LMFDB - Embedded newform 800.3.m.b.657.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,3,Mod(593,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("800.593");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	800.m (of order $4$, degree $2$, 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:	$21.7984211488$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 $ x^20 - x^18 - 3x^16 + 11x^14 + x^12 - 40x^10 + 4x^8 + 176x^6 - 192x^4 - 256*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			657.6
Root			$0.541828 - 1.30630i$ of defining polynomial
Character	$\chi$	$=$	800.657
Dual form			800.3.m.b.593.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.130791 + 0.130791i) q^{3} +(-1.59713 - 1.59713i) q^{7} -8.96579i q^{9} -11.9427i q^{11} +(9.59714 + 9.59714i) q^{13} +(-0.857288 - 0.857288i) q^{17} -20.5611 q^{19} -0.417782i q^{21} +(-22.1560 + 22.1560i) q^{23} +(2.34977 - 2.34977i) q^{27} +27.3404 q^{29} -40.0267 q^{31} +(1.56200 - 1.56200i) q^{33} +(-1.57131 + 1.57131i) q^{37} +2.51044i q^{39} -37.5504 q^{41} +(-49.2602 - 49.2602i) q^{43} +(-34.0876 - 34.0876i) q^{47} -43.8983i q^{49} -0.224252i q^{51} +(-28.8002 - 28.8002i) q^{53} +(-2.68921 - 2.68921i) q^{57} +92.7071 q^{59} -4.82618i q^{61} +(-14.3196 + 14.3196i) q^{63} +(54.6048 - 54.6048i) q^{67} -5.79564 q^{69} -59.2198 q^{71} +(-34.1124 + 34.1124i) q^{73} +(-19.0740 + 19.0740i) q^{77} -96.2455i q^{79} -80.0774 q^{81} +(63.6959 + 63.6959i) q^{83} +(3.57588 + 3.57588i) q^{87} +3.68406i q^{89} -30.6558i q^{91} +(-5.23514 - 5.23514i) q^{93} +(-46.0410 - 46.0410i) q^{97} -107.075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{7} + 12 q^{17} - 4 q^{23} + 136 q^{31} - 32 q^{33} - 8 q^{41} + 188 q^{47} + 40 q^{57} + 228 q^{63} - 248 q^{71} + 124 q^{73} + 132 q^{81} - 488 q^{87} - 100 q^{97}+O(q^{100}) $ 20 * q - 4 * q^7 + 12 * q^17 - 4 * q^23 + 136 * q^31 - 32 * q^33 - 8 * q^41 + 188 * q^47 + 40 * q^57 + 228 * q^63 - 248 * q^71 + 124 * q^73 + 132 * q^81 - 488 * q^87 - 100 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	0.130791	+	0.130791i	0.0435971	+	0.0435971i	0.728569	−	0.684972i	$-0.240186\pi$
$3$	0.130791	+	0.130791i	0.0435971	+	0.0435971i	−0.684972	+	0.728569i	$0.740186\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.59713	−	1.59713i	−0.228162	−	0.228162i	0.583763	−	0.811924i	$-0.301580\pi$
$7$	−1.59713	−	1.59713i	−0.228162	−	0.228162i	−0.811924	+	0.583763i	$0.801580\pi$
$8$		0			0
$9$		−	8.96579i		−	0.996199i
$10$		0			0
$11$		−	11.9427i		−	1.08570i	−0.839831	−	0.542849i	$-0.817346\pi$
$11$		−	11.9427i		−	1.08570i	0.839831	−	0.542849i	$-0.182654\pi$
$12$		0			0
$13$	9.59714	+	9.59714i	0.738241	+	0.738241i	0.972238	−	0.233996i	$-0.0751803\pi$
$13$	9.59714	+	9.59714i	0.738241	+	0.738241i	−0.233996	+	0.972238i	$0.575180\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.857288	−	0.857288i	−0.0504287	−	0.0504287i	0.681443	−	0.731871i	$-0.261353\pi$
$17$	−0.857288	−	0.857288i	−0.0504287	−	0.0504287i	−0.731871	+	0.681443i	$0.761353\pi$
$18$		0			0
$19$	−20.5611			−1.08216			−0.541082	−	0.840970i	$-0.681985\pi$
$19$	−20.5611			−1.08216			−0.541082	+	0.840970i	$0.681985\pi$
$20$		0			0
$21$		−	0.417782i		−	0.0198944i
$22$		0			0
$23$	−22.1560	+	22.1560i	−0.963306	+	0.963306i	−0.999350	−	0.0360441i	$-0.988524\pi$
$23$	−22.1560	+	22.1560i	−0.963306	+	0.963306i	0.0360441	+	0.999350i	$0.488524\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.34977	−	2.34977i	0.0870285	−	0.0870285i
$28$		0			0
$29$	27.3404			0.942771			0.471385	−	0.881927i	$-0.343754\pi$
$29$	27.3404			0.942771			0.471385	+	0.881927i	$0.343754\pi$
$30$		0			0
$31$	−40.0267			−1.29118			−0.645591	−	0.763683i	$-0.723389\pi$
$31$	−40.0267			−1.29118			−0.645591	+	0.763683i	$0.723389\pi$
$32$		0			0
$33$	1.56200	−	1.56200i	0.0473333	−	0.0473333i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.57131	+	1.57131i	−0.0424677	+	0.0424677i	−0.728022	−	0.685554i	$-0.759560\pi$
$37$	−1.57131	+	1.57131i	−0.0424677	+	0.0424677i	0.685554	+	0.728022i	$0.259560\pi$
$38$		0			0
$39$			2.51044i			0.0643704i
$40$		0			0
$41$	−37.5504			−0.915864			−0.457932	−	0.888987i	$-0.651410\pi$
$41$	−37.5504			−0.915864			−0.457932	+	0.888987i	$0.651410\pi$
$42$		0			0
$43$	−49.2602	−	49.2602i	−1.14558	−	1.14558i	−0.987411	−	0.158174i	$-0.949439\pi$
$43$	−49.2602	−	49.2602i	−1.14558	−	1.14558i	−0.158174	−	0.987411i	$-0.550561\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−34.0876	−	34.0876i	−0.725268	−	0.725268i	0.244405	−	0.969673i	$-0.421407\pi$
$47$	−34.0876	−	34.0876i	−0.725268	−	0.725268i	−0.969673	+	0.244405i	$0.921407\pi$
$48$		0			0
$49$		−	43.8983i		−	0.895884i
$50$		0			0
$51$		−	0.224252i		−	0.00439709i
$52$		0			0
$53$	−28.8002	−	28.8002i	−0.543401	−	0.543401i	0.381124	−	0.924524i	$-0.375537\pi$
$53$	−28.8002	−	28.8002i	−0.543401	−	0.543401i	−0.924524	+	0.381124i	$0.875537\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.68921	−	2.68921i	−0.0471792	−	0.0471792i
$58$		0			0
$59$	92.7071			1.57131			0.785653	−	0.618667i	$-0.212327\pi$
$59$	92.7071			1.57131			0.785653	+	0.618667i	$0.212327\pi$
$60$		0			0
$61$		−	4.82618i		−	0.0791176i	−0.999217	−	0.0395588i	$-0.987405\pi$
$61$		−	4.82618i		−	0.0791176i	0.999217	−	0.0395588i	$-0.0125952\pi$
$62$		0			0
$63$	−14.3196	+	14.3196i	−0.227294	+	0.227294i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	54.6048	−	54.6048i	0.814997	−	0.814997i	−0.170381	−	0.985378i	$-0.554500\pi$
$67$	54.6048	−	54.6048i	0.814997	−	0.814997i	0.985378	+	0.170381i	$0.0544998\pi$
$68$		0			0
$69$	−5.79564			−0.0839947
$70$		0			0
$71$	−59.2198			−0.834082			−0.417041	−	0.908888i	$-0.636933\pi$
$71$	−59.2198			−0.834082			−0.417041	+	0.908888i	$0.636933\pi$
$72$		0			0
$73$	−34.1124	+	34.1124i	−0.467293	+	0.467293i	−0.901037	−	0.433743i	$-0.857193\pi$
$73$	−34.1124	+	34.1124i	−0.467293	+	0.467293i	0.433743	+	0.901037i	$0.357193\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−19.0740	+	19.0740i	−0.247715	+	0.247715i
