LMFDB - Embedded newform 816.2.bf.b.47.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(47,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))

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

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

chi := DirichletCharacter("816.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.bf (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:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.2
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	816.47
Dual form			816.2.bf.b.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(0.707107 - 0.707107i) q^{5} +(1.73205 + 1.73205i) q^{7} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 - 1.22474i) q^{3} +(0.707107 - 0.707107i) q^{5} +(1.73205 + 1.73205i) q^{7} +3.00000i q^{9} +(-3.67423 + 3.67423i) q^{11} +3.00000 q^{13} -1.73205 q^{15} +(-3.53553 + 2.12132i) q^{17} -5.19615 q^{19} -4.24264i q^{21} +(-1.22474 + 1.22474i) q^{23} +4.00000i q^{25} +(3.67423 - 3.67423i) q^{27} +(2.82843 - 2.82843i) q^{29} +(-6.92820 + 6.92820i) q^{31} +9.00000 q^{33} +2.44949 q^{35} +(5.00000 + 5.00000i) q^{37} +(-3.67423 - 3.67423i) q^{39} +(3.53553 + 3.53553i) q^{41} +5.19615 q^{43} +(2.12132 + 2.12132i) q^{45} +9.79796 q^{47} -1.00000i q^{49} +(6.92820 + 1.73205i) q^{51} -11.3137 q^{53} +5.19615i q^{55} +(6.36396 + 6.36396i) q^{57} +2.44949i q^{59} +(-1.00000 + 1.00000i) q^{61} +(-5.19615 + 5.19615i) q^{63} +(2.12132 - 2.12132i) q^{65} -13.8564i q^{67} +3.00000 q^{69} +(4.89898 + 4.89898i) q^{71} +(8.00000 + 8.00000i) q^{73} +(4.89898 - 4.89898i) q^{75} -12.7279 q^{77} +(-1.73205 - 1.73205i) q^{79} -9.00000 q^{81} +9.79796i q^{83} +(-1.00000 + 4.00000i) q^{85} -6.92820 q^{87} +9.89949i q^{89} +(5.19615 + 5.19615i) q^{91} +16.9706 q^{93} +(-3.67423 + 3.67423i) q^{95} +(-8.00000 - 8.00000i) q^{97} +(-11.0227 - 11.0227i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{13} + 72 q^{33} + 40 q^{37} - 8 q^{61} + 24 q^{69} + 64 q^{73} - 72 q^{81} - 8 q^{85} - 64 q^{97}+O(q^{100}) $ 8 * q + 24 * q^13 + 72 * q^33 + 40 * q^37 - 8 * q^61 + 24 * q^69 + 64 * q^73 - 72 * q^81 - 8 * q^85 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$545$	$613$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$	$1$

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
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$4$		0			0
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i	−0.531089	−	0.847316i	$-0.678217\pi$
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i	0.847316	+	0.531089i	$0.178217\pi$
$6$		0			0
$7$	1.73205	+	1.73205i	0.654654	+	0.654654i	0.954110	−	0.299456i	$-0.0968053\pi$
$7$	1.73205	+	1.73205i	0.654654	+	0.654654i	−0.299456	+	0.954110i	$0.596805\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	−3.67423	+	3.67423i	−1.10782	+	1.10782i	−0.114387	+	0.993436i	$0.536490\pi$
$11$	−3.67423	+	3.67423i	−1.10782	+	1.10782i	−0.993436	+	0.114387i	$0.963510\pi$
$12$		0			0
$13$	3.00000			0.832050			0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000			0.832050			0.416025	+	0.909353i	$0.363423\pi$
$14$		0			0
$15$	−1.73205			−0.447214
$16$		0			0
$17$	−3.53553	+	2.12132i	−0.857493	+	0.514496i
$17$	−3.53553	+	2.12132i	−0.857493	+	0.514496i
$18$		0			0
$19$	−5.19615			−1.19208			−0.596040	−	0.802955i	$-0.703260\pi$
$19$	−5.19615			−1.19208			−0.596040	+	0.802955i	$0.703260\pi$
$20$		0			0
$21$		−	4.24264i		−	0.925820i
$22$		0			0
$23$	−1.22474	+	1.22474i	−0.255377	+	0.255377i	−0.823171	−	0.567794i	$-0.807797\pi$
$23$	−1.22474	+	1.22474i	−0.255377	+	0.255377i	0.567794	+	0.823171i	$0.307797\pi$
$24$		0			0
$25$			4.00000i			0.800000i
$26$		0			0
$27$	3.67423	−	3.67423i	0.707107	−	0.707107i
$28$		0			0
$29$	2.82843	−	2.82843i	0.525226	−	0.525226i	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	2.82843	−	2.82843i	0.525226	−	0.525226i	0.919145	+	0.393919i	$0.128881\pi$
$30$		0			0
$31$	−6.92820	+	6.92820i	−1.24434	+	1.24434i	−0.286160	+	0.958182i	$0.592379\pi$
$31$	−6.92820	+	6.92820i	−1.24434	+	1.24434i	−0.958182	+	0.286160i	$0.907621\pi$
$32$		0			0
$33$	9.00000			1.56670
$34$		0			0
$35$	2.44949			0.414039
$36$		0			0
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	−3.67423	−	3.67423i	−0.588348	−	0.588348i
$40$		0			0
$41$	3.53553	+	3.53553i	0.552158	+	0.552158i	0.927063	−	0.374905i	$-0.122325\pi$
$41$	3.53553	+	3.53553i	0.552158	+	0.552158i	−0.374905	+	0.927063i	$0.622325\pi$
$42$		0			0
$43$	5.19615			0.792406			0.396203	−	0.918163i	$-0.370328\pi$
$43$	5.19615			0.792406			0.396203	+	0.918163i	$0.370328\pi$
$44$		0			0
$45$	2.12132	+	2.12132i	0.316228	+	0.316228i
$46$		0			0
$47$	9.79796			1.42918			0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796			1.42918			0.714590	+	0.699544i	$0.246613\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$	6.92820	+	1.73205i	0.970143	+	0.242536i
$52$		0			0
$53$	−11.3137			−1.55406			−0.777029	−	0.629465i	$-0.783274\pi$
$53$	−11.3137			−1.55406			−0.777029	+	0.629465i	$0.783274\pi$
$54$		0			0
$55$			5.19615i			0.700649i
$56$		0			0
$57$	6.36396	+	6.36396i	0.842927	+	0.842927i
$58$		0			0
$59$			2.44949i			0.318896i	0.987206	+	0.159448i	$0.0509715\pi$
$59$			2.44949i			0.318896i	−0.987206	+	0.159448i	$0.949029\pi$
$60$		0			0
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$	−5.19615	+	5.19615i	−0.654654	+	0.654654i
$64$		0			0
$65$	2.12132	−	2.12132i	0.263117	−	0.263117i
$66$		0			0
$67$		−	13.8564i		−	1.69283i	−0.532524	−	0.846415i	$-0.678756\pi$
$67$		−	13.8564i		−	1.69283i	0.532524	−	0.846415i	$-0.321244\pi$
$68$		0			0
$69$	3.00000			0.361158
$70$		0			0
$71$	4.89898	+	4.89898i	0.581402	+	0.581402i	0.935288	−	0.353886i	$-0.115140\pi$
$71$	4.89898	+	4.89898i	0.581402	+	0.581402i	−0.353886	+	0.935288i	$0.615140\pi$
$72$		0			0
$73$	8.00000	+	8.00000i	0.936329	+	0.936329i	0.998091	−	0.0617617i	$-0.0196719\pi$
$73$	8.00000	+	8.00000i	0.936329	+	0.936329i	−0.0617617	+	0.998091i	$0.519672\pi$
$74$		0			0
$75$	4.89898	−	4.89898i	0.565685	−	0.565685i
$76$		0			0
