LMFDB - Embedded newform 4080.2.m.o.2449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4080,2,Mod(2449,4080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4080.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.m (of order $2$, degree $1$, not 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:	$32.5789640247$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	4080.2449
Dual form			4080.2.m.o.2449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-0.224745 - 2.22474i) q^{5} -2.44949i q^{7} -1.00000 q^{9} +4.89898 q^{11} +6.00000i q^{13} +(-2.22474 + 0.224745i) q^{15} -1.00000i q^{17} +6.89898 q^{19} -2.44949 q^{21} -6.44949i q^{23} +(-4.89898 + 1.00000i) q^{25} +1.00000i q^{27} +9.34847 q^{29} +6.44949 q^{31} -4.89898i q^{33} +(-5.44949 + 0.550510i) q^{35} +0.449490i q^{37} +6.00000 q^{39} -1.10102 q^{41} +2.89898i q^{43} +(0.224745 + 2.22474i) q^{45} +4.89898i q^{47} +1.00000 q^{49} -1.00000 q^{51} -1.10102i q^{53} +(-1.10102 - 10.8990i) q^{55} -6.89898i q^{57} +5.79796 q^{59} +13.3485 q^{61} +2.44949i q^{63} +(13.3485 - 1.34847i) q^{65} +4.00000i q^{67} -6.44949 q^{69} -2.44949 q^{71} +14.8990i q^{73} +(1.00000 + 4.89898i) q^{75} -12.0000i q^{77} -1.55051 q^{79} +1.00000 q^{81} -2.89898i q^{83} +(-2.22474 + 0.224745i) q^{85} -9.34847i q^{87} +1.79796 q^{89} +14.6969 q^{91} -6.44949i q^{93} +(-1.55051 - 15.3485i) q^{95} +3.79796i q^{97} -4.89898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{9} - 4 q^{15} + 8 q^{19} + 8 q^{29} + 16 q^{31} - 12 q^{35} + 24 q^{39} - 24 q^{41} - 4 q^{45} + 4 q^{49} - 4 q^{51} - 24 q^{55} - 16 q^{59} + 24 q^{61} + 24 q^{65} - 16 q^{69} + 4 q^{75}+ \cdots - 16 q^{95}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^9 - 4 * q^15 + 8 * q^19 + 8 * q^29 + 16 * q^31 - 12 * q^35 + 24 * q^39 - 24 * q^41 - 4 * q^45 + 4 * q^49 - 4 * q^51 - 24 * q^55 - 16 * q^59 + 24 * q^61 + 24 * q^65 - 16 * q^69 + 4 * q^75 - 16 * q^79 + 4 * q^81 - 4 * q^85 - 32 * q^89 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1361$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−0.224745	−	2.22474i	−0.100509	−	0.994936i
$5$	−0.224745	−	2.22474i	−0.100509	−	0.994936i
$6$		0			0
$7$		−	2.44949i		−	0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$		−	2.44949i		−	0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	−2.22474	+	0.224745i	−0.574427	+	0.0580289i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$	6.89898			1.58273			0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898			1.58273			0.791367	+	0.611341i	$0.209370\pi$
$20$		0			0
$21$	−2.44949			−0.534522
$22$		0			0
$23$		−	6.44949i		−	1.34481i	−0.740183	−	0.672406i	$-0.765261\pi$
$23$		−	6.44949i		−	1.34481i	0.740183	−	0.672406i	$-0.234739\pi$
$24$		0			0
$25$	−4.89898	+	1.00000i	−0.979796	+	0.200000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	9.34847			1.73597			0.867984	−	0.496593i	$-0.165416\pi$
$29$	9.34847			1.73597			0.867984	+	0.496593i	$0.165416\pi$
$30$		0			0
$31$	6.44949			1.15836			0.579181	−	0.815199i	$-0.303372\pi$
$31$	6.44949			1.15836			0.579181	+	0.815199i	$0.303372\pi$
$32$		0			0
$33$		−	4.89898i		−	0.852803i
$34$		0			0
$35$	−5.44949	+	0.550510i	−0.921132	+	0.0930532i
$36$		0			0
$37$			0.449490i			0.0738957i	0.999317	+	0.0369478i	$0.0117635\pi$
$37$			0.449490i			0.0738957i	−0.999317	+	0.0369478i	$0.988236\pi$
$38$		0			0
$39$	6.00000			0.960769
$40$		0			0
$41$	−1.10102			−0.171951			−0.0859753	−	0.996297i	$-0.527401\pi$
$41$	−1.10102			−0.171951			−0.0859753	+	0.996297i	$0.527401\pi$
$42$		0			0
$43$			2.89898i			0.442090i	0.975264	+	0.221045i	$0.0709468\pi$
$43$			2.89898i			0.442090i	−0.975264	+	0.221045i	$0.929053\pi$
$44$		0			0
$45$	0.224745	+	2.22474i	0.0335030	+	0.331645i
$46$		0			0
$47$			4.89898i			0.714590i	0.933992	+	0.357295i	$0.116301\pi$
$47$			4.89898i			0.714590i	−0.933992	+	0.357295i	$0.883699\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$		−	1.10102i		−	0.151237i	−0.997137	−	0.0756184i	$-0.975907\pi$
$53$		−	1.10102i		−	0.151237i	0.997137	−	0.0756184i	$-0.0240931\pi$
$54$		0			0
$55$	−1.10102	−	10.8990i	−0.148462	−	1.46962i
$56$		0			0
$57$		−	6.89898i		−	0.913792i
$58$		0			0
$59$	5.79796			0.754830			0.377415	−	0.926044i	$-0.376813\pi$
$59$	5.79796			0.754830			0.377415	+	0.926044i	$0.376813\pi$
$60$		0			0
$61$	13.3485			1.70910			0.854548	−	0.519372i	$-0.173834\pi$
$61$	13.3485			1.70910			0.854548	+	0.519372i	$0.173834\pi$
$62$		0			0
$63$			2.44949i			0.308607i
$64$		0			0
$65$	13.3485	−	1.34847i	1.65567	−	0.167257i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$	−6.44949			−0.776427
$70$		0			0
$71$	−2.44949			−0.290701			−0.145350	−	0.989380i	$-0.546431\pi$
$71$	−2.44949			−0.290701			−0.145350	+	0.989380i	$0.546431\pi$
$72$		0			0
$73$			14.8990i			1.74379i	0.489690	+	0.871897i	$0.337110\pi$
$73$			14.8990i			1.74379i	−0.489690	+	0.871897i	$0.662890\pi$
$74$		0			0
$75$	1.00000	+	4.89898i	0.115470	+	0.565685i
$76$		0			0
$77$		−	12.0000i		−	1.36753i
$78$		0			0
$79$	−1.55051			−0.174446			−0.0872230	−	0.996189i	$-0.527799\pi$
$79$	−1.55051			−0.174446			−0.0872230	+	0.996189i	$0.527799\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	2.89898i		−	0.318204i	−0.987262	−	0.159102i	$-0.949140\pi$
$83$		−	2.89898i		−	0.318204i	0.987262	−	0.159102i	$-0.0508599\pi$
$84$		0			0
