LMFDB - Embedded newform 8048.2.a.x.1.7

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:	$1$
Dimension:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-2.60958 q^{3} -1.79981 q^{5} +2.96082 q^{7} +3.80991 q^{9} +O(q^{10})$	$q-2.60958 q^{3} -1.79981 q^{5} +2.96082 q^{7} +3.80991 q^{9} +0.556169 q^{11} +3.76956 q^{13} +4.69674 q^{15} +1.41609 q^{17} -1.17955 q^{19} -7.72649 q^{21} -0.626692 q^{23} -1.76069 q^{25} -2.11352 q^{27} -0.654195 q^{29} -1.67221 q^{31} -1.45137 q^{33} -5.32890 q^{35} -2.59709 q^{37} -9.83697 q^{39} +6.68824 q^{41} -11.7396 q^{43} -6.85710 q^{45} -13.0111 q^{47} +1.76644 q^{49} -3.69540 q^{51} +9.66280 q^{53} -1.00100 q^{55} +3.07812 q^{57} -5.84207 q^{59} -6.37379 q^{61} +11.2804 q^{63} -6.78448 q^{65} -9.80452 q^{67} +1.63540 q^{69} +14.7673 q^{71} +15.5674 q^{73} +4.59467 q^{75} +1.64671 q^{77} +2.17637 q^{79} -5.91432 q^{81} -2.48063 q^{83} -2.54869 q^{85} +1.70717 q^{87} -1.95542 q^{89} +11.1610 q^{91} +4.36376 q^{93} +2.12296 q^{95} +15.4557 q^{97} +2.11895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * 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
$3$	−2.60958	−1.50664	−0.753321	−	0.657653i	$-0.771549\pi$
$3$	−2.60958	−1.50664	−0.753321	+	0.657653i	$0.771549\pi$
$4$	0	0
$5$	−1.79981	−0.804898	−0.402449	−	0.915442i	$-0.631841\pi$
$5$	−1.79981	−0.804898	−0.402449	+	0.915442i	$0.631841\pi$
$6$	0	0
$7$	2.96082	1.11908	0.559542	−	0.828802i	$-0.310977\pi$
$7$	2.96082	1.11908	0.559542	+	0.828802i	$0.310977\pi$
$8$	0	0
$9$	3.80991	1.26997
$10$	0	0
$11$	0.556169	0.167691	0.0838456	−	0.996479i	$-0.473280\pi$
$11$	0.556169	0.167691	0.0838456	+	0.996479i	$0.473280\pi$
$12$	0	0
$13$	3.76956	1.04549	0.522744	−	0.852490i	$-0.324908\pi$
$13$	3.76956	1.04549	0.522744	+	0.852490i	$0.324908\pi$
$14$	0	0
$15$	4.69674	1.21269
$16$	0	0
$17$	1.41609	0.343452	0.171726	−	0.985145i	$-0.445066\pi$
$17$	1.41609	0.343452	0.171726	+	0.985145i	$0.445066\pi$
$18$	0	0
$19$	−1.17955	−0.270607	−0.135303	−	0.990804i	$-0.543201\pi$
$19$	−1.17955	−0.270607	−0.135303	+	0.990804i	$0.543201\pi$
$20$	0	0
$21$	−7.72649	−1.68606
$22$	0	0
$23$	−0.626692	−0.130674	−0.0653372	−	0.997863i	$-0.520812\pi$
$23$	−0.626692	−0.130674	−0.0653372	+	0.997863i	$0.520812\pi$
$24$	0	0
$25$	−1.76069	−0.352139
$26$	0	0
$27$	−2.11352	−0.406747
$28$	0	0
$29$	−0.654195	−0.121481	−0.0607405	−	0.998154i	$-0.519346\pi$
$29$	−0.654195	−0.121481	−0.0607405	+	0.998154i	$0.519346\pi$
$30$	0	0
$31$	−1.67221	−0.300337	−0.150169	−	0.988660i	$-0.547982\pi$
$31$	−1.67221	−0.300337	−0.150169	+	0.988660i	$0.547982\pi$
$32$	0	0
$33$	−1.45137	−0.252651
$34$	0	0
$35$	−5.32890	−0.900748
$36$	0	0
$37$	−2.59709	−0.426958	−0.213479	−	0.976948i	$-0.568480\pi$
$37$	−2.59709	−0.426958	−0.213479	+	0.976948i	$0.568480\pi$
$38$	0	0
$39$	−9.83697	−1.57518
$40$	0	0
$41$	6.68824	1.04453	0.522264	−	0.852784i	$-0.325088\pi$
$41$	6.68824	1.04453	0.522264	+	0.852784i	$0.325088\pi$
$42$	0	0
$43$	−11.7396	−1.79028	−0.895138	−	0.445789i	$-0.852923\pi$
$43$	−11.7396	−1.79028	−0.895138	+	0.445789i	$0.852923\pi$
$44$	0	0
$45$	−6.85710	−1.02220
$46$	0	0
$47$	−13.0111	−1.89786	−0.948930	−	0.315487i	$-0.897832\pi$
$47$	−13.0111	−1.89786	−0.948930	+	0.315487i	$0.897832\pi$
$48$	0	0
$49$	1.76644	0.252348
$50$	0	0
$51$	−3.69540	−0.517459
$52$	0	0
$53$	9.66280	1.32729	0.663644	−	0.748049i	$-0.269009\pi$
$53$	9.66280	1.32729	0.663644	+	0.748049i	$0.269009\pi$
$54$	0	0
$55$	−1.00100	−0.134974
$56$	0	0
$57$	3.07812	0.407707
$58$	0	0
$59$	−5.84207	−0.760573	−0.380287	−	0.924869i	$-0.624175\pi$
$59$	−5.84207	−0.760573	−0.380287	+	0.924869i	$0.624175\pi$
$60$	0	0
$61$	−6.37379	−0.816080	−0.408040	−	0.912964i	$-0.633788\pi$
$61$	−6.37379	−0.816080	−0.408040	+	0.912964i	$0.633788\pi$
$62$	0	0
$63$	11.2804	1.42120
$64$	0	0
$65$	−6.78448	−0.841511
$66$	0	0
$67$	−9.80452	−1.19781	−0.598906	−	0.800819i	$-0.704398\pi$
$67$	−9.80452	−1.19781	−0.598906	+	0.800819i	$0.704398\pi$
$68$	0	0
$69$	1.63540	0.196879
$70$	0	0
$71$	14.7673	1.75255	0.876277	−	0.481807i	$-0.160019\pi$
$71$	14.7673	1.75255	0.876277	+	0.481807i	$0.160019\pi$
$72$	0	0
$73$	15.5674	1.82203	0.911013	−	0.412378i	$-0.135302\pi$
$73$	15.5674	1.82203	0.911013	+	0.412378i	$0.135302\pi$
$74$	0	0
$75$	4.59467	0.530547
$76$	0	0
$77$	1.64671	0.187660
$78$	0	0
$79$	2.17637	0.244860	0.122430	−	0.992477i	$-0.460931\pi$
$79$	2.17637	0.244860	0.122430	+	0.992477i	$0.460931\pi$
$80$	0	0
$81$	−5.91432	−0.657147
$82$	0	0