$77$	−12.7279			−1.45048
$78$		0			0
$79$	−1.73205	−	1.73205i	−0.194871	−	0.194871i	0.602926	−	0.797797i	$-0.294001\pi$
$79$	−1.73205	−	1.73205i	−0.194871	−	0.194871i	−0.797797	+	0.602926i	$0.794001\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			9.79796i			1.07547i	0.843115	+	0.537733i	$0.180719\pi$
$83$			9.79796i			1.07547i	−0.843115	+	0.537733i	$0.819281\pi$
$84$		0			0
$85$	−1.00000	+	4.00000i	−0.108465	+	0.433861i
$86$		0			0
$87$	−6.92820			−0.742781
$88$		0			0
$89$			9.89949i			1.04934i	0.851304	+	0.524672i	$0.175812\pi$
$89$			9.89949i			1.04934i	−0.851304	+	0.524672i	$0.824188\pi$
$90$		0			0
$91$	5.19615	+	5.19615i	0.544705	+	0.544705i
$92$		0			0
$93$	16.9706			1.75977
$94$		0			0
$95$	−3.67423	+	3.67423i	−0.376969	+	0.376969i
$96$		0			0
$97$	−8.00000	−	8.00000i	−0.812277	−	0.812277i	0.172698	−	0.984975i	$-0.444752\pi$
$97$	−8.00000	−	8.00000i	−0.812277	−	0.812277i	−0.984975	+	0.172698i	$0.944752\pi$
$98$		0			0
$99$	−11.0227	−	11.0227i	−1.10782	−	1.10782i
$100$		0			0
$101$		−	5.65685i		−	0.562878i	−0.959579	−	0.281439i	$-0.909188\pi$
$101$		−	5.65685i		−	0.562878i	0.959579	−	0.281439i	$-0.0908117\pi$
$102$		0			0
$103$			1.73205i			0.170664i	0.996353	+	0.0853320i	$0.0271951\pi$
$103$			1.73205i			0.170664i	−0.996353	+	0.0853320i	$0.972805\pi$
$104$		0			0
$105$	−3.00000	−	3.00000i	−0.292770	−	0.292770i
$106$		0			0
$107$	−1.22474	−	1.22474i	−0.118401	−	0.118401i	0.645424	−	0.763825i	$-0.276681\pi$
$107$	−1.22474	−	1.22474i	−0.118401	−	0.118401i	−0.763825	+	0.645424i	$0.776681\pi$
$108$		0			0
$109$	4.00000	−	4.00000i	0.383131	−	0.383131i	−0.489098	−	0.872229i	$-0.662674\pi$
$109$	4.00000	−	4.00000i	0.383131	−	0.383131i	0.872229	+	0.489098i	$0.162674\pi$
$110$		0			0
$111$		−	12.2474i		−	1.16248i
$112$		0			0
$113$	−4.94975	−	4.94975i	−0.465633	−	0.465633i	0.434863	−	0.900496i	$-0.356797\pi$
$113$	−4.94975	−	4.94975i	−0.465633	−	0.465633i	−0.900496	+	0.434863i	$0.856797\pi$
$114$		0			0
$115$			1.73205i			0.161515i
$116$		0			0
$117$			9.00000i			0.832050i
$118$		0			0
$119$	−9.79796	−	2.44949i	−0.898177	−	0.224544i
$120$		0			0
$121$		−	16.0000i		−	1.45455i
$122$		0			0
$123$		−	8.66025i		−	0.780869i
$124$		0			0
$125$	6.36396	+	6.36396i	0.569210	+	0.569210i
$126$		0			0
$127$	−8.66025			−0.768473			−0.384237	−	0.923235i	$-0.625535\pi$
$127$	−8.66025			−0.768473			−0.384237	+	0.923235i	$0.625535\pi$
$128$		0			0
$129$	−6.36396	−	6.36396i	−0.560316	−	0.560316i
$130$		0			0
$131$	6.12372	+	6.12372i	0.535032	+	0.535032i	0.922066	−	0.387033i	$-0.126500\pi$
$131$	6.12372	+	6.12372i	0.535032	+	0.535032i	−0.387033	+	0.922066i	$0.626500\pi$
$132$		0			0
$133$	−9.00000	−	9.00000i	−0.780399	−	0.780399i
$134$		0			0
$135$		−	5.19615i		−	0.447214i
$136$		0			0
$137$		−	7.07107i		−	0.604122i	−0.953289	−	0.302061i	$-0.902325\pi$
$137$		−	7.07107i		−	0.604122i	0.953289	−	0.302061i	$-0.0976746\pi$
$138$		0			0
$139$	15.5885	−	15.5885i	1.32220	−	1.32220i	0.410200	−	0.911996i	$-0.365459\pi$
$139$	15.5885	−	15.5885i	1.32220	−	1.32220i	0.911996	−	0.410200i	$-0.134541\pi$
$140$		0			0
$141$	−12.0000	−	12.0000i	−1.01058	−	1.01058i
$142$		0			0
$143$	−11.0227	+	11.0227i	−0.921765	+	0.921765i
$144$		0			0
$145$		−	4.00000i		−	0.332182i
$146$		0			0
$147$	−1.22474	+	1.22474i	−0.101015	+	0.101015i
$148$		0			0
$149$		−	12.7279i		−	1.04271i	−0.853339	−	0.521356i	$-0.825426\pi$
$149$		−	12.7279i		−	1.04271i	0.853339	−	0.521356i	$-0.174574\pi$
$150$		0			0
$151$	6.92820			0.563809			0.281905	−	0.959442i	$-0.409034\pi$
$151$	6.92820			0.563809			0.281905	+	0.959442i	$0.409034\pi$
$152$		0			0
$153$	−6.36396	−	10.6066i	−0.514496	−	0.857493i
$154$		0			0
$155$			9.79796i			0.786991i
$156$		0			0
$157$	−7.00000			−0.558661			−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000			−0.558661			−0.279330	+	0.960195i	$0.590112\pi$
$158$		0			0
$159$	13.8564	+	13.8564i	1.09888	+	1.09888i
$160$		0			0
$161$	−4.24264			−0.334367
$162$		0			0
$163$	6.92820	+	6.92820i	0.542659	+	0.542659i	0.924307	−	0.381649i	$-0.124644\pi$
$163$	6.92820	+	6.92820i	0.542659	+	0.542659i	−0.381649	+	0.924307i	$0.624644\pi$
$164$		0			0
$165$	6.36396	−	6.36396i	0.495434	−	0.495434i
$166$		0			0
$167$	3.67423	+	3.67423i	0.284321	+	0.284321i	0.834829	−	0.550509i	$-0.185566\pi$
$167$	3.67423	+	3.67423i	0.284321	+	0.284321i	−0.550509	+	0.834829i	$0.685566\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$		−	15.5885i		−	1.19208i
$172$		0			0
$173$	−17.6777	+	17.6777i	−1.34401	+	1.34401i	−0.451981	+	0.892028i	$0.649282\pi$
$173$	−17.6777	+	17.6777i	−1.34401	+	1.34401i	−0.892028	+	0.451981i	$0.850718\pi$
$174$		0			0
$175$	−6.92820	+	6.92820i	−0.523723	+	0.523723i
$176$		0			0
$177$	3.00000	−	3.00000i	0.225494	−	0.225494i
$178$		0			0
$179$		−	9.79796i		−	0.732334i	−0.930549	−	0.366167i	$-0.880670\pi$
$179$		−	9.79796i		−	0.732334i	0.930549	−	0.366167i	$-0.119330\pi$
$180$		0			0
$181$	−11.0000	+	11.0000i	−0.817624	+	0.817624i	−0.985763	−	0.168140i	$-0.946224\pi$
$181$	−11.0000	+	11.0000i	−0.817624	+	0.817624i	0.168140	+	0.985763i	$0.446224\pi$
$182$		0			0
$183$	2.44949			0.181071
$184$		0			0
$185$	7.07107			0.519875
$186$		0			0
$187$	5.19615	−	20.7846i	0.379980	−	1.51992i
$188$		0			0
$189$	12.7279			0.925820
$190$		0			0
$191$	−12.2474			−0.886194			−0.443097	−	0.896474i	$-0.646120\pi$
$191$	−12.2474			−0.886194			−0.443097	+	0.896474i	$0.646120\pi$
$192$		0			0
$193$	−8.00000	+	8.00000i	−0.575853	+	0.575853i	−0.933758	−	0.357905i	$-0.883491\pi$
$193$	−8.00000	+	8.00000i	−0.575853	+	0.575853i	0.357905	+	0.933758i	$0.383491\pi$
$194$		0			0
$195$	−5.19615			−0.372104
$196$		0			0
$197$	9.19239	+	9.19239i	0.654931	+	0.654931i	0.954176	−	0.299246i	$-0.0967350\pi$
$197$	9.19239	+	9.19239i	0.654931	+	0.654931i	−0.299246	+	0.954176i	$0.596735\pi$
$198$		0			0
$199$	6.92820	−	6.92820i	0.491127	−	0.491127i	−0.417534	−	0.908661i	$-0.637106\pi$
$199$	6.92820	−	6.92820i	0.491127	−	0.491127i	0.908661	+	0.417534i	$0.137106\pi$
$200$		0			0
$201$	−16.9706	+	16.9706i	−1.19701	+	1.19701i
