LMFDB - Embedded newform 810.4.c.b.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(649,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("810.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$810 = 2 \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	810.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:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - 4 x^{9} + 8 x^{8} + 326 x^{7} + 17389 x^{6} - 26726 x^{5} + 20930 x^{4} + 60664 x^{3} + \cdots + 5604552$ x^10 - 4x^9 + 8x^8 + 326x^7 + 17389x^6 - 26726x^5 + 20930x^4 + 60664x^3 + 4452100x^2 - 7064280*x + 5604552
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{8}\cdot 3^{2}\cdot 5$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$2.81091 - 2.81091i$ of defining polynomial
Character	$\chi$	$=$	810.649
Dual form			810.4.c.b.649.6

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +(-9.35710 - 6.11921i) q^{5} +15.3362i q^{7} +8.00000i q^{8} +(-12.2384 + 18.7142i) q^{10} +22.0754 q^{11} +7.07434i q^{13} +30.6725 q^{14} +16.0000 q^{16} -43.9694i q^{17} -107.012 q^{19} +(37.4284 + 24.4768i) q^{20} -44.1508i q^{22} +117.609i q^{23} +(50.1106 + 114.516i) q^{25} +14.1487 q^{26} -61.3449i q^{28} +168.170 q^{29} +96.5359 q^{31} -32.0000i q^{32} -87.9388 q^{34} +(93.8456 - 143.503i) q^{35} -48.5614i q^{37} +214.024i q^{38} +(48.9537 - 74.8568i) q^{40} +251.025 q^{41} -159.323i q^{43} -88.3016 q^{44} +235.219 q^{46} -202.889i q^{47} +107.800 q^{49} +(229.032 - 100.221i) q^{50} -28.2974i q^{52} -594.696i q^{53} +(-206.562 - 135.084i) q^{55} -122.690 q^{56} -336.340i q^{58} +184.900 q^{59} -743.245 q^{61} -193.072i q^{62} -64.0000 q^{64} +(43.2894 - 66.1953i) q^{65} -753.434i q^{67} +175.878i q^{68} +(-287.005 - 187.691i) q^{70} -930.606 q^{71} +228.974i q^{73} -97.1227 q^{74} +428.049 q^{76} +338.553i q^{77} -466.376 q^{79} +(-149.714 - 97.9073i) q^{80} -502.050i q^{82} -208.466i q^{83} +(-269.058 + 411.426i) q^{85} -318.647 q^{86} +176.603i q^{88} +484.281 q^{89} -108.494 q^{91} -470.437i q^{92} -405.778 q^{94} +(1001.32 + 654.830i) q^{95} -878.085i q^{97} -215.600i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 40 q^{4} + 22 q^{5} - 4 q^{10} - 28 q^{11} - 4 q^{14} + 160 q^{16} - 42 q^{19} - 88 q^{20} + 96 q^{25} - 108 q^{26} + 216 q^{29} + 88 q^{31} - 64 q^{34} - 22 q^{35} + 16 q^{40} + 38 q^{41} + 112 q^{44}+ \cdots + 2472 q^{95}+O(q^{100})$ 10 * q - 40 * q^4 + 22 * q^5 - 4 * q^10 - 28 * q^11 - 4 * q^14 + 160 * q^16 - 42 * q^19 - 88 * q^20 + 96 * q^25 - 108 * q^26 + 216 * q^29 + 88 * q^31 - 64 * q^34 - 22 * q^35 + 16 * q^40 + 38 * q^41 + 112 * q^44 + 172 * q^46 - 1452 * q^49 + 256 * q^50 + 20 * q^55 + 16 * q^56 - 2050 * q^59 + 1064 * q^61 - 640 * q^64 - 2268 * q^65 + 464 * q^70 - 2844 * q^71 + 584 * q^74 + 168 * q^76 - 1608 * q^79 + 352 * q^80 - 526 * q^85 + 184 * q^86 + 1132 * q^89 + 3402 * q^91 - 1268 * q^94 + 2472 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$ .

$n$	$487$	$731$
$\chi(n)$	$-1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		−	2.00000i		−	0.707107i
$2$		−	2.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−4.00000			−0.500000
$5$	−9.35710	−	6.11921i	−0.836924	−	0.547319i
$5$	−9.35710	−	6.11921i	−0.836924	−	0.547319i
$6$		0			0
$7$			15.3362i			0.828079i	0.910259	+	0.414039i	$0.135882\pi$
$7$			15.3362i			0.828079i	−0.910259	+	0.414039i	$0.864118\pi$
$8$			8.00000i			0.353553i
$9$		0			0
$10$	−12.2384	+	18.7142i	−0.387013	+	0.591795i
$11$	22.0754			0.605089			0.302545	−	0.953135i	$-0.402164\pi$
$11$	22.0754			0.605089			0.302545	+	0.953135i	$0.402164\pi$
$12$		0			0
$13$			7.07434i			0.150928i	0.997149	+	0.0754642i	$0.0240439\pi$
$13$			7.07434i			0.150928i	−0.997149	+	0.0754642i	$0.975956\pi$
$14$	30.6725			0.585540
$15$		0			0
$16$	16.0000			0.250000
$17$		−	43.9694i		−	0.627303i	−0.949538	−	0.313651i	$-0.898448\pi$
$17$		−	43.9694i		−	0.627303i	0.949538	−	0.313651i	$-0.101552\pi$
$18$		0			0
$19$	−107.012			−1.29212			−0.646060	−	0.763287i	$-0.723584\pi$
$19$	−107.012			−1.29212			−0.646060	+	0.763287i	$0.723584\pi$
$20$	37.4284	+	24.4768i	0.418462	+	0.273659i
$21$		0			0
$22$		−	44.1508i		−	0.427863i
$23$			117.609i			1.06623i	0.846044	+	0.533114i	$0.178978\pi$
$23$			117.609i			1.06623i	−0.846044	+	0.533114i	$0.821022\pi$
$24$		0			0
$25$	50.1106	+	114.516i	0.400885	+	0.916128i
$26$	14.1487			0.106723
$27$		0			0
$28$		−	61.3449i		−	0.414039i
$29$	168.170			1.07684			0.538421	−	0.842676i	$-0.319021\pi$
$29$	168.170			1.07684			0.538421	+	0.842676i	$0.319021\pi$
$30$		0			0
$31$	96.5359			0.559302			0.279651	−	0.960102i	$-0.409781\pi$
$31$	96.5359			0.559302			0.279651	+	0.960102i	$0.409781\pi$
$32$		−	32.0000i		−	0.176777i
$33$		0			0
$34$	−87.9388			−0.443570
$35$	93.8456	−	143.503i	0.453223	−	0.693039i
$36$		0			0
$37$		−	48.5614i		−	0.215769i	−0.994163	−	0.107884i	$-0.965592\pi$
$37$		−	48.5614i		−	0.215769i	0.994163	−	0.107884i	$-0.0344076\pi$
$38$			214.024i			0.913667i
$39$		0			0
$40$	48.9537	−	74.8568i	0.193506	−	0.295897i
$41$	251.025			0.956184			0.478092	−	0.878310i	$-0.341329\pi$
$41$	251.025			0.956184			0.478092	+	0.878310i	$0.341329\pi$
$42$		0			0
$43$		−	159.323i		−	0.565037i	−0.959262	−	0.282518i	$-0.908830\pi$
$43$		−	159.323i		−	0.565037i	0.959262	−	0.282518i	$-0.0911698\pi$
$44$	−88.3016			−0.302545
$45$		0			0
$46$	235.219			0.753936
$47$		−	202.889i		−	0.629668i	−0.949147	−	0.314834i	$-0.898051\pi$
$47$		−	202.889i		−	0.629668i	0.949147	−	0.314834i	$-0.101949\pi$
$48$		0			0
$49$	107.800			0.314286
$50$	229.032	−	100.221i	0.647801	−	0.283468i
$51$		0			0
$52$		−	28.2974i		−	0.0754642i
$53$		−	594.696i		−	1.54128i	−0.637271	−	0.770640i	$-0.719937\pi$
$53$		−	594.696i		−	1.54128i	0.637271	−	0.770640i	$-0.280063\pi$
$54$		0			0
$55$	−206.562	−	135.084i	−0.506414	−	0.331176i
