LMFDB - Embedded newform 1368.2.f.a.1025.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(1025,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1368.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.f (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:	$10.9235349965$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	1368.1025
Dual form			1368.2.f.a.1025.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} +2.00000 q^{7} +4.24264i q^{11} +2.82843i q^{13} +7.07107i q^{17} +(-1.00000 - 4.24264i) q^{19} +1.41421i q^{23} +3.00000 q^{25} -10.0000 q^{29} +2.82843i q^{31} -2.82843i q^{35} +5.65685i q^{37} +10.0000 q^{41} +12.0000 q^{43} -1.41421i q^{47} -3.00000 q^{49} +10.0000 q^{53} +6.00000 q^{55} -12.0000 q^{59} +8.00000 q^{61} +4.00000 q^{65} -14.1421i q^{67} -8.00000 q^{71} +6.00000 q^{73} +8.48528i q^{77} +11.3137i q^{79} +4.24264i q^{83} +10.0000 q^{85} +6.00000 q^{89} +5.65685i q^{91} +(-6.00000 + 1.41421i) q^{95} +8.48528i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} - 2 q^{19} + 6 q^{25} - 20 q^{29} + 20 q^{41} + 24 q^{43} - 6 q^{49} + 20 q^{53} + 12 q^{55} - 24 q^{59} + 16 q^{61} + 8 q^{65} - 16 q^{71} + 12 q^{73} + 20 q^{85} + 12 q^{89} - 12 q^{95}+O(q^{100}) $ 2 * q + 4 * q^7 - 2 * q^19 + 6 * q^25 - 20 * q^29 + 20 * q^41 + 24 * q^43 - 6 * q^49 + 20 * q^53 + 12 * q^55 - 24 * q^59 + 16 * q^61 + 8 * q^65 - 16 * q^71 + 12 * q^73 + 20 * q^85 + 12 * q^89 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$		0			0
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.24264i			1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$			4.24264i			1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$		0			0
$13$			2.82843i			0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$			2.82843i			0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			7.07107i			1.71499i	0.514496	+	0.857493i	$0.327979\pi$
$17$			7.07107i			1.71499i	−0.514496	+	0.857493i	$0.672021\pi$
$18$		0			0
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$20$		0			0
$21$		0			0
$22$		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
$27$		0			0
$28$		0			0
$29$	−10.0000			−1.85695			−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000			−1.85695			−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$			2.82843i			0.508001i	0.967204	+	0.254000i	$0.0817464\pi$
$31$			2.82843i			0.508001i	−0.967204	+	0.254000i	$0.918254\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.82843i		−	0.478091i
$36$		0			0
$37$			5.65685i			0.929981i	0.885316	+	0.464991i	$0.153942\pi$
$37$			5.65685i			0.929981i	−0.885316	+	0.464991i	$0.846058\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$	12.0000			1.82998			0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000			1.82998			0.914991	+	0.403473i	$0.132197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	1.41421i		−	0.206284i	−0.994667	−	0.103142i	$-0.967110\pi$
$47$		−	1.41421i		−	0.206284i	0.994667	−	0.103142i	$-0.0328896\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.0000			1.37361			0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000			1.37361			0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000			0.496139
$66$		0			0
$67$		−	14.1421i		−	1.72774i	−0.503718	−	0.863868i	$-0.668035\pi$
$67$		−	14.1421i		−	1.72774i	0.503718	−	0.863868i	$-0.331965\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			8.48528i			0.966988i
$78$		0			0
$79$			11.3137i			1.27289i	0.771321	+	0.636446i	$0.219596\pi$
$79$			11.3137i			1.27289i	−0.771321	+	0.636446i	$0.780404\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			4.24264i			0.465690i	0.972514	+	0.232845i	$0.0748035\pi$
$83$			4.24264i			0.465690i	−0.972514	+	0.232845i	$0.925196\pi$
$84$		0			0
$85$	10.0000			1.08465
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$			5.65685i			0.592999i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.00000	+	1.41421i	−0.615587	+	0.145095i
$96$		0			0
$97$			8.48528i			0.861550i	0.902459	+	0.430775i	$0.141760\pi$