$202$		0			0
$203$	9.79796			0.687682
$204$		0			0
$205$	5.00000			0.349215
$206$		0			0
$207$	−3.67423	−	3.67423i	−0.255377	−	0.255377i
$208$		0			0
$209$	19.0919	−	19.0919i	1.32061	−	1.32061i
$210$		0			0
$211$	−6.92820	−	6.92820i	−0.476957	−	0.476957i	0.427200	−	0.904157i	$-0.359500\pi$
$211$	−6.92820	−	6.92820i	−0.476957	−	0.476957i	−0.904157	+	0.427200i	$0.859500\pi$
$212$		0			0
$213$		−	12.0000i		−	0.822226i
$214$		0			0
$215$	3.67423	−	3.67423i	0.250581	−	0.250581i
$216$		0			0
$217$	−24.0000			−1.62923
$218$		0			0
$219$		−	19.5959i		−	1.32417i
$220$		0			0
$221$	−10.6066	+	6.36396i	−0.713477	+	0.428086i
$222$		0			0
$223$	−12.1244			−0.811907			−0.405953	−	0.913894i	$-0.633060\pi$
$223$	−12.1244			−0.811907			−0.405953	+	0.913894i	$0.633060\pi$
$224$		0			0
$225$	−12.0000			−0.800000
$226$		0			0
$227$	−6.12372	+	6.12372i	−0.406446	+	0.406446i	−0.880497	−	0.474051i	$-0.842791\pi$
$227$	−6.12372	+	6.12372i	−0.406446	+	0.406446i	0.474051	+	0.880497i	$0.342791\pi$
$228$		0			0
$229$		−	28.0000i		−	1.85029i	−0.379611	−	0.925146i	$-0.623942\pi$
$229$		−	28.0000i		−	1.85029i	0.379611	−	0.925146i	$-0.376058\pi$
$230$		0			0
$231$	15.5885	+	15.5885i	1.02565	+	1.02565i
$232$		0			0
$233$	−0.707107	+	0.707107i	−0.0463241	+	0.0463241i	−0.729889	−	0.683565i	$-0.760428\pi$
$233$	−0.707107	+	0.707107i	−0.0463241	+	0.0463241i	0.683565	+	0.729889i	$0.260428\pi$
$234$		0			0
$235$	6.92820	−	6.92820i	0.451946	−	0.451946i
$236$		0			0
$237$			4.24264i			0.275589i
$238$		0			0
$239$	29.3939			1.90133			0.950666	−	0.310217i	$-0.100402\pi$
$239$	29.3939			1.90133			0.950666	+	0.310217i	$0.100402\pi$
$240$		0			0
$241$	11.0000	+	11.0000i	0.708572	+	0.708572i	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	11.0000	+	11.0000i	0.708572	+	0.708572i	−0.257663	+	0.966235i	$0.582952\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.707107	+	0.707107i
$244$		0			0
$245$	−0.707107	−	0.707107i	−0.0451754	−	0.0451754i
$246$		0			0
$247$	−15.5885			−0.991870
$248$		0			0
$249$	12.0000	−	12.0000i	0.760469	−	0.760469i
$250$		0			0
$251$	2.44949			0.154610			0.0773052	−	0.997007i	$-0.475368\pi$
$251$	2.44949			0.154610			0.0773052	+	0.997007i	$0.475368\pi$
$252$		0			0
$253$		−	9.00000i		−	0.565825i
$254$		0			0
$255$	6.12372	−	3.67423i	0.383482	−	0.230089i
$256$		0			0
$257$	21.2132			1.32324			0.661622	−	0.749838i	$-0.269869\pi$
$257$	21.2132			1.32324			0.661622	+	0.749838i	$0.269869\pi$
$258$		0			0
$259$			17.3205i			1.07624i
$260$		0			0
$261$	8.48528	+	8.48528i	0.525226	+	0.525226i
$262$		0			0
$263$		−	2.44949i		−	0.151042i	−0.997144	−	0.0755210i	$-0.975938\pi$
$263$		−	2.44949i		−	0.151042i	0.997144	−	0.0755210i	$-0.0240620\pi$
$264$		0			0
$265$	−8.00000	+	8.00000i	−0.491436	+	0.491436i
$266$		0			0
$267$	12.1244	−	12.1244i	0.741999	−	0.741999i
$268$		0			0
$269$	−17.6777	+	17.6777i	−1.07783	+	1.07783i	−0.0811224	+	0.996704i	$0.525850\pi$
$269$	−17.6777	+	17.6777i	−1.07783	+	1.07783i	−0.996704	+	0.0811224i	$0.974150\pi$
$270$		0			0
$271$			5.19615i			0.315644i	0.987468	+	0.157822i	$0.0504472\pi$
$271$			5.19615i			0.315644i	−0.987468	+	0.157822i	$0.949553\pi$
$272$		0			0
$273$		−	12.7279i		−	0.770329i
$274$		0			0
$275$	−14.6969	−	14.6969i	−0.886259	−	0.886259i
$276$		0			0
$277$	−4.00000	−	4.00000i	−0.240337	−	0.240337i	0.576653	−	0.816989i	$-0.304359\pi$
$277$	−4.00000	−	4.00000i	−0.240337	−	0.240337i	−0.816989	+	0.576653i	$0.804359\pi$
$278$		0			0
$279$	−20.7846	−	20.7846i	−1.24434	−	1.24434i
$280$		0			0
$281$	−18.3848			−1.09674			−0.548372	−	0.836235i	$-0.684752\pi$
$281$	−18.3848			−1.09674			−0.548372	+	0.836235i	$0.684752\pi$
$282$		0			0
$283$	8.66025	+	8.66025i	0.514799	+	0.514799i	0.915993	−	0.401194i	$-0.131405\pi$
$283$	8.66025	+	8.66025i	0.514799	+	0.514799i	−0.401194	+	0.915993i	$0.631405\pi$
$284$		0			0
$285$	9.00000			0.533114
$286$		0			0
$287$			12.2474i			0.722944i
$288$		0			0
$289$	8.00000	−	15.0000i	0.470588	−	0.882353i
$290$		0			0
$291$			19.5959i			1.14873i
$292$		0			0
$293$			29.6985i			1.73500i	0.497434	+	0.867502i	$0.334276\pi$
$293$			29.6985i			1.73500i	−0.497434	+	0.867502i	$0.665724\pi$
$294$		0			0
$295$	1.73205	+	1.73205i	0.100844	+	0.100844i
$296$		0			0
$297$			27.0000i			1.56670i
$298$		0			0
$299$	−3.67423	+	3.67423i	−0.212486	+	0.212486i
$300$		0			0
$301$	9.00000	+	9.00000i	0.518751	+	0.518751i
$302$		0			0
$303$	−6.92820	+	6.92820i	−0.398015	+	0.398015i
$304$		0			0
$305$			1.41421i			0.0809776i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	2.12132	−	2.12132i	0.120678	−	0.120678i
$310$		0			0
$311$	14.6969	+	14.6969i	0.833387	+	0.833387i	0.987978	−	0.154592i	$-0.0494062\pi$
$311$	14.6969	+	14.6969i	0.833387	+	0.833387i	−0.154592	+	0.987978i	$0.549406\pi$
$312$		0			0
$313$	16.0000	−	16.0000i	0.904373	−	0.904373i	−0.0914374	−	0.995811i	$-0.529146\pi$
$313$	16.0000	−	16.0000i	0.904373	−	0.904373i	0.995811	+	0.0914374i	$0.0291461\pi$
$314$		0			0
$315$			7.34847i			0.414039i
$316$		0			0
$317$	−14.1421	−	14.1421i	−0.794301	−	0.794301i	0.187889	−	0.982190i	$-0.439836\pi$
$317$	−14.1421	−	14.1421i	−0.794301	−	0.794301i	−0.982190	+	0.187889i	$0.939836\pi$
$318$		0			0
$319$			20.7846i			1.16371i
$320$		0			0
$321$			3.00000i			0.167444i
$322$		0			0
$323$	18.3712	−	11.0227i	1.02220	−	0.613320i
$324$		0			0
$325$			12.0000i			0.665640i
$326$		0			0
$327$	−9.79796			−0.541828
$328$		0			0
$329$	16.9706	+	16.9706i	0.935617	+	0.935617i
$330$		0			0
$331$	−12.1244			−0.666415			−0.333207	−	0.942854i	$-0.608131\pi$
$331$	−12.1244			−0.666415			−0.333207	+	0.942854i	$0.608131\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$		0			0
$335$	−9.79796	−	9.79796i	−0.535320	−	0.535320i
$336$		0			0
$337$	17.0000	+	17.0000i	0.926049	+	0.926049i	0.997448	−	0.0713988i	$-0.0227463\pi$
$337$	17.0000	+	17.0000i	0.926049	+	0.926049i	−0.0713988	+	0.997448i	$0.522746\pi$
$338$		0			0
$339$			12.1244i			0.658505i
$340$		0			0
$341$		−	50.9117i		−	2.75702i
$342$		0			0
$343$	13.8564	−	13.8564i	0.748176	−	0.748176i
