LMFDB - Embedded newform 800.2.o.h.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(143,800)]

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([2, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("800.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.3317760000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 16 $ x^8 - 7*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.1
Root			$-0.178197 + 1.40294i$ of defining polynomial
Character	$\chi$	$=$	800.143
Dual form			800.2.o.h.207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(-3.16228 - 3.16228i) q^{7} +O(q^{10})$	$q+(-1.22474 - 1.22474i) q^{3} +(-3.16228 - 3.16228i) q^{7} +1.00000 q^{11} +(3.16228 - 3.16228i) q^{13} +(-3.67423 + 3.67423i) q^{17} +3.00000i q^{19} +7.74597i q^{21} +(-3.67423 + 3.67423i) q^{27} -7.74597 q^{29} +(-1.22474 - 1.22474i) q^{33} +(-3.16228 - 3.16228i) q^{37} -7.74597 q^{39} -1.00000 q^{41} +(2.44949 + 2.44949i) q^{43} +(-3.16228 - 3.16228i) q^{47} +13.0000i q^{49} +9.00000 q^{51} +(-6.32456 + 6.32456i) q^{53} +(3.67423 - 3.67423i) q^{57} +4.00000i q^{59} -7.74597i q^{61} +(3.67423 - 3.67423i) q^{67} +7.74597i q^{71} +(1.22474 + 1.22474i) q^{73} +(-3.16228 - 3.16228i) q^{77} -7.74597 q^{79} +9.00000 q^{81} +(-1.22474 - 1.22474i) q^{83} +(9.48683 + 9.48683i) q^{87} -13.0000i q^{89} -20.0000 q^{91} +(-4.89898 + 4.89898i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} - 8 q^{41} + 72 q^{51} + 72 q^{81} - 160 q^{91}+O(q^{100}) $ 8 * q + 8 * q^11 - 8 * q^41 + 72 * q^51 + 72 * q^81 - 160 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.16228	−	3.16228i	−1.19523	−	1.19523i	−0.975579	−	0.219650i	$-0.929509\pi$
$7$	−3.16228	−	3.16228i	−1.19523	−	1.19523i	−0.219650	−	0.975579i	$-0.570491\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000			0.301511			0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000			0.301511			0.150756	+	0.988571i	$0.451829\pi$
$12$		0			0
$13$	3.16228	−	3.16228i	0.877058	−	0.877058i	−0.116171	−	0.993229i	$-0.537062\pi$
$13$	3.16228	−	3.16228i	0.877058	−	0.877058i	0.993229	+	0.116171i	$0.0370621\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.67423	+	3.67423i	−0.891133	+	0.891133i	−0.994630	−	0.103497i	$-0.966997\pi$
$17$	−3.67423	+	3.67423i	−0.891133	+	0.891133i	0.103497	+	0.994630i	$0.466997\pi$
$18$		0			0
$19$			3.00000i			0.688247i	0.938924	+	0.344124i	$0.111824\pi$
$19$			3.00000i			0.688247i	−0.938924	+	0.344124i	$0.888176\pi$
$20$		0			0
$21$			7.74597i			1.69031i
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$	−7.74597			−1.43839			−0.719195	−	0.694808i	$-0.755489\pi$
$29$	−7.74597			−1.43839			−0.719195	+	0.694808i	$0.755489\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−1.22474	−	1.22474i	−0.213201	−	0.213201i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.16228	−	3.16228i	−0.519875	−	0.519875i	0.397658	−	0.917534i	$-0.369823\pi$
$37$	−3.16228	−	3.16228i	−0.519875	−	0.519875i	−0.917534	+	0.397658i	$0.869823\pi$
$38$		0			0
$39$	−7.74597			−1.24035
$40$		0			0
$41$	−1.00000			−0.156174			−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000			−0.156174			−0.0780869	+	0.996947i	$0.524881\pi$
$42$		0			0
$43$	2.44949	+	2.44949i	0.373544	+	0.373544i	0.868766	−	0.495222i	$-0.164913\pi$
$43$	2.44949	+	2.44949i	0.373544	+	0.373544i	−0.495222	+	0.868766i	$0.664913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.16228	−	3.16228i	−0.461266	−	0.461266i	0.437805	−	0.899070i	$-0.355756\pi$
$47$	−3.16228	−	3.16228i	−0.461266	−	0.461266i	−0.899070	+	0.437805i	$0.855756\pi$
$48$		0			0
$49$			13.0000i			1.85714i
$50$		0			0
$51$	9.00000			1.26025
$52$		0			0
$53$	−6.32456	+	6.32456i	−0.868744	+	0.868744i	−0.992333	−	0.123589i	$-0.960560\pi$
$53$	−6.32456	+	6.32456i	−0.868744	+	0.868744i	0.123589	+	0.992333i	$0.460560\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.67423	−	3.67423i	0.486664	−	0.486664i
$58$		0			0
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		0			0
$61$		−	7.74597i		−	0.991769i	−0.868388	−	0.495885i	$-0.834844\pi$
$61$		−	7.74597i		−	0.991769i	0.868388	−	0.495885i	$-0.165156\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.67423	−	3.67423i	0.448879	−	0.448879i	−0.446103	−	0.894982i	$-0.647188\pi$
$67$	3.67423	−	3.67423i	0.448879	−	0.448879i	0.894982	+	0.446103i	$0.147188\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.74597i			0.919277i	0.888106	+	0.459639i	$0.152021\pi$
$71$			7.74597i			0.919277i	−0.888106	+	0.459639i	$0.847979\pi$
$72$		0			0
$73$	1.22474	+	1.22474i	0.143346	+	0.143346i	0.775138	−	0.631792i	$-0.217680\pi$
$73$	1.22474	+	1.22474i	0.143346	+	0.143346i	−0.631792	+	0.775138i	$0.717680\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.16228	−	3.16228i	−0.360375	−	0.360375i
$78$		0			0
$79$	−7.74597			−0.871489			−0.435745	−	0.900070i	$-0.643515\pi$
$79$	−7.74597			−0.871489			−0.435745	+	0.900070i	$0.643515\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	−1.22474	−	1.22474i	−0.134433	−	0.134433i	0.636688	−	0.771121i	$-0.280304\pi$
$83$	−1.22474	−	1.22474i	−0.134433	−	0.134433i	−0.771121	+	0.636688i	$0.780304\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	9.48683	+	9.48683i	1.01710	+	1.01710i
$88$		0			0
