LMFDB - Embedded newform 768.2.j.b.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(193,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("768.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.j (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:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	768.577
Dual form			768.2.j.b.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{3} -1.41421i q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{3} -1.41421i q^{7} +1.00000i q^{9} +(-1.00000 - 1.00000i) q^{13} +6.00000 q^{17} +(-4.24264 - 4.24264i) q^{19} +(-1.00000 + 1.00000i) q^{21} -8.48528i q^{23} +5.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(-6.00000 - 6.00000i) q^{29} -1.41421 q^{31} +(-5.00000 + 5.00000i) q^{37} +1.41421i q^{39} -6.00000i q^{41} +(4.24264 - 4.24264i) q^{43} -8.48528 q^{47} +5.00000 q^{49} +(-4.24264 - 4.24264i) q^{51} +(6.00000 - 6.00000i) q^{53} +6.00000i q^{57} +(-5.00000 - 5.00000i) q^{61} +1.41421 q^{63} +(-2.82843 - 2.82843i) q^{67} +(-6.00000 + 6.00000i) q^{69} -8.48528i q^{71} +(3.53553 - 3.53553i) q^{75} +1.41421 q^{79} -1.00000 q^{81} +(8.48528 + 8.48528i) q^{83} +8.48528i q^{87} -6.00000i q^{89} +(-1.41421 + 1.41421i) q^{91} +(1.00000 + 1.00000i) q^{93} -12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{13} + 24 q^{17} - 4 q^{21} - 24 q^{29} - 20 q^{37} + 20 q^{49} + 24 q^{53} - 20 q^{61} - 24 q^{69} - 4 q^{81} + 4 q^{93} - 48 q^{97}+O(q^{100}) $ 4 * q - 4 * q^13 + 24 * q^17 - 4 * q^21 - 24 * q^29 - 20 * q^37 + 20 * q^49 + 24 * q^53 - 20 * q^61 - 24 * q^69 - 4 * q^81 + 4 * q^93 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$		−	1.41421i		−	0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$7$		−	1.41421i		−	0.534522i	0.963624	−	0.267261i	$-0.0861187\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		0			0
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	0.554700	−	0.832050i	$-0.312833\pi$
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	−4.24264	−	4.24264i	−0.973329	−	0.973329i	0.0263249	−	0.999653i	$-0.491620\pi$
$19$	−4.24264	−	4.24264i	−0.973329	−	0.973329i	−0.999653	+	0.0263249i	$0.991620\pi$
$20$		0			0
$21$	−1.00000	+	1.00000i	−0.218218	+	0.218218i
$22$		0			0
$23$		−	8.48528i		−	1.76930i	−0.466252	−	0.884652i	$-0.654396\pi$
$23$		−	8.48528i		−	1.76930i	0.466252	−	0.884652i	$-0.345604\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$	0.707107	−	0.707107i	0.136083	−	0.136083i
$28$		0			0
$29$	−6.00000	−	6.00000i	−1.11417	−	1.11417i	−0.992580	−	0.121592i	$-0.961200\pi$
$29$	−6.00000	−	6.00000i	−1.11417	−	1.11417i	−0.121592	−	0.992580i	$-0.538800\pi$
$30$		0			0
$31$	−1.41421			−0.254000			−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421			−0.254000			−0.127000	+	0.991903i	$0.540535\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$			1.41421i			0.226455i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$	4.24264	−	4.24264i	0.646997	−	0.646997i	−0.305269	−	0.952266i	$-0.598747\pi$
$43$	4.24264	−	4.24264i	0.646997	−	0.646997i	0.952266	+	0.305269i	$0.0987465\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.48528			−1.23771			−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528			−1.23771			−0.618853	+	0.785507i	$0.712402\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$	−4.24264	−	4.24264i	−0.594089	−	0.594089i
$52$		0			0
$53$	6.00000	−	6.00000i	0.824163	−	0.824163i	−0.162539	−	0.986702i	$-0.551968\pi$
$53$	6.00000	−	6.00000i	0.824163	−	0.824163i	0.986702	+	0.162539i	$0.0519683\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			6.00000i			0.794719i
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	−5.00000	−	5.00000i	−0.640184	−	0.640184i	0.310416	−	0.950601i	$-0.399532\pi$
$61$	−5.00000	−	5.00000i	−0.640184	−	0.640184i	−0.950601	+	0.310416i	$0.899532\pi$
$62$		0			0
$63$	1.41421			0.178174
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.82843	−	2.82843i	−0.345547	−	0.345547i	0.512901	−	0.858448i	$-0.328571\pi$
$67$	−2.82843	−	2.82843i	−0.345547	−	0.345547i	−0.858448	+	0.512901i	$0.828571\pi$
$68$		0			0
$69$	−6.00000	+	6.00000i	−0.722315	+	0.722315i
$70$		0			0
$71$		−	8.48528i		−	1.00702i	−0.863990	−	0.503509i	$-0.832042\pi$
$71$		−	8.48528i		−	1.00702i	0.863990	−	0.503509i	$-0.167958\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	3.53553	−	3.53553i	0.408248	−	0.408248i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.41421			0.159111			0.0795557	−	0.996830i	$-0.474650\pi$
$79$	1.41421			0.159111			0.0795557	+	0.996830i	$0.474650\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	0.997792	−	0.0664117i	$-0.0211551\pi$
$83$	8.48528	+	8.48528i	0.931381	+	0.931381i	−0.0664117	+	0.997792i	$0.521155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			8.48528i			0.909718i
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$	−1.41421	+	1.41421i	−0.148250	+	0.148250i
$92$		0			0
$93$	1.00000	+	1.00000i	0.103695	+	0.103695i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.0000			−1.21842			−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000			−1.21842			−0.609208	+	0.793011i	$0.708512\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000	−	6.00000i	0.597022	−	0.597022i	−0.342497	−	0.939519i	$-0.611273\pi$