$56$	−122.690			−0.292770
$57$		0			0
$58$		−	336.340i		−	0.761442i
$59$	184.900			0.407999			0.204000	−	0.978971i	$-0.434606\pi$
$59$	184.900			0.407999			0.204000	+	0.978971i	$0.434606\pi$
$60$		0			0
$61$	−743.245			−1.56005			−0.780023	−	0.625751i	$-0.784793\pi$
$61$	−743.245			−1.56005			−0.780023	+	0.625751i	$0.784793\pi$
$62$		−	193.072i		−	0.395486i
$63$		0			0
$64$	−64.0000			−0.125000
$65$	43.2894	−	66.1953i	0.0826060	−	0.126316i
$66$		0			0
$67$		−	753.434i		−	1.37383i	−0.726737	−	0.686915i	$-0.758964\pi$
$67$		−	753.434i		−	1.37383i	0.726737	−	0.686915i	$-0.241036\pi$
$68$			175.878i			0.313651i
$69$		0			0
$70$	−287.005	−	187.691i	−0.490053	−	0.320477i
$71$	−930.606			−1.55553			−0.777765	−	0.628556i	$-0.783646\pi$
$71$	−930.606			−1.55553			−0.777765	+	0.628556i	$0.783646\pi$
$72$		0			0
$73$			228.974i			0.367114i	0.983009	+	0.183557i	$0.0587612\pi$
$73$			228.974i			0.367114i	−0.983009	+	0.183557i	$0.941239\pi$
$74$	−97.1227			−0.152571
$75$		0			0
$76$	428.049			0.646060
$77$			338.553i			0.501061i
$78$		0			0
$79$	−466.376			−0.664195			−0.332097	−	0.943245i	$-0.607756\pi$
$79$	−466.376			−0.664195			−0.332097	+	0.943245i	$0.607756\pi$
$80$	−149.714	−	97.9073i	−0.209231	−	0.136830i
$81$		0			0
$82$		−	502.050i		−	0.676124i
$83$		−	208.466i		−	0.275688i	−0.990454	−	0.137844i	$-0.955983\pi$
$83$		−	208.466i		−	0.275688i	0.990454	−	0.137844i	$-0.0440172\pi$
$84$		0			0
$85$	−269.058	+	411.426i	−0.343334	+	0.525005i
$86$	−318.647			−0.399541
$87$		0			0
$88$			176.603i			0.213931i
$89$	484.281			0.576783			0.288391	−	0.957513i	$-0.406880\pi$
$89$	484.281			0.576783			0.288391	+	0.957513i	$0.406880\pi$
$90$		0			0
$91$	−108.494			−0.124981
$92$		−	470.437i		−	0.533114i
$93$		0			0
$94$	−405.778			−0.445243
$95$	1001.32	+	654.830i	1.08141	+	0.707201i
$96$		0			0
$97$		−	878.085i		−	0.919135i	−0.888143	−	0.459567i	$-0.848005\pi$
$97$		−	878.085i		−	0.919135i	0.888143	−	0.459567i	$-0.151995\pi$
$98$		−	215.600i		−	0.222233i
$99$		0			0
$100$	−200.442	−	458.064i	−0.200442	−	0.458064i
$101$	616.460			0.607327			0.303663	−	0.952779i	$-0.401790\pi$
$101$	616.460			0.607327			0.303663	+	0.952779i	$0.401790\pi$
$102$		0			0
$103$		−	1656.06i		−	1.58424i	−0.610368	−	0.792118i	$-0.708978\pi$
$103$		−	1656.06i		−	1.58424i	0.610368	−	0.792118i	$-0.291022\pi$
$104$	−56.5948			−0.0533613
$105$		0			0
$106$	−1189.39			−1.08985
$107$		−	1738.82i		−	1.57101i	−0.618855	−	0.785506i	$-0.712403\pi$
$107$		−	1738.82i		−	1.57101i	0.618855	−	0.785506i	$-0.287597\pi$
$108$		0			0
$109$	−1349.20			−1.18560			−0.592799	−	0.805350i	$-0.701977\pi$
$109$	−1349.20			−1.18560			−0.592799	+	0.805350i	$0.701977\pi$
$110$	−270.168	+	413.123i	−0.234177	+	0.358089i
$111$		0			0
$112$			245.380i			0.207020i
$113$			1321.97i			1.10054i	0.834987	+	0.550269i	$0.185475\pi$
$113$			1321.97i			1.10054i	−0.834987	+	0.550269i	$0.814525\pi$
$114$		0			0
$115$	719.675	−	1100.48i	0.583566	−	0.892352i
$116$	−672.681			−0.538421
$117$		0			0
$118$		−	369.800i		−	0.288499i
$119$	674.325			0.519456
$120$		0			0
$121$	−843.677			−0.633867
$122$			1486.49i			1.10312i
$123$		0			0
$124$	−386.143			−0.279651
$125$	231.857	−	1378.18i	0.165904	−	0.986142i
$126$		0			0
$127$		−	1311.76i		−	0.916537i	−0.888814	−	0.458269i	$-0.848470\pi$
$127$		−	1311.76i		−	0.916537i	0.888814	−	0.458269i	$-0.151530\pi$
$128$			128.000i			0.0883883i
$129$		0			0
$130$	−132.391	−	86.5788i	−0.0893187	−	0.0584112i
$131$	385.182			0.256897			0.128448	−	0.991716i	$-0.459000\pi$
$131$	385.182			0.256897			0.128448	+	0.991716i	$0.459000\pi$
$132$		0			0
$133$		−	1641.16i		−	1.06998i
$134$	−1506.87			−0.971445
$135$		0			0
$136$	351.755			0.221785
$137$			1476.86i			0.920998i	0.887660	+	0.460499i	$0.152330\pi$
$137$			1476.86i			0.920998i	−0.887660	+	0.460499i	$0.847670\pi$
$138$		0			0
$139$	−1246.21			−0.760448			−0.380224	−	0.924894i	$-0.624153\pi$
$139$	−1246.21			−0.760448			−0.380224	+	0.924894i	$0.624153\pi$
$140$	−375.382	+	574.011i	−0.226611	+	0.346520i
$141$		0			0
$142$			1861.21i			1.09993i
$143$			156.169i			0.0913252i
$144$		0			0
$145$	−1573.58	−	1029.07i	−0.901235	−	0.589375i
$146$	457.947			0.259589
$147$		0			0
$148$			194.245i			0.107884i
$149$	3068.44			1.68709			0.843545	−	0.537059i	$-0.180465\pi$
$149$	3068.44			1.68709			0.843545	+	0.537059i	$0.180465\pi$
$150$		0			0
$151$	3333.12			1.79633			0.898164	−	0.439661i	$-0.144901\pi$
$151$	3333.12			1.79633			0.898164	+	0.439661i	$0.144901\pi$
$152$		−	856.097i		−	0.456833i
$153$		0			0
$154$	677.107			0.354304
$155$	−903.296	−	590.723i	−0.468093	−	0.306116i
$156$		0			0
$157$		−	1378.63i		−	0.700805i	−0.936599	−	0.350403i	$-0.886045\pi$
$157$		−	1378.63i		−	0.700805i	0.936599	−	0.350403i	$-0.113955\pi$
$158$			932.752i			0.469657i
$159$		0			0
$160$	−195.815	+	299.427i	−0.0967532	+	0.147949i
$161$	−1803.68			−0.882920
$162$		0			0
$163$			257.789i			0.123875i	0.998080	+	0.0619374i	$0.0197279\pi$
$163$			257.789i			0.123875i	−0.998080	+	0.0619374i	$0.980272\pi$
$164$	−1004.10			−0.478092
$165$		0			0
$166$	−416.931			−0.194941
$167$		−	2279.52i		−	1.05626i	−0.849165	−	0.528128i	$-0.822894\pi$
$167$		−	2279.52i		−	1.05626i	0.849165	−	0.528128i	$-0.177106\pi$
$168$		0			0
$169$	2146.95			0.977221
$170$	822.852	+	538.116i	0.371234	+	0.242774i
$171$		0			0
$172$			637.293i			0.282518i
$173$			2692.95i			1.18347i	0.806131	+	0.591737i	$0.201558\pi$
$173$			2692.95i			1.18347i	−0.806131	+	0.591737i	$0.798442\pi$
$174$		0			0
$175$	−1756.24	+	768.508i	−0.758627	+	0.331964i
$176$	353.206			0.151272
$177$		0			0
$178$		−	968.562i		−	0.407847i
$179$	−1259.53			−0.525929			−0.262965	−	0.964805i	$-0.584700\pi$
$179$	−1259.53			−0.525929			−0.262965	+	0.964805i	$0.584700\pi$
$180$		0			0
$181$	337.957			0.138785			0.0693926	−	0.997589i	$-0.477894\pi$
