LMFDB - Embedded newform 4704.2.b.e.1567.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1567,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("4704.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.b (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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 $ x^16 + 24x^14 + 218x^12 + 968x^10 + 2241x^8 + 2672x^6 + 1512x^4 + 320*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.3
Root			$-1.60698i$ of defining polynomial
Character	$\chi$	$=$	4704.1567
Dual form			4704.2.b.e.1567.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.70012i q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.70012i q^{5} +1.00000 q^{9} +1.41762i q^{11} +5.69489i q^{13} -2.70012i q^{15} -1.95966i q^{17} +0.723292 q^{19} +0.649746i q^{23} -2.29063 q^{25} +1.00000 q^{27} +6.06459 q^{29} +4.99520 q^{31} +1.41762i q^{33} +1.15370 q^{37} +5.69489i q^{39} +1.58141i q^{41} +10.8193i q^{43} -2.70012i q^{45} -9.62969 q^{47} -1.95966i q^{51} +12.5241 q^{53} +3.82774 q^{55} +0.723292 q^{57} -0.868280 q^{59} -13.6101i q^{61} +15.3769 q^{65} +0.0530875i q^{67} +0.649746i q^{69} -14.0593i q^{71} +15.6484i q^{73} -2.29063 q^{75} +11.4936i q^{79} +1.00000 q^{81} +2.95860 q^{83} -5.29130 q^{85} +6.06459 q^{87} +8.24185i q^{89} +4.99520 q^{93} -1.95297i q^{95} -1.01880i q^{97} +1.41762i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 16 * q^9	$ 16 q + 16 q^{3} + 16 q^{9} - 8 q^{19} - 8 q^{25} + 16 q^{27} - 8 q^{31} - 8 q^{37} - 16 q^{47} - 16 q^{53} - 16 q^{55} - 8 q^{57} - 8 q^{59} - 16 q^{65} - 8 q^{75} + 16 q^{81} + 8 q^{83} + 32 q^{85} - 8 q^{93}+O(q^{100}) $ 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^19 - 8 * q^25 + 16 * q^27 - 8 * q^31 - 8 * q^37 - 16 * q^47 - 16 * q^53 - 16 * q^55 - 8 * q^57 - 8 * q^59 - 16 * q^65 - 8 * q^75 + 16 * q^81 + 8 * q^83 + 32 * q^85 - 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	− 2.70012i	− 1.20753i	−0.797163	−	0.603764i	$-0.793667\pi$
$5$	− 2.70012i	− 1.20753i	0.797163	−	0.603764i	$-0.206333\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.41762i	0.427428i	0.976896	+	0.213714i	$0.0685561\pi$
$11$	1.41762i	0.427428i	−0.976896	+	0.213714i	$0.931444\pi$
$12$	0	0
$13$	5.69489i	1.57948i	0.613442	+	0.789739i	$0.289784\pi$
$13$	5.69489i	1.57948i	−0.613442	+	0.789739i	$0.710216\pi$
$14$	0	0
$15$	− 2.70012i	− 0.697167i
$16$	0	0
$17$	− 1.95966i	− 0.475286i	−0.971353	−	0.237643i	$-0.923625\pi$
$17$	− 1.95966i	− 0.475286i	0.971353	−	0.237643i	$-0.0763749\pi$
$18$	0	0
$19$	0.723292	0.165934	0.0829672	−	0.996552i	$-0.473560\pi$
$19$	0.723292	0.165934	0.0829672	+	0.996552i	$0.473560\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.649746i	0.135481i	0.997703	+	0.0677407i	$0.0215791\pi$
$23$	0.649746i	0.135481i	−0.997703	+	0.0677407i	$0.978421\pi$
$24$	0	0
$25$	−2.29063	−0.458126
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.06459	1.12617	0.563083	−	0.826400i	$-0.309615\pi$
$29$	6.06459	1.12617	0.563083	+	0.826400i	$0.309615\pi$
$30$	0	0
$31$	4.99520	0.897164	0.448582	−	0.893742i	$-0.351929\pi$
$31$	4.99520	0.897164	0.448582	+	0.893742i	$0.351929\pi$
$32$	0	0
$33$	1.41762i	0.246776i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.15370	0.189667	0.0948336	−	0.995493i	$-0.469768\pi$
$37$	1.15370	0.189667	0.0948336	+	0.995493i	$0.469768\pi$
$38$	0	0
$39$	5.69489i	0.911913i
$40$	0	0
$41$	1.58141i	0.246975i	0.992346	+	0.123487i	$0.0394078\pi$
$41$	1.58141i	0.246975i	−0.992346	+	0.123487i	$0.960592\pi$
$42$	0	0
$43$	10.8193i	1.64993i	0.565185	+	0.824965i	$0.308805\pi$
$43$	10.8193i	1.64993i	−0.565185	+	0.824965i	$0.691195\pi$
$44$	0	0
$45$	− 2.70012i	− 0.402510i
$46$	0	0
$47$	−9.62969	−1.40464	−0.702318	−	0.711864i	$-0.747851\pi$
$47$	−9.62969	−1.40464	−0.702318	+	0.711864i	$0.747851\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 1.95966i	− 0.274407i
$52$	0	0
$53$	12.5241	1.72032	0.860158	−	0.510027i	$-0.170365\pi$
$53$	12.5241	1.72032	0.860158	+	0.510027i	$0.170365\pi$
$54$	0	0
$55$	3.82774	0.516132
$56$	0	0
$57$	0.723292	0.0958023
$58$	0	0
$59$	−0.868280	−0.113040	−0.0565202	−	0.998401i	$-0.518001\pi$
$59$	−0.868280	−0.113040	−0.0565202	+	0.998401i	$0.518001\pi$
$60$	0	0
$61$	− 13.6101i	− 1.74259i	−0.490756	−	0.871297i	$-0.663279\pi$
$61$	− 13.6101i	− 1.74259i	0.490756	−	0.871297i	$-0.336721\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	15.3769	1.90727
$66$	0	0
$67$	0.0530875i	0.00648567i	0.999995	+	0.00324283i	$0.00103223\pi$
$67$	0.0530875i	0.00648567i	−0.999995	+	0.00324283i	$0.998968\pi$
$68$	0	0
$69$	0.649746i	0.0782202i
$70$	0	0
$71$	− 14.0593i	− 1.66853i	−0.551362	−	0.834266i	$-0.685892\pi$
$71$	− 14.0593i	− 1.66853i	0.551362	−	0.834266i	$-0.314108\pi$
$72$	0	0
$73$	15.6484i	1.83151i	0.401735	+	0.915756i	$0.368407\pi$