$78$		0			0
$79$		−	96.2455i		−	1.21830i	−0.793056	−	0.609149i	$-0.791511\pi$
$79$		−	96.2455i		−	1.21830i	0.793056	−	0.609149i	$-0.208489\pi$
$80$		0			0
$81$	−80.0774			−0.988610
$82$		0			0
$83$	63.6959	+	63.6959i	0.767421	+	0.767421i	0.977652	−	0.210231i	$-0.0674216\pi$
$83$	63.6959	+	63.6959i	0.767421	+	0.767421i	−0.210231	+	0.977652i	$0.567422\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.57588	+	3.57588i	0.0411021	+	0.0411021i
$88$		0			0
$89$			3.68406i			0.0413939i	0.999786	+	0.0206970i	$0.00658852\pi$
$89$			3.68406i			0.0413939i	−0.999786	+	0.0206970i	$0.993411\pi$
$90$		0			0
$91$		−	30.6558i		−	0.336877i
$92$		0			0
$93$	−5.23514	−	5.23514i	−0.0562918	−	0.0562918i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−46.0410	−	46.0410i	−0.474650	−	0.474650i	0.428766	−	0.903416i	$-0.358949\pi$
$97$	−46.0410	−	46.0410i	−0.474650	−	0.474650i	−0.903416	+	0.428766i	$0.858949\pi$
$98$		0			0
$99$	−107.075			−1.08157
$100$		0			0
$101$		−	108.173i		−	1.07102i	−0.844529	−	0.535510i	$-0.820120\pi$
$101$		−	108.173i		−	1.07102i	0.844529	−	0.535510i	$-0.179880\pi$
$102$		0			0
$103$	−50.9112	+	50.9112i	−0.494284	+	0.494284i	−0.909653	−	0.415369i	$-0.863652\pi$
$103$	−50.9112	+	50.9112i	−0.494284	+	0.494284i	0.415369	+	0.909653i	$0.363652\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−95.4147	+	95.4147i	−0.891726	+	0.891726i	−0.994686	−	0.102960i	$-0.967169\pi$
$107$	−95.4147	+	95.4147i	−0.891726	+	0.891726i	0.102960	+	0.994686i	$0.467169\pi$
$108$		0			0
$109$	−46.5432			−0.427002			−0.213501	−	0.976943i	$-0.568487\pi$
$109$	−46.5432			−0.427002			−0.213501	+	0.976943i	$0.568487\pi$
$110$		0			0
$111$	−0.411026			−0.00370294
$112$		0			0
$113$	−88.0400	+	88.0400i	−0.779115	+	0.779115i	−0.979680	−	0.200565i	$-0.935722\pi$
$113$	−88.0400	+	88.0400i	−0.779115	+	0.779115i	0.200565	+	0.979680i	$0.435722\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	86.0459	−	86.0459i	0.735435	−	0.735435i
$118$		0			0
$119$			2.73841i			0.0230118i
$120$		0			0
$121$	−21.6274			−0.178739
$122$		0			0
$123$	−4.91127	−	4.91127i	−0.0399290	−	0.0399290i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	44.3246	+	44.3246i	0.349012	+	0.349012i	0.859742	−	0.510729i	$-0.170625\pi$
$127$	44.3246	+	44.3246i	0.349012	+	0.349012i	−0.510729	+	0.859742i	$0.670625\pi$
$128$		0			0
$129$		−	12.8856i		−	0.0998884i
$130$		0			0
$131$		−	82.8025i		−	0.632080i	−0.948746	−	0.316040i	$-0.897647\pi$
$131$		−	82.8025i		−	0.632080i	0.948746	−	0.316040i	$-0.102353\pi$
$132$		0			0
$133$	32.8388	+	32.8388i	0.246908	+	0.246908i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−25.0030	−	25.0030i	−0.182503	−	0.182503i	0.609942	−	0.792446i	$-0.291193\pi$
$137$	−25.0030	−	25.0030i	−0.182503	−	0.182503i	−0.792446	+	0.609942i	$0.791193\pi$
$138$		0			0
$139$	104.684			0.753121			0.376561	−	0.926392i	$-0.377107\pi$
$139$	104.684			0.753121			0.376561	+	0.926392i	$0.377107\pi$
$140$		0			0
$141$		−	8.91673i		−	0.0632392i
$142$		0			0
$143$	114.615	−	114.615i	0.801507	−	0.801507i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.74152	−	5.74152i	0.0390580	−	0.0390580i
$148$		0			0
$149$	49.3481			0.331196			0.165598	−	0.986193i	$-0.447045\pi$
$149$	49.3481			0.331196			0.165598	+	0.986193i	$0.447045\pi$
$150$		0			0
$151$	22.1078			0.146409			0.0732047	−	0.997317i	$-0.476677\pi$
$151$	22.1078			0.146409			0.0732047	+	0.997317i	$0.476677\pi$
$152$		0			0
$153$	−7.68626	+	7.68626i	−0.0502370	+	0.0502370i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	64.7264	−	64.7264i	0.412270	−	0.412270i	−0.470258	−	0.882529i	$-0.655839\pi$
$157$	64.7264	−	64.7264i	0.412270	−	0.412270i	0.882529	+	0.470258i	$0.155839\pi$
$158$		0			0
$159$		−	7.53364i		−	0.0473814i
$160$		0			0
$161$	70.7723			0.439579
$162$		0			0
$163$	25.7888	+	25.7888i	0.158213	+	0.158213i	0.781775	−	0.623561i	$-0.214315\pi$
$163$	25.7888	+	25.7888i	0.158213	+	0.158213i	−0.623561	+	0.781775i	$0.714315\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−56.4754	−	56.4754i	−0.338176	−	0.338176i	0.517504	−	0.855681i	$-0.326861\pi$
$167$	−56.4754	−	56.4754i	−0.338176	−	0.338176i	−0.855681	+	0.517504i	$0.826861\pi$
$168$		0			0
$169$			15.2101i			0.0900005i
$170$		0			0
$171$			184.346i			1.07805i
$172$		0			0
$173$	−135.496	−	135.496i	−0.783216	−	0.783216i	0.197156	−	0.980372i	$-0.436829\pi$
$173$	−135.496	−	135.496i	−0.783216	−	0.783216i	−0.980372	+	0.197156i	$0.936829\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	12.1253	+	12.1253i	0.0685044	+	0.0685044i
$178$		0			0
$179$	−15.5036			−0.0866121			−0.0433061	−	0.999062i	$-0.513789\pi$
$179$	−15.5036			−0.0866121			−0.0433061	+	0.999062i	$0.513789\pi$
$180$		0			0
$181$			263.672i			1.45675i	0.685179	+	0.728375i	$0.259724\pi$
$181$			263.672i			1.45675i	−0.685179	+	0.728375i	$0.740276\pi$
$182$		0			0
$183$	0.631222	−	0.631222i	0.00344930	−	0.00344930i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.2383	+	10.2383i	−0.0547503	+	0.0547503i
$188$		0			0
$189$	−7.50579			−0.0397132
$190$		0			0
$191$	120.714			0.632010			0.316005	−	0.948758i	$-0.397658\pi$
$191$	120.714			0.632010			0.316005	+	0.948758i	$0.397658\pi$
$192$		0			0
$193$	182.404	−	182.404i	0.945098	−	0.945098i	−0.0534716	−	0.998569i	$-0.517029\pi$
$193$	182.404	−	182.404i	0.945098	−	0.945098i	0.998569	+	0.0534716i	$0.0170287\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−224.288	+	224.288i	−1.13852	+	1.13852i	−0.149799	+	0.988716i	$0.547863\pi$
$197$	−224.288	+	224.288i	−1.13852	+	1.13852i	−0.988716	+	0.149799i	$0.952137\pi$
$198$		0			0
$199$			87.9089i			0.441753i	0.975302	+	0.220877i	$0.0708918\pi$
$199$			87.9089i			0.441753i	−0.975302	+	0.220877i	$0.929108\pi$
$200$		0			0
$201$	14.2837			0.0710631
$202$		0			0
$203$	−43.6662	−	43.6662i	−0.215104	−	0.215104i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	198.646	+	198.646i	0.959644	+	0.959644i