$85$	−2.22474	+	0.224745i	−0.241307	+	0.0243770i
$86$		0			0
$87$		−	9.34847i		−	1.00226i
$88$		0			0
$89$	1.79796			0.190583			0.0952916	−	0.995449i	$-0.469622\pi$
$89$	1.79796			0.190583			0.0952916	+	0.995449i	$0.469622\pi$
$90$		0			0
$91$	14.6969			1.54066
$92$		0			0
$93$		−	6.44949i		−	0.668781i
$94$		0			0
$95$	−1.55051	−	15.3485i	−0.159079	−	1.57472i
$96$		0			0
$97$			3.79796i			0.385624i	0.981236	+	0.192812i	$0.0617608\pi$
$97$			3.79796i			0.385624i	−0.981236	+	0.192812i	$0.938239\pi$
$98$		0			0
$99$	−4.89898			−0.492366
$100$		0			0
$101$	3.79796			0.377911			0.188956	−	0.981986i	$-0.439490\pi$
$101$	3.79796			0.377911			0.188956	+	0.981986i	$0.439490\pi$
$102$		0			0
$103$		−	16.8990i		−	1.66511i	−0.553945	−	0.832553i	$-0.686878\pi$
$103$		−	16.8990i		−	1.66511i	0.553945	−	0.832553i	$-0.313122\pi$
$104$		0			0
$105$	0.550510	+	5.44949i	0.0537243	+	0.531816i
$106$		0			0
$107$			3.10102i			0.299787i	0.988702	+	0.149893i	$0.0478931\pi$
$107$			3.10102i			0.299787i	−0.988702	+	0.149893i	$0.952107\pi$
$108$		0			0
$109$	−18.2474			−1.74779			−0.873894	−	0.486116i	$-0.838413\pi$
$109$	−18.2474			−1.74779			−0.873894	+	0.486116i	$0.838413\pi$
$110$		0			0
$111$	0.449490			0.0426637
$112$		0			0
$113$			16.6969i			1.57072i	0.619042	+	0.785358i	$0.287521\pi$
$113$			16.6969i			1.57072i	−0.619042	+	0.785358i	$0.712479\pi$
$114$		0			0
$115$	−14.3485	+	1.44949i	−1.33800	+	0.135166i
$116$		0			0
$117$		−	6.00000i		−	0.554700i
$118$		0			0
$119$	−2.44949			−0.224544
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$			1.10102i			0.0992757i
$124$		0			0
$125$	3.32577	+	10.6742i	0.297465	+	0.954733i
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		0			0
$129$	2.89898			0.255241
$130$		0			0
$131$	−4.89898			−0.428026			−0.214013	−	0.976831i	$-0.568653\pi$
$131$	−4.89898			−0.428026			−0.214013	+	0.976831i	$0.568653\pi$
$132$		0			0
$133$		−	16.8990i		−	1.46533i
$134$		0			0
$135$	2.22474	−	0.224745i	0.191476	−	0.0193430i
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$	−16.8990			−1.43335			−0.716676	−	0.697406i	$-0.754338\pi$
$139$	−16.8990			−1.43335			−0.716676	+	0.697406i	$0.754338\pi$
$140$		0			0
$141$	4.89898			0.412568
$142$		0			0
$143$			29.3939i			2.45804i
$144$		0			0
$145$	−2.10102	−	20.7980i	−0.174480	−	1.72718i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−12.6969			−1.04017			−0.520087	−	0.854113i	$-0.674100\pi$
$149$	−12.6969			−1.04017			−0.520087	+	0.854113i	$0.674100\pi$
$150$		0			0
$151$	−13.7980			−1.12286			−0.561431	−	0.827524i	$-0.689749\pi$
$151$	−13.7980			−1.12286			−0.561431	+	0.827524i	$0.689749\pi$
$152$		0			0
$153$			1.00000i			0.0808452i
$154$		0			0
$155$	−1.44949	−	14.3485i	−0.116426	−	1.15250i
$156$		0			0
$157$		−	16.6969i		−	1.33256i	−0.745701	−	0.666280i	$-0.767885\pi$
$157$		−	16.6969i		−	1.33256i	0.745701	−	0.666280i	$-0.232115\pi$
$158$		0			0
$159$	−1.10102			−0.0873166
$160$		0			0
$161$	−15.7980			−1.24505
$162$		0			0
$163$		−	5.79796i		−	0.454131i	−0.973879	−	0.227066i	$-0.927087\pi$
$163$		−	5.79796i		−	0.454131i	0.973879	−	0.227066i	$-0.0729132\pi$
$164$		0			0
$165$	−10.8990	+	1.10102i	−0.848484	+	0.0857143i
$166$		0			0
$167$			4.65153i			0.359946i	0.983672	+	0.179973i	$0.0576011\pi$
$167$			4.65153i			0.359946i	−0.983672	+	0.179973i	$0.942399\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$	−6.89898			−0.527578
$172$		0			0
$173$		−	13.3485i		−	1.01487i	−0.861691	−	0.507433i	$-0.830595\pi$
$173$		−	13.3485i		−	1.01487i	0.861691	−	0.507433i	$-0.169405\pi$
$174$		0			0
$175$	2.44949	+	12.0000i	0.185164	+	0.907115i
$176$		0			0
$177$		−	5.79796i		−	0.435801i
$178$		0			0
$179$	−20.6969			−1.54696			−0.773481	−	0.633820i	$-0.781486\pi$
$179$	−20.6969			−1.54696			−0.773481	+	0.633820i	$0.781486\pi$
$180$		0			0
$181$	4.44949			0.330728			0.165364	−	0.986233i	$-0.447120\pi$
$181$	4.44949			0.330728			0.165364	+	0.986233i	$0.447120\pi$
$182$		0			0
$183$		−	13.3485i		−	0.986747i
$184$		0			0
$185$	1.00000	−	0.101021i	0.0735215	−	0.00742718i
$186$		0			0
$187$		−	4.89898i		−	0.358249i
$188$		0			0
$189$	2.44949			0.178174
$190$		0			0
$191$	16.8990			1.22277			0.611384	−	0.791334i	$-0.290613\pi$
$191$	16.8990			1.22277			0.611384	+	0.791334i	$0.290613\pi$
$192$		0			0
$193$			14.0000i			1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$	−1.34847	−	13.3485i	−0.0965659	−	0.955904i
$196$		0			0
$197$		−	19.1464i		−	1.36413i	−0.731293	−	0.682063i	$-0.761083\pi$
$197$		−	19.1464i		−	1.36413i	0.731293	−	0.682063i	$-0.238917\pi$
$198$		0			0
$199$	−18.0454			−1.27921			−0.639603	−	0.768706i	$-0.720901\pi$
$199$	−18.0454			−1.27921			−0.639603	+	0.768706i	$0.720901\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$		−	22.8990i		−	1.60719i
$204$		0			0
$205$	0.247449	+	2.44949i	0.0172826	+	0.171080i
$206$		0			0
$207$			6.44949i			0.448271i
$208$		0			0
$209$	33.7980			2.33785
$210$		0			0
$211$	3.10102			0.213483			0.106742	−	0.994287i	$-0.465958\pi$
$211$	3.10102			0.213483			0.106742	+	0.994287i	$0.465958\pi$
$212$		0			0
$213$			2.44949i			0.167836i
$214$		0			0
$215$	6.44949	−	0.651531i	0.439852	−	0.0444340i
$216$		0			0
$217$		−	15.7980i		−	1.07244i
$218$		0			0
$219$	14.8990			1.00678
$220$		0			0
$221$	6.00000			0.403604
$222$		0			0
$223$		−	4.89898i		−	0.328060i	−0.986455	−	0.164030i	$-0.947551\pi$
