LMFDB - Embedded newform 7448.2.a.w.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -5.27492 q^{11} +2.00000 q^{13} +2.00000 q^{15} -0.274917 q^{17} -1.00000 q^{19} +7.54983 q^{23} -4.00000 q^{25} +4.00000 q^{27} -4.00000 q^{29} -4.00000 q^{31} +10.5498 q^{33} +2.00000 q^{37} -4.00000 q^{39} +10.5498 q^{41} -1.00000 q^{43} -1.00000 q^{45} +5.27492 q^{47} +0.549834 q^{51} +8.54983 q^{53} +5.27492 q^{55} +2.00000 q^{57} -4.00000 q^{59} +4.72508 q^{61} -2.00000 q^{65} -15.0997 q^{69} -4.54983 q^{71} +11.2749 q^{73} +8.00000 q^{75} -14.5498 q^{79} -11.0000 q^{81} -3.54983 q^{83} +0.274917 q^{85} +8.00000 q^{87} +1.45017 q^{89} +8.00000 q^{93} +1.00000 q^{95} -6.54983 q^{97} -5.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.27492	−1.59045	−0.795224	−	0.606316i	$-0.792647\pi$
$11$	−5.27492	−1.59045	−0.795224	+	0.606316i	$0.792647\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−0.274917	−0.0666772	−0.0333386	−	0.999444i	$-0.510614\pi$
$17$	−0.274917	−0.0666772	−0.0333386	+	0.999444i	$0.510614\pi$
$18$	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	$-0.211569\pi$
$23$	7.54983	1.57425	0.787125	+	0.616794i	$0.211569\pi$
$24$	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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	10.5498	1.83649
$34$	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
$41$	10.5498	1.64761	0.823804	−	0.566875i	$-0.191848\pi$
$41$	10.5498	1.64761	0.823804	+	0.566875i	$0.191848\pi$
$42$	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
$47$	5.27492	0.769426	0.384713	−	0.923036i	$-0.374300\pi$
$47$	5.27492	0.769426	0.384713	+	0.923036i	$0.374300\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.549834	0.0769922
$52$	0	0
$53$	8.54983	1.17441	0.587205	−	0.809438i	$-0.300228\pi$
$53$	8.54983	1.17441	0.587205	+	0.809438i	$0.300228\pi$
$54$	0	0
$55$	5.27492	0.711270
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	4.72508	0.604985	0.302492	−	0.953152i	$-0.402181\pi$
$61$	4.72508	0.604985	0.302492	+	0.953152i	$0.402181\pi$
$62$	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
$71$	−4.54983	−0.539966	−0.269983	−	0.962865i	$-0.587018\pi$
$71$	−4.54983	−0.539966	−0.269983	+	0.962865i	$0.587018\pi$
$72$	0	0
$73$	11.2749	1.31963	0.659815	−	0.751428i	$-0.270635\pi$
$73$	11.2749	1.31963	0.659815	+	0.751428i	$0.270635\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.5498	−1.63698	−0.818492	−	0.574518i	$-0.805190\pi$
$79$	−14.5498	−1.63698	−0.818492	+	0.574518i	$0.805190\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−3.54983	−0.389645	−0.194822	−	0.980839i	$-0.562413\pi$
$83$	−3.54983	−0.389645	−0.194822	+	0.980839i	$0.562413\pi$
$84$	0	0
$85$	0.274917	0.0298190
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	1.45017	0.153717	0.0768586	−	0.997042i	$-0.475511\pi$
$89$	1.45017	0.153717	0.0768586	+	0.997042i	$0.475511\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−6.54983	−0.665035	−0.332517	−	0.943097i	$-0.607898\pi$
$97$	−6.54983	−0.665035	−0.332517	+	0.943097i	$0.607898\pi$
$98$	0	0
$99$	−5.27492	−0.530149
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.5498	−1.79328	−0.896640	−	0.442760i	$-0.853999\pi$
$107$	−18.5498	−1.79328	−0.896640	+	0.442760i	$0.853999\pi$
$108$	0	0
$109$	4.54983	0.435795	0.217898	−	0.975972i	$-0.430080\pi$
$109$	4.54983	0.435795	0.217898	+	0.975972i	$0.430080\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	3.45017	0.324564	0.162282	−	0.986744i	$-0.448115\pi$
$113$	3.45017	0.324564	0.162282	+	0.986744i	$0.448115\pi$
$114$	0	0
$115$	−7.54983	−0.704026
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.8248	1.52952
$122$	0	0
$123$	−21.0997	−1.90249