$208$		0			0
$209$			245.554i			1.17490i
$210$		0			0
$211$			119.117i			0.564534i	0.959336	+	0.282267i	$0.0910864\pi$
$211$			119.117i			0.564534i	−0.959336	+	0.282267i	$0.908914\pi$
$212$		0			0
$213$	−7.74544	−	7.74544i	−0.0363636	−	0.0363636i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	63.9279	+	63.9279i	0.294599	+	0.294599i
$218$		0			0
$219$	−8.92322			−0.0407453
$220$		0			0
$221$		−	16.4550i		−	0.0744571i
$222$		0			0
$223$	111.993	−	111.993i	0.502212	−	0.502212i	−0.409913	−	0.912125i	$-0.634441\pi$
$223$	111.993	−	111.993i	0.502212	−	0.502212i	0.912125	+	0.409913i	$0.134441\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	85.8837	−	85.8837i	0.378342	−	0.378342i	−0.492162	−	0.870504i	$-0.663793\pi$
$227$	85.8837	−	85.8837i	0.378342	−	0.378342i	0.870504	+	0.492162i	$0.163793\pi$
$228$		0			0
$229$	−26.5156			−0.115789			−0.0578943	−	0.998323i	$-0.518439\pi$
$229$	−26.5156			−0.115789			−0.0578943	+	0.998323i	$0.518439\pi$
$230$		0			0
$231$	−4.98944			−0.0215993
$232$		0			0
$233$	71.6569	−	71.6569i	0.307541	−	0.307541i	−0.536414	−	0.843955i	$-0.680222\pi$
$233$	71.6569	−	71.6569i	0.307541	−	0.307541i	0.843955	+	0.536414i	$0.180222\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	12.5881	−	12.5881i	0.0531143	−	0.0531143i
$238$		0			0
$239$			114.913i			0.480809i	0.970673	+	0.240405i	$0.0772801\pi$
$239$			114.913i			0.480809i	−0.970673	+	0.240405i	$0.922720\pi$
$240$		0			0
$241$	219.080			0.909045			0.454522	−	0.890735i	$-0.349810\pi$
$241$	219.080			0.909045			0.454522	+	0.890735i	$0.349810\pi$
$242$		0			0
$243$	−31.6214	−	31.6214i	−0.130129	−	0.130129i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−197.328	−	197.328i	−0.798897	−	0.798897i
$248$		0			0
$249$			16.6618i			0.0669147i
$250$		0			0
$251$		−	95.0552i		−	0.378706i	−0.981909	−	0.189353i	$-0.939361\pi$
$251$		−	95.0552i		−	0.378706i	0.981909	−	0.189353i	$-0.0606390\pi$
$252$		0			0
$253$	264.602	+	264.602i	1.04586	+	1.04586i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	249.128	+	249.128i	0.969369	+	0.969369i	0.999545	−	0.0301755i	$-0.00960661\pi$
$257$	249.128	+	249.128i	0.969369	+	0.969369i	−0.0301755	+	0.999545i	$0.509607\pi$
$258$		0			0
$259$	5.01917			0.0193790
$260$		0			0
$261$		−	245.128i		−	0.939187i
$262$		0			0
$263$	−85.6095	+	85.6095i	−0.325511	+	0.325511i	−0.850877	−	0.525365i	$-0.823929\pi$
$263$	−85.6095	+	85.6095i	−0.325511	+	0.325511i	0.525365	+	0.850877i	$0.323929\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.481843	+	0.481843i	−0.00180465	+	0.00180465i
$268$		0			0
$269$	−192.119			−0.714196			−0.357098	−	0.934067i	$-0.616234\pi$
$269$	−192.119			−0.714196			−0.357098	+	0.934067i	$0.616234\pi$
$270$		0			0
$271$	485.780			1.79255			0.896274	−	0.443501i	$-0.146264\pi$
$271$	485.780			1.79255			0.896274	+	0.443501i	$0.146264\pi$
$272$		0			0
$273$	4.00951	−	4.00951i	0.0146869	−	0.0146869i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	88.8551	−	88.8551i	0.320777	−	0.320777i	−0.528288	−	0.849065i	$-0.677166\pi$
$277$	88.8551	−	88.8551i	0.320777	−	0.320777i	0.849065	+	0.528288i	$0.177166\pi$
$278$		0			0
$279$			358.871i			1.28627i
$280$		0			0
$281$	390.006			1.38792			0.693961	−	0.720012i	$-0.255864\pi$
$281$	390.006			1.38792			0.693961	+	0.720012i	$0.255864\pi$
$282$		0			0
$283$	121.062	+	121.062i	0.427779	+	0.427779i	0.887871	−	0.460092i	$-0.152184\pi$
$283$	121.062	+	121.062i	0.427779	+	0.427779i	−0.460092	+	0.887871i	$0.652184\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	59.9730	+	59.9730i	0.208965	+	0.208965i
$288$		0			0
$289$		−	287.530i		−	0.994914i
$290$		0			0
$291$		−	12.0435i		−	0.0413867i
$292$		0			0
$293$	244.956	+	244.956i	0.836026	+	0.836026i	0.988333	−	0.152307i	$-0.0486702\pi$
$293$	244.956	+	244.956i	0.836026	+	0.836026i	−0.152307	+	0.988333i	$0.548670\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−28.0625	−	28.0625i	−0.0944866	−	0.0944866i
$298$		0			0
$299$	−425.269			−1.42230
$300$		0			0
$301$			157.350i			0.522757i
$302$		0			0
$303$	14.1481	−	14.1481i	0.0466934	−	0.0466934i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−135.324	+	135.324i	−0.440796	+	0.440796i	−0.892280	−	0.451483i	$-0.850895\pi$
$307$	−135.324	+	135.324i	−0.440796	+	0.440796i	0.451483	+	0.892280i	$0.350895\pi$
$308$		0			0
$309$	−13.3175			−0.0430987
$310$		0			0
$311$	−225.951			−0.726531			−0.363266	−	0.931686i	$-0.618338\pi$
$311$	−225.951			−0.726531			−0.363266	+	0.931686i	$0.618338\pi$
$312$		0			0
$313$	230.622	−	230.622i	0.736812	−	0.736812i	−0.235148	−	0.971960i	$-0.575557\pi$
$313$	230.622	−	230.622i	0.736812	−	0.736812i	0.971960	+	0.235148i	$0.0755575\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−221.563	+	221.563i	−0.698938	+	0.698938i	−0.964182	−	0.265243i	$-0.914548\pi$
$317$	−221.563	+	221.563i	−0.698938	+	0.698938i	0.265243	+	0.964182i	$0.414548\pi$
$318$		0			0
$319$		−	326.517i		−	1.02356i
$320$		0			0
$321$	−24.9588			−0.0777534
$322$		0			0
$323$	17.6268	+	17.6268i	0.0545721	+	0.0545721i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.08745	−	6.08745i	−0.0186161	−	0.0186161i
$328$		0			0
$329$			108.885i			0.330957i
$330$		0			0
$331$		−	141.212i		−	0.426624i	−0.976984	−	0.213312i	$-0.931575\pi$
$331$		−	141.212i		−	0.426624i	0.976984	−	0.213312i	$-0.0684250\pi$
$332$		0			0
$333$	14.0880	+	14.0880i	0.0423063	+	0.0423063i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−225.465	−	225.465i	−0.669035	−	0.669035i	0.288458	−	0.957493i	$-0.406858\pi$
$337$	−225.465	−	225.465i	−0.669035	−	0.669035i	−0.957493	+	0.288458i	$0.906858\pi$
$338$		0			0
$339$	−23.0297			−0.0679343
$340$		0			0
$341$			478.025i			1.40183i
$342$		0			0
$343$	−148.371	+	148.371i	−0.432568	+	0.432568i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	346.013	−	346.013i	0.997155	−	0.997155i	−0.00284067	−	0.999996i	$-0.500904\pi$