$101$	6.00000	−	6.00000i	0.597022	−	0.597022i	0.939519	+	0.342497i	$0.111273\pi$
$102$		0			0
$103$			9.89949i			0.975426i	0.873004	+	0.487713i	$0.162169\pi$
$103$			9.89949i			0.975426i	−0.873004	+	0.487713i	$0.837831\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	−0.986151	−	0.165848i	$-0.946964\pi$
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	0.165848	+	0.986151i	$0.446964\pi$
$108$		0			0
$109$	13.0000	+	13.0000i	1.24517	+	1.24517i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	13.0000	+	13.0000i	1.24517	+	1.24517i	0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$	7.07107			0.671156
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.00000	−	1.00000i	0.0924500	−	0.0924500i
$118$		0			0
$119$		−	8.48528i		−	0.777844i
$120$		0			0
$121$			11.0000i			1.00000i
$122$		0			0
$123$	−4.24264	+	4.24264i	−0.382546	+	0.382546i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	9.89949			0.878438			0.439219	−	0.898380i	$-0.355255\pi$
$127$	9.89949			0.878438			0.439219	+	0.898380i	$0.355255\pi$
$128$		0			0
$129$	−6.00000			−0.528271
$130$		0			0
$131$	8.48528	+	8.48528i	0.741362	+	0.741362i	0.972840	−	0.231478i	$-0.0743560\pi$
$131$	8.48528	+	8.48528i	0.741362	+	0.741362i	−0.231478	+	0.972840i	$0.574356\pi$
$132$		0			0
$133$	−6.00000	+	6.00000i	−0.520266	+	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	−2.82843	+	2.82843i	−0.239904	+	0.239904i	−0.816810	−	0.576906i	$-0.804260\pi$
$139$	−2.82843	+	2.82843i	−0.239904	+	0.239904i	0.576906	+	0.816810i	$0.304260\pi$
$140$		0			0
$141$	6.00000	+	6.00000i	0.505291	+	0.505291i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−3.53553	−	3.53553i	−0.291606	−	0.291606i
$148$		0			0
$149$	12.0000	−	12.0000i	0.983078	−	0.983078i	−0.0167809	−	0.999859i	$-0.505342\pi$
$149$	12.0000	−	12.0000i	0.983078	−	0.983078i	0.999859	+	0.0167809i	$0.00534177\pi$
$150$		0			0
$151$			18.3848i			1.49613i	0.663624	+	0.748066i	$0.269017\pi$
$151$			18.3848i			1.49613i	−0.663624	+	0.748066i	$0.730983\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	0.877896	−	0.478852i	$-0.158947\pi$
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$	−8.48528			−0.672927
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	4.24264	+	4.24264i	0.332309	+	0.332309i	0.853463	−	0.521154i	$-0.174498\pi$
$163$	4.24264	+	4.24264i	0.332309	+	0.332309i	−0.521154	+	0.853463i	$0.674498\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			16.9706i			1.31322i	0.754230	+	0.656611i	$0.228011\pi$
$167$			16.9706i			1.31322i	−0.754230	+	0.656611i	$0.771989\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$	4.24264	−	4.24264i	0.324443	−	0.324443i
$172$		0			0
$173$	12.0000	+	12.0000i	0.912343	+	0.912343i	0.996456	−	0.0841131i	$-0.0268057\pi$
$173$	12.0000	+	12.0000i	0.912343	+	0.912343i	−0.0841131	+	0.996456i	$0.526806\pi$
$174$		0			0
$175$	7.07107			0.534522
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.9706	+	16.9706i	1.26844	+	1.26844i	0.946894	+	0.321545i	$0.104202\pi$
$179$	16.9706	+	16.9706i	1.26844	+	1.26844i	0.321545	+	0.946894i	$0.395798\pi$
$180$		0			0
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	−0.168140	−	0.985763i	$-0.553776\pi$
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	0.985763	+	0.168140i	$0.0537759\pi$
$182$		0			0
$183$			7.07107i			0.522708i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.00000	−	1.00000i	−0.0727393	−	0.0727393i
$190$		0			0
$191$	16.9706			1.22795			0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706			1.22795			0.613973	+	0.789327i	$0.289570\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000	+	6.00000i	−0.427482	+	0.427482i	−0.887770	−	0.460288i	$-0.847746\pi$
$197$	−6.00000	+	6.00000i	−0.427482	+	0.427482i	0.460288	+	0.887770i	$0.347746\pi$
$198$		0			0
$199$		−	9.89949i		−	0.701757i	−0.936421	−	0.350878i	$-0.885883\pi$
$199$		−	9.89949i		−	0.701757i	0.936421	−	0.350878i	$-0.114117\pi$
$200$		0			0
$201$			4.00000i			0.282138i
$202$		0			0
$203$	−8.48528	+	8.48528i	−0.595550	+	0.595550i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	8.48528			0.589768
$208$		0			0
$209$		0			0
$210$		0			0
$211$	2.82843	+	2.82843i	0.194717	+	0.194717i	0.797731	−	0.603014i	$-0.206034\pi$
$211$	2.82843	+	2.82843i	0.194717	+	0.194717i	−0.603014	+	0.797731i	$0.706034\pi$
$212$		0			0
$213$	−6.00000	+	6.00000i	−0.411113	+	0.411113i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	−	6.00000i	−0.403604	−	0.403604i
$222$		0			0
$223$	−26.8701			−1.79935			−0.899676	−	0.436558i	$-0.856197\pi$
$223$	−26.8701			−1.79935			−0.899676	+	0.436558i	$0.856197\pi$
$224$		0			0
$225$	−5.00000			−0.333333
$226$		0			0
$227$	−8.48528	−	8.48528i	−0.563188	−	0.563188i	0.367024	−	0.930212i	$-0.380377\pi$
$227$	−8.48528	−	8.48528i	−0.563188	−	0.563188i	−0.930212	+	0.367024i	$0.880377\pi$
$228$		0			0
$229$	−1.00000	+	1.00000i	−0.0660819	+	0.0660819i	−0.739375	−	0.673293i	$-0.764879\pi$
