LMFDB - Embedded newform 160.6.f.a.49.11

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			49.11
Character	$\chi$	$=$	160.49
Dual form			160.6.f.a.49.12

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.17847 q^{3} +(1.28331 + 55.8870i) q^{5} -146.905i q^{7} -191.470 q^{9} -42.4351i q^{11} +605.383 q^{13} +(-9.21223 - 401.183i) q^{15} +409.138i q^{17} -2096.86i q^{19} +1054.55i q^{21} +3048.59i q^{23} +(-3121.71 + 143.441i) q^{25} +3118.83 q^{27} +4594.96i q^{29} +5118.83 q^{31} +304.619i q^{33} +(8210.06 - 188.525i) q^{35} +11248.5 q^{37} -4345.73 q^{39} +11002.0 q^{41} +8413.16 q^{43} +(-245.715 - 10700.7i) q^{45} -3975.72i q^{47} -4774.02 q^{49} -2936.99i q^{51} +38273.5 q^{53} +(2371.57 - 54.4575i) q^{55} +15052.2i q^{57} +36611.0i q^{59} -3181.32i q^{61} +28127.8i q^{63} +(776.897 + 33833.0i) q^{65} +41894.6 q^{67} -21884.2i q^{69} -72161.6 q^{71} -48275.5i q^{73} +(22409.1 - 1029.69i) q^{75} -6233.92 q^{77} +39428.9 q^{79} +24138.7 q^{81} -46504.5 q^{83} +(-22865.5 + 525.053i) q^{85} -32984.8i q^{87} -38217.3 q^{89} -88933.7i q^{91} -36745.3 q^{93} +(117187. - 2690.93i) q^{95} -95979.0i q^{97} +8125.03i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$28 q + 1940 q^{9} + 488 q^{15} + 1556 q^{25} - 4368 q^{31} - 23360 q^{39} - 2480 q^{41} - 38420 q^{49} + 48776 q^{55} + 37200 q^{65} + 69232 q^{71} + 35984 q^{79} + 122596 q^{81} - 178744 q^{89} - 89416 q^{95}+O(q^{100})$ 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
$3$	−7.17847			−0.460499			−0.230250	−	0.973132i	$-0.573954\pi$
$3$	−7.17847			−0.460499			−0.230250	+	0.973132i	$0.573954\pi$
$4$		0			0
$5$	1.28331	+	55.8870i	0.0229566	+	0.999736i
$5$	1.28331	+	55.8870i	0.0229566	+	0.999736i
$6$		0			0
$7$		−	146.905i		−	1.13316i	−0.824007	−	0.566580i	$-0.808266\pi$
$7$		−	146.905i		−	1.13316i	0.824007	−	0.566580i	$-0.191734\pi$
$8$		0			0
$9$	−191.470			−0.787941
$10$		0			0
$11$		−	42.4351i		−	0.105741i	−0.998601	−	0.0528705i	$-0.983163\pi$
$11$		−	42.4351i		−	0.105741i	0.998601	−	0.0528705i	$-0.0168371\pi$
$12$		0			0
$13$	605.383			0.993510			0.496755	−	0.867891i	$-0.334525\pi$
$13$	605.383			0.993510			0.496755	+	0.867891i	$0.334525\pi$
$14$		0			0
$15$	−9.21223	−	401.183i	−0.0105715	−	0.460378i
$16$		0			0
$17$			409.138i			0.343359i	0.985153	+	0.171679i	$0.0549193\pi$
$17$			409.138i			0.343359i	−0.985153	+	0.171679i	$0.945081\pi$
$18$		0			0
$19$		−	2096.86i		−	1.33256i	−0.745704	−	0.666278i	$-0.767886\pi$
$19$		−	2096.86i		−	1.33256i	0.745704	−	0.666278i	$-0.232114\pi$
$20$		0			0
$21$			1054.55i			0.521819i
$22$		0			0
$23$			3048.59i			1.20166i	0.799379	+	0.600828i	$0.205162\pi$
$23$			3048.59i			1.20166i	−0.799379	+	0.600828i	$0.794838\pi$
$24$		0			0
$25$	−3121.71	+	143.441i	−0.998946	+	0.0459011i
$26$		0			0
$27$	3118.83			0.823345
$28$		0			0
$29$			4594.96i			1.01458i	0.861775	+	0.507290i	$0.169353\pi$
$29$			4594.96i			1.01458i	−0.861775	+	0.507290i	$0.830647\pi$
$30$		0			0
$31$	5118.83			0.956679			0.478339	−	0.878175i	$-0.341239\pi$
$31$	5118.83			0.956679			0.478339	+	0.878175i	$0.341239\pi$
$32$		0			0
$33$			304.619i			0.0486937i
$34$		0			0
$35$	8210.06	−	188.525i	1.13286	−	0.0260135i
$36$		0			0
$37$	11248.5			1.35080			0.675398	−	0.737454i	$-0.263972\pi$
$37$	11248.5			1.35080			0.675398	+	0.737454i	$0.263972\pi$
$38$		0			0
$39$	−4345.73			−0.457510
$40$		0			0
$41$	11002.0			1.02215			0.511073	−	0.859537i	$-0.329248\pi$
$41$	11002.0			1.02215			0.511073	+	0.859537i	$0.329248\pi$
$42$		0			0
$43$	8413.16			0.693886			0.346943	−	0.937886i	$-0.387220\pi$
$43$	8413.16			0.693886			0.346943	+	0.937886i	$0.387220\pi$
$44$		0			0
$45$	−245.715	−	10700.7i	−0.0180884	−	0.787733i
$46$		0			0
$47$		−	3975.72i		−	0.262525i	−0.991348	−	0.131263i	$-0.958097\pi$
$47$		−	3975.72i		−	0.262525i	0.991348	−	0.131263i	$-0.0419031\pi$
$48$		0			0
$49$	−4774.02			−0.284049
$50$		0			0
$51$		−	2936.99i		−	0.158116i
$52$		0			0
$53$	38273.5			1.87158			0.935790	−	0.352558i	$-0.114688\pi$
$53$	38273.5			1.87158			0.935790	+	0.352558i	$0.114688\pi$
$54$		0			0
$55$	2371.57	−	54.4575i	0.105713	−	0.00242746i
$56$		0			0
$57$			15052.2i			0.613641i
$58$		0			0
$59$			36611.0i			1.36924i	0.728898	+	0.684622i	$0.240033\pi$
$59$			36611.0i			1.36924i	−0.728898	+	0.684622i	$0.759967\pi$
$60$		0			0
$61$		−	3181.32i		−	0.109467i	−0.998501	−	0.0547335i	$-0.982569\pi$
$61$		−	3181.32i		−	0.109467i	0.998501	−	0.0547335i	$-0.0174309\pi$
$62$		0			0
$63$			28127.8i			0.892862i
$64$		0			0
$65$	776.897	+	33833.0i	0.0228076	+	0.993248i
$66$		0			0
$67$	41894.6			1.14017			0.570087	−	0.821584i	$-0.306909\pi$
$67$	41894.6			1.14017			0.570087	+	0.821584i	$0.306909\pi$
$68$		0			0
$69$		−	21884.2i		−	0.553361i
$70$		0			0
$71$	−72161.6			−1.69887			−0.849435	−	0.527693i	$-0.823057\pi$
$71$	−72161.6			−1.69887			−0.849435	+	0.527693i	$0.823057\pi$
$72$		0			0
$73$		−	48275.5i		−	1.06028i	−0.847911	−	0.530138i	$-0.822140\pi$
$73$		−	48275.5i		−	1.06028i	0.847911	−	0.530138i	$-0.177860\pi$
$74$		0			0
$75$	22409.1	−	1029.69i	0.460014	−	0.0211374i
$76$		0			0
$77$	−6233.92			−0.119821
$78$		0			0
$79$	39428.9			0.710798			0.355399	−	0.934715i	$-0.384345\pi$
$79$	39428.9			0.710798			0.355399	+	0.934715i	$0.384345\pi$
$80$		0			0
$81$	24138.7			0.408791
$82$		0			0
$83$	−46504.5			−0.740968			−0.370484	−	0.928839i	$-0.620808\pi$
$83$	−46504.5			−0.740968			−0.370484	+	0.928839i	$0.620808\pi$
$84$		0			0
