LMFDB - Embedded newform 3432.2.g.d.1585.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3432,2,Mod(1585,3432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3432.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.g (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:	$27.4046579737$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 25x^{12} + 236x^{10} + 1040x^{8} + 2124x^{6} + 1676x^{4} + 340x^{2} + 16 $ x^14 + 25x^12 + 236x^10 + 1040x^8 + 2124x^6 + 1676x^4 + 340x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.3
Root			$-2.15782i$ of defining polynomial
Character	$\chi$	$=$	3432.1585
Dual form			3432.2.g.d.1585.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.15782i q^{5} -3.35467i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.15782i q^{5} -3.35467i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(1.55466 - 3.25316i) q^{13} -2.15782i q^{15} +3.52753 q^{17} -0.243564i q^{19} -3.35467i q^{21} -4.97180 q^{23} +0.343829 q^{25} +1.00000 q^{27} +5.65697 q^{29} -1.52056i q^{31} +1.00000i q^{33} -7.23876 q^{35} -4.12766i q^{37} +(1.55466 - 3.25316i) q^{39} +1.73126i q^{41} +6.16479 q^{43} -2.15782i q^{45} -6.52136i q^{47} -4.25381 q^{49} +3.52753 q^{51} -5.94781 q^{53} +2.15782 q^{55} -0.243564i q^{57} +3.23041i q^{59} -5.29551 q^{61} -3.35467i q^{63} +(-7.01972 - 3.35467i) q^{65} +2.04164i q^{67} -4.97180 q^{69} -1.57233i q^{71} +5.64365i q^{73} +0.343829 q^{75} +3.35467 q^{77} -7.28784 q^{79} +1.00000 q^{81} -2.31356i q^{83} -7.61177i q^{85} +5.65697 q^{87} +13.8162i q^{89} +(-10.9133 - 5.21537i) q^{91} -1.52056i q^{93} -0.525567 q^{95} -15.7748i q^{97} +1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) $ 14 * q + 14 * q^3 + 14 * q^9	$ 14 q + 14 q^{3} + 14 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{23} + 20 q^{25} + 14 q^{27} - 10 q^{29} - 14 q^{35} - 4 q^{39} + 22 q^{43} + 8 q^{49} - 4 q^{51} - 20 q^{53} + 2 q^{55} - 2 q^{61} - 20 q^{65} + 2 q^{69} + 20 q^{75} + 2 q^{77} - 40 q^{79} + 14 q^{81} - 10 q^{87} - 44 q^{91} + 4 q^{95}+O(q^{100}) $ 14 * q + 14 * q^3 + 14 * q^9 - 4 * q^13 - 4 * q^17 + 2 * q^23 + 20 * q^25 + 14 * q^27 - 10 * q^29 - 14 * q^35 - 4 * q^39 + 22 * q^43 + 8 * q^49 - 4 * q^51 - 20 * q^53 + 2 * q^55 - 2 * q^61 - 20 * q^65 + 2 * q^69 + 20 * q^75 + 2 * q^77 - 40 * q^79 + 14 * q^81 - 10 * q^87 - 44 * q^91 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3432\mathbb{Z}\right)^\times$.

$n$	$937$	$1145$	$1717$	$2575$	$2641$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	2.15782i		−	0.965005i	−0.875895	−	0.482502i	$-0.839728\pi$
$5$		−	2.15782i		−	0.965005i	0.875895	−	0.482502i	$-0.160272\pi$
$6$		0			0
$7$		−	3.35467i		−	1.26795i	−0.773355	−	0.633973i	$-0.781423\pi$
$7$		−	3.35467i		−	1.26795i	0.773355	−	0.633973i	$-0.218577\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	1.55466	−	3.25316i	0.431185	−	0.902264i
$13$	1.55466	−	3.25316i	0.431185	−	0.902264i
$14$		0			0
$15$		−	2.15782i		−	0.557146i
$16$		0			0
$17$	3.52753			0.855552			0.427776	−	0.903885i	$-0.359297\pi$
$17$	3.52753			0.855552			0.427776	+	0.903885i	$0.359297\pi$
$18$		0			0
$19$		−	0.243564i		−	0.0558775i	−0.999610	−	0.0279388i	$-0.991106\pi$
$19$		−	0.243564i		−	0.0558775i	0.999610	−	0.0279388i	$-0.00889434\pi$
$20$		0			0
$21$		−	3.35467i		−	0.732049i
$22$		0			0
$23$	−4.97180			−1.03669			−0.518346	−	0.855171i	$-0.673452\pi$
$23$	−4.97180			−1.03669			−0.518346	+	0.855171i	$0.673452\pi$
$24$		0			0
$25$	0.343829			0.0687658
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	5.65697			1.05047			0.525237	−	0.850956i	$-0.323977\pi$
$29$	5.65697			1.05047			0.525237	+	0.850956i	$0.323977\pi$
$30$		0			0
$31$		−	1.52056i		−	0.273100i	−0.990633	−	0.136550i	$-0.956398\pi$
$31$		−	1.52056i		−	0.273100i	0.990633	−	0.136550i	$-0.0436015\pi$
$32$		0			0
$33$			1.00000i			0.174078i
$34$		0			0
$35$	−7.23876			−1.22357
$36$		0			0
$37$		−	4.12766i		−	0.678582i	−0.940681	−	0.339291i	$-0.889813\pi$
$37$		−	4.12766i		−	0.678582i	0.940681	−	0.339291i	$-0.110187\pi$
$38$		0			0
$39$	1.55466	−	3.25316i	0.248945	−	0.520922i
$40$		0			0
$41$			1.73126i			0.270377i	0.990820	+	0.135188i	$0.0431640\pi$
$41$			1.73126i			0.270377i	−0.990820	+	0.135188i	$0.956836\pi$
$42$		0			0
$43$	6.16479			0.940122			0.470061	−	0.882634i	$-0.344232\pi$
$43$	6.16479			0.940122			0.470061	+	0.882634i	$0.344232\pi$
$44$		0			0
$45$		−	2.15782i		−	0.321668i
$46$		0			0
$47$		−	6.52136i		−	0.951238i	−0.879651	−	0.475619i	$-0.842224\pi$
$47$		−	6.52136i		−	0.951238i	0.879651	−	0.475619i	$-0.157776\pi$
$48$		0			0
$49$	−4.25381			−0.607687
$50$		0			0
$51$	3.52753			0.493953
$52$		0			0
$53$	−5.94781			−0.816995			−0.408498	−	0.912759i	$-0.633947\pi$
$53$	−5.94781			−0.816995			−0.408498	+	0.912759i	$0.633947\pi$
$54$		0			0
$55$	2.15782			0.290960
$56$		0			0
$57$		−	0.243564i		−	0.0322609i
$58$		0			0
$59$			3.23041i			0.420564i	0.977641	+	0.210282i	$0.0674382\pi$
$59$			3.23041i			0.420564i	−0.977641	+	0.210282i	$0.932562\pi$
$60$		0			0
$61$	−5.29551			−0.678021			−0.339010	−	0.940783i	$-0.610092\pi$
$61$	−5.29551			−0.678021			−0.339010	+	0.940783i	$0.610092\pi$
$62$		0			0
$63$		−	3.35467i		−	0.422649i
$64$		0			0
$65$	−7.01972	−	3.35467i	−0.870689	−	0.416096i
$66$		0			0
$67$			2.04164i			0.249425i	0.992193	+	0.124713i	$0.0398009\pi$
$67$			2.04164i			0.249425i	−0.992193	+	0.124713i	$0.960199\pi$
