi$
$547$	−6.52487e6			−0.932403			−0.466201	+	0.884679i	$0.654378\pi$
$548$		0			0
$549$			1.17781e7i			1.66781i
$550$		0			0
$551$	−6.14232e6			−0.861893
$552$		0			0
$553$			8.48824e6i			1.18033i
$554$		0			0
$555$	353333.	+	151198.i	0.0486913	+	0.0208359i
$556$		0			0
$557$	−1.09214e7			−1.49156			−0.745778	−	0.666194i	$-0.767922\pi$
$557$	−1.09214e7			−1.49156			−0.745778	+	0.666194i	$0.767922\pi$
$558$		0			0
$559$	1.03891e7			1.40621
$560$		0			0
$561$	−70081.7			−0.00940150
$562$		0			0
$563$	3.64784e6			0.485025			0.242513	−	0.970148i	$-0.422028\pi$
$563$	3.64784e6			0.485025			0.242513	+	0.970148i	$0.422028\pi$
$564$		0			0
$565$	−81484.5	+	190420.i	−0.0107388	+	0.0250953i
$566$		0			0
$567$		−	9.85302e6i		−	1.28710i
$568$		0			0
$569$	7.78255e6			1.00772			0.503861	−	0.863784i	$-0.331912\pi$
$569$	7.78255e6			1.00772			0.503861	+	0.863784i	$0.331912\pi$
$570$		0			0
$571$			1.52835e7i			1.96170i	0.194753	+	0.980852i	$0.437609\pi$
$571$			1.52835e7i			1.96170i	−0.194753	+	0.980852i	$0.562391\pi$
$572$		0			0
$573$	−159785.			−0.0203306
$574$		0			0
$575$	3.64088e6	−	3.47517e6i	0.459237	−	0.438335i
$576$		0			0
$577$		−	1.01412e7i		−	1.26809i	−0.773296	−	0.634046i	$-0.781393\pi$
$577$		−	1.01412e7i		−	1.26809i	0.773296	−	0.634046i	$-0.218607\pi$
$578$		0			0
$579$		−	895016.i		−	0.110952i
$580$		0			0
$581$			2.37517e6i			0.291914i
$582$		0			0
$583$		−	753811.i		−	0.0918526i
$584$		0			0
$585$	6.66284e6	+	2.85116e6i	0.804951	+	0.344454i
$586$		0			0
$587$	3.15239e6			0.377612			0.188806	−	0.982014i	$-0.439538\pi$
$587$	3.15239e6			0.377612			0.188806	+	0.982014i	$0.439538\pi$
$588$		0			0
$589$			9.98839e6i			1.18634i
$590$		0			0
$591$	−91889.3			−0.0108217
$592$		0			0
$593$		−	2.82025e6i		−	0.329345i	−0.986348	−	0.164673i	$-0.947343\pi$
$593$		−	2.82025e6i		−	0.329345i	0.986348	−	0.164673i	$-0.0526568\pi$
$594$		0			0
$595$	1.18736e7	+	5.08094e6i	1.37496	+	0.588372i
$596$		0			0
$597$	408387.			0.0468961
$598$		0			0
$599$	−1.25796e7			−1.43252			−0.716261	−	0.697833i	$-0.754148\pi$
$599$	−1.25796e7			−1.43252			−0.716261	+	0.697833i	$0.754148\pi$
$600$		0			0
$601$	3.63021e6			0.409964			0.204982	−	0.978766i	$-0.434286\pi$
$601$	3.63021e6			0.409964			0.204982	+	0.978766i	$0.434286\pi$
$602$		0			0
$603$	−212899.			−0.0238440
$604$		0			0
$605$	8.19556e6	+	3.50704e6i	0.910312	+	0.389540i
$606$		0			0
$607$		−	322555.i		−	0.0355331i	−0.999842	−	0.0177665i	$-0.994344\pi$
$607$		−	322555.i		−	0.0355331i	0.999842	−	0.0177665i	$-0.00565556\pi$
$608$		0			0
$609$	1.30747e6			0.142853
$610$		0			0
$611$		−	4.24112e6i		−	0.459598i
$612$		0			0
$613$	−1.51954e6			−0.163329			−0.0816643	−	0.996660i	$-0.526024\pi$
$613$	−1.51954e6			−0.163329			−0.0816643	+	0.996660i	$0.526024\pi$
$614$		0			0
