LMFDB - Embedded newform 7448.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.a (trivial)

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:	yes
Analytic conductor:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +2.27492 q^{11} -2.00000 q^{13} +2.00000 q^{15} -7.27492 q^{17} +1.00000 q^{19} -7.54983 q^{23} -4.00000 q^{25} -4.00000 q^{27} -4.00000 q^{29} +4.00000 q^{31} +4.54983 q^{33} +2.00000 q^{37} -4.00000 q^{39} +4.54983 q^{41} -1.00000 q^{43} +1.00000 q^{45} +2.27492 q^{47} -14.5498 q^{51} -6.54983 q^{53} +2.27492 q^{55} +2.00000 q^{57} +4.00000 q^{59} -12.2749 q^{61} -2.00000 q^{65} -15.0997 q^{69} +10.5498 q^{71} -3.72508 q^{73} -8.00000 q^{75} +0.549834 q^{79} -11.0000 q^{81} -11.5498 q^{83} -7.27492 q^{85} -8.00000 q^{87} -16.5498 q^{89} +8.00000 q^{93} +1.00000 q^{95} -8.54983 q^{97} +2.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q + 4 q^{3} + 2 q^{5} + 2 q^{9} - 3 q^{11} - 4 q^{13} + 4 q^{15} - 7 q^{17} + 2 q^{19} - 8 q^{25} - 8 q^{27} - 8 q^{29} + 8 q^{31} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 6 q^{41} - 2 q^{43} + 2 q^{45} - 3 q^{47} - 14 q^{51} + 2 q^{53} - 3 q^{55} + 4 q^{57} + 8 q^{59} - 17 q^{61} - 4 q^{65} + 6 q^{71} - 15 q^{73} - 16 q^{75} - 14 q^{79} - 22 q^{81} - 8 q^{83} - 7 q^{85} - 16 q^{87} - 18 q^{89} + 16 q^{93} + 2 q^{95} - 2 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^5 + 2 * q^9 - 3 * q^11 - 4 * q^13 + 4 * q^15 - 7 * q^17 + 2 * q^19 - 8 * q^25 - 8 * q^27 - 8 * q^29 + 8 * q^31 - 6 * q^33 + 4 * q^37 - 8 * q^39 - 6 * q^41 - 2 * q^43 + 2 * q^45 - 3 * q^47 - 14 * q^51 + 2 * q^53 - 3 * q^55 + 4 * q^57 + 8 * q^59 - 17 * q^61 - 4 * q^65 + 6 * q^71 - 15 * q^73 - 16 * q^75 - 14 * q^79 - 22 * q^81 - 8 * q^83 - 7 * q^85 - 16 * q^87 - 18 * q^89 + 16 * q^93 + 2 * q^95 - 2 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.27492	0.685913	0.342957	−	0.939351i	$-0.388572\pi$
$11$	2.27492	0.685913	0.342957	+	0.939351i	$0.388572\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−7.27492	−1.76443	−0.882213	−	0.470850i	$-0.843947\pi$
$17$	−7.27492	−1.76443	−0.882213	+	0.470850i	$0.843947\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.54983	−1.57425	−0.787125	−	0.616794i	$-0.788431\pi$
$23$	−7.54983	−1.57425	−0.787125	+	0.616794i	$0.788431\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	4.54983	0.792025
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	4.54983	0.710565	0.355282	−	0.934759i	$-0.384385\pi$
$41$	4.54983	0.710565	0.355282	+	0.934759i	$0.384385\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.27492	0.331831	0.165915	−	0.986140i	$-0.446942\pi$
$47$	2.27492	0.331831	0.165915	+	0.986140i	$0.446942\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−14.5498	−2.03738
$52$	0	0
$53$	−6.54983	−0.899689	−0.449844	−	0.893107i	$-0.648521\pi$
$53$	−6.54983	−0.899689	−0.449844	+	0.893107i	$0.648521\pi$
$54$	0	0
$55$	2.27492	0.306750
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−12.2749	−1.57164	−0.785821	−	0.618454i	$-0.787759\pi$
$61$	−12.2749	−1.57164	−0.785821	+	0.618454i	$0.787759\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−15.0997	−1.81779
$70$	0	0
$71$	10.5498	1.25204	0.626018	−	0.779809i	$-0.284684\pi$
$71$	10.5498	1.25204	0.626018	+	0.779809i	$0.284684\pi$
$72$	0	0
$73$	−3.72508	−0.435988	−0.217994	−	0.975950i	$-0.569951\pi$
$73$	−3.72508	−0.435988	−0.217994	+	0.975950i	$0.569951\pi$
$74$	0	0
$75$	−8.00000	−0.923760
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.549834	0.0618612	0.0309306	−	0.999522i	$-0.490153\pi$
$79$	0.549834	0.0618612	0.0309306	+	0.999522i	$0.490153\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−11.5498	−1.26776	−0.633880	−	0.773432i	$-0.718538\pi$
$83$	−11.5498	−1.26776	−0.633880	+	0.773432i	$0.718538\pi$
$84$	0	0
$85$	−7.27492	−0.789076
$86$	0	0
$87$	−8.00000	−0.857690
$88$	0	0
$89$	−16.5498	−1.75428	−0.877139	−	0.480236i	$-0.840551\pi$
$89$	−16.5498	−1.75428	−0.877139	+	0.480236i	$0.840551\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−8.54983	−0.868104	−0.434052	−	0.900888i	$-0.642917\pi$