$83$	−2.48063	−0.272285	−0.136142	−	0.990689i	$-0.543470\pi$
$83$	−2.48063	−0.272285	−0.136142	+	0.990689i	$0.543470\pi$
$84$	0	0
$85$	−2.54869	−0.276444
$86$	0	0
$87$	1.70717	0.183028
$88$	0	0
$89$	−1.95542	−0.207274	−0.103637	−	0.994615i	$-0.533048\pi$
$89$	−1.95542	−0.207274	−0.103637	+	0.994615i	$0.533048\pi$
$90$	0	0
$91$	11.1610	1.16999
$92$	0	0
$93$	4.36376	0.452501
$94$	0	0
$95$	2.12296	0.217811
$96$	0	0
$97$	15.4557	1.56928	0.784642	−	0.619949i	$-0.212847\pi$
$97$	15.4557	1.56928	0.784642	+	0.619949i	$0.212847\pi$
$98$	0	0
$99$	2.11895	0.212963
$100$	0	0
$101$	−8.17209	−0.813153	−0.406577	−	0.913617i	$-0.633278\pi$
$101$	−8.17209	−0.813153	−0.406577	+	0.913617i	$0.633278\pi$
$102$	0	0
$103$	−17.4480	−1.71920	−0.859601	−	0.510965i	$-0.829288\pi$
$103$	−17.4480	−1.71920	−0.859601	+	0.510965i	$0.829288\pi$
$104$	0	0
$105$	13.9062	1.35711
$106$	0	0
$107$	13.8402	1.33799	0.668994	−	0.743268i	$-0.266725\pi$
$107$	13.8402	1.33799	0.668994	+	0.743268i	$0.266725\pi$
$108$	0	0
$109$	−1.27978	−0.122581	−0.0612904	−	0.998120i	$-0.519522\pi$
$109$	−1.27978	−0.122581	−0.0612904	+	0.998120i	$0.519522\pi$
$110$	0	0
$111$	6.77731	0.643273
$112$	0	0
$113$	10.7715	1.01330	0.506648	−	0.862153i	$-0.330884\pi$
$113$	10.7715	1.01330	0.506648	+	0.862153i	$0.330884\pi$
$114$	0	0
$115$	1.12792	0.105180
$116$	0	0
$117$	14.3617	1.32774
$118$	0	0
$119$	4.19278	0.384351
$120$	0	0
$121$	−10.6907	−0.971880
$122$	0	0
$123$	−17.4535	−1.57373
$124$	0	0
$125$	12.1679	1.08833
$126$	0	0
$127$	15.7660	1.39900	0.699501	−	0.714632i	$-0.253406\pi$
$127$	15.7660	1.39900	0.699501	+	0.714632i	$0.253406\pi$
$128$	0	0
$129$	30.6355	2.69730
$130$	0	0
$131$	−20.8656	−1.82304	−0.911518	−	0.411260i	$-0.865089\pi$
$131$	−20.8656	−1.82304	−0.911518	+	0.411260i	$0.865089\pi$
$132$	0	0
$133$	−3.49242	−0.302831
$134$	0	0
$135$	3.80393	0.327390
$136$	0	0
$137$	−9.99541	−0.853965	−0.426983	−	0.904260i	$-0.640423\pi$
$137$	−9.99541	−0.853965	−0.426983	+	0.904260i	$0.640423\pi$
$138$	0	0
$139$	−0.122649	−0.0104029	−0.00520147	−	0.999986i	$-0.501656\pi$
$139$	−0.122649	−0.0104029	−0.00520147	+	0.999986i	$0.501656\pi$
$140$	0	0
$141$	33.9534	2.85939
$142$	0	0
$143$	2.09651	0.175319
$144$	0	0
$145$	1.17743	0.0977798
$146$	0	0
$147$	−4.60966	−0.380198
$148$	0	0
$149$	−2.60628	−0.213515	−0.106757	−	0.994285i	$-0.534047\pi$
$149$	−2.60628	−0.213515	−0.106757	+	0.994285i	$0.534047\pi$
$150$	0	0
$151$	17.5638	1.42932	0.714659	−	0.699473i	$-0.246582\pi$
$151$	17.5638	1.42932	0.714659	+	0.699473i	$0.246582\pi$
$152$	0	0
$153$	5.39517	0.436173
$154$	0	0
$155$	3.00965	0.241741
$156$	0	0
$157$	−16.6468	−1.32856	−0.664281	−	0.747483i	$-0.731262\pi$
$157$	−16.6468	−1.32856	−0.664281	+	0.747483i	$0.731262\pi$
$158$	0	0
$159$	−25.2159	−1.99975
$160$	0	0
$161$	−1.85552	−0.146235
$162$	0	0
$163$	−15.9897	−1.25241	−0.626206	−	0.779658i	$-0.715393\pi$
$163$	−15.9897	−1.25241	−0.626206	+	0.779658i	$0.715393\pi$
$164$	0	0
$165$	2.61218	0.203358
$166$	0	0
$167$	−2.01901	−0.156236	−0.0781179	−	0.996944i	$-0.524891\pi$
$167$	−2.01901	−0.156236	−0.0781179	+	0.996944i	$0.524891\pi$
$168$	0	0
$169$	1.20958	0.0930448
$170$	0	0
$171$	−4.49397	−0.343662
$172$	0	0
$173$	20.6171	1.56749	0.783745	−	0.621083i	$-0.213307\pi$
$173$	20.6171	1.56749	0.783745	+	0.621083i	$0.213307\pi$
$174$	0	0
$175$	−5.21309	−0.394073
$176$	0	0
$177$	15.2454	1.14591
$178$	0	0
$179$	6.13044	0.458211	0.229105	−	0.973402i	$-0.426420\pi$
$179$	6.13044	0.458211	0.229105	+	0.973402i	$0.426420\pi$
$180$	0	0
$181$	−1.40854	−0.104696	−0.0523478	−	0.998629i	$-0.516670\pi$
$181$	−1.40854	−0.104696	−0.0523478	+	0.998629i	$0.516670\pi$
$182$	0	0
$183$	16.6329	1.22954
$184$	0	0
$185$	4.67426	0.343658
$186$	0	0
$187$	0.787584	0.0575939
$188$	0	0
$189$	−6.25775	−0.455184
$190$	0	0
$191$	−14.3272	−1.03668	−0.518342	−	0.855174i	$-0.673450\pi$
$191$	−14.3272	−1.03668	−0.518342	+	0.855174i	$0.673450\pi$
$192$	0	0
$193$	−12.1991	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$193$	−12.1991	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$194$	0	0
$195$	17.7046	1.26786
$196$	0	0
$197$	1.77757	0.126647	0.0633234	−	0.997993i	$-0.479830\pi$
$197$	1.77757	0.126647	0.0633234	+	0.997993i	$0.479830\pi$
$198$	0	0
$199$	12.8198	0.908774	0.454387	−	0.890804i	$-0.349858\pi$
$199$	12.8198	0.908774	0.454387	+	0.890804i	$0.349858\pi$
$200$	0	0
$201$	25.5857	1.80467
$202$	0	0
$203$	−1.93695	−0.135947
$204$	0	0
$205$	−12.0375	−0.840739
