LMFDB - Embedded newform 8048.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -4.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -4.00000 q^{5} +3.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} +1.00000 q^{13} +4.00000 q^{15} -8.00000 q^{19} -3.00000 q^{21} -9.00000 q^{23} +11.0000 q^{25} +5.00000 q^{27} -6.00000 q^{29} +2.00000 q^{31} +5.00000 q^{33} -12.0000 q^{35} +2.00000 q^{37} -1.00000 q^{39} -10.0000 q^{41} -5.00000 q^{43} +8.00000 q^{45} +1.00000 q^{47} +2.00000 q^{49} -6.00000 q^{53} +20.0000 q^{55} +8.00000 q^{57} -4.00000 q^{59} +5.00000 q^{61} -6.00000 q^{63} -4.00000 q^{65} -13.0000 q^{67} +9.00000 q^{69} -6.00000 q^{71} +6.00000 q^{73} -11.0000 q^{75} -15.0000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -9.00000 q^{83} +6.00000 q^{87} -6.00000 q^{89} +3.00000 q^{91} -2.00000 q^{93} +32.0000 q^{95} +10.0000 q^{97} +10.0000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	−12.0000	−2.02837
$36$	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
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	8.00000	1.19257
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	20.0000	2.69680
$56$	0	0
$57$	8.00000	1.05963
$58$	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
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	0	0
$69$	9.00000	1.08347
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	−15.0000	−1.70941
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	32.0000	3.28313
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	0	0
$115$	36.0000	3.35702
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	10.0000	0.901670
$124$	0	0
$125$	−24.0000	−2.14663
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	5.00000	0.440225
$130$	0	0
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	0	0
$135$	−20.0000	−1.72133
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	0	0
$143$	−5.00000	−0.418121
$144$	0	0
$145$	24.0000	1.99309
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−27.0000	−2.12790
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	−20.0000	−1.55700
$166$	0	0
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	16.0000	1.22355
$172$	0	0
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	33.0000	2.49457
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	15.0000	1.09109
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	−5.00000	−0.356235	−0.178118	−	0.984009i	$-0.557001\pi$
$197$	−5.00000	−0.356235	−0.178118	+	0.984009i	$0.557001\pi$
$198$	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
$201$	13.0000	0.916949
$202$	0	0
$203$	−18.0000	−1.26335
$204$	0	0
$205$	40.0000	2.79372
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	40.0000	2.76686
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	20.0000	1.36399
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5.00000	−0.334825	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000	−0.334825	−0.167412	+	0.985887i	$0.553541\pi$
$224$	0	0
$225$	−22.0000	−1.46667
$226$	0	0
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	15.0000	0.986928
$232$	0	0
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	28.0000	1.81117	0.905585	−	0.424165i	$-0.139432\pi$
$239$	28.0000	1.81117	0.905585	+	0.424165i	$0.139432\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−8.00000	−0.511101
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	9.00000	0.570352
$250$	0	0
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	0	0
$253$	45.0000	2.82913
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.0000	−0.810918	−0.405459	−	0.914113i	$-0.632888\pi$
$257$	−13.0000	−0.810918	−0.405459	+	0.914113i	$0.632888\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	−13.0000	−0.801614	−0.400807	−	0.916162i	$-0.631270\pi$
$263$	−13.0000	−0.801614	−0.400807	+	0.916162i	$0.631270\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	21.0000	1.27566	0.637830	−	0.770178i	$-0.279832\pi$
$271$	21.0000	1.27566	0.637830	+	0.770178i	$0.279832\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	0	0
$275$	−55.0000	−3.31662
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	−32.0000	−1.89552
$286$	0	0
$287$	−30.0000	−1.77084
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−15.0000	−0.876309	−0.438155	−	0.898900i	$-0.644368\pi$
$293$	−15.0000	−0.876309	−0.438155	+	0.898900i	$0.644368\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	−25.0000	−1.45065
$298$	0	0
$299$	−9.00000	−0.520483
$300$	0	0
$301$	−15.0000	−0.864586
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−22.0000	−1.24751	−0.623753	−	0.781622i	$-0.714393\pi$
$311$	−22.0000	−1.24751	−0.623753	+	0.781622i	$0.714393\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	24.0000	1.35225
$316$	0	0
$317$	27.0000	1.51647	0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000	1.51647	0.758236	+	0.651981i	$0.226062\pi$
$318$	0	0
$319$	30.0000	1.67968
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	0	0
$324$	0	0
$325$	11.0000	0.610170
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	52.0000	2.84106
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	11.0000	0.597438
