LMFDB - Embedded newform 800.2.o.h.143.4

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			143.4
Root			$0.178197 - 1.40294i$ of defining polynomial
Character	$\chi$	$=$	800.143
Dual form			800.2.o.h.207.4

$q$-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	0.965926	−	0.258819i	$-0.0833333\pi$
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.16228	+	3.16228i	1.19523	+	1.19523i	0.975579	+	0.219650i	$0.0704915\pi$
$7$	3.16228	+	3.16228i	1.19523	+	1.19523i	0.219650	+	0.975579i	$0.429509\pi$
$8$		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	−0.993229	−	0.116171i	$-0.962938\pi$
$13$	−3.16228	+	3.16228i	−0.877058	+	0.877058i	0.116171	+	0.993229i	$0.462938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.67423	−	3.67423i	0.891133	−	0.891133i	−0.103497	−	0.994630i	$-0.533003\pi$
$17$	3.67423	−	3.67423i	0.891133	−	0.891133i	0.994630	+	0.103497i	$0.0330032\pi$
$18$		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	0.917534	−	0.397658i	$-0.130177\pi$
$37$	3.16228	+	3.16228i	0.519875	+	0.519875i	−0.397658	+	0.917534i	$0.630177\pi$
$38$		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	0.495222	−	0.868766i	$-0.335087\pi$
$43$	−2.44949	−	2.44949i	−0.373544	−	0.373544i	−0.868766	+	0.495222i	$0.835087\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.16228	+	3.16228i	0.461266	+	0.461266i	0.899070	−	0.437805i	$-0.144244\pi$
$47$	3.16228	+	3.16228i	0.461266	+	0.461266i	−0.437805	+	0.899070i	$0.644244\pi$
$48$		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	−0.123589	−	0.992333i	$-0.539440\pi$
$53$	6.32456	−	6.32456i	0.868744	−	0.868744i	0.992333	+	0.123589i	$0.0394404\pi$
$54$		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	−0.894982	−	0.446103i	$-0.852812\pi$
$67$	−3.67423	+	3.67423i	−0.448879	+	0.448879i	0.446103	+	0.894982i	$0.352812\pi$
$68$		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	0.631792	−	0.775138i	$-0.282320\pi$
$73$	−1.22474	−	1.22474i	−0.143346	−	0.143346i	−0.775138	+	0.631792i	$0.782320\pi$
$74$		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	0.771121	−	0.636688i	$-0.219696\pi$
$83$	1.22474	+	1.22474i	0.134433	+	0.134433i	−0.636688	+	0.771121i	$0.719696\pi$
$84$		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	−0.413217	−	0.910633i	$-0.635595\pi$
$97$	4.89898	−	4.89898i	0.497416	−	0.497416i	0.910633	+	0.413217i	$0.135595\pi$
$98$		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	−0.845525	−	0.533936i	$-0.820712\pi$
$103$	−3.16228	+	3.16228i	−0.311588	+	0.311588i	0.533936	+	0.845525i	$0.320712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.57321	−	8.57321i	0.828804	−	0.828804i	−0.158547	−	0.987351i	$-0.550681\pi$
$107$	8.57321	−	8.57321i	0.828804	−	0.828804i	0.987351	+	0.158547i	$0.0506811\pi$
$108$		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	0.647149	−	0.762363i	$-0.275961\pi$
$113$	−1.22474	−	1.22474i	−0.115214	−	0.115214i	−0.762363	+	0.647149i	$0.775961\pi$
$114$		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	0.368439	−	0.929652i	$-0.379892\pi$
$127$	−6.32456	−	6.32456i	−0.561214	−	0.561214i	−0.929652	+	0.368439i	$0.879892\pi$
$128$		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	−0.971106	−	0.238647i	$-0.923296\pi$
$137$	−8.57321	+	8.57321i	−0.732459	+	0.732459i	0.238647	+	0.971106i	$0.423296\pi$
