LMFDB - Embedded newform 3432.2.g.d.1585.9

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			1585.9
Root			$0.450909i$ of defining polynomial
Character	$\chi$	$=$	3432.1585
Dual form			3432.2.g.d.1585.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.450909i q^{5} -1.57703i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.450909i q^{5} -1.57703i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(-3.49745 - 0.876263i) q^{13} +0.450909i q^{15} -1.67575 q^{17} +0.817096i q^{19} -1.57703i q^{21} +4.69850 q^{23} +4.79668 q^{25} +1.00000 q^{27} +2.60806 q^{29} +5.45408i q^{31} +1.00000i q^{33} +0.711099 q^{35} +3.10523i q^{37} +(-3.49745 - 0.876263i) q^{39} -9.37643i q^{41} +5.32743 q^{43} +0.450909i q^{45} -0.950657i q^{47} +4.51297 q^{49} -1.67575 q^{51} +7.97922 q^{53} -0.450909 q^{55} +0.817096i q^{57} -8.31746i q^{59} +12.1805 q^{61} -1.57703i q^{63} +(0.395115 - 1.57703i) q^{65} -12.9256i q^{67} +4.69850 q^{69} +7.38430i q^{71} +6.54090i q^{73} +4.79668 q^{75} +1.57703 q^{77} +4.85663 q^{79} +1.00000 q^{81} -10.7208i q^{83} -0.755609i q^{85} +2.60806 q^{87} -2.12764i q^{89} +(-1.38190 + 5.51560i) q^{91} +5.45408i q^{93} -0.368436 q^{95} +8.21369i q^{97} +1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			0.450909i			0.201653i	0.994904	+	0.100826i	$0.0321487\pi$
$5$			0.450909i			0.201653i	−0.994904	+	0.100826i	$0.967851\pi$
$6$		0			0
$7$		−	1.57703i		−	0.596062i	−0.954556	−	0.298031i	$-0.903670\pi$
$7$		−	1.57703i		−	0.596062i	0.954556	−	0.298031i	$-0.0963299\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−3.49745	−	0.876263i	−0.970018	−	0.243032i
$13$	−3.49745	−	0.876263i	−0.970018	−	0.243032i
$14$		0			0
$15$			0.450909i			0.116424i
$16$		0			0
$17$	−1.67575			−0.406428			−0.203214	−	0.979134i	$-0.565139\pi$
$17$	−1.67575			−0.406428			−0.203214	+	0.979134i	$0.565139\pi$
$18$		0			0
$19$			0.817096i			0.187455i	0.995598	+	0.0937273i	$0.0298782\pi$
$19$			0.817096i			0.187455i	−0.995598	+	0.0937273i	$0.970122\pi$
$20$		0			0
$21$		−	1.57703i		−	0.344137i
$22$		0			0
$23$	4.69850			0.979705			0.489852	−	0.871805i	$-0.337051\pi$
$23$	4.69850			0.979705			0.489852	+	0.871805i	$0.337051\pi$
$24$		0			0
$25$	4.79668			0.959336
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	2.60806			0.484304			0.242152	−	0.970238i	$-0.422147\pi$
$29$	2.60806			0.484304			0.242152	+	0.970238i	$0.422147\pi$
$30$		0			0
$31$			5.45408i			0.979582i	0.871840	+	0.489791i	$0.162927\pi$
$31$			5.45408i			0.979582i	−0.871840	+	0.489791i	$0.837073\pi$
$32$		0			0
$33$			1.00000i			0.174078i
$34$		0			0
$35$	0.711099			0.120198
$36$		0			0
$37$			3.10523i			0.510496i	0.966876	+	0.255248i	$0.0821572\pi$
$37$			3.10523i			0.510496i	−0.966876	+	0.255248i	$0.917843\pi$
$38$		0			0
$39$	−3.49745	−	0.876263i	−0.560040	−	0.140314i
$40$		0			0
$41$		−	9.37643i		−	1.46435i	−0.681115	−	0.732176i	$-0.738505\pi$
$41$		−	9.37643i		−	1.46435i	0.681115	−	0.732176i	$-0.261495\pi$
$42$		0			0
$43$	5.32743			0.812425			0.406213	−	0.913779i	$-0.366849\pi$
$43$	5.32743			0.812425			0.406213	+	0.913779i	$0.366849\pi$
$44$		0			0
$45$			0.450909i			0.0672176i
$46$		0			0
$47$		−	0.950657i		−	0.138668i	−0.997594	−	0.0693338i	$-0.977913\pi$
$47$		−	0.950657i		−	0.138668i	0.997594	−	0.0693338i	$-0.0220873\pi$
$48$		0			0
$49$	4.51297			0.644710
$50$		0			0
$51$	−1.67575			−0.234651
$52$		0			0
$53$	7.97922			1.09603			0.548015	−	0.836468i	$-0.315384\pi$
$53$	7.97922			1.09603			0.548015	+	0.836468i	$0.315384\pi$
$54$		0			0
$55$	−0.450909			−0.0608006
$56$		0			0
$57$			0.817096i			0.108227i
$58$		0			0
$59$		−	8.31746i		−	1.08284i	−0.840752	−	0.541421i	$-0.817887\pi$
$59$		−	8.31746i		−	1.08284i	0.840752	−	0.541421i	$-0.182113\pi$
$60$		0			0
$61$	12.1805			1.55956			0.779778	−	0.626056i	$-0.215332\pi$
$61$	12.1805			1.55956			0.779778	+	0.626056i	$0.215332\pi$
$62$		0			0
$63$		−	1.57703i		−	0.198687i
$64$		0			0
$65$	0.395115	−	1.57703i	0.0490080	−	0.195607i
$66$		0			0
$67$		−	12.9256i		−	1.57911i	−0.613677	−	0.789557i	$-0.710310\pi$
$67$		−	12.9256i		−	1.57911i	0.613677	−	0.789557i	$-0.289690\pi$
$68$		0			0
$69$	4.69850			0.565633
$70$		0			0
$71$			7.38430i			0.876355i	0.898888	+	0.438178i	$0.144376\pi$
$71$			7.38430i			0.876355i	−0.898888	+	0.438178i	$0.855624\pi$
$72$		0			0
$73$			6.54090i			0.765555i	0.923841	+	0.382777i	$0.125032\pi$
$73$			6.54090i			0.765555i	−0.923841	+	0.382777i	$0.874968\pi$
$74$		0			0
$75$	4.79668			0.553873
$76$		0			0
$77$	1.57703			0.179720
$78$		0			0
$79$	4.85663			0.546414			0.273207	−	0.961955i	$-0.411916\pi$
$79$	4.85663			0.546414			0.273207	+	0.961955i	$0.411916\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	10.7208i		−	1.17676i	−0.808584	−	0.588381i	$-0.799766\pi$
$83$		−	10.7208i		−	1.17676i	0.808584	−	0.588381i	$-0.200234\pi$
$84$		0			0
$85$		−	0.755609i		−	0.0819573i
$86$		0			0
$87$	2.60806			0.279613
$88$		0			0
$89$		−	2.12764i		−	0.225529i	−0.993622	−	0.112765i	$-0.964029\pi$
$89$		−	2.12764i		−	0.225529i	0.993622	−	0.112765i	$-0.0359706\pi$
$90$		0			0
$91$	−1.38190	+	5.51560i	−0.144862	+	0.578191i
$92$		0			0
