LMFDB - Embedded newform 1344.2.c.h.673.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,2,Mod(673,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.673"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,4,0,-4,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.4
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.2.c.h.673.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +0.732051i q^{5} +1.00000 q^{7} -1.00000 q^{9} -2.73205i q^{11} -4.00000i q^{13} -0.732051 q^{15} +4.19615 q^{17} +3.46410i q^{19} +1.00000i q^{21} +4.73205 q^{23} +4.46410 q^{25} -1.00000i q^{27} -7.46410i q^{29} -2.00000 q^{31} +2.73205 q^{33} +0.732051i q^{35} -2.00000i q^{37} +4.00000 q^{39} +9.66025 q^{41} +8.92820i q^{43} -0.732051i q^{45} -4.00000 q^{47} +1.00000 q^{49} +4.19615i q^{51} -3.46410i q^{53} +2.00000 q^{55} -3.46410 q^{57} +8.00000i q^{59} +8.39230i q^{61} -1.00000 q^{63} +2.92820 q^{65} +6.39230i q^{67} +4.73205i q^{69} -3.66025 q^{71} -8.53590 q^{73} +4.46410i q^{75} -2.73205i q^{77} +12.3923 q^{79} +1.00000 q^{81} -2.53590i q^{83} +3.07180i q^{85} +7.46410 q^{87} +12.1962 q^{89} -4.00000i q^{91} -2.00000i q^{93} -2.53590 q^{95} -2.39230 q^{97} +2.73205i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 4 q^{9} + 4 q^{15} - 4 q^{17} + 12 q^{23} + 4 q^{25} - 8 q^{31} + 4 q^{33} + 16 q^{39} + 4 q^{41} - 16 q^{47} + 4 q^{49} + 8 q^{55} - 4 q^{63} - 16 q^{65} + 20 q^{71} - 48 q^{73} + 8 q^{79}+ \cdots + 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 4 * q^9 + 4 * q^15 - 4 * q^17 + 12 * q^23 + 4 * q^25 - 8 * q^31 + 4 * q^33 + 16 * q^39 + 4 * q^41 - 16 * q^47 + 4 * q^49 + 8 * q^55 - 4 * q^63 - 16 * q^65 + 20 * q^71 - 48 * q^73 + 8 * q^79 + 4 * q^81 + 16 * q^87 + 28 * q^89 - 24 * q^95 + 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	0.732051i	0.327383i	0.986512	+	0.163692i	$0.0523402\pi$
$5$	0.732051i	0.327383i	−0.986512	+	0.163692i	$0.947660\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.73205i	− 0.823744i	−0.911242	−	0.411872i	$-0.864875\pi$
$11$	− 2.73205i	− 0.823744i	0.911242	−	0.411872i	$-0.135125\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	4.19615	1.01772	0.508858	−	0.860850i	$-0.330068\pi$
$17$	4.19615	1.01772	0.508858	+	0.860850i	$0.330068\pi$
$18$	0	0
$19$	3.46410i	0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.46410i	0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	4.73205	0.986701	0.493350	−	0.869831i	$-0.335772\pi$
$23$	4.73205	0.986701	0.493350	+	0.869831i	$0.335772\pi$
$24$	0	0
$25$	4.46410	0.892820
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 7.46410i	− 1.38605i	−0.720914	−	0.693024i	$-0.756278\pi$
$29$	− 7.46410i	− 1.38605i	0.720914	−	0.693024i	$-0.243722\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	2.73205	0.475589
$34$	0	0
$35$	0.732051i	0.123739i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	9.66025	1.50868	0.754339	−	0.656485i	$-0.227957\pi$
$41$	9.66025	1.50868	0.754339	+	0.656485i	$0.227957\pi$
$42$	0	0
$43$	8.92820i	1.36154i	0.732498	+	0.680769i	$0.238354\pi$
$43$	8.92820i	1.36154i	−0.732498	+	0.680769i	$0.761646\pi$
$44$	0	0
$45$	− 0.732051i	− 0.109128i
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.19615i	0.587579i
$52$	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
$55$	2.00000	0.269680
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	8.39230i	1.07452i	0.843415	+	0.537262i	$0.180541\pi$
$61$	8.39230i	1.07452i	−0.843415	+	0.537262i	$0.819459\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	2.92820	0.363199
$66$	0	0
$67$	6.39230i	0.780944i	0.920615	+	0.390472i	$0.127688\pi$
$67$	6.39230i	0.780944i	−0.920615	+	0.390472i	$0.872312\pi$
$68$	0	0
$69$	4.73205i	0.569672i
$70$	0	0
$71$	−3.66025	−0.434392	−0.217196	−	0.976128i	$-0.569691\pi$
$71$	−3.66025	−0.434392	−0.217196	+	0.976128i	$0.569691\pi$
$72$	0	0
$73$	−8.53590	−0.999051	−0.499526	−	0.866299i	$-0.666492\pi$
$73$	−8.53590	−0.999051	−0.499526	+	0.866299i	$0.666492\pi$
$74$	0	0
$75$	4.46410i	0.515470i
$76$	0	0
$77$	− 2.73205i	− 0.311346i
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.53590i	− 0.278351i	−0.990268	−	0.139176i	$-0.955555\pi$
$83$	− 2.53590i	− 0.278351i	0.990268	−	0.139176i	$-0.0444452\pi$
$84$	0	0
$85$	3.07180i	0.333183i
$86$	0	0
$87$	7.46410	0.800236
$88$	0	0
$89$	12.1962	1.29279	0.646395	−	0.763003i	$-0.276276\pi$
$89$	12.1962	1.29279	0.646395	+	0.763003i	$0.276276\pi$
$90$	0	0
$91$	− 4.00000i	− 0.419314i
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	−2.53590	−0.260178
$96$	0	0
$97$	−2.39230	−0.242902	−0.121451	−	0.992597i	$-0.538755\pi$
