LMFDB - Embedded newform 648.5.e.b.161.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,5,Mod(161,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	648.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$66.9837360783$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 160x^{10} + 9733x^{8} + 278004x^{6} + 3678300x^{4} + 18632592x^{2} + 25765776 $ x^12 + 160x^10 + 9733x^8 + 278004x^6 + 3678300x^4 + 18632592*x^2 + 25765776
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$1.47426i$ of defining polynomial
Character	$\chi$	$=$	648.161
Dual form			648.5.e.b.161.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-33.0209i q^{5} +51.5500 q^{7} -165.647i q^{11} +307.599 q^{13} -475.873i q^{17} +436.081 q^{19} +604.867i q^{23} -465.380 q^{25} +892.900i q^{29} -665.817 q^{31} -1702.23i q^{35} +1587.03 q^{37} -1534.49i q^{41} +3074.28 q^{43} +812.510i q^{47} +256.399 q^{49} +694.609i q^{53} -5469.80 q^{55} +2494.00i q^{59} +1664.40 q^{61} -10157.2i q^{65} -6935.90 q^{67} +7010.97i q^{71} +4089.49 q^{73} -8539.07i q^{77} +276.871 q^{79} +1397.82i q^{83} -15713.8 q^{85} -4544.24i q^{89} +15856.7 q^{91} -14399.8i q^{95} -9814.76 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 96 q^{7} - 72 q^{13} + 336 q^{19} + 84 q^{25} - 1536 q^{31} - 492 q^{37} + 10128 q^{43} - 6828 q^{49} - 13968 q^{55} + 26268 q^{61} - 20784 q^{67} - 9984 q^{73} + 44592 q^{79} - 45348 q^{85} - 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) $ 12 * q + 96 * q^7 - 72 * q^13 + 336 * q^19 + 84 * q^25 - 1536 * q^31 - 492 * q^37 + 10128 * q^43 - 6828 * q^49 - 13968 * q^55 + 26268 * q^61 - 20784 * q^67 - 9984 * q^73 + 44592 * q^79 - 45348 * q^85 - 11808 * q^91 + 76608 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 33.0209i	− 1.32084i	−0.750898	−	0.660418i	$-0.770379\pi$
$5$	− 33.0209i	− 1.32084i	0.750898	−	0.660418i	$-0.229621\pi$
$6$	0	0
$7$	51.5500	1.05204	0.526020	−	0.850472i	$-0.323684\pi$
$7$	51.5500	1.05204	0.526020	+	0.850472i	$0.323684\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 165.647i	− 1.36898i	−0.729022	−	0.684490i	$-0.760025\pi$
$11$	− 165.647i	− 1.36898i	0.729022	−	0.684490i	$-0.239975\pi$
$12$	0	0
$13$	307.599	1.82011	0.910057	−	0.414484i	$-0.136038\pi$
$13$	307.599	1.82011	0.910057	+	0.414484i	$0.136038\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 475.873i	− 1.64662i	−0.567592	−	0.823310i	$-0.692125\pi$
$17$	− 475.873i	− 1.64662i	0.567592	−	0.823310i	$-0.307875\pi$
$18$	0	0
$19$	436.081	1.20798	0.603991	−	0.796991i	$-0.293576\pi$
$19$	436.081	1.20798	0.603991	+	0.796991i	$0.293576\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	604.867i	1.14342i	0.820457	+	0.571708i	$0.193719\pi$
$23$	604.867i	1.14342i	−0.820457	+	0.571708i	$0.806281\pi$
$24$	0	0
$25$	−465.380	−0.744608
$26$	0	0
$27$	0	0
$28$	0	0
$29$	892.900i	1.06171i	0.847462	+	0.530856i	$0.178130\pi$
$29$	892.900i	1.06171i	−0.847462	+	0.530856i	$0.821870\pi$
$30$	0	0
$31$	−665.817	−0.692838	−0.346419	−	0.938080i	$-0.612602\pi$
$31$	−665.817	−0.692838	−0.346419	+	0.938080i	$0.612602\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1702.23i	− 1.38957i
$36$	0	0
$37$	1587.03	1.15926	0.579630	−	0.814880i	$-0.303197\pi$
$37$	1587.03	1.15926	0.579630	+	0.814880i	$0.303197\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1534.49i	− 0.912846i	−0.889763	−	0.456423i	$-0.849130\pi$
$41$	− 1534.49i	− 0.912846i	0.889763	−	0.456423i	$-0.150870\pi$
$42$	0	0
$43$	3074.28	1.66267	0.831336	−	0.555771i	$-0.187577\pi$
$43$	3074.28	1.66267	0.831336	+	0.555771i	$0.187577\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	812.510i	0.367818i	0.982943	+	0.183909i	$0.0588752\pi$
$47$	812.510i	0.367818i	−0.982943	+	0.183909i	$0.941125\pi$
$48$	0	0
$49$	256.399	0.106788
$50$	0	0
$51$	0	0
$52$	0	0
$53$	694.609i	0.247280i	0.992327	+	0.123640i	$0.0394568\pi$
$53$	694.609i	0.247280i	−0.992327	+	0.123640i	$0.960543\pi$
$54$	0	0
$55$	−5469.80	−1.80820
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2494.00i	0.716461i	0.933633	+	0.358231i	$0.116620\pi$
$59$	2494.00i	0.716461i	−0.933633	+	0.358231i	$0.883380\pi$
$60$	0	0
$61$	1664.40	0.447300	0.223650	−	0.974669i	$-0.428203\pi$
$61$	1664.40	0.447300	0.223650	+	0.974669i	$0.428203\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10157.2i	− 2.40407i
$66$	0	0
$67$	−6935.90	−1.54509	−0.772544	−	0.634961i	$-0.781016\pi$
$67$	−6935.90	−1.54509	−0.772544	+	0.634961i	$0.781016\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7010.97i	1.39079i	0.718628	+	0.695395i	$0.244770\pi$
$71$	7010.97i	1.39079i	−0.718628	+	0.695395i	$0.755230\pi$
$72$	0	0
