LMFDB - Embedded newform 3040.2.j.c.2431.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(2431,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("3040.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.j (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:	$24.2745222145$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3040.2431
Dual form			3040.2.j.c.2431.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{3} +1.00000 q^{5} -2.00000i q^{7} +5.00000 q^{9} +O(q^{10})$	$q+2.82843 q^{3} +1.00000 q^{5} -2.00000i q^{7} +5.00000 q^{9} -2.00000i q^{11} -2.82843i q^{13} +2.82843 q^{15} +2.00000 q^{17} +(-4.24264 - 1.00000i) q^{19} -5.65685i q^{21} -2.00000i q^{23} +1.00000 q^{25} +5.65685 q^{27} -2.82843i q^{29} -5.65685i q^{33} -2.00000i q^{35} -8.48528i q^{37} -8.00000i q^{39} +5.65685i q^{41} +6.00000i q^{43} +5.00000 q^{45} -2.00000i q^{47} +3.00000 q^{49} +5.65685 q^{51} +2.82843i q^{53} -2.00000i q^{55} +(-12.0000 - 2.82843i) q^{57} -2.82843 q^{59} -2.00000 q^{61} -10.0000i q^{63} -2.82843i q^{65} +14.1421 q^{67} -5.65685i q^{69} +5.65685 q^{71} +2.00000 q^{73} +2.82843 q^{75} -4.00000 q^{77} -11.3137 q^{79} +1.00000 q^{81} +14.0000i q^{83} +2.00000 q^{85} -8.00000i q^{87} +5.65685i q^{89} -5.65685 q^{91} +(-4.24264 - 1.00000i) q^{95} -10.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 20 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 20 * q^9	$ 4 q + 4 q^{5} + 20 q^{9} + 8 q^{17} + 4 q^{25} + 20 q^{45} + 12 q^{49} - 48 q^{57} - 8 q^{61} + 8 q^{73} - 16 q^{77} + 4 q^{81} + 8 q^{85}+O(q^{100}) $ 4 * q + 4 * q^5 + 20 * q^9 + 8 * q^17 + 4 * q^25 + 20 * q^45 + 12 * q^49 - 48 * q^57 - 8 * q^61 + 8 * q^73 - 16 * q^77 + 4 * q^81 + 8 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$.

$n$	$191$	$1217$	$1921$	$2661$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	2.82843			1.63299			0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843			1.63299			0.816497	+	0.577350i	$0.195913\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	5.00000			1.66667
$10$		0			0
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$		−	2.82843i		−	0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$		−	2.82843i		−	0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$		0			0
$15$	2.82843			0.730297
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	−4.24264	−	1.00000i	−0.973329	−	0.229416i
$19$	−4.24264	−	1.00000i	−0.973329	−	0.229416i
$20$		0			0
$21$		−	5.65685i		−	1.23443i
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	5.65685			1.08866
$28$		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
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		−	5.65685i		−	0.984732i
$34$		0			0
$35$		−	2.00000i		−	0.338062i
$36$		0			0
$37$		−	8.48528i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$		−	8.48528i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$		0			0
$39$		−	8.00000i		−	1.28103i
$40$		0			0
$41$			5.65685i			0.883452i	0.897150	+	0.441726i	$0.145634\pi$
$41$			5.65685i			0.883452i	−0.897150	+	0.441726i	$0.854366\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$		0			0
$45$	5.00000			0.745356
$46$		0			0
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	5.65685			0.792118
$52$		0			0
$53$			2.82843i			0.388514i	0.980951	+	0.194257i	$0.0622296\pi$
$53$			2.82843i			0.388514i	−0.980951	+	0.194257i	$0.937770\pi$
$54$		0			0
$55$		−	2.00000i		−	0.269680i
$56$		0			0
$57$	−12.0000	−	2.82843i	−1.58944	−	0.374634i
$58$		0			0
$59$	−2.82843			−0.368230			−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843			−0.368230			−0.184115	+	0.982905i	$0.558942\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		−	10.0000i		−	1.25988i
$64$		0			0
$65$		−	2.82843i		−	0.350823i
$66$		0			0
$67$	14.1421			1.72774			0.863868	−	0.503718i	$-0.168035\pi$
$67$	14.1421			1.72774			0.863868	+	0.503718i	$0.168035\pi$
$68$		0			0
$69$		−	5.65685i		−	0.681005i
$70$		0			0
$71$	5.65685			0.671345			0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685			0.671345			0.335673	+	0.941979i	$0.391036\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	2.82843			0.326599
$76$		0			0
$77$	−4.00000			−0.455842
$78$		0			0
$79$	−11.3137			−1.27289			−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137			−1.27289			−0.636446	+	0.771321i	$0.719596\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			14.0000i			1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$			14.0000i			1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$		0			0
$85$	2.00000			0.216930
$86$		0			0
$87$		−	8.00000i		−	0.857690i
$88$		0			0
$89$			5.65685i			0.599625i	0.953998	+	0.299813i	$0.0969242\pi$