$73$	15.6484i	1.83151i	−0.401735	+	0.915756i	$0.631593\pi$
$74$	0	0
$75$	−2.29063	−0.264499
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.4936i	1.29313i	0.762861	+	0.646563i	$0.223794\pi$
$79$	11.4936i	1.29313i	−0.762861	+	0.646563i	$0.776206\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.95860	0.324749	0.162375	−	0.986729i	$-0.448085\pi$
$83$	2.95860	0.324749	0.162375	+	0.986729i	$0.448085\pi$
$84$	0	0
$85$	−5.29130	−0.573922
$86$	0	0
$87$	6.06459	0.650192
$88$	0	0
$89$	8.24185i	0.873634i	0.899550	+	0.436817i	$0.143894\pi$
$89$	8.24185i	0.873634i	−0.899550	+	0.436817i	$0.856106\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.99520	0.517978
$94$	0	0
$95$	− 1.95297i	− 0.200371i
$96$	0	0
$97$	− 1.01880i	− 0.103444i	−0.998662	−	0.0517218i	$-0.983529\pi$
$97$	− 1.01880i	− 0.103444i	0.998662	−	0.0517218i	$-0.0164709\pi$
$98$	0	0
$99$	1.41762i	0.142476i
$100$	0	0
$101$	− 11.9466i	− 1.18873i	−0.804196	−	0.594364i	$-0.797404\pi$
$101$	− 11.9466i	− 1.18873i	0.804196	−	0.594364i	$-0.202596\pi$
$102$	0	0
$103$	17.5903	1.73322	0.866610	−	0.498986i	$-0.166294\pi$
$103$	17.5903	1.73322	0.866610	+	0.498986i	$0.166294\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.5752i	1.40903i	0.709687	+	0.704517i	$0.248836\pi$
$107$	14.5752i	1.40903i	−0.709687	+	0.704517i	$0.751164\pi$
$108$	0	0
$109$	5.41674	0.518829	0.259415	−	0.965766i	$-0.416470\pi$
$109$	5.41674	0.518829	0.259415	+	0.965766i	$0.416470\pi$
$110$	0	0
$111$	1.15370	0.109504
$112$	0	0
$113$	−5.65726	−0.532190	−0.266095	−	0.963947i	$-0.585733\pi$
$113$	−5.65726	−0.532190	−0.266095	+	0.963947i	$0.585733\pi$
$114$	0	0
$115$	1.75439	0.163598
$116$	0	0
$117$	5.69489i	0.526493i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.99035	0.817305
$122$	0	0
$123$	1.58141i	0.142591i
$124$	0	0
$125$	− 7.31562i	− 0.654329i
$126$	0	0
$127$	11.7616i	1.04367i	0.853046	+	0.521835i	$0.174753\pi$
$127$	11.7616i	1.04367i	−0.853046	+	0.521835i	$0.825247\pi$
$128$	0	0
$129$	10.8193i	0.952587i
$130$	0	0
$131$	20.6203	1.80161	0.900804	−	0.434226i	$-0.142978\pi$
$131$	20.6203	1.80161	0.900804	+	0.434226i	$0.142978\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 2.70012i	− 0.232389i
$136$	0	0
$137$	19.5698	1.67196	0.835979	−	0.548762i	$-0.184901\pi$
$137$	19.5698	1.67196	0.835979	+	0.548762i	$0.184901\pi$
$138$	0	0
$139$	−19.4109	−1.64641	−0.823205	−	0.567745i	$-0.807816\pi$
$139$	−19.4109	−1.64641	−0.823205	+	0.567745i	$0.807816\pi$
$140$	0	0
$141$	−9.62969	−0.810967
$142$	0	0
$143$	−8.07319	−0.675114
$144$	0	0
$145$	− 16.3751i	− 1.35988i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.2112	−1.57385	−0.786923	−	0.617051i	$-0.788327\pi$
$149$	−19.2112	−1.57385	−0.786923	+	0.617051i	$0.788327\pi$
$150$	0	0
$151$	1.67704i	0.136475i	0.997669	+	0.0682376i	$0.0217376\pi$
$151$	1.67704i	0.136475i	−0.997669	+	0.0682376i	$0.978262\pi$
$152$	0	0
$153$	− 1.95966i	− 0.158429i
$154$	0	0
$155$	− 13.4876i	− 1.08335i
$156$	0	0
$157$	− 7.87226i	− 0.628275i	−0.949378	−	0.314137i	$-0.898285\pi$
$157$	− 7.87226i	− 0.628275i	0.949378	−	0.314137i	$-0.101715\pi$
$158$	0	0
$159$	12.5241	0.993225
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.56484i	− 0.279220i	−0.990207	−	0.139610i	$-0.955415\pi$
$163$	− 3.56484i	− 0.279220i	0.990207	−	0.139610i	$-0.0445849\pi$
$164$	0	0
$165$	3.82774	0.297989
$166$	0	0
$167$	10.9594	0.848063	0.424031	−	0.905647i	$-0.360615\pi$
$167$	10.9594	0.848063	0.424031	+	0.905647i	$0.360615\pi$
$168$	0	0
$169$	−19.4318	−1.49475
$170$	0	0
$171$	0.723292	0.0553115
$172$	0	0
$173$	− 18.9366i	− 1.43972i	−0.694118	−	0.719861i	$-0.744205\pi$
$173$	− 18.9366i	− 1.43972i	0.694118	−	0.719861i	$-0.255795\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.868280	−0.0652640
$178$	0	0
$179$	− 5.90622i	− 0.441451i	−0.975336	−	0.220726i	$-0.929157\pi$
$179$	− 5.90622i	− 0.441451i	0.975336	−	0.220726i	$-0.0708425\pi$
$180$	0	0
$181$	− 24.5302i	− 1.82331i	−0.410953	−	0.911656i	$-0.634804\pi$
$181$	− 24.5302i	− 1.82331i	0.410953	−	0.911656i	$-0.365196\pi$
$182$	0	0
$183$	− 13.6101i	− 1.00609i
$184$	0	0
$185$	− 3.11513i	− 0.229029i
$186$	0	0
$187$	2.77805	0.203151
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.88475i	0.570520i	0.958450	+	0.285260i	$0.0920800\pi$
$191$	7.88475i	0.570520i	−0.958450	+	0.285260i	$0.907920\pi$
$192$	0	0
$193$	−10.7493	−0.773750	−0.386875	−	0.922132i	$-0.626446\pi$
$193$	−10.7493	−0.773750	−0.386875	+	0.922132i	$0.626446\pi$
$194$	0	0
$195$	15.3769	1.10116
$196$	0	0
$197$	6.22489	0.443505	0.221753	−	0.975103i	$-0.428822\pi$
$197$	6.22489	0.443505	0.221753	+	0.975103i	$0.428822\pi$
$198$	0	0
$199$	4.28223	0.303559	0.151780	−	0.988414i	$-0.451500\pi$
$199$	4.28223	0.303559	0.151780	+	0.988414i	$0.451500\pi$
$200$	0	0
$201$	0.0530875i	0.00374450i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.26999	0.298229