$181$	337.957			0.138785			0.0693926	+	0.997589i	$0.477894\pi$
$182$			216.988i			0.0883747i
$183$		0			0
$184$	−940.874			−0.376968
$185$	−297.157	+	454.393i	−0.118094	+	0.180582i
$186$		0			0
$187$		−	970.641i		−	0.379574i
$188$			811.556i			0.314834i
$189$		0			0
$190$	1309.66	−	2002.65i	0.500067	−	0.764670i
$191$	2809.28			1.06425			0.532127	−	0.846665i	$-0.321393\pi$
$191$	2809.28			1.06425			0.532127	+	0.846665i	$0.321393\pi$
$192$		0			0
$193$			3497.97i			1.30461i	0.757958	+	0.652303i	$0.226197\pi$
$193$			3497.97i			1.30461i	−0.757958	+	0.652303i	$0.773803\pi$
$194$	−1756.17			−0.649926
$195$		0			0
$196$	−431.200			−0.157143
$197$		−	3934.58i		−	1.42298i	−0.702696	−	0.711490i	$-0.748020\pi$
$197$		−	3934.58i		−	1.42298i	0.702696	−	0.711490i	$-0.251980\pi$
$198$		0			0
$199$	−1420.01			−0.505840			−0.252920	−	0.967487i	$-0.581391\pi$
$199$	−1420.01			−0.505840			−0.252920	+	0.967487i	$0.581391\pi$
$200$	−916.128	+	400.885i	−0.323900	+	0.141734i
$201$		0			0
$202$		−	1232.92i		−	0.429445i
$203$			2579.10i			0.891710i
$204$		0			0
$205$	−2348.87	−	1536.08i	−0.800254	−	0.523337i
$206$	−3312.12			−1.12022
$207$		0			0
$208$			113.190i			0.0377321i
$209$	−2362.34			−0.781848
$210$		0			0
$211$	1679.13			0.547850			0.273925	−	0.961751i	$-0.411678\pi$
$211$	1679.13			0.547850			0.273925	+	0.961751i	$0.411678\pi$
$212$			2378.79i			0.770640i
$213$		0			0
$214$	−3477.64			−1.11087
$215$	−974.933	+	1490.80i	−0.309255	+	0.472893i
$216$		0			0
$217$			1480.50i			0.463146i
$218$			2698.41i			0.838345i
$219$		0			0
$220$	826.246	+	540.335i	0.253207	+	0.165588i
$221$	311.055			0.0946778
$222$		0			0
$223$		−	3486.29i		−	1.04690i	−0.852056	−	0.523451i	$-0.824644\pi$
$223$		−	3486.29i		−	1.04690i	0.852056	−	0.523451i	$-0.175356\pi$
$224$	490.759			0.146385
$225$		0			0
$226$	2643.95			0.778198
$227$		−	3779.57i		−	1.10511i	−0.833478	−	0.552553i	$-0.813654\pi$
$227$		−	3779.57i		−	1.10511i	0.833478	−	0.552553i	$-0.186346\pi$
$228$		0			0
$229$	−1067.54			−0.308058			−0.154029	−	0.988066i	$-0.549225\pi$
$229$	−1067.54			−0.308058			−0.154029	+	0.988066i	$0.549225\pi$
$230$	−2200.96	−	1439.35i	−0.630988	−	0.412643i
$231$		0			0
$232$			1345.36i			0.380721i
$233$		−	819.730i		−	0.230482i	−0.993338	−	0.115241i	$-0.963236\pi$
$233$		−	819.730i		−	0.230482i	0.993338	−	0.115241i	$-0.0367640\pi$
$234$		0			0
$235$	−1241.52	+	1898.45i	−0.344629	+	0.526985i
$236$	−739.601			−0.204000
$237$		0			0
$238$		−	1348.65i		−	0.367311i
$239$	−4811.58			−1.30224			−0.651119	−	0.758975i	$-0.725700\pi$
$239$	−4811.58			−1.30224			−0.651119	+	0.758975i	$0.725700\pi$
$240$		0			0
$241$	4503.16			1.20363			0.601813	−	0.798637i	$-0.294445\pi$
$241$	4503.16			1.20363			0.601813	+	0.798637i	$0.294445\pi$
$242$			1687.35i			0.448212i
$243$		0			0
$244$	2972.98			0.780023
$245$	−1008.69	−	659.650i	−0.263033	−	0.172014i
$246$		0			0
$247$		−	757.041i		−	0.195018i
$248$			772.287i			0.197743i
$249$		0			0
$250$	−2756.35	−	463.715i	−0.697308	−	0.117312i
$251$	1659.33			0.417274			0.208637	−	0.977993i	$-0.433097\pi$
$251$	1659.33			0.417274			0.208637	+	0.977993i	$0.433097\pi$
$252$		0			0
$253$			2596.27i			0.645162i
$254$	−2623.53			−0.648090
$255$		0			0
$256$	256.000			0.0625000
$257$		−	5112.49i		−	1.24089i	−0.784250	−	0.620444i	$-0.786952\pi$
$257$		−	5112.49i		−	1.24089i	0.784250	−	0.620444i	$-0.213048\pi$
$258$		0			0
$259$	744.748			0.178673
$260$	−173.158	+	264.781i	−0.0413030	+	0.0631579i
$261$		0			0
$262$		−	770.364i		−	0.181653i
$263$		−	4759.53i		−	1.11591i	−0.829870	−	0.557956i	$-0.811586\pi$
$263$		−	4759.53i		−	1.11591i	0.829870	−	0.557956i	$-0.188414\pi$
$264$		0			0
$265$	−3639.07	+	5564.63i	−0.843571	+	1.28993i
$266$	−3282.33			−0.756588
$267$		0			0
$268$			3013.74i			0.686915i
$269$	6293.34			1.42644			0.713219	−	0.700942i	$-0.247237\pi$
$269$	6293.34			1.42644			0.713219	+	0.700942i	$0.247237\pi$
$270$		0			0
$271$	−4412.77			−0.989138			−0.494569	−	0.869138i	$-0.664674\pi$
$271$	−4412.77			−0.989138			−0.494569	+	0.869138i	$0.664674\pi$
$272$		−	703.510i		−	0.156826i
$273$		0			0
$274$	2953.72			0.651244
$275$	1106.21	+	2527.99i	0.242571	+	0.554339i
$276$		0			0
$277$			5450.49i			1.18227i	0.806574	+	0.591134i	$0.201319\pi$
$277$			5450.49i			1.18227i	−0.806574	+	0.591134i	$0.798681\pi$
$278$			2492.42i			0.537718i
$279$		0			0
$280$	1148.02	+	750.765i	0.245026	+	0.160238i
$281$	8226.63			1.74647			0.873237	−	0.487295i	$-0.162016\pi$
$281$	8226.63			1.74647			0.873237	+	0.487295i	$0.162016\pi$
$282$		0			0
$283$		−	2350.25i		−	0.493666i	−0.969058	−	0.246833i	$-0.920610\pi$
$283$		−	2350.25i		−	0.493666i	0.969058	−	0.246833i	$-0.0793900\pi$
$284$	3722.42			0.777765
$285$		0			0
$286$	312.338			0.0645767
$287$			3849.78i			0.791796i
$288$		0			0
$289$	2979.69			0.606492
$290$	−2058.14	+	3147.17i	−0.416751	+	0.637270i
$291$		0			0
$292$		−	915.894i		−	0.183557i
$293$		−	4060.90i		−	0.809693i	−0.914385	−	0.404847i	$-0.867325\pi$
$293$		−	4060.90i		−	0.809693i	0.914385	−	0.404847i	$-0.132675\pi$
$294$		0			0
$295$	−1730.13	−	1131.44i	−0.341464	−	0.223306i
$296$	388.491			0.0762857
$297$		0			0
$298$		−	6136.88i		−	1.19295i
$299$	−832.008			−0.160924
$300$		0			0
$301$	2443.42			0.467895
$302$		−	6666.24i		−	1.27020i
$303$		0			0
$304$	−1712.19			−0.323030
$305$	6954.62	+	4548.07i	1.30564	+	0.853842i
$306$		0			0
$307$			3369.41i			0.626392i	0.949689	+	0.313196i	$0.101400\pi$
$307$			3369.41i			0.626392i	−0.949689	+	0.313196i	$0.898600\pi$
$308$		−	1354.21i		−	0.250531i
$309$		0			0
$310$	−1181.45	+	1806.59i	−0.216457	+	0.330992i
$311$	2277.32			0.415225			0.207612	−	0.978211i	$-0.433431\pi$
$311$	2277.32			0.415225			0.207612	+	0.978211i	$0.433431\pi$
$312$		0			0
$313$			2475.56i			0.447051i	0.974698	+	0.223525i	$0.0717566\pi$