$89$			5.65685i			0.599625i	−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$	−5.65685			−0.592999
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.24264	−	1.00000i	−0.435286	−	0.102598i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		−	10.0000i		−	1.00504i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	−5.65685			−0.557386			−0.278693	−	0.960380i	$-0.589901\pi$
$103$	−5.65685			−0.557386			−0.278693	+	0.960380i	$0.589901\pi$
$104$		0			0
$105$		−	5.65685i		−	0.552052i
$106$		0			0
$107$	−2.82843			−0.273434			−0.136717	−	0.990610i	$-0.543655\pi$
$107$	−2.82843			−0.273434			−0.136717	+	0.990610i	$0.543655\pi$
$108$		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
$111$		−	24.0000i		−	2.27798i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	2.00000i		−	0.186501i
$116$		0			0
$117$		−	14.1421i		−	1.30744i
$118$		0			0
$119$		−	4.00000i		−	0.366679i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$			16.0000i			1.44267i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	11.3137			1.00393			0.501965	−	0.864888i	$-0.332611\pi$
$127$	11.3137			1.00393			0.501965	+	0.864888i	$0.332611\pi$
$128$		0			0
$129$			16.9706i			1.49417i
$130$		0			0
$131$			6.00000i			0.524222i	0.965038	+	0.262111i	$0.0844187\pi$
$131$			6.00000i			0.524222i	−0.965038	+	0.262111i	$0.915581\pi$
$132$		0			0
$133$	−2.00000	+	8.48528i	−0.173422	+	0.735767i
$134$		0			0
$135$	5.65685			0.486864
$136$		0			0
$137$	10.0000			0.854358			0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000			0.854358			0.427179	+	0.904167i	$0.359507\pi$
$138$		0			0
$139$		−	10.0000i		−	0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$		−	10.0000i		−	0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$		0			0
$141$		−	5.65685i		−	0.476393i
$142$		0			0
$143$	−5.65685			−0.473050
$144$		0			0
$145$		−	2.82843i		−	0.234888i
$146$		0			0
$147$	8.48528			0.699854
$148$		0			0
$149$	22.0000			1.80231			0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000			1.80231			0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$	−5.65685			−0.460348			−0.230174	−	0.973149i	$-0.573930\pi$
$151$	−5.65685			−0.460348			−0.230174	+	0.973149i	$0.573930\pi$
$152$		0			0
$153$	10.0000			0.808452
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		0			0
$159$			8.00000i			0.634441i
$160$		0			0
$161$	−4.00000			−0.315244
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$		0			0
$165$		−	5.65685i		−	0.440386i
$166$		0			0
$167$	−5.65685			−0.437741			−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685			−0.437741			−0.218870	+	0.975754i	$0.570237\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−21.2132	−	5.00000i	−1.62221	−	0.382360i
$172$		0			0
$173$			19.7990i			1.50529i	0.658427	+	0.752645i	$0.271222\pi$
$173$			19.7990i			1.50529i	−0.658427	+	0.752645i	$0.728778\pi$
$174$		0			0
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$	−8.00000			−0.601317
$178$		0			0
$179$	−19.7990			−1.47985			−0.739923	−	0.672692i	$-0.765138\pi$
$179$	−19.7990			−1.47985			−0.739923	+	0.672692i	$0.765138\pi$
$180$		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
$183$	−5.65685			−0.418167
$184$		0			0
$185$		−	8.48528i		−	0.623850i
$186$		0			0
$187$		−	4.00000i		−	0.292509i
$188$		0			0
$189$		−	11.3137i		−	0.822951i
$190$		0			0
$191$		−	18.0000i		−	1.30243i	−0.758891	−	0.651217i	$-0.774259\pi$
$191$		−	18.0000i		−	1.30243i	0.758891	−	0.651217i	$-0.225741\pi$
$192$		0			0
$193$			22.6274i			1.62876i	0.580334	+	0.814379i	$0.302922\pi$
$193$			22.6274i			1.62876i	−0.580334	+	0.814379i	$0.697078\pi$
$194$		0			0
$195$		−	8.00000i		−	0.572892i
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$			14.0000i			0.992434i	0.868199	+	0.496217i	$0.165278\pi$
$199$			14.0000i			0.992434i	−0.868199	+	0.496217i	$0.834722\pi$
$200$		0			0
$201$	40.0000			2.82138
$202$		0			0
$203$	−5.65685			−0.397033
$204$		0			0
$205$			5.65685i			0.395092i
$206$		0			0
$207$		−	10.0000i		−	0.695048i
$208$		0			0
$209$	−2.00000	+	8.48528i	−0.138343	+	0.586939i
$210$		0			0
$211$	−19.7990			−1.36302			−0.681509	−	0.731809i	$-0.738676\pi$
$211$	−19.7990			−1.36302			−0.681509	+	0.731809i	$0.738676\pi$
$212$		0			0
$213$	16.0000			1.09630
$214$		0			0
$215$			6.00000i			0.409197i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	5.65685			0.382255
$220$		0			0
$221$		−	5.65685i		−	0.380521i
$222$		0			0
$223$	−11.3137			−0.757622			−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137			−0.757622			−0.378811	+	0.925474i	$0.623667\pi$
$224$		0			0
$225$	5.00000			0.333333
$226$		0			0