$138$		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	0.408157	−	0.912912i	$-0.366172\pi$
$157$	−6.32456	−	6.32456i	−0.504754	−	0.504754i	−0.912912	+	0.408157i	$0.866172\pi$
$158$		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	$0.0181026\pi$
$163$	13.4722	+	13.4722i	1.05522	+	1.05522i	0.0568404	+	0.998383i	$0.481897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.48683	+	9.48683i	0.734113	+	0.734113i	0.971432	−	0.237319i	$-0.0762686\pi$
$167$	9.48683	+	9.48683i	0.734113	+	0.734113i	−0.237319	+	0.971432i	$0.576269\pi$
$168$		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	−0.0375988	−	0.999293i	$-0.511971\pi$
$173$	12.6491	−	12.6491i	0.961694	−	0.961694i	0.999293	+	0.0375988i	$0.0119709\pi$
$174$		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	0.982050	−	0.188619i	$-0.0604011\pi$
$193$	11.0227	+	11.0227i	0.793432	+	0.793432i	−0.188619	+	0.982050i	$0.560401\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.9737	−	18.9737i	−1.35182	−	1.35182i	−0.883628	−	0.468190i	$-0.844906\pi$
$197$	−18.9737	−	18.9737i	−1.35182	−	1.35182i	−0.468190	−	0.883628i	$-0.655094\pi$
$198$		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	−0.462891	−	0.886415i	$-0.653188\pi$
$223$	6.32456	−	6.32456i	0.423524	−	0.423524i	0.886415	+	0.462891i	$0.153188\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.1464	+	17.1464i	−1.13805	+	1.13805i	−0.149249	+	0.988800i	$0.547686\pi$
$227$	−17.1464	+	17.1464i	−1.13805	+	1.13805i	−0.988800	+	0.149249i	$0.952314\pi$
$228$		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	0.309133	−	0.951019i	$-0.399961\pi$
$233$	−9.79796	−	9.79796i	−0.641886	−	0.641886i	−0.951019	+	0.309133i	$0.899961\pi$
$234$		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	−0.537606	−	0.843196i	$-0.680671\pi$
$257$	4.89898	−	4.89898i	0.305590	−	0.305590i	0.843196	+	0.537606i	$0.180671\pi$
$258$		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	0.795697	−	0.605694i	$-0.207105\pi$
$277$	3.16228	+	3.16228i	0.190003	+	0.190003i	−0.605694	+	0.795697i	$0.707105\pi$
$278$		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	0.983227	−	0.182388i	$-0.0583827\pi$
$283$	13.4722	+	13.4722i	0.800839	+	0.800839i	−0.182388	+	0.983227i	$0.558383\pi$
$284$		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	−0.373431	−	0.927658i	$-0.621819\pi$
$293$	9.48683	−	9.48683i	0.554227	−	0.554227i	0.927658	+	0.373431i	$0.121819\pi$
$294$		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	−0.804140	−	0.594440i	$-0.797374\pi$
$307$	−3.67423	+	3.67423i	−0.209700	+	0.209700i	0.594440	+	0.804140i	$0.297374\pi$
$308$		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	0.987615	−	0.156895i	$-0.0501485\pi$
$313$	14.6969	+	14.6969i	0.830720	+	0.830720i	−0.156895	+	0.987615i	$0.550148\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.9737	+	18.9737i	1.06567	+	1.06567i	0.997687	+	0.0679806i	$0.0216556\pi$
$317$	18.9737	+	18.9737i	1.06567	+	1.06567i	0.0679806	+	0.997687i	$0.478344\pi$
$318$		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	$0.546221\pi$
$337$	−20.8207	+	20.8207i	−1.13417	+	1.13417i	−0.989476	+	0.144698i	$0.953779\pi$
$338$		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	−0.438498	−	0.898732i	$-0.644489\pi$