$313$			2475.56i			0.447051i	−0.974698	+	0.223525i	$0.928243\pi$
$314$	−2757.25			−0.495544
$315$		0			0
$316$	1865.50			0.332097
$317$		−	8208.78i		−	1.45442i	−0.686415	−	0.727210i	$-0.740817\pi$
$317$		−	8208.78i		−	1.45442i	0.686415	−	0.727210i	$-0.259183\pi$
$318$		0			0
$319$	3712.42			0.651585
$320$	598.854	+	391.629i	0.104616	+	0.0684148i
$321$		0			0
$322$			3607.37i			0.624319i
$323$			4705.26i			0.810550i
$324$		0			0
$325$	−810.126	+	354.500i	−0.138270	+	0.0605049i
$326$	515.578			0.0875927
$327$		0			0
$328$			2008.20i			0.338062i
$329$	3111.55			0.521415
$330$		0			0
$331$	6701.98			1.11291			0.556457	−	0.830877i	$-0.312161\pi$
$331$	6701.98			1.11291			0.556457	+	0.830877i	$0.312161\pi$
$332$			833.862i			0.137844i
$333$		0			0
$334$	−4559.04			−0.746885
$335$	−4610.42	+	7049.96i	−0.751923	+	1.14979i
$336$		0			0
$337$		−	10642.7i		−	1.72032i	−0.510026	−	0.860159i	$-0.670364\pi$
$337$		−	10642.7i		−	1.72032i	0.510026	−	0.860159i	$-0.329636\pi$
$338$		−	4293.91i		−	0.690999i
$339$		0			0
$340$	1076.23	−	1645.70i	0.171667	−	0.262502i
$341$	2131.07			0.338427
$342$		0			0
$343$			6913.57i			1.08833i
$344$	1274.59			0.199771
$345$		0			0
$346$	5385.90			0.836843
$347$		−	3765.84i		−	0.582596i	−0.956632	−	0.291298i	$-0.905913\pi$
$347$		−	3765.84i		−	0.582596i	0.956632	−	0.291298i	$-0.0940871\pi$
$348$		0			0
$349$	4664.11			0.715369			0.357685	−	0.933842i	$-0.383566\pi$
$349$	4664.11			0.715369			0.357685	+	0.933842i	$0.383566\pi$
$350$	1537.02	+	3512.49i	0.234734	+	0.536430i
$351$		0			0
$352$		−	706.412i		−	0.106966i
$353$			577.000i			0.0869989i	0.999053	+	0.0434994i	$0.0138507\pi$
$353$			577.000i			0.0869989i	−0.999053	+	0.0434994i	$0.986149\pi$
$354$		0			0
$355$	8707.77	+	5694.57i	1.30186	+	0.851370i
$356$	−1937.12			−0.288391
$357$		0			0
$358$			2519.05i			0.371888i
$359$	10754.2			1.58102			0.790511	−	0.612448i	$-0.209815\pi$
$359$	10754.2			1.58102			0.790511	+	0.612448i	$0.209815\pi$
$360$		0			0
$361$	4592.61			0.669574
$362$		−	675.914i		−	0.0981360i
$363$		0			0
$364$	433.975			0.0624903
$365$	1401.14	−	2142.53i	0.200928	−	0.307247i
$366$		0			0
$367$			9630.18i			1.36973i	0.728670	+	0.684865i	$0.240139\pi$
$367$			9630.18i			1.36973i	−0.728670	+	0.684865i	$0.759861\pi$
$368$			1881.75i			0.266557i
$369$		0			0
$370$	908.787	+	594.314i	0.127691	+	0.0835052i
$371$	9120.40			1.27630
$372$		0			0
$373$			13200.8i			1.83247i	0.400645	+	0.916234i	$0.368786\pi$
$373$			13200.8i			1.83247i	−0.400645	+	0.916234i	$0.631214\pi$
$374$	−1941.28			−0.268399
$375$		0			0
$376$	1623.11			0.222621
$377$			1189.69i			0.162526i
$378$		0			0
$379$	−6581.79			−0.892042			−0.446021	−	0.895022i	$-0.647159\pi$
$379$	−6581.79			−0.892042			−0.446021	+	0.895022i	$0.647159\pi$
$380$	−4005.29	−	2619.32i	−0.540703	−	0.353601i
$381$		0			0
$382$		−	5618.56i		−	0.752541i
$383$		−	8383.18i		−	1.11843i	−0.829021	−	0.559217i	$-0.811102\pi$
$383$		−	8383.18i		−	1.11843i	0.829021	−	0.559217i	$-0.188898\pi$
$384$		0			0
$385$	2071.68	−	3167.88i	0.274240	−	0.419351i
$386$	6995.93			0.922496
$387$		0			0
$388$			3512.34i			0.459567i
$389$	−222.395			−0.0289868			−0.0144934	−	0.999895i	$-0.504614\pi$
$389$	−222.395			−0.0289868			−0.0144934	+	0.999895i	$0.504614\pi$
$390$		0			0
$391$	5171.21			0.668847
$392$			862.400i			0.111117i
$393$		0			0
$394$	−7869.16			−1.00620
$395$	4363.92	+	2853.85i	0.555881	+	0.363526i
$396$		0			0
$397$			7702.09i			0.973695i	0.873487	+	0.486848i	$0.161853\pi$
$397$			7702.09i			0.973695i	−0.873487	+	0.486848i	$0.838147\pi$
$398$			2840.03i			0.357683i
$399$		0			0
$400$	801.770	+	1832.26i	0.100221	+	0.229032i
$401$	−2831.08			−0.352562			−0.176281	−	0.984340i	$-0.556407\pi$
$401$	−2831.08			−0.352562			−0.176281	+	0.984340i	$0.556407\pi$
$402$		0			0
$403$			682.928i			0.0844145i
$404$	−2465.84			−0.303663
$405$		0			0
$406$	5158.19			0.630534
$407$		−	1072.01i		−	0.130559i
$408$		0			0
$409$	−321.757			−0.0388994			−0.0194497	−	0.999811i	$-0.506191\pi$
$409$	−321.757			−0.0388994			−0.0194497	+	0.999811i	$0.506191\pi$
$410$	−3072.15	+	4697.74i	−0.370055	+	0.565865i
$411$		0			0
$412$			6624.23i			0.792118i
$413$			2835.67i			0.337855i
$414$		0			0
$415$	−1275.64	+	1950.63i	−0.150889	+	0.230730i
$416$	226.379			0.0266806
$417$		0			0
$418$			4724.67i			0.552850i
$419$	−7459.03			−0.869684			−0.434842	−	0.900507i	$-0.643196\pi$
$419$	−7459.03			−0.869684			−0.434842	+	0.900507i	$0.643196\pi$
$420$		0			0
$421$	−6315.85			−0.731154			−0.365577	−	0.930781i	$-0.619128\pi$
$421$	−6315.85			−0.731154			−0.365577	+	0.930781i	$0.619128\pi$
$422$		−	3358.27i		−	0.387389i
$423$		0			0
$424$	4757.57			0.544925
$425$	5035.20	−	2203.33i	0.574690	−	0.251476i
$426$		0			0
$427$		−	11398.6i		−	1.29184i
$428$			6955.29i			0.785506i
$429$		0			0
$430$	2981.61	+	1949.87i	0.334386	+	0.218676i
$431$	−11806.3			−1.31947			−0.659734	−	0.751499i	$-0.729331\pi$
$431$	−11806.3			−1.31947			−0.659734	+	0.751499i	$0.729331\pi$
$432$		0			0
$433$		−	9553.15i		−	1.06027i	−0.847915	−	0.530133i	$-0.822142\pi$
$433$		−	9553.15i		−	1.06027i	0.847915	−	0.530133i	$-0.177858\pi$
$434$	2960.99			0.327494
$435$		0			0
$436$	5396.81			0.592799
$437$		−	12585.6i		−	1.37769i
$438$		0			0
$439$	1955.52			0.212601			0.106301	−	0.994334i	$-0.466099\pi$
$439$	1955.52			0.212601			0.106301	+	0.994334i	$0.466099\pi$
$440$	1080.67	−	1652.49i	0.117089	−	0.179044i
$441$		0			0
$442$		−	622.109i		−	0.0669473i
$443$			5006.58i			0.536952i	0.963286	+	0.268476i	$0.0865200\pi$
$443$			5006.58i			0.536952i	−0.963286	+	0.268476i	$0.913480\pi$
$444$		0			0
$445$	−4531.46	−	2963.41i	−0.482724	−	0.315684i
$446$	−6972.58			−0.740272
$447$		0			0
$448$		−	981.519i		−	0.103510i
$449$	4878.58			0.512771			0.256386	−	0.966575i	$-0.417468\pi$
$449$	4878.58			0.512771			0.256386	+	0.966575i	$0.417468\pi$