$223$		−	4.89898i		−	0.328060i	0.986455	−	0.164030i	$-0.0524494\pi$
$224$		0			0
$225$	4.89898	−	1.00000i	0.326599	−	0.0666667i
$226$		0			0
$227$		−	19.5959i		−	1.30063i	−0.759666	−	0.650313i	$-0.774638\pi$
$227$		−	19.5959i		−	1.30063i	0.759666	−	0.650313i	$-0.225362\pi$
$228$		0			0
$229$	−18.0000			−1.18947			−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000			−1.18947			−0.594737	+	0.803921i	$0.702744\pi$
$230$		0			0
$231$	−12.0000			−0.789542
$232$		0			0
$233$		−	1.10102i		−	0.0721303i	−0.999349	−	0.0360651i	$-0.988518\pi$
$233$		−	1.10102i		−	0.0721303i	0.999349	−	0.0360651i	$-0.0114824\pi$
$234$		0			0
$235$	10.8990	−	1.10102i	0.710971	−	0.0718227i
$236$		0			0
$237$			1.55051i			0.100716i
$238$		0			0
$239$	8.89898			0.575627			0.287814	−	0.957686i	$-0.407072\pi$
$239$	8.89898			0.575627			0.287814	+	0.957686i	$0.407072\pi$
$240$		0			0
$241$	−6.89898			−0.444402			−0.222201	−	0.975001i	$-0.571324\pi$
$241$	−6.89898			−0.444402			−0.222201	+	0.975001i	$0.571324\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−0.224745	−	2.22474i	−0.0143584	−	0.142134i
$246$		0			0
$247$			41.3939i			2.63383i
$248$		0			0
$249$	−2.89898			−0.183715
$250$		0			0
$251$	20.6969			1.30638			0.653190	−	0.757194i	$-0.273430\pi$
$251$	20.6969			1.30638			0.653190	+	0.757194i	$0.273430\pi$
$252$		0			0
$253$		−	31.5959i		−	1.98642i
$254$		0			0
$255$	0.224745	+	2.22474i	0.0140741	+	0.139319i
$256$		0			0
$257$			20.0000i			1.24757i	0.781598	+	0.623783i	$0.214405\pi$
$257$			20.0000i			1.24757i	−0.781598	+	0.623783i	$0.785595\pi$
$258$		0			0
$259$	1.10102			0.0684141
$260$		0			0
$261$	−9.34847			−0.578656
$262$		0			0
$263$			26.6969i			1.64620i	0.567894	+	0.823102i	$0.307758\pi$
$263$			26.6969i			1.64620i	−0.567894	+	0.823102i	$0.692242\pi$
$264$		0			0
$265$	−2.44949	+	0.247449i	−0.150471	+	0.0152007i
$266$		0			0
$267$		−	1.79796i		−	0.110033i
$268$		0			0
$269$	−23.5505			−1.43590			−0.717950	−	0.696095i	$-0.754919\pi$
$269$	−23.5505			−1.43590			−0.717950	+	0.696095i	$0.754919\pi$
$270$		0			0
$271$	7.59592			0.461419			0.230710	−	0.973023i	$-0.425895\pi$
$271$	7.59592			0.461419			0.230710	+	0.973023i	$0.425895\pi$
$272$		0			0
$273$		−	14.6969i		−	0.889499i
$274$		0			0
$275$	−24.0000	+	4.89898i	−1.44725	+	0.295420i
$276$		0			0
$277$			28.4495i			1.70936i	0.519152	+	0.854682i	$0.326248\pi$
$277$			28.4495i			1.70936i	−0.519152	+	0.854682i	$0.673752\pi$
$278$		0			0
$279$	−6.44949			−0.386121
$280$		0			0
$281$	9.79796			0.584497			0.292249	−	0.956342i	$-0.405597\pi$
$281$	9.79796			0.584497			0.292249	+	0.956342i	$0.405597\pi$
$282$		0			0
$283$			6.69694i			0.398092i	0.979990	+	0.199046i	$0.0637843\pi$
$283$			6.69694i			0.398092i	−0.979990	+	0.199046i	$0.936216\pi$
$284$		0			0
$285$	−15.3485	+	1.55051i	−0.909165	+	0.0918443i
$286$		0			0
$287$			2.69694i			0.159195i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	3.79796			0.222640
$292$		0			0
$293$		−	10.0000i		−	0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$		−	10.0000i		−	0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$		0			0
$295$	−1.30306	−	12.8990i	−0.0758672	−	0.751008i
$296$		0			0
$297$			4.89898i			0.284268i
$298$		0			0
$299$	38.6969			2.23790
$300$		0			0
$301$	7.10102			0.409296
$302$		0			0
$303$		−	3.79796i		−	0.218187i
$304$		0			0
$305$	−3.00000	−	29.6969i	−0.171780	−	1.70044i
$306$		0			0
$307$			10.8990i			0.622038i	0.950404	+	0.311019i	$0.100670\pi$
$307$			10.8990i			0.622038i	−0.950404	+	0.311019i	$0.899330\pi$
$308$		0			0
$309$	−16.8990			−0.961349
$310$		0			0
$311$	23.3485			1.32397			0.661985	−	0.749517i	$-0.269714\pi$
$311$	23.3485			1.32397			0.661985	+	0.749517i	$0.269714\pi$
$312$		0			0
$313$			2.00000i			0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$			2.00000i			0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$		0			0
$315$	5.44949	−	0.550510i	0.307044	−	0.0310177i
$316$		0			0
$317$			8.44949i			0.474571i	0.971440	+	0.237285i	$0.0762576\pi$
$317$			8.44949i			0.474571i	−0.971440	+	0.237285i	$0.923742\pi$
$318$		0			0
$319$	45.7980			2.56419
$320$		0			0
$321$	3.10102			0.173082
$322$		0			0
$323$		−	6.89898i		−	0.383869i
$324$		0			0
$325$	−6.00000	−	29.3939i	−0.332820	−	1.63048i
$326$		0			0
$327$			18.2474i			1.00909i
$328$		0			0
$329$	12.0000			0.661581
$330$		0			0
$331$	−33.3939			−1.83549			−0.917747	−	0.397166i	$-0.869994\pi$
$331$	−33.3939			−1.83549			−0.917747	+	0.397166i	$0.869994\pi$
$332$		0			0
$333$		−	0.449490i		−	0.0246319i
$334$		0			0
$335$	8.89898	−	0.898979i	0.486203	−	0.0491165i
$336$		0			0
$337$		−	2.89898i		−	0.157917i	−0.996878	−	0.0789587i	$-0.974840\pi$
$337$		−	2.89898i		−	0.157917i	0.996878	−	0.0789587i	$-0.0251595\pi$
$338$		0			0
$339$	16.6969			0.906853
$340$		0			0
$341$	31.5959			1.71101
$342$		0			0
$343$		−	19.5959i		−	1.05808i
$344$		0			0
$345$	1.44949	+	14.3485i	0.0780379	+	0.772496i
$346$		0			0
$347$		−	13.7980i		−	0.740713i	−0.928890	−	0.370357i	$-0.879235\pi$
$347$		−	13.7980i		−	0.740713i	0.928890	−	0.370357i	$-0.120765\pi$
$348$		0			0
$349$	−7.30306			−0.390924			−0.195462	−	0.980711i	$-0.562621\pi$
$349$	−7.30306			−0.390924			−0.195462	+	0.980711i	$0.562621\pi$
$350$		0			0
$351$	−6.00000			−0.320256
$352$		0			0
$353$			10.0000i			0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$			10.0000i			0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$		0			0