$73$	4089.49	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$73$	4089.49	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 8539.07i	− 1.44022i
$78$	0	0
$79$	276.871	0.0443632	0.0221816	−	0.999754i	$-0.492939\pi$
$79$	276.871	0.0443632	0.0221816	+	0.999754i	$0.492939\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1397.82i	0.202906i	0.994840	+	0.101453i	$0.0323492\pi$
$83$	1397.82i	0.202906i	−0.994840	+	0.101453i	$0.967651\pi$
$84$	0	0
$85$	−15713.8	−2.17492
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4544.24i	− 0.573696i	−0.957976	−	0.286848i	$-0.907393\pi$
$89$	− 4544.24i	− 0.573696i	0.957976	−	0.286848i	$-0.0926074\pi$
$90$	0	0
$91$	15856.7	1.91483
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 14399.8i	− 1.59555i
$96$	0	0
$97$	−9814.76	−1.04313	−0.521563	−	0.853213i	$-0.674651\pi$
$97$	−9814.76	−1.04313	−0.521563	+	0.853213i	$0.674651\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2566.88i	0.251630i	0.992054	+	0.125815i	$0.0401546\pi$
$101$	2566.88i	0.251630i	−0.992054	+	0.125815i	$0.959845\pi$
$102$	0	0
$103$	−1153.06	−0.108687	−0.0543437	−	0.998522i	$-0.517307\pi$
$103$	−1153.06	−0.108687	−0.0543437	+	0.998522i	$0.517307\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 21258.4i	− 1.85679i	−0.371596	−	0.928394i	$-0.621189\pi$
$107$	− 21258.4i	− 1.85679i	0.371596	−	0.928394i	$-0.378811\pi$
$108$	0	0
$109$	−14999.1	−1.26245	−0.631223	−	0.775602i	$-0.717447\pi$
$109$	−14999.1	−1.26245	−0.631223	+	0.775602i	$0.717447\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 3100.17i	− 0.242789i	−0.992604	−	0.121394i	$-0.961263\pi$
$113$	− 3100.17i	− 0.242789i	0.992604	−	0.121394i	$-0.0387366\pi$
$114$	0	0
$115$	19973.3	1.51027
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 24531.2i	− 1.73231i
$120$	0	0
$121$	−12797.8	−0.874105
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 5270.79i	− 0.337331i
$126$	0	0
$127$	12283.3	0.761565	0.380783	−	0.924665i	$-0.375655\pi$
$127$	12283.3	0.761565	0.380783	+	0.924665i	$0.375655\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1593.24i	0.0928405i	0.998922	+	0.0464203i	$0.0147813\pi$
$131$	1593.24i	0.0928405i	−0.998922	+	0.0464203i	$0.985219\pi$
$132$	0	0
$133$	22480.0	1.27085
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16216.9i	− 0.864024i	−0.901868	−	0.432012i	$-0.857804\pi$
$137$	− 16216.9i	− 0.864024i	0.901868	−	0.432012i	$-0.142196\pi$
$138$	0	0
$139$	−9907.97	−0.512808	−0.256404	−	0.966570i	$-0.582538\pi$
$139$	−9907.97	−0.512808	−0.256404	+	0.966570i	$0.582538\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 50952.7i	− 2.49170i
$144$	0	0
$145$	29484.4	1.40235
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16890.9i	0.760817i	0.924819	+	0.380408i	$0.124217\pi$
$149$	16890.9i	0.760817i	−0.924819	+	0.380408i	$0.875783\pi$
$150$	0	0
$151$	33178.1	1.45512	0.727559	−	0.686045i	$-0.240655\pi$
$151$	33178.1	1.45512	0.727559	+	0.686045i	$0.240655\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	21985.9i	0.915125i
$156$	0	0
$157$	−38922.3	−1.57906	−0.789530	−	0.613711i	$-0.789676\pi$
$157$	−38922.3	−1.57906	−0.789530	+	0.613711i	$0.789676\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	31180.9i	1.20292i
$162$	0	0
$163$	45088.3	1.69703	0.848513	−	0.529174i	$-0.177498\pi$
$163$	45088.3	1.69703	0.848513	+	0.529174i	$0.177498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14912.7i	− 0.534717i	−0.963597	−	0.267359i	$-0.913849\pi$
$167$	− 14912.7i	− 0.534717i	0.963597	−	0.267359i	$-0.0861509\pi$
$168$	0	0
$169$	66056.2	2.31281
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3498.12i	0.116881i	0.998291	+	0.0584403i	$0.0186127\pi$
$173$	3498.12i	0.116881i	−0.998291	+	0.0584403i	$0.981387\pi$
$174$	0	0
$175$	−23990.3	−0.783358
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 675.912i	− 0.0210952i	−0.999944	−	0.0105476i	$-0.996643\pi$
$179$	− 675.912i	− 0.0210952i	0.999944	−	0.0105476i	$-0.00335747\pi$
$180$	0	0
$181$	−34209.3	−1.04421	−0.522104	−	0.852882i	$-0.674853\pi$
$181$	−34209.3	−1.04421	−0.522104	+	0.852882i	$0.674853\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 52405.1i	− 1.53119i
$186$	0	0
$187$	−78826.8	−2.25419
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 3351.90i	− 0.0918806i	−0.998944	−	0.0459403i	$-0.985372\pi$
$191$	− 3351.90i	− 0.0918806i	0.998944	−	0.0459403i	$-0.0146284\pi$
$192$	0	0
$193$	−32775.4	−0.879901	−0.439950	−	0.898022i	$-0.645004\pi$
$193$	−32775.4	−0.879901	−0.439950	+	0.898022i	$0.645004\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3193.55i	0.0822890i	0.999153	+	0.0411445i	$0.0131004\pi$
$197$	3193.55i	0.0822890i	−0.999153	+	0.0411445i	$0.986900\pi$