$93$			5.45408i			0.565562i
$94$		0			0
$95$	−0.368436			−0.0378007
$96$		0			0
$97$			8.21369i			0.833974i	0.908912	+	0.416987i	$0.136914\pi$
$97$			8.21369i			0.833974i	−0.908912	+	0.416987i	$0.863086\pi$
$98$		0			0
$99$			1.00000i			0.100504i
$100$		0			0
$101$	5.48976			0.546251			0.273126	−	0.961978i	$-0.411943\pi$
$101$	5.48976			0.546251			0.273126	+	0.961978i	$0.411943\pi$
$102$		0			0
$103$	17.5184			1.72614			0.863070	−	0.505085i	$-0.168539\pi$
$103$	17.5184			1.72614			0.863070	+	0.505085i	$0.168539\pi$
$104$		0			0
$105$	0.711099			0.0693961
$106$		0			0
$107$	−3.41565			−0.330203			−0.165101	−	0.986277i	$-0.552795\pi$
$107$	−3.41565			−0.330203			−0.165101	+	0.986277i	$0.552795\pi$
$108$		0			0
$109$		−	17.5127i		−	1.67741i	−0.544583	−	0.838707i	$-0.683312\pi$
$109$		−	17.5127i		−	1.67741i	0.544583	−	0.838707i	$-0.316688\pi$
$110$		0			0
$111$			3.10523i			0.294735i
$112$		0			0
$113$	−12.9265			−1.21602			−0.608012	−	0.793928i	$-0.708033\pi$
$113$	−12.9265			−1.21602			−0.608012	+	0.793928i	$0.708033\pi$
$114$		0			0
$115$			2.11860i			0.197560i
$116$		0			0
$117$	−3.49745	−	0.876263i	−0.323339	−	0.0810105i
$118$		0			0
$119$			2.64271i			0.242256i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	9.37643i		−	0.845444i
$124$		0			0
$125$			4.41741i			0.395105i
$126$		0			0
$127$	−8.22838			−0.730150			−0.365075	−	0.930978i	$-0.618957\pi$
$127$	−8.22838			−0.730150			−0.365075	+	0.930978i	$0.618957\pi$
$128$		0			0
$129$	5.32743			0.469054
$130$		0			0
$131$	−12.2131			−1.06707			−0.533533	−	0.845779i	$-0.679136\pi$
$131$	−12.2131			−1.06707			−0.533533	+	0.845779i	$0.679136\pi$
$132$		0			0
$133$	1.28859			0.111735
$134$		0			0
$135$			0.450909i			0.0388081i
$136$		0			0
$137$			13.9981i			1.19594i	0.801519	+	0.597969i	$0.204026\pi$
$137$			13.9981i			1.19594i	−0.801519	+	0.597969i	$0.795974\pi$
$138$		0			0
$139$	−11.5592			−0.980439			−0.490220	−	0.871599i	$-0.663083\pi$
$139$	−11.5592			−0.980439			−0.490220	+	0.871599i	$0.663083\pi$
$140$		0			0
$141$		−	0.950657i		−	0.0800597i
$142$		0			0
$143$	0.876263	−	3.49745i	0.0732768	−	0.292472i
$144$		0			0
$145$			1.17600i			0.0976613i
$146$		0			0
$147$	4.51297			0.372223
$148$		0			0
$149$			5.64119i			0.462144i	0.972937	+	0.231072i	$0.0742233\pi$
$149$			5.64119i			0.462144i	−0.972937	+	0.231072i	$0.925777\pi$
$150$		0			0
$151$		−	11.5645i		−	0.941108i	−0.882371	−	0.470554i	$-0.844054\pi$
$151$		−	11.5645i		−	0.941108i	0.882371	−	0.470554i	$-0.155946\pi$
$152$		0			0
$153$	−1.67575			−0.135476
$154$		0			0
$155$	−2.45930			−0.197535
$156$		0			0
$157$	19.2374			1.53531			0.767654	−	0.640864i	$-0.221424\pi$
$157$	19.2374			1.53531			0.767654	+	0.640864i	$0.221424\pi$
$158$		0			0
$159$	7.97922			0.632793
$160$		0			0
$161$		−	7.40969i		−	0.583965i
$162$		0			0
$163$		−	9.63619i		−	0.754765i	−0.926058	−	0.377382i	$-0.876824\pi$
$163$		−	9.63619i		−	0.754765i	0.926058	−	0.377382i	$-0.123176\pi$
$164$		0			0
$165$	−0.450909			−0.0351032
$166$		0			0
$167$			6.81102i			0.527052i	0.964652	+	0.263526i	$0.0848856\pi$
$167$			6.81102i			0.527052i	−0.964652	+	0.263526i	$0.915114\pi$
$168$		0			0
$169$	11.4643	+	6.12937i	0.881871	+	0.471490i
$170$		0			0
$171$			0.817096i			0.0624849i
$172$		0			0
$173$	−7.61596			−0.579031			−0.289516	−	0.957173i	$-0.593494\pi$
$173$	−7.61596			−0.579031			−0.289516	+	0.957173i	$0.593494\pi$
$174$		0			0
$175$		−	7.56452i		−	0.571824i
$176$		0			0
$177$		−	8.31746i		−	0.625179i
$178$		0			0
$179$	16.3459			1.22175			0.610877	−	0.791726i	$-0.290817\pi$
$179$	16.3459			1.22175			0.610877	+	0.791726i	$0.290817\pi$
$180$		0			0
$181$	−5.79839			−0.430991			−0.215495	−	0.976505i	$-0.569137\pi$
$181$	−5.79839			−0.430991			−0.215495	+	0.976505i	$0.569137\pi$
$182$		0			0
$183$	12.1805			0.900410
$184$		0			0
$185$	−1.40018			−0.102943
$186$		0			0
$187$		−	1.67575i		−	0.122543i
$188$		0			0
$189$		−	1.57703i		−	0.114712i
$190$		0			0
$191$	−1.13135			−0.0818615			−0.0409307	−	0.999162i	$-0.513032\pi$
$191$	−1.13135			−0.0818615			−0.0409307	+	0.999162i	$0.513032\pi$
$192$		0			0
$193$		−	2.88305i		−	0.207527i	−0.994602	−	0.103763i	$-0.966912\pi$
$193$		−	2.88305i		−	0.207527i	0.994602	−	0.103763i	$-0.0330884\pi$
$194$		0			0
$195$	0.395115	−	1.57703i	0.0282948	−	0.112934i
$196$		0			0
$197$			13.6638i			0.973504i	0.873540	+	0.486752i	$0.161818\pi$
$197$			13.6638i			0.973504i	−0.873540	+	0.486752i	$0.838182\pi$
$198$		0			0
$199$	−1.24590			−0.0883193			−0.0441596	−	0.999024i	$-0.514061\pi$
$199$	−1.24590			−0.0883193			−0.0441596	+	0.999024i	$0.514061\pi$
$200$		0			0
$201$		−	12.9256i		−	0.911702i
$202$		0			0
$203$		−	4.11299i		−	0.288676i
$204$		0			0
$205$	4.22792			0.295291
$206$		0			0
$207$	4.69850			0.326568
$208$		0			0
$209$	−0.817096			−0.0565197
$210$		0			0
$211$	21.2317			1.46165			0.730825	−	0.682565i	$-0.239136\pi$
$211$	21.2317			1.46165			0.730825	+	0.682565i	$0.239136\pi$
$212$		0			0
$213$			7.38430i			0.505964i
$214$		0			0
$215$			2.40219i			0.163828i
$216$		0			0
$217$	8.60127			0.583892
$218$		0			0
$219$			6.54090i			0.441993i
$220$		0			0
