LMFDB - Embedded newform 800.3.p.i.257.3

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:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.3534400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 $ x^6 - 2x^5 - 3x^4 + 16x^3 + x^2 - 12x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			257.3
Root			$2.19082 - 1.44755i$ of defining polynomial
Character	$\chi$	$=$	800.257
Dual form			800.3.p.i.193.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.89511 - 2.89511i) q^{3} +(-5.86818 - 5.86818i) q^{7} -7.76328i q^{9} +O(q^{10})$	$q+(2.89511 - 2.89511i) q^{3} +(-5.86818 - 5.86818i) q^{7} -7.76328i q^{9} -7.84407 q^{11} +(15.5535 - 15.5535i) q^{13} +(-13.3976 - 13.3976i) q^{17} +21.1070i q^{19} -33.9780 q^{21} +(-2.28775 + 2.28775i) q^{23} +(3.58043 + 3.58043i) q^{27} -7.68814i q^{29} -32.1559 q^{31} +(-22.7094 + 22.7094i) q^{33} +(-38.2629 - 38.2629i) q^{37} -90.0581i q^{39} +20.8710 q^{41} +(-57.9000 + 57.9000i) q^{43} +(-35.1906 - 35.1906i) q^{47} +19.8710i q^{49} -77.5748 q^{51} +(22.8441 - 22.8441i) q^{53} +(61.1070 + 61.1070i) q^{57} -69.7307i q^{59} -11.9681 q^{61} +(-45.5563 + 45.5563i) q^{63} +(27.5825 + 27.5825i) q^{67} +13.2466i q^{69} +52.2720 q^{71} +(4.73070 - 4.73070i) q^{73} +(46.0304 + 46.0304i) q^{77} -31.3763i q^{79} +90.6010 q^{81} +(-33.3090 + 33.3090i) q^{83} +(-22.2580 - 22.2580i) q^{87} +0.623711i q^{89} -182.541 q^{91} +(-93.0948 + 93.0948i) q^{93} +(83.1714 + 83.1714i) q^{97} +60.8958i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{7}+O(q^{10}) $ 6 * q - 8 * q^7	$ 6 q - 8 q^{7} - 32 q^{11} + 14 q^{13} + 14 q^{17} - 40 q^{21} - 56 q^{23} - 48 q^{27} - 208 q^{31} - 72 q^{33} - 86 q^{37} + 120 q^{41} - 176 q^{43} - 104 q^{47} - 352 q^{51} + 122 q^{53} + 208 q^{57} - 216 q^{61} - 216 q^{63} - 80 q^{67} - 336 q^{71} - 70 q^{73} - 264 q^{77} + 242 q^{81} - 208 q^{83} - 144 q^{87} - 544 q^{91} + 72 q^{93} + 250 q^{97}+O(q^{100}) $ 6 * q - 8 * q^7 - 32 * q^11 + 14 * q^13 + 14 * q^17 - 40 * q^21 - 56 * q^23 - 48 * q^27 - 208 * q^31 - 72 * q^33 - 86 * q^37 + 120 * q^41 - 176 * q^43 - 104 * q^47 - 352 * q^51 + 122 * q^53 + 208 * q^57 - 216 * q^61 - 216 * q^63 - 80 * q^67 - 336 * q^71 - 70 * q^73 - 264 * q^77 + 242 * q^81 - 208 * q^83 - 144 * q^87 - 544 * q^91 + 72 * q^93 + 250 * q^97

Display 100 coefficients 10 coefficients

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
$3$	2.89511	−	2.89511i	0.965036	−	0.965036i	−0.0343735	−	0.999409i	$-0.510944\pi$
$3$	2.89511	−	2.89511i	0.965036	−	0.965036i	0.999409	+	0.0343735i	$0.0109436\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−5.86818	−	5.86818i	−0.838311	−	0.838311i	0.150326	−	0.988637i	$-0.451968\pi$
$7$	−5.86818	−	5.86818i	−0.838311	−	0.838311i	−0.988637	+	0.150326i	$0.951968\pi$
$8$		0			0
$9$		−	7.76328i		−	0.862587i
$10$		0			0
$11$	−7.84407			−0.713097			−0.356549	−	0.934277i	$-0.616047\pi$
$11$	−7.84407			−0.713097			−0.356549	+	0.934277i	$0.616047\pi$
$12$		0			0
$13$	15.5535	−	15.5535i	1.19642	−	1.19642i	0.221193	−	0.975230i	$-0.429005\pi$
$13$	15.5535	−	15.5535i	1.19642	−	1.19642i	0.975230	−	0.221193i	$-0.0709950\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−13.3976	−	13.3976i	−0.788092	−	0.788092i	0.193089	−	0.981181i	$-0.438149\pi$
$17$	−13.3976	−	13.3976i	−0.788092	−	0.788092i	−0.981181	+	0.193089i	$0.938149\pi$
$18$		0			0
$19$			21.1070i			1.11089i	0.831552	+	0.555447i	$0.187453\pi$
$19$			21.1070i			1.11089i	−0.831552	+	0.555447i	$0.812547\pi$
$20$		0			0
$21$	−33.9780			−1.61800
$22$		0			0
$23$	−2.28775	+	2.28775i	−0.0994674	+	0.0994674i	−0.755089	−	0.655622i	$-0.772407\pi$
$23$	−2.28775	+	2.28775i	−0.0994674	+	0.0994674i	0.655622	+	0.755089i	$0.272407\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.58043	+	3.58043i	0.132608	+	0.132608i
$28$		0			0
$29$		−	7.68814i		−	0.265108i	−0.991176	−	0.132554i	$-0.957682\pi$
$29$		−	7.68814i		−	0.265108i	0.991176	−	0.132554i	$-0.0423179\pi$
$30$		0			0
$31$	−32.1559			−1.03729			−0.518644	−	0.854990i	$-0.673563\pi$
$31$	−32.1559			−1.03729			−0.518644	+	0.854990i	$0.673563\pi$
$32$		0			0
$33$	−22.7094	+	22.7094i	−0.688164	+	0.688164i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−38.2629	−	38.2629i	−1.03413	−	1.03413i	−0.999397	−	0.0347365i	$-0.988941\pi$
$37$	−38.2629	−	38.2629i	−1.03413	−	1.03413i	−0.0347365	−	0.999397i	$-0.511059\pi$
$38$		0			0
$39$		−	90.0581i		−	2.30918i
$40$		0			0
$41$	20.8710			0.509049			0.254524	−	0.967066i	$-0.418081\pi$
$41$	20.8710			0.509049			0.254524	+	0.967066i	$0.418081\pi$
$42$		0			0
$43$	−57.9000	+	57.9000i	−1.34651	+	1.34651i	−0.457094	+	0.889418i	$0.651110\pi$
$43$	−57.9000	+	57.9000i	−1.34651	+	1.34651i	−0.889418	+	0.457094i	$0.848890\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−35.1906	−	35.1906i	−0.748736	−	0.748736i	0.225506	−	0.974242i	$-0.427597\pi$
$47$	−35.1906	−	35.1906i	−0.748736	−	0.748736i	−0.974242	+	0.225506i	$0.927597\pi$
$48$		0			0
$49$			19.8710i			0.405531i
$50$		0			0
$51$	−77.5748			−1.52107
$52$		0			0
$53$	22.8441	−	22.8441i	0.431020	−	0.431020i	−0.457955	−	0.888975i	$-0.651418\pi$
$53$	22.8441	−	22.8441i	0.431020	−	0.431020i	0.888975	+	0.457955i	$0.151418\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	61.1070	+	61.1070i	1.07205	+	1.07205i
$58$		0			0
$59$		−	69.7307i		−	1.18188i	−0.806717	−	0.590938i	$-0.798758\pi$
$59$		−	69.7307i		−	1.18188i	0.806717	−	0.590938i	$-0.201242\pi$
$60$		0			0
$61$	−11.9681			−0.196199			−0.0980996	−	0.995177i	$-0.531276\pi$
$61$	−11.9681			−0.196199			−0.0980996	+	0.995177i	$0.531276\pi$
$62$		0			0
$63$	−45.5563	+	45.5563i	−0.723116	+	0.723116i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	27.5825	+	27.5825i	0.411679	+	0.411679i	0.882323	−	0.470644i	$-0.155978\pi$
$67$	27.5825	+	27.5825i	0.411679	+	0.411679i	−0.470644	+	0.882323i	$0.655978\pi$
$68$		0			0
$69$			13.2466i			0.191979i
$70$		0			0
$71$	52.2720			0.736226			0.368113	−	0.929781i	$-0.380004\pi$