$97$	−2.39230	−0.242902	−0.121451	+	0.992597i	$0.538755\pi$
$98$	0	0
$99$	2.73205i	0.274581i
$100$	0	0
$101$	− 11.2679i	− 1.12120i	−0.828086	−	0.560601i	$-0.810570\pi$
$101$	− 11.2679i	− 1.12120i	0.828086	−	0.560601i	$-0.189430\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	−0.732051	−0.0714408
$106$	0	0
$107$	− 8.19615i	− 0.792352i	−0.918175	−	0.396176i	$-0.870337\pi$
$107$	− 8.19615i	− 0.792352i	0.918175	−	0.396176i	$-0.129663\pi$
$108$	0	0
$109$	− 10.9282i	− 1.04673i	−0.852108	−	0.523366i	$-0.824676\pi$
$109$	− 10.9282i	− 1.04673i	0.852108	−	0.523366i	$-0.175324\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	6.39230	0.601337	0.300669	−	0.953729i	$-0.402790\pi$
$113$	6.39230	0.601337	0.300669	+	0.953729i	$0.402790\pi$
$114$	0	0
$115$	3.46410i	0.323029i
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	4.19615	0.384661
$120$	0	0
$121$	3.53590	0.321445
$122$	0	0
$123$	9.66025i	0.871036i
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	−19.3205	−1.71442	−0.857209	−	0.514969i	$-0.827804\pi$
$127$	−19.3205	−1.71442	−0.857209	+	0.514969i	$0.827804\pi$
$128$	0	0
$129$	−8.92820	−0.786084
$130$	0	0
$131$	− 4.39230i	− 0.383757i	−0.981419	−	0.191879i	$-0.938542\pi$
$131$	− 4.39230i	− 0.383757i	0.981419	−	0.191879i	$-0.0614580\pi$
$132$	0	0
$133$	3.46410i	0.300376i
$134$	0	0
$135$	0.732051	0.0630049
$136$	0	0
$137$	−3.46410	−0.295958	−0.147979	−	0.988990i	$-0.547277\pi$
$137$	−3.46410	−0.295958	−0.147979	+	0.988990i	$0.547277\pi$
$138$	0	0
$139$	10.9282i	0.926918i	0.886118	+	0.463459i	$0.153392\pi$
$139$	10.9282i	0.926918i	−0.886118	+	0.463459i	$0.846608\pi$
$140$	0	0
$141$	− 4.00000i	− 0.336861i
$142$	0	0
$143$	−10.9282	−0.913862
$144$	0	0
$145$	5.46410	0.453769
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	2.39230i	0.195985i	0.995187	+	0.0979926i	$0.0312422\pi$
$149$	2.39230i	0.195985i	−0.995187	+	0.0979926i	$0.968758\pi$
$150$	0	0
$151$	1.07180	0.0872216	0.0436108	−	0.999049i	$-0.486114\pi$
$151$	1.07180	0.0872216	0.0436108	+	0.999049i	$0.486114\pi$
$152$	0	0
$153$	−4.19615	−0.339239
$154$	0	0
$155$	− 1.46410i	− 0.117599i
$156$	0	0
$157$	− 8.39230i	− 0.669779i	−0.942257	−	0.334889i	$-0.891301\pi$
$157$	− 8.39230i	− 0.669779i	0.942257	−	0.334889i	$-0.108699\pi$
$158$	0	0
$159$	3.46410	0.274721
$160$	0	0
$161$	4.73205	0.372938
$162$	0	0
$163$	4.53590i	0.355279i	0.984096	+	0.177639i	$0.0568461\pi$
$163$	4.53590i	0.355279i	−0.984096	+	0.177639i	$0.943154\pi$
$164$	0	0
$165$	2.00000i	0.155700i
$166$	0	0
$167$	16.3923	1.26847	0.634237	−	0.773138i	$-0.281314\pi$
$167$	16.3923	1.26847	0.634237	+	0.773138i	$0.281314\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	− 3.46410i	− 0.264906i
$172$	0	0
$173$	16.0526i	1.22045i	0.792227	+	0.610227i	$0.208922\pi$
$173$	16.0526i	1.22045i	−0.792227	+	0.610227i	$0.791078\pi$
$174$	0	0
$175$	4.46410	0.337454
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	− 20.5885i	− 1.53885i	−0.638735	−	0.769427i	$-0.720542\pi$
$179$	− 20.5885i	− 1.53885i	0.638735	−	0.769427i	$-0.279458\pi$
$180$	0	0
$181$	18.9282i	1.40692i	0.710734	+	0.703461i	$0.248363\pi$
$181$	18.9282i	1.40692i	−0.710734	+	0.703461i	$0.751637\pi$
$182$	0	0
$183$	−8.39230	−0.620377
$184$	0	0
$185$	1.46410	0.107643
$186$	0	0
$187$	− 11.4641i	− 0.838338i
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	−24.7321	−1.78955	−0.894774	−	0.446519i	$-0.852664\pi$
$191$	−24.7321	−1.78955	−0.894774	+	0.446519i	$0.852664\pi$
$192$	0	0
$193$	−24.3923	−1.75580	−0.877898	−	0.478847i	$-0.841055\pi$
$193$	−24.3923	−1.75580	−0.877898	+	0.478847i	$0.841055\pi$
$194$	0	0
$195$	2.92820i	0.209693i
$196$	0	0
$197$	− 26.7846i	− 1.90832i	−0.299290	−	0.954162i	$-0.596750\pi$
$197$	− 26.7846i	− 1.90832i	0.299290	−	0.954162i	$-0.403250\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−6.39230	−0.450878
$202$	0	0
$203$	− 7.46410i	− 0.523877i
$204$	0	0
$205$	7.07180i	0.493916i
$206$	0	0
$207$	−4.73205	−0.328900
$208$	0	0
$209$	9.46410	0.654646
$210$	0	0
$211$	0.143594i	0.00988539i	0.999988	+	0.00494269i	$0.00157331\pi$
$211$	0.143594i	0.00988539i	−0.999988	+	0.00494269i	$0.998427\pi$
$212$	0	0
$213$	− 3.66025i	− 0.250796i
$214$	0	0
$215$	−6.53590	−0.445745
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	− 8.53590i	− 0.576803i
$220$	0	0
$221$	− 16.7846i	− 1.12906i
$222$	0	0
$223$	−5.07180	−0.339633	−0.169816	−	0.985476i	$-0.554317\pi$
$223$	−5.07180	−0.339633	−0.169816	+	0.985476i	$0.554317\pi$
$224$	0	0
$225$	−4.46410	−0.297607
$226$	0	0
$227$	− 11.3205i	− 0.751369i	−0.926748	−	0.375684i	$-0.877408\pi$
$227$	− 11.3205i	− 0.751369i	0.926748	−	0.375684i	$-0.122592\pi$