$615$	807639.	+	345604.i	0.0861052	+	0.0368461i
$616$		0			0
$617$		−	8.51821e6i		−	0.900815i	−0.892823	−	0.450408i	$-0.851279\pi$
$617$		−	8.51821e6i		−	0.900815i	0.892823	−	0.450408i	$-0.148721\pi$
$618$		0			0
$619$		−	6.08130e6i		−	0.637926i	−0.947767	−	0.318963i	$-0.896665\pi$
$619$		−	6.08130e6i		−	0.637926i	0.947767	−	0.318963i	$-0.103335\pi$
$620$		0			0
$621$		−	1.01264e6i		−	0.105372i
$622$		0			0
$623$		−	1.29188e6i		−	0.133353i
$624$		0			0
$625$	−454587.	+	9.75504e6i	−0.0465497	+	0.998916i
$626$		0			0
$627$	53716.0			0.00545676
$628$		0			0
$629$		−	7.18026e6i		−	0.723625i
$630$		0			0
$631$	1.52443e7			1.52417			0.762086	−	0.647476i	$-0.224175\pi$
$631$	1.52443e7			1.52417			0.762086	+	0.647476i	$0.224175\pi$
$632$		0			0
$633$			1.41965e6i			0.140823i
$634$		0			0
$635$	−1.70606e6	+	3.98686e6i	−0.167903	+	0.392371i
$636$		0			0
$637$	6.56972e6			0.641503
$638$		0			0
$639$	−2.12810e6			−0.206177
$640$		0			0
$641$	1.58871e7			1.52722			0.763609	−	0.645679i	$-0.223426\pi$
$641$	1.58871e7			1.52722			0.763609	+	0.645679i	$0.223426\pi$
$642$		0			0
$643$	3.42866e6			0.327037			0.163519	−	0.986540i	$-0.447716\pi$
$643$	3.42866e6			0.327037			0.163519	+	0.986540i	$0.447716\pi$
$644$		0			0
$645$	1.29022e6	+	552109.i	0.122113	+	0.0522547i
$646$		0			0
$647$		−	3.99640e6i		−	0.375326i	−0.982234	−	0.187663i	$-0.939909\pi$
$647$		−	3.99640e6i		−	0.375326i	0.982234	−	0.187663i	$-0.0600912\pi$
$648$		0			0
$649$	−1.28696e6			−0.119937
$650$		0			0
$651$		−	2.12616e6i		−	0.196627i
$652$		0			0
$653$	6.78426e6			0.622615			0.311308	−	0.950309i	$-0.399233\pi$
$653$	6.78426e6			0.622615			0.311308	+	0.950309i	$0.399233\pi$
$654$		0			0
$655$	−7.25801e6	+	1.69612e7i	−0.661019	+	1.54473i
$656$		0			0
$657$			3.47020e6i			0.313647i
$658$		0			0
$659$			1.38518e7i			1.24249i	0.783617	+	0.621244i	$0.213372\pi$
$659$			1.38518e7i			1.24249i	−0.783617	+	0.621244i	$0.786628\pi$
$660$		0			0
$661$		−	7.99240e6i		−	0.711498i	−0.934582	−	0.355749i	$-0.884226\pi$
$661$		−	7.99240e6i		−	0.711498i	0.934582	−	0.355749i	$-0.115774\pi$
$662$		0			0
$663$			945593.i			0.0835450i
$664$		0			0
$665$	−9.10084e6	−	3.89443e6i	−0.798045	−	0.341499i
$666$		0			0
$667$	−9.51964e6			−0.828526
$668$		0			0
$669$			1.19164e6i			0.102939i
$670$		0			0
$671$	−1.94338e6			−0.166629
$672$		0			0
$673$		−	1.10858e7i		−	0.943471i	−0.881740	−	0.471735i	$-0.843628\pi$
$673$		−	1.10858e7i		−	0.943471i	0.881740	−	0.471735i	$-0.156372\pi$
$674$		0			0
$675$	1.35659e6	+	1.42128e6i	0.114601	+	0.120066i
$676$		0			0
$677$	−1.70523e7			−1.42992			−0.714958	−	0.699167i	$-0.753554\pi$
$677$	−1.70523e7			−1.42992			−0.714958	+	0.699167i	$0.753554\pi$
$678$		0			0
$679$	1.13742e7			0.946772
$680$		0			0
$681$	−1.56808e6			−0.129569
$682$		0			0