$97$	−8.54983	−0.868104	−0.434052	+	0.900888i	$0.642917\pi$
$98$	0	0
$99$	2.27492	0.228638
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.45017	−0.333540	−0.166770	−	0.985996i	$-0.553334\pi$
$107$	−3.45017	−0.333540	−0.166770	+	0.985996i	$0.553334\pi$
$108$	0	0
$109$	−10.5498	−1.01049	−0.505245	−	0.862976i	$-0.668598\pi$
$109$	−10.5498	−1.01049	−0.505245	+	0.862976i	$0.668598\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	18.5498	1.74502	0.872511	−	0.488595i	$-0.162490\pi$
$113$	18.5498	1.74502	0.872511	+	0.488595i	$0.162490\pi$
$114$	0	0
$115$	−7.54983	−0.704026
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.82475	−0.529523
$122$	0	0
$123$	9.09967	0.820490
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−15.0997	−1.33988	−0.669939	−	0.742416i	$-0.733680\pi$
$127$	−15.0997	−1.33988	−0.669939	+	0.742416i	$0.733680\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	15.8248	1.38261	0.691307	−	0.722561i	$-0.257035\pi$
$131$	15.8248	1.38261	0.691307	+	0.722561i	$0.257035\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	3.72508	0.318255	0.159128	−	0.987258i	$-0.449132\pi$
$137$	3.72508	0.318255	0.159128	+	0.987258i	$0.449132\pi$
$138$	0	0
$139$	10.8248	0.918143	0.459072	−	0.888399i	$-0.348182\pi$
$139$	10.8248	0.918143	0.459072	+	0.888399i	$0.348182\pi$
$140$	0	0
$141$	4.54983	0.383165
$142$	0	0
$143$	−4.54983	−0.380476
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0997	1.15509	0.577545	−	0.816359i	$-0.304011\pi$
$149$	14.0997	1.15509	0.577545	+	0.816359i	$0.304011\pi$
$150$	0	0
$151$	−8.54983	−0.695776	−0.347888	−	0.937536i	$-0.613101\pi$
$151$	−8.54983	−0.695776	−0.347888	+	0.937536i	$0.613101\pi$
$152$	0	0
$153$	−7.27492	−0.588142
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	16.0997	1.28489	0.642447	−	0.766330i	$-0.277919\pi$
$157$	16.0997	1.28489	0.642447	+	0.766330i	$0.277919\pi$
$158$	0	0
$159$	−13.0997	−1.03887
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.45017	−0.505216	−0.252608	−	0.967569i	$-0.581288\pi$
$163$	−6.45017	−0.505216	−0.252608	+	0.967569i	$0.581288\pi$
$164$	0	0
$165$	4.54983	0.354204
$166$	0	0
$167$	17.0997	1.32321	0.661606	−	0.749852i	$-0.269875\pi$
$167$	17.0997	1.32321	0.661606	+	0.749852i	$0.269875\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−5.45017	−0.414368	−0.207184	−	0.978302i	$-0.566430\pi$
$173$	−5.45017	−0.414368	−0.207184	+	0.978302i	$0.566430\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	15.0997	1.12860	0.564301	−	0.825569i	$-0.309146\pi$
$179$	15.0997	1.12860	0.564301	+	0.825569i	$0.309146\pi$
$180$	0	0
$181$	14.5498	1.08148	0.540740	−	0.841190i	$-0.318144\pi$
$181$	14.5498	1.08148	0.540740	+	0.841190i	$0.318144\pi$
$182$	0	0
$183$	−24.5498	−1.81478
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	−16.5498	−1.21024
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	0	0
$193$	27.0997	1.95068	0.975338	−	0.220715i	$-0.0708390\pi$
$193$	27.0997	1.95068	0.975338	+	0.220715i	$0.0708390\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−18.0997	−1.28955	−0.644774	−	0.764373i	$-0.723049\pi$
$197$	−18.0997	−1.28955	−0.644774	+	0.764373i	$0.723049\pi$
$198$	0	0
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.54983	0.317774
$206$	0	0
$207$	−7.54983	−0.524750
$208$	0	0
$209$	2.27492	0.157359
$210$	0	0
$211$	19.6495	1.35273	0.676364	−	0.736568i	$-0.263555\pi$
$211$	19.6495	1.35273	0.676364	+	0.736568i	$0.263555\pi$
$212$	0	0
$213$	21.0997	1.44573
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.45017	−0.503436
$220$	0	0
$221$	14.5498	0.978728
$222$	0	0
$223$	−13.0997	−0.877219	−0.438609	−	0.898678i	$-0.644529\pi$
$223$	−13.0997	−0.877219	−0.438609	+	0.898678i	$0.644529\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	24.5498	1.62943	0.814715	−	0.579862i	$-0.196893\pi$
$227$	24.5498	1.62943	0.814715	+	0.579862i	$0.196893\pi$
$228$	0	0
$229$	8.72508	0.576570	0.288285	−	0.957545i	$-0.406915\pi$