$228$	0	0
$229$	10.9282i	0.722156i	0.932536	+	0.361078i	$0.117591\pi$
$229$	10.9282i	0.722156i	−0.932536	+	0.361078i	$0.882409\pi$
$230$	0	0
$231$	2.73205	0.179756
$232$	0	0
$233$	12.9282	0.846955	0.423477	−	0.905907i	$-0.360809\pi$
$233$	12.9282	0.846955	0.423477	+	0.905907i	$0.360809\pi$
$234$	0	0
$235$	− 2.92820i	− 0.191015i
$236$	0	0
$237$	12.3923i	0.804967i
$238$	0	0
$239$	−27.6603	−1.78919	−0.894597	−	0.446875i	$-0.852537\pi$
$239$	−27.6603	−1.78919	−0.894597	+	0.446875i	$0.852537\pi$
$240$	0	0
$241$	−7.46410	−0.480805	−0.240403	−	0.970673i	$-0.577279\pi$
$241$	−7.46410	−0.480805	−0.240403	+	0.970673i	$0.577279\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0.732051i	0.0467690i
$246$	0	0
$247$	13.8564	0.881662
$248$	0	0
$249$	2.53590	0.160706
$250$	0	0
$251$	− 23.3205i	− 1.47198i	−0.676994	−	0.735989i	$-0.736718\pi$
$251$	− 23.3205i	− 1.47198i	0.676994	−	0.735989i	$-0.263282\pi$
$252$	0	0
$253$	− 12.9282i	− 0.812789i
$254$	0	0
$255$	−3.07180	−0.192363
$256$	0	0
$257$	3.80385	0.237277	0.118639	−	0.992937i	$-0.462147\pi$
$257$	3.80385	0.237277	0.118639	+	0.992937i	$0.462147\pi$
$258$	0	0
$259$	− 2.00000i	− 0.124274i
$260$	0	0
$261$	7.46410i	0.462016i
$262$	0	0
$263$	6.19615	0.382071	0.191036	−	0.981583i	$-0.438815\pi$
$263$	6.19615	0.382071	0.191036	+	0.981583i	$0.438815\pi$
$264$	0	0
$265$	2.53590	0.155779
$266$	0	0
$267$	12.1962i	0.746392i
$268$	0	0
$269$	19.2679i	1.17479i	0.809301	+	0.587394i	$0.199846\pi$
$269$	19.2679i	1.17479i	−0.809301	+	0.587394i	$0.800154\pi$
$270$	0	0
$271$	−15.0718	−0.915546	−0.457773	−	0.889069i	$-0.651353\pi$
$271$	−15.0718	−0.915546	−0.457773	+	0.889069i	$0.651353\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	− 12.1962i	− 0.735456i
$276$	0	0
$277$	16.9282i	1.01712i	0.861027	+	0.508559i	$0.169821\pi$
$277$	16.9282i	1.01712i	−0.861027	+	0.508559i	$0.830179\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−28.9282	−1.72571	−0.862856	−	0.505450i	$-0.831327\pi$
$281$	−28.9282	−1.72571	−0.862856	+	0.505450i	$0.831327\pi$
$282$	0	0
$283$	2.39230i	0.142208i	0.997469	+	0.0711039i	$0.0226522\pi$
$283$	2.39230i	0.142208i	−0.997469	+	0.0711039i	$0.977348\pi$
$284$	0	0
$285$	− 2.53590i	− 0.150214i
$286$	0	0
$287$	9.66025	0.570227
$288$	0	0
$289$	0.607695	0.0357468
$290$	0	0
$291$	− 2.39230i	− 0.140239i
$292$	0	0
$293$	− 13.8038i	− 0.806429i	−0.915105	−	0.403215i	$-0.867893\pi$
$293$	− 13.8038i	− 0.806429i	0.915105	−	0.403215i	$-0.132107\pi$
$294$	0	0
$295$	−5.85641	−0.340973
$296$	0	0
$297$	−2.73205	−0.158530
$298$	0	0
$299$	− 18.9282i	− 1.09465i
$300$	0	0
$301$	8.92820i	0.514613i
$302$	0	0
$303$	11.2679	0.647327
$304$	0	0
$305$	−6.14359	−0.351781
$306$	0	0
$307$	− 25.3205i	− 1.44512i	−0.691309	−	0.722559i	$-0.742966\pi$
$307$	− 25.3205i	− 1.44512i	0.691309	−	0.722559i	$-0.257034\pi$
$308$	0	0
$309$	14.0000i	0.796432i
$310$	0	0
$311$	−31.7128	−1.79827	−0.899134	−	0.437673i	$-0.855803\pi$
$311$	−31.7128	−1.79827	−0.899134	+	0.437673i	$0.855803\pi$
$312$	0	0
$313$	−0.928203	−0.0524651	−0.0262326	−	0.999656i	$-0.508351\pi$
$313$	−0.928203	−0.0524651	−0.0262326	+	0.999656i	$0.508351\pi$
$314$	0	0
$315$	− 0.732051i	− 0.0412464i
$316$	0	0
$317$	18.3923i	1.03301i	0.856283	+	0.516507i	$0.172768\pi$
$317$	18.3923i	1.03301i	−0.856283	+	0.516507i	$0.827232\pi$
$318$	0	0
$319$	−20.3923	−1.14175
$320$	0	0
$321$	8.19615	0.457465
$322$	0	0
$323$	14.5359i	0.808799i
$324$	0	0
$325$	− 17.8564i	− 0.990495i
$326$	0	0
$327$	10.9282	0.604331
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	− 12.9282i	− 0.710598i	−0.934753	−	0.355299i	$-0.884379\pi$
$331$	− 12.9282i	− 0.710598i	0.934753	−	0.355299i	$-0.115621\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−4.67949	−0.255668
$336$	0	0
$337$	−33.7128	−1.83645	−0.918227	−	0.396055i	$-0.870379\pi$
$337$	−33.7128	−1.83645	−0.918227	+	0.396055i	$0.870379\pi$
$338$	0	0
$339$	6.39230i	0.347182i
$340$	0	0
$341$	5.46410i	0.295898i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.46410	−0.186501
$346$	0	0
$347$	13.2679i	0.712261i	0.934436	+	0.356130i	$0.115904\pi$
$347$	13.2679i	0.712261i	−0.934436	+	0.356130i	$0.884096\pi$
$348$	0	0
$349$	15.3205i	0.820088i	0.912066	+	0.410044i	$0.134487\pi$
$349$	15.3205i	0.820088i	−0.912066	+	0.410044i	$0.865513\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	18.3397	0.976126	0.488063	−	0.872808i	$-0.337704\pi$
$353$	18.3397	0.976126	0.488063	+	0.872808i	$0.337704\pi$
$354$	0	0
$355$	− 2.67949i	− 0.142213i
$356$	0	0
$357$	4.19615i	0.222084i
$358$	0	0
$359$	21.8038	1.15076	0.575382	−	0.817885i	$-0.304854\pi$