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	15.0997	1.33988	0.669939	−	0.742416i	$-0.266320\pi$
$127$	15.0997	1.33988	0.669939	+	0.742416i	$0.266320\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	6.82475	0.596281	0.298141	−	0.954522i	$-0.403634\pi$
$131$	6.82475	0.596281	0.298141	+	0.954522i	$0.403634\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	11.2749	0.963281	0.481641	−	0.876369i	$-0.340041\pi$
$137$	11.2749	0.963281	0.481641	+	0.876369i	$0.340041\pi$
$138$	0	0
$139$	11.8248	1.00296	0.501481	−	0.865169i	$-0.332789\pi$
$139$	11.8248	1.00296	0.501481	+	0.865169i	$0.332789\pi$
$140$	0	0
$141$	−10.5498	−0.888456
$142$	0	0
$143$	−10.5498	−0.882221
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0997	−1.31894	−0.659468	−	0.751733i	$-0.729218\pi$
$149$	−16.0997	−1.31894	−0.659468	+	0.751733i	$0.729218\pi$
$150$	0	0
$151$	6.54983	0.533018	0.266509	−	0.963832i	$-0.414130\pi$
$151$	6.54983	0.533018	0.266509	+	0.963832i	$0.414130\pi$
$152$	0	0
$153$	−0.274917	−0.0222257
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	14.0997	1.12528	0.562638	−	0.826703i	$-0.309786\pi$
$157$	14.0997	1.12528	0.562638	+	0.826703i	$0.309786\pi$
$158$	0	0
$159$	−17.0997	−1.35609
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−21.5498	−1.68791	−0.843957	−	0.536411i	$-0.819780\pi$
$163$	−21.5498	−1.68791	−0.843957	+	0.536411i	$0.819780\pi$
$164$	0	0
$165$	−10.5498	−0.821303
$166$	0	0
$167$	13.0997	1.01368	0.506841	−	0.862039i	$-0.330813\pi$
$167$	13.0997	1.01368	0.506841	+	0.862039i	$0.330813\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	20.5498	1.56237	0.781187	−	0.624296i	$-0.214614\pi$
$173$	20.5498	1.56237	0.781187	+	0.624296i	$0.214614\pi$
$174$	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	$-0.690854\pi$
$179$	−15.0997	−1.12860	−0.564301	+	0.825569i	$0.690854\pi$
$180$	0	0
$181$	0.549834	0.0408689	0.0204344	−	0.999791i	$-0.493495\pi$
$181$	0.549834	0.0408689	0.0204344	+	0.999791i	$0.493495\pi$
$182$	0	0
$183$	−9.45017	−0.698576
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	1.45017	0.106047
$188$	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
$193$	−3.09967	−0.223119	−0.111560	−	0.993758i	$-0.535585\pi$
$193$	−3.09967	−0.223119	−0.111560	+	0.993758i	$0.535585\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	12.0997	0.862066	0.431033	−	0.902336i	$-0.358149\pi$
$197$	12.0997	0.862066	0.431033	+	0.902336i	$0.358149\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−10.5498	−0.736832
$206$	0	0
$207$	7.54983	0.524750
$208$	0	0
$209$	5.27492	0.364874
$210$	0	0
$211$	−25.6495	−1.76578	−0.882892	−	0.469576i	$-0.844407\pi$
$211$	−25.6495	−1.76578	−0.882892	+	0.469576i	$0.844407\pi$
$212$	0	0
$213$	9.09967	0.623499
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−22.5498	−1.52378
$220$	0	0
$221$	−0.549834	−0.0369859
$222$	0	0
$223$	−17.0997	−1.14508	−0.572539	−	0.819877i	$-0.694042\pi$
$223$	−17.0997	−1.14508	−0.572539	+	0.819877i	$0.694042\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−9.45017	−0.627230	−0.313615	−	0.949550i	$-0.601540\pi$
$227$	−9.45017	−0.627230	−0.313615	+	0.949550i	$0.601540\pi$
$228$	0	0
$229$	−16.2749	−1.07548	−0.537738	−	0.843112i	$-0.680721\pi$
$229$	−16.2749	−1.07548	−0.537738	+	0.843112i	$0.680721\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.27492	0.280059	0.140030	−	0.990147i	$-0.455280\pi$
$233$	4.27492	0.280059	0.140030	+	0.990147i	$0.455280\pi$
$234$	0	0
$235$	−5.27492	−0.344098
$236$	0	0
$237$	29.0997	1.89023
$238$	0	0
$239$	23.3746	1.51198	0.755988	−	0.654585i	$-0.227157\pi$
$239$	23.3746	1.51198	0.755988	+	0.654585i	$0.227157\pi$
$240$	0	0
$241$	10.5498	0.679575	0.339787	−	0.940502i	$-0.389645\pi$