$227$	14.1421			0.938647			0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421			0.938647			0.469323	+	0.883026i	$0.344498\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$	−11.3137			−0.744387
$232$		0			0
$233$	26.0000			1.70332			0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000			1.70332			0.851658	+	0.524097i	$0.175597\pi$
$234$		0			0
$235$		−	2.00000i		−	0.130466i
$236$		0			0
$237$	−32.0000			−2.07862
$238$		0			0
$239$			30.0000i			1.94054i	0.242028	+	0.970269i	$0.422188\pi$
$239$			30.0000i			1.94054i	−0.242028	+	0.970269i	$0.577812\pi$
$240$		0			0
$241$			22.6274i			1.45756i	0.684748	+	0.728780i	$0.259912\pi$
$241$			22.6274i			1.45756i	−0.684748	+	0.728780i	$0.740088\pi$
$242$		0			0
$243$	−14.1421			−0.907218
$244$		0			0
$245$	3.00000			0.191663
$246$		0			0
$247$	−2.82843	+	12.0000i	−0.179969	+	0.763542i
$248$		0			0
$249$			39.5980i			2.50942i
$250$		0			0
$251$		−	10.0000i		−	0.631194i	−0.948893	−	0.315597i	$-0.897795\pi$
$251$		−	10.0000i		−	0.631194i	0.948893	−	0.315597i	$-0.102205\pi$
$252$		0			0
$253$	−4.00000			−0.251478
$254$		0			0
$255$	5.65685			0.354246
$256$		0			0
$257$		−	11.3137i		−	0.705730i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$		−	11.3137i		−	0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$		0			0
$259$	−16.9706			−1.05450
$260$		0			0
$261$		−	14.1421i		−	0.875376i
$262$		0			0
$263$		−	18.0000i		−	1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$		−	18.0000i		−	1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$		0			0
$265$			2.82843i			0.173749i
$266$		0			0
$267$			16.0000i			0.979184i
$268$		0			0
$269$			8.48528i			0.517357i	0.965964	+	0.258678i	$0.0832870\pi$
$269$			8.48528i			0.517357i	−0.965964	+	0.258678i	$0.916713\pi$
$270$		0			0
$271$		−	2.00000i		−	0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$		−	2.00000i		−	0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$		0			0
$273$	−16.0000			−0.968364
$274$		0			0
$275$		−	2.00000i		−	0.120605i
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			28.2843i			1.68730i	0.536895	+	0.843649i	$0.319597\pi$
$281$			28.2843i			1.68730i	−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$			6.00000i			0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$			6.00000i			0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$		0			0
$285$	−12.0000	−	2.82843i	−0.710819	−	0.167542i
$286$		0			0
$287$	11.3137			0.667827
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.48528i		−	0.495715i	−0.968796	−	0.247858i	$-0.920273\pi$
$293$		−	8.48528i		−	0.495715i	0.968796	−	0.247858i	$-0.0797265\pi$
$294$		0			0
$295$	−2.82843			−0.164677
$296$		0			0
$297$		−	11.3137i		−	0.656488i
$298$		0			0
$299$	−5.65685			−0.327144
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$	16.9706			0.974933
$304$		0			0
$305$	−2.00000			−0.114520
$306$		0			0
$307$	2.82843			0.161427			0.0807134	−	0.996737i	$-0.474280\pi$
$307$	2.82843			0.161427			0.0807134	+	0.996737i	$0.474280\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		0			0
$311$		−	18.0000i		−	1.02069i	−0.859971	−	0.510343i	$-0.829518\pi$
$311$		−	18.0000i		−	1.02069i	0.859971	−	0.510343i	$-0.170482\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$		0			0
$315$		−	10.0000i		−	0.563436i
$316$		0			0
$317$		−	14.1421i		−	0.794301i	−0.917753	−	0.397151i	$-0.869999\pi$
$317$		−	14.1421i		−	0.794301i	0.917753	−	0.397151i	$-0.130001\pi$
$318$		0			0
$319$	−5.65685			−0.316723
$320$		0			0
$321$	−8.00000			−0.446516
$322$		0			0
$323$	−8.48528	−	2.00000i	−0.472134	−	0.111283i
$324$		0			0
$325$		−	2.82843i		−	0.156893i
$326$		0			0
$327$			24.0000i			1.32720i
$328$		0			0
$329$	−4.00000			−0.220527
$330$		0			0
$331$	8.48528			0.466393			0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528			0.466393			0.233197	+	0.972430i	$0.425081\pi$
$332$		0			0
$333$		−	42.4264i		−	2.32495i
$334$		0			0
$335$	14.1421			0.772667
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$		−	5.65685i		−	0.304555i
$346$		0			0
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		−	16.0000i		−	0.854017i
$352$		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
$355$	5.65685			0.300235
$356$		0			0
$357$		−	11.3137i		−	0.598785i
$358$		0			0
$359$			30.0000i			1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$			30.0000i			1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$		0			0
$361$	17.0000	+	8.48528i	0.894737	+	0.446594i
$362$		0			0
$363$	19.7990			1.03918
$364$		0			0
$365$	2.00000			0.104685
$366$		0			0
$367$			30.0000i			1.56599i	0.622030	+	0.782994i	$0.286308\pi$
$367$			30.0000i			1.56599i	−0.622030	+	0.782994i	$0.713692\pi$
$368$		0			0
$369$			28.2843i			1.47242i
$370$		0			0
