LMFDB - Embedded newform 800.3.p.n.257.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,3,Mod(193,800)]

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, 0, 3]))

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

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

chi := DirichletCharacter("800.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	800.p (of order $4$, degree $2$, 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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.151613669376.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 256 $ x^8 - 7*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			257.2
Root			$0.662382 + 1.88713i$ of defining polynomial
Character	$\chi$	$=$	800.257
Dual form			800.3.p.n.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.54951 + 1.54951i) q^{3} +(-2.00000 - 2.00000i) q^{7} +4.19804i q^{9} +1.10347 q^{11} +(-10.0405 + 10.0405i) q^{13} +(-3.91678 - 3.91678i) q^{17} -23.3914i q^{19} +6.19804 q^{21} +(5.29706 - 5.29706i) q^{23} +(-20.4505 - 20.4505i) q^{27} +32.5941i q^{29} +42.3690 q^{31} +(-1.70984 + 1.70984i) q^{33} +(-22.2880 - 22.2880i) q^{37} -31.1157i q^{39} -15.0000 q^{41} +(-32.5941 + 32.5941i) q^{43} +(-55.2971 - 55.2971i) q^{47} -41.0000i q^{49} +12.1382 q^{51} +(-16.6613 + 16.6613i) q^{53} +(36.2452 + 36.2452i) q^{57} -111.440i q^{59} +5.40588 q^{61} +(8.39608 - 8.39608i) q^{63} +(-36.0544 - 36.0544i) q^{67} +16.4157i q^{69} +71.2777 q^{71} +(93.0686 - 93.0686i) q^{73} +(-2.20694 - 2.20694i) q^{77} -118.061i q^{79} +25.5941 q^{81} +(-6.64853 + 6.64853i) q^{83} +(-50.5049 - 50.5049i) q^{87} -126.782i q^{89} +40.1620 q^{91} +(-65.6511 + 65.6511i) q^{93} +(73.4847 + 73.4847i) q^{97} +4.63242i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 16 q^{7} - 32 q^{21} - 80 q^{23} - 184 q^{27} - 120 q^{41} - 16 q^{43} - 320 q^{47} + 288 q^{61} - 96 q^{63} - 472 q^{67} - 40 q^{81} + 8 q^{83} - 608 q^{87}+O(q^{100}) $ 8 * q + 8 * q^3 - 16 * q^7 - 32 * q^21 - 80 * q^23 - 184 * q^27 - 120 * q^41 - 16 * q^43 - 320 * q^47 + 288 * q^61 - 96 * q^63 - 472 * q^67 - 40 * q^81 + 8 * q^83 - 608 * q^87

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
$3$	−1.54951	+	1.54951i	−0.516503	+	0.516503i	−0.916512	−	0.400008i	$-0.869007\pi$
$3$	−1.54951	+	1.54951i	−0.516503	+	0.516503i	0.400008	+	0.916512i	$0.369007\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.00000	−	2.00000i	−0.285714	−	0.285714i	0.549669	−	0.835383i	$-0.314754\pi$
$7$	−2.00000	−	2.00000i	−0.285714	−	0.285714i	−0.835383	+	0.549669i	$0.814754\pi$
$8$		0			0
$9$			4.19804i			0.466449i
$10$		0			0
$11$	1.10347			0.100316			0.0501578	−	0.998741i	$-0.484028\pi$
$11$	1.10347			0.100316			0.0501578	+	0.998741i	$0.484028\pi$
$12$		0			0
$13$	−10.0405	+	10.0405i	−0.772347	+	0.772347i	−0.978516	−	0.206170i	$-0.933900\pi$
$13$	−10.0405	+	10.0405i	−0.772347	+	0.772347i	0.206170	+	0.978516i	$0.433900\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.91678	−	3.91678i	−0.230399	−	0.230399i	0.582460	−	0.812859i	$-0.302090\pi$
$17$	−3.91678	−	3.91678i	−0.230399	−	0.230399i	−0.812859	+	0.582460i	$0.802090\pi$
$18$		0			0
$19$		−	23.3914i		−	1.23113i	−0.788087	−	0.615564i	$-0.788928\pi$
$19$		−	23.3914i		−	1.23113i	0.788087	−	0.615564i	$-0.211072\pi$
$20$		0			0
$21$	6.19804			0.295145
$22$		0			0
$23$	5.29706	−	5.29706i	0.230307	−	0.230307i	−0.582514	−	0.812821i	$-0.697931\pi$
$23$	5.29706	−	5.29706i	0.230307	−	0.230307i	0.812821	+	0.582514i	$0.197931\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−20.4505	−	20.4505i	−0.757426	−	0.757426i
$28$		0			0
$29$			32.5941i			1.12394i	0.827159	+	0.561968i	$0.189955\pi$
$29$			32.5941i			1.12394i	−0.827159	+	0.561968i	$0.810045\pi$
$30$		0			0
$31$	42.3690			1.36674			0.683370	−	0.730072i	$-0.260513\pi$
$31$	42.3690			1.36674			0.683370	+	0.730072i	$0.260513\pi$
$32$		0			0
$33$	−1.70984	+	1.70984i	−0.0518133	+	0.0518133i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−22.2880	−	22.2880i	−0.602377	−	0.602377i	0.338566	−	0.940943i	$-0.390058\pi$
$37$	−22.2880	−	22.2880i	−0.602377	−	0.602377i	−0.940943	+	0.338566i	$0.890058\pi$
$38$		0			0
$39$		−	31.1157i		−	0.797839i
$40$		0			0
$41$	−15.0000			−0.365854			−0.182927	−	0.983127i	$-0.558557\pi$
$41$	−15.0000			−0.365854			−0.182927	+	0.983127i	$0.558557\pi$
$42$		0			0
$43$	−32.5941	+	32.5941i	−0.758003	+	0.758003i	−0.975959	−	0.217956i	$-0.930061\pi$
$43$	−32.5941	+	32.5941i	−0.758003	+	0.758003i	0.217956	+	0.975959i	$0.430061\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−55.2971	−	55.2971i	−1.17653	−	1.17653i	−0.980622	−	0.195912i	$-0.937233\pi$
$47$	−55.2971	−	55.2971i	−1.17653	−	1.17653i	−0.195912	−	0.980622i	$-0.562767\pi$
$48$		0			0
$49$		−	41.0000i		−	0.836735i
$50$		0			0
$51$	12.1382			0.238004
$52$		0			0
$53$	−16.6613	+	16.6613i	−0.314365	+	0.314365i	−0.846598	−	0.532233i	$-0.821353\pi$
$53$	−16.6613	+	16.6613i	−0.314365	+	0.314365i	0.532233	+	0.846598i	$0.321353\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	36.2452	+	36.2452i	0.635881	+	0.635881i
$58$		0			0
$59$		−	111.440i		−	1.88881i	−0.328785	−	0.944405i	$-0.606639\pi$
$59$		−	111.440i		−	1.88881i	0.328785	−	0.944405i	$-0.393361\pi$
$60$		0			0
$61$	5.40588			0.0886210			0.0443105	−	0.999018i	$-0.485891\pi$
$61$	5.40588			0.0886210			0.0443105	+	0.999018i	$0.485891\pi$
$62$		0			0
$63$	8.39608	−	8.39608i	0.133271	−	0.133271i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−36.0544	−	36.0544i	−0.538126	−	0.538126i	0.384853	−	0.922978i	$-0.374252\pi$
$67$	−36.0544	−	36.0544i	−0.538126	−	0.538126i	−0.922978	+	0.384853i	$0.874252\pi$
$68$		0			0
$69$			16.4157i			0.237909i
$70$		0			0
$71$	71.2777			1.00391			0.501956	−	0.864893i	$-0.332614\pi$
$71$	71.2777			1.00391			0.501956	+	0.864893i	$0.332614\pi$
$72$		0			0