$221$	5.86084	+	1.46839i	0.394243	+	0.0987748i
$222$		0			0
$223$			7.23706i			0.484630i	0.970198	+	0.242315i	$0.0779067\pi$
$223$			7.23706i			0.484630i	−0.970198	+	0.242315i	$0.922093\pi$
$224$		0			0
$225$	4.79668			0.319779
$226$		0			0
$227$		−	7.78736i		−	0.516865i	−0.966029	−	0.258432i	$-0.916794\pi$
$227$		−	7.78736i		−	0.516865i	0.966029	−	0.258432i	$-0.0832059\pi$
$228$		0			0
$229$			6.28345i			0.415222i	0.978211	+	0.207611i	$0.0665688\pi$
$229$			6.28345i			0.415222i	−0.978211	+	0.207611i	$0.933431\pi$
$230$		0			0
$231$	1.57703			0.103761
$232$		0			0
$233$	2.66678			0.174706			0.0873532	−	0.996177i	$-0.472159\pi$
$233$	2.66678			0.174706			0.0873532	+	0.996177i	$0.472159\pi$
$234$		0			0
$235$	0.428660			0.0279627
$236$		0			0
$237$	4.85663			0.315472
$238$		0			0
$239$			6.50123i			0.420529i	0.977644	+	0.210265i	$0.0674326\pi$
$239$			6.50123i			0.420529i	−0.977644	+	0.210265i	$0.932567\pi$
$240$		0			0
$241$			16.2062i			1.04393i	0.852966	+	0.521967i	$0.174802\pi$
$241$			16.2062i			1.04393i	−0.852966	+	0.521967i	$0.825198\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			2.03494i			0.130007i
$246$		0			0
$247$	0.715991	−	2.85775i	0.0455574	−	0.181834i
$248$		0			0
$249$		−	10.7208i		−	0.679404i
$250$		0			0
$251$	9.53127			0.601608			0.300804	−	0.953686i	$-0.402745\pi$
$251$	9.53127			0.601608			0.300804	+	0.953686i	$0.402745\pi$
$252$		0			0
$253$			4.69850i			0.295392i
$254$		0			0
$255$		−	0.755609i		−	0.0473181i
$256$		0			0
$257$	−15.8528			−0.988870			−0.494435	−	0.869215i	$-0.664625\pi$
$257$	−15.8528			−0.988870			−0.494435	+	0.869215i	$0.664625\pi$
$258$		0			0
$259$	4.89705			0.304288
$260$		0			0
$261$	2.60806			0.161435
$262$		0			0
$263$	−9.32064			−0.574735			−0.287368	−	0.957820i	$-0.592780\pi$
$263$	−9.32064			−0.574735			−0.287368	+	0.957820i	$0.592780\pi$
$264$		0			0
$265$			3.59790i			0.221017i
$266$		0			0
$267$		−	2.12764i		−	0.130209i
$268$		0			0
$269$	−16.0781			−0.980300			−0.490150	−	0.871638i	$-0.663058\pi$
$269$	−16.0781			−0.980300			−0.490150	+	0.871638i	$0.663058\pi$
$270$		0			0
$271$		−	5.09967i		−	0.309783i	−0.987931	−	0.154892i	$-0.950497\pi$
$271$		−	5.09967i		−	0.309783i	0.987931	−	0.154892i	$-0.0495028\pi$
$272$		0			0
$273$	−1.38190	+	5.51560i	−0.0836361	+	0.333819i
$274$		0			0
$275$			4.79668i			0.289251i
$276$		0			0
$277$	8.60022			0.516737			0.258368	−	0.966046i	$-0.416815\pi$
$277$	8.60022			0.516737			0.258368	+	0.966046i	$0.416815\pi$
$278$		0			0
$279$			5.45408i			0.326527i
$280$		0			0
$281$		−	16.4366i		−	0.980523i	−0.871575	−	0.490262i	$-0.836901\pi$
$281$		−	16.4366i		−	0.980523i	0.871575	−	0.490262i	$-0.163099\pi$
$282$		0			0
$283$	8.07109			0.479776			0.239888	−	0.970801i	$-0.422889\pi$
$283$	8.07109			0.479776			0.239888	+	0.970801i	$0.422889\pi$
$284$		0			0
$285$	−0.368436			−0.0218243
$286$		0			0
$287$	−14.7869			−0.872846
$288$		0			0
$289$	−14.1919			−0.834816
$290$		0			0
$291$			8.21369i			0.481495i
$292$		0			0
$293$			21.4865i			1.25525i	0.778514	+	0.627627i	$0.215974\pi$
$293$			21.4865i			1.25525i	−0.778514	+	0.627627i	$0.784026\pi$
$294$		0			0
$295$	3.75042			0.218358
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	−16.4328	−	4.11712i	−0.950332	−	0.238099i
$300$		0			0
$301$		−	8.40153i		−	0.484256i
$302$		0			0
$303$	5.48976			0.315378
$304$		0			0
$305$			5.49231i			0.314489i
$306$		0			0
$307$		−	4.68263i		−	0.267252i	−0.991032	−	0.133626i	$-0.957338\pi$
$307$		−	4.68263i		−	0.267252i	0.991032	−	0.133626i	$-0.0426621\pi$
$308$		0			0
$309$	17.5184			0.996587
$310$		0			0
$311$	−8.71420			−0.494137			−0.247068	−	0.968998i	$-0.579467\pi$
$311$	−8.71420			−0.494137			−0.247068	+	0.968998i	$0.579467\pi$
$312$		0			0
$313$	−24.0084			−1.35704			−0.678518	−	0.734584i	$-0.737377\pi$
$313$	−24.0084			−1.35704			−0.678518	+	0.734584i	$0.737377\pi$
$314$		0			0
$315$	0.711099			0.0400659
$316$		0			0
$317$			15.7982i			0.887315i	0.896197	+	0.443657i	$0.146319\pi$
$317$			15.7982i			0.887315i	−0.896197	+	0.443657i	$0.853681\pi$
$318$		0			0
$319$			2.60806i			0.146023i
$320$		0			0
$321$	−3.41565			−0.190643
$322$		0			0
$323$		−	1.36924i		−	0.0761868i
$324$		0			0
$325$	−16.7762	−	4.20315i	−0.930574	−	0.233149i
$326$		0			0
$327$		−	17.5127i		−	0.968455i
$328$		0			0
$329$	−1.49922			−0.0826545
$330$		0			0
$331$		−	33.3420i		−	1.83264i	−0.400444	−	0.916321i	$-0.631144\pi$
$331$		−	33.3420i		−	1.83264i	0.400444	−	0.916321i	$-0.368856\pi$
$332$		0			0
$333$			3.10523i			0.170165i
$334$		0			0
$335$	5.82828			0.318433
$336$		0			0
$337$	23.4018			1.27478			0.637388	−	0.770543i	$-0.280015\pi$
$337$	23.4018			1.27478			0.637388	+	0.770543i	$0.280015\pi$
$338$		0			0
$339$	−12.9265			−0.702072
$340$		0			0
$341$	−5.45408			−0.295355
$342$		0			0
$343$		−	18.1563i		−	0.980350i
$344$		0			0
$345$			2.11860i			0.114061i
$346$		0			0
$347$	−2.97110			−0.159497			−0.0797486	−	0.996815i	$-0.525412\pi$
$347$	−2.97110			−0.159497			−0.0797486	+	0.996815i	$0.525412\pi$
$348$		0			0
$349$			19.6156i			1.05000i	0.851103	+	0.524999i	$0.175934\pi$