$371$	5.65685			0.293689
$372$		0			0
$373$			2.82843i			0.146450i	0.997315	+	0.0732252i	$0.0233292\pi$
$373$			2.82843i			0.146450i	−0.997315	+	0.0732252i	$0.976671\pi$
$374$		0			0
$375$	2.82843			0.146059
$376$		0			0
$377$	−8.00000			−0.412021
$378$		0			0
$379$	8.48528			0.435860			0.217930	−	0.975964i	$-0.430070\pi$
$379$	8.48528			0.435860			0.217930	+	0.975964i	$0.430070\pi$
$380$		0			0
$381$	32.0000			1.63941
$382$		0			0
$383$	−11.3137			−0.578103			−0.289052	−	0.957313i	$-0.593340\pi$
$383$	−11.3137			−0.578103			−0.289052	+	0.957313i	$0.593340\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$			30.0000i			1.52499i
$388$		0			0
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$		−	4.00000i		−	0.202289i
$392$		0			0
$393$			16.9706i			0.856052i
$394$		0			0
$395$	−11.3137			−0.569254
$396$		0			0
$397$	−34.0000			−1.70641			−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000			−1.70641			−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$	−5.65685	+	24.0000i	−0.283197	+	1.20150i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−16.9706			−0.841200
$408$		0			0
$409$			16.9706i			0.839140i	0.907723	+	0.419570i	$0.137819\pi$
$409$			16.9706i			0.839140i	−0.907723	+	0.419570i	$0.862181\pi$
$410$		0			0
$411$	28.2843			1.39516
$412$		0			0
$413$			5.65685i			0.278356i
$414$		0			0
$415$			14.0000i			0.687233i
$416$		0			0
$417$		−	28.2843i		−	1.38509i
$418$		0			0
$419$		−	26.0000i		−	1.27018i	−0.772437	−	0.635092i	$-0.780962\pi$
$419$		−	26.0000i		−	1.27018i	0.772437	−	0.635092i	$-0.219038\pi$
$420$		0			0
$421$		−	19.7990i		−	0.964944i	−0.875911	−	0.482472i	$-0.839739\pi$
$421$		−	19.7990i		−	0.964944i	0.875911	−	0.482472i	$-0.160261\pi$
$422$		0			0
$423$		−	10.0000i		−	0.486217i
$424$		0			0
$425$	2.00000			0.0970143
$426$		0			0
$427$			4.00000i			0.193574i
$428$		0			0
$429$	−16.0000			−0.772487
$430$		0			0
$431$	11.3137			0.544962			0.272481	−	0.962161i	$-0.412156\pi$
$431$	11.3137			0.544962			0.272481	+	0.962161i	$0.412156\pi$
$432$		0			0
$433$			11.3137i			0.543702i	0.962339	+	0.271851i	$0.0876358\pi$
$433$			11.3137i			0.543702i	−0.962339	+	0.271851i	$0.912364\pi$
$434$		0			0
$435$		−	8.00000i		−	0.383571i
$436$		0			0
$437$	−2.00000	+	8.48528i	−0.0956730	+	0.405906i
$438$		0			0
$439$	−5.65685			−0.269987			−0.134993	−	0.990846i	$-0.543101\pi$
$439$	−5.65685			−0.269987			−0.134993	+	0.990846i	$0.543101\pi$
$440$		0			0
$441$	15.0000			0.714286
$442$		0			0
$443$		−	18.0000i		−	0.855206i	−0.903967	−	0.427603i	$-0.859358\pi$
$443$		−	18.0000i		−	0.855206i	0.903967	−	0.427603i	$-0.140642\pi$
$444$		0			0
$445$			5.65685i			0.268161i
$446$		0			0
$447$	62.2254			2.94316
$448$		0			0
$449$			11.3137i			0.533927i	0.963707	+	0.266963i	$0.0860203\pi$
$449$			11.3137i			0.533927i	−0.963707	+	0.266963i	$0.913980\pi$
$450$		0			0
$451$	11.3137			0.532742
$452$		0			0
$453$	−16.0000			−0.751746
$454$		0			0
$455$	−5.65685			−0.265197
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$	11.3137			0.528079
$460$		0			0
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$			14.0000i			0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$			14.0000i			0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	18.0000i		−	0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$		−	18.0000i		−	0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$		0			0
$469$		−	28.2843i		−	1.30605i
$470$		0			0
$471$	−5.65685			−0.260654
$472$		0			0
$473$	12.0000			0.551761
$474$		0			0
$475$	−4.24264	−	1.00000i	−0.194666	−	0.0458831i
$476$		0			0
$477$			14.1421i			0.647524i
$478$		0			0
$479$			30.0000i			1.37073i	0.728197	+	0.685367i	$0.240358\pi$
$479$			30.0000i			1.37073i	−0.728197	+	0.685367i	$0.759642\pi$
$480$		0			0
$481$	−24.0000			−1.09431
$482$		0			0
$483$	−11.3137			−0.514792
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−28.2843			−1.28168			−0.640841	−	0.767673i	$-0.721414\pi$
$487$	−28.2843			−1.28168			−0.640841	+	0.767673i	$0.721414\pi$
$488$		0			0
$489$		−	28.2843i		−	1.27906i
$490$		0			0
$491$		−	34.0000i		−	1.53440i	−0.641409	−	0.767199i	$-0.721650\pi$
$491$		−	34.0000i		−	1.53440i	0.641409	−	0.767199i	$-0.278350\pi$
$492$		0			0
$493$		−	5.65685i		−	0.254772i
$494$		0			0
$495$		−	10.0000i		−	0.449467i
$496$		0			0
$497$		−	11.3137i		−	0.507489i
$498$		0			0
$499$		−	26.0000i		−	1.16392i	−0.813217	−	0.581960i	$-0.802286\pi$
$499$		−	26.0000i		−	1.16392i	0.813217	−	0.581960i	$-0.197714\pi$
$500$		0			0
$501$	−16.0000			−0.714827
$502$		0			0
$503$			14.0000i			0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$			14.0000i			0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$	14.1421			0.628074