$71$	52.2720			0.736226			0.368113	+	0.929781i	$0.380004\pi$
$72$		0			0
$73$	4.73070	−	4.73070i	0.0648042	−	0.0648042i	−0.673962	−	0.738766i	$-0.735409\pi$
$73$	4.73070	−	4.73070i	0.0648042	−	0.0648042i	0.738766	+	0.673962i	$0.235409\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	46.0304	+	46.0304i	0.597797	+	0.597797i
$78$		0			0
$79$		−	31.3763i		−	0.397168i	−0.980084	−	0.198584i	$-0.936366\pi$
$79$		−	31.3763i		−	0.397168i	0.980084	−	0.198584i	$-0.0636343\pi$
$80$		0			0
$81$	90.6010			1.11853
$82$		0			0
$83$	−33.3090	+	33.3090i	−0.401314	+	0.401314i	−0.878696	−	0.477382i	$-0.841586\pi$
$83$	−33.3090	+	33.3090i	−0.401314	+	0.401314i	0.477382	+	0.878696i	$0.341586\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−22.2580	−	22.2580i	−0.255839	−	0.255839i
$88$		0			0
$89$			0.623711i			0.00700799i	0.999994	+	0.00350400i	$0.00111536\pi$
$89$			0.623711i			0.00700799i	−0.999994	+	0.00350400i	$0.998885\pi$
$90$		0			0
$91$	−182.541			−2.00595
$92$		0			0
$93$	−93.0948	+	93.0948i	−1.00102	+	1.00102i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	83.1714	+	83.1714i	0.857437	+	0.857437i	0.991036	−	0.133598i	$-0.0426532\pi$
$97$	83.1714	+	83.1714i	0.857437	+	0.857437i	−0.133598	+	0.991036i	$0.542653\pi$
$98$		0			0
$99$			60.8958i			0.615109i
$100$		0			0
$101$	103.223			1.02201			0.511005	−	0.859577i	$-0.329273\pi$
$101$	103.223			1.02201			0.511005	+	0.859577i	$0.329273\pi$
$102$		0			0
$103$	63.0871	−	63.0871i	0.612496	−	0.612496i	−0.331100	−	0.943596i	$-0.607420\pi$
$103$	63.0871	−	63.0871i	0.612496	−	0.612496i	0.943596	+	0.331100i	$0.107420\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−145.533	−	145.533i	−1.36012	−	1.36012i	−0.873756	−	0.486364i	$-0.838323\pi$
$107$	−145.533	−	145.533i	−1.36012	−	1.36012i	−0.486364	−	0.873756i	$-0.661677\pi$
$108$		0			0
$109$		−	102.816i		−	0.943263i	−0.881796	−	0.471632i	$-0.843665\pi$
$109$		−	102.816i		−	0.943263i	0.881796	−	0.471632i	$-0.156335\pi$
$110$		0			0
$111$	−221.550			−1.99595
$112$		0			0
$113$	−25.2261	+	25.2261i	−0.223240	+	0.223240i	−0.809861	−	0.586621i	$-0.800458\pi$
$113$	−25.2261	+	25.2261i	−0.223240	+	0.223240i	0.586621	+	0.809861i	$0.300458\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−120.746	−	120.746i	−1.03202	−	1.03202i
$118$		0			0
$119$			157.239i			1.32133i
$120$		0			0
$121$	−59.4705			−0.491492
$122$		0			0
$123$	60.4238	−	60.4238i	0.491250	−	0.491250i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	43.0347	+	43.0347i	0.338856	+	0.338856i	0.855937	−	0.517081i	$-0.172981\pi$
$127$	43.0347	+	43.0347i	0.338856	+	0.338856i	−0.517081	+	0.855937i	$0.672981\pi$
$128$		0			0
$129$			335.254i			2.59886i
$130$		0			0
$131$	241.012			1.83979			0.919893	−	0.392170i	$-0.128276\pi$
$131$	241.012			1.83979			0.919893	+	0.392170i	$0.128276\pi$
$132$		0			0
$133$	123.860	−	123.860i	0.931275	−	0.931275i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	110.688	+	110.688i	0.807943	+	0.807943i	0.984322	−	0.176380i	$-0.0564386\pi$
$137$	110.688	+	110.688i	0.807943	+	0.807943i	−0.176380	+	0.984322i	$0.556439\pi$
$138$		0			0
$139$		−	88.2875i		−	0.635162i	−0.948231	−	0.317581i	$-0.897129\pi$
$139$		−	88.2875i		−	0.635162i	0.948231	−	0.317581i	$-0.102871\pi$
$140$		0			0
$141$	−203.761			−1.44511
$142$		0			0
$143$	−122.003	+	122.003i	−0.853166	+	0.853166i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	57.5287	+	57.5287i	0.391351	+	0.391351i
$148$		0			0
$149$		−	82.3437i		−	0.552642i	−0.961065	−	0.276321i	$-0.910885\pi$
$149$		−	82.3437i		−	0.552642i	0.961065	−	0.276321i	$-0.0891153\pi$
$150$		0			0
$151$	−0.779639			−0.00516317			−0.00258159	−	0.999997i	$-0.500822\pi$
$151$	−0.779639			−0.00516317			−0.00258159	+	0.999997i	$0.500822\pi$
$152$		0			0
$153$	−104.009	+	104.009i	−0.679798	+	0.679798i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−167.767	−	167.767i	−1.06858	−	1.06858i	−0.997468	−	0.0711145i	$-0.977344\pi$
$157$	−167.767	−	167.767i	−1.06858	−	1.06858i	−0.0711145	−	0.997468i	$-0.522656\pi$
$158$		0			0
$159$		−	132.272i		−	0.831900i
$160$		0			0
$161$	26.8499			0.166769
$162$		0			0
$163$	64.7859	−	64.7859i	0.397460	−	0.397460i	−0.479876	−	0.877336i	$-0.659318\pi$
$163$	64.7859	−	64.7859i	0.397460	−	0.397460i	0.877336	+	0.479876i	$0.159318\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	36.4422	+	36.4422i	0.218217	+	0.218217i	0.807747	−	0.589530i	$-0.200687\pi$
$167$	36.4422	+	36.4422i	0.218217	+	0.218217i	−0.589530	+	0.807747i	$0.700687\pi$
$168$		0			0
$169$		−	314.823i		−	1.86286i
$170$		0			0
$171$	163.860			0.958243
$172$		0			0
$173$	97.0027	−	97.0027i	0.560710	−	0.560710i	−0.368799	−	0.929509i	$-0.620231\pi$
$173$	97.0027	−	97.0027i	0.560710	−	0.560710i	0.929509	+	0.368799i	$0.120231\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−201.878	−	201.878i	−1.14055	−	1.14055i
$178$		0			0
$179$		−	164.996i		−	0.921767i	−0.887461	−	0.460884i	$-0.847533\pi$
$179$		−	164.996i		−	0.921767i	0.887461	−	0.460884i	$-0.152467\pi$
$180$		0			0
$181$	−67.5052			−0.372957			−0.186478	−	0.982459i	$-0.559707\pi$
$181$	−67.5052			−0.372957			−0.186478	+	0.982459i	$0.559707\pi$
$182$		0			0
$183$	−34.6491	+	34.6491i	−0.189339	+	0.189339i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	105.091	+	105.091i	0.561987	+	0.561987i
$188$		0			0
$189$		−	42.0211i		−	0.222334i
$190$		0			0
$191$	74.7924			0.391583			0.195792	−	0.980646i	$-0.437272\pi$
$191$	74.7924			0.391583			0.195792	+	0.980646i	$0.437272\pi$
$192$		0			0
$193$	33.2444	−	33.2444i	0.172251	−	0.172251i	−0.615717	−	0.787968i	$-0.711133\pi$
$193$	33.2444	−	33.2444i	0.172251	−	0.172251i	0.787968	+	0.615717i	$0.211133\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.46778	−	1.46778i	−0.00745068	−	0.00745068i	0.703372	−	0.710822i	$-0.251677\pi$
$197$	−1.46778	−	1.46778i	−0.00745068	−	0.00745068i	−0.710822	+	0.703372i	$0.751677\pi$
$198$		0			0