$683$	−4.24549e6			−0.348238			−0.174119	−	0.984725i	$-0.555708\pi$
$683$	−4.24549e6			−0.348238			−0.174119	+	0.984725i	$0.555708\pi$
$684$		0			0
$685$	−2.96929e6	+	6.93889e6i	−0.241783	+	0.565020i
$686$		0			0
$687$		−	420154.i		−	0.0339639i
$688$		0			0
$689$	−1.01710e7			−0.816234
$690$		0			0
$691$		−	8.87637e6i		−	0.707196i	−0.935397	−	0.353598i	$-0.884958\pi$
$691$		−	8.87637e6i		−	0.707196i	0.935397	−	0.353598i	$-0.115042\pi$
$692$		0			0
$693$	1.63725e6			0.129503
$694$		0			0
$695$	−2.86627e6	+	6.69816e6i	−0.225090	+	0.526009i
$696$		0			0
$697$		−	1.64125e7i		−	1.27965i
$698$		0			0
$699$			1.74890e6i			0.135386i
$700$		0			0
$701$			1.25460e7i			0.964298i	0.876089	+	0.482149i	$0.160144\pi$
$701$			1.25460e7i			0.964298i	−0.876089	+	0.482149i	$0.839856\pi$
$702$		0			0
$703$			5.50351e6i			0.420002i
$704$		0			0
$705$	225386.	−	526702.i	0.0170787	−	0.0399110i
$706$		0			0
$707$	−1.13303e7			−0.852495
$708$		0			0
$709$		−	6.92792e6i		−	0.517592i	−0.965932	−	0.258796i	$-0.916674\pi$
$709$		−	6.92792e6i		−	0.517592i	0.965932	−	0.258796i	$-0.0833257\pi$
$710$		0			0
$711$	1.20208e7			0.891786
$712$		0			0
$713$			1.54805e7i			1.14041i
$714$		0			0
$715$	−470437.	+	1.09936e6i	−0.0344141	+	0.0804219i
$716$		0			0
$717$	1.29493e6			0.0940694
$718$		0			0
$719$	−5.47581e6			−0.395026			−0.197513	−	0.980300i	$-0.563287\pi$
$719$	−5.47581e6			−0.395026			−0.197513	+	0.980300i	$0.563287\pi$
$720$		0			0
$721$	−2.83391e7			−2.03024
$722$		0			0
$723$	−792016.			−0.0563492
$724$		0			0
$725$	1.33611e7	−	1.27530e7i	0.944058	−	0.901089i
$726$		0			0
$727$		−	1.16133e7i		−	0.814926i	−0.913222	−	0.407463i	$-0.866414\pi$
$727$		−	1.16133e7i		−	0.814926i	0.913222	−	0.407463i	$-0.133586\pi$
$728$		0			0
$729$	−1.37553e7			−0.958628
$730$		0			0
$731$		−	2.62192e7i		−	1.81479i
$732$		0			0
$733$	−1.43863e7			−0.988984			−0.494492	−	0.869182i	$-0.664646\pi$
$733$	−1.43863e7			−0.988984			−0.494492	+	0.869182i	$0.664646\pi$
$734$		0			0
$735$	815889.	+	349135.i	0.0557074	+	0.0238383i
$736$		0			0
$737$		−	35128.0i		−	0.00238223i
$738$		0			0
$739$		−	8.91435e6i		−	0.600452i	−0.953868	−	0.300226i	$-0.902938\pi$
$739$		−	8.91435e6i		−	0.600452i	0.953868	−	0.300226i	$-0.0970621\pi$
$740$		0			0
$741$		−	724775.i		−	0.0484906i
$742$		0			0
$743$			9.84971e6i			0.654563i	0.944927	+	0.327281i	$0.106133\pi$
$743$			9.84971e6i			0.654563i	−0.944927	+	0.327281i	$0.893867\pi$
$744$		0			0
$745$	1.24760e6	−	2.91551e6i	0.0823543	−	0.192453i
$746$		0			0
$747$	3.36366e6			0.220552
$748$		0			0
$749$		−	2.07284e7i		−	1.35009i
$750$		0			0
$751$	−2.31729e7			−1.49928			−0.749638	−	0.661849i	$-0.769772\pi$
$751$	−2.31729e7			−1.49928			−0.749638	+	0.661849i	$0.769772\pi$
$752$		0			0
$753$			897165.i			0.0576614i
$754$		0			0