$206$	0	0
$207$	−2.38764	−0.165952
$208$	0	0
$209$	−0.656027	−0.0453783
$210$	0	0
$211$	−16.1612	−1.11259	−0.556293	−	0.830986i	$-0.687777\pi$
$211$	−16.1612	−1.11259	−0.556293	+	0.830986i	$0.687777\pi$
$212$	0	0
$213$	−38.5364	−2.64047
$214$	0	0
$215$	21.1291	1.44099
$216$	0	0
$217$	−4.95110	−0.336103
$218$	0	0
$219$	−40.6244	−2.74514
$220$	0	0
$221$	5.33803	0.359075
$222$	0	0
$223$	−3.94572	−0.264225	−0.132113	−	0.991235i	$-0.542176\pi$
$223$	−3.94572	−0.264225	−0.132113	+	0.991235i	$0.542176\pi$
$224$	0	0
$225$	−6.70808	−0.447205
$226$	0	0
$227$	−22.7538	−1.51022	−0.755110	−	0.655598i	$-0.772417\pi$
$227$	−22.7538	−1.51022	−0.755110	+	0.655598i	$0.772417\pi$
$228$	0	0
$229$	4.37027	0.288796	0.144398	−	0.989520i	$-0.453875\pi$
$229$	4.37027	0.288796	0.144398	+	0.989520i	$0.453875\pi$
$230$	0	0
$231$	−4.29723	−0.282737
$232$	0	0
$233$	18.6088	1.21910	0.609552	−	0.792746i	$-0.291349\pi$
$233$	18.6088	1.21910	0.609552	+	0.792746i	$0.291349\pi$
$234$	0	0
$235$	23.4174	1.52758
$236$	0	0
$237$	−5.67940	−0.368917
$238$	0	0
$239$	5.09446	0.329533	0.164767	−	0.986333i	$-0.447313\pi$
$239$	5.09446	0.329533	0.164767	+	0.986333i	$0.447313\pi$
$240$	0	0
$241$	16.7744	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$241$	16.7744	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$242$	0	0
$243$	21.7745	1.39683
$244$	0	0
$245$	−3.17924	−0.203114
$246$	0	0
$247$	−4.44637	−0.282916
$248$	0	0
$249$	6.47340	0.410235
$250$	0	0
$251$	5.11995	0.323169	0.161584	−	0.986859i	$-0.448340\pi$
$251$	5.11995	0.323169	0.161584	+	0.986859i	$0.448340\pi$
$252$	0	0
$253$	−0.348547	−0.0219129
$254$	0	0
$255$	6.65100	0.416502
$256$	0	0
$257$	24.5001	1.52828	0.764138	−	0.645053i	$-0.223165\pi$
$257$	24.5001	1.52828	0.764138	+	0.645053i	$0.223165\pi$
$258$	0	0
$259$	−7.68950	−0.477802
$260$	0	0
$261$	−2.49242	−0.154277
$262$	0	0
$263$	−18.0183	−1.11106	−0.555529	−	0.831497i	$-0.687484\pi$
$263$	−18.0183	−1.11106	−0.555529	+	0.831497i	$0.687484\pi$
$264$	0	0
$265$	−17.3912	−1.06833
$266$	0	0
$267$	5.10283	0.312288
$268$	0	0
$269$	−2.96396	−0.180716	−0.0903581	−	0.995909i	$-0.528801\pi$
$269$	−2.96396	−0.180716	−0.0903581	+	0.995909i	$0.528801\pi$
$270$	0	0
$271$	19.2891	1.17173	0.585864	−	0.810409i	$-0.300755\pi$
$271$	19.2891	1.17173	0.585864	+	0.810409i	$0.300755\pi$
$272$	0	0
$273$	−29.1255	−1.76275
$274$	0	0
$275$	−0.979243	−0.0590506
$276$	0	0
$277$	2.24357	0.134803	0.0674014	−	0.997726i	$-0.478529\pi$
$277$	2.24357	0.134803	0.0674014	+	0.997726i	$0.478529\pi$
$278$	0	0
$279$	−6.37096	−0.381419
$280$	0	0
$281$	−10.0037	−0.596772	−0.298386	−	0.954445i	$-0.596448\pi$
$281$	−10.0037	−0.596772	−0.298386	+	0.954445i	$0.596448\pi$
$282$	0	0
$283$	3.97169	0.236093	0.118046	−	0.993008i	$-0.462337\pi$
$283$	3.97169	0.236093	0.118046	+	0.993008i	$0.462337\pi$
$284$	0	0
$285$	−5.54003	−0.328163
$286$	0	0
$287$	19.8027	1.16891
$288$	0	0
$289$	−14.9947	−0.882041
$290$	0	0
$291$	−40.3328	−2.36435
$292$	0	0
$293$	−9.07806	−0.530346	−0.265173	−	0.964201i	$-0.585429\pi$
$293$	−9.07806	−0.530346	−0.265173	+	0.964201i	$0.585429\pi$
$294$	0	0
$295$	10.5146	0.612184
$296$	0	0
$297$	−1.17548	−0.0682080
$298$	0	0
$299$	−2.36235	−0.136618
$300$	0	0
$301$	−34.7589	−2.00347
$302$	0	0
$303$	21.3257	1.22513
$304$	0	0
$305$	11.4716	0.656862
$306$	0	0
$307$	−4.14787	−0.236731	−0.118366	−	0.992970i	$-0.537765\pi$
$307$	−4.14787	−0.236731	−0.118366	+	0.992970i	$0.537765\pi$
$308$	0	0
$309$	45.5320	2.59022
$310$	0	0
$311$	−1.77044	−0.100393	−0.0501963	−	0.998739i	$-0.515985\pi$
$311$	−1.77044	−0.100393	−0.0501963	+	0.998739i	$0.515985\pi$
$312$	0	0
$313$	−7.56980	−0.427870	−0.213935	−	0.976848i	$-0.568628\pi$
$313$	−7.56980	−0.427870	−0.213935	+	0.976848i	$0.568628\pi$
$314$	0	0
$315$	−20.3026	−1.14392
$316$	0	0
$317$	−5.00364	−0.281032	−0.140516	−	0.990078i	$-0.544876\pi$
$317$	−5.00364	−0.281032	−0.140516	+	0.990078i	$0.544876\pi$
$318$	0	0
$319$	−0.363843	−0.0203713
$320$	0	0
$321$	−36.1172	−2.01587
$322$	0	0
$323$	−1.67034	−0.0929403
$324$	0	0
$325$	−6.63704	−0.368157
$326$	0	0
$327$	3.33969	0.184685
$328$	0	0
$329$	−38.5234	−2.12386
$330$	0	0
$331$	−19.1833	−1.05441	−0.527205	−	0.849738i	$-0.676760\pi$
$331$	−19.1833	−1.05441	−0.527205	+	0.849738i	$0.676760\pi$
$332$	0	0
$333$	−9.89466	−0.542224
$334$	0	0
$335$	17.6462	0.964117
$336$	0	0
$337$	−28.3529	−1.54448	−0.772240	−	0.635331i	$-0.780864\pi$
$337$	−28.3529	−1.54448	−0.772240	+	0.635331i	$0.780864\pi$