$344$		0			0
$345$	2.12132	−	2.12132i	0.114208	−	0.114208i
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$			15.0000i			0.802932i	0.915874	+	0.401466i	$0.131499\pi$
$349$			15.0000i			0.802932i	−0.915874	+	0.401466i	$0.868501\pi$
$350$		0			0
$351$	11.0227	−	11.0227i	0.588348	−	0.588348i
$352$		0			0
$353$		−	29.6985i		−	1.58069i	−0.612661	−	0.790345i	$-0.709901\pi$
$353$		−	29.6985i		−	1.58069i	0.612661	−	0.790345i	$-0.290099\pi$
$354$		0			0
$355$	6.92820			0.367711
$356$		0			0
$357$	9.00000	+	15.0000i	0.476331	+	0.793884i
$358$		0			0
$359$		−	36.7423i		−	1.93919i	−0.244721	−	0.969593i	$-0.578697\pi$
$359$		−	36.7423i		−	1.93919i	0.244721	−	0.969593i	$-0.421303\pi$
$360$		0			0
$361$	8.00000			0.421053
$362$		0			0
$363$	−19.5959	+	19.5959i	−1.02852	+	1.02852i
$364$		0			0
$365$	11.3137			0.592187
$366$		0			0
$367$	−1.73205	−	1.73205i	−0.0904123	−	0.0904123i	0.660454	−	0.750866i	$-0.270364\pi$
$367$	−1.73205	−	1.73205i	−0.0904123	−	0.0904123i	−0.750866	+	0.660454i	$0.770364\pi$
$368$		0			0
$369$	−10.6066	+	10.6066i	−0.552158	+	0.552158i
$370$		0			0
$371$	−19.5959	−	19.5959i	−1.01737	−	1.01737i
$372$		0			0
$373$	12.0000			0.621336			0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000			0.621336			0.310668	+	0.950518i	$0.399447\pi$
$374$		0			0
$375$		−	15.5885i		−	0.804984i
$376$		0			0
$377$	8.48528	−	8.48528i	0.437014	−	0.437014i
$378$		0			0
$379$	−5.19615	+	5.19615i	−0.266908	+	0.266908i	−0.827853	−	0.560945i	$-0.810438\pi$
$379$	−5.19615	+	5.19615i	−0.266908	+	0.266908i	0.560945	+	0.827853i	$0.310438\pi$
$380$		0			0
$381$	10.6066	+	10.6066i	0.543393	+	0.543393i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	−9.00000	+	9.00000i	−0.458682	+	0.458682i
$386$		0			0
$387$			15.5885i			0.792406i
$388$		0			0
$389$	4.24264			0.215110			0.107555	−	0.994199i	$-0.465698\pi$
$389$	4.24264			0.215110			0.107555	+	0.994199i	$0.465698\pi$
$390$		0			0
$391$	1.73205	−	6.92820i	0.0875936	−	0.350374i
$392$		0			0
$393$		−	15.0000i		−	0.756650i
$394$		0			0
$395$	−2.44949			−0.123247
$396$		0			0
$397$	−16.0000	+	16.0000i	−0.803017	+	0.803017i	−0.983566	−	0.180549i	$-0.942213\pi$
$397$	−16.0000	+	16.0000i	−0.803017	+	0.803017i	0.180549	+	0.983566i	$0.442213\pi$
$398$		0			0
$399$			22.0454i			1.10365i
$400$		0			0
$401$	3.53553	+	3.53553i	0.176556	+	0.176556i	0.789853	−	0.613297i	$-0.210157\pi$
$401$	3.53553	+	3.53553i	0.176556	+	0.176556i	−0.613297	+	0.789853i	$0.710157\pi$
$402$		0			0
$403$	−20.7846	+	20.7846i	−1.03536	+	1.03536i
$404$		0			0
$405$	−6.36396	+	6.36396i	−0.316228	+	0.316228i
$406$		0			0
$407$	−36.7423			−1.82125
$408$		0			0
$409$	11.0000			0.543915			0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000			0.543915			0.271957	+	0.962309i	$0.412329\pi$
$410$		0			0
$411$	−8.66025	+	8.66025i	−0.427179	+	0.427179i
$412$		0			0
$413$	−4.24264	+	4.24264i	−0.208767	+	0.208767i
$414$		0			0
$415$	6.92820	+	6.92820i	0.340092	+	0.340092i
$416$		0			0
$417$	−38.1838			−1.86987
$418$		0			0
$419$	19.5959	−	19.5959i	0.957323	−	0.957323i	−0.0418027	−	0.999126i	$-0.513310\pi$
$419$	19.5959	−	19.5959i	0.957323	−	0.957323i	0.999126	+	0.0418027i	$0.0133101\pi$
$420$		0			0
$421$	33.0000			1.60832			0.804161	−	0.594412i	$-0.202615\pi$
$421$	33.0000			1.60832			0.804161	+	0.594412i	$0.202615\pi$
$422$		0			0
$423$			29.3939i			1.42918i
$424$		0			0
$425$	−8.48528	−	14.1421i	−0.411597	−	0.685994i
$426$		0			0
$427$	−3.46410			−0.167640
$428$		0			0
$429$	27.0000			1.30357
$430$		0			0
$431$	−14.6969	+	14.6969i	−0.707927	+	0.707927i	−0.966099	−	0.258172i	$-0.916880\pi$
$431$	−14.6969	+	14.6969i	−0.707927	+	0.707927i	0.258172	+	0.966099i	$0.416880\pi$
$432$		0			0
$433$		−	7.00000i		−	0.336399i	−0.985753	−	0.168199i	$-0.946205\pi$
$433$		−	7.00000i		−	0.336399i	0.985753	−	0.168199i	$-0.0537952\pi$
$434$		0			0
$435$	−4.89898	+	4.89898i	−0.234888	+	0.234888i
$436$		0			0
$437$	6.36396	−	6.36396i	0.304430	−	0.304430i
$438$		0			0
$439$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$439$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	−9.79796			−0.465515			−0.232758	−	0.972535i	$-0.574775\pi$
$443$	−9.79796			−0.465515			−0.232758	+	0.972535i	$0.574775\pi$
$444$		0			0
$445$	7.00000	+	7.00000i	0.331832	+	0.331832i
$446$		0			0
$447$	−15.5885	+	15.5885i	−0.737309	+	0.737309i
$448$		0			0
$449$	24.0416	+	24.0416i	1.13459	+	1.13459i	0.989404	+	0.145191i	$0.0463797\pi$
$449$	24.0416	+	24.0416i	1.13459	+	1.13459i	0.145191	+	0.989404i	$0.453620\pi$
$450$		0			0
$451$	−25.9808			−1.22339
$452$		0			0
$453$	−8.48528	−	8.48528i	−0.398673	−	0.398673i
$454$		0			0
$455$	7.34847			0.344502
$456$		0			0
$457$			13.0000i			0.608114i	0.952654	+	0.304057i	$0.0983414\pi$
$457$			13.0000i			0.608114i	−0.952654	+	0.304057i	$0.901659\pi$
$458$		0			0
$459$	−5.19615	+	20.7846i	−0.242536	+	0.970143i
$460$		0			0
$461$	24.0416			1.11973			0.559865	−	0.828584i	$-0.310853\pi$
$461$	24.0416			1.11973			0.559865	+	0.828584i	$0.310853\pi$
$462$		0			0
$463$		−	6.92820i		−	0.321981i	−0.986956	−	0.160990i	$-0.948531\pi$
$463$		−	6.92820i		−	0.321981i	0.986956	−	0.160990i	$-0.0514688\pi$
$464$		0			0
$465$	12.0000	−	12.0000i	0.556487	−	0.556487i
$466$		0			0
$467$			26.9444i			1.24684i	0.781888	+	0.623419i	$0.214257\pi$
$467$			26.9444i			1.24684i	−0.781888	+	0.623419i	$0.785743\pi$
$468$		0			0
$469$	24.0000	−	24.0000i	1.10822	−	1.10822i
$470$		0			0
$471$	8.57321	+	8.57321i	0.395033	+	0.395033i
$472$		0			0
$473$	−19.0919	+	19.0919i	−0.877846	+	0.877846i
$474$		0			0
$475$		−	20.7846i		−	0.953663i
$476$		0			0
$477$		−	33.9411i		−	1.55406i
$478$		0			0
$479$	−11.0227	−	11.0227i	−0.503640	−	0.503640i	0.408927	−	0.912567i	$-0.365903\pi$
$479$	−11.0227	−	11.0227i	−0.503640	−	0.503640i	−0.912567	+	0.408927i	$0.865903\pi$
$480$		0			0
$481$	15.0000	+	15.0000i	0.683941	+	0.683941i
$482$		0			0
$483$	5.19615	+	5.19615i	0.236433	+	0.236433i
$484$		0			0
$485$	−11.3137			−0.513729
$486$		0			0
$487$	−5.19615	−	5.19615i	−0.235460	−	0.235460i	0.579507	−	0.814967i	$-0.303245\pi$