$199$			233.590i			1.17382i	0.809652	+	0.586910i	$0.199656\pi$
$199$			233.590i			1.17382i	−0.809652	+	0.586910i	$0.800344\pi$
$200$		0			0
$201$	159.709			0.794571
$202$		0			0
$203$	−45.1154	+	45.1154i	−0.222243	+	0.222243i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	17.7605	+	17.7605i	0.0857993	+	0.0857993i
$208$		0			0
$209$		−	165.565i		−	0.792176i
$210$		0			0
$211$	86.2848			0.408933			0.204466	−	0.978874i	$-0.434454\pi$
$211$	86.2848			0.408933			0.204466	+	0.978874i	$0.434454\pi$
$212$		0			0
$213$	151.333	−	151.333i	0.710484	−	0.710484i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	188.697	+	188.697i	0.869570	+	0.869570i
$218$		0			0
$219$		−	27.3918i		−	0.125077i
$220$		0			0
$221$	−416.758			−1.88578
$222$		0			0
$223$	53.1212	−	53.1212i	0.238212	−	0.238212i	−0.577897	−	0.816109i	$-0.696127\pi$
$223$	53.1212	−	53.1212i	0.238212	−	0.238212i	0.816109	+	0.577897i	$0.196127\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−103.694	−	103.694i	−0.456804	−	0.456804i	0.440801	−	0.897605i	$-0.354694\pi$
$227$	−103.694	−	103.694i	−0.456804	−	0.456804i	−0.897605	+	0.440801i	$0.854694\pi$
$228$		0			0
$229$			187.382i			0.818261i	0.912476	+	0.409131i	$0.134168\pi$
$229$			187.382i			0.818261i	−0.912476	+	0.409131i	$0.865832\pi$
$230$		0			0
$231$	266.526			1.15379
$232$		0			0
$233$	155.881	−	155.881i	0.669017	−	0.669017i	−0.288472	−	0.957488i	$-0.593147\pi$
$233$	155.881	−	155.881i	0.669017	−	0.669017i	0.957488	+	0.288472i	$0.0931472\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−90.8377	−	90.8377i	−0.383281	−	0.383281i
$238$		0			0
$239$			164.428i			0.687983i	0.938973	+	0.343992i	$0.111779\pi$
$239$			164.428i			0.687983i	−0.938973	+	0.343992i	$0.888221\pi$
$240$		0			0
$241$	203.663			0.845075			0.422537	−	0.906346i	$-0.361140\pi$
$241$	203.663			0.845075			0.422537	+	0.906346i	$0.361140\pi$
$242$		0			0
$243$	230.076	−	230.076i	0.946813	−	0.946813i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	328.288	+	328.288i	1.32910	+	1.32910i
$248$		0			0
$249$			192.866i			0.774564i
$250$		0			0
$251$	−30.8289			−0.122824			−0.0614121	−	0.998112i	$-0.519560\pi$
$251$	−30.8289			−0.122824			−0.0614121	+	0.998112i	$0.519560\pi$
$252$		0			0
$253$	17.9453	−	17.9453i	0.0709300	−	0.0709300i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.15015	+	6.15015i	0.0239305	+	0.0239305i	0.718971	−	0.695040i	$-0.244613\pi$
$257$	6.15015	+	6.15015i	0.0239305	+	0.0239305i	−0.695040	+	0.718971i	$0.744613\pi$
$258$		0			0
$259$			449.067i			1.73385i
$260$		0			0
$261$	−59.6852			−0.228679
$262$		0			0
$263$	293.848	−	293.848i	1.11729	−	1.11729i	0.125156	−	0.992137i	$-0.460057\pi$
$263$	293.848	−	293.848i	1.11729	−	1.11729i	0.992137	−	0.125156i	$-0.0399431\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.80571	+	1.80571i	0.00676296	+	0.00676296i
$268$		0			0
$269$			76.9052i			0.285893i	0.989730	+	0.142946i	$0.0456577\pi$
$269$			76.9052i			0.285893i	−0.989730	+	0.142946i	$0.954342\pi$
$270$		0			0
$271$	135.391			0.499596			0.249798	−	0.968298i	$-0.419636\pi$
$271$	135.391			0.499596			0.249798	+	0.968298i	$0.419636\pi$
$272$		0			0
$273$	−528.477	+	528.477i	−1.93581	+	1.93581i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	208.153	+	208.153i	0.751454	+	0.751454i	0.974751	−	0.223296i	$-0.0716817\pi$
$277$	208.153	+	208.153i	0.751454	+	0.751454i	−0.223296	+	0.974751i	$0.571682\pi$
$278$		0			0
$279$			249.636i			0.894751i
$280$		0			0
$281$	325.512			1.15840			0.579202	−	0.815184i	$-0.303364\pi$
$281$	325.512			1.15840			0.579202	+	0.815184i	$0.303364\pi$
$282$		0			0
$283$	−91.9950	+	91.9950i	−0.325071	+	0.325071i	−0.850708	−	0.525638i	$-0.823827\pi$
$283$	−91.9950	+	91.9950i	−0.325071	+	0.325071i	0.525638	+	0.850708i	$0.323827\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−122.475	−	122.475i	−0.426741	−	0.426741i
$288$		0			0
$289$			69.9897i			0.242179i
$290$		0			0
$291$	481.580			1.65492
$292$		0			0
$293$	−152.734	+	152.734i	−0.521277	+	0.521277i	−0.917957	−	0.396680i	$-0.870162\pi$
$293$	−152.734	+	152.734i	−0.521277	+	0.521277i	0.396680	+	0.917957i	$0.370162\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−28.0851	−	28.0851i	−0.0945627	−	0.0945627i
$298$		0			0
$299$			71.1651i			0.238010i
$300$		0			0
$301$	679.535			2.25759
$302$		0			0
$303$	298.842	−	298.842i	0.986277	−	0.986277i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−12.0914	−	12.0914i	−0.0393857	−	0.0393857i	0.687140	−	0.726525i	$-0.258866\pi$
$307$	−12.0914	−	12.0914i	−0.0393857	−	0.0393857i	−0.726525	+	0.687140i	$0.758866\pi$
$308$		0			0
$309$		−	365.288i		−	1.18216i
$310$		0			0
$311$	292.272			0.939781			0.469891	−	0.882725i	$-0.344293\pi$
$311$	292.272			0.939781			0.469891	+	0.882725i	$0.344293\pi$
$312$		0			0
$313$	−38.9325	+	38.9325i	−0.124385	+	0.124385i	−0.766559	−	0.642174i	$-0.778033\pi$
$313$	−38.9325	+	38.9325i	−0.124385	+	0.124385i	0.642174	+	0.766559i	$0.278033\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−46.5444	−	46.5444i	−0.146828	−	0.146828i	0.629872	−	0.776699i	$-0.283107\pi$
$317$	−46.5444	−	46.5444i	−0.146828	−	0.146828i	−0.776699	+	0.629872i	$0.783107\pi$
$318$		0			0
$319$			60.3064i			0.189048i
$320$		0			0
$321$	−842.666			−2.62513
$322$		0			0
$323$	282.782	−	282.782i	0.875487	−	0.875487i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−297.662	−	297.662i	−0.910283	−	0.910283i
$328$		0			0
$329$			413.009i			1.25535i
$330$		0			0
$331$	−536.932			−1.62215			−0.811076	−	0.584941i	$-0.801118\pi$
$331$	−536.932			−1.62215			−0.811076	+	0.584941i	$0.801118\pi$
$332$		0			0
$333$	−297.046	+	297.046i	−0.892030	+	0.892030i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	230.498	+	230.498i	0.683970	+	0.683970i	0.960892	−	0.276922i	$-0.0893145\pi$
$337$	230.498	+	230.498i	0.683970	+	0.683970i	−0.276922	+	0.960892i	$0.589314\pi$
$338$		0			0
$339$			146.065i			0.430869i
$340$		0			0
$341$	252.233			0.739687