$508$		0			0
$509$		−	2.82843i		−	0.125368i	−0.998033	−	0.0626839i	$-0.980034\pi$
$509$		−	2.82843i		−	0.125368i	0.998033	−	0.0626839i	$-0.0199660\pi$
$510$		0			0
$511$		−	4.00000i		−	0.176950i
$512$		0			0
$513$	−24.0000	−	5.65685i	−1.05963	−	0.249756i
$514$		0			0
$515$	−5.65685			−0.249271
$516$		0			0
$517$	−4.00000			−0.175920
$518$		0			0
$519$			56.0000i			2.45813i
$520$		0			0
$521$		−	39.5980i		−	1.73482i	−0.497595	−	0.867409i	$-0.665783\pi$
$521$		−	39.5980i		−	1.73482i	0.497595	−	0.867409i	$-0.334217\pi$
$522$		0			0
$523$	8.48528			0.371035			0.185518	−	0.982641i	$-0.440604\pi$
$523$	8.48528			0.371035			0.185518	+	0.982641i	$0.440604\pi$
$524$		0			0
$525$		−	5.65685i		−	0.246885i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	−14.1421			−0.613716
$532$		0			0
$533$	16.0000			0.693037
$534$		0			0
$535$	−2.82843			−0.122284
$536$		0			0
$537$	−56.0000			−2.41658
$538$		0			0
$539$		−	6.00000i		−	0.258438i
$540$		0			0
$541$	−18.0000			−0.773880			−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000			−0.773880			−0.386940	+	0.922105i	$0.626468\pi$
$542$		0			0
$543$		−	24.0000i		−	1.02994i
$544$		0			0
$545$			8.48528i			0.363470i
$546$		0			0
$547$	25.4558			1.08841			0.544207	−	0.838951i	$-0.316831\pi$
$547$	25.4558			1.08841			0.544207	+	0.838951i	$0.316831\pi$
$548$		0			0
$549$	−10.0000			−0.426790
$550$		0			0
$551$	−2.82843	+	12.0000i	−0.120495	+	0.511217i
$552$		0			0
$553$			22.6274i			0.962216i
$554$		0			0
$555$		−	24.0000i		−	1.01874i
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	16.9706			0.717778
$560$		0			0
$561$		−	11.3137i		−	0.477665i
$562$		0			0
$563$	−31.1127			−1.31124			−0.655622	−	0.755089i	$-0.727593\pi$
$563$	−31.1127			−1.31124			−0.655622	+	0.755089i	$0.727593\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$		−	28.2843i		−	1.18574i	−0.805299	−	0.592869i	$-0.797995\pi$
$569$		−	28.2843i		−	1.18574i	0.805299	−	0.592869i	$-0.202005\pi$
$570$		0			0
$571$		−	2.00000i		−	0.0836974i	−0.999124	−	0.0418487i	$-0.986675\pi$
$571$		−	2.00000i		−	0.0836974i	0.999124	−	0.0418487i	$-0.0133247\pi$
$572$		0			0
$573$		−	50.9117i		−	2.12687i
$574$		0			0
$575$		−	2.00000i		−	0.0834058i
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$			64.0000i			2.65975i
$580$		0			0
$581$	28.0000			1.16164
$582$		0			0
$583$	5.65685			0.234283
$584$		0			0
$585$		−	14.1421i		−	0.584705i
$586$		0			0
$587$			38.0000i			1.56843i	0.620491	+	0.784214i	$0.286934\pi$
$587$			38.0000i			1.56843i	−0.620491	+	0.784214i	$0.713066\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	16.9706			0.698076
$592$		0			0
$593$	10.0000			0.410651			0.205325	−	0.978694i	$-0.434175\pi$
$593$	10.0000			0.410651			0.205325	+	0.978694i	$0.434175\pi$
$594$		0			0
$595$		−	4.00000i		−	0.163984i
$596$		0			0
$597$			39.5980i			1.62064i
$598$		0			0
$599$	16.9706			0.693398			0.346699	−	0.937976i	$-0.387302\pi$
$599$	16.9706			0.693398			0.346699	+	0.937976i	$0.387302\pi$
$600$		0			0
$601$		−	39.5980i		−	1.61524i	−0.589706	−	0.807618i	$-0.700757\pi$
$601$		−	39.5980i		−	1.61524i	0.589706	−	0.807618i	$-0.299243\pi$
$602$		0			0
$603$	70.7107			2.87956
$604$		0			0
$605$	7.00000			0.284590
$606$		0			0
$607$	33.9411			1.37763			0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411			1.37763			0.688814	+	0.724938i	$0.258132\pi$
$608$		0			0
$609$	−16.0000			−0.648353
$610$		0			0
$611$	−5.65685			−0.228852
$612$		0			0
$613$	22.0000			0.888572			0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000			0.888572			0.444286	+	0.895885i	$0.353457\pi$
$614$		0			0
$615$			16.0000i			0.645182i
$616$		0			0
$617$	2.00000			0.0805170			0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000			0.0805170			0.0402585	+	0.999189i	$0.487182\pi$
$618$		0			0
$619$		−	34.0000i		−	1.36658i	−0.730149	−	0.683288i	$-0.760549\pi$
$619$		−	34.0000i		−	1.36658i	0.730149	−	0.683288i	$-0.239451\pi$
$620$		0			0
$621$		−	11.3137i		−	0.454003i
$622$		0			0
$623$	11.3137			0.453274
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−5.65685	+	24.0000i	−0.225913	+	0.958468i
$628$		0			0
$629$		−	16.9706i		−	0.676661i
$630$		0			0
$631$			30.0000i			1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$			30.0000i			1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$		0			0
$633$	−56.0000			−2.22580
$634$		0			0
$635$	11.3137			0.448971
$636$		0			0
$637$		−	8.48528i		−	0.336199i
$638$		0			0
$639$	28.2843			1.11891
$640$		0			0
$641$		−	33.9411i		−	1.34059i	−0.742093	−	0.670297i	$-0.766167\pi$
$641$		−	33.9411i		−	1.34059i	0.742093	−	0.670297i	$-0.233833\pi$
$642$		0			0
