LMFDB - Embedded newform 1920.2.d.b.1729.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(1729,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1920.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.d (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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
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			1729.4
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	1920.1729
Dual form			1920.2.d.b.1729.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(1.73205 + 1.41421i) q^{5} +1.00000 q^{9} +3.46410 q^{13} +(1.73205 + 1.41421i) q^{15} +4.89898i q^{17} +4.89898i q^{19} -2.82843i q^{23} +(1.00000 + 4.89898i) q^{25} +1.00000 q^{27} -2.82843i q^{29} -6.92820 q^{31} -3.46410 q^{37} +3.46410 q^{39} +6.00000 q^{41} +4.00000 q^{43} +(1.73205 + 1.41421i) q^{45} +2.82843i q^{47} +7.00000 q^{49} +4.89898i q^{51} -3.46410 q^{53} +4.89898i q^{57} -9.79796i q^{59} +(6.00000 + 4.89898i) q^{65} +4.00000 q^{67} -2.82843i q^{69} +13.8564 q^{71} +9.79796i q^{73} +(1.00000 + 4.89898i) q^{75} -6.92820 q^{79} +1.00000 q^{81} -12.0000 q^{83} +(-6.92820 + 8.48528i) q^{85} -2.82843i q^{87} +6.00000 q^{89} -6.92820 q^{93} +(-6.92820 + 8.48528i) q^{95} +9.79796i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9} + 4 q^{25} + 4 q^{27} + 24 q^{41} + 16 q^{43} + 28 q^{49} + 24 q^{65} + 16 q^{67} + 4 q^{75} + 4 q^{81} - 48 q^{83} + 24 q^{89}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 4 * q^25 + 4 * q^27 + 24 * q^41 + 16 * q^43 + 28 * q^49 + 24 * q^65 + 16 * q^67 + 4 * q^75 + 4 * q^81 - 48 * q^83 + 24 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	1.73205	+	1.41421i	0.774597	+	0.632456i
$5$	1.73205	+	1.41421i	0.774597	+	0.632456i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	3.46410			0.960769			0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410			0.960769			0.480384	+	0.877058i	$0.340497\pi$
$14$		0			0
$15$	1.73205	+	1.41421i	0.447214	+	0.365148i
$16$		0			0
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$			4.89898i			1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$			4.89898i			1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$	1.00000	+	4.89898i	0.200000	+	0.979796i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	2.82843i		−	0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$		−	2.82843i		−	0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$		0			0
$31$	−6.92820			−1.24434			−0.622171	−	0.782881i	$-0.713749\pi$
$31$	−6.92820			−1.24434			−0.622171	+	0.782881i	$0.713749\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.46410			−0.569495			−0.284747	−	0.958603i	$-0.591910\pi$
$37$	−3.46410			−0.569495			−0.284747	+	0.958603i	$0.591910\pi$
$38$		0			0
$39$	3.46410			0.554700
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	1.73205	+	1.41421i	0.258199	+	0.210819i
$46$		0			0
$47$			2.82843i			0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$			2.82843i			0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$			4.89898i			0.685994i
$52$		0			0
$53$	−3.46410			−0.475831			−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−3.46410			−0.475831			−0.237915	+	0.971286i	$0.576464\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			4.89898i			0.648886i
$58$		0			0
$59$		−	9.79796i		−	1.27559i	−0.770208	−	0.637793i	$-0.779848\pi$
$59$		−	9.79796i		−	1.27559i	0.770208	−	0.637793i	$-0.220152\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	+	4.89898i	0.744208	+	0.607644i
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$		−	2.82843i		−	0.340503i
$70$		0			0
$71$	13.8564			1.64445			0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564			1.64445			0.822226	+	0.569160i	$0.192732\pi$
$72$		0			0
$73$			9.79796i			1.14676i	0.819288	+	0.573382i	$0.194369\pi$
$73$			9.79796i			1.14676i	−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$	1.00000	+	4.89898i	0.115470	+	0.565685i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.92820			−0.779484			−0.389742	−	0.920924i	$-0.627436\pi$
$79$	−6.92820			−0.779484			−0.389742	+	0.920924i	$0.627436\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	−6.92820	+	8.48528i	−0.751469	+	0.920358i
$86$		0			0
$87$		−	2.82843i		−	0.303239i
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6.92820			−0.718421
$94$		0			0
$95$	−6.92820	+	8.48528i	−0.710819	+	0.870572i
$96$		0			0
$97$			9.79796i			0.994832i	0.867512	+	0.497416i	$0.165718\pi$