$229$	−1.00000	+	1.00000i	−0.0660819	+	0.0660819i	0.673293	+	0.739375i	$0.264879\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.00000	−	1.00000i	−0.0649570	−	0.0649570i
$238$		0			0
$239$	−16.9706			−1.09773			−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706			−1.09773			−0.548867	+	0.835910i	$0.684941\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$	0.707107	+	0.707107i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			8.48528i			0.539906i
$248$		0			0
$249$		−	12.0000i		−	0.760469i
$250$		0			0
$251$	16.9706	−	16.9706i	1.07117	−	1.07117i	0.0739073	−	0.997265i	$-0.476453\pi$
$251$	16.9706	−	16.9706i	1.07117	−	1.07117i	0.997265	−	0.0739073i	$-0.0235469\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$	7.07107	+	7.07107i	0.439375	+	0.439375i
$260$		0			0
$261$	6.00000	−	6.00000i	0.371391	−	0.371391i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.24264	+	4.24264i	−0.259645	+	0.259645i
$268$		0			0
$269$	−6.00000	−	6.00000i	−0.365826	−	0.365826i	0.500126	−	0.865953i	$-0.333287\pi$
$269$	−6.00000	−	6.00000i	−0.365826	−	0.365826i	−0.865953	+	0.500126i	$0.833287\pi$
$270$		0			0
$271$	26.8701			1.63224			0.816120	−	0.577883i	$-0.196121\pi$
$271$	26.8701			1.63224			0.816120	+	0.577883i	$0.196121\pi$
$272$		0			0
$273$	2.00000			0.121046
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	−0.955598	−	0.294672i	$-0.904789\pi$
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	0.294672	+	0.955598i	$0.404789\pi$
$278$		0			0
$279$		−	1.41421i		−	0.0846668i
$280$		0			0
$281$			18.0000i			1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		0			0
$283$	−2.82843	+	2.82843i	−0.168133	+	0.168133i	−0.786158	−	0.618026i	$-0.787933\pi$
$283$	−2.82843	+	2.82843i	−0.168133	+	0.168133i	0.618026	+	0.786158i	$0.287933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.48528			−0.500870
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$	8.48528	+	8.48528i	0.497416	+	0.497416i
$292$		0			0
$293$	−6.00000	+	6.00000i	−0.350524	+	0.350524i	−0.860304	−	0.509781i	$-0.829727\pi$
$293$	−6.00000	+	6.00000i	−0.350524	+	0.350524i	0.509781	+	0.860304i	$0.329727\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.48528	+	8.48528i	−0.490716	+	0.490716i
$300$		0			0
$301$	−6.00000	−	6.00000i	−0.345834	−	0.345834i
$302$		0			0
$303$	−8.48528			−0.487467
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−14.1421	−	14.1421i	−0.807134	−	0.807134i	0.177065	−	0.984199i	$-0.443340\pi$
$307$	−14.1421	−	14.1421i	−0.807134	−	0.807134i	−0.984199	+	0.177065i	$0.943340\pi$
$308$		0			0
$309$	7.00000	−	7.00000i	0.398216	−	0.398216i
$310$		0			0
$311$		−	16.9706i		−	0.962312i	−0.876635	−	0.481156i	$-0.840217\pi$
$311$		−	16.9706i		−	0.962312i	0.876635	−	0.481156i	$-0.159783\pi$
$312$		0			0
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	+	6.00000i	0.336994	+	0.336994i	0.855235	−	0.518241i	$-0.173413\pi$
$317$	6.00000	+	6.00000i	0.336994	+	0.336994i	−0.518241	+	0.855235i	$0.673413\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$	−25.4558	−	25.4558i	−1.41640	−	1.41640i
$324$		0			0
$325$	5.00000	−	5.00000i	0.277350	−	0.277350i
$326$		0			0
$327$		−	18.3848i		−	1.01668i
$328$		0			0
$329$			12.0000i			0.661581i
$330$		0			0
$331$	19.7990	−	19.7990i	1.08825	−	1.08825i	0.0925421	−	0.995709i	$-0.470501\pi$
$331$	19.7990	−	19.7990i	1.08825	−	1.08825i	0.995709	−	0.0925421i	$-0.0294993\pi$
$332$		0			0
$333$	−5.00000	−	5.00000i	−0.273998	−	0.273998i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.0000			1.30736			0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000			1.30736			0.653682	+	0.756770i	$0.273224\pi$
$338$		0			0
$339$	−4.24264	−	4.24264i	−0.230429	−	0.230429i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.48528	−	8.48528i	0.455514	−	0.455514i	−0.441666	−	0.897180i	$-0.645612\pi$
$347$	8.48528	−	8.48528i	0.455514	−	0.455514i	0.897180	+	0.441666i	$0.145612\pi$
$348$		0			0
$349$	−19.0000	−	19.0000i	−1.01705	−	1.01705i	−0.999852	−	0.0171945i	$-0.994527\pi$
$349$	−19.0000	−	19.0000i	−1.01705	−	1.01705i	−0.0171945	−	0.999852i	$-0.505473\pi$
$350$		0			0
$351$	−1.41421			−0.0754851
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−6.00000	+	6.00000i	−0.317554	+	0.317554i
$358$		0			0
$359$			25.4558i			1.34351i	0.740774	+	0.671754i	$0.234459\pi$
$359$			25.4558i			1.34351i	−0.740774	+	0.671754i	$0.765541\pi$
$360$		0			0
$361$			17.0000i			0.894737i
$362$		0			0
$363$	7.77817	−	7.77817i	0.408248	−	0.408248i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−32.5269			−1.69789			−0.848945	−	0.528480i	$-0.822762\pi$
$367$	−32.5269			−1.69789			−0.848945	+	0.528480i	$0.822762\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	−8.48528	−	8.48528i	−0.440534	−	0.440534i
$372$		0			0
$373$	5.00000	−	5.00000i	0.258890	−	0.258890i	−0.565712	−	0.824603i	$-0.691399\pi$
$373$	5.00000	−	5.00000i	0.258890	−	0.258890i	0.824603	+	0.565712i	$0.191399\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			12.0000i			0.618031i
$378$		0			0
$379$	−21.2132	+	21.2132i	−1.08965	+	1.08965i	−0.0940849	+	0.995564i	$0.529993\pi$
