LMFDB - Embedded newform 8048.2.a.u.1.10

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:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-1.34769 q^{3} +4.23270 q^{5} +3.43664 q^{7} -1.18373 q^{9} +O(q^{10})$	$q-1.34769 q^{3} +4.23270 q^{5} +3.43664 q^{7} -1.18373 q^{9} -5.77016 q^{11} +1.41129 q^{13} -5.70437 q^{15} +5.54572 q^{17} +2.00551 q^{19} -4.63154 q^{21} +3.79879 q^{23} +12.9157 q^{25} +5.63837 q^{27} -10.1941 q^{29} -4.53890 q^{31} +7.77639 q^{33} +14.5463 q^{35} +4.04066 q^{37} -1.90198 q^{39} +6.27255 q^{41} +0.618226 q^{43} -5.01037 q^{45} -4.16333 q^{47} +4.81053 q^{49} -7.47392 q^{51} +6.53494 q^{53} -24.4234 q^{55} -2.70281 q^{57} -6.82079 q^{59} -5.36740 q^{61} -4.06805 q^{63} +5.97356 q^{65} -0.00814770 q^{67} -5.11960 q^{69} +13.8914 q^{71} -0.481274 q^{73} -17.4064 q^{75} -19.8300 q^{77} +10.8740 q^{79} -4.04760 q^{81} -5.05289 q^{83} +23.4734 q^{85} +13.7385 q^{87} +4.64626 q^{89} +4.85010 q^{91} +6.11703 q^{93} +8.48873 q^{95} +3.07039 q^{97} +6.83030 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	−1.34769	−0.778090	−0.389045	−	0.921219i	$-0.627195\pi$
$3$	−1.34769	−0.778090	−0.389045	+	0.921219i	$0.627195\pi$
$4$	0	0
$5$	4.23270	1.89292	0.946460	−	0.322820i	$-0.104631\pi$
$5$	4.23270	1.89292	0.946460	+	0.322820i	$0.104631\pi$
$6$	0	0
$7$	3.43664	1.29893	0.649465	−	0.760392i	$-0.274993\pi$
$7$	3.43664	1.29893	0.649465	+	0.760392i	$0.274993\pi$
$8$	0	0
$9$	−1.18373	−0.394576
$10$	0	0
$11$	−5.77016	−1.73977	−0.869884	−	0.493256i	$-0.835807\pi$
$11$	−5.77016	−1.73977	−0.869884	+	0.493256i	$0.835807\pi$
$12$	0	0
$13$	1.41129	0.391421	0.195711	−	0.980662i	$-0.437299\pi$
$13$	1.41129	0.391421	0.195711	+	0.980662i	$0.437299\pi$
$14$	0	0
$15$	−5.70437	−1.47286
$16$	0	0
$17$	5.54572	1.34504	0.672518	−	0.740081i	$-0.265213\pi$
$17$	5.54572	1.34504	0.672518	+	0.740081i	$0.265213\pi$
$18$	0	0
$19$	2.00551	0.460096	0.230048	−	0.973179i	$-0.426112\pi$
$19$	2.00551	0.460096	0.230048	+	0.973179i	$0.426112\pi$
$20$	0	0
$21$	−4.63154	−1.01068
$22$	0	0
$23$	3.79879	0.792103	0.396051	−	0.918228i	$-0.370380\pi$
$23$	3.79879	0.792103	0.396051	+	0.918228i	$0.370380\pi$
$24$	0	0
$25$	12.9157	2.58315
$26$	0	0
$27$	5.63837	1.08511
$28$	0	0
$29$	−10.1941	−1.89300	−0.946500	−	0.322705i	$-0.895408\pi$
$29$	−10.1941	−1.89300	−0.946500	+	0.322705i	$0.895408\pi$
$30$	0	0
$31$	−4.53890	−0.815210	−0.407605	−	0.913158i	$-0.633636\pi$
$31$	−4.53890	−0.815210	−0.407605	+	0.913158i	$0.633636\pi$
$32$	0	0
$33$	7.77639	1.35370
$34$	0	0
$35$	14.5463	2.45877
$36$	0	0
$37$	4.04066	0.664281	0.332140	−	0.943230i	$-0.392229\pi$
$37$	4.04066	0.664281	0.332140	+	0.943230i	$0.392229\pi$
$38$	0	0
$39$	−1.90198	−0.304561
$40$	0	0
$41$	6.27255	0.979608	0.489804	−	0.871832i	$-0.337068\pi$
$41$	6.27255	0.979608	0.489804	+	0.871832i	$0.337068\pi$
$42$	0	0
$43$	0.618226	0.0942786	0.0471393	−	0.998888i	$-0.484990\pi$
$43$	0.618226	0.0942786	0.0471393	+	0.998888i	$0.484990\pi$
$44$	0	0
$45$	−5.01037	−0.746901
$46$	0	0
$47$	−4.16333	−0.607285	−0.303642	−	0.952786i	$-0.598203\pi$
$47$	−4.16333	−0.607285	−0.303642	+	0.952786i	$0.598203\pi$
$48$	0	0
$49$	4.81053	0.687218
$50$	0	0
$51$	−7.47392	−1.04656
$52$	0	0
$53$	6.53494	0.897643	0.448822	−	0.893621i	$-0.351844\pi$
$53$	6.53494	0.897643	0.448822	+	0.893621i	$0.351844\pi$
$54$	0	0
$55$	−24.4234	−3.29324
$56$	0	0
$57$	−2.70281	−0.357996
$58$	0	0
$59$	−6.82079	−0.887991	−0.443995	−	0.896029i	$-0.646439\pi$
$59$	−6.82079	−0.887991	−0.443995	+	0.896029i	$0.646439\pi$
$60$	0	0
$61$	−5.36740	−0.687226	−0.343613	−	0.939111i	$-0.611651\pi$
$61$	−5.36740	−0.687226	−0.343613	+	0.939111i	$0.611651\pi$
$62$	0	0
$63$	−4.06805	−0.512527
$64$	0	0
$65$	5.97356	0.740929
$66$	0	0