$73$	93.0686	−	93.0686i	1.27491	−	1.27491i	0.331434	−	0.943478i	$-0.392468\pi$
$73$	93.0686	−	93.0686i	1.27491	−	1.27491i	0.943478	−	0.331434i	$-0.107532\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.20694	−	2.20694i	−0.0286616	−	0.0286616i
$78$		0			0
$79$		−	118.061i		−	1.49444i	−0.664578	−	0.747219i	$-0.731389\pi$
$79$		−	118.061i		−	1.49444i	0.664578	−	0.747219i	$-0.268611\pi$
$80$		0			0
$81$	25.5941			0.315977
$82$		0			0
$83$	−6.64853	+	6.64853i	−0.0801028	+	0.0801028i	−0.746023	−	0.665920i	$-0.768039\pi$
$83$	−6.64853	+	6.64853i	−0.0801028	+	0.0801028i	0.665920	+	0.746023i	$0.268039\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−50.5049	−	50.5049i	−0.580516	−	0.580516i
$88$		0			0
$89$		−	126.782i		−	1.42452i	−0.701915	−	0.712260i	$-0.747672\pi$
$89$		−	126.782i		−	1.42452i	0.701915	−	0.712260i	$-0.252328\pi$
$90$		0			0
$91$	40.1620			0.441341
$92$		0			0
$93$	−65.6511	+	65.6511i	−0.705926	+	0.705926i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	73.4847	+	73.4847i	0.757574	+	0.757574i	0.975880	−	0.218306i	$-0.0700531\pi$
$97$	73.4847	+	73.4847i	0.757574	+	0.757574i	−0.218306	+	0.975880i	$0.570053\pi$
$98$		0			0
$99$			4.63242i			0.0467921i
$100$		0			0
$101$	−22.5941			−0.223704			−0.111852	−	0.993725i	$-0.535678\pi$
$101$	−22.5941			−0.223704			−0.111852	+	0.993725i	$0.535678\pi$
$102$		0			0
$103$	−40.7029	+	40.7029i	−0.395174	+	0.395174i	−0.876527	−	0.481353i	$-0.840146\pi$
$103$	−40.7029	+	40.7029i	−0.395174	+	0.395174i	0.481353	+	0.876527i	$0.340146\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−75.8368	−	75.8368i	−0.708755	−	0.708755i	0.257519	−	0.966273i	$-0.417095\pi$
$107$	−75.8368	−	75.8368i	−0.708755	−	0.708755i	−0.966273	+	0.257519i	$0.917095\pi$
$108$		0			0
$109$			65.4059i			0.600054i	0.953931	+	0.300027i	$0.0969957\pi$
$109$			65.4059i			0.600054i	−0.953931	+	0.300027i	$0.903004\pi$
$110$		0			0
$111$	69.0708			0.622260
$112$		0			0
$113$	95.2755	−	95.2755i	0.843146	−	0.843146i	−0.146120	−	0.989267i	$-0.546679\pi$
$113$	95.2755	−	95.2755i	0.843146	−	0.843146i	0.989267	+	0.146120i	$0.0466787\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−42.1504	−	42.1504i	−0.360260	−	0.360260i
$118$		0			0
$119$			15.6671i			0.131657i
$120$		0			0
$121$	−119.782			−0.989937
$122$		0			0
$123$	23.2426	−	23.2426i	0.188965	−	0.188965i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−23.8912	−	23.8912i	−0.188119	−	0.188119i	0.606763	−	0.794883i	$-0.292468\pi$
$127$	−23.8912	−	23.8912i	−0.188119	−	0.188119i	−0.794883	+	0.606763i	$0.792468\pi$
$128$		0			0
$129$		−	101.010i		−	0.783022i
$130$		0			0
$131$	66.8639			0.510411			0.255206	−	0.966887i	$-0.417857\pi$
$131$	66.8639			0.510411			0.255206	+	0.966887i	$0.417857\pi$
$132$		0			0
$133$	−46.7829	+	46.7829i	−0.351751	+	0.351751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−54.1193	−	54.1193i	−0.395031	−	0.395031i	0.481445	−	0.876476i	$-0.340112\pi$
$137$	−54.1193	−	54.1193i	−0.395031	−	0.395031i	−0.876476	+	0.481445i	$0.840112\pi$
$138$		0			0
$139$		−	83.6345i		−	0.601687i	−0.953674	−	0.300843i	$-0.902732\pi$
$139$		−	83.6345i		−	0.601687i	0.953674	−	0.300843i	$-0.0972681\pi$
$140$		0			0
$141$	171.367			1.21537
$142$		0			0
$143$	−11.0794	+	11.0794i	−0.0774784	+	0.0774784i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	63.5299	+	63.5299i	0.432176	+	0.432176i
$148$		0			0
$149$			242.971i			1.63068i	0.578986	+	0.815338i	$0.303449\pi$
$149$			242.971i			1.63068i	−0.578986	+	0.815338i	$0.696551\pi$
$150$		0			0
$151$	−274.295			−1.81652			−0.908261	−	0.418404i	$-0.862590\pi$
$151$	−274.295			−1.81652			−0.908261	+	0.418404i	$0.862590\pi$
$152$		0			0
$153$	16.4428	−	16.4428i	0.107469	−	0.107469i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−88.1576	−	88.1576i	−0.561513	−	0.561513i	0.368224	−	0.929737i	$-0.379966\pi$
$157$	−88.1576	−	88.1576i	−0.561513	−	0.561513i	−0.929737	+	0.368224i	$0.879966\pi$
$158$		0			0
$159$		−	51.6338i		−	0.324741i
$160$		0			0
$161$	−21.1882			−0.131604
$162$		0			0
$163$	34.4309	−	34.4309i	0.211232	−	0.211232i	−0.593558	−	0.804791i	$-0.702277\pi$
$163$	34.4309	−	34.4309i	0.211232	−	0.211232i	0.804791	+	0.593558i	$0.202277\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	153.674	+	153.674i	0.920201	+	0.920201i	0.997043	−	0.0768425i	$-0.0244839\pi$
$167$	153.674	+	153.674i	0.920201	+	0.920201i	−0.0768425	+	0.997043i	$0.524484\pi$
$168$		0			0
$169$		−	32.6235i		−	0.193039i
$170$		0			0
$171$	98.1981			0.574258
$172$		0			0
$173$	−182.718	+	182.718i	−1.05617	+	1.05617i	−0.0578451	+	0.998326i	$0.518423\pi$
$173$	−182.718	+	182.718i	−1.05617	+	1.05617i	−0.998326	+	0.0578451i	$0.981577\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	172.677	+	172.677i	0.975576	+	0.975576i
$178$		0			0
$179$			241.857i			1.35116i	0.737288	+	0.675578i	$0.236106\pi$
$179$			241.857i			1.35116i	−0.737288	+	0.675578i	$0.763894\pi$
$180$		0			0
$181$	−111.188			−0.614300			−0.307150	−	0.951661i	$-0.599375\pi$
$181$	−111.188			−0.614300			−0.307150	+	0.951661i	$0.599375\pi$
$182$		0			0
$183$	−8.37647	+	8.37647i	−0.0457731	+	0.0457731i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−4.32206	−	4.32206i	−0.0231126	−	0.0231126i
$188$		0			0
$189$			81.8020i			0.432815i
$190$		0			0
$191$	−42.3690			−0.221827			−0.110914	−	0.993830i	$-0.535378\pi$
$191$	−42.3690			−0.221827			−0.110914	+	0.993830i	$0.535378\pi$
$192$		0			0
$193$	−11.7503	+	11.7503i	−0.0608826	+	0.0608826i	−0.736893	−	0.676010i	$-0.763708\pi$
$193$	−11.7503	+	11.7503i	−0.0608826	+	0.0608826i	0.676010	+	0.736893i	$0.263708\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−152.815	−	152.815i	−0.775708	−	0.775708i	0.203390	−	0.979098i	$-0.434804\pi$
$197$	−152.815	−	152.815i	−0.775708	−	0.775708i	−0.979098	+	0.203390i	$0.934804\pi$
$198$		0			0