$347$	8.57321	−	8.57321i	0.460234	−	0.460234i	0.898732	+	0.438498i	$0.144489\pi$
$348$		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	0.980208	−	0.197969i	$-0.0634346\pi$
$353$	14.6969	+	14.6969i	0.782239	+	0.782239i	−0.197969	+	0.980208i	$0.563435\pi$
$354$		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	0.852639	−	0.522500i	$-0.175001\pi$
$367$	6.32456	+	6.32456i	0.330139	+	0.330139i	−0.522500	+	0.852639i	$0.675001\pi$
$368$		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	−0.851625	−	0.524152i	$-0.824382\pi$
$373$	−6.32456	+	6.32456i	−0.327473	+	0.327473i	0.524152	+	0.851625i	$0.324382\pi$
$374$		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	−0.984319	−	0.176395i	$-0.943556\pi$
$383$	−15.8114	+	15.8114i	−0.807924	+	0.807924i	0.176395	+	0.984319i	$0.443556\pi$
$384$		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	0.847776	−	0.530355i	$-0.177941\pi$
$397$	6.32456	+	6.32456i	0.317420	+	0.317420i	−0.530355	+	0.847776i	$0.677941\pi$
$398$		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	0.677065	−	0.735923i	$-0.263251\pi$
$433$	−1.22474	−	1.22474i	−0.0588575	−	0.0588575i	−0.735923	+	0.677065i	$0.763251\pi$
$434$		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	0.735603	−	0.677413i	$-0.236899\pi$
$443$	1.22474	+	1.22474i	0.0581894	+	0.0581894i	−0.677413	+	0.735603i	$0.736899\pi$
$444$		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	−0.615929	−	0.787802i	$-0.711219\pi$
$457$	3.67423	−	3.67423i	0.171873	−	0.171873i	0.787802	+	0.615929i	$0.211219\pi$
$458$		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	$0.562432\pi$
$463$	−25.2982	+	25.2982i	−1.17571	+	1.17571i	−0.980827	+	0.194881i	$0.937568\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.34847	−	7.34847i	0.340047	−	0.340047i	−0.516338	−	0.856385i	$-0.672705\pi$
$467$	7.34847	−	7.34847i	0.340047	−	0.340047i	0.856385	+	0.516338i	$0.172705\pi$
$468$		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	0.359831	−	0.933017i	$-0.382834\pi$
$487$	−12.6491	−	12.6491i	−0.573186	−	0.573186i	−0.933017	+	0.359831i	$0.882834\pi$
$488$		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	$-0.456097\pi$
$503$	25.2982	−	25.2982i	1.12799	−	1.12799i	0.990503	−	0.137489i	$-0.0439030\pi$
$504$		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	0.423777	−	0.905766i	$-0.360704\pi$
$523$	−11.0227	−	11.0227i	−0.481989	−	0.481989i	−0.905766	+	0.423777i	$0.860704\pi$
$524$		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	$0.574402\pi$
$547$	−28.1691	+	28.1691i	−1.20443	+	1.20443i	−0.972807	+	0.231618i	$0.925598\pi$
$548$		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	0.636931	−	0.770921i	$-0.280204\pi$
$557$	−3.16228	−	3.16228i	−0.133990	−	0.133990i	−0.770921	+	0.636931i	$0.780204\pi$
$558$		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	0.997648	−	0.0685449i	$-0.0218356\pi$
$563$	22.0454	+	22.0454i	0.929103	+	0.929103i	−0.0685449	+	0.997648i	$0.521836\pi$
$564$		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	−0.862672	−	0.505764i	$-0.831211\pi$
$577$	−8.57321	+	8.57321i	−0.356908	+	0.356908i	0.505764	+	0.862672i	$0.331211\pi$
$578$		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	$0.557213\pi$
$587$	−28.1691	+	28.1691i	−1.16266	+	1.16266i	−0.983890	+	0.178774i	$0.942787\pi$