$97$			9.79796i			0.994832i	−0.867512	+	0.497416i	$0.834282\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			2.82843i			0.281439i	0.990050	+	0.140720i	$0.0449416\pi$
$101$			2.82843i			0.281439i	−0.990050	+	0.140720i	$0.955058\pi$
$102$		0			0
$103$		−	16.9706i		−	1.67216i	−0.548608	−	0.836080i	$-0.684842\pi$
$103$		−	16.9706i		−	1.67216i	0.548608	−	0.836080i	$-0.315158\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$		−	16.9706i		−	1.62549i	−0.582623	−	0.812743i	$-0.697974\pi$
$109$		−	16.9706i		−	1.62549i	0.582623	−	0.812743i	$-0.302026\pi$
$110$		0			0
$111$	−3.46410			−0.328798
$112$		0			0
$113$			14.6969i			1.38257i	0.722581	+	0.691286i	$0.242955\pi$
$113$			14.6969i			1.38257i	−0.722581	+	0.691286i	$0.757045\pi$
$114$		0			0
$115$	4.00000	−	4.89898i	0.373002	−	0.456832i
$116$		0			0
$117$	3.46410			0.320256
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	6.00000			0.541002
$124$		0			0
$125$	−5.19615	+	9.89949i	−0.464758	+	0.885438i
$126$		0			0
$127$			16.9706i			1.50589i	0.658081	+	0.752947i	$0.271368\pi$
$127$			16.9706i			1.50589i	−0.658081	+	0.752947i	$0.728632\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$		−	19.5959i		−	1.71210i	−0.516890	−	0.856052i	$-0.672910\pi$
$131$		−	19.5959i		−	1.71210i	0.516890	−	0.856052i	$-0.327090\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.73205	+	1.41421i	0.149071	+	0.121716i
$136$		0			0
$137$			4.89898i			0.418548i	0.977857	+	0.209274i	$0.0671101\pi$
$137$			4.89898i			0.418548i	−0.977857	+	0.209274i	$0.932890\pi$
$138$		0			0
$139$			14.6969i			1.24658i	0.781992	+	0.623289i	$0.214204\pi$
$139$			14.6969i			1.24658i	−0.781992	+	0.623289i	$0.785796\pi$
$140$		0			0
$141$			2.82843i			0.238197i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000	−	4.89898i	0.332182	−	0.406838i
$146$		0			0
$147$	7.00000			0.577350
$148$		0			0
$149$		−	19.7990i		−	1.62200i	−0.585049	−	0.810998i	$-0.698925\pi$
$149$		−	19.7990i		−	1.62200i	0.585049	−	0.810998i	$-0.301075\pi$
$150$		0			0
$151$	6.92820			0.563809			0.281905	−	0.959442i	$-0.409034\pi$
$151$	6.92820			0.563809			0.281905	+	0.959442i	$0.409034\pi$
$152$		0			0
$153$			4.89898i			0.396059i
$154$		0			0
$155$	−12.0000	−	9.79796i	−0.963863	−	0.786991i
$156$		0			0
$157$	10.3923			0.829396			0.414698	−	0.909959i	$-0.363887\pi$
$157$	10.3923			0.829396			0.414698	+	0.909959i	$0.363887\pi$
$158$		0			0
$159$	−3.46410			−0.274721
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.1421i		−	1.09435i	−0.837018	−	0.547176i	$-0.815703\pi$
$167$		−	14.1421i		−	1.09435i	0.837018	−	0.547176i	$-0.184297\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$			4.89898i			0.374634i
$172$		0			0
$173$	3.46410			0.263371			0.131685	−	0.991292i	$-0.457961\pi$
$173$	3.46410			0.263371			0.131685	+	0.991292i	$0.457961\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	9.79796i		−	0.736460i
$178$		0			0
$179$		−	19.5959i		−	1.46467i	−0.680946	−	0.732334i	$-0.738431\pi$
$179$		−	19.5959i		−	1.46467i	0.680946	−	0.732334i	$-0.261569\pi$
$180$		0			0
$181$			16.9706i			1.26141i	0.776022	+	0.630706i	$0.217235\pi$
$181$			16.9706i			1.26141i	−0.776022	+	0.630706i	$0.782765\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	−	4.89898i	−0.441129	−	0.360180i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.8564			−1.00261			−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564			−1.00261			−0.501307	+	0.865269i	$0.667147\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	6.00000	+	4.89898i	0.429669	+	0.350823i
$196$		0			0
$197$	10.3923			0.740421			0.370211	−	0.928948i	$-0.379286\pi$
$197$	10.3923			0.740421			0.370211	+	0.928948i	$0.379286\pi$
$198$		0			0
$199$	20.7846			1.47338			0.736691	−	0.676230i	$-0.236387\pi$
$199$	20.7846			1.47338			0.736691	+	0.676230i	$0.236387\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$		0			0
$204$		0			0
$205$	10.3923	+	8.48528i	0.725830	+	0.592638i
$206$		0			0
$207$		−	2.82843i		−	0.196589i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			14.6969i			1.01178i	0.862598	+	0.505889i	$0.168836\pi$
$211$			14.6969i			1.01178i	−0.862598	+	0.505889i	$0.831164\pi$
$212$		0			0
$213$	13.8564			0.949425
$214$		0			0
$215$	6.92820	+	5.65685i	0.472500	+	0.385794i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			9.79796i			0.662085i
$220$		0			0
$221$			16.9706i			1.14156i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	1.00000	+	4.89898i	0.0666667	+	0.326599i
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$		−	16.9706i		−	1.12145i	−0.828003	−	0.560723i	$-0.810523\pi$
