LMFDB - Embedded newform 68.2.b.a.33.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,2,Mod(33,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("68.33");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	68.b (of order $2$, degree $1$, 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:	$0.542982733745$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	68.33
Dual form			68.2.b.a.33.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} -2.82843i q^{5} +4.24264i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.41421i q^{3} -2.82843i q^{5} +4.24264i q^{7} +1.00000 q^{9} +1.41421i q^{11} -4.00000 q^{13} -4.00000 q^{15} +(3.00000 + 2.82843i) q^{17} -4.00000 q^{19} +6.00000 q^{21} +1.41421i q^{23} -3.00000 q^{25} -5.65685i q^{27} -2.82843i q^{29} -4.24264i q^{31} +2.00000 q^{33} +12.0000 q^{35} +8.48528i q^{37} +5.65685i q^{39} -11.3137i q^{41} +8.00000 q^{43} -2.82843i q^{45} -12.0000 q^{47} -11.0000 q^{49} +(4.00000 - 4.24264i) q^{51} -6.00000 q^{53} +4.00000 q^{55} +5.65685i q^{57} +8.48528i q^{61} +4.24264i q^{63} +11.3137i q^{65} -4.00000 q^{67} +2.00000 q^{69} -7.07107i q^{71} +4.24264i q^{75} -6.00000 q^{77} -4.24264i q^{79} -5.00000 q^{81} +(8.00000 - 8.48528i) q^{85} -4.00000 q^{87} +12.0000 q^{89} -16.9706i q^{91} -6.00000 q^{93} +11.3137i q^{95} +1.41421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^9	$ 2 q + 2 q^{9} - 8 q^{13} - 8 q^{15} + 6 q^{17} - 8 q^{19} + 12 q^{21} - 6 q^{25} + 4 q^{33} + 24 q^{35} + 16 q^{43} - 24 q^{47} - 22 q^{49} + 8 q^{51} - 12 q^{53} + 8 q^{55} - 8 q^{67} + 4 q^{69} - 12 q^{77} - 10 q^{81} + 16 q^{85} - 8 q^{87} + 24 q^{89} - 12 q^{93}+O(q^{100}) $ 2 * q + 2 * q^9 - 8 * q^13 - 8 * q^15 + 6 * q^17 - 8 * q^19 + 12 * q^21 - 6 * q^25 + 4 * q^33 + 24 * q^35 + 16 * q^43 - 24 * q^47 - 22 * q^49 + 8 * q^51 - 12 * q^53 + 8 * q^55 - 8 * q^67 + 4 * q^69 - 12 * q^77 - 10 * q^81 + 16 * q^85 - 8 * q^87 + 24 * q^89 - 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$1$	$-1$

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
$3$		−	1.41421i		−	0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$		−	1.41421i		−	0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$		0			0
$5$		−	2.82843i		−	1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$		−	2.82843i		−	1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$		0			0
$7$			4.24264i			1.60357i	0.597614	+	0.801784i	$0.296115\pi$
$7$			4.24264i			1.60357i	−0.597614	+	0.801784i	$0.703885\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	−4.00000			−1.03280
$16$		0			0
$17$	3.00000	+	2.82843i	0.727607	+	0.685994i
$17$	3.00000	+	2.82843i	0.727607	+	0.685994i
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	6.00000			1.30931
$22$		0			0
$23$			1.41421i			0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$			1.41421i			0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$		−	5.65685i		−	1.08866i
$28$		0			0
$29$		−	2.82843i		−	0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$		−	2.82843i		−	0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$		0			0
$31$		−	4.24264i		−	0.762001i	−0.924575	−	0.381000i	$-0.875580\pi$
$31$		−	4.24264i		−	0.762001i	0.924575	−	0.381000i	$-0.124420\pi$
$32$		0			0
$33$	2.00000			0.348155
$34$		0			0
$35$	12.0000			2.02837
$36$		0			0
$37$			8.48528i			1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$			8.48528i			1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$		0			0
$39$			5.65685i			0.905822i
$40$		0			0
$41$		−	11.3137i		−	1.76690i	−0.468521	−	0.883452i	$-0.655213\pi$
$41$		−	11.3137i		−	1.76690i	0.468521	−	0.883452i	$-0.344787\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		−	2.82843i		−	0.421637i
$46$		0			0
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$		0			0
$49$	−11.0000			−1.57143
$50$		0			0
$51$	4.00000	−	4.24264i	0.560112	−	0.594089i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$			5.65685i			0.749269i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			8.48528i			1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$			8.48528i			1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$		0			0
$63$			4.24264i			0.534522i
$64$		0			0
$65$			11.3137i			1.40329i
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$	2.00000			0.240772
$70$		0			0
$71$		−	7.07107i		−	0.839181i	−0.907713	−	0.419591i	$-0.862174\pi$
$71$		−	7.07107i		−	0.839181i	0.907713	−	0.419591i	$-0.137826\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			4.24264i			0.489898i
$76$		0			0
$77$	−6.00000			−0.683763
$78$		0			0
$79$		−	4.24264i		−	0.477334i	−0.971101	−	0.238667i	$-0.923290\pi$