$342$		0			0
$343$	−170.934	+	170.934i	−0.498350	+	0.498350i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	248.171	+	248.171i	0.715189	+	0.715189i	0.967616	−	0.252427i	$-0.0812287\pi$
$347$	248.171	+	248.171i	0.715189	+	0.715189i	−0.252427	+	0.967616i	$0.581229\pi$
$348$		0			0
$349$			549.768i			1.57527i	0.616144	+	0.787634i	$0.288694\pi$
$349$			549.768i			1.57527i	−0.616144	+	0.787634i	$0.711306\pi$
$350$		0			0
$351$	111.376			0.317311
$352$		0			0
$353$	365.531	−	365.531i	1.03550	−	1.03550i	0.0361533	−	0.999346i	$-0.488490\pi$
$353$	365.531	−	365.531i	1.03550	−	1.03550i	0.999346	−	0.0361533i	$-0.0115105\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	455.223	+	455.223i	1.27513	+	1.27513i
$358$		0			0
$359$		−	657.884i		−	1.83255i	−0.400553	−	0.916274i	$-0.631182\pi$
$359$		−	657.884i		−	1.83255i	0.400553	−	0.916274i	$-0.368818\pi$
$360$		0			0
$361$	−84.5052			−0.234086
$362$		0			0
$363$	−172.174	+	172.174i	−0.474307	+	0.474307i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−154.382	−	154.382i	−0.420661	−	0.420661i	0.464771	−	0.885431i	$-0.346137\pi$
$367$	−154.382	−	154.382i	−0.420661	−	0.420661i	−0.885431	+	0.464771i	$0.846137\pi$
$368$		0			0
$369$		−	162.028i		−	0.439099i
$370$		0			0
$371$	−268.106			−0.722658
$372$		0			0
$373$	146.049	−	146.049i	0.391552	−	0.391552i	−0.483688	−	0.875240i	$-0.660703\pi$
$373$	146.049	−	146.049i	0.391552	−	0.391552i	0.875240	+	0.483688i	$0.160703\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−119.578	−	119.578i	−0.317182	−	0.317182i
$378$		0			0
$379$			499.125i			1.31695i	0.752601	+	0.658477i	$0.228799\pi$
$379$			499.125i			1.31695i	−0.752601	+	0.658477i	$0.771201\pi$
$380$		0			0
$381$	249.180			0.654016
$382$		0			0
$383$	−51.0942	+	51.0942i	−0.133405	+	0.133405i	−0.770656	−	0.637251i	$-0.780071\pi$
$383$	−51.0942	+	51.0942i	−0.133405	+	0.133405i	0.637251	+	0.770656i	$0.280071\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	449.494	+	449.494i	1.16148	+	1.16148i
$388$		0			0
$389$		−	727.728i		−	1.87077i	−0.353637	−	0.935383i	$-0.615055\pi$
$389$		−	727.728i		−	1.87077i	0.353637	−	0.935383i	$-0.384945\pi$
$390$		0			0
$391$	61.3006			0.156779
$392$		0			0
$393$	697.755	−	697.755i	1.77546	−	1.77546i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−109.303	−	109.303i	−0.275323	−	0.275323i	0.555916	−	0.831238i	$-0.312368\pi$
$397$	−109.303	−	109.303i	−0.275323	−	0.275323i	−0.831238	+	0.555916i	$0.812368\pi$
$398$		0			0
$399$		−	717.173i		−	1.79743i
$400$		0			0
$401$	225.951			0.563470			0.281735	−	0.959492i	$-0.409090\pi$
$401$	225.951			0.563470			0.281735	+	0.959492i	$0.409090\pi$
$402$		0			0
$403$	−500.137	+	500.137i	−1.24104	+	1.24104i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	300.137	+	300.137i	0.737438	+	0.737438i
$408$		0			0
$409$		−	488.563i		−	1.19453i	−0.802044	−	0.597266i	$-0.796254\pi$
$409$		−	488.563i		−	1.19453i	0.802044	−	0.597266i	$-0.203746\pi$
$410$		0			0
$411$	640.908			1.55939
$412$		0			0
$413$	−409.192	+	409.192i	−0.990780	+	0.990780i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−255.602	−	255.602i	−0.612954	−	0.612954i
$418$		0			0
$419$		−	442.955i		−	1.05717i	−0.848880	−	0.528586i	$-0.822722\pi$
$419$		−	442.955i		−	1.05717i	0.848880	−	0.528586i	$-0.177278\pi$
$420$		0			0
$421$	−61.8074			−0.146811			−0.0734055	−	0.997302i	$-0.523387\pi$
$421$	−61.8074			−0.146811			−0.0734055	+	0.997302i	$0.523387\pi$
$422$		0			0
$423$	−273.195	+	273.195i	−0.645850	+	0.645850i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	70.2312	+	70.2312i	0.164476	+	0.164476i
$428$		0			0
$429$			706.422i			1.64667i
$430$		0			0
$431$	−199.702			−0.463347			−0.231673	−	0.972794i	$-0.574420\pi$
$431$	−199.702			−0.463347			−0.231673	+	0.972794i	$0.574420\pi$
$432$		0			0
$433$	−318.138	+	318.138i	−0.734730	+	0.734730i	−0.971553	−	0.236823i	$-0.923894\pi$
$433$	−318.138	+	318.138i	−0.734730	+	0.734730i	0.236823	+	0.971553i	$0.423894\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−48.2875	−	48.2875i	−0.110498	−	0.110498i
$438$		0			0
$439$			426.276i			0.971016i	0.874232	+	0.485508i	$0.161365\pi$
$439$			426.276i			0.971016i	−0.874232	+	0.485508i	$0.838635\pi$
$440$		0			0
$441$	154.264			0.349805
$442$		0			0
$443$	47.7415	−	47.7415i	0.107769	−	0.107769i	−0.651166	−	0.758935i	$-0.725720\pi$
$443$	47.7415	−	47.7415i	0.107769	−	0.107769i	0.758935	+	0.651166i	$0.225720\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−238.394	−	238.394i	−0.533319	−	0.533319i
$448$		0			0
$449$		−	112.227i		−	0.249950i	−0.992160	−	0.124975i	$-0.960115\pi$
$449$		−	112.227i		−	0.249950i	0.992160	−	0.124975i	$-0.0398850\pi$
$450$		0			0
$451$	−163.714			−0.363001
$452$		0			0
$453$	−2.25714	+	2.25714i	−0.00498265	+	0.00498265i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−436.951	−	436.951i	−0.956129	−	0.956129i	0.0429487	−	0.999077i	$-0.486325\pi$
$457$	−436.951	−	436.951i	−0.956129	−	0.956129i	−0.999077	+	0.0429487i	$0.986325\pi$
$458$		0			0
$459$		−	95.9380i		−	0.209015i
$460$		0			0
$461$	−491.174			−1.06545			−0.532727	−	0.846287i	$-0.678833\pi$
$461$	−491.174			−1.06545			−0.532727	+	0.846287i	$0.678833\pi$
$462$		0			0
$463$	−375.171	+	375.171i	−0.810304	+	0.810304i	−0.984679	−	0.174375i	$-0.944210\pi$
$463$	−375.171	+	375.171i	−0.810304	+	0.810304i	0.174375	+	0.984679i	$0.444210\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−383.352	−	383.352i	−0.820881	−	0.820881i	0.165353	−	0.986234i	$-0.447124\pi$
$467$	−383.352	−	383.352i	−0.820881	−	0.820881i	−0.986234	+	0.165353i	$0.947124\pi$
$468$		0			0
$469$		−	323.718i		−	0.690231i
$470$		0			0
$471$	−971.409			−2.06244
$472$		0			0
$473$	454.172	−	454.172i	0.960195	−	0.960195i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−177.345	−	177.345i	−0.371792	−	0.371792i
$478$		0			0
$479$		−	346.948i		−	0.724318i	−0.932116	−	0.362159i	$-0.882040\pi$
