LMFDB - Embedded newform 3840.2.f.j.769.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(769,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3840.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 1920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.8
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3840.769
Dual form			3840.2.f.j.769.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.41421 + 1.73205i) q^{5} -1.00000 q^{9} -3.46410i q^{13} +(-1.73205 + 1.41421i) q^{15} -4.89898i q^{17} +4.89898 q^{19} +2.82843i q^{23} +(-1.00000 + 4.89898i) q^{25} -1.00000i q^{27} +2.82843 q^{29} +6.92820 q^{31} -3.46410i q^{37} +3.46410 q^{39} -6.00000 q^{41} -4.00000i q^{43} +(-1.41421 - 1.73205i) q^{45} +2.82843i q^{47} +7.00000 q^{49} +4.89898 q^{51} -3.46410i q^{53} +4.89898i q^{57} +9.79796 q^{59} +(6.00000 - 4.89898i) q^{65} +4.00000i q^{67} -2.82843 q^{69} +13.8564 q^{71} +9.79796i q^{73} +(-4.89898 - 1.00000i) q^{75} +6.92820 q^{79} +1.00000 q^{81} -12.0000i q^{83} +(8.48528 - 6.92820i) q^{85} +2.82843i q^{87} -6.00000 q^{89} +6.92820i q^{93} +(6.92820 + 8.48528i) q^{95} -9.79796i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 8 q^{25} - 48 q^{41} + 56 q^{49} + 48 q^{65} + 8 q^{81} - 48 q^{89}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^25 - 48 * q^41 + 56 * q^49 + 48 * q^65 + 8 * q^81 - 48 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.41421	+	1.73205i	0.632456	+	0.774597i
$5$	1.41421	+	1.73205i	0.632456	+	0.774597i
$6$		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			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$	−1.73205	+	1.41421i	−0.447214	+	0.365148i
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$	4.89898			1.12390			0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898			1.12390			0.561951	+	0.827170i	$0.310051\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.82843i			0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$			2.82843i			0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$		0			0
$25$	−1.00000	+	4.89898i	−0.200000	+	0.979796i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.82843			0.525226			0.262613	−	0.964901i	$-0.415416\pi$
$29$	2.82843			0.525226			0.262613	+	0.964901i	$0.415416\pi$
$30$		0			0
$31$	6.92820			1.24434			0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820			1.24434			0.622171	+	0.782881i	$0.286251\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	3.46410i		−	0.569495i	−0.958603	−	0.284747i	$-0.908090\pi$
$37$		−	3.46410i		−	0.569495i	0.958603	−	0.284747i	$-0.0919097\pi$
$38$		0			0
$39$	3.46410			0.554700
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−1.41421	−	1.73205i	−0.210819	−	0.258199i
$46$		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
$51$	4.89898			0.685994
$52$		0			0
$53$		−	3.46410i		−	0.475831i	−0.971286	−	0.237915i	$-0.923536\pi$
$53$		−	3.46410i		−	0.475831i	0.971286	−	0.237915i	$-0.0764641\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			4.89898i			0.648886i
$58$		0			0
$59$	9.79796			1.27559			0.637793	−	0.770208i	$-0.279848\pi$
$59$	9.79796			1.27559			0.637793	+	0.770208i	$0.279848\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	−	4.89898i	0.744208	−	0.607644i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$	−2.82843			−0.340503
$70$		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
$75$	−4.89898	−	1.00000i	−0.565685	−	0.115470i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.92820			0.779484			0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820			0.779484			0.389742	+	0.920924i	$0.372564\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	8.48528	−	6.92820i	0.920358	−	0.751469i
$86$		0			0
$87$			2.82843i			0.303239i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			6.92820i			0.718421i
$94$		0			0