$355$	0.550510	+	5.44949i	0.0292181	+	0.289229i
$356$		0			0
$357$			2.44949i			0.129641i
$358$		0			0
$359$	−6.20204			−0.327331			−0.163666	−	0.986516i	$-0.552332\pi$
$359$	−6.20204			−0.327331			−0.163666	+	0.986516i	$0.552332\pi$
$360$		0			0
$361$	28.5959			1.50505
$362$		0			0
$363$		−	13.0000i		−	0.682323i
$364$		0			0
$365$	33.1464	−	3.34847i	1.73496	−	0.175267i
$366$		0			0
$367$		−	14.0454i		−	0.733164i	−0.930386	−	0.366582i	$-0.880528\pi$
$367$		−	14.0454i		−	0.733164i	0.930386	−	0.366582i	$-0.119472\pi$
$368$		0			0
$369$	1.10102			0.0573168
$370$		0			0
$371$	−2.69694			−0.140018
$372$		0			0
$373$		−	11.7980i		−	0.610875i	−0.952212	−	0.305438i	$-0.901197\pi$
$373$		−	11.7980i		−	0.610875i	0.952212	−	0.305438i	$-0.0988027\pi$
$374$		0			0
$375$	10.6742	−	3.32577i	0.551215	−	0.171742i
$376$		0			0
$377$			56.0908i			2.88882i
$378$		0			0
$379$	−10.2020			−0.524044			−0.262022	−	0.965062i	$-0.584389\pi$
$379$	−10.2020			−0.524044			−0.262022	+	0.965062i	$0.584389\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$		0			0
$383$		−	16.8990i		−	0.863498i	−0.901994	−	0.431749i	$-0.857897\pi$
$383$		−	16.8990i		−	0.863498i	0.901994	−	0.431749i	$-0.142103\pi$
$384$		0			0
$385$	−26.6969	+	2.69694i	−1.36060	+	0.137449i
$386$		0			0
$387$		−	2.89898i		−	0.147363i
$388$		0			0
$389$	−3.30306			−0.167472			−0.0837359	−	0.996488i	$-0.526685\pi$
$389$	−3.30306			−0.167472			−0.0837359	+	0.996488i	$0.526685\pi$
$390$		0			0
$391$	−6.44949			−0.326165
$392$		0			0
$393$			4.89898i			0.247121i
$394$		0			0
$395$	0.348469	+	3.44949i	0.0175334	+	0.173563i
$396$		0			0
$397$			36.0454i			1.80907i	0.426402	+	0.904534i	$0.359781\pi$
$397$			36.0454i			1.80907i	−0.426402	+	0.904534i	$0.640219\pi$
$398$		0			0
$399$	−16.8990			−0.846007
$400$		0			0
$401$	−0.696938			−0.0348034			−0.0174017	−	0.999849i	$-0.505539\pi$
$401$	−0.696938			−0.0348034			−0.0174017	+	0.999849i	$0.505539\pi$
$402$		0			0
$403$			38.6969i			1.92763i
$404$		0			0
$405$	−0.224745	−	2.22474i	−0.0111677	−	0.110548i
$406$		0			0
$407$			2.20204i			0.109151i
$408$		0			0
$409$	−15.5959			−0.771169			−0.385584	−	0.922673i	$-0.626000\pi$
$409$	−15.5959			−0.771169			−0.385584	+	0.922673i	$0.626000\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$		−	14.2020i		−	0.698837i
$414$		0			0
$415$	−6.44949	+	0.651531i	−0.316593	+	0.0319824i
$416$		0			0
$417$			16.8990i			0.827547i
$418$		0			0
$419$	7.59592			0.371085			0.185542	−	0.982636i	$-0.440596\pi$
$419$	7.59592			0.371085			0.185542	+	0.982636i	$0.440596\pi$
$420$		0			0
$421$	−0.696938			−0.0339667			−0.0169834	−	0.999856i	$-0.505406\pi$
$421$	−0.696938			−0.0339667			−0.0169834	+	0.999856i	$0.505406\pi$
$422$		0			0
$423$		−	4.89898i		−	0.238197i
$424$		0			0
$425$	1.00000	+	4.89898i	0.0485071	+	0.237635i
$426$		0			0
$427$		−	32.6969i		−	1.58232i
$428$		0			0
$429$	29.3939			1.41915
$430$		0			0
$431$	−25.1464			−1.21126			−0.605630	−	0.795746i	$-0.707079\pi$
$431$	−25.1464			−1.21126			−0.605630	+	0.795746i	$0.707079\pi$
$432$		0			0
$433$		−	10.0000i		−	0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$		−	10.0000i		−	0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$		0			0
$435$	−20.7980	+	2.10102i	−0.997186	+	0.100736i
$436$		0			0
$437$		−	44.4949i		−	2.12848i
$438$		0			0
$439$	−27.3485			−1.30527			−0.652636	−	0.757672i	$-0.726337\pi$
$439$	−27.3485			−1.30527			−0.652636	+	0.757672i	$0.726337\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$			10.2020i			0.484714i	0.970187	+	0.242357i	$0.0779205\pi$
$443$			10.2020i			0.484714i	−0.970187	+	0.242357i	$0.922080\pi$
$444$		0			0
$445$	−0.404082	−	4.00000i	−0.0191553	−	0.189618i
$446$		0			0
$447$			12.6969i			0.600545i
$448$		0			0
$449$	32.6969			1.54306			0.771532	−	0.636191i	$-0.219491\pi$
$449$	32.6969			1.54306			0.771532	+	0.636191i	$0.219491\pi$
$450$		0			0
$451$	−5.39388			−0.253988
$452$		0			0
$453$			13.7980i			0.648285i
$454$		0			0
$455$	−3.30306	−	32.6969i	−0.154850	−	1.53286i
$456$		0			0
$457$		−	1.59592i		−	0.0746539i	−0.999303	−	0.0373269i	$-0.988116\pi$
$457$		−	1.59592i		−	0.0746539i	0.999303	−	0.0373269i	$-0.0118843\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$	−21.5959			−1.00582			−0.502911	−	0.864338i	$-0.667738\pi$
$461$	−21.5959			−1.00582			−0.502911	+	0.864338i	$0.667738\pi$
$462$		0			0
$463$			5.30306i			0.246454i	0.992378	+	0.123227i	$0.0393244\pi$
$463$			5.30306i			0.246454i	−0.992378	+	0.123227i	$0.960676\pi$
$464$		0			0
$465$	−14.3485	+	1.44949i	−0.665394	+	0.0672185i
$466$		0			0
$467$		−	0.404082i		−	0.0186987i	−0.999956	−	0.00934934i	$-0.997024\pi$
$467$		−	0.404082i		−	0.0186987i	0.999956	−	0.00934934i	$-0.00297603\pi$
$468$		0			0
$469$	9.79796			0.452428
$470$		0			0
$471$	−16.6969			−0.769354
$472$		0			0
$473$			14.2020i			0.653011i
$474$		0			0
$475$	−33.7980	+	6.89898i	−1.55076	+	0.316547i
$476$		0			0
$477$			1.10102i			0.0504123i
$478$		0			0
$479$	15.3485			0.701289			0.350645	−	0.936509i	$-0.385962\pi$
$479$	15.3485			0.701289			0.350645	+	0.936509i	$0.385962\pi$
$480$		0			0
$481$	−2.69694			−0.122970
$482$		0			0
$483$			15.7980i			0.718832i
$484$		0			0
$485$	8.44949	−	0.853572i	0.383672	−	0.0387587i
$486$		0			0
$487$		−	15.3485i		−	0.695506i	−0.937586	−	0.347753i	$-0.886945\pi$