$85$	−22865.5	+	525.053i	−0.343268	+	0.00788235i
$86$		0			0
$87$		−	32984.8i		−	0.467213i
$88$		0			0
$89$	−38217.3			−0.511428			−0.255714	−	0.966752i	$-0.582311\pi$
$89$	−38217.3			−0.511428			−0.255714	+	0.966752i	$0.582311\pi$
$90$		0			0
$91$		−	88933.7i		−	1.12580i
$92$		0			0
$93$	−36745.3			−0.440550
$94$		0			0
$95$	117187.	−	2690.93i	1.33220	−	0.0305910i
$96$		0			0
$97$		−	95979.0i		−	1.03573i	−0.855462	−	0.517865i	$-0.826727\pi$
$97$		−	95979.0i		−	1.03573i	0.855462	−	0.517865i	$-0.173273\pi$
$98$		0			0
$99$			8125.03i			0.0833177i
$100$		0			0
$101$			83900.0i			0.818386i	0.912448	+	0.409193i	$0.134190\pi$
$101$			83900.0i			0.818386i	−0.912448	+	0.409193i	$0.865810\pi$
$102$		0			0
$103$		−	10693.4i		−	0.0993165i	−0.998766	−	0.0496583i	$-0.984187\pi$
$103$		−	10693.4i		−	0.0993165i	0.998766	−	0.0496583i	$-0.0158132\pi$
$104$		0			0
$105$	−58935.7	+	1353.32i	−0.521681	+	0.0119792i
$106$		0			0
$107$	−109794.			−0.927081			−0.463540	−	0.886076i	$-0.653421\pi$
$107$	−109794.			−0.927081			−0.463540	+	0.886076i	$0.653421\pi$
$108$		0			0
$109$			135615.i			1.09330i	0.837361	+	0.546651i	$0.184097\pi$
$109$			135615.i			1.09330i	−0.837361	+	0.546651i	$0.815903\pi$
$110$		0			0
$111$	−80746.9			−0.622040
$112$		0			0
$113$			195440.i			1.43985i	0.694050	+	0.719927i	$0.255825\pi$
$113$			195440.i			1.43985i	−0.694050	+	0.719927i	$0.744175\pi$
$114$		0			0
$115$	−170377.	+	3912.30i	−1.20134	+	0.0275859i
$116$		0			0
$117$	−115913.			−0.782827
$118$		0			0
$119$	60104.4			0.389080
$120$		0			0
$121$	159250.			0.988819
$122$		0			0
$123$	−78977.7			−0.470697
$124$		0			0
$125$	−12022.6	−	174279.i	−0.0688214	−	0.997629i
$126$		0			0
$127$		−	171245.i		−	0.942126i	−0.882100	−	0.471063i	$-0.843870\pi$
$127$		−	171245.i		−	0.942126i	0.882100	−	0.471063i	$-0.156130\pi$
$128$		0			0
$129$	−60393.6			−0.319534
$130$		0			0
$131$		−	111420.i		−	0.567265i	−0.958933	−	0.283632i	$-0.908460\pi$
$131$		−	111420.i		−	0.567265i	0.958933	−	0.283632i	$-0.0915395\pi$
$132$		0			0
$133$	−308039.			−1.51000
$134$		0			0
$135$	4002.43	+	174302.i	0.0189012	+	0.823128i
$136$		0			0
$137$		−	282243.i		−	1.28476i	−0.766387	−	0.642379i	$-0.777948\pi$
$137$		−	282243.i		−	1.28476i	0.766387	−	0.642379i	$-0.222052\pi$
$138$		0			0
$139$			44060.1i			0.193423i	0.995312	+	0.0967117i	$0.0308325\pi$
$139$			44060.1i			0.193423i	−0.995312	+	0.0967117i	$0.969168\pi$
$140$		0			0
$141$			28539.6i			0.120893i
$142$		0			0
$143$		−	25689.5i		−	0.105055i
$144$		0			0
$145$	−256798.	+	5896.77i	−1.01431	+	0.0232913i
$146$		0			0
$147$	34270.1			0.130804
$148$		0			0
$149$		−	251552.i		−	0.928244i	−0.885771	−	0.464122i	$-0.846370\pi$
$149$		−	251552.i		−	0.928244i	0.885771	−	0.464122i	$-0.153630\pi$
$150$		0			0
$151$	−95790.0			−0.341883			−0.170942	−	0.985281i	$-0.554681\pi$
$151$	−95790.0			−0.341883			−0.170942	+	0.985281i	$0.554681\pi$
$152$		0			0
$153$		−	78337.6i		−	0.270546i
$154$		0			0
$155$	6569.06	+	286076.i	0.0219621	+	0.956427i
$156$		0			0
$157$	178462.			0.577824			0.288912	−	0.957356i	$-0.406706\pi$
$157$	178462.			0.577824			0.288912	+	0.957356i	$0.406706\pi$
$158$		0			0
$159$	−274745.			−0.861861
$160$		0			0
$161$	447853.			1.36167
$162$		0			0
$163$	147330.			0.434332			0.217166	−	0.976135i	$-0.430319\pi$
$163$	147330.			0.434332			0.217166	+	0.976135i	$0.430319\pi$
$164$		0			0
$165$	−17024.2	+	390.922i	−0.0486808	+	0.00111784i
$166$		0			0
$167$		−	310989.i		−	0.862888i	−0.902140	−	0.431444i	$-0.858004\pi$
$167$		−	310989.i		−	0.862888i	0.902140	−	0.431444i	$-0.141996\pi$
$168$		0			0
$169$	−4803.84			−0.0129381
$170$		0			0
$171$			401485.i			1.04997i
$172$		0			0
$173$	120278.			0.305542			0.152771	−	0.988262i	$-0.451180\pi$
$173$	120278.			0.305542			0.152771	+	0.988262i	$0.451180\pi$
$174$		0			0
$175$	21072.2	+	458594.i	0.0520133	+	1.13196i
$176$		0			0
$177$		−	262811.i		−	0.630536i
$178$		0			0
$179$		−	180544.i		−	0.421162i	−0.977576	−	0.210581i	$-0.932464\pi$
$179$		−	180544.i		−	0.421162i	0.977576	−	0.210581i	$-0.0675356\pi$
$180$		0			0
$181$			149699.i			0.339643i	0.985475	+	0.169821i	$0.0543191\pi$
$181$			149699.i			0.339643i	−0.985475	+	0.169821i	$0.945681\pi$
$182$		0			0
$183$			22837.0i			0.0504095i
$184$		0			0
$185$	14435.3	+	628643.i	0.0310097	+	1.35044i
$186$		0			0
$187$	17361.8			0.0363071
$188$		0			0
$189$		−	458171.i		−	0.932981i
$190$		0			0
$191$	−213909.			−0.424273			−0.212136	−	0.977240i	$-0.568042\pi$
$191$	−213909.			−0.424273			−0.212136	+	0.977240i	$0.568042\pi$
$192$		0			0
$193$			729587.i			1.40988i	0.709265	+	0.704942i	$0.249027\pi$
$193$			729587.i			1.40988i	−0.709265	+	0.704942i	$0.750973\pi$
$194$		0			0
$195$	−5576.93	−	242869.i	−0.0105029	−	0.457390i
$196$		0			0
$197$	922110.			1.69285			0.846423	−	0.532512i	$-0.178752\pi$
$197$	922110.			1.69285			0.846423	+	0.532512i	$0.178752\pi$
$198$		0			0
$199$	985328.			1.76379			0.881897	−	0.471442i	$-0.156266\pi$
$199$	985328.			1.76379			0.881897	+	0.471442i	$0.156266\pi$
$200$		0			0
$201$	−300739.			−0.525049
$202$		0			0
$203$	675021.			1.14968
$204$		0			0
$205$	14119.0	+	614870.i	0.0234650	+	1.02188i
$206$		0			0
$207$		−	583713.i		−	0.946833i
$208$		0			0
$209$	−88980.5			−0.140906
$210$		0			0
$211$		−	932526.i		−	1.44197i	−0.692953	−	0.720983i	$-0.743691\pi$
$211$		−	932526.i		−	1.44197i	0.692953	−	0.720983i	$-0.256309\pi$
$212$		0			0
$213$	518010.			0.782328
$214$		0			0