$199$			209.638i			1.05346i	0.850034	+	0.526728i	$0.176581\pi$
$199$			209.638i			1.05346i	−0.850034	+	0.526728i	$0.823419\pi$
$200$		0			0
$201$	111.733			0.555887
$202$		0			0
$203$	65.1882	−	65.1882i	0.321124	−	0.321124i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	22.2373	+	22.2373i	0.107426	+	0.107426i
$208$		0			0
$209$		−	25.8118i		−	0.123501i
$210$		0			0
$211$	−257.087			−1.21842			−0.609211	−	0.793008i	$-0.708514\pi$
$211$	−257.087			−1.21842			−0.609211	+	0.793008i	$0.708514\pi$
$212$		0			0
$213$	−110.446	+	110.446i	−0.518524	+	0.518524i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−84.7379	−	84.7379i	−0.390497	−	0.390497i
$218$		0			0
$219$			288.421i			1.31699i
$220$		0			0
$221$	78.6529			0.355896
$222$		0			0
$223$	−117.406	+	117.406i	−0.526484	+	0.526484i	−0.919522	−	0.393038i	$-0.871424\pi$
$223$	−117.406	+	117.406i	−0.526484	+	0.526484i	0.393038	+	0.919522i	$0.371424\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	171.188	+	171.188i	0.754133	+	0.754133i	0.975248	−	0.221115i	$-0.0709695\pi$
$227$	171.188	+	171.188i	0.754133	+	0.754133i	−0.221115	+	0.975248i	$0.570969\pi$
$228$		0			0
$229$			139.565i			0.609453i	0.952440	+	0.304726i	$0.0985650\pi$
$229$			139.565i			0.609453i	−0.952440	+	0.304726i	$0.901435\pi$
$230$		0			0
$231$	6.83936			0.0296076
$232$		0			0
$233$	167.050	−	167.050i	0.716954	−	0.716954i	−0.251026	−	0.967980i	$-0.580768\pi$
$233$	167.050	−	167.050i	0.716954	−	0.716954i	0.967980	+	0.251026i	$0.0807679\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	182.936	+	182.936i	0.771882	+	0.771882i
$238$		0			0
$239$			186.913i			0.782062i	0.920377	+	0.391031i	$0.127882\pi$
$239$			186.913i			0.782062i	−0.920377	+	0.391031i	$0.872118\pi$
$240$		0			0
$241$	170.406			0.707078			0.353539	−	0.935420i	$-0.384978\pi$
$241$	170.406			0.707078			0.353539	+	0.935420i	$0.384978\pi$
$242$		0			0
$243$	144.396	−	144.396i	0.594223	−	0.594223i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	234.862	+	234.862i	0.950857	+	0.950857i
$248$		0			0
$249$		−	20.6039i		−	0.0827467i
$250$		0			0
$251$	7.94283			0.0316447			0.0158224	−	0.999875i	$-0.494963\pi$
$251$	7.94283			0.0316447			0.0158224	+	0.999875i	$0.494963\pi$
$252$		0			0
$253$	5.84515	−	5.84515i	0.0231034	−	0.0231034i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	69.0708	+	69.0708i	0.268758	+	0.268758i	0.828600	−	0.559842i	$-0.189138\pi$
$257$	69.0708	+	69.0708i	0.268758	+	0.268758i	−0.559842	+	0.828600i	$0.689138\pi$
$258$		0			0
$259$			89.1518i			0.344216i
$260$		0			0
$261$	−136.831			−0.524258
$262$		0			0
$263$	262.268	−	262.268i	0.997215	−	0.997215i	−0.00278076	−	0.999996i	$-0.500885\pi$
$263$	262.268	−	262.268i	0.997215	−	0.997215i	0.999996	+	0.00278076i	$0.000885144\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	196.450	+	196.450i	0.735770	+	0.735770i
$268$		0			0
$269$		−	196.753i		−	0.731424i	−0.930728	−	0.365712i	$-0.880826\pi$
$269$		−	196.753i		−	0.731424i	0.930728	−	0.365712i	$-0.119174\pi$
$270$		0			0
$271$	−345.136			−1.27356			−0.636781	−	0.771044i	$-0.719735\pi$
$271$	−345.136			−1.27356			−0.636781	+	0.771044i	$0.719735\pi$
$272$		0			0
$273$	−62.2314	+	62.2314i	−0.227954	+	0.227954i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−210.414	−	210.414i	−0.759616	−	0.759616i	0.216637	−	0.976252i	$-0.430491\pi$
$277$	−210.414	−	210.414i	−0.759616	−	0.759616i	−0.976252	+	0.216637i	$0.930491\pi$
$278$		0			0
$279$			177.867i			0.637515i
$280$		0			0
$281$	−515.941			−1.83609			−0.918045	−	0.396477i	$-0.870233\pi$
$281$	−515.941			−1.83609			−0.918045	+	0.396477i	$0.870233\pi$
$282$		0			0
$283$	−170.916	+	170.916i	−0.603944	+	0.603944i	−0.941357	−	0.337413i	$-0.890448\pi$
$283$	−170.916	+	170.916i	−0.603944	+	0.603944i	0.337413	+	0.941357i	$0.390448\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	30.0000	+	30.0000i	0.104530	+	0.104530i
$288$		0			0
$289$		−	258.318i		−	0.893833i
$290$		0			0
$291$	−227.730			−0.782579
$292$		0			0
$293$	265.249	−	265.249i	0.905285	−	0.905285i	−0.0906021	−	0.995887i	$-0.528879\pi$
$293$	265.249	−	265.249i	0.905285	−	0.905285i	0.995887	+	0.0906021i	$0.0288792\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−22.5665	−	22.5665i	−0.0759816	−	0.0759816i
$298$		0			0
$299$			106.370i			0.355754i
$300$		0			0
$301$	130.376			0.433144
$302$		0			0
$303$	35.0098	−	35.0098i	0.115544	−	0.115544i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	231.243	+	231.243i	0.753233	+	0.753233i	0.975081	−	0.221848i	$-0.0712088\pi$
$307$	231.243	+	231.243i	0.753233	+	0.753233i	−0.221848	+	0.975081i	$0.571209\pi$
$308$		0			0
$309$		−	126.139i		−	0.408218i
$310$		0			0
$311$	55.3921			0.178110			0.0890548	−	0.996027i	$-0.471615\pi$
$311$	55.3921			0.178110			0.0890548	+	0.996027i	$0.471615\pi$
$312$		0			0
$313$	91.5773	−	91.5773i	0.292579	−	0.292579i	−0.545519	−	0.838098i	$-0.683667\pi$
$313$	91.5773	−	91.5773i	0.292579	−	0.292579i	0.838098	+	0.545519i	$0.183667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	309.606	+	309.606i	0.976675	+	0.976675i	0.999734	−	0.0230593i	$-0.00734067\pi$
$317$	309.606	+	309.606i	0.976675	+	0.976675i	−0.0230593	+	0.999734i	$0.507341\pi$
$318$		0			0
$319$			35.9667i			0.112748i
$320$		0			0
$321$	235.020			0.732148
$322$		0			0
$323$	−91.6191	+	91.6191i	−0.283651	+	0.283651i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−101.347	−	101.347i	−0.309930	−	0.309930i
$328$		0			0
$329$			221.188i			0.672305i
$330$		0			0
$331$	−370.952			−1.12070			−0.560351	−	0.828255i	$-0.689334\pi$
$331$	−370.952			−1.12070			−0.560351	+	0.828255i	$0.689334\pi$
$332$		0			0
$333$	93.5657	−	93.5657i	0.280978	−	0.280978i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	97.2640	+	97.2640i	0.288617	+	0.288617i	0.836533	−	0.547916i	$-0.184579\pi$