$487$	−5.19615	−	5.19615i	−0.235460	−	0.235460i	−0.814967	+	0.579507i	$0.803245\pi$
$488$		0			0
$489$		−	16.9706i		−	0.767435i
$490$		0			0
$491$		−	19.5959i		−	0.884351i	−0.896928	−	0.442176i	$-0.854207\pi$
$491$		−	19.5959i		−	0.884351i	0.896928	−	0.442176i	$-0.145793\pi$
$492$		0			0
$493$	−4.00000	+	16.0000i	−0.180151	+	0.720604i
$494$		0			0
$495$	−15.5885			−0.700649
$496$		0			0
$497$			16.9706i			0.761234i
$498$		0			0
$499$	13.8564	+	13.8564i	0.620298	+	0.620298i	0.945608	−	0.325310i	$-0.105469\pi$
$499$	13.8564	+	13.8564i	0.620298	+	0.620298i	−0.325310	+	0.945608i	$0.605469\pi$
$500$		0			0
$501$		−	9.00000i		−	0.402090i
$502$		0			0
$503$	13.4722	−	13.4722i	0.600695	−	0.600695i	−0.339802	−	0.940497i	$-0.610360\pi$
$503$	13.4722	−	13.4722i	0.600695	−	0.600695i	0.940497	+	0.339802i	$0.110360\pi$
$504$		0			0
$505$	−4.00000	−	4.00000i	−0.177998	−	0.177998i
$506$		0			0
$507$	4.89898	+	4.89898i	0.217571	+	0.217571i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$			27.7128i			1.22594i
$512$		0			0
$513$	−19.0919	+	19.0919i	−0.842927	+	0.842927i
$514$		0			0
$515$	1.22474	+	1.22474i	0.0539687	+	0.0539687i
$516$		0			0
$517$	−36.0000	+	36.0000i	−1.58328	+	1.58328i
$518$		0			0
$519$	43.3013			1.90071
$520$		0			0
$521$	16.2635	+	16.2635i	0.712515	+	0.712515i	0.967061	−	0.254546i	$-0.0819260\pi$
$521$	16.2635	+	16.2635i	0.712515	+	0.712515i	−0.254546	+	0.967061i	$0.581926\pi$
$522$		0			0
$523$		−	13.8564i		−	0.605898i	−0.953007	−	0.302949i	$-0.902029\pi$
$523$		−	13.8564i		−	0.605898i	0.953007	−	0.302949i	$-0.0979712\pi$
$524$		0			0
$525$	16.9706			0.740656
$526$		0			0
$527$	9.79796	−	39.1918i	0.426806	−	1.70722i
$528$		0			0
$529$			20.0000i			0.869565i
$530$		0			0
$531$	−7.34847			−0.318896
$532$		0			0
$533$	10.6066	+	10.6066i	0.459423	+	0.459423i
$534$		0			0
$535$	−1.73205			−0.0748831
$536$		0			0
$537$	−12.0000	+	12.0000i	−0.517838	+	0.517838i
$538$		0			0
$539$	3.67423	+	3.67423i	0.158260	+	0.158260i
$540$		0			0
$541$	−11.0000	−	11.0000i	−0.472927	−	0.472927i	0.429934	−	0.902861i	$-0.358537\pi$
$541$	−11.0000	−	11.0000i	−0.472927	−	0.472927i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$	26.9444			1.15629
$544$		0			0
$545$		−	5.65685i		−	0.242313i
$546$		0			0
$547$	−19.0526	+	19.0526i	−0.814629	+	0.814629i	−0.985324	−	0.170695i	$-0.945399\pi$
$547$	−19.0526	+	19.0526i	−0.814629	+	0.814629i	0.170695	+	0.985324i	$0.445399\pi$
$548$		0			0
$549$	−3.00000	−	3.00000i	−0.128037	−	0.128037i
$550$		0			0
$551$	−14.6969	+	14.6969i	−0.626111	+	0.626111i
$552$		0			0
$553$		−	6.00000i		−	0.255146i
$554$		0			0
$555$	−8.66025	−	8.66025i	−0.367607	−	0.367607i
$556$		0			0
$557$		−	15.5563i		−	0.659144i	−0.944131	−	0.329572i	$-0.893096\pi$
$557$		−	15.5563i		−	0.659144i	0.944131	−	0.329572i	$-0.106904\pi$
$558$		0			0
$559$	15.5885			0.659321
$560$		0			0
$561$	−31.8198	+	19.0919i	−1.34343	+	0.806060i
$562$		0			0
$563$			9.79796i			0.412935i	0.978453	+	0.206467i	$0.0661967\pi$
$563$			9.79796i			0.412935i	−0.978453	+	0.206467i	$0.933803\pi$
$564$		0			0
$565$	−7.00000			−0.294492
$566$		0			0
$567$	−15.5885	−	15.5885i	−0.654654	−	0.654654i
$568$		0			0
$569$	39.5980			1.66003			0.830017	−	0.557738i	$-0.188331\pi$
$569$	39.5980			1.66003			0.830017	+	0.557738i	$0.188331\pi$
$570$		0			0
$571$	−20.7846	−	20.7846i	−0.869809	−	0.869809i	0.122642	−	0.992451i	$-0.460863\pi$
$571$	−20.7846	−	20.7846i	−0.869809	−	0.869809i	−0.992451	+	0.122642i	$0.960863\pi$
$572$		0			0
$573$	15.0000	+	15.0000i	0.626634	+	0.626634i
$574$		0			0
$575$	−4.89898	−	4.89898i	−0.204302	−	0.204302i
$576$		0			0
$577$	3.00000			0.124892			0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000			0.124892			0.0624458	+	0.998048i	$0.480110\pi$
$578$		0			0
$579$	19.5959			0.814379
$580$		0			0
$581$	−16.9706	+	16.9706i	−0.704058	+	0.704058i
$582$		0			0
$583$	41.5692	−	41.5692i	1.72162	−	1.72162i
$584$		0			0
$585$	6.36396	+	6.36396i	0.263117	+	0.263117i
$586$		0			0
$587$		−	39.1918i		−	1.61762i	−0.588070	−	0.808810i	$-0.700112\pi$
$587$		−	39.1918i		−	1.61762i	0.588070	−	0.808810i	$-0.299888\pi$
$588$		0			0
$589$	36.0000	−	36.0000i	1.48335	−	1.48335i
$590$		0			0
$591$		−	22.5167i		−	0.926212i
$592$		0			0
$593$	39.5980			1.62609			0.813047	−	0.582198i	$-0.197807\pi$
$593$	39.5980			1.62609			0.813047	+	0.582198i	$0.197807\pi$
$594$		0			0
$595$	−8.66025	+	5.19615i	−0.355036	+	0.213021i
$596$		0			0
$597$	−16.9706			−0.694559
$598$		0			0
$599$	31.8434			1.30108			0.650542	−	0.759470i	$-0.274542\pi$
$599$	31.8434			1.30108			0.650542	+	0.759470i	$0.274542\pi$
$600$		0			0
$601$	4.00000	−	4.00000i	0.163163	−	0.163163i	−0.620803	−	0.783967i	$-0.713193\pi$
$601$	4.00000	−	4.00000i	0.163163	−	0.163163i	0.783967	+	0.620803i	$0.213193\pi$
$602$		0			0
$603$	41.5692			1.69283
$604$		0			0
$605$	−11.3137	−	11.3137i	−0.459968	−	0.459968i
$606$		0			0
$607$	−15.5885	+	15.5885i	−0.632716	+	0.632716i	−0.948748	−	0.316032i	$-0.897649\pi$
$607$	−15.5885	+	15.5885i	−0.632716	+	0.632716i	0.316032	+	0.948748i	$0.397649\pi$
$608$		0			0
$609$	−12.0000	−	12.0000i	−0.486265	−	0.486265i
$610$		0			0
$611$	29.3939			1.18915
$612$		0			0
$613$	−37.0000			−1.49442			−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−37.0000			−1.49442			−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$	−6.12372	−	6.12372i	−0.246932	−	0.246932i
$616$		0			0
$617$	5.65685	−	5.65685i	0.227736	−	0.227736i	−0.584010	−	0.811746i	$-0.698517\pi$
$617$	5.65685	−	5.65685i	0.227736	−	0.227736i	0.811746	+	0.584010i	$0.198517\pi$
$618$		0			0
$619$	19.0526	+	19.0526i	0.765787	+	0.765787i	0.977362	−	0.211575i	$-0.0678592\pi$
$619$	19.0526	+	19.0526i	0.765787	+	0.765787i	−0.211575	+	0.977362i	$0.567859\pi$
$620$		0			0
$621$			9.00000i			0.361158i
$622$		0			0
$623$	−17.1464	+	17.1464i	−0.686957	+	0.686957i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−46.7654			−1.86763
$628$		0			0
$629$	−28.2843	−	7.07107i	−1.12777	−	0.281942i
$630$		0			0