$215$	10796.7	+	470186.i	0.0159293	+	0.693703i
$216$		0			0
$217$		−	751980.i		−	1.08407i
$218$		0			0
$219$			346544.i			0.488257i
$220$		0			0
$221$			247686.i			0.341130i
$222$		0			0
$223$			748564.i			1.00801i	0.863699	+	0.504007i	$0.168142\pi$
$223$			748564.i			1.00801i	−0.863699	+	0.504007i	$0.831858\pi$
$224$		0			0
$225$	597712.	−	27464.6i	0.787110	−	0.0361674i
$226$		0			0
$227$	1.47117e6			1.89495			0.947475	−	0.319829i	$-0.103626\pi$
$227$	1.47117e6			1.89495			0.947475	+	0.319829i	$0.103626\pi$
$228$		0			0
$229$			835029.i			1.05223i	0.850412	+	0.526117i	$0.176353\pi$
$229$			835029.i			1.05223i	−0.850412	+	0.526117i	$0.823647\pi$
$230$		0			0
$231$	44750.0			0.0551777
$232$		0			0
$233$		−	284718.i		−	0.343577i	−0.985134	−	0.171789i	$-0.945045\pi$
$233$		−	284718.i		−	0.343577i	0.985134	−	0.171789i	$-0.0549546\pi$
$234$		0			0
$235$	222191.	−	5102.09i	0.262456	−	0.00602669i
$236$		0			0
$237$	−283039.			−0.327322
$238$		0			0
$239$	−800635.			−0.906650			−0.453325	−	0.891345i	$-0.649762\pi$
$239$	−800635.			−0.906650			−0.453325	+	0.891345i	$0.649762\pi$
$240$		0			0
$241$	−1.50596e6			−1.67021			−0.835107	−	0.550088i	$-0.814594\pi$
$241$	−1.50596e6			−1.67021			−0.835107	+	0.550088i	$0.814594\pi$
$242$		0			0
$243$	−931154.			−1.01159
$244$		0			0
$245$	−6126.56	−	266805.i	−0.00652081	−	0.283975i
$246$		0			0
$247$		−	1.26940e6i		−	1.32391i
$248$		0			0
$249$	333831.			0.341215
$250$		0			0
$251$			1.10255e6i			1.10463i	0.833637	+	0.552313i	$0.186255\pi$
$251$			1.10255e6i			1.10463i	−0.833637	+	0.552313i	$0.813745\pi$
$252$		0			0
$253$	129367.			0.127064
$254$		0			0
$255$	164139.	−	3769.08i	0.158075	−	0.00362981i
$256$		0			0
$257$		−	1.41038e6i		−	1.33200i	−0.745953	−	0.665999i	$-0.768006\pi$
$257$		−	1.41038e6i		−	1.33200i	0.745953	−	0.665999i	$-0.231994\pi$
$258$		0			0
$259$		−	1.65246e6i		−	1.53067i
$260$		0			0
$261$		−	879794.i		−	0.799429i
$262$		0			0
$263$			1.68879e6i			1.50552i	0.658298	+	0.752758i	$0.271277\pi$
$263$			1.68879e6i			1.50552i	−0.658298	+	0.752758i	$0.728723\pi$
$264$		0			0
$265$	49116.9	+	2.13899e6i	0.0429651	+	1.87109i
$266$		0			0
$267$	274342.			0.235512
$268$		0			0
$269$		−	305848.i		−	0.257706i	−0.991664	−	0.128853i	$-0.958870\pi$
$269$		−	305848.i		−	0.257706i	0.991664	−	0.128853i	$-0.0411295\pi$
$270$		0			0
$271$	1.25325e6			1.03661			0.518306	−	0.855195i	$-0.326563\pi$
$271$	1.25325e6			1.03661			0.518306	+	0.855195i	$0.326563\pi$
$272$		0			0
$273$			638408.i			0.518432i
$274$		0			0
$275$	6086.93	+	132470.i	0.00485363	+	0.105630i
$276$		0			0
$277$	−483396.			−0.378533			−0.189267	−	0.981926i	$-0.560611\pi$
$277$	−483396.			−0.378533			−0.189267	+	0.981926i	$0.560611\pi$
$278$		0			0
$279$	−980100.			−0.753806
$280$		0			0
$281$	88721.8			0.0670293			0.0335147	−	0.999438i	$-0.489330\pi$
$281$	88721.8			0.0670293			0.0335147	+	0.999438i	$0.489330\pi$
$282$		0			0
$283$	−1.76445e6			−1.30961			−0.654807	−	0.755796i	$-0.727250\pi$
$283$	−1.76445e6			−1.30961			−0.654807	+	0.755796i	$0.727250\pi$
$284$		0			0
$285$	−841224.	+	19316.7i	−0.613479	+	0.0140871i
$286$		0			0
$287$		−	1.61625e6i		−	1.15825i
$288$		0			0
$289$	1.25246e6			0.882105
$290$		0			0
$291$			688982.i			0.476953i
$292$		0			0
$293$	9124.96			0.00620957			0.00310479	−	0.999995i	$-0.499012\pi$
$293$	9124.96			0.00620957			0.00310479	+	0.999995i	$0.499012\pi$
$294$		0			0
$295$	−2.04608e6	+	46983.3i	−1.36888	+	0.0314332i
$296$		0			0
$297$		−	132348.i		−	0.0870614i
$298$		0			0
$299$			1.84557e6i			1.19386i
$300$		0			0
$301$		−	1.23593e6i		−	0.786283i
$302$		0			0
$303$		−	602273.i		−	0.376866i
$304$		0			0
$305$	177795.	−	4082.64i	0.109438	−	0.00251299i
$306$		0			0
$307$	−2.64214e6			−1.59996			−0.799980	−	0.600026i	$-0.795157\pi$
$307$	−2.64214e6			−1.59996			−0.799980	+	0.600026i	$0.795157\pi$
$308$		0			0
$309$			76762.0i			0.0457352i
$310$		0			0
$311$	1.40253e6			0.822262			0.411131	−	0.911576i	$-0.365134\pi$
$311$	1.40253e6			0.822262			0.411131	+	0.911576i	$0.365134\pi$
$312$		0			0
$313$		−	1.35907e6i		−	0.784115i	−0.919941	−	0.392058i	$-0.871763\pi$
$313$		−	1.35907e6i		−	0.784115i	0.919941	−	0.392058i	$-0.128237\pi$
$314$		0			0
$315$	−1.57198e6	+	36096.8i	−0.892627	+	0.0204971i
$316$		0			0
$317$	−300344.			−0.167869			−0.0839344	−	0.996471i	$-0.526749\pi$
$317$	−300344.			−0.167869			−0.0839344	+	0.996471i	$0.526749\pi$
$318$		0			0
$319$	194987.			0.107283
$320$		0			0
$321$	788150.			0.426920
$322$		0			0
$323$	857906.			0.457545
$324$		0			0
$325$	−1.88983e6	+	86836.8i	−0.992463	+	0.0456032i
$326$		0			0
$327$		−	973505.i		−	0.503464i
$328$		0			0
$329$	−584052.			−0.297483
$330$		0			0
$331$			2.37543e6i			1.19172i	0.803090	+	0.595858i	$0.203188\pi$
$331$			2.37543e6i			1.19172i	−0.803090	+	0.595858i	$0.796812\pi$
$332$		0			0
$333$	−2.15374e6			−1.06435
$334$		0			0
$335$	53763.9	+	2.34136e6i	0.0261745	+	1.13987i
$336$		0			0
$337$		−	3.69929e6i		−	1.77437i	−0.461416	−	0.887184i	$-0.652659\pi$
$337$		−	3.69929e6i		−	1.77437i	0.461416	−	0.887184i	$-0.347341\pi$
$338$		0			0
$339$		−	1.40296e6i		−	0.663051i
$340$		0			0
$341$		−	217218.i		−	0.101160i
$342$		0			0
$343$		−	1.76770e6i		−	0.811286i
$344$		0			0
$345$	1.22304e6	−	28084.3i	0.553215	−	0.0127033i
$346$		0			0
$347$	1.98355e6			0.884339			0.442170	−	0.896931i	$-0.354209\pi$
$347$	1.98355e6			0.884339			0.442170	+	0.896931i	$0.354209\pi$
$348$		0			0
$349$			1.51829e6i			0.667256i	0.942705	+	0.333628i	$0.108273\pi$