$359$	21.8038	1.15076	0.575382	+	0.817885i	$0.304854\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	3.53590i	0.185587i
$364$	0	0
$365$	− 6.24871i	− 0.327072i
$366$	0	0
$367$	−13.8564	−0.723299	−0.361649	−	0.932314i	$-0.617786\pi$
$367$	−13.8564	−0.723299	−0.361649	+	0.932314i	$0.617786\pi$
$368$	0	0
$369$	−9.66025	−0.502893
$370$	0	0
$371$	− 3.46410i	− 0.179847i
$372$	0	0
$373$	12.0000i	0.621336i	0.950518	+	0.310668i	$0.100553\pi$
$373$	12.0000i	0.621336i	−0.950518	+	0.310668i	$0.899447\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	−29.8564	−1.53768
$378$	0	0
$379$	15.0718i	0.774186i	0.922041	+	0.387093i	$0.126521\pi$
$379$	15.0718i	0.774186i	−0.922041	+	0.387093i	$0.873479\pi$
$380$	0	0
$381$	− 19.3205i	− 0.989820i
$382$	0	0
$383$	28.3923	1.45078	0.725390	−	0.688339i	$-0.241660\pi$
$383$	28.3923	1.45078	0.725390	+	0.688339i	$0.241660\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	− 8.92820i	− 0.453846i
$388$	0	0
$389$	− 27.4641i	− 1.39249i	−0.717807	−	0.696243i	$-0.754854\pi$
$389$	− 27.4641i	− 1.39249i	0.717807	−	0.696243i	$-0.245146\pi$
$390$	0	0
$391$	19.8564	1.00418
$392$	0	0
$393$	4.39230	0.221562
$394$	0	0
$395$	9.07180i	0.456452i
$396$	0	0
$397$	15.3205i	0.768914i	0.923143	+	0.384457i	$0.125611\pi$
$397$	15.3205i	0.768914i	−0.923143	+	0.384457i	$0.874389\pi$
$398$	0	0
$399$	−3.46410	−0.173422
$400$	0	0
$401$	−13.3205	−0.665194	−0.332597	−	0.943069i	$-0.607925\pi$
$401$	−13.3205	−0.665194	−0.332597	+	0.943069i	$0.607925\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0.732051i	0.0363759i
$406$	0	0
$407$	−5.46410	−0.270845
$408$	0	0
$409$	−40.2487	−1.99017	−0.995085	−	0.0990210i	$-0.968429\pi$
$409$	−40.2487	−1.99017	−0.995085	+	0.0990210i	$0.968429\pi$
$410$	0	0
$411$	− 3.46410i	− 0.170872i
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	1.85641	0.0911274
$416$	0	0
$417$	−10.9282	−0.535156
$418$	0	0
$419$	− 22.9282i	− 1.12012i	−0.828453	−	0.560058i	$-0.810779\pi$
$419$	− 22.9282i	− 1.12012i	0.828453	−	0.560058i	$-0.189221\pi$
$420$	0	0
$421$	− 38.7846i	− 1.89025i	−0.326714	−	0.945123i	$-0.605942\pi$
$421$	− 38.7846i	− 1.89025i	0.326714	−	0.945123i	$-0.394058\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	18.7321	0.908638
$426$	0	0
$427$	8.39230i	0.406132i
$428$	0	0
$429$	− 10.9282i	− 0.527619i
$430$	0	0
$431$	6.19615	0.298458	0.149229	−	0.988803i	$-0.452321\pi$
$431$	6.19615	0.298458	0.149229	+	0.988803i	$0.452321\pi$
$432$	0	0
$433$	34.7846	1.67164	0.835821	−	0.549002i	$-0.184992\pi$
$433$	34.7846	1.67164	0.835821	+	0.549002i	$0.184992\pi$
$434$	0	0
$435$	5.46410i	0.261984i
$436$	0	0
$437$	16.3923i	0.784150i
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	10.0526i	0.477611i	0.971067	+	0.238806i	$0.0767559\pi$
$443$	10.0526i	0.477611i	−0.971067	+	0.238806i	$0.923244\pi$
$444$	0	0
$445$	8.92820i	0.423237i
$446$	0	0
$447$	−2.39230	−0.113152
$448$	0	0
$449$	−16.5359	−0.780377	−0.390189	−	0.920735i	$-0.627590\pi$
$449$	−16.5359	−0.780377	−0.390189	+	0.920735i	$0.627590\pi$
$450$	0	0
$451$	− 26.3923i	− 1.24277i
$452$	0	0
$453$	1.07180i	0.0503574i
$454$	0	0
$455$	2.92820	0.137276
$456$	0	0
$457$	−10.7846	−0.504483	−0.252241	−	0.967664i	$-0.581168\pi$
$457$	−10.7846	−0.504483	−0.252241	+	0.967664i	$0.581168\pi$
$458$	0	0
$459$	− 4.19615i	− 0.195860i
$460$	0	0
$461$	20.7321i	0.965588i	0.875734	+	0.482794i	$0.160378\pi$
$461$	20.7321i	0.965588i	−0.875734	+	0.482794i	$0.839622\pi$
$462$	0	0
$463$	−21.1769	−0.984175	−0.492087	−	0.870546i	$-0.663766\pi$
$463$	−21.1769	−0.984175	−0.492087	+	0.870546i	$0.663766\pi$
$464$	0	0
$465$	1.46410	0.0678961
$466$	0	0
$467$	20.3923i	0.943643i	0.881694	+	0.471822i	$0.156403\pi$
$467$	20.3923i	0.943643i	−0.881694	+	0.471822i	$0.843597\pi$
$468$	0	0
$469$	6.39230i	0.295169i
$470$	0	0
$471$	8.39230	0.386697
$472$	0	0
$473$	24.3923	1.12156
$474$	0	0
$475$	15.4641i	0.709542i
$476$	0	0
$477$	3.46410i	0.158610i
$478$	0	0
$479$	−30.6410	−1.40002	−0.700012	−	0.714131i	$-0.746822\pi$
$479$	−30.6410	−1.40002	−0.700012	+	0.714131i	$0.746822\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	4.73205i	0.215316i
$484$	0	0
$485$	− 1.75129i	− 0.0795219i
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	−4.53590	−0.205120
$490$	0	0
$491$	− 27.1244i	− 1.22411i	−0.790817	−	0.612053i	$-0.790344\pi$
$491$	− 27.1244i	− 1.22411i	0.790817	−	0.612053i	$-0.209656\pi$
$492$	0	0
$493$	− 31.3205i	− 1.41060i
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	−3.66025	−0.164185
$498$	0	0
$499$	11.8564i	0.530766i	0.964143	+	0.265383i	$0.0854983\pi$
$499$	11.8564i	0.530766i	−0.964143	+	0.265383i	$0.914502\pi$