$338$	0	0
$339$	−28.1091	−1.52667
$340$	0	0
$341$	−0.930030	−0.0503639
$342$	0	0
$343$	−15.4956	−0.836685
$344$	0	0
$345$	−2.94341	−0.158468
$346$	0	0
$347$	25.9864	1.39502	0.697512	−	0.716573i	$-0.254290\pi$
$347$	25.9864	1.39502	0.697512	+	0.716573i	$0.254290\pi$
$348$	0	0
$349$	−5.04597	−0.270104	−0.135052	−	0.990838i	$-0.543120\pi$
$349$	−5.04597	−0.270104	−0.135052	+	0.990838i	$0.543120\pi$
$350$	0	0
$351$	−7.96705	−0.425250
$352$	0	0
$353$	0.970056	0.0516309	0.0258154	−	0.999667i	$-0.491782\pi$
$353$	0.970056	0.0516309	0.0258154	+	0.999667i	$0.491782\pi$
$354$	0	0
$355$	−26.5783	−1.41063
$356$	0	0
$357$	−10.9414	−0.579080
$358$	0	0
$359$	−8.25636	−0.435754	−0.217877	−	0.975976i	$-0.569913\pi$
$359$	−8.25636	−0.435754	−0.217877	+	0.975976i	$0.569913\pi$
$360$	0	0
$361$	−17.6087	−0.926772
$362$	0	0
$363$	27.8982	1.46427
$364$	0	0
$365$	−28.0183	−1.46655
$366$	0	0
$367$	5.64701	0.294772	0.147386	−	0.989079i	$-0.452914\pi$
$367$	5.64701	0.294772	0.147386	+	0.989079i	$0.452914\pi$
$368$	0	0
$369$	25.4816	1.32652
$370$	0	0
$371$	28.6098	1.48535
$372$	0	0
$373$	−30.4077	−1.57445	−0.787226	−	0.616665i	$-0.788484\pi$
$373$	−30.4077	−1.57445	−0.787226	+	0.616665i	$0.788484\pi$
$374$	0	0
$375$	−31.7532	−1.63973
$376$	0	0
$377$	−2.46603	−0.127007
$378$	0	0
$379$	4.11553	0.211401	0.105700	−	0.994398i	$-0.466292\pi$
$379$	4.11553	0.211401	0.105700	+	0.994398i	$0.466292\pi$
$380$	0	0
$381$	−41.1425	−2.10779
$382$	0	0
$383$	23.2452	1.18777	0.593887	−	0.804549i	$-0.297593\pi$
$383$	23.2452	1.18777	0.593887	+	0.804549i	$0.297593\pi$
$384$	0	0
$385$	−2.96377	−0.151048
$386$	0	0
$387$	−44.7269	−2.27360
$388$	0	0
$389$	−8.20992	−0.416260	−0.208130	−	0.978101i	$-0.566738\pi$
$389$	−8.20992	−0.416260	−0.208130	+	0.978101i	$0.566738\pi$
$390$	0	0
$391$	−0.887451	−0.0448803
$392$	0	0
$393$	54.4505	2.74666
$394$	0	0
$395$	−3.91704	−0.197088
$396$	0	0
$397$	−1.21987	−0.0612236	−0.0306118	−	0.999531i	$-0.509746\pi$
$397$	−1.21987	−0.0612236	−0.0306118	+	0.999531i	$0.509746\pi$
$398$	0	0
$399$	9.11375	0.456258
$400$	0	0
$401$	−22.0356	−1.10040	−0.550202	−	0.835032i	$-0.685449\pi$
$401$	−22.0356	−1.10040	−0.550202	+	0.835032i	$0.685449\pi$
$402$	0	0
$403$	−6.30349	−0.313999
$404$	0	0
$405$	10.6446	0.528936
$406$	0	0
$407$	−1.44442	−0.0715972
$408$	0	0
$409$	33.1713	1.64022	0.820109	−	0.572208i	$-0.193913\pi$
$409$	33.1713	1.64022	0.820109	+	0.572208i	$0.193913\pi$
$410$	0	0
$411$	26.0838	1.28662
$412$	0	0
$413$	−17.2973	−0.851145
$414$	0	0
$415$	4.46466	0.219161
$416$	0	0
$417$	0.320062	0.0156735
$418$	0	0
$419$	15.0663	0.736035	0.368017	−	0.929819i	$-0.380037\pi$
$419$	15.0663	0.736035	0.368017	+	0.929819i	$0.380037\pi$
$420$	0	0
$421$	3.23366	0.157599	0.0787994	−	0.996890i	$-0.474891\pi$
$421$	3.23366	0.157599	0.0787994	+	0.996890i	$0.474891\pi$
$422$	0	0
$423$	−49.5710	−2.41022
$424$	0	0
$425$	−2.49330	−0.120943
$426$	0	0
$427$	−18.8716	−0.913262
$428$	0	0
$429$	−5.47102	−0.264143
$430$	0	0
$431$	−11.7054	−0.563828	−0.281914	−	0.959440i	$-0.590969\pi$
$431$	−11.7054	−0.563828	−0.281914	+	0.959440i	$0.590969\pi$
$432$	0	0
$433$	−36.2313	−1.74116	−0.870582	−	0.492023i	$-0.836257\pi$
$433$	−36.2313	−1.74116	−0.870582	+	0.492023i	$0.836257\pi$
$434$	0	0
$435$	−3.07259	−0.147319
$436$	0	0
$437$	0.739212	0.0353613
$438$	0	0
$439$	−41.0159	−1.95758	−0.978791	−	0.204863i	$-0.934325\pi$
$439$	−41.0159	−1.95758	−0.978791	+	0.204863i	$0.934325\pi$
$440$	0	0
$441$	6.72996	0.320474
$442$	0	0
$443$	−14.0937	−0.669614	−0.334807	−	0.942287i	$-0.608671\pi$
$443$	−14.0937	−0.669614	−0.334807	+	0.942287i	$0.608671\pi$
$444$	0	0
$445$	3.51939	0.166835
$446$	0	0
$447$	6.80129	0.321690
$448$	0	0
$449$	5.72344	0.270106	0.135053	−	0.990838i	$-0.456880\pi$
$449$	5.72344	0.270106	0.135053	+	0.990838i	$0.456880\pi$
$450$	0	0
$451$	3.71979	0.175158
$452$	0	0
$453$	−45.8340	−2.15347
$454$	0	0
$455$	−20.0876	−0.941722
$456$	0	0
$457$	−39.7796	−1.86081	−0.930405	−	0.366533i	$-0.880545\pi$
$457$	−39.7796	−1.86081	−0.930405	+	0.366533i	$0.880545\pi$
$458$	0	0
$459$	−2.99293	−0.139698
$460$	0	0
$461$	−27.4806	−1.27990	−0.639950	−	0.768417i	$-0.721045\pi$
$461$	−27.4806	−1.27990	−0.639950	+	0.768417i	$0.721045\pi$
$462$	0	0
$463$	−18.6810	−0.868181	−0.434090	−	0.900869i	$-0.642930\pi$
$463$	−18.6810	−0.868181	−0.434090	+	0.900869i	$0.642930\pi$
$464$	0	0
$465$	−7.85393	−0.364217
$466$	0	0
$467$	−7.67859	−0.355323	−0.177661	−	0.984092i	$-0.556853\pi$