$337$	97.2640	+	97.2640i	0.288617	+	0.288617i	−0.547916	+	0.836533i	$0.684579\pi$
$338$		0			0
$339$			295.261i			0.870976i
$340$		0			0
$341$	46.7529			0.137105
$342$		0			0
$343$	−180.000	+	180.000i	−0.524781	+	0.524781i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−145.243	−	145.243i	−0.418567	−	0.418567i	0.466143	−	0.884709i	$-0.345643\pi$
$347$	−145.243	−	145.243i	−0.418567	−	0.418567i	−0.884709	+	0.466143i	$0.845643\pi$
$348$		0			0
$349$		−	24.2176i		−	0.0693915i	−0.999398	−	0.0346958i	$-0.988954\pi$
$349$		−	24.2176i		−	0.0693915i	0.999398	−	0.0346958i	$-0.0110462\pi$
$350$		0			0
$351$	410.667			1.16999
$352$		0			0
$353$	−95.5541	+	95.5541i	−0.270692	+	0.270692i	−0.829379	−	0.558687i	$-0.811305\pi$
$353$	−95.5541	+	95.5541i	−0.270692	+	0.270692i	0.558687	+	0.829379i	$0.311305\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−24.2764	−	24.2764i	−0.0680010	−	0.0680010i
$358$		0			0
$359$			466.277i			1.29882i	0.760438	+	0.649411i	$0.224985\pi$
$359$			466.277i			1.29882i	−0.760438	+	0.649411i	$0.775015\pi$
$360$		0			0
$361$	−186.159			−0.515675
$362$		0			0
$363$	185.604	−	185.604i	0.511306	−	0.511306i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	244.971	+	244.971i	0.667495	+	0.667495i	0.957135	−	0.289641i	$-0.0935358\pi$
$367$	244.971	+	244.971i	0.667495	+	0.667495i	−0.289641	+	0.957135i	$0.593536\pi$
$368$		0			0
$369$		−	62.9706i		−	0.170652i
$370$		0			0
$371$	66.6453			0.179637
$372$		0			0
$373$	159.217	−	159.217i	0.426855	−	0.426855i	−0.460701	−	0.887556i	$-0.652402\pi$
$373$	159.217	−	159.217i	0.426855	−	0.426855i	0.887556	+	0.460701i	$0.152402\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−327.261	−	327.261i	−0.868067	−	0.868067i
$378$		0			0
$379$		−	515.933i		−	1.36130i	−0.732608	−	0.680651i	$-0.761697\pi$
$379$		−	515.933i		−	1.36130i	0.732608	−	0.680651i	$-0.238303\pi$
$380$		0			0
$381$	74.0392			0.194329
$382$		0			0
$383$	343.941	−	343.941i	0.898019	−	0.898019i	−0.0972420	−	0.995261i	$-0.531002\pi$
$383$	343.941	−	343.941i	0.898019	−	0.898019i	0.995261	+	0.0972420i	$0.0310021\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−136.831	−	136.831i	−0.353569	−	0.353569i
$388$		0			0
$389$		−	537.565i		−	1.38191i	−0.722896	−	0.690957i	$-0.757189\pi$
$389$		−	537.565i		−	1.38191i	0.722896	−	0.690957i	$-0.242811\pi$
$390$		0			0
$391$	−41.4948			−0.106125
$392$		0			0
$393$	−103.606	+	103.606i	−0.263629	+	0.263629i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	289.525	+	289.525i	0.729282	+	0.729282i	0.970477	−	0.241195i	$-0.0775393\pi$
$397$	289.525	+	289.525i	0.729282	+	0.729282i	−0.241195	+	0.970477i	$0.577539\pi$
$398$		0			0
$399$		−	144.981i		−	0.363361i
$400$		0			0
$401$	441.912			1.10202			0.551012	−	0.834497i	$-0.314242\pi$
$401$	441.912			1.10202			0.551012	+	0.834497i	$0.314242\pi$
$402$		0			0
$403$	−425.406	+	425.406i	−1.05560	+	1.05560i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.5941	−	24.5941i	−0.0604278	−	0.0604278i
$408$		0			0
$409$			492.565i			1.20431i	0.798377	+	0.602157i	$0.205692\pi$
$409$			492.565i			1.20431i	−0.798377	+	0.602157i	$0.794308\pi$
$410$		0			0
$411$	167.717			0.408070
$412$		0			0
$413$	−222.880	+	222.880i	−0.539660	+	0.539660i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	129.592	+	129.592i	0.310773	+	0.310773i
$418$		0			0
$419$		−	404.712i		−	0.965900i	−0.875648	−	0.482950i	$-0.839565\pi$
$419$		−	404.712i		−	0.965900i	0.875648	−	0.482950i	$-0.160435\pi$
$420$		0			0
$421$	−731.882			−1.73844			−0.869219	−	0.494427i	$-0.835378\pi$
$421$	−731.882			−1.73844			−0.869219	+	0.494427i	$0.835378\pi$
$422$		0			0
$423$	232.139	−	232.139i	0.548792	−	0.548792i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−10.8118	−	10.8118i	−0.0253203	−	0.0253203i
$428$		0			0
$429$		−	34.3353i		−	0.0800357i
$430$		0			0
$431$	117.624			0.272908			0.136454	−	0.990646i	$-0.456429\pi$
$431$	117.624			0.272908			0.136454	+	0.990646i	$0.456429\pi$
$432$		0			0
$433$	−558.909	+	558.909i	−1.29078	+	1.29078i	−0.356479	+	0.934303i	$0.616023\pi$
$433$	−558.909	+	558.909i	−1.29078	+	1.29078i	−0.934303	+	0.356479i	$0.883977\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−123.906	−	123.906i	−0.283537	−	0.283537i
$438$		0			0
$439$		−	570.441i		−	1.29941i	−0.760187	−	0.649704i	$-0.774893\pi$
$439$		−	570.441i		−	1.29941i	0.760187	−	0.649704i	$-0.225107\pi$
$440$		0			0
$441$	172.120			0.390294
$442$		0			0
$443$	159.569	−	159.569i	0.360201	−	0.360201i	−0.503686	−	0.863887i	$-0.668023\pi$
$443$	159.569	−	159.569i	0.360201	−	0.360201i	0.863887	+	0.503686i	$0.168023\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−376.485	−	376.485i	−0.842249	−	0.842249i
$448$		0			0
$449$			640.665i			1.42687i	0.700721	+	0.713435i	$0.252862\pi$
$449$			640.665i			1.42687i	−0.700721	+	0.713435i	$0.747138\pi$
$450$		0			0
$451$	−16.5521			−0.0367008
$452$		0			0
$453$	425.022	−	425.022i	0.938240	−	0.938240i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−344.638	−	344.638i	−0.754132	−	0.754132i	0.221115	−	0.975248i	$-0.429030\pi$
$457$	−344.638	−	344.638i	−0.754132	−	0.754132i	−0.975248	+	0.221115i	$0.929030\pi$
$458$		0			0
$459$			160.200i			0.349020i
$460$		0			0
$461$	475.565			1.03159			0.515797	−	0.856711i	$-0.327496\pi$
$461$	475.565			1.03159			0.515797	+	0.856711i	$0.327496\pi$
$462$		0			0
$463$	−330.159	+	330.159i	−0.713086	+	0.713086i	−0.967180	−	0.254094i	$-0.918223\pi$
$463$	−330.159	+	330.159i	−0.713086	+	0.713086i	0.254094	+	0.967180i	$0.418223\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−291.724	−	291.724i	−0.624676	−	0.624676i	0.322048	−	0.946723i	$-0.395629\pi$
$467$	−291.724	−	291.724i	−0.624676	−	0.624676i	−0.946723	+	0.322048i	$0.895629\pi$
$468$		0			0
$469$			144.218i			0.307500i
$470$		0			0
$471$	273.202			0.580047
$472$		0			0