$68$		0			0
$69$	−4.97180			−0.598535
$70$		0			0
$71$		−	1.57233i		−	0.186601i	−0.995638	−	0.0933004i	$-0.970258\pi$
$71$		−	1.57233i		−	0.186601i	0.995638	−	0.0933004i	$-0.0297417\pi$
$72$		0			0
$73$			5.64365i			0.660539i	0.943887	+	0.330269i	$0.107140\pi$
$73$			5.64365i			0.660539i	−0.943887	+	0.330269i	$0.892860\pi$
$74$		0			0
$75$	0.343829			0.0397019
$76$		0			0
$77$	3.35467			0.382300
$78$		0			0
$79$	−7.28784			−0.819946			−0.409973	−	0.912098i	$-0.634462\pi$
$79$	−7.28784			−0.819946			−0.409973	+	0.912098i	$0.634462\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	2.31356i		−	0.253947i	−0.991906	−	0.126973i	$-0.959474\pi$
$83$		−	2.31356i		−	0.253947i	0.991906	−	0.126973i	$-0.0405263\pi$
$84$		0			0
$85$		−	7.61177i		−	0.825612i
$86$		0			0
$87$	5.65697			0.606491
$88$		0			0
$89$			13.8162i			1.46451i	0.681030	+	0.732256i	$0.261532\pi$
$89$			13.8162i			1.46451i	−0.681030	+	0.732256i	$0.738468\pi$
$90$		0			0
$91$	−10.9133	−	5.21537i	−1.14402	−	0.546719i
$92$		0			0
$93$		−	1.52056i		−	0.157675i
$94$		0			0
$95$	−0.525567			−0.0539221
$96$		0			0
$97$		−	15.7748i		−	1.60169i	−0.598872	−	0.800845i	$-0.704384\pi$
$97$		−	15.7748i		−	1.60169i	0.598872	−	0.800845i	$-0.295616\pi$
$98$		0			0
$99$			1.00000i			0.100504i
$100$		0			0
$101$	−2.87655			−0.286227			−0.143114	−	0.989706i	$-0.545711\pi$
$101$	−2.87655			−0.286227			−0.143114	+	0.989706i	$0.545711\pi$
$102$		0			0
$103$	−8.11332			−0.799429			−0.399714	−	0.916640i	$-0.630891\pi$
$103$	−8.11332			−0.799429			−0.399714	+	0.916640i	$0.630891\pi$
$104$		0			0
$105$	−7.23876			−0.706431
$106$		0			0
$107$	2.91478			0.281783			0.140891	−	0.990025i	$-0.455003\pi$
$107$	2.91478			0.281783			0.140891	+	0.990025i	$0.455003\pi$
$108$		0			0
$109$		−	4.60391i		−	0.440975i	−0.975390	−	0.220487i	$-0.929235\pi$
$109$		−	4.60391i		−	0.440975i	0.975390	−	0.220487i	$-0.0707648\pi$
$110$		0			0
$111$		−	4.12766i		−	0.391780i
$112$		0			0
$113$	3.70426			0.348468			0.174234	−	0.984704i	$-0.444255\pi$
$113$	3.70426			0.348468			0.174234	+	0.984704i	$0.444255\pi$
$114$		0			0
$115$			10.7282i			1.00041i
$116$		0			0
$117$	1.55466	−	3.25316i	0.143728	−	0.300755i
$118$		0			0
$119$		−	11.8337i		−	1.08479i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			1.73126i			0.156102i
$124$		0			0
$125$		−	11.5310i		−	1.03136i
$126$		0			0
$127$	−9.67231			−0.858278			−0.429139	−	0.903238i	$-0.641183\pi$
$127$	−9.67231			−0.858278			−0.429139	+	0.903238i	$0.641183\pi$
$128$		0			0
$129$	6.16479			0.542779
$130$		0			0
$131$	−9.74030			−0.851014			−0.425507	−	0.904955i	$-0.639904\pi$
$131$	−9.74030			−0.851014			−0.425507	+	0.904955i	$0.639904\pi$
$132$		0			0
$133$	−0.817078			−0.0708497
$134$		0			0
$135$		−	2.15782i		−	0.185715i
$136$		0			0
$137$			3.68846i			0.315126i	0.987509	+	0.157563i	$0.0503638\pi$
$137$			3.68846i			0.315126i	−0.987509	+	0.157563i	$0.949636\pi$
$138$		0			0
$139$	−17.2978			−1.46718			−0.733590	−	0.679593i	$-0.762156\pi$
$139$	−17.2978			−1.46718			−0.733590	+	0.679593i	$0.762156\pi$
$140$		0			0
$141$		−	6.52136i		−	0.549198i
$142$		0			0
$143$	3.25316	+	1.55466i	0.272043	+	0.130007i
$144$		0			0
$145$		−	12.2067i		−	1.01371i
$146$		0			0
$147$	−4.25381			−0.350848
$148$		0			0
$149$			11.4950i			0.941703i	0.882212	+	0.470852i	$0.156053\pi$
$149$			11.4950i			0.941703i	−0.882212	+	0.470852i	$0.843947\pi$
$150$		0			0
$151$		−	5.15343i		−	0.419380i	−0.977768	−	0.209690i	$-0.932754\pi$
$151$		−	5.15343i		−	0.419380i	0.977768	−	0.209690i	$-0.0672455\pi$
$152$		0			0
$153$	3.52753			0.285184
$154$		0			0
$155$	−3.28109			−0.263543
$156$		0			0
$157$	1.28466			0.102527			0.0512635	−	0.998685i	$-0.483675\pi$
$157$	1.28466			0.102527			0.0512635	+	0.998685i	$0.483675\pi$
$158$		0			0
$159$	−5.94781			−0.471692
$160$		0			0
$161$			16.6788i			1.31447i
$162$		0			0
$163$			4.58825i			0.359379i	0.983723	+	0.179690i	$0.0575094\pi$
$163$			4.58825i			0.359379i	−0.983723	+	0.179690i	$0.942491\pi$
$164$		0			0
$165$	2.15782			0.167986
$166$		0			0
$167$		−	4.92266i		−	0.380927i	−0.981694	−	0.190463i	$-0.939001\pi$
$167$		−	4.92266i		−	0.380927i	0.981694	−	0.190463i	$-0.0609991\pi$
$168$		0			0
$169$	−8.16607	−	10.1151i	−0.628159	−	0.778085i
$170$		0			0
$171$		−	0.243564i		−	0.0186258i
$172$		0			0
$173$	−5.86834			−0.446162			−0.223081	−	0.974800i	$-0.571611\pi$
$173$	−5.86834			−0.446162			−0.223081	+	0.974800i	$0.571611\pi$
$174$		0			0
$175$		−	1.15343i		−	0.0871913i
$176$		0			0
$177$			3.23041i			0.242813i
$178$		0			0
$179$	10.2659			0.767306			0.383653	−	0.923477i	$-0.374666\pi$
$179$	10.2659			0.767306			0.383653	+	0.923477i	$0.374666\pi$
$180$		0			0
$181$	3.11563			0.231583			0.115791	−	0.993274i	$-0.463060\pi$
$181$	3.11563			0.231583			0.115791	+	0.993274i	$0.463060\pi$
$182$		0			0
$183$	−5.29551			−0.391455
$184$		0			0
$185$	−8.90672			−0.654835
$186$		0			0
$187$			3.52753i			0.257959i
$188$		0			0
$189$		−	3.35467i		−	0.244016i
$190$		0			0
$191$	0.163674			0.0118430			0.00592152	−	0.999982i	$-0.498115\pi$
$191$	0.163674			0.0118430			0.00592152	+	0.999982i	$0.498115\pi$
$192$		0			0
$193$			8.25958i			0.594537i	0.954794	+	0.297269i	$0.0960757\pi$
$193$			8.25958i			0.594537i	−0.954794	+	0.297269i	$0.903924\pi$
$194$		0			0
$195$	−7.01972	−	3.35467i	−0.502692	−	0.240233i
$196$		0			0