$588$		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	$-0.981574\pi$
$593$	−25.7196	−	25.7196i	−1.05618	−	1.05618i	−0.0578541	−	0.998325i	$-0.518426\pi$
$594$		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	0.915570	−	0.402159i	$-0.131740\pi$
$607$	12.6491	+	12.6491i	0.513412	+	0.513412i	−0.402159	+	0.915570i	$0.631740\pi$
$608$		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	−0.914800	−	0.403907i	$-0.867652\pi$
$613$	−12.6491	+	12.6491i	−0.510893	+	0.510893i	0.403907	+	0.914800i	$0.367652\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.89898	−	4.89898i	0.197225	−	0.197225i	−0.601584	−	0.798810i	$-0.705464\pi$
$617$	4.89898	−	4.89898i	0.197225	−	0.197225i	0.798810	+	0.601584i	$0.205464\pi$
$618$		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	0.657156	−	0.753755i	$-0.271759\pi$
$643$	−2.44949	−	2.44949i	−0.0965984	−	0.0965984i	−0.753755	+	0.657156i	$0.771759\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−31.6228	−	31.6228i	−1.24322	−	1.24322i	−0.958658	−	0.284562i	$-0.908152\pi$
$647$	−31.6228	−	31.6228i	−1.24322	−	1.24322i	−0.284562	−	0.958658i	$-0.591848\pi$
$648$		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	−0.867932	−	0.496683i	$-0.834551\pi$
$653$	−9.48683	+	9.48683i	−0.371248	+	0.371248i	0.496683	+	0.867932i	$0.334551\pi$
$654$		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	0.931153	−	0.364628i	$-0.118804\pi$
$673$	14.6969	+	14.6969i	0.566525	+	0.566525i	−0.364628	+	0.931153i	$0.618804\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.9737	−	18.9737i	−0.729217	−	0.729217i	0.241247	−	0.970464i	$-0.422444\pi$
$677$	−18.9737	−	18.9737i	−0.729217	−	0.729217i	−0.970464	+	0.241247i	$0.922444\pi$
$678$		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	0.464041	−	0.885813i	$-0.346399\pi$
$683$	−11.0227	−	11.0227i	−0.421772	−	0.421772i	−0.885813	+	0.464041i	$0.846399\pi$
$684$		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	0.936658	−	0.350246i	$-0.113902\pi$
$727$	15.8114	+	15.8114i	0.586412	+	0.586412i	−0.350246	+	0.936658i	$0.613902\pi$
$728$		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	−0.168151	−	0.985761i	$-0.553780\pi$
$733$	22.1359	−	22.1359i	0.817610	−	0.817610i	0.985761	+	0.168151i	$0.0537798\pi$
$734$		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	−0.813538	−	0.581512i	$-0.802461\pi$
$743$	−6.32456	+	6.32456i	−0.232025	+	0.232025i	0.581512	+	0.813538i	$0.302461\pi$
$744$		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	0.812638	−	0.582768i	$-0.198031\pi$
$757$	6.32456	+	6.32456i	0.229870	+	0.229870i	−0.582768	+	0.812638i	$0.698031\pi$
$758$		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	$0.507621\pi$
$773$	−28.4605	+	28.4605i	−1.02365	+	1.02365i	−0.999713	+	0.0239396i	$0.992379\pi$
$774$		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	−0.943260	−	0.332056i	$-0.892258\pi$
$787$	−17.1464	+	17.1464i	−0.611204	+	0.611204i	0.332056	+	0.943260i	$0.392258\pi$
$788$		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	0.995089	−	0.0989804i	$-0.0315581\pi$
$797$	25.2982	+	25.2982i	0.896109	+	0.896109i	−0.0989804	+	0.995089i	$0.531558\pi$
$798$		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	−0.451401	−	0.892321i	$-0.649076\pi$