$67$	−0.00814770	−0.000995401 0	−0.000497700	−	1.00000i	$-0.500158\pi$
$67$	−0.00814770	−0.000995401 0	−0.000497700		1.00000i	$0.500158\pi$
$68$	0	0
$69$	−5.11960	−0.616327
$70$	0	0
$71$	13.8914	1.64861	0.824306	−	0.566145i	$-0.191566\pi$
$71$	13.8914	1.64861	0.824306	+	0.566145i	$0.191566\pi$
$72$	0	0
$73$	−0.481274	−0.0563289	−0.0281644	−	0.999603i	$-0.508966\pi$
$73$	−0.481274	−0.0563289	−0.0281644	+	0.999603i	$0.508966\pi$
$74$	0	0
$75$	−17.4064	−2.00992
$76$	0	0
$77$	−19.8300	−2.25984
$78$	0	0
$79$	10.8740	1.22342	0.611709	−	0.791083i	$-0.290482\pi$
$79$	10.8740	1.22342	0.611709	+	0.791083i	$0.290482\pi$
$80$	0	0
$81$	−4.04760	−0.449734
$82$	0	0
$83$	−5.05289	−0.554627	−0.277313	−	0.960780i	$-0.589444\pi$
$83$	−5.05289	−0.554627	−0.277313	+	0.960780i	$0.589444\pi$
$84$	0	0
$85$	23.4734	2.54605
$86$	0	0
$87$	13.7385	1.47292
$88$	0	0
$89$	4.64626	0.492502	0.246251	−	0.969206i	$-0.420801\pi$
$89$	4.64626	0.492502	0.246251	+	0.969206i	$0.420801\pi$
$90$	0	0
$91$	4.85010	0.508429
$92$	0	0
$93$	6.11703	0.634307
$94$	0	0
$95$	8.48873	0.870925
$96$	0	0
$97$	3.07039	0.311751	0.155875	−	0.987777i	$-0.450180\pi$
$97$	3.07039	0.311751	0.155875	+	0.987777i	$0.450180\pi$
$98$	0	0
$99$	6.83030	0.686471
$100$	0	0
$101$	−1.78619	−0.177733	−0.0888664	−	0.996044i	$-0.528324\pi$
$101$	−1.78619	−0.177733	−0.0888664	+	0.996044i	$0.528324\pi$
$102$	0	0
$103$	−8.05778	−0.793957	−0.396978	−	0.917828i	$-0.629941\pi$
$103$	−8.05778	−0.793957	−0.396978	+	0.917828i	$0.629941\pi$
$104$	0	0
$105$	−19.6039	−1.91314
$106$	0	0
$107$	5.12440	0.495395	0.247697	−	0.968837i	$-0.420326\pi$
$107$	5.12440	0.495395	0.247697	+	0.968837i	$0.420326\pi$
$108$	0	0
$109$	1.18839	0.113827	0.0569137	−	0.998379i	$-0.481874\pi$
$109$	1.18839	0.113827	0.0569137	+	0.998379i	$0.481874\pi$
$110$	0	0
$111$	−5.44557	−0.516870
$112$	0	0
$113$	12.6876	1.19355	0.596774	−	0.802409i	$-0.296449\pi$
$113$	12.6876	1.19355	0.596774	+	0.802409i	$0.296449\pi$
$114$	0	0
$115$	16.0791	1.49939
$116$	0	0
$117$	−1.67058	−0.154445
$118$	0	0
$119$	19.0587	1.74711
$120$	0	0
$121$	22.2948	2.02680
$122$	0	0
$123$	−8.45347	−0.762223
$124$	0	0
$125$	33.5050	2.99677
$126$	0	0
$127$	16.0129	1.42092	0.710459	−	0.703739i	$-0.248487\pi$
$127$	16.0129	1.42092	0.710459	+	0.703739i	$0.248487\pi$
$128$	0	0
$129$	−0.833178	−0.0733572
$130$	0	0
$131$	17.2786	1.50964	0.754819	−	0.655933i	$-0.227725\pi$
$131$	17.2786	1.50964	0.754819	+	0.655933i	$0.227725\pi$
$132$	0	0
$133$	6.89223	0.597632
$134$	0	0
$135$	23.8655	2.05402
$136$	0	0
$137$	−14.5279	−1.24120	−0.620601	−	0.784127i	$-0.713111\pi$
$137$	−14.5279	−1.24120	−0.620601	+	0.784127i	$0.713111\pi$
$138$	0	0
$139$	19.3784	1.64365	0.821826	−	0.569739i	$-0.192956\pi$
$139$	19.3784	1.64365	0.821826	+	0.569739i	$0.192956\pi$
$140$	0	0
$141$	5.61089	0.472522
$142$	0	0
$143$	−8.14337	−0.680983
$144$	0	0
$145$	−43.1486	−3.58330
$146$	0	0
$147$	−6.48310	−0.534717
$148$	0	0
$149$	−8.86742	−0.726448	−0.363224	−	0.931702i	$-0.618324\pi$
$149$	−8.86742	−0.726448	−0.363224	+	0.931702i	$0.618324\pi$
$150$	0	0
$151$	10.9320	0.889630	0.444815	−	0.895622i	$-0.353269\pi$
$151$	10.9320	0.889630	0.444815	+	0.895622i	$0.353269\pi$
$152$	0	0
$153$	−6.56463	−0.530719
$154$	0	0
$155$	−19.2118	−1.54313
$156$	0	0
$157$	−12.7588	−1.01826	−0.509131	−	0.860689i	$-0.670033\pi$
$157$	−12.7588	−1.01826	−0.509131	+	0.860689i	$0.670033\pi$
$158$	0	0
$159$	−8.80708	−0.698447
$160$	0	0
$161$	13.0551	1.02889
$162$	0	0
$163$	10.1694	0.796531	0.398265	−	0.917270i	$-0.369612\pi$
$163$	10.1694	0.796531	0.398265	+	0.917270i	$0.369612\pi$
$164$	0	0
$165$	32.9151	2.56244
$166$	0	0
$167$	−9.77649	−0.756528	−0.378264	−	0.925698i	$-0.623479\pi$
$167$	−9.77649	−0.756528	−0.378264	+	0.925698i	$0.623479\pi$
$168$	0	0
$169$	−11.0083	−0.846789