$349$			19.6156i			1.05000i	−0.851103	+	0.524999i	$0.824066\pi$
$350$		0			0
$351$	−3.49745	−	0.876263i	−0.186680	−	0.0467714i
$352$		0			0
$353$		−	1.49976i		−	0.0798241i	−0.999203	−	0.0399120i	$-0.987292\pi$
$353$		−	1.49976i		−	0.0798241i	0.999203	−	0.0399120i	$-0.0127078\pi$
$354$		0			0
$355$	−3.32965			−0.176719
$356$		0			0
$357$			2.64271i			0.139867i
$358$		0			0
$359$		−	14.7075i		−	0.776232i	−0.921611	−	0.388116i	$-0.873126\pi$
$359$		−	14.7075i		−	0.776232i	0.921611	−	0.388116i	$-0.126874\pi$
$360$		0			0
$361$	18.3324			0.964861
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	−2.94935			−0.154376
$366$		0			0
$367$	−32.5069			−1.69685			−0.848424	−	0.529317i	$-0.822448\pi$
$367$	−32.5069			−1.69685			−0.848424	+	0.529317i	$0.822448\pi$
$368$		0			0
$369$		−	9.37643i		−	0.488118i
$370$		0			0
$371$		−	12.5835i		−	0.653302i
$372$		0			0
$373$	21.1414			1.09466			0.547331	−	0.836916i	$-0.315644\pi$
$373$	21.1414			1.09466			0.547331	+	0.836916i	$0.315644\pi$
$374$		0			0
$375$			4.41741i			0.228114i
$376$		0			0
$377$	−9.12156	−	2.28534i	−0.469784	−	0.117701i
$378$		0			0
$379$		−	22.9022i		−	1.17641i	−0.808712	−	0.588205i	$-0.799835\pi$
$379$		−	22.9022i		−	1.17641i	0.808712	−	0.588205i	$-0.200165\pi$
$380$		0			0
$381$	−8.22838			−0.421552
$382$		0			0
$383$			32.4446i			1.65784i	0.559367	+	0.828920i	$0.311044\pi$
$383$			32.4446i			1.65784i	−0.559367	+	0.828920i	$0.688956\pi$
$384$		0			0
$385$			0.711099i			0.0362409i
$386$		0			0
$387$	5.32743			0.270808
$388$		0			0
$389$	29.5664			1.49907			0.749537	−	0.661962i	$-0.230276\pi$
$389$	29.5664			1.49907			0.749537	+	0.661962i	$0.230276\pi$
$390$		0			0
$391$	−7.87349			−0.398179
$392$		0			0
$393$	−12.2131			−0.616071
$394$		0			0
$395$			2.18990i			0.110186i
$396$		0			0
$397$			15.5715i			0.781510i	0.920495	+	0.390755i	$0.127786\pi$
$397$			15.5715i			0.781510i	−0.920495	+	0.390755i	$0.872214\pi$
$398$		0			0
$399$	1.28859			0.0645100
$400$		0			0
$401$			16.8103i			0.839469i	0.907647	+	0.419734i	$0.137877\pi$
$401$			16.8103i			0.839469i	−0.907647	+	0.419734i	$0.862123\pi$
$402$		0			0
$403$	4.77921	−	19.0754i	0.238069	−	0.950213i
$404$		0			0
$405$			0.450909i			0.0224059i
$406$		0			0
$407$	−3.10523			−0.153920
$408$		0			0
$409$			38.4535i			1.90140i	0.310106	+	0.950702i	$0.399636\pi$
$409$			38.4535i			1.90140i	−0.310106	+	0.950702i	$0.600364\pi$
$410$		0			0
$411$			13.9981i			0.690475i
$412$		0			0
$413$	−13.1169			−0.645441
$414$		0			0
$415$	4.83411			0.237297
$416$		0			0
$417$	−11.5592			−0.566057
$418$		0			0
$419$	11.4622			0.559967			0.279984	−	0.960005i	$-0.409671\pi$
$419$	11.4622			0.559967			0.279984	+	0.960005i	$0.409671\pi$
$420$		0			0
$421$			22.0087i			1.07264i	0.844016	+	0.536318i	$0.180185\pi$
$421$			22.0087i			1.07264i	−0.844016	+	0.536318i	$0.819815\pi$
$422$		0			0
$423$		−	0.950657i		−	0.0462225i
$424$		0			0
$425$	−8.03801			−0.389901
$426$		0			0
$427$		−	19.2091i		−	0.929593i
$428$		0			0
$429$	0.876263	−	3.49745i	0.0423064	−	0.168859i
$430$		0			0
$431$			8.39740i			0.404489i	0.979335	+	0.202244i	$0.0648235\pi$
$431$			8.39740i			0.404489i	−0.979335	+	0.202244i	$0.935176\pi$
$432$		0			0
$433$	21.4756			1.03205			0.516027	−	0.856572i	$-0.327410\pi$
$433$	21.4756			1.03205			0.516027	+	0.856572i	$0.327410\pi$
$434$		0			0
$435$			1.17600i			0.0563848i
$436$		0			0
$437$			3.83912i			0.183650i
$438$		0			0
$439$	2.29266			0.109423			0.0547113	−	0.998502i	$-0.482576\pi$
$439$	2.29266			0.109423			0.0547113	+	0.998502i	$0.482576\pi$
$440$		0			0
$441$	4.51297			0.214903
$442$		0			0
$443$	−20.1791			−0.958740			−0.479370	−	0.877613i	$-0.659135\pi$
$443$	−20.1791			−0.958740			−0.479370	+	0.877613i	$0.659135\pi$
$444$		0			0
$445$	0.959373			0.0454786
$446$		0			0
$447$			5.64119i			0.266819i
$448$		0			0
$449$		−	27.8560i		−	1.31461i	−0.753626	−	0.657303i	$-0.771697\pi$
$449$		−	27.8560i		−	1.31461i	0.753626	−	0.657303i	$-0.228303\pi$
$450$		0			0
$451$	9.37643			0.441519
$452$		0			0
$453$		−	11.5645i		−	0.543349i
$454$		0			0
$455$	−2.48703	−	0.623109i	−0.116594	−	0.0292118i
$456$		0			0
$457$			4.61719i			0.215983i	0.994152	+	0.107992i	$0.0344420\pi$
$457$			4.61719i			0.215983i	−0.994152	+	0.107992i	$0.965558\pi$
$458$		0			0
$459$	−1.67575			−0.0782171
$460$		0			0
$461$		−	10.4322i		−	0.485875i	−0.970042	−	0.242937i	$-0.921889\pi$
$461$		−	10.4322i		−	0.485875i	0.970042	−	0.242937i	$-0.0781110\pi$
$462$		0			0
$463$			7.02337i			0.326404i	0.986593	+	0.163202i	$0.0521822\pi$
$463$			7.02337i			0.326404i	−0.986593	+	0.163202i	$0.947818\pi$
$464$		0			0
$465$	−2.45930			−0.114047
$466$		0			0
$467$	10.4014			0.481321			0.240661	−	0.970609i	$-0.422636\pi$
$467$	10.4014			0.481321			0.240661	+	0.970609i	$0.422636\pi$
$468$		0			0
$469$	−20.3841			−0.941251
$470$		0			0
$471$	19.2374			0.886411
$472$		0			0
$473$			5.32743i			0.244955i
$474$		0			0
$475$			3.91935i			0.179832i
$476$		0			0
$477$	7.97922			0.365343
$478$		0			0
$479$		−	12.7581i		−	0.582931i	−0.956581	−	0.291466i	$-0.905857\pi$
$479$		−	12.7581i		−	0.582931i	0.956581	−	0.291466i	$-0.0941429\pi$
$480$		0			0