$97$			8.48528i			0.861550i	−0.902459	+	0.430775i	$0.858240\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			9.89949i			0.985037i	0.870302	+	0.492518i	$0.163924\pi$
$101$			9.89949i			0.985037i	−0.870302	+	0.492518i	$0.836076\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		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
$111$		0			0
$112$		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
$115$	2.00000			0.186501
$116$		0			0
$117$		0			0
$118$		0			0
$119$			14.1421i			1.29641i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$		−	19.7990i		−	1.75688i	−0.477856	−	0.878438i	$-0.658586\pi$
$127$		−	19.7990i		−	1.75688i	0.477856	−	0.878438i	$-0.341414\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	9.89949i		−	0.864923i	−0.901652	−	0.432461i	$-0.857645\pi$
$131$		−	9.89949i		−	0.864923i	0.901652	−	0.432461i	$-0.142355\pi$
$132$		0			0
$133$	−2.00000	−	8.48528i	−0.173422	−	0.735767i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.89949i			0.845771i	0.906183	+	0.422885i	$0.138983\pi$
$137$			9.89949i			0.845771i	−0.906183	+	0.422885i	$0.861017\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000			−1.00349
$144$		0			0
$145$			14.1421i			1.17444i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			4.24264i			0.347571i	0.984784	+	0.173785i	$0.0555999\pi$
$149$			4.24264i			0.347571i	−0.984784	+	0.173785i	$0.944400\pi$
$150$		0			0
$151$		−	11.3137i		−	0.920697i	−0.887738	−	0.460348i	$-0.847725\pi$
$151$		−	11.3137i		−	0.920697i	0.887738	−	0.460348i	$-0.152275\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	−24.0000			−1.91541			−0.957704	−	0.287754i	$-0.907091\pi$
$157$	−24.0000			−1.91541			−0.957704	+	0.287754i	$0.907091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.82843i			0.222911i
$162$		0			0
$163$	14.0000			1.09656			0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000			1.09656			0.548282	+	0.836293i	$0.315282\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.0000			1.23812			0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000			1.23812			0.619059	+	0.785345i	$0.287514\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.00000			−0.152057			−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000			−0.152057			−0.0760286	+	0.997106i	$0.524224\pi$
$174$		0			0
$175$	6.00000			0.453557
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.00000			0.597948			0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000			0.597948			0.298974	+	0.954261i	$0.403356\pi$
$180$		0			0
$181$		−	14.1421i		−	1.05118i	−0.850739	−	0.525588i	$-0.823845\pi$
$181$		−	14.1421i		−	1.05118i	0.850739	−	0.525588i	$-0.176155\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.00000			0.588172
$186$		0			0
$187$	−30.0000			−2.19382
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	15.5563i		−	1.12562i	−0.826587	−	0.562809i	$-0.809721\pi$
$191$		−	15.5563i		−	1.12562i	0.826587	−	0.562809i	$-0.190279\pi$
$192$		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
$195$		0			0
$196$		0			0
$197$		−	21.2132i		−	1.51138i	−0.654931	−	0.755689i	$-0.727302\pi$
$197$		−	21.2132i		−	1.51138i	0.654931	−	0.755689i	$-0.272698\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.0000			−1.40372
$204$		0			0
$205$		−	14.1421i		−	0.987730i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	18.0000	−	4.24264i	1.24509	−	0.293470i
$210$		0			0
$211$			14.1421i			0.973585i	0.873518	+	0.486792i	$0.161833\pi$
$211$			14.1421i			0.973585i	−0.873518	+	0.486792i	$0.838167\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	16.9706i		−	1.15738i
$216$		0			0
$217$			5.65685i			0.384012i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−20.0000			−1.34535
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.00000			−0.530979			−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000			−0.530979			−0.265489	+	0.964114i	$0.585534\pi$
$228$		0			0
$229$	4.00000			0.264327			0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000			0.264327			0.132164	+	0.991228i	$0.457808\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.7279i		−	0.833834i	−0.908945	−	0.416917i	$-0.863111\pi$