$487$		−	15.3485i		−	0.695506i	0.937586	−	0.347753i	$-0.113055\pi$
$488$		0			0
$489$	−5.79796			−0.262193
$490$		0			0
$491$	14.8990			0.672382			0.336191	−	0.941794i	$-0.390861\pi$
$491$	14.8990			0.672382			0.336191	+	0.941794i	$0.390861\pi$
$492$		0			0
$493$		−	9.34847i		−	0.421034i
$494$		0			0
$495$	1.10102	+	10.8990i	0.0494872	+	0.489873i
$496$		0			0
$497$			6.00000i			0.269137i
$498$		0			0
$499$	0.898979			0.0402438			0.0201219	−	0.999798i	$-0.493595\pi$
$499$	0.898979			0.0402438			0.0201219	+	0.999798i	$0.493595\pi$
$500$		0			0
$501$	4.65153			0.207815
$502$		0			0
$503$		−	33.5505i		−	1.49594i	−0.663731	−	0.747972i	$-0.731028\pi$
$503$		−	33.5505i		−	1.49594i	0.663731	−	0.747972i	$-0.268972\pi$
$504$		0			0
$505$	−0.853572	−	8.44949i	−0.0379834	−	0.375997i
$506$		0			0
$507$			23.0000i			1.02147i
$508$		0			0
$509$	10.8990			0.483089			0.241544	−	0.970390i	$-0.422346\pi$
$509$	10.8990			0.483089			0.241544	+	0.970390i	$0.422346\pi$
$510$		0			0
$511$	36.4949			1.61444
$512$		0			0
$513$			6.89898i			0.304597i
$514$		0			0
$515$	−37.5959	+	3.79796i	−1.65667	+	0.167358i
$516$		0			0
$517$			24.0000i			1.05552i
$518$		0			0
$519$	−13.3485			−0.585933
$520$		0			0
$521$	−16.6969			−0.731506			−0.365753	−	0.930712i	$-0.619189\pi$
$521$	−16.6969			−0.731506			−0.365753	+	0.930712i	$0.619189\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$		0			0
$525$	12.0000	−	2.44949i	0.523723	−	0.106904i
$526$		0			0
$527$		−	6.44949i		−	0.280944i
$528$		0			0
$529$	−18.5959			−0.808518
$530$		0			0
$531$	−5.79796			−0.251610
$532$		0			0
$533$		−	6.60612i		−	0.286143i
$534$		0			0
$535$	6.89898	−	0.696938i	0.298269	−	0.0301313i
$536$		0			0
$537$			20.6969i			0.893139i
$538$		0			0
$539$	4.89898			0.211014
$540$		0			0
$541$	31.1464			1.33909			0.669545	−	0.742772i	$-0.266489\pi$
$541$	31.1464			1.33909			0.669545	+	0.742772i	$0.266489\pi$
$542$		0			0
$543$		−	4.44949i		−	0.190946i
$544$		0			0
$545$	4.10102	+	40.5959i	0.175668	+	1.73894i
$546$		0			0
$547$			18.6969i			0.799423i	0.916641	+	0.399712i	$0.130890\pi$
$547$			18.6969i			0.799423i	−0.916641	+	0.399712i	$0.869110\pi$
$548$		0			0
$549$	−13.3485			−0.569699
$550$		0			0
$551$	64.4949			2.74758
$552$		0			0
$553$			3.79796i			0.161506i
$554$		0			0
$555$	−0.101021	−	1.00000i	−0.00428808	−	0.0424476i
$556$		0			0
$557$			2.89898i			0.122834i	0.998112	+	0.0614169i	$0.0195619\pi$
$557$			2.89898i			0.122834i	−0.998112	+	0.0614169i	$0.980438\pi$
$558$		0			0
$559$	−17.3939			−0.735683
$560$		0			0
$561$	−4.89898			−0.206835
$562$		0			0
$563$		−	6.89898i		−	0.290757i	−0.989376	−	0.145379i	$-0.953560\pi$
$563$		−	6.89898i		−	0.290757i	0.989376	−	0.145379i	$-0.0464400\pi$
$564$		0			0
$565$	37.1464	−	3.75255i	1.56276	−	0.157871i
$566$		0			0
$567$		−	2.44949i		−	0.102869i
$568$		0			0
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$	21.3939			0.895306			0.447653	−	0.894207i	$-0.352260\pi$
$571$	21.3939			0.895306			0.447653	+	0.894207i	$0.352260\pi$
$572$		0			0
$573$		−	16.8990i		−	0.705965i
$574$		0			0
$575$	6.44949	+	31.5959i	0.268962	+	1.31764i
$576$		0			0
$577$			10.2020i			0.424717i	0.977192	+	0.212358i	$0.0681144\pi$
$577$			10.2020i			0.424717i	−0.977192	+	0.212358i	$0.931886\pi$
$578$		0			0
$579$	14.0000			0.581820
$580$		0			0
$581$	−7.10102			−0.294600
$582$		0			0
$583$		−	5.39388i		−	0.223392i
$584$		0			0
$585$	−13.3485	+	1.34847i	−0.551891	+	0.0557523i
$586$		0			0
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	44.4949			1.83338
$590$		0			0
$591$	−19.1464			−0.787579
$592$		0			0
$593$			1.79796i			0.0738333i	0.999318	+	0.0369167i	$0.0117536\pi$
$593$			1.79796i			0.0738333i	−0.999318	+	0.0369167i	$0.988246\pi$
$594$		0			0
$595$	0.550510	+	5.44949i	0.0225687	+	0.223407i
$596$		0			0
$597$			18.0454i			0.738549i
$598$		0			0
$599$	13.7980			0.563769			0.281885	−	0.959448i	$-0.409040\pi$
$599$	13.7980			0.563769			0.281885	+	0.959448i	$0.409040\pi$
$600$		0			0
$601$	0.202041			0.00824143			0.00412071	−	0.999992i	$-0.498688\pi$
$601$	0.202041			0.00824143			0.00412071	+	0.999992i	$0.498688\pi$
$602$		0			0
$603$		−	4.00000i		−	0.162893i
$604$		0			0
$605$	−2.92168	−	28.9217i	−0.118783	−	1.17583i
$606$		0			0
$607$		−	26.4495i		−	1.07355i	−0.843725	−	0.536776i	$-0.819642\pi$
$607$		−	26.4495i		−	1.07355i	0.843725	−	0.536776i	$-0.180358\pi$
$608$		0			0
$609$	−22.8990			−0.927913
$610$		0			0
$611$	−29.3939			−1.18915
$612$		0			0
$613$		−	48.2929i		−	1.95053i	−0.221039	−	0.975265i	$-0.570945\pi$
$613$		−	48.2929i		−	1.95053i	0.221039	−	0.975265i	$-0.429055\pi$
$614$		0			0
$615$	2.44949	−	0.247449i	0.0987730	−	0.00997810i
$616$		0			0
$617$			26.4949i			1.06664i	0.845912	+	0.533322i	$0.179057\pi$
$617$			26.4949i			1.06664i	−0.845912	+	0.533322i	$0.820943\pi$
$618$		0			0
$619$	−16.0000			−0.643094			−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000			−0.643094			−0.321547	+	0.946894i	$0.604203\pi$
$620$		0			0
$621$	6.44949			0.258809
$622$		0			0
$623$		−	4.40408i		−	0.176446i
$624$		0			0
$625$	23.0000	−	9.79796i	0.920000	−	0.391918i
$626$		0			0
$627$		−	33.7980i		−	1.34976i
$628$		0			0
$629$	0.449490			0.0179223
$630$		0			0
$631$	−24.4949			−0.975126			−0.487563	−	0.873088i	$-0.662114\pi$
$631$	−24.4949			−0.975126			−0.487563	+	0.873088i	$0.662114\pi$
$632$		0			0