$823$	12.6491	−	12.6491i	0.440920	−	0.440920i	0.892321	+	0.451401i	$0.149076\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.9217	+	15.9217i	−0.553651	+	0.553651i	−0.927493	−	0.373841i	$-0.878040\pi$
$827$	−15.9217	+	15.9217i	−0.553651	+	0.553651i	0.373841	+	0.927493i	$0.378040\pi$
$828$		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	−0.125854	−	0.992049i	$-0.540167\pi$
$853$	25.2982	−	25.2982i	0.866195	−	0.866195i	0.992049	+	0.125854i	$0.0401671\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.8207	+	20.8207i	−0.711220	+	0.711220i	−0.966790	−	0.255570i	$-0.917737\pi$
$857$	−20.8207	+	20.8207i	−0.711220	+	0.711220i	0.255570	+	0.966790i	$0.417737\pi$
$858$		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	−0.849892	−	0.526957i	$-0.823333\pi$
$863$	−9.48683	+	9.48683i	−0.322936	+	0.322936i	0.526957	+	0.849892i	$0.323333\pi$
$864$		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	0.887650	−	0.460519i	$-0.152337\pi$
$877$	12.6491	+	12.6491i	0.427130	+	0.427130i	−0.460519	+	0.887650i	$0.652337\pi$
$878$		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	0.197250	−	0.980353i	$-0.436799\pi$
$883$	−23.2702	−	23.2702i	−0.783103	−	0.783103i	−0.980353	+	0.197250i	$0.936799\pi$
$884$		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	−0.931943	−	0.362605i	$-0.881887\pi$
$907$	−17.1464	+	17.1464i	−0.569338	+	0.569338i	0.362605	+	0.931943i	$0.381887\pi$
$908$		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	−0.644539	−	0.764571i	$-0.722951\pi$
$937$	3.67423	−	3.67423i	0.120032	−	0.120032i	0.764571	+	0.644539i	$0.222951\pi$
$938$		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	$-0.488730\pi$
$947$	31.8434	−	31.8434i	1.03477	−	1.03477i	0.999373	−	0.0353971i	$-0.0112696\pi$
$948$		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	$-0.914343\pi$
$953$	−37.9671	−	37.9671i	−1.22987	−	1.22987i	−0.265864	−	0.964010i	$-0.585657\pi$
$954$		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	0.801448	−	0.598064i	$-0.204063\pi$
$967$	6.32456	+	6.32456i	0.203384	+	0.203384i	−0.598064	+	0.801448i	$0.704063\pi$
$968$		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	−0.645885	−	0.763434i	$-0.723512\pi$
$977$	3.67423	−	3.67423i	0.117549	−	0.117549i	0.763434	+	0.645885i	$0.223512\pi$
$978$		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	−0.912773	−	0.408468i	$-0.866063\pi$
$983$	−15.8114	+	15.8114i	−0.504305	+	0.504305i	0.408468	+	0.912773i	$0.366063\pi$
$984$		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	$-0.999521\pi$
$997$	−31.6228	−	31.6228i	−1.00150	−	1.00150i	−0.00150452	−	0.999999i	$-0.500479\pi$
$998$		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.143.4		8
4.3	odd	2		200.2.k.g.43.3	yes	8
5.2	odd	4	inner	800.2.o.h.207.3		8
5.3	odd	4	inner	800.2.o.h.207.2		8
5.4	even	2	inner	800.2.o.h.143.1		8
8.3	odd	2	inner	800.2.o.h.143.3		8
8.5	even	2		200.2.k.g.43.1	✓	8
20.3	even	4		200.2.k.g.107.4	yes	8
20.7	even	4		200.2.k.g.107.1	yes	8
20.19	odd	2		200.2.k.g.43.2	yes	8
40.3	even	4	inner	800.2.o.h.207.1		8
40.13	odd	4		200.2.k.g.107.2	yes	8
40.19	odd	2	inner	800.2.o.h.143.2		8
40.27	even	4	inner	800.2.o.h.207.4		8
40.29	even	2		200.2.k.g.43.4	yes	8
40.37	odd	4		200.2.k.g.107.3	yes	8

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