$206$	0	0
$207$	0.649746i	0.0451605i
$208$	0	0
$209$	1.02535i	0.0709251i
$210$	0	0
$211$	− 17.2858i	− 1.19001i	−0.803723	−	0.595003i	$-0.797151\pi$
$211$	− 17.2858i	− 1.19001i	0.803723	−	0.595003i	$-0.202849\pi$
$212$	0	0
$213$	− 14.0593i	− 0.963328i
$214$	0	0
$215$	29.2134	1.99234
$216$	0	0
$217$	0	0
$218$	0	0
$219$	15.6484i	1.05742i
$220$	0	0
$221$	11.1600	0.750705
$222$	0	0
$223$	4.40635	0.295071	0.147535	−	0.989057i	$-0.452866\pi$
$223$	4.40635	0.295071	0.147535	+	0.989057i	$0.452866\pi$
$224$	0	0
$225$	−2.29063	−0.152709
$226$	0	0
$227$	6.22077	0.412887	0.206444	−	0.978458i	$-0.433811\pi$
$227$	6.22077	0.412887	0.206444	+	0.978458i	$0.433811\pi$
$228$	0	0
$229$	− 11.5332i	− 0.762138i	−0.924547	−	0.381069i	$-0.875556\pi$
$229$	− 11.5332i	− 0.762138i	0.924547	−	0.381069i	$-0.124444\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.27693	0.607752	0.303876	−	0.952712i	$-0.401719\pi$
$233$	9.27693	0.607752	0.303876	+	0.952712i	$0.401719\pi$
$234$	0	0
$235$	26.0013i	1.69614i
$236$	0	0
$237$	11.4936i	0.746587i
$238$	0	0
$239$	− 3.55540i	− 0.229980i	−0.993367	−	0.114990i	$-0.963316\pi$
$239$	− 3.55540i	− 0.229980i	0.993367	−	0.114990i	$-0.0366835\pi$
$240$	0	0
$241$	8.59044i	0.553359i	0.960962	+	0.276679i	$0.0892340\pi$
$241$	8.59044i	0.553359i	−0.960962	+	0.276679i	$0.910766\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.11907i	0.262090i
$248$	0	0
$249$	2.95860	0.187494
$250$	0	0
$251$	10.6135	0.669918	0.334959	−	0.942233i	$-0.391277\pi$
$251$	10.6135	0.669918	0.334959	+	0.942233i	$0.391277\pi$
$252$	0	0
$253$	−0.921093	−0.0579086
$254$	0	0
$255$	−5.29130	−0.331354
$256$	0	0
$257$	− 15.8261i	− 0.987203i	−0.869688	−	0.493601i	$-0.835680\pi$
$257$	− 15.8261i	− 0.987203i	0.869688	−	0.493601i	$-0.164320\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.06459	0.375389
$262$	0	0
$263$	0.437527i	0.0269790i	0.999909	+	0.0134895i	$0.00429398\pi$
$263$	0.437527i	0.0269790i	−0.999909	+	0.0134895i	$0.995706\pi$
$264$	0	0
$265$	− 33.8165i	− 2.07733i
$266$	0	0
$267$	8.24185i	0.504393i
$268$	0	0
$269$	0.496114i	0.0302486i	0.999886	+	0.0151243i	$0.00481440\pi$
$269$	0.496114i	0.0302486i	−0.999886	+	0.0151243i	$0.995186\pi$
$270$	0	0
$271$	−3.37679	−0.205125	−0.102563	−	0.994727i	$-0.532704\pi$
$271$	−3.37679	−0.205125	−0.102563	+	0.994727i	$0.532704\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.24724i	− 0.195816i
$276$	0	0
$277$	10.9336	0.656934	0.328467	−	0.944515i	$-0.393468\pi$
$277$	10.9336	0.656934	0.328467	+	0.944515i	$0.393468\pi$
$278$	0	0
$279$	4.99520	0.299055
$280$	0	0
$281$	−26.4932	−1.58045	−0.790226	−	0.612815i	$-0.790037\pi$
$281$	−26.4932	−1.58045	−0.790226	+	0.612815i	$0.790037\pi$
$282$	0	0
$283$	2.17861	0.129505	0.0647524	−	0.997901i	$-0.479374\pi$
$283$	2.17861	0.129505	0.0647524	+	0.997901i	$0.479374\pi$
$284$	0	0
$285$	− 1.95297i	− 0.115684i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.1598	0.774103
$290$	0	0
$291$	− 1.01880i	− 0.0597231i
$292$	0	0
$293$	− 6.82696i	− 0.398835i	−0.979915	−	0.199418i	$-0.936095\pi$
$293$	− 6.82696i	− 0.398835i	0.979915	−	0.199418i	$-0.0639050\pi$
$294$	0	0
$295$	2.34446i	0.136500i
$296$	0	0
$297$	1.41762i	0.0822586i
$298$	0	0
$299$	−3.70023	−0.213990
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 11.9466i	− 0.686312i
$304$	0	0
$305$	−36.7489	−2.10423
$306$	0	0
$307$	−12.9855	−0.741122	−0.370561	−	0.928808i	$-0.620835\pi$
$307$	−12.9855	−0.741122	−0.370561	+	0.928808i	$0.620835\pi$
$308$	0	0
$309$	17.5903	1.00068
$310$	0	0
$311$	−32.7636	−1.85786	−0.928928	−	0.370261i	$-0.879268\pi$
$311$	−32.7636	−1.85786	−0.928928	+	0.370261i	$0.879268\pi$
$312$	0	0
$313$	− 9.15065i	− 0.517225i	−0.965981	−	0.258613i	$-0.916735\pi$
$313$	− 9.15065i	− 0.517225i	0.965981	−	0.258613i	$-0.0832653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.5872	1.38096	0.690479	−	0.723353i	$-0.257400\pi$
$317$	24.5872	1.38096	0.690479	+	0.723353i	$0.257400\pi$
$318$	0	0
$319$	8.59728i	0.481355i
$320$	0	0
$321$	14.5752i	0.813506i
$322$	0	0
$323$	− 1.41740i	− 0.0788664i
$324$	0	0
$325$	− 13.0449i	− 0.723600i
$326$	0	0
$327$	5.41674	0.299546
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.2118i	1.11094i	0.831537	+	0.555470i	$0.187461\pi$
$331$	20.2118i	1.11094i	−0.831537	+	0.555470i	$0.812539\pi$
$332$	0	0
$333$	1.15370	0.0632224
$334$	0	0
$335$	0.143342	0.00783163
$336$	0	0
$337$	−15.4425	−0.841209	−0.420605	−	0.907244i	$-0.638182\pi$
$337$	−15.4425	−0.841209	−0.420605	+	0.907244i	$0.638182\pi$
$338$	0	0
$339$	−5.65726	−0.307260
$340$	0	0
$341$	7.08129i	0.383474i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.75439	0.0944532
$346$	0	0
$347$	22.9682i	1.23300i	0.787356	+	0.616498i	$0.211449\pi$
$347$	22.9682i	1.23300i	−0.787356	+	0.616498i	$0.788551\pi$
$348$	0	0