$229$		−	16.9706i		−	1.12145i	0.828003	−	0.560723i	$-0.189477\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.89898i		−	0.320943i	−0.987040	−	0.160471i	$-0.948699\pi$
$233$		−	4.89898i		−	0.320943i	0.987040	−	0.160471i	$-0.0513014\pi$
$234$		0			0
$235$	−4.00000	+	4.89898i	−0.260931	+	0.319574i
$236$		0			0
$237$	−6.92820			−0.450035
$238$		0			0
$239$	−13.8564			−0.896296			−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564			−0.896296			−0.448148	+	0.893959i	$0.647916\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	12.1244	+	9.89949i	0.774597	+	0.632456i
$246$		0			0
$247$			16.9706i			1.07981i
$248$		0			0
$249$	−12.0000			−0.760469
$250$		0			0
$251$		−	9.79796i		−	0.618442i	−0.950990	−	0.309221i	$-0.899932\pi$
$251$		−	9.79796i		−	0.618442i	0.950990	−	0.309221i	$-0.100068\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−6.92820	+	8.48528i	−0.433861	+	0.531369i
$256$		0			0
$257$		−	14.6969i		−	0.916770i	−0.888754	−	0.458385i	$-0.848428\pi$
$257$		−	14.6969i		−	0.916770i	0.888754	−	0.458385i	$-0.151572\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		−	2.82843i		−	0.175075i
$262$		0			0
$263$		−	14.1421i		−	0.872041i	−0.899937	−	0.436021i	$-0.856387\pi$
$263$		−	14.1421i		−	0.872041i	0.899937	−	0.436021i	$-0.143613\pi$
$264$		0			0
$265$	−6.00000	−	4.89898i	−0.368577	−	0.300942i
$266$		0			0
$267$	6.00000			0.367194
$268$		0			0
$269$		−	19.7990i		−	1.20717i	−0.797300	−	0.603583i	$-0.793739\pi$
$269$		−	19.7990i		−	1.20717i	0.797300	−	0.603583i	$-0.206261\pi$
$270$		0			0
$271$	6.92820			0.420858			0.210429	−	0.977609i	$-0.432514\pi$
$271$	6.92820			0.420858			0.210429	+	0.977609i	$0.432514\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−17.3205			−1.04069			−0.520344	−	0.853957i	$-0.674196\pi$
$277$	−17.3205			−1.04069			−0.520344	+	0.853957i	$0.674196\pi$
$278$		0			0
$279$	−6.92820			−0.414781
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	28.0000			1.66443			0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000			1.66443			0.832214	+	0.554455i	$0.187073\pi$
$284$		0			0
$285$	−6.92820	+	8.48528i	−0.410391	+	0.502625i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$			9.79796i			0.574367i
$292$		0			0
$293$	3.46410			0.202375			0.101187	−	0.994867i	$-0.467736\pi$
$293$	3.46410			0.202375			0.101187	+	0.994867i	$0.467736\pi$
$294$		0			0
$295$	13.8564	−	16.9706i	0.806751	−	0.988064i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	9.79796i		−	0.566631i
$300$		0			0
$301$		0			0
$302$		0			0
$303$			2.82843i			0.162489i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		−	16.9706i		−	0.965422i
$310$		0			0
$311$	−27.7128			−1.57145			−0.785725	−	0.618576i	$-0.787710\pi$
$311$	−27.7128			−1.57145			−0.785725	+	0.618576i	$0.787710\pi$
$312$		0			0
$313$		−	19.5959i		−	1.10763i	−0.832641	−	0.553813i	$-0.813172\pi$
$313$		−	19.5959i		−	1.10763i	0.832641	−	0.553813i	$-0.186828\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.1769			−1.75107			−0.875535	−	0.483155i	$-0.839491\pi$
$317$	−31.1769			−1.75107			−0.875535	+	0.483155i	$0.839491\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$	−24.0000			−1.33540
$324$		0			0
$325$	3.46410	+	16.9706i	0.192154	+	0.941357i
$326$		0			0
$327$		−	16.9706i		−	0.938474i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			4.89898i			0.269272i	0.990895	+	0.134636i	$0.0429866\pi$
$331$			4.89898i			0.269272i	−0.990895	+	0.134636i	$0.957013\pi$
$332$		0			0
$333$	−3.46410			−0.189832
$334$		0			0
$335$	6.92820	+	5.65685i	0.378528	+	0.309067i
$336$		0			0
$337$		−	9.79796i		−	0.533729i	−0.963734	−	0.266864i	$-0.914012\pi$
$337$		−	9.79796i		−	0.533729i	0.963734	−	0.266864i	$-0.0859876\pi$
$338$		0			0
$339$			14.6969i			0.798228i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	4.00000	−	4.89898i	0.215353	−	0.263752i
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$		−	33.9411i		−	1.81683i	−0.418073	−	0.908413i	$-0.637294\pi$
$349$		−	33.9411i		−	1.81683i	0.418073	−	0.908413i	$-0.362706\pi$
$350$		0			0
$351$	3.46410			0.184900
$352$		0			0
$353$		−	34.2929i		−	1.82522i	−0.408826	−	0.912612i	$-0.634062\pi$
$353$		−	34.2929i		−	1.82522i	0.408826	−	0.912612i	$-0.365938\pi$
$354$		0			0
$355$	24.0000	+	19.5959i	1.27379	+	1.04004i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	27.7128			1.46263			0.731313	−	0.682042i	$-0.238908\pi$
$359$	27.7128			1.46263			0.731313	+	0.682042i	$0.238908\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$	11.0000			0.577350