$233$		−	12.7279i		−	0.833834i	0.908945	−	0.416917i	$-0.136889\pi$
$234$		0			0
$235$	−2.00000			−0.130466
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.24264i		−	0.274434i	−0.990541	−	0.137217i	$-0.956184\pi$
$239$		−	4.24264i		−	0.274434i	0.990541	−	0.137217i	$-0.0438157\pi$
$240$		0			0
$241$		−	8.48528i		−	0.546585i	−0.961931	−	0.273293i	$-0.911887\pi$
$241$		−	8.48528i		−	0.546585i	0.961931	−	0.273293i	$-0.0881127\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			4.24264i			0.271052i
$246$		0			0
$247$	12.0000	−	2.82843i	0.763542	−	0.179969i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.5563i		−	0.981908i	−0.871185	−	0.490954i	$-0.836648\pi$
$251$		−	15.5563i		−	0.981908i	0.871185	−	0.490954i	$-0.163352\pi$
$252$		0			0
$253$	−6.00000			−0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$			11.3137i			0.703000i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7.07107i			0.436021i	0.975946	+	0.218010i	$0.0699567\pi$
$263$			7.07107i			0.436021i	−0.975946	+	0.218010i	$0.930043\pi$
$264$		0			0
$265$		−	14.1421i		−	0.868744i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			12.7279i			0.767523i
$276$		0			0
$277$	−20.0000			−1.20168			−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000			−1.20168			−0.600842	+	0.799368i	$0.705168\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$	−20.0000			−1.18888			−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000			−1.18888			−0.594438	+	0.804141i	$0.702626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	20.0000			1.18056
$288$		0			0
$289$	−33.0000			−1.94118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		0			0
$295$			16.9706i			0.988064i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.00000			−0.231326
$300$		0			0
$301$	24.0000			1.38334
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	11.3137i		−	0.647821i
$306$		0			0
$307$		−	33.9411i		−	1.93712i	−0.248776	−	0.968561i	$-0.580028\pi$
$307$		−	33.9411i		−	1.93712i	0.248776	−	0.968561i	$-0.419972\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			15.5563i			0.882120i	0.897478	+	0.441060i	$0.145397\pi$
$311$			15.5563i			0.882120i	−0.897478	+	0.441060i	$0.854603\pi$
$312$		0			0
$313$	−4.00000			−0.226093			−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000			−0.226093			−0.113047	+	0.993590i	$0.536061\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.0000			−0.786318			−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000			−0.786318			−0.393159	+	0.919470i	$0.628618\pi$
$318$		0			0
$319$		−	42.4264i		−	2.37542i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	30.0000	−	7.07107i	1.66924	−	0.393445i
$324$		0			0
$325$			8.48528i			0.470679i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	2.82843i		−	0.155936i
$330$		0			0
$331$			5.65685i			0.310929i	0.987841	+	0.155464i	$0.0496874\pi$
$331$			5.65685i			0.310929i	−0.987841	+	0.155464i	$0.950313\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−20.0000			−1.09272
$336$		0			0
$337$			25.4558i			1.38667i	0.720616	+	0.693334i	$0.243859\pi$
$337$			25.4558i			1.38667i	−0.720616	+	0.693334i	$0.756141\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.07107i		−	0.379595i	−0.981823	−	0.189797i	$-0.939217\pi$
$347$		−	7.07107i		−	0.379595i	0.981823	−	0.189797i	$-0.0607831\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.3848i			0.978523i	0.872137	+	0.489261i	$0.162734\pi$
$353$			18.3848i			0.978523i	−0.872137	+	0.489261i	$0.837266\pi$
$354$		0			0
$355$			11.3137i			0.600469i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	32.5269i		−	1.71670i	−0.513061	−	0.858352i	$-0.671488\pi$
$359$		−	32.5269i		−	1.71670i	0.513061	−	0.858352i	$-0.328512\pi$
$360$		0			0
$361$	−17.0000	+	8.48528i	−0.894737	+	0.446594i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	8.48528i		−	0.444140i
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	20.0000			1.03835
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	28.2843i		−	1.45671i
$378$		0			0
$379$		−	11.3137i		−	0.581146i	−0.956853	−	0.290573i	$-0.906154\pi$