$755$	−1.40218e7	−	6.00021e6i	−0.895235	−	0.383089i
$756$		0			0
$757$	1.04310e7			0.661586			0.330793	−	0.943703i	$-0.392684\pi$
$757$	1.04310e7			0.661586			0.330793	+	0.943703i	$0.392684\pi$
$758$		0			0
$759$	83251.5			0.00524551
$760$		0			0
$761$	7.98132e6			0.499589			0.249795	−	0.968299i	$-0.419637\pi$
$761$	7.98132e6			0.499589			0.249795	+	0.968299i	$0.419637\pi$
$762$		0			0
$763$	2.90995e7			1.80956
$764$		0			0
$765$	7.19551e6	−	1.68151e7i	0.444537	−	1.03883i
$766$		0			0
$767$			1.73646e7i			1.06580i
$768$		0			0
$769$	1.28302e7			0.782378			0.391189	−	0.920310i	$-0.372064\pi$
$769$	1.28302e7			0.782378			0.391189	+	0.920310i	$0.372064\pi$
$770$		0			0
$771$		−	981167.i		−	0.0594439i
$772$		0			0
$773$	−1.70361e7			−1.02547			−0.512733	−	0.858548i	$-0.671367\pi$
$773$	−1.70361e7			−1.02547			−0.512733	+	0.858548i	$0.671367\pi$
$774$		0			0
$775$	−2.07385e7	−	2.17274e7i	−1.24029	−	1.29943i
$776$		0			0
$777$		−	1.17149e6i		−	0.0696124i
$778$		0			0
$779$			1.25798e7i			0.742727i
$780$		0			0
$781$		−	351134.i		−	0.0205989i
$782$		0			0
$783$		−	3.71615e6i		−	0.216615i
$784$		0			0
$785$	1.80632e7	+	7.72960e6i	1.04621	+	0.447695i
$786$		0			0
$787$	−2.17976e7			−1.25450			−0.627252	−	0.778817i	$-0.715820\pi$
$787$	−2.17976e7			−1.25450			−0.627252	+	0.778817i	$0.715820\pi$
$788$		0			0
$789$		−	411167.i		−	0.0235139i
$790$		0			0
$791$	631348.			0.0358779
$792$		0			0
$793$			2.62215e7i			1.48072i
$794$		0			0
$795$	−1.26313e6	−	540516.i	−0.0708808	−	0.0303313i
$796$		0			0
$797$	−7.68922e6			−0.428782			−0.214391	−	0.976748i	$-0.568777\pi$
$797$	−7.68922e6			−0.428782			−0.214391	+	0.976748i	$0.568777\pi$
$798$		0			0
$799$	−1.07034e7			−0.593136
$800$		0			0
$801$	−1.82953e6			−0.100753
$802$		0			0
$803$	−572578.			−0.0313362
$804$		0			0
$805$	−1.41049e7	−	6.03576e6i	−0.767150	−	0.328278i
$806$		0			0
$807$			1.94834e6i			0.105313i
$808$		0			0
$809$	4.70149e6			0.252560			0.126280	−	0.991995i	$-0.459696\pi$
$809$	4.70149e6			0.252560			0.126280	+	0.991995i	$0.459696\pi$
$810$		0			0
$811$		−	1.31175e7i		−	0.700322i	−0.936689	−	0.350161i	$-0.886127\pi$
$811$		−	1.31175e7i		−	0.700322i	0.936689	−	0.350161i	$-0.113873\pi$
$812$		0			0
$813$	−857847.			−0.0455180
$814$		0			0
$815$	2.23192e7	+	9.55080e6i	1.17702	+	0.503670i
$816$		0			0
$817$			2.00964e7i			1.05333i
$818$		0			0
$819$		−	2.20910e7i		−	1.15081i
$820$		0			0
$821$			2.69837e7i			1.39715i	0.715537	+	0.698575i	$0.246182\pi$
$821$			2.69837e7i			1.39715i	−0.715537	+	0.698575i	$0.753818\pi$
$822$		0			0
$823$		−	279830.i		−	0.0144011i	−0.999974	−	0.00720053i	$-0.997708\pi$
$823$		−	279830.i		−	0.0144011i	0.999974	−	0.00720053i	$-0.00229202\pi$
$824$		0			0
$825$	−116846.	+	111528.i	−0.00597696	+	0.00570492i
$826$		0			0