$79$		−	4.24264i		−	0.477334i	0.971101	−	0.238667i	$-0.0767105\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	8.00000	−	8.48528i	0.867722	−	0.920358i
$86$		0			0
$87$	−4.00000			−0.428845
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$		−	16.9706i		−	1.77900i
$92$		0			0
$93$	−6.00000			−0.622171
$94$		0			0
$95$			11.3137i			1.16076i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$			1.41421i			0.142134i
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$		−	16.9706i		−	1.65616i
$106$		0			0
$107$		−	7.07107i		−	0.683586i	−0.939775	−	0.341793i	$-0.888966\pi$
$107$		−	7.07107i		−	0.683586i	0.939775	−	0.341793i	$-0.111034\pi$
$108$		0			0
$109$			8.48528i			0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$			8.48528i			0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$		0			0
$111$	12.0000			1.13899
$112$		0			0
$113$			5.65685i			0.532152i	0.963952	+	0.266076i	$0.0857272\pi$
$113$			5.65685i			0.532152i	−0.963952	+	0.266076i	$0.914273\pi$
$114$		0			0
$115$	4.00000			0.373002
$116$		0			0
$117$	−4.00000			−0.369800
$118$		0			0
$119$	−12.0000	+	12.7279i	−1.10004	+	1.16677i
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$	−16.0000			−1.44267
$124$		0			0
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		−	11.3137i		−	0.996116i
$130$		0			0
$131$			9.89949i			0.864923i	0.901652	+	0.432461i	$0.142355\pi$
$131$			9.89949i			0.864923i	−0.901652	+	0.432461i	$0.857645\pi$
$132$		0			0
$133$		−	16.9706i		−	1.47153i
$134$		0			0
$135$	−16.0000			−1.37706
$136$		0			0
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$			12.7279i			1.07957i	0.841803	+	0.539784i	$0.181494\pi$
$139$			12.7279i			1.07957i	−0.841803	+	0.539784i	$0.818506\pi$
$140$		0			0
$141$			16.9706i			1.42918i
$142$		0			0
$143$		−	5.65685i		−	0.473050i
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$			15.5563i			1.28307i
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	3.00000	+	2.82843i	0.242536	+	0.228665i
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$			8.48528i			0.672927i
$160$		0			0
$161$	−6.00000			−0.472866
$162$		0			0
$163$		−	4.24264i		−	0.332309i	−0.986100	−	0.166155i	$-0.946865\pi$
$163$		−	4.24264i		−	0.332309i	0.986100	−	0.166155i	$-0.0531351\pi$
$164$		0			0
$165$		−	5.65685i		−	0.440386i
$166$		0			0
$167$		−	7.07107i		−	0.547176i	−0.961847	−	0.273588i	$-0.911790\pi$
$167$		−	7.07107i		−	0.547176i	0.961847	−	0.273588i	$-0.0882104\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−4.00000			−0.305888
$172$		0			0
$173$			14.1421i			1.07521i	0.843198	+	0.537603i	$0.180670\pi$
$173$			14.1421i			1.07521i	−0.843198	+	0.537603i	$0.819330\pi$
$174$		0			0
$175$		−	12.7279i		−	0.962140i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$		−	8.48528i		−	0.630706i	−0.948974	−	0.315353i	$-0.897877\pi$
$181$		−	8.48528i		−	0.630706i	0.948974	−	0.315353i	$-0.102123\pi$
$182$		0			0
$183$	12.0000			0.887066
$184$		0			0
$185$	24.0000			1.76452
$186$		0			0
$187$	−4.00000	+	4.24264i	−0.292509	+	0.310253i
$188$		0			0
$189$	24.0000			1.74574
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$			16.9706i			1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$193$			16.9706i			1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$194$		0			0
$195$	16.0000			1.14578
$196$		0			0
$197$		−	19.7990i		−	1.41062i	−0.708899	−	0.705310i	$-0.750808\pi$
$197$		−	19.7990i		−	1.41062i	0.708899	−	0.705310i	$-0.249192\pi$
$198$		0			0
$199$		−	21.2132i		−	1.50376i	−0.659298	−	0.751882i	$-0.729146\pi$
$199$		−	21.2132i		−	1.50376i	0.659298	−	0.751882i	$-0.270854\pi$
$200$		0			0
$201$			5.65685i			0.399004i
$202$		0			0
$203$	12.0000			0.842235
$204$		0			0
$205$	−32.0000			−2.23498
$206$		0			0
$207$			1.41421i			0.0982946i
$208$		0			0
$209$		−	5.65685i		−	0.391293i
$210$		0			0
$211$		−	12.7279i		−	0.876226i	−0.898920	−	0.438113i	$-0.855647\pi$
$211$		−	12.7279i		−	0.876226i	0.898920	−	0.438113i	$-0.144353\pi$
$212$		0			0
$213$	−10.0000			−0.685189
$214$		0			0
$215$		−	22.6274i		−	1.54318i
$216$		0			0
$217$	18.0000			1.22192
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000	−	11.3137i	−0.807207	−	0.761042i
$222$		0			0
$223$	20.0000			1.33930			0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000			1.33930			0.669650	+	0.742677i	$0.266444\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$			18.3848i			1.22024i	0.792309	+	0.610120i	$0.208879\pi$
$227$			18.3848i			1.22024i	−0.792309	+	0.610120i	$0.791121\pi$
$228$		0			0
$229$	−4.00000			−0.264327			−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000			−0.264327			−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$			8.48528i			0.558291i