$241$	10.5498	0.679575	0.339787	+	0.940502i	$0.389645\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	7.09967	0.449923
$250$	0	0
$251$	−28.0997	−1.77364	−0.886818	−	0.462119i	$-0.847089\pi$
$251$	−28.0997	−1.77364	−0.886818	+	0.462119i	$0.847089\pi$
$252$	0	0
$253$	−39.8248	−2.50376
$254$	0	0
$255$	−0.549834	−0.0344320
$256$	0	0
$257$	−22.5498	−1.40662	−0.703310	−	0.710883i	$-0.748295\pi$
$257$	−22.5498	−1.40662	−0.703310	+	0.710883i	$0.748295\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	22.8248	1.40743	0.703717	−	0.710480i	$-0.251522\pi$
$263$	22.8248	1.40743	0.703717	+	0.710480i	$0.251522\pi$
$264$	0	0
$265$	−8.54983	−0.525212
$266$	0	0
$267$	−2.90033	−0.177497
$268$	0	0
$269$	−24.5498	−1.49683	−0.748415	−	0.663231i	$-0.769185\pi$
$269$	−24.5498	−1.49683	−0.748415	+	0.663231i	$0.769185\pi$
$270$	0	0
$271$	17.5498	1.06608	0.533038	−	0.846091i	$-0.321050\pi$
$271$	17.5498	1.06608	0.533038	+	0.846091i	$0.321050\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.0997	1.27236
$276$	0	0
$277$	−0.725083	−0.0435660	−0.0217830	−	0.999763i	$-0.506934\pi$
$277$	−0.725083	−0.0435660	−0.0217830	+	0.999763i	$0.506934\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	0.549834	0.0328004	0.0164002	−	0.999866i	$-0.494779\pi$
$281$	0.549834	0.0328004	0.0164002	+	0.999866i	$0.494779\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9244	−0.995554
$290$	0	0
$291$	13.0997	0.767916
$292$	0	0
$293$	27.6495	1.61530	0.807651	−	0.589661i	$-0.200739\pi$
$293$	27.6495	1.61530	0.807651	+	0.589661i	$0.200739\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−21.0997	−1.22433
$298$	0	0
$299$	15.0997	0.873236
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−4.72508	−0.270557
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	0	0
$311$	−10.2749	−0.582637	−0.291319	−	0.956626i	$-0.594094\pi$
$311$	−10.2749	−0.582637	−0.291319	+	0.956626i	$0.594094\pi$
$312$	0	0
$313$	−31.1993	−1.76349	−0.881745	−	0.471726i	$-0.843631\pi$
$313$	−31.1993	−1.76349	−0.881745	+	0.471726i	$0.843631\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.09967	0.511088	0.255544	−	0.966797i	$-0.417745\pi$
$317$	9.09967	0.511088	0.255544	+	0.966797i	$0.417745\pi$
$318$	0	0
$319$	21.0997	1.18135
$320$	0	0
$321$	37.0997	2.07070
$322$	0	0
$323$	0.274917	0.0152968
$324$	0	0
$325$	−8.00000	−0.443760
$326$	0	0
$327$	−9.09967	−0.503213
$328$	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
$337$	9.09967	0.495691	0.247845	−	0.968800i	$-0.420278\pi$
$337$	9.09967	0.495691	0.247845	+	0.968800i	$0.420278\pi$
$338$	0	0
$339$	−6.90033	−0.374775
$340$	0	0
$341$	21.0997	1.14261
$342$	0	0
$343$	0	0
$344$	0	0
$345$	15.0997	0.812939
$346$	0	0
$347$	27.1993	1.46014	0.730068	−	0.683374i	$-0.239488\pi$
$347$	27.1993	1.46014	0.730068	+	0.683374i	$0.239488\pi$
$348$	0	0
$349$	−9.37459	−0.501810	−0.250905	−	0.968012i	$-0.580728\pi$
$349$	−9.37459	−0.501810	−0.250905	+	0.968012i	$0.580728\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	−7.09967	−0.377877	−0.188939	−	0.981989i	$-0.560505\pi$
$353$	−7.09967	−0.377877	−0.188939	+	0.981989i	$0.560505\pi$
$354$	0	0
$355$	4.54983	0.241480
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.2749	−1.33396	−0.666980	−	0.745076i	$-0.732413\pi$
$359$	−25.2749	−1.33396	−0.666980	+	0.745076i	$0.732413\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−33.6495	−1.76614
$364$	0	0
$365$	−11.2749	−0.590156
$366$	0	0
$367$	34.1993	1.78519	0.892595	−	0.450858i	$-0.148882\pi$
$367$	34.1993	1.78519	0.892595	+	0.450858i	$0.148882\pi$
$368$	0	0
$369$	10.5498	0.549202
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.54983	0.442694	0.221347	−	0.975195i	$-0.428955\pi$