$364$		0			0
$365$	−13.8564	+	16.9706i	−0.725277	+	0.888280i
$366$		0			0
$367$			16.9706i			0.885856i	0.896557	+	0.442928i	$0.146060\pi$
$367$			16.9706i			0.885856i	−0.896557	+	0.442928i	$0.853940\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−31.1769			−1.61428			−0.807140	−	0.590360i	$-0.798986\pi$
$373$	−31.1769			−1.61428			−0.807140	+	0.590360i	$0.798986\pi$
$374$		0			0
$375$	−5.19615	+	9.89949i	−0.268328	+	0.511208i
$376$		0			0
$377$		−	9.79796i		−	0.504621i
$378$		0			0
$379$		−	4.89898i		−	0.251644i	−0.992053	−	0.125822i	$-0.959843\pi$
$379$		−	4.89898i		−	0.251644i	0.992053	−	0.125822i	$-0.0401568\pi$
$380$		0			0
$381$			16.9706i			0.869428i
$382$		0			0
$383$			14.1421i			0.722629i	0.932444	+	0.361315i	$0.117672\pi$
$383$			14.1421i			0.722629i	−0.932444	+	0.361315i	$0.882328\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.00000			0.203331
$388$		0			0
$389$		−	31.1127i		−	1.57748i	−0.614729	−	0.788738i	$-0.710735\pi$
$389$		−	31.1127i		−	1.57748i	0.614729	−	0.788738i	$-0.289265\pi$
$390$		0			0
$391$	13.8564			0.700749
$392$		0			0
$393$		−	19.5959i		−	0.988483i
$394$		0			0
$395$	−12.0000	−	9.79796i	−0.603786	−	0.492989i
$396$		0			0
$397$	17.3205			0.869291			0.434646	−	0.900602i	$-0.356874\pi$
$397$	17.3205			0.869291			0.434646	+	0.900602i	$0.356874\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$	−24.0000			−1.19553
$404$		0			0
$405$	1.73205	+	1.41421i	0.0860663	+	0.0702728i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$			4.89898i			0.241649i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−20.7846	−	16.9706i	−1.02028	−	0.833052i
$416$		0			0
$417$			14.6969i			0.719712i
$418$		0			0
$419$			9.79796i			0.478662i	0.970938	+	0.239331i	$0.0769280\pi$
$419$			9.79796i			0.478662i	−0.970938	+	0.239331i	$0.923072\pi$
$420$		0			0
$421$			16.9706i			0.827095i	0.910483	+	0.413547i	$0.135710\pi$
$421$			16.9706i			0.827095i	−0.910483	+	0.413547i	$0.864290\pi$
$422$		0			0
$423$			2.82843i			0.137523i
$424$		0			0
$425$	−24.0000	+	4.89898i	−1.16417	+	0.237635i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.8564			0.667440			0.333720	−	0.942672i	$-0.391696\pi$
$431$	13.8564			0.667440			0.333720	+	0.942672i	$0.391696\pi$
$432$		0			0
$433$		−	19.5959i		−	0.941720i	−0.882208	−	0.470860i	$-0.843944\pi$
$433$		−	19.5959i		−	0.941720i	0.882208	−	0.470860i	$-0.156056\pi$
$434$		0			0
$435$	4.00000	−	4.89898i	0.191785	−	0.234888i
$436$		0			0
$437$	13.8564			0.662842
$438$		0			0
$439$	−34.6410			−1.65333			−0.826663	−	0.562698i	$-0.809764\pi$
$439$	−34.6410			−1.65333			−0.826663	+	0.562698i	$0.809764\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$	12.0000			0.570137			0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000			0.570137			0.285069	+	0.958507i	$0.407984\pi$
$444$		0			0
$445$	10.3923	+	8.48528i	0.492642	+	0.402241i
$446$		0			0
$447$		−	19.7990i		−	0.936460i
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	6.92820			0.325515
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	29.3939i		−	1.37499i	−0.726190	−	0.687494i	$-0.758711\pi$
$457$		−	29.3939i		−	1.37499i	0.726190	−	0.687494i	$-0.241289\pi$
$458$		0			0
$459$			4.89898i			0.228665i
$460$		0			0
$461$			14.1421i			0.658665i	0.944214	+	0.329332i	$0.106824\pi$
$461$			14.1421i			0.658665i	−0.944214	+	0.329332i	$0.893176\pi$
$462$		0			0
$463$		−	16.9706i		−	0.788689i	−0.918963	−	0.394344i	$-0.870972\pi$
$463$		−	16.9706i		−	0.788689i	0.918963	−	0.394344i	$-0.129028\pi$
$464$		0			0
$465$	−12.0000	−	9.79796i	−0.556487	−	0.454369i
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	10.3923			0.478852
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−24.0000	+	4.89898i	−1.10120	+	0.224781i
$476$		0			0
$477$	−3.46410			−0.158610
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−13.8564	+	16.9706i	−0.629187	+	0.770594i
$486$		0			0
$487$			16.9706i			0.769010i	0.923123	+	0.384505i	$0.125628\pi$
$487$			16.9706i			0.769010i	−0.923123	+	0.384505i	$0.874372\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$		−	29.3939i		−	1.32653i	−0.748386	−	0.663264i	$-0.769171\pi$
$491$		−	29.3939i		−	1.32653i	0.748386	−	0.663264i	$-0.230829\pi$
$492$		0			0
$493$	13.8564			0.624061
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			4.89898i			0.219308i	0.993970	+	0.109654i	$0.0349744\pi$
$499$			4.89898i			0.219308i	−0.993970	+	0.109654i	$0.965026\pi$
$500$		0			0