$95$	6.92820	+	8.48528i	0.710819	+	0.870572i
$96$		0			0
$97$		−	9.79796i		−	0.994832i	−0.867512	−	0.497416i	$-0.834282\pi$
$97$		−	9.79796i		−	0.994832i	0.867512	−	0.497416i	$-0.165718\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.82843			0.281439			0.140720	−	0.990050i	$-0.455058\pi$
$101$	2.82843			0.281439			0.140720	+	0.990050i	$0.455058\pi$
$102$		0			0
$103$			16.9706i			1.67216i	0.548608	+	0.836080i	$0.315158\pi$
$103$			16.9706i			1.67216i	−0.548608	+	0.836080i	$0.684842\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	16.9706			1.62549			0.812743	−	0.582623i	$-0.197974\pi$
$109$	16.9706			1.62549			0.812743	+	0.582623i	$0.197974\pi$
$110$		0			0
$111$	3.46410			0.328798
$112$		0			0
$113$		−	14.6969i		−	1.38257i	−0.722581	−	0.691286i	$-0.757045\pi$
$113$		−	14.6969i		−	1.38257i	0.722581	−	0.691286i	$-0.242955\pi$
$114$		0			0
$115$	−4.89898	+	4.00000i	−0.456832	+	0.373002i
$116$		0			0
$117$			3.46410i			0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$	−9.89949	+	5.19615i	−0.885438	+	0.464758i
$126$		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
$131$	−19.5959			−1.71210			−0.856052	−	0.516890i	$-0.827090\pi$
$131$	−19.5959			−1.71210			−0.856052	+	0.516890i	$0.827090\pi$
$132$		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
$139$	−14.6969			−1.24658			−0.623289	−	0.781992i	$-0.714204\pi$
$139$	−14.6969			−1.24658			−0.623289	+	0.781992i	$0.714204\pi$
$140$		0			0
$141$	−2.82843			−0.238197
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000	+	4.89898i	0.332182	+	0.406838i
$146$		0			0
$147$			7.00000i			0.577350i
$148$		0			0
$149$	−19.7990			−1.62200			−0.810998	−	0.585049i	$-0.801075\pi$
$149$	−19.7990			−1.62200			−0.810998	+	0.585049i	$0.801075\pi$
$150$		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
$155$	9.79796	+	12.0000i	0.786991	+	0.963863i
$156$		0			0
$157$		−	10.3923i		−	0.829396i	−0.909959	−	0.414698i	$-0.863887\pi$
$157$		−	10.3923i		−	0.829396i	0.909959	−	0.414698i	$-0.136113\pi$
$158$		0			0
$159$	3.46410			0.274721
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.1421i			1.09435i	0.837018	+	0.547176i	$0.184297\pi$
$167$			14.1421i			1.09435i	−0.837018	+	0.547176i	$0.815703\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−4.89898			−0.374634
$172$		0			0
$173$		−	3.46410i		−	0.263371i	−0.991292	−	0.131685i	$-0.957961\pi$
$173$		−	3.46410i		−	0.263371i	0.991292	−	0.131685i	$-0.0420389\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			9.79796i			0.736460i
$178$		0			0
$179$	−19.5959			−1.46467			−0.732334	−	0.680946i	$-0.761569\pi$
$179$	−19.5959			−1.46467			−0.732334	+	0.680946i	$0.761569\pi$
$180$		0			0
$181$	16.9706			1.26141			0.630706	−	0.776022i	$-0.282765\pi$
$181$	16.9706			1.26141			0.630706	+	0.776022i	$0.282765\pi$
$182$		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	$-0.332853\pi$
$191$	13.8564			1.00261			0.501307	+	0.865269i	$0.332853\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	4.89898	+	6.00000i	0.350823	+	0.429669i
$196$		0			0
$197$			10.3923i			0.740421i	0.928948	+	0.370211i	$0.120714\pi$
$197$			10.3923i			0.740421i	−0.928948	+	0.370211i	$0.879286\pi$
$198$		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
$205$	−8.48528	−	10.3923i	−0.592638	−	0.725830i
$206$		0			0
$207$		−	2.82843i		−	0.196589i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	14.6969			1.01178			0.505889	−	0.862598i	$-0.331164\pi$
$211$	14.6969			1.01178			0.505889	+	0.862598i	$0.331164\pi$
$212$		0			0
$213$			13.8564i			0.949425i
$214$		0			0
$215$	6.92820	−	5.65685i	0.472500	−	0.385794i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−9.79796			−0.662085
$220$		0			0
$221$	−16.9706			−1.14156
$222$		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
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	−16.9706			−1.12145			−0.560723	−	0.828003i	$-0.689477\pi$