$170$	0	0
$171$	−2.37398	−0.181543
$172$	0	0
$173$	−10.6985	−0.813393	−0.406696	−	0.913563i	$-0.633319\pi$
$173$	−10.6985	−0.813393	−0.406696	+	0.913563i	$0.633319\pi$
$174$	0	0
$175$	44.3868	3.35533
$176$	0	0
$177$	9.19232	0.690937
$178$	0	0
$179$	−14.3364	−1.07155	−0.535777	−	0.844360i	$-0.679981\pi$
$179$	−14.3364	−1.07155	−0.535777	+	0.844360i	$0.679981\pi$
$180$	0	0
$181$	0.313374	0.0232929	0.0116465	−	0.999932i	$-0.496293\pi$
$181$	0.313374	0.0232929	0.0116465	+	0.999932i	$0.496293\pi$
$182$	0	0
$183$	7.23360	0.534723
$184$	0	0
$185$	17.1029	1.25743
$186$	0	0
$187$	−31.9997	−2.34005
$188$	0	0
$189$	19.3771	1.40948
$190$	0	0
$191$	4.85740	0.351469	0.175734	−	0.984438i	$-0.443770\pi$
$191$	4.85740	0.351469	0.175734	+	0.984438i	$0.443770\pi$
$192$	0	0
$193$	−3.39724	−0.244539	−0.122269	−	0.992497i	$-0.539017\pi$
$193$	−3.39724	−0.244539	−0.122269	+	0.992497i	$0.539017\pi$
$194$	0	0
$195$	−8.05052	−0.576510
$196$	0	0
$197$	−5.11964	−0.364759	−0.182380	−	0.983228i	$-0.558380\pi$
$197$	−5.11964	−0.364759	−0.182380	+	0.983228i	$0.558380\pi$
$198$	0	0
$199$	19.5514	1.38596	0.692981	−	0.720956i	$-0.256297\pi$
$199$	19.5514	1.38596	0.692981	+	0.720956i	$0.256297\pi$
$200$	0	0
$201$	0.0109806	0.000774511 0
$202$	0	0
$203$	−35.0335	−2.45887
$204$	0	0
$205$	26.5498	1.85432
$206$	0	0
$207$	−4.49674	−0.312545
$208$	0	0
$209$	−11.5721	−0.800460
$210$	0	0
$211$	−16.4119	−1.12984	−0.564922	−	0.825144i	$-0.691094\pi$
$211$	−16.4119	−1.12984	−0.564922	+	0.825144i	$0.691094\pi$
$212$	0	0
$213$	−18.7214	−1.28277
$214$	0	0
$215$	2.61676	0.178462
$216$	0	0
$217$	−15.5986	−1.05890
$218$	0	0
$219$	0.648609	0.0438289
$220$	0	0
$221$	7.82662	0.526476
$222$	0	0
$223$	−18.2279	−1.22063	−0.610314	−	0.792160i	$-0.708957\pi$
$223$	−18.2279	−1.22063	−0.610314	+	0.792160i	$0.708957\pi$
$224$	0	0
$225$	−15.2887	−1.01925
$226$	0	0
$227$	6.46624	0.429179	0.214590	−	0.976704i	$-0.431159\pi$
$227$	6.46624	0.429179	0.214590	+	0.976704i	$0.431159\pi$
$228$	0	0
$229$	16.9870	1.12253	0.561266	−	0.827635i	$-0.310314\pi$
$229$	16.9870	1.12253	0.561266	+	0.827635i	$0.310314\pi$
$230$	0	0
$231$	26.7247	1.75836
$232$	0	0
$233$	−19.5840	−1.28299	−0.641495	−	0.767128i	$-0.721685\pi$
$233$	−19.5840	−1.28299	−0.641495	+	0.767128i	$0.721685\pi$
$234$	0	0
$235$	−17.6221	−1.14954
$236$	0	0
$237$	−14.6548	−0.951929
$238$	0	0
$239$	6.46160	0.417966	0.208983	−	0.977919i	$-0.432985\pi$
$239$	6.46160	0.417966	0.208983	+	0.977919i	$0.432985\pi$
$240$	0	0
$241$	−10.2891	−0.662779	−0.331390	−	0.943494i	$-0.607517\pi$
$241$	−10.2891	−0.662779	−0.331390	+	0.943494i	$0.607517\pi$
$242$	0	0
$243$	−11.4602	−0.735172
$244$	0	0
$245$	20.3615	1.30085
$246$	0	0
$247$	2.83036	0.180091
$248$	0	0
$249$	6.80973	0.431549
$250$	0	0
$251$	22.1524	1.39825	0.699123	−	0.715002i	$-0.253574\pi$
$251$	22.1524	1.39825	0.699123	+	0.715002i	$0.253574\pi$
$252$	0	0
$253$	−21.9196	−1.37808
$254$	0	0
$255$	−31.6349	−1.98105
$256$	0	0
$257$	−24.6888	−1.54004	−0.770022	−	0.638017i	$-0.779755\pi$
$257$	−24.6888	−1.54004	−0.770022	+	0.638017i	$0.779755\pi$
$258$	0	0
$259$	13.8863	0.862854
$260$	0	0
$261$	12.0671	0.746932
$262$	0	0
$263$	15.3803	0.948392	0.474196	−	0.880419i	$-0.342739\pi$
$263$	15.3803	0.948392	0.474196	+	0.880419i	$0.342739\pi$
$264$	0	0
$265$	27.6604	1.69917
$266$	0	0
$267$	−6.26172	−0.383211
$268$	0	0
$269$	−4.33299	−0.264187	−0.132094	−	0.991237i	$-0.542170\pi$
$269$	−4.33299	−0.264187	−0.132094	+	0.991237i	$0.542170\pi$
$270$	0	0
$271$	−14.7999	−0.899028	−0.449514	−	0.893273i	$-0.648403\pi$
$271$	−14.7999	−0.899028	−0.449514	+	0.893273i	$0.648403\pi$
$272$	0	0
$273$	−6.53644	−0.395603
$274$	0	0
$275$	−74.5259	−4.49408
$276$	0	0
$277$	20.9007	1.25580	0.627900	−	0.778294i	$-0.283915\pi$