$89$		−	13.0000i		−	1.37800i	−0.724763	−	0.688999i	$-0.758051\pi$
$89$		−	13.0000i		−	1.37800i	0.724763	−	0.688999i	$-0.241949\pi$
$90$		0			0
$91$	−20.0000			−2.09657
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	−0.910633	−	0.413217i	$-0.864405\pi$
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	0.413217	+	0.910633i	$0.364405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	7.74597i		−	0.770752i	−0.922760	−	0.385376i	$-0.874072\pi$
$101$		−	7.74597i		−	0.770752i	0.922760	−	0.385376i	$-0.125928\pi$
$102$		0			0
$103$	3.16228	−	3.16228i	0.311588	−	0.311588i	−0.533936	−	0.845525i	$-0.679288\pi$
$103$	3.16228	−	3.16228i	0.311588	−	0.311588i	0.845525	+	0.533936i	$0.179288\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.57321	+	8.57321i	−0.828804	+	0.828804i	−0.987351	−	0.158547i	$-0.949319\pi$
$107$	−8.57321	+	8.57321i	−0.828804	+	0.828804i	0.158547	+	0.987351i	$0.449319\pi$
$108$		0			0
$109$	7.74597			0.741929			0.370965	−	0.928647i	$-0.379027\pi$
$109$	7.74597			0.741929			0.370965	+	0.928647i	$0.379027\pi$
$110$		0			0
$111$			7.74597i			0.735215i
$112$		0			0
$113$	1.22474	+	1.22474i	0.115214	+	0.115214i	0.762363	−	0.647149i	$-0.224039\pi$
$113$	1.22474	+	1.22474i	0.115214	+	0.115214i	−0.647149	+	0.762363i	$0.724039\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	23.2379			2.13021
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$	1.22474	+	1.22474i	0.110432	+	0.110432i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.32456	+	6.32456i	0.561214	+	0.561214i	0.929652	−	0.368439i	$-0.120108\pi$
$127$	6.32456	+	6.32456i	0.561214	+	0.561214i	−0.368439	+	0.929652i	$0.620108\pi$
$128$		0			0
$129$		−	6.00000i		−	0.528271i
$130$		0			0
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$	9.48683	−	9.48683i	0.822613	−	0.822613i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.57321	−	8.57321i	0.732459	−	0.732459i	−0.238647	−	0.971106i	$-0.576704\pi$
$137$	8.57321	−	8.57321i	0.732459	−	0.732459i	0.971106	+	0.238647i	$0.0767040\pi$
$138$		0			0
$139$		−	17.0000i		−	1.44192i	−0.692976	−	0.720961i	$-0.743701\pi$
$139$		−	17.0000i		−	1.44192i	0.692976	−	0.720961i	$-0.256299\pi$
$140$		0			0
$141$			7.74597i			0.652328i
$142$		0			0
$143$	3.16228	−	3.16228i	0.264443	−	0.264443i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	15.9217	−	15.9217i	1.31320	−	1.31320i
$148$		0			0
$149$	7.74597			0.634574			0.317287	−	0.948330i	$-0.397228\pi$
$149$	7.74597			0.634574			0.317287	+	0.948330i	$0.397228\pi$
$150$		0			0
$151$		−	23.2379i		−	1.89107i	−0.325515	−	0.945537i	$-0.605538\pi$
$151$		−	23.2379i		−	1.89107i	0.325515	−	0.945537i	$-0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.32456	+	6.32456i	0.504754	+	0.504754i	0.912912	−	0.408157i	$-0.133828\pi$
$157$	6.32456	+	6.32456i	0.504754	+	0.504754i	−0.408157	+	0.912912i	$0.633828\pi$
$158$		0			0
$159$	15.4919			1.22859
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−13.4722	−	13.4722i	−1.05522	−	1.05522i	−0.998383	−	0.0568404i	$-0.981897\pi$
$163$	−13.4722	−	13.4722i	−1.05522	−	1.05522i	−0.0568404	−	0.998383i	$-0.518103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.48683	−	9.48683i	−0.734113	−	0.734113i	0.237319	−	0.971432i	$-0.423731\pi$
$167$	−9.48683	−	9.48683i	−0.734113	−	0.734113i	−0.971432	+	0.237319i	$0.923731\pi$
$168$		0			0
$169$		−	7.00000i		−	0.538462i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.6491	+	12.6491i	−0.961694	+	0.961694i	−0.999293	−	0.0375988i	$-0.988029\pi$
$173$	−12.6491	+	12.6491i	−0.961694	+	0.961694i	0.0375988	+	0.999293i	$0.488029\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.89898	−	4.89898i	0.368230	−	0.368230i
$178$		0			0
$179$		−	7.00000i		−	0.523205i	−0.965176	−	0.261602i	$-0.915749\pi$
$179$		−	7.00000i		−	0.523205i	0.965176	−	0.261602i	$-0.0842509\pi$
$180$		0			0
$181$		−	15.4919i		−	1.15151i	−0.817624	−	0.575753i	$-0.804709\pi$
$181$		−	15.4919i		−	1.15151i	0.817624	−	0.575753i	$-0.195291\pi$
$182$		0			0
$183$	−9.48683	+	9.48683i	−0.701287	+	0.701287i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.67423	+	3.67423i	−0.268687	+	0.268687i
$188$		0			0
$189$	23.2379			1.69031
$190$		0			0
$191$		−	15.4919i		−	1.12096i	−0.828169	−	0.560478i	$-0.810617\pi$
$191$		−	15.4919i		−	1.12096i	0.828169	−	0.560478i	$-0.189383\pi$
$192$		0			0
$193$	−11.0227	−	11.0227i	−0.793432	−	0.793432i	0.188619	−	0.982050i	$-0.439599\pi$
$193$	−11.0227	−	11.0227i	−0.793432	−	0.793432i	−0.982050	+	0.188619i	$0.939599\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.9737	+	18.9737i	1.35182	+	1.35182i	0.883628	+	0.468190i	$0.155094\pi$
$197$	18.9737	+	18.9737i	1.35182	+	1.35182i	0.468190	+	0.883628i	$0.344906\pi$
$198$		0			0
$199$	−15.4919			−1.09819			−0.549097	−	0.835759i	$-0.685028\pi$
$199$	−15.4919			−1.09819			−0.549097	+	0.835759i	$0.685028\pi$
$200$		0			0
$201$	−9.00000			−0.634811
$202$		0			0
$203$	24.4949	+	24.4949i	1.71920	+	1.71920i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.00000i			0.207514i
$210$		0			0
$211$	11.0000			0.757271			0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000			0.757271			0.378636	+	0.925546i	$0.376393\pi$
$212$		0			0