$501$		−	14.1421i		−	0.631824i
$502$		0			0
$503$			36.7696i			1.63947i	0.572741	+	0.819737i	$0.305880\pi$
$503$			36.7696i			1.63947i	−0.572741	+	0.819737i	$0.694120\pi$
$504$		0			0
$505$	−4.00000	+	4.89898i	−0.177998	+	0.218002i
$506$		0			0
$507$	−1.00000			−0.0444116
$508$		0			0
$509$		−	14.1421i		−	0.626839i	−0.949615	−	0.313420i	$-0.898525\pi$
$509$		−	14.1421i		−	0.626839i	0.949615	−	0.313420i	$-0.101475\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			4.89898i			0.216295i
$514$		0			0
$515$	24.0000	−	29.3939i	1.05757	−	1.29525i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	3.46410			0.152057
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	33.9411i		−	1.47850i
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		−	9.79796i		−	0.425195i
$532$		0			0
$533$	20.7846			0.900281
$534$		0			0
$535$	−20.7846	−	16.9706i	−0.898597	−	0.733701i
$536$		0			0
$537$		−	19.5959i		−	0.845626i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	16.9706i		−	0.729621i	−0.931082	−	0.364811i	$-0.881134\pi$
$541$		−	16.9706i		−	0.729621i	0.931082	−	0.364811i	$-0.118866\pi$
$542$		0			0
$543$			16.9706i			0.728277i
$544$		0			0
$545$	24.0000	−	29.3939i	1.02805	−	1.25910i
$546$		0			0
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.8564			0.590303
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−6.00000	−	4.89898i	−0.254686	−	0.207950i
$556$		0			0
$557$	31.1769			1.32101			0.660504	−	0.750822i	$-0.270343\pi$
$557$	31.1769			1.32101			0.660504	+	0.750822i	$0.270343\pi$
$558$		0			0
$559$	13.8564			0.586064
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$	−20.7846	+	25.4558i	−0.874415	+	1.07094i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$			24.4949i			1.02508i	0.858663	+	0.512540i	$0.171295\pi$
$571$			24.4949i			1.02508i	−0.858663	+	0.512540i	$0.828705\pi$
$572$		0			0
$573$	−13.8564			−0.578860
$574$		0			0
$575$	13.8564	−	2.82843i	0.577852	−	0.117954i
$576$		0			0
$577$		−	19.5959i		−	0.815789i	−0.913029	−	0.407894i	$-0.866263\pi$
$577$		−	19.5959i		−	0.815789i	0.913029	−	0.407894i	$-0.133737\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	6.00000	+	4.89898i	0.248069	+	0.202548i
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$		−	33.9411i		−	1.39852i
$590$		0			0
$591$	10.3923			0.427482
$592$		0			0
$593$		−	4.89898i		−	0.201177i	−0.994928	−	0.100588i	$-0.967927\pi$
$593$		−	4.89898i		−	0.201177i	0.994928	−	0.100588i	$-0.0320726\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	20.7846			0.850657
$598$		0			0
$599$	−27.7128			−1.13231			−0.566157	−	0.824297i	$-0.691571\pi$
$599$	−27.7128			−1.13231			−0.566157	+	0.824297i	$0.691571\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$	4.00000			0.162893
$604$		0			0
$605$	19.0526	+	15.5563i	0.774597	+	0.632456i
$606$		0			0
$607$			33.9411i			1.37763i	0.724938	+	0.688814i	$0.241868\pi$
$607$			33.9411i			1.37763i	−0.724938	+	0.688814i	$0.758132\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			9.79796i			0.396383i
$612$		0			0
$613$	3.46410			0.139914			0.0699569	−	0.997550i	$-0.477714\pi$
$613$	3.46410			0.139914			0.0699569	+	0.997550i	$0.477714\pi$
$614$		0			0
$615$	10.3923	+	8.48528i	0.419058	+	0.342160i
$616$		0			0
$617$			4.89898i			0.197225i	0.995126	+	0.0986127i	$0.0314405\pi$
$617$			4.89898i			0.197225i	−0.995126	+	0.0986127i	$0.968559\pi$
$618$		0			0
$619$		−	44.0908i		−	1.77216i	−0.463533	−	0.886080i	$-0.653418\pi$
$619$		−	44.0908i		−	1.77216i	0.463533	−	0.886080i	$-0.346582\pi$
$620$		0			0
$621$		−	2.82843i		−	0.113501i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	16.9706i		−	0.676661i
$630$		0			0
$631$	−48.4974			−1.93065			−0.965326	−	0.261048i	$-0.915932\pi$
$631$	−48.4974			−1.93065			−0.965326	+	0.261048i	$0.915932\pi$
$632$		0			0
$633$			14.6969i			0.584151i
$634$		0			0
$635$	−24.0000	+	29.3939i	−0.952411	+	1.16646i
$636$		0			0
$637$	24.2487			0.960769
$638$		0			0
$639$	13.8564			0.548151
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$		0			0
$645$	6.92820	+	5.65685i	0.272798	+	0.222738i
$646$		0			0
$647$		−	31.1127i		−	1.22317i	−0.791180	−	0.611583i	$-0.790533\pi$
$647$		−	31.1127i		−	1.22317i	0.791180	−	0.611583i	$-0.209467\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.1769			−1.22005			−0.610023	−	0.792383i	$-0.708840\pi$
$653$	−31.1769			−1.22005			−0.610023	+	0.792383i	$0.708840\pi$
$654$		0			0
$655$	27.7128	−	33.9411i	1.08283	−	1.32619i
$656$		0			0
$657$			9.79796i			0.382255i
$658$		0			0