$229$	−16.9706			−1.12145			−0.560723	+	0.828003i	$0.689477\pi$
$230$		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
$235$	−4.89898	+	4.00000i	−0.319574	+	0.260931i
$236$		0			0
$237$			6.92820i			0.450035i
$238$		0			0
$239$	13.8564			0.896296			0.448148	−	0.893959i	$-0.352084\pi$
$239$	13.8564			0.896296			0.448148	+	0.893959i	$0.352084\pi$
$240$		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
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	9.89949	+	12.1244i	0.632456	+	0.774597i
$246$		0			0
$247$		−	16.9706i		−	1.07981i
$248$		0			0
$249$	12.0000			0.760469
$250$		0			0
$251$	9.79796			0.618442			0.309221	−	0.950990i	$-0.399932\pi$
$251$	9.79796			0.618442			0.309221	+	0.950990i	$0.399932\pi$
$252$		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	$0.151572\pi$
$257$			14.6969i			0.916770i	−0.888754	+	0.458385i	$0.848428\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.82843			−0.175075
$262$		0			0
$263$			14.1421i			0.872041i	0.899937	+	0.436021i	$0.143613\pi$
$263$			14.1421i			0.872041i	−0.899937	+	0.436021i	$0.856387\pi$
$264$		0			0
$265$	6.00000	−	4.89898i	0.368577	−	0.300942i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$	19.7990			1.20717			0.603583	−	0.797300i	$-0.293739\pi$
$269$	19.7990			1.20717			0.603583	+	0.797300i	$0.293739\pi$
$270$		0			0
$271$	−6.92820			−0.420858			−0.210429	−	0.977609i	$-0.567486\pi$
$271$	−6.92820			−0.420858			−0.210429	+	0.977609i	$0.567486\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	17.3205i		−	1.04069i	−0.853957	−	0.520344i	$-0.825804\pi$
$277$		−	17.3205i		−	1.04069i	0.853957	−	0.520344i	$-0.174196\pi$
$278$		0			0
$279$	−6.92820			−0.414781
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	28.0000i		−	1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$		−	28.0000i		−	1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$		0			0
$285$	−8.48528	+	6.92820i	−0.502625	+	0.410391i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	9.79796			0.574367
$292$		0			0
$293$			3.46410i			0.202375i	0.994867	+	0.101187i	$0.0322642\pi$
$293$			3.46410i			0.202375i	−0.994867	+	0.101187i	$0.967736\pi$
$294$		0			0
$295$	13.8564	+	16.9706i	0.806751	+	0.988064i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.79796			0.566631
$300$		0			0
$301$		0			0
$302$		0			0
$303$			2.82843i			0.162489i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			20.0000i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$			20.0000i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$		0			0
$309$	−16.9706			−0.965422
$310$		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
$317$			31.1769i			1.75107i	0.483155	+	0.875535i	$0.339491\pi$
$317$			31.1769i			1.75107i	−0.483155	+	0.875535i	$0.660509\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$	16.9706	+	3.46410i	0.941357	+	0.192154i
$326$		0			0
$327$			16.9706i			0.938474i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.89898			−0.269272			−0.134636	−	0.990895i	$-0.542987\pi$
$331$	−4.89898			−0.269272			−0.134636	+	0.990895i	$0.542987\pi$
$332$		0			0
$333$			3.46410i			0.189832i
$334$		0			0
$335$	−6.92820	+	5.65685i	−0.378528	+	0.309067i
$336$		0			0
$337$			9.79796i			0.533729i	0.963734	+	0.266864i	$0.0859876\pi$
$337$			9.79796i			0.533729i	−0.963734	+	0.266864i	$0.914012\pi$
$338$		0			0
$339$	14.6969			0.798228
$340$		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
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$	33.9411			1.81683			0.908413	−	0.418073i	$-0.137294\pi$
$349$	33.9411			1.81683			0.908413	+	0.418073i	$0.137294\pi$
$350$		0			0
$351$	−3.46410			−0.184900
$352$		0			0
$353$			34.2929i			1.82522i	0.408826	+	0.912612i	$0.365938\pi$
$353$			34.2929i			1.82522i	−0.408826	+	0.912612i	$0.634062\pi$