$373$	8.54983	0.442694	0.221347	+	0.975195i	$0.428955\pi$
$374$	0	0
$375$	−18.0000	−0.929516
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−27.0997	−1.39202	−0.696008	−	0.718034i	$-0.745042\pi$
$379$	−27.0997	−1.39202	−0.696008	+	0.718034i	$0.745042\pi$
$380$	0	0
$381$	−30.1993	−1.54716
$382$	0	0
$383$	22.0000	1.12415	0.562074	−	0.827087i	$-0.310004\pi$
$383$	22.0000	1.12415	0.562074	+	0.827087i	$0.310004\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	15.1752	0.769416	0.384708	−	0.923038i	$-0.374302\pi$
$389$	15.1752	0.769416	0.384708	+	0.923038i	$0.374302\pi$
$390$	0	0
$391$	−2.07558	−0.104967
$392$	0	0
$393$	−13.6495	−0.688526
$394$	0	0
$395$	14.5498	0.732082
$396$	0	0
$397$	−28.2749	−1.41908	−0.709539	−	0.704666i	$-0.751097\pi$
$397$	−28.2749	−1.41908	−0.709539	+	0.704666i	$0.751097\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.09967	−0.454416	−0.227208	−	0.973846i	$-0.572960\pi$
$401$	−9.09967	−0.454416	−0.227208	+	0.973846i	$0.572960\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	11.0000	0.546594
$406$	0	0
$407$	−10.5498	−0.522936
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−22.5498	−1.11230
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.54983	0.174255
$416$	0	0
$417$	−23.6495	−1.15812
$418$	0	0
$419$	4.45017	0.217405	0.108702	−	0.994074i	$-0.465330\pi$
$419$	4.45017	0.217405	0.108702	+	0.994074i	$0.465330\pi$
$420$	0	0
$421$	3.45017	0.168151	0.0840754	−	0.996459i	$-0.473206\pi$
$421$	3.45017	0.168151	0.0840754	+	0.996459i	$0.473206\pi$
$422$	0	0
$423$	5.27492	0.256475
$424$	0	0
$425$	1.09967	0.0533418
$426$	0	0
$427$	0	0
$428$	0	0
$429$	21.0997	1.01870
$430$	0	0
$431$	−21.0997	−1.01634	−0.508168	−	0.861258i	$-0.669677\pi$
$431$	−21.0997	−1.01634	−0.508168	+	0.861258i	$0.669677\pi$
$432$	0	0
$433$	−5.09967	−0.245074	−0.122537	−	0.992464i	$-0.539103\pi$
$433$	−5.09967	−0.245074	−0.122537	+	0.992464i	$0.539103\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	0	0
$437$	−7.54983	−0.361158
$438$	0	0
$439$	−26.1993	−1.25043	−0.625213	−	0.780454i	$-0.714988\pi$
$439$	−26.1993	−1.25043	−0.625213	+	0.780454i	$0.714988\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.8248	1.08444	0.542218	−	0.840238i	$-0.317585\pi$
$443$	22.8248	1.08444	0.542218	+	0.840238i	$0.317585\pi$
$444$	0	0
$445$	−1.45017	−0.0687444
$446$	0	0
$447$	32.1993	1.52298
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	−55.6495	−2.62043
$452$	0	0
$453$	−13.0997	−0.615476
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.72508	0.221030	0.110515	−	0.993874i	$-0.464750\pi$
$457$	4.72508	0.221030	0.110515	+	0.993874i	$0.464750\pi$
$458$	0	0
$459$	−1.09967	−0.0513281
$460$	0	0
$461$	23.2749	1.08402	0.542010	−	0.840372i	$-0.317663\pi$
$461$	23.2749	1.08402	0.542010	+	0.840372i	$0.317663\pi$
$462$	0	0
$463$	−19.8248	−0.921334	−0.460667	−	0.887573i	$-0.652390\pi$
$463$	−19.8248	−0.921334	−0.460667	+	0.887573i	$0.652390\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−10.7251	−0.496298	−0.248149	−	0.968722i	$-0.579822\pi$
$467$	−10.7251	−0.496298	−0.248149	+	0.968722i	$0.579822\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−28.1993	−1.29936
$472$	0	0
$473$	5.27492	0.242541
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	8.54983	0.391470
$478$	0	0
$479$	−14.4502	−0.660245	−0.330122	−	0.943938i	$-0.607090\pi$
$479$	−14.4502	−0.660245	−0.330122	+	0.943938i	$0.607090\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.54983	0.297413
$486$	0	0
$487$	7.09967	0.321717	0.160858	−	0.986978i	$-0.448574\pi$
$487$	7.09967	0.321717	0.160858	+	0.986978i	$0.448574\pi$
$488$	0	0
$489$	43.0997	1.94903
$490$	0	0
$491$	5.54983	0.250461	0.125230	−	0.992128i	$-0.460033\pi$