$659$		−	9.79796i		−	0.381674i	−0.981622	−	0.190837i	$-0.938880\pi$
$659$		−	9.79796i		−	0.381674i	0.981622	−	0.190837i	$-0.0611202\pi$
$660$		0			0
$661$			33.9411i			1.32016i	0.751197	+	0.660078i	$0.229477\pi$
$661$			33.9411i			1.32016i	−0.751197	+	0.660078i	$0.770523\pi$
$662$		0			0
$663$			16.9706i			0.659082i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000			−0.309761
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			39.1918i			1.51073i	0.655302	+	0.755367i	$0.272541\pi$
$673$			39.1918i			1.51073i	−0.655302	+	0.755367i	$0.727459\pi$
$674$		0			0
$675$	1.00000	+	4.89898i	0.0384900	+	0.188562i
$676$		0			0
$677$	−17.3205			−0.665681			−0.332841	−	0.942983i	$-0.608007\pi$
$677$	−17.3205			−0.665681			−0.332841	+	0.942983i	$0.608007\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$	36.0000			1.37750			0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$684$		0			0
$685$	−6.92820	+	8.48528i	−0.264713	+	0.324206i
$686$		0			0
$687$		−	16.9706i		−	0.647467i
$688$		0			0
$689$	−12.0000			−0.457164
$690$		0			0
$691$		−	34.2929i		−	1.30456i	−0.757977	−	0.652281i	$-0.773812\pi$
$691$		−	34.2929i		−	1.30456i	0.757977	−	0.652281i	$-0.226188\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.7846	+	25.4558i	−0.788405	+	0.965595i
$696$		0			0
$697$			29.3939i			1.11337i
$698$		0			0
$699$		−	4.89898i		−	0.185296i
$700$		0			0
$701$			2.82843i			0.106828i	0.998572	+	0.0534141i	$0.0170103\pi$
$701$			2.82843i			0.106828i	−0.998572	+	0.0534141i	$0.982990\pi$
$702$		0			0
$703$		−	16.9706i		−	0.640057i
$704$		0			0
$705$	−4.00000	+	4.89898i	−0.150649	+	0.184506i
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	16.9706i		−	0.637343i	−0.947865	−	0.318671i	$-0.896763\pi$
$709$		−	16.9706i		−	0.637343i	0.947865	−	0.318671i	$-0.103237\pi$
$710$		0			0
$711$	−6.92820			−0.259828
$712$		0			0
$713$			19.5959i			0.733873i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−13.8564			−0.517477
$718$		0			0
$719$	41.5692			1.55027			0.775135	−	0.631795i	$-0.217682\pi$
$719$	41.5692			1.55027			0.775135	+	0.631795i	$0.217682\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−14.0000			−0.520666
$724$		0			0
$725$	13.8564	−	2.82843i	0.514614	−	0.105045i
$726$		0			0
$727$			50.9117i			1.88821i	0.329645	+	0.944105i	$0.393071\pi$
$727$			50.9117i			1.88821i	−0.329645	+	0.944105i	$0.606929\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			19.5959i			0.724781i
$732$		0			0
$733$	51.9615			1.91924			0.959621	−	0.281295i	$-0.0907639\pi$
$733$	51.9615			1.91924			0.959621	+	0.281295i	$0.0907639\pi$
$734$		0			0
$735$	12.1244	+	9.89949i	0.447214	+	0.365148i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	14.6969i		−	0.540636i	−0.962771	−	0.270318i	$-0.912871\pi$
$739$		−	14.6969i		−	0.540636i	0.962771	−	0.270318i	$-0.0871288\pi$
$740$		0			0
$741$			16.9706i			0.623429i
$742$		0			0
$743$			14.1421i			0.518825i	0.965767	+	0.259412i	$0.0835289\pi$
$743$			14.1421i			0.518825i	−0.965767	+	0.259412i	$0.916471\pi$
$744$		0			0
$745$	28.0000	−	34.2929i	1.02584	−	1.25639i
$746$		0			0
$747$	−12.0000			−0.439057
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.92820			−0.252814			−0.126407	−	0.991978i	$-0.540344\pi$
$751$	−6.92820			−0.252814			−0.126407	+	0.991978i	$0.540344\pi$
$752$		0			0
$753$		−	9.79796i		−	0.357057i
$754$		0			0
$755$	12.0000	+	9.79796i	0.436725	+	0.356584i
$756$		0			0
$757$	−45.0333			−1.63676			−0.818382	−	0.574675i	$-0.805129\pi$
$757$	−45.0333			−1.63676			−0.818382	+	0.574675i	$0.805129\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−6.92820	+	8.48528i	−0.250490	+	0.306786i
$766$		0			0
$767$		−	33.9411i		−	1.22554i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		−	14.6969i		−	0.529297i
$772$		0			0
$773$	−17.3205			−0.622975			−0.311488	−	0.950250i	$-0.600827\pi$
$773$	−17.3205			−0.622975			−0.311488	+	0.950250i	$0.600827\pi$
$774$		0			0
$775$	−6.92820	−	33.9411i	−0.248868	−	1.21920i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			29.3939i			1.05314i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	2.82843i		−	0.101080i
$784$		0			0
$785$	18.0000	+	14.6969i	0.642448	+	0.524556i
$786$		0			0
$787$	44.0000			1.56843			0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000			1.56843			0.784215	+	0.620489i	$0.213066\pi$
$788$		0			0
$789$		−	14.1421i		−	0.503473i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−6.00000	−	4.89898i	−0.212798	−	0.173749i
$796$		0			0