$349$			1.51829e6i			0.667256i	−0.942705	+	0.333628i	$0.891727\pi$
$350$		0			0
$351$	1.88809e6			0.818001
$352$		0			0
$353$			2.06194e6i			0.880724i	0.897820	+	0.440362i	$0.145150\pi$
$353$			2.06194e6i			0.880724i	−0.897820	+	0.440362i	$0.854850\pi$
$354$		0			0
$355$	−92605.9	−	4.03289e6i	−0.0390003	−	1.69842i
$356$		0			0
$357$	−431458.			−0.179171
$358$		0			0
$359$	−1.33255e6			−0.545691			−0.272846	−	0.962058i	$-0.587965\pi$
$359$	−1.33255e6			−0.545691			−0.272846	+	0.962058i	$0.587965\pi$
$360$		0			0
$361$	−1.92072e6			−0.775705
$362$		0			0
$363$	−1.14317e6			−0.455350
$364$		0			0
$365$	2.69797e6	−	61952.6i	1.06000	−	0.0243404i
$366$		0			0
$367$			313330.i			0.121433i	0.998155	+	0.0607166i	$0.0193386\pi$
$367$			313330.i			0.121433i	−0.998155	+	0.0607166i	$0.980661\pi$
$368$		0			0
$369$	−2.10655e6			−0.805390
$370$		0			0
$371$		−	5.62256e6i		−	2.12080i
$372$		0			0
$373$	−2.93069e6			−1.09068			−0.545341	−	0.838215i	$-0.683600\pi$
$373$	−2.93069e6			−1.09068			−0.545341	+	0.838215i	$0.683600\pi$
$374$		0			0
$375$	86303.9	+	1.25105e6i	0.0316922	+	0.459407i
$376$		0			0
$377$			2.78171e6i			1.00800i
$378$		0			0
$379$			4.10811e6i			1.46907i	0.678569	+	0.734537i	$0.262601\pi$
$379$			4.10811e6i			1.46907i	−0.678569	+	0.734537i	$0.737399\pi$
$380$		0			0
$381$			1.22928e6i			0.433848i
$382$		0			0
$383$			958991.i			0.334055i	0.985952	+	0.167027i	$0.0534168\pi$
$383$			958991.i			0.334055i	−0.985952	+	0.167027i	$0.946583\pi$
$384$		0			0
$385$	−8000.07	−	348395.i	−0.00275069	−	0.119790i
$386$		0			0
$387$	−1.61086e6			−0.546741
$388$		0			0
$389$			647634.i			0.216998i	0.994097	+	0.108499i	$0.0346044\pi$
$389$			647634.i			0.216998i	−0.994097	+	0.108499i	$0.965396\pi$
$390$		0			0
$391$	−1.24730e6			−0.412599
$392$		0			0
$393$			799827.i			0.261225i
$394$		0			0
$395$	50599.6	+	2.20356e6i	0.0163175	+	0.710611i
$396$		0			0
$397$	2.06284e6			0.656885			0.328443	−	0.944524i	$-0.393476\pi$
$397$	2.06284e6			0.656885			0.328443	+	0.944524i	$0.393476\pi$
$398$		0			0
$399$	2.21125e6			0.695352
$400$		0			0
$401$	−1.92225e6			−0.596964			−0.298482	−	0.954415i	$-0.596480\pi$
$401$	−1.92225e6			−0.596964			−0.298482	+	0.954415i	$0.596480\pi$
$402$		0			0
$403$	3.09885e6			0.950470
$404$		0			0
$405$	30977.5	+	1.34904e6i	0.00938446	+	0.408683i
$406$		0			0
$407$		−	477330.i		−	0.142834i
$408$		0			0
$409$	1.40471e6			0.415221			0.207610	−	0.978212i	$-0.433431\pi$
$409$	1.40471e6			0.415221			0.207610	+	0.978212i	$0.433431\pi$
$410$		0			0
$411$			2.02607e6i			0.591630i
$412$		0			0
$413$	5.37833e6			1.55157
$414$		0			0
$415$	−59679.8	−	2.59899e6i	−0.0170101	−	0.740773i
$416$		0			0
$417$		−	316284.i		−	0.0890712i
$418$		0			0
$419$		−	4.03594e6i		−	1.12308i	−0.827451	−	0.561538i	$-0.810210\pi$
$419$		−	4.03594e6i		−	1.12308i	0.827451	−	0.561538i	$-0.189790\pi$
$420$		0			0
$421$			594910.i			0.163586i	0.996649	+	0.0817929i	$0.0260646\pi$
$421$			594910.i			0.163586i	−0.996649	+	0.0817929i	$0.973935\pi$
$422$		0			0
$423$			761229.i			0.206854i
$424$		0			0
$425$	−58687.2	−	1.27721e6i	−0.0157605	−	0.342997i
$426$		0			0
$427$	−467352.			−0.124044
$428$		0			0
$429$			184411.i			0.0483776i
$430$		0			0
$431$	−4.15801e6			−1.07818			−0.539091	−	0.842247i	$-0.681232\pi$
$431$	−4.15801e6			−1.07818			−0.539091	+	0.842247i	$0.681232\pi$
$432$		0			0
$433$			2.27636e6i			0.583474i	0.956499	+	0.291737i	$0.0942333\pi$
$433$			2.27636e6i			0.583474i	−0.956499	+	0.291737i	$0.905767\pi$
$434$		0			0
$435$	1.84342e6	−	42329.8i	0.467090	−	0.0107256i
$436$		0			0
$437$	6.39247e6			1.60127
$438$		0			0
$439$	1.28126e6			0.317305			0.158653	−	0.987334i	$-0.449285\pi$
$439$	1.28126e6			0.317305			0.158653	+	0.987334i	$0.449285\pi$
$440$		0			0
$441$	914079.			0.223814
$442$		0			0
$443$	6.01266e6			1.45565			0.727825	−	0.685763i	$-0.240531\pi$
$443$	6.01266e6			1.45565			0.727825	+	0.685763i	$0.240531\pi$
$444$		0			0
$445$	−49044.7	−	2.13585e6i	−0.0117407	−	0.511294i
$446$		0			0
$447$			1.80576e6i			0.427455i
$448$		0			0
$449$	2.11543e6			0.495202			0.247601	−	0.968862i	$-0.420358\pi$
$449$	2.11543e6			0.495202			0.247601	+	0.968862i	$0.420358\pi$
$450$		0			0
$451$		−	466872.i		−	0.108083i
$452$		0			0
$453$	687625.			0.157437
$454$		0			0
$455$	4.97024e6	−	114130.i	1.12551	−	0.0258447i
$456$		0			0
$457$		−	1.35964e6i		−	0.304533i	−0.988339	−	0.152267i	$-0.951343\pi$
$457$		−	1.35964e6i		−	0.304533i	0.988339	−	0.152267i	$-0.0486573\pi$
$458$		0			0
$459$			1.27603e6i			0.282703i
$460$		0			0
$461$		−	2.06438e6i		−	0.452415i	−0.974079	−	0.226208i	$-0.927367\pi$
$461$		−	2.06438e6i		−	0.452415i	0.974079	−	0.226208i	$-0.0726328\pi$
$462$		0			0
$463$		−	1.48917e6i		−	0.322843i	−0.986886	−	0.161421i	$-0.948392\pi$
$463$		−	1.48917e6i		−	0.322843i	0.986886	−	0.161421i	$-0.0516078\pi$
$464$		0			0
$465$	−47155.8	−	2.05359e6i	−0.0101135	−	0.440434i
$466$		0			0
$467$	7.05856e6			1.49770			0.748849	−	0.662741i	$-0.230607\pi$
$467$	7.05856e6			1.49770			0.748849	+	0.662741i	$0.230607\pi$
$468$		0			0
$469$		−	6.15452e6i		−	1.29200i
$470$		0			0
$471$	−1.28108e6			−0.266087
$472$		0			0
$473$		−	357014.i		−	0.0733722i
$474$		0			0
$475$	300776.	+	6.54578e6i	0.0611658	+	1.33115i
$476$		0			0
$477$	−7.32821e6			−1.47469
$478$		0			0
$479$	−1.47482e6			−0.293697			−0.146849	−	0.989159i	$-0.546913\pi$
$479$	−1.47482e6			−0.293697			−0.146849	+	0.989159i	$0.546913\pi$