$198$	0	0
$199$	−28346.2	−0.715796	−0.357898	−	0.933761i	$-0.616506\pi$
$199$	−28346.2	−0.715796	−0.357898	+	0.933761i	$0.616506\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	46029.0i	1.11696i
$204$	0	0
$205$	−50670.4	−1.20572
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 72235.4i	− 1.65370i
$210$	0	0
$211$	−15568.1	−0.349680	−0.174840	−	0.984597i	$-0.555941\pi$
$211$	−15568.1	−0.349680	−0.174840	+	0.984597i	$0.555941\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 101515.i	− 2.19612i
$216$	0	0
$217$	−34322.9	−0.728893
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 146378.i	− 2.99704i
$222$	0	0
$223$	86140.7	1.73220	0.866101	−	0.499869i	$-0.166619\pi$
$223$	86140.7	1.73220	0.866101	+	0.499869i	$0.166619\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 48393.8i	− 0.939156i	−0.882891	−	0.469578i	$-0.844406\pi$
$227$	− 48393.8i	− 0.939156i	0.882891	−	0.469578i	$-0.155594\pi$
$228$	0	0
$229$	−66630.2	−1.27058	−0.635288	−	0.772276i	$-0.719118\pi$
$229$	−66630.2	−1.27058	−0.635288	+	0.772276i	$0.719118\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	85009.8i	1.56587i	0.622101	+	0.782937i	$0.286279\pi$
$233$	85009.8i	1.56587i	−0.622101	+	0.782937i	$0.713721\pi$
$234$	0	0
$235$	26829.8	0.485827
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8528.96i	0.149314i	0.997209	+	0.0746570i	$0.0237862\pi$
$239$	8528.96i	0.149314i	−0.997209	+	0.0746570i	$0.976214\pi$
$240$	0	0
$241$	−88488.1	−1.52353	−0.761764	−	0.647854i	$-0.775667\pi$
$241$	−88488.1	−1.52353	−0.761764	+	0.647854i	$0.775667\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 8466.52i	− 0.141050i
$246$	0	0
$247$	134138.	2.19866
$248$	0	0
$249$	0	0
$250$	0	0
$251$	95847.5i	1.52137i	0.649124	+	0.760683i	$0.275136\pi$
$251$	95847.5i	1.52137i	−0.649124	+	0.760683i	$0.724864\pi$
$252$	0	0
$253$	100194.	1.56531
$254$	0	0
$255$	0	0
$256$	0	0
$257$	47664.4i	0.721652i	0.932633	+	0.360826i	$0.117505\pi$
$257$	47664.4i	0.721652i	−0.932633	+	0.360826i	$0.882495\pi$
$258$	0	0
$259$	81811.2	1.21959
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27496.1i	0.397521i	0.980048	+	0.198760i	$0.0636916\pi$
$263$	27496.1i	0.397521i	−0.980048	+	0.198760i	$0.936308\pi$
$264$	0	0
$265$	22936.6	0.326616
$266$	0	0
$267$	0	0
$268$	0	0
$269$	57584.9i	0.795800i	0.917429	+	0.397900i	$0.130261\pi$
$269$	57584.9i	0.795800i	−0.917429	+	0.397900i	$0.869739\pi$
$270$	0	0
$271$	24915.5	0.339258	0.169629	−	0.985508i	$-0.445743\pi$
$271$	24915.5	0.339258	0.169629	+	0.985508i	$0.445743\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	77088.6i	1.01935i
$276$	0	0
$277$	−136189.	−1.77494	−0.887469	−	0.460867i	$-0.847539\pi$
$277$	−136189.	−1.77494	−0.887469	+	0.460867i	$0.847539\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	67597.1i	0.856082i	0.903759	+	0.428041i	$0.140796\pi$
$281$	67597.1i	0.856082i	−0.903759	+	0.428041i	$0.859204\pi$
$282$	0	0
$283$	−122973.	−1.53546	−0.767728	−	0.640776i	$-0.778613\pi$
$283$	−122973.	−1.53546	−0.767728	+	0.640776i	$0.778613\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 79103.1i	− 0.960351i
$288$	0	0
$289$	−142934.	−1.71136
$290$	0	0
$291$	0	0
$292$	0	0
$293$	91384.6i	1.06448i	0.846593	+	0.532240i	$0.178650\pi$
$293$	91384.6i	1.06448i	−0.846593	+	0.532240i	$0.821350\pi$
$294$	0	0
$295$	82354.2	0.946328
$296$	0	0
$297$	0	0
$298$	0	0
$299$	186057.i	2.08115i
$300$	0	0
$301$	158479.	1.74920
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 54960.1i	− 0.590810i
$306$	0	0
$307$	39161.7	0.415513	0.207757	−	0.978181i	$-0.433384\pi$
$307$	39161.7	0.415513	0.207757	+	0.978181i	$0.433384\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	54760.0i	0.566164i	0.959096	+	0.283082i	$0.0913569\pi$
$311$	54760.0i	0.566164i	−0.959096	+	0.283082i	$0.908643\pi$
$312$	0	0
$313$	51694.4	0.527660	0.263830	−	0.964569i	$-0.415014\pi$
$313$	51694.4	0.527660	0.263830	+	0.964569i	$0.415014\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 118103.i	− 1.17529i	−0.809120	−	0.587643i	$-0.800056\pi$
$317$	− 118103.i	− 1.17529i	0.809120	−	0.587643i	$-0.199944\pi$
$318$	0	0
$319$	147906.	1.45346
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 207520.i	− 1.98909i
$324$	0	0
$325$	−143151.	−1.35527
$326$	0	0
$327$	0	0
$328$	0	0
$329$	41884.9i	0.386959i
$330$	0	0
$331$	42783.7	0.390501	0.195251	−	0.980753i	$-0.437448\pi$
$331$	42783.7	0.390501	0.195251	+	0.980753i	$0.437448\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	229030.i	2.04081i
$336$	0	0
$337$	−31621.0	−0.278430	−0.139215	−	0.990262i	$-0.544458\pi$