$479$		−	346.948i		−	0.724318i	0.932116	−	0.362159i	$-0.117960\pi$
$480$		0			0
$481$	−1190.24			−2.47452
$482$		0			0
$483$	77.7332	−	77.7332i	0.160938	−	0.160938i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−237.686	−	237.686i	−0.488062	−	0.488062i	0.419632	−	0.907694i	$-0.362159\pi$
$487$	−237.686	−	237.686i	−0.488062	−	0.488062i	−0.907694	+	0.419632i	$0.862159\pi$
$488$		0			0
$489$		−	375.124i		−	0.767126i
$490$		0			0
$491$	−590.829			−1.20332			−0.601659	−	0.798753i	$-0.705493\pi$
$491$	−590.829			−1.20332			−0.601659	+	0.798753i	$0.705493\pi$
$492$		0			0
$493$	−103.002	+	103.002i	−0.208930	+	0.208930i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−306.742	−	306.742i	−0.617186	−	0.617186i
$498$		0			0
$499$		−	313.486i		−	0.628229i	−0.949385	−	0.314115i	$-0.898292\pi$
$499$		−	313.486i		−	0.628229i	0.949385	−	0.314115i	$-0.101708\pi$
$500$		0			0
$501$	211.008			0.421174
$502$		0			0
$503$	545.919	−	545.919i	1.08533	−	1.08533i	0.0893233	−	0.996003i	$-0.471530\pi$
$503$	545.919	−	545.919i	1.08533	−	1.08533i	0.996003	−	0.0893233i	$-0.0284704\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−911.445	−	911.445i	−1.79772	−	1.79772i
$508$		0			0
$509$			589.253i			1.15767i	0.815445	+	0.578834i	$0.196492\pi$
$509$			589.253i			1.15767i	−0.815445	+	0.578834i	$0.803508\pi$
$510$		0			0
$511$	−55.5212			−0.108652
$512$		0			0
$513$	−75.5720	+	75.5720i	−0.147314	+	0.147314i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	276.038	+	276.038i	0.533922	+	0.533922i
$518$		0			0
$519$		−	561.667i		−	1.08221i
$520$		0			0
$521$	−726.868			−1.39514			−0.697570	−	0.716516i	$-0.745735\pi$
$521$	−726.868			−1.39514			−0.697570	+	0.716516i	$0.745735\pi$
$522$		0			0
$523$	333.243	−	333.243i	0.637176	−	0.637176i	−0.312682	−	0.949858i	$-0.601227\pi$
$523$	333.243	−	333.243i	0.637176	−	0.637176i	0.949858	+	0.312682i	$0.101227\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	430.811	+	430.811i	0.817479	+	0.817479i
$528$		0			0
$529$			518.532i			0.980212i
$530$		0			0
$531$	−541.339			−1.01947
$532$		0			0
$533$	324.617	−	324.617i	0.609038	−	0.609038i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−477.682	−	477.682i	−0.889538	−	0.889538i
$538$		0			0
$539$		−	155.870i		−	0.289183i
$540$		0			0
$541$	119.223			0.220375			0.110188	−	0.993911i	$-0.464855\pi$
$541$	119.223			0.220375			0.110188	+	0.993911i	$0.464855\pi$
$542$		0			0
$543$	−195.435	+	195.435i	−0.359916	+	0.359916i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	61.7713	+	61.7713i	0.112927	+	0.112927i	0.761313	−	0.648385i	$-0.224555\pi$
$547$	61.7713	+	61.7713i	0.112927	+	0.112927i	−0.648385	+	0.761313i	$0.724555\pi$
$548$		0			0
$549$			92.9121i			0.169239i
$550$		0			0
$551$	162.274			0.294507
$552$		0			0
$553$	−184.122	+	184.122i	−0.332950	+	0.332950i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	33.7681	+	33.7681i	0.0606249	+	0.0606249i	0.736769	−	0.676144i	$-0.236350\pi$
$557$	33.7681	+	33.7681i	0.0606249	+	0.0606249i	−0.676144	+	0.736769i	$0.736350\pi$
$558$		0			0
$559$			1801.10i			3.22200i
$560$		0			0
$561$	608.502			1.08467
$562$		0			0
$563$	−639.744	+	639.744i	−1.13631	+	1.13631i	−0.147207	+	0.989106i	$0.547028\pi$
$563$	−639.744	+	639.744i	−1.13631	+	1.13631i	−0.989106	+	0.147207i	$0.952972\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−531.663	−	531.663i	−0.937677	−	0.937677i
$568$		0			0
$569$		−	901.316i		−	1.58404i	−0.610498	−	0.792018i	$-0.709031\pi$
$569$		−	901.316i		−	1.58404i	0.610498	−	0.792018i	$-0.290969\pi$
$570$		0			0
$571$	−87.3796			−0.153029			−0.0765145	−	0.997068i	$-0.524379\pi$
$571$	−87.3796			−0.153029			−0.0765145	+	0.997068i	$0.524379\pi$
$572$		0			0
$573$	216.532	−	216.532i	0.377892	−	0.377892i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−93.0790	−	93.0790i	−0.161315	−	0.161315i	0.621834	−	0.783149i	$-0.286388\pi$
$577$	−93.0790	−	93.0790i	−0.161315	−	0.161315i	−0.783149	+	0.621834i	$0.786388\pi$
$578$		0			0
$579$		−	192.492i		−	0.332456i
$580$		0			0
$581$	390.927			0.672851
$582$		0			0
$583$	−179.191	+	179.191i	−0.307359	+	0.307359i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	487.674	+	487.674i	0.830791	+	0.830791i	0.987625	−	0.156834i	$-0.0501288\pi$
$587$	487.674	+	487.674i	0.830791	+	0.830791i	−0.156834	+	0.987625i	$0.550129\pi$
$588$		0			0
$589$		−	678.715i		−	1.15232i
$590$		0			0
$591$	−8.49878			−0.0143803
$592$		0			0
$593$	−687.372	+	687.372i	−1.15914	+	1.15914i	−0.174483	+	0.984660i	$0.555826\pi$
$593$	−687.372	+	687.372i	−1.15914	+	1.15914i	−0.984660	+	0.174483i	$0.944174\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	676.269	+	676.269i	1.13278	+	1.13278i
$598$		0			0
$599$			301.235i			0.502897i	0.967871	+	0.251448i	$0.0809069\pi$
$599$			301.235i			0.502897i	−0.967871	+	0.251448i	$0.919093\pi$
$600$		0			0
$601$	949.845			1.58044			0.790221	−	0.612823i	$-0.209966\pi$
$601$	949.845			1.58044			0.790221	+	0.612823i	$0.209966\pi$
$602$		0			0
$603$	214.131	−	214.131i	0.355109	−	0.355109i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	582.388	+	582.388i	0.959453	+	0.959453i	0.999209	−	0.0397563i	$-0.0126581\pi$
$607$	582.388	+	582.388i	0.959453	+	0.959453i	−0.0397563	+	0.999209i	$0.512658\pi$
$608$		0			0
$609$			261.228i			0.428945i
$610$		0			0
$611$	−1094.67			−1.79161
$612$		0			0
$613$	349.087	−	349.087i	0.569474	−	0.569474i	−0.362507	−	0.931981i	$-0.618079\pi$
$613$	349.087	−	349.087i	0.569474	−	0.569474i	0.931981	+	0.362507i	$0.118079\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	674.131	+	674.131i	1.09260	+	1.09260i	0.995251	+	0.0973446i	$0.0310349\pi$
$617$	674.131	+	674.131i	1.09260	+	1.09260i	0.0973446	+	0.995251i	$0.468965\pi$
$618$		0			0
$619$		−	463.613i		−	0.748971i	−0.927233	−	0.374485i	$-0.877819\pi$
$619$		−	463.613i		−	0.748971i	0.927233	−	0.374485i	$-0.122181\pi$
$620$		0			0
$621$	−16.3822			−0.0263804
$622$		0			0