$347$	346.013	−	346.013i	0.997155	−	0.997155i	0.999996	+	0.00284067i	$0.000904214\pi$
$348$		0			0
$349$	428.411			1.22754			0.613769	−	0.789486i	$-0.289653\pi$
$349$	428.411			1.22754			0.613769	+	0.789486i	$0.289653\pi$
$350$		0			0
$351$	45.1021			0.128496
$352$		0			0
$353$	−25.9724	+	25.9724i	−0.0735762	+	0.0735762i	−0.742937	−	0.669361i	$-0.766568\pi$
$353$	−25.9724	+	25.9724i	−0.0735762	+	0.0735762i	0.669361	+	0.742937i	$0.266568\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−0.358160	+	0.358160i	−0.00100325	+	0.00100325i
$358$		0			0
$359$			288.043i			0.802348i	0.916002	+	0.401174i	$0.131398\pi$
$359$			288.043i			0.802348i	−0.916002	+	0.401174i	$0.868602\pi$
$360$		0			0
$361$	61.7587			0.171077
$362$		0			0
$363$	−2.82868	−	2.82868i	−0.00779249	−	0.00779249i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	160.854	+	160.854i	0.438295	+	0.438295i	0.891438	−	0.453143i	$-0.149697\pi$
$367$	160.854	+	160.854i	0.438295	+	0.438295i	−0.453143	+	0.891438i	$0.649697\pi$
$368$		0			0
$369$			336.669i			0.912382i
$370$		0			0
$371$			91.9956i			0.247966i
$372$		0			0
$373$	−360.121	−	360.121i	−0.965471	−	0.965471i	0.0339529	−	0.999423i	$-0.489190\pi$
$373$	−360.121	−	360.121i	−0.965471	−	0.965471i	−0.999423	+	0.0339529i	$0.989190\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	262.389	+	262.389i	0.695992	+	0.695992i
$378$		0			0
$379$	−64.1572			−0.169280			−0.0846402	−	0.996412i	$-0.526974\pi$
$379$	−64.1572			−0.169280			−0.0846402	+	0.996412i	$0.526974\pi$
$380$		0			0
$381$			11.5945i			0.0304319i
$382$		0			0
$383$	376.877	−	376.877i	0.984014	−	0.984014i	−0.0158604	−	0.999874i	$-0.505049\pi$
$383$	376.877	−	376.877i	0.984014	−	0.984014i	0.999874	+	0.0158604i	$0.00504875\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−441.656	+	441.656i	−1.14123	+	1.14123i
$388$		0			0
$389$	−10.3794			−0.0266823			−0.0133411	−	0.999911i	$-0.504247\pi$
$389$	−10.3794			−0.0266823			−0.0133411	+	0.999911i	$0.504247\pi$
$390$		0			0
$391$	37.9882			0.0971566
$392$		0			0
$393$	10.8298	−	10.8298i	0.0275569	−	0.0275569i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	184.458	−	184.458i	0.464630	−	0.464630i	−0.435540	−	0.900169i	$-0.643442\pi$
$397$	184.458	−	184.458i	0.464630	−	0.464630i	0.900169	+	0.435540i	$0.143442\pi$
$398$		0			0
$399$			8.59006i			0.0215290i
$400$		0			0
$401$	−262.943			−0.655719			−0.327859	−	0.944727i	$-0.606327\pi$
$401$	−262.943			−0.655719			−0.327859	+	0.944727i	$0.606327\pi$
$402$		0			0
$403$	−384.141	−	384.141i	−0.953204	−	0.953204i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.7656	+	18.7656i	0.0461071	+	0.0461071i
$408$		0			0
$409$		−	245.618i		−	0.600532i	−0.953855	−	0.300266i	$-0.902925\pi$
$409$		−	245.618i		−	0.600532i	0.953855	−	0.300266i	$-0.0970755\pi$
$410$		0			0
$411$		−	6.54034i		−	0.0159132i
$412$		0			0
$413$	−148.065	−	148.065i	−0.358512	−	0.358512i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	13.6917	+	13.6917i	0.0328339	+	0.0328339i
$418$		0			0
$419$	673.011			1.60623			0.803116	−	0.595823i	$-0.203174\pi$
$419$	673.011			1.60623			0.803116	+	0.595823i	$0.203174\pi$
$420$		0			0
$421$		−	701.227i		−	1.66562i	−0.553557	−	0.832811i	$-0.686730\pi$
$421$		−	701.227i		−	1.66562i	0.553557	−	0.832811i	$-0.313270\pi$
$422$		0			0
$423$	−305.622	+	305.622i	−0.722511	+	0.722511i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.70804	+	7.70804i	−0.0180516	+	0.0180516i
$428$		0			0
$429$	29.9814			0.0698868
$430$		0			0
$431$	−405.559			−0.940973			−0.470486	−	0.882407i	$-0.655922\pi$
$431$	−405.559			−0.940973			−0.470486	+	0.882407i	$0.655922\pi$
$432$		0			0
$433$	−145.764	+	145.764i	−0.336638	+	0.336638i	−0.855100	−	0.518462i	$-0.826505\pi$
$433$	−145.764	+	145.764i	−0.336638	+	0.336638i	0.518462	+	0.855100i	$0.326505\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	455.553	−	455.553i	1.04245	−	1.04245i
$438$		0			0
$439$		−	62.3888i		−	0.142116i	−0.997472	−	0.0710579i	$-0.977362\pi$
$439$		−	62.3888i		−	0.142116i	0.997472	−	0.0710579i	$-0.0226375\pi$
$440$		0			0
$441$	−393.583			−0.892479
$442$		0			0
$443$	169.632	+	169.632i	0.382918	+	0.382918i	0.872152	−	0.489235i	$-0.162724\pi$
$443$	169.632	+	169.632i	0.382918	+	0.382918i	−0.489235	+	0.872152i	$0.662724\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	6.45431	+	6.45431i	0.0144392	+	0.0144392i
$448$		0			0
$449$			17.0056i			0.0378743i	0.999821	+	0.0189371i	$0.00602824\pi$
$449$			17.0056i			0.0378743i	−0.999821	+	0.0189371i	$0.993972\pi$
$450$		0			0
$451$			448.452i			0.994351i
$452$		0			0
$453$	2.89151	+	2.89151i	0.00638303	+	0.00638303i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−559.163	−	559.163i	−1.22355	−	1.22355i	−0.966360	−	0.257191i	$-0.917203\pi$
$457$	−559.163	−	559.163i	−1.22355	−	1.22355i	−0.257191	−	0.966360i	$-0.582797\pi$
$458$		0			0
$459$	−4.02886			−0.00877747
$460$		0			0
$461$			327.625i			0.710683i	0.934737	+	0.355341i	$0.115635\pi$
$461$			327.625i			0.710683i	−0.934737	+	0.355341i	$0.884365\pi$
$462$		0			0
$463$	605.934	−	605.934i	1.30871	−	1.30871i	0.386367	−	0.922345i	$-0.373730\pi$
$463$	605.934	−	605.934i	1.30871	−	1.30871i	0.922345	−	0.386367i	$-0.126270\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	83.9231	−	83.9231i	0.179707	−	0.179707i	−0.611521	−	0.791228i	$-0.709442\pi$
$467$	83.9231	−	83.9231i	0.179707	−	0.179707i	0.791228	+	0.611521i	$0.209442\pi$
$468$		0			0
$469$	−174.422			−0.371903
$470$		0			0
$471$	16.9313			0.0359476
$472$		0			0
$473$	−588.298	+	588.298i	−1.24376	+	1.24376i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−258.217	+	258.217i	−0.541335	+	0.541335i
$478$		0			0
$479$		−	831.992i		−	1.73694i	−0.495746	−	0.868468i	$-0.665105\pi$
$479$		−	831.992i		−	1.73694i	0.495746	−	0.868468i	$-0.334895\pi$
$480$		0			0
$481$	−30.1601			−0.0627028
$482$		0			0