$643$			22.0000i			0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$			22.0000i			0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$		0			0
$645$			16.9706i			0.668215i
$646$		0			0
$647$		−	18.0000i		−	0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$		−	18.0000i		−	0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$		0			0
$649$			5.65685i			0.222051i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−34.0000			−1.33052			−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000			−1.33052			−0.665261	+	0.746611i	$0.731680\pi$
$654$		0			0
$655$			6.00000i			0.234439i
$656$		0			0
$657$	10.0000			0.390137
$658$		0			0
$659$	−42.4264			−1.65270			−0.826349	−	0.563158i	$-0.809586\pi$
$659$	−42.4264			−1.65270			−0.826349	+	0.563158i	$0.809586\pi$
$660$		0			0
$661$			36.7696i			1.43017i	0.699038	+	0.715085i	$0.253612\pi$
$661$			36.7696i			1.43017i	−0.699038	+	0.715085i	$0.746388\pi$
$662$		0			0
$663$		−	16.0000i		−	0.621389i
$664$		0			0
$665$	−2.00000	+	8.48528i	−0.0775567	+	0.329045i
$666$		0			0
$667$	−5.65685			−0.219034
$668$		0			0
$669$	−32.0000			−1.23719
$670$		0			0
$671$			4.00000i			0.154418i
$672$		0			0
$673$			45.2548i			1.74445i	0.489109	+	0.872223i	$0.337322\pi$
$673$			45.2548i			1.74445i	−0.489109	+	0.872223i	$0.662678\pi$
$674$		0			0
$675$	5.65685			0.217732
$676$		0			0
$677$			36.7696i			1.41317i	0.707629	+	0.706584i	$0.249765\pi$
$677$			36.7696i			1.41317i	−0.707629	+	0.706584i	$0.750235\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	40.0000			1.53280
$682$		0			0
$683$	−2.82843			−0.108227			−0.0541134	−	0.998535i	$-0.517233\pi$
$683$	−2.82843			−0.108227			−0.0541134	+	0.998535i	$0.517233\pi$
$684$		0			0
$685$	10.0000			0.382080
$686$		0			0
$687$	16.9706			0.647467
$688$		0			0
$689$	8.00000			0.304776
$690$		0			0
$691$			30.0000i			1.14125i	0.821209	+	0.570627i	$0.193300\pi$
$691$			30.0000i			1.14125i	−0.821209	+	0.570627i	$0.806700\pi$
$692$		0			0
$693$	−20.0000			−0.759737
$694$		0			0
$695$		−	10.0000i		−	0.379322i
$696$		0			0
$697$			11.3137i			0.428537i
$698$		0			0
$699$	73.5391			2.78150
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$	−8.48528	+	36.0000i	−0.320028	+	1.35777i
$704$		0			0
$705$		−	5.65685i		−	0.213049i
$706$		0			0
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$	−56.5685			−2.12149
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−5.65685			−0.211554
$716$		0			0
$717$			84.8528i			3.16889i
$718$		0			0
$719$		−	18.0000i		−	0.671287i	−0.941989	−	0.335643i	$-0.891046\pi$
$719$		−	18.0000i		−	0.671287i	0.941989	−	0.335643i	$-0.108954\pi$
$720$		0			0
$721$			11.3137i			0.421345i
$722$		0			0
$723$			64.0000i			2.38019i
$724$		0			0
$725$		−	2.82843i		−	0.105045i
$726$		0			0
$727$			46.0000i			1.70605i	0.521874	+	0.853023i	$0.325233\pi$
$727$			46.0000i			1.70605i	−0.521874	+	0.853023i	$0.674767\pi$
$728$		0			0
$729$	−43.0000			−1.59259
$730$		0			0
$731$			12.0000i			0.443836i
$732$		0			0
$733$	−50.0000			−1.84679			−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000			−1.84679			−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$	8.48528			0.312984
$736$		0			0
$737$		−	28.2843i		−	1.04186i
$738$		0			0
$739$		−	2.00000i		−	0.0735712i	−0.999323	−	0.0367856i	$-0.988288\pi$
$739$		−	2.00000i		−	0.0735712i	0.999323	−	0.0367856i	$-0.0117119\pi$
$740$		0			0
$741$	−8.00000	+	33.9411i	−0.293887	+	1.24686i
$742$		0			0
$743$	−28.2843			−1.03765			−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843			−1.03765			−0.518825	+	0.854881i	$0.673630\pi$
$744$		0			0
$745$	22.0000			0.806018
$746$		0			0
$747$			70.0000i			2.56117i
$748$		0			0
$749$			5.65685i			0.206697i
$750$		0			0
$751$	−11.3137			−0.412843			−0.206422	−	0.978463i	$-0.566182\pi$
$751$	−11.3137			−0.412843			−0.206422	+	0.978463i	$0.566182\pi$
$752$		0			0
$753$		−	28.2843i		−	1.03074i
$754$		0			0
$755$	−5.65685			−0.205874
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$	−11.3137			−0.410662
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$	16.9706			0.614376
$764$		0			0
$765$	10.0000			0.361551
$766$		0			0
$767$			8.00000i			0.288863i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$		−	32.0000i		−	1.15245i
$772$		0			0
$773$			14.1421i			0.508657i	0.967118	+	0.254329i	$0.0818545\pi$
$773$			14.1421i			0.508657i	−0.967118	+	0.254329i	$0.918146\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−48.0000			−1.72199
$778$		0			0
$779$	5.65685	−	24.0000i	0.202678	−	0.859889i
$780$		0			0
$781$		−	11.3137i		−	0.404836i
$782$		0			0
$783$		−	16.0000i		−	0.571793i
$784$		0			0
$785$	−2.00000			−0.0713831
$786$		0			0
$787$	−42.4264			−1.51234			−0.756169	−	0.654376i	$-0.772931\pi$