$631$	39.8372			1.58589			0.792946	−	0.609292i	$-0.208546\pi$
$631$	39.8372			1.58589			0.792946	+	0.609292i	$0.208546\pi$
$632$		0			0
$633$			16.9706i			0.674519i
$634$		0			0
$635$	−6.12372	+	6.12372i	−0.243013	+	0.243013i
$636$		0			0
$637$		−	3.00000i		−	0.118864i
$638$		0			0
$639$	−14.6969	+	14.6969i	−0.581402	+	0.581402i
$640$		0			0
$641$	21.9203	−	21.9203i	0.865800	−	0.865800i	−0.126204	−	0.992004i	$-0.540279\pi$
$641$	21.9203	−	21.9203i	0.865800	−	0.865800i	0.992004	+	0.126204i	$0.0402794\pi$
$642$		0			0
$643$	−20.7846	+	20.7846i	−0.819665	+	0.819665i	−0.986059	−	0.166394i	$-0.946788\pi$
$643$	−20.7846	+	20.7846i	−0.819665	+	0.819665i	0.166394	+	0.986059i	$0.446788\pi$
$644$		0			0
$645$	−9.00000			−0.354375
$646$		0			0
$647$	41.6413			1.63709			0.818545	−	0.574443i	$-0.194781\pi$
$647$	41.6413			1.63709			0.818545	+	0.574443i	$0.194781\pi$
$648$		0			0
$649$	−9.00000	−	9.00000i	−0.353281	−	0.353281i
$650$		0			0
$651$	29.3939	+	29.3939i	1.15204	+	1.15204i
$652$		0			0
$653$	−3.53553	−	3.53553i	−0.138356	−	0.138356i	0.634537	−	0.772893i	$-0.281191\pi$
$653$	−3.53553	−	3.53553i	−0.138356	−	0.138356i	−0.772893	+	0.634537i	$0.781191\pi$
$654$		0			0
$655$	8.66025			0.338384
$656$		0			0
$657$	−24.0000	+	24.0000i	−0.936329	+	0.936329i
$658$		0			0
$659$	−12.2474			−0.477093			−0.238546	−	0.971131i	$-0.576671\pi$
$659$	−12.2474			−0.477093			−0.238546	+	0.971131i	$0.576671\pi$
$660$		0			0
$661$		−	15.0000i		−	0.583432i	−0.956505	−	0.291716i	$-0.905774\pi$
$661$		−	15.0000i		−	0.583432i	0.956505	−	0.291716i	$-0.0942263\pi$
$662$		0			0
$663$	20.7846	+	5.19615i	0.807207	+	0.201802i
$664$		0			0
$665$	−12.7279			−0.493568
$666$		0			0
$667$			6.92820i			0.268261i
$668$		0			0
$669$	14.8492	+	14.8492i	0.574105	+	0.574105i
$670$		0			0
$671$		−	7.34847i		−	0.283685i
$672$		0			0
$673$	32.0000	−	32.0000i	1.23351	−	1.23351i	0.270903	−	0.962607i	$-0.412678\pi$
$673$	32.0000	−	32.0000i	1.23351	−	1.23351i	0.962607	−	0.270903i	$-0.0873221\pi$
$674$		0			0
$675$	14.6969	+	14.6969i	0.565685	+	0.565685i
$676$		0			0
$677$	−28.9914	+	28.9914i	−1.11423	+	1.11423i	−0.121657	+	0.992572i	$0.538821\pi$
$677$	−28.9914	+	28.9914i	−1.11423	+	1.11423i	−0.992572	+	0.121657i	$0.961179\pi$
$678$		0			0
$679$		−	27.7128i		−	1.06352i
$680$		0			0
$681$	15.0000			0.574801
$682$		0			0
$683$	1.22474	+	1.22474i	0.0468636	+	0.0468636i	0.730150	−	0.683287i	$-0.239450\pi$
$683$	1.22474	+	1.22474i	0.0468636	+	0.0468636i	−0.683287	+	0.730150i	$0.739450\pi$
$684$		0			0
$685$	−5.00000	−	5.00000i	−0.191040	−	0.191040i
$686$		0			0
$687$	−34.2929	+	34.2929i	−1.30835	+	1.30835i
$688$		0			0
$689$	−33.9411			−1.29305
$690$		0			0
$691$	13.8564	+	13.8564i	0.527123	+	0.527123i	0.919713	−	0.392591i	$-0.128421\pi$
$691$	13.8564	+	13.8564i	0.527123	+	0.527123i	−0.392591	+	0.919713i	$0.628421\pi$
$692$		0			0
$693$		−	38.1838i		−	1.45048i
$694$		0			0
$695$		−	22.0454i		−	0.836230i
$696$		0			0
$697$	−20.0000	−	5.00000i	−0.757554	−	0.189389i
$698$		0			0
$699$	1.73205			0.0655122
$700$		0			0
$701$		−	39.5980i		−	1.49560i	−0.663927	−	0.747798i	$-0.731111\pi$
$701$		−	39.5980i		−	1.49560i	0.663927	−	0.747798i	$-0.268889\pi$
$702$		0			0
$703$	−25.9808	−	25.9808i	−0.979883	−	0.979883i
$704$		0			0
$705$	−16.9706			−0.639148
$706$		0			0
$707$	9.79796	−	9.79796i	0.368490	−	0.368490i
$708$		0			0
$709$	−4.00000	−	4.00000i	−0.150223	−	0.150223i	0.627995	−	0.778218i	$-0.283876\pi$
$709$	−4.00000	−	4.00000i	−0.150223	−	0.150223i	−0.778218	+	0.627995i	$0.783876\pi$
$710$		0			0
$711$	5.19615	−	5.19615i	0.194871	−	0.194871i
$712$		0			0
$713$		−	16.9706i		−	0.635553i
$714$		0			0
$715$			15.5885i			0.582975i
$716$		0			0
$717$	−36.0000	−	36.0000i	−1.34444	−	1.34444i
$718$		0			0
$719$	−25.7196	−	25.7196i	−0.959181	−	0.959181i	0.0400181	−	0.999199i	$-0.487258\pi$
$719$	−25.7196	−	25.7196i	−0.959181	−	0.959181i	−0.999199	+	0.0400181i	$0.987258\pi$
$720$		0			0
$721$	−3.00000	+	3.00000i	−0.111726	+	0.111726i
$722$		0			0
$723$		−	26.9444i		−	1.00207i
$724$		0			0
$725$	11.3137	+	11.3137i	0.420181	+	0.420181i
$726$		0			0
$727$		−	24.2487i		−	0.899335i	−0.893196	−	0.449667i	$-0.851542\pi$
$727$		−	24.2487i		−	0.899335i	0.893196	−	0.449667i	$-0.148458\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	−18.3712	+	11.0227i	−0.679482	+	0.407689i
$732$		0			0
$733$		−	36.0000i		−	1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$		−	36.0000i		−	1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$		0			0
$735$			1.73205i			0.0638877i
$736$		0			0
$737$	50.9117	+	50.9117i	1.87536	+	1.87536i
$738$		0			0
$739$	−1.73205			−0.0637145			−0.0318573	−	0.999492i	$-0.510142\pi$
$739$	−1.73205			−0.0637145			−0.0318573	+	0.999492i	$0.510142\pi$
$740$		0			0
$741$	19.0919	+	19.0919i	0.701358	+	0.701358i
$742$		0			0
$743$	−2.44949	−	2.44949i	−0.0898631	−	0.0898631i	0.660746	−	0.750609i	$-0.270240\pi$
$743$	−2.44949	−	2.44949i	−0.0898631	−	0.0898631i	−0.750609	+	0.660746i	$0.770240\pi$
$744$		0			0
$745$	−9.00000	−	9.00000i	−0.329734	−	0.329734i
$746$		0			0
$747$	−29.3939			−1.07547
$748$		0			0
$749$		−	4.24264i		−	0.155023i
$750$		0			0
$751$	20.7846	−	20.7846i	0.758441	−	0.758441i	−0.217597	−	0.976039i	$-0.569822\pi$
$751$	20.7846	−	20.7846i	0.758441	−	0.758441i	0.976039	+	0.217597i	$0.0698220\pi$
$752$		0			0
$753$	−3.00000	−	3.00000i	−0.109326	−	0.109326i
$754$		0			0
$755$	4.89898	−	4.89898i	0.178292	−	0.178292i
$756$		0			0
$757$			5.00000i			0.181728i	0.995863	+	0.0908640i	$0.0289629\pi$
$757$			5.00000i			0.181728i	−0.995863	+	0.0908640i	$0.971037\pi$
$758$		0			0
$759$	−11.0227	+	11.0227i	−0.400099	+	0.400099i
$760$		0			0
$761$		−	5.65685i		−	0.205061i	−0.994730	−	0.102530i	$-0.967306\pi$
$761$		−	5.65685i		−	0.205061i	0.994730	−	0.102530i	$-0.0326939\pi$
$762$		0			0
$763$	13.8564			0.501636
$764$		0			0
$765$	−12.0000	−	3.00000i	−0.433861	−	0.108465i
$766$		0			0
$767$			7.34847i			0.265338i
$768$		0			0
$769$	−33.0000			−1.19001			−0.595005	−	0.803722i	$-0.702850\pi$