$481$	2.72100	−	10.8604i	0.124067	−	0.495191i
$482$		0			0
$483$		−	7.40969i		−	0.337152i
$484$		0			0
$485$	−3.70363			−0.168173
$486$		0			0
$487$		−	12.7148i		−	0.576165i	−0.957606	−	0.288082i	$-0.906982\pi$
$487$		−	12.7148i		−	0.576165i	0.957606	−	0.288082i	$-0.0930177\pi$
$488$		0			0
$489$		−	9.63619i		−	0.435764i
$490$		0			0
$491$	−37.3101			−1.68378			−0.841891	−	0.539647i	$-0.818558\pi$
$491$	−37.3101			−1.68378			−0.841891	+	0.539647i	$0.818558\pi$
$492$		0			0
$493$	−4.37044			−0.196835
$494$		0			0
$495$	−0.450909			−0.0202669
$496$		0			0
$497$	11.6453			0.522362
$498$		0			0
$499$		−	41.7515i		−	1.86906i	−0.355890	−	0.934528i	$-0.615822\pi$
$499$		−	41.7515i		−	1.86906i	0.355890	−	0.934528i	$-0.384178\pi$
$500$		0			0
$501$			6.81102i			0.304294i
$502$		0			0
$503$	−35.4832			−1.58212			−0.791059	−	0.611740i	$-0.790470\pi$
$503$	−35.4832			−1.58212			−0.791059	+	0.611740i	$0.790470\pi$
$504$		0			0
$505$			2.47538i			0.110153i
$506$		0			0
$507$	11.4643	+	6.12937i	0.509149	+	0.272215i
$508$		0			0
$509$		−	33.1922i		−	1.47122i	−0.677407	−	0.735609i	$-0.736896\pi$
$509$		−	33.1922i		−	1.47122i	0.677407	−	0.735609i	$-0.263104\pi$
$510$		0			0
$511$	10.3152			0.456318
$512$		0			0
$513$			0.817096i			0.0360757i
$514$		0			0
$515$			7.89921i			0.348081i
$516$		0			0
$517$	0.950657			0.0418098
$518$		0			0
$519$	−7.61596			−0.334304
$520$		0			0
$521$	−16.7836			−0.735302			−0.367651	−	0.929964i	$-0.619838\pi$
$521$	−16.7836			−0.735302			−0.367651	+	0.929964i	$0.619838\pi$
$522$		0			0
$523$	1.01714			0.0444765			0.0222382	−	0.999753i	$-0.492921\pi$
$523$	1.01714			0.0444765			0.0222382	+	0.999753i	$0.492921\pi$
$524$		0			0
$525$		−	7.56452i		−	0.330143i
$526$		0			0
$527$		−	9.13965i		−	0.398129i
$528$		0			0
$529$	−0.924104			−0.0401784
$530$		0			0
$531$		−	8.31746i		−	0.360947i
$532$		0			0
$533$	−8.21622	+	32.7936i	−0.355884	+	1.42045i
$534$		0			0
$535$		−	1.54015i		−	0.0665863i
$536$		0			0
$537$	16.3459			0.705380
$538$		0			0
$539$			4.51297i			0.194387i
$540$		0			0
$541$			30.4385i			1.30865i	0.756212	+	0.654327i	$0.227048\pi$
$541$			30.4385i			1.30865i	−0.756212	+	0.654327i	$0.772952\pi$
$542$		0			0
$543$	−5.79839			−0.248833
$544$		0			0
$545$	7.89664			0.338255
$546$		0			0
$547$	−30.2865			−1.29496			−0.647480	−	0.762083i	$-0.724177\pi$
$547$	−30.2865			−1.29496			−0.647480	+	0.762083i	$0.724177\pi$
$548$		0			0
$549$	12.1805			0.519852
$550$		0			0
$551$			2.13103i			0.0907851i
$552$		0			0
$553$		−	7.65907i		−	0.325697i
$554$		0			0
$555$	−1.40018			−0.0594341
$556$		0			0
$557$			5.14095i			0.217829i	0.994051	+	0.108915i	$0.0347375\pi$
$557$			5.14095i			0.217829i	−0.994051	+	0.108915i	$0.965262\pi$
$558$		0			0
$559$	−18.6324	−	4.66823i	−0.788067	−	0.197445i
$560$		0			0
$561$		−	1.67575i		−	0.0707500i
$562$		0			0
$563$	−28.6958			−1.20938			−0.604692	−	0.796459i	$-0.706704\pi$
$563$	−28.6958			−1.20938			−0.604692	+	0.796459i	$0.706704\pi$
$564$		0			0
$565$		−	5.82868i		−	0.245214i
$566$		0			0
$567$		−	1.57703i		−	0.0662292i
$568$		0			0
$569$	36.3518			1.52395			0.761973	−	0.647609i	$-0.224231\pi$
$569$	36.3518			1.52395			0.761973	+	0.647609i	$0.224231\pi$
$570$		0			0
$571$	8.93649			0.373981			0.186990	−	0.982362i	$-0.440127\pi$
$571$	8.93649			0.373981			0.186990	+	0.982362i	$0.440127\pi$
$572$		0			0
$573$	−1.13135			−0.0472627
$574$		0			0
$575$	22.5372			0.939866
$576$		0			0
$577$			43.5168i			1.81163i	0.423678	+	0.905813i	$0.360739\pi$
$577$			43.5168i			1.81163i	−0.423678	+	0.905813i	$0.639261\pi$
$578$		0			0
$579$		−	2.88305i		−	0.119816i
$580$		0			0
$581$	−16.9071			−0.701424
$582$		0			0
$583$			7.97922i			0.330465i
$584$		0			0
$585$	0.395115	−	1.57703i	0.0163360	−	0.0652023i
$586$		0			0
$587$			27.1699i			1.12142i	0.828012	+	0.560711i	$0.189472\pi$
$587$			27.1699i			1.12142i	−0.828012	+	0.560711i	$0.810528\pi$
$588$		0			0
$589$	−4.45651			−0.183627
$590$		0			0
$591$			13.6638i			0.562053i
$592$		0			0
$593$		−	18.6333i		−	0.765177i	−0.923919	−	0.382588i	$-0.875033\pi$
$593$		−	18.6333i		−	0.765177i	0.923919	−	0.382588i	$-0.124967\pi$
$594$		0			0
$595$	−1.19162			−0.0488517
$596$		0			0
$597$	−1.24590			−0.0509912
$598$		0			0
$599$	44.6523			1.82444			0.912222	−	0.409696i	$-0.134365\pi$
$599$	44.6523			1.82444			0.912222	+	0.409696i	$0.134365\pi$
$600$		0			0
$601$	3.23725			0.132050			0.0660251	−	0.997818i	$-0.478968\pi$
$601$	3.23725			0.132050			0.0660251	+	0.997818i	$0.478968\pi$
$602$		0			0
$603$		−	12.9256i		−	0.526372i
$604$		0			0
$605$		−	0.450909i		−	0.0183321i
$606$		0			0
$607$	−5.77847			−0.234541			−0.117270	−	0.993100i	$-0.537414\pi$
$607$	−5.77847			−0.234541			−0.117270	+	0.993100i	$0.537414\pi$
$608$		0			0
$609$		−	4.11299i		−	0.166667i
$610$		0			0
$611$	−0.833025	+	3.32488i	−0.0337006	+	0.134510i
$612$		0			0
$613$			12.7252i			0.513966i	0.966416	+	0.256983i	$0.0827285\pi$
$613$			12.7252i			0.513966i	−0.966416	+	0.256983i	$0.917272\pi$
$614$		0			0
$615$	4.22792			0.170486
$616$		0			0
$617$			17.8798i			0.719814i	0.932988	+	0.359907i	$0.117192\pi$