$379$	−21.2132	+	21.2132i	−1.08965	+	1.08965i	−0.995564	+	0.0940849i	$0.970007\pi$
$380$		0			0
$381$	−7.00000	−	7.00000i	−0.358621	−	0.358621i
$382$		0			0
$383$	33.9411			1.73431			0.867155	−	0.498038i	$-0.165946\pi$
$383$	33.9411			1.73431			0.867155	+	0.498038i	$0.165946\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.24264	+	4.24264i	0.215666	+	0.215666i
$388$		0			0
$389$	−12.0000	+	12.0000i	−0.608424	+	0.608424i	−0.942534	−	0.334110i	$-0.891564\pi$
$389$	−12.0000	+	12.0000i	−0.608424	+	0.608424i	0.334110	+	0.942534i	$0.391564\pi$
$390$		0			0
$391$		−	50.9117i		−	2.57471i
$392$		0			0
$393$		−	12.0000i		−	0.605320i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.00000	+	7.00000i	0.351320	+	0.351320i	0.860601	−	0.509281i	$-0.170088\pi$
$397$	7.00000	+	7.00000i	0.351320	+	0.351320i	−0.509281	+	0.860601i	$0.670088\pi$
$398$		0			0
$399$	8.48528			0.424795
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$	1.41421	+	1.41421i	0.0704470	+	0.0704470i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$	−12.7279	+	12.7279i	−0.627822	+	0.627822i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.00000			0.195881
$418$		0			0
$419$	−25.4558	−	25.4558i	−1.24360	−	1.24360i	−0.958497	−	0.285102i	$-0.907972\pi$
$419$	−25.4558	−	25.4558i	−1.24360	−	1.24360i	−0.285102	−	0.958497i	$-0.592028\pi$
$420$		0			0
$421$	11.0000	−	11.0000i	0.536107	−	0.536107i	−0.386276	−	0.922383i	$-0.626239\pi$
$421$	11.0000	−	11.0000i	0.536107	−	0.536107i	0.922383	+	0.386276i	$0.126239\pi$
$422$		0			0
$423$		−	8.48528i		−	0.412568i
$424$		0			0
$425$			30.0000i			1.45521i
$426$		0			0
$427$	−7.07107	+	7.07107i	−0.342193	+	0.342193i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.48528			−0.408722			−0.204361	−	0.978896i	$-0.565512\pi$
$431$	−8.48528			−0.408722			−0.204361	+	0.978896i	$0.565512\pi$
$432$		0			0
$433$	24.0000			1.15337			0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000			1.15337			0.576683	+	0.816968i	$0.304347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−36.0000	+	36.0000i	−1.72211	+	1.72211i
$438$		0			0
$439$		−	7.07107i		−	0.337484i	−0.985660	−	0.168742i	$-0.946030\pi$
$439$		−	7.07107i		−	0.337484i	0.985660	−	0.168742i	$-0.0539704\pi$
$440$		0			0
$441$			5.00000i			0.238095i
$442$		0			0
$443$	8.48528	−	8.48528i	0.403148	−	0.403148i	−0.476193	−	0.879341i	$-0.657984\pi$
$443$	8.48528	−	8.48528i	0.403148	−	0.403148i	0.879341	+	0.476193i	$0.157984\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−16.9706			−0.802680
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	13.0000	−	13.0000i	0.610793	−	0.610793i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		0			0
$459$	4.24264	−	4.24264i	0.198030	−	0.198030i
$460$		0			0
$461$	−24.0000	−	24.0000i	−1.11779	−	1.11779i	−0.992065	−	0.125726i	$-0.959874\pi$
$461$	−24.0000	−	24.0000i	−1.11779	−	1.11779i	−0.125726	−	0.992065i	$-0.540126\pi$
$462$		0			0
$463$	7.07107			0.328620			0.164310	−	0.986409i	$-0.447460\pi$
$463$	7.07107			0.328620			0.164310	+	0.986409i	$0.447460\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.48528	−	8.48528i	−0.392652	−	0.392652i	0.482980	−	0.875632i	$-0.339555\pi$
$467$	−8.48528	−	8.48528i	−0.392652	−	0.392652i	−0.875632	+	0.482980i	$0.839555\pi$
$468$		0			0
$469$	−4.00000	+	4.00000i	−0.184703	+	0.184703i
$470$		0			0
$471$		−	7.07107i		−	0.325818i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	21.2132	−	21.2132i	0.973329	−	0.973329i
$476$		0			0
$477$	6.00000	+	6.00000i	0.274721	+	0.274721i
$478$		0			0
$479$	−8.48528			−0.387702			−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528			−0.387702			−0.193851	+	0.981031i	$0.562098\pi$
$480$		0			0
$481$	10.0000			0.455961
$482$		0			0
$483$	8.48528	+	8.48528i	0.386094	+	0.386094i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			18.3848i			0.833094i	0.909114	+	0.416547i	$0.136760\pi$
$487$			18.3848i			0.833094i	−0.909114	+	0.416547i	$0.863240\pi$
$488$		0			0
$489$		−	6.00000i		−	0.271329i
$490$		0			0
$491$	16.9706	−	16.9706i	0.765871	−	0.765871i	−0.211506	−	0.977377i	$-0.567837\pi$
$491$	16.9706	−	16.9706i	0.765871	−	0.765871i	0.977377	+	0.211506i	$0.0678368\pi$
$492$		0			0
$493$	−36.0000	−	36.0000i	−1.62136	−	1.62136i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000			−0.538274
$498$		0			0
$499$	−2.82843	−	2.82843i	−0.126618	−	0.126618i	0.640958	−	0.767576i	$-0.278537\pi$
$499$	−2.82843	−	2.82843i	−0.126618	−	0.126618i	−0.767576	+	0.640958i	$0.778537\pi$
$500$		0			0
$501$	12.0000	−	12.0000i	0.536120	−	0.536120i
$502$		0			0
$503$		−	8.48528i		−	0.378340i	−0.981944	−	0.189170i	$-0.939420\pi$
$503$		−	8.48528i		−	0.378340i	0.981944	−	0.189170i	$-0.0605797\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.77817	+	7.77817i	−0.345441	+	0.345441i
$508$		0			0
$509$	18.0000	+	18.0000i	0.797836	+	0.797836i	0.982754	−	0.184918i	$-0.0592019\pi$
$509$	18.0000	+	18.0000i	0.797836	+	0.797836i	−0.184918	+	0.982754i	$0.559202\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−6.00000			−0.264906