$349$	− 2.24157i	− 0.119988i	−0.998199	−	0.0599941i	$-0.980892\pi$
$349$	− 2.24157i	− 0.119988i	0.998199	−	0.0599941i	$-0.0191082\pi$
$350$	0	0
$351$	5.69489i	0.303971i
$352$	0	0
$353$	− 12.7735i	− 0.679863i	−0.940450	−	0.339932i	$-0.889596\pi$
$353$	− 12.7735i	− 0.679863i	0.940450	−	0.339932i	$-0.110404\pi$
$354$	0	0
$355$	−37.9618	−2.01480
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.4512i	1.18493i	0.805597	+	0.592464i	$0.201845\pi$
$359$	22.4512i	1.18493i	−0.805597	+	0.592464i	$0.798155\pi$
$360$	0	0
$361$	−18.4768	−0.972466
$362$	0	0
$363$	8.99035	0.471871
$364$	0	0
$365$	42.2526	2.21160
$366$	0	0
$367$	3.97020	0.207243	0.103621	−	0.994617i	$-0.466957\pi$
$367$	3.97020	0.207243	0.103621	+	0.994617i	$0.466957\pi$
$368$	0	0
$369$	1.58141i	0.0823249i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.0774	0.936014	0.468007	−	0.883725i	$-0.344972\pi$
$373$	18.0774	0.936014	0.468007	+	0.883725i	$0.344972\pi$
$374$	0	0
$375$	− 7.31562i	− 0.377777i
$376$	0	0
$377$	34.5372i	1.77876i
$378$	0	0
$379$	− 9.69206i	− 0.497848i	−0.968523	−	0.248924i	$-0.919923\pi$
$379$	− 9.69206i	− 0.497848i	0.968523	−	0.248924i	$-0.0800769\pi$
$380$	0	0
$381$	11.7616i	0.602564i
$382$	0	0
$383$	−13.8602	−0.708222	−0.354111	−	0.935203i	$-0.615217\pi$
$383$	−13.8602	−0.708222	−0.354111	+	0.935203i	$0.615217\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.8193i	0.549976i
$388$	0	0
$389$	19.3052	0.978814	0.489407	−	0.872055i	$-0.337213\pi$
$389$	19.3052	0.978814	0.489407	+	0.872055i	$0.337213\pi$
$390$	0	0
$391$	1.27328	0.0643925
$392$	0	0
$393$	20.6203	1.04016
$394$	0	0
$395$	31.0339	1.56149
$396$	0	0
$397$	0.531536i	0.0266770i	0.999911	+	0.0133385i	$0.00424591\pi$
$397$	0.531536i	0.0266770i	−0.999911	+	0.0133385i	$0.995754\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.749230	−0.0374147	−0.0187074	−	0.999825i	$-0.505955\pi$
$401$	−0.749230	−0.0374147	−0.0187074	+	0.999825i	$0.505955\pi$
$402$	0	0
$403$	28.4471i	1.41705i
$404$	0	0
$405$	− 2.70012i	− 0.134170i
$406$	0	0
$407$	1.63551i	0.0810692i
$408$	0	0
$409$	10.0341i	0.496156i	0.968740	+	0.248078i	$0.0797990\pi$
$409$	10.0341i	0.496156i	−0.968740	+	0.248078i	$0.920201\pi$
$410$	0	0
$411$	19.5698	0.965305
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 7.98858i	− 0.392144i
$416$	0	0
$417$	−19.4109	−0.950555
$418$	0	0
$419$	2.20727	0.107832	0.0539160	−	0.998545i	$-0.482830\pi$
$419$	2.20727	0.107832	0.0539160	+	0.998545i	$0.482830\pi$
$420$	0	0
$421$	−21.3916	−1.04256	−0.521280	−	0.853386i	$-0.674545\pi$
$421$	−21.3916	−1.04256	−0.521280	+	0.853386i	$0.674545\pi$
$422$	0	0
$423$	−9.62969	−0.468212
$424$	0	0
$425$	4.48884i	0.217741i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8.07319	−0.389777
$430$	0	0
$431$	32.7031i	1.57525i	0.616153	+	0.787626i	$0.288690\pi$
$431$	32.7031i	1.57525i	−0.616153	+	0.787626i	$0.711310\pi$
$432$	0	0
$433$	3.18695i	0.153155i	0.997064	+	0.0765775i	$0.0243993\pi$
$433$	3.18695i	0.153155i	−0.997064	+	0.0765775i	$0.975601\pi$
$434$	0	0
$435$	− 16.3751i	− 0.785126i
$436$	0	0
$437$	0.469956i	0.0224810i
$438$	0	0
$439$	−10.6702	−0.509262	−0.254631	−	0.967038i	$-0.581954\pi$
$439$	−10.6702	−0.509262	−0.254631	+	0.967038i	$0.581954\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 26.2559i	− 1.24746i	−0.781641	−	0.623729i	$-0.785617\pi$
$443$	− 26.2559i	− 1.24746i	0.781641	−	0.623729i	$-0.214383\pi$
$444$	0	0
$445$	22.2539	1.05494
$446$	0	0
$447$	−19.2112	−0.908661
$448$	0	0
$449$	10.3240	0.487220	0.243610	−	0.969873i	$-0.421668\pi$
$449$	10.3240	0.487220	0.243610	+	0.969873i	$0.421668\pi$
$450$	0	0
$451$	−2.24184	−0.105564
$452$	0	0
$453$	1.67704i	0.0787940i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.2002	−1.17881	−0.589407	−	0.807836i	$-0.700638\pi$
$457$	−25.2002	−1.17881	−0.589407	+	0.807836i	$0.700638\pi$
$458$	0	0
$459$	− 1.95966i	− 0.0914689i
$460$	0	0
$461$	20.7224i	0.965140i	0.875857	+	0.482570i	$0.160297\pi$
$461$	20.7224i	0.965140i	−0.875857	+	0.482570i	$0.839703\pi$
$462$	0	0
$463$	39.6594i	1.84313i	0.388227	+	0.921564i	$0.373088\pi$
$463$	39.6594i	1.84313i	−0.388227	+	0.921564i	$0.626912\pi$
$464$	0	0
$465$	− 13.4876i	− 0.625474i
$466$	0	0
$467$	33.1962	1.53614	0.768068	−	0.640369i	$-0.221219\pi$
$467$	33.1962	1.53614	0.768068	+	0.640369i	$0.221219\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 7.87226i	− 0.362735i
$472$	0	0
$473$	−15.3377	−0.705227
$474$	0	0
$475$	−1.65679	−0.0760189
$476$	0	0
$477$	12.5241	0.573439
$478$	0	0
$479$	18.1873	0.831000	0.415500	−	0.909593i	$-0.363607\pi$
$479$	18.1873	0.831000	0.415500	+	0.909593i	$0.363607\pi$
$480$	0	0
$481$	6.57020i	0.299575i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.75088	−0.124911
$486$	0	0
$487$	23.9614i	1.08579i	0.839799	+	0.542897i	$0.182673\pi$
$487$	23.9614i	1.08579i	−0.839799	+	0.542897i	$0.817327\pi$