$197$			6.57065i			0.468139i	0.972220	+	0.234070i	$0.0752044\pi$
$197$			6.57065i			0.468139i	−0.972220	+	0.234070i	$0.924796\pi$
$198$		0			0
$199$	12.4626			0.883451			0.441726	−	0.897150i	$-0.354366\pi$
$199$	12.4626			0.883451			0.441726	+	0.897150i	$0.354366\pi$
$200$		0			0
$201$			2.04164i			0.144006i
$202$		0			0
$203$		−	18.9773i		−	1.33194i
$204$		0			0
$205$	3.73574			0.260915
$206$		0			0
$207$	−4.97180			−0.345564
$208$		0			0
$209$	0.243564			0.0168477
$210$		0			0
$211$	−8.58400			−0.590947			−0.295473	−	0.955351i	$-0.595477\pi$
$211$	−8.58400			−0.590947			−0.295473	+	0.955351i	$0.595477\pi$
$212$		0			0
$213$		−	1.57233i		−	0.107734i
$214$		0			0
$215$		−	13.3025i		−	0.907222i
$216$		0			0
$217$	−5.10097			−0.346277
$218$		0			0
$219$			5.64365i			0.381362i
$220$		0			0
$221$	5.48411	−	11.4756i	0.368901	−	0.771934i
$222$		0			0
$223$			4.52872i			0.303266i	0.988437	+	0.151633i	$0.0484531\pi$
$223$			4.52872i			0.303266i	−0.988437	+	0.151633i	$0.951547\pi$
$224$		0			0
$225$	0.343829			0.0229219
$226$		0			0
$227$			22.6980i			1.50652i	0.657725	+	0.753258i	$0.271519\pi$
$227$			22.6980i			1.50652i	−0.657725	+	0.753258i	$0.728481\pi$
$228$		0			0
$229$		−	19.8836i		−	1.31394i	−0.753915	−	0.656972i	$-0.771837\pi$
$229$		−	19.8836i		−	1.31394i	0.753915	−	0.656972i	$-0.228163\pi$
$230$		0			0
$231$	3.35467			0.220721
$232$		0			0
$233$	26.6568			1.74634			0.873171	−	0.487413i	$-0.162059\pi$
$233$	26.6568			1.74634			0.873171	+	0.487413i	$0.162059\pi$
$234$		0			0
$235$	−14.0719			−0.917950
$236$		0			0
$237$	−7.28784			−0.473396
$238$		0			0
$239$			3.66781i			0.237251i	0.992939	+	0.118625i	$0.0378487\pi$
$239$			3.66781i			0.237251i	−0.992939	+	0.118625i	$0.962151\pi$
$240$		0			0
$241$		−	0.709972i		−	0.0457333i	−0.999739	−	0.0228667i	$-0.992721\pi$
$241$		−	0.709972i		−	0.0457333i	0.999739	−	0.0228667i	$-0.00727932\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			9.17893i			0.586421i
$246$		0			0
$247$	−0.792354	−	0.378660i	−0.0504162	−	0.0240935i
$248$		0			0
$249$		−	2.31356i		−	0.146616i
$250$		0			0
$251$	16.1009			1.01628			0.508141	−	0.861274i	$-0.330333\pi$
$251$	16.1009			1.01628			0.508141	+	0.861274i	$0.330333\pi$
$252$		0			0
$253$		−	4.97180i		−	0.312575i
$254$		0			0
$255$		−	7.61177i		−	0.476667i
$256$		0			0
$257$	23.6242			1.47364			0.736819	−	0.676090i	$-0.236327\pi$
$257$	23.6242			1.47364			0.736819	+	0.676090i	$0.236327\pi$
$258$		0			0
$259$	−13.8469			−0.860406
$260$		0			0
$261$	5.65697			0.350158
$262$		0			0
$263$	6.59316			0.406552			0.203276	−	0.979122i	$-0.434841\pi$
$263$	6.59316			0.406552			0.203276	+	0.979122i	$0.434841\pi$
$264$		0			0
$265$			12.8343i			0.788404i
$266$		0			0
$267$			13.8162i			0.845536i
$268$		0			0
$269$	22.5436			1.37451			0.687254	−	0.726417i	$-0.258816\pi$
$269$	22.5436			1.37451			0.687254	+	0.726417i	$0.258816\pi$
$270$		0			0
$271$			16.3040i			0.990397i	0.868780	+	0.495198i	$0.164905\pi$
$271$			16.3040i			0.990397i	−0.868780	+	0.495198i	$0.835095\pi$
$272$		0			0
$273$	−10.9133	−	5.21537i	−0.660501	−	0.315648i
$274$		0			0
$275$			0.343829i			0.0207337i
$276$		0			0
$277$	13.6586			0.820664			0.410332	−	0.911936i	$-0.365413\pi$
$277$	13.6586			0.820664			0.410332	+	0.911936i	$0.365413\pi$
$278$		0			0
$279$		−	1.52056i		−	0.0910335i
$280$		0			0
$281$		−	24.6144i		−	1.46837i	−0.678950	−	0.734185i	$-0.737565\pi$
$281$		−	24.6144i		−	1.46837i	0.678950	−	0.734185i	$-0.262435\pi$
$282$		0			0
$283$	2.00467			0.119165			0.0595825	−	0.998223i	$-0.481023\pi$
$283$	2.00467			0.119165			0.0595825	+	0.998223i	$0.481023\pi$
$284$		0			0
$285$	−0.525567			−0.0311319
$286$		0			0
$287$	5.80780			0.342823
$288$		0			0
$289$	−4.55652			−0.268030
$290$		0			0
$291$		−	15.7748i		−	0.924736i
$292$		0			0
$293$			11.5156i			0.672749i	0.941728	+	0.336375i	$0.109201\pi$
$293$			11.5156i			0.672749i	−0.941728	+	0.336375i	$0.890799\pi$
$294$		0			0
$295$	6.97064			0.405846
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	−7.72946	+	16.1741i	−0.447006	+	0.935370i
$300$		0			0
$301$		−	20.6808i		−	1.19202i
$302$		0			0
$303$	−2.87655			−0.165253
$304$		0			0
$305$			11.4267i			0.654293i
$306$		0			0
$307$		−	28.2595i		−	1.61285i	−0.591334	−	0.806427i	$-0.701399\pi$
$307$		−	28.2595i		−	1.61285i	0.591334	−	0.806427i	$-0.298601\pi$
$308$		0			0
$309$	−8.11332			−0.461551
$310$		0			0
$311$	−10.5329			−0.597266			−0.298633	−	0.954368i	$-0.596531\pi$
$311$	−10.5329			−0.597266			−0.298633	+	0.954368i	$0.596531\pi$
$312$		0			0
$313$	32.8602			1.85737			0.928685	−	0.370870i	$-0.120940\pi$
$313$	32.8602			1.85737			0.928685	+	0.370870i	$0.120940\pi$
$314$		0			0
$315$	−7.23876			−0.407858
$316$		0			0
$317$		−	5.47914i		−	0.307739i	−0.988091	−	0.153870i	$-0.950826\pi$
$317$		−	5.47914i		−	0.307739i	0.988091	−	0.153870i	$-0.0491735\pi$
$318$		0			0
$319$			5.65697i			0.316730i
$320$		0			0
$321$	2.91478			0.162687
$322$		0			0
$323$		−	0.859181i		−	0.0478061i
$324$		0			0
$325$	0.534537	−	1.11853i	0.0296508	−	0.0620448i
$326$		0			0
$327$		−	4.60391i		−	0.254597i
$328$		0			0
$329$	−21.8770			−1.20612
$330$		0			0
$331$			3.91067i			0.214950i	0.994208	+	0.107475i	$0.0342766\pi$
$331$			3.91067i			0.214950i	−0.994208	+	0.107475i	$0.965723\pi$
$332$		0			0
$333$		−	4.12766i		−	0.226194i
$334$		0			0
$335$	4.40547			0.240697