$229$	8.72508	0.576570	0.288285	+	0.957545i	$0.406915\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.27492	−0.214547	−0.107273	−	0.994230i	$-0.534212\pi$
$233$	−3.27492	−0.214547	−0.107273	+	0.994230i	$0.534212\pi$
$234$	0	0
$235$	2.27492	0.148399
$236$	0	0
$237$	1.09967	0.0714312
$238$	0	0
$239$	−14.3746	−0.929815	−0.464907	−	0.885359i	$-0.653912\pi$
$239$	−14.3746	−0.929815	−0.464907	+	0.885359i	$0.653912\pi$
$240$	0	0
$241$	4.54983	0.293081	0.146540	−	0.989205i	$-0.453186\pi$
$241$	4.54983	0.293081	0.146540	+	0.989205i	$0.453186\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−23.0997	−1.46388
$250$	0	0
$251$	−2.09967	−0.132530	−0.0662650	−	0.997802i	$-0.521108\pi$
$251$	−2.09967	−0.132530	−0.0662650	+	0.997802i	$0.521108\pi$
$252$	0	0
$253$	−17.1752	−1.07980
$254$	0	0
$255$	−14.5498	−0.911146
$256$	0	0
$257$	7.45017	0.464729	0.232364	−	0.972629i	$-0.425354\pi$
$257$	7.45017	0.464729	0.232364	+	0.972629i	$0.425354\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	0.175248	0.0108063	0.00540314	−	0.999985i	$-0.498280\pi$
$263$	0.175248	0.0108063	0.00540314	+	0.999985i	$0.498280\pi$
$264$	0	0
$265$	−6.54983	−0.402353
$266$	0	0
$267$	−33.0997	−2.02567
$268$	0	0
$269$	9.45017	0.576187	0.288093	−	0.957602i	$-0.406979\pi$
$269$	9.45017	0.576187	0.288093	+	0.957602i	$0.406979\pi$
$270$	0	0
$271$	−2.45017	−0.148837	−0.0744185	−	0.997227i	$-0.523710\pi$
$271$	−2.45017	−0.148837	−0.0744185	+	0.997227i	$0.523710\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.09967	−0.548731
$276$	0	0
$277$	−8.27492	−0.497192	−0.248596	−	0.968607i	$-0.579969\pi$
$277$	−8.27492	−0.497192	−0.248596	+	0.968607i	$0.579969\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−14.5498	−0.867970	−0.433985	−	0.900920i	$-0.642893\pi$
$281$	−14.5498	−0.867970	−0.433985	+	0.900920i	$0.642893\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	0	0
$288$	0	0
$289$	35.9244	2.11320
$290$	0	0
$291$	−17.0997	−1.00240
$292$	0	0
$293$	17.6495	1.03109	0.515547	−	0.856861i	$-0.327589\pi$
$293$	17.6495	1.03109	0.515547	+	0.856861i	$0.327589\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−9.09967	−0.528016
$298$	0	0
$299$	15.0997	0.873236
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−12.2749	−0.702860
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	0	0
$311$	2.72508	0.154525	0.0772626	−	0.997011i	$-0.475382\pi$
$311$	2.72508	0.154525	0.0772626	+	0.997011i	$0.475382\pi$
$312$	0	0
$313$	−29.1993	−1.65044	−0.825222	−	0.564808i	$-0.808950\pi$
$313$	−29.1993	−1.65044	−0.825222	+	0.564808i	$0.808950\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.0997	−1.18508	−0.592538	−	0.805543i	$-0.701874\pi$
$317$	−21.0997	−1.18508	−0.592538	+	0.805543i	$0.701874\pi$
$318$	0	0
$319$	−9.09967	−0.509484
$320$	0	0
$321$	−6.90033	−0.385139
$322$	0	0
$323$	−7.27492	−0.404787
$324$	0	0
$325$	8.00000	0.443760
$326$	0	0
$327$	−21.0997	−1.16681
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.0997	−1.14937	−0.574686	−	0.818374i	$-0.694876\pi$
$337$	−21.0997	−1.14937	−0.574686	+	0.818374i	$0.694876\pi$
$338$	0	0
$339$	37.0997	2.01498
$340$	0	0
$341$	9.09967	0.492775
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−15.0997	−0.812939
$346$	0	0
$347$	−33.1993	−1.78223	−0.891117	−	0.453774i	$-0.850077\pi$
$347$	−33.1993	−1.78223	−0.891117	+	0.453774i	$0.850077\pi$
$348$	0	0
$349$	−28.3746	−1.51886	−0.759428	−	0.650591i	$-0.774521\pi$
$349$	−28.3746	−1.51886	−0.759428	+	0.650591i	$0.774521\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	−23.0997	−1.22947	−0.614736	−	0.788733i	$-0.710737\pi$
$353$	−23.0997	−1.22947	−0.614736	+	0.788733i	$0.710737\pi$
$354$	0	0
$355$	10.5498	0.559927
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.7251	−0.935494	−0.467747	−	0.883862i	$-0.654934\pi$
$359$	−17.7251	−0.935494	−0.467747	+	0.883862i	$0.654934\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−11.6495	−0.611440
$364$	0	0
$365$	−3.72508	−0.194980
$366$	0	0