$450$		0			0
$451$	5541.48			0.578577
$452$		−	5287.90i		−	0.550269i
$453$		0			0
$454$	−7559.14			−0.781428
$455$	1015.19	+	663.896i	0.104599	+	0.0684042i
$456$		0			0
$457$			14790.9i			1.51398i	0.653425	+	0.756991i	$0.273332\pi$
$457$			14790.9i			1.51398i	−0.653425	+	0.756991i	$0.726668\pi$
$458$			2135.09i			0.217830i
$459$		0			0
$460$	−2878.70	+	4401.93i	−0.291783	+	0.446176i
$461$	−9128.46			−0.922245			−0.461122	−	0.887336i	$-0.652553\pi$
$461$	−9128.46			−0.922245			−0.461122	+	0.887336i	$0.652553\pi$
$462$		0			0
$463$			15865.8i			1.59254i	0.604939	+	0.796272i	$0.293198\pi$
$463$			15865.8i			1.59254i	−0.604939	+	0.796272i	$0.706802\pi$
$464$	2690.72			0.269210
$465$		0			0
$466$	−1639.46			−0.162975
$467$		−	11496.2i		−	1.13914i	−0.821941	−	0.569572i	$-0.807109\pi$
$467$		−	11496.2i		−	1.13914i	0.821941	−	0.569572i	$-0.192891\pi$
$468$		0			0
$469$	11554.8			1.13764
$470$	3796.90	+	2483.04i	0.372634	+	0.243690i
$471$		0			0
$472$			1479.20i			0.144249i
$473$		−	3517.13i		−	0.341898i
$474$		0			0
$475$	−5362.45	−	12254.6i	−0.517991	−	1.18375i
$476$	−2697.30			−0.259728
$477$		0			0
$478$			9623.15i			0.920822i
$479$	−3142.86			−0.299793			−0.149897	−	0.988702i	$-0.547894\pi$
$479$	−3142.86			−0.299793			−0.149897	+	0.988702i	$0.547894\pi$
$480$		0			0
$481$	343.540			0.0325656
$482$		−	9006.31i		−	0.851092i
$483$		0			0
$484$	3374.71			0.316934
$485$	−5373.19	+	8216.33i	−0.503059	+	0.769246i
$486$		0			0
$487$			16790.6i			1.56233i	0.624323	+	0.781167i	$0.285375\pi$
$487$			16790.6i			1.56233i	−0.624323	+	0.781167i	$0.714625\pi$
$488$		−	5945.96i		−	0.551560i
$489$		0			0
$490$	−1319.30	+	2017.39i	−0.121632	+	0.185993i
$491$	−2638.80			−0.242541			−0.121270	−	0.992620i	$-0.538697\pi$
$491$	−2638.80			−0.242541			−0.121270	+	0.992620i	$0.538697\pi$
$492$		0			0
$493$		−	7394.34i		−	0.675506i
$494$	−1514.08			−0.137898
$495$		0			0
$496$	1544.57			0.139825
$497$		−	14272.0i		−	1.28810i
$498$		0			0
$499$	6907.72			0.619704			0.309852	−	0.950785i	$-0.399721\pi$
$499$	6907.72			0.619704			0.309852	+	0.950785i	$0.399721\pi$
$500$	−927.430	+	5512.70i	−0.0829519	+	0.493071i
$501$		0			0
$502$		−	3318.65i		−	0.295057i
$503$			14516.8i			1.28683i	0.765519	+	0.643413i	$0.222482\pi$
$503$			14516.8i			1.28683i	−0.765519	+	0.643413i	$0.777518\pi$
$504$		0			0
$505$	−5768.27	−	3772.24i	−0.508287	−	0.332401i
$506$	5192.54			0.456199
$507$		0			0
$508$			5247.06i			0.458269i
$509$	−16442.8			−1.43186			−0.715929	−	0.698173i	$-0.753997\pi$
$509$	−16442.8			−1.43186			−0.715929	+	0.698173i	$0.753997\pi$
$510$		0			0
$511$	−3511.59			−0.303999
$512$		−	512.000i		−	0.0441942i
$513$		0			0
$514$	−10225.0			−0.877441
$515$	−10133.8	+	15495.9i	−0.867082	+	1.32589i
$516$		0			0
$517$		−	4478.85i		−	0.381005i
$518$		−	1489.50i		−	0.126341i
$519$		0			0
$520$	529.563	+	346.315i	0.0446594	+	0.0292056i
$521$	−8210.59			−0.690427			−0.345213	−	0.938524i	$-0.612193\pi$
$521$	−8210.59			−0.690427			−0.345213	+	0.938524i	$0.612193\pi$
$522$		0			0
$523$		−	19135.0i		−	1.59984i	−0.600108	−	0.799919i	$-0.704876\pi$
$523$		−	19135.0i		−	1.59984i	0.600108	−	0.799919i	$-0.295124\pi$
$524$	−1540.73			−0.128448
$525$		0			0
$526$	−9519.06			−0.789070
$527$		−	4244.62i		−	0.350851i
$528$		0			0
$529$	−1664.94			−0.136840
$530$	11129.3	+	7278.14i	0.912121	+	0.596495i
$531$		0			0
$532$			6564.66i			0.534989i
$533$			1775.84i			0.144315i
$534$		0			0
$535$	−10640.2	+	16270.3i	−0.859844	+	1.31482i
$536$	6027.47			0.485722
$537$		0			0
$538$		−	12586.7i		−	1.00864i
$539$	2379.73			0.190171
$540$		0			0
$541$	16141.8			1.28279			0.641395	−	0.767211i	$-0.278356\pi$
$541$	16141.8			1.28279			0.641395	+	0.767211i	$0.278356\pi$
$542$			8825.53i			0.699426i
$543$		0			0
$544$	−1407.02			−0.110892
$545$	12624.6	+	8256.06i	0.992257	+	0.648900i
$546$		0			0
$547$			14220.2i			1.11154i	0.831337	+	0.555768i	$0.187576\pi$
$547$			14220.2i			1.11154i	−0.831337	+	0.555768i	$0.812424\pi$
$548$		−	5907.44i		−	0.460499i
$549$		0			0
$550$	5055.97	−	2212.42i	0.391977	−	0.171524i
$551$	−17996.3			−1.39141
$552$		0			0
$553$		−	7152.45i		−	0.550005i
$554$	10901.0			0.835989
$555$		0			0
$556$	4984.85			0.380224
$557$		−	13464.3i		−	1.02424i	−0.858914	−	0.512119i	$-0.828861\pi$
$557$		−	13464.3i		−	1.02424i	0.858914	−	0.512119i	$-0.171139\pi$
$558$		0			0
$559$	1127.11			0.0852802
$560$	1501.53	−	2296.04i	0.113306	−	0.173260i
$561$		0			0
$562$		−	16453.3i		−	1.23494i
$563$			19033.2i			1.42479i	0.701781	+	0.712393i	$0.252388\pi$
$563$			19033.2i			1.42479i	−0.701781	+	0.712393i	$0.747612\pi$
$564$		0			0
$565$	8089.43	−	12369.8i	0.602345	−	0.921068i
$566$	−4700.49			−0.349075
$567$		0			0
$568$		−	7444.85i		−	0.549963i
$569$	5997.95			0.441911			0.220955	−	0.975284i	$-0.429082\pi$
$569$	5997.95			0.441911			0.220955	+	0.975284i	$0.429082\pi$
$570$		0			0
$571$	3153.17			0.231097			0.115548	−	0.993302i	$-0.463137\pi$
$571$	3153.17			0.231097			0.115548	+	0.993302i	$0.463137\pi$
$572$		−	624.676i		−	0.0456626i
$573$		0			0
$574$	7699.56			0.559884
$575$	−13468.1	+	5893.47i	−0.976801	+	0.427434i
$576$		0			0
$577$		−	25188.8i		−	1.81737i	−0.417481	−	0.908686i	$-0.637087\pi$
$577$		−	25188.8i		−	1.81737i	0.417481	−	0.908686i	$-0.362913\pi$
$578$		−	5959.39i		−	0.428854i
$579$		0			0
$580$	6294.34	+	4116.27i	0.450618	+	0.294688i
$581$	3197.08			0.228291
$582$		0			0
$583$		−	13128.2i		−	0.932611i
$584$	−1831.79			−0.129794
$585$		0			0
$586$	−8121.80			−0.572540
$587$			8516.07i			0.598800i	0.954128	+	0.299400i	$0.0967865\pi$
$587$			8516.07i			0.598800i	−0.954128	+	0.299400i	$0.903213\pi$
$588$		0			0
$589$	−10330.5			−0.722685
$590$	−2262.88	+	3460.26i	−0.157901	+	0.241452i
$591$		0			0
$592$		−	776.982i		−	0.0539421i
$593$		−	11235.8i		−	0.778078i	−0.921221	−	0.389039i	$-0.872807\pi$