$769$	−33.0000			−1.19001			−0.595005	+	0.803722i	$0.702850\pi$
$770$		0			0
$771$	−25.9808	−	25.9808i	−0.935674	−	0.935674i
$772$		0			0
$773$	33.9411			1.22078			0.610389	−	0.792102i	$-0.291013\pi$
$773$	33.9411			1.22078			0.610389	+	0.792102i	$0.291013\pi$
$774$		0			0
$775$	−27.7128	−	27.7128i	−0.995474	−	0.995474i
$776$		0			0
$777$	21.2132	−	21.2132i	0.761019	−	0.761019i
$778$		0			0
$779$	−18.3712	−	18.3712i	−0.658216	−	0.658216i
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$		−	20.7846i		−	0.742781i
$784$		0			0
$785$	−4.94975	+	4.94975i	−0.176664	+	0.176664i
$786$		0			0
$787$	13.8564	−	13.8564i	0.493928	−	0.493928i	−0.415614	−	0.909541i	$-0.636433\pi$
$787$	13.8564	−	13.8564i	0.493928	−	0.493928i	0.909541	+	0.415614i	$0.136433\pi$
$788$		0			0
$789$	−3.00000	+	3.00000i	−0.106803	+	0.106803i
$790$		0			0
$791$		−	17.1464i		−	0.609657i
$792$		0			0
$793$	−3.00000	+	3.00000i	−0.106533	+	0.106533i
$794$		0			0
$795$	19.5959			0.694996
$796$		0			0
$797$	−28.2843			−1.00188			−0.500940	−	0.865482i	$-0.667012\pi$
$797$	−28.2843			−1.00188			−0.500940	+	0.865482i	$0.667012\pi$
$798$		0			0
$799$	−34.6410	+	20.7846i	−1.22551	+	0.735307i
$800$		0			0
$801$	−29.6985			−1.04934
$802$		0			0
$803$	−58.7878			−2.07457
$804$		0			0
$805$	−3.00000	+	3.00000i	−0.105736	+	0.105736i
$806$		0			0
$807$	43.3013			1.52428
$808$		0			0
$809$	−28.9914	−	28.9914i	−1.01928	−	1.01928i	−0.999810	−	0.0194722i	$-0.993801\pi$
$809$	−28.9914	−	28.9914i	−1.01928	−	1.01928i	−0.0194722	−	0.999810i	$-0.506199\pi$
$810$		0			0
$811$	−6.92820	+	6.92820i	−0.243282	+	0.243282i	−0.818207	−	0.574924i	$-0.805031\pi$
$811$	−6.92820	+	6.92820i	−0.243282	+	0.243282i	0.574924	+	0.818207i	$0.305031\pi$
$812$		0			0
$813$	6.36396	−	6.36396i	0.223194	−	0.223194i
$814$		0			0
$815$	9.79796			0.343208
$816$		0			0
$817$	−27.0000			−0.944610
$818$		0			0
$819$	−15.5885	+	15.5885i	−0.544705	+	0.544705i
$820$		0			0
$821$	28.9914	−	28.9914i	1.01181	−	1.01181i	0.0118766	−	0.999929i	$-0.496219\pi$
$821$	28.9914	−	28.9914i	1.01181	−	1.01181i	0.999929	−	0.0118766i	$-0.00378053\pi$
$822$		0			0
$823$	19.0526	+	19.0526i	0.664130	+	0.664130i	0.956351	−	0.292221i	$-0.0943941\pi$
$823$	19.0526	+	19.0526i	0.664130	+	0.664130i	−0.292221	+	0.956351i	$0.594394\pi$
$824$		0			0
$825$			36.0000i			1.25336i
$826$		0			0
$827$	11.0227	−	11.0227i	0.383297	−	0.383297i	−0.488992	−	0.872289i	$-0.662635\pi$
$827$	11.0227	−	11.0227i	0.383297	−	0.383297i	0.872289	+	0.488992i	$0.162635\pi$
$828$		0			0
$829$	4.00000			0.138926			0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000			0.138926			0.0694629	+	0.997585i	$0.477871\pi$
$830$		0			0
$831$			9.79796i			0.339887i
$832$		0			0
$833$	2.12132	+	3.53553i	0.0734994	+	0.122499i
$834$		0			0
$835$	5.19615			0.179820
$836$		0			0
$837$			50.9117i			1.75977i
$838$		0			0
$839$	−37.9671	+	37.9671i	−1.31077	+	1.31077i	−0.389921	+	0.920848i	$0.627498\pi$
$839$	−37.9671	+	37.9671i	−1.31077	+	1.31077i	−0.920848	+	0.389921i	$0.872502\pi$
$840$		0			0
$841$			13.0000i			0.448276i
$842$		0			0
$843$	22.5167	+	22.5167i	0.775515	+	0.775515i
$844$		0			0
$845$	−2.82843	+	2.82843i	−0.0973009	+	0.0973009i
$846$		0			0
$847$	27.7128	−	27.7128i	0.952224	−	0.952224i
$848$		0			0
$849$		−	21.2132i		−	0.728035i
$850$		0			0
$851$	−12.2474			−0.419837
$852$		0			0
$853$	−1.00000	−	1.00000i	−0.0342393	−	0.0342393i	0.689780	−	0.724019i	$-0.257707\pi$
$853$	−1.00000	−	1.00000i	−0.0342393	−	0.0342393i	−0.724019	+	0.689780i	$0.757707\pi$
$854$		0			0
$855$	−11.0227	−	11.0227i	−0.376969	−	0.376969i
$856$		0			0
$857$	−22.6274	−	22.6274i	−0.772938	−	0.772938i	0.205681	−	0.978619i	$-0.434059\pi$
$857$	−22.6274	−	22.6274i	−0.772938	−	0.772938i	−0.978619	+	0.205681i	$0.934059\pi$
$858$		0			0
$859$	27.7128			0.945549			0.472774	−	0.881183i	$-0.343253\pi$
$859$	27.7128			0.945549			0.472774	+	0.881183i	$0.343253\pi$
$860$		0			0
$861$	15.0000	−	15.0000i	0.511199	−	0.511199i
$862$		0			0
$863$	31.8434			1.08396			0.541980	−	0.840391i	$-0.317675\pi$
$863$	31.8434			1.08396			0.541980	+	0.840391i	$0.317675\pi$
$864$		0			0
$865$			25.0000i			0.850026i
$866$		0			0
$867$	−28.1691	+	8.57321i	−0.956674	+	0.291162i
$868$		0			0
$869$	12.7279			0.431765
$870$		0			0
$871$		−	41.5692i		−	1.40852i
$872$		0			0
$873$	24.0000	−	24.0000i	0.812277	−	0.812277i
$874$		0			0
$875$			22.0454i			0.745271i
$876$		0			0
$877$	−19.0000	+	19.0000i	−0.641584	+	0.641584i	−0.950945	−	0.309360i	$-0.899885\pi$
$877$	−19.0000	+	19.0000i	−0.641584	+	0.641584i	0.309360	+	0.950945i	$0.399885\pi$
$878$		0			0
$879$	36.3731	−	36.3731i	1.22683	−	1.22683i
$880$		0			0
$881$	−9.89949	+	9.89949i	−0.333522	+	0.333522i	−0.853923	−	0.520400i	$-0.825783\pi$
$881$	−9.89949	+	9.89949i	−0.333522	+	0.333522i	0.520400	+	0.853923i	$0.325783\pi$
$882$		0			0
$883$			50.2295i			1.69036i	0.534485	+	0.845178i	$0.320506\pi$
$883$			50.2295i			1.69036i	−0.534485	+	0.845178i	$0.679494\pi$
$884$		0			0
$885$		−	4.24264i		−	0.142615i
$886$		0			0
$887$	23.2702	+	23.2702i	0.781335	+	0.781335i	0.980056	−	0.198721i	$-0.0636787\pi$
$887$	23.2702	+	23.2702i	0.781335	+	0.781335i	−0.198721	+	0.980056i	$0.563679\pi$
$888$		0			0
$889$	−15.0000	−	15.0000i	−0.503084	−	0.503084i
$890$		0			0
$891$	33.0681	−	33.0681i	1.10782	−	1.10782i
$892$		0			0
$893$	−50.9117			−1.70369
$894$		0			0
$895$	−6.92820	−	6.92820i	−0.231584	−	0.231584i
$896$		0			0
$897$	9.00000			0.300501
$898$		0			0
$899$			39.1918i			1.30712i
$900$		0			0
$901$	40.0000	−	24.0000i	1.33259	−	0.799556i
$902$		0			0
$903$		−	22.0454i		−	0.733625i
$904$		0			0
$905$			15.5563i			0.517111i
$906$		0			0
$907$	−12.1244	−	12.1244i	−0.402583	−	0.402583i	0.476560	−	0.879142i	$-0.341884\pi$
$907$	−12.1244	−	12.1244i	−0.402583	−	0.402583i	−0.879142	+	0.476560i	$0.841884\pi$
$908$		0			0
$909$	16.9706			0.562878
$910$		0			0
$911$	−6.12372	+	6.12372i	−0.202888	+	0.202888i	−0.801236	−	0.598348i	$-0.795824\pi$
$911$	−6.12372	+	6.12372i	−0.202888	+	0.202888i	0.598348	+	0.801236i	$0.295824\pi$