$277$	20.9007	1.25580	0.627900	+	0.778294i	$0.283915\pi$
$278$	0	0
$279$	5.37282	0.321662
$280$	0	0
$281$	6.41669	0.382788	0.191394	−	0.981513i	$-0.438699\pi$
$281$	6.41669	0.382788	0.191394	+	0.981513i	$0.438699\pi$
$282$	0	0
$283$	−15.7102	−0.933874	−0.466937	−	0.884291i	$-0.654642\pi$
$283$	−15.7102	−0.933874	−0.466937	+	0.884291i	$0.654642\pi$
$284$	0	0
$285$	−11.4402	−0.677658
$286$	0	0
$287$	21.5565	1.27244
$288$	0	0
$289$	13.7551	0.809121
$290$	0	0
$291$	−4.13794	−0.242570
$292$	0	0
$293$	17.6324	1.03010	0.515048	−	0.857161i	$-0.327774\pi$
$293$	17.6324	1.03010	0.515048	+	0.857161i	$0.327774\pi$
$294$	0	0
$295$	−28.8703	−1.68090
$296$	0	0
$297$	−32.5343	−1.88783
$298$	0	0
$299$	5.36119	0.310046
$300$	0	0
$301$	2.12462	0.122461
$302$	0	0
$303$	2.40724	0.138292
$304$	0	0
$305$	−22.7186	−1.30086
$306$	0	0
$307$	17.2730	0.985821	0.492910	−	0.870080i	$-0.335933\pi$
$307$	17.2730	0.985821	0.492910	+	0.870080i	$0.335933\pi$
$308$	0	0
$309$	10.8594	0.617770
$310$	0	0
$311$	13.0297	0.738845	0.369422	−	0.929262i	$-0.379556\pi$
$311$	13.0297	0.738845	0.369422	+	0.929262i	$0.379556\pi$
$312$	0	0
$313$	29.5011	1.66750	0.833752	−	0.552140i	$-0.186188\pi$
$313$	29.5011	1.66750	0.833752	+	0.552140i	$0.186188\pi$
$314$	0	0
$315$	−17.2188	−0.970172
$316$	0	0
$317$	1.65127	0.0927444	0.0463722	−	0.998924i	$-0.485234\pi$
$317$	1.65127	0.0927444	0.0463722	+	0.998924i	$0.485234\pi$
$318$	0	0
$319$	58.8217	3.29338
$320$	0	0
$321$	−6.90611	−0.385462
$322$	0	0
$323$	11.1220	0.618845
$324$	0	0
$325$	18.2279	1.01110
$326$	0	0
$327$	−1.60159	−0.0885680
$328$	0	0
$329$	−14.3079	−0.788820
$330$	0	0
$331$	27.7113	1.52315	0.761577	−	0.648075i	$-0.224426\pi$
$331$	27.7113	1.52315	0.761577	+	0.648075i	$0.224426\pi$
$332$	0	0
$333$	−4.78305	−0.262109
$334$	0	0
$335$	−0.0344868	−0.00188421
$336$	0	0
$337$	21.8036	1.18772	0.593858	−	0.804570i	$-0.297604\pi$
$337$	21.8036	1.18772	0.593858	+	0.804570i	$0.297604\pi$
$338$	0	0
$339$	−17.0990	−0.928688
$340$	0	0
$341$	26.1902	1.41828
$342$	0	0
$343$	−7.52444	−0.406282
$344$	0	0
$345$	−21.6697	−1.16666
$346$	0	0
$347$	−6.73604	−0.361609	−0.180805	−	0.983519i	$-0.557870\pi$
$347$	−6.73604	−0.361609	−0.180805	+	0.983519i	$0.557870\pi$
$348$	0	0
$349$	−15.7685	−0.844067	−0.422034	−	0.906580i	$-0.638684\pi$
$349$	−15.7685	−0.844067	−0.422034	+	0.906580i	$0.638684\pi$
$350$	0	0
$351$	7.95738	0.424733
$352$	0	0
$353$	14.7706	0.786158	0.393079	−	0.919505i	$-0.371410\pi$
$353$	14.7706	0.786158	0.393079	+	0.919505i	$0.371410\pi$
$354$	0	0
$355$	58.7983	3.12069
$356$	0	0
$357$	−25.6852	−1.35941
$358$	0	0
$359$	−16.6709	−0.879855	−0.439928	−	0.898033i	$-0.644996\pi$
$359$	−16.6709	−0.879855	−0.439928	+	0.898033i	$0.644996\pi$
$360$	0	0
$361$	−14.9779	−0.788312
$362$	0	0
$363$	−30.0464	−1.57703
$364$	0	0
$365$	−2.03709	−0.106626
$366$	0	0
$367$	−25.6489	−1.33886	−0.669430	−	0.742875i	$-0.733462\pi$
$367$	−25.6489	−1.33886	−0.669430	+	0.742875i	$0.733462\pi$
$368$	0	0
$369$	−7.42500	−0.386530
$370$	0	0
$371$	22.4583	1.16598
$372$	0	0
$373$	−0.752388	−0.0389572	−0.0194786	−	0.999810i	$-0.506201\pi$
$373$	−0.752388	−0.0389572	−0.0194786	+	0.999810i	$0.506201\pi$
$374$	0	0
$375$	−45.1543	−2.33176
$376$	0	0
$377$	−14.3868	−0.740960
$378$	0	0
$379$	28.1396	1.44543	0.722717	−	0.691145i	$-0.242893\pi$
$379$	28.1396	1.44543	0.722717	+	0.691145i	$0.242893\pi$
$380$	0	0
$381$	−21.5805	−1.10560
$382$	0	0
$383$	−7.68839	−0.392858	−0.196429	−	0.980518i	$-0.562935\pi$
$383$	−7.68839	−0.392858	−0.196429	+	0.980518i	$0.562935\pi$
$384$	0	0
$385$	−83.9344	−4.27769
$386$	0	0
$387$	−0.731812	−0.0372001
$388$	0	0
$389$	−35.4754	−1.79867	−0.899337	−	0.437256i	$-0.855950\pi$
$389$	−35.4754	−1.79867	−0.899337	+	0.437256i	$0.855950\pi$
$390$	0	0
$391$	21.0670	1.06541