$354$		0			0
$355$	19.5959	+	24.0000i	1.04004	+	1.27379i
$356$		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
$363$		−	11.0000i		−	0.577350i
$364$		0			0
$365$	−16.9706	+	13.8564i	−0.888280	+	0.725277i
$366$		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
$373$		−	31.1769i		−	1.61428i	−0.590360	−	0.807140i	$-0.701014\pi$
$373$		−	31.1769i		−	1.61428i	0.590360	−	0.807140i	$-0.298986\pi$
$374$		0			0
$375$	−5.19615	−	9.89949i	−0.268328	−	0.511208i
$376$		0			0
$377$		−	9.79796i		−	0.504621i
$378$		0			0
$379$	4.89898			0.251644			0.125822	−	0.992053i	$-0.459843\pi$
$379$	4.89898			0.251644			0.125822	+	0.992053i	$0.459843\pi$
$380$		0			0
$381$	−16.9706			−0.869428
$382$		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
$387$			4.00000i			0.203331i
$388$		0			0
$389$	−31.1127			−1.57748			−0.788738	−	0.614729i	$-0.789265\pi$
$389$	−31.1127			−1.57748			−0.788738	+	0.614729i	$0.789265\pi$
$390$		0			0
$391$	13.8564			0.700749
$392$		0			0
$393$		−	19.5959i		−	0.988483i
$394$		0			0
$395$	9.79796	+	12.0000i	0.492989	+	0.603786i
$396$		0			0
$397$		−	17.3205i		−	0.869291i	−0.900602	−	0.434646i	$-0.856874\pi$
$397$		−	17.3205i		−	0.869291i	0.900602	−	0.434646i	$-0.143126\pi$
$398$		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
$403$		−	24.0000i		−	1.19553i
$404$		0			0
$405$	1.41421	+	1.73205i	0.0702728	+	0.0860663i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$	−4.89898			−0.241649
$412$		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
$419$	9.79796			0.478662			0.239331	−	0.970938i	$-0.423072\pi$
$419$	9.79796			0.478662			0.239331	+	0.970938i	$0.423072\pi$
$420$		0			0
$421$	16.9706			0.827095			0.413547	−	0.910483i	$-0.364290\pi$
$421$	16.9706			0.827095			0.413547	+	0.910483i	$0.364290\pi$
$422$		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	$-0.608304\pi$
$431$	−13.8564			−0.667440			−0.333720	+	0.942672i	$0.608304\pi$
$432$		0			0
$433$			19.5959i			0.941720i	0.882208	+	0.470860i	$0.156056\pi$
$433$			19.5959i			0.941720i	−0.882208	+	0.470860i	$0.843944\pi$
$434$		0			0
$435$	−4.89898	+	4.00000i	−0.234888	+	0.191785i
$436$		0			0
$437$			13.8564i			0.662842i
$438$		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
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$	−8.48528	−	10.3923i	−0.402241	−	0.492642i
$446$		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
$453$			6.92820i			0.325515i
$454$		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
$459$	−4.89898			−0.228665
$460$		0			0
$461$	−14.1421			−0.658665			−0.329332	−	0.944214i	$-0.606824\pi$
$461$	−14.1421			−0.658665			−0.329332	+	0.944214i	$0.606824\pi$
$462$		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
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	10.3923			0.478852
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.89898	+	24.0000i	−0.224781	+	1.10120i
$476$		0			0
$477$			3.46410i			0.158610i
$478$		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
$485$	16.9706	−	13.8564i	0.770594	−	0.629187i
$486$		0			0
$487$		−	16.9706i		−	0.769010i	−0.923123	−	0.384505i	$-0.874372\pi$
$487$		−	16.9706i		−	0.769010i	0.923123	−	0.384505i	$-0.125628\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	29.3939			1.32653			0.663264	−	0.748386i	$-0.269171\pi$
$491$	29.3939			1.32653			0.663264	+	0.748386i	$0.269171\pi$
$492$		0			0
$493$		−	13.8564i		−	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4.89898			0.219308			0.109654	−	0.993970i	$-0.465026\pi$
$499$	4.89898			0.219308			0.109654	+	0.993970i	$0.465026\pi$
$500$		0			0
$501$	−14.1421			−0.631824
$502$		0			0
$503$		−	36.7696i		−	1.63947i	−0.572741	−	0.819737i	$-0.694120\pi$
$503$		−	36.7696i		−	1.63947i	0.572741	−	0.819737i	$-0.305880\pi$
$504$		0			0
$505$	4.00000	+	4.89898i	0.177998	+	0.218002i
$506$		0			0