$500$	0	0
$501$	16.3923i	0.732354i
$502$	0	0
$503$	16.3923	0.730897	0.365448	−	0.930832i	$-0.380916\pi$
$503$	16.3923	0.730897	0.365448	+	0.930832i	$0.380916\pi$
$504$	0	0
$505$	8.24871	0.367063
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	0	0
$509$	30.5885i	1.35581i	0.735150	+	0.677905i	$0.237112\pi$
$509$	30.5885i	1.35581i	−0.735150	+	0.677905i	$0.762888\pi$
$510$	0	0
$511$	−8.53590	−0.377606
$512$	0	0
$513$	3.46410	0.152944
$514$	0	0
$515$	10.2487i	0.451612i
$516$	0	0
$517$	10.9282i	0.480622i
$518$	0	0
$519$	−16.0526	−0.704629
$520$	0	0
$521$	3.80385	0.166650	0.0833248	−	0.996522i	$-0.473446\pi$
$521$	3.80385	0.166650	0.0833248	+	0.996522i	$0.473446\pi$
$522$	0	0
$523$	− 14.9282i	− 0.652765i	−0.945238	−	0.326382i	$-0.894170\pi$
$523$	− 14.9282i	− 0.652765i	0.945238	−	0.326382i	$-0.105830\pi$
$524$	0	0
$525$	4.46410i	0.194829i
$526$	0	0
$527$	−8.39230	−0.365575
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	0	0
$531$	− 8.00000i	− 0.347170i
$532$	0	0
$533$	− 38.6410i	− 1.67373i
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	20.5885	0.888458
$538$	0	0
$539$	− 2.73205i	− 0.117678i
$540$	0	0
$541$	− 26.0000i	− 1.11783i	−0.829226	−	0.558914i	$-0.811218\pi$
$541$	− 26.0000i	− 1.11783i	0.829226	−	0.558914i	$-0.188782\pi$
$542$	0	0
$543$	−18.9282	−0.812287
$544$	0	0
$545$	8.00000	0.342682
$546$	0	0
$547$	44.2487i	1.89194i	0.324257	+	0.945969i	$0.394886\pi$
$547$	44.2487i	1.89194i	−0.324257	+	0.945969i	$0.605114\pi$
$548$	0	0
$549$	− 8.39230i	− 0.358175i
$550$	0	0
$551$	25.8564	1.10152
$552$	0	0
$553$	12.3923	0.526974
$554$	0	0
$555$	1.46410i	0.0621477i
$556$	0	0
$557$	24.5359i	1.03962i	0.854282	+	0.519810i	$0.173997\pi$
$557$	24.5359i	1.03962i	−0.854282	+	0.519810i	$0.826003\pi$
$558$	0	0
$559$	35.7128	1.51049
$560$	0	0
$561$	11.4641	0.484015
$562$	0	0
$563$	46.6410i	1.96568i	0.184448	+	0.982842i	$0.440950\pi$
$563$	46.6410i	1.96568i	−0.184448	+	0.982842i	$0.559050\pi$
$564$	0	0
$565$	4.67949i	0.196868i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	35.8564	1.50318	0.751589	−	0.659631i	$-0.229288\pi$
$569$	35.8564	1.50318	0.751589	+	0.659631i	$0.229288\pi$
$570$	0	0
$571$	1.60770i	0.0672799i	0.999434	+	0.0336400i	$0.0107100\pi$
$571$	1.60770i	0.0672799i	−0.999434	+	0.0336400i	$0.989290\pi$
$572$	0	0
$573$	− 24.7321i	− 1.03320i
$574$	0	0
$575$	21.1244	0.880947
$576$	0	0
$577$	44.6410	1.85843	0.929215	−	0.369540i	$-0.120485\pi$
$577$	44.6410	1.85843	0.929215	+	0.369540i	$0.120485\pi$
$578$	0	0
$579$	− 24.3923i	− 1.01371i
$580$	0	0
$581$	− 2.53590i	− 0.105207i
$582$	0	0
$583$	−9.46410	−0.391963
$584$	0	0
$585$	−2.92820	−0.121066
$586$	0	0
$587$	27.3205i	1.12764i	0.825898	+	0.563819i	$0.190668\pi$
$587$	27.3205i	1.12764i	−0.825898	+	0.563819i	$0.809332\pi$
$588$	0	0
$589$	− 6.92820i	− 0.285472i
$590$	0	0
$591$	26.7846	1.10177
$592$	0	0
$593$	1.66025	0.0681785	0.0340892	−	0.999419i	$-0.489147\pi$
$593$	1.66025	0.0681785	0.0340892	+	0.999419i	$0.489147\pi$
$594$	0	0
$595$	3.07180i	0.125931i
$596$	0	0
$597$	8.00000i	0.327418i
$598$	0	0
$599$	2.19615	0.0897324	0.0448662	−	0.998993i	$-0.485714\pi$
$599$	2.19615	0.0897324	0.0448662	+	0.998993i	$0.485714\pi$
$600$	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
$603$	− 6.39230i	− 0.260315i
$604$	0	0
$605$	2.58846i	0.105236i
$606$	0	0
$607$	46.6410	1.89310	0.946550	−	0.322556i	$-0.104542\pi$
$607$	46.6410	1.89310	0.946550	+	0.322556i	$0.104542\pi$
$608$	0	0
$609$	7.46410	0.302461
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	− 14.1436i	− 0.571254i	−0.958341	−	0.285627i	$-0.907798\pi$
$613$	− 14.1436i	− 0.571254i	0.958341	−	0.285627i	$-0.0922019\pi$
$614$	0	0
$615$	−7.07180	−0.285162
$616$	0	0
$617$	−28.5359	−1.14881	−0.574406	−	0.818571i	$-0.694767\pi$
$617$	−28.5359	−1.14881	−0.574406	+	0.818571i	$0.694767\pi$
$618$	0	0
$619$	9.85641i	0.396162i	0.980186	+	0.198081i	$0.0634710\pi$
$619$	9.85641i	0.396162i	−0.980186	+	0.198081i	$0.936529\pi$
$620$	0	0
$621$	− 4.73205i	− 0.189891i
$622$	0	0
$623$	12.1962	0.488629
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	9.46410i	0.377960i
$628$	0	0
$629$	− 8.39230i	− 0.334623i
$630$	0	0
$631$	−29.4641	−1.17295	−0.586474	−	0.809968i	$-0.699484\pi$
$631$	−29.4641	−1.17295	−0.586474	+	0.809968i	$0.699484\pi$
$632$	0	0
$633$	−0.143594	−0.00570733
$634$	0	0
$635$	− 14.1436i	− 0.561271i
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	3.66025	0.144797
$640$	0	0
$641$	−43.4641	−1.71673	−0.858364	−	0.513041i	$-0.828519\pi$
$641$	−43.4641	−1.71673	−0.858364	+	0.513041i	$0.828519\pi$
$642$	0	0