$617$			17.8798i			0.719814i	−0.932988	+	0.359907i	$0.882808\pi$
$618$		0			0
$619$		−	31.3044i		−	1.25823i	−0.777312	−	0.629115i	$-0.783417\pi$
$619$		−	31.3044i		−	1.25823i	0.777312	−	0.629115i	$-0.216583\pi$
$620$		0			0
$621$	4.69850			0.188544
$622$		0			0
$623$	−3.35536			−0.134430
$624$		0			0
$625$	21.9916			0.879662
$626$		0			0
$627$	−0.817096			−0.0326317
$628$		0			0
$629$		−	5.20357i		−	0.207480i
$630$		0			0
$631$			1.92595i			0.0766708i	0.999265	+	0.0383354i	$0.0122055\pi$
$631$			1.92595i			0.0766708i	−0.999265	+	0.0383354i	$0.987794\pi$
$632$		0			0
$633$	21.2317			0.843884
$634$		0			0
$635$		−	3.71025i		−	0.147237i
$636$		0			0
$637$	−15.7839	−	3.95455i	−0.625380	−	0.156685i
$638$		0			0
$639$			7.38430i			0.292118i
$640$		0			0
$641$	−27.9165			−1.10264			−0.551318	−	0.834295i	$-0.685875\pi$
$641$	−27.9165			−1.10264			−0.551318	+	0.834295i	$0.685875\pi$
$642$		0			0
$643$			22.4238i			0.884309i	0.896939	+	0.442155i	$0.145786\pi$
$643$			22.4238i			0.884309i	−0.896939	+	0.442155i	$0.854214\pi$
$644$		0			0
$645$			2.40219i			0.0945860i
$646$		0			0
$647$	−34.9098			−1.37245			−0.686223	−	0.727391i	$-0.740733\pi$
$647$	−34.9098			−1.37245			−0.686223	+	0.727391i	$0.740733\pi$
$648$		0			0
$649$	8.31746			0.326489
$650$		0			0
$651$	8.60127			0.337110
$652$		0			0
$653$	−15.4250			−0.603625			−0.301813	−	0.953367i	$-0.597592\pi$
$653$	−15.4250			−0.603625			−0.301813	+	0.953367i	$0.597592\pi$
$654$		0			0
$655$		−	5.50701i		−	0.215177i
$656$		0			0
$657$			6.54090i			0.255185i
$658$		0			0
$659$	−46.6172			−1.81595			−0.907974	−	0.419026i	$-0.862372\pi$
$659$	−46.6172			−1.81595			−0.907974	+	0.419026i	$0.862372\pi$
$660$		0			0
$661$		−	48.3873i		−	1.88205i	−0.338339	−	0.941024i	$-0.609865\pi$
$661$		−	48.3873i		−	1.88205i	0.338339	−	0.941024i	$-0.390135\pi$
$662$		0			0
$663$	5.86084	+	1.46839i	0.227616	+	0.0570277i
$664$		0			0
$665$			0.581036i			0.0225316i
$666$		0			0
$667$	12.2540			0.474475
$668$		0			0
$669$			7.23706i			0.279801i
$670$		0			0
$671$			12.1805i			0.470224i
$672$		0			0
$673$	−7.55415			−0.291191			−0.145595	−	0.989344i	$-0.546510\pi$
$673$	−7.55415			−0.291191			−0.145595	+	0.989344i	$0.546510\pi$
$674$		0			0
$675$	4.79668			0.184624
$676$		0			0
$677$	−36.5944			−1.40644			−0.703219	−	0.710973i	$-0.748255\pi$
$677$	−36.5944			−1.40644			−0.703219	+	0.710973i	$0.748255\pi$
$678$		0			0
$679$	12.9533			0.497100
$680$		0			0
$681$		−	7.78736i		−	0.298412i
$682$		0			0
$683$		−	11.1641i		−	0.427182i	−0.976923	−	0.213591i	$-0.931484\pi$
$683$		−	11.1641i		−	0.427182i	0.976923	−	0.213591i	$-0.0685160\pi$
$684$		0			0
$685$	−6.31187			−0.241164
$686$		0			0
$687$			6.28345i			0.239729i
$688$		0			0
$689$	−27.9069	−	6.99189i	−1.06317	−	0.266370i
$690$		0			0
$691$			44.5904i			1.69630i	0.529757	+	0.848149i	$0.322283\pi$
$691$			44.5904i			1.69630i	−0.529757	+	0.848149i	$0.677717\pi$
$692$		0			0
$693$	1.57703			0.0599065
$694$		0			0
$695$		−	5.21215i		−	0.197708i
$696$		0			0
$697$			15.7125i			0.595154i
$698$		0			0
$699$	2.66678			0.100867
$700$		0			0
$701$	7.63901			0.288521			0.144261	−	0.989540i	$-0.453920\pi$
$701$	7.63901			0.288521			0.144261	+	0.989540i	$0.453920\pi$
$702$		0			0
$703$	−2.53727			−0.0956949
$704$		0			0
$705$	0.428660			0.0161443
$706$		0			0
$707$		−	8.65753i		−	0.325600i
$708$		0			0
$709$			47.7904i			1.79480i	0.441213	+	0.897402i	$0.354548\pi$
$709$			47.7904i			1.79480i	−0.441213	+	0.897402i	$0.645452\pi$
$710$		0			0
$711$	4.85663			0.182138
$712$		0			0
$713$			25.6260i			0.959701i
$714$		0			0
$715$	1.57703	+	0.395115i	0.0589777	+	0.0147765i
$716$		0			0
$717$			6.50123i			0.242793i
$718$		0			0
$719$	−19.4015			−0.723554			−0.361777	−	0.932265i	$-0.617830\pi$
$719$	−19.4015			−0.723554			−0.361777	+	0.932265i	$0.617830\pi$
$720$		0			0
$721$		−	27.6271i		−	1.02889i
$722$		0			0
$723$			16.2062i			0.602716i
$724$		0			0
$725$	12.5100			0.464611
$726$		0			0
$727$	23.8411			0.884217			0.442109	−	0.896962i	$-0.354231\pi$
$727$	23.8411			0.884217			0.442109	+	0.896962i	$0.354231\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−8.92741			−0.330192
$732$		0			0
$733$			8.61689i			0.318272i	0.987257	+	0.159136i	$0.0508709\pi$
$733$			8.61689i			0.318272i	−0.987257	+	0.159136i	$0.949129\pi$
$734$		0			0
$735$			2.03494i			0.0750598i
$736$		0			0
$737$	12.9256			0.476121
$738$		0			0
$739$			26.3789i			0.970365i	0.874413	+	0.485182i	$0.161247\pi$
$739$			26.3789i			0.970365i	−0.874413	+	0.485182i	$0.838753\pi$
$740$		0			0
$741$	0.715991	−	2.85775i	0.0263026	−	0.104982i
$742$		0			0
$743$		−	29.6486i		−	1.08770i	−0.839182	−	0.543851i	$-0.816966\pi$
$743$		−	29.6486i		−	1.08770i	0.839182	−	0.543851i	$-0.183034\pi$
$744$		0			0
$745$	−2.54366			−0.0931926
$746$		0			0
$747$		−	10.7208i		−	0.392254i
$748$		0			0
$749$			5.38659i			0.196822i
$750$		0			0
$751$	13.1445			0.479650			0.239825	−	0.970816i	$-0.422910\pi$
$751$	13.1445			0.479650			0.239825	+	0.970816i	$0.422910\pi$
$752$		0			0
$753$	9.53127			0.347339
$754$		0			0
$755$	5.21455			0.189777
$756$		0			0
$757$	17.5007			0.636074			0.318037	−	0.948078i	$-0.396976\pi$