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−36.0000	−1.93817
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−24.0000	−1.25622
$366$	0	0
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	0	0
$369$	20.0000	1.04116
$370$	0	0
$371$	−18.0000	−0.934513
$372$	0	0
$373$	17.0000	0.880227	0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000	0.880227	0.440113	+	0.897942i	$0.354938\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	60.0000	3.05788
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.00000	0.0504433
$394$	0	0
$395$	64.0000	3.22019
$396$	0	0
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0	0
$405$	−4.00000	−0.198762
$406$	0	0
$407$	−10.0000	−0.495682
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	10.0000	0.489702
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.0000	0.725901
$428$	0	0
$429$	5.00000	0.241402
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	−24.0000	−1.15071
$436$	0	0
$437$	72.0000	3.44423
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−5.00000	−0.237557	−0.118779	−	0.992921i	$-0.537898\pi$
$443$	−5.00000	−0.237557	−0.118779	+	0.992921i	$0.537898\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	50.0000	2.35441
$452$	0	0
$453$	−12.0000	−0.563809
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−11.0000	−0.511213	−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000	−0.511213	−0.255607	+	0.966781i	$0.582275\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−39.0000	−1.80085
$470$	0	0
$471$	0	0
$472$	0	0
$473$	25.0000	1.14950
$474$	0	0
$475$	−88.0000	−4.03772
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	27.0000	1.22854
$484$	0	0
$485$	−40.0000	−1.81631
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	0	0
$489$	10.0000	0.452216
$490$	0	0
$491$	40.0000	1.80517	0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000	1.80517	0.902587	+	0.430507i	$0.141665\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−40.0000	−1.79787
$496$	0	0
$497$	−18.0000	−0.807410
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−32.0000	−1.42398
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−27.0000	−1.19675	−0.598377	−	0.801215i	$-0.704187\pi$
$509$	−27.0000	−1.19675	−0.598377	+	0.801215i	$0.704187\pi$
$510$	0	0
$511$	18.0000	0.796273
$512$	0	0
$513$	−40.0000	−1.76604
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−5.00000	−0.219900
$518$	0	0
$519$	−7.00000	−0.307266
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	−30.0000	−1.31181	−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	0	0
$525$	−33.0000	−1.44024
$526$	0	0
$527$	0	0
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−10.0000	−0.433148
$534$	0	0
$535$	32.0000	1.38348
$536$	0	0
$537$	−2.00000	−0.0863064
$538$	0	0
$539$	−10.0000	−0.430730
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	−48.0000	−2.04117
$554$	0	0
$555$	8.00000	0.339581
$556$	0	0
$557$	25.0000	1.05928	0.529642	−	0.848221i	$-0.322326\pi$
$557$	25.0000	1.05928	0.529642	+	0.848221i	$0.322326\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	44.0000	1.85109
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−6.00000	−0.251092	−0.125546	−	0.992088i	$-0.540068\pi$
$571$	−6.00000	−0.251092	−0.125546	+	0.992088i	$0.540068\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	−99.0000	−4.12859
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	0	0
$579$	22.0000	0.914289
$580$	0	0
$581$	−27.0000	−1.12015
$582$	0	0
$583$	30.0000	1.24247
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	5.00000	0.205673
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	26.0000	1.05880
$604$	0	0
$605$	−56.0000	−2.27672
$606$	0	0
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	0	0
$609$	18.0000	0.729397
$610$	0	0
$611$	1.00000	0.0404557
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	−40.0000	−1.61296
$616$	0	0
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	−45.0000	−1.80579
$622$	0	0
$623$	−18.0000	−0.721155
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−40.0000	−1.59745
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	0	0
$633$	22.0000	0.874421
$634$	0	0
$635$	24.0000	0.952411
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	17.0000	0.671460	0.335730	−	0.941958i	$-0.391017\pi$
$641$	17.0000	0.671460	0.335730	+	0.941958i	$0.391017\pi$
$642$	0	0
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	−20.0000	−0.787499
$646$	0	0
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	−51.0000	−1.99578	−0.997892	−	0.0648948i	$-0.979329\pi$
$653$	−51.0000	−1.99578	−0.997892	+	0.0648948i	$0.979329\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−12.0000	−0.468165
$658$	0	0
$659$	23.0000	0.895953	0.447976	−	0.894045i	$-0.352145\pi$
$659$	23.0000	0.895953	0.447976	+	0.894045i	$0.352145\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	96.0000	3.72272
$666$	0	0
$667$	54.0000	2.09089
$668$	0	0
$669$	5.00000	0.193311
$670$	0	0
$671$	−25.0000	−0.965114
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	55.0000	2.11695
$676$	0	0