$379$		−	11.3137i		−	0.581146i	0.956853	−	0.290573i	$-0.0938459\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	12.0000			0.611577
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	24.0416i		−	1.21896i	−0.792802	−	0.609480i	$-0.791378\pi$
$389$		−	24.0416i		−	1.21896i	0.792802	−	0.609480i	$-0.208622\pi$
$390$		0			0
$391$	−10.0000			−0.505722
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	−14.0000			−0.702640			−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000			−0.702640			−0.351320	+	0.936255i	$0.614267\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.00000			0.299626			0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000			0.299626			0.149813	+	0.988714i	$0.452133\pi$
$402$		0			0
$403$	−8.00000			−0.398508
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$		−	8.48528i		−	0.419570i	−0.977748	−	0.209785i	$-0.932724\pi$
$409$		−	8.48528i		−	0.419570i	0.977748	−	0.209785i	$-0.0672764\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−24.0000			−1.18096
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$		0			0
$418$		0			0
$419$			15.5563i			0.759977i	0.924991	+	0.379989i	$0.124072\pi$
$419$			15.5563i			0.759977i	−0.924991	+	0.379989i	$0.875928\pi$
$420$		0			0
$421$		−	8.48528i		−	0.413547i	−0.978389	−	0.206774i	$-0.933704\pi$
$421$		−	8.48528i		−	0.413547i	0.978389	−	0.206774i	$-0.0662964\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			21.2132i			1.02899i
$426$		0			0
$427$	16.0000			0.774294
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.00000	−	1.41421i	0.287019	−	0.0676510i
$438$		0			0
$439$			16.9706i			0.809961i	0.914325	+	0.404980i	$0.132722\pi$
$439$			16.9706i			0.809961i	−0.914325	+	0.404980i	$0.867278\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			7.07107i			0.335957i	0.985791	+	0.167978i	$0.0537239\pi$
$443$			7.07107i			0.335957i	−0.985791	+	0.167978i	$0.946276\pi$
$444$		0			0
$445$		−	8.48528i		−	0.402241i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$			42.4264i			1.99778i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.00000			0.375046
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	18.3848i		−	0.856264i	−0.903716	−	0.428132i	$-0.859172\pi$
$461$		−	18.3848i		−	0.856264i	0.903716	−	0.428132i	$-0.140828\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.41421i		−	0.0654420i	−0.999465	−	0.0327210i	$-0.989583\pi$
$467$		−	1.41421i		−	0.0654420i	0.999465	−	0.0327210i	$-0.0104173\pi$
$468$		0			0
$469$		−	28.2843i		−	1.30605i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			50.9117i			2.34092i
$474$		0			0
$475$	−3.00000	−	12.7279i	−0.137649	−	0.583997i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			15.5563i			0.710788i	0.934717	+	0.355394i	$0.115653\pi$
$479$			15.5563i			0.710788i	−0.934717	+	0.355394i	$0.884347\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000			0.544892
$486$		0			0
$487$			31.1127i			1.40985i	0.709281	+	0.704925i	$0.249020\pi$
$487$			31.1127i			1.40985i	−0.709281	+	0.704925i	$0.750980\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			12.7279i			0.574403i	0.957870	+	0.287202i	$0.0927249\pi$
$491$			12.7279i			0.574403i	−0.957870	+	0.287202i	$0.907275\pi$
$492$		0			0
$493$		−	70.7107i		−	3.18465i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.0000			−0.717698
$498$		0			0
$499$	−30.0000			−1.34298			−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000			−1.34298			−0.671492	+	0.741012i	$0.734346\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15.5563i		−	0.693623i	−0.937935	−	0.346812i	$-0.887264\pi$
$503$		−	15.5563i		−	0.693623i	0.937935	−	0.346812i	$-0.112736\pi$
$504$		0			0
$505$	14.0000			0.622992
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.00000			0.263880
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$		−	14.1421i		−	0.618392i	−0.950998	−	0.309196i	$-0.899940\pi$
$523$		−	14.1421i		−	0.618392i	0.950998	−	0.309196i	$-0.100060\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.0000			−0.871214
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$			28.2843i			1.22513i