$336$		0			0
$337$	17.5877			0.958062			0.479031	−	0.877798i	$-0.340988\pi$
$337$	17.5877			0.958062			0.479031	+	0.877798i	$0.340988\pi$
$338$		0			0
$339$	3.70426			0.201188
$340$		0			0
$341$	1.52056			0.0823429
$342$		0			0
$343$		−	9.21257i		−	0.497432i
$344$		0			0
$345$			10.7282i			0.577589i
$346$		0			0
$347$	−7.39389			−0.396925			−0.198462	−	0.980108i	$-0.563595\pi$
$347$	−7.39389			−0.396925			−0.198462	+	0.980108i	$0.563595\pi$
$348$		0			0
$349$		−	25.6330i		−	1.37210i	−0.727554	−	0.686051i	$-0.759343\pi$
$349$		−	25.6330i		−	1.37210i	0.727554	−	0.686051i	$-0.240657\pi$
$350$		0			0
$351$	1.55466	−	3.25316i	0.0829816	−	0.173641i
$352$		0			0
$353$			3.79863i			0.202181i	0.994877	+	0.101090i	$0.0322331\pi$
$353$			3.79863i			0.202181i	−0.994877	+	0.101090i	$0.967767\pi$
$354$		0			0
$355$	−3.39279			−0.180071
$356$		0			0
$357$		−	11.8337i		−	0.626306i
$358$		0			0
$359$			23.0600i			1.21706i	0.793530	+	0.608531i	$0.208241\pi$
$359$			23.0600i			1.21706i	−0.793530	+	0.608531i	$0.791759\pi$
$360$		0			0
$361$	18.9407			0.996878
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	12.1779			0.637423
$366$		0			0
$367$	6.59629			0.344323			0.172162	−	0.985069i	$-0.444925\pi$
$367$	6.59629			0.344323			0.172162	+	0.985069i	$0.444925\pi$
$368$		0			0
$369$			1.73126i			0.0901257i
$370$		0			0
$371$			19.9529i			1.03591i
$372$		0			0
$373$	−12.1203			−0.627563			−0.313782	−	0.949495i	$-0.601596\pi$
$373$	−12.1203			−0.627563			−0.313782	+	0.949495i	$0.601596\pi$
$374$		0			0
$375$		−	11.5310i		−	0.595458i
$376$		0			0
$377$	8.79467	−	18.4030i	0.452948	−	0.947804i
$378$		0			0
$379$			15.4756i			0.794929i	0.917618	+	0.397464i	$0.130110\pi$
$379$			15.4756i			0.794929i	−0.917618	+	0.397464i	$0.869890\pi$
$380$		0			0
$381$	−9.67231			−0.495527
$382$		0			0
$383$		−	24.8929i		−	1.27197i	−0.771703	−	0.635983i	$-0.780595\pi$
$383$		−	24.8929i		−	1.27197i	0.771703	−	0.635983i	$-0.219405\pi$
$384$		0			0
$385$		−	7.23876i		−	0.368921i
$386$		0			0
$387$	6.16479			0.313374
$388$		0			0
$389$	20.5468			1.04177			0.520883	−	0.853628i	$-0.325603\pi$
$389$	20.5468			1.04177			0.520883	+	0.853628i	$0.325603\pi$
$390$		0			0
$391$	−17.5382			−0.886945
$392$		0			0
$393$	−9.74030			−0.491333
$394$		0			0
$395$			15.7258i			0.791251i
$396$		0			0
$397$			3.17834i			0.159516i	0.996814	+	0.0797581i	$0.0254148\pi$
$397$			3.17834i			0.159516i	−0.996814	+	0.0797581i	$0.974585\pi$
$398$		0			0
$399$	−0.817078			−0.0409051
$400$		0			0
$401$			12.9557i			0.646975i	0.946232	+	0.323487i	$0.104855\pi$
$401$			12.9557i			0.646975i	−0.946232	+	0.323487i	$0.895145\pi$
$402$		0			0
$403$	−4.94662	−	2.36395i	−0.246409	−	0.117757i
$404$		0			0
$405$		−	2.15782i		−	0.107223i
$406$		0			0
$407$	4.12766			0.204600
$408$		0			0
$409$		−	29.5400i		−	1.46066i	−0.683095	−	0.730329i	$-0.739367\pi$
$409$		−	29.5400i		−	1.46066i	0.683095	−	0.730329i	$-0.260633\pi$
$410$		0			0
$411$			3.68846i			0.181938i
$412$		0			0
$413$	10.8370			0.533253
$414$		0			0
$415$	−4.99225			−0.245060
$416$		0			0
$417$	−17.2978			−0.847076
$418$		0			0
$419$	1.70265			0.0831798			0.0415899	−	0.999135i	$-0.486758\pi$
$419$	1.70265			0.0831798			0.0415899	+	0.999135i	$0.486758\pi$
$420$		0			0
$421$			27.5960i			1.34495i	0.740120	+	0.672474i	$0.234768\pi$
$421$			27.5960i			1.34495i	−0.740120	+	0.672474i	$0.765232\pi$
$422$		0			0
$423$		−	6.52136i		−	0.317079i
$424$		0			0
$425$	1.21287			0.0588327
$426$		0			0
$427$			17.7647i			0.859693i
$428$		0			0
$429$	3.25316	+	1.55466i	0.157064	+	0.0750597i
$430$		0			0
$431$			13.5488i			0.652621i	0.945263	+	0.326311i	$0.105806\pi$
$431$			13.5488i			0.652621i	−0.945263	+	0.326311i	$0.894194\pi$
$432$		0			0
$433$	−12.6948			−0.610072			−0.305036	−	0.952341i	$-0.598668\pi$
$433$	−12.6948			−0.610072			−0.305036	+	0.952341i	$0.598668\pi$
$434$		0			0
$435$		−	12.2067i		−	0.585267i
$436$		0			0
$437$			1.21095i			0.0579278i
$438$		0			0
$439$	4.46589			0.213145			0.106573	−	0.994305i	$-0.466012\pi$
$439$	4.46589			0.213145			0.106573	+	0.994305i	$0.466012\pi$
$440$		0			0
$441$	−4.25381			−0.202562
$442$		0			0
$443$	−6.38874			−0.303538			−0.151769	−	0.988416i	$-0.548497\pi$
$443$	−6.38874			−0.303538			−0.151769	+	0.988416i	$0.548497\pi$
$444$		0			0
$445$	29.8128			1.41326
$446$		0			0
$447$			11.4950i			0.543693i
$448$		0			0
$449$		−	1.03468i		−	0.0488295i	−0.999702	−	0.0244147i	$-0.992228\pi$
$449$		−	1.03468i		−	0.0488295i	0.999702	−	0.0244147i	$-0.00777222\pi$
$450$		0			0
$451$	−1.73126			−0.0815217
$452$		0			0
$453$		−	5.15343i		−	0.242129i
$454$		0			0
$455$	−11.2538	+	23.5488i	−0.527587	+	1.10399i
$456$		0			0
$457$		−	23.3556i		−	1.09253i	−0.837612	−	0.546265i	$-0.816049\pi$
$457$		−	23.3556i		−	1.09253i	0.837612	−	0.546265i	$-0.183951\pi$
$458$		0			0
$459$	3.52753			0.164651
$460$		0			0
$461$			38.6321i			1.79928i	0.436635	+	0.899639i	$0.356170\pi$
$461$			38.6321i			1.79928i	−0.436635	+	0.899639i	$0.643830\pi$
$462$		0			0
$463$			28.3035i			1.31538i	0.753290	+	0.657688i	$0.228466\pi$
$463$			28.3035i			1.31538i	−0.753290	+	0.657688i	$0.771534\pi$
$464$		0			0
$465$	−3.28109			−0.152157
$466$		0			0
$467$	29.1490			1.34886			0.674428	−	0.738341i	$-0.264390\pi$
$467$	29.1490			1.34886			0.674428	+	0.738341i	$0.264390\pi$
$468$		0			0
$469$	6.84901			0.316258
$470$		0			0
$471$	1.28466			0.0591940