$467$	−7.67859	−0.355323	−0.177661	+	0.984092i	$0.556853\pi$
$468$	0	0
$469$	−29.0294	−1.34045
$470$	0	0
$471$	43.4412	2.00167
$472$	0	0
$473$	−6.52921	−0.300214
$474$	0	0
$475$	2.07682	0.0952911
$476$	0	0
$477$	36.8144	1.68562
$478$	0	0
$479$	4.30239	0.196581	0.0982905	−	0.995158i	$-0.468663\pi$
$479$	4.30239	0.196581	0.0982905	+	0.995158i	$0.468663\pi$
$480$	0	0
$481$	−9.78987	−0.446380
$482$	0	0
$483$	4.84213	0.220324
$484$	0	0
$485$	−27.8172	−1.26311
$486$	0	0
$487$	13.5347	0.613314	0.306657	−	0.951820i	$-0.400789\pi$
$487$	13.5347	0.613314	0.306657	+	0.951820i	$0.400789\pi$
$488$	0	0
$489$	41.7265	1.88694
$490$	0	0
$491$	5.64844	0.254911	0.127455	−	0.991844i	$-0.459319\pi$
$491$	5.64844	0.254911	0.127455	+	0.991844i	$0.459319\pi$
$492$	0	0
$493$	−0.926398	−0.0417229
$494$	0	0
$495$	−3.81371	−0.171413
$496$	0	0
$497$	43.7232	1.96126
$498$	0	0
$499$	−3.55401	−0.159099	−0.0795497	−	0.996831i	$-0.525348\pi$
$499$	−3.55401	−0.159099	−0.0795497	+	0.996831i	$0.525348\pi$
$500$	0	0
$501$	5.26877	0.235391
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	14.7082	0.654506
$506$	0	0
$507$	−3.15650	−0.140185
$508$	0	0
$509$	3.88595	0.172242	0.0861210	−	0.996285i	$-0.472553\pi$
$509$	3.88595	0.172242	0.0861210	+	0.996285i	$0.472553\pi$
$510$	0	0
$511$	46.0922	2.03900
$512$	0	0
$513$	2.49300	0.110069
$514$	0	0
$515$	31.4030	1.38378
$516$	0	0
$517$	−7.23635	−0.318254
$518$	0	0
$519$	−53.8020	−2.36165
$520$	0	0
$521$	11.7710	0.515697	0.257848	−	0.966185i	$-0.416986\pi$
$521$	11.7710	0.515697	0.257848	+	0.966185i	$0.416986\pi$
$522$	0	0
$523$	11.1788	0.488813	0.244406	−	0.969673i	$-0.421407\pi$
$523$	11.1788	0.488813	0.244406	+	0.969673i	$0.421407\pi$
$524$	0	0
$525$	13.6040	0.593726
$526$	0	0
$527$	−2.36799	−0.103151
$528$	0	0
$529$	−22.6073	−0.982924
$530$	0	0
$531$	−22.2578	−0.965905
$532$	0	0
$533$	25.2117	1.09204
$534$	0	0
$535$	−24.9098	−1.07694
$536$	0	0
$537$	−15.9979	−0.690360
$538$	0	0
$539$	0.982437	0.0423165
$540$	0	0
$541$	22.3291	0.960002	0.480001	−	0.877268i	$-0.340636\pi$
$541$	22.3291	0.960002	0.480001	+	0.877268i	$0.340636\pi$
$542$	0	0
$543$	3.67569	0.157739
$544$	0	0
$545$	2.30336	0.0986650
$546$	0	0
$547$	−34.3993	−1.47081	−0.735404	−	0.677628i	$-0.763008\pi$
$547$	−34.3993	−1.47081	−0.735404	+	0.677628i	$0.763008\pi$
$548$	0	0
$549$	−24.2836	−1.03640
$550$	0	0
$551$	0.771654	0.0328735
$552$	0	0
$553$	6.44382	0.274019
$554$	0	0
$555$	−12.1978	−0.517770
$556$	0	0
$557$	−1.27841	−0.0541681	−0.0270840	−	0.999633i	$-0.508622\pi$
$557$	−1.27841	−0.0541681	−0.0270840	+	0.999633i	$0.508622\pi$
$558$	0	0
$559$	−44.2532	−1.87171
$560$	0	0
$561$	−2.05526	−0.0867733
$562$	0	0
$563$	30.8888	1.30181	0.650904	−	0.759160i	$-0.274390\pi$
$563$	30.8888	1.30181	0.650904	+	0.759160i	$0.274390\pi$
$564$	0	0
$565$	−19.3866	−0.815600
$566$	0	0
$567$	−17.5112	−0.735402
$568$	0	0
$569$	26.1015	1.09423	0.547116	−	0.837056i	$-0.315725\pi$
$569$	26.1015	1.09423	0.547116	+	0.837056i	$0.315725\pi$
$570$	0	0
$571$	17.3941	0.727921	0.363961	−	0.931414i	$-0.381424\pi$
$571$	17.3941	0.727921	0.363961	+	0.931414i	$0.381424\pi$
$572$	0	0
$573$	37.3881	1.56191
$574$	0	0
$575$	1.10341	0.0460155
$576$	0	0
$577$	−10.3304	−0.430060	−0.215030	−	0.976607i	$-0.568985\pi$
$577$	−10.3304	−0.430060	−0.215030	+	0.976607i	$0.568985\pi$
$578$	0	0
$579$	31.8347	1.32300
$580$	0	0
$581$	−7.34469	−0.304709
$582$	0	0
$583$	5.37415	0.222575
$584$	0	0
$585$	−25.8483	−1.06869
$586$	0	0
$587$	−14.0116	−0.578320	−0.289160	−	0.957281i	$-0.593376\pi$
$587$	−14.0116	−0.578320	−0.289160	+	0.957281i	$0.593376\pi$
$588$	0	0
$589$	1.97245	0.0812732
$590$	0	0
$591$	−4.63872	−0.190811
$592$	0	0
$593$	32.0928	1.31789	0.658947	−	0.752189i	$-0.271002\pi$
$593$	32.0928	1.31789	0.658947	+	0.752189i	$0.271002\pi$
$594$	0	0
$595$	−7.54619	−0.309364
$596$	0	0
$597$	−33.4544	−1.36920
$598$	0	0
$599$	−21.7082	−0.886972	−0.443486	−	0.896281i	$-0.646258\pi$
$599$	−21.7082	−0.886972	−0.443486	+	0.896281i	$0.646258\pi$
$600$	0	0
$601$	−20.8258	−0.849501	−0.424751	−	0.905310i	$-0.639638\pi$
$601$	−20.8258	−0.849501	−0.424751	+	0.905310i	$0.639638\pi$
$602$	0	0
$603$	−37.3543	−1.52119
$604$	0	0
$605$	19.2412	0.782264
$606$	0	0
$607$	5.01526	0.203563	0.101782	−	0.994807i	$-0.467546\pi$
$607$	5.01526	0.203563	0.101782	+	0.994807i	$0.467546\pi$
$608$	0	0
$609$	5.05463	0.204824
$610$	0	0
$611$	−49.0460	−1.98419
$612$	0	0
$613$	10.8523	0.438322	0.219161	−	0.975689i	$-0.429668\pi$