$232$		0			0
$233$			5.65685i			0.370593i	0.982683	+	0.185296i	$0.0593245\pi$
$233$			5.65685i			0.370593i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$			33.9411i			2.21407i
$236$		0			0
$237$	−6.00000			−0.389742
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		−	9.89949i		−	0.635053i
$244$		0			0
$245$			31.1127i			1.98772i
$246$		0			0
$247$	16.0000			1.01806
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	−2.00000			−0.125739
$254$		0			0
$255$	−12.0000	−	11.3137i	−0.751469	−	0.708492i
$256$		0			0
$257$	−12.0000			−0.748539			−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000			−0.748539			−0.374270	+	0.927320i	$0.622107\pi$
$258$		0			0
$259$	−36.0000			−2.23693
$260$		0			0
$261$		−	2.82843i		−	0.175075i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$			16.9706i			1.04249i
$266$		0			0
$267$		−	16.9706i		−	1.03858i
$268$		0			0
$269$		−	2.82843i		−	0.172452i	−0.996276	−	0.0862261i	$-0.972519\pi$
$269$		−	2.82843i		−	0.172452i	0.996276	−	0.0862261i	$-0.0274808\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$	−24.0000			−1.45255
$274$		0			0
$275$		−	4.24264i		−	0.255841i
$276$		0			0
$277$		−	8.48528i		−	0.509831i	−0.966963	−	0.254916i	$-0.917952\pi$
$277$		−	8.48528i		−	0.509831i	0.966963	−	0.254916i	$-0.0820477\pi$
$278$		0			0
$279$		−	4.24264i		−	0.254000i
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$			12.7279i			0.756596i	0.925684	+	0.378298i	$0.123491\pi$
$283$			12.7279i			0.756596i	−0.925684	+	0.378298i	$0.876509\pi$
$284$		0			0
$285$	16.0000			0.947758
$286$		0			0
$287$	48.0000			2.83335
$288$		0			0
$289$	1.00000	+	16.9706i	0.0588235	+	0.998268i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	8.00000			0.464207
$298$		0			0
$299$		−	5.65685i		−	0.327144i
$300$		0			0
$301$			33.9411i			1.95633i
$302$		0			0
$303$		−	16.9706i		−	0.974933i
$304$		0			0
$305$	24.0000			1.37424
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$		0			0
$309$		−	11.3137i		−	0.643614i
$310$		0			0
$311$			26.8701i			1.52366i	0.647776	+	0.761831i	$0.275699\pi$
$311$			26.8701i			1.52366i	−0.647776	+	0.761831i	$0.724301\pi$
$312$		0			0
$313$		−	16.9706i		−	0.959233i	−0.877478	−	0.479616i	$-0.840776\pi$
$313$		−	16.9706i		−	0.959233i	0.877478	−	0.479616i	$-0.159224\pi$
$314$		0			0
$315$	12.0000			0.676123
$316$		0			0
$317$		−	2.82843i		−	0.158860i	−0.996840	−	0.0794301i	$-0.974690\pi$
$317$		−	2.82843i		−	0.158860i	0.996840	−	0.0794301i	$-0.0253101\pi$
$318$		0			0
$319$	4.00000			0.223957
$320$		0			0
$321$	−10.0000			−0.558146
$322$		0			0
$323$	−12.0000	−	11.3137i	−0.667698	−	0.629512i
$324$		0			0
$325$	12.0000			0.665640
$326$		0			0
$327$	12.0000			0.663602
$328$		0			0
$329$		−	50.9117i		−	2.80685i
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$			8.48528i			0.464991i
$334$		0			0
$335$			11.3137i			0.618134i
$336$		0			0
$337$		−	16.9706i		−	0.924445i	−0.886764	−	0.462223i	$-0.847052\pi$
$337$		−	16.9706i		−	0.924445i	0.886764	−	0.462223i	$-0.152948\pi$
$338$		0			0
$339$	8.00000			0.434500
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		−	5.65685i		−	0.304555i
$346$		0			0
$347$			1.41421i			0.0759190i	0.999279	+	0.0379595i	$0.0120858\pi$
$347$			1.41421i			0.0759190i	−0.999279	+	0.0379595i	$0.987914\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$			22.6274i			1.20776i
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$	−20.0000			−1.06149
$356$		0			0
$357$	18.0000	+	16.9706i	0.952661	+	0.898177i
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	12.7279i		−	0.668043i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	12.7279i		−	0.664392i	−0.943210	−	0.332196i	$-0.892210\pi$
$367$		−	12.7279i		−	0.664392i	0.943210	−	0.332196i	$-0.107790\pi$
$368$		0			0
$369$		−	11.3137i		−	0.588968i
$370$		0			0
$371$		−	25.4558i		−	1.32160i
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		0			0
$375$	−8.00000			−0.413118
$376$		0			0
$377$			11.3137i			0.582686i
$378$		0			0
$379$			38.1838i			1.96137i	0.195598	+	0.980684i	$0.437335\pi$
$379$			38.1838i			1.96137i	−0.195598	+	0.980684i	$0.562665\pi$
$380$		0			0
$381$		−	11.3137i		−	0.579619i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$			16.9706i			0.864900i
$386$		0			0
$387$	8.00000			0.406663
$388$		0			0
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$	−4.00000	+	4.24264i	−0.202289	+	0.214560i
$392$		0			0
$393$	14.0000			0.706207
$394$		0			0
$395$	−12.0000			−0.603786
$396$		0			0
$397$		−	8.48528i		−	0.425864i	−0.977067	−	0.212932i	$-0.931699\pi$
$397$		−	8.48528i		−	0.425864i	0.977067	−	0.212932i	$-0.0683013\pi$
$398$		0			0