$491$	5.54983	0.250461	0.125230	+	0.992128i	$0.460033\pi$
$492$	0	0
$493$	1.09967	0.0495266
$494$	0	0
$495$	5.27492	0.237090
$496$	0	0
$497$	0	0
$498$	0	0
$499$	31.1993	1.39667	0.698337	−	0.715769i	$-0.253924\pi$
$499$	31.1993	1.39667	0.698337	+	0.715769i	$0.253924\pi$
$500$	0	0
$501$	−26.1993	−1.17050
$502$	0	0
$503$	−10.6495	−0.474838	−0.237419	−	0.971407i	$-0.576301\pi$
$503$	−10.6495	−0.474838	−0.237419	+	0.971407i	$0.576301\pi$
$504$	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	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−10.0000	−0.440653
$516$	0	0
$517$	−27.8248	−1.22373
$518$	0	0
$519$	−41.0997	−1.80408
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	3.64950	0.159582	0.0797908	−	0.996812i	$-0.474575\pi$
$523$	3.64950	0.159582	0.0797908	+	0.996812i	$0.474575\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.09967	0.0479023
$528$	0	0
$529$	34.0000	1.47826
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	21.0997	0.913928
$534$	0	0
$535$	18.5498	0.801979
$536$	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
$543$	−1.09967	−0.0471913
$544$	0	0
$545$	−4.54983	−0.194893
$546$	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
$549$	4.72508	0.201662
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−28.0997	−1.19062	−0.595311	−	0.803496i	$-0.702971\pi$
$557$	−28.0997	−1.19062	−0.595311	+	0.803496i	$0.702971\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	−2.90033	−0.122452
$562$	0	0
$563$	−26.5498	−1.11894	−0.559471	−	0.828850i	$-0.688996\pi$
$563$	−26.5498	−1.11894	−0.559471	+	0.828850i	$0.688996\pi$
$564$	0	0
$565$	−3.45017	−0.145150
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.1993	1.85293	0.926466	−	0.376378i	$-0.122830\pi$
$569$	44.1993	1.85293	0.926466	+	0.376378i	$0.122830\pi$
$570$	0	0
$571$	−35.5498	−1.48771	−0.743857	−	0.668339i	$-0.767006\pi$
$571$	−35.5498	−1.48771	−0.743857	+	0.668339i	$0.767006\pi$
$572$	0	0
$573$	42.0000	1.75458
$574$	0	0
$575$	−30.1993	−1.25940
$576$	0	0
$577$	−24.3746	−1.01473	−0.507364	−	0.861732i	$-0.669380\pi$
$577$	−24.3746	−1.01473	−0.507364	+	0.861732i	$0.669380\pi$
$578$	0	0
$579$	6.19934	0.257636
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−45.0997	−1.86784
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−10.2749	−0.424091	−0.212046	−	0.977260i	$-0.568013\pi$
$587$	−10.2749	−0.424091	−0.212046	+	0.977260i	$0.568013\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−24.1993	−0.995428
$592$	0	0
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.00000	−0.245564
$598$	0	0
$599$	−29.6495	−1.21145	−0.605723	−	0.795676i	$-0.707116\pi$
$599$	−29.6495	−1.21145	−0.605723	+	0.795676i	$0.707116\pi$
$600$	0	0
$601$	−26.5498	−1.08299	−0.541495	−	0.840704i	$-0.682142\pi$
$601$	−26.5498	−1.08299	−0.541495	+	0.840704i	$0.682142\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16.8248	−0.684023
$606$	0	0
$607$	−19.6495	−0.797549	−0.398774	−	0.917049i	$-0.630564\pi$
$607$	−19.6495	−0.797549	−0.398774	+	0.917049i	$0.630564\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.5498	0.426801
$612$	0	0
$613$	−15.1752	−0.612923	−0.306461	−	0.951883i	$-0.599145\pi$
$613$	−15.1752	−0.612923	−0.306461	+	0.951883i	$0.599145\pi$
$614$	0	0
$615$	21.0997	0.850821
$616$	0	0
$617$	6.09967	0.245563	0.122782	−	0.992434i	$-0.460818\pi$
$617$	6.09967	0.245563	0.122782	+	0.992434i	$0.460818\pi$
$618$	0	0
$619$	−3.54983	−0.142680	−0.0713399	−	0.997452i	$-0.522728\pi$
$619$	−3.54983	−0.142680	−0.0713399	+	0.997452i	$0.522728\pi$
$620$	0	0
$621$	30.1993	1.21186
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−10.5498	−0.421320
$628$	0	0
$629$	−0.549834	−0.0219233