$613$	10.8523	0.438322	0.219161	+	0.975689i	$0.429668\pi$
$614$	0	0
$615$	31.4129	1.26669
$616$	0	0
$617$	−40.3077	−1.62273	−0.811363	−	0.584542i	$-0.801274\pi$
$617$	−40.3077	−1.62273	−0.811363	+	0.584542i	$0.801274\pi$
$618$	0	0
$619$	−1.09183	−0.0438844	−0.0219422	−	0.999759i	$-0.506985\pi$
$619$	−1.09183	−0.0438844	−0.0219422	+	0.999759i	$0.506985\pi$
$620$	0	0
$621$	1.32453	0.0531514
$622$	0	0
$623$	−5.78965	−0.231957
$624$	0	0
$625$	−13.0965	−0.523860
$626$	0	0
$627$	1.71196	0.0683689
$628$	0	0
$629$	−3.67770	−0.146640
$630$	0	0
$631$	−8.88157	−0.353570	−0.176785	−	0.984250i	$-0.556570\pi$
$631$	−8.88157	−0.353570	−0.176785	+	0.984250i	$0.556570\pi$
$632$	0	0
$633$	42.1740	1.67627
$634$	0	0
$635$	−28.3757	−1.12605
$636$	0	0
$637$	6.65869	0.263827
$638$	0	0
$639$	56.2620	2.22569
$640$	0	0
$641$	−25.8395	−1.02060	−0.510300	−	0.859997i	$-0.670466\pi$
$641$	−25.8395	−1.02060	−0.510300	+	0.859997i	$0.670466\pi$
$642$	0	0
$643$	15.5850	0.614612	0.307306	−	0.951611i	$-0.400573\pi$
$643$	15.5850	0.614612	0.307306	+	0.951611i	$0.400573\pi$
$644$	0	0
$645$	−55.1380	−2.17106
$646$	0	0
$647$	−42.7676	−1.68137	−0.840683	−	0.541527i	$-0.817846\pi$
$647$	−42.7676	−1.68137	−0.840683	+	0.541527i	$0.817846\pi$
$648$	0	0
$649$	−3.24918	−0.127541
$650$	0	0
$651$	12.9203	0.506386
$652$	0	0
$653$	−1.42196	−0.0556456	−0.0278228	−	0.999613i	$-0.508857\pi$
$653$	−1.42196	−0.0556456	−0.0278228	+	0.999613i	$0.508857\pi$
$654$	0	0
$655$	37.5541	1.46736
$656$	0	0
$657$	59.3104	2.31392
$658$	0	0
$659$	46.1363	1.79722	0.898608	−	0.438753i	$-0.144580\pi$
$659$	46.1363	1.79722	0.898608	+	0.438753i	$0.144580\pi$
$660$	0	0
$661$	7.57700	0.294711	0.147355	−	0.989084i	$-0.452924\pi$
$661$	7.57700	0.294711	0.147355	+	0.989084i	$0.452924\pi$
$662$	0	0
$663$	−13.9300	−0.540997
$664$	0	0
$665$	6.28569	0.243748
$666$	0	0
$667$	0.409979	0.0158744
$668$	0	0
$669$	10.2967	0.398093
$670$	0	0
$671$	−3.54490	−0.136850
$672$	0	0
$673$	8.27947	0.319150	0.159575	−	0.987186i	$-0.448988\pi$
$673$	8.27947	0.319150	0.159575	+	0.987186i	$0.448988\pi$
$674$	0	0
$675$	3.72126	0.143232
$676$	0	0
$677$	−19.3771	−0.744724	−0.372362	−	0.928088i	$-0.621452\pi$
$677$	−19.3771	−0.744724	−0.372362	+	0.928088i	$0.621452\pi$
$678$	0	0
$679$	45.7614	1.75616
$680$	0	0
$681$	59.3778	2.27536
$682$	0	0
$683$	−6.53322	−0.249987	−0.124993	−	0.992158i	$-0.539891\pi$
$683$	−6.53322	−0.249987	−0.124993	+	0.992158i	$0.539891\pi$
$684$	0	0
$685$	17.9898	0.687355
$686$	0	0
$687$	−11.4046	−0.435112
$688$	0	0
$689$	36.4245	1.38766
$690$	0	0
$691$	−37.1091	−1.41170	−0.705848	−	0.708364i	$-0.749434\pi$
$691$	−37.1091	−1.41170	−0.705848	+	0.708364i	$0.749434\pi$
$692$	0	0
$693$	6.27383	0.238323
$694$	0	0
$695$	0.220744	0.00837331
$696$	0	0
$697$	9.47114	0.358745
$698$	0	0
$699$	−48.5612	−1.83675
$700$	0	0
$701$	16.0494	0.606179	0.303090	−	0.952962i	$-0.401982\pi$
$701$	16.0494	0.606179	0.303090	+	0.952962i	$0.401982\pi$
$702$	0	0
$703$	3.06338	0.115538
$704$	0	0
$705$	−61.1096	−2.30152
$706$	0	0
$707$	−24.1961	−0.909987
$708$	0	0
$709$	28.3712	1.06550	0.532752	−	0.846272i	$-0.321158\pi$
$709$	28.3712	1.06550	0.532752	+	0.846272i	$0.321158\pi$
$710$	0	0
$711$	8.29176	0.310965
$712$	0	0
$713$	1.04796	0.0392464
$714$	0	0
$715$	−3.77332	−0.141114
$716$	0	0
$717$	−13.2944	−0.496488
$718$	0	0
$719$	30.5561	1.13955	0.569776	−	0.821800i	$-0.307030\pi$
$719$	30.5561	1.13955	0.569776	+	0.821800i	$0.307030\pi$
$720$	0	0
$721$	−51.6603	−1.92393
$722$	0	0
$723$	−43.7742	−1.62798
$724$	0	0
$725$	1.15184	0.0427782
$726$	0	0
$727$	−10.1731	−0.377298	−0.188649	−	0.982045i	$-0.560411\pi$
$727$	−10.1731	−0.377298	−0.188649	+	0.982045i	$0.560411\pi$
$728$	0	0
$729$	−39.0792	−1.44738
$730$	0	0
$731$	−16.6243	−0.614874
$732$	0	0
$733$	−33.2995	−1.22995	−0.614973	−	0.788548i	$-0.710833\pi$
$733$	−33.2995	−1.22995	−0.614973	+	0.788548i	$0.710833\pi$
$734$	0	0
$735$	8.29649	0.306021
$736$	0	0
$737$	−5.45297	−0.200863
$738$	0	0
$739$	−29.7130	−1.09301	−0.546506	−	0.837455i	$-0.684042\pi$
$739$	−29.7130	−1.09301	−0.546506	+	0.837455i	$0.684042\pi$
$740$	0	0
$741$	11.6032	0.426253
$742$	0	0
$743$	−4.26066	−0.156309	−0.0781543	−	0.996941i	$-0.524903\pi$
$743$	−4.26066	−0.156309	−0.0781543	+	0.996941i	$0.524903\pi$
$744$	0	0
$745$	4.69080	0.171857
$746$	0	0
$747$	−9.45098	−0.345793
$748$	0	0
$749$	40.9784	1.49732
$750$	0	0
$751$	34.8393	1.27130	0.635652	−	0.771976i	$-0.280732\pi$