$213$	9.48683	−	9.48683i	0.650027	−	0.650027i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	3.00000i		−	0.202721i
$220$		0			0
$221$			23.2379i			1.56315i
$222$		0			0
$223$	−6.32456	+	6.32456i	−0.423524	+	0.423524i	−0.886415	−	0.462891i	$-0.846812\pi$
$223$	−6.32456	+	6.32456i	−0.423524	+	0.423524i	0.462891	+	0.886415i	$0.346812\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.1464	−	17.1464i	1.13805	−	1.13805i	0.149249	−	0.988800i	$-0.452314\pi$
$227$	17.1464	−	17.1464i	1.13805	−	1.13805i	0.988800	−	0.149249i	$-0.0476855\pi$
$228$		0			0
$229$	−15.4919			−1.02374			−0.511868	−	0.859064i	$-0.671046\pi$
$229$	−15.4919			−1.02374			−0.511868	+	0.859064i	$0.671046\pi$
$230$		0			0
$231$			7.74597i			0.509647i
$232$		0			0
$233$	9.79796	+	9.79796i	0.641886	+	0.641886i	0.951019	−	0.309133i	$-0.100039\pi$
$233$	9.79796	+	9.79796i	0.641886	+	0.641886i	−0.309133	+	0.951019i	$0.600039\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	9.48683	+	9.48683i	0.616236	+	0.616236i
$238$		0			0
$239$	7.74597			0.501045			0.250522	−	0.968111i	$-0.419398\pi$
$239$	7.74597			0.501045			0.250522	+	0.968111i	$0.419398\pi$
$240$		0			0
$241$	−21.0000			−1.35273			−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000			−1.35273			−0.676364	+	0.736567i	$0.736446\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	9.48683	+	9.48683i	0.603633	+	0.603633i
$248$		0			0
$249$			3.00000i			0.190117i
$250$		0			0
$251$	−29.0000			−1.83046			−0.915232	−	0.402928i	$-0.867993\pi$
$251$	−29.0000			−1.83046			−0.915232	+	0.402928i	$0.867993\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.89898	+	4.89898i	−0.305590	+	0.305590i	−0.843196	−	0.537606i	$-0.819329\pi$
$257$	−4.89898	+	4.89898i	−0.305590	+	0.305590i	0.537606	+	0.843196i	$0.319329\pi$
$258$		0			0
$259$			20.0000i			1.24274i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−15.9217	+	15.9217i	−0.974391	+	0.974391i
$268$		0			0
$269$	−15.4919			−0.944560			−0.472280	−	0.881449i	$-0.656569\pi$
$269$	−15.4919			−0.944560			−0.472280	+	0.881449i	$0.656569\pi$
$270$		0			0
$271$		−	15.4919i		−	0.941068i	−0.882382	−	0.470534i	$-0.844061\pi$
$271$		−	15.4919i		−	0.941068i	0.882382	−	0.470534i	$-0.155939\pi$
$272$		0			0
$273$	24.4949	+	24.4949i	1.48250	+	1.48250i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.16228	−	3.16228i	−0.190003	−	0.190003i	0.605694	−	0.795697i	$-0.292895\pi$
$277$	−3.16228	−	3.16228i	−0.190003	−	0.190003i	−0.795697	+	0.605694i	$0.792895\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$	−13.4722	−	13.4722i	−0.800839	−	0.800839i	0.182388	−	0.983227i	$-0.441617\pi$
$283$	−13.4722	−	13.4722i	−0.800839	−	0.800839i	−0.983227	+	0.182388i	$0.941617\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.16228	+	3.16228i	0.186663	+	0.186663i
$288$		0			0
$289$		−	10.0000i		−	0.588235i
$290$		0			0
$291$	12.0000			0.703452
$292$		0			0
$293$	−9.48683	+	9.48683i	−0.554227	+	0.554227i	−0.927658	−	0.373431i	$-0.878181\pi$
$293$	−9.48683	+	9.48683i	−0.554227	+	0.554227i	0.373431	+	0.927658i	$0.378181\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.67423	+	3.67423i	−0.213201	+	0.213201i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	15.4919i		−	0.892940i
$302$		0			0
$303$	−9.48683	+	9.48683i	−0.545004	+	0.545004i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	3.67423	−	3.67423i	0.209700	−	0.209700i	−0.594440	−	0.804140i	$-0.702626\pi$
$307$	3.67423	−	3.67423i	0.209700	−	0.209700i	0.804140	+	0.594440i	$0.202626\pi$
$308$		0			0
$309$	−7.74597			−0.440653
$310$		0			0
$311$			23.2379i			1.31770i	0.752274	+	0.658850i	$0.228957\pi$
$311$			23.2379i			1.31770i	−0.752274	+	0.658850i	$0.771043\pi$
$312$		0			0
$313$	−14.6969	−	14.6969i	−0.830720	−	0.830720i	0.156895	−	0.987615i	$-0.449852\pi$
$313$	−14.6969	−	14.6969i	−0.830720	−	0.830720i	−0.987615	+	0.156895i	$0.949852\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.9737	−	18.9737i	−1.06567	−	1.06567i	−0.997687	−	0.0679806i	$-0.978344\pi$
$317$	−18.9737	−	18.9737i	−1.06567	−	1.06567i	−0.0679806	−	0.997687i	$-0.521656\pi$
$318$		0			0
$319$	−7.74597			−0.433691
$320$		0			0
$321$	21.0000			1.17211
$322$		0			0
$323$	−11.0227	−	11.0227i	−0.613320	−	0.613320i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−9.48683	−	9.48683i	−0.524623	−	0.524623i
$328$		0			0
$329$			20.0000i			1.10264i
$330$		0			0
$331$	−9.00000			−0.494685			−0.247342	−	0.968928i	$-0.579557\pi$
$331$	−9.00000			−0.494685			−0.247342	+	0.968928i	$0.579557\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.8207	−	20.8207i	1.13417	−	1.13417i	0.144698	−	0.989476i	$-0.453779\pi$
$337$	20.8207	−	20.8207i	1.13417	−	1.13417i	0.989476	−	0.144698i	$-0.0462210\pi$
$338$		0			0
$339$		−	3.00000i		−	0.162938i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18.9737	−	18.9737i	1.02448	−	1.02448i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.57321	+	8.57321i	−0.460234	+	0.460234i	−0.898732	−	0.438498i	$-0.855511\pi$
$347$	−8.57321	+	8.57321i	−0.460234	+	0.460234i	0.438498	+	0.898732i	$0.355511\pi$
$348$		0			0
$349$	−7.74597			−0.414632			−0.207316	−	0.978274i	$-0.566473\pi$
$349$	−7.74597			−0.414632			−0.207316	+	0.978274i	$0.566473\pi$
$350$		0			0