$483$	9.25640	+	9.25640i	0.0191644	+	0.0191644i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−373.676	−	373.676i	−0.767301	−	0.767301i	0.210329	−	0.977631i	$-0.432546\pi$
$487$	−373.676	−	373.676i	−0.767301	−	0.767301i	−0.977631	+	0.210329i	$0.932546\pi$
$488$		0			0
$489$			6.74590i			0.0137953i
$490$		0			0
$491$			42.8051i			0.0871795i	0.999050	+	0.0435898i	$0.0138794\pi$
$491$			42.8051i			0.0871795i	−0.999050	+	0.0435898i	$0.986121\pi$
$492$		0			0
$493$	−23.4386	−	23.4386i	−0.0475427	−	0.0475427i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	94.5819	+	94.5819i	0.190306	+	0.190306i
$498$		0			0
$499$	485.100			0.972143			0.486072	−	0.873919i	$-0.338429\pi$
$499$	485.100			0.972143			0.486072	+	0.873919i	$0.338429\pi$
$500$		0			0
$501$		−	14.7730i		−	0.0294870i
$502$		0			0
$503$	−666.926	+	666.926i	−1.32590	+	1.32590i	−0.416980	+	0.908916i	$0.636912\pi$
$503$	−666.926	+	666.926i	−1.32590	+	1.32590i	−0.908916	+	0.416980i	$0.863088\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.98935	+	1.98935i	−0.00392376	+	0.00392376i
$508$		0			0
$509$	−779.971			−1.53236			−0.766180	−	0.642626i	$-0.777845\pi$
$509$	−779.971			−1.53236			−0.766180	+	0.642626i	$0.777845\pi$
$510$		0			0
$511$	108.964			0.213237
$512$		0			0
$513$	−48.3138	+	48.3138i	−0.0941790	+	0.0941790i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−407.097	+	407.097i	−0.787422	+	0.787422i
$518$		0			0
$519$		−	35.4435i		−	0.0682919i
$520$		0			0
$521$	−736.972			−1.41453			−0.707267	−	0.706947i	$-0.750072\pi$
$521$	−736.972			−1.41453			−0.707267	+	0.706947i	$0.750072\pi$
$522$		0			0
$523$	671.690	+	671.690i	1.28430	+	1.28430i	0.938195	+	0.346108i	$0.112497\pi$
$523$	671.690	+	671.690i	1.28430	+	1.28430i	0.346108	+	0.938195i	$0.387503\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	34.3144	+	34.3144i	0.0651127	+	0.0651127i
$528$		0			0
$529$		−	452.780i		−	0.855917i
$530$		0			0
$531$		−	831.192i		−	1.56533i
$532$		0			0
$533$	−360.377	−	360.377i	−0.676129	−	0.676129i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.02773	−	2.02773i	−0.00377604	−	0.00377604i
$538$		0			0
$539$	−524.263			−0.972659
$540$		0			0
$541$		−	1039.61i		−	1.92164i	−0.277168	−	0.960822i	$-0.589396\pi$
$541$		−	1039.61i		−	1.92164i	0.277168	−	0.960822i	$-0.410604\pi$
$542$		0			0
$543$	−34.4860	+	34.4860i	−0.0635101	+	0.0635101i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	238.506	−	238.506i	0.436026	−	0.436026i	−0.454646	−	0.890672i	$-0.650234\pi$
$547$	238.506	−	238.506i	0.436026	−	0.436026i	0.890672	+	0.454646i	$0.150234\pi$
$548$		0			0
$549$	−43.2705			−0.0788169
$550$		0			0
$551$	−562.148			−1.02023
$552$		0			0
$553$	−153.717	+	153.717i	−0.277969	+	0.277969i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	263.387	−	263.387i	0.472867	−	0.472867i	−0.429974	−	0.902841i	$-0.641477\pi$
$557$	263.387	−	263.387i	0.472867	−	0.472867i	0.902841	+	0.429974i	$0.141477\pi$
$558$		0			0
$559$		−	945.513i		−	1.69144i
$560$		0			0
$561$	−2.67816			−0.00477391
$562$		0			0
$563$	160.698	+	160.698i	0.285431	+	0.285431i	0.835270	−	0.549839i	$-0.185311\pi$
$563$	160.698	+	160.698i	0.285431	+	0.285431i	−0.549839	+	0.835270i	$0.685311\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	127.894	+	127.894i	0.225563	+	0.225563i
$568$		0			0
$569$			407.195i			0.715633i	0.933792	+	0.357817i	$0.116479\pi$
$569$			407.195i			0.715633i	−0.933792	+	0.357817i	$0.883521\pi$
$570$		0			0
$571$			131.146i			0.229678i	0.993384	+	0.114839i	$0.0366351\pi$
$571$			131.146i			0.229678i	−0.993384	+	0.114839i	$0.963365\pi$
$572$		0			0
$573$	15.7883	+	15.7883i	0.0275538	+	0.0275538i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−417.933	−	417.933i	−0.724321	−	0.724321i	0.245162	−	0.969482i	$-0.421159\pi$
$577$	−417.933	−	417.933i	−0.724321	−	0.724321i	−0.969482	+	0.245162i	$0.921159\pi$
$578$		0			0
$579$	47.7137			0.0824071
$580$		0			0
$581$		−	203.462i		−	0.350192i
$582$		0			0
$583$	−343.952	+	343.952i	−0.589969	+	0.589969i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	212.882	−	212.882i	0.362660	−	0.362660i	−0.502131	−	0.864791i	$-0.667451\pi$
$587$	212.882	−	212.882i	0.362660	−	0.362660i	0.864791	+	0.502131i	$0.167451\pi$
$588$		0			0
$589$	822.992			1.39727
$590$		0			0
$591$	−58.6697			−0.0992720
$592$		0			0
$593$	−97.2530	+	97.2530i	−0.164002	+	0.164002i	−0.784337	−	0.620335i	$-0.786997\pi$
$593$	−97.2530	+	97.2530i	−0.164002	+	0.164002i	0.620335	+	0.784337i	$0.286997\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−11.4977	+	11.4977i	−0.0192592	+	0.0192592i
$598$		0			0
$599$			712.479i			1.18945i	0.803930	+	0.594723i	$0.202738\pi$
$599$			712.479i			1.18945i	−0.803930	+	0.594723i	$0.797262\pi$
$600$		0			0
$601$	217.550			0.361980			0.180990	−	0.983485i	$-0.442070\pi$
$601$	217.550			0.361980			0.180990	+	0.983485i	$0.442070\pi$
$602$		0			0
$603$	−489.575	−	489.575i	−0.811899	−	0.811899i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	12.6560	+	12.6560i	0.0208501	+	0.0208501i	0.717455	−	0.696605i	$-0.245307\pi$
$607$	12.6560	+	12.6560i	0.0208501	+	0.0208501i	−0.696605	+	0.717455i	$0.745307\pi$
$608$		0			0
$609$		−	11.4223i		−	0.0187558i
$610$		0			0
$611$		−	654.287i		−	1.07085i
$612$		0			0
$613$	523.841	+	523.841i	0.854552	+	0.854552i	0.990690	−	0.136138i	$-0.0434689\pi$
$613$	523.841	+	523.841i	0.854552	+	0.854552i	−0.136138	+	0.990690i	$0.543469\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	331.035	+	331.035i	0.536523	+	0.536523i	0.922506	−	0.385983i	$-0.126138\pi$
$617$	331.035	+	331.035i	0.536523	+	0.536523i	−0.385983	+	0.922506i	$0.626138\pi$
$618$		0			0
$619$	277.907			0.448961			0.224481	−	0.974479i	$-0.427931\pi$
$619$	277.907			0.448961			0.224481	+	0.974479i	$0.427931\pi$
$620$		0			0
$621$			104.123i			0.167670i
$622$		0			0
$623$	5.88393	−	5.88393i	0.00944451	−	0.00944451i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−32.1164	+	32.1164i	−0.0512223	+	0.0512223i
$628$		0			0