$827$	2.31889e7			1.17901			0.589504	−	0.807765i	$-0.299323\pi$
$827$	2.31889e7			1.17901			0.589504	+	0.807765i	$0.299323\pi$
$828$		0			0
$829$		−	5.30543e6i		−	0.268123i	−0.990973	−	0.134062i	$-0.957198\pi$
$829$		−	5.30543e6i		−	0.268123i	0.990973	−	0.134062i	$-0.0428020\pi$
$830$		0			0
$831$	−2.04551e6			−0.102754
$832$		0			0
$833$		−	1.65801e7i		−	0.827894i
$834$		0			0
$835$	−1.27637e7	+	2.98273e7i	−0.633520	+	1.48047i
$836$		0			0
$837$	−6.04305e6			−0.298155
$838$		0			0
$839$	−1.39887e7			−0.686077			−0.343039	−	0.939321i	$-0.611456\pi$
$839$	−1.39887e7			−0.686077			−0.343039	+	0.939321i	$0.611456\pi$
$840$		0			0
$841$	−1.44236e7			−0.703207
$842$		0			0
$843$	1.89183e6			0.0916882
$844$		0			0
$845$	−4.24884e6	−	1.81816e6i	−0.204705	−	0.0875973i
$846$		0			0
$847$		−	2.71728e7i		−	1.30144i
$848$		0			0
$849$	1.79593e6			0.0855105
$850$		0			0
$851$			8.52958e6i			0.403742i
$852$		0			0
$853$	1.48401e7			0.698335			0.349168	−	0.937060i	$-0.386464\pi$
$853$	1.48401e7			0.698335			0.349168	+	0.937060i	$0.386464\pi$
$854$		0			0
$855$	−5.51519e6	+	1.28884e7i	−0.258015	+	0.602953i
$856$		0			0
$857$			8.97091e6i			0.417239i	0.977997	+	0.208619i	$0.0668970\pi$
$857$			8.97091e6i			0.417239i	−0.977997	+	0.208619i	$0.933103\pi$
$858$		0			0
$859$		−	2.21880e7i		−	1.02597i	−0.858398	−	0.512985i	$-0.828540\pi$
$859$		−	2.21880e7i		−	1.02597i	0.858398	−	0.512985i	$-0.171460\pi$
$860$		0			0
$861$		−	2.67777e6i		−	0.123102i
$862$		0			0
$863$		−	2.60169e7i		−	1.18913i	−0.804047	−	0.594565i	$-0.797324\pi$
$863$		−	2.60169e7i		−	1.18913i	0.804047	−	0.594565i	$-0.202676\pi$
$864$		0			0
$865$	−1.25280e7	−	5.36097e6i	−0.569300	−	0.243614i
$866$		0			0
$867$	543172.			0.0245408
$868$		0			0
$869$			1.98342e6i			0.0890975i
$870$		0			0
$871$	−473972.			−0.0211694
$872$		0			0
$873$		−	1.61078e7i		−	0.715322i
$874$		0			0
$875$	2.78825e7	−	1.04243e7i	1.23115	−	0.460284i
$876$		0			0
$877$	−5.75323e6			−0.252588			−0.126294	−	0.991993i	$-0.540308\pi$
$877$	−5.75323e6			−0.252588			−0.126294	+	0.991993i	$0.540308\pi$
$878$		0			0
$879$	−706892.			−0.0308589
$880$		0			0
$881$	−1.39231e7			−0.604362			−0.302181	−	0.953251i	$-0.597715\pi$
$881$	−1.39231e7			−0.604362			−0.302181	+	0.953251i	$0.597715\pi$
$882$		0			0
$883$	−4.47703e7			−1.93236			−0.966180	−	0.257867i	$-0.916980\pi$
$883$	−4.47703e7			−1.93236			−0.966180	+	0.257867i	$0.916980\pi$
$884$		0			0
$885$	−922808.	+	2.15650e6i	−0.0396053	+	0.0925531i
$886$		0			0
$887$			1.25115e7i			0.533949i	0.963704	+	0.266974i	$0.0860239\pi$
$887$			1.25115e7i			0.533949i	−0.963704	+	0.266974i	$0.913976\pi$
$888$		0			0
$889$	1.32186e7			0.560960
$890$		0			0
$891$		−	2.30232e6i		−	0.0971566i
$892$		0			0
$893$	8.20390e6			0.344264
$894$		0			0