$751$	34.8393	1.27130	0.635652	+	0.771976i	$0.280732\pi$
$752$	0	0
$753$	−13.3609	−0.486899
$754$	0	0
$755$	−31.6114	−1.15046
$756$	0	0
$757$	−31.1452	−1.13199	−0.565996	−	0.824408i	$-0.691508\pi$
$757$	−31.1452	−1.13199	−0.565996	+	0.824408i	$0.691508\pi$
$758$	0	0
$759$	0.909560	0.0330149
$760$	0	0
$761$	−18.4463	−0.668676	−0.334338	−	0.942453i	$-0.608513\pi$
$761$	−18.4463	−0.668676	−0.334338	+	0.942453i	$0.608513\pi$
$762$	0	0
$763$	−3.78919	−0.137178
$764$	0	0
$765$	−9.71026	−0.351075
$766$	0	0
$767$	−22.0221	−0.795170
$768$	0	0
$769$	−31.5664	−1.13831	−0.569157	−	0.822229i	$-0.692730\pi$
$769$	−31.5664	−1.13831	−0.569157	+	0.822229i	$0.692730\pi$
$770$	0	0
$771$	−63.9350	−2.30256
$772$	0	0
$773$	8.87791	0.319316	0.159658	−	0.987172i	$-0.448961\pi$
$773$	8.87791	0.319316	0.159658	+	0.987172i	$0.448961\pi$
$774$	0	0
$775$	2.94424	0.105760
$776$	0	0
$777$	20.0664	0.719877
$778$	0	0
$779$	−7.88909	−0.282656
$780$	0	0
$781$	8.21311	0.293888
$782$	0	0
$783$	1.38266	0.0494121
$784$	0	0
$785$	29.9611	1.06936
$786$	0	0
$787$	−46.8385	−1.66961	−0.834806	−	0.550544i	$-0.814420\pi$
$787$	−46.8385	−1.66961	−0.834806	+	0.550544i	$0.814420\pi$
$788$	0	0
$789$	47.0203	1.67397
$790$	0	0
$791$	31.8924	1.13396
$792$	0	0
$793$	−24.0264	−0.853202
$794$	0	0
$795$	45.3837	1.60959
$796$	0	0
$797$	10.2988	0.364801	0.182400	−	0.983224i	$-0.441613\pi$
$797$	10.2988	0.364801	0.182400	+	0.983224i	$0.441613\pi$
$798$	0	0
$799$	−18.4248	−0.651823
$800$	0	0
$801$	−7.44999	−0.263232
$802$	0	0
$803$	8.65810	0.305538
$804$	0	0
$805$	3.33958	0.117705
$806$	0	0
$807$	7.73470	0.272274
$808$	0	0
$809$	16.1438	0.567585	0.283793	−	0.958886i	$-0.408407\pi$
$809$	16.1438	0.567585	0.283793	+	0.958886i	$0.408407\pi$
$810$	0	0
$811$	−6.06725	−0.213050	−0.106525	−	0.994310i	$-0.533972\pi$
$811$	−6.06725	−0.213050	−0.106525	+	0.994310i	$0.533972\pi$
$812$	0	0
$813$	−50.3364	−1.76537
$814$	0	0
$815$	28.7784	1.00806
$816$	0	0
$817$	13.8474	0.484460
$818$	0	0
$819$	42.5223	1.48585
$820$	0	0
$821$	−51.4788	−1.79662	−0.898311	−	0.439360i	$-0.855205\pi$
$821$	−51.4788	−1.79662	−0.898311	+	0.439360i	$0.855205\pi$
$822$	0	0
$823$	10.0549	0.350490	0.175245	−	0.984525i	$-0.443928\pi$
$823$	10.0549	0.350490	0.175245	+	0.984525i	$0.443928\pi$
$824$	0	0
$825$	2.55541	0.0889681
$826$	0	0
$827$	21.0562	0.732195	0.366097	−	0.930577i	$-0.380694\pi$
$827$	21.0562	0.732195	0.366097	+	0.930577i	$0.380694\pi$
$828$	0	0
$829$	21.9761	0.763263	0.381631	−	0.924315i	$-0.375362\pi$
$829$	21.9761	0.763263	0.381631	+	0.924315i	$0.375362\pi$
$830$	0	0
$831$	−5.85477	−0.203100
$832$	0	0
$833$	2.50143	0.0866694
$834$	0	0
$835$	3.63383	0.125754
$836$	0	0
$837$	3.53425	0.122161
$838$	0	0
$839$	19.9675	0.689356	0.344678	−	0.938721i	$-0.387988\pi$
$839$	19.9675	0.689356	0.344678	+	0.938721i	$0.387988\pi$
$840$	0	0
$841$	−28.5720	−0.985242
$842$	0	0
$843$	26.1055	0.899122
$844$	0	0
$845$	−2.17701	−0.0748916
$846$	0	0
$847$	−31.6531	−1.08761
$848$	0	0
$849$	−10.3645	−0.355707
$850$	0	0
$851$	1.62757	0.0557925
$852$	0	0
$853$	−47.8390	−1.63798	−0.818988	−	0.573811i	$-0.805464\pi$
$853$	−47.8390	−1.63798	−0.818988	+	0.573811i	$0.805464\pi$
$854$	0	0
$855$	8.08827	0.276613
$856$	0	0
$857$	10.1197	0.345683	0.172842	−	0.984950i	$-0.444705\pi$
$857$	10.1197	0.345683	0.172842	+	0.984950i	$0.444705\pi$
$858$	0	0
$859$	−9.08765	−0.310067	−0.155033	−	0.987909i	$-0.549548\pi$
$859$	−9.08765	−0.310067	−0.155033	+	0.987909i	$0.549548\pi$
$860$	0	0
$861$	−51.6766	−1.76113
$862$	0	0
$863$	−29.4843	−1.00366	−0.501828	−	0.864967i	$-0.667339\pi$
$863$	−29.4843	−1.00366	−0.501828	+	0.864967i	$0.667339\pi$
$864$	0	0
$865$	−37.1068	−1.26167
$866$	0	0
$867$	39.1299	1.32892
$868$	0	0
$869$	1.21043	0.0410609
$870$	0	0
$871$	−36.9587	−1.25230
$872$	0	0
$873$	58.8846	1.99294
$874$	0	0
$875$	36.0271	1.21794
$876$	0	0
$877$	−5.90777	−0.199491	−0.0997457	−	0.995013i	$-0.531803\pi$
$877$	−5.90777	−0.199491	−0.0997457	+	0.995013i	$0.531803\pi$
$878$	0	0
$879$	23.6899	0.799042
$880$	0	0
$881$	6.18210	0.208280	0.104140	−	0.994563i	$-0.466791\pi$
$881$	6.18210	0.208280	0.104140	+	0.994563i	$0.466791\pi$
$882$	0	0
$883$	53.4280	1.79800	0.898998	−	0.437952i	$-0.144296\pi$
$883$	53.4280	1.79800	0.898998	+	0.437952i	$0.144296\pi$
$884$	0	0
$885$	−27.4387	−0.922342
$886$	0	0
$887$	−9.69232	−0.325436	−0.162718	−	0.986673i	$-0.552026\pi$
$887$	−9.69232	−0.325436	−0.162718	+	0.986673i	$0.552026\pi$
$888$	0	0
$889$	46.6801	1.56560
$890$	0	0