$757$	17.5007			0.636074			0.318037	+	0.948078i	$0.396976\pi$
$758$		0			0
$759$			4.69850i			0.170545i
$760$		0			0
$761$			15.1248i			0.548275i	0.961690	+	0.274138i	$0.0883924\pi$
$761$			15.1248i			0.548275i	−0.961690	+	0.274138i	$0.911608\pi$
$762$		0			0
$763$	−27.6181			−0.999843
$764$		0			0
$765$		−	0.755609i		−	0.0273191i
$766$		0			0
$767$	−7.28828	+	29.0899i	−0.263165	+	1.05038i
$768$		0			0
$769$		−	11.8296i		−	0.426586i	−0.976988	−	0.213293i	$-0.931581\pi$
$769$		−	11.8296i		−	0.426586i	0.976988	−	0.213293i	$-0.0684189\pi$
$770$		0			0
$771$	−15.8528			−0.570924
$772$		0			0
$773$		−	15.9549i		−	0.573859i	−0.957952	−	0.286929i	$-0.907366\pi$
$773$		−	15.9549i		−	0.573859i	0.957952	−	0.286929i	$-0.0926345\pi$
$774$		0			0
$775$			26.1615i			0.939749i
$776$		0			0
$777$	4.89705			0.175681
$778$		0			0
$779$	7.66144			0.274500
$780$		0			0
$781$	−7.38430			−0.264231
$782$		0			0
$783$	2.60806			0.0932044
$784$		0			0
$785$			8.67430i			0.309599i
$786$		0			0
$787$		−	39.9335i		−	1.42347i	−0.702446	−	0.711737i	$-0.747909\pi$
$787$		−	39.9335i		−	1.42347i	0.702446	−	0.711737i	$-0.252091\pi$
$788$		0			0
$789$	−9.32064			−0.331823
$790$		0			0
$791$			20.3855i			0.724826i
$792$		0			0
$793$	−42.6008	−	10.6733i	−1.51280	−	0.379021i
$794$		0			0
$795$			3.59790i			0.127604i
$796$		0			0
$797$	−43.4062			−1.53753			−0.768763	−	0.639533i	$-0.779128\pi$
$797$	−43.4062			−1.53753			−0.768763	+	0.639533i	$0.779128\pi$
$798$		0			0
$799$			1.59306i			0.0563584i
$800$		0			0
$801$		−	2.12764i		−	0.0751765i
$802$		0			0
$803$	−6.54090			−0.230823
$804$		0			0
$805$	3.34110			0.117758
$806$		0			0
$807$	−16.0781			−0.565976
$808$		0			0
$809$	−32.8507			−1.15497			−0.577485	−	0.816401i	$-0.695966\pi$
$809$	−32.8507			−1.15497			−0.577485	+	0.816401i	$0.695966\pi$
$810$		0			0
$811$		−	28.5547i		−	1.00269i	−0.865248	−	0.501345i	$-0.832839\pi$
$811$		−	28.5547i		−	1.00269i	0.865248	−	0.501345i	$-0.167161\pi$
$812$		0			0
$813$		−	5.09967i		−	0.178853i
$814$		0			0
$815$	4.34505			0.152200
$816$		0			0
$817$			4.35302i			0.152293i
$818$		0			0
$819$	−1.38190	+	5.51560i	−0.0482873	+	0.192730i
$820$		0			0
$821$		−	42.0586i		−	1.46786i	−0.679227	−	0.733928i	$-0.737685\pi$
$821$		−	42.0586i		−	1.46786i	0.679227	−	0.733928i	$-0.262315\pi$
$822$		0			0
$823$	−20.3677			−0.709973			−0.354986	−	0.934871i	$-0.615514\pi$
$823$	−20.3677			−0.709973			−0.354986	+	0.934871i	$0.615514\pi$
$824$		0			0
$825$			4.79668i			0.166999i
$826$		0			0
$827$			20.1758i			0.701583i	0.936454	+	0.350791i	$0.114087\pi$
$827$			20.1758i			0.701583i	−0.936454	+	0.350791i	$0.885913\pi$
$828$		0			0
$829$	25.5449			0.887211			0.443605	−	0.896222i	$-0.353699\pi$
$829$	25.5449			0.887211			0.443605	+	0.896222i	$0.353699\pi$
$830$		0			0
$831$	8.60022			0.298338
$832$		0			0
$833$	−7.56258			−0.262028
$834$		0			0
$835$	−3.07115			−0.106282
$836$		0			0
$837$			5.45408i			0.188521i
$838$		0			0
$839$		−	3.40525i		−	0.117562i	−0.998271	−	0.0587812i	$-0.981279\pi$
$839$		−	3.40525i		−	0.117562i	0.998271	−	0.0587812i	$-0.0187214\pi$
$840$		0			0
$841$	−22.1980			−0.765449
$842$		0			0
$843$		−	16.4366i		−	0.566105i
$844$		0			0
$845$	−2.76379	+	5.16937i	−0.0950773	+	0.177832i
$846$		0			0
$847$			1.57703i			0.0541875i
$848$		0			0
$849$	8.07109			0.276999
$850$		0			0
$851$			14.5899i			0.500136i
$852$		0			0
$853$		−	34.0974i		−	1.16747i	−0.811943	−	0.583736i	$-0.801590\pi$
$853$		−	34.0974i		−	1.16747i	0.811943	−	0.583736i	$-0.198410\pi$
$854$		0			0
$855$	−0.368436			−0.0126002
$856$		0			0
$857$	−16.0324			−0.547655			−0.273828	−	0.961779i	$-0.588290\pi$
$857$	−16.0324			−0.547655			−0.273828	+	0.961779i	$0.588290\pi$
$858$		0			0
$859$	−18.1090			−0.617872			−0.308936	−	0.951083i	$-0.599973\pi$
$859$	−18.1090			−0.617872			−0.308936	+	0.951083i	$0.599973\pi$
$860$		0			0
$861$	−14.7869			−0.503938
$862$		0			0
$863$			38.3435i			1.30523i	0.757691	+	0.652614i	$0.226327\pi$
$863$			38.3435i			1.30523i	−0.757691	+	0.652614i	$0.773673\pi$
$864$		0			0
$865$		−	3.43411i		−	0.116763i
$866$		0			0
$867$	−14.1919			−0.481981
$868$		0			0
$869$			4.85663i			0.164750i
$870$		0			0
$871$	−11.3262	+	45.2067i	−0.383775	+	1.53177i
$872$		0			0
$873$			8.21369i			0.277991i
$874$		0			0
$875$	6.96641			0.235508
$876$		0			0
$877$		−	13.9353i		−	0.470562i	−0.971927	−	0.235281i	$-0.924399\pi$
$877$		−	13.9353i		−	0.470562i	0.971927	−	0.235281i	$-0.0756010\pi$
$878$		0			0
$879$			21.4865i			0.724721i
$880$		0			0
$881$	−46.6331			−1.57111			−0.785554	−	0.618793i	$-0.787622\pi$
$881$	−46.6331			−1.57111			−0.785554	+	0.618793i	$0.787622\pi$
$882$		0			0
$883$	−2.72293			−0.0916339			−0.0458170	−	0.998950i	$-0.514589\pi$
$883$	−2.72293			−0.0916339			−0.0458170	+	0.998950i	$0.514589\pi$
$884$		0			0
$885$	3.75042			0.126069
$886$		0			0
$887$	47.6307			1.59928			0.799641	−	0.600479i	$-0.205023\pi$
$887$	47.6307			1.59928			0.799641	+	0.600479i	$0.205023\pi$
$888$		0			0
$889$			12.9764i			0.435215i
$890$		0			0
$891$			1.00000i			0.0335013i
$892$		0			0
$893$	0.776778			0.0259939
$894$		0			0