$633$		−	3.10102i		−	0.123255i
$634$		0			0
$635$	−17.7980	+	1.79796i	−0.706290	+	0.0713498i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$	2.44949			0.0969003
$640$		0			0
$641$	8.69694			0.343508			0.171754	−	0.985140i	$-0.445057\pi$
$641$	8.69694			0.343508			0.171754	+	0.985140i	$0.445057\pi$
$642$		0			0
$643$			8.00000i			0.315489i	0.987480	+	0.157745i	$0.0504223\pi$
$643$			8.00000i			0.315489i	−0.987480	+	0.157745i	$0.949578\pi$
$644$		0			0
$645$	−0.651531	−	6.44949i	−0.0256540	−	0.253948i
$646$		0			0
$647$			30.2929i			1.19094i	0.803379	+	0.595468i	$0.203033\pi$
$647$			30.2929i			1.19094i	−0.803379	+	0.595468i	$0.796967\pi$
$648$		0			0
$649$	28.4041			1.11496
$650$		0			0
$651$	−15.7980			−0.619171
$652$		0			0
$653$			20.4495i			0.800250i	0.916460	+	0.400125i	$0.131033\pi$
$653$			20.4495i			0.800250i	−0.916460	+	0.400125i	$0.868967\pi$
$654$		0			0
$655$	1.10102	+	10.8990i	0.0430204	+	0.425858i
$656$		0			0
$657$		−	14.8990i		−	0.581265i
$658$		0			0
$659$	−15.5959			−0.607531			−0.303765	−	0.952747i	$-0.598244\pi$
$659$	−15.5959			−0.607531			−0.303765	+	0.952747i	$0.598244\pi$
$660$		0			0
$661$	−46.4949			−1.80844			−0.904221	−	0.427065i	$-0.859548\pi$
$661$	−46.4949			−1.80844			−0.904221	+	0.427065i	$0.859548\pi$
$662$		0			0
$663$		−	6.00000i		−	0.233021i
$664$		0			0
$665$	−37.5959	+	3.79796i	−1.45791	+	0.147279i
$666$		0			0
$667$		−	60.2929i		−	2.33455i
$668$		0			0
$669$	−4.89898			−0.189405
$670$		0			0
$671$	65.3939			2.52450
$672$		0			0
$673$			34.0000i			1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$			34.0000i			1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$		0			0
$675$	−1.00000	−	4.89898i	−0.0384900	−	0.188562i
$676$		0			0
$677$		−	47.6413i		−	1.83100i	−0.402312	−	0.915502i	$-0.631793\pi$
$677$		−	47.6413i		−	1.83100i	0.402312	−	0.915502i	$-0.368207\pi$
$678$		0			0
$679$	9.30306			0.357019
$680$		0			0
$681$	−19.5959			−0.750917
$682$		0			0
$683$			9.79796i			0.374908i	0.982273	+	0.187454i	$0.0600236\pi$
$683$			9.79796i			0.374908i	−0.982273	+	0.187454i	$0.939976\pi$
$684$		0			0
$685$	−26.6969	+	2.69694i	−1.02004	+	0.103045i
$686$		0			0
$687$			18.0000i			0.686743i
$688$		0			0
$689$	6.60612			0.251673
$690$		0			0
$691$	50.2929			1.91323			0.956615	−	0.291354i	$-0.0941059\pi$
$691$	50.2929			1.91323			0.956615	+	0.291354i	$0.0941059\pi$
$692$		0			0
$693$			12.0000i			0.455842i
$694$		0			0
$695$	3.79796	+	37.5959i	0.144065	+	1.42609i
$696$		0			0
$697$			1.10102i			0.0417041i
$698$		0			0
$699$	−1.10102			−0.0416444
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$			3.10102i			0.116957i
$704$		0			0
$705$	−1.10102	−	10.8990i	−0.0414668	−	0.410479i
$706$		0			0
$707$		−	9.30306i		−	0.349878i
$708$		0			0
$709$	24.4495			0.918220			0.459110	−	0.888379i	$-0.348168\pi$
$709$	24.4495			0.918220			0.459110	+	0.888379i	$0.348168\pi$
$710$		0			0
$711$	1.55051			0.0581487
$712$		0			0
$713$		−	41.5959i		−	1.55778i
$714$		0			0
$715$	65.3939	−	6.60612i	2.44559	−	0.247055i
$716$		0			0
$717$		−	8.89898i		−	0.332338i
$718$		0			0
$719$	12.2474			0.456753			0.228376	−	0.973573i	$-0.426658\pi$
$719$	12.2474			0.456753			0.228376	+	0.973573i	$0.426658\pi$
$720$		0			0
$721$	−41.3939			−1.54159
$722$		0			0
$723$			6.89898i			0.256576i
$724$		0			0
$725$	−45.7980	+	9.34847i	−1.70089	+	0.347193i
$726$		0			0
$727$		−	32.0000i		−	1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$		−	32.0000i		−	1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	2.89898			0.107223
$732$		0			0
$733$			19.7980i			0.731254i	0.930761	+	0.365627i	$0.119145\pi$
$733$			19.7980i			0.731254i	−0.930761	+	0.365627i	$0.880855\pi$
$734$		0			0
$735$	−2.22474	+	0.224745i	−0.0820610	+	0.00828984i
$736$		0			0
$737$			19.5959i			0.721825i
$738$		0			0
$739$	−17.1010			−0.629071			−0.314536	−	0.949246i	$-0.601849\pi$
$739$	−17.1010			−0.629071			−0.314536	+	0.949246i	$0.601849\pi$
$740$		0			0
$741$	41.3939			1.52064
$742$		0			0
$743$		−	14.4495i		−	0.530100i	−0.964235	−	0.265050i	$-0.914611\pi$
$743$		−	14.4495i		−	0.530100i	0.964235	−	0.265050i	$-0.0853885\pi$
$744$		0			0
$745$	2.85357	+	28.2474i	0.104547	+	1.03491i
$746$		0			0
$747$			2.89898i			0.106068i
$748$		0			0
$749$	7.59592			0.277549
$750$		0			0
$751$	−8.24745			−0.300954			−0.150477	−	0.988614i	$-0.548081\pi$
$751$	−8.24745			−0.300954			−0.150477	+	0.988614i	$0.548081\pi$
$752$		0			0
$753$		−	20.6969i		−	0.754238i
$754$		0			0
$755$	3.10102	+	30.6969i	0.112858	+	1.11718i
$756$		0			0
$757$		−	5.50510i		−	0.200086i	−0.994983	−	0.100043i	$-0.968102\pi$
$757$		−	5.50510i		−	0.200086i	0.994983	−	0.100043i	$-0.0318981\pi$
$758$		0			0
$759$	−31.5959			−1.14686
$760$		0			0
$761$	9.59592			0.347852			0.173926	−	0.984759i	$-0.444355\pi$
$761$	9.59592			0.347852			0.173926	+	0.984759i	$0.444355\pi$
$762$		0			0
$763$			44.6969i			1.61814i
$764$		0			0
$765$	2.22474	−	0.224745i	0.0804358	−	0.00812567i
$766$		0			0
$767$			34.7878i			1.25611i
$768$		0			0
$769$	−27.3939			−0.987848			−0.493924	−	0.869505i	$-0.664438\pi$
$769$	−27.3939			−0.987848			−0.493924	+	0.869505i	$0.664438\pi$
$770$		0			0
$771$	20.0000			0.720282
$772$		0			0
$773$			42.0000i			1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$			42.0000i			1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$		0			0