$473$	−35.9667	+	35.9667i	−0.0760395	+	0.0760395i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−69.9449	−	69.9449i	−0.146635	−	0.146635i
$478$		0			0
$479$			100.624i			0.210070i	0.994469	+	0.105035i	$0.0334955\pi$
$479$			100.624i			0.210070i	−0.994469	+	0.105035i	$0.966505\pi$
$480$		0			0
$481$	447.565			0.930488
$482$		0			0
$483$	32.8314	−	32.8314i	0.0679739	−	0.0679739i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−385.297	−	385.297i	−0.791164	−	0.791164i	0.190519	−	0.981683i	$-0.438983\pi$
$487$	−385.297	−	385.297i	−0.791164	−	0.791164i	−0.981683	+	0.190519i	$0.938983\pi$
$488$		0			0
$489$			106.702i			0.218204i
$490$		0			0
$491$	454.368			0.925394			0.462697	−	0.886517i	$-0.346882\pi$
$491$	454.368			0.925394			0.462697	+	0.886517i	$0.346882\pi$
$492$		0			0
$493$	127.664	−	127.664i	0.258953	−	0.258953i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−142.555	−	142.555i	−0.286832	−	0.286832i
$498$		0			0
$499$		−	753.158i		−	1.50933i	−0.656108	−	0.754667i	$-0.727798\pi$
$499$		−	753.158i		−	1.50933i	0.656108	−	0.754667i	$-0.272202\pi$
$500$		0			0
$501$	−476.237			−0.950573
$502$		0			0
$503$	−450.694	+	450.694i	−0.896012	+	0.896012i	−0.995081	−	0.0990685i	$-0.968414\pi$
$503$	−450.694	+	450.694i	−0.896012	+	0.896012i	0.0990685	+	0.995081i	$0.468414\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	50.5505	+	50.5505i	0.0997051	+	0.0997051i
$508$		0			0
$509$		−	332.812i		−	0.653854i	−0.945050	−	0.326927i	$-0.893987\pi$
$509$		−	332.812i		−	0.653854i	0.945050	−	0.326927i	$-0.106013\pi$
$510$		0			0
$511$	−372.274			−0.728521
$512$		0			0
$513$	−478.366	+	478.366i	−0.932488	+	0.932488i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−61.0187	−	61.0187i	−0.118025	−	0.118025i
$518$		0			0
$519$		−	566.245i		−	1.09103i
$520$		0			0
$521$	−976.506			−1.87429			−0.937146	−	0.348938i	$-0.886542\pi$
$521$	−976.506			−1.87429			−0.937146	+	0.348938i	$0.886542\pi$
$522$		0			0
$523$	−530.649	+	530.649i	−1.01462	+	1.01462i	−0.0147329	+	0.999891i	$0.504690\pi$
$523$	−530.649	+	530.649i	−1.01462	+	1.01462i	−0.999891	+	0.0147329i	$0.995310\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−165.950	−	165.950i	−0.314896	−	0.314896i
$528$		0			0
$529$			472.882i			0.893917i
$530$		0			0
$531$	467.829			0.881033
$532$		0			0
$533$	150.608	−	150.608i	0.282566	−	0.282566i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−374.760	−	374.760i	−0.697877	−	0.697877i
$538$		0			0
$539$		−	45.2423i		−	0.0839375i
$540$		0			0
$541$	558.259			1.03190			0.515951	−	0.856618i	$-0.327439\pi$
$541$	558.259			1.03190			0.515951	+	0.856618i	$0.327439\pi$
$542$		0			0
$543$	172.287	−	172.287i	0.317288	−	0.317288i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−572.857	−	572.857i	−1.04727	−	1.04727i	−0.998826	−	0.0484454i	$-0.984573\pi$
$547$	−572.857	−	572.857i	−1.04727	−	1.04727i	−0.0484454	−	0.998826i	$-0.515427\pi$
$548$		0			0
$549$			22.6941i			0.0413372i
$550$		0			0
$551$	762.423			1.38371
$552$		0			0
$553$	−236.121	+	236.121i	−0.426982	+	0.426982i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	693.909	+	693.909i	1.24580	+	1.24580i	0.957558	+	0.288239i	$0.0930698\pi$
$557$	693.909	+	693.909i	1.24580	+	1.24580i	0.288239	+	0.957558i	$0.406930\pi$
$558$		0			0
$559$		−	654.523i		−	1.17088i
$560$		0			0
$561$	13.3941			0.0238755
$562$		0			0
$563$	709.347	−	709.347i	1.25994	−	1.25994i	0.308821	−	0.951120i	$-0.400065\pi$
$563$	709.347	−	709.347i	1.25994	−	1.25994i	0.951120	−	0.308821i	$-0.0999346\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−51.1882	−	51.1882i	−0.0902791	−	0.0902791i
$568$		0			0
$569$			19.8706i			0.0349220i	0.999848	+	0.0174610i	$0.00555828\pi$
$569$			19.8706i			0.0349220i	−0.999848	+	0.0174610i	$0.994442\pi$
$570$		0			0
$571$	−445.104			−0.779516			−0.389758	−	0.920917i	$-0.627441\pi$
$571$	−445.104			−0.779516			−0.389758	+	0.920917i	$0.627441\pi$
$572$		0			0
$573$	65.6511	−	65.6511i	0.114574	−	0.114574i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	401.462	+	401.462i	0.695774	+	0.695774i	0.963496	−	0.267722i	$-0.0862709\pi$
$577$	401.462	+	401.462i	0.695774	+	0.695774i	−0.267722	+	0.963496i	$0.586271\pi$
$578$		0			0
$579$		−	36.4146i		−	0.0628921i
$580$		0			0
$581$	26.5941			0.0457730
$582$		0			0
$583$	−18.3853	+	18.3853i	−0.0315357	+	0.0315357i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−356.649	−	356.649i	−0.607578	−	0.607578i	0.334734	−	0.942313i	$-0.391353\pi$
$587$	−356.649	−	356.649i	−0.607578	−	0.607578i	−0.942313	+	0.334734i	$0.891353\pi$
$588$		0			0
$589$		−	991.071i		−	1.68263i
$590$		0			0
$591$	473.575			0.801312
$592$		0			0
$593$	−209.916	+	209.916i	−0.353991	+	0.353991i	−0.861592	−	0.507601i	$-0.830532\pi$
$593$	−209.916	+	209.916i	−0.353991	+	0.353991i	0.507601	+	0.861592i	$0.330532\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−324.836	−	324.836i	−0.544114	−	0.544114i
$598$		0			0
$599$		−	880.702i		−	1.47029i	−0.677911	−	0.735144i	$-0.737115\pi$
$599$		−	880.702i		−	1.47029i	0.677911	−	0.735144i	$-0.262885\pi$
$600$		0			0
$601$	580.665			0.966164			0.483082	−	0.875575i	$-0.339517\pi$
$601$	580.665			0.966164			0.483082	+	0.875575i	$0.339517\pi$
$602$		0			0
$603$	151.358	−	151.358i	0.251008	−	0.251008i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	81.1882	+	81.1882i	0.133753	+	0.133753i	0.770814	−	0.637061i	$-0.219850\pi$
$607$	81.1882	+	81.1882i	0.133753	+	0.133753i	−0.637061	+	0.770814i	$0.719850\pi$
$608$		0			0
$609$			202.020i			0.331723i
$610$		0			0
$611$	1110.42			1.81738
$612$		0			0
$613$	756.359	−	756.359i	1.23386	−	1.23386i	0.271398	−	0.962467i	$-0.412514\pi$
$613$	756.359	−	756.359i	1.23386	−	1.23386i	0.962467	−	0.271398i	$-0.0874858\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−614.798	−	614.798i	−0.996431	−	0.996431i	0.00356268	−	0.999994i	$-0.498866\pi$