$507$			1.00000i			0.0444116i
$508$		0			0
$509$	14.1421			0.626839			0.313420	−	0.949615i	$-0.398525\pi$
$509$	14.1421			0.626839			0.313420	+	0.949615i	$0.398525\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	4.89898i		−	0.216295i
$514$		0			0
$515$	−29.3939	+	24.0000i	−1.29525	+	1.05757i
$516$		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	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$		−	20.0000i		−	0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$		−	20.0000i		−	0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	33.9411i		−	1.47850i
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$	−9.79796			−0.425195
$532$		0			0
$533$			20.7846i			0.900281i
$534$		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
$541$	16.9706			0.729621			0.364811	−	0.931082i	$-0.381134\pi$
$541$	16.9706			0.729621			0.364811	+	0.931082i	$0.381134\pi$
$542$		0			0
$543$			16.9706i			0.728277i
$544$		0			0
$545$	24.0000	+	29.3939i	1.02805	+	1.25910i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.8564			0.590303
$552$		0			0
$553$		0			0
$554$		0			0
$555$	4.89898	+	6.00000i	0.207950	+	0.254686i
$556$		0			0
$557$		−	31.1769i		−	1.32101i	−0.750822	−	0.660504i	$-0.770343\pi$
$557$		−	31.1769i		−	1.32101i	0.750822	−	0.660504i	$-0.229657\pi$
$558$		0			0
$559$	−13.8564			−0.586064
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	36.0000i		−	1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$		−	36.0000i		−	1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$		0			0
$565$	25.4558	−	20.7846i	1.07094	−	0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	−24.4949			−1.02508			−0.512540	−	0.858663i	$-0.671295\pi$
$571$	−24.4949			−1.02508			−0.512540	+	0.858663i	$0.671295\pi$
$572$		0			0
$573$			13.8564i			0.578860i
$574$		0			0
$575$	−13.8564	−	2.82843i	−0.577852	−	0.117954i
$576$		0			0
$577$			19.5959i			0.815789i	0.913029	+	0.407894i	$0.133737\pi$
$577$			19.5959i			0.815789i	−0.913029	+	0.407894i	$0.866263\pi$
$578$		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
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	33.9411			1.39852
$590$		0			0
$591$	−10.3923			−0.427482
$592$		0			0
$593$			4.89898i			0.201177i	0.994928	+	0.100588i	$0.0320726\pi$
$593$			4.89898i			0.201177i	−0.994928	+	0.100588i	$0.967927\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			20.7846i			0.850657i
$598$		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	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$		−	4.00000i		−	0.162893i
$604$		0			0
$605$	−15.5563	−	19.0526i	−0.632456	−	0.774597i
$606$		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
$611$	9.79796			0.396383
$612$		0			0
$613$			3.46410i			0.139914i	0.997550	+	0.0699569i	$0.0222862\pi$
$613$			3.46410i			0.139914i	−0.997550	+	0.0699569i	$0.977714\pi$
$614$		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
$619$	44.0908			1.77216			0.886080	−	0.463533i	$-0.153418\pi$
$619$	44.0908			1.77216			0.886080	+	0.463533i	$0.153418\pi$
$620$		0			0
$621$	2.82843			0.113501
$622$		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
$629$	−16.9706			−0.676661
$630$		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
$635$	−29.3939	+	24.0000i	−1.16646	+	0.952411i
$636$		0			0
$637$		−	24.2487i		−	0.960769i
$638$		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
$643$		−	4.00000i		−	0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$		−	4.00000i		−	0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$		0			0
$645$	5.65685	+	6.92820i	0.222738	+	0.272798i
$646$		0			0
$647$			31.1127i			1.22317i	0.791180	+	0.611583i	$0.209467\pi$
$647$			31.1127i			1.22317i	−0.791180	+	0.611583i	$0.790533\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			31.1769i			1.22005i	0.792383	+	0.610023i	$0.208840\pi$