$399$	−24.0000			−1.20150
$400$		0			0
$401$			22.6274i			1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$			22.6274i			1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$		0			0
$403$			16.9706i			0.845364i
$404$		0			0
$405$			14.1421i			0.702728i
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$			16.9706i			0.837096i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	18.0000			0.881464
$418$		0			0
$419$		−	24.0416i		−	1.17451i	−0.809402	−	0.587255i	$-0.800208\pi$
$419$		−	24.0416i		−	1.17451i	0.809402	−	0.587255i	$-0.199792\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$		0			0
$423$	−12.0000			−0.583460
$424$		0			0
$425$	−9.00000	−	8.48528i	−0.436564	−	0.411597i
$426$		0			0
$427$	−36.0000			−1.74216
$428$		0			0
$429$	−8.00000			−0.386244
$430$		0			0
$431$			18.3848i			0.885564i	0.896629	+	0.442782i	$0.146008\pi$
$431$			18.3848i			0.885564i	−0.896629	+	0.442782i	$0.853992\pi$
$432$		0			0
$433$	−4.00000			−0.192228			−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000			−0.192228			−0.0961139	+	0.995370i	$0.530641\pi$
$434$		0			0
$435$			11.3137i			0.542451i
$436$		0			0
$437$		−	5.65685i		−	0.270604i
$438$		0			0
$439$			38.1838i			1.82241i	0.411951	+	0.911206i	$0.364847\pi$
$439$			38.1838i			1.82241i	−0.411951	+	0.911206i	$0.635153\pi$
$440$		0			0
$441$	−11.0000			−0.523810
$442$		0			0
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$		0			0
$445$		−	33.9411i		−	1.60896i
$446$		0			0
$447$		−	8.48528i		−	0.401340i
$448$		0			0
$449$			5.65685i			0.266963i	0.991051	+	0.133482i	$0.0426157\pi$
$449$			5.65685i			0.266963i	−0.991051	+	0.133482i	$0.957384\pi$
$450$		0			0
$451$	16.0000			0.753411
$452$		0			0
$453$		−	11.3137i		−	0.531564i
$454$		0			0
$455$	−48.0000			−2.25027
$456$		0			0
$457$	−4.00000			−0.187112			−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000			−0.187112			−0.0935561	+	0.995614i	$0.529823\pi$
$458$		0			0
$459$	16.0000	−	16.9706i	0.746816	−	0.792118i
$460$		0			0
$461$	−30.0000			−1.39724			−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000			−1.39724			−0.698620	+	0.715493i	$0.746202\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$		0			0
$465$			16.9706i			0.786991i
$466$		0			0
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$		−	16.9706i		−	0.783628i
$470$		0			0
$471$		−	2.82843i		−	0.130327i
$472$		0			0
$473$			11.3137i			0.520205i
$474$		0			0
$475$	12.0000			0.550598
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$			18.3848i			0.840022i	0.907519	+	0.420011i	$0.137974\pi$
$479$			18.3848i			0.840022i	−0.907519	+	0.420011i	$0.862026\pi$
$480$		0			0
$481$		−	33.9411i		−	1.54758i
$482$		0			0
$483$			8.48528i			0.386094i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	29.6985i		−	1.34577i	−0.739749	−	0.672883i	$-0.765056\pi$
$487$		−	29.6985i		−	1.34577i	0.739749	−	0.672883i	$-0.234944\pi$
$488$		0			0
$489$	−6.00000			−0.271329
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	8.00000	−	8.48528i	0.360302	−	0.382158i
$494$		0			0
$495$	4.00000			0.179787
$496$		0			0
$497$	30.0000			1.34568
$498$		0			0
$499$			21.2132i			0.949633i	0.880085	+	0.474817i	$0.157486\pi$
$499$			21.2132i			0.949633i	−0.880085	+	0.474817i	$0.842514\pi$
$500$		0			0
$501$	−10.0000			−0.446767
$502$		0			0
$503$		−	7.07107i		−	0.315283i	−0.987496	−	0.157642i	$-0.949611\pi$
$503$		−	7.07107i		−	0.315283i	0.987496	−	0.157642i	$-0.0503891\pi$
$504$		0			0
$505$		−	33.9411i		−	1.51036i
$506$		0			0
$507$		−	4.24264i		−	0.188422i
$508$		0			0
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			22.6274i			0.999025i
$514$		0			0
$515$		−	22.6274i		−	0.997083i
$516$		0			0
$517$		−	16.9706i		−	0.746364i
$518$		0			0
$519$	20.0000			0.877903
$520$		0			0
$521$			22.6274i			0.991325i	0.868515	+	0.495663i	$0.165075\pi$
$521$			22.6274i			0.991325i	−0.868515	+	0.495663i	$0.834925\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$	−18.0000			−0.785584
$526$		0			0
$527$	12.0000	−	12.7279i	0.522728	−	0.554437i
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$			45.2548i			1.96020i
$534$		0			0
$535$	−20.0000			−0.864675
$536$		0			0
$537$			16.9706i			0.732334i
$538$		0			0
$539$		−	15.5563i		−	0.670059i
$540$		0			0
$541$			8.48528i			0.364811i	0.983223	+	0.182405i	$0.0583883\pi$
$541$			8.48528i			0.364811i	−0.983223	+	0.182405i	$0.941612\pi$
$542$		0			0
$543$	−12.0000			−0.514969
$544$		0			0
$545$	24.0000			1.02805
$546$		0			0
$547$		−	12.7279i		−	0.544207i	−0.962268	−	0.272103i	$-0.912281\pi$
$547$		−	12.7279i		−	0.544207i	0.962268	−	0.272103i	$-0.0877193\pi$
$548$		0			0
$549$			8.48528i			0.362143i
$550$		0			0