$623$	3.66005	−	3.66005i	0.00587488	−	0.00587488i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−479.328	−	479.328i	−0.764478	−	0.764478i
$628$		0			0
$629$			1025.26i			1.62998i
$630$		0			0
$631$	−1105.31			−1.75167			−0.875837	−	0.482607i	$-0.839690\pi$
$631$	−1105.31			−1.75167			−0.875837	+	0.482607i	$0.839690\pi$
$632$		0			0
$633$	249.804	−	249.804i	0.394635	−	0.394635i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	309.064	+	309.064i	0.485186	+	0.485186i
$638$		0			0
$639$		−	405.803i		−	0.635059i
$640$		0			0
$641$	737.758			1.15095			0.575474	−	0.817820i	$-0.304818\pi$
$641$	737.758			1.15095			0.575474	+	0.817820i	$0.304818\pi$
$642$		0			0
$643$	−470.964	+	470.964i	−0.732448	+	0.732448i	−0.971104	−	0.238656i	$-0.923293\pi$
$643$	−470.964	+	470.964i	−0.732448	+	0.732448i	0.238656	+	0.971104i	$0.423293\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−571.622	−	571.622i	−0.883497	−	0.883497i	0.110391	−	0.993888i	$-0.464790\pi$
$647$	−571.622	−	571.622i	−0.883497	−	0.883497i	−0.993888	+	0.110391i	$0.964790\pi$
$648$		0			0
$649$			546.973i			0.842793i
$650$		0			0
$651$	1092.59			1.67833
$652$		0			0
$653$	−872.172	+	872.172i	−1.33564	+	1.33564i	−0.435402	+	0.900236i	$0.643394\pi$
$653$	−872.172	+	872.172i	−1.33564	+	1.33564i	−0.900236	+	0.435402i	$0.856606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−36.7258	−	36.7258i	−0.0558992	−	0.0558992i
$658$		0			0
$659$		−	767.847i		−	1.16517i	−0.812770	−	0.582585i	$-0.802041\pi$
$659$		−	767.847i		−	1.16517i	0.812770	−	0.582585i	$-0.197959\pi$
$660$		0			0
$661$	−390.199			−0.590316			−0.295158	−	0.955448i	$-0.595372\pi$
$661$	−390.199			−0.590316			−0.295158	+	0.955448i	$0.595372\pi$
$662$		0			0
$663$	−1206.56	+	1206.56i	−1.81985	+	1.81985i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.5886	+	17.5886i	0.0263697	+	0.0263697i
$668$		0			0
$669$		−	307.583i		−	0.459766i
$670$		0			0
$671$	93.8790			0.139909
$672$		0			0
$673$	425.311	−	425.311i	0.631963	−	0.631963i	−0.316597	−	0.948560i	$-0.602540\pi$
$673$	425.311	−	425.311i	0.631963	−	0.631963i	0.948560	+	0.316597i	$0.102540\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	420.208	+	420.208i	0.620691	+	0.620691i	0.945708	−	0.325017i	$-0.105370\pi$
$677$	420.208	+	420.208i	0.620691	+	0.620691i	−0.325017	+	0.945708i	$0.605370\pi$
$678$		0			0
$679$		−	976.129i		−	1.43760i
$680$		0			0
$681$	−600.413			−0.881664
$682$		0			0
$683$	132.246	−	132.246i	0.193625	−	0.193625i	−0.603636	−	0.797260i	$-0.706282\pi$
$683$	132.246	−	132.246i	0.193625	−	0.193625i	0.797260	+	0.603636i	$0.206282\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	542.490	+	542.490i	0.789651	+	0.789651i
$688$		0			0
$689$		−	710.610i		−	1.03136i
$690$		0			0
$691$	95.1407			0.137686			0.0688428	−	0.997628i	$-0.478069\pi$
$691$	95.1407			0.137686			0.0688428	+	0.997628i	$0.478069\pi$
$692$		0			0
$693$	357.347	−	357.347i	0.515652	−	0.515652i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−279.621	−	279.621i	−0.401177	−	0.401177i
$698$		0			0
$699$		−	902.583i		−	1.29125i
$700$		0			0
$701$	81.3946			0.116112			0.0580560	−	0.998313i	$-0.481510\pi$
$701$	81.3946			0.116112			0.0580560	+	0.998313i	$0.481510\pi$
$702$		0			0
$703$	807.615	−	807.615i	1.14881	−	1.14881i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−605.731	−	605.731i	−0.856763	−	0.856763i
$708$		0			0
$709$			205.828i			0.290307i	0.989409	+	0.145154i	$0.0463677\pi$
$709$			205.828i			0.290307i	−0.989409	+	0.145154i	$0.953632\pi$
$710$		0			0
$711$	−243.583			−0.342592
$712$		0			0
$713$	73.5648	−	73.5648i	0.103176	−	0.103176i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	476.036	+	476.036i	0.663928	+	0.663928i
$718$		0			0
$719$			26.4486i			0.0367852i	0.999831	+	0.0183926i	$0.00585488\pi$
$719$			26.4486i			0.0367852i	−0.999831	+	0.0183926i	$0.994145\pi$
$720$		0			0
$721$	−740.412			−1.02692
$722$		0			0
$723$	589.626	−	589.626i	0.815527	−	0.815527i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	942.528	+	942.528i	1.29646	+	1.29646i	0.930714	+	0.365748i	$0.119187\pi$
$727$	942.528	+	942.528i	1.29646	+	1.29646i	0.365748	+	0.930714i	$0.380813\pi$
$728$		0			0
$729$		−	516.778i		−	0.708886i
$730$		0			0
$731$	1551.44			2.12235
$732$		0			0
$733$	−849.970	+	849.970i	−1.15958	+	1.15958i	−0.175010	+	0.984567i	$0.555996\pi$
$733$	−849.970	+	849.970i	−1.15958	+	1.15958i	−0.984567	+	0.175010i	$0.944004\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−216.359	−	216.359i	−0.293568	−	0.293568i
$738$		0			0
$739$			425.070i			0.575197i	0.957751	+	0.287598i	$0.0928568\pi$
$739$			425.070i			0.575197i	−0.957751	+	0.287598i	$0.907143\pi$
$740$		0			0
$741$	1900.85			2.56526
$742$		0			0
$743$	−162.191	+	162.191i	−0.218293	+	0.218293i	−0.807779	−	0.589486i	$-0.799330\pi$
$743$	−162.191	+	162.191i	−0.218293	+	0.218293i	0.589486	+	0.807779i	$0.299330\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	258.587	+	258.587i	0.346168	+	0.346168i
$748$		0			0
$749$			1708.03i			2.28041i
$750$		0			0
$751$	−232.043			−0.308979			−0.154489	−	0.987994i	$-0.549373\pi$
$751$	−232.043			−0.308979			−0.154489	+	0.987994i	$0.549373\pi$
$752$		0			0
$753$	−89.2529	+	89.2529i	−0.118530	+	0.118530i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−620.397	−	620.397i	−0.819546	−	0.819546i	0.166496	−	0.986042i	$-0.446755\pi$
$757$	−620.397	−	620.397i	−0.819546	−	0.819546i	−0.986042	+	0.166496i	$0.946755\pi$
$758$		0			0
$759$		−	103.907i		−	0.136900i
$760$		0			0
$761$	−405.339			−0.532639			−0.266320	−	0.963885i	$-0.585808\pi$
$761$	−405.339			−0.532639			−0.266320	+	0.963885i	$0.585808\pi$
$762$		0			0
$763$	−603.341	+	603.341i	−0.790748	+	0.790748i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1084.56	−	1084.56i	−1.41402	−	1.41402i
$768$		0			0
$769$			1181.58i			1.53651i	0.640144	+	0.768255i	$0.278875\pi$
$769$			1181.58i			1.53651i	−0.640144	+	0.768255i	$0.721125\pi$
$770$		0			0
$771$	35.6107			0.0461876