$797$	−17.3205			−0.613524			−0.306762	−	0.951786i	$-0.599246\pi$
$797$	−17.3205			−0.613524			−0.306762	+	0.951786i	$0.599246\pi$
$798$		0			0
$799$	−13.8564			−0.490204
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	19.7990i		−	0.696957i
$808$		0			0
$809$	42.0000			1.47664			0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000			1.47664			0.738321	+	0.674450i	$0.235619\pi$
$810$		0			0
$811$		−	4.89898i		−	0.172026i	−0.996294	−	0.0860132i	$-0.972587\pi$
$811$		−	4.89898i		−	0.172026i	0.996294	−	0.0860132i	$-0.0274127\pi$
$812$		0			0
$813$	6.92820			0.242983
$814$		0			0
$815$	−6.92820	−	5.65685i	−0.242684	−	0.198151i
$816$		0			0
$817$			19.5959i			0.685574i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2.82843i		−	0.0987128i	−0.998781	−	0.0493564i	$-0.984283\pi$
$821$		−	2.82843i		−	0.0987128i	0.998781	−	0.0493564i	$-0.0157170\pi$
$822$		0			0
$823$			33.9411i			1.18311i	0.806263	+	0.591557i	$0.201486\pi$
$823$			33.9411i			1.18311i	−0.806263	+	0.591557i	$0.798514\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$		−	16.9706i		−	0.589412i	−0.955588	−	0.294706i	$-0.904778\pi$
$829$		−	16.9706i		−	0.589412i	0.955588	−	0.294706i	$-0.0952217\pi$
$830$		0			0
$831$	−17.3205			−0.600842
$832$		0			0
$833$			34.2929i			1.18818i
$834$		0			0
$835$	20.0000	−	24.4949i	0.692129	−	0.847681i
$836$		0			0
$837$	−6.92820			−0.239474
$838$		0			0
$839$	−27.7128			−0.956753			−0.478376	−	0.878155i	$-0.658774\pi$
$839$	−27.7128			−0.956753			−0.478376	+	0.878155i	$0.658774\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	−6.00000			−0.206651
$844$		0			0
$845$	−1.73205	−	1.41421i	−0.0595844	−	0.0486504i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$			9.79796i			0.335870i
$852$		0			0
$853$	−24.2487			−0.830260			−0.415130	−	0.909762i	$-0.636264\pi$
$853$	−24.2487			−0.830260			−0.415130	+	0.909762i	$0.636264\pi$
$854$		0			0
$855$	−6.92820	+	8.48528i	−0.236940	+	0.290191i
$856$		0			0
$857$		−	4.89898i		−	0.167346i	−0.996493	−	0.0836730i	$-0.973335\pi$
$857$		−	4.89898i		−	0.167346i	0.996493	−	0.0836730i	$-0.0266651\pi$
$858$		0			0
$859$			53.8888i			1.83866i	0.393486	+	0.919331i	$0.371269\pi$
$859$			53.8888i			1.83866i	−0.393486	+	0.919331i	$0.628731\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			19.7990i			0.673965i	0.941511	+	0.336983i	$0.109406\pi$
$863$			19.7990i			0.673965i	−0.941511	+	0.336983i	$0.890594\pi$
$864$		0			0
$865$	6.00000	+	4.89898i	0.204006	+	0.166570i
$866$		0			0
$867$	−7.00000			−0.237732
$868$		0			0
$869$		0			0
$870$		0			0
$871$	13.8564			0.469506
$872$		0			0
$873$			9.79796i			0.331611i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−45.0333			−1.52067			−0.760334	−	0.649533i	$-0.774965\pi$
$877$	−45.0333			−1.52067			−0.760334	+	0.649533i	$0.774965\pi$
$878$		0			0
$879$	3.46410			0.116841
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$	13.8564	−	16.9706i	0.465778	−	0.570459i
$886$		0			0
$887$			19.7990i			0.664785i	0.943141	+	0.332393i	$0.107856\pi$
$887$			19.7990i			0.664785i	−0.943141	+	0.332393i	$0.892144\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.8564			−0.463687
$894$		0			0
$895$	27.7128	−	33.9411i	0.926337	−	1.13453i
$896$		0			0
$897$		−	9.79796i		−	0.327144i
$898$		0			0
$899$			19.5959i			0.653560i
$900$		0			0
$901$		−	16.9706i		−	0.565371i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−24.0000	+	29.3939i	−0.797787	+	0.977086i
$906$		0			0
$907$	−44.0000			−1.46100			−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000			−1.46100			−0.730498	+	0.682915i	$0.760712\pi$
$908$		0			0
$909$			2.82843i			0.0938130i
$910$		0			0
$911$	27.7128			0.918166			0.459083	−	0.888393i	$-0.348178\pi$
$911$	27.7128			0.918166			0.459083	+	0.888393i	$0.348178\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	6.92820			0.228540			0.114270	−	0.993450i	$-0.463547\pi$
$919$	6.92820			0.228540			0.114270	+	0.993450i	$0.463547\pi$
$920$		0			0
$921$	20.0000			0.659022
$922$		0			0
$923$	48.0000			1.57994
$924$		0			0
$925$	−3.46410	−	16.9706i	−0.113899	−	0.557989i
$926$		0			0
$927$		−	16.9706i		−	0.557386i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$			34.2929i			1.12390i
$932$		0			0
$933$	−27.7128			−0.907277
$934$		0			0
$935$		0			0
$936$		0			0
$937$			58.7878i			1.92051i	0.279120	+	0.960256i	$0.409957\pi$
$937$			58.7878i			1.92051i	−0.279120	+	0.960256i	$0.590043\pi$
$938$		0			0
$939$		−	19.5959i		−	0.639489i
$940$		0			0
$941$			2.82843i			0.0922041i	0.998937	+	0.0461020i	$0.0146799\pi$