$895$	1.29720e7	−	3.03141e7i	0.541313	−	1.26499i
$896$		0			0
$897$		−	1.12329e6i		−	0.0466134i
$898$		0			0
$899$			5.68095e7i			2.34435i
$900$		0			0
$901$			2.56686e7i			1.05339i
$902$		0			0
$903$		−	4.27778e6i		−	0.174582i
$904$		0			0
$905$	6.73831e6	−	1.57467e7i	0.273482	−	0.639098i
$906$		0			0
$907$	3.38418e7			1.36595			0.682975	−	0.730442i	$-0.260686\pi$
$907$	3.38418e7			1.36595			0.682975	+	0.730442i	$0.260686\pi$
$908$		0			0
$909$			1.60457e7i			0.644092i
$910$		0			0
$911$	3.39018e7			1.35340			0.676701	−	0.736258i	$-0.263409\pi$
$911$	3.39018e7			1.35340			0.676701	+	0.736258i	$0.263409\pi$
$912$		0			0
$913$			554999.i			0.0220351i
$914$		0			0
$915$	−1.39349e6	+	3.25642e6i	−0.0550237	+	0.128584i
$916$		0			0
$917$	5.62355e7			2.20845
$918$		0			0
$919$	2.44903e7			0.956544			0.478272	−	0.878212i	$-0.341263\pi$
$919$	2.44903e7			0.956544			0.478272	+	0.878212i	$0.341263\pi$
$920$		0			0
$921$	−3.51245e6			−0.136446
$922$		0			0
$923$	−4.73776e6			−0.183049
$924$		0			0
$925$	−1.14267e7	−	1.19716e7i	−0.439102	−	0.460041i
$926$		0			0
$927$			4.01331e7i			1.53392i
$928$		0			0
$929$	4.91159e6			0.186717			0.0933583	−	0.995633i	$-0.470240\pi$
$929$	4.91159e6			0.186717			0.0933583	+	0.995633i	$0.470240\pi$
$930$		0			0
$931$			1.27083e7i			0.480521i
$932$		0			0
$933$	−402034.			−0.0151202
$934$		0			0
$935$	2.77447e6	+	1.18725e6i	0.103789	+	0.0444132i
$936$		0			0
$937$			2.63398e7i			0.980083i	0.871699	+	0.490042i	$0.163018\pi$
$937$			2.63398e7i			0.980083i	−0.871699	+	0.490042i	$0.836982\pi$
$938$		0			0
$939$			2.46284e6i			0.0911532i
$940$		0			0
$941$			3.47907e6i			0.128082i	0.997947	+	0.0640411i	$0.0203989\pi$
$941$			3.47907e6i			0.128082i	−0.997947	+	0.0640411i	$0.979601\pi$
$942$		0			0
$943$			1.94967e7i			0.713973i
$944$		0			0
$945$	2.35616e6	−	5.50608e6i	0.0858271	−	0.200568i
$946$		0			0
$947$	1.24338e7			0.450537			0.225268	−	0.974297i	$-0.427674\pi$
$947$	1.24338e7			0.450537			0.225268	+	0.974297i	$0.427674\pi$
$948$		0			0
$949$			7.72564e6i			0.278464i
$950$		0			0
$951$	−708891.			−0.0254172
$952$		0			0
$953$		−	4.64549e7i		−	1.65691i	−0.560056	−	0.828455i	$-0.689220\pi$
$953$		−	4.64549e7i		−	1.65691i	0.560056	−	0.828455i	$-0.310780\pi$
$954$		0			0
$955$	6.32574e6	+	2.70691e6i	0.224441	+	0.0960428i
$956$		0			0
$957$	305512.			0.0107832
$958$		0			0
$959$	2.30062e7			0.807791
$960$		0			0
$961$	6.37523e7			2.22683
$962$		0			0
$963$	−2.93551e7			−1.02004
$964$		0			0
$965$	−1.51624e7	+	3.54329e7i	−0.524143	+	1.22486i
$966$		0			0
$967$			4.66797e7i			1.60532i	0.596435	+	0.802661i	$0.296583\pi$
$967$			4.66797e7i			1.60532i	−0.596435	+	0.802661i	$0.703417\pi$
$968$		0			0
$969$	−1.82913e6			−0.0625798
$970$		0			0
$971$			1.61987e7i			0.551357i	0.961250	+	0.275678i	$0.0889025\pi$