$630$	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
$633$	51.2990	2.03895
$634$	0	0
$635$	−15.0997	−0.599212
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.54983	−0.179989
$640$	0	0
$641$	31.4502	1.24221	0.621103	−	0.783729i	$-0.286685\pi$
$641$	31.4502	1.24221	0.621103	+	0.783729i	$0.286685\pi$
$642$	0	0
$643$	−22.8248	−0.900120	−0.450060	−	0.892998i	$-0.648597\pi$
$643$	−22.8248	−0.900120	−0.450060	+	0.892998i	$0.648597\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	16.0997	0.632943	0.316472	−	0.948602i	$-0.397502\pi$
$647$	16.0997	0.632943	0.316472	+	0.948602i	$0.397502\pi$
$648$	0	0
$649$	21.0997	0.828234
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.27492	−0.323823	−0.161911	−	0.986805i	$-0.551766\pi$
$653$	−8.27492	−0.323823	−0.161911	+	0.986805i	$0.551766\pi$
$654$	0	0
$655$	−6.82475	−0.266665
$656$	0	0
$657$	11.2749	0.439876
$658$	0	0
$659$	−40.5498	−1.57960	−0.789799	−	0.613366i	$-0.789815\pi$
$659$	−40.5498	−1.57960	−0.789799	+	0.613366i	$0.789815\pi$
$660$	0	0
$661$	−29.4502	−1.14548	−0.572739	−	0.819738i	$-0.694119\pi$
$661$	−29.4502	−1.14548	−0.572739	+	0.819738i	$0.694119\pi$
$662$	0	0
$663$	1.09967	0.0427076
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−30.1993	−1.16932
$668$	0	0
$669$	34.1993	1.32222
$670$	0	0
$671$	−24.9244	−0.962197
$672$	0	0
$673$	−4.54983	−0.175383	−0.0876916	−	0.996148i	$-0.527949\pi$
$673$	−4.54983	−0.175383	−0.0876916	+	0.996148i	$0.527949\pi$
$674$	0	0
$675$	−16.0000	−0.615840
$676$	0	0
$677$	6.90033	0.265201	0.132601	−	0.991170i	$-0.457667\pi$
$677$	6.90033	0.265201	0.132601	+	0.991170i	$0.457667\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	18.9003	0.724262
$682$	0	0
$683$	27.6495	1.05798	0.528989	−	0.848628i	$-0.322571\pi$
$683$	27.6495	1.05798	0.528989	+	0.848628i	$0.322571\pi$
$684$	0	0
$685$	−11.2749	−0.430792
$686$	0	0
$687$	32.5498	1.24185
$688$	0	0
$689$	17.0997	0.651446
$690$	0	0
$691$	15.9244	0.605794	0.302897	−	0.953023i	$-0.402046\pi$
$691$	15.9244	0.605794	0.302897	+	0.953023i	$0.402046\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.8248	−0.448538
$696$	0	0
$697$	−2.90033	−0.109858
$698$	0	0
$699$	−8.54983	−0.323384
$700$	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
$705$	10.5498	0.397330
$706$	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
$711$	−14.5498	−0.545661
$712$	0	0
$713$	−30.1993	−1.13097
$714$	0	0
$715$	10.5498	0.394541
$716$	0	0
$717$	−46.7492	−1.74588
$718$	0	0
$719$	34.2749	1.27824	0.639119	−	0.769108i	$-0.279299\pi$
$719$	34.2749	1.27824	0.639119	+	0.769108i	$0.279299\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−21.0997	−0.784705
$724$	0	0
$725$	16.0000	0.594225
$726$	0	0
$727$	−44.0997	−1.63557	−0.817783	−	0.575527i	$-0.804797\pi$
$727$	−44.0997	−1.63557	−0.817783	+	0.575527i	$0.804797\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0.274917	0.0101682
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−26.8248	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$739$	−26.8248	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	28.7492	1.05470	0.527352	−	0.849647i	$-0.323185\pi$
$743$	28.7492	1.05470	0.527352	+	0.849647i	$0.323185\pi$
$744$	0	0
$745$	16.0997	0.589846
$746$	0	0
$747$	−3.54983	−0.129882
$748$	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
$753$	56.1993	2.04802
$754$	0	0
$755$	−6.54983	−0.238373
$756$	0	0
$757$	−12.3746	−0.449762	−0.224881	−	0.974386i	$-0.572199\pi$
$757$	−12.3746	−0.449762	−0.224881	+	0.974386i	$0.572199\pi$
$758$	0	0
$759$	79.6495	2.89109
$760$	0	0
$761$	−43.0000	−1.55875	−0.779374	−	0.626559i	$-0.784463\pi$
$761$	−43.0000	−1.55875	−0.779374	+	0.626559i	$0.784463\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.274917	0.00993965