$392$	0	0
$393$	−23.2862	−1.17463
$394$	0	0
$395$	46.0263	2.31583
$396$	0	0
$397$	3.53032	0.177182	0.0885908	−	0.996068i	$-0.471764\pi$
$397$	3.53032	0.177182	0.0885908	+	0.996068i	$0.471764\pi$
$398$	0	0
$399$	−9.28860	−0.465012
$400$	0	0
$401$	−27.3948	−1.36803	−0.684016	−	0.729467i	$-0.739768\pi$
$401$	−27.3948	−1.36803	−0.684016	+	0.729467i	$0.739768\pi$
$402$	0	0
$403$	−6.40570	−0.319091
$404$	0	0
$405$	−17.1323	−0.851310
$406$	0	0
$407$	−23.3153	−1.15570
$408$	0	0
$409$	8.50584	0.420587	0.210293	−	0.977638i	$-0.432558\pi$
$409$	8.50584	0.420587	0.210293	+	0.977638i	$0.432558\pi$
$410$	0	0
$411$	19.5791	0.965767
$412$	0	0
$413$	−23.4406	−1.15344
$414$	0	0
$415$	−21.3874	−1.04986
$416$	0	0
$417$	−26.1161	−1.27891
$418$	0	0
$419$	−25.2785	−1.23494	−0.617468	−	0.786596i	$-0.711842\pi$
$419$	−25.2785	−1.23494	−0.617468	+	0.786596i	$0.711842\pi$
$420$	0	0
$421$	34.8639	1.69916	0.849581	−	0.527459i	$-0.176855\pi$
$421$	34.8639	1.69916	0.849581	+	0.527459i	$0.176855\pi$
$422$	0	0
$423$	4.92826	0.239620
$424$	0	0
$425$	71.6271	3.47443
$426$	0	0
$427$	−18.4459	−0.892658
$428$	0	0
$429$	10.9747	0.529866
$430$	0	0
$431$	11.0190	0.530768	0.265384	−	0.964143i	$-0.414501\pi$
$431$	11.0190	0.530768	0.265384	+	0.964143i	$0.414501\pi$
$432$	0	0
$433$	23.8824	1.14772	0.573859	−	0.818954i	$-0.305446\pi$
$433$	23.8824	1.14772	0.573859	+	0.818954i	$0.305446\pi$
$434$	0	0
$435$	58.1510	2.78813
$436$	0	0
$437$	7.61852	0.364443
$438$	0	0
$439$	−31.7408	−1.51490	−0.757452	−	0.652891i	$-0.773556\pi$
$439$	−31.7408	−1.51490	−0.757452	+	0.652891i	$0.773556\pi$
$440$	0	0
$441$	−5.69436	−0.271160
$442$	0	0
$443$	−0.805878	−0.0382884	−0.0191442	−	0.999817i	$-0.506094\pi$
$443$	−0.805878	−0.0382884	−0.0191442	+	0.999817i	$0.506094\pi$
$444$	0	0
$445$	19.6662	0.932268
$446$	0	0
$447$	11.9505	0.565241
$448$	0	0
$449$	−17.1047	−0.807220	−0.403610	−	0.914931i	$-0.632245\pi$
$449$	−17.1047	−0.807220	−0.403610	+	0.914931i	$0.632245\pi$
$450$	0	0
$451$	−36.1936	−1.70429
$452$	0	0
$453$	−14.7329	−0.692212
$454$	0	0
$455$	20.5290	0.962415
$456$	0	0
$457$	26.7058	1.24924	0.624622	−	0.780928i	$-0.285253\pi$
$457$	26.7058	1.24924	0.624622	+	0.780928i	$0.285253\pi$
$458$	0	0
$459$	31.2689	1.45951
$460$	0	0
$461$	14.9168	0.694743	0.347371	−	0.937728i	$-0.387074\pi$
$461$	14.9168	0.694743	0.347371	+	0.937728i	$0.387074\pi$
$462$	0	0
$463$	23.5385	1.09392	0.546962	−	0.837157i	$-0.315784\pi$
$463$	23.5385	1.09392	0.546962	+	0.837157i	$0.315784\pi$
$464$	0	0
$465$	25.8916	1.20069
$466$	0	0
$467$	5.07184	0.234697	0.117348	−	0.993091i	$-0.462561\pi$
$467$	5.07184	0.234697	0.117348	+	0.993091i	$0.462561\pi$
$468$	0	0
$469$	−0.0280008	−0.00129296
$470$	0	0
$471$	17.1949	0.792300
$472$	0	0
$473$	−3.56726	−0.164023
$474$	0	0
$475$	25.9027	1.18850
$476$	0	0
$477$	−7.73559	−0.354188
$478$	0	0
$479$	−0.246063	−0.0112429	−0.00562145	−	0.999984i	$-0.501789\pi$
$479$	−0.246063	−0.0112429	−0.00562145	+	0.999984i	$0.501789\pi$
$480$	0	0
$481$	5.70254	0.260014
$482$	0	0
$483$	−17.5942	−0.800566
$484$	0	0
$485$	12.9960	0.590120
$486$	0	0
$487$	17.0344	0.771902	0.385951	−	0.922519i	$-0.373873\pi$
$487$	17.0344	0.771902	0.385951	+	0.922519i	$0.373873\pi$
$488$	0	0
$489$	−13.7052	−0.619773
$490$	0	0
$491$	8.45703	0.381660	0.190830	−	0.981623i	$-0.438882\pi$
$491$	8.45703	0.381660	0.190830	+	0.981623i	$0.438882\pi$
$492$	0	0
$493$	−56.5337	−2.54615
$494$	0	0
$495$	28.9106	1.29944
$496$	0	0
$497$	47.7400	2.14143
$498$	0	0
$499$	−14.5367	−0.650753	−0.325377	−	0.945585i	$-0.605491\pi$
$499$	−14.5367	−0.650753	−0.325377	+	0.945585i	$0.605491\pi$
$500$	0	0
$501$	13.1757	0.588647
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−7.56041	−0.336434
$506$	0	0
$507$	14.8357	0.658878
$508$	0	0