$337$	−31621.0	−0.278430	−0.139215	+	0.990262i	$0.544458\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	110290.i	0.948481i
$342$	0	0
$343$	−110554.	−0.939694
$344$	0	0
$345$	0	0
$346$	0	0
$347$	60323.1i	0.500985i	0.968118	+	0.250493i	$0.0805926\pi$
$347$	60323.1i	0.500985i	−0.968118	+	0.250493i	$0.919407\pi$
$348$	0	0
$349$	−33773.3	−0.277283	−0.138641	−	0.990343i	$-0.544273\pi$
$349$	−33773.3	−0.277283	−0.138641	+	0.990343i	$0.544273\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 25417.9i	− 0.203982i	−0.994785	−	0.101991i	$-0.967479\pi$
$353$	− 25417.9i	− 0.203982i	0.994785	−	0.101991i	$-0.0325212\pi$
$354$	0	0
$355$	231508.	1.83700
$356$	0	0
$357$	0	0
$358$	0	0
$359$	127120.i	0.986338i	0.869933	+	0.493169i	$0.164162\pi$
$359$	127120.i	0.986338i	−0.869933	+	0.493169i	$0.835838\pi$
$360$	0	0
$361$	59846.1	0.459220
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 135039.i	− 1.01361i
$366$	0	0
$367$	−150735.	−1.11913	−0.559565	−	0.828786i	$-0.689032\pi$
$367$	−150735.	−1.11913	−0.559565	+	0.828786i	$0.689032\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35807.1i	0.260148i
$372$	0	0
$373$	122041.	0.877181	0.438591	−	0.898687i	$-0.355478\pi$
$373$	122041.	0.877181	0.438591	+	0.898687i	$0.355478\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	274655.i	1.93244i
$378$	0	0
$379$	124950.	0.869878	0.434939	−	0.900460i	$-0.356770\pi$
$379$	124950.	0.869878	0.434939	+	0.900460i	$0.356770\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	104609.i	0.713131i	0.934270	+	0.356566i	$0.116052\pi$
$383$	104609.i	0.713131i	−0.934270	+	0.356566i	$0.883948\pi$
$384$	0	0
$385$	−281968.	−1.90230
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24860.3i	0.164288i	0.996620	+	0.0821442i	$0.0261768\pi$
$389$	24860.3i	0.164288i	−0.996620	+	0.0821442i	$0.973823\pi$
$390$	0	0
$391$	287840.	1.88277
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 9142.52i	− 0.0585965i
$396$	0	0
$397$	15508.2	0.0983968	0.0491984	−	0.998789i	$-0.484333\pi$
$397$	15508.2	0.0983968	0.0491984	+	0.998789i	$0.484333\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 12584.9i	− 0.0782640i	−0.999234	−	0.0391320i	$-0.987541\pi$
$401$	− 12584.9i	− 0.0782640i	0.999234	−	0.0391320i	$-0.0124593\pi$
$402$	0	0
$403$	−204805.	−1.26104
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 262886.i	− 1.58700i
$408$	0	0
$409$	−153706.	−0.918848	−0.459424	−	0.888217i	$-0.651944\pi$
$409$	−153706.	−0.918848	−0.459424	+	0.888217i	$0.651944\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	128566.i	0.753746i
$414$	0	0
$415$	46157.4	0.268006
$416$	0	0
$417$	0	0
$418$	0	0
$419$	267743.i	1.52507i	0.646947	+	0.762535i	$0.276046\pi$
$419$	267743.i	1.52507i	−0.646947	+	0.762535i	$0.723954\pi$
$420$	0	0
$421$	34512.1	0.194718	0.0973591	−	0.995249i	$-0.468960\pi$
$421$	34512.1	0.194718	0.0973591	+	0.995249i	$0.468960\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	221462.i	1.22609i
$426$	0	0
$427$	85800.0	0.470578
$428$	0	0
$429$	0	0
$430$	0	0
$431$	206321.i	1.11068i	0.831623	+	0.555340i	$0.187412\pi$
$431$	206321.i	1.11068i	−0.831623	+	0.555340i	$0.812588\pi$
$432$	0	0
$433$	−227991.	−1.21602	−0.608011	−	0.793929i	$-0.708032\pi$
$433$	−227991.	−1.21602	−0.608011	+	0.793929i	$0.708032\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	263771.i	1.38123i
$438$	0	0
$439$	−152231.	−0.789904	−0.394952	−	0.918702i	$-0.629239\pi$
$439$	−152231.	−0.789904	−0.394952	+	0.918702i	$0.629239\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 206153.i	− 1.05047i	−0.850958	−	0.525234i	$-0.823978\pi$
$443$	− 206153.i	− 1.05047i	0.850958	−	0.525234i	$-0.176022\pi$
$444$	0	0
$445$	−150055.	−0.757758
$446$	0	0
$447$	0	0
$448$	0	0
$449$	76495.2i	0.379439i	0.981838	+	0.189719i	$0.0607578\pi$
$449$	76495.2i	0.379439i	−0.981838	+	0.189719i	$0.939242\pi$
$450$	0	0
$451$	−254184.	−1.24967
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 523603.i	− 2.52918i
$456$	0	0
$457$	18230.0	0.0872880	0.0436440	−	0.999047i	$-0.486103\pi$
$457$	18230.0	0.0872880	0.0436440	+	0.999047i	$0.486103\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	126985.i	0.597516i	0.954329	+	0.298758i	$0.0965724\pi$
$461$	126985.i	0.597516i	−0.954329	+	0.298758i	$0.903428\pi$
$462$	0	0
$463$	−210154.	−0.980337	−0.490168	−	0.871628i	$-0.663065\pi$
$463$	−210154.	−0.980337	−0.490168	+	0.871628i	$0.663065\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 335490.i	− 1.53832i	−0.639058	−	0.769159i	$-0.720675\pi$
$467$	− 335490.i	− 1.53832i	0.639058	−	0.769159i	$-0.279325\pi$
$468$	0	0
$469$	−357546.	−1.62550
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 509244.i	− 2.27616i
$474$	0	0