$643$	8.24871i	0.325297i	0.986684	+	0.162649i	$0.0520037\pi$
$643$	8.24871i	0.325297i	−0.986684	+	0.162649i	$0.947996\pi$
$644$	0	0
$645$	− 6.53590i	− 0.257351i
$646$	0	0
$647$	−44.1051	−1.73395	−0.866976	−	0.498351i	$-0.833939\pi$
$647$	−44.1051	−1.73395	−0.866976	+	0.498351i	$0.833939\pi$
$648$	0	0
$649$	21.8564	0.857939
$650$	0	0
$651$	− 2.00000i	− 0.0783862i
$652$	0	0
$653$	31.8564i	1.24664i	0.781968	+	0.623319i	$0.214216\pi$
$653$	31.8564i	1.24664i	−0.781968	+	0.623319i	$0.785784\pi$
$654$	0	0
$655$	3.21539	0.125636
$656$	0	0
$657$	8.53590	0.333017
$658$	0	0
$659$	19.1244i	0.744979i	0.928036	+	0.372490i	$0.121496\pi$
$659$	19.1244i	0.744979i	−0.928036	+	0.372490i	$0.878504\pi$
$660$	0	0
$661$	19.6077i	0.762651i	0.924441	+	0.381325i	$0.124532\pi$
$661$	19.6077i	0.762651i	−0.924441	+	0.381325i	$0.875468\pi$
$662$	0	0
$663$	16.7846	0.651860
$664$	0	0
$665$	−2.53590	−0.0983379
$666$	0	0
$667$	− 35.3205i	− 1.36762i
$668$	0	0
$669$	− 5.07180i	− 0.196087i
$670$	0	0
$671$	22.9282	0.885133
$672$	0	0
$673$	25.1769	0.970499	0.485249	−	0.874376i	$-0.338729\pi$
$673$	25.1769	0.970499	0.485249	+	0.874376i	$0.338729\pi$
$674$	0	0
$675$	− 4.46410i	− 0.171823i
$676$	0	0
$677$	− 31.2679i	− 1.20172i	−0.799352	−	0.600862i	$-0.794824\pi$
$677$	− 31.2679i	− 1.20172i	0.799352	−	0.600862i	$-0.205176\pi$
$678$	0	0
$679$	−2.39230	−0.0918082
$680$	0	0
$681$	11.3205	0.433803
$682$	0	0
$683$	31.9090i	1.22096i	0.792031	+	0.610481i	$0.209024\pi$
$683$	31.9090i	1.22096i	−0.792031	+	0.610481i	$0.790976\pi$
$684$	0	0
$685$	− 2.53590i	− 0.0968917i
$686$	0	0
$687$	−10.9282	−0.416937
$688$	0	0
$689$	−13.8564	−0.527887
$690$	0	0
$691$	− 24.7846i	− 0.942851i	−0.881906	−	0.471425i	$-0.843740\pi$
$691$	− 24.7846i	− 0.942851i	0.881906	−	0.471425i	$-0.156260\pi$
$692$	0	0
$693$	2.73205i	0.103782i
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	40.5359	1.53541
$698$	0	0
$699$	12.9282i	0.488990i
$700$	0	0
$701$	− 33.7128i	− 1.27332i	−0.771147	−	0.636658i	$-0.780316\pi$
$701$	− 33.7128i	− 1.27332i	0.771147	−	0.636658i	$-0.219684\pi$
$702$	0	0
$703$	6.92820	0.261302
$704$	0	0
$705$	2.92820	0.110283
$706$	0	0
$707$	− 11.2679i	− 0.423775i
$708$	0	0
$709$	2.92820i	0.109971i	0.998487	+	0.0549855i	$0.0175113\pi$
$709$	2.92820i	0.109971i	−0.998487	+	0.0549855i	$0.982489\pi$
$710$	0	0
$711$	−12.3923	−0.464748
$712$	0	0
$713$	−9.46410	−0.354433
$714$	0	0
$715$	− 8.00000i	− 0.299183i
$716$	0	0
$717$	− 27.6603i	− 1.03299i
$718$	0	0
$719$	−17.4641	−0.651301	−0.325651	−	0.945490i	$-0.605583\pi$
$719$	−17.4641	−0.651301	−0.325651	+	0.945490i	$0.605583\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	0	0
$723$	− 7.46410i	− 0.277593i
$724$	0	0
$725$	− 33.3205i	− 1.23749i
$726$	0	0
$727$	−24.6410	−0.913885	−0.456942	−	0.889496i	$-0.651055\pi$
$727$	−24.6410	−0.913885	−0.456942	+	0.889496i	$0.651055\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	37.4641i	1.38566i
$732$	0	0
$733$	− 46.9282i	− 1.73333i	−0.498888	−	0.866666i	$-0.666258\pi$
$733$	− 46.9282i	− 1.73333i	0.498888	−	0.866666i	$-0.333742\pi$
$734$	0	0
$735$	−0.732051	−0.0270021
$736$	0	0
$737$	17.4641	0.643298
$738$	0	0
$739$	− 14.3923i	− 0.529429i	−0.964327	−	0.264715i	$-0.914722\pi$
$739$	− 14.3923i	− 0.529429i	0.964327	−	0.264715i	$-0.0852778\pi$
$740$	0	0
$741$	13.8564i	0.509028i
$742$	0	0
$743$	29.8038	1.09340	0.546699	−	0.837329i	$-0.315884\pi$
$743$	29.8038	1.09340	0.546699	+	0.837329i	$0.315884\pi$
$744$	0	0
$745$	−1.75129	−0.0641623
$746$	0	0
$747$	2.53590i	0.0927837i
$748$	0	0
$749$	− 8.19615i	− 0.299481i
$750$	0	0
$751$	5.07180	0.185072	0.0925362	−	0.995709i	$-0.470503\pi$
$751$	5.07180	0.185072	0.0925362	+	0.995709i	$0.470503\pi$
$752$	0	0
$753$	23.3205	0.849847
$754$	0	0
$755$	0.784610i	0.0285549i
$756$	0	0
$757$	16.0000i	0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$	16.0000i	0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$	0	0
$759$	12.9282	0.469264
$760$	0	0
$761$	13.2679	0.480963	0.240481	−	0.970654i	$-0.422695\pi$
$761$	13.2679	0.480963	0.240481	+	0.970654i	$0.422695\pi$
$762$	0	0
$763$	− 10.9282i	− 0.395628i
$764$	0	0
$765$	− 3.07180i	− 0.111061i
$766$	0	0
$767$	32.0000	1.15545
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	3.80385i	0.136992i
$772$	0	0
$773$	27.3731i	0.984541i	0.870442	+	0.492270i	$0.163833\pi$
$773$	27.3731i	0.984541i	−0.870442	+	0.492270i	$0.836167\pi$
$774$	0	0
$775$	−8.92820	−0.320711
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	33.4641i	1.19898i
$780$	0	0
$781$	10.0000i	0.357828i
$782$	0	0
$783$	−7.46410	−0.266745
$784$	0	0
$785$	6.14359	0.219274
$786$	0	0