$677$	−4.00000	−0.153732	−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000	−0.153732	−0.0768662	+	0.997041i	$0.524491\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.0000	0.495981
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	30.0000	1.13961
$694$	0	0
$695$	40.0000	1.51729
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−27.0000	−1.02123
$700$	0	0
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	24.0000	0.902613
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	32.0000	1.20009
$712$	0	0
$713$	−18.0000	−0.674105
$714$	0	0
$715$	20.0000	0.747958
$716$	0	0
$717$	−28.0000	−1.04568
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	−12.0000	−0.446285
$724$	0	0
$725$	−66.0000	−2.45118
$726$	0	0
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	8.00000	0.295084
$736$	0	0
$737$	65.0000	2.39431
$738$	0	0
$739$	37.0000	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$739$	37.0000	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.0000	0.658586
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	0	0
$753$	−26.0000	−0.947493
$754$	0	0
$755$	−48.0000	−1.74690
$756$	0	0
$757$	−32.0000	−1.16306	−0.581530	−	0.813525i	$-0.697546\pi$
$757$	−32.0000	−1.16306	−0.581530	+	0.813525i	$0.697546\pi$
$758$	0	0
$759$	−45.0000	−1.63340
$760$	0	0
$761$	−17.0000	−0.616250	−0.308125	−	0.951346i	$-0.599701\pi$
$761$	−17.0000	−0.616250	−0.308125	+	0.951346i	$0.599701\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	0	0
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	22.0000	0.790263
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	80.0000	2.86630
$780$	0	0
$781$	30.0000	1.07348
$782$	0	0
$783$	−30.0000	−1.07211
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	0	0
$789$	13.0000	0.462812
$790$	0	0
$791$	−33.0000	−1.17334
$792$	0	0
$793$	5.00000	0.177555
$794$	0	0
$795$	−24.0000	−0.851192
$796$	0	0
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−30.0000	−1.05868
$804$	0	0
$805$	108.000	3.80650
$806$	0	0
$807$	−10.0000	−0.352017
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	0	0
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	0	0
$813$	−21.0000	−0.736502
$814$	0	0
$815$	40.0000	1.40114
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	55.0000	1.91485
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−7.00000	−0.241093
$844$	0	0
$845$	48.0000	1.65125
$846$	0	0
$847$	42.0000	1.44314
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	−18.0000	−0.617032
$852$	0	0
$853$	−31.0000	−1.06142	−0.530710	−	0.847554i	$-0.678075\pi$
$853$	−31.0000	−1.06142	−0.530710	+	0.847554i	$0.678075\pi$
$854$	0	0
$855$	−64.0000	−2.18875
$856$	0	0
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	30.0000	1.02240
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	80.0000	2.71381
$870$	0	0
$871$	−13.0000	−0.440488
$872$	0	0
$873$	−20.0000	−0.676897
$874$	0	0
$875$	−72.0000	−2.43404
$876$	0	0
$877$	−16.0000	−0.540282	−0.270141	−	0.962821i	$-0.587070\pi$
$877$	−16.0000	−0.540282	−0.270141	+	0.962821i	$0.587070\pi$
$878$	0	0
$879$	15.0000	0.505937
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	0	0
$902$	0	0
$903$	15.0000	0.499169
$904$	0	0
$905$	48.0000	1.59557
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	0	0
$909$	−16.0000	−0.530687
$910$	0	0
$911$	−42.0000	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$912$	0	0
$913$	45.0000	1.48928
$914$	0	0
$915$	20.0000	0.661180
$916$	0	0
$917$	−3.00000	−0.0990687
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	−6.00000	−0.197492
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	0	0
$933$	22.0000	0.720248
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	−59.0000	−1.92335	−0.961673	−	0.274201i	$-0.911587\pi$
$941$	−59.0000	−1.92335	−0.961673	+	0.274201i	$0.911587\pi$
$942$	0	0
$943$	90.0000	2.93080
$944$	0	0
$945$	−60.0000	−1.95180
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	−27.0000	−0.875535
$952$	0	0
$953$	−39.0000	−1.26333	−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000	−1.26333	−0.631667	+	0.775240i	$0.717629\pi$
$954$	0	0
$955$	72.0000	2.32987
$956$	0	0
$957$	−30.0000	−0.969762
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	88.0000	2.83282
$966$	0	0
$967$	−46.0000	−1.47926	−0.739630	−	0.673014i	$-0.765000\pi$
$967$	−46.0000	−1.47926	−0.739630	+	0.673014i	$0.765000\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.0000	−1.12320	−0.561602	−	0.827408i	$-0.689815\pi$
$971$	−35.0000	−1.12320	−0.561602	+	0.827408i	$0.689815\pi$
$972$	0	0
$973$	−30.0000	−0.961756
$974$	0	0
$975$	−11.0000	−0.352282
$976$	0	0
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	0	0
$989$	45.0000	1.43092
$990$	0	0
$991$	−21.0000	−0.667087	−0.333543	−	0.942735i	$-0.608244\pi$
$991$	−21.0000	−0.667087	−0.333543	+	0.942735i	$0.608244\pi$
$992$	0	0
$993$	−26.0000	−0.825085
$994$	0	0
$995$	−32.0000	−1.01447
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.c.1.1		1
4.3	odd	2		503.2.a.a.1.1	✓	1
12.11	even	2		4527.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.a.1.1	✓	1	4.3	odd	2
4527.2.a.g.1.1		1	12.11	even	2
8048.2.a.c.1.1		1	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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