$593$		−	11235.8i		−	0.778078i	0.921221	−	0.389039i	$-0.127193\pi$
$594$		0			0
$595$	−6309.72	−	4126.33i	−0.434745	−	0.284308i
$596$	−12273.8			−0.843545
$597$		0			0
$598$			1664.02i			0.113790i
$599$	9840.08			0.671210			0.335605	−	0.942003i	$-0.391059\pi$
$599$	9840.08			0.671210			0.335605	+	0.942003i	$0.391059\pi$
$600$		0			0
$601$	−2244.19			−0.152317			−0.0761584	−	0.997096i	$-0.524265\pi$
$601$	−2244.19			−0.152317			−0.0761584	+	0.997096i	$0.524265\pi$
$602$		−	4886.84i		−	0.330852i
$603$		0			0
$604$	−13332.5			−0.898164
$605$	7894.37	+	5162.64i	0.530499	+	0.346927i
$606$		0			0
$607$		−	25483.2i		−	1.70400i	−0.523538	−	0.852002i	$-0.675388\pi$
$607$		−	25483.2i		−	1.70400i	0.523538	−	0.852002i	$-0.324612\pi$
$608$			3424.39i			0.228417i
$609$		0			0
$610$	9096.14	−	13909.2i	0.603758	−	0.923227i
$611$	1435.31			0.0950349
$612$		0			0
$613$		−	1326.50i		−	0.0874007i	−0.999045	−	0.0437004i	$-0.986085\pi$
$613$		−	1326.50i		−	0.0874007i	0.999045	−	0.0437004i	$-0.0139147\pi$
$614$	6738.81			0.442926
$615$		0			0
$616$	−2708.43			−0.177152
$617$			18518.0i			1.20828i	0.796880	+	0.604138i	$0.206482\pi$
$617$			18518.0i			1.20828i	−0.796880	+	0.604138i	$0.793518\pi$
$618$		0			0
$619$	11665.5			0.757473			0.378736	−	0.925505i	$-0.376359\pi$
$619$	11665.5			0.757473			0.378736	+	0.925505i	$0.376359\pi$
$620$	3613.18	+	2362.89i	0.234047	+	0.153058i
$621$		0			0
$622$		−	4554.64i		−	0.293608i
$623$			7427.04i			0.477622i
$624$		0			0
$625$	−10602.9	+	11476.9i	−0.678583	+	0.734524i
$626$	4951.12			0.316113
$627$		0			0
$628$			5514.51i			0.350403i
$629$	−2135.21			−0.135352
$630$		0			0
$631$	−30284.6			−1.91063			−0.955317	−	0.295583i	$-0.904486\pi$
$631$	−30284.6			−1.91063			−0.955317	+	0.295583i	$0.904486\pi$
$632$		−	3731.01i		−	0.234828i
$633$		0			0
$634$	−16417.6			−1.02843
$635$	−8026.96	+	12274.3i	−0.501638	+	0.767072i
$636$		0			0
$637$			762.614i			0.0474346i
$638$		−	7424.84i		−	0.460740i
$639$		0			0
$640$	783.259	−	1197.71i	0.0483766	−	0.0739744i
$641$	15593.6			0.960856			0.480428	−	0.877034i	$-0.340481\pi$
$641$	15593.6			0.960856			0.480428	+	0.877034i	$0.340481\pi$
$642$		0			0
$643$		−	15898.1i		−	0.975052i	−0.873108	−	0.487526i	$-0.837899\pi$
$643$		−	15898.1i		−	0.975052i	0.873108	−	0.487526i	$-0.162101\pi$
$644$	7214.73			0.441460
$645$		0			0
$646$	9410.52			0.573145
$647$		−	8147.23i		−	0.495055i	−0.968881	−	0.247528i	$-0.920382\pi$
$647$		−	8147.23i		−	0.495055i	0.968881	−	0.247528i	$-0.0796181\pi$
$648$		0			0
$649$	4081.74			0.246876
$650$	708.999	+	1620.25i	0.0427835	+	0.0977716i
$651$		0			0
$652$		−	1031.16i		−	0.0619374i
$653$			14735.9i			0.883096i	0.897238	+	0.441548i	$0.145571\pi$
$653$			14735.9i			0.883096i	−0.897238	+	0.441548i	$0.854429\pi$
$654$		0			0
$655$	−3604.18	−	2357.01i	−0.215003	−	0.140604i
$656$	4016.40			0.239046
$657$		0			0
$658$		−	6223.11i		−	0.368696i
$659$	−25732.7			−1.52110			−0.760548	−	0.649282i	$-0.775070\pi$
$659$	−25732.7			−1.52110			−0.760548	+	0.649282i	$0.775070\pi$
$660$		0			0
$661$	−31073.4			−1.82847			−0.914234	−	0.405187i	$-0.867206\pi$
$661$	−31073.4			−1.82847			−0.914234	+	0.405187i	$0.867206\pi$
$662$		−	13404.0i		−	0.786948i
$663$		0			0
$664$	1667.72			0.0974703
$665$	−10042.6	+	15356.5i	−0.585618	+	0.895490i
$666$		0			0
$667$			19778.4i			1.14816i
$668$			9118.08i			0.528128i
$669$		0			0
$670$	14099.9	+	9220.84i	0.813026	+	0.531690i
$671$	−16407.4			−0.943967
$672$		0			0
$673$			10636.5i			0.609220i	0.952477	+	0.304610i	$0.0985262\pi$
$673$			10636.5i			0.609220i	−0.952477	+	0.304610i	$0.901474\pi$
$674$	−21285.5			−1.21645
$675$		0			0
$676$	−8587.81			−0.488610
$677$		−	29438.9i		−	1.67124i	−0.549307	−	0.835621i	$-0.685108\pi$
$677$		−	29438.9i		−	1.67124i	0.549307	−	0.835621i	$-0.314892\pi$
$678$		0			0
$679$	13466.5			0.761116
$680$	−3291.41	−	2152.46i	−0.185617	−	0.121387i
$681$		0			0
$682$		−	4262.13i		−	0.239304i
$683$		−	13446.6i		−	0.753323i	−0.926351	−	0.376661i	$-0.877072\pi$
$683$		−	13446.6i		−	0.753323i	0.926351	−	0.376661i	$-0.122928\pi$
$684$		0			0
$685$	9037.22	−	13819.1i	0.504079	−	0.770806i
$686$	13827.1			0.769567
$687$		0			0
$688$		−	2549.17i		−	0.141259i
$689$	4207.09			0.232623
$690$		0			0
$691$	−26140.7			−1.43913			−0.719564	−	0.694426i	$-0.755658\pi$
$691$	−26140.7			−1.43913			−0.719564	+	0.694426i	$0.755658\pi$
$692$		−	10771.8i		−	0.591737i
$693$		0			0
$694$	−7531.67			−0.411957
$695$	11660.9	+	7625.83i	0.636437	+	0.416207i
$696$		0			0
$697$		−	11037.4i		−	0.599817i
$698$		−	9328.21i		−	0.505843i
$699$		0			0
$700$	7024.98	−	3074.03i	0.379313	−	0.165982i
$701$	706.817			0.0380829			0.0190414	−	0.999819i	$-0.493939\pi$
$701$	706.817			0.0380829			0.0190414	+	0.999819i	$0.493939\pi$
$702$		0			0
$703$			5196.66i			0.278799i
$704$	−1412.82			−0.0756361
$705$		0			0
$706$	1154.00			0.0615175
$707$			9454.17i			0.502915i
$708$		0			0
$709$	−27882.0			−1.47691			−0.738455	−	0.674302i	$-0.764444\pi$
$709$	−27882.0			−1.47691			−0.738455	+	0.674302i	$0.764444\pi$
$710$	11389.1	−	17415.5i	0.602009	−	0.920554i
$711$		0			0
$712$			3874.25i			0.203923i
$713$			11353.5i			0.596343i
$714$		0			0
$715$	955.630	−	1461.29i	0.0499840	−	0.0764323i
$716$	5038.10			0.262965
$717$		0			0
$718$		−	21508.5i		−	1.11795i
$719$	−28953.3			−1.50178			−0.750888	−	0.660429i	$-0.770374\pi$
$719$	−28953.3			−1.50178			−0.750888	+	0.660429i	$0.770374\pi$
$720$		0			0
$721$	25397.7			1.31187
$722$		−	9185.21i		−	0.473460i
$723$		0			0
$724$	−1351.83			−0.0693926
$725$	8427.11	+	19258.2i	0.431690	+	0.986525i
$726$		0			0
$727$			36050.9i			1.83914i	0.392926	+	0.919570i	$0.371463\pi$
$727$			36050.9i			1.83914i	−0.392926	+	0.919570i	$0.628537\pi$
$728$		−	867.950i		−	0.0441873i
$729$		0			0
$730$	−4285.06	−	2802.27i	−0.217256	−	0.142078i