$351$			23.2379i			1.24035i
$352$		0			0
$353$	−14.6969	−	14.6969i	−0.782239	−	0.782239i	0.197969	−	0.980208i	$-0.436565\pi$
$353$	−14.6969	−	14.6969i	−0.782239	−	0.782239i	−0.980208	+	0.197969i	$0.936565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−28.4605	−	28.4605i	−1.50629	−	1.50629i
$358$		0			0
$359$	−7.74597			−0.408816			−0.204408	−	0.978886i	$-0.565527\pi$
$359$	−7.74597			−0.408816			−0.204408	+	0.978886i	$0.565527\pi$
$360$		0			0
$361$	10.0000			0.526316
$362$		0			0
$363$	12.2474	+	12.2474i	0.642824	+	0.642824i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−6.32456	−	6.32456i	−0.330139	−	0.330139i	0.522500	−	0.852639i	$-0.324999\pi$
$367$	−6.32456	−	6.32456i	−0.330139	−	0.330139i	−0.852639	+	0.522500i	$0.824999\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	40.0000			2.07670
$372$		0			0
$373$	6.32456	−	6.32456i	0.327473	−	0.327473i	−0.524152	−	0.851625i	$-0.675618\pi$
$373$	6.32456	−	6.32456i	0.327473	−	0.327473i	0.851625	+	0.524152i	$0.175618\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.4949	+	24.4949i	−1.26155	+	1.26155i
$378$		0			0
$379$			23.0000i			1.18143i	0.806880	+	0.590715i	$0.201154\pi$
$379$			23.0000i			1.18143i	−0.806880	+	0.590715i	$0.798846\pi$
$380$		0			0
$381$		−	15.4919i		−	0.793676i
$382$		0			0
$383$	15.8114	−	15.8114i	0.807924	−	0.807924i	−0.176395	−	0.984319i	$-0.556444\pi$
$383$	15.8114	−	15.8114i	0.807924	−	0.807924i	0.984319	+	0.176395i	$0.0564437\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	30.9839			1.57094			0.785472	−	0.618897i	$-0.212420\pi$
$389$	30.9839			1.57094			0.785472	+	0.618897i	$0.212420\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−9.79796	−	9.79796i	−0.494242	−	0.494242i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−6.32456	−	6.32456i	−0.317420	−	0.317420i	0.530355	−	0.847776i	$-0.322059\pi$
$397$	−6.32456	−	6.32456i	−0.317420	−	0.317420i	−0.847776	+	0.530355i	$0.822059\pi$
$398$		0			0
$399$	−23.2379			−1.16335
$400$		0			0
$401$	−11.0000			−0.549314			−0.274657	−	0.961542i	$-0.588564\pi$
$401$	−11.0000			−0.549314			−0.274657	+	0.961542i	$0.588564\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.16228	−	3.16228i	−0.156748	−	0.156748i
$408$		0			0
$409$			17.0000i			0.840596i	0.907386	+	0.420298i	$0.138074\pi$
$409$			17.0000i			0.840596i	−0.907386	+	0.420298i	$0.861926\pi$
$410$		0			0
$411$	−21.0000			−1.03585
$412$		0			0
$413$	12.6491	−	12.6491i	0.622422	−	0.622422i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−20.8207	+	20.8207i	−1.01959	+	1.01959i
$418$		0			0
$419$		−	7.00000i		−	0.341972i	−0.985273	−	0.170986i	$-0.945305\pi$
$419$		−	7.00000i		−	0.341972i	0.985273	−	0.170986i	$-0.0546954\pi$
$420$		0			0
$421$			30.9839i			1.51006i	0.655690	+	0.755031i	$0.272378\pi$
$421$			30.9839i			1.51006i	−0.655690	+	0.755031i	$0.727622\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−24.4949	+	24.4949i	−1.18539	+	1.18539i
$428$		0			0
$429$	−7.74597			−0.373979
$430$		0			0
$431$			38.7298i			1.86555i	0.360459	+	0.932775i	$0.382620\pi$
$431$			38.7298i			1.86555i	−0.360459	+	0.932775i	$0.617380\pi$
$432$		0			0
$433$	1.22474	+	1.22474i	0.0588575	+	0.0588575i	0.735923	−	0.677065i	$-0.236749\pi$
$433$	1.22474	+	1.22474i	0.0588575	+	0.0588575i	−0.677065	+	0.735923i	$0.736749\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−7.74597			−0.369695			−0.184847	−	0.982767i	$-0.559179\pi$
$439$	−7.74597			−0.369695			−0.184847	+	0.982767i	$0.559179\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.22474	−	1.22474i	−0.0581894	−	0.0581894i	0.677413	−	0.735603i	$-0.263101\pi$
$443$	−1.22474	−	1.22474i	−0.0581894	−	0.0581894i	−0.735603	+	0.677413i	$0.763101\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−9.48683	−	9.48683i	−0.448712	−	0.448712i
$448$		0			0
$449$		−	23.0000i		−	1.08544i	−0.839915	−	0.542719i	$-0.817395\pi$
$449$		−	23.0000i		−	1.08544i	0.839915	−	0.542719i	$-0.182605\pi$
$450$		0			0
$451$	−1.00000			−0.0470882
$452$		0			0
$453$	−28.4605	+	28.4605i	−1.33719	+	1.33719i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.67423	+	3.67423i	−0.171873	+	0.171873i	−0.787802	−	0.615929i	$-0.788781\pi$
$457$	−3.67423	+	3.67423i	−0.171873	+	0.171873i	0.615929	+	0.787802i	$0.288781\pi$
$458$		0			0
$459$		−	27.0000i		−	1.26025i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	25.2982	−	25.2982i	1.17571	−	1.17571i	0.194881	−	0.980827i	$-0.437568\pi$
$463$	25.2982	−	25.2982i	1.17571	−	1.17571i	0.980827	−	0.194881i	$-0.0624321\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.34847	+	7.34847i	−0.340047	+	0.340047i	−0.856385	−	0.516338i	$-0.827295\pi$
$467$	−7.34847	+	7.34847i	−0.340047	+	0.340047i	0.516338	+	0.856385i	$0.327295\pi$
$468$		0			0
$469$	−23.2379			−1.07303
$470$		0			0
$471$		−	15.4919i		−	0.713831i
$472$		0			0
$473$	2.44949	+	2.44949i	0.112628	+	0.112628i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.4919			−0.707845			−0.353922	−	0.935275i	$-0.615152\pi$
$479$	−15.4919			−0.707845			−0.353922	+	0.935275i	$0.615152\pi$
$480$		0			0
$481$	−20.0000			−0.911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	12.6491	+	12.6491i	0.573186	+	0.573186i	0.933017	−	0.359831i	$-0.117166\pi$