$775$	−31.5959	+	6.44949i	−1.13496	+	0.231673i
$776$		0			0
$777$		−	1.10102i		−	0.0394989i
$778$		0			0
$779$	−7.59592			−0.272152
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$			9.34847i			0.334087i
$784$		0			0
$785$	−37.1464	+	3.75255i	−1.32581	+	0.133934i
$786$		0			0
$787$		−	17.3031i		−	0.616788i	−0.951259	−	0.308394i	$-0.900209\pi$
$787$		−	17.3031i		−	0.616788i	0.951259	−	0.308394i	$-0.0997914\pi$
$788$		0			0
$789$	26.6969			0.950436
$790$		0			0
$791$	40.8990			1.45420
$792$		0			0
$793$			80.0908i			2.84411i
$794$		0			0
$795$	0.247449	+	2.44949i	0.00877610	+	0.0868744i
$796$		0			0
$797$			22.8990i			0.811123i	0.914068	+	0.405562i	$0.132924\pi$
$797$			22.8990i			0.811123i	−0.914068	+	0.405562i	$0.867076\pi$
$798$		0			0
$799$	4.89898			0.173313
$800$		0			0
$801$	−1.79796			−0.0635278
$802$		0			0
$803$			72.9898i			2.57575i
$804$		0			0
$805$	3.55051	+	35.1464i	0.125139	+	1.23875i
$806$		0			0
$807$			23.5505i			0.829017i
$808$		0			0
$809$	−42.0908			−1.47983			−0.739917	−	0.672698i	$-0.765135\pi$
$809$	−42.0908			−1.47983			−0.739917	+	0.672698i	$0.765135\pi$
$810$		0			0
$811$	−1.79796			−0.0631349			−0.0315674	−	0.999502i	$-0.510050\pi$
$811$	−1.79796			−0.0631349			−0.0315674	+	0.999502i	$0.510050\pi$
$812$		0			0
$813$		−	7.59592i		−	0.266400i
$814$		0			0
$815$	−12.8990	+	1.30306i	−0.451832	+	0.0456443i
$816$		0			0
$817$			20.0000i			0.699711i
$818$		0			0
$819$	−14.6969			−0.513553
$820$		0			0
$821$	35.6413			1.24389			0.621945	−	0.783061i	$-0.286343\pi$
$821$	35.6413			1.24389			0.621945	+	0.783061i	$0.286343\pi$
$822$		0			0
$823$			26.4495i			0.921971i	0.887408	+	0.460986i	$0.152504\pi$
$823$			26.4495i			0.921971i	−0.887408	+	0.460986i	$0.847496\pi$
$824$		0			0
$825$	4.89898	+	24.0000i	0.170561	+	0.835573i
$826$		0			0
$827$			16.4949i			0.573584i	0.957993	+	0.286792i	$0.0925888\pi$
$827$			16.4949i			0.573584i	−0.957993	+	0.286792i	$0.907411\pi$
$828$		0			0
$829$	−8.20204			−0.284869			−0.142434	−	0.989804i	$-0.545493\pi$
$829$	−8.20204			−0.284869			−0.142434	+	0.989804i	$0.545493\pi$
$830$		0			0
$831$	28.4495			0.986902
$832$		0			0
$833$		−	1.00000i		−	0.0346479i
$834$		0			0
$835$	10.3485	−	1.04541i	0.358124	−	0.0361778i
$836$		0			0
$837$			6.44949i			0.222927i
$838$		0			0
$839$	−39.3485			−1.35846			−0.679230	−	0.733925i	$-0.737686\pi$
$839$	−39.3485			−1.35846			−0.679230	+	0.733925i	$0.737686\pi$
$840$		0			0
$841$	58.3939			2.01358
$842$		0			0
$843$		−	9.79796i		−	0.337460i
$844$		0			0
$845$	5.16913	+	51.1691i	0.177824	+	1.76027i
$846$		0			0
$847$		−	31.8434i		−	1.09415i
$848$		0			0
$849$	6.69694			0.229838
$850$		0			0
$851$	2.89898			0.0993757
$852$		0			0
$853$			10.6515i			0.364701i	0.983234	+	0.182351i	$0.0583706\pi$
$853$			10.6515i			0.364701i	−0.983234	+	0.182351i	$0.941629\pi$
$854$		0			0
$855$	1.55051	+	15.3485i	0.0530263	+	0.524907i
$856$		0			0
$857$		−	6.00000i		−	0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$		−	6.00000i		−	0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$		0			0
$859$	−26.2020			−0.894002			−0.447001	−	0.894533i	$-0.647508\pi$
$859$	−26.2020			−0.894002			−0.447001	+	0.894533i	$0.647508\pi$
$860$		0			0
$861$	2.69694			0.0919114
$862$		0			0
$863$		−	23.5959i		−	0.803214i	−0.915812	−	0.401607i	$-0.868452\pi$
$863$		−	23.5959i		−	0.803214i	0.915812	−	0.401607i	$-0.131548\pi$
$864$		0			0
$865$	−29.6969	+	3.00000i	−1.00973	+	0.102003i
$866$		0			0
$867$			1.00000i			0.0339618i
$868$		0			0
$869$	−7.59592			−0.257674
$870$		0			0
$871$	−24.0000			−0.813209
$872$		0			0
$873$		−	3.79796i		−	0.128541i
$874$		0			0
$875$	26.1464	−	8.14643i	0.883911	−	0.275400i
$876$		0			0
$877$		−	18.6515i		−	0.629817i	−0.949122	−	0.314909i	$-0.898026\pi$
$877$		−	18.6515i		−	0.629817i	0.949122	−	0.314909i	$-0.101974\pi$
$878$		0			0
$879$	−10.0000			−0.337292
$880$		0			0
$881$	−37.1918			−1.25302			−0.626512	−	0.779411i	$-0.715518\pi$
$881$	−37.1918			−1.25302			−0.626512	+	0.779411i	$0.715518\pi$
$882$		0			0
$883$			10.2020i			0.343326i	0.985156	+	0.171663i	$0.0549140\pi$
$883$			10.2020i			0.343326i	−0.985156	+	0.171663i	$0.945086\pi$
$884$		0			0
$885$	−12.8990	+	1.30306i	−0.433594	+	0.0438019i
$886$		0			0
$887$			40.2474i			1.35138i	0.737187	+	0.675689i	$0.236154\pi$
$887$			40.2474i			1.35138i	−0.737187	+	0.675689i	$0.763846\pi$
$888$		0			0
$889$	−19.5959			−0.657226
$890$		0			0
$891$	4.89898			0.164122
$892$		0			0
$893$			33.7980i			1.13101i
$894$		0			0
$895$	4.65153	+	46.0454i	0.155484	+	1.53913i
$896$		0			0
$897$		−	38.6969i		−	1.29205i
$898$		0			0
$899$	60.2929			2.01088
$900$		0			0
$901$	−1.10102			−0.0366803
$902$		0			0
$903$		−	7.10102i		−	0.236307i
$904$		0			0
$905$	−1.00000	−	9.89898i	−0.0332411	−	0.329053i
$906$		0			0
$907$			50.6969i			1.68336i	0.539973	+	0.841682i	$0.318434\pi$
$907$			50.6969i			1.68336i	−0.539973	+	0.841682i	$0.681566\pi$
$908$		0			0
$909$	−3.79796			−0.125970
$910$		0			0
$911$	32.6515			1.08179			0.540897	−	0.841089i	$-0.318085\pi$
$911$	32.6515			1.08179			0.540897	+	0.841089i	$0.318085\pi$
$912$		0			0
$913$		−	14.2020i		−	0.470019i
$914$		0			0
$915$	−29.6969	+	3.00000i	−0.981751	+	0.0991769i
$916$		0			0
$917$			12.0000i			0.396275i
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$	10.8990			0.359134
$922$		0			0
$923$		−	14.6969i		−	0.483756i