$472$		0			0
$473$			6.16479i			0.283457i
$474$		0			0
$475$		−	0.0837445i		−	0.00384246i
$476$		0			0
$477$	−5.94781			−0.272332
$478$		0			0
$479$			24.9353i			1.13932i	0.821879	+	0.569662i	$0.192926\pi$
$479$			24.9353i			1.13932i	−0.821879	+	0.569662i	$0.807074\pi$
$480$		0			0
$481$	−13.4279	−	6.41710i	−0.612260	−	0.292594i
$482$		0			0
$483$			16.6788i			0.758910i
$484$		0			0
$485$	−34.0392			−1.54564
$486$		0			0
$487$		−	2.15784i		−	0.0977809i	−0.998804	−	0.0488905i	$-0.984431\pi$
$487$		−	2.15784i		−	0.0977809i	0.998804	−	0.0488905i	$-0.0155685\pi$
$488$		0			0
$489$			4.58825i			0.207488i
$490$		0			0
$491$	−17.4713			−0.788468			−0.394234	−	0.919010i	$-0.628990\pi$
$491$	−17.4713			−0.788468			−0.394234	+	0.919010i	$0.628990\pi$
$492$		0			0
$493$	19.9552			0.898735
$494$		0			0
$495$	2.15782			0.0969866
$496$		0			0
$497$	−5.27463			−0.236600
$498$		0			0
$499$		−	6.20808i		−	0.277912i	−0.990299	−	0.138956i	$-0.955625\pi$
$499$		−	6.20808i		−	0.277912i	0.990299	−	0.138956i	$-0.0443746\pi$
$500$		0			0
$501$		−	4.92266i		−	0.219928i
$502$		0			0
$503$	−22.9759			−1.02444			−0.512222	−	0.858853i	$-0.671178\pi$
$503$	−22.9759			−1.02444			−0.512222	+	0.858853i	$0.671178\pi$
$504$		0			0
$505$			6.20706i			0.276211i
$506$		0			0
$507$	−8.16607	−	10.1151i	−0.362668	−	0.449228i
$508$		0			0
$509$			17.0924i			0.757609i	0.925477	+	0.378805i	$0.123665\pi$
$509$			17.0924i			0.757609i	−0.925477	+	0.378805i	$0.876335\pi$
$510$		0			0
$511$	18.9326			0.837527
$512$		0			0
$513$		−	0.243564i		−	0.0107536i
$514$		0			0
$515$			17.5070i			0.771453i
$516$		0			0
$517$	6.52136			0.286809
$518$		0			0
$519$	−5.86834			−0.257592
$520$		0			0
$521$	17.7217			0.776403			0.388202	−	0.921574i	$-0.373096\pi$
$521$	17.7217			0.776403			0.388202	+	0.921574i	$0.373096\pi$
$522$		0			0
$523$	−15.1106			−0.660741			−0.330370	−	0.943851i	$-0.607174\pi$
$523$	−15.1106			−0.660741			−0.330370	+	0.943851i	$0.607174\pi$
$524$		0			0
$525$		−	1.15343i		−	0.0503399i
$526$		0			0
$527$		−	5.36382i		−	0.233652i
$528$		0			0
$529$	1.71883			0.0747319
$530$		0			0
$531$			3.23041i			0.140188i
$532$		0			0
$533$	5.63205	+	2.69152i	0.243951	+	0.116582i
$534$		0			0
$535$		−	6.28956i		−	0.271922i
$536$		0			0
$537$	10.2659			0.443004
$538$		0			0
$539$		−	4.25381i		−	0.183224i
$540$		0			0
$541$		−	12.7438i		−	0.547900i	−0.961744	−	0.273950i	$-0.911670\pi$
$541$		−	12.7438i		−	0.547900i	0.961744	−	0.273950i	$-0.0883303\pi$
$542$		0			0
$543$	3.11563			0.133704
$544$		0			0
$545$	−9.93440			−0.425543
$546$		0			0
$547$	−18.6653			−0.798070			−0.399035	−	0.916936i	$-0.630655\pi$
$547$	−18.6653			−0.798070			−0.399035	+	0.916936i	$0.630655\pi$
$548$		0			0
$549$	−5.29551			−0.226007
$550$		0			0
$551$		−	1.37784i		−	0.0586979i
$552$		0			0
$553$			24.4483i			1.03965i
$554$		0			0
$555$	−8.90672			−0.378069
$556$		0			0
$557$			4.83740i			0.204967i	0.994735	+	0.102484i	$0.0326789\pi$
$557$			4.83740i			0.204967i	−0.994735	+	0.102484i	$0.967321\pi$
$558$		0			0
$559$	9.58415	−	20.0550i	0.405366	−	0.848237i
$560$		0			0
$561$			3.52753i			0.148933i
$562$		0			0
$563$	−3.29440			−0.138843			−0.0694213	−	0.997587i	$-0.522115\pi$
$563$	−3.29440			−0.138843			−0.0694213	+	0.997587i	$0.522115\pi$
$564$		0			0
$565$		−	7.99312i		−	0.336273i
$566$		0			0
$567$		−	3.35467i		−	0.140883i
$568$		0			0
$569$	−40.2810			−1.68867			−0.844333	−	0.535819i	$-0.820003\pi$
$569$	−40.2810			−1.68867			−0.844333	+	0.535819i	$0.820003\pi$
$570$		0			0
$571$	−14.4230			−0.603584			−0.301792	−	0.953374i	$-0.597585\pi$
$571$	−14.4230			−0.603584			−0.301792	+	0.953374i	$0.597585\pi$
$572$		0			0
$573$	0.163674			0.00683759
$574$		0			0
$575$	−1.70945			−0.0712890
$576$		0			0
$577$		−	19.2186i		−	0.800081i	−0.916497	−	0.400041i	$-0.868996\pi$
$577$		−	19.2186i		−	0.800081i	0.916497	−	0.400041i	$-0.131004\pi$
$578$		0			0
$579$			8.25958i			0.343256i
$580$		0			0
$581$	−7.76124			−0.321991
$582$		0			0
$583$		−	5.94781i		−	0.246333i
$584$		0			0
$585$	−7.01972	−	3.35467i	−0.290230	−	0.138699i
$586$		0			0
$587$			40.9008i			1.68816i	0.536218	+	0.844079i	$0.319852\pi$
$587$			40.9008i			1.68816i	−0.536218	+	0.844079i	$0.680148\pi$
$588$		0			0
$589$	−0.370354			−0.0152602
$590$		0			0
$591$			6.57065i			0.270280i
$592$		0			0
$593$		−	41.4899i		−	1.70378i	−0.523717	−	0.851892i	$-0.675455\pi$
$593$		−	41.4899i		−	1.70378i	0.523717	−	0.851892i	$-0.324545\pi$
$594$		0			0
$595$	−25.5350			−1.04683
$596$		0			0
$597$	12.4626			0.510061
$598$		0			0
$599$	−22.4566			−0.917552			−0.458776	−	0.888552i	$-0.651712\pi$
$599$	−22.4566			−0.917552			−0.458776	+	0.888552i	$0.651712\pi$
$600$		0			0
$601$	47.2748			1.92838			0.964189	−	0.265216i	$-0.0854431\pi$
$601$	47.2748			1.92838			0.964189	+	0.265216i	$0.0854431\pi$
$602$		0			0
$603$			2.04164i			0.0831418i
$604$		0			0
$605$			2.15782i			0.0877277i
$606$		0			0
$607$	−17.0271			−0.691108			−0.345554	−	0.938399i	$-0.612309\pi$
$607$	−17.0271			−0.691108			−0.345554	+	0.938399i	$0.612309\pi$
$608$		0			0
$609$		−	18.9773i		−	0.768998i
$610$		0			0
$611$	−21.2150	−	10.1385i	−0.858268	−	0.410160i
$612$		0			0
$613$		−	2.76405i		−	0.111639i	−0.998441	−	0.0558195i	$-0.982223\pi$
$613$		−	2.76405i		−	0.111639i	0.998441	−	0.0558195i	$-0.0177771\pi$
$614$		0			0
$615$	3.73574			0.150639
$616$		0			0