$653$			31.1769i			1.22005i	−0.792383	+	0.610023i	$0.791160\pi$
$654$		0			0
$655$	−27.7128	−	33.9411i	−1.08283	−	1.32619i
$656$		0			0
$657$		−	9.79796i		−	0.382255i
$658$		0			0
$659$	−9.79796			−0.381674			−0.190837	−	0.981622i	$-0.561120\pi$
$659$	−9.79796			−0.381674			−0.190837	+	0.981622i	$0.561120\pi$
$660$		0			0
$661$	33.9411			1.32016			0.660078	−	0.751197i	$-0.270523\pi$
$661$	33.9411			1.32016			0.660078	+	0.751197i	$0.270523\pi$
$662$		0			0
$663$		−	16.9706i		−	0.659082i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			8.00000i			0.309761i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	39.1918i		−	1.51073i	−0.655302	−	0.755367i	$-0.727459\pi$
$673$		−	39.1918i		−	1.51073i	0.655302	−	0.755367i	$-0.272541\pi$
$674$		0			0
$675$	4.89898	+	1.00000i	0.188562	+	0.0384900i
$676$		0			0
$677$		−	17.3205i		−	0.665681i	−0.942983	−	0.332841i	$-0.891993\pi$
$677$		−	17.3205i		−	0.665681i	0.942983	−	0.332841i	$-0.108007\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$		−	36.0000i		−	1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$		−	36.0000i		−	1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$		0			0
$685$	−8.48528	+	6.92820i	−0.324206	+	0.264713i
$686$		0			0
$687$		−	16.9706i		−	0.647467i
$688$		0			0
$689$	−12.0000			−0.457164
$690$		0			0
$691$	−34.2929			−1.30456			−0.652281	−	0.757977i	$-0.726188\pi$
$691$	−34.2929			−1.30456			−0.652281	+	0.757977i	$0.726188\pi$
$692$		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
$699$	4.89898			0.185296
$700$		0			0
$701$	−2.82843			−0.106828			−0.0534141	−	0.998572i	$-0.517010\pi$
$701$	−2.82843			−0.106828			−0.0534141	+	0.998572i	$0.517010\pi$
$702$		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
$709$	−16.9706			−0.637343			−0.318671	−	0.947865i	$-0.603237\pi$
$709$	−16.9706			−0.637343			−0.318671	+	0.947865i	$0.603237\pi$
$710$		0			0
$711$	−6.92820			−0.259828
$712$		0			0
$713$			19.5959i			0.733873i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			13.8564i			0.517477i
$718$		0			0
$719$	−41.5692			−1.55027			−0.775135	−	0.631795i	$-0.782318\pi$
$719$	−41.5692			−1.55027			−0.775135	+	0.631795i	$0.782318\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	14.0000i		−	0.520666i
$724$		0			0
$725$	−2.82843	+	13.8564i	−0.105045	+	0.514614i
$726$		0			0
$727$		−	50.9117i		−	1.88821i	−0.329645	−	0.944105i	$-0.606929\pi$
$727$		−	50.9117i		−	1.88821i	0.329645	−	0.944105i	$-0.393071\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−19.5959			−0.724781
$732$		0			0
$733$		−	51.9615i		−	1.91924i	−0.281295	−	0.959621i	$-0.590764\pi$
$733$		−	51.9615i		−	1.91924i	0.281295	−	0.959621i	$-0.409236\pi$
$734$		0			0
$735$	−12.1244	+	9.89949i	−0.447214	+	0.365148i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−14.6969			−0.540636			−0.270318	−	0.962771i	$-0.587129\pi$
$739$	−14.6969			−0.540636			−0.270318	+	0.962771i	$0.587129\pi$
$740$		0			0
$741$	16.9706			0.623429
$742$		0			0
$743$		−	14.1421i		−	0.518825i	−0.965767	−	0.259412i	$-0.916471\pi$
$743$		−	14.1421i		−	0.518825i	0.965767	−	0.259412i	$-0.0835289\pi$
$744$		0			0
$745$	−28.0000	−	34.2929i	−1.02584	−	1.25639i
$746$		0			0
$747$			12.0000i			0.439057i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6.92820			0.252814			0.126407	−	0.991978i	$-0.459656\pi$
$751$	6.92820			0.252814			0.126407	+	0.991978i	$0.459656\pi$
$752$		0			0
$753$			9.79796i			0.357057i
$754$		0			0
$755$	9.79796	+	12.0000i	0.356584	+	0.436725i
$756$		0			0
$757$		−	45.0333i		−	1.63676i	−0.574675	−	0.818382i	$-0.694871\pi$
$757$		−	45.0333i		−	1.63676i	0.574675	−	0.818382i	$-0.305129\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−8.48528	+	6.92820i	−0.306786	+	0.250490i
$766$		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
$771$	−14.6969			−0.529297
$772$		0			0