$787$	13.0718i	0.465959i	0.972482	+	0.232980i	$0.0748475\pi$
$787$	13.0718i	0.465959i	−0.972482	+	0.232980i	$0.925152\pi$
$788$	0	0
$789$	6.19615i	0.220589i
$790$	0	0
$791$	6.39230	0.227284
$792$	0	0
$793$	33.5692	1.19208
$794$	0	0
$795$	2.53590i	0.0899390i
$796$	0	0
$797$	4.33975i	0.153722i	0.997042	+	0.0768608i	$0.0244897\pi$
$797$	4.33975i	0.153722i	−0.997042	+	0.0768608i	$0.975510\pi$
$798$	0	0
$799$	−16.7846	−0.593797
$800$	0	0
$801$	−12.1962	−0.430930
$802$	0	0
$803$	23.3205i	0.822963i
$804$	0	0
$805$	3.46410i	0.122094i
$806$	0	0
$807$	−19.2679	−0.678264
$808$	0	0
$809$	6.67949	0.234838	0.117419	−	0.993082i	$-0.462538\pi$
$809$	6.67949	0.234838	0.117419	+	0.993082i	$0.462538\pi$
$810$	0	0
$811$	− 20.7846i	− 0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$	− 20.7846i	− 0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$	0	0
$813$	− 15.0718i	− 0.528591i
$814$	0	0
$815$	−3.32051	−0.116312
$816$	0	0
$817$	−30.9282	−1.08204
$818$	0	0
$819$	4.00000i	0.139771i
$820$	0	0
$821$	53.3205i	1.86090i	0.366421	+	0.930449i	$0.380583\pi$
$821$	53.3205i	1.86090i	−0.366421	+	0.930449i	$0.619417\pi$
$822$	0	0
$823$	−0.392305	−0.0136749	−0.00683744	−	0.999977i	$-0.502176\pi$
$823$	−0.392305	−0.0136749	−0.00683744	+	0.999977i	$0.502176\pi$
$824$	0	0
$825$	12.1962	0.424616
$826$	0	0
$827$	4.98076i	0.173198i	0.996243	+	0.0865990i	$0.0275999\pi$
$827$	4.98076i	0.173198i	−0.996243	+	0.0865990i	$0.972400\pi$
$828$	0	0
$829$	− 32.7846i	− 1.13866i	−0.822110	−	0.569328i	$-0.807203\pi$
$829$	− 32.7846i	− 1.13866i	0.822110	−	0.569328i	$-0.192797\pi$
$830$	0	0
$831$	−16.9282	−0.587233
$832$	0	0
$833$	4.19615	0.145388
$834$	0	0
$835$	12.0000i	0.415277i
$836$	0	0
$837$	2.00000i	0.0691301i
$838$	0	0
$839$	32.1051	1.10839	0.554196	−	0.832386i	$-0.313026\pi$
$839$	32.1051	1.10839	0.554196	+	0.832386i	$0.313026\pi$
$840$	0	0
$841$	−26.7128	−0.921131
$842$	0	0
$843$	− 28.9282i	− 0.996340i
$844$	0	0
$845$	− 2.19615i	− 0.0755499i
$846$	0	0
$847$	3.53590	0.121495
$848$	0	0
$849$	−2.39230	−0.0821037
$850$	0	0
$851$	− 9.46410i	− 0.324425i
$852$	0	0
$853$	18.2487i	0.624824i	0.949947	+	0.312412i	$0.101137\pi$
$853$	18.2487i	0.624824i	−0.949947	+	0.312412i	$0.898863\pi$
$854$	0	0
$855$	2.53590	0.0867259
$856$	0	0
$857$	−2.73205	−0.0933251	−0.0466625	−	0.998911i	$-0.514859\pi$
$857$	−2.73205	−0.0933251	−0.0466625	+	0.998911i	$0.514859\pi$
$858$	0	0
$859$	51.1769i	1.74613i	0.487600	+	0.873067i	$0.337872\pi$
$859$	51.1769i	1.74613i	−0.487600	+	0.873067i	$0.662128\pi$
$860$	0	0
$861$	9.66025i	0.329221i
$862$	0	0
$863$	26.8756	0.914858	0.457429	−	0.889246i	$-0.348770\pi$
$863$	26.8756	0.914858	0.457429	+	0.889246i	$0.348770\pi$
$864$	0	0
$865$	−11.7513	−0.399556
$866$	0	0
$867$	0.607695i	0.0206384i
$868$	0	0
$869$	− 33.8564i	− 1.14850i
$870$	0	0
$871$	25.5692	0.866380
$872$	0	0
$873$	2.39230	0.0809673
$874$	0	0
$875$	6.92820i	0.234216i
$876$	0	0
$877$	− 9.07180i	− 0.306333i	−0.988200	−	0.153166i	$-0.951053\pi$
$877$	− 9.07180i	− 0.306333i	0.988200	−	0.153166i	$-0.0489471\pi$
$878$	0	0
$879$	13.8038	0.465592
$880$	0	0
$881$	8.87564	0.299028	0.149514	−	0.988760i	$-0.452229\pi$
$881$	8.87564	0.299028	0.149514	+	0.988760i	$0.452229\pi$
$882$	0	0
$883$	− 12.9282i	− 0.435069i	−0.976053	−	0.217534i	$-0.930199\pi$
$883$	− 12.9282i	− 0.435069i	0.976053	−	0.217534i	$-0.0698014\pi$
$884$	0	0
$885$	− 5.85641i	− 0.196861i
$886$	0	0
$887$	6.24871	0.209811	0.104906	−	0.994482i	$-0.466546\pi$
$887$	6.24871	0.209811	0.104906	+	0.994482i	$0.466546\pi$
$888$	0	0
$889$	−19.3205	−0.647989
$890$	0	0
$891$	− 2.73205i	− 0.0915271i
$892$	0	0
$893$	− 13.8564i	− 0.463687i
$894$	0	0
$895$	15.0718	0.503795
$896$	0	0
$897$	18.9282	0.631994
$898$	0	0
$899$	14.9282i	0.497883i
$900$	0	0
$901$	− 14.5359i	− 0.484261i
$902$	0	0
$903$	−8.92820	−0.297112
$904$	0	0
$905$	−13.8564	−0.460603
$906$	0	0
$907$	15.8564i	0.526503i	0.964727	+	0.263252i	$0.0847950\pi$
$907$	15.8564i	0.526503i	−0.964727	+	0.263252i	$0.915205\pi$
$908$	0	0
$909$	11.2679i	0.373734i
$910$	0	0
$911$	33.9090	1.12345	0.561727	−	0.827323i	$-0.310137\pi$
$911$	33.9090	1.12345	0.561727	+	0.827323i	$0.310137\pi$
$912$	0	0
$913$	−6.92820	−0.229290
$914$	0	0
$915$	− 6.14359i	− 0.203101i
$916$	0	0
$917$	− 4.39230i	− 0.145047i
$918$	0	0
$919$	−10.9282	−0.360488	−0.180244	−	0.983622i	$-0.557689\pi$
$919$	−10.9282	−0.360488	−0.180244	+	0.983622i	$0.557689\pi$
$920$	0	0
$921$	25.3205	0.834339
$922$	0	0
$923$	14.6410i	0.481915i
$924$	0	0
$925$	− 8.92820i	− 0.293558i
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	−44.5885	−1.46290	−0.731450	−	0.681895i	$-0.761156\pi$