$487$	12.6491	+	12.6491i	0.573186	+	0.573186i	−0.359831	+	0.933017i	$0.617166\pi$
$488$		0			0
$489$			33.0000i			1.49231i
$490$		0			0
$491$	28.0000			1.26362			0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000			1.26362			0.631811	+	0.775122i	$0.282312\pi$
$492$		0			0
$493$	28.4605	−	28.4605i	1.28180	−	1.28180i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.4949	−	24.4949i	1.09875	−	1.09875i
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$		0			0
$501$			23.2379i			1.03819i
$502$		0			0
$503$	−25.2982	+	25.2982i	−1.12799	+	1.12799i	−0.137489	+	0.990503i	$0.543903\pi$
$503$	−25.2982	+	25.2982i	−1.12799	+	1.12799i	−0.990503	+	0.137489i	$0.956097\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8.57321	+	8.57321i	−0.380750	+	0.380750i
$508$		0			0
$509$	−30.9839			−1.37334			−0.686668	−	0.726971i	$-0.740927\pi$
$509$	−30.9839			−1.37334			−0.686668	+	0.726971i	$0.740927\pi$
$510$		0			0
$511$		−	7.74597i		−	0.342661i
$512$		0			0
$513$	−11.0227	−	11.0227i	−0.486664	−	0.486664i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−3.16228	−	3.16228i	−0.139077	−	0.139077i
$518$		0			0
$519$	30.9839			1.36004
$520$		0			0
$521$	19.0000			0.832405			0.416203	−	0.909272i	$-0.363361\pi$
$521$	19.0000			0.832405			0.416203	+	0.909272i	$0.363361\pi$
$522$		0			0
$523$	11.0227	+	11.0227i	0.481989	+	0.481989i	0.905766	−	0.423777i	$-0.139296\pi$
$523$	11.0227	+	11.0227i	0.481989	+	0.481989i	−0.423777	+	0.905766i	$0.639296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.16228	+	3.16228i	−0.136973	+	0.136973i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−8.57321	+	8.57321i	−0.369961	+	0.369961i
$538$		0			0
$539$			13.0000i			0.559950i
$540$		0			0
$541$		−	15.4919i		−	0.666050i	−0.942918	−	0.333025i	$-0.891931\pi$
$541$		−	15.4919i		−	0.666050i	0.942918	−	0.333025i	$-0.108069\pi$
$542$		0			0
$543$	−18.9737	+	18.9737i	−0.814238	+	0.814238i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	28.1691	−	28.1691i	1.20443	−	1.20443i	0.231618	−	0.972807i	$-0.425598\pi$
$547$	28.1691	−	28.1691i	1.20443	−	1.20443i	0.972807	−	0.231618i	$-0.0744021\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	23.2379i		−	0.989968i
$552$		0			0
$553$	24.4949	+	24.4949i	1.04163	+	1.04163i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.16228	+	3.16228i	0.133990	+	0.133990i	0.770921	−	0.636931i	$-0.219796\pi$
$557$	3.16228	+	3.16228i	0.133990	+	0.133990i	−0.636931	+	0.770921i	$0.719796\pi$
$558$		0			0
$559$	15.4919			0.655239
$560$		0			0
$561$	9.00000			0.379980
$562$		0			0
$563$	−22.0454	−	22.0454i	−0.929103	−	0.929103i	0.0685449	−	0.997648i	$-0.478164\pi$
$563$	−22.0454	−	22.0454i	−0.929103	−	0.929103i	−0.997648	+	0.0685449i	$0.978164\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−28.4605	−	28.4605i	−1.19523	−	1.19523i
$568$		0			0
$569$		−	13.0000i		−	0.544988i	−0.962157	−	0.272494i	$-0.912151\pi$
$569$		−	13.0000i		−	0.544988i	0.962157	−	0.272494i	$-0.0878485\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$	−18.9737	+	18.9737i	−0.792636	+	0.792636i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.57321	−	8.57321i	0.356908	−	0.356908i	−0.505764	−	0.862672i	$-0.668789\pi$
$577$	8.57321	−	8.57321i	0.356908	−	0.356908i	0.862672	+	0.505764i	$0.168789\pi$
$578$		0			0
$579$			27.0000i			1.12208i
$580$		0			0
$581$			7.74597i			0.321357i
$582$		0			0
$583$	−6.32456	+	6.32456i	−0.261936	+	0.261936i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.1691	−	28.1691i	1.16266	−	1.16266i	0.178774	−	0.983890i	$-0.442787\pi$
$587$	28.1691	−	28.1691i	1.16266	−	1.16266i	0.983890	−	0.178774i	$-0.0572132\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	46.4758i		−	1.91176i
$592$		0			0
$593$	25.7196	+	25.7196i	1.05618	+	1.05618i	0.998325	+	0.0578541i	$0.0184258\pi$
$593$	25.7196	+	25.7196i	1.05618	+	1.05618i	0.0578541	+	0.998325i	$0.481574\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	18.9737	+	18.9737i	0.776540	+	0.776540i
$598$		0			0
$599$	−38.7298			−1.58246			−0.791229	−	0.611520i	$-0.790558\pi$
$599$	−38.7298			−1.58246			−0.791229	+	0.611520i	$0.790558\pi$
$600$		0			0
$601$	9.00000			0.367118			0.183559	−	0.983009i	$-0.441238\pi$
$601$	9.00000			0.367118			0.183559	+	0.983009i	$0.441238\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−12.6491	−	12.6491i	−0.513412	−	0.513412i	0.402159	−	0.915570i	$-0.368260\pi$
$607$	−12.6491	−	12.6491i	−0.513412	−	0.513412i	−0.915570	+	0.402159i	$0.868260\pi$
$608$		0			0
$609$		−	60.0000i		−	2.43132i
$610$		0			0
$611$	−20.0000			−0.809113
$612$		0			0
$613$	12.6491	−	12.6491i	0.510893	−	0.510893i	−0.403907	−	0.914800i	$-0.632348\pi$
$613$	12.6491	−	12.6491i	0.510893	−	0.510893i	0.914800	+	0.403907i	$0.132348\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.89898	+	4.89898i	−0.197225	+	0.197225i	−0.798810	−	0.601584i	$-0.794536\pi$
$617$	−4.89898	+	4.89898i	−0.197225	+	0.197225i	0.601584	+	0.798810i	$0.294536\pi$
$618$		0			0
$619$			4.00000i			0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$			4.00000i			0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−41.1096	+	41.1096i	−1.64702	+	1.64702i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	3.67423	−	3.67423i	0.146735	−	0.146735i