$475$	−202944.	−0.899473
$476$	0	0
$477$	0	0
$478$	0	0
$479$	259372.i	1.13045i	0.824937	+	0.565225i	$0.191211\pi$
$479$	259372.i	1.13045i	−0.824937	+	0.565225i	$0.808789\pi$
$480$	0	0
$481$	488168.	2.10999
$482$	0	0
$483$	0	0
$484$	0	0
$485$	324092.i	1.37780i
$486$	0	0
$487$	33863.5	0.142782	0.0713912	−	0.997448i	$-0.477256\pi$
$487$	33863.5	0.142782	0.0713912	+	0.997448i	$0.477256\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	304452.i	1.26286i	0.775433	+	0.631430i	$0.217532\pi$
$491$	304452.i	1.26286i	−0.775433	+	0.631430i	$0.782468\pi$
$492$	0	0
$493$	424907.	1.74824
$494$	0	0
$495$	0	0
$496$	0	0
$497$	361415.i	1.46317i
$498$	0	0
$499$	295499.	1.18674	0.593369	−	0.804930i	$-0.297797\pi$
$499$	295499.	1.18674	0.593369	+	0.804930i	$0.297797\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	145108.i	0.573530i	0.958001	+	0.286765i	$0.0925798\pi$
$503$	145108.i	0.573530i	−0.958001	+	0.286765i	$0.907420\pi$
$504$	0	0
$505$	84760.6	0.332362
$506$	0	0
$507$	0	0
$508$	0	0
$509$	301907.i	1.16530i	0.812724	+	0.582649i	$0.197984\pi$
$509$	301907.i	1.16530i	−0.812724	+	0.582649i	$0.802016\pi$
$510$	0	0
$511$	210813.	0.807338
$512$	0	0
$513$	0	0
$514$	0	0
$515$	38075.2i	0.143558i
$516$	0	0
$517$	134589.	0.503535
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 405316.i	− 1.49320i	−0.665273	−	0.746600i	$-0.731685\pi$
$521$	− 405316.i	− 1.49320i	0.665273	−	0.746600i	$-0.268315\pi$
$522$	0	0
$523$	−118055.	−0.431598	−0.215799	−	0.976438i	$-0.569236\pi$
$523$	−118055.	−0.431598	−0.215799	+	0.976438i	$0.569236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	316845.i	1.14084i
$528$	0	0
$529$	−86023.6	−0.307402
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 472009.i	− 1.66148i
$534$	0	0
$535$	−701971.	−2.45251
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 42471.6i	− 0.146191i
$540$	0	0
$541$	416332.	1.42248	0.711238	−	0.702952i	$-0.248135\pi$
$541$	416332.	1.42248	0.711238	+	0.702952i	$0.248135\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	495284.i	1.66748i
$546$	0	0
$547$	366447.	1.22472	0.612359	−	0.790580i	$-0.290221\pi$
$547$	366447.	1.22472	0.612359	+	0.790580i	$0.290221\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	389377.i	1.28253i
$552$	0	0
$553$	14272.7	0.0466719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 91968.7i	− 0.296435i	−0.988955	−	0.148218i	$-0.952646\pi$
$557$	− 91968.7i	− 0.296435i	0.988955	−	0.148218i	$-0.0473536\pi$
$558$	0	0
$559$	945646.	3.02625
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 226931.i	− 0.715940i	−0.933733	−	0.357970i	$-0.883469\pi$
$563$	− 226931.i	− 0.715940i	0.933733	−	0.357970i	$-0.116531\pi$
$564$	0	0
$565$	−102370.	−0.320684
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 514190.i	− 1.58818i	−0.607802	−	0.794088i	$-0.707949\pi$
$569$	− 514190.i	− 1.58818i	0.607802	−	0.794088i	$-0.292051\pi$
$570$	0	0
$571$	−101793.	−0.312210	−0.156105	−	0.987740i	$-0.549894\pi$
$571$	−101793.	−0.312210	−0.156105	+	0.987740i	$0.549894\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 281493.i	− 0.851397i
$576$	0	0
$577$	185485.	0.557131	0.278565	−	0.960417i	$-0.410141\pi$
$577$	185485.	0.557131	0.278565	+	0.960417i	$0.410141\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	72057.7i	0.213466i
$582$	0	0
$583$	115060.	0.338521
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 357079.i	− 1.03631i	−0.855288	−	0.518153i	$-0.826620\pi$
$587$	− 357079.i	− 1.03631i	0.855288	−	0.518153i	$-0.173380\pi$
$588$	0	0
$589$	−290351.	−0.836936
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 405138.i	− 1.15211i	−0.817411	−	0.576055i	$-0.804591\pi$
$593$	− 405138.i	− 1.15211i	0.817411	−	0.576055i	$-0.195409\pi$
$594$	0	0
$595$	−810044.	−2.28810
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 515042.i	− 1.43545i	−0.696325	−	0.717727i	$-0.745183\pi$
$599$	− 515042.i	− 1.43545i	0.696325	−	0.717727i	$-0.254817\pi$
$600$	0	0
$601$	−136403.	−0.377636	−0.188818	−	0.982012i	$-0.560466\pi$
$601$	−136403.	−0.377636	−0.188818	+	0.982012i	$0.560466\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	422594.i	1.15455i
$606$	0	0
$607$	701196.	1.90310	0.951551	−	0.307492i	$-0.0994899\pi$
$607$	701196.	1.90310	0.951551	+	0.307492i	$0.0994899\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	249927.i	0.669470i
$612$	0	0
$613$	468081.	1.24566	0.622831	−	0.782357i	$-0.285983\pi$
$613$	468081.	1.24566	0.622831	+	0.782357i	$0.285983\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 272513.i	− 0.715842i	−0.933752	−	0.357921i	$-0.883486\pi$
$617$	− 272513.i	− 0.715842i	0.933752	−	0.357921i	$-0.116514\pi$