$509$	29.5484	1.30971	0.654856	−	0.755754i	$-0.272729\pi$
$509$	29.5484	1.30971	0.654856	+	0.755754i	$0.272729\pi$
$510$	0	0
$511$	−1.65397	−0.0731672
$512$	0	0
$513$	11.3078	0.499253
$514$	0	0
$515$	−34.1062	−1.50290
$516$	0	0
$517$	24.0231	1.05653
$518$	0	0
$519$	14.4183	0.632893
$520$	0	0
$521$	32.6735	1.43145	0.715726	−	0.698381i	$-0.246096\pi$
$521$	32.6735	1.43145	0.715726	+	0.698381i	$0.246096\pi$
$522$	0	0
$523$	−2.38576	−0.104322	−0.0521610	−	0.998639i	$-0.516611\pi$
$523$	−2.38576	−0.104322	−0.0521610	+	0.998639i	$0.516611\pi$
$524$	0	0
$525$	−59.8197	−2.61075
$526$	0	0
$527$	−25.1715	−1.09649
$528$	0	0
$529$	−8.56919	−0.372573
$530$	0	0
$531$	8.07396	0.350380
$532$	0	0
$533$	8.85239	0.383440
$534$	0	0
$535$	21.6901	0.937743
$536$	0	0
$537$	19.3211	0.833765
$538$	0	0
$539$	−27.7575	−1.19560
$540$	0	0
$541$	−36.0449	−1.54969	−0.774846	−	0.632150i	$-0.782173\pi$
$541$	−36.0449	−1.54969	−0.774846	+	0.632150i	$0.782173\pi$
$542$	0	0
$543$	−0.422332	−0.0181240
$544$	0	0
$545$	5.03011	0.215466
$546$	0	0
$547$	−14.9474	−0.639107	−0.319553	−	0.947568i	$-0.603533\pi$
$547$	−14.9474	−0.639107	−0.319553	+	0.947568i	$0.603533\pi$
$548$	0	0
$549$	6.35355	0.271163
$550$	0	0
$551$	−20.4444	−0.870961
$552$	0	0
$553$	37.3700	1.58913
$554$	0	0
$555$	−23.0494	−0.978394
$556$	0	0
$557$	1.18008	0.0500016	0.0250008	−	0.999687i	$-0.492041\pi$
$557$	1.18008	0.0500016	0.0250008	+	0.999687i	$0.492041\pi$
$558$	0	0
$559$	0.872496	0.0369026
$560$	0	0
$561$	43.1257	1.82077
$562$	0	0
$563$	−23.0454	−0.971246	−0.485623	−	0.874168i	$-0.661407\pi$
$563$	−23.0454	−0.971246	−0.485623	+	0.874168i	$0.661407\pi$
$564$	0	0
$565$	53.7028	2.25929
$566$	0	0
$567$	−13.9102	−0.584172
$568$	0	0
$569$	−20.9738	−0.879268	−0.439634	−	0.898177i	$-0.644892\pi$
$569$	−20.9738	−0.879268	−0.439634	+	0.898177i	$0.644892\pi$
$570$	0	0
$571$	−19.5693	−0.818949	−0.409475	−	0.912321i	$-0.634288\pi$
$571$	−19.5693	−0.818949	−0.409475	+	0.912321i	$0.634288\pi$
$572$	0	0
$573$	−6.54627	−0.273474
$574$	0	0
$575$	49.0642	2.04612
$576$	0	0
$577$	23.9183	0.995731	0.497865	−	0.867254i	$-0.334117\pi$
$577$	23.9183	0.995731	0.497865	+	0.867254i	$0.334117\pi$
$578$	0	0
$579$	4.57843	0.190273
$580$	0	0
$581$	−17.3650	−0.720421
$582$	0	0
$583$	−37.7077	−1.56169
$584$	0	0
$585$	−7.07108	−0.292353
$586$	0	0
$587$	−9.68432	−0.399715	−0.199857	−	0.979825i	$-0.564048\pi$
$587$	−9.68432	−0.399715	−0.199857	+	0.979825i	$0.564048\pi$
$588$	0	0
$589$	−9.10281	−0.375075
$590$	0	0
$591$	6.89969	0.283815
$592$	0	0
$593$	−9.30964	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$593$	−9.30964	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$594$	0	0
$595$	80.6697	3.30713
$596$	0	0
$597$	−26.3492	−1.07840
$598$	0	0
$599$	5.32081	0.217402	0.108701	−	0.994074i	$-0.465331\pi$
$599$	5.32081	0.217402	0.108701	+	0.994074i	$0.465331\pi$
$600$	0	0
$601$	26.5574	1.08330	0.541649	−	0.840605i	$-0.317800\pi$
$601$	26.5574	1.08330	0.541649	+	0.840605i	$0.317800\pi$
$602$	0	0
$603$	0.00964467	0.000392761 0
$604$	0	0
$605$	94.3670	3.83656
$606$	0	0
$607$	−8.50362	−0.345151	−0.172576	−	0.984996i	$-0.555209\pi$
$607$	−8.50362	−0.345151	−0.172576	+	0.984996i	$0.555209\pi$
$608$	0	0
$609$	47.2144	1.91322
$610$	0	0
$611$	−5.87567	−0.237704
$612$	0	0
$613$	−30.3814	−1.22709	−0.613546	−	0.789659i	$-0.710258\pi$
$613$	−30.3814	−1.22709	−0.613546	+	0.789659i	$0.710258\pi$
$614$	0	0
$615$	−35.7810	−1.44283
$616$	0	0
$617$	23.6702	0.952927	0.476463	−	0.879194i	$-0.341918\pi$
$617$	23.6702	0.952927	0.476463	+	0.879194i	$0.341918\pi$
$618$	0	0
$619$	−32.2482	−1.29617	−0.648083	−	0.761570i	$-0.724429\pi$
$619$	−32.2482	−1.29617	−0.648083	+	0.761570i	$0.724429\pi$
$620$	0	0
$621$	21.4190	0.859515
$622$	0	0
$623$	15.9675	0.639726