$891$	−3.28936	−0.110198
$892$	0	0
$893$	15.3472	0.513573
$894$	0	0
$895$	−11.0336	−0.368813
$896$	0	0
$897$	6.16475	0.205835
$898$	0	0
$899$	1.09395	0.0364853
$900$	0	0
$901$	13.6834	0.455859
$902$	0	0
$903$	90.7061	3.01851
$904$	0	0
$905$	2.53509	0.0842694
$906$	0	0
$907$	37.9880	1.26137	0.630686	−	0.776038i	$-0.282774\pi$
$907$	37.9880	1.26137	0.630686	+	0.776038i	$0.282774\pi$
$908$	0	0
$909$	−31.1349	−1.03268
$910$	0	0
$911$	−27.6776	−0.916999	−0.458499	−	0.888695i	$-0.651613\pi$
$911$	−27.6776	−0.916999	−0.458499	+	0.888695i	$0.651613\pi$
$912$	0	0
$913$	−1.37965	−0.0456597
$914$	0	0
$915$	−29.9360	−0.989655
$916$	0	0
$917$	−61.7792	−2.04013
$918$	0	0
$919$	35.8523	1.18266	0.591328	−	0.806431i	$-0.298604\pi$
$919$	35.8523	1.18266	0.591328	+	0.806431i	$0.298604\pi$
$920$	0	0
$921$	10.8242	0.356669
$922$	0	0
$923$	55.6662	1.83227
$924$	0	0
$925$	4.57267	0.150349
$926$	0	0
$927$	−66.4753	−2.18333
$928$	0	0
$929$	19.9529	0.654634	0.327317	−	0.944915i	$-0.393855\pi$
$929$	19.9529	0.654634	0.327317	+	0.944915i	$0.393855\pi$
$930$	0	0
$931$	−2.08359	−0.0682870
$932$	0	0
$933$	4.62011	0.151256
$934$	0	0
$935$	−1.41750	−0.0463572
$936$	0	0
$937$	−47.3525	−1.54694	−0.773469	−	0.633835i	$-0.781480\pi$
$937$	−47.3525	−1.54694	−0.773469	+	0.633835i	$0.781480\pi$
$938$	0	0
$939$	19.7540	0.644647
$940$	0	0
$941$	27.3298	0.890926	0.445463	−	0.895300i	$-0.353039\pi$
$941$	27.3298	0.890926	0.445463	+	0.895300i	$0.353039\pi$
$942$	0	0
$943$	−4.19147	−0.136493
$944$	0	0
$945$	11.2627	0.366377
$946$	0	0
$947$	15.6080	0.507193	0.253597	−	0.967310i	$-0.418386\pi$
$947$	15.6080	0.507193	0.253597	+	0.967310i	$0.418386\pi$
$948$	0	0
$949$	58.6822	1.90491
$950$	0	0
$951$	13.0574	0.423415
$952$	0	0
$953$	21.8768	0.708659	0.354329	−	0.935121i	$-0.384709\pi$
$953$	21.8768	0.708659	0.354329	+	0.935121i	$0.384709\pi$
$954$	0	0
$955$	25.7863	0.834425
$956$	0	0
$957$	0.949477	0.0306922
$958$	0	0
$959$	−29.5946	−0.955659
$960$	0	0
$961$	−28.2037	−0.909798
$962$	0	0
$963$	52.7301	1.69920
$964$	0	0
$965$	21.9561	0.706792
$966$	0	0
$967$	35.3634	1.13721	0.568605	−	0.822611i	$-0.307483\pi$
$967$	35.3634	1.13721	0.568605	+	0.822611i	$0.307483\pi$
$968$	0	0
$969$	4.35889	0.140028
$970$	0	0
$971$	−25.6834	−0.824219	−0.412110	−	0.911134i	$-0.635208\pi$
$971$	−25.6834	−0.824219	−0.412110	+	0.911134i	$0.635208\pi$
$972$	0	0
$973$	−0.363141	−0.0116418
$974$	0	0
$975$	17.3199	0.554680
$976$	0	0
$977$	−45.7636	−1.46411	−0.732053	−	0.681248i	$-0.761438\pi$
$977$	−45.7636	−1.46411	−0.732053	+	0.681248i	$0.761438\pi$
$978$	0	0
$979$	−1.08755	−0.0347581
$980$	0	0
$981$	−4.87585	−0.155674
$982$	0	0
$983$	−14.2255	−0.453723	−0.226861	−	0.973927i	$-0.572846\pi$
$983$	−14.2255	−0.453723	−0.226861	+	0.973927i	$0.572846\pi$
$984$	0	0
$985$	−3.19929	−0.101938
$986$	0	0
$987$	100.530	3.19990
$988$	0	0
$989$	7.35713	0.233943
$990$	0	0
$991$	12.1156	0.384864	0.192432	−	0.981310i	$-0.438363\pi$
$991$	12.1156	0.384864	0.192432	+	0.981310i	$0.438363\pi$
$992$	0	0
$993$	50.0604	1.58862
$994$	0	0
$995$	−23.0732	−0.731470
$996$	0	0
$997$	59.3899	1.88090	0.940449	−	0.339935i	$-0.110405\pi$
$997$	59.3899	1.88090	0.940449	+	0.339935i	$0.110405\pi$
$998$	0	0
$999$	5.48900	0.173664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.7		33
4.3	odd	2		4024.2.a.g.1.27	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.27	✓	33	4.3	odd	2
8048.2.a.x.1.7		33	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)

\(f(q)\)	\(=\)	\(q-2.60958 q^{3} -1.79981 q^{5} +2.96082 q^{7} +3.80991 q^{9} +O(q^{10})\)	\(q-2.60958 q^{3} -1.79981 q^{5} +2.96082 q^{7} +3.80991 q^{9} +0.556169 q^{11} +3.76956 q^{13} +4.69674 q^{15} +1.41609 q^{17} -1.17955 q^{19} -7.72649 q^{21} -0.626692 q^{23} -1.76069 q^{25} -2.11352 q^{27} -0.654195 q^{29} -1.67221 q^{31} -1.45137 q^{33} -5.32890 q^{35} -2.59709 q^{37} -9.83697 q^{39} +6.68824 q^{41} -11.7396 q^{43} -6.85710 q^{45} -13.0111 q^{47} +1.76644 q^{49} -3.69540 q^{51} +9.66280 q^{53} -1.00100 q^{55} +3.07812 q^{57} -5.84207 q^{59} -6.37379 q^{61} +11.2804 q^{63} -6.78448 q^{65} -9.80452 q^{67} +1.63540 q^{69} +14.7673 q^{71} +15.5674 q^{73} +4.59467 q^{75} +1.64671 q^{77} +2.17637 q^{79} -5.91432 q^{81} -2.48063 q^{83} -2.54869 q^{85} +1.70717 q^{87} -1.95542 q^{89} +11.1610 q^{91} +4.36376 q^{93} +2.12296 q^{95} +15.4557 q^{97} +2.11895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * q^99

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