$618$	0	0
$619$	−21965.3	−0.0573266	−0.0286633	−	0.999589i	$-0.509125\pi$
$619$	−21965.3	−0.0573266	−0.0286633	+	0.999589i	$0.509125\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 234256.i	− 0.603551i
$624$	0	0
$625$	−464909.	−1.19017
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 755224.i	− 1.90886i
$630$	0	0
$631$	−158922.	−0.399139	−0.199570	−	0.979884i	$-0.563954\pi$
$631$	−158922.	−0.399139	−0.199570	+	0.979884i	$0.563954\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 405605.i	− 1.00590i
$636$	0	0
$637$	78868.1	0.194367
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 416685.i	− 1.01412i	−0.861909	−	0.507062i	$-0.830731\pi$
$641$	− 416685.i	− 1.01412i	0.861909	−	0.507062i	$-0.169269\pi$
$642$	0	0
$643$	−556364.	−1.34566	−0.672832	−	0.739795i	$-0.734922\pi$
$643$	−556364.	−1.34566	−0.672832	+	0.739795i	$0.734922\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	529664.i	1.26530i	0.774440	+	0.632648i	$0.218032\pi$
$647$	529664.i	1.26530i	−0.774440	+	0.632648i	$0.781968\pi$
$648$	0	0
$649$	413123.	0.980821
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 722214.i	− 1.69371i	−0.531821	−	0.846857i	$-0.678492\pi$
$653$	− 722214.i	− 1.69371i	0.531821	−	0.846857i	$-0.321508\pi$
$654$	0	0
$655$	52610.1	0.122627
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 376232.i	− 0.866334i	−0.901314	−	0.433167i	$-0.857396\pi$
$659$	− 376232.i	− 0.866334i	0.901314	−	0.433167i	$-0.142604\pi$
$660$	0	0
$661$	236955.	0.542330	0.271165	−	0.962533i	$-0.412591\pi$
$661$	236955.	0.542330	0.271165	+	0.962533i	$0.412591\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 742309.i	− 1.67858i
$666$	0	0
$667$	−540086.	−1.21398
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 275703.i	− 0.612345i
$672$	0	0
$673$	416556.	0.919694	0.459847	−	0.887998i	$-0.347904\pi$
$673$	416556.	0.919694	0.459847	+	0.887998i	$0.347904\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	559542.i	1.22083i	0.792082	+	0.610415i	$0.208997\pi$
$677$	559542.i	1.22083i	−0.792082	+	0.610415i	$0.791003\pi$
$678$	0	0
$679$	−505951.	−1.09741
$680$	0	0
$681$	0	0
$682$	0	0
$683$	651830.i	1.39731i	0.715458	+	0.698655i	$0.246218\pi$
$683$	651830.i	1.39731i	−0.715458	+	0.698655i	$0.753782\pi$
$684$	0	0
$685$	−535496.	−1.14123
$686$	0	0
$687$	0	0
$688$	0	0
$689$	213661.i	0.450077i
$690$	0	0
$691$	−20233.2	−0.0423748	−0.0211874	−	0.999776i	$-0.506745\pi$
$691$	−20233.2	−0.0423748	−0.0211874	+	0.999776i	$0.506745\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	327170.i	0.677336i
$696$	0	0
$697$	−730225.	−1.50311
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 657191.i	− 1.33738i	−0.743541	−	0.668691i	$-0.766855\pi$
$701$	− 657191.i	− 1.33738i	0.743541	−	0.668691i	$-0.233145\pi$
$702$	0	0
$703$	692074.	1.40037
$704$	0	0
$705$	0	0
$706$	0	0
$707$	132322.i	0.264725i
$708$	0	0
$709$	−364079.	−0.724274	−0.362137	−	0.932125i	$-0.617953\pi$
$709$	−364079.	−0.724274	−0.362137	+	0.932125i	$0.617953\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 402731.i	− 0.792202i
$714$	0	0
$715$	−1.68251e6	−3.29113
$716$	0	0
$717$	0	0
$718$	0	0
$719$	996163.i	1.92696i	0.267781	+	0.963480i	$0.413710\pi$
$719$	996163.i	1.92696i	−0.267781	+	0.963480i	$0.586290\pi$
$720$	0	0
$721$	−59440.4	−0.114343
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 415538.i	− 0.790560i
$726$	0	0
$727$	634137.	1.19982	0.599908	−	0.800069i	$-0.295204\pi$
$727$	634137.	1.19982	0.599908	+	0.800069i	$0.295204\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.46297e6i	− 2.73779i
$732$	0	0
$733$	−642819.	−1.19641	−0.598206	−	0.801343i	$-0.704119\pi$
$733$	−642819.	−1.19641	−0.598206	+	0.801343i	$0.704119\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.14891e6i	2.11519i
$738$	0	0
$739$	2203.60	0.00403499	0.00201750	−	0.999998i	$-0.499358\pi$
$739$	2203.60	0.00403499	0.00201750	+	0.999998i	$0.499358\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	271477.i	0.491762i	0.969300	+	0.245881i	$0.0790773\pi$
$743$	271477.i	0.491762i	−0.969300	+	0.245881i	$0.920923\pi$
$744$	0	0
$745$	557752.	1.00491
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.09587e6i	− 1.95342i
$750$	0	0
$751$	−234201.	−0.415249	−0.207624	−	0.978209i	$-0.566573\pi$
$751$	−234201.	−0.415249	−0.207624	+	0.978209i	$0.566573\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.09557e6i	− 1.92197i
$756$	0	0
$757$	774350.	1.35128	0.675640	−	0.737232i	$-0.263867\pi$
$757$	774350.	1.35128	0.675640	+	0.737232i	$0.263867\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	669442.i	1.15596i	0.816050	+	0.577981i	$0.196159\pi$