$367$	26.1993	1.36759	0.683797	−	0.729672i	$-0.260327\pi$
$367$	26.1993	1.36759	0.683797	+	0.729672i	$0.260327\pi$
$368$	0	0
$369$	4.54983	0.236855
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.54983	−0.339138	−0.169569	−	0.985518i	$-0.554237\pi$
$373$	−6.54983	−0.339138	−0.169569	+	0.985518i	$0.554237\pi$
$374$	0	0
$375$	−18.0000	−0.929516
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	3.09967	0.159219	0.0796096	−	0.996826i	$-0.474633\pi$
$379$	3.09967	0.159219	0.0796096	+	0.996826i	$0.474633\pi$
$380$	0	0
$381$	−30.1993	−1.54716
$382$	0	0
$383$	−22.0000	−1.12415	−0.562074	−	0.827087i	$-0.689996\pi$
$383$	−22.0000	−1.12415	−0.562074	+	0.827087i	$0.689996\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	37.8248	1.91779	0.958896	−	0.283759i	$-0.0915817\pi$
$389$	37.8248	1.91779	0.958896	+	0.283759i	$0.0915817\pi$
$390$	0	0
$391$	54.9244	2.77765
$392$	0	0
$393$	31.6495	1.59651
$394$	0	0
$395$	0.549834	0.0276652
$396$	0	0
$397$	20.7251	1.04016	0.520081	−	0.854117i	$-0.325902\pi$
$397$	20.7251	1.04016	0.520081	+	0.854117i	$0.325902\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.0997	1.05367	0.526834	−	0.849968i	$-0.323379\pi$
$401$	21.0997	1.05367	0.526834	+	0.849968i	$0.323379\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	4.54983	0.225527
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	7.45017	0.367490
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.5498	−0.566959
$416$	0	0
$417$	21.6495	1.06018
$418$	0	0
$419$	−19.5498	−0.955072	−0.477536	−	0.878612i	$-0.658470\pi$
$419$	−19.5498	−0.955072	−0.477536	+	0.878612i	$0.658470\pi$
$420$	0	0
$421$	18.5498	0.904064	0.452032	−	0.892002i	$-0.350699\pi$
$421$	18.5498	0.904064	0.452032	+	0.892002i	$0.350699\pi$
$422$	0	0
$423$	2.27492	0.110610
$424$	0	0
$425$	29.0997	1.41154
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.09967	−0.439336
$430$	0	0
$431$	9.09967	0.438316	0.219158	−	0.975689i	$-0.429669\pi$
$431$	9.09967	0.438316	0.219158	+	0.975689i	$0.429669\pi$
$432$	0	0
$433$	−25.0997	−1.20621	−0.603107	−	0.797661i	$-0.706071\pi$
$433$	−25.0997	−1.20621	−0.603107	+	0.797661i	$0.706071\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	0	0
$437$	−7.54983	−0.361158
$438$	0	0
$439$	−34.1993	−1.63225	−0.816123	−	0.577879i	$-0.803881\pi$
$439$	−34.1993	−1.63225	−0.816123	+	0.577879i	$0.803881\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.175248	0.00832630	0.00416315	−	0.999991i	$-0.498675\pi$
$443$	0.175248	0.00832630	0.00416315	+	0.999991i	$0.498675\pi$
$444$	0	0
$445$	−16.5498	−0.784537
$446$	0	0
$447$	28.1993	1.33378
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	10.3505	0.487386
$452$	0	0
$453$	−17.0997	−0.803413
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.2749	0.574196	0.287098	−	0.957901i	$-0.407309\pi$
$457$	12.2749	0.574196	0.287098	+	0.957901i	$0.407309\pi$
$458$	0	0
$459$	29.0997	1.35826
$460$	0	0
$461$	−15.7251	−0.732390	−0.366195	−	0.930538i	$-0.619340\pi$
$461$	−15.7251	−0.732390	−0.366195	+	0.930538i	$0.619340\pi$
$462$	0	0
$463$	2.82475	0.131277	0.0656387	−	0.997843i	$-0.479092\pi$
$463$	2.82475	0.131277	0.0656387	+	0.997843i	$0.479092\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	18.2749	0.845662	0.422831	−	0.906208i	$-0.361036\pi$
$467$	18.2749	0.845662	0.422831	+	0.906208i	$0.361036\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	32.1993	1.48367
$472$	0	0
$473$	−2.27492	−0.104601
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−6.54983	−0.299896
$478$	0	0
$479$	29.5498	1.35017	0.675083	−	0.737742i	$-0.264108\pi$
$479$	29.5498	1.35017	0.675083	+	0.737742i	$0.264108\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.54983	−0.388228
$486$	0	0
$487$	−23.0997	−1.04675	−0.523373	−	0.852104i	$-0.675327\pi$
$487$	−23.0997	−1.04675	−0.523373	+	0.852104i	$0.675327\pi$
$488$	0	0
$489$	−12.9003	−0.583373
$490$	0	0
$491$	−9.54983	−0.430978	−0.215489	−	0.976506i	$-0.569135\pi$
$491$	−9.54983	−0.430978	−0.215489	+	0.976506i	$0.569135\pi$
$492$	0	0