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	16.9706i		−	0.744925i
$520$		0			0
$521$			30.0000i			1.31432i	0.753749	+	0.657162i	$0.228243\pi$
$521$			30.0000i			1.31432i	−0.753749	+	0.657162i	$0.771757\pi$
$522$		0			0
$523$	−12.7279	+	12.7279i	−0.556553	+	0.556553i	−0.928324	−	0.371771i	$-0.878751\pi$
$523$	−12.7279	+	12.7279i	−0.556553	+	0.556553i	0.371771	+	0.928324i	$0.378751\pi$
$524$		0			0
$525$	−5.00000	−	5.00000i	−0.218218	−	0.218218i
$526$		0			0
$527$	−8.48528			−0.369625
$528$		0			0
$529$	−49.0000			−2.13043
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.00000	+	6.00000i	−0.259889	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	24.0000i		−	1.03568i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	25.0000	+	25.0000i	1.07483	+	1.07483i	0.996963	+	0.0778705i	$0.0248121\pi$
$541$	25.0000	+	25.0000i	1.07483	+	1.07483i	0.0778705	+	0.996963i	$0.475188\pi$
$542$		0			0
$543$	−15.5563			−0.667587
$544$		0			0
$545$		0			0
$546$		0			0
$547$	12.7279	+	12.7279i	0.544207	+	0.544207i	0.924759	−	0.380553i	$-0.124266\pi$
$547$	12.7279	+	12.7279i	0.544207	+	0.544207i	−0.380553	+	0.924759i	$0.624266\pi$
$548$		0			0
$549$	5.00000	−	5.00000i	0.213395	−	0.213395i
$550$		0			0
$551$			50.9117i			2.16891i
$552$		0			0
$553$		−	2.00000i		−	0.0850487i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.0000	+	12.0000i	0.508456	+	0.508456i	0.914052	−	0.405596i	$-0.132936\pi$
$557$	12.0000	+	12.0000i	0.508456	+	0.508456i	−0.405596	+	0.914052i	$0.632936\pi$
$558$		0			0
$559$	−8.48528			−0.358889
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			1.41421i			0.0593914i
$568$		0			0
$569$		−	18.0000i		−	0.754599i	−0.926091	−	0.377300i	$-0.876853\pi$
$569$		−	18.0000i		−	0.754599i	0.926091	−	0.377300i	$-0.123147\pi$
$570$		0			0
$571$	2.82843	−	2.82843i	0.118366	−	0.118366i	−0.645443	−	0.763809i	$-0.723327\pi$
$571$	2.82843	−	2.82843i	0.118366	−	0.118366i	0.763809	+	0.645443i	$0.223327\pi$
$572$		0			0
$573$	−12.0000	−	12.0000i	−0.501307	−	0.501307i
$574$		0			0
$575$	42.4264			1.76930
$576$		0			0
$577$	36.0000			1.49870			0.749350	−	0.662174i	$-0.230366\pi$
$577$	36.0000			1.49870			0.749350	+	0.662174i	$0.230366\pi$
$578$		0			0
$579$	9.89949	+	9.89949i	0.411409	+	0.411409i
$580$		0			0
$581$	12.0000	−	12.0000i	0.497844	−	0.497844i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$	6.00000	+	6.00000i	0.247226	+	0.247226i
$590$		0			0
$591$	8.48528			0.349038
$592$		0			0
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.00000	+	7.00000i	−0.286491	+	0.286491i
$598$		0			0
$599$			25.4558i			1.04010i	0.854137	+	0.520049i	$0.174086\pi$
$599$			25.4558i			1.04010i	−0.854137	+	0.520049i	$0.825914\pi$
$600$		0			0
$601$		−	12.0000i		−	0.489490i	−0.969587	−	0.244745i	$-0.921296\pi$
$601$		−	12.0000i		−	0.489490i	0.969587	−	0.244745i	$-0.0787043\pi$
$602$		0			0
$603$	2.82843	−	2.82843i	0.115182	−	0.115182i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−24.0416			−0.975820			−0.487910	−	0.872894i	$-0.662241\pi$
$607$	−24.0416			−0.975820			−0.487910	+	0.872894i	$0.662241\pi$
$608$		0			0
$609$	12.0000			0.486265
$610$		0			0
$611$	8.48528	+	8.48528i	0.343278	+	0.343278i
$612$		0			0
$613$	19.0000	−	19.0000i	0.767403	−	0.767403i	−0.210246	−	0.977649i	$-0.567426\pi$
$613$	19.0000	−	19.0000i	0.767403	−	0.767403i	0.977649	+	0.210246i	$0.0674264\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6.00000i			0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$			6.00000i			0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$		0			0
$619$	19.7990	−	19.7990i	0.795789	−	0.795789i	−0.186640	−	0.982428i	$-0.559760\pi$
$619$	19.7990	−	19.7990i	0.795789	−	0.795789i	0.982428	+	0.186640i	$0.0597597\pi$
$620$		0			0
$621$	−6.00000	−	6.00000i	−0.240772	−	0.240772i
$622$		0			0
$623$	−8.48528			−0.339956
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−30.0000	+	30.0000i	−1.19618	+	1.19618i
$630$		0			0
$631$		−	18.3848i		−	0.731886i	−0.930637	−	0.365943i	$-0.880746\pi$
$631$		−	18.3848i		−	0.731886i	0.930637	−	0.365943i	$-0.119254\pi$
$632$		0			0
$633$		−	4.00000i		−	0.158986i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.00000	−	5.00000i	−0.198107	−	0.198107i
$638$		0			0
$639$	8.48528			0.335673
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	12.7279	+	12.7279i	0.501940	+	0.501940i	0.912040	−	0.410100i	$-0.134506\pi$
$643$	12.7279	+	12.7279i	0.501940	+	0.501940i	−0.410100	+	0.912040i	$0.634506\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			8.48528i			0.333591i	0.985992	+	0.166795i	$0.0533419\pi$
$647$			8.48528i			0.333591i	−0.985992	+	0.166795i	$0.946658\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.41421	−	1.41421i	0.0554274	−	0.0554274i
$652$		0			0
$653$	12.0000	+	12.0000i	0.469596	+	0.469596i	0.901784	−	0.432187i	$-0.142258\pi$
$653$	12.0000	+	12.0000i	0.469596	+	0.469596i	−0.432187	+	0.901784i	$0.642258\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	7.00000	−	7.00000i	0.272268	−	0.272268i	−0.557744	−	0.830013i	$-0.688333\pi$