$628$		0			0
$629$	23.2379			0.926556
$630$		0			0
$631$			30.9839i			1.23345i	0.787179	+	0.616724i	$0.211541\pi$
$631$			30.9839i			1.23345i	−0.787179	+	0.616724i	$0.788459\pi$
$632$		0			0
$633$	−13.4722	−	13.4722i	−0.535472	−	0.535472i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	41.1096	+	41.1096i	1.62882	+	1.62882i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$	2.44949	+	2.44949i	0.0965984	+	0.0965984i	0.753755	−	0.657156i	$-0.228241\pi$
$643$	2.44949	+	2.44949i	0.0965984	+	0.0965984i	−0.657156	+	0.753755i	$0.728241\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.6228	+	31.6228i	1.24322	+	1.24322i	0.958658	+	0.284562i	$0.0918482\pi$
$647$	31.6228	+	31.6228i	1.24322	+	1.24322i	0.284562	+	0.958658i	$0.408152\pi$
$648$		0			0
$649$			4.00000i			0.157014i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.48683	−	9.48683i	0.371248	−	0.371248i	−0.496683	−	0.867932i	$-0.665449\pi$
$653$	9.48683	−	9.48683i	0.371248	−	0.371248i	0.867932	+	0.496683i	$0.165449\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	37.0000i		−	1.44132i	−0.693291	−	0.720658i	$-0.743840\pi$
$659$		−	37.0000i		−	1.44132i	0.693291	−	0.720658i	$-0.256160\pi$
$660$		0			0
$661$		−	23.2379i		−	0.903850i	−0.892056	−	0.451925i	$-0.850738\pi$
$661$		−	23.2379i		−	0.903850i	0.892056	−	0.451925i	$-0.149262\pi$
$662$		0			0
$663$	28.4605	−	28.4605i	1.10531	−	1.10531i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	15.4919			0.598953
$670$		0			0
$671$		−	7.74597i		−	0.299030i
$672$		0			0
$673$	−14.6969	−	14.6969i	−0.566525	−	0.566525i	0.364628	−	0.931153i	$-0.381196\pi$
$673$	−14.6969	−	14.6969i	−0.566525	−	0.566525i	−0.931153	+	0.364628i	$0.881196\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.9737	+	18.9737i	0.729217	+	0.729217i	0.970464	−	0.241247i	$-0.0775563\pi$
$677$	18.9737	+	18.9737i	0.729217	+	0.729217i	−0.241247	+	0.970464i	$0.577556\pi$
$678$		0			0
$679$	30.9839			1.18905
$680$		0			0
$681$	−42.0000			−1.60944
$682$		0			0
$683$	11.0227	+	11.0227i	0.421772	+	0.421772i	0.885813	−	0.464041i	$-0.153601\pi$
$683$	11.0227	+	11.0227i	0.421772	+	0.421772i	−0.464041	+	0.885813i	$0.653601\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	18.9737	+	18.9737i	0.723891	+	0.723891i
$688$		0			0
$689$			40.0000i			1.52388i
$690$		0			0
$691$	−9.00000			−0.342376			−0.171188	−	0.985238i	$-0.554761\pi$
$691$	−9.00000			−0.342376			−0.171188	+	0.985238i	$0.554761\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.67423	−	3.67423i	0.139172	−	0.139172i
$698$		0			0
$699$		−	24.0000i		−	0.907763i
$700$		0			0
$701$			23.2379i			0.877683i	0.898564	+	0.438842i	$0.144611\pi$
$701$			23.2379i			0.877683i	−0.898564	+	0.438842i	$0.855389\pi$
$702$		0			0
$703$	9.48683	−	9.48683i	0.357803	−	0.357803i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.4949	+	24.4949i	−0.921225	+	0.921225i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−9.48683	−	9.48683i	−0.354292	−	0.354292i
$718$		0			0
$719$	38.7298			1.44438			0.722190	−	0.691695i	$-0.243136\pi$
$719$	38.7298			1.44438			0.722190	+	0.691695i	$0.243136\pi$
$720$		0			0
$721$	−20.0000			−0.744839
$722$		0			0
$723$	25.7196	+	25.7196i	0.956524	+	0.956524i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.8114	−	15.8114i	−0.586412	−	0.586412i	0.350246	−	0.936658i	$-0.386098\pi$
$727$	−15.8114	−	15.8114i	−0.586412	−	0.586412i	−0.936658	+	0.350246i	$0.886098\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	−18.0000			−0.665754
$732$		0			0
$733$	−22.1359	+	22.1359i	−0.817610	+	0.817610i	−0.985761	−	0.168151i	$-0.946220\pi$
$733$	−22.1359	+	22.1359i	−0.817610	+	0.817610i	0.168151	+	0.985761i	$0.446220\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.67423	−	3.67423i	0.135342	−	0.135342i
$738$		0			0
$739$		−	36.0000i		−	1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$		−	36.0000i		−	1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$		0			0
$741$		−	23.2379i		−	0.853666i
$742$		0			0
$743$	6.32456	−	6.32456i	0.232025	−	0.232025i	−0.581512	−	0.813538i	$-0.697539\pi$
$743$	6.32456	−	6.32456i	0.232025	−	0.232025i	0.813538	+	0.581512i	$0.197539\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	54.2218			1.98122
$750$		0			0
$751$		−	7.74597i		−	0.282654i	−0.989963	−	0.141327i	$-0.954863\pi$
$751$		−	7.74597i		−	0.282654i	0.989963	−	0.141327i	$-0.0451370\pi$
$752$		0			0
$753$	35.5176	+	35.5176i	1.29433	+	1.29433i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−6.32456	−	6.32456i	−0.229870	−	0.229870i	0.582768	−	0.812638i	$-0.301969\pi$
$757$	−6.32456	−	6.32456i	−0.229870	−	0.229870i	−0.812638	+	0.582768i	$0.801969\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.0000			−0.398750			−0.199375	−	0.979923i	$-0.563891\pi$
$761$	−11.0000			−0.398750			−0.199375	+	0.979923i	$0.563891\pi$
$762$		0			0
$763$	−24.4949	−	24.4949i	−0.886775	−	0.886775i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.6491	+	12.6491i	0.456733	+	0.456733i
$768$		0			0
$769$		−	33.0000i		−	1.19001i	−0.803722	−	0.595005i	$-0.797150\pi$
$769$		−	33.0000i		−	1.19001i	0.803722	−	0.595005i	$-0.202850\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$		0			0