$493$	29.0997	1.31058
$494$	0	0
$495$	2.27492	0.102250
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−29.1993	−1.30714	−0.653571	−	0.756865i	$-0.726730\pi$
$499$	−29.1993	−1.30714	−0.653571	+	0.756865i	$0.726730\pi$
$500$	0	0
$501$	34.1993	1.52791
$502$	0	0
$503$	−34.6495	−1.54494	−0.772472	−	0.635048i	$-0.780980\pi$
$503$	−34.6495	−1.54494	−0.772472	+	0.635048i	$0.780980\pi$
$504$	0	0
$505$	3.00000	0.133498
$506$	0	0
$507$	−18.0000	−0.799408
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−10.0000	−0.440653
$516$	0	0
$517$	5.17525	0.227607
$518$	0	0
$519$	−10.9003	−0.478471
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	41.6495	1.82121	0.910603	−	0.413283i	$-0.135618\pi$
$523$	41.6495	1.82121	0.910603	+	0.413283i	$0.135618\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.0997	−1.26760
$528$	0	0
$529$	34.0000	1.47826
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−9.09967	−0.394150
$534$	0	0
$535$	−3.45017	−0.149164
$536$	0	0
$537$	30.1993	1.30320
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	0	0
$543$	29.0997	1.24879
$544$	0	0
$545$	−10.5498	−0.451905
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	0	0
$549$	−12.2749	−0.523881
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	2.09967	0.0889658	0.0444829	−	0.999010i	$-0.485836\pi$
$557$	2.09967	0.0889658	0.0444829	+	0.999010i	$0.485836\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	−33.0997	−1.39747
$562$	0	0
$563$	11.4502	0.482567	0.241283	−	0.970455i	$-0.422432\pi$
$563$	11.4502	0.482567	0.241283	+	0.970455i	$0.422432\pi$
$564$	0	0
$565$	18.5498	0.780397
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.1993	−0.679112	−0.339556	−	0.940586i	$-0.610277\pi$
$569$	−16.1993	−0.679112	−0.339556	+	0.940586i	$0.610277\pi$
$570$	0	0
$571$	−20.4502	−0.855813	−0.427906	−	0.903823i	$-0.640749\pi$
$571$	−20.4502	−0.855813	−0.427906	+	0.903823i	$0.640749\pi$
$572$	0	0
$573$	−42.0000	−1.75458
$574$	0	0
$575$	30.1993	1.25940
$576$	0	0
$577$	−13.3746	−0.556791	−0.278396	−	0.960466i	$-0.589803\pi$
$577$	−13.3746	−0.556791	−0.278396	+	0.960466i	$0.589803\pi$
$578$	0	0
$579$	54.1993	2.25245
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−14.9003	−0.617109
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	2.72508	0.112476	0.0562381	−	0.998417i	$-0.482089\pi$
$587$	2.72508	0.112476	0.0562381	+	0.998417i	$0.482089\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−36.1993	−1.48904
$592$	0	0
$593$	−3.00000	−0.123195	−0.0615976	−	0.998101i	$-0.519620\pi$
$593$	−3.00000	−0.123195	−0.0615976	+	0.998101i	$0.519620\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.00000	−0.245564
$598$	0	0
$599$	15.6495	0.639421	0.319711	−	0.947515i	$-0.396414\pi$
$599$	15.6495	0.639421	0.319711	+	0.947515i	$0.396414\pi$
$600$	0	0
$601$	11.4502	0.467062	0.233531	−	0.972349i	$-0.424972\pi$
$601$	11.4502	0.467062	0.233531	+	0.972349i	$0.424972\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.82475	−0.236810
$606$	0	0
$607$	−25.6495	−1.04108	−0.520541	−	0.853837i	$-0.674270\pi$
$607$	−25.6495	−1.04108	−0.520541	+	0.853837i	$0.674270\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.54983	−0.184067
$612$	0	0
$613$	−37.8248	−1.52773	−0.763864	−	0.645378i	$-0.776700\pi$
$613$	−37.8248	−1.52773	−0.763864	+	0.645378i	$0.776700\pi$
$614$	0	0
$615$	9.09967	0.366934
$616$	0	0
$617$	−24.0997	−0.970216	−0.485108	−	0.874454i	$-0.661220\pi$
$617$	−24.0997	−0.970216	−0.485108	+	0.874454i	$0.661220\pi$
$618$	0	0
$619$	−11.5498	−0.464227	−0.232114	−	0.972689i	$-0.574564\pi$
$619$	−11.5498	−0.464227	−0.232114	+	0.972689i	$0.574564\pi$
$620$	0	0
$621$	30.1993	1.21186
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	4.54983	0.181703
$628$	0	0
$629$	−14.5498	−0.580140
$630$	0	0
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	0	0
$633$	39.2990	1.56200
$634$	0	0
$635$	−15.0997	−0.599212
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.5498	0.417345
$640$	0	0