$551$			11.3137i			0.481980i
$552$		0			0
$553$	18.0000			0.765438
$554$		0			0
$555$		−	33.9411i		−	1.44072i
$556$		0			0
$557$	−36.0000			−1.52537			−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000			−1.52537			−0.762684	+	0.646771i	$0.776119\pi$
$558$		0			0
$559$	−32.0000			−1.35346
$560$		0			0
$561$	6.00000	+	5.65685i	0.253320	+	0.238833i
$562$		0			0
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$	16.0000			0.673125
$566$		0			0
$567$		−	21.2132i		−	0.890871i
$568$		0			0
$569$	−42.0000			−1.76073			−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000			−1.76073			−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$		−	29.6985i		−	1.24284i	−0.783476	−	0.621421i	$-0.786555\pi$
$571$		−	29.6985i		−	1.24284i	0.783476	−	0.621421i	$-0.213445\pi$
$572$		0			0
$573$			16.9706i			0.708955i
$574$		0			0
$575$		−	4.24264i		−	0.176930i
$576$		0			0
$577$	−28.0000			−1.16566			−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000			−1.16566			−0.582828	+	0.812596i	$0.698054\pi$
$578$		0			0
$579$	24.0000			0.997406
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	8.48528i		−	0.351424i
$584$		0			0
$585$			11.3137i			0.467764i
$586$		0			0
$587$	−36.0000			−1.48588			−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000			−1.48588			−0.742940	+	0.669359i	$0.766569\pi$
$588$		0			0
$589$			16.9706i			0.699260i
$590$		0			0
$591$	−28.0000			−1.15177
$592$		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
$595$	36.0000	+	33.9411i	1.47586	+	1.39145i
$596$		0			0
$597$	−30.0000			−1.22782
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$			16.9706i			0.692244i	0.938190	+	0.346122i	$0.112502\pi$
$601$			16.9706i			0.692244i	−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	−4.00000			−0.162893
$604$		0			0
$605$		−	25.4558i		−	1.03493i
$606$		0			0
$607$			29.6985i			1.20542i	0.797959	+	0.602712i	$0.205913\pi$
$607$			29.6985i			1.20542i	−0.797959	+	0.602712i	$0.794087\pi$
$608$		0			0
$609$		−	16.9706i		−	0.687682i
$610$		0			0
$611$	48.0000			1.94187
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$			45.2548i			1.82485i
$616$		0			0
$617$			39.5980i			1.59415i	0.603877	+	0.797077i	$0.293622\pi$
$617$			39.5980i			1.59415i	−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$			12.7279i			0.511578i	0.966733	+	0.255789i	$0.0823353\pi$
$619$			12.7279i			0.511578i	−0.966733	+	0.255789i	$0.917665\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$			50.9117i			2.03973i
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	−8.00000			−0.319489
$628$		0			0
$629$	−24.0000	+	25.4558i	−0.956943	+	1.01499i
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$	−18.0000			−0.715436
$634$		0			0
$635$		−	22.6274i		−	0.897942i
$636$		0			0
$637$	44.0000			1.74334
$638$		0			0
$639$		−	7.07107i		−	0.279727i
$640$		0			0
$641$		−	45.2548i		−	1.78746i	−0.448607	−	0.893729i	$-0.648080\pi$
$641$		−	45.2548i		−	1.78746i	0.448607	−	0.893729i	$-0.351920\pi$
$642$		0			0
$643$		−	46.6690i		−	1.84045i	−0.391393	−	0.920224i	$-0.628007\pi$
$643$		−	46.6690i		−	1.84045i	0.391393	−	0.920224i	$-0.371993\pi$
$644$		0			0
$645$	−32.0000			−1.26000
$646$		0			0
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		−	25.4558i		−	0.997693i
$652$		0			0
$653$		−	19.7990i		−	0.774794i	−0.921913	−	0.387397i	$-0.873374\pi$
$653$		−	19.7990i		−	0.774794i	0.921913	−	0.387397i	$-0.126626\pi$
$654$		0			0
$655$	28.0000			1.09405
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$		0			0
$663$	−16.0000	+	16.9706i	−0.621389	+	0.659082i
$664$		0			0
$665$	−48.0000			−1.86136
$666$		0			0
$667$	4.00000			0.154881
$668$		0			0
$669$		−	28.2843i		−	1.09353i
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$		−	16.9706i		−	0.654167i	−0.944995	−	0.327084i	$-0.893934\pi$
$673$		−	16.9706i		−	0.654167i	0.944995	−	0.327084i	$-0.106066\pi$
$674$		0			0
$675$			16.9706i			0.653197i
$676$		0			0
$677$		−	2.82843i		−	0.108705i	−0.998522	−	0.0543526i	$-0.982690\pi$
$677$		−	2.82843i		−	0.108705i	0.998522	−	0.0543526i	$-0.0173095\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	26.0000			0.996322
$682$		0			0
$683$		−	41.0122i		−	1.56929i	−0.619947	−	0.784644i	$-0.712846\pi$
$683$		−	41.0122i		−	1.56929i	0.619947	−	0.784644i	$-0.287154\pi$
$684$		0			0
$685$			33.9411i			1.29682i
$686$		0			0
$687$			5.65685i			0.215822i
$688$		0			0
$689$	24.0000			0.914327
$690$		0			0
$691$		−	4.24264i		−	0.161398i	−0.996739	−	0.0806988i	$-0.974285\pi$
$691$		−	4.24264i		−	0.161398i	0.996739	−	0.0806988i	$-0.0257152\pi$
$692$		0			0
$693$	−6.00000			−0.227921
$694$		0			0
$695$	36.0000			1.36556