$773$		−	17.3205i		−	0.622975i	−0.950250	−	0.311488i	$-0.899173\pi$
$773$		−	17.3205i		−	0.622975i	0.950250	−	0.311488i	$-0.100827\pi$
$774$		0			0
$775$	−6.92820	+	33.9411i	−0.248868	+	1.21920i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−29.3939			−1.05314
$780$		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
$787$			44.0000i			1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$			44.0000i			1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$		0			0
$789$	−14.1421			−0.503473
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	4.89898	+	6.00000i	0.173749	+	0.212798i
$796$		0			0
$797$			17.3205i			0.613524i	0.951786	+	0.306762i	$0.0992455\pi$
$797$			17.3205i			0.613524i	−0.951786	+	0.306762i	$0.900754\pi$
$798$		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	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	4.89898			0.172026			0.0860132	−	0.996294i	$-0.472587\pi$
$811$	4.89898			0.172026			0.0860132	+	0.996294i	$0.472587\pi$
$812$		0			0
$813$		−	6.92820i		−	0.242983i
$814$		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
$821$	−2.82843			−0.0987128			−0.0493564	−	0.998781i	$-0.515717\pi$
$821$	−2.82843			−0.0987128			−0.0493564	+	0.998781i	$0.515717\pi$
$822$		0			0
$823$		−	33.9411i		−	1.18311i	−0.806263	−	0.591557i	$-0.798514\pi$
$823$		−	33.9411i		−	1.18311i	0.806263	−	0.591557i	$-0.201486\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		0			0
$829$	16.9706			0.589412			0.294706	−	0.955588i	$-0.404778\pi$
$829$	16.9706			0.589412			0.294706	+	0.955588i	$0.404778\pi$
$830$		0			0
$831$	17.3205			0.600842
$832$		0			0
$833$		−	34.2929i		−	1.18818i
$834$		0			0
$835$	−24.4949	+	20.0000i	−0.847681	+	0.692129i
$836$		0			0
$837$		−	6.92820i		−	0.239474i
$838$		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
$843$			6.00000i			0.206651i
$844$		0			0
$845$	1.41421	+	1.73205i	0.0486504	+	0.0595844i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$	9.79796			0.335870
$852$		0			0
$853$		−	24.2487i		−	0.830260i	−0.909762	−	0.415130i	$-0.863736\pi$
$853$		−	24.2487i		−	0.830260i	0.909762	−	0.415130i	$-0.136264\pi$
$854$		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
$859$	−53.8888			−1.83866			−0.919331	−	0.393486i	$-0.871269\pi$
$859$	−53.8888			−1.83866			−0.919331	+	0.393486i	$0.871269\pi$
$860$		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
$867$		−	7.00000i		−	0.237732i
$868$		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
$877$			45.0333i			1.52067i	0.649533	+	0.760334i	$0.274965\pi$
$877$			45.0333i			1.52067i	−0.649533	+	0.760334i	$0.725035\pi$
$878$		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
$883$		−	52.0000i		−	1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$		−	52.0000i		−	1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$		0			0
$885$	−16.9706	+	13.8564i	−0.570459	+	0.465778i
$886$		0			0
$887$		−	19.7990i		−	0.664785i	−0.943141	−	0.332393i	$-0.892144\pi$
$887$		−	19.7990i		−	0.664785i	0.943141	−	0.332393i	$-0.107856\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			13.8564i			0.463687i
$894$		0			0
$895$	−27.7128	−	33.9411i	−0.926337	−	1.13453i
$896$		0			0
$897$			9.79796i			0.327144i
$898$		0			0
$899$	19.5959			0.653560
$900$		0			0
$901$	−16.9706			−0.565371
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24.0000	+	29.3939i	0.797787	+	0.977086i
$906$		0			0
$907$			44.0000i			1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$			44.0000i			1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$		0			0
$909$	−2.82843			−0.0938130
$910$		0			0
$911$	−27.7128			−0.918166			−0.459083	−	0.888393i	$-0.651822\pi$
$911$	−27.7128			−0.918166			−0.459083	+	0.888393i	$0.651822\pi$
$912$		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
$923$		−	48.0000i		−	1.57994i
$924$		0			0