$624$	0	0
$625$	77.2377	3.08951
$626$	0	0
$627$	15.5956	0.622830
$628$	0	0
$629$	22.4084	0.893481
$630$	0	0
$631$	29.5271	1.17546	0.587728	−	0.809058i	$-0.300022\pi$
$631$	29.5271	1.17546	0.587728	+	0.809058i	$0.300022\pi$
$632$	0	0
$633$	22.1182	0.879120
$634$	0	0
$635$	67.7779	2.68968
$636$	0	0
$637$	6.78905	0.268992
$638$	0	0
$639$	−16.4437	−0.650503
$640$	0	0
$641$	−45.1904	−1.78491	−0.892456	−	0.451134i	$-0.851020\pi$
$641$	−45.1904	−1.78491	−0.892456	+	0.451134i	$0.851020\pi$
$642$	0	0
$643$	22.4443	0.885118	0.442559	−	0.896739i	$-0.354071\pi$
$643$	22.4443	0.885118	0.442559	+	0.896739i	$0.354071\pi$
$644$	0	0
$645$	−3.52659	−0.138859
$646$	0	0
$647$	−27.9941	−1.10056	−0.550282	−	0.834979i	$-0.685480\pi$
$647$	−27.9941	−1.10056	−0.550282	+	0.834979i	$0.685480\pi$
$648$	0	0
$649$	39.3570	1.54490
$650$	0	0
$651$	21.0221	0.823920
$652$	0	0
$653$	−36.3882	−1.42398	−0.711990	−	0.702190i	$-0.752206\pi$
$653$	−36.3882	−1.42398	−0.711990	+	0.702190i	$0.752206\pi$
$654$	0	0
$655$	73.1351	2.85763
$656$	0	0
$657$	0.569698	0.0222260
$658$	0	0
$659$	29.2202	1.13826	0.569130	−	0.822248i	$-0.307280\pi$
$659$	29.2202	1.13826	0.569130	+	0.822248i	$0.307280\pi$
$660$	0	0
$661$	3.59496	0.139828	0.0699139	−	0.997553i	$-0.477728\pi$
$661$	3.59496	0.139828	0.0699139	+	0.997553i	$0.477728\pi$
$662$	0	0
$663$	−10.5479	−0.409645
$664$	0	0
$665$	29.1727	1.13127
$666$	0	0
$667$	−38.7253	−1.49945
$668$	0	0
$669$	24.5655	0.949758
$670$	0	0
$671$	30.9708	1.19561
$672$	0	0
$673$	7.10264	0.273786	0.136893	−	0.990586i	$-0.456288\pi$
$673$	7.10264	0.273786	0.136893	+	0.990586i	$0.456288\pi$
$674$	0	0
$675$	72.8238	2.80299
$676$	0	0
$677$	46.2992	1.77942	0.889711	−	0.456524i	$-0.150906\pi$
$677$	46.2992	1.77942	0.889711	+	0.456524i	$0.150906\pi$
$678$	0	0
$679$	10.5518	0.404942
$680$	0	0
$681$	−8.71449	−0.333940
$682$	0	0
$683$	−36.9029	−1.41205	−0.706024	−	0.708188i	$-0.749513\pi$
$683$	−36.9029	−1.41205	−0.706024	+	0.708188i	$0.749513\pi$
$684$	0	0
$685$	−61.4922	−2.34950
$686$	0	0
$687$	−22.8932	−0.873431
$688$	0	0
$689$	9.22269	0.351357
$690$	0	0
$691$	−21.0788	−0.801875	−0.400937	−	0.916105i	$-0.631315\pi$
$691$	−21.0788	−0.801875	−0.400937	+	0.916105i	$0.631315\pi$
$692$	0	0
$693$	23.4733	0.891678
$694$	0	0
$695$	82.0228	3.11130
$696$	0	0
$697$	34.7859	1.31761
$698$	0	0
$699$	26.3932	0.998281
$700$	0	0
$701$	32.2867	1.21945	0.609727	−	0.792612i	$-0.291279\pi$
$701$	32.2867	1.21945	0.609727	+	0.792612i	$0.291279\pi$
$702$	0	0
$703$	8.10359	0.305633
$704$	0	0
$705$	23.7492	0.894447
$706$	0	0
$707$	−6.13851	−0.230862
$708$	0	0
$709$	−3.57881	−0.134405	−0.0672026	−	0.997739i	$-0.521407\pi$
$709$	−3.57881	−0.134405	−0.0672026	+	0.997739i	$0.521407\pi$
$710$	0	0
$711$	−12.8718	−0.482732
$712$	0	0
$713$	−17.2423	−0.645730
$714$	0	0
$715$	−34.4684	−1.28905
$716$	0	0
$717$	−8.70824	−0.325215
$718$	0	0
$719$	−36.8095	−1.37276	−0.686382	−	0.727242i	$-0.740802\pi$
$719$	−36.8095	−1.37276	−0.686382	+	0.727242i	$0.740802\pi$
$720$	0	0
$721$	−27.6917	−1.03129
$722$	0	0
$723$	13.8665	0.515702
$724$	0	0
$725$	−131.665	−4.88990
$726$	0	0
$727$	42.6149	1.58050	0.790249	−	0.612786i	$-0.209951\pi$
$727$	42.6149	1.58050	0.790249	+	0.612786i	$0.209951\pi$
$728$	0	0
$729$	27.5876	1.02176
$730$	0	0
$731$	3.42851	0.126808
$732$	0	0
$733$	−24.2407	−0.895350	−0.447675	−	0.894196i	$-0.647748\pi$
$733$	−24.2407	−0.895350	−0.447675	+	0.894196i	$0.647748\pi$
$734$	0	0
$735$	−27.4410	−1.01218
$736$	0	0
$737$	0.0470136	0.00173177
$738$	0	0
$739$	21.1857	0.779327	0.389664	−	0.920957i	$-0.372591\pi$
$739$	21.1857	0.779327	0.389664	+	0.920957i	$0.372591\pi$
$740$	0	0
$741$	−3.81445	−0.140127
$742$	0	0
$743$	26.5767	0.975003	0.487502	−	0.873122i	$-0.337908\pi$