$488$	0	0
$489$	− 3.56484i	− 0.161208i
$490$	0	0
$491$	11.5449i	0.521016i	0.965472	+	0.260508i	$0.0838900\pi$
$491$	11.5449i	0.521016i	−0.965472	+	0.260508i	$0.916110\pi$
$492$	0	0
$493$	− 11.8845i	− 0.535251i
$494$	0	0
$495$	3.82774	0.172044
$496$	0	0
$497$	0	0
$498$	0	0
$499$	22.1819i	0.993000i	0.868037	+	0.496500i	$0.165382\pi$
$499$	22.1819i	0.993000i	−0.868037	+	0.496500i	$0.834618\pi$
$500$	0	0
$501$	10.9594	0.489629
$502$	0	0
$503$	1.53547	0.0684631	0.0342316	−	0.999414i	$-0.489102\pi$
$503$	1.53547	0.0684631	0.0342316	+	0.999414i	$0.489102\pi$
$504$	0	0
$505$	−32.2571	−1.43542
$506$	0	0
$507$	−19.4318	−0.862997
$508$	0	0
$509$	9.70502i	0.430167i	0.976596	+	0.215084i	$0.0690024\pi$
$509$	9.70502i	0.430167i	−0.976596	+	0.215084i	$0.930998\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.723292	0.0319341
$514$	0	0
$515$	− 47.4958i	− 2.09291i
$516$	0	0
$517$	− 13.6512i	− 0.600381i
$518$	0	0
$519$	− 18.9366i	− 0.831224i
$520$	0	0
$521$	26.6360i	1.16695i	0.812133	+	0.583473i	$0.198306\pi$
$521$	26.6360i	1.16695i	−0.812133	+	0.583473i	$0.801694\pi$
$522$	0	0
$523$	−9.48803	−0.414882	−0.207441	−	0.978247i	$-0.566514\pi$
$523$	−9.48803	−0.414882	−0.207441	+	0.978247i	$0.566514\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.78887i	− 0.426410i
$528$	0	0
$529$	22.5778	0.981645
$530$	0	0
$531$	−0.868280	−0.0376802
$532$	0	0
$533$	−9.00595	−0.390091
$534$	0	0
$535$	39.3546	1.70145
$536$	0	0
$537$	− 5.90622i	− 0.254872i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−27.0579	−1.16331	−0.581654	−	0.813436i	$-0.697594\pi$
$541$	−27.0579	−1.16331	−0.581654	+	0.813436i	$0.697594\pi$
$542$	0	0
$543$	− 24.5302i	− 1.05269i
$544$	0	0
$545$	− 14.6258i	− 0.626501i
$546$	0	0
$547$	32.3241i	1.38208i	0.722818	+	0.691039i	$0.242847\pi$
$547$	32.3241i	1.38208i	−0.722818	+	0.691039i	$0.757153\pi$
$548$	0	0
$549$	− 13.6101i	− 0.580865i
$550$	0	0
$551$	4.38647	0.186870
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 3.11513i	− 0.132230i
$556$	0	0
$557$	−4.13372	−0.175151	−0.0875757	−	0.996158i	$-0.527912\pi$
$557$	−4.13372	−0.175151	−0.0875757	+	0.996158i	$0.527912\pi$
$558$	0	0
$559$	−61.6148	−2.60603
$560$	0	0
$561$	2.77805	0.117289
$562$	0	0
$563$	−32.1077	−1.35318	−0.676589	−	0.736361i	$-0.736542\pi$
$563$	−32.1077	−1.35318	−0.676589	+	0.736361i	$0.736542\pi$
$564$	0	0
$565$	15.2753i	0.642635i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.83749	0.370487	0.185243	−	0.982693i	$-0.440693\pi$
$569$	8.83749	0.370487	0.185243	+	0.982693i	$0.440693\pi$
$570$	0	0
$571$	− 26.6970i	− 1.11724i	−0.829425	−	0.558618i	$-0.811332\pi$
$571$	− 26.6970i	− 1.11724i	0.829425	−	0.558618i	$-0.188668\pi$
$572$	0	0
$573$	7.88475i	0.329390i
$574$	0	0
$575$	− 1.48833i	− 0.0620675i
$576$	0	0
$577$	− 20.6269i	− 0.858710i	−0.903136	−	0.429355i	$-0.858741\pi$
$577$	− 20.6269i	− 0.858710i	0.903136	−	0.429355i	$-0.141259\pi$
$578$	0	0
$579$	−10.7493	−0.446725
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.7544i	0.735312i
$584$	0	0
$585$	15.3769	0.635755
$586$	0	0
$587$	−30.8831	−1.27468	−0.637341	−	0.770582i	$-0.719966\pi$
$587$	−30.8831	−1.27468	−0.637341	+	0.770582i	$0.719966\pi$
$588$	0	0
$589$	3.61299	0.148871
$590$	0	0
$591$	6.22489	0.256058
$592$	0	0
$593$	30.6091i	1.25697i	0.777823	+	0.628483i	$0.216324\pi$
$593$	30.6091i	1.25697i	−0.777823	+	0.628483i	$0.783676\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.28223	0.175260
$598$	0	0
$599$	− 2.72452i	− 0.111321i	−0.998450	−	0.0556605i	$-0.982274\pi$
$599$	− 2.72452i	− 0.111321i	0.998450	−	0.0556605i	$-0.0177264\pi$
$600$	0	0
$601$	28.9310i	1.18012i	0.807360	+	0.590059i	$0.200896\pi$
$601$	28.9310i	1.18012i	−0.807360	+	0.590059i	$0.799104\pi$
$602$	0	0
$603$	0.0530875i	0.00216189i
$604$	0	0
$605$	− 24.2750i	− 0.986919i
$606$	0	0
$607$	−33.3046	−1.35179	−0.675896	−	0.736997i	$-0.736243\pi$
$607$	−33.3046	−1.35179	−0.675896	+	0.736997i	$0.736243\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 54.8401i	− 2.21859i
$612$	0	0
$613$	−1.85667	−0.0749902	−0.0374951	−	0.999297i	$-0.511938\pi$
$613$	−1.85667	−0.0749902	−0.0374951	+	0.999297i	$0.511938\pi$
$614$	0	0
$615$	4.26999	0.172183
$616$	0	0
$617$	27.4264	1.10415	0.552073	−	0.833796i	$-0.313837\pi$
$617$	27.4264	1.10415	0.552073	+	0.833796i	$0.313837\pi$
$618$	0	0
$619$	2.20947	0.0888061	0.0444031	−	0.999014i	$-0.485861\pi$
$619$	2.20947	0.0888061	0.0444031	+	0.999014i	$0.485861\pi$
$620$	0	0
$621$	0.649746i	0.0260734i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.2062	−1.24825
$626$	0	0
$627$	1.02535i	0.0409486i
$628$	0	0
$629$	− 2.26086i	− 0.0901462i
$630$	0	0
$631$	− 19.7387i	− 0.785784i	−0.919585	−	0.392892i	$-0.871475\pi$
$631$	− 19.7387i	− 0.785784i	0.919585	−	0.392892i	$-0.128525\pi$
$632$	0	0
$633$	− 17.2858i	− 0.687051i