$787$	−42.4264			−1.51234			−0.756169	+	0.654376i	$0.772931\pi$
$788$		0			0
$789$		−	50.9117i		−	1.81250i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			5.65685i			0.200881i
$794$		0			0
$795$			8.00000i			0.283731i
$796$		0			0
$797$			53.7401i			1.90357i	0.306762	+	0.951786i	$0.400754\pi$
$797$			53.7401i			1.90357i	−0.306762	+	0.951786i	$0.599246\pi$
$798$		0			0
$799$		−	4.00000i		−	0.141510i
$800$		0			0
$801$			28.2843i			0.999376i
$802$		0			0
$803$		−	4.00000i		−	0.141157i
$804$		0			0
$805$	−4.00000			−0.140981
$806$		0			0
$807$			24.0000i			0.844840i
$808$		0			0
$809$	50.0000			1.75791			0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000			1.75791			0.878953	+	0.476908i	$0.158243\pi$
$810$		0			0
$811$	−2.82843			−0.0993195			−0.0496598	−	0.998766i	$-0.515814\pi$
$811$	−2.82843			−0.0993195			−0.0496598	+	0.998766i	$0.515814\pi$
$812$		0			0
$813$		−	5.65685i		−	0.198395i
$814$		0			0
$815$		−	10.0000i		−	0.350285i
$816$		0			0
$817$	6.00000	−	25.4558i	0.209913	−	0.890587i
$818$		0			0
$819$	−28.2843			−0.988332
$820$		0			0
$821$	−10.0000			−0.349002			−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000			−0.349002			−0.174501	+	0.984657i	$0.555831\pi$
$822$		0			0
$823$		−	2.00000i		−	0.0697156i	−0.999392	−	0.0348578i	$-0.988902\pi$
$823$		−	2.00000i		−	0.0697156i	0.999392	−	0.0348578i	$-0.0110978\pi$
$824$		0			0
$825$		−	5.65685i		−	0.196946i
$826$		0			0
$827$	−25.4558			−0.885186			−0.442593	−	0.896723i	$-0.645941\pi$
$827$	−25.4558			−0.885186			−0.442593	+	0.896723i	$0.645941\pi$
$828$		0			0
$829$		−	2.82843i		−	0.0982353i	−0.998793	−	0.0491177i	$-0.984359\pi$
$829$		−	2.82843i		−	0.0982353i	0.998793	−	0.0491177i	$-0.0156409\pi$
$830$		0			0
$831$	62.2254			2.15858
$832$		0			0
$833$	6.00000			0.207888
$834$		0			0
$835$	−5.65685			−0.195764
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28.2843			0.976481			0.488241	−	0.872709i	$-0.337639\pi$
$839$	28.2843			0.976481			0.488241	+	0.872709i	$0.337639\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$			80.0000i			2.75535i
$844$		0			0
$845$	5.00000			0.172005
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$		0			0
$849$			16.9706i			0.582428i
$850$		0			0
$851$	−16.9706			−0.581743
$852$		0			0
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$	−21.2132	−	5.00000i	−0.725476	−	0.170996i
$856$		0			0
$857$			28.2843i			0.966172i	0.875573	+	0.483086i	$0.160484\pi$
$857$			28.2843i			0.966172i	−0.875573	+	0.483086i	$0.839516\pi$
$858$		0			0
$859$		−	10.0000i		−	0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$859$		−	10.0000i		−	0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$860$		0			0
$861$	32.0000			1.09056
$862$		0			0
$863$	−11.3137			−0.385123			−0.192562	−	0.981285i	$-0.561680\pi$
$863$	−11.3137			−0.385123			−0.192562	+	0.981285i	$0.561680\pi$
$864$		0			0
$865$			19.7990i			0.673186i
$866$		0			0
$867$	−36.7696			−1.24876
$868$		0			0
$869$			22.6274i			0.767583i
$870$		0			0
$871$		−	40.0000i		−	1.35535i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	2.00000i		−	0.0676123i
$876$		0			0
$877$		−	2.82843i		−	0.0955092i	−0.998859	−	0.0477546i	$-0.984793\pi$
$877$		−	2.82843i		−	0.0955092i	0.998859	−	0.0477546i	$-0.0152065\pi$
$878$		0			0
$879$		−	24.0000i		−	0.809500i
$880$		0			0
$881$	10.0000			0.336909			0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000			0.336909			0.168454	+	0.985709i	$0.446122\pi$
$882$		0			0
$883$			46.0000i			1.54802i	0.633171	+	0.774012i	$0.281753\pi$
$883$			46.0000i			1.54802i	−0.633171	+	0.774012i	$0.718247\pi$
$884$		0			0
$885$	−8.00000			−0.268917
$886$		0			0
$887$	−16.9706			−0.569816			−0.284908	−	0.958555i	$-0.591963\pi$
$887$	−16.9706			−0.569816			−0.284908	+	0.958555i	$0.591963\pi$
$888$		0			0
$889$		−	22.6274i		−	0.758899i
$890$		0			0
$891$		−	2.00000i		−	0.0670025i
$892$		0			0
$893$	−2.00000	+	8.48528i	−0.0669274	+	0.283949i
$894$		0			0
$895$	−19.7990			−0.661807
$896$		0			0
$897$	−16.0000			−0.534224
$898$		0			0
$899$		0			0
$900$		0			0
$901$			5.65685i			0.188457i
$902$		0			0
$903$	33.9411			1.12949
$904$		0			0
$905$		−	8.48528i		−	0.282060i
$906$		0			0
$907$	−36.7696			−1.22091			−0.610456	−	0.792050i	$-0.709014\pi$
$907$	−36.7696			−1.22091			−0.610456	+	0.792050i	$0.709014\pi$
$908$		0			0
$909$	30.0000			0.995037
$910$		0			0
$911$	−56.5685			−1.87420			−0.937100	−	0.349062i	$-0.886500\pi$
$911$	−56.5685			−1.87420			−0.937100	+	0.349062i	$0.886500\pi$
$912$		0			0
$913$	28.0000			0.926665
$914$		0			0
$915$	−5.65685			−0.187010
$916$		0			0
$917$	12.0000			0.396275
$918$		0			0