$617$			3.82115i			0.153834i	0.997038	+	0.0769168i	$0.0245076\pi$
$617$			3.82115i			0.153834i	−0.997038	+	0.0769168i	$0.975492\pi$
$618$		0			0
$619$			8.71841i			0.350422i	0.984531	+	0.175211i	$0.0560608\pi$
$619$			8.71841i			0.350422i	−0.984531	+	0.175211i	$0.943939\pi$
$620$		0			0
$621$	−4.97180			−0.199512
$622$		0			0
$623$	46.3487			1.85692
$624$		0			0
$625$	−23.1626			−0.926506
$626$		0			0
$627$	0.243564			0.00972703
$628$		0			0
$629$		−	14.5604i		−	0.580563i
$630$		0			0
$631$		−	1.88974i		−	0.0752295i	−0.999292	−	0.0376147i	$-0.988024\pi$
$631$		−	1.88974i		−	0.0752295i	0.999292	−	0.0376147i	$-0.0119760\pi$
$632$		0			0
$633$	−8.58400			−0.341183
$634$		0			0
$635$			20.8711i			0.828243i
$636$		0			0
$637$	−6.61322	+	13.8383i	−0.262025	+	0.548294i
$638$		0			0
$639$		−	1.57233i		−	0.0622003i
$640$		0			0
$641$	−29.1226			−1.15027			−0.575136	−	0.818058i	$-0.695051\pi$
$641$	−29.1226			−1.15027			−0.575136	+	0.818058i	$0.695051\pi$
$642$		0			0
$643$		−	48.5984i		−	1.91654i	−0.285872	−	0.958268i	$-0.592283\pi$
$643$		−	48.5984i		−	1.91654i	0.285872	−	0.958268i	$-0.407717\pi$
$644$		0			0
$645$		−	13.3025i		−	0.523785i
$646$		0			0
$647$	42.8626			1.68510			0.842552	−	0.538615i	$-0.181052\pi$
$647$	42.8626			1.68510			0.842552	+	0.538615i	$0.181052\pi$
$648$		0			0
$649$	−3.23041			−0.126805
$650$		0			0
$651$	−5.10097			−0.199923
$652$		0			0
$653$	22.7625			0.890765			0.445382	−	0.895340i	$-0.353068\pi$
$653$	22.7625			0.890765			0.445382	+	0.895340i	$0.353068\pi$
$654$		0			0
$655$			21.0178i			0.821233i
$656$		0			0
$657$			5.64365i			0.220180i
$658$		0			0
$659$	10.5386			0.410526			0.205263	−	0.978707i	$-0.434195\pi$
$659$	10.5386			0.410526			0.205263	+	0.978707i	$0.434195\pi$
$660$		0			0
$661$			31.0675i			1.20838i	0.796839	+	0.604192i	$0.206504\pi$
$661$			31.0675i			1.20838i	−0.796839	+	0.604192i	$0.793496\pi$
$662$		0			0
$663$	5.48411	−	11.4756i	0.212985	−	0.445676i
$664$		0			0
$665$			1.76310i			0.0683703i
$666$		0			0
$667$	−28.1254			−1.08902
$668$		0			0
$669$			4.52872i			0.175090i
$670$		0			0
$671$		−	5.29551i		−	0.204431i
$672$		0			0
$673$	34.8752			1.34434			0.672171	−	0.740396i	$-0.265362\pi$
$673$	34.8752			1.34434			0.672171	+	0.740396i	$0.265362\pi$
$674$		0			0
$675$	0.343829			0.0132340
$676$		0			0
$677$	14.9584			0.574897			0.287449	−	0.957796i	$-0.407193\pi$
$677$	14.9584			0.574897			0.287449	+	0.957796i	$0.407193\pi$
$678$		0			0
$679$	−52.9193			−2.03086
$680$		0			0
$681$			22.6980i			0.869788i
$682$		0			0
$683$			7.65756i			0.293008i	0.989210	+	0.146504i	$0.0468022\pi$
$683$			7.65756i			0.293008i	−0.989210	+	0.146504i	$0.953198\pi$
$684$		0			0
$685$	7.95902			0.304098
$686$		0			0
$687$		−	19.8836i		−	0.758606i
$688$		0			0
$689$	−9.24682	+	19.3492i	−0.352276	+	0.737145i
$690$		0			0
$691$			7.72767i			0.293974i	0.989138	+	0.146987i	$0.0469576\pi$
$691$			7.72767i			0.293974i	−0.989138	+	0.146987i	$0.953042\pi$
$692$		0			0
$693$	3.35467			0.127433
$694$		0			0
$695$			37.3254i			1.41583i
$696$		0			0
$697$			6.10707i			0.231322i
$698$		0			0
$699$	26.6568			1.00825
$700$		0			0
$701$	38.5325			1.45535			0.727675	−	0.685922i	$-0.240601\pi$
$701$	38.5325			1.45535			0.727675	+	0.685922i	$0.240601\pi$
$702$		0			0
$703$	−1.00535			−0.0379175
$704$		0			0
$705$	−14.0719			−0.529978
$706$		0			0
$707$			9.64987i			0.362921i
$708$		0			0
$709$			12.5780i			0.472376i	0.971707	+	0.236188i	$0.0758981\pi$
$709$			12.5780i			0.472376i	−0.971707	+	0.236188i	$0.924102\pi$
$710$		0			0
$711$	−7.28784			−0.273315
$712$		0			0
$713$			7.55992i			0.283121i
$714$		0			0
$715$	3.35467	−	7.01972i	0.125458	−	0.262523i
$716$		0			0
$717$			3.66781i			0.136977i
$718$		0			0
$719$	40.4450			1.50835			0.754173	−	0.656676i	$-0.228038\pi$
$719$	40.4450			1.50835			0.754173	+	0.656676i	$0.228038\pi$
$720$		0			0
$721$			27.2175i			1.01363i
$722$		0			0
$723$		−	0.709972i		−	0.0264042i
$724$		0			0
$725$	1.94503			0.0722366
$726$		0			0
$727$	43.8912			1.62783			0.813917	−	0.580981i	$-0.197331\pi$
$727$	43.8912			1.62783			0.813917	+	0.580981i	$0.197331\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	21.7465			0.804323
$732$		0			0
$733$			0.665638i			0.0245859i	0.999924	+	0.0122930i	$0.00391307\pi$
$733$			0.665638i			0.0245859i	−0.999924	+	0.0122930i	$0.996087\pi$
$734$		0			0
$735$			9.17893i			0.338570i
$736$		0			0
$737$	−2.04164			−0.0752046
$738$		0			0
$739$			8.81221i			0.324162i	0.986777	+	0.162081i	$0.0518206\pi$
$739$			8.81221i			0.324162i	−0.986777	+	0.162081i	$0.948179\pi$
$740$		0			0
$741$	−0.792354	−	0.378660i	−0.0291078	−	0.0139104i
$742$		0			0
$743$			33.3394i			1.22311i	0.791204	+	0.611553i	$0.209455\pi$
$743$			33.3394i			1.22311i	−0.791204	+	0.611553i	$0.790545\pi$
$744$		0			0
$745$	24.8040			0.908748
$746$		0			0
$747$		−	2.31356i		−	0.0846489i
$748$		0			0
$749$		−	9.77813i		−	0.357285i
$750$		0			0
$751$	47.0267			1.71603			0.858015	−	0.513625i	$-0.171698\pi$
$751$	47.0267			1.71603			0.858015	+	0.513625i	$0.171698\pi$
$752$		0			0
$753$	16.1009			0.586751
$754$		0			0
$755$	−11.1202			−0.404704
$756$		0			0
$757$	36.6623			1.33251			0.666257	−	0.745722i	$-0.267895\pi$
$757$	36.6623			1.33251			0.666257	+	0.745722i	$0.267895\pi$
$758$		0			0
$759$		−	4.97180i		−	0.180465i
$760$		0			0
$761$			34.9488i			1.26689i	0.773787	+	0.633446i	$0.218360\pi$