$941$			2.82843i			0.0922041i	−0.998937	+	0.0461020i	$0.985320\pi$
$942$		0			0
$943$		−	16.9706i		−	0.552638i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	60.0000			1.94974			0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000			1.94974			0.974869	+	0.222779i	$0.0715128\pi$
$948$		0			0
$949$			33.9411i			1.10178i
$950$		0			0
$951$	−31.1769			−1.01098
$952$		0			0
$953$			14.6969i			0.476081i	0.971255	+	0.238040i	$0.0765050\pi$
$953$			14.6969i			0.476081i	−0.971255	+	0.238040i	$0.923495\pi$
$954$		0			0
$955$	−24.0000	−	19.5959i	−0.776622	−	0.634109i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$		0			0
$966$		0			0
$967$			33.9411i			1.09147i	0.837957	+	0.545737i	$0.183750\pi$
$967$			33.9411i			1.09147i	−0.837957	+	0.545737i	$0.816250\pi$
$968$		0			0
$969$	−24.0000			−0.770991
$970$		0			0
$971$			29.3939i			0.943294i	0.881787	+	0.471647i	$0.156340\pi$
$971$			29.3939i			0.943294i	−0.881787	+	0.471647i	$0.843660\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	3.46410	+	16.9706i	0.110940	+	0.543493i
$976$		0			0
$977$			24.4949i			0.783661i	0.920037	+	0.391831i	$0.128158\pi$
$977$			24.4949i			0.783661i	−0.920037	+	0.391831i	$0.871842\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	16.9706i		−	0.541828i
$982$		0			0
$983$			14.1421i			0.451064i	0.974236	+	0.225532i	$0.0724120\pi$
$983$			14.1421i			0.451064i	−0.974236	+	0.225532i	$0.927588\pi$
$984$		0			0
$985$	18.0000	+	14.6969i	0.573528	+	0.468283i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	11.3137i		−	0.359755i
$990$		0			0
$991$	20.7846			0.660245			0.330122	−	0.943938i	$-0.392910\pi$
$991$	20.7846			0.660245			0.330122	+	0.943938i	$0.392910\pi$
$992$		0			0
$993$			4.89898i			0.155464i
$994$		0			0
$995$	36.0000	+	29.3939i	1.14128	+	0.931849i
$996$		0			0
$997$	31.1769			0.987383			0.493691	−	0.869637i	$-0.335647\pi$
$997$	31.1769			0.987383			0.493691	+	0.869637i	$0.335647\pi$
$998$		0			0
$999$	−3.46410			−0.109599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.d.b.1729.4	yes	4
4.3	odd	2		1920.2.d.a.1729.4	yes	4
5.4	even	2		1920.2.d.a.1729.2	yes	4
8.3	odd	2	inner	1920.2.d.b.1729.1	yes	4
8.5	even	2		1920.2.d.a.1729.1	✓	4
16.3	odd	4		3840.2.f.j.769.3		8
16.5	even	4		3840.2.f.j.769.2		8
16.11	odd	4		3840.2.f.j.769.6		8
16.13	even	4		3840.2.f.j.769.7		8
20.19	odd	2	inner	1920.2.d.b.1729.2	yes	4
40.19	odd	2		1920.2.d.a.1729.3	yes	4
40.29	even	2	inner	1920.2.d.b.1729.3	yes	4
80.19	odd	4		3840.2.f.j.769.8		8
80.29	even	4		3840.2.f.j.769.4		8
80.59	odd	4		3840.2.f.j.769.1		8
80.69	even	4		3840.2.f.j.769.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.d.a.1729.1	✓	4	8.5	even	2
1920.2.d.a.1729.2	yes	4	5.4	even	2
1920.2.d.a.1729.3	yes	4	40.19	odd	2
1920.2.d.a.1729.4	yes	4	4.3	odd	2
1920.2.d.b.1729.1	yes	4	8.3	odd	2	inner
1920.2.d.b.1729.2	yes	4	20.19	odd	2	inner
1920.2.d.b.1729.3	yes	4	40.29	even	2	inner
1920.2.d.b.1729.4	yes	4	1.1	even	1	trivial
3840.2.f.j.769.1		8	80.59	odd	4
3840.2.f.j.769.2		8	16.5	even	4
3840.2.f.j.769.3		8	16.3	odd	4
3840.2.f.j.769.4		8	80.29	even	4
3840.2.f.j.769.5		8	80.69	even	4
3840.2.f.j.769.6		8	16.11	odd	4
3840.2.f.j.769.7		8	16.13	even	4
3840.2.f.j.769.8		8	80.19	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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(1.73205 + 1.41421i) q^{5} +1.00000 q^{9} +3.46410 q^{13} +(1.73205 + 1.41421i) q^{15} +4.89898i q^{17} +4.89898i q^{19} -2.82843i q^{23} +(1.00000 + 4.89898i) q^{25} +1.00000 q^{27} -2.82843i q^{29} -6.92820 q^{31} -3.46410 q^{37} +3.46410 q^{39} +6.00000 q^{41} +4.00000 q^{43} +(1.73205 + 1.41421i) q^{45} +2.82843i q^{47} +7.00000 q^{49} +4.89898i q^{51} -3.46410 q^{53} +4.89898i q^{57} -9.79796i q^{59} +(6.00000 + 4.89898i) q^{65} +4.00000 q^{67} -2.82843i q^{69} +13.8564 q^{71} +9.79796i q^{73} +(1.00000 + 4.89898i) q^{75} -6.92820 q^{79} +1.00000 q^{81} -12.0000 q^{83} +(-6.92820 + 8.48528i) q^{85} -2.82843i q^{87} +6.00000 q^{89} -6.92820 q^{93} +(-6.92820 + 8.48528i) q^{95} +9.79796i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9} + 4 q^{25} + 4 q^{27} + 24 q^{41} + 16 q^{43} + 28 q^{49} + 24 q^{65} + 16 q^{67} + 4 q^{75} + 4 q^{81} - 48 q^{83} + 24 q^{89}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 4 * q^25 + 4 * q^27 + 24 * q^41 + 16 * q^43 + 28 * q^49 + 24 * q^65 + 16 * q^67 + 4 * q^75 + 4 * q^81 - 48 * q^83 + 24 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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