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	17.1993	0.620224	0.310112	−	0.950700i	$-0.399633\pi$
$769$	17.1993	0.620224	0.310112	+	0.950700i	$0.399633\pi$
$770$	0	0
$771$	45.0997	1.62422
$772$	0	0
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.5498	−0.377987
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	−16.0000	−0.571793
$784$	0	0
$785$	−14.0997	−0.503239
$786$	0	0
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	0	0
$789$	−45.6495	−1.62517
$790$	0	0
$791$	0	0
$792$	0	0
$793$	9.45017	0.335585
$794$	0	0
$795$	17.0997	0.606463
$796$	0	0
$797$	−1.45017	−0.0513675	−0.0256837	−	0.999670i	$-0.508176\pi$
$797$	−1.45017	−0.0513675	−0.0256837	+	0.999670i	$0.508176\pi$
$798$	0	0
$799$	−1.45017	−0.0513032
$800$	0	0
$801$	1.45017	0.0512391
$802$	0	0
$803$	−59.4743	−2.09880
$804$	0	0
$805$	0	0
$806$	0	0
$807$	49.0997	1.72839
$808$	0	0
$809$	12.0997	0.425402	0.212701	−	0.977117i	$-0.431774\pi$
$809$	12.0997	0.425402	0.212701	+	0.977117i	$0.431774\pi$
$810$	0	0
$811$	52.7492	1.85227	0.926137	−	0.377187i	$-0.123109\pi$
$811$	52.7492	1.85227	0.926137	+	0.377187i	$0.123109\pi$
$812$	0	0
$813$	−35.0997	−1.23100
$814$	0	0
$815$	21.5498	0.754858
$816$	0	0
$817$	1.00000	0.0349856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.9244	−0.939669	−0.469834	−	0.882755i	$-0.655686\pi$
$821$	−26.9244	−0.939669	−0.469834	+	0.882755i	$0.655686\pi$
$822$	0	0
$823$	4.92442	0.171655	0.0858273	−	0.996310i	$-0.472647\pi$
$823$	4.92442	0.171655	0.0858273	+	0.996310i	$0.472647\pi$
$824$	0	0
$825$	−42.1993	−1.46919
$826$	0	0
$827$	−41.6495	−1.44830	−0.724148	−	0.689645i	$-0.757767\pi$
$827$	−41.6495	−1.44830	−0.724148	+	0.689645i	$0.757767\pi$
$828$	0	0
$829$	−14.9003	−0.517510	−0.258755	−	0.965943i	$-0.583312\pi$
$829$	−14.9003	−0.517510	−0.258755	+	0.965943i	$0.583312\pi$
$830$	0	0
$831$	1.45017	0.0503057
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.0997	−0.453333
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	37.6495	1.29981	0.649903	−	0.760018i	$-0.274810\pi$
$839$	37.6495	1.29981	0.649903	+	0.760018i	$0.274810\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−1.09967	−0.0378746
$844$	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
$853$	−42.0997	−1.44147	−0.720733	−	0.693213i	$-0.756194\pi$
$853$	−42.0997	−1.44147	−0.720733	+	0.693213i	$0.756194\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	41.6495	1.42272	0.711360	−	0.702828i	$-0.248080\pi$
$857$	41.6495	1.42272	0.711360	+	0.702828i	$0.248080\pi$
$858$	0	0
$859$	−10.0997	−0.344596	−0.172298	−	0.985045i	$-0.555119\pi$
$859$	−10.0997	−0.344596	−0.172298	+	0.985045i	$0.555119\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.90033	−0.0987284	−0.0493642	−	0.998781i	$-0.515720\pi$
$863$	−2.90033	−0.0987284	−0.0493642	+	0.998781i	$0.515720\pi$
$864$	0	0
$865$	−20.5498	−0.698715
$866$	0	0
$867$	33.8488	1.14957
$868$	0	0
$869$	76.7492	2.60354
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.54983	−0.221678
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.7492	1.10586	0.552930	−	0.833227i	$-0.313510\pi$
$877$	32.7492	1.10586	0.552930	+	0.833227i	$0.313510\pi$
$878$	0	0
$879$	−55.2990	−1.86519
$880$	0	0
$881$	49.9244	1.68200	0.840998	−	0.541038i	$-0.181968\pi$
$881$	49.9244	1.68200	0.840998	+	0.541038i	$0.181968\pi$
$882$	0	0
$883$	−41.0241	−1.38057	−0.690285	−	0.723537i	$-0.742515\pi$
$883$	−41.0241	−1.38057	−0.690285	+	0.723537i	$0.742515\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	−52.1993	−1.75268	−0.876341	−	0.481691i	$-0.840023\pi$
$887$	−52.1993	−1.75268	−0.876341	+	0.481691i	$0.840023\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	58.0241	1.94388