$617$	−614.798	−	614.798i	−0.996431	−	0.996431i	−0.999994	+	0.00356268i	$0.998866\pi$
$618$		0			0
$619$		−	231.052i		−	0.373266i	−0.982430	−	0.186633i	$-0.940242\pi$
$619$		−	231.052i		−	0.373266i	0.982430	−	0.186633i	$-0.0597575\pi$
$620$		0			0
$621$	−216.655			−0.348881
$622$		0			0
$623$	−253.565	+	253.565i	−0.407006	+	0.407006i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	39.9956	+	39.9956i	0.0637888	+	0.0637888i
$628$		0			0
$629$			174.594i			0.277574i
$630$		0			0
$631$	−1058.79			−1.67795			−0.838976	−	0.544169i	$-0.816845\pi$
$631$	−1058.79			−1.67795			−0.838976	+	0.544169i	$0.816845\pi$
$632$		0			0
$633$	398.359	−	398.359i	0.629319	−	0.629319i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	411.661	+	411.661i	0.646249	+	0.646249i
$638$		0			0
$639$			299.227i			0.468274i
$640$		0			0
$641$	434.318			0.677563			0.338781	−	0.940865i	$-0.389985\pi$
$641$	434.318			0.677563			0.338781	+	0.940865i	$0.389985\pi$
$642$		0			0
$643$	−130.653	+	130.653i	−0.203193	+	0.203193i	−0.801366	−	0.598174i	$-0.795893\pi$
$643$	−130.653	+	130.653i	−0.203193	+	0.203193i	0.598174	+	0.801366i	$0.295893\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−524.753	−	524.753i	−0.811056	−	0.811056i	0.173737	−	0.984792i	$-0.444416\pi$
$647$	−524.753	−	524.753i	−0.811056	−	0.811056i	−0.984792	+	0.173737i	$0.944416\pi$
$648$		0			0
$649$		−	122.971i		−	0.189477i
$650$		0			0
$651$	262.605			0.403386
$652$		0			0
$653$	−182.499	+	182.499i	−0.279478	+	0.279478i	−0.832901	−	0.553423i	$-0.813321\pi$
$653$	−182.499	+	182.499i	−0.279478	+	0.279478i	0.553423	+	0.832901i	$0.313321\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	390.706	+	390.706i	0.594681	+	0.594681i
$658$		0			0
$659$			20.5289i			0.0311516i	0.999879	+	0.0155758i	$0.00495813\pi$
$659$			20.5289i			0.0311516i	−0.999879	+	0.0155758i	$0.995042\pi$
$660$		0			0
$661$	−662.971			−1.00298			−0.501491	−	0.865163i	$-0.667215\pi$
$661$	−662.971			−1.00298			−0.501491	+	0.865163i	$0.667215\pi$
$662$		0			0
$663$	−121.874	+	121.874i	−0.183821	+	0.183821i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	172.653	+	172.653i	0.258850	+	0.258850i
$668$		0			0
$669$		−	363.843i		−	0.543861i
$670$		0			0
$671$	5.96524			0.00889007
$672$		0			0
$673$	31.7713	−	31.7713i	0.0472085	−	0.0472085i	−0.683108	−	0.730317i	$-0.739372\pi$
$673$	31.7713	−	31.7713i	0.0472085	−	0.0472085i	0.730317	+	0.683108i	$0.239372\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	740.135	+	740.135i	1.09326	+	1.09326i	0.995179	+	0.0980782i	$0.0312695\pi$
$677$	740.135	+	740.135i	1.09326	+	1.09326i	0.0980782	+	0.995179i	$0.468730\pi$
$678$		0			0
$679$		−	293.939i		−	0.432900i
$680$		0			0
$681$	−530.516			−0.779024
$682$		0			0
$683$	14.9750	−	14.9750i	0.0219253	−	0.0219253i	−0.696059	−	0.717984i	$-0.745065\pi$
$683$	14.9750	−	14.9750i	0.0219253	−	0.0219253i	0.717984	+	0.696059i	$0.245065\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−216.257	−	216.257i	−0.314784	−	0.314784i
$688$		0			0
$689$		−	334.576i		−	0.485597i
$690$		0			0
$691$	−203.028			−0.293817			−0.146909	−	0.989150i	$-0.546932\pi$
$691$	−203.028			−0.293817			−0.146909	+	0.989150i	$0.546932\pi$
$692$		0			0
$693$	9.26483	−	9.26483i	0.0133692	−	0.0133692i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	58.7517	+	58.7517i	0.0842923	+	0.0842923i
$698$		0			0
$699$			517.692i			0.740619i
$700$		0			0
$701$	624.594			0.891004			0.445502	−	0.895281i	$-0.353025\pi$
$701$	624.594			0.891004			0.445502	+	0.895281i	$0.353025\pi$
$702$		0			0
$703$	−521.347	+	521.347i	−0.741603	+	0.741603i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	45.1882	+	45.1882i	0.0639155	+	0.0639155i
$708$		0			0
$709$		−	925.506i		−	1.30537i	−0.757630	−	0.652684i	$-0.773643\pi$
$709$		−	925.506i		−	1.30537i	0.757630	−	0.652684i	$-0.226357\pi$
$710$		0			0
$711$	495.623			0.697079
$712$		0			0
$713$	224.431	−	224.431i	0.314770	−	0.314770i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−289.623	−	289.623i	−0.403938	−	0.403938i
$718$		0			0
$719$			487.472i			0.677987i	0.940789	+	0.338993i	$0.110086\pi$
$719$			487.472i			0.677987i	−0.940789	+	0.338993i	$0.889914\pi$
$720$		0			0
$721$	162.812			0.225814
$722$		0			0
$723$	−264.046	+	264.046i	−0.365208	+	0.365208i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	98.5441	+	98.5441i	0.135549	+	0.135549i	0.771626	−	0.636077i	$-0.219444\pi$
$727$	98.5441	+	98.5441i	0.135549	+	0.135549i	−0.636077	+	0.771626i	$0.719444\pi$
$728$		0			0
$729$			677.833i			0.929813i
$730$		0			0
$731$	255.328			0.349286
$732$		0			0
$733$	−132.515	+	132.515i	−0.180784	+	0.180784i	−0.791698	−	0.610913i	$-0.790802\pi$
$733$	−132.515	+	132.515i	−0.180784	+	0.180784i	0.610913	+	0.791698i	$0.290802\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−39.7850	−	39.7850i	−0.0539824	−	0.0539824i
$738$		0			0
$739$		−	543.957i		−	0.736072i	−0.929812	−	0.368036i	$-0.880030\pi$
$739$		−	543.957i		−	0.736072i	0.929812	−	0.368036i	$-0.119970\pi$
$740$		0			0
$741$	−727.841			−0.982242
$742$		0			0
$743$	447.991	−	447.991i	0.602949	−	0.602949i	−0.338145	−	0.941094i	$-0.609799\pi$
$743$	447.991	−	447.991i	0.602949	−	0.602949i	0.941094	+	0.338145i	$0.109799\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−27.9108	−	27.9108i	−0.0373638	−	0.0373638i
$748$		0			0
$749$			303.347i			0.405003i
$750$		0			0
$751$	−981.326			−1.30669			−0.653346	−	0.757059i	$-0.726635\pi$
$751$	−981.326			−1.30669			−0.653346	+	0.757059i	$0.726635\pi$
$752$		0			0
$753$	−12.3075	+	12.3075i	−0.0163446	+	0.0163446i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−965.003	−	965.003i	−1.27477	−	1.27477i	−0.943554	−	0.331219i	$-0.892540\pi$
$757$	−965.003	−	965.003i	−1.27477	−	1.27477i	−0.331219	−	0.943554i	$-0.607460\pi$
$758$		0			0
$759$			18.1142i			0.0238659i