$661$	7.00000	−	7.00000i	0.272268	−	0.272268i	0.830013	+	0.557744i	$0.188333\pi$
$662$		0			0
$663$			8.48528i			0.329541i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−50.9117	+	50.9117i	−1.97131	+	1.97131i
$668$		0			0
$669$	19.0000	+	19.0000i	0.734582	+	0.734582i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$	3.53553	+	3.53553i	0.136083	+	0.136083i
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$			16.9706i			0.651270i
$680$		0			0
$681$			12.0000i			0.459841i
$682$		0			0
$683$	−25.4558	+	25.4558i	−0.974041	+	0.974041i	−0.999671	−	0.0256307i	$-0.991841\pi$
$683$	−25.4558	+	25.4558i	−0.974041	+	0.974041i	0.0256307	+	0.999671i	$0.491841\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.41421			0.0539556
$688$		0			0
$689$	−12.0000			−0.457164
$690$		0			0
$691$	29.6985	+	29.6985i	1.12978	+	1.12978i	0.990212	+	0.139572i	$0.0445726\pi$
$691$	29.6985	+	29.6985i	1.12978	+	1.12978i	0.139572	+	0.990212i	$0.455427\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	36.0000i		−	1.36360i
$698$		0			0
$699$	−4.24264	+	4.24264i	−0.160471	+	0.160471i
$700$		0			0
$701$	18.0000	+	18.0000i	0.679851	+	0.679851i	0.959966	−	0.280116i	$-0.0903728\pi$
$701$	18.0000	+	18.0000i	0.679851	+	0.679851i	−0.280116	+	0.959966i	$0.590373\pi$
$702$		0			0
$703$	42.4264			1.60014
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.48528	−	8.48528i	−0.319122	−	0.319122i
$708$		0			0
$709$	1.00000	−	1.00000i	0.0375558	−	0.0375558i	−0.688080	−	0.725635i	$-0.741546\pi$
$709$	1.00000	−	1.00000i	0.0375558	−	0.0375558i	0.725635	+	0.688080i	$0.241546\pi$
$710$		0			0
$711$			1.41421i			0.0530372i
$712$		0			0
$713$			12.0000i			0.449404i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	12.0000	+	12.0000i	0.448148	+	0.448148i
$718$		0			0
$719$	8.48528			0.316448			0.158224	−	0.987403i	$-0.449423\pi$
$719$	8.48528			0.316448			0.158224	+	0.987403i	$0.449423\pi$
$720$		0			0
$721$	14.0000			0.521387
$722$		0			0
$723$	7.07107	+	7.07107i	0.262976	+	0.262976i
$724$		0			0
$725$	30.0000	−	30.0000i	1.11417	−	1.11417i
$726$		0			0
$727$			32.5269i			1.20636i	0.797606	+	0.603178i	$0.206099\pi$
$727$			32.5269i			1.20636i	−0.797606	+	0.603178i	$0.793901\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$	25.4558	−	25.4558i	0.941518	−	0.941518i
$732$		0			0
$733$	35.0000	+	35.0000i	1.29275	+	1.29275i	0.933076	+	0.359678i	$0.117113\pi$
$733$	35.0000	+	35.0000i	1.29275	+	1.29275i	0.359678	+	0.933076i	$0.382887\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	14.1421	+	14.1421i	0.520227	+	0.520227i	0.917640	−	0.397413i	$-0.130092\pi$
$739$	14.1421	+	14.1421i	0.520227	+	0.520227i	−0.397413	+	0.917640i	$0.630092\pi$
$740$		0			0
$741$	6.00000	−	6.00000i	0.220416	−	0.220416i
$742$		0			0
$743$			16.9706i			0.622590i	0.950313	+	0.311295i	$0.100763\pi$
$743$			16.9706i			0.622590i	−0.950313	+	0.311295i	$0.899237\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−8.48528	+	8.48528i	−0.310460	+	0.310460i
$748$		0			0
$749$	12.0000	+	12.0000i	0.438470	+	0.438470i
$750$		0			0
$751$	49.4975			1.80619			0.903094	−	0.429442i	$-0.141290\pi$
$751$	49.4975			1.80619			0.903094	+	0.429442i	$0.141290\pi$
$752$		0			0
$753$	−24.0000			−0.874609
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.00000	+	1.00000i	−0.0363456	+	0.0363456i	−0.725046	−	0.688700i	$-0.758182\pi$
$757$	−1.00000	+	1.00000i	−0.0363456	+	0.0363456i	0.688700	+	0.725046i	$0.258182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.00000i			0.217500i	0.994069	+	0.108750i	$0.0346848\pi$
$761$			6.00000i			0.217500i	−0.994069	+	0.108750i	$0.965315\pi$
$762$		0			0
$763$	18.3848	−	18.3848i	0.665574	−	0.665574i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	4.24264	+	4.24264i	0.152795	+	0.152795i
$772$		0			0
$773$	18.0000	−	18.0000i	0.647415	−	0.647415i	−0.304953	−	0.952368i	$-0.598641\pi$
$773$	18.0000	−	18.0000i	0.647415	−	0.647415i	0.952368	+	0.304953i	$0.0986406\pi$
$774$		0			0
$775$		−	7.07107i		−	0.254000i
$776$		0			0
$777$		−	10.0000i		−	0.358748i
$778$		0			0
$779$	−25.4558	+	25.4558i	−0.912050	+	0.912050i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−8.48528			−0.303239
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−12.7279	−	12.7279i	−0.453701	−	0.453701i	0.442880	−	0.896581i	$-0.353957\pi$
$787$	−12.7279	−	12.7279i	−0.453701	−	0.453701i	−0.896581	+	0.442880i	$0.853957\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	8.48528i		−	0.301702i
$792$		0			0
$793$			10.0000i			0.355110i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.0000	−	12.0000i	−0.425062	−	0.425062i	0.461880	−	0.886942i	$-0.347175\pi$
$797$	−12.0000	−	12.0000i	−0.425062	−	0.425062i	−0.886942	+	0.461880i	$0.847175\pi$
$798$		0			0
$799$	−50.9117			−1.80113
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$			8.48528i			0.298696i
$808$		0			0
$809$		−	18.0000i		−	0.632846i	−0.948618	−	0.316423i	$-0.897518\pi$
$809$		−	18.0000i		−	0.632846i	0.948618	−	0.316423i	$-0.102482\pi$
$810$		0			0
$811$	−4.24264	+	4.24264i	−0.148979	+	0.148979i	−0.777662	−	0.628683i	$-0.783594\pi$