$629$	2.69412			0.00428318
$630$		0			0
$631$	−425.237			−0.673910			−0.336955	−	0.941521i	$-0.609397\pi$
$631$	−425.237			−0.673910			−0.336955	+	0.941521i	$0.609397\pi$
$632$		0			0
$633$	−15.5794	+	15.5794i	−0.0246121	+	0.0246121i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	421.298	−	421.298i	0.661379	−	0.661379i
$638$		0			0
$639$			530.952i			0.830911i
$640$		0			0
$641$	−474.056			−0.739558			−0.369779	−	0.929120i	$-0.620567\pi$
$641$	−474.056			−0.739558			−0.369779	+	0.929120i	$0.620567\pi$
$642$		0			0
$643$	−490.408	−	490.408i	−0.762687	−	0.762687i	0.214121	−	0.976807i	$-0.431311\pi$
$643$	−490.408	−	490.408i	−0.762687	−	0.762687i	−0.976807	+	0.214121i	$0.931311\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−319.187	−	319.187i	−0.493334	−	0.493334i	0.416021	−	0.909355i	$-0.363424\pi$
$647$	−319.187	−	319.187i	−0.493334	−	0.493334i	−0.909355	+	0.416021i	$0.863424\pi$
$648$		0			0
$649$		−	1107.17i		−	1.70596i
$650$		0			0
$651$			16.7224i			0.0256873i
$652$		0			0
$653$	−95.6351	−	95.6351i	−0.146455	−	0.146455i	0.630077	−	0.776532i	$-0.283023\pi$
$653$	−95.6351	−	95.6351i	−0.146455	−	0.146455i	−0.776532	+	0.630077i	$0.783023\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	305.845	+	305.845i	0.465517	+	0.465517i
$658$		0			0
$659$	−295.703			−0.448715			−0.224358	−	0.974507i	$-0.572028\pi$
$659$	−295.703			−0.448715			−0.224358	+	0.974507i	$0.572028\pi$
$660$		0			0
$661$			1223.87i			1.85154i	0.378088	+	0.925770i	$0.376582\pi$
$661$			1223.87i			1.85154i	−0.378088	+	0.925770i	$0.623418\pi$
$662$		0			0
$663$	2.15217	−	2.15217i	0.00324611	−	0.00324611i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−605.754	+	605.754i	−0.908177	+	0.908177i
$668$		0			0
$669$	29.2955			0.0437900
$670$		0			0
$671$	−57.6374			−0.0858978
$672$		0			0
$673$	95.3663	−	95.3663i	0.141703	−	0.141703i	−0.632697	−	0.774400i	$-0.718052\pi$
$673$	95.3663	−	95.3663i	0.141703	−	0.141703i	0.774400	+	0.632697i	$0.218052\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−208.006	+	208.006i	−0.307247	+	0.307247i	−0.843841	−	0.536594i	$-0.819711\pi$
$677$	−208.006	+	208.006i	−0.307247	+	0.307247i	0.536594	+	0.843841i	$0.319711\pi$
$678$		0			0
$679$			147.067i			0.216594i
$680$		0			0
$681$	22.4657			0.0329892
$682$		0			0
$683$	−704.881	−	704.881i	−1.03204	−	1.03204i	−0.999470	−	0.0325668i	$-0.989632\pi$
$683$	−704.881	−	704.881i	−1.03204	−	1.03204i	−0.0325668	−	0.999470i	$-0.510368\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−3.46801	−	3.46801i	−0.00504805	−	0.00504805i
$688$		0			0
$689$		−	552.800i		−	0.802321i
$690$		0			0
$691$			814.437i			1.17864i	0.807901	+	0.589318i	$0.200603\pi$
$691$			814.437i			1.17864i	−0.807901	+	0.589318i	$0.799397\pi$
$692$		0			0
$693$	171.014	+	171.014i	0.246773	+	0.246773i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	32.1915	+	32.1915i	0.0461858	+	0.0461858i
$698$		0			0
$699$	18.7442			0.0268158
$700$		0			0
$701$			39.4607i			0.0562920i	0.999604	+	0.0281460i	$0.00896033\pi$
$701$			39.4607i			0.0562920i	−0.999604	+	0.0281460i	$0.991040\pi$
$702$		0			0
$703$	32.3078	−	32.3078i	0.0459570	−	0.0459570i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−172.767	+	172.767i	−0.244366	+	0.244366i
$708$		0			0
$709$	1177.86			1.66130			0.830648	−	0.556798i	$-0.187970\pi$
$709$	1177.86			1.66130			0.830648	+	0.556798i	$0.187970\pi$
$710$		0			0
$711$	−862.917			−1.21367
$712$		0			0
$713$	886.832	−	886.832i	1.24380	−	1.24380i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−15.0297	+	15.0297i	−0.0209619	+	0.0209619i
$718$		0			0
$719$		−	242.835i		−	0.337740i	−0.985638	−	0.168870i	$-0.945988\pi$
$719$		−	242.835i		−	0.337740i	0.985638	−	0.168870i	$-0.0540119\pi$
$720$		0			0
$721$	162.624			0.225553
$722$		0			0
$723$	28.6537	+	28.6537i	0.0396317	+	0.0396317i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−619.622	−	619.622i	−0.852300	−	0.852300i	0.138116	−	0.990416i	$-0.455895\pi$
$727$	−619.622	−	619.622i	−0.852300	−	0.852300i	−0.990416	+	0.138116i	$0.955895\pi$
$728$		0			0
$729$			712.425i			0.977264i
$730$		0			0
$731$			84.4603i			0.115541i
$732$		0			0
$733$	−281.127	−	281.127i	−0.383529	−	0.383529i	0.488843	−	0.872372i	$-0.337419\pi$
$733$	−281.127	−	281.127i	−0.383529	−	0.383529i	−0.872372	+	0.488843i	$0.837419\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−652.127	−	652.127i	−0.884841	−	0.884841i
$738$		0			0
$739$	−1109.74			−1.50168			−0.750839	−	0.660485i	$-0.770351\pi$
$739$	−1109.74			−1.50168			−0.750839	+	0.660485i	$0.770351\pi$
$740$		0			0
$741$		−	51.6175i		−	0.0696592i
$742$		0			0
$743$	−339.850	+	339.850i	−0.457403	+	0.457403i	−0.897802	−	0.440399i	$-0.854837\pi$
$743$	−339.850	+	339.850i	−0.457403	+	0.457403i	0.440399	+	0.897802i	$0.354837\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	571.084	−	571.084i	0.764504	−	0.764504i
$748$		0			0
$749$	304.780			0.406916
$750$		0			0
$751$	980.557			1.30567			0.652834	−	0.757501i	$-0.273580\pi$
$751$	980.557			1.30567			0.652834	+	0.757501i	$0.273580\pi$
$752$		0			0
$753$	12.4324	−	12.4324i	0.0165105	−	0.0165105i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	101.564	−	101.564i	0.134166	−	0.134166i	−0.636834	−	0.771001i	$-0.719757\pi$
$757$	101.564	−	101.564i	0.134166	−	0.134166i	0.771001	+	0.636834i	$0.219757\pi$
$758$		0			0
$759$			69.2154i			0.0911929i
$760$		0			0
$761$	767.675			1.00877			0.504386	−	0.863478i	$-0.331719\pi$
$761$	767.675			1.00877			0.504386	+	0.863478i	$0.331719\pi$
$762$		0			0
$763$	74.3357	+	74.3357i	0.0974256	+	0.0974256i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	889.722	+	889.722i	1.16000	+	1.16000i
$768$		0			0
$769$			331.840i			0.431522i	0.976446	+	0.215761i	$0.0692232\pi$
$769$			331.840i			0.431522i	−0.976446	+	0.215761i	$0.930777\pi$
$770$		0			0
$771$			65.1675i			0.0845234i
$772$		0			0
$773$	180.328	+	180.328i	0.233283	+	0.233283i	0.814061	−	0.580779i	$-0.197252\pi$