$971$			1.61987e7i			0.551357i	−0.961250	+	0.275678i	$0.911098\pi$
$972$		0			0
$973$	2.22081e7			0.752019
$974$		0			0
$975$	1.50482e6	+	1.57658e6i	0.0506959	+	0.0531133i
$976$		0			0
$977$		−	1.46888e7i		−	0.492322i	−0.969229	−	0.246161i	$-0.920831\pi$
$977$		−	1.46888e7i		−	0.492322i	0.969229	−	0.246161i	$-0.0791693\pi$
$978$		0			0
$979$		−	301869.i		−	0.0100661i
$980$		0			0
$981$		−	4.12100e7i		−	1.36719i
$982$		0			0
$983$		−	4.22739e6i		−	0.139537i	−0.997563	−	0.0697684i	$-0.977774\pi$
$983$		−	4.22739e6i		−	0.139537i	0.997563	−	0.0697684i	$-0.0222260\pi$
$984$		0			0
$985$	3.63781e6	+	1.55669e6i	0.119467	+	0.0511224i
$986$		0			0
$987$	−1.74631e6			−0.0570594
$988$		0			0
$989$			3.11463e7i			1.01255i
$990$		0			0
$991$	−1.49618e7			−0.483948			−0.241974	−	0.970283i	$-0.577795\pi$
$991$	−1.49618e7			−0.483948			−0.241974	+	0.970283i	$0.577795\pi$
$992$		0			0
$993$		−	2.04432e6i		−	0.0657925i
$994$		0			0
$995$	−1.61677e7	−	6.91846e6i	−0.517714	−	0.221540i
$996$		0			0
$997$	−3.45075e7			−1.09945			−0.549724	−	0.835346i	$-0.685267\pi$
$997$	−3.45075e7			−1.09945			−0.549724	+	0.835346i	$0.685267\pi$
$998$		0			0
$999$	−3.32966e6			−0.105557

Display

a_p

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p

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a_n

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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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.f.a.49.14		28
4.3	odd	2		40.6.f.a.29.10	yes	28
5.2	odd	4		800.6.d.e.401.4		28
5.3	odd	4		800.6.d.e.401.25		28
5.4	even	2	inner	160.6.f.a.49.15		28
8.3	odd	2		40.6.f.a.29.20	yes	28
8.5	even	2	inner	160.6.f.a.49.16		28
20.3	even	4		200.6.d.e.101.24		28
20.7	even	4		200.6.d.e.101.5		28
20.19	odd	2		40.6.f.a.29.19	yes	28
40.3	even	4		200.6.d.e.101.23		28
40.13	odd	4		800.6.d.e.401.26		28
40.19	odd	2		40.6.f.a.29.9	✓	28
40.27	even	4		200.6.d.e.101.6		28
40.29	even	2	inner	160.6.f.a.49.13		28
40.37	odd	4		800.6.d.e.401.3		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.f.a.29.9	✓	28	40.19	odd	2
40.6.f.a.29.10	yes	28	4.3	odd	2
40.6.f.a.29.19	yes	28	20.19	odd	2
40.6.f.a.29.20	yes	28	8.3	odd	2
160.6.f.a.49.13		28	40.29	even	2	inner
160.6.f.a.49.14		28	1.1	even	1	trivial
160.6.f.a.49.15		28	5.4	even	2	inner
160.6.f.a.49.16		28	8.5	even	2	inner
200.6.d.e.101.5		28	20.7	even	4
200.6.d.e.101.6		28	40.27	even	4
200.6.d.e.101.23		28	40.3	even	4
200.6.d.e.101.24		28	20.3	even	4
800.6.d.e.401.3		28	40.37	odd	4
800.6.d.e.401.4		28	5.2	odd	4
800.6.d.e.401.25		28	5.3	odd	4
800.6.d.e.401.26		28	40.13	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	160.6.f.a.49.14

Level	$160$
Weight	$6$
Character	160.49
Analytic conductor	$25.661$
Analytic rank	$0$
Dimension	$28$
Inner twists	$4$