$761$	669442.i	1.15596i	−0.816050	+	0.577981i	$0.803841\pi$
$762$	0	0
$763$	−773204.	−1.32814
$764$	0	0
$765$	0	0
$766$	0	0
$767$	767153.i	1.30404i
$768$	0	0
$769$	1.05084e6	1.77699	0.888497	−	0.458882i	$-0.151750\pi$
$769$	1.05084e6	1.77699	0.888497	+	0.458882i	$0.151750\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	334785.i	0.560283i	0.959959	+	0.280141i	$0.0903813\pi$
$773$	334785.i	0.560283i	−0.959959	+	0.280141i	$0.909619\pi$
$774$	0	0
$775$	309858.	0.515893
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 669165.i	− 1.10270i
$780$	0	0
$781$	1.16134e6	1.90396
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.28525e6i	2.08568i
$786$	0	0
$787$	−566522.	−0.914676	−0.457338	−	0.889293i	$-0.651197\pi$
$787$	−566522.	−0.914676	−0.457338	+	0.889293i	$0.651197\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 159814.i	− 0.255424i
$792$	0	0
$793$	511969.	0.814137
$794$	0	0
$795$	0	0
$796$	0	0
$797$	558221.i	0.878799i	0.898292	+	0.439399i	$0.144809\pi$
$797$	558221.i	0.878799i	−0.898292	+	0.439399i	$0.855191\pi$
$798$	0	0
$799$	386652.	0.605656
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 677410.i	− 1.05056i
$804$	0	0
$805$	1.02962e6	1.58886
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 354802.i	− 0.542112i	−0.962564	−	0.271056i	$-0.912627\pi$
$809$	− 354802.i	− 0.542112i	0.962564	−	0.271056i	$-0.0873729\pi$
$810$	0	0
$811$	−878257.	−1.33530	−0.667652	−	0.744474i	$-0.732700\pi$
$811$	−878257.	−1.33530	−0.667652	+	0.744474i	$0.732700\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.48886e6i	− 2.24149i
$816$	0	0
$817$	1.34064e6	2.00848
$818$	0	0
$819$	0	0
$820$	0	0
$821$	883479.i	1.31072i	0.755317	+	0.655360i	$0.227483\pi$
$821$	883479.i	1.31072i	−0.755317	+	0.655360i	$0.772517\pi$
$822$	0	0
$823$	363017.	0.535953	0.267977	−	0.963425i	$-0.413645\pi$
$823$	363017.	0.535953	0.267977	+	0.963425i	$0.413645\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 321961.i	− 0.470752i	−0.971904	−	0.235376i	$-0.924368\pi$
$827$	− 321961.i	− 0.470752i	0.971904	−	0.235376i	$-0.0756322\pi$
$828$	0	0
$829$	−26246.1	−0.0381905	−0.0190953	−	0.999818i	$-0.506079\pi$
$829$	−26246.1	−0.0381905	−0.0190953	+	0.999818i	$0.506079\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 122013.i	− 0.175840i
$834$	0	0
$835$	−492432.	−0.706274
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 268612.i	− 0.381594i	−0.981630	−	0.190797i	$-0.938893\pi$
$839$	− 268612.i	− 0.381594i	0.981630	−	0.190797i	$-0.0611072\pi$
$840$	0	0
$841$	−89989.3	−0.127233
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.18124e6i	− 3.05485i
$846$	0	0
$847$	−659725.	−0.919594
$848$	0	0
$849$	0	0
$850$	0	0
$851$	959942.i	1.32552i
$852$	0	0
$853$	−528476.	−0.726318	−0.363159	−	0.931727i	$-0.618302\pi$
$853$	−528476.	−0.726318	−0.363159	+	0.931727i	$0.618302\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 343142.i	− 0.467210i	−0.972332	−	0.233605i	$-0.924948\pi$
$857$	− 343142.i	− 0.467210i	0.972332	−	0.233605i	$-0.0750523\pi$
$858$	0	0
$859$	613924.	0.832009	0.416005	−	0.909362i	$-0.363430\pi$
$859$	613924.	0.832009	0.416005	+	0.909362i	$0.363430\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 653461.i	− 0.877401i	−0.898633	−	0.438700i	$-0.855439\pi$
$863$	− 653461.i	− 0.877401i	0.898633	−	0.438700i	$-0.144561\pi$
$864$	0	0
$865$	115511.	0.154380
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 45862.7i	− 0.0607323i
$870$	0	0
$871$	−2.13348e6	−2.81224
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 271709.i	− 0.354886i
$876$	0	0
$877$	−1.33452e6	−1.73511	−0.867553	−	0.497345i	$-0.834308\pi$
$877$	−1.33452e6	−1.73511	−0.867553	+	0.497345i	$0.834308\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22500.8i	0.0289899i	0.999895	+	0.0144949i	$0.00461404\pi$
$881$	22500.8i	0.0289899i	−0.999895	+	0.0144949i	$0.995386\pi$
$882$	0	0
$883$	866942.	1.11191	0.555954	−	0.831213i	$-0.312353\pi$
$883$	866942.	1.11191	0.555954	+	0.831213i	$0.312353\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.44231e6i	− 1.83320i	−0.399802	−	0.916602i	$-0.630921\pi$
$887$	− 1.44231e6i	− 1.83320i	0.399802	−	0.916602i	$-0.369079\pi$
$888$	0	0
$889$	633203.	0.801197
$890$	0	0
$891$	0	0
$892$	0	0
$893$	354320.i	0.444317i
$894$	0	0
$895$	−22319.2	−0.0278633
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 594508.i	− 0.735594i
$900$	0	0
$901$	330546.	0.407176
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.12962e6i	1.37923i
$906$	0	0
$907$	88929.1	0.108101	0.0540504	−	0.998538i	$-0.482787\pi$
$907$	88929.1	0.108101	0.0540504	+	0.998538i	$0.482787\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 801731.i	− 0.966033i	−0.875611	−	0.483016i	$-0.839541\pi$