$480$		0			0
$481$	6.80964e6			1.34203
$482$		0			0
$483$	−3.21490e6			−0.627046
$484$		0			0
$485$	5.36397e6	−	123171.i	1.03546	−	0.0237769i
$486$		0			0
$487$			9.13416e6i			1.74520i	0.488433	+	0.872602i	$0.337569\pi$
$487$			9.13416e6i			1.74520i	−0.488433	+	0.872602i	$0.662431\pi$
$488$		0			0
$489$	−1.05760e6			−0.200009
$490$		0			0
$491$			4.74188e6i			0.887660i	0.896111	+	0.443830i	$0.146381\pi$
$491$			4.74188e6i			0.887660i	−0.896111	+	0.443830i	$0.853619\pi$
$492$		0			0
$493$	−1.87997e6			−0.348365
$494$		0			0
$495$	−454083.	+	10427.0i	−0.0832957	+	0.00191269i
$496$		0			0
$497$			1.06009e7i			1.92509i
$498$		0			0
$499$		−	539009.i		−	0.0969046i	−0.998825	−	0.0484523i	$-0.984571\pi$
$499$		−	539009.i		−	0.0969046i	0.998825	−	0.0484523i	$-0.0154289\pi$
$500$		0			0
$501$			2.23243e6i			0.397359i
$502$		0			0
$503$		−	2.21657e6i		−	0.390627i	−0.980741	−	0.195313i	$-0.937428\pi$
$503$		−	2.21657e6i		−	0.390627i	0.980741	−	0.195313i	$-0.0625724\pi$
$504$		0			0
$505$	−4.68891e6	+	107670.i	−0.818171	+	0.0187874i
$506$		0			0
$507$	34484.2			0.00595800
$508$		0			0
$509$			3.35132e6i			0.573352i	0.958028	+	0.286676i	$0.0925503\pi$
$509$			3.35132e6i			0.573352i	−0.958028	+	0.286676i	$0.907450\pi$
$510$		0			0
$511$	−7.09190e6			−1.20146
$512$		0			0
$513$		−	6.53974e6i		−	1.09715i
$514$		0			0
$515$	597620.	−	13722.9i	0.0992903	−	0.00227997i
$516$		0			0
$517$	−168710.			−0.0277597
$518$		0			0
$519$	−863411.			−0.140702
$520$		0			0
$521$	5.01924e6			0.810109			0.405054	−	0.914293i	$-0.367253\pi$
$521$	5.01924e6			0.810109			0.405054	+	0.914293i	$0.367253\pi$
$522$		0			0
$523$	−3.44581e6			−0.550855			−0.275428	−	0.961322i	$-0.588819\pi$
$523$	−3.44581e6			−0.550855			−0.275428	+	0.961322i	$0.588819\pi$
$524$		0			0
$525$	−151266.	−	3.29200e6i	−0.0239521	−	0.521269i
$526$		0			0
$527$			2.09431e6i			0.328484i
$528$		0			0
$529$	−2.85758e6			−0.443975
$530$		0			0
$531$		−	7.00988e6i		−	1.07888i
$532$		0			0
$533$	6.66044e6			1.01551
$534$		0			0
$535$	−140900.	−	6.13603e6i	−0.0212826	−	0.926836i
$536$		0			0
$537$			1.29603e6i			0.193945i
$538$		0			0
$539$			202586.i			0.0300357i
$540$		0			0
$541$			3.97667e6i			0.584153i	0.956395	+	0.292076i	$0.0943461\pi$
$541$			3.97667e6i			0.584153i	−0.956395	+	0.292076i	$0.905654\pi$
$542$		0			0
$543$		−	1.07461e6i		−	0.156405i
$544$		0			0
$545$	−7.57908e6	+	174036.i	−1.09301	+	0.0250985i
$546$		0			0
$547$	−9.20664e6			−1.31563			−0.657814	−	0.753181i	$-0.728519\pi$
$547$	−9.20664e6			−1.31563			−0.657814	+	0.753181i	$0.728519\pi$
$548$		0			0
$549$			609127.i			0.0862535i
$550$		0			0
$551$	9.63498e6			1.35198
$552$		0			0
$553$		−	5.79229e6i		−	0.805448i
$554$		0			0
$555$	−103624.	−	4.51270e6i	−0.0142799	−	0.621876i
$556$		0			0
$557$	−6.00805e6			−0.820531			−0.410266	−	0.911966i	$-0.634564\pi$
$557$	−6.00805e6			−0.820531			−0.410266	+	0.911966i	$0.634564\pi$
$558$		0			0
$559$	5.09319e6			0.689383
$560$		0			0
$561$	−124631.			−0.0167194
$562$		0			0
$563$	−3.22469e6			−0.428763			−0.214381	−	0.976750i	$-0.568774\pi$
$563$	−3.22469e6			−0.428763			−0.214381	+	0.976750i	$0.568774\pi$
$564$		0			0
$565$	−1.09226e7	+	250811.i	−1.43947	+	0.0330542i
$566$		0			0
$567$		−	3.54609e6i		−	0.463225i
$568$		0			0
$569$	−3.74802e6			−0.485312			−0.242656	−	0.970112i	$-0.578019\pi$
$569$	−3.74802e6			−0.485312			−0.242656	+	0.970112i	$0.578019\pi$
$570$		0			0
$571$		−	4.37966e6i		−	0.562148i	−0.959686	−	0.281074i	$-0.909309\pi$
$571$		−	4.37966e6i		−	0.562148i	0.959686	−	0.281074i	$-0.0906906\pi$
$572$		0			0
$573$	1.53554e6			0.195377
$574$		0			0
$575$	−437293.	−	9.51681e6i	−0.0551573	−	1.20039i
$576$		0			0
$577$			1.03145e7i			1.28976i	0.764284	+	0.644880i	$0.223093\pi$
$577$			1.03145e7i			1.28976i	−0.764284	+	0.644880i	$0.776907\pi$
$578$		0			0
$579$		−	5.23732e6i		−	0.649251i
$580$		0			0
$581$			6.83173e6i			0.839635i
$582$		0			0
$583$		−	1.62414e6i		−	0.197903i
$584$		0			0
$585$	−148752.	−	6.47800e6i	−0.0179710	−	0.782620i
$586$		0			0
$587$	−5.88489e6			−0.704925			−0.352463	−	0.935826i	$-0.614656\pi$
$587$	−5.88489e6			−0.704925			−0.352463	+	0.935826i	$0.614656\pi$
$588$		0			0
$589$		−	1.07335e7i		−	1.27483i
$590$		0			0
$591$	−6.61934e6			−0.779554
$592$		0			0
$593$		−	1.07458e7i		−	1.25488i	−0.778664	−	0.627442i	$-0.784102\pi$
$593$		−	1.07458e7i		−	1.25488i	0.778664	−	0.627442i	$-0.215898\pi$
$594$		0			0
$595$	77132.8	+	3.35905e6i	0.00893196	+	0.388977i
$596$		0			0
$597$	−7.07314e6			−0.812226
$598$		0			0
$599$	−1.01249e6			−0.115299			−0.0576493	−	0.998337i	$-0.518361\pi$
$599$	−1.01249e6			−0.115299			−0.0576493	+	0.998337i	$0.518361\pi$
$600$		0			0
$601$	7.16049e6			0.808643			0.404321	−	0.914617i	$-0.367508\pi$
$601$	7.16049e6			0.808643			0.404321	+	0.914617i	$0.367508\pi$
$602$		0			0
$603$	−8.02154e6			−0.898390
$604$		0			0
$605$	204368.	+	8.90001e6i	0.0226999	+	0.988558i
$606$		0			0
$607$			1.22036e7i			1.34437i	0.740385	+	0.672183i	$0.234643\pi$
$607$			1.22036e7i			1.34437i	−0.740385	+	0.672183i	$0.765357\pi$
$608$		0			0
$609$	−4.84562e6			−0.529427
$610$		0			0
$611$		−	2.40683e6i		−	0.260821i
$612$		0			0
$613$	−1.15888e7			−1.24563			−0.622815	−	0.782369i	$-0.714011\pi$
$613$	−1.15888e7			−1.24563			−0.622815	+	0.782369i	$0.714011\pi$
$614$		0			0
$615$	−101353.	−	4.41382e6i	−0.0108056	−	0.470573i
$616$		0			0
$617$			1.47499e6i			0.155983i	0.996954	+	0.0779915i	$0.0248507\pi$