$696$		0			0
$697$	32.0000	−	33.9411i	1.21209	−	1.28561i
$698$		0			0
$699$	8.00000			0.302588
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$		−	33.9411i		−	1.28011i
$704$		0			0
$705$	48.0000			1.80778
$706$		0			0
$707$			50.9117i			1.91473i
$708$		0			0
$709$			8.48528i			0.318671i	0.987224	+	0.159336i	$0.0509352\pi$
$709$			8.48528i			0.318671i	−0.987224	+	0.159336i	$0.949065\pi$
$710$		0			0
$711$		−	4.24264i		−	0.159111i
$712$		0			0
$713$	6.00000			0.224702
$714$		0			0
$715$	−16.0000			−0.598366
$716$		0			0
$717$		−	16.9706i		−	0.633777i
$718$		0			0
$719$		−	24.0416i		−	0.896602i	−0.893883	−	0.448301i	$-0.852029\pi$
$719$		−	24.0416i		−	0.896602i	0.893883	−	0.448301i	$-0.147971\pi$
$720$		0			0
$721$			33.9411i			1.26403i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			8.48528i			0.315135i
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	24.0000	+	22.6274i	0.887672	+	0.836905i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$		0			0
$735$	44.0000			1.62296
$736$		0			0
$737$		−	5.65685i		−	0.208373i
$738$		0			0
$739$	−52.0000			−1.91285			−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000			−1.91285			−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$		−	22.6274i		−	0.831239i
$742$		0			0
$743$			1.41421i			0.0518825i	0.999663	+	0.0259412i	$0.00825828\pi$
$743$			1.41421i			0.0518825i	−0.999663	+	0.0259412i	$0.991742\pi$
$744$		0			0
$745$		−	16.9706i		−	0.621753i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	30.0000			1.09618
$750$		0			0
$751$		−	12.7279i		−	0.464448i	−0.972662	−	0.232224i	$-0.925400\pi$
$751$		−	12.7279i		−	0.464448i	0.972662	−	0.232224i	$-0.0746003\pi$
$752$		0			0
$753$		−	16.9706i		−	0.618442i
$754$		0			0
$755$		−	22.6274i		−	0.823496i
$756$		0			0
$757$	−4.00000			−0.145382			−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000			−0.145382			−0.0726912	+	0.997354i	$0.523159\pi$
$758$		0			0
$759$			2.82843i			0.102665i
$760$		0			0
$761$	36.0000			1.30500			0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000			1.30500			0.652499	+	0.757789i	$0.273720\pi$
$762$		0			0
$763$	−36.0000			−1.30329
$764$		0			0
$765$	8.00000	−	8.48528i	0.289241	−	0.306786i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−28.0000			−1.00971			−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000			−1.00971			−0.504853	+	0.863205i	$0.668453\pi$
$770$		0			0
$771$			16.9706i			0.611180i
$772$		0			0
$773$	−36.0000			−1.29483			−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000			−1.29483			−0.647415	+	0.762138i	$0.724150\pi$
$774$		0			0
$775$			12.7279i			0.457200i
$776$		0			0
$777$			50.9117i			1.82645i
$778$		0			0
$779$			45.2548i			1.62142i
$780$		0			0
$781$	10.0000			0.357828
$782$		0			0
$783$	−16.0000			−0.571793
$784$		0			0
$785$		−	5.65685i		−	0.201902i
$786$		0			0
$787$			29.6985i			1.05864i	0.848423	+	0.529318i	$0.177552\pi$
$787$			29.6985i			1.05864i	−0.848423	+	0.529318i	$0.822448\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$		−	33.9411i		−	1.20528i
$794$		0			0
$795$	24.0000			0.851192
$796$		0			0
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$		0			0
$799$	−36.0000	−	33.9411i	−1.27359	−	1.20075i
$800$		0			0
$801$	12.0000			0.423999
$802$		0			0
$803$		0			0
$804$		0			0
$805$			16.9706i			0.598134i
$806$		0			0
$807$	−4.00000			−0.140807
$808$		0			0
$809$			39.5980i			1.39219i	0.717949	+	0.696095i	$0.245081\pi$
$809$			39.5980i			1.39219i	−0.717949	+	0.696095i	$0.754919\pi$
$810$		0			0
$811$		−	21.2132i		−	0.744896i	−0.928053	−	0.372448i	$-0.878518\pi$
$811$		−	21.2132i		−	0.744896i	0.928053	−	0.372448i	$-0.121482\pi$
$812$		0			0
$813$			39.5980i			1.38876i
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$	−32.0000			−1.11954
$818$		0			0
$819$		−	16.9706i		−	0.592999i
$820$		0			0
$821$		−	36.7696i		−	1.28327i	−0.767012	−	0.641633i	$-0.778257\pi$
$821$		−	36.7696i		−	1.28327i	0.767012	−	0.641633i	$-0.221743\pi$
$822$		0			0
$823$			12.7279i			0.443667i	0.975085	+	0.221834i	$0.0712042\pi$
$823$			12.7279i			0.443667i	−0.975085	+	0.221834i	$0.928796\pi$
$824$		0			0
$825$	−6.00000			−0.208893
$826$		0			0
$827$			26.8701i			0.934363i	0.884161	+	0.467182i	$0.154731\pi$
$827$			26.8701i			0.934363i	−0.884161	+	0.467182i	$0.845269\pi$
$828$		0			0
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	−12.0000			−0.416275
$832$		0			0
$833$	−33.0000	−	31.1127i	−1.14338	−	1.07799i
$834$		0			0
$835$	−20.0000			−0.692129
$836$		0			0
$837$	−24.0000			−0.829561
$838$		0			0
$839$		−	7.07107i		−	0.244120i	−0.992523	−	0.122060i	$-0.961050\pi$