$641$	46.5498	1.83861	0.919304	−	0.393548i	$-0.128753\pi$
$641$	46.5498	1.83861	0.919304	+	0.393548i	$0.128753\pi$
$642$	0	0
$643$	0.175248	0.00691112	0.00345556	−	0.999994i	$-0.498900\pi$
$643$	0.175248	0.00691112	0.00345556	+	0.999994i	$0.498900\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	14.0997	0.554315	0.277158	−	0.960824i	$-0.410608\pi$
$647$	14.0997	0.554315	0.277158	+	0.960824i	$0.410608\pi$
$648$	0	0
$649$	9.09967	0.357193
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.725083	−0.0283747	−0.0141873	−	0.999899i	$-0.504516\pi$
$653$	−0.725083	−0.0283747	−0.0141873	+	0.999899i	$0.504516\pi$
$654$	0	0
$655$	15.8248	0.618324
$656$	0	0
$657$	−3.72508	−0.145329
$658$	0	0
$659$	−25.4502	−0.991398	−0.495699	−	0.868494i	$-0.665088\pi$
$659$	−25.4502	−0.991398	−0.495699	+	0.868494i	$0.665088\pi$
$660$	0	0
$661$	44.5498	1.73279	0.866394	−	0.499361i	$-0.166432\pi$
$661$	44.5498	1.73279	0.866394	+	0.499361i	$0.166432\pi$
$662$	0	0
$663$	29.0997	1.13014
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.1993	1.16932
$668$	0	0
$669$	−26.1993	−1.01292
$670$	0	0
$671$	−27.9244	−1.07801
$672$	0	0
$673$	10.5498	0.406666	0.203333	−	0.979110i	$-0.434823\pi$
$673$	10.5498	0.406666	0.203333	+	0.979110i	$0.434823\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	0	0
$677$	−37.0997	−1.42586	−0.712928	−	0.701237i	$-0.752631\pi$
$677$	−37.0997	−1.42586	−0.712928	+	0.701237i	$0.752631\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	49.0997	1.88150
$682$	0	0
$683$	−17.6495	−0.675339	−0.337670	−	0.941265i	$-0.609639\pi$
$683$	−17.6495	−0.675339	−0.337670	+	0.941265i	$0.609639\pi$
$684$	0	0
$685$	3.72508	0.142328
$686$	0	0
$687$	17.4502	0.665765
$688$	0	0
$689$	13.0997	0.499058
$690$	0	0
$691$	36.9244	1.40467	0.702336	−	0.711846i	$-0.252141\pi$
$691$	36.9244	1.40467	0.702336	+	0.711846i	$0.252141\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.8248	0.410606
$696$	0	0
$697$	−33.0997	−1.25374
$698$	0	0
$699$	−6.54983	−0.247737
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	4.54983	0.171357
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15.0000	−0.563337	−0.281668	−	0.959512i	$-0.590888\pi$
$709$	−15.0000	−0.563337	−0.281668	+	0.959512i	$0.590888\pi$
$710$	0	0
$711$	0.549834	0.0206204
$712$	0	0
$713$	−30.1993	−1.13097
$714$	0	0
$715$	−4.54983	−0.170154
$716$	0	0
$717$	−28.7492	−1.07366
$718$	0	0
$719$	−26.7251	−0.996677	−0.498339	−	0.866982i	$-0.666056\pi$
$719$	−26.7251	−0.996677	−0.498339	+	0.866982i	$0.666056\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	9.09967	0.338420
$724$	0	0
$725$	16.0000	0.594225
$726$	0	0
$727$	13.9003	0.515535	0.257767	−	0.966207i	$-0.417013\pi$
$727$	13.9003	0.515535	0.257767	+	0.966207i	$0.417013\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	7.27492	0.269073
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.17525	−0.153589	−0.0767945	−	0.997047i	$-0.524469\pi$
$739$	−4.17525	−0.153589	−0.0767945	+	0.997047i	$0.524469\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−46.7492	−1.71506	−0.857530	−	0.514433i	$-0.828002\pi$
$743$	−46.7492	−1.71506	−0.857530	+	0.514433i	$0.828002\pi$
$744$	0	0
$745$	14.0997	0.516572
$746$	0	0
$747$	−11.5498	−0.422586
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	−4.19934	−0.153032
$754$	0	0
$755$	−8.54983	−0.311160
$756$	0	0
$757$	25.3746	0.922255	0.461128	−	0.887334i	$-0.347445\pi$
$757$	25.3746	0.922255	0.461128	+	0.887334i	$0.347445\pi$
$758$	0	0
$759$	−34.3505	−1.24684
$760$	0	0
$761$	43.0000	1.55875	0.779374	−	0.626559i	$-0.215537\pi$
$761$	43.0000	1.55875	0.779374	+	0.626559i	$0.215537\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−7.27492	−0.263025
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	43.1993	1.55781	0.778904	−	0.627143i	$-0.215776\pi$
$769$	43.1993	1.55781	0.778904	+	0.627143i	$0.215776\pi$
$770$	0	0
$771$	14.9003	0.536622
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.54983	0.163015
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	16.0000	0.571793