$634$	0	0
$635$	31.7576	1.26026
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 14.0593i	− 0.556177i
$640$	0	0
$641$	−18.9726	−0.749372	−0.374686	−	0.927152i	$-0.622249\pi$
$641$	−18.9726	−0.749372	−0.374686	+	0.927152i	$0.622249\pi$
$642$	0	0
$643$	17.5988	0.694027	0.347013	−	0.937860i	$-0.387196\pi$
$643$	17.5988	0.694027	0.347013	+	0.937860i	$0.387196\pi$
$644$	0	0
$645$	29.2134	1.15028
$646$	0	0
$647$	6.03295	0.237180	0.118590	−	0.992943i	$-0.462163\pi$
$647$	6.03295	0.237180	0.118590	+	0.992943i	$0.462163\pi$
$648$	0	0
$649$	− 1.23089i	− 0.0483167i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.59067	−0.0622478	−0.0311239	−	0.999516i	$-0.509909\pi$
$653$	−1.59067	−0.0622478	−0.0311239	+	0.999516i	$0.509909\pi$
$654$	0	0
$655$	− 55.6773i	− 2.17549i
$656$	0	0
$657$	15.6484i	0.610504i
$658$	0	0
$659$	8.83213i	0.344051i	0.985093	+	0.172025i	$0.0550311\pi$
$659$	8.83213i	0.344051i	−0.985093	+	0.172025i	$0.944969\pi$
$660$	0	0
$661$	− 1.46020i	− 0.0567953i	−0.999597	−	0.0283976i	$-0.990960\pi$
$661$	− 1.46020i	− 0.0567953i	0.999597	−	0.0283976i	$-0.00904047\pi$
$662$	0	0
$663$	11.1600	0.433419
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.94044i	0.152575i
$668$	0	0
$669$	4.40635	0.170359
$670$	0	0
$671$	19.2939	0.744835
$672$	0	0
$673$	−31.7279	−1.22302	−0.611510	−	0.791237i	$-0.709438\pi$
$673$	−31.7279	−1.22302	−0.611510	+	0.791237i	$0.709438\pi$
$674$	0	0
$675$	−2.29063	−0.0881664
$676$	0	0
$677$	6.31504i	0.242707i	0.992609	+	0.121353i	$0.0387234\pi$
$677$	6.31504i	0.242707i	−0.992609	+	0.121353i	$0.961277\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6.22077	0.238381
$682$	0	0
$683$	3.46801i	0.132700i	0.997796	+	0.0663499i	$0.0211354\pi$
$683$	3.46801i	0.132700i	−0.997796	+	0.0663499i	$0.978865\pi$
$684$	0	0
$685$	− 52.8406i	− 2.01894i
$686$	0	0
$687$	− 11.5332i	− 0.440020i
$688$	0	0
$689$	71.3234i	2.71720i
$690$	0	0
$691$	−8.57217	−0.326101	−0.163050	−	0.986618i	$-0.552133\pi$
$691$	−8.57217	−0.326101	−0.163050	+	0.986618i	$0.552133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	52.4116i	1.98809i
$696$	0	0
$697$	3.09902	0.117384
$698$	0	0
$699$	9.27693	0.350886
$700$	0	0
$701$	−6.08683	−0.229896	−0.114948	−	0.993371i	$-0.536670\pi$
$701$	−6.08683	−0.229896	−0.114948	+	0.993371i	$0.536670\pi$
$702$	0	0
$703$	0.834462	0.0314723
$704$	0	0
$705$	26.0013i	0.979266i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.8567	−1.38418	−0.692092	−	0.721810i	$-0.743311\pi$
$709$	−36.8567	−1.38418	−0.692092	+	0.721810i	$0.743311\pi$
$710$	0	0
$711$	11.4936i	0.431042i
$712$	0	0
$713$	3.24561i	0.121549i
$714$	0	0
$715$	21.7986i	0.815220i
$716$	0	0
$717$	− 3.55540i	− 0.132779i
$718$	0	0
$719$	−45.5835	−1.69998	−0.849988	−	0.526802i	$-0.823391\pi$
$719$	−45.5835	−1.69998	−0.849988	+	0.526802i	$0.823391\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	8.59044i	0.319482i
$724$	0	0
$725$	−13.8917	−0.515926
$726$	0	0
$727$	9.02124	0.334579	0.167290	−	0.985908i	$-0.446498\pi$
$727$	9.02124	0.334579	0.167290	+	0.985908i	$0.446498\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	21.2021	0.784188
$732$	0	0
$733$	− 31.0705i	− 1.14762i	−0.818990	−	0.573808i	$-0.805465\pi$
$733$	− 31.0705i	− 1.14762i	0.818990	−	0.573808i	$-0.194535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.0752578	−0.00277216
$738$	0	0
$739$	− 37.7370i	− 1.38818i	−0.719890	−	0.694089i	$-0.755808\pi$
$739$	− 37.7370i	− 1.38818i	0.719890	−	0.694089i	$-0.244192\pi$
$740$	0	0
$741$	4.11907i	0.151318i
$742$	0	0
$743$	43.6664i	1.60197i	0.598687	+	0.800983i	$0.295689\pi$
$743$	43.6664i	1.60197i	−0.598687	+	0.800983i	$0.704311\pi$
$744$	0	0
$745$	51.8726i	1.90047i
$746$	0	0
$747$	2.95860	0.108250
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 46.8611i	− 1.70999i	−0.518639	−	0.854993i	$-0.673561\pi$
$751$	− 46.8611i	− 1.70999i	0.518639	−	0.854993i	$-0.326439\pi$
$752$	0	0
$753$	10.6135	0.386778
$754$	0	0
$755$	4.52819	0.164798
$756$	0	0
$757$	20.1824	0.733543	0.366771	−	0.930311i	$-0.380463\pi$
$757$	20.1824	0.733543	0.366771	+	0.930311i	$0.380463\pi$
$758$	0	0
$759$	−0.921093	−0.0334336
$760$	0	0
$761$	4.46892i	0.161998i	0.996714	+	0.0809991i	$0.0258111\pi$
$761$	4.46892i	0.161998i	−0.996714	+	0.0809991i	$0.974189\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5.29130	−0.191307
$766$	0	0
$767$	− 4.94476i	− 0.178545i
$768$	0	0
$769$	− 25.5449i	− 0.921173i	−0.887615	−	0.460587i	$-0.847639\pi$
$769$	− 25.5449i	− 0.921173i	0.887615	−	0.460587i	$-0.152361\pi$
$770$	0	0
$771$	− 15.8261i	− 0.569962i
$772$	0	0
$773$	− 18.1119i	− 0.651442i	−0.945466	−	0.325721i	$-0.894393\pi$
$773$	− 18.1119i	− 0.651442i	0.945466	−	0.325721i	$-0.105607\pi$
$774$	0	0
$775$	−11.4422	−0.411014
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.14382i	0.0409816i
$780$	0	0