$534$		0			0
$535$			11.3137i			0.489134i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	12.7279i		−	0.548230i
$540$		0			0
$541$	34.0000			1.46177			0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000			1.46177			0.730887	+	0.682498i	$0.239107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$			16.9706i			0.725609i	0.931865	+	0.362804i	$0.118181\pi$
$547$			16.9706i			0.725609i	−0.931865	+	0.362804i	$0.881819\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.0000	+	42.4264i	0.426014	+	1.80743i
$552$		0			0
$553$			22.6274i			0.962216i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	18.3848i		−	0.778988i	−0.921029	−	0.389494i	$-0.872650\pi$
$557$		−	18.3848i		−	0.778988i	0.921029	−	0.389494i	$-0.127350\pi$
$558$		0			0
$559$			33.9411i			1.43556i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	24.0000			1.01148			0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000			1.01148			0.505740	+	0.862686i	$0.331220\pi$
$564$		0			0
$565$		−	8.48528i		−	0.356978i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	44.0000			1.84134			0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000			1.84134			0.920671	+	0.390339i	$0.127642\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			4.24264i			0.176930i
$576$		0			0
$577$	44.0000			1.83174			0.915872	−	0.401470i	$-0.131501\pi$
$577$	44.0000			1.83174			0.915872	+	0.401470i	$0.131501\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			8.48528i			0.352029i
$582$		0			0
$583$			42.4264i			1.75712i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.3848i		−	0.758821i	−0.925228	−	0.379410i	$-0.876127\pi$
$587$		−	18.3848i		−	0.758821i	0.925228	−	0.379410i	$-0.123873\pi$
$588$		0			0
$589$	12.0000	−	2.82843i	0.494451	−	0.116543i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	43.8406i		−	1.80032i	−0.435561	−	0.900159i	$-0.643450\pi$
$593$		−	43.8406i		−	1.80032i	0.435561	−	0.900159i	$-0.356550\pi$
$594$		0			0
$595$	20.0000			0.819920
$596$		0			0
$597$		0			0
$598$		0			0
$599$	44.0000			1.79779			0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000			1.79779			0.898896	+	0.438163i	$0.144371\pi$
$600$		0			0
$601$			33.9411i			1.38449i	0.721664	+	0.692244i	$0.243378\pi$
$601$			33.9411i			1.38449i	−0.721664	+	0.692244i	$0.756622\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			9.89949i			0.402472i
$606$		0			0
$607$		−	31.1127i		−	1.26283i	−0.775447	−	0.631413i	$-0.782475\pi$
$607$		−	31.1127i		−	1.26283i	0.775447	−	0.631413i	$-0.217525\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000			0.161823
$612$		0			0
$613$	24.0000			0.969351			0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000			0.969351			0.484675	+	0.874694i	$0.338938\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.7279i		−	0.512407i	−0.966623	−	0.256203i	$-0.917528\pi$
$617$		−	12.7279i		−	0.512407i	0.966623	−	0.256203i	$-0.0824717\pi$
$618$		0			0
$619$	34.0000			1.36658			0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000			1.36658			0.683288	+	0.730149i	$0.260549\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0000			0.480770
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−40.0000			−1.59490
$630$		0			0
$631$	34.0000			1.35352			0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000			1.35352			0.676759	+	0.736204i	$0.263384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−28.0000			−1.11115
$636$		0			0
$637$		−	8.48528i		−	0.336199i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−46.0000			−1.81689			−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000			−1.81689			−0.908445	+	0.418004i	$0.862730\pi$
$642$		0			0
$643$	−12.0000			−0.473234			−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000			−0.473234			−0.236617	+	0.971603i	$0.576039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	18.3848i		−	0.722780i	−0.932415	−	0.361390i	$-0.882302\pi$
$647$		−	18.3848i		−	0.722780i	0.932415	−	0.361390i	$-0.117698\pi$
$648$		0			0
$649$		−	50.9117i		−	1.99846i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	4.24264i		−	0.166027i	−0.996548	−	0.0830137i	$-0.973545\pi$