$839$		−	7.07107i		−	0.244120i	0.992523	−	0.122060i	$-0.0389501\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$			42.4264i			1.46124i
$844$		0			0
$845$		−	8.48528i		−	0.291903i
$846$		0			0
$847$			38.1838i			1.31201i
$848$		0			0
$849$	18.0000			0.617758
$850$		0			0
$851$	−12.0000			−0.411355
$852$		0			0
$853$			42.4264i			1.45265i	0.687350	+	0.726326i	$0.258774\pi$
$853$			42.4264i			1.45265i	−0.687350	+	0.726326i	$0.741226\pi$
$854$		0			0
$855$			11.3137i			0.386921i
$856$		0			0
$857$		−	28.2843i		−	0.966172i	−0.875573	−	0.483086i	$-0.839516\pi$
$857$		−	28.2843i		−	0.966172i	0.875573	−	0.483086i	$-0.160484\pi$
$858$		0			0
$859$	−40.0000			−1.36478			−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000			−1.36478			−0.682391	+	0.730987i	$0.739060\pi$
$860$		0			0
$861$		−	67.8823i		−	2.31342i
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	40.0000			1.36004
$866$		0			0
$867$	24.0000	−	1.41421i	0.815083	−	0.0480292i
$868$		0			0
$869$	6.00000			0.203536
$870$		0			0
$871$	16.0000			0.542139
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.0000			0.811348
$876$		0			0
$877$			42.4264i			1.43264i	0.697773	+	0.716319i	$0.254174\pi$
$877$			42.4264i			1.43264i	−0.697773	+	0.716319i	$0.745826\pi$
$878$		0			0
$879$		−	8.48528i		−	0.286201i
$880$		0			0
$881$			5.65685i			0.190584i	0.995449	+	0.0952921i	$0.0303785\pi$
$881$			5.65685i			0.190584i	−0.995449	+	0.0952921i	$0.969621\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	32.5269i		−	1.09215i	−0.837737	−	0.546073i	$-0.816122\pi$
$887$		−	32.5269i		−	1.09215i	0.837737	−	0.546073i	$-0.183878\pi$
$888$		0			0
$889$			33.9411i			1.13835i
$890$		0			0
$891$		−	7.07107i		−	0.236890i
$892$		0			0
$893$	48.0000			1.60626
$894$		0			0
$895$			33.9411i			1.13453i
$896$		0			0
$897$	−8.00000			−0.267112
$898$		0			0
$899$	−12.0000			−0.400222
$900$		0			0
$901$	−18.0000	−	16.9706i	−0.599667	−	0.565371i
$902$		0			0
$903$	48.0000			1.59734
$904$		0			0
$905$	−24.0000			−0.797787
$906$		0			0
$907$			12.7279i			0.422624i	0.977419	+	0.211312i	$0.0677736\pi$
$907$			12.7279i			0.422624i	−0.977419	+	0.211312i	$0.932226\pi$
$908$		0			0
$909$	12.0000			0.398015
$910$		0			0
$911$		−	57.9828i		−	1.92105i	−0.278186	−	0.960527i	$-0.589733\pi$
$911$		−	57.9828i		−	1.92105i	0.278186	−	0.960527i	$-0.410267\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	33.9411i		−	1.12206i
$916$		0			0
$917$	−42.0000			−1.38696
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$		0			0
$921$			22.6274i			0.745599i
$922$		0			0
$923$			28.2843i			0.930988i
$924$		0			0
$925$		−	25.4558i		−	0.836983i
$926$		0			0
$927$	8.00000			0.262754
$928$		0			0
$929$		−	45.2548i		−	1.48476i	−0.669977	−	0.742381i	$-0.733696\pi$
$929$		−	45.2548i		−	1.48476i	0.669977	−	0.742381i	$-0.266304\pi$
$930$		0			0
$931$	44.0000			1.44204
$932$		0			0
$933$	38.0000			1.24406
$934$		0			0
$935$	12.0000	+	11.3137i	0.392442	+	0.369998i
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−24.0000			−0.783210
$940$		0			0
$941$			14.1421i			0.461020i	0.973070	+	0.230510i	$0.0740395\pi$
$941$			14.1421i			0.461020i	−0.973070	+	0.230510i	$0.925960\pi$
$942$		0			0
$943$	16.0000			0.521032
$944$		0			0
$945$		−	67.8823i		−	2.20821i
$946$		0			0
$947$		−	15.5563i		−	0.505513i	−0.967530	−	0.252757i	$-0.918663\pi$
$947$		−	15.5563i		−	0.505513i	0.967530	−	0.252757i	$-0.0813372\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−4.00000			−0.129709
$952$		0			0
$953$	12.0000			0.388718			0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000			0.388718			0.194359	+	0.980930i	$0.437737\pi$
$954$		0			0
$955$			33.9411i			1.09831i
$956$		0			0
$957$		−	5.65685i		−	0.182860i
$958$		0			0
$959$		−	50.9117i		−	1.64402i
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$		−	7.07107i		−	0.227862i
$964$		0			0
$965$	48.0000			1.54517
$966$		0			0
$967$	44.0000			1.41494			0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000			1.41494			0.707472	+	0.706741i	$0.249835\pi$
$968$		0			0
$969$	−16.0000	+	16.9706i	−0.513994	+	0.545173i
$970$		0			0
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$	−54.0000			−1.73116
$974$		0			0
$975$		−	16.9706i		−	0.543493i
$976$		0			0
$977$	30.0000			0.959785			0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000			0.959785			0.479893	+	0.877327i	$0.340676\pi$
$978$		0			0
$979$			16.9706i			0.542382i
$980$		0			0
$981$			8.48528i			0.270914i
$982$		0			0
$983$		−	7.07107i		−	0.225532i	−0.993622	−	0.112766i	$-0.964029\pi$
$983$		−	7.07107i		−	0.225532i	0.993622	−	0.112766i	$-0.0359711\pi$
$984$		0			0
$985$	−56.0000			−1.78431
$986$		0			0
$987$	−72.0000			−2.29179
$988$		0			0