$781$	19.9307	0.713178
$782$	0	0
$783$	6.06459	0.216731
$784$	0	0
$785$	−21.2560	−0.758660
$786$	0	0
$787$	−44.6272	−1.59079	−0.795394	−	0.606093i	$-0.792736\pi$
$787$	−44.6272	−1.59079	−0.795394	+	0.606093i	$0.792736\pi$
$788$	0	0
$789$	0.437527i	0.0155764i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	77.5081	2.75239
$794$	0	0
$795$	− 33.8165i	− 1.19935i
$796$	0	0
$797$	24.9941i	0.885335i	0.896686	+	0.442668i	$0.145968\pi$
$797$	24.9941i	0.885335i	−0.896686	+	0.442668i	$0.854032\pi$
$798$	0	0
$799$	18.8709i	0.667604i
$800$	0	0
$801$	8.24185i	0.291211i
$802$	0	0
$803$	−22.1835	−0.782840
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.496114i	0.0174640i
$808$	0	0
$809$	26.6868	0.938258	0.469129	−	0.883130i	$-0.344568\pi$
$809$	26.6868	0.938258	0.469129	+	0.883130i	$0.344568\pi$
$810$	0	0
$811$	−12.1884	−0.427994	−0.213997	−	0.976834i	$-0.568648\pi$
$811$	−12.1884	−0.427994	−0.213997	+	0.976834i	$0.568648\pi$
$812$	0	0
$813$	−3.37679	−0.118429
$814$	0	0
$815$	−9.62549	−0.337166
$816$	0	0
$817$	7.82551i	0.273780i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.0914	0.561594	0.280797	−	0.959767i	$-0.409401\pi$
$821$	16.0914	0.561594	0.280797	+	0.959767i	$0.409401\pi$
$822$	0	0
$823$	18.4743i	0.643973i	0.946744	+	0.321986i	$0.104351\pi$
$823$	18.4743i	0.643973i	−0.946744	+	0.321986i	$0.895649\pi$
$824$	0	0
$825$	− 3.24724i	− 0.113054i
$826$	0	0
$827$	− 50.1954i	− 1.74547i	−0.488198	−	0.872733i	$-0.662346\pi$
$827$	− 50.1954i	− 1.74547i	0.488198	−	0.872733i	$-0.337654\pi$
$828$	0	0
$829$	20.4162i	0.709084i	0.935040	+	0.354542i	$0.115363\pi$
$829$	20.4162i	0.709084i	−0.935040	+	0.354542i	$0.884637\pi$
$830$	0	0
$831$	10.9336	0.379281
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 29.5916i	− 1.02406i
$836$	0	0
$837$	4.99520	0.172659
$838$	0	0
$839$	20.8855	0.721048	0.360524	−	0.932750i	$-0.382598\pi$
$839$	20.8855	0.721048	0.360524	+	0.932750i	$0.382598\pi$
$840$	0	0
$841$	7.77923	0.268249
$842$	0	0
$843$	−26.4932	−0.912475
$844$	0	0
$845$	52.4681i	1.80496i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.17861	0.0747696
$850$	0	0
$851$	0.749613i	0.0256964i
$852$	0	0
$853$	23.6666i	0.810329i	0.914244	+	0.405165i	$0.132786\pi$
$853$	23.6666i	0.810329i	−0.914244	+	0.405165i	$0.867214\pi$
$854$	0	0
$855$	− 1.95297i	− 0.0667902i
$856$	0	0
$857$	− 3.13263i	− 0.107009i	−0.998568	−	0.0535043i	$-0.982961\pi$
$857$	− 3.13263i	− 0.107009i	0.998568	−	0.0535043i	$-0.0170391\pi$
$858$	0	0
$859$	18.3010	0.624423	0.312211	−	0.950013i	$-0.398930\pi$
$859$	18.3010	0.624423	0.312211	+	0.950013i	$0.398930\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.92199i	− 0.167546i	−0.996485	−	0.0837732i	$-0.973303\pi$
$863$	− 4.92199i	− 0.167546i	0.996485	−	0.0837732i	$-0.0266971\pi$
$864$	0	0
$865$	−51.1310	−1.73851
$866$	0	0
$867$	13.1598	0.446929
$868$	0	0
$869$	−16.2935	−0.552719
$870$	0	0
$871$	−0.302327	−0.0102440
$872$	0	0
$873$	− 1.01880i	− 0.0344812i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−39.7764	−1.34316	−0.671578	−	0.740934i	$-0.734383\pi$
$877$	−39.7764	−1.34316	−0.671578	+	0.740934i	$0.734383\pi$
$878$	0	0
$879$	− 6.82696i	− 0.230268i
$880$	0	0
$881$	− 21.2750i	− 0.716774i	−0.933573	−	0.358387i	$-0.883327\pi$
$881$	− 21.2750i	− 0.716774i	0.933573	−	0.358387i	$-0.116673\pi$
$882$	0	0
$883$	22.6054i	0.760733i	0.924836	+	0.380366i	$0.124202\pi$
$883$	22.6054i	0.760733i	−0.924836	+	0.380366i	$0.875798\pi$
$884$	0	0
$885$	2.34446i	0.0788081i
$886$	0	0
$887$	−31.6324	−1.06211	−0.531057	−	0.847336i	$-0.678205\pi$
$887$	−31.6324	−1.06211	−0.531057	+	0.847336i	$0.678205\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.41762i	0.0474920i
$892$	0	0
$893$	−6.96508	−0.233077
$894$	0	0
$895$	−15.9475	−0.533065
$896$	0	0
$897$	−3.70023	−0.123547
$898$	0	0
$899$	30.2938	1.01036
$900$	0	0
$901$	− 24.5429i	− 0.817643i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−66.2343	−2.20170
$906$	0	0
$907$	17.9573i	0.596264i	0.954525	+	0.298132i	$0.0963635\pi$
$907$	17.9573i	0.596264i	−0.954525	+	0.298132i	$0.903636\pi$
$908$	0	0
$909$	− 11.9466i	− 0.396242i
$910$	0	0
$911$	− 38.5199i	− 1.27622i	−0.769945	−	0.638110i	$-0.779716\pi$
$911$	− 38.5199i	− 1.27622i	0.769945	−	0.638110i	$-0.220284\pi$
$912$	0	0
$913$	4.19418i	0.138807i
$914$	0	0
$915$	−36.7489	−1.21488
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 3.77085i	− 0.124389i	−0.998064	−	0.0621944i	$-0.980190\pi$
$919$	− 3.77085i	− 0.124389i	0.998064	−	0.0621944i	$-0.0198099\pi$
$920$	0	0
$921$	−12.9855	−0.427887
$922$	0	0
$923$	80.0662	2.63541
$924$	0	0
$925$	−2.64270	−0.0868915
$926$	0	0
$927$	17.5903	0.577740
$928$	0	0
$929$	11.0167i	0.361448i	0.983534	+	0.180724i	$0.0578440\pi$
$929$	11.0167i	0.361448i	−0.983534	+	0.180724i	$0.942156\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−32.7636	−1.07263