$772$		0			0
$773$	−898.811	+	898.811i	−1.16276	+	1.16276i	−0.178887	+	0.983870i	$0.557250\pi$
$773$	−898.811	+	898.811i	−1.16276	+	1.16276i	−0.983870	+	0.178887i	$0.942750\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1300.10	+	1300.10i	1.67323	+	1.67323i
$778$		0			0
$779$			440.524i			0.565499i
$780$		0			0
$781$	−410.026			−0.525001
$782$		0			0
$783$	27.5268	−	27.5268i	0.0351556	−	0.0351556i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	772.073	+	772.073i	0.981033	+	0.981033i	0.999823	−	0.0187904i	$-0.00598152\pi$
$787$	772.073	+	772.073i	0.981033	+	0.981033i	−0.0187904	+	0.999823i	$0.505982\pi$
$788$		0			0
$789$		−	1701.44i		−	2.15645i
$790$		0			0
$791$	296.063			0.374289
$792$		0			0
$793$	−186.147	+	186.147i	−0.234737	+	0.234737i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1063.35	−	1063.35i	−1.33419	−	1.33419i	−0.901580	−	0.432613i	$-0.857592\pi$
$797$	−1063.35	−	1063.35i	−1.33419	−	1.33419i	−0.432613	−	0.901580i	$-0.642408\pi$
$798$		0			0
$799$			942.937i			1.18015i
$800$		0			0
$801$	4.84205			0.00604500
$802$		0			0
$803$	−37.1080	+	37.1080i	−0.0462117	+	0.0462117i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	222.649	+	222.649i	0.275897	+	0.275897i
$808$		0			0
$809$			479.496i			0.592702i	0.955079	+	0.296351i	$0.0957697\pi$
$809$			479.496i			0.592702i	−0.955079	+	0.296351i	$0.904230\pi$
$810$		0			0
$811$	−576.200			−0.710481			−0.355241	−	0.934775i	$-0.615601\pi$
$811$	−576.200			−0.710481			−0.355241	+	0.934775i	$0.615601\pi$
$812$		0			0
$813$	391.970	−	391.970i	0.482128	−	0.482128i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1222.10	−	1222.10i	−1.49583	−	1.49583i
$818$		0			0
$819$			1417.12i			1.73031i
$820$		0			0
$821$	−259.667			−0.316281			−0.158141	−	0.987417i	$-0.550550\pi$
$821$	−259.667			−0.316281			−0.158141	+	0.987417i	$0.550550\pi$
$822$		0			0
$823$	−346.123	+	346.123i	−0.420562	+	0.420562i	−0.885397	−	0.464835i	$-0.846114\pi$
$823$	−346.123	+	346.123i	−0.420562	+	0.420562i	0.464835	+	0.885397i	$0.346114\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	72.3467	+	72.3467i	0.0874809	+	0.0874809i	0.749493	−	0.662012i	$-0.230297\pi$
$827$	72.3467	+	72.3467i	0.0874809	+	0.0874809i	−0.662012	+	0.749493i	$0.730297\pi$
$828$		0			0
$829$		−	1293.19i		−	1.55994i	−0.625820	−	0.779968i	$-0.715235\pi$
$829$		−	1293.19i		−	1.55994i	0.625820	−	0.779968i	$-0.284765\pi$
$830$		0			0
$831$	1205.25			1.45036
$832$		0			0
$833$	266.223	−	266.223i	0.319596	−	0.319596i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−115.132	−	115.132i	−0.137553	−	0.137553i
$838$		0			0
$839$		−	648.648i		−	0.773120i	−0.922264	−	0.386560i	$-0.873663\pi$
$839$		−	648.648i		−	0.773120i	0.922264	−	0.386560i	$-0.126337\pi$
$840$		0			0
$841$	781.892			0.929718
$842$		0			0
$843$	942.390	−	942.390i	1.11790	−	1.11790i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	348.984	+	348.984i	0.412023	+	0.412023i
$848$		0			0
$849$			532.671i			0.627409i
$850$		0			0
$851$	175.072			0.205725
$852$		0			0
$853$	568.257	−	568.257i	0.666186	−	0.666186i	−0.290645	−	0.956831i	$-0.593870\pi$
$853$	568.257	−	568.257i	0.666186	−	0.666186i	0.956831	+	0.290645i	$0.0938698\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	989.843	+	989.843i	1.15501	+	1.15501i	0.985534	+	0.169475i	$0.0542072\pi$
$857$	989.843	+	989.843i	1.15501	+	1.15501i	0.169475	+	0.985534i	$0.445793\pi$
$858$		0			0
$859$			1433.95i			1.66933i	0.550761	+	0.834663i	$0.314337\pi$
$859$			1433.95i			1.66933i	−0.550761	+	0.834663i	$0.685663\pi$
$860$		0			0
$861$	−709.155			−0.823641
$862$		0			0
$863$	581.248	−	581.248i	0.673520	−	0.673520i	−0.285006	−	0.958526i	$-0.591996\pi$
$863$	581.248	−	581.248i	0.673520	−	0.673520i	0.958526	+	0.285006i	$0.0919955\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	202.628	+	202.628i	0.233711	+	0.233711i
$868$		0			0
$869$			246.118i			0.283220i
$870$		0			0
$871$	858.009			0.985085
$872$		0			0
$873$	645.683	−	645.683i	0.739614	−	0.739614i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−197.335	−	197.335i	−0.225011	−	0.225011i	0.585594	−	0.810605i	$-0.300861\pi$
$877$	−197.335	−	197.335i	−0.225011	−	0.225011i	−0.810605	+	0.585594i	$0.800861\pi$
$878$		0			0
$879$			884.363i			1.00610i
$880$		0			0
$881$	−1314.80			−1.49240			−0.746200	−	0.665722i	$-0.768124\pi$
$881$	−1314.80			−1.49240			−0.746200	+	0.665722i	$0.768124\pi$
$882$		0			0
$883$	803.073	−	803.073i	0.909482	−	0.909482i	−0.0867479	−	0.996230i	$-0.527647\pi$
$883$	803.073	−	803.073i	0.909482	−	0.909482i	0.996230	+	0.0867479i	$0.0276475\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−503.101	−	503.101i	−0.567193	−	0.567193i	0.364148	−	0.931341i	$-0.381360\pi$
$887$	−503.101	−	503.101i	−0.567193	−	0.567193i	−0.931341	+	0.364148i	$0.881360\pi$
$888$		0			0
$889$		−	505.070i		−	0.568133i
$890$		0			0
$891$	−710.681			−0.797621
$892$		0			0
$893$	742.768	−	742.768i	0.831767	−	0.831767i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	206.030	+	206.030i	0.229688	+	0.229688i
$898$		0			0
$899$			247.219i			0.274994i
$900$		0			0
$901$	−612.110			−0.679367
$902$		0			0
$903$	1967.33	−	1967.33i	2.17866	−	2.17866i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−127.617	−	127.617i	−0.140702	−	0.140702i	0.633248	−	0.773949i	$-0.281722\pi$
$907$	−127.617	−	127.617i	−0.140702	−	0.140702i	−0.773949	+	0.633248i	$0.781722\pi$
$908$		0			0
$909$		−	801.350i		−	0.881573i
$910$		0			0
$911$	1014.55			1.11366			0.556831	−	0.830626i	$-0.312017\pi$
$911$	1014.55			1.11366			0.556831	+	0.830626i	$0.312017\pi$
$912$		0			0
$913$	261.278	−	261.278i	0.286176	−	0.286176i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1414.30	−	1414.30i	−1.54231	−	1.54231i
$918$		0			0
$919$			1263.16i			1.37450i	0.726422	+	0.687249i	$0.241182\pi$
$919$			1263.16i			1.37450i	−0.726422	+	0.687249i	$0.758818\pi$
$920$		0			0
$921$	−70.0119			−0.0760173
$922$		0			0