$811$	−4.24264	+	4.24264i	−0.148979	+	0.148979i	0.628683	+	0.777662i	$0.283594\pi$
$812$		0			0
$813$	−19.0000	−	19.0000i	−0.666359	−	0.666359i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−36.0000			−1.25948
$818$		0			0
$819$	−1.41421	−	1.41421i	−0.0494166	−	0.0494166i
$820$		0			0
$821$	−6.00000	+	6.00000i	−0.209401	+	0.209401i	−0.804013	−	0.594612i	$-0.797306\pi$
$821$	−6.00000	+	6.00000i	−0.209401	+	0.209401i	0.594612	+	0.804013i	$0.297306\pi$
$822$		0			0
$823$			32.5269i			1.13382i	0.823781	+	0.566908i	$0.191861\pi$
$823$			32.5269i			1.13382i	−0.823781	+	0.566908i	$0.808139\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8.48528	+	8.48528i	−0.295062	+	0.295062i	−0.839076	−	0.544014i	$-0.816904\pi$
$827$	−8.48528	+	8.48528i	−0.295062	+	0.295062i	0.544014	+	0.839076i	$0.316904\pi$
$828$		0			0
$829$	11.0000	+	11.0000i	0.382046	+	0.382046i	0.871839	−	0.489793i	$-0.162928\pi$
$829$	11.0000	+	11.0000i	0.382046	+	0.382046i	−0.489793	+	0.871839i	$0.662928\pi$
$830$		0			0
$831$	15.5563			0.539644
$832$		0			0
$833$	30.0000			1.03944
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.00000	+	1.00000i	−0.0345651	+	0.0345651i
$838$		0			0
$839$		−	25.4558i		−	0.878833i	−0.898283	−	0.439417i	$-0.855185\pi$
$839$		−	25.4558i		−	0.878833i	0.898283	−	0.439417i	$-0.144815\pi$
$840$		0			0
$841$			43.0000i			1.48276i
$842$		0			0
$843$	12.7279	−	12.7279i	0.438373	−	0.438373i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	15.5563			0.534522
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$	42.4264	+	42.4264i	1.45436	+	1.45436i
$852$		0			0
$853$	−31.0000	+	31.0000i	−1.06142	+	1.06142i	−0.0634337	+	0.997986i	$0.520205\pi$
$853$	−31.0000	+	31.0000i	−1.06142	+	1.06142i	−0.997986	+	0.0634337i	$0.979795\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	42.0000i		−	1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$		−	42.0000i		−	1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$		0			0
$859$	4.24264	−	4.24264i	0.144757	−	0.144757i	−0.631014	−	0.775771i	$-0.717361\pi$
$859$	4.24264	−	4.24264i	0.144757	−	0.144757i	0.775771	+	0.631014i	$0.217361\pi$
$860$		0			0
$861$	6.00000	+	6.00000i	0.204479	+	0.204479i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−13.4350	−	13.4350i	−0.456278	−	0.456278i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			5.65685i			0.191675i
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.00000	+	5.00000i	0.168838	+	0.168838i	0.786468	−	0.617630i	$-0.211907\pi$
$877$	5.00000	+	5.00000i	0.168838	+	0.168838i	−0.617630	+	0.786468i	$0.711907\pi$
$878$		0			0
$879$	8.48528			0.286201
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	−21.2132	−	21.2132i	−0.713881	−	0.713881i	0.253464	−	0.967345i	$-0.418430\pi$
$883$	−21.2132	−	21.2132i	−0.713881	−	0.713881i	−0.967345	+	0.253464i	$0.918430\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			16.9706i			0.569816i	0.958555	+	0.284908i	$0.0919630\pi$
$887$			16.9706i			0.569816i	−0.958555	+	0.284908i	$0.908037\pi$
$888$		0			0
$889$		−	14.0000i		−	0.469545i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	36.0000	+	36.0000i	1.20469	+	1.20469i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	12.0000			0.400668
$898$		0			0
$899$	8.48528	+	8.48528i	0.283000	+	0.283000i
$900$		0			0
$901$	36.0000	−	36.0000i	1.19933	−	1.19933i
$902$		0			0
$903$			8.48528i			0.282372i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−4.24264	+	4.24264i	−0.140875	+	0.140875i	−0.774027	−	0.633153i	$-0.781761\pi$
$907$	−4.24264	+	4.24264i	−0.140875	+	0.140875i	0.633153	+	0.774027i	$0.281761\pi$
$908$		0			0
$909$	6.00000	+	6.00000i	0.199007	+	0.199007i
$910$		0			0
$911$	16.9706			0.562260			0.281130	−	0.959670i	$-0.409291\pi$
$911$	16.9706			0.562260			0.281130	+	0.959670i	$0.409291\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.0000	−	12.0000i	0.396275	−	0.396275i
$918$		0			0
$919$		−	1.41421i		−	0.0466506i	−0.999728	−	0.0233253i	$-0.992575\pi$
$919$		−	1.41421i		−	0.0466506i	0.999728	−	0.0233253i	$-0.00742535\pi$
$920$		0			0
$921$			20.0000i			0.659022i
$922$		0			0
$923$	−8.48528	+	8.48528i	−0.279296	+	0.279296i
$924$		0			0
$925$	−25.0000	−	25.0000i	−0.821995	−	0.821995i
$926$		0			0
$927$	−9.89949			−0.325142
$928$		0			0
$929$	6.00000			0.196854			0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000			0.196854			0.0984268	+	0.995144i	$0.468619\pi$
$930$		0			0
$931$	−21.2132	−	21.2132i	−0.695235	−	0.695235i
$932$		0			0
$933$	−12.0000	+	12.0000i	−0.392862	+	0.392862i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			2.00000i			0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$			2.00000i			0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$		0			0
$939$	−7.07107	+	7.07107i	−0.230756	+	0.230756i
$940$		0			0
$941$	−36.0000	−	36.0000i	−1.17357	−	1.17357i	−0.981353	−	0.192213i	$-0.938433\pi$
$941$	−36.0000	−	36.0000i	−1.17357	−	1.17357i	−0.192213	−	0.981353i	$-0.561567\pi$
$942$		0			0
$943$	−50.9117			−1.65791
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.9706	+	16.9706i	0.551469	+	0.551469i	0.926865	−	0.375396i	$-0.122493\pi$