$934$	0	0
$935$	− 7.50105i	− 0.245310i
$936$	0	0
$937$	− 5.09549i	− 0.166462i	−0.996530	−	0.0832312i	$-0.973476\pi$
$937$	− 5.09549i	− 0.166462i	0.996530	−	0.0832312i	$-0.0265240\pi$
$938$	0	0
$939$	− 9.15065i	− 0.298620i
$940$	0	0
$941$	10.3499i	0.337397i	0.985668	+	0.168698i	$0.0539563\pi$
$941$	10.3499i	0.337397i	−0.985668	+	0.168698i	$0.946044\pi$
$942$	0	0
$943$	−1.02751	−0.0334605
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 46.8596i	− 1.52273i	−0.648321	−	0.761367i	$-0.724529\pi$
$947$	− 46.8596i	− 1.52273i	0.648321	−	0.761367i	$-0.275471\pi$
$948$	0	0
$949$	−89.1162	−2.89284
$950$	0	0
$951$	24.5872	0.797296
$952$	0	0
$953$	41.3560	1.33965	0.669825	−	0.742519i	$-0.266369\pi$
$953$	41.3560	1.33965	0.669825	+	0.742519i	$0.266369\pi$
$954$	0	0
$955$	21.2897	0.688920
$956$	0	0
$957$	8.59728i	0.277911i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.04797	−0.195096
$962$	0	0
$963$	14.5752i	0.469678i
$964$	0	0
$965$	29.0243i	0.934326i
$966$	0	0
$967$	− 23.6169i	− 0.759467i	−0.925096	−	0.379733i	$-0.876016\pi$
$967$	− 23.6169i	− 0.759467i	0.925096	−	0.379733i	$-0.123984\pi$
$968$	0	0
$969$	− 1.41740i	− 0.0455335i
$970$	0	0
$971$	46.3328	1.48689	0.743446	−	0.668796i	$-0.233190\pi$
$971$	46.3328	1.48689	0.743446	+	0.668796i	$0.233190\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 13.0449i	− 0.417771i
$976$	0	0
$977$	−12.3404	−0.394804	−0.197402	−	0.980323i	$-0.563250\pi$
$977$	−12.3404	−0.394804	−0.197402	+	0.980323i	$0.563250\pi$
$978$	0	0
$979$	−11.6838	−0.373416
$980$	0	0
$981$	5.41674	0.172943
$982$	0	0
$983$	−19.1901	−0.612069	−0.306034	−	0.952020i	$-0.599002\pi$
$983$	−19.1901	−0.612069	−0.306034	+	0.952020i	$0.599002\pi$
$984$	0	0
$985$	− 16.8079i	− 0.535545i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.02980	−0.223535
$990$	0	0
$991$	− 9.99448i	− 0.317485i	−0.987320	−	0.158743i	$-0.949256\pi$
$991$	− 9.99448i	− 0.317485i	0.987320	−	0.158743i	$-0.0507440\pi$
$992$	0	0
$993$	20.2118i	0.641401i
$994$	0	0
$995$	− 11.5625i	− 0.366556i
$996$	0	0
$997$	38.5449i	1.22073i	0.792120	+	0.610365i	$0.208977\pi$
$997$	38.5449i	1.22073i	−0.792120	+	0.610365i	$0.791023\pi$
$998$	0	0
$999$	1.15370	0.0365015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.b.e.1567.3		16
4.3	odd	2		4704.2.b.d.1567.3		16
7.4	even	3		672.2.bl.a.607.7	yes	16
7.5	odd	6		672.2.bl.b.31.7	yes	16
7.6	odd	2		4704.2.b.d.1567.14		16
21.5	even	6		2016.2.cs.c.703.2		16
21.11	odd	6		2016.2.cs.a.1279.2		16
28.11	odd	6		672.2.bl.b.607.7	yes	16
28.19	even	6		672.2.bl.a.31.7	✓	16
28.27	even	2	inner	4704.2.b.e.1567.14		16
56.5	odd	6		1344.2.bl.k.703.2		16
56.11	odd	6		1344.2.bl.k.1279.2		16
56.19	even	6		1344.2.bl.l.703.2		16
56.53	even	6		1344.2.bl.l.1279.2		16
84.11	even	6		2016.2.cs.c.1279.2		16
84.47	odd	6		2016.2.cs.a.703.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bl.a.31.7	✓	16	28.19	even	6
672.2.bl.a.607.7	yes	16	7.4	even	3
672.2.bl.b.31.7	yes	16	7.5	odd	6
672.2.bl.b.607.7	yes	16	28.11	odd	6
1344.2.bl.k.703.2		16	56.5	odd	6
1344.2.bl.k.1279.2		16	56.11	odd	6
1344.2.bl.l.703.2		16	56.19	even	6
1344.2.bl.l.1279.2		16	56.53	even	6
2016.2.cs.a.703.2		16	84.47	odd	6
2016.2.cs.a.1279.2		16	21.11	odd	6
2016.2.cs.c.703.2		16	21.5	even	6
2016.2.cs.c.1279.2		16	84.11	even	6
4704.2.b.d.1567.3		16	4.3	odd	2
4704.2.b.d.1567.14		16	7.6	odd	2
4704.2.b.e.1567.3		16	1.1	even	1	trivial
4704.2.b.e.1567.14		16	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.70012i q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.70012i q^{5} +1.00000 q^{9} +1.41762i q^{11} +5.69489i q^{13} -2.70012i q^{15} -1.95966i q^{17} +0.723292 q^{19} +0.649746i q^{23} -2.29063 q^{25} +1.00000 q^{27} +6.06459 q^{29} +4.99520 q^{31} +1.41762i q^{33} +1.15370 q^{37} +5.69489i q^{39} +1.58141i q^{41} +10.8193i q^{43} -2.70012i q^{45} -9.62969 q^{47} -1.95966i q^{51} +12.5241 q^{53} +3.82774 q^{55} +0.723292 q^{57} -0.868280 q^{59} -13.6101i q^{61} +15.3769 q^{65} +0.0530875i q^{67} +0.649746i q^{69} -14.0593i q^{71} +15.6484i q^{73} -2.29063 q^{75} +11.4936i q^{79} +1.00000 q^{81} +2.95860 q^{83} -5.29130 q^{85} +6.06459 q^{87} +8.24185i q^{89} +4.99520 q^{93} -1.95297i q^{95} -1.01880i q^{97} +1.41762i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 16 * q^9	\( 16 q + 16 q^{3} + 16 q^{9} - 8 q^{19} - 8 q^{25} + 16 q^{27} - 8 q^{31} - 8 q^{37} - 16 q^{47} - 16 q^{53} - 16 q^{55} - 8 q^{57} - 8 q^{59} - 16 q^{65} - 8 q^{75} + 16 q^{81} + 8 q^{83} + 32 q^{85} - 8 q^{93}+O(q^{100}) \) 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^19 - 8 * q^25 + 16 * q^27 - 8 * q^31 - 8 * q^37 - 16 * q^47 - 16 * q^53 - 16 * q^55 - 8 * q^57 - 8 * q^59 - 16 * q^65 - 8 * q^75 + 16 * q^81 + 8 * q^83 + 32 * q^85 - 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more