$760$		0			0
$761$	−459.494			−0.603803			−0.301902	−	0.953339i	$-0.597621\pi$
$761$	−459.494			−0.603803			−0.301902	+	0.953339i	$0.597621\pi$
$762$		0			0
$763$	130.812	−	130.812i	0.171444	−	0.171444i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1118.91	+	1118.91i	1.45882	+	1.45882i
$768$		0			0
$769$		−	757.535i		−	0.985091i	−0.870287	−	0.492546i	$-0.836066\pi$
$769$		−	757.535i		−	0.985091i	0.870287	−	0.492546i	$-0.163934\pi$
$770$		0			0
$771$	−214.052			−0.277629
$772$		0			0
$773$	285.887	−	285.887i	0.369840	−	0.369840i	−0.497578	−	0.867419i	$-0.665777\pi$
$773$	285.887	−	285.887i	0.369840	−	0.369840i	0.867419	+	0.497578i	$0.165777\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−138.142	−	138.142i	−0.177788	−	0.177788i
$778$		0			0
$779$			350.871i			0.450413i
$780$		0			0
$781$	78.6529			0.100708
$782$		0			0
$783$	666.566	−	666.566i	0.851297	−	0.851297i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−320.159	−	320.159i	−0.406809	−	0.406809i	0.473815	−	0.880624i	$-0.342876\pi$
$787$	−320.159	−	320.159i	−0.406809	−	0.406809i	−0.880624	+	0.473815i	$0.842876\pi$
$788$		0			0
$789$			812.773i			1.03013i
$790$		0			0
$791$	−381.102			−0.481798
$792$		0			0
$793$	−54.2778	+	54.2778i	−0.0684462	+	0.0684462i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	587.757	+	587.757i	0.737462	+	0.737462i	0.972086	−	0.234624i	$-0.0753859\pi$
$797$	587.757	+	587.757i	0.737462	+	0.737462i	−0.234624	+	0.972086i	$0.575386\pi$
$798$		0			0
$799$			433.173i			0.542144i
$800$		0			0
$801$	532.237			0.664466
$802$		0			0
$803$	102.699	−	102.699i	0.127894	−	0.127894i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	304.871	+	304.871i	0.377783	+	0.377783i
$808$		0			0
$809$			163.941i			0.202647i	0.994854	+	0.101323i	$0.0323077\pi$
$809$			163.941i			0.202647i	−0.994854	+	0.101323i	$0.967692\pi$
$810$		0			0
$811$	−1194.07			−1.47234			−0.736169	−	0.676798i	$-0.763367\pi$
$811$	−1194.07			−1.47234			−0.736169	+	0.676798i	$0.763367\pi$
$812$		0			0
$813$	534.791	−	534.791i	0.657799	−	0.657799i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	762.423	+	762.423i	0.933198	+	0.933198i
$818$		0			0
$819$			168.602i			0.205863i
$820$		0			0
$821$	722.435			0.879946			0.439973	−	0.898011i	$-0.354988\pi$
$821$	722.435			0.879946			0.439973	+	0.898011i	$0.354988\pi$
$822$		0			0
$823$	−728.218	+	728.218i	−0.884833	+	0.884833i	−0.994021	−	0.109188i	$-0.965175\pi$
$823$	−728.218	+	728.218i	−0.884833	+	0.884833i	0.109188	+	0.994021i	$0.465175\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	635.560	+	635.560i	0.768513	+	0.768513i	0.977845	−	0.209332i	$-0.0671288\pi$
$827$	635.560	+	635.560i	0.768513	+	0.768513i	−0.209332	+	0.977845i	$0.567129\pi$
$828$		0			0
$829$			1051.61i			1.26852i	0.773118	+	0.634262i	$0.218696\pi$
$829$			1051.61i			1.26852i	−0.773118	+	0.634262i	$0.781304\pi$
$830$		0			0
$831$	652.076			0.784688
$832$		0			0
$833$	−160.588	+	160.588i	−0.192783	+	0.192783i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−866.466	−	866.466i	−1.03520	−	1.03520i
$838$		0			0
$839$			1052.17i			1.25407i	0.778990	+	0.627036i	$0.215732\pi$
$839$			1052.17i			1.25407i	−0.778990	+	0.627036i	$0.784268\pi$
$840$		0			0
$841$	−221.376			−0.263230
$842$		0			0
$843$	799.456	−	799.456i	0.948346	−	0.948346i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	239.565	+	239.565i	0.282839	+	0.282839i
$848$		0			0
$849$		−	529.673i		−	0.623878i
$850$		0			0
$851$	−236.121			−0.277463
$852$		0			0
$853$	−832.706	+	832.706i	−0.976209	+	0.976209i	−0.999724	−	0.0235143i	$-0.992514\pi$
$853$	−832.706	+	832.706i	−0.976209	+	0.976209i	0.0235143	+	0.999724i	$0.492514\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−492.045	−	492.045i	−0.574148	−	0.574148i	0.359137	−	0.933285i	$-0.383071\pi$
$857$	−492.045	−	492.045i	−0.574148	−	0.574148i	−0.933285	+	0.359137i	$0.883071\pi$
$858$		0			0
$859$			362.125i			0.421565i	0.977533	+	0.210783i	$0.0676012\pi$
$859$			362.125i			0.421565i	−0.977533	+	0.210783i	$0.932399\pi$
$860$		0			0
$861$	−92.9706			−0.107980
$862$		0			0
$863$	968.268	−	968.268i	1.12198	−	1.12198i	0.130535	−	0.991444i	$-0.458330\pi$
$863$	968.268	−	968.268i	1.12198	−	1.12198i	0.991444	−	0.130535i	$-0.0416695\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	400.266	+	400.266i	0.461667	+	0.461667i
$868$		0			0
$869$		−	130.276i		−	0.149915i
$870$		0			0
$871$	724.009			0.831239
$872$		0			0
$873$	−308.492	+	308.492i	−0.353370	+	0.353370i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	198.385	+	198.385i	0.226208	+	0.226208i	0.811107	−	0.584898i	$-0.198866\pi$
$877$	198.385	+	198.385i	0.226208	+	0.226208i	−0.584898	+	0.811107i	$0.698866\pi$
$878$		0			0
$879$			822.010i			0.935165i
$880$		0			0
$881$	−127.306			−0.144502			−0.0722508	−	0.997386i	$-0.523018\pi$
$881$	−127.306			−0.144502			−0.0722508	+	0.997386i	$0.523018\pi$
$882$		0			0
$883$	−711.451	+	711.451i	−0.805721	+	0.805721i	−0.983983	−	0.178262i	$-0.942952\pi$
$883$	−711.451	+	711.451i	−0.805721	+	0.805721i	0.178262	+	0.983983i	$0.442952\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−392.000	−	392.000i	−0.441939	−	0.441939i	0.450724	−	0.892663i	$-0.351166\pi$
$887$	−392.000	−	392.000i	−0.441939	−	0.441939i	−0.892663	+	0.450724i	$0.851166\pi$
$888$		0			0
$889$			95.5647i			0.107497i
$890$		0			0
$891$	28.2424			0.0316974
$892$		0			0
$893$	−1293.48	+	1293.48i	−1.44846	+	1.44846i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−164.822	−	164.822i	−0.183748	−	0.183748i
$898$		0			0
$899$			1380.98i			1.53613i
$900$		0			0
$901$	130.518			0.144859
$902$		0			0
$903$	−202.020	+	202.020i	−0.223720	+	0.223720i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1259.61	+	1259.61i	1.38876	+	1.38876i	0.827932	+	0.560829i	$0.189517\pi$