$784$	0	0
$785$	16.0997	0.574622
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	0	0
$789$	0.350497	0.0124780
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.5498	0.871790
$794$	0	0
$795$	−13.0997	−0.464597
$796$	0	0
$797$	16.5498	0.586225	0.293113	−	0.956078i	$-0.405309\pi$
$797$	16.5498	0.586225	0.293113	+	0.956078i	$0.405309\pi$
$798$	0	0
$799$	−16.5498	−0.585491
$800$	0	0
$801$	−16.5498	−0.584760
$802$	0	0
$803$	−8.47425	−0.299050
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.9003	0.665323
$808$	0	0
$809$	−18.0997	−0.636350	−0.318175	−	0.948032i	$-0.603070\pi$
$809$	−18.0997	−0.636350	−0.318175	+	0.948032i	$0.603070\pi$
$810$	0	0
$811$	22.7492	0.798831	0.399416	−	0.916770i	$-0.369213\pi$
$811$	22.7492	0.798831	0.399416	+	0.916770i	$0.369213\pi$
$812$	0	0
$813$	−4.90033	−0.171862
$814$	0	0
$815$	−6.45017	−0.225939
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.9244	0.904769	0.452384	−	0.891823i	$-0.350574\pi$
$821$	25.9244	0.904769	0.452384	+	0.891823i	$0.350574\pi$
$822$	0	0
$823$	−47.9244	−1.67054	−0.835270	−	0.549840i	$-0.814689\pi$
$823$	−47.9244	−1.67054	−0.835270	+	0.549840i	$0.814689\pi$
$824$	0	0
$825$	−18.1993	−0.633620
$826$	0	0
$827$	3.64950	0.126906	0.0634528	−	0.997985i	$-0.479789\pi$
$827$	3.64950	0.126906	0.0634528	+	0.997985i	$0.479789\pi$
$828$	0	0
$829$	45.0997	1.56638	0.783188	−	0.621785i	$-0.213592\pi$
$829$	45.0997	1.56638	0.783188	+	0.621785i	$0.213592\pi$
$830$	0	0
$831$	−16.5498	−0.574107
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17.0997	0.591758
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	7.64950	0.264090	0.132045	−	0.991244i	$-0.457846\pi$
$839$	7.64950	0.264090	0.132045	+	0.991244i	$0.457846\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−29.0997	−1.00225
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	34.0000	1.16688
$850$	0	0
$851$	−15.0997	−0.517610
$852$	0	0
$853$	11.9003	0.407460	0.203730	−	0.979027i	$-0.434694\pi$
$853$	11.9003	0.407460	0.203730	+	0.979027i	$0.434694\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	3.64950	0.124665	0.0623323	−	0.998055i	$-0.480146\pi$
$857$	3.64950	0.124665	0.0623323	+	0.998055i	$0.480146\pi$
$858$	0	0
$859$	−20.0997	−0.685792	−0.342896	−	0.939373i	$-0.611408\pi$
$859$	−20.0997	−0.685792	−0.342896	+	0.939373i	$0.611408\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.0997	−1.12673	−0.563363	−	0.826210i	$-0.690493\pi$
$863$	−33.0997	−1.12673	−0.563363	+	0.826210i	$0.690493\pi$
$864$	0	0
$865$	−5.45017	−0.185311
$866$	0	0
$867$	71.8488	2.44011
$868$	0	0
$869$	1.25083	0.0424314
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−8.54983	−0.289368
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.7492	−1.44354	−0.721768	−	0.692135i	$-0.756670\pi$
$877$	−42.7492	−1.44354	−0.721768	+	0.692135i	$0.756670\pi$
$878$	0	0
$879$	35.2990	1.19061
$880$	0	0
$881$	2.92442	0.0985262	0.0492631	−	0.998786i	$-0.484313\pi$
$881$	2.92442	0.0985262	0.0492631	+	0.998786i	$0.484313\pi$
$882$	0	0
$883$	42.0241	1.41422	0.707112	−	0.707102i	$-0.249998\pi$
$883$	42.0241	1.41422	0.707112	+	0.707102i	$0.249998\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	−8.19934	−0.275307	−0.137653	−	0.990480i	$-0.543956\pi$
$887$	−8.19934	−0.275307	−0.137653	+	0.990480i	$0.543956\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−25.0241	−0.838339
$892$	0	0
$893$	2.27492	0.0761272
$894$	0	0
$895$	15.0997	0.504726
$896$	0	0
$897$	30.1993	1.00833
$898$	0	0
$899$	−16.0000	−0.533630
$900$	0	0
$901$	47.6495	1.58744
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.5498	0.483653
$906$	0	0
$907$	37.0997	1.23187	0.615937	−	0.787795i	$-0.288778\pi$
$907$	37.0997	1.23187	0.615937	+	0.787795i	$0.288778\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	26.5498	0.879635	0.439818	−	0.898087i	$-0.355043\pi$
$911$	26.5498	0.879635	0.439818	+	0.898087i	$0.355043\pi$
$912$	0	0
$913$	−26.2749	−0.869573
$914$	0	0
$915$	−24.5498	−0.811592