$653$		−	4.24264i		−	0.166027i	0.996548	−	0.0830137i	$-0.0264545\pi$
$654$		0			0
$655$	−14.0000			−0.547025
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$		−	22.6274i		−	0.880105i	−0.897972	−	0.440052i	$-0.854960\pi$
$661$		−	22.6274i		−	0.880105i	0.897972	−	0.440052i	$-0.145040\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.0000	+	2.82843i	−0.465340	+	0.109682i
$666$		0			0
$667$		−	14.1421i		−	0.547586i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			33.9411i			1.31028i
$672$		0			0
$673$			22.6274i			0.872223i	0.899893	+	0.436111i	$0.143645\pi$
$673$			22.6274i			0.872223i	−0.899893	+	0.436111i	$0.856355\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$			16.9706i			0.651270i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	44.0000			1.68361			0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000			1.68361			0.841807	+	0.539779i	$0.181492\pi$
$684$		0			0
$685$	14.0000			0.534913
$686$		0			0
$687$		0			0
$688$		0			0
$689$			28.2843i			1.07754i
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	2.82843i		−	0.107288i
$696$		0			0
$697$			70.7107i			2.67836i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			12.7279i			0.480727i	0.970683	+	0.240363i	$0.0772666\pi$
$701$			12.7279i			0.480727i	−0.970683	+	0.240363i	$0.922733\pi$
$702$		0			0
$703$	24.0000	−	5.65685i	0.905177	−	0.213352i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			19.7990i			0.744618i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000			−0.149801
$714$		0			0
$715$			16.9706i			0.634663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			41.0122i			1.52950i	0.644329	+	0.764748i	$0.277137\pi$
$719$			41.0122i			1.52950i	−0.644329	+	0.764748i	$0.722863\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−30.0000			−1.11417
$726$		0			0
$727$	−32.0000			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			84.8528i			3.13839i
$732$		0			0
$733$	−14.0000			−0.517102			−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000			−0.517102			−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	60.0000			2.21013
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.0000			−0.440237			−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000			−0.440237			−0.220119	+	0.975473i	$0.570644\pi$
$744$		0			0
$745$	6.00000			0.219823
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16.0000			−0.584627
$750$		0			0
$751$		−	45.2548i		−	1.65137i	−0.564130	−	0.825686i	$-0.690788\pi$
$751$		−	45.2548i		−	1.65137i	0.564130	−	0.825686i	$-0.309212\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	−20.0000			−0.726912			−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000			−0.726912			−0.363456	+	0.931611i	$0.618403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1.41421i			0.0512652i	0.999671	+	0.0256326i	$0.00816000\pi$
$761$			1.41421i			0.0512652i	−0.999671	+	0.0256326i	$0.991840\pi$
$762$		0			0
$763$			16.9706i			0.614376i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	33.9411i		−	1.22554i
$768$		0			0
$769$	−20.0000			−0.721218			−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000			−0.721218			−0.360609	+	0.932717i	$0.617431\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$			8.48528i			0.304800i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10.0000	−	42.4264i	−0.358287	−	1.52008i
$780$		0			0
$781$		−	33.9411i		−	1.21451i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			33.9411i			1.21141i
$786$		0			0
$787$			8.48528i			0.302468i	0.988498	+	0.151234i	$0.0483246\pi$
$787$			8.48528i			0.302468i	−0.988498	+	0.151234i	$0.951675\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$			22.6274i			0.803523i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	42.0000			1.48772			0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000			1.48772			0.743858	+	0.668338i	$0.232994\pi$
$798$		0			0
$799$	10.0000			0.353775
$800$		0			0
$801$		0			0
$802$		0			0
$803$			25.4558i			0.898317i
$804$		0			0
$805$	4.00000			0.140981