$761$			34.9488i			1.26689i	−0.773787	+	0.633446i	$0.781640\pi$
$762$		0			0
$763$	−15.4446			−0.559132
$764$		0			0
$765$		−	7.61177i		−	0.275204i
$766$		0			0
$767$	10.5090	+	5.02219i	0.379460	+	0.181341i
$768$		0			0
$769$		−	22.1271i		−	0.797925i	−0.916967	−	0.398963i	$-0.869370\pi$
$769$		−	22.1271i		−	0.797925i	0.916967	−	0.398963i	$-0.130630\pi$
$770$		0			0
$771$	23.6242			0.850806
$772$		0			0
$773$			17.8259i			0.641151i	0.947223	+	0.320576i	$0.103876\pi$
$773$			17.8259i			0.641151i	−0.947223	+	0.320576i	$0.896124\pi$
$774$		0			0
$775$		−	0.522812i		−	0.0187800i
$776$		0			0
$777$	−13.8469			−0.496755
$778$		0			0
$779$	0.421673			0.0151080
$780$		0			0
$781$	1.57233			0.0562622
$782$		0			0
$783$	5.65697			0.202164
$784$		0			0
$785$		−	2.77206i		−	0.0989390i
$786$		0			0
$787$		−	30.6230i		−	1.09159i	−0.837918	−	0.545796i	$-0.816227\pi$
$787$		−	30.6230i		−	1.09159i	0.837918	−	0.545796i	$-0.183773\pi$
$788$		0			0
$789$	6.59316			0.234723
$790$		0			0
$791$		−	12.4266i		−	0.441838i
$792$		0			0
$793$	−8.23271	+	17.2271i	−0.292352	+	0.611753i
$794$		0			0
$795$			12.8343i			0.455185i
$796$		0			0
$797$	5.99532			0.212365			0.106183	−	0.994347i	$-0.466137\pi$
$797$	5.99532			0.212365			0.106183	+	0.994347i	$0.466137\pi$
$798$		0			0
$799$		−	23.0043i		−	0.813834i
$800$		0			0
$801$			13.8162i			0.488170i
$802$		0			0
$803$	−5.64365			−0.199160
$804$		0			0
$805$	35.9897			1.26847
$806$		0			0
$807$	22.5436			0.793573
$808$		0			0
$809$	−26.6643			−0.937465			−0.468733	−	0.883340i	$-0.655289\pi$
$809$	−26.6643			−0.937465			−0.468733	+	0.883340i	$0.655289\pi$
$810$		0			0
$811$			1.48717i			0.0522217i	0.999659	+	0.0261109i	$0.00831229\pi$
$811$			1.48717i			0.0522217i	−0.999659	+	0.0261109i	$0.991688\pi$
$812$		0			0
$813$			16.3040i			0.571806i
$814$		0			0
$815$	9.90060			0.346803
$816$		0			0
$817$		−	1.50152i		−	0.0525317i
$818$		0			0
$819$	−10.9133	−	5.21537i	−0.381340	−	0.182240i
$820$		0			0
$821$			25.4427i			0.887958i	0.896037	+	0.443979i	$0.146434\pi$
$821$			25.4427i			0.887958i	−0.896037	+	0.443979i	$0.853566\pi$
$822$		0			0
$823$	−22.4925			−0.784039			−0.392019	−	0.919957i	$-0.628223\pi$
$823$	−22.4925			−0.784039			−0.392019	+	0.919957i	$0.628223\pi$
$824$		0			0
$825$			0.343829i			0.0119706i
$826$		0			0
$827$		−	2.54567i		−	0.0885215i	−0.999020	−	0.0442607i	$-0.985907\pi$
$827$		−	2.54567i		−	0.0885215i	0.999020	−	0.0442607i	$-0.0140932\pi$
$828$		0			0
$829$	−14.8272			−0.514970			−0.257485	−	0.966282i	$-0.582894\pi$
$829$	−14.8272			−0.514970			−0.257485	+	0.966282i	$0.582894\pi$
$830$		0			0
$831$	13.6586			0.473810
$832$		0			0
$833$	−15.0054			−0.519908
$834$		0			0
$835$	−10.6222			−0.367596
$836$		0			0
$837$		−	1.52056i		−	0.0525582i
$838$		0			0
$839$		−	40.1300i		−	1.38544i	−0.721206	−	0.692720i	$-0.756412\pi$
$839$		−	40.1300i		−	1.38544i	0.721206	−	0.692720i	$-0.243588\pi$
$840$		0			0
$841$	3.00135			0.103495
$842$		0			0
$843$		−	24.6144i		−	0.847764i
$844$		0			0
$845$	−21.8265	+	17.6209i	−0.750856	+	0.606177i
$846$		0			0
$847$			3.35467i			0.115268i
$848$		0			0
$849$	2.00467			0.0688000
$850$		0			0
$851$			20.5219i			0.703481i
$852$		0			0
$853$			43.0050i			1.47246i	0.676729	+	0.736232i	$0.263397\pi$
$853$			43.0050i			1.47246i	−0.676729	+	0.736232i	$0.736603\pi$
$854$		0			0
$855$	−0.525567			−0.0179740
$856$		0			0
$857$	18.8317			0.643280			0.321640	−	0.946862i	$-0.395766\pi$
$857$	18.8317			0.643280			0.321640	+	0.946862i	$0.395766\pi$
$858$		0			0
$859$	−41.1951			−1.40556			−0.702780	−	0.711407i	$-0.748058\pi$
$859$	−41.1951			−1.40556			−0.702780	+	0.711407i	$0.748058\pi$
$860$		0			0
$861$	5.80780			0.197929
$862$		0			0
$863$		−	34.1910i		−	1.16387i	−0.813234	−	0.581937i	$-0.802295\pi$
$863$		−	34.1910i		−	1.16387i	0.813234	−	0.581937i	$-0.197705\pi$
$864$		0			0
$865$			12.6628i			0.430548i
$866$		0			0
$867$	−4.55652			−0.154747
$868$		0			0
$869$		−	7.28784i		−	0.247223i
$870$		0			0
$871$	6.64176	+	3.17405i	0.225047	+	0.107548i
$872$		0			0
$873$		−	15.7748i		−	0.533897i
$874$		0			0
$875$	−38.6827			−1.30771
$876$		0			0
$877$		−	25.5632i		−	0.863209i	−0.902063	−	0.431605i	$-0.857948\pi$
$877$		−	25.5632i		−	0.863209i	0.902063	−	0.431605i	$-0.142052\pi$
$878$		0			0
$879$			11.5156i			0.388412i
$880$		0			0
$881$	6.15871			0.207492			0.103746	−	0.994604i	$-0.466917\pi$
$881$	6.15871			0.207492			0.103746	+	0.994604i	$0.466917\pi$
$882$		0			0
$883$	−15.6180			−0.525588			−0.262794	−	0.964852i	$-0.584644\pi$
$883$	−15.6180			−0.525588			−0.262794	+	0.964852i	$0.584644\pi$
$884$		0			0
$885$	6.97064			0.234316
$886$		0			0
$887$	15.7935			0.530295			0.265148	−	0.964208i	$-0.414579\pi$
$887$	15.7935			0.530295			0.265148	+	0.964208i	$0.414579\pi$
$888$		0			0
$889$			32.4474i			1.08825i
$890$		0			0
$891$			1.00000i			0.0335013i
$892$		0			0
$893$	−1.58837			−0.0531528
$894$		0			0
$895$		−	22.1518i		−	0.740454i
$896$		0			0
$897$	−7.72946	+	16.1741i	−0.258079	+	0.540036i
$898$		0			0
$899$		−	8.60176i		−	0.286885i
$900$		0			0
$901$	−20.9811			−0.698982
$902$		0			0
$903$		−	20.6808i		−	0.688215i
$904$		0			0
$905$		−	6.72295i		−	0.223478i
$906$		0			0
$907$	55.1476			1.83115			0.915573	−	0.402152i	$-0.131738\pi$
$907$	55.1476			1.83115			0.915573	+	0.402152i	$0.131738\pi$
$908$		0			0
$909$	−2.87655			−0.0954091