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−24.4502	−0.806537	−0.403268	−	0.915082i	$-0.632126\pi$
$919$	−24.4502	−0.806537	−0.403268	+	0.915082i	$0.632126\pi$
$920$	0	0
$921$	52.0000	1.71346
$922$	0	0
$923$	−21.0997	−0.694504
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	−10.0000	−0.328443
$928$	0	0
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	5.45017	0.178430
$934$	0	0
$935$	−16.5498	−0.541237
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	−58.3987	−1.90577
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	0	0
$943$	−34.3505	−1.11861
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	7.45017	0.241843
$950$	0	0
$951$	−42.1993	−1.36841
$952$	0	0
$953$	12.9003	0.417883	0.208941	−	0.977928i	$-0.432998\pi$
$953$	12.9003	0.417883	0.208941	+	0.977928i	$0.432998\pi$
$954$	0	0
$955$	−21.0000	−0.679544
$956$	0	0
$957$	−18.1993	−0.588301
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−3.45017	−0.111180
$964$	0	0
$965$	27.0997	0.872369
$966$	0	0
$967$	−18.1993	−0.585251	−0.292626	−	0.956227i	$-0.594529\pi$
$967$	−18.1993	−0.585251	−0.292626	+	0.956227i	$0.594529\pi$
$968$	0	0
$969$	−14.5498	−0.467408
$970$	0	0
$971$	−36.5498	−1.17294	−0.586470	−	0.809971i	$-0.699483\pi$
$971$	−36.5498	−1.17294	−0.586470	+	0.809971i	$0.699483\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	16.0000	0.512410
$976$	0	0
$977$	55.2990	1.76917	0.884586	−	0.466377i	$-0.154441\pi$
$977$	55.2990	1.76917	0.884586	+	0.466377i	$0.154441\pi$
$978$	0	0
$979$	−37.6495	−1.20328
$980$	0	0
$981$	−10.5498	−0.336830
$982$	0	0
$983$	−8.19934	−0.261518	−0.130759	−	0.991414i	$-0.541741\pi$
$983$	−8.19934	−0.261518	−0.130759	+	0.991414i	$0.541741\pi$
$984$	0	0
$985$	−18.0997	−0.576703
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.54983	0.240071
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−3.00000	−0.0951064
$996$	0	0
$997$	29.8248	0.944559	0.472280	−	0.881449i	$-0.343431\pi$
$997$	29.8248	0.944559	0.472280	+	0.881449i	$0.343431\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.be.1.2		2
7.3	odd	6		1064.2.q.l.457.2	yes	4
7.5	odd	6		1064.2.q.l.305.2	✓	4
7.6	odd	2		7448.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.l.305.2	✓	4	7.5	odd	6
1064.2.q.l.457.2	yes	4	7.3	odd	6
7448.2.a.w.1.2		2	7.6	odd	2
7448.2.a.be.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +2.27492 q^{11} -2.00000 q^{13} +2.00000 q^{15} -7.27492 q^{17} +1.00000 q^{19} -7.54983 q^{23} -4.00000 q^{25} -4.00000 q^{27} -4.00000 q^{29} +4.00000 q^{31} +4.54983 q^{33} +2.00000 q^{37} -4.00000 q^{39} +4.54983 q^{41} -1.00000 q^{43} +1.00000 q^{45} +2.27492 q^{47} -14.5498 q^{51} -6.54983 q^{53} +2.27492 q^{55} +2.00000 q^{57} +4.00000 q^{59} -12.2749 q^{61} -2.00000 q^{65} -15.0997 q^{69} +10.5498 q^{71} -3.72508 q^{73} -8.00000 q^{75} +0.549834 q^{79} -11.0000 q^{81} -11.5498 q^{83} -7.27492 q^{85} -8.00000 q^{87} -16.5498 q^{89} +8.00000 q^{93} +1.00000 q^{95} -8.54983 q^{97} +2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q + 4 q^{3} + 2 q^{5} + 2 q^{9} - 3 q^{11} - 4 q^{13} + 4 q^{15} - 7 q^{17} + 2 q^{19} - 8 q^{25} - 8 q^{27} - 8 q^{29} + 8 q^{31} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 6 q^{41} - 2 q^{43} + 2 q^{45} - 3 q^{47} - 14 q^{51} + 2 q^{53} - 3 q^{55} + 4 q^{57} + 8 q^{59} - 17 q^{61} - 4 q^{65} + 6 q^{71} - 15 q^{73} - 16 q^{75} - 14 q^{79} - 22 q^{81} - 8 q^{83} - 7 q^{85} - 16 q^{87} - 18 q^{89} + 16 q^{93} + 2 q^{95} - 2 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^5 + 2 * q^9 - 3 * q^11 - 4 * q^13 + 4 * q^15 - 7 * q^17 + 2 * q^19 - 8 * q^25 - 8 * q^27 - 8 * q^29 + 8 * q^31 - 6 * q^33 + 4 * q^37 - 8 * q^39 - 6 * q^41 - 2 * q^43 + 2 * q^45 - 3 * q^47 - 14 * q^51 + 2 * q^53 - 3 * q^55 + 4 * q^57 + 8 * q^59 - 17 * q^61 - 4 * q^65 + 6 * q^71 - 15 * q^73 - 16 * q^75 - 14 * q^79 - 22 * q^81 - 8 * q^83 - 7 * q^85 - 16 * q^87 - 18 * q^89 + 16 * q^93 + 2 * q^95 - 2 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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