$731$	−7005.35			−0.354449
$732$		0			0
$733$			32024.9i			1.61373i	0.590733	+	0.806867i	$0.298839\pi$
$733$			32024.9i			1.61373i	−0.590733	+	0.806867i	$0.701161\pi$
$734$	19260.4			0.968546
$735$		0			0
$736$	3763.50			0.188484
$737$		−	16632.4i		−	0.831290i
$738$		0			0
$739$	4853.58			0.241599			0.120800	−	0.992677i	$-0.461454\pi$
$739$	4853.58			0.241599			0.120800	+	0.992677i	$0.461454\pi$
$740$	1188.63	−	1817.57i	0.0590471	−	0.0902910i
$741$		0			0
$742$		−	18240.8i		−	0.902481i
$743$			26375.1i			1.30230i	0.758949	+	0.651150i	$0.225713\pi$
$743$			26375.1i			1.30230i	−0.758949	+	0.651150i	$0.774287\pi$
$744$		0			0
$745$	−28711.7	−	18776.4i	−1.41197	−	0.923376i
$746$	26401.6			1.29575
$747$		0			0
$748$			3882.57i			0.189787i
$749$	26667.0			1.30092
$750$		0			0
$751$	−6479.28			−0.314823			−0.157412	−	0.987533i	$-0.550315\pi$
$751$	−6479.28			−0.314823			−0.157412	+	0.987533i	$0.550315\pi$
$752$		−	3246.22i		−	0.157417i
$753$		0			0
$754$	2379.39			0.114923
$755$	−31188.3	−	20396.1i	−1.50339	−	0.983163i
$756$		0			0
$757$		−	34618.7i		−	1.66214i	−0.556169	−	0.831069i	$-0.687729\pi$
$757$		−	34618.7i		−	1.66214i	0.556169	−	0.831069i	$-0.312271\pi$
$758$			13163.6i			0.630769i
$759$		0			0
$760$	−5238.64	+	8010.59i	−0.250033	+	0.382335i
$761$	33438.9			1.59285			0.796425	−	0.604737i	$-0.206722\pi$
$761$	33438.9			1.59285			0.796425	+	0.604737i	$0.206722\pi$
$762$		0			0
$763$		−	20691.7i		−	0.981769i
$764$	−11237.1			−0.532127
$765$		0			0
$766$	−16766.4			−0.790853
$767$			1308.05i			0.0615787i
$768$		0			0
$769$	7469.80			0.350283			0.175142	−	0.984543i	$-0.443962\pi$
$769$	7469.80			0.350283			0.175142	+	0.984543i	$0.443962\pi$
$770$	−6335.75	−	4143.36i	−0.296526	−	0.193917i
$771$		0			0
$772$		−	13991.9i		−	0.652303i
$773$		−	3658.78i		−	0.170242i	−0.996371	−	0.0851212i	$-0.972872\pi$
$773$		−	3658.78i		−	0.170242i	0.996371	−	0.0851212i	$-0.0271277\pi$
$774$		0			0
$775$	4837.47	+	11054.9i	0.224216	+	0.512392i
$776$	7024.68			0.324963
$777$		0			0
$778$			444.789i			0.0204967i
$779$	−26862.8			−1.23550
$780$		0			0
$781$	−20543.5			−0.941234
$782$		−	10342.4i		−	0.472946i
$783$		0			0
$784$	1724.80			0.0785714
$785$	−8436.10	+	12899.9i	−0.383564	+	0.586521i
$786$		0			0
$787$		−	40483.7i		−	1.83366i	−0.399283	−	0.916828i	$-0.630741\pi$
$787$		−	40483.7i		−	1.83366i	0.399283	−	0.916828i	$-0.369259\pi$
$788$			15738.3i			0.711490i
$789$		0			0
$790$	5707.70	−	8727.85i	0.257052	−	0.393067i
$791$	−20274.1			−0.911333
$792$		0			0
$793$		−	5257.97i		−	0.235455i
$794$	15404.2			0.688506
$795$		0			0
$796$	5680.06			0.252920
$797$		−	30644.1i		−	1.36194i	−0.732310	−	0.680971i	$-0.761558\pi$
$797$		−	30644.1i		−	1.36194i	0.732310	−	0.680971i	$-0.238442\pi$
$798$		0			0
$799$	−8920.90			−0.394992
$800$	3664.51	−	1603.54i	0.161950	−	0.0708671i
$801$		0			0
$802$			5662.16i			0.249299i
$803$			5054.68i			0.222137i
$804$		0			0
$805$	16877.2	+	11037.1i	0.738937	+	0.483239i
$806$	1365.86			0.0596901
$807$		0			0
$808$			4931.68i			0.214723i
$809$	35800.6			1.55585			0.777924	−	0.628358i	$-0.216273\pi$
$809$	35800.6			1.55585			0.777924	+	0.628358i	$0.216273\pi$
$810$		0			0
$811$	−39786.3			−1.72267			−0.861336	−	0.508036i	$-0.830372\pi$
$811$	−39786.3			−1.72267			−0.861336	+	0.508036i	$0.830372\pi$
$812$		−	10316.4i		−	0.445855i
$813$		0			0
$814$	−2144.02			−0.0923193
$815$	1577.46	−	2412.16i	0.0677990	−	0.103674i
$816$		0			0
$817$			17049.5i			0.730095i
$818$			643.515i			0.0275061i
$819$		0			0
$820$	9395.47	+	6144.30i	0.400127	+	0.261669i
$821$	−24472.6			−1.04032			−0.520158	−	0.854070i	$-0.674127\pi$
$821$	−24472.6			−1.04032			−0.520158	+	0.854070i	$0.674127\pi$
$822$		0			0
$823$		−	17746.4i		−	0.751641i	−0.926692	−	0.375821i	$-0.877361\pi$
$823$		−	17746.4i		−	0.751641i	0.926692	−	0.375821i	$-0.122639\pi$
$824$	13248.5			0.560112
$825$		0			0
$826$	5671.34			0.238900
$827$		−	39143.8i		−	1.64591i	−0.568110	−	0.822953i	$-0.692325\pi$
$827$		−	39143.8i		−	1.64591i	0.568110	−	0.822953i	$-0.307675\pi$
$828$		0			0
$829$	47622.1			1.99516			0.997578	−	0.0695580i	$-0.0221589\pi$
$829$	47622.1			1.99516			0.997578	+	0.0695580i	$0.0221589\pi$
$830$	3901.27	+	2551.29i	0.163151	+	0.106695i
$831$		0			0
$832$		−	452.758i		−	0.0188661i
$833$		−	4739.90i		−	0.197152i
$834$		0			0
$835$	−13948.9	+	21329.7i	−0.578108	+	0.884006i
$836$	9449.34			0.390924
$837$		0			0
$838$			14918.1i			0.614959i
$839$	−11444.1			−0.470913			−0.235456	−	0.971885i	$-0.575658\pi$
$839$	−11444.1			−0.470913			−0.235456	+	0.971885i	$0.575658\pi$
$840$		0			0
$841$	3892.20			0.159588
$842$			12631.7i			0.517004i
$843$		0			0
$844$	−6716.54			−0.273925
$845$	−20089.3	−	13137.7i	−0.817860	−	0.534851i
$846$		0			0
$847$		−	12938.8i		−	0.524892i
$848$		−	9515.14i		−	0.385320i
$849$		0			0
$850$	−4406.67	−	10070.4i	−0.177820	−	0.406367i
$851$	5711.27			0.230058
$852$		0			0
$853$		−	18442.6i		−	0.740283i	−0.928975	−	0.370142i	$-0.879309\pi$
$853$		−	18442.6i		−	0.740283i	0.928975	−	0.370142i	$-0.120691\pi$
$854$	−22797.2			−0.913470
$855$		0			0
$856$	13910.6			0.555436
$857$		−	4761.51i		−	0.189790i	−0.995487	−	0.0948949i	$-0.969748\pi$
$857$		−	4761.51i		−	0.189790i	0.995487	−	0.0948949i	$-0.0302515\pi$
$858$		0			0
$859$	49730.8			1.97531			0.987655	−	0.156643i	$-0.0500673\pi$
$859$	49730.8			1.97531			0.987655	+	0.156643i	$0.0500673\pi$
$860$	3899.73	−	5963.22i	0.154628	−	0.236447i
$861$		0			0
$862$			23612.6i			0.933004i
$863$			29555.7i			1.16580i	0.812544	+	0.582900i	$0.198082\pi$
$863$			29555.7i			1.16580i	−0.812544	+	0.582900i	$0.801918\pi$
$864$		0			0
$865$	16478.7	−	25198.2i	0.647737	−	0.990479i
$866$	−19106.3			−0.749721
$867$		0			0
$868$		−	5921.99i		−	0.231573i
$869$	−10295.4			−0.401897
$870$		0			0
$871$	5330.05			0.207350
$872$		−	10793.6i		−	0.419172i