$892$	0	0
$893$	−5.27492	−0.176518
$894$	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
$901$	−2.35050	−0.0783064
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.549834	−0.0182771
$906$	0	0
$907$	6.90033	0.229122	0.114561	−	0.993416i	$-0.463454\pi$
$907$	6.90033	0.229122	0.114561	+	0.993416i	$0.463454\pi$
$908$	0	0
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	11.4502	0.379361	0.189680	−	0.981846i	$-0.439255\pi$
$911$	11.4502	0.379361	0.189680	+	0.981846i	$0.439255\pi$
$912$	0	0
$913$	18.7251	0.619710
$914$	0	0
$915$	9.45017	0.312413
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−39.5498	−1.30463	−0.652314	−	0.757949i	$-0.726202\pi$
$919$	−39.5498	−1.30463	−0.652314	+	0.757949i	$0.726202\pi$
$920$	0	0
$921$	52.0000	1.71346
$922$	0	0
$923$	−9.09967	−0.299519
$924$	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	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	20.5498	0.672771
$934$	0	0
$935$	−1.45017	−0.0474255
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	0	0
$939$	62.3987	2.03630
$940$	0	0
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	0	0
$943$	79.6495	2.59374
$944$	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
$949$	22.5498	0.731999
$950$	0	0
$951$	−18.1993	−0.590154
$952$	0	0
$953$	43.0997	1.39614	0.698068	−	0.716032i	$-0.254043\pi$
$953$	43.0997	1.39614	0.698068	+	0.716032i	$0.254043\pi$
$954$	0	0
$955$	21.0000	0.679544
$956$	0	0
$957$	−42.1993	−1.36411
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−18.5498	−0.597760
$964$	0	0
$965$	3.09967	0.0997819
$966$	0	0
$967$	42.1993	1.35704	0.678520	−	0.734582i	$-0.262622\pi$
$967$	42.1993	1.35704	0.678520	+	0.734582i	$0.262622\pi$
$968$	0	0
$969$	−0.549834	−0.0176632
$970$	0	0
$971$	21.4502	0.688369	0.344184	−	0.938902i	$-0.388156\pi$
$971$	21.4502	0.688369	0.344184	+	0.938902i	$0.388156\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	16.0000	0.512410
$976$	0	0
$977$	−35.2990	−1.12932	−0.564658	−	0.825325i	$-0.690992\pi$
$977$	−35.2990	−1.12932	−0.564658	+	0.825325i	$0.690992\pi$
$978$	0	0
$979$	−7.64950	−0.244479
$980$	0	0
$981$	4.54983	0.145265
$982$	0	0
$983$	−52.1993	−1.66490	−0.832450	−	0.554100i	$-0.813063\pi$
$983$	−52.1993	−1.66490	−0.832450	+	0.554100i	$0.813063\pi$
$984$	0	0
$985$	−12.0997	−0.385528
$986$	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
$997$	−7.17525	−0.227242	−0.113621	−	0.993524i	$-0.536245\pi$
$997$	−7.17525	−0.227242	−0.113621	+	0.993524i	$0.536245\pi$
$998$	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.w.1.1		2
7.2	even	3		1064.2.q.l.305.1	✓	4
7.4	even	3		1064.2.q.l.457.1	yes	4
7.6	odd	2		7448.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.l.305.1	✓	4	7.2	even	3
1064.2.q.l.457.1	yes	4	7.4	even	3
7448.2.a.w.1.1		2	1.1	even	1	trivial
7448.2.a.be.1.1		2	7.6	odd	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -5.27492 q^{11} +2.00000 q^{13} +2.00000 q^{15} -0.274917 q^{17} -1.00000 q^{19} +7.54983 q^{23} -4.00000 q^{25} +4.00000 q^{27} -4.00000 q^{29} -4.00000 q^{31} +10.5498 q^{33} +2.00000 q^{37} -4.00000 q^{39} +10.5498 q^{41} -1.00000 q^{43} -1.00000 q^{45} +5.27492 q^{47} +0.549834 q^{51} +8.54983 q^{53} +5.27492 q^{55} +2.00000 q^{57} -4.00000 q^{59} +4.72508 q^{61} -2.00000 q^{65} -15.0997 q^{69} -4.54983 q^{71} +11.2749 q^{73} +8.00000 q^{75} -14.5498 q^{79} -11.0000 q^{81} -3.54983 q^{83} +0.274917 q^{85} +8.00000 q^{87} +1.45017 q^{89} +8.00000 q^{93} +1.00000 q^{95} -6.54983 q^{97} -5.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 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

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