$919$			14.0000i			0.461817i	0.972975	+	0.230909i	$0.0741699\pi$
$919$			14.0000i			0.461817i	−0.972975	+	0.230909i	$0.925830\pi$
$920$		0			0
$921$	8.00000			0.263609
$922$		0			0
$923$		−	16.0000i		−	0.526646i
$924$		0			0
$925$		−	8.48528i		−	0.278994i
$926$		0			0
$927$	−28.2843			−0.928977
$928$		0			0
$929$	10.0000			0.328089			0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000			0.328089			0.164045	+	0.986453i	$0.447546\pi$
$930$		0			0
$931$	−12.7279	−	3.00000i	−0.417141	−	0.0983210i
$932$		0			0
$933$		−	50.9117i		−	1.66677i
$934$		0			0
$935$		−	4.00000i		−	0.130814i
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$	−16.9706			−0.553813
$940$		0			0
$941$			8.48528i			0.276612i	0.990390	+	0.138306i	$0.0441658\pi$
$941$			8.48528i			0.276612i	−0.990390	+	0.138306i	$0.955834\pi$
$942$		0			0
$943$	11.3137			0.368425
$944$		0			0
$945$		−	11.3137i		−	0.368035i
$946$		0			0
$947$		−	42.0000i		−	1.36482i	−0.730971	−	0.682408i	$-0.760933\pi$
$947$		−	42.0000i		−	1.36482i	0.730971	−	0.682408i	$-0.239067\pi$
$948$		0			0
$949$		−	5.65685i		−	0.183629i
$950$		0			0
$951$		−	40.0000i		−	1.29709i
$952$		0			0
$953$			28.2843i			0.916217i	0.888896	+	0.458109i	$0.151473\pi$
$953$			28.2843i			0.916217i	−0.888896	+	0.458109i	$0.848527\pi$
$954$		0			0
$955$		−	18.0000i		−	0.582466i
$956$		0			0
$957$	−16.0000			−0.517207
$958$		0			0
$959$		−	20.0000i		−	0.645834i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	−14.1421			−0.455724
$964$		0			0
$965$			22.6274i			0.728402i
$966$		0			0
$967$		−	2.00000i		−	0.0643157i	−0.999483	−	0.0321578i	$-0.989762\pi$
$967$		−	2.00000i		−	0.0643157i	0.999483	−	0.0321578i	$-0.0102379\pi$
$968$		0			0
$969$	−24.0000	−	5.65685i	−0.770991	−	0.181724i
$970$		0			0
$971$	31.1127			0.998454			0.499227	−	0.866471i	$-0.333617\pi$
$971$	31.1127			0.998454			0.499227	+	0.866471i	$0.333617\pi$
$972$		0			0
$973$	−20.0000			−0.641171
$974$		0			0
$975$		−	8.00000i		−	0.256205i
$976$		0			0
$977$			33.9411i			1.08587i	0.839774	+	0.542936i	$0.182688\pi$
$977$			33.9411i			1.08587i	−0.839774	+	0.542936i	$0.817312\pi$
$978$		0			0
$979$	11.3137			0.361588
$980$		0			0
$981$			42.4264i			1.35457i
$982$		0			0
$983$	−16.9706			−0.541277			−0.270638	−	0.962681i	$-0.587235\pi$
$983$	−16.9706			−0.541277			−0.270638	+	0.962681i	$0.587235\pi$
$984$		0			0
$985$	6.00000			0.191176
$986$		0			0
$987$	−11.3137			−0.360119
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$	22.6274			0.718784			0.359392	−	0.933187i	$-0.382984\pi$
$991$	22.6274			0.718784			0.359392	+	0.933187i	$0.382984\pi$
$992$		0			0
$993$	24.0000			0.761617
$994$		0			0
$995$			14.0000i			0.443830i
$996$		0			0
$997$	6.00000			0.190022			0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000			0.190022			0.0950110	+	0.995476i	$0.469711\pi$
$998$		0			0
$999$		−	48.0000i		−	1.51865i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.j.c.2431.3	yes	4
4.3	odd	2	inner	3040.2.j.c.2431.2	yes	4
19.18	odd	2	inner	3040.2.j.c.2431.1	✓	4
76.75	even	2	inner	3040.2.j.c.2431.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.j.c.2431.1	✓	4	19.18	odd	2	inner
3040.2.j.c.2431.2	yes	4	4.3	odd	2	inner
3040.2.j.c.2431.3	yes	4	1.1	even	1	trivial
3040.2.j.c.2431.4	yes	4	76.75	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{3} +1.00000 q^{5} -2.00000i q^{7} +5.00000 q^{9} +O(q^{10})\)	\(q+2.82843 q^{3} +1.00000 q^{5} -2.00000i q^{7} +5.00000 q^{9} -2.00000i q^{11} -2.82843i q^{13} +2.82843 q^{15} +2.00000 q^{17} +(-4.24264 - 1.00000i) q^{19} -5.65685i q^{21} -2.00000i q^{23} +1.00000 q^{25} +5.65685 q^{27} -2.82843i q^{29} -5.65685i q^{33} -2.00000i q^{35} -8.48528i q^{37} -8.00000i q^{39} +5.65685i q^{41} +6.00000i q^{43} +5.00000 q^{45} -2.00000i q^{47} +3.00000 q^{49} +5.65685 q^{51} +2.82843i q^{53} -2.00000i q^{55} +(-12.0000 - 2.82843i) q^{57} -2.82843 q^{59} -2.00000 q^{61} -10.0000i q^{63} -2.82843i q^{65} +14.1421 q^{67} -5.65685i q^{69} +5.65685 q^{71} +2.00000 q^{73} +2.82843 q^{75} -4.00000 q^{77} -11.3137 q^{79} +1.00000 q^{81} +14.0000i q^{83} +2.00000 q^{85} -8.00000i q^{87} +5.65685i q^{89} -5.65685 q^{91} +(-4.24264 - 1.00000i) q^{95} -10.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 20 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 20 * q^9	\( 4 q + 4 q^{5} + 20 q^{9} + 8 q^{17} + 4 q^{25} + 20 q^{45} + 12 q^{49} - 48 q^{57} - 8 q^{61} + 8 q^{73} - 16 q^{77} + 4 q^{81} + 8 q^{85}+O(q^{100}) \) 4 * q + 4 * q^5 + 20 * q^9 + 8 * q^17 + 4 * q^25 + 20 * q^45 + 12 * q^49 - 48 * q^57 - 8 * q^61 + 8 * q^73 - 16 * q^77 + 4 * q^81 + 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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