$617$			1.47499e6i			0.155983i	−0.996954	+	0.0779915i	$0.975149\pi$
$618$		0			0
$619$			6.94510e6i			0.728537i	0.931294	+	0.364269i	$0.118681\pi$
$619$			6.94510e6i			0.728537i	−0.931294	+	0.364269i	$0.881319\pi$
$620$		0			0
$621$			9.50803e6i			0.989377i
$622$		0			0
$623$			5.61430e6i			0.579530i
$624$		0			0
$625$	9.72447e6	−	895561.i	0.995786	−	0.0917055i
$626$		0			0
$627$	638744.			0.0648870
$628$		0			0
$629$			4.60218e6i			0.463807i
$630$		0			0
$631$	−3.38551e6			−0.338494			−0.169247	−	0.985574i	$-0.554133\pi$
$631$	−3.38551e6			−0.338494			−0.169247	+	0.985574i	$0.554133\pi$
$632$		0			0
$633$			6.69411e6i			0.664024i
$634$		0			0
$635$	9.57037e6	−	219761.i	0.941878	−	0.0216280i
$636$		0			0
$637$	−2.89011e6			−0.282206
$638$		0			0
$639$	1.38167e7			1.33861
$640$		0			0
$641$	−6.43478e6			−0.618570			−0.309285	−	0.950969i	$-0.600090\pi$
$641$	−6.43478e6			−0.618570			−0.309285	+	0.950969i	$0.600090\pi$
$642$		0			0
$643$	819506.			0.0781672			0.0390836	−	0.999236i	$-0.487556\pi$
$643$	819506.			0.0781672			0.0390836	+	0.999236i	$0.487556\pi$
$644$		0			0
$645$	−77504.0	−	3.37522e6i	−0.00733541	−	0.319450i
$646$		0			0
$647$			1.00673e7i			0.945482i	0.881201	+	0.472741i	$0.156735\pi$
$647$			1.00673e7i			0.945482i	−0.881201	+	0.472741i	$0.843265\pi$
$648$		0			0
$649$	1.55359e6			0.144785
$650$		0			0
$651$			5.39807e6i			0.499213i
$652$		0			0
$653$	−3.73524e6			−0.342796			−0.171398	−	0.985202i	$-0.554828\pi$
$653$	−3.73524e6			−0.342796			−0.171398	+	0.985202i	$0.554828\pi$
$654$		0			0
$655$	6.22694e6	−	142987.i	0.567115	−	0.0130225i
$656$		0			0
$657$			9.24328e6i			0.835435i
$658$		0			0
$659$		−	2.11585e6i		−	0.189789i	−0.995487	−	0.0948947i	$-0.969749\pi$
$659$		−	2.11585e6i		−	0.189789i	0.995487	−	0.0948947i	$-0.0302514\pi$
$660$		0			0
$661$		−	1.89096e7i		−	1.68337i	−0.539970	−	0.841684i	$-0.681565\pi$
$661$		−	1.89096e7i		−	1.68337i	0.539970	−	0.841684i	$-0.318435\pi$
$662$		0			0
$663$		−	1.77800e6i		−	0.157090i
$664$		0			0
$665$	−395310.	−	1.72154e7i	−0.0346644	−	1.50960i
$666$		0			0
$667$	−1.40082e7			−1.21918
$668$		0			0
$669$		−	5.37355e6i		−	0.464190i
$670$		0			0
$671$	−135000.			−0.0115752
$672$		0			0
$673$		−	2.10683e6i		−	0.179304i	−0.995973	−	0.0896522i	$-0.971424\pi$
$673$		−	2.10683e6i		−	0.179304i	0.995973	−	0.0896522i	$-0.0285755\pi$
$674$		0			0
$675$	−9.73606e6	+	447368.i	−0.822477	+	0.0377925i
$676$		0			0
$677$	5.17934e6			0.434313			0.217156	−	0.976137i	$-0.430322\pi$
$677$	5.17934e6			0.434313			0.217156	+	0.976137i	$0.430322\pi$
$678$		0			0
$679$	−1.40998e7			−1.17365
$680$		0			0
$681$	−1.05607e7			−0.872623
$682$		0			0
$683$	8.80956e6			0.722608			0.361304	−	0.932448i	$-0.382332\pi$
$683$	8.80956e6			0.722608			0.361304	+	0.932448i	$0.382332\pi$
$684$		0			0
$685$	1.57737e7	−	362206.i	1.28442	−	0.0294937i
$686$		0			0
$687$		−	5.99423e6i		−	0.484553i
$688$		0			0
$689$	2.31701e7			1.85943
$690$		0			0
$691$			5.78539e6i			0.460933i	0.973080	+	0.230466i	$0.0740252\pi$
$691$			5.78539e6i			0.460933i	−0.973080	+	0.230466i	$0.925975\pi$
$692$		0			0
$693$	1.19361e6			0.0944122
$694$		0			0
$695$	−2.46239e6	+	56543.0i	−0.193372	+	0.00444034i
$696$		0			0
$697$			4.50135e6i			0.350963i
$698$		0			0
$699$			2.04384e6i			0.158217i
$700$		0			0
$701$		−	2.01716e7i		−	1.55041i	−0.631711	−	0.775204i	$-0.717647\pi$
$701$		−	2.01716e7i		−	1.55041i	0.631711	−	0.775204i	$-0.282353\pi$
$702$		0			0
$703$		−	2.35865e7i		−	1.80001i
$704$		0			0
$705$	−1.59499e6	+	36625.2i	−0.120861	+	0.00277528i
$706$		0			0
$707$	1.23253e7			0.927362
$708$		0			0
$709$			1.71661e7i			1.28250i	0.767334	+	0.641248i	$0.221583\pi$
$709$			1.71661e7i			1.28250i	−0.767334	+	0.641248i	$0.778417\pi$
$710$		0			0
$711$	−7.54943e6			−0.560067
$712$		0			0
$713$			1.56052e7i			1.14960i
$714$		0			0
$715$	1.43571e6	−	32967.7i	0.105027	−	0.00241170i
$716$		0			0
$717$	5.74733e6			0.417511
$718$		0			0
$719$	−1.37927e7			−0.995009			−0.497505	−	0.867461i	$-0.665750\pi$
$719$	−1.37927e7			−0.995009			−0.497505	+	0.867461i	$0.665750\pi$
$720$		0			0
$721$	−1.57091e6			−0.112541
$722$		0			0
$723$	1.08105e7			0.769132
$724$		0			0
$725$	−659105.	−	1.43441e7i	−0.0465704	−	1.01351i
$726$		0			0
$727$			9.10175e6i			0.638688i	0.947639	+	0.319344i	$0.103463\pi$
$727$			9.10175e6i			0.638688i	−0.947639	+	0.319344i	$0.896537\pi$
$728$		0			0
$729$	818554.			0.0570465
$730$		0			0
$731$			3.44215e6i			0.238252i
$732$		0			0
$733$	1.79637e7			1.23491			0.617456	−	0.786605i	$-0.288163\pi$
$733$	1.79637e7			1.23491			0.617456	+	0.786605i	$0.288163\pi$
$734$		0			0
$735$	43979.3	+	1.91525e6i	0.00300283	+	0.130770i
$736$		0			0
$737$		−	1.77780e6i		−	0.120563i
$738$		0			0
$739$			1.38839e7i			0.935192i	0.883942	+	0.467596i	$0.154880\pi$
$739$			1.38839e7i			0.935192i	−0.883942	+	0.467596i	$0.845120\pi$
$740$		0			0
$741$			9.11238e6i			0.609658i
$742$		0			0
$743$		−	2.37521e6i		−	0.157845i	−0.996881	−	0.0789225i	$-0.974852\pi$
$743$		−	2.37521e6i		−	0.157845i	0.996881	−	0.0789225i	$-0.0251480\pi$
$744$		0			0
$745$	1.40585e7	−	322820.i	0.927999	−	0.0213093i
$746$		0			0
$747$	8.90419e6			0.583839
$748$		0			0
$749$			1.61292e7i			1.05053i
$750$		0			0
$751$	1.99260e7			1.28920			0.644600	−	0.764520i	$-0.277024\pi$
$751$	1.99260e7			1.28920			0.644600	+	0.764520i	$0.277024\pi$
$752$		0			0
$753$		−	7.91465e6i		−	0.508679i
$754$		0			0
$755$	−122929.	−	5.35341e6i	−0.00784848	−	0.341793i