$773$	180.328	+	180.328i	0.233283	+	0.233283i	−0.580779	+	0.814061i	$0.697252\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	0.656463	+	0.656463i	0.000844869	+	0.000844869i
$778$		0			0
$779$	772.078			0.991114
$780$		0			0
$781$			707.243i			0.905561i
$782$		0			0
$783$	64.2435	−	64.2435i	0.0820479	−	0.0820479i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	945.029	−	945.029i	1.20080	−	1.20080i	0.226876	−	0.973924i	$-0.427149\pi$
$787$	945.029	−	945.029i	1.20080	−	1.20080i	0.973924	−	0.226876i	$-0.0728513\pi$
$788$		0			0
$789$	−22.3940			−0.0283827
$790$		0			0
$791$	281.223			0.355529
$792$		0			0
$793$	46.3175	−	46.3175i	0.0584079	−	0.0584079i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−680.774	+	680.774i	−0.854171	+	0.854171i	−0.990644	−	0.136473i	$-0.956423\pi$
$797$	−680.774	+	680.774i	−0.854171	+	0.854171i	0.136473	+	0.990644i	$0.456423\pi$
$798$		0			0
$799$			58.4458i			0.0731487i
$800$		0			0
$801$	33.0305			0.0412366
$802$		0			0
$803$	407.393	+	407.393i	0.507339	+	0.507339i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−25.1275	−	25.1275i	−0.0311369	−	0.0311369i
$808$		0			0
$809$			427.952i			0.528989i	0.964387	+	0.264495i	$0.0852051\pi$
$809$			427.952i			0.528989i	−0.964387	+	0.264495i	$0.914795\pi$
$810$		0			0
$811$		−	1222.46i		−	1.50735i	−0.657245	−	0.753677i	$-0.728278\pi$
$811$		−	1222.46i		−	1.50735i	0.657245	−	0.753677i	$-0.271722\pi$
$812$		0			0
$813$	63.5359	+	63.5359i	0.0781499	+	0.0781499i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1012.84	+	1012.84i	1.23971	+	1.23971i
$818$		0			0
$819$	−274.853			−0.335596
$820$		0			0
$821$		−	1019.26i		−	1.24148i	−0.784015	−	0.620742i	$-0.786831\pi$
$821$		−	1019.26i		−	1.24148i	0.784015	−	0.620742i	$-0.213169\pi$
$822$		0			0
$823$	−158.477	+	158.477i	−0.192560	+	0.192560i	−0.796801	−	0.604241i	$-0.793476\pi$
$823$	−158.477	+	158.477i	−0.192560	+	0.192560i	0.604241	+	0.796801i	$0.293476\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−573.534	+	573.534i	−0.693511	+	0.693511i	−0.963003	−	0.269492i	$-0.913144\pi$
$827$	−573.534	+	573.534i	−0.693511	+	0.693511i	0.269492	+	0.963003i	$0.413144\pi$
$828$		0			0
$829$	−1133.71			−1.36756			−0.683782	−	0.729686i	$-0.739666\pi$
$829$	−1133.71			−1.36756			−0.683782	+	0.729686i	$0.739666\pi$
$830$		0			0
$831$	23.2430			0.0279699
$832$		0			0
$833$	−37.6335	+	37.6335i	−0.0451783	+	0.0451783i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−94.0534	+	94.0534i	−0.112370	+	0.112370i
$838$		0			0
$839$			1109.49i			1.32240i	0.750212	+	0.661198i	$0.229952\pi$
$839$			1109.49i			1.32240i	−0.750212	+	0.661198i	$0.770048\pi$
$840$		0			0
$841$	−93.5052			−0.111183
$842$		0			0
$843$	51.0094	+	51.0094i	0.0605094	+	0.0605094i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	34.5418	+	34.5418i	0.0407814	+	0.0407814i
$848$		0			0
$849$			31.6676i			0.0372999i
$850$		0			0
$851$		−	69.6278i		−	0.0818188i
$852$		0			0
$853$	1.51055	+	1.51055i	0.00177087	+	0.00177087i	0.707992	−	0.706221i	$-0.249601\pi$
$853$	1.51055	+	1.51055i	0.00177087	+	0.00177087i	−0.706221	+	0.707992i	$0.749601\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	396.892	+	396.892i	0.463118	+	0.463118i	0.899676	−	0.436558i	$-0.143803\pi$
$857$	396.892	+	396.892i	0.463118	+	0.463118i	−0.436558	+	0.899676i	$0.643803\pi$
$858$		0			0
$859$	941.956			1.09657			0.548286	−	0.836291i	$-0.315280\pi$
$859$	941.956			1.09657			0.548286	+	0.836291i	$0.315280\pi$
$860$		0			0
$861$			15.6879i			0.0182206i
$862$		0			0
$863$	833.800	−	833.800i	0.966164	−	0.966164i	−0.0332818	−	0.999446i	$-0.510596\pi$
$863$	833.800	−	833.800i	0.966164	−	0.966164i	0.999446	+	0.0332818i	$0.0105959\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	37.6064	−	37.6064i	0.0433754	−	0.0433754i
$868$		0			0
$869$	−1149.43			−1.32270
$870$		0			0
$871$	1048.10			1.20333
$872$		0			0
$873$	−412.794	+	412.794i	−0.472846	+	0.472846i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	33.6015	−	33.6015i	0.0383142	−	0.0383142i	−0.687690	−	0.726004i	$-0.741375\pi$
$877$	33.6015	−	33.6015i	0.0383142	−	0.0383142i	0.726004	+	0.687690i	$0.241375\pi$
$878$		0			0
$879$			64.0762i			0.0728967i
$880$		0			0
$881$	1084.55			1.23104			0.615521	−	0.788121i	$-0.288946\pi$
$881$	1084.55			1.23104			0.615521	+	0.788121i	$0.288946\pi$
$882$		0			0
$883$	1165.53	+	1165.53i	1.31997	+	1.31997i	0.913796	+	0.406174i	$0.133137\pi$
$883$	1165.53	+	1165.53i	1.31997	+	1.31997i	0.406174	+	0.913796i	$0.366863\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−381.705	−	381.705i	−0.430333	−	0.430333i	0.458409	−	0.888742i	$-0.348420\pi$
$887$	−381.705	−	381.705i	−0.430333	−	0.430333i	−0.888742	+	0.458409i	$0.848420\pi$
$888$		0			0
$889$		−	141.584i		−	0.159263i
$890$		0			0
$891$			956.338i			1.07333i
$892$		0			0
$893$	700.879	+	700.879i	0.784859	+	0.784859i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−55.6215	−	55.6215i	−0.0620084	−	0.0620084i
$898$		0			0
$899$	−1094.34			−1.21729
$900$		0			0
$901$			49.3802i			0.0548060i
$902$		0			0
$903$	−20.5800	+	20.5800i	−0.0227907	+	0.0227907i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−457.511	+	457.511i	−0.504422	+	0.504422i	−0.912809	−	0.408387i	$-0.866092\pi$
$907$	−457.511	+	457.511i	−0.504422	+	0.504422i	0.408387	+	0.912809i	$0.366092\pi$
$908$		0			0
$909$	−969.856			−1.06695
$910$		0			0
$911$	620.530			0.681152			0.340576	−	0.940217i	$-0.389378\pi$
$911$	620.530			0.681152			0.340576	+	0.940217i	$0.389378\pi$
$912$		0			0
$913$	760.699	−	760.699i	0.833187	−	0.833187i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−132.247	+	132.247i	−0.144216	+	0.144216i
$918$		0			0
$919$		−	981.064i		−	1.06753i	−0.845631	−	0.533767i	$-0.820776\pi$
$919$		−	981.064i		−	1.06753i	0.845631	−	0.533767i	$-0.179224\pi$
$920$		0			0
$921$	−35.3985			−0.0384349
$922$		0			0