$907$	1259.61	+	1259.61i	1.38876	+	1.38876i	0.560829	+	0.827932i	$0.310483\pi$
$908$		0			0
$909$		−	94.8510i		−	0.104347i
$910$		0			0
$911$	−770.595			−0.845878			−0.422939	−	0.906158i	$-0.639002\pi$
$911$	−770.595			−0.845878			−0.422939	+	0.906158i	$0.639002\pi$
$912$		0			0
$913$	−7.33646	+	7.33646i	−0.00803555	+	0.00803555i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−133.728	−	133.728i	−0.145832	−	0.145832i
$918$		0			0
$919$			594.935i			0.647373i	0.946164	+	0.323686i	$0.104922\pi$
$919$			594.935i			0.647373i	−0.946164	+	0.323686i	$0.895078\pi$
$920$		0			0
$921$	−716.625			−0.778095
$922$		0			0
$923$	−715.665	+	715.665i	−0.775368	+	0.775368i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−170.873	−	170.873i	−0.184329	−	0.184329i
$928$		0			0
$929$			68.3765i			0.0736022i	0.999323	+	0.0368011i	$0.0117168\pi$
$929$			68.3765i			0.0736022i	−0.999323	+	0.0368011i	$0.988283\pi$
$930$		0			0
$931$	−959.048			−1.03013
$932$		0			0
$933$	−85.8306	+	85.8306i	−0.0919942	+	0.0919942i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	636.370	+	636.370i	0.679157	+	0.679157i	0.959809	−	0.280652i	$-0.0905508\pi$
$937$	636.370	+	636.370i	0.679157	+	0.679157i	−0.280652	+	0.959809i	$0.590551\pi$
$938$		0			0
$939$			283.800i			0.302236i
$940$		0			0
$941$	690.971			0.734294			0.367147	−	0.930163i	$-0.380335\pi$
$941$	690.971			0.734294			0.367147	+	0.930163i	$0.380335\pi$
$942$		0			0
$943$	−79.4559	+	79.4559i	−0.0842586	+	0.0842586i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−84.1588	−	84.1588i	−0.0888689	−	0.0888689i	0.661275	−	0.750144i	$-0.270016\pi$
$947$	−84.1588	−	84.1588i	−0.0888689	−	0.0888689i	−0.750144	+	0.661275i	$0.770016\pi$
$948$		0			0
$949$			1868.91i			1.96935i
$950$		0			0
$951$	−959.475			−1.00891
$952$		0			0
$953$	1046.60	−	1046.60i	1.09822	−	1.09822i	0.103596	−	0.994619i	$-0.466965\pi$
$953$	1046.60	−	1046.60i	1.09822	−	1.09822i	0.994619	−	0.103596i	$-0.0330350\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−55.7307	−	55.7307i	−0.0582348	−	0.0582348i
$958$		0			0
$959$			216.477i			0.225732i
$960$		0			0
$961$	834.129			0.867981
$962$		0			0
$963$	318.366	−	318.366i	0.330598	−	0.330598i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1204.97	+	1204.97i	1.24609	+	1.24609i	0.957432	+	0.288660i	$0.0932097\pi$
$967$	1204.97	+	1204.97i	1.24609	+	1.24609i	0.288660	+	0.957432i	$0.406790\pi$
$968$		0			0
$969$		−	283.929i		−	0.293013i
$970$		0			0
$971$	−475.094			−0.489283			−0.244642	−	0.969614i	$-0.578670\pi$
$971$	−475.094			−0.489283			−0.244642	+	0.969614i	$0.578670\pi$
$972$		0			0
$973$	−167.269	+	167.269i	−0.171911	+	0.171911i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−452.101	−	452.101i	−0.462744	−	0.462744i	0.436810	−	0.899554i	$-0.356108\pi$
$977$	−452.101	−	452.101i	−0.462744	−	0.462744i	−0.899554	+	0.436810i	$0.856108\pi$
$978$		0			0
$979$		−	139.901i		−	0.142902i
$980$		0			0
$981$	−274.576			−0.279894
$982$		0			0
$983$	1014.97	−	1014.97i	1.03252	−	1.03252i	0.0330705	−	0.999453i	$-0.489471\pi$
$983$	1014.97	−	1014.97i	1.03252	−	1.03252i	0.999453	−	0.0330705i	$-0.0105286\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−342.733	−	342.733i	−0.347248	−	0.347248i
$988$		0			0
$989$			345.306i			0.349147i
$990$		0			0
$991$	−362.135			−0.365424			−0.182712	−	0.983166i	$-0.558488\pi$
$991$	−362.135			−0.365424			−0.182712	+	0.983166i	$0.558488\pi$
$992$		0			0
$993$	574.794	−	574.794i	0.578846	−	0.578846i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−236.121	−	236.121i	−0.236832	−	0.236832i	0.578705	−	0.815537i	$-0.303558\pi$
$997$	−236.121	−	236.121i	−0.236832	−	0.236832i	−0.815537	+	0.578705i	$0.803558\pi$
$998$		0			0
$999$			911.599i			0.912512i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.3.p.n.257.2	yes	8
4.3	odd	2		800.3.p.k.257.3	yes	8
5.2	odd	4		800.3.p.k.193.4	yes	8
5.3	odd	4	inner	800.3.p.n.193.2	yes	8
5.4	even	2		800.3.p.k.257.4	yes	8
20.3	even	4		800.3.p.k.193.3	✓	8
20.7	even	4	inner	800.3.p.n.193.1	yes	8
20.19	odd	2	inner	800.3.p.n.257.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.3.p.k.193.3	✓	8	20.3	even	4
800.3.p.k.193.4	yes	8	5.2	odd	4
800.3.p.k.257.3	yes	8	4.3	odd	2
800.3.p.k.257.4	yes	8	5.4	even	2
800.3.p.n.193.1	yes	8	20.7	even	4	inner
800.3.p.n.193.2	yes	8	5.3	odd	4	inner
800.3.p.n.257.1	yes	8	20.19	odd	2	inner
800.3.p.n.257.2	yes	8	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.p (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.54951 + 1.54951i) q^{3} +(-2.00000 - 2.00000i) q^{7} +4.19804i q^{9} +1.10347 q^{11} +(-10.0405 + 10.0405i) q^{13} +(-3.91678 - 3.91678i) q^{17} -23.3914i q^{19} +6.19804 q^{21} +(5.29706 - 5.29706i) q^{23} +(-20.4505 - 20.4505i) q^{27} +32.5941i q^{29} +42.3690 q^{31} +(-1.70984 + 1.70984i) q^{33} +(-22.2880 - 22.2880i) q^{37} -31.1157i q^{39} -15.0000 q^{41} +(-32.5941 + 32.5941i) q^{43} +(-55.2971 - 55.2971i) q^{47} -41.0000i q^{49} +12.1382 q^{51} +(-16.6613 + 16.6613i) q^{53} +(36.2452 + 36.2452i) q^{57} -111.440i q^{59} +5.40588 q^{61} +(8.39608 - 8.39608i) q^{63} +(-36.0544 - 36.0544i) q^{67} +16.4157i q^{69} +71.2777 q^{71} +(93.0686 - 93.0686i) q^{73} +(-2.20694 - 2.20694i) q^{77} -118.061i q^{79} +25.5941 q^{81} +(-6.64853 + 6.64853i) q^{83} +(-50.5049 - 50.5049i) q^{87} -126.782i q^{89} +40.1620 q^{91} +(-65.6511 + 65.6511i) q^{93} +(73.4847 + 73.4847i) q^{97} +4.63242i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 16 q^{7} - 32 q^{21} - 80 q^{23} - 184 q^{27} - 120 q^{41} - 16 q^{43} - 320 q^{47} + 288 q^{61} - 96 q^{63} - 472 q^{67} - 40 q^{81} + 8 q^{83} - 608 q^{87}+O(q^{100}) \) 8 * q + 8 * q^3 - 16 * q^7 - 32 * q^21 - 80 * q^23 - 184 * q^27 - 120 * q^41 - 16 * q^43 - 320 * q^47 + 288 * q^61 - 96 * q^63 - 472 * q^67 - 40 * q^81 + 8 * q^83 - 608 * q^87

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