$756$		0			0
$757$	1.30771e7			0.829413			0.414707	−	0.909955i	$-0.363884\pi$
$757$	1.30771e7			0.829413			0.414707	+	0.909955i	$0.363884\pi$
$758$		0			0
$759$	−928660.			−0.0585130
$760$		0			0
$761$	−1.75251e7			−1.09698			−0.548492	−	0.836156i	$-0.684798\pi$
$761$	−1.75251e7			−1.09698			−0.548492	+	0.836156i	$0.684798\pi$
$762$		0			0
$763$	1.99224e7			1.23888
$764$		0			0
$765$	4.37805e6	−	100532.i	0.270475	−	0.00621082i
$766$		0			0
$767$			2.21637e7i			1.36036i
$768$		0			0
$769$	−3.13820e7			−1.91366			−0.956830	−	0.290647i	$-0.906129\pi$
$769$	−3.13820e7			−1.91366			−0.956830	+	0.290647i	$0.906129\pi$
$770$		0			0
$771$			1.01244e7i			0.613384i
$772$		0			0
$773$	−103265.			−0.00621590			−0.00310795	−	0.999995i	$-0.500989\pi$
$773$	−103265.			−0.00621590			−0.00310795	+	0.999995i	$0.500989\pi$
$774$		0			0
$775$	−1.59795e7	+	734250.i	−0.955671	+	0.0439126i
$776$		0			0
$777$			1.18621e7i			0.704870i
$778$		0			0
$779$		−	2.30697e7i		−	1.36207i
$780$		0			0
$781$			3.06218e6i			0.179640i
$782$		0			0
$783$			1.43309e7i			0.835350i
$784$		0			0
$785$	229022.	+	9.97368e6i	0.0132649	+	0.577672i
$786$		0			0
$787$	5.01823e6			0.288811			0.144405	−	0.989519i	$-0.453873\pi$
$787$	5.01823e6			0.288811			0.144405	+	0.989519i	$0.453873\pi$
$788$		0			0
$789$		−	1.21229e7i		−	0.693288i
$790$		0			0
$791$	2.87111e7			1.63158
$792$		0			0
$793$		−	1.92592e6i		−	0.108757i
$794$		0			0
$795$	−352584.	−	1.53547e7i	−0.0197854	−	0.861634i
$796$		0			0
$797$	−9.61578e6			−0.536215			−0.268107	−	0.963389i	$-0.586398\pi$
$797$	−9.61578e6			−0.536215			−0.268107	+	0.963389i	$0.586398\pi$
$798$		0			0
$799$	1.62662e6			0.0901403
$800$		0			0
$801$	7.31745e6			0.402975
$802$		0			0
$803$	−2.04857e6			−0.112115
$804$		0			0
$805$	574736.	+	2.50291e7i	0.0312592	+	1.36131i
$806$		0			0
$807$			2.19552e6i			0.118673i
$808$		0			0
$809$	2.06325e7			1.10836			0.554181	−	0.832396i	$-0.313032\pi$
$809$	2.06325e7			1.10836			0.554181	+	0.832396i	$0.313032\pi$
$810$		0			0
$811$		−	3.18795e7i		−	1.70200i	−0.525164	−	0.851001i	$-0.675996\pi$
$811$		−	3.18795e7i		−	1.70200i	0.525164	−	0.851001i	$-0.324004\pi$
$812$		0			0
$813$	−8.99645e6			−0.477359
$814$		0			0
$815$	189070.	+	8.23381e6i	0.00997078	+	0.434217i
$816$		0			0
$817$		−	1.76412e7i		−	0.924642i
$818$		0			0
$819$			1.70281e7i			0.887067i
$820$		0			0
$821$		−	7.88522e6i		−	0.408278i	−0.978942	−	0.204139i	$-0.934561\pi$
$821$		−	7.88522e6i		−	0.408278i	0.978942	−	0.204139i	$-0.0654394\pi$
$822$		0			0
$823$		−	51764.9i		−	0.00266401i	−0.999999	−	0.00133200i	$-0.999576\pi$
$823$		−	51764.9i		−	0.00266401i	0.999999	−	0.00133200i	$-0.000423990\pi$
$824$		0			0
$825$	−43694.9	−	950932.i	−0.00223509	−	0.0486423i
$826$		0			0
$827$	5.24793e6			0.266824			0.133412	−	0.991061i	$-0.457407\pi$
$827$	5.24793e6			0.266824			0.133412	+	0.991061i	$0.457407\pi$
$828$		0			0
$829$			6.76109e6i			0.341689i	0.985298	+	0.170844i	$0.0546495\pi$
$829$			6.76109e6i			0.341689i	−0.985298	+	0.170844i	$0.945350\pi$
$830$		0			0
$831$	3.47004e6			0.174314
$832$		0			0
$833$		−	1.95323e6i		−	0.0975308i
$834$		0			0
$835$	1.73802e7	−	399097.i	0.862660	−	0.0198090i
$836$		0			0
$837$	1.59647e7			0.787677
$838$		0			0
$839$	5.32681e6			0.261254			0.130627	−	0.991432i	$-0.458301\pi$
$839$	5.32681e6			0.261254			0.130627	+	0.991432i	$0.458301\pi$
$840$		0			0
$841$	−602476.			−0.0293731
$842$		0			0
$843$	−636887.			−0.0308669
$844$		0			0
$845$	−6164.83	−	268472.i	−0.000297016	−	0.0129347i
$846$		0			0
$847$		−	2.33946e7i		−	1.12049i
$848$		0			0
$849$	1.26660e7			0.603076
$850$		0			0
$851$			3.42920e7i			1.62319i
$852$		0			0
$853$	8.20548e6			0.386128			0.193064	−	0.981186i	$-0.438157\pi$
$853$	8.20548e6			0.386128			0.193064	+	0.981186i	$0.438157\pi$
$854$		0			0
$855$	−2.24378e7	+	515231.i	−1.04970	+	0.0241039i
$856$		0			0
$857$		−	5.45868e6i		−	0.253884i	−0.991910	−	0.126942i	$-0.959484\pi$
$857$		−	5.45868e6i		−	0.253884i	0.991910	−	0.126942i	$-0.0405162\pi$
$858$		0			0
$859$		−	3.18230e7i		−	1.47149i	−0.677257	−	0.735747i	$-0.736831\pi$
$859$		−	3.18230e7i		−	1.47149i	0.677257	−	0.735747i	$-0.263169\pi$
$860$		0			0
$861$			1.16022e7i			0.533375i
$862$		0			0
$863$		−	9.28914e6i		−	0.424569i	−0.977208	−	0.212285i	$-0.931910\pi$
$863$		−	9.28914e6i		−	0.424569i	0.977208	−	0.212285i	$-0.0680904\pi$
$864$		0			0
$865$	154354.	+	6.72196e6i	0.00701420	+	0.305461i
$866$		0			0
$867$	−8.99077e6			−0.406208
$868$		0			0
$869$		−	1.67317e6i		−	0.0751606i
$870$		0			0
$871$	2.53623e7			1.13277
$872$		0			0
$873$			1.83771e7i			0.816094i
$874$		0			0
$875$	−2.56024e7	+	1.76618e6i	−1.13047	+	0.0779856i
$876$		0			0
$877$	−1.60806e7			−0.705997			−0.352999	−	0.935624i	$-0.614838\pi$
$877$	−1.60806e7			−0.705997			−0.352999	+	0.935624i	$0.614838\pi$
$878$		0			0
$879$	−65503.2			−0.00285950
$880$		0			0
$881$	−1.74471e7			−0.757327			−0.378663	−	0.925535i	$-0.623616\pi$
$881$	−1.74471e7			−0.757327			−0.378663	+	0.925535i	$0.623616\pi$
$882$		0			0
$883$	−2.18862e7			−0.944645			−0.472323	−	0.881426i	$-0.656584\pi$
$883$	−2.18862e7			−0.944645			−0.472323	+	0.881426i	$0.656584\pi$
$884$		0			0
$885$	1.46877e7	−	337268.i	0.630370	−	0.0144750i
$886$		0			0
$887$		−	4.00556e7i		−	1.70944i	−0.519088	−	0.854721i	$-0.673728\pi$
$887$		−	4.00556e7i		−	1.70944i	0.519088	−	0.854721i	$-0.326272\pi$
$888$		0			0
$889$	−2.51567e7			−1.06758
$890$		0			0