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	15.5563i		−	0.546932i	−0.961882	−	0.273466i	$-0.911830\pi$
$809$		−	15.5563i		−	0.546932i	0.961882	−	0.273466i	$-0.0881701\pi$
$810$		0			0
$811$		−	16.9706i		−	0.595917i	−0.954579	−	0.297959i	$-0.903694\pi$
$811$		−	16.9706i		−	0.595917i	0.954579	−	0.297959i	$-0.0963057\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	19.7990i		−	0.693528i
$816$		0			0
$817$	−12.0000	−	50.9117i	−0.419827	−	1.78117i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			46.6690i			1.62876i	0.580331	+	0.814380i	$0.302923\pi$
$821$			46.6690i			1.62876i	−0.580331	+	0.814380i	$0.697077\pi$
$822$		0			0
$823$	26.0000			0.906303			0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000			0.906303			0.453152	+	0.891434i	$0.350300\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$			48.0833i			1.67000i	0.550249	+	0.835000i	$0.314533\pi$
$829$			48.0833i			1.67000i	−0.550249	+	0.835000i	$0.685467\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	21.2132i		−	0.734994i
$834$		0			0
$835$		−	22.6274i		−	0.783054i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	32.0000			1.10476			0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000			1.10476			0.552381	+	0.833592i	$0.313719\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	7.07107i		−	0.243252i
$846$		0			0
$847$	−14.0000			−0.481046
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00000			−0.274236
$852$		0			0
$853$	2.00000			0.0684787			0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000			0.0684787			0.0342393	+	0.999414i	$0.489099\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.0000			0.614868			0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000			0.614868			0.307434	+	0.951569i	$0.400530\pi$
$858$		0			0
$859$	6.00000			0.204717			0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000			0.204717			0.102359	+	0.994748i	$0.467361\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20.0000			−0.680808			−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000			−0.680808			−0.340404	+	0.940279i	$0.610564\pi$
$864$		0			0
$865$			2.82843i			0.0961694i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−48.0000			−1.62829
$870$		0			0
$871$	40.0000			1.35535
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	22.6274i		−	0.764946i
$876$		0			0
$877$		−	28.2843i		−	0.955092i	−0.878607	−	0.477546i	$-0.841526\pi$
$877$		−	28.2843i		−	0.955092i	0.878607	−	0.477546i	$-0.158474\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			49.4975i			1.66761i	0.552057	+	0.833806i	$0.313843\pi$
$881$			49.4975i			1.66761i	−0.552057	+	0.833806i	$0.686157\pi$
$882$		0			0
$883$	2.00000			0.0673054			0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000			0.0673054			0.0336527	+	0.999434i	$0.489286\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	36.0000			1.20876			0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000			1.20876			0.604381	+	0.796696i	$0.293421\pi$
$888$		0			0
$889$		−	39.5980i		−	1.32807i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.00000	+	1.41421i	−0.200782	+	0.0473249i
$894$		0			0
$895$		−	11.3137i		−	0.378176i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	28.2843i		−	0.943333i
$900$		0			0
$901$			70.7107i			2.35571i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.0000			−0.664822
$906$		0			0
$907$			42.4264i			1.40875i	0.709830	+	0.704373i	$0.248772\pi$
$907$			42.4264i			1.40875i	−0.709830	+	0.704373i	$0.751228\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	20.0000			0.662630			0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000			0.662630			0.331315	+	0.943520i	$0.392508\pi$
$912$		0			0
$913$	−18.0000			−0.595713
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	19.7990i		−	0.653820i
$918$		0			0
$919$	−26.0000			−0.857661			−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000			−0.857661			−0.428830	+	0.903385i	$0.641074\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	22.6274i		−	0.744791i
$924$		0			0
$925$			16.9706i			0.557989i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	18.3848i		−	0.603185i	−0.953437	−	0.301592i	$-0.902482\pi$