$911$	− 801731.i	− 0.966033i	0.875611	−	0.483016i	$-0.160459\pi$
$912$	0	0
$913$	231544.	0.277775
$914$	0	0
$915$	0	0
$916$	0	0
$917$	82131.3i	0.0976720i
$918$	0	0
$919$	860214.	1.01853	0.509267	−	0.860609i	$-0.329917\pi$
$919$	860214.	1.01853	0.509267	+	0.860609i	$0.329917\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.15657e6i	2.53139i
$924$	0	0
$925$	−738571.	−0.863195
$926$	0	0
$927$	0	0
$928$	0	0
$929$	854907.i	0.990575i	0.868729	+	0.495288i	$0.164937\pi$
$929$	854907.i	0.990575i	−0.868729	+	0.495288i	$0.835063\pi$
$930$	0	0
$931$	111811.	0.128998
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.60293e6i	2.97742i
$936$	0	0
$937$	41248.1	0.0469813	0.0234907	−	0.999724i	$-0.492522\pi$
$937$	41248.1	0.0469813	0.0234907	+	0.999724i	$0.492522\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	252432.i	0.285079i	0.989789	+	0.142539i	$0.0455267\pi$
$941$	252432.i	0.285079i	−0.989789	+	0.142539i	$0.954473\pi$
$942$	0	0
$943$	928166.	1.04376
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.11557e6i	1.24393i	0.783043	+	0.621967i	$0.213666\pi$
$947$	1.11557e6i	1.24393i	−0.783043	+	0.621967i	$0.786334\pi$
$948$	0	0
$949$	1.25792e6	1.39676
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.14945e6i	− 1.26562i	−0.774306	−	0.632811i	$-0.781901\pi$
$953$	− 1.14945e6i	− 1.26562i	0.774306	−	0.632811i	$-0.218099\pi$
$954$	0	0
$955$	−110683.	−0.121359
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 835979.i	− 0.908988i
$960$	0	0
$961$	−480209.	−0.519976
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.08227e6i	1.16220i
$966$	0	0
$967$	1.27170e6	1.35998	0.679990	−	0.733221i	$-0.261984\pi$
$967$	1.27170e6	1.35998	0.679990	+	0.733221i	$0.261984\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.67377e6i	1.77524i	0.460576	+	0.887620i	$0.347643\pi$
$971$	1.67377e6i	1.77524i	−0.460576	+	0.887620i	$0.652357\pi$
$972$	0	0
$973$	−510756.	−0.539495
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 670102.i	− 0.702023i	−0.936371	−	0.351012i	$-0.885838\pi$
$977$	− 670102.i	− 0.702023i	0.936371	−	0.351012i	$-0.114162\pi$
$978$	0	0
$979$	−752738.	−0.785378
$980$	0	0
$981$	0	0
$982$	0	0
$983$	542692.i	0.561625i	0.959763	+	0.280813i	$0.0906039\pi$
$983$	542692.i	0.561625i	−0.959763	+	0.280813i	$0.909396\pi$
$984$	0	0
$985$	105454.	0.108690
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.85953e6i	1.90113i
$990$	0	0
$991$	−819270.	−0.834219	−0.417109	−	0.908856i	$-0.636957\pi$
$991$	−819270.	−0.834219	−0.417109	+	0.908856i	$0.636957\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	936018.i	0.945449i
$996$	0	0
$997$	−659791.	−0.663767	−0.331884	−	0.943320i	$-0.607684\pi$
$997$	−659791.	−0.663767	−0.331884	+	0.943320i	$0.607684\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.5.e.b.161.2	✓	12
3.2	odd	2	inner	648.5.e.b.161.11	yes	12
4.3	odd	2		1296.5.e.h.161.2		12
9.2	odd	6		648.5.m.g.377.2		24
9.4	even	3		648.5.m.g.593.2		24
9.5	odd	6		648.5.m.g.593.11		24
9.7	even	3		648.5.m.g.377.11		24
12.11	even	2		1296.5.e.h.161.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.5.e.b.161.2	✓	12	1.1	even	1	trivial
648.5.e.b.161.11	yes	12	3.2	odd	2	inner
648.5.m.g.377.2		24	9.2	odd	6
648.5.m.g.377.11		24	9.7	even	3
648.5.m.g.593.2		24	9.4	even	3
648.5.m.g.593.11		24	9.5	odd	6
1296.5.e.h.161.2		12	4.3	odd	2
1296.5.e.h.161.11		12	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	648.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-33.0209i q^{5} +51.5500 q^{7} -165.647i q^{11} +307.599 q^{13} -475.873i q^{17} +436.081 q^{19} +604.867i q^{23} -465.380 q^{25} +892.900i q^{29} -665.817 q^{31} -1702.23i q^{35} +1587.03 q^{37} -1534.49i q^{41} +3074.28 q^{43} +812.510i q^{47} +256.399 q^{49} +694.609i q^{53} -5469.80 q^{55} +2494.00i q^{59} +1664.40 q^{61} -10157.2i q^{65} -6935.90 q^{67} +7010.97i q^{71} +4089.49 q^{73} -8539.07i q^{77} +276.871 q^{79} +1397.82i q^{83} -15713.8 q^{85} -4544.24i q^{89} +15856.7 q^{91} -14399.8i q^{95} -9814.76 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 96 q^{7} - 72 q^{13} + 336 q^{19} + 84 q^{25} - 1536 q^{31} - 492 q^{37} + 10128 q^{43} - 6828 q^{49} - 13968 q^{55} + 26268 q^{61} - 20784 q^{67} - 9984 q^{73} + 44592 q^{79} - 45348 q^{85} - 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) \) 12 * q + 96 * q^7 - 72 * q^13 + 336 * q^19 + 84 * q^25 - 1536 * q^31 - 492 * q^37 + 10128 * q^43 - 6828 * q^49 - 13968 * q^55 + 26268 * q^61 - 20784 * q^67 - 9984 * q^73 + 44592 * q^79 - 45348 * q^85 - 11808 * q^91 + 76608 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more