$925$	16.9706	+	3.46410i	0.557989	+	0.113899i
$926$		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
$931$	34.2929			1.12390
$932$		0			0
$933$		−	27.7128i		−	0.907277i
$934$		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
$939$	19.5959			0.639489
$940$		0			0
$941$	−2.82843			−0.0922041			−0.0461020	−	0.998937i	$-0.514680\pi$
$941$	−2.82843			−0.0922041			−0.0461020	+	0.998937i	$0.514680\pi$
$942$		0			0
$943$		−	16.9706i		−	0.552638i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			60.0000i			1.94974i	0.222779	+	0.974869i	$0.428487\pi$
$947$			60.0000i			1.94974i	−0.222779	+	0.974869i	$0.571513\pi$
$948$		0			0
$949$	33.9411			1.10178
$950$		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
$955$	19.5959	+	24.0000i	0.634109	+	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	33.9411i		−	1.09147i	−0.837957	−	0.545737i	$-0.816250\pi$
$967$		−	33.9411i		−	1.09147i	0.837957	−	0.545737i	$-0.183750\pi$
$968$		0			0
$969$	24.0000			0.770991
$970$		0			0
$971$	−29.3939			−0.943294			−0.471647	−	0.881787i	$-0.656340\pi$
$971$	−29.3939			−0.943294			−0.471647	+	0.881787i	$0.656340\pi$
$972$		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	$-0.871842\pi$
$977$		−	24.4949i		−	0.783661i	0.920037	−	0.391831i	$-0.128158\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−16.9706			−0.541828
$982$		0			0
$983$		−	14.1421i		−	0.451064i	−0.974236	−	0.225532i	$-0.927588\pi$
$983$		−	14.1421i		−	0.451064i	0.974236	−	0.225532i	$-0.0724120\pi$
$984$		0			0
$985$	−18.0000	+	14.6969i	−0.573528	+	0.468283i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11.3137			0.359755
$990$		0			0
$991$	−20.7846			−0.660245			−0.330122	−	0.943938i	$-0.607090\pi$
$991$	−20.7846			−0.660245			−0.330122	+	0.943938i	$0.607090\pi$
$992$		0			0
$993$		−	4.89898i		−	0.155464i
$994$		0			0
$995$	29.3939	+	36.0000i	0.931849	+	1.14128i
$996$		0			0
$997$			31.1769i			0.987383i	0.869637	+	0.493691i	$0.164353\pi$
$997$			31.1769i			0.987383i	−0.869637	+	0.493691i	$0.835647\pi$
$998$		0			0
$999$	−3.46410			−0.109599

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.d.a.1729.1	✓	4	80.19	odd	4
1920.2.d.a.1729.2	yes	4	16.11	odd	4
1920.2.d.a.1729.3	yes	4	16.13	even	4
1920.2.d.a.1729.4	yes	4	80.69	even	4
1920.2.d.b.1729.1	yes	4	80.29	even	4
1920.2.d.b.1729.2	yes	4	16.5	even	4
1920.2.d.b.1729.3	yes	4	16.3	odd	4
1920.2.d.b.1729.4	yes	4	80.59	odd	4
3840.2.f.j.769.1		8	8.5	even	2	inner
3840.2.f.j.769.2		8	40.19	odd	2	inner
3840.2.f.j.769.3		8	5.4	even	2	inner
3840.2.f.j.769.4		8	4.3	odd	2	inner
3840.2.f.j.769.5		8	8.3	odd	2	inner
3840.2.f.j.769.6		8	40.29	even	2	inner
3840.2.f.j.769.7		8	20.19	odd	2	inner
3840.2.f.j.769.8		8	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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.41421 + 1.73205i) q^{5} -1.00000 q^{9} -3.46410i q^{13} +(-1.73205 + 1.41421i) q^{15} -4.89898i q^{17} +4.89898 q^{19} +2.82843i q^{23} +(-1.00000 + 4.89898i) q^{25} -1.00000i q^{27} +2.82843 q^{29} +6.92820 q^{31} -3.46410i q^{37} +3.46410 q^{39} -6.00000 q^{41} -4.00000i q^{43} +(-1.41421 - 1.73205i) q^{45} +2.82843i q^{47} +7.00000 q^{49} +4.89898 q^{51} -3.46410i q^{53} +4.89898i q^{57} +9.79796 q^{59} +(6.00000 - 4.89898i) q^{65} +4.00000i q^{67} -2.82843 q^{69} +13.8564 q^{71} +9.79796i q^{73} +(-4.89898 - 1.00000i) q^{75} +6.92820 q^{79} +1.00000 q^{81} -12.0000i q^{83} +(8.48528 - 6.92820i) q^{85} +2.82843i q^{87} -6.00000 q^{89} +6.92820i q^{93} +(6.92820 + 8.48528i) q^{95} -9.79796i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{25} - 48 q^{41} + 56 q^{49} + 48 q^{65} + 8 q^{81} - 48 q^{89}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^25 - 48 * q^41 + 56 * q^49 + 48 * q^65 + 8 * q^81 - 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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