$924$		0			0
$925$	−0.449490	−	2.20204i	−0.0147791	−	0.0724027i
$926$		0			0
$927$			16.8990i			0.555035i
$928$		0			0
$929$	36.6969			1.20399			0.601994	−	0.798501i	$-0.294373\pi$
$929$	36.6969			1.20399			0.601994	+	0.798501i	$0.294373\pi$
$930$		0			0
$931$	6.89898			0.226105
$932$		0			0
$933$		−	23.3485i		−	0.764395i
$934$		0			0
$935$	−10.8990	+	1.10102i	−0.356435	+	0.0360072i
$936$		0			0
$937$		−	22.0000i		−	0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$		−	22.0000i		−	0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$		0			0
$939$	2.00000			0.0652675
$940$		0			0
$941$	9.75255			0.317924			0.158962	−	0.987285i	$-0.449185\pi$
$941$	9.75255			0.317924			0.158962	+	0.987285i	$0.449185\pi$
$942$		0			0
$943$			7.10102i			0.231241i
$944$		0			0
$945$	−0.550510	−	5.44949i	−0.0179081	−	0.177272i
$946$		0			0
$947$			31.1010i			1.01065i	0.862930	+	0.505324i	$0.168627\pi$
$947$			31.1010i			1.01065i	−0.862930	+	0.505324i	$0.831373\pi$
$948$		0			0
$949$	−89.3939			−2.90185
$950$		0			0
$951$	8.44949			0.273993
$952$		0			0
$953$		−	33.5959i		−	1.08828i	−0.838995	−	0.544139i	$-0.816856\pi$
$953$		−	33.5959i		−	1.08828i	0.838995	−	0.544139i	$-0.183144\pi$
$954$		0			0
$955$	−3.79796	−	37.5959i	−0.122899	−	1.21658i
$956$		0			0
$957$		−	45.7980i		−	1.48044i
$958$		0			0
$959$	−29.3939			−0.949178
$960$		0			0
$961$	10.5959			0.341804
$962$		0			0
$963$		−	3.10102i		−	0.0999290i
$964$		0			0
$965$	31.1464	−	3.14643i	1.00264	−	0.101287i
$966$		0			0
$967$		−	42.6969i		−	1.37304i	−0.727110	−	0.686520i	$-0.759137\pi$
$967$		−	42.6969i		−	1.37304i	0.727110	−	0.686520i	$-0.240863\pi$
$968$		0			0
$969$	−6.89898			−0.221627
$970$		0			0
$971$	44.0000			1.41203			0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000			1.41203			0.706014	+	0.708198i	$0.250492\pi$
$972$		0			0
$973$			41.3939i			1.32703i
$974$		0			0
$975$	−29.3939	+	6.00000i	−0.941357	+	0.192154i
$976$		0			0
$977$		−	0.404082i		−	0.0129277i	−0.999979	−	0.00646387i	$-0.997942\pi$
$977$		−	0.404082i		−	0.0129277i	0.999979	−	0.00646387i	$-0.00205753\pi$
$978$		0			0
$979$	8.80816			0.281510
$980$		0			0
$981$	18.2474			0.582596
$982$		0			0
$983$		−	18.8536i		−	0.601336i	−0.953729	−	0.300668i	$-0.902790\pi$
$983$		−	18.8536i		−	0.601336i	0.953729	−	0.300668i	$-0.0972095\pi$
$984$		0			0
$985$	−42.5959	+	4.30306i	−1.35722	+	0.137107i
$986$		0			0
$987$		−	12.0000i		−	0.381964i
$988$		0			0
$989$	18.6969			0.594528
$990$		0			0
$991$	−32.2474			−1.02437			−0.512187	−	0.858874i	$-0.671165\pi$
$991$	−32.2474			−1.02437			−0.512187	+	0.858874i	$0.671165\pi$
$992$		0			0
$993$			33.3939i			1.05972i
$994$		0			0
$995$	4.05561	+	40.1464i	0.128572	+	1.27273i
$996$		0			0
$997$		−	46.7423i		−	1.48034i	−0.672417	−	0.740172i	$-0.734744\pi$
$997$		−	46.7423i		−	1.48034i	0.672417	−	0.740172i	$-0.265256\pi$
$998$		0			0
$999$	−0.449490			−0.0142212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.m.o.2449.1		4
4.3	odd	2		510.2.d.c.409.3	yes	4
5.4	even	2	inner	4080.2.m.o.2449.3		4
12.11	even	2		1530.2.d.e.919.2		4
20.3	even	4		2550.2.a.bj.1.2		2
20.7	even	4		2550.2.a.bi.1.1		2
20.19	odd	2		510.2.d.c.409.1	✓	4
60.23	odd	4		7650.2.a.ct.1.2		2
60.47	odd	4		7650.2.a.dg.1.1		2
60.59	even	2		1530.2.d.e.919.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.c.409.1	✓	4	20.19	odd	2
510.2.d.c.409.3	yes	4	4.3	odd	2
1530.2.d.e.919.2		4	12.11	even	2
1530.2.d.e.919.4		4	60.59	even	2
2550.2.a.bi.1.1		2	20.7	even	4
2550.2.a.bj.1.2		2	20.3	even	4
4080.2.m.o.2449.1		4	1.1	even	1	trivial
4080.2.m.o.2449.3		4	5.4	even	2	inner
7650.2.a.ct.1.2		2	60.23	odd	4
7650.2.a.dg.1.1		2	60.47	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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-0.224745 - 2.22474i) q^{5} -2.44949i q^{7} -1.00000 q^{9} +4.89898 q^{11} +6.00000i q^{13} +(-2.22474 + 0.224745i) q^{15} -1.00000i q^{17} +6.89898 q^{19} -2.44949 q^{21} -6.44949i q^{23} +(-4.89898 + 1.00000i) q^{25} +1.00000i q^{27} +9.34847 q^{29} +6.44949 q^{31} -4.89898i q^{33} +(-5.44949 + 0.550510i) q^{35} +0.449490i q^{37} +6.00000 q^{39} -1.10102 q^{41} +2.89898i q^{43} +(0.224745 + 2.22474i) q^{45} +4.89898i q^{47} +1.00000 q^{49} -1.00000 q^{51} -1.10102i q^{53} +(-1.10102 - 10.8990i) q^{55} -6.89898i q^{57} +5.79796 q^{59} +13.3485 q^{61} +2.44949i q^{63} +(13.3485 - 1.34847i) q^{65} +4.00000i q^{67} -6.44949 q^{69} -2.44949 q^{71} +14.8990i q^{73} +(1.00000 + 4.89898i) q^{75} -12.0000i q^{77} -1.55051 q^{79} +1.00000 q^{81} -2.89898i q^{83} +(-2.22474 + 0.224745i) q^{85} -9.34847i q^{87} +1.79796 q^{89} +14.6969 q^{91} -6.44949i q^{93} +(-1.55051 - 15.3485i) q^{95} +3.79796i q^{97} -4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{9} - 4 q^{15} + 8 q^{19} + 8 q^{29} + 16 q^{31} - 12 q^{35} + 24 q^{39} - 24 q^{41} - 4 q^{45} + 4 q^{49} - 4 q^{51} - 24 q^{55} - 16 q^{59} + 24 q^{61} + 24 q^{65} - 16 q^{69} + 4 q^{75}+ \cdots - 16 q^{95}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^9 - 4 * q^15 + 8 * q^19 + 8 * q^29 + 16 * q^31 - 12 * q^35 + 24 * q^39 - 24 * q^41 - 4 * q^45 + 4 * q^49 - 4 * q^51 - 24 * q^55 - 16 * q^59 + 24 * q^61 + 24 * q^65 - 16 * q^69 + 4 * q^75 - 16 * q^79 + 4 * q^81 - 4 * q^85 - 32 * q^89 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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