$895$			7.37054i			0.246370i
$896$		0			0
$897$	−16.4328	−	4.11712i	−0.548674	−	0.137467i
$898$		0			0
$899$			14.2246i			0.474416i
$900$		0			0
$901$	−13.3711			−0.445457
$902$		0			0
$903$		−	8.40153i		−	0.279585i
$904$		0			0
$905$		−	2.61455i		−	0.0869104i
$906$		0			0
$907$	27.3442			0.907950			0.453975	−	0.891014i	$-0.350006\pi$
$907$	27.3442			0.907950			0.453975	+	0.891014i	$0.350006\pi$
$908$		0			0
$909$	5.48976			0.182084
$910$		0			0
$911$	−56.7134			−1.87900			−0.939500	−	0.342549i	$-0.888710\pi$
$911$	−56.7134			−1.87900			−0.939500	+	0.342549i	$0.888710\pi$
$912$		0			0
$913$	10.7208			0.354807
$914$		0			0
$915$			5.49231i			0.181570i
$916$		0			0
$917$			19.2605i			0.636038i
$918$		0			0
$919$	−24.6253			−0.812313			−0.406156	−	0.913804i	$-0.633131\pi$
$919$	−24.6253			−0.812313			−0.406156	+	0.913804i	$0.633131\pi$
$920$		0			0
$921$		−	4.68263i		−	0.154298i
$922$		0			0
$923$	6.47059	−	25.8262i	0.212982	−	0.850080i
$924$		0			0
$925$			14.8948i			0.489738i
$926$		0			0
$927$	17.5184			0.575380
$928$		0			0
$929$			2.48728i			0.0816051i	0.999167	+	0.0408025i	$0.0129915\pi$
$929$			2.48728i			0.0816051i	−0.999167	+	0.0408025i	$0.987009\pi$
$930$		0			0
$931$			3.68753i			0.120854i
$932$		0			0
$933$	−8.71420			−0.285290
$934$		0			0
$935$	0.755609			0.0247111
$936$		0			0
$937$	4.57401			0.149426			0.0747132	−	0.997205i	$-0.476196\pi$
$937$	4.57401			0.149426			0.0747132	+	0.997205i	$0.476196\pi$
$938$		0			0
$939$	−24.0084			−0.783485
$940$		0			0
$941$			15.6530i			0.510274i	0.966905	+	0.255137i	$0.0821206\pi$
$941$			15.6530i			0.510274i	−0.966905	+	0.255137i	$0.917879\pi$
$942$		0			0
$943$		−	44.0552i		−	1.43463i
$944$		0			0
$945$	0.711099			0.0231320
$946$		0			0
$947$		−	39.4617i		−	1.28233i	−0.767401	−	0.641167i	$-0.778451\pi$
$947$		−	39.4617i		−	1.28233i	0.767401	−	0.641167i	$-0.221549\pi$
$948$		0			0
$949$	5.73155	−	22.8765i	0.186054	−	0.742602i
$950$		0			0
$951$			15.7982i			0.512291i
$952$		0			0
$953$	0.757641			0.0245424			0.0122712	−	0.999925i	$-0.496094\pi$
$953$	0.757641			0.0245424			0.0122712	+	0.999925i	$0.496094\pi$
$954$		0			0
$955$		−	0.510135i		−	0.0165076i
$956$		0			0
$957$			2.60806i			0.0843066i
$958$		0			0
$959$	22.0755			0.712854
$960$		0			0
$961$	1.25298			0.0404188
$962$		0			0
$963$	−3.41565			−0.110068
$964$		0			0
$965$	1.29999			0.0418483
$966$		0			0
$967$			12.8129i			0.412034i	0.978548	+	0.206017i	$0.0660503\pi$
$967$			12.8129i			0.412034i	−0.978548	+	0.206017i	$0.933950\pi$
$968$		0			0
$969$		−	1.36924i		−	0.0439865i
$970$		0			0
$971$	−0.265182			−0.00851010			−0.00425505	−	0.999991i	$-0.501354\pi$
$971$	−0.265182			−0.00851010			−0.00425505	+	0.999991i	$0.501354\pi$
$972$		0			0
$973$			18.2293i			0.584403i
$974$		0			0
$975$	−16.7762	−	4.20315i	−0.537267	−	0.134609i
$976$		0			0
$977$		−	36.1762i		−	1.15738i	−0.815548	−	0.578690i	$-0.803564\pi$
$977$		−	36.1762i		−	1.15738i	0.815548	−	0.578690i	$-0.196436\pi$
$978$		0			0
$979$	2.12764			0.0679997
$980$		0			0
$981$		−	17.5127i		−	0.559138i
$982$		0			0
$983$			28.7731i			0.917720i	0.888509	+	0.458860i	$0.151742\pi$
$983$			28.7731i			0.917720i	−0.888509	+	0.458860i	$0.848258\pi$
$984$		0			0
$985$	−6.16112			−0.196310
$986$		0			0
$987$	−1.49922			−0.0477206
$988$		0			0
$989$	25.0309			0.795937
$990$		0			0
$991$	−16.9876			−0.539628			−0.269814	−	0.962912i	$-0.586962\pi$
$991$	−16.9876			−0.539628			−0.269814	+	0.962912i	$0.586962\pi$
$992$		0			0
$993$		−	33.3420i		−	1.05808i
$994$		0			0
$995$		−	0.561786i		−	0.0178098i
$996$		0			0
$997$	−53.3104			−1.68836			−0.844179	−	0.536061i	$-0.819912\pi$
$997$	−53.3104			−1.68836			−0.844179	+	0.536061i	$0.819912\pi$
$998$		0			0
$999$			3.10523i			0.0982450i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.g.d.1585.9	yes	14
13.12	even	2	inner	3432.2.g.d.1585.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.g.d.1585.6	✓	14	13.12	even	2	inner
3432.2.g.d.1585.9	yes	14	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.450909i q^{5} -1.57703i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.450909i q^{5} -1.57703i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(-3.49745 - 0.876263i) q^{13} +0.450909i q^{15} -1.67575 q^{17} +0.817096i q^{19} -1.57703i q^{21} +4.69850 q^{23} +4.79668 q^{25} +1.00000 q^{27} +2.60806 q^{29} +5.45408i q^{31} +1.00000i q^{33} +0.711099 q^{35} +3.10523i q^{37} +(-3.49745 - 0.876263i) q^{39} -9.37643i q^{41} +5.32743 q^{43} +0.450909i q^{45} -0.950657i q^{47} +4.51297 q^{49} -1.67575 q^{51} +7.97922 q^{53} -0.450909 q^{55} +0.817096i q^{57} -8.31746i q^{59} +12.1805 q^{61} -1.57703i q^{63} +(0.395115 - 1.57703i) q^{65} -12.9256i q^{67} +4.69850 q^{69} +7.38430i q^{71} +6.54090i q^{73} +4.79668 q^{75} +1.57703 q^{77} +4.85663 q^{79} +1.00000 q^{81} -10.7208i q^{83} -0.755609i q^{85} +2.60806 q^{87} -2.12764i q^{89} +(-1.38190 + 5.51560i) q^{91} +5.45408i q^{93} -0.368436 q^{95} +8.21369i q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) \) 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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