$989$			11.3137i			0.359755i
$990$		0			0
$991$		−	12.7279i		−	0.404316i	−0.979353	−	0.202158i	$-0.935205\pi$
$991$		−	12.7279i		−	0.404316i	0.979353	−	0.202158i	$-0.0647954\pi$
$992$		0			0
$993$		−	11.3137i		−	0.359030i
$994$		0			0
$995$	−60.0000			−1.90213
$996$		0			0
$997$			25.4558i			0.806195i	0.915157	+	0.403097i	$0.132066\pi$
$997$			25.4558i			0.806195i	−0.915157	+	0.403097i	$0.867934\pi$
$998$		0			0
$999$	48.0000			1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.2.b.a.33.1	✓	2
3.2	odd	2		612.2.b.a.577.2		2
4.3	odd	2		272.2.b.c.33.2		2
5.2	odd	4		1700.2.g.a.849.2		4
5.3	odd	4		1700.2.g.a.849.4		4
5.4	even	2		1700.2.c.a.101.2		2
7.6	odd	2		3332.2.b.a.2549.2		2
8.3	odd	2		1088.2.b.f.577.1		2
8.5	even	2		1088.2.b.e.577.2		2
12.11	even	2		2448.2.c.d.577.2		2
17.2	even	8		1156.2.e.a.829.1		2
17.3	odd	16		1156.2.h.d.977.2		8
17.4	even	4		1156.2.a.c.1.1		2
17.5	odd	16		1156.2.h.d.757.2		8
17.6	odd	16		1156.2.h.d.1001.1		8
17.7	odd	16		1156.2.h.d.733.1		8
17.8	even	8		1156.2.e.a.905.1		2
17.9	even	8		1156.2.e.b.905.1		2
17.10	odd	16		1156.2.h.d.733.2		8
17.11	odd	16		1156.2.h.d.1001.2		8
17.12	odd	16		1156.2.h.d.757.1		8
17.13	even	4		1156.2.a.c.1.2		2
17.14	odd	16		1156.2.h.d.977.1		8
17.15	even	8		1156.2.e.b.829.1		2
17.16	even	2	inner	68.2.b.a.33.2	yes	2
51.50	odd	2		612.2.b.a.577.1		2
68.47	odd	4		4624.2.a.n.1.1		2
68.55	odd	4		4624.2.a.n.1.2		2
68.67	odd	2		272.2.b.c.33.1		2
85.33	odd	4		1700.2.g.a.849.1		4
85.67	odd	4		1700.2.g.a.849.3		4
85.84	even	2		1700.2.c.a.101.1		2
119.118	odd	2		3332.2.b.a.2549.1		2
136.67	odd	2		1088.2.b.f.577.2		2
136.101	even	2		1088.2.b.e.577.1		2
204.203	even	2		2448.2.c.d.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.b.a.33.1	✓	2	1.1	even	1	trivial
68.2.b.a.33.2	yes	2	17.16	even	2	inner
272.2.b.c.33.1		2	68.67	odd	2
272.2.b.c.33.2		2	4.3	odd	2
612.2.b.a.577.1		2	51.50	odd	2
612.2.b.a.577.2		2	3.2	odd	2
1088.2.b.e.577.1		2	136.101	even	2
1088.2.b.e.577.2		2	8.5	even	2
1088.2.b.f.577.1		2	8.3	odd	2
1088.2.b.f.577.2		2	136.67	odd	2
1156.2.a.c.1.1		2	17.4	even	4
1156.2.a.c.1.2		2	17.13	even	4
1156.2.e.a.829.1		2	17.2	even	8
1156.2.e.a.905.1		2	17.8	even	8
1156.2.e.b.829.1		2	17.15	even	8
1156.2.e.b.905.1		2	17.9	even	8
1156.2.h.d.733.1		8	17.7	odd	16
1156.2.h.d.733.2		8	17.10	odd	16
1156.2.h.d.757.1		8	17.12	odd	16
1156.2.h.d.757.2		8	17.5	odd	16
1156.2.h.d.977.1		8	17.14	odd	16
1156.2.h.d.977.2		8	17.3	odd	16
1156.2.h.d.1001.1		8	17.6	odd	16
1156.2.h.d.1001.2		8	17.11	odd	16
1700.2.c.a.101.1		2	85.84	even	2
1700.2.c.a.101.2		2	5.4	even	2
1700.2.g.a.849.1		4	85.33	odd	4
1700.2.g.a.849.2		4	5.2	odd	4
1700.2.g.a.849.3		4	85.67	odd	4
1700.2.g.a.849.4		4	5.3	odd	4
2448.2.c.d.577.1		2	204.203	even	2
2448.2.c.d.577.2		2	12.11	even	2
3332.2.b.a.2549.1		2	119.118	odd	2
3332.2.b.a.2549.2		2	7.6	odd	2
4624.2.a.n.1.1		2	68.47	odd	4
4624.2.a.n.1.2		2	68.55	odd	4

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:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	68.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} -2.82843i q^{5} +4.24264i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{3} -2.82843i q^{5} +4.24264i q^{7} +1.00000 q^{9} +1.41421i q^{11} -4.00000 q^{13} -4.00000 q^{15} +(3.00000 + 2.82843i) q^{17} -4.00000 q^{19} +6.00000 q^{21} +1.41421i q^{23} -3.00000 q^{25} -5.65685i q^{27} -2.82843i q^{29} -4.24264i q^{31} +2.00000 q^{33} +12.0000 q^{35} +8.48528i q^{37} +5.65685i q^{39} -11.3137i q^{41} +8.00000 q^{43} -2.82843i q^{45} -12.0000 q^{47} -11.0000 q^{49} +(4.00000 - 4.24264i) q^{51} -6.00000 q^{53} +4.00000 q^{55} +5.65685i q^{57} +8.48528i q^{61} +4.24264i q^{63} +11.3137i q^{65} -4.00000 q^{67} +2.00000 q^{69} -7.07107i q^{71} +4.24264i q^{75} -6.00000 q^{77} -4.24264i q^{79} -5.00000 q^{81} +(8.00000 - 8.48528i) q^{85} -4.00000 q^{87} +12.0000 q^{89} -16.9706i q^{91} -6.00000 q^{93} +11.3137i q^{95} +1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^9	\( 2 q + 2 q^{9} - 8 q^{13} - 8 q^{15} + 6 q^{17} - 8 q^{19} + 12 q^{21} - 6 q^{25} + 4 q^{33} + 24 q^{35} + 16 q^{43} - 24 q^{47} - 22 q^{49} + 8 q^{51} - 12 q^{53} + 8 q^{55} - 8 q^{67} + 4 q^{69} - 12 q^{77} - 10 q^{81} + 16 q^{85} - 8 q^{87} + 24 q^{89} - 12 q^{93}+O(q^{100}) \) 2 * q + 2 * q^9 - 8 * q^13 - 8 * q^15 + 6 * q^17 - 8 * q^19 + 12 * q^21 - 6 * q^25 + 4 * q^33 + 24 * q^35 + 16 * q^43 - 24 * q^47 - 22 * q^49 + 8 * q^51 - 12 * q^53 + 8 * q^55 - 8 * q^67 + 4 * q^69 - 12 * q^77 - 10 * q^81 + 16 * q^85 - 8 * q^87 + 24 * q^89 - 12 * q^93

Introduction

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Database

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