$947$	16.9706	+	16.9706i	0.551469	+	0.551469i	−0.375396	+	0.926865i	$0.622493\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		−	8.48528i		−	0.275154i
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−25.4558			−0.822012
$960$		0			0
$961$	−29.0000			−0.935484
$962$		0			0
$963$	−8.48528	−	8.48528i	−0.273434	−	0.273434i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	41.0122i		−	1.31886i	−0.751765	−	0.659432i	$-0.770797\pi$
$967$		−	41.0122i		−	1.31886i	0.751765	−	0.659432i	$-0.229203\pi$
$968$		0			0
$969$			36.0000i			1.15649i
$970$		0			0
$971$	42.4264	−	42.4264i	1.36153	−	1.36153i	0.489557	−	0.871971i	$-0.337159\pi$
$971$	42.4264	−	42.4264i	1.36153	−	1.36153i	0.871971	−	0.489557i	$-0.162841\pi$
$972$		0			0
$973$	4.00000	+	4.00000i	0.128234	+	0.128234i
$974$		0			0
$975$	−7.07107			−0.226455
$976$		0			0
$977$	−6.00000			−0.191957			−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000			−0.191957			−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−13.0000	+	13.0000i	−0.415058	+	0.415058i
$982$		0			0
$983$		−	33.9411i		−	1.08255i	−0.840844	−	0.541277i	$-0.817941\pi$
$983$		−	33.9411i		−	1.08255i	0.840844	−	0.541277i	$-0.182059\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	8.48528	−	8.48528i	0.270089	−	0.270089i
$988$		0			0
$989$	−36.0000	−	36.0000i	−1.14473	−	1.14473i
$990$		0			0
$991$	52.3259			1.66219			0.831094	−	0.556133i	$-0.187715\pi$
$991$	52.3259			1.66219			0.831094	+	0.556133i	$0.187715\pi$
$992$		0			0
$993$	−28.0000			−0.888553
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−41.0000	+	41.0000i	−1.29848	+	1.29848i	−0.369089	+	0.929394i	$0.620330\pi$
$997$	−41.0000	+	41.0000i	−1.29848	+	1.29848i	−0.929394	+	0.369089i	$0.879670\pi$
$998$		0			0
$999$			7.07107i			0.223719i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.j.b.577.1	yes	4
3.2	odd	2		2304.2.k.b.577.1		4
4.3	odd	2	inner	768.2.j.b.577.2	yes	4
8.3	odd	2		768.2.j.c.577.1	yes	4
8.5	even	2		768.2.j.c.577.2	yes	4
12.11	even	2		2304.2.k.b.577.2		4
16.3	odd	4		768.2.j.c.193.1	yes	4
16.5	even	4	inner	768.2.j.b.193.1	✓	4
16.11	odd	4	inner	768.2.j.b.193.2	yes	4
16.13	even	4		768.2.j.c.193.2	yes	4
24.5	odd	2		2304.2.k.c.577.1		4
24.11	even	2		2304.2.k.c.577.2		4
32.3	odd	8		3072.2.d.a.1537.4		4
32.5	even	8		3072.2.a.g.1.2		2
32.11	odd	8		3072.2.a.g.1.1		2
32.13	even	8		3072.2.d.a.1537.3		4
32.19	odd	8		3072.2.d.a.1537.2		4
32.21	even	8		3072.2.a.a.1.2		2
32.27	odd	8		3072.2.a.a.1.1		2
32.29	even	8		3072.2.d.a.1537.1		4
48.5	odd	4		2304.2.k.b.1729.2		4
48.11	even	4		2304.2.k.b.1729.1		4
48.29	odd	4		2304.2.k.c.1729.2		4
48.35	even	4		2304.2.k.c.1729.1		4
96.5	odd	8		9216.2.a.n.1.2		2
96.11	even	8		9216.2.a.n.1.1		2
96.53	odd	8		9216.2.a.o.1.2		2
96.59	even	8		9216.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.b.193.1	✓	4	16.5	even	4	inner
768.2.j.b.193.2	yes	4	16.11	odd	4	inner
768.2.j.b.577.1	yes	4	1.1	even	1	trivial
768.2.j.b.577.2	yes	4	4.3	odd	2	inner
768.2.j.c.193.1	yes	4	16.3	odd	4
768.2.j.c.193.2	yes	4	16.13	even	4
768.2.j.c.577.1	yes	4	8.3	odd	2
768.2.j.c.577.2	yes	4	8.5	even	2
2304.2.k.b.577.1		4	3.2	odd	2
2304.2.k.b.577.2		4	12.11	even	2
2304.2.k.b.1729.1		4	48.11	even	4
2304.2.k.b.1729.2		4	48.5	odd	4
2304.2.k.c.577.1		4	24.5	odd	2
2304.2.k.c.577.2		4	24.11	even	2
2304.2.k.c.1729.1		4	48.35	even	4
2304.2.k.c.1729.2		4	48.29	odd	4
3072.2.a.a.1.1		2	32.27	odd	8
3072.2.a.a.1.2		2	32.21	even	8
3072.2.a.g.1.1		2	32.11	odd	8
3072.2.a.g.1.2		2	32.5	even	8
3072.2.d.a.1537.1		4	32.29	even	8
3072.2.d.a.1537.2		4	32.19	odd	8
3072.2.d.a.1537.3		4	32.13	even	8
3072.2.d.a.1537.4		4	32.3	odd	8
9216.2.a.n.1.1		2	96.11	even	8
9216.2.a.n.1.2		2	96.5	odd	8
9216.2.a.o.1.1		2	96.59	even	8
9216.2.a.o.1.2		2	96.53	odd	8

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{3} -1.41421i q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{3} -1.41421i q^{7} +1.00000i q^{9} +(-1.00000 - 1.00000i) q^{13} +6.00000 q^{17} +(-4.24264 - 4.24264i) q^{19} +(-1.00000 + 1.00000i) q^{21} -8.48528i q^{23} +5.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(-6.00000 - 6.00000i) q^{29} -1.41421 q^{31} +(-5.00000 + 5.00000i) q^{37} +1.41421i q^{39} -6.00000i q^{41} +(4.24264 - 4.24264i) q^{43} -8.48528 q^{47} +5.00000 q^{49} +(-4.24264 - 4.24264i) q^{51} +(6.00000 - 6.00000i) q^{53} +6.00000i q^{57} +(-5.00000 - 5.00000i) q^{61} +1.41421 q^{63} +(-2.82843 - 2.82843i) q^{67} +(-6.00000 + 6.00000i) q^{69} -8.48528i q^{71} +(3.53553 - 3.53553i) q^{75} +1.41421 q^{79} -1.00000 q^{81} +(8.48528 + 8.48528i) q^{83} +8.48528i q^{87} -6.00000i q^{89} +(-1.41421 + 1.41421i) q^{91} +(1.00000 + 1.00000i) q^{93} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{13} + 24 q^{17} - 4 q^{21} - 24 q^{29} - 20 q^{37} + 20 q^{49} + 24 q^{53} - 20 q^{61} - 24 q^{69} - 4 q^{81} + 4 q^{93} - 48 q^{97}+O(q^{100}) \) 4 * q - 4 * q^13 + 24 * q^17 - 4 * q^21 - 24 * q^29 - 20 * q^37 + 20 * q^49 + 24 * q^53 - 20 * q^61 - 24 * q^69 - 4 * q^81 + 4 * q^93 - 48 * q^97

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