$873$		0			0
$874$	−25171.2			−0.974176
$875$	21136.0	+	3555.82i	0.816603	+	0.137381i
$876$		0			0
$877$		−	31861.1i		−	1.22677i	−0.789785	−	0.613383i	$-0.789808\pi$
$877$		−	31861.1i		−	1.22677i	0.789785	−	0.613383i	$-0.210192\pi$
$878$		−	3911.04i		−	0.150332i
$879$		0			0
$880$	−3304.99	−	2161.34i	−0.126603	−	0.0827941i
$881$	6124.16			0.234198			0.117099	−	0.993120i	$-0.462641\pi$
$881$	6124.16			0.234198			0.117099	+	0.993120i	$0.462641\pi$
$882$		0			0
$883$			17206.8i			0.655780i	0.944716	+	0.327890i	$0.106338\pi$
$883$			17206.8i			0.655780i	−0.944716	+	0.327890i	$0.893662\pi$
$884$	−1244.22			−0.0473389
$885$		0			0
$886$	10013.2			0.379683
$887$		−	2765.97i		−	0.104704i	−0.998629	−	0.0523519i	$-0.983328\pi$
$887$		−	2765.97i		−	0.104704i	0.998629	−	0.0523519i	$-0.0166717\pi$
$888$		0			0
$889$	20117.5			0.758965
$890$	−5926.83	+	9062.93i	−0.223222	+	0.341337i
$891$		0			0
$892$			13945.2i			0.523451i
$893$			21711.6i			0.813607i
$894$		0			0
$895$	11785.5	+	7707.30i	0.440163	+	0.287851i
$896$	−1963.04			−0.0731925
$897$		0			0
$898$		−	9757.15i		−	0.362584i
$899$	16234.4			0.602279
$900$		0			0
$901$	−26148.4			−0.966849
$902$		−	11083.0i		−	0.409115i
$903$		0			0
$904$	−10575.8			−0.389099
$905$	−3162.30	−	2068.03i	−0.116153	−	0.0759597i
$906$		0			0
$907$		−	21624.1i		−	0.791640i	−0.918328	−	0.395820i	$-0.870460\pi$
$907$		−	21624.1i		−	0.791640i	0.918328	−	0.395820i	$-0.129540\pi$
$908$			15118.3i			0.552553i
$909$		0			0
$910$	1327.79	−	2030.37i	0.0483691	−	0.0739629i
$911$	−14608.6			−0.531290			−0.265645	−	0.964071i	$-0.585585\pi$
$911$	−14608.6			−0.531290			−0.265645	+	0.964071i	$0.585585\pi$
$912$		0			0
$913$		−	4601.96i		−	0.166816i
$914$	29581.8			1.07055
$915$		0			0
$916$	4270.17			0.154029
$917$			5907.24i			0.212731i
$918$		0			0
$919$	9860.22			0.353927			0.176963	−	0.984217i	$-0.443373\pi$
$919$	9860.22			0.353927			0.176963	+	0.984217i	$0.443373\pi$
$920$	8803.85	+	5757.40i	0.315494	+	0.206322i
$921$		0			0
$922$			18256.9i			0.652126i
$923$		−	6583.43i		−	0.234774i
$924$		0			0
$925$	5561.05	−	2433.44i	0.197672	−	0.0864984i
$926$	31731.7			1.12610
$927$		0			0
$928$		−	5381.45i		−	0.190361i
$929$	−5685.81			−0.200802			−0.100401	−	0.994947i	$-0.532013\pi$
$929$	−5685.81			−0.200802			−0.100401	+	0.994947i	$0.532013\pi$
$930$		0			0
$931$	−11535.9			−0.406095
$932$			3278.92i			0.115241i
$933$		0			0
$934$	−22992.4			−0.805497
$935$	−5939.56	+	9082.39i	−0.207748	+	0.317675i
$936$		0			0
$937$			15841.5i			0.552316i	0.961112	+	0.276158i	$0.0890613\pi$
$937$			15841.5i			0.552316i	−0.961112	+	0.276158i	$0.910939\pi$
$938$		−	23109.7i		−	0.804433i
$939$		0			0
$940$	4966.08	−	7593.81i	0.172315	−	0.263492i
$941$	51898.1			1.79791			0.898954	−	0.438043i	$-0.144328\pi$
$941$	51898.1			1.79791			0.898954	+	0.438043i	$0.144328\pi$
$942$		0			0
$943$			29522.9i			1.01951i
$944$	2958.40			0.102000
$945$		0			0
$946$	−7034.25			−0.241758
$947$		−	10163.2i		−	0.348742i	−0.984680	−	0.174371i	$-0.944211\pi$
$947$		−	10163.2i		−	0.348742i	0.984680	−	0.174371i	$-0.0557892\pi$
$948$		0			0
$949$	−1619.84			−0.0554080
$950$	−24509.2	+	10724.9i	−0.837036	+	0.366275i
$951$		0			0
$952$			5394.60i			0.183655i
$953$		−	3728.55i		−	0.126736i	−0.997990	−	0.0633680i	$-0.979816\pi$
$953$		−	3728.55i		−	0.126736i	0.997990	−	0.0633680i	$-0.0201842\pi$
$954$		0			0
$955$	−26286.7	−	17190.6i	−0.890699	−	0.582485i
$956$	19246.3			0.651119
$957$		0			0
$958$			6285.72i			0.211986i
$959$	−22649.5			−0.762659
$960$		0			0
$961$	−20471.8			−0.687182
$962$		−	687.080i		−	0.0230274i
$963$		0			0
$964$	−18012.6			−0.601813
$965$	21404.8	−	32730.8i	0.714035	−	1.09186i
$966$		0			0
$967$		−	520.743i		−	0.0173174i	−0.999963	−	0.00865872i	$-0.997244\pi$
$967$		−	520.743i		−	0.0173174i	0.999963	−	0.00865872i	$-0.00275619\pi$
$968$		−	6749.42i		−	0.224106i
$969$		0			0
$970$	16432.7	+	10746.4i	0.543939	+	0.355717i
$971$	20023.8			0.661787			0.330893	−	0.943668i	$-0.392650\pi$
$971$	20023.8			0.661787			0.330893	+	0.943668i	$0.392650\pi$
$972$		0			0
$973$		−	19112.2i		−	0.629711i
$974$	33581.3			1.10474
$975$		0			0
$976$	−11891.9			−0.390012
$977$		−	38502.3i		−	1.26080i	−0.776272	−	0.630398i	$-0.782892\pi$
$977$		−	38502.3i		−	1.26080i	0.776272	−	0.630398i	$-0.217108\pi$
$978$		0			0
$979$	10690.7			0.349005
$980$	4034.78	+	2638.60i	0.131517	+	0.0860072i
$981$		0			0
$982$			5277.61i			0.171502i
$983$		−	8713.22i		−	0.282715i	−0.989959	−	0.141357i	$-0.954853\pi$
$983$		−	8713.22i		−	0.282715i	0.989959	−	0.141357i	$-0.0451466\pi$
$984$		0			0
$985$	−24076.5	+	36816.3i	−0.778824	+	1.19093i
$986$	−14788.7			−0.477655
$987$		0			0
$988$			3028.16i			0.0975088i
$989$	18737.9			0.602458
$990$		0			0
$991$	−9655.88			−0.309515			−0.154757	−	0.987953i	$-0.549460\pi$
$991$	−9655.88			−0.309515			−0.154757	+	0.987953i	$0.549460\pi$
$992$		−	3089.15i		−	0.0988715i
$993$		0			0
$994$	−28544.0			−0.910825
$995$	13287.2	+	8689.36i	0.423350	+	0.276856i
$996$		0			0
$997$			26015.6i			0.826401i	0.910640	+	0.413201i	$0.135589\pi$
$997$			26015.6i			0.826401i	−0.910640	+	0.413201i	$0.864411\pi$
$998$		−	13815.4i		−	0.438197i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.c.b.649.1	yes	10
3.2	odd	2		810.4.c.a.649.10	yes	10
5.4	even	2	inner	810.4.c.b.649.6	yes	10
15.14	odd	2		810.4.c.a.649.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.4.c.a.649.5	✓	10	15.14	odd	2
810.4.c.a.649.10	yes	10	3.2	odd	2
810.4.c.b.649.1	yes	10	1.1	even	1	trivial
810.4.c.b.649.6	yes	10	5.4	even	2	inner

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

Learn more

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	810.4.c.b.649.1

Level	$810$
Weight	$4$
Character	810.649
Analytic conductor	$47.792$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$