$929$		−	18.3848i		−	0.603185i	0.953437	−	0.301592i	$-0.0975182\pi$
$930$		0			0
$931$	3.00000	+	12.7279i	0.0983210	+	0.417141i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			42.4264i			1.38749i
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.0000			−0.325991			−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000			−0.325991			−0.162995	+	0.986627i	$0.552116\pi$
$942$		0			0
$943$			14.1421i			0.460531i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.41421i			0.0459558i	0.999736	+	0.0229779i	$0.00731473\pi$
$947$			1.41421i			0.0459558i	−0.999736	+	0.0229779i	$0.992685\pi$
$948$		0			0
$949$			16.9706i			0.550888i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	46.0000			1.49009			0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000			1.49009			0.745043	+	0.667016i	$0.232429\pi$
$954$		0			0
$955$	−22.0000			−0.711903
$956$		0			0
$957$		0			0
$958$		0			0
$959$			19.7990i			0.639343i
$960$		0			0
$961$	23.0000			0.741935
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.0000			0.772587
$966$		0			0
$967$	50.0000			1.60789			0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000			1.60789			0.803946	+	0.594703i	$0.202730\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36.0000			−1.15529			−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		0			0
$973$	4.00000			0.128234
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.00000			0.191957			0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000			0.191957			0.0959785	+	0.995383i	$0.469402\pi$
$978$		0			0
$979$			25.4558i			0.813572i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.00000			−0.127580			−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000			−0.127580			−0.0637901	+	0.997963i	$0.520319\pi$
$984$		0			0
$985$	−30.0000			−0.955879
$986$		0			0
$987$		0			0
$988$		0			0
$989$			16.9706i			0.539633i
$990$		0			0
$991$		−	31.1127i		−	0.988327i	−0.869369	−	0.494164i	$-0.835474\pi$
$991$		−	31.1127i		−	0.988327i	0.869369	−	0.494164i	$-0.164526\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			33.9411i			1.07601i
$996$		0			0
$997$	16.0000			0.506725			0.253363	−	0.967371i	$-0.418463\pi$
$997$	16.0000			0.506725			0.253363	+	0.967371i	$0.418463\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.f.a.1025.1	✓	2
3.2	odd	2		1368.2.f.b.1025.2	yes	2
4.3	odd	2		2736.2.f.c.1025.1		2
12.11	even	2		2736.2.f.d.1025.2		2
19.18	odd	2		1368.2.f.b.1025.1	yes	2
57.56	even	2	inner	1368.2.f.a.1025.2	yes	2
76.75	even	2		2736.2.f.d.1025.1		2
228.227	odd	2		2736.2.f.c.1025.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.f.a.1025.1	✓	2	1.1	even	1	trivial
1368.2.f.a.1025.2	yes	2	57.56	even	2	inner
1368.2.f.b.1025.1	yes	2	19.18	odd	2
1368.2.f.b.1025.2	yes	2	3.2	odd	2
2736.2.f.c.1025.1		2	4.3	odd	2
2736.2.f.c.1025.2		2	228.227	odd	2
2736.2.f.d.1025.1		2	76.75	even	2
2736.2.f.d.1025.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} +2.00000 q^{7} +4.24264i q^{11} +2.82843i q^{13} +7.07107i q^{17} +(-1.00000 - 4.24264i) q^{19} +1.41421i q^{23} +3.00000 q^{25} -10.0000 q^{29} +2.82843i q^{31} -2.82843i q^{35} +5.65685i q^{37} +10.0000 q^{41} +12.0000 q^{43} -1.41421i q^{47} -3.00000 q^{49} +10.0000 q^{53} +6.00000 q^{55} -12.0000 q^{59} +8.00000 q^{61} +4.00000 q^{65} -14.1421i q^{67} -8.00000 q^{71} +6.00000 q^{73} +8.48528i q^{77} +11.3137i q^{79} +4.24264i q^{83} +10.0000 q^{85} +6.00000 q^{89} +5.65685i q^{91} +(-6.00000 + 1.41421i) q^{95} +8.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} - 2 q^{19} + 6 q^{25} - 20 q^{29} + 20 q^{41} + 24 q^{43} - 6 q^{49} + 20 q^{53} + 12 q^{55} - 24 q^{59} + 16 q^{61} + 8 q^{65} - 16 q^{71} + 12 q^{73} + 20 q^{85} + 12 q^{89} - 12 q^{95}+O(q^{100}) \) 2 * q + 4 * q^7 - 2 * q^19 + 6 * q^25 - 20 * q^29 + 20 * q^41 + 24 * q^43 - 6 * q^49 + 20 * q^53 + 12 * q^55 - 24 * q^59 + 16 * q^61 + 8 * q^65 - 16 * q^71 + 12 * q^73 + 20 * q^85 + 12 * q^89 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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