$891$		−	1.02433e6i		−	0.0432260i
$892$		0			0
$893$	−8.33653e6			−0.349830
$894$		0			0
$895$	1.00900e7	−	231694.i	0.421051	−	0.00966846i
$896$		0			0
$897$		−	1.32484e7i		−	0.549770i
$898$		0			0
$899$			2.35208e7i			0.970628i
$900$		0			0
$901$			1.56592e7i			0.642623i
$902$		0			0
$903$			8.87211e6i			0.362083i
$904$		0			0
$905$	−8.36622e6	+	192111.i	−0.339553	+	0.00779704i
$906$		0			0
$907$	1.61797e7			0.653060			0.326530	−	0.945187i	$-0.394121\pi$
$907$	1.61797e7			0.653060			0.326530	+	0.945187i	$0.394121\pi$
$908$		0			0
$909$		−	1.60643e7i		−	0.644840i
$910$		0			0
$911$	−3.81429e7			−1.52271			−0.761356	−	0.648334i	$-0.775466\pi$
$911$	−3.81429e7			−1.52271			−0.761356	+	0.648334i	$0.775466\pi$
$912$		0			0
$913$			1.97342e6i			0.0783508i
$914$		0			0
$915$	−1.27629e6	+	29307.1i	−0.0503962	+	0.00115723i
$916$		0			0
$917$	−1.63682e7			−0.642801
$918$		0			0
$919$	5.75041e6			0.224600			0.112300	−	0.993674i	$-0.464178\pi$
$919$	5.75041e6			0.224600			0.112300	+	0.993674i	$0.464178\pi$
$920$		0			0
$921$	1.89665e7			0.736780
$922$		0			0
$923$	−4.36854e7			−1.68784
$924$		0			0
$925$	−3.51144e7	+	1.61349e6i	−1.34937	+	0.0620030i
$926$		0			0
$927$			2.04745e6i			0.0782555i
$928$		0			0
$929$	1.58033e7			0.600771			0.300386	−	0.953818i	$-0.402885\pi$
$929$	1.58033e7			0.600771			0.300386	+	0.953818i	$0.402885\pi$
$930$		0			0
$931$			1.00104e7i			0.378512i
$932$		0			0
$933$	−1.00680e7			−0.378651
$934$		0			0
$935$	22280.7	+	970300.i	0.000833488	+	0.0362975i
$936$		0			0
$937$		−	1.61564e7i		−	0.601167i	−0.953756	−	0.300583i	$-0.902819\pi$
$937$		−	1.61564e7i		−	0.601167i	0.953756	−	0.300583i	$-0.0971814\pi$
$938$		0			0
$939$			9.75602e6i			0.361084i
$940$		0			0
$941$		−	4.39614e7i		−	1.61844i	−0.587504	−	0.809221i	$-0.699889\pi$
$941$		−	4.39614e7i		−	1.61844i	0.587504	−	0.809221i	$-0.300111\pi$
$942$		0			0
$943$			3.35407e7i			1.22827i
$944$		0			0
$945$	2.56058e7	−	587976.i	0.932735	−	0.0214181i
$946$		0			0
$947$	−4.52665e7			−1.64022			−0.820111	−	0.572205i	$-0.806088\pi$
$947$	−4.52665e7			−1.64022			−0.820111	+	0.572205i	$0.806088\pi$
$948$		0			0
$949$		−	2.92252e7i		−	1.05340i
$950$		0			0
$951$	2.15601e6			0.0773035
$952$		0			0
$953$		−	3.21216e6i		−	0.114569i	−0.998358	−	0.0572843i	$-0.981756\pi$
$953$		−	3.21216e6i		−	0.114569i	0.998358	−	0.0572843i	$-0.0182441\pi$
$954$		0			0
$955$	−274512.	−	1.19547e7i	−0.00973986	−	0.424161i
$956$		0			0
$957$	−1.39971e6			−0.0494036
$958$		0			0
$959$	−4.14628e7			−1.45583
$960$		0			0
$961$	−2.42676e6			−0.0847654
$962$		0			0
$963$	2.10221e7			0.730485
$964$		0			0
$965$	−4.07744e7	+	936288.i	−1.40951	+	0.0323662i
$966$		0			0
$967$		−	4.06241e6i		−	0.139707i	−0.997557	−	0.0698533i	$-0.977747\pi$
$967$		−	4.06241e6i		−	0.139707i	0.997557	−	0.0698533i	$-0.0222531\pi$
$968$		0			0
$969$	−6.15845e6			−0.210699
$970$		0			0
$971$		−	2.41409e7i		−	0.821685i	−0.911706	−	0.410842i	$-0.865235\pi$
$971$		−	2.41409e7i		−	0.821685i	0.911706	−	0.410842i	$-0.134765\pi$
$972$		0			0
$973$	6.47264e6			0.219179
$974$		0			0
$975$	1.35661e7	−	623355.i	0.457028	−	0.0210002i
$976$		0			0
$977$		−	1.48416e7i		−	0.497443i	−0.968575	−	0.248722i	$-0.919990\pi$
$977$		−	1.48416e7i		−	0.497443i	0.968575	−	0.248722i	$-0.0800104\pi$
$978$		0			0
$979$			1.62175e6i			0.0540790i
$980$		0			0
$981$		−	2.59661e7i		−	0.861457i
$982$		0			0
$983$			3.34107e7i			1.10281i	0.834237	+	0.551406i	$0.185908\pi$
$983$			3.34107e7i			1.10281i	−0.834237	+	0.551406i	$0.814092\pi$
$984$		0			0
$985$	1.18336e6	+	5.15339e7i	0.0388620	+	1.69240i
$986$		0			0
$987$	4.19260e6			0.136991
$988$		0			0
$989$			2.56483e7i			0.833812i
$990$		0			0
$991$	−5.15057e7			−1.66599			−0.832993	−	0.553284i	$-0.813374\pi$
$991$	−5.15057e7			−1.66599			−0.832993	+	0.553284i	$0.813374\pi$
$992$		0			0
$993$		−	1.70520e7i		−	0.548784i
$994$		0			0
$995$	1.26448e6	+	5.50670e7i	0.0404907	+	1.76333i
$996$		0			0
$997$	1.18985e7			0.379102			0.189551	−	0.981871i	$-0.439297\pi$
$997$	1.18985e7			0.379102			0.189551	+	0.981871i	$0.439297\pi$
$998$		0			0
$999$	3.50821e7			1.11217

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

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.f.a.49.11		28
4.3	odd	2		40.6.f.a.29.11	✓	28
5.2	odd	4		800.6.d.e.401.22		28
5.3	odd	4		800.6.d.e.401.7		28
5.4	even	2	inner	160.6.f.a.49.18		28
8.3	odd	2		40.6.f.a.29.17	yes	28
8.5	even	2	inner	160.6.f.a.49.17		28
20.3	even	4		200.6.d.e.101.4		28
20.7	even	4		200.6.d.e.101.25		28
20.19	odd	2		40.6.f.a.29.18	yes	28
40.3	even	4		200.6.d.e.101.3		28
40.13	odd	4		800.6.d.e.401.8		28
40.19	odd	2		40.6.f.a.29.12	yes	28
40.27	even	4		200.6.d.e.101.26		28
40.29	even	2	inner	160.6.f.a.49.12		28
40.37	odd	4		800.6.d.e.401.21		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.f.a.29.11	✓	28	4.3	odd	2
40.6.f.a.29.12	yes	28	40.19	odd	2
40.6.f.a.29.17	yes	28	8.3	odd	2
40.6.f.a.29.18	yes	28	20.19	odd	2
160.6.f.a.49.11		28	1.1	even	1	trivial
160.6.f.a.49.12		28	40.29	even	2	inner
160.6.f.a.49.17		28	8.5	even	2	inner
160.6.f.a.49.18		28	5.4	even	2	inner
200.6.d.e.101.3		28	40.3	even	4
200.6.d.e.101.4		28	20.3	even	4
200.6.d.e.101.25		28	20.7	even	4
200.6.d.e.101.26		28	40.27	even	4
800.6.d.e.401.7		28	5.3	odd	4
800.6.d.e.401.8		28	40.13	odd	4
800.6.d.e.401.21		28	40.37	odd	4
800.6.d.e.401.22		28	5.2	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.49.11

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