$912$		0			0
$913$	−36.0000	−	36.0000i	−1.19143	−	1.19143i
$914$		0			0
$915$	1.73205	−	1.73205i	0.0572598	−	0.0572598i
$916$		0			0
$917$			21.2132i			0.700522i
$918$		0			0
$919$			36.3731i			1.19984i	0.800061	+	0.599918i	$0.204800\pi$
$919$			36.3731i			1.19984i	−0.800061	+	0.599918i	$0.795200\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	14.6969	+	14.6969i	0.483756	+	0.483756i
$924$		0			0
$925$	−20.0000	+	20.0000i	−0.657596	+	0.657596i
$926$		0			0
$927$	−5.19615			−0.170664
$928$		0			0
$929$	−24.7487	−	24.7487i	−0.811980	−	0.811980i	0.172951	−	0.984930i	$-0.444670\pi$
$929$	−24.7487	−	24.7487i	−0.811980	−	0.811980i	−0.984930	+	0.172951i	$0.944670\pi$
$930$		0			0
$931$			5.19615i			0.170297i
$932$		0			0
$933$		−	36.0000i		−	1.17859i
$934$		0			0
$935$	−11.0227	−	18.3712i	−0.360481	−	0.600802i
$936$		0			0
$937$			54.0000i			1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$937$			54.0000i			1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$938$		0			0
$939$	−39.1918			−1.27898
$940$		0			0
$941$	31.1127	+	31.1127i	1.01424	+	1.01424i	0.999897	+	0.0143479i	$0.00456724\pi$
$941$	31.1127	+	31.1127i	1.01424	+	1.01424i	0.0143479	+	0.999897i	$0.495433\pi$
$942$		0			0
$943$	−8.66025			−0.282017
$944$		0			0
$945$	9.00000	−	9.00000i	0.292770	−	0.292770i
$946$		0			0
$947$	19.5959	+	19.5959i	0.636782	+	0.636782i	0.949760	−	0.312978i	$-0.101327\pi$
$947$	19.5959	+	19.5959i	0.636782	+	0.636782i	−0.312978	+	0.949760i	$0.601327\pi$
$948$		0			0
$949$	24.0000	+	24.0000i	0.779073	+	0.779073i
$950$		0			0
$951$			34.6410i			1.12331i
$952$		0			0
$953$		−	22.6274i		−	0.732974i	−0.930423	−	0.366487i	$-0.880560\pi$
$953$		−	22.6274i		−	0.732974i	0.930423	−	0.366487i	$-0.119440\pi$
$954$		0			0
$955$	−8.66025	+	8.66025i	−0.280239	+	0.280239i
$956$		0			0
$957$	25.4558	−	25.4558i	0.822871	−	0.822871i
$958$		0			0
$959$	12.2474	−	12.2474i	0.395491	−	0.395491i
$960$		0			0
$961$		−	65.0000i		−	2.09677i
$962$		0			0
$963$	3.67423	−	3.67423i	0.118401	−	0.118401i
$964$		0			0
$965$			11.3137i			0.364201i
$966$		0			0
$967$	1.73205			0.0556990			0.0278495	−	0.999612i	$-0.491134\pi$
$967$	1.73205			0.0556990			0.0278495	+	0.999612i	$0.491134\pi$
$968$		0			0
$969$	−36.0000	−	9.00000i	−1.15649	−	0.289122i
$970$		0			0
$971$			29.3939i			0.943294i	0.881787	+	0.471647i	$0.156340\pi$
$971$			29.3939i			0.943294i	−0.881787	+	0.471647i	$0.843660\pi$
$972$		0			0
$973$	54.0000			1.73116
$974$		0			0
$975$	14.6969	−	14.6969i	0.470679	−	0.470679i
$976$		0			0
$977$	43.8406			1.40259			0.701293	−	0.712873i	$-0.252607\pi$
$977$	43.8406			1.40259			0.701293	+	0.712873i	$0.252607\pi$
$978$		0			0
$979$	−36.3731	−	36.3731i	−1.16249	−	1.16249i
$980$		0			0
$981$	12.0000	+	12.0000i	0.383131	+	0.383131i
$982$		0			0
$983$	−23.2702	−	23.2702i	−0.742203	−	0.742203i	0.230799	−	0.973001i	$-0.425866\pi$
$983$	−23.2702	−	23.2702i	−0.742203	−	0.742203i	−0.973001	+	0.230799i	$0.925866\pi$
$984$		0			0
$985$	13.0000			0.414214
$986$		0			0
$987$		−	41.5692i		−	1.32316i
$988$		0			0
$989$	−6.36396	+	6.36396i	−0.202362	+	0.202362i
$990$		0			0
$991$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$991$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$992$		0			0
$993$	14.8492	+	14.8492i	0.471226	+	0.471226i
$994$		0			0
$995$		−	9.79796i		−	0.310616i
$996$		0			0
$997$	−17.0000	+	17.0000i	−0.538395	+	0.538395i	−0.923057	−	0.384662i	$-0.874318\pi$
$997$	−17.0000	+	17.0000i	−0.538395	+	0.538395i	0.384662	+	0.923057i	$0.374318\pi$
$998$		0			0
$999$	36.7423			1.16248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.bf.b.47.2	yes	8
3.2	odd	2	inner	816.2.bf.b.47.3	yes	8
4.3	odd	2	inner	816.2.bf.b.47.4	yes	8
12.11	even	2	inner	816.2.bf.b.47.1	✓	8
17.4	even	4	inner	816.2.bf.b.191.1	yes	8
51.38	odd	4	inner	816.2.bf.b.191.4	yes	8
68.55	odd	4	inner	816.2.bf.b.191.3	yes	8
204.191	even	4	inner	816.2.bf.b.191.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
816.2.bf.b.47.1	✓	8	12.11	even	2	inner
816.2.bf.b.47.2	yes	8	1.1	even	1	trivial
816.2.bf.b.47.3	yes	8	3.2	odd	2	inner
816.2.bf.b.47.4	yes	8	4.3	odd	2	inner
816.2.bf.b.191.1	yes	8	17.4	even	4	inner
816.2.bf.b.191.2	yes	8	204.191	even	4	inner
816.2.bf.b.191.3	yes	8	68.55	odd	4	inner
816.2.bf.b.191.4	yes	8	51.38	odd	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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.bf (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(0.707107 - 0.707107i) q^{5} +(1.73205 + 1.73205i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(0.707107 - 0.707107i) q^{5} +(1.73205 + 1.73205i) q^{7} +3.00000i q^{9} +(-3.67423 + 3.67423i) q^{11} +3.00000 q^{13} -1.73205 q^{15} +(-3.53553 + 2.12132i) q^{17} -5.19615 q^{19} -4.24264i q^{21} +(-1.22474 + 1.22474i) q^{23} +4.00000i q^{25} +(3.67423 - 3.67423i) q^{27} +(2.82843 - 2.82843i) q^{29} +(-6.92820 + 6.92820i) q^{31} +9.00000 q^{33} +2.44949 q^{35} +(5.00000 + 5.00000i) q^{37} +(-3.67423 - 3.67423i) q^{39} +(3.53553 + 3.53553i) q^{41} +5.19615 q^{43} +(2.12132 + 2.12132i) q^{45} +9.79796 q^{47} -1.00000i q^{49} +(6.92820 + 1.73205i) q^{51} -11.3137 q^{53} +5.19615i q^{55} +(6.36396 + 6.36396i) q^{57} +2.44949i q^{59} +(-1.00000 + 1.00000i) q^{61} +(-5.19615 + 5.19615i) q^{63} +(2.12132 - 2.12132i) q^{65} -13.8564i q^{67} +3.00000 q^{69} +(4.89898 + 4.89898i) q^{71} +(8.00000 + 8.00000i) q^{73} +(4.89898 - 4.89898i) q^{75} -12.7279 q^{77} +(-1.73205 - 1.73205i) q^{79} -9.00000 q^{81} +9.79796i q^{83} +(-1.00000 + 4.00000i) q^{85} -6.92820 q^{87} +9.89949i q^{89} +(5.19615 + 5.19615i) q^{91} +16.9706 q^{93} +(-3.67423 + 3.67423i) q^{95} +(-8.00000 - 8.00000i) q^{97} +(-11.0227 - 11.0227i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{13} + 72 q^{33} + 40 q^{37} - 8 q^{61} + 24 q^{69} + 64 q^{73} - 72 q^{81} - 8 q^{85} - 64 q^{97}+O(q^{100}) \) 8 * q + 24 * q^13 + 72 * q^33 + 40 * q^37 - 8 * q^61 + 24 * q^69 + 64 * q^73 - 72 * q^81 - 8 * q^85 - 64 * q^97

Introduction

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