$929$	−44.5885	−1.46290	−0.731450	+	0.681895i	$0.761156\pi$
$930$	0	0
$931$	3.46410i	0.113531i
$932$	0	0
$933$	− 31.7128i	− 1.03823i
$934$	0	0
$935$	8.39230	0.274458
$936$	0	0
$937$	−12.1436	−0.396714	−0.198357	−	0.980130i	$-0.563561\pi$
$937$	−12.1436	−0.396714	−0.198357	+	0.980130i	$0.563561\pi$
$938$	0	0
$939$	− 0.928203i	− 0.0302908i
$940$	0	0
$941$	57.2295i	1.86563i	0.360359	+	0.932814i	$0.382654\pi$
$941$	57.2295i	1.86563i	−0.360359	+	0.932814i	$0.617346\pi$
$942$	0	0
$943$	45.7128	1.48861
$944$	0	0
$945$	0.732051	0.0238136
$946$	0	0
$947$	4.58846i	0.149105i	0.997217	+	0.0745524i	$0.0237528\pi$
$947$	4.58846i	0.149105i	−0.997217	+	0.0745524i	$0.976247\pi$
$948$	0	0
$949$	34.1436i	1.10835i
$950$	0	0
$951$	−18.3923	−0.596411
$952$	0	0
$953$	−26.7846	−0.867639	−0.433819	−	0.901000i	$-0.642834\pi$
$953$	−26.7846	−0.867639	−0.433819	+	0.901000i	$0.642834\pi$
$954$	0	0
$955$	− 18.1051i	− 0.585868i
$956$	0	0
$957$	− 20.3923i	− 0.659190i
$958$	0	0
$959$	−3.46410	−0.111862
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	8.19615i	0.264117i
$964$	0	0
$965$	− 17.8564i	− 0.574818i
$966$	0	0
$967$	−34.5359	−1.11060	−0.555300	−	0.831650i	$-0.687396\pi$
$967$	−34.5359	−1.11060	−0.555300	+	0.831650i	$0.687396\pi$
$968$	0	0
$969$	−14.5359	−0.466960
$970$	0	0
$971$	8.78461i	0.281912i	0.990016	+	0.140956i	$0.0450175\pi$
$971$	8.78461i	0.281912i	−0.990016	+	0.140956i	$0.954982\pi$
$972$	0	0
$973$	10.9282i	0.350342i
$974$	0	0
$975$	17.8564	0.571863
$976$	0	0
$977$	−54.4974	−1.74353	−0.871764	−	0.489927i	$-0.837023\pi$
$977$	−54.4974	−1.74353	−0.871764	+	0.489927i	$0.837023\pi$
$978$	0	0
$979$	− 33.3205i	− 1.06493i
$980$	0	0
$981$	10.9282i	0.348911i
$982$	0	0
$983$	−4.78461	−0.152605	−0.0763027	−	0.997085i	$-0.524312\pi$
$983$	−4.78461	−0.152605	−0.0763027	+	0.997085i	$0.524312\pi$
$984$	0	0
$985$	19.6077	0.624753
$986$	0	0
$987$	− 4.00000i	− 0.127321i
$988$	0	0
$989$	42.2487i	1.34343i
$990$	0	0
$991$	−52.7846	−1.67676	−0.838379	−	0.545087i	$-0.816496\pi$
$991$	−52.7846	−1.67676	−0.838379	+	0.545087i	$0.816496\pi$
$992$	0	0
$993$	12.9282	0.410264
$994$	0	0
$995$	5.85641i	0.185661i
$996$	0	0
$997$	− 31.3205i	− 0.991930i	−0.868342	−	0.495965i	$-0.834814\pi$
$997$	− 31.3205i	− 0.991930i	0.868342	−	0.495965i	$-0.165186\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.c.h.673.4	yes	4
3.2	odd	2		4032.2.c.o.2017.2		4
4.3	odd	2		1344.2.c.e.673.2	✓	4
8.3	odd	2		1344.2.c.e.673.3	yes	4
8.5	even	2	inner	1344.2.c.h.673.1	yes	4
12.11	even	2		4032.2.c.l.2017.2		4
16.3	odd	4		5376.2.a.z.1.2		2
16.5	even	4		5376.2.a.bd.1.1		2
16.11	odd	4		5376.2.a.t.1.1		2
16.13	even	4		5376.2.a.n.1.2		2
24.5	odd	2		4032.2.c.o.2017.3		4
24.11	even	2		4032.2.c.l.2017.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.c.e.673.2	✓	4	4.3	odd	2
1344.2.c.e.673.3	yes	4	8.3	odd	2
1344.2.c.h.673.1	yes	4	8.5	even	2	inner
1344.2.c.h.673.4	yes	4	1.1	even	1	trivial
4032.2.c.l.2017.2		4	12.11	even	2
4032.2.c.l.2017.3		4	24.11	even	2
4032.2.c.o.2017.2		4	3.2	odd	2
4032.2.c.o.2017.3		4	24.5	odd	2
5376.2.a.n.1.2		2	16.13	even	4
5376.2.a.t.1.1		2	16.11	odd	4
5376.2.a.z.1.2		2	16.3	odd	4
5376.2.a.bd.1.1		2	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +0.732051i q^{5} +1.00000 q^{7} -1.00000 q^{9} -2.73205i q^{11} -4.00000i q^{13} -0.732051 q^{15} +4.19615 q^{17} +3.46410i q^{19} +1.00000i q^{21} +4.73205 q^{23} +4.46410 q^{25} -1.00000i q^{27} -7.46410i q^{29} -2.00000 q^{31} +2.73205 q^{33} +0.732051i q^{35} -2.00000i q^{37} +4.00000 q^{39} +9.66025 q^{41} +8.92820i q^{43} -0.732051i q^{45} -4.00000 q^{47} +1.00000 q^{49} +4.19615i q^{51} -3.46410i q^{53} +2.00000 q^{55} -3.46410 q^{57} +8.00000i q^{59} +8.39230i q^{61} -1.00000 q^{63} +2.92820 q^{65} +6.39230i q^{67} +4.73205i q^{69} -3.66025 q^{71} -8.53590 q^{73} +4.46410i q^{75} -2.73205i q^{77} +12.3923 q^{79} +1.00000 q^{81} -2.53590i q^{83} +3.07180i q^{85} +7.46410 q^{87} +12.1962 q^{89} -4.00000i q^{91} -2.00000i q^{93} -2.53590 q^{95} -2.39230 q^{97} +2.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 4 q^{9} + 4 q^{15} - 4 q^{17} + 12 q^{23} + 4 q^{25} - 8 q^{31} + 4 q^{33} + 16 q^{39} + 4 q^{41} - 16 q^{47} + 4 q^{49} + 8 q^{55} - 4 q^{63} - 16 q^{65} + 20 q^{71} - 48 q^{73} + 8 q^{79}+ \cdots + 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 4 * q^9 + 4 * q^15 - 4 * q^17 + 12 * q^23 + 4 * q^25 - 8 * q^31 + 4 * q^33 + 16 * q^39 + 4 * q^41 - 16 * q^47 + 4 * q^49 + 8 * q^55 - 4 * q^63 - 16 * q^65 + 20 * q^71 - 48 * q^73 + 8 * q^79 + 4 * q^81 + 16 * q^87 + 28 * q^89 - 24 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more