$773$	28.4605	−	28.4605i	1.02365	−	1.02365i	0.0239396	−	0.999713i	$-0.492379\pi$
$773$	28.4605	−	28.4605i	1.02365	−	1.02365i	0.999713	−	0.0239396i	$-0.00762094\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	24.4949	−	24.4949i	0.878750	−	0.878750i
$778$		0			0
$779$		−	3.00000i		−	0.107486i
$780$		0			0
$781$			7.74597i			0.277172i
$782$		0			0
$783$	28.4605	−	28.4605i	1.01710	−	1.01710i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	17.1464	−	17.1464i	0.611204	−	0.611204i	−0.332056	−	0.943260i	$-0.607742\pi$
$787$	17.1464	−	17.1464i	0.611204	−	0.611204i	0.943260	+	0.332056i	$0.107742\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	7.74597i		−	0.275415i
$792$		0			0
$793$	−24.4949	−	24.4949i	−0.869839	−	0.869839i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.2982	−	25.2982i	−0.896109	−	0.896109i	0.0989804	−	0.995089i	$-0.468442\pi$
$797$	−25.2982	−	25.2982i	−0.896109	−	0.896109i	−0.995089	+	0.0989804i	$0.968442\pi$
$798$		0			0
$799$	23.2379			0.822098
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.22474	+	1.22474i	0.0432203	+	0.0432203i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	18.9737	+	18.9737i	0.667905	+	0.667905i
$808$		0			0
$809$		−	14.0000i		−	0.492214i	−0.969243	−	0.246107i	$-0.920849\pi$
$809$		−	14.0000i		−	0.492214i	0.969243	−	0.246107i	$-0.0791514\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$	−18.9737	+	18.9737i	−0.665436	+	0.665436i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−7.34847	+	7.34847i	−0.257090	+	0.257090i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	30.9839i		−	1.08134i	−0.841233	−	0.540672i	$-0.818170\pi$
$821$		−	30.9839i		−	1.08134i	0.841233	−	0.540672i	$-0.181830\pi$
$822$		0			0
$823$	−12.6491	+	12.6491i	−0.440920	+	0.440920i	−0.892321	−	0.451401i	$-0.850924\pi$
$823$	−12.6491	+	12.6491i	−0.440920	+	0.440920i	0.451401	+	0.892321i	$0.350924\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.9217	−	15.9217i	0.553651	−	0.553651i	−0.373841	−	0.927493i	$-0.621960\pi$
$827$	15.9217	−	15.9217i	0.553651	−	0.553651i	0.927493	+	0.373841i	$0.121960\pi$
$828$		0			0
$829$	23.2379			0.807086			0.403543	−	0.914961i	$-0.367779\pi$
$829$	23.2379			0.807086			0.403543	+	0.914961i	$0.367779\pi$
$830$		0			0
$831$			7.74597i			0.268705i
$832$		0			0
$833$	−47.7650	−	47.7650i	−1.65496	−	1.65496i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.4919			0.534841			0.267420	−	0.963580i	$-0.413829\pi$
$839$	15.4919			0.534841			0.267420	+	0.963580i	$0.413829\pi$
$840$		0			0
$841$	31.0000			1.06897
$842$		0			0
$843$	−26.9444	−	26.9444i	−0.928014	−	0.928014i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	31.6228	+	31.6228i	1.08657	+	1.08657i
$848$		0			0
$849$			33.0000i			1.13256i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−25.2982	+	25.2982i	−0.866195	+	0.866195i	−0.992049	−	0.125854i	$-0.959833\pi$
$853$	−25.2982	+	25.2982i	−0.866195	+	0.866195i	0.125854	+	0.992049i	$0.459833\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.8207	−	20.8207i	0.711220	−	0.711220i	−0.255570	−	0.966790i	$-0.582263\pi$
$857$	20.8207	−	20.8207i	0.711220	−	0.711220i	0.966790	+	0.255570i	$0.0822632\pi$
$858$		0			0
$859$		−	17.0000i		−	0.580033i	−0.957022	−	0.290016i	$-0.906339\pi$
$859$		−	17.0000i		−	0.580033i	0.957022	−	0.290016i	$-0.0936607\pi$
$860$		0			0
$861$		−	7.74597i		−	0.263982i
$862$		0			0
$863$	9.48683	−	9.48683i	0.322936	−	0.322936i	−0.526957	−	0.849892i	$-0.676667\pi$
$863$	9.48683	−	9.48683i	0.322936	−	0.322936i	0.849892	+	0.526957i	$0.176667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−12.2474	+	12.2474i	−0.415945	+	0.415945i
$868$		0			0
$869$	−7.74597			−0.262764
$870$		0			0
$871$		−	23.2379i		−	0.787386i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−12.6491	−	12.6491i	−0.427130	−	0.427130i	0.460519	−	0.887650i	$-0.347663\pi$
$877$	−12.6491	−	12.6491i	−0.427130	−	0.427130i	−0.887650	+	0.460519i	$0.847663\pi$
$878$		0			0
$879$	23.2379			0.783795
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$	23.2702	+	23.2702i	0.783103	+	0.783103i	0.980353	−	0.197250i	$-0.0632011\pi$
$883$	23.2702	+	23.2702i	0.783103	+	0.783103i	−0.197250	+	0.980353i	$0.563201\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		−	40.0000i		−	1.34156i
$890$		0			0
$891$	9.00000			0.301511
$892$		0			0
$893$	9.48683	−	9.48683i	0.317465	−	0.317465i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	46.4758i		−	1.54833i
$902$		0			0
$903$	−18.9737	+	18.9737i	−0.631404	+	0.631404i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	17.1464	−	17.1464i	0.569338	−	0.569338i	−0.362605	−	0.931943i	$-0.618113\pi$
$907$	17.1464	−	17.1464i	0.569338	−	0.569338i	0.931943	+	0.362605i	$0.118113\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			15.4919i			0.513271i	0.966508	+	0.256635i	$0.0826139\pi$
$911$			15.4919i			0.513271i	−0.966508	+	0.256635i	$0.917386\pi$
$912$		0			0
$913$	−1.22474	−	1.22474i	−0.0405331	−	0.0405331i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−25.2982	−	25.2982i	−0.835421	−	0.835421i
$918$		0			0
$919$	−30.9839			−1.02206			−0.511032	−	0.859562i	$-0.670737\pi$
$919$	−30.9839			−1.02206			−0.511032	+	0.859562i	$0.670737\pi$
$920$		0			0