$910$		0			0
$911$	31.4939			1.04344			0.521720	−	0.853117i	$-0.325291\pi$
$911$	31.4939			1.04344			0.521720	+	0.853117i	$0.325291\pi$
$912$		0			0
$913$	2.31356			0.0765678
$914$		0			0
$915$			11.4267i			0.377756i
$916$		0			0
$917$			32.6755i			1.07904i
$918$		0			0
$919$	−29.0822			−0.959333			−0.479666	−	0.877451i	$-0.659242\pi$
$919$	−29.0822			−0.959333			−0.479666	+	0.877451i	$0.659242\pi$
$920$		0			0
$921$		−	28.2595i		−	0.931181i
$922$		0			0
$923$	−5.11502	−	2.44443i	−0.168363	−	0.0804594i
$924$		0			0
$925$		−	1.41921i		−	0.0466632i
$926$		0			0
$927$	−8.11332			−0.266476
$928$		0			0
$929$		−	15.5399i		−	0.509849i	−0.966961	−	0.254924i	$-0.917949\pi$
$929$		−	15.5399i		−	0.509849i	0.966961	−	0.254924i	$-0.0820506\pi$
$930$		0			0
$931$			1.03608i			0.0339560i
$932$		0			0
$933$	−10.5329			−0.344832
$934$		0			0
$935$	7.61177			0.248931
$936$		0			0
$937$	27.1441			0.886760			0.443380	−	0.896334i	$-0.353779\pi$
$937$	27.1441			0.886760			0.443380	+	0.896334i	$0.353779\pi$
$938$		0			0
$939$	32.8602			1.07235
$940$		0			0
$941$			36.7156i			1.19689i	0.801162	+	0.598447i	$0.204215\pi$
$941$			36.7156i			1.19689i	−0.801162	+	0.598447i	$0.795785\pi$
$942$		0			0
$943$		−	8.60747i		−	0.280298i
$944$		0			0
$945$	−7.23876			−0.235477
$946$		0			0
$947$			20.8653i			0.678031i	0.940781	+	0.339016i	$0.110094\pi$
$947$			20.8653i			0.678031i	−0.940781	+	0.339016i	$0.889906\pi$
$948$		0			0
$949$	18.3597	+	8.77395i	0.595980	+	0.284814i
$950$		0			0
$951$		−	5.47914i		−	0.177673i
$952$		0			0
$953$	−2.40010			−0.0777469			−0.0388734	−	0.999244i	$-0.512377\pi$
$953$	−2.40010			−0.0777469			−0.0388734	+	0.999244i	$0.512377\pi$
$954$		0			0
$955$		−	0.353179i		−	0.0114286i
$956$		0			0
$957$			5.65697i			0.182864i
$958$		0			0
$959$	12.3736			0.399563
$960$		0			0
$961$	28.6879			0.925416
$962$		0			0
$963$	2.91478			0.0939275
$964$		0			0
$965$	17.8227			0.573732
$966$		0			0
$967$			35.4652i			1.14049i	0.821476	+	0.570243i	$0.193151\pi$
$967$			35.4652i			1.14049i	−0.821476	+	0.570243i	$0.806849\pi$
$968$		0			0
$969$		−	0.859181i		−	0.0276009i
$970$		0			0
$971$	−51.2382			−1.64431			−0.822156	−	0.569262i	$-0.807229\pi$
$971$	−51.2382			−1.64431			−0.822156	+	0.569262i	$0.807229\pi$
$972$		0			0
$973$			58.0284i			1.86030i
$974$		0			0
$975$	0.534537	−	1.11853i	0.0171189	−	0.0358216i
$976$		0			0
$977$			12.6989i			0.406274i	0.979150	+	0.203137i	$0.0651137\pi$
$977$			12.6989i			0.406274i	−0.979150	+	0.203137i	$0.934886\pi$
$978$		0			0
$979$	−13.8162			−0.441567
$980$		0			0
$981$		−	4.60391i		−	0.146992i
$982$		0			0
$983$		−	9.32322i		−	0.297364i	−0.988885	−	0.148682i	$-0.952497\pi$
$983$		−	9.32322i		−	0.297364i	0.988885	−	0.148682i	$-0.0475032\pi$
$984$		0			0
$985$	14.1783			0.451757
$986$		0			0
$987$	−21.8770			−0.696353
$988$		0			0
$989$	−30.6501			−0.974617
$990$		0			0
$991$	12.9380			0.410990			0.205495	−	0.978658i	$-0.434120\pi$
$991$	12.9380			0.410990			0.205495	+	0.978658i	$0.434120\pi$
$992$		0			0
$993$			3.91067i			0.124101i
$994$		0			0
$995$		−	26.8920i		−	0.852535i
$996$		0			0
$997$	−43.9684			−1.39249			−0.696246	−	0.717803i	$-0.745148\pi$
$997$	−43.9684			−1.39249			−0.696246	+	0.717803i	$0.745148\pi$
$998$		0			0
$999$		−	4.12766i		−	0.130593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.g.d.1585.3	✓	14
13.12	even	2	inner	3432.2.g.d.1585.12	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.g.d.1585.3	✓	14	1.1	even	1	trivial
3432.2.g.d.1585.12	yes	14	13.12	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:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.15782i q^{5} -3.35467i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.15782i q^{5} -3.35467i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(1.55466 - 3.25316i) q^{13} -2.15782i q^{15} +3.52753 q^{17} -0.243564i q^{19} -3.35467i q^{21} -4.97180 q^{23} +0.343829 q^{25} +1.00000 q^{27} +5.65697 q^{29} -1.52056i q^{31} +1.00000i q^{33} -7.23876 q^{35} -4.12766i q^{37} +(1.55466 - 3.25316i) q^{39} +1.73126i q^{41} +6.16479 q^{43} -2.15782i q^{45} -6.52136i q^{47} -4.25381 q^{49} +3.52753 q^{51} -5.94781 q^{53} +2.15782 q^{55} -0.243564i q^{57} +3.23041i q^{59} -5.29551 q^{61} -3.35467i q^{63} +(-7.01972 - 3.35467i) q^{65} +2.04164i q^{67} -4.97180 q^{69} -1.57233i q^{71} +5.64365i q^{73} +0.343829 q^{75} +3.35467 q^{77} -7.28784 q^{79} +1.00000 q^{81} -2.31356i q^{83} -7.61177i q^{85} +5.65697 q^{87} +13.8162i q^{89} +(-10.9133 - 5.21537i) q^{91} -1.52056i q^{93} -0.525567 q^{95} -15.7748i q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) \) 14 * q + 14 * q^3 + 14 * q^9	\( 14 q + 14 q^{3} + 14 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{23} + 20 q^{25} + 14 q^{27} - 10 q^{29} - 14 q^{35} - 4 q^{39} + 22 q^{43} + 8 q^{49} - 4 q^{51} - 20 q^{53} + 2 q^{55} - 2 q^{61} - 20 q^{65} + 2 q^{69} + 20 q^{75} + 2 q^{77} - 40 q^{79} + 14 q^{81} - 10 q^{87} - 44 q^{91} + 4 q^{95}+O(q^{100}) \) 14 * q + 14 * q^3 + 14 * q^9 - 4 * q^13 - 4 * q^17 + 2 * q^23 + 20 * q^25 + 14 * q^27 - 10 * q^29 - 14 * q^35 - 4 * q^39 + 22 * q^43 + 8 * q^49 - 4 * q^51 - 20 * q^53 + 2 * q^55 - 2 * q^61 - 20 * q^65 + 2 * q^69 + 20 * q^75 + 2 * q^77 - 40 * q^79 + 14 * q^81 - 10 * q^87 - 44 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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