$923$	−568.341	−	568.341i	−0.615754	−	0.615754i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	456.459	+	456.459i	0.492405	+	0.492405i
$928$		0			0
$929$			776.484i			0.835827i	0.908487	+	0.417914i	$0.137239\pi$
$929$			776.484i			0.835827i	−0.908487	+	0.417914i	$0.862761\pi$
$930$		0			0
$931$			902.598i			0.969493i
$932$		0			0
$933$	−29.5525	−	29.5525i	−0.0316747	−	0.0316747i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−759.800	−	759.800i	−0.810885	−	0.810885i	0.173881	−	0.984767i	$-0.444369\pi$
$937$	−759.800	−	759.800i	−0.810885	−	0.810885i	−0.984767	+	0.173881i	$0.944369\pi$
$938$		0			0
$939$	60.3267			0.0642457
$940$		0			0
$941$		−	759.500i		−	0.807120i	−0.914953	−	0.403560i	$-0.867773\pi$
$941$		−	759.500i		−	0.807120i	0.914953	−	0.403560i	$-0.132227\pi$
$942$		0			0
$943$	831.969	−	831.969i	0.882257	−	0.882257i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−822.618	+	822.618i	−0.868657	+	0.868657i	−0.992324	−	0.123667i	$-0.960535\pi$
$947$	−822.618	+	822.618i	−0.868657	+	0.868657i	0.123667	+	0.992324i	$0.460535\pi$
$948$		0			0
$949$	−654.763			−0.689951
$950$		0			0
$951$	−57.9572			−0.0609434
$952$		0			0
$953$	−659.139	+	659.139i	−0.691647	+	0.691647i	−0.962594	−	0.270947i	$-0.912663\pi$
$953$	−659.139	+	659.139i	−0.691647	+	0.691647i	0.270947	+	0.962594i	$0.412663\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	42.7056	−	42.7056i	0.0446244	−	0.0446244i
$958$		0			0
$959$			79.8661i			0.0832806i
$960$		0			0
$961$	641.133			0.667152
$962$		0			0
$963$	855.468	+	855.468i	0.888336	+	0.888336i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	332.005	+	332.005i	0.343335	+	0.343335i	0.857620	−	0.514284i	$-0.171942\pi$
$967$	332.005	+	332.005i	0.343335	+	0.343335i	−0.514284	+	0.857620i	$0.671942\pi$
$968$		0			0
$969$			4.61086i			0.00475837i
$970$		0			0
$971$			960.961i			0.989661i	0.868989	+	0.494831i	$0.164770\pi$
$971$			960.961i			0.989661i	−0.868989	+	0.494831i	$0.835230\pi$
$972$		0			0
$973$	−167.194	−	167.194i	−0.171834	−	0.171834i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−69.4628	−	69.4628i	−0.0710980	−	0.0710980i	0.670664	−	0.741762i	$-0.266009\pi$
$977$	−69.4628	−	69.4628i	−0.0710980	−	0.0710980i	−0.741762	+	0.670664i	$0.766009\pi$
$978$		0			0
$979$	43.9975			0.0449413
$980$		0			0
$981$			417.297i			0.425379i
$982$		0			0
$983$	234.943	−	234.943i	0.239006	−	0.239006i	−0.577433	−	0.816438i	$-0.695945\pi$
$983$	234.943	−	234.943i	0.239006	−	0.239006i	0.816438	+	0.577433i	$0.195945\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−14.2412	+	14.2412i	−0.0144288	+	0.0144288i
$988$		0			0
$989$	2182.82			2.20710
$990$		0			0
$991$	919.592			0.927944			0.463972	−	0.885850i	$-0.346424\pi$
$991$	919.592			0.927944			0.463972	+	0.885850i	$0.346424\pi$
$992$		0			0
$993$	18.4694	−	18.4694i	0.0185996	−	0.0185996i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	835.935	−	835.935i	0.838450	−	0.838450i	−0.150205	−	0.988655i	$-0.547993\pi$
$997$	835.935	−	835.935i	0.838450	−	0.838450i	0.988655	+	0.150205i	$0.0479932\pi$
$998$		0			0
$999$			7.38441i			0.00739180i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.3.m.b.657.6		20
4.3	odd	2		200.3.i.b.157.2		20
5.2	odd	4		160.3.m.a.113.6		20
5.3	odd	4	inner	800.3.m.b.593.5		20
5.4	even	2		160.3.m.a.17.5		20
8.3	odd	2		200.3.i.b.157.4		20
8.5	even	2	inner	800.3.m.b.657.5		20
20.3	even	4		200.3.i.b.93.4		20
20.7	even	4		40.3.i.a.13.7	✓	20
20.19	odd	2		40.3.i.a.37.9	yes	20
40.3	even	4		200.3.i.b.93.2		20
40.13	odd	4	inner	800.3.m.b.593.6		20
40.19	odd	2		40.3.i.a.37.7	yes	20
40.27	even	4		40.3.i.a.13.9	yes	20
40.29	even	2		160.3.m.a.17.6		20
40.37	odd	4		160.3.m.a.113.5		20
60.47	odd	4		360.3.u.b.253.4		20
60.59	even	2		360.3.u.b.37.2		20
120.59	even	2		360.3.u.b.37.4		20
120.107	odd	4		360.3.u.b.253.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.i.a.13.7	✓	20	20.7	even	4
40.3.i.a.13.9	yes	20	40.27	even	4
40.3.i.a.37.7	yes	20	40.19	odd	2
40.3.i.a.37.9	yes	20	20.19	odd	2
160.3.m.a.17.5		20	5.4	even	2
160.3.m.a.17.6		20	40.29	even	2
160.3.m.a.113.5		20	40.37	odd	4
160.3.m.a.113.6		20	5.2	odd	4
200.3.i.b.93.2		20	40.3	even	4
200.3.i.b.93.4		20	20.3	even	4
200.3.i.b.157.2		20	4.3	odd	2
200.3.i.b.157.4		20	8.3	odd	2
360.3.u.b.37.2		20	60.59	even	2
360.3.u.b.37.4		20	120.59	even	2
360.3.u.b.253.2		20	120.107	odd	4
360.3.u.b.253.4		20	60.47	odd	4
800.3.m.b.593.5		20	5.3	odd	4	inner
800.3.m.b.593.6		20	40.13	odd	4	inner
800.3.m.b.657.5		20	8.5	even	2	inner
800.3.m.b.657.6		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.130791 + 0.130791i) q^{3} +(-1.59713 - 1.59713i) q^{7} -8.96579i q^{9} -11.9427i q^{11} +(9.59714 + 9.59714i) q^{13} +(-0.857288 - 0.857288i) q^{17} -20.5611 q^{19} -0.417782i q^{21} +(-22.1560 + 22.1560i) q^{23} +(2.34977 - 2.34977i) q^{27} +27.3404 q^{29} -40.0267 q^{31} +(1.56200 - 1.56200i) q^{33} +(-1.57131 + 1.57131i) q^{37} +2.51044i q^{39} -37.5504 q^{41} +(-49.2602 - 49.2602i) q^{43} +(-34.0876 - 34.0876i) q^{47} -43.8983i q^{49} -0.224252i q^{51} +(-28.8002 - 28.8002i) q^{53} +(-2.68921 - 2.68921i) q^{57} +92.7071 q^{59} -4.82618i q^{61} +(-14.3196 + 14.3196i) q^{63} +(54.6048 - 54.6048i) q^{67} -5.79564 q^{69} -59.2198 q^{71} +(-34.1124 + 34.1124i) q^{73} +(-19.0740 + 19.0740i) q^{77} -96.2455i q^{79} -80.0774 q^{81} +(63.6959 + 63.6959i) q^{83} +(3.57588 + 3.57588i) q^{87} +3.68406i q^{89} -30.6558i q^{91} +(-5.23514 - 5.23514i) q^{93} +(-46.0410 - 46.0410i) q^{97} -107.075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{7} + 12 q^{17} - 4 q^{23} + 136 q^{31} - 32 q^{33} - 8 q^{41} + 188 q^{47} + 40 q^{57} + 228 q^{63} - 248 q^{71} + 124 q^{73} + 132 q^{81} - 488 q^{87} - 100 q^{97}+O(q^{100}) \) 20 * q - 4 * q^7 + 12 * q^17 - 4 * q^23 + 136 * q^31 - 32 * q^33 - 8 * q^41 + 188 * q^47 + 40 * q^57 + 228 * q^63 - 248 * q^71 + 124 * q^73 + 132 * q^81 - 488 * q^87 - 100 * q^97

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