$923$	813.013	−	813.013i	0.880838	−	0.880838i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−489.763	−	489.763i	−0.528331	−	0.528331i
$928$		0			0
$929$		−	23.8641i		−	0.0256880i	−0.999918	−	0.0128440i	$-0.995912\pi$
$929$		−	23.8641i		−	0.0256880i	0.999918	−	0.0128440i	$-0.00408848\pi$
$930$		0			0
$931$	−419.417			−0.450502
$932$		0			0
$933$	846.159	−	846.159i	0.906923	−	0.906923i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−514.151	−	514.151i	−0.548720	−	0.548720i	0.377350	−	0.926071i	$-0.376835\pi$
$937$	−514.151	−	514.151i	−0.548720	−	0.548720i	−0.926071	+	0.377350i	$0.876835\pi$
$938$		0			0
$939$			225.428i			0.240072i
$940$		0			0
$941$	−73.6230			−0.0782391			−0.0391196	−	0.999235i	$-0.512455\pi$
$941$	−73.6230			−0.0782391			−0.0391196	+	0.999235i	$0.512455\pi$
$942$		0			0
$943$	−47.7477	+	47.7477i	−0.0506338	+	0.0506338i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	406.273	+	406.273i	0.429010	+	0.429010i	0.888291	−	0.459281i	$-0.151893\pi$
$947$	406.273	+	406.273i	0.429010	+	0.429010i	−0.459281	+	0.888291i	$0.651893\pi$
$948$		0			0
$949$		−	147.158i		−	0.155066i
$950$		0			0
$951$	−269.502			−0.283388
$952$		0			0
$953$	504.112	−	504.112i	0.528974	−	0.528974i	−0.391293	−	0.920266i	$-0.627972\pi$
$953$	504.112	−	504.112i	0.528974	−	0.528974i	0.920266	+	0.391293i	$0.127972\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	174.593	+	174.593i	0.182438	+	0.182438i
$958$		0			0
$959$		−	1299.08i		−	1.35461i
$960$		0			0
$961$	73.0037			0.0759664
$962$		0			0
$963$	−1129.81	+	1129.81i	−1.17322	+	1.17322i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−796.532	−	796.532i	−0.823714	−	0.823714i	0.162924	−	0.986639i	$-0.447907\pi$
$967$	−796.532	−	796.532i	−0.823714	−	0.823714i	−0.986639	+	0.162924i	$0.947907\pi$
$968$		0			0
$969$		−	1637.37i		−	1.68975i
$970$		0			0
$971$	1638.59			1.68753			0.843764	−	0.536714i	$-0.180335\pi$
$971$	1638.59			1.68753			0.843764	+	0.536714i	$0.180335\pi$
$972$		0			0
$973$	−518.087	+	518.087i	−0.532463	+	0.532463i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−318.602	−	318.602i	−0.326103	−	0.326103i	0.525000	−	0.851102i	$-0.324065\pi$
$977$	−318.602	−	318.602i	−0.326103	−	0.326103i	−0.851102	+	0.525000i	$0.824065\pi$
$978$		0			0
$979$		−	4.89244i		−	0.00499738i
$980$		0			0
$981$	−798.187			−0.813647
$982$		0			0
$983$	−396.649	+	396.649i	−0.403508	+	0.403508i	−0.879467	−	0.475959i	$-0.842101\pi$
$983$	−396.649	+	396.649i	−0.403508	+	0.403508i	0.475959	+	0.879467i	$0.342101\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1195.71	+	1195.71i	1.21146	+	1.21146i
$988$		0			0
$989$		−	264.922i		−	0.267868i
$990$		0			0
$991$	34.6794			0.0349943			0.0174972	−	0.999847i	$-0.494430\pi$
$991$	34.6794			0.0349943			0.0174972	+	0.999847i	$0.494430\pi$
$992$		0			0
$993$	−1554.48	+	1554.48i	−1.56543	+	1.56543i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1115.88	+	1115.88i	1.11924	+	1.11924i	0.991853	+	0.127389i	$0.0406597\pi$
$997$	1115.88	+	1115.88i	1.11924	+	1.11924i	0.127389	+	0.991853i	$0.459340\pi$
$998$		0			0
$999$		−	273.995i		−	0.274269i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.3.p.i.257.3		6
4.3	odd	2		800.3.p.j.257.1		6
5.2	odd	4		160.3.p.f.33.1	yes	6
5.3	odd	4	inner	800.3.p.i.193.3		6
5.4	even	2		160.3.p.f.97.1	yes	6
20.3	even	4		800.3.p.j.193.1		6
20.7	even	4		160.3.p.e.33.3	✓	6
20.19	odd	2		160.3.p.e.97.3	yes	6
40.19	odd	2		320.3.p.m.257.1		6
40.27	even	4		320.3.p.m.193.1		6
40.29	even	2		320.3.p.n.257.3		6
40.37	odd	4		320.3.p.n.193.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.p.e.33.3	✓	6	20.7	even	4
160.3.p.e.97.3	yes	6	20.19	odd	2
160.3.p.f.33.1	yes	6	5.2	odd	4
160.3.p.f.97.1	yes	6	5.4	even	2
320.3.p.m.193.1		6	40.27	even	4
320.3.p.m.257.1		6	40.19	odd	2
320.3.p.n.193.3		6	40.37	odd	4
320.3.p.n.257.3		6	40.29	even	2
800.3.p.i.193.3		6	5.3	odd	4	inner
800.3.p.i.257.3		6	1.1	even	1	trivial
800.3.p.j.193.1		6	20.3	even	4
800.3.p.j.257.1		6	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q+(2.89511 - 2.89511i) q^{3} +(-5.86818 - 5.86818i) q^{7} -7.76328i q^{9} +O(q^{10})\)	\(q+(2.89511 - 2.89511i) q^{3} +(-5.86818 - 5.86818i) q^{7} -7.76328i q^{9} -7.84407 q^{11} +(15.5535 - 15.5535i) q^{13} +(-13.3976 - 13.3976i) q^{17} +21.1070i q^{19} -33.9780 q^{21} +(-2.28775 + 2.28775i) q^{23} +(3.58043 + 3.58043i) q^{27} -7.68814i q^{29} -32.1559 q^{31} +(-22.7094 + 22.7094i) q^{33} +(-38.2629 - 38.2629i) q^{37} -90.0581i q^{39} +20.8710 q^{41} +(-57.9000 + 57.9000i) q^{43} +(-35.1906 - 35.1906i) q^{47} +19.8710i q^{49} -77.5748 q^{51} +(22.8441 - 22.8441i) q^{53} +(61.1070 + 61.1070i) q^{57} -69.7307i q^{59} -11.9681 q^{61} +(-45.5563 + 45.5563i) q^{63} +(27.5825 + 27.5825i) q^{67} +13.2466i q^{69} +52.2720 q^{71} +(4.73070 - 4.73070i) q^{73} +(46.0304 + 46.0304i) q^{77} -31.3763i q^{79} +90.6010 q^{81} +(-33.3090 + 33.3090i) q^{83} +(-22.2580 - 22.2580i) q^{87} +0.623711i q^{89} -182.541 q^{91} +(-93.0948 + 93.0948i) q^{93} +(83.1714 + 83.1714i) q^{97} +60.8958i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{7}+O(q^{10}) \) 6 * q - 8 * q^7	\( 6 q - 8 q^{7} - 32 q^{11} + 14 q^{13} + 14 q^{17} - 40 q^{21} - 56 q^{23} - 48 q^{27} - 208 q^{31} - 72 q^{33} - 86 q^{37} + 120 q^{41} - 176 q^{43} - 104 q^{47} - 352 q^{51} + 122 q^{53} + 208 q^{57} - 216 q^{61} - 216 q^{63} - 80 q^{67} - 336 q^{71} - 70 q^{73} - 264 q^{77} + 242 q^{81} - 208 q^{83} - 144 q^{87} - 544 q^{91} + 72 q^{93} + 250 q^{97}+O(q^{100}) \) 6 * q - 8 * q^7 - 32 * q^11 + 14 * q^13 + 14 * q^17 - 40 * q^21 - 56 * q^23 - 48 * q^27 - 208 * q^31 - 72 * q^33 - 86 * q^37 + 120 * q^41 - 176 * q^43 - 104 * q^47 - 352 * q^51 + 122 * q^53 + 208 * q^57 - 216 * q^61 - 216 * q^63 - 80 * q^67 - 336 * q^71 - 70 * q^73 - 264 * q^77 + 242 * q^81 - 208 * q^83 - 144 * q^87 - 544 * q^91 + 72 * q^93 + 250 * q^97

Introduction

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