$921$	−9.00000			−0.296560
$922$		0			0
$923$	24.4949	+	24.4949i	0.806259	+	0.806259i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	14.0000i		−	0.459325i	−0.973270	−	0.229663i	$-0.926238\pi$
$929$		−	14.0000i		−	0.459325i	0.973270	−	0.229663i	$-0.0737623\pi$
$930$		0			0
$931$	−39.0000			−1.27817
$932$		0			0
$933$	28.4605	−	28.4605i	0.931755	−	0.931755i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−3.67423	+	3.67423i	−0.120032	+	0.120032i	−0.764571	−	0.644539i	$-0.777049\pi$
$937$	−3.67423	+	3.67423i	−0.120032	+	0.120032i	0.644539	+	0.764571i	$0.277049\pi$
$938$		0			0
$939$			36.0000i			1.17482i
$940$		0			0
$941$			7.74597i			0.252511i	0.991998	+	0.126256i	$0.0402960\pi$
$941$			7.74597i			0.252511i	−0.991998	+	0.126256i	$0.959704\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.8434	+	31.8434i	−1.03477	+	1.03477i	−0.0353971	+	0.999373i	$0.511270\pi$
$947$	−31.8434	+	31.8434i	−1.03477	+	1.03477i	−0.999373	+	0.0353971i	$0.988730\pi$
$948$		0			0
$949$	7.74597			0.251445
$950$		0			0
$951$			46.4758i			1.50708i
$952$		0			0
$953$	37.9671	+	37.9671i	1.22987	+	1.22987i	0.964010	+	0.265864i	$0.0856573\pi$
$953$	37.9671	+	37.9671i	1.22987	+	1.22987i	0.265864	+	0.964010i	$0.414343\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	9.48683	+	9.48683i	0.306666	+	0.306666i
$958$		0			0
$959$	−54.2218			−1.75091
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−6.32456	−	6.32456i	−0.203384	−	0.203384i	0.598064	−	0.801448i	$-0.295937\pi$
$967$	−6.32456	−	6.32456i	−0.203384	−	0.203384i	−0.801448	+	0.598064i	$0.795937\pi$
$968$		0			0
$969$			27.0000i			0.867365i
$970$		0			0
$971$	−29.0000			−0.930654			−0.465327	−	0.885139i	$-0.654063\pi$
$971$	−29.0000			−0.930654			−0.465327	+	0.885139i	$0.654063\pi$
$972$		0			0
$973$	−53.7587	+	53.7587i	−1.72343	+	1.72343i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.67423	+	3.67423i	−0.117549	+	0.117549i	−0.763434	−	0.645885i	$-0.776488\pi$
$977$	−3.67423	+	3.67423i	−0.117549	+	0.117549i	0.645885	+	0.763434i	$0.276488\pi$
$978$		0			0
$979$		−	13.0000i		−	0.415482i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.8114	−	15.8114i	0.504305	−	0.504305i	−0.408468	−	0.912773i	$-0.633937\pi$
$983$	15.8114	−	15.8114i	0.504305	−	0.504305i	0.912773	+	0.408468i	$0.133937\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	24.4949	−	24.4949i	0.779681	−	0.779681i
$988$		0			0
$989$		0			0
$990$		0			0
$991$			54.2218i			1.72241i	0.508257	+	0.861206i	$0.330290\pi$
$991$			54.2218i			1.72241i	−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$	11.0227	+	11.0227i	0.349795	+	0.349795i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	31.6228	+	31.6228i	1.00150	+	1.00150i	0.999999	+	0.00150452i	$0.000478902\pi$
$997$	31.6228	+	31.6228i	1.00150	+	1.00150i	0.00150452	+	0.999999i	$0.499521\pi$
$998$		0			0
$999$	23.2379			0.735215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.o.h.143.1		8
4.3	odd	2		200.2.k.g.43.2	yes	8
5.2	odd	4	inner	800.2.o.h.207.2		8
5.3	odd	4	inner	800.2.o.h.207.3		8
5.4	even	2	inner	800.2.o.h.143.4		8
8.3	odd	2	inner	800.2.o.h.143.2		8
8.5	even	2		200.2.k.g.43.4	yes	8
20.3	even	4		200.2.k.g.107.1	yes	8
20.7	even	4		200.2.k.g.107.4	yes	8
20.19	odd	2		200.2.k.g.43.3	yes	8
40.3	even	4	inner	800.2.o.h.207.4		8
40.13	odd	4		200.2.k.g.107.3	yes	8
40.19	odd	2	inner	800.2.o.h.143.3		8
40.27	even	4	inner	800.2.o.h.207.1		8
40.29	even	2		200.2.k.g.43.1	✓	8
40.37	odd	4		200.2.k.g.107.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.k.g.43.1	✓	8	40.29	even	2
200.2.k.g.43.2	yes	8	4.3	odd	2
200.2.k.g.43.3	yes	8	20.19	odd	2
200.2.k.g.43.4	yes	8	8.5	even	2
200.2.k.g.107.1	yes	8	20.3	even	4
200.2.k.g.107.2	yes	8	40.37	odd	4
200.2.k.g.107.3	yes	8	40.13	odd	4
200.2.k.g.107.4	yes	8	20.7	even	4
800.2.o.h.143.1		8	1.1	even	1	trivial
800.2.o.h.143.2		8	8.3	odd	2	inner
800.2.o.h.143.3		8	40.19	odd	2	inner
800.2.o.h.143.4		8	5.4	even	2	inner
800.2.o.h.207.1		8	40.27	even	4	inner
800.2.o.h.207.2		8	5.2	odd	4	inner
800.2.o.h.207.3		8	5.3	odd	4	inner
800.2.o.h.207.4		8	40.3	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(-3.16228 - 3.16228i) q^{7} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(-3.16228 - 3.16228i) q^{7} +1.00000 q^{11} +(3.16228 - 3.16228i) q^{13} +(-3.67423 + 3.67423i) q^{17} +3.00000i q^{19} +7.74597i q^{21} +(-3.67423 + 3.67423i) q^{27} -7.74597 q^{29} +(-1.22474 - 1.22474i) q^{33} +(-3.16228 - 3.16228i) q^{37} -7.74597 q^{39} -1.00000 q^{41} +(2.44949 + 2.44949i) q^{43} +(-3.16228 - 3.16228i) q^{47} +13.0000i q^{49} +9.00000 q^{51} +(-6.32456 + 6.32456i) q^{53} +(3.67423 - 3.67423i) q^{57} +4.00000i q^{59} -7.74597i q^{61} +(3.67423 - 3.67423i) q^{67} +7.74597i q^{71} +(1.22474 + 1.22474i) q^{73} +(-3.16228 - 3.16228i) q^{77} -7.74597 q^{79} +9.00000 q^{81} +(-1.22474 - 1.22474i) q^{83} +(9.48683 + 9.48683i) q^{87} -13.0000i q^{89} -20.0000 q^{91} +(-4.89898 + 4.89898i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} - 8 q^{41} + 72 q^{51} + 72 q^{81} - 160 q^{91}+O(q^{100}) \) 8 * q + 8 * q^11 - 8 * q^41 + 72 * q^51 + 72 * q^81 - 160 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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