$743$	26.5767	0.975003	0.487502	+	0.873122i	$0.337908\pi$
$744$	0	0
$745$	−37.5331	−1.37511
$746$	0	0
$747$	5.98125	0.218842
$748$	0	0
$749$	17.6107	0.643483
$750$	0	0
$751$	9.50773	0.346942	0.173471	−	0.984839i	$-0.444502\pi$
$751$	9.50773	0.346942	0.173471	+	0.984839i	$0.444502\pi$
$752$	0	0
$753$	−29.8546	−1.08796
$754$	0	0
$755$	46.2717	1.68400
$756$	0	0
$757$	−8.78214	−0.319192	−0.159596	−	0.987182i	$-0.551019\pi$
$757$	−8.78214	−0.319192	−0.159596	+	0.987182i	$0.551019\pi$
$758$	0	0
$759$	29.5409	1.07227
$760$	0	0
$761$	19.3228	0.700451	0.350225	−	0.936665i	$-0.386105\pi$
$761$	19.3228	0.700451	0.350225	+	0.936665i	$0.386105\pi$
$762$	0	0
$763$	4.08409	0.147854
$764$	0	0
$765$	−27.7861	−1.00461
$766$	0	0
$767$	−9.62611	−0.347579
$768$	0	0
$769$	−38.7557	−1.39757	−0.698784	−	0.715333i	$-0.746275\pi$
$769$	−38.7557	−1.39757	−0.698784	+	0.715333i	$0.746275\pi$
$770$	0	0
$771$	33.2729	1.19829
$772$	0	0
$773$	−29.3487	−1.05560	−0.527800	−	0.849368i	$-0.676983\pi$
$773$	−29.3487	−1.05560	−0.527800	+	0.849368i	$0.676983\pi$
$774$	0	0
$775$	−58.6232	−2.10581
$776$	0	0
$777$	−18.7145	−0.671378
$778$	0	0
$779$	12.5797	0.450714
$780$	0	0
$781$	−80.1559	−2.86820
$782$	0	0
$783$	−57.4782	−2.05410
$784$	0	0
$785$	−54.0041	−1.92749
$786$	0	0
$787$	−22.3645	−0.797207	−0.398604	−	0.917123i	$-0.630505\pi$
$787$	−22.3645	−0.797207	−0.398604	+	0.917123i	$0.630505\pi$
$788$	0	0
$789$	−20.7279	−0.737934
$790$	0	0
$791$	43.6027	1.55034
$792$	0	0
$793$	−7.57496	−0.268995
$794$	0	0
$795$	−37.2777	−1.32210
$796$	0	0
$797$	53.9304	1.91031	0.955156	−	0.296104i	$-0.0956874\pi$
$797$	53.9304	1.91031	0.955156	+	0.296104i	$0.0956874\pi$
$798$	0	0
$799$	−23.0887	−0.816820
$800$	0	0
$801$	−5.49991	−0.194330
$802$	0	0
$803$	2.77703	0.0979992
$804$	0	0
$805$	55.2583	1.94760
$806$	0	0
$807$	5.83953	0.205561
$808$	0	0
$809$	15.5460	0.546569	0.273285	−	0.961933i	$-0.411890\pi$
$809$	15.5460	0.546569	0.273285	+	0.961933i	$0.411890\pi$
$810$	0	0
$811$	−26.0501	−0.914744	−0.457372	−	0.889275i	$-0.651209\pi$
$811$	−26.0501	−0.914744	−0.457372	+	0.889275i	$0.651209\pi$
$812$	0	0
$813$	19.9457	0.699525
$814$	0	0
$815$	43.0441	1.50777
$816$	0	0
$817$	1.23986	0.0433772
$818$	0	0
$819$	−5.74120	−0.200614
$820$	0	0
$821$	−7.57886	−0.264504	−0.132252	−	0.991216i	$-0.542221\pi$
$821$	−7.57886	−0.264504	−0.132252	+	0.991216i	$0.542221\pi$
$822$	0	0
$823$	14.8769	0.518577	0.259289	−	0.965800i	$-0.416512\pi$
$823$	14.8769	0.518577	0.259289	+	0.965800i	$0.416512\pi$
$824$	0	0
$825$	100.438	3.49680
$826$	0	0
$827$	−17.2505	−0.599860	−0.299930	−	0.953961i	$-0.596963\pi$
$827$	−17.2505	−0.599860	−0.299930	+	0.953961i	$0.596963\pi$
$828$	0	0
$829$	−42.4087	−1.47291	−0.736457	−	0.676484i	$-0.763503\pi$
$829$	−42.4087	−1.47291	−0.736457	+	0.676484i	$0.763503\pi$
$830$	0	0
$831$	−28.1677	−0.977125
$832$	0	0
$833$	26.6779	0.924333
$834$	0	0
$835$	−41.3809	−1.43205
$836$	0	0
$837$	−25.5920	−0.884589
$838$	0	0
$839$	−11.5875	−0.400047	−0.200023	−	0.979791i	$-0.564102\pi$
$839$	−11.5875	−0.400047	−0.200023	+	0.979791i	$0.564102\pi$
$840$	0	0
$841$	74.9200	2.58345
$842$	0	0
$843$	−8.64772	−0.297843
$844$	0	0
$845$	−46.5947	−1.60291
$846$	0	0
$847$	76.6191	2.63266
$848$	0	0
$849$	21.1725	0.726638
$850$	0	0
$851$	15.3496	0.526179
$852$	0	0
$853$	18.2112	0.623538	0.311769	−	0.950158i	$-0.399078\pi$
$853$	18.2112	0.623538	0.311769	+	0.950158i	$0.399078\pi$
$854$	0	0
$855$	−10.0483	−0.343646
$856$	0	0
$857$	12.4746	0.426126	0.213063	−	0.977038i	$-0.431656\pi$
$857$	12.4746	0.426126	0.213063	+	0.977038i	$0.431656\pi$
$858$	0	0
$859$	14.9918	0.511512	0.255756	−	0.966741i	$-0.417676\pi$
$859$	14.9918	0.511512	0.255756	+	0.966741i	$0.417676\pi$
$860$	0	0
$861$	−29.0516	−0.990075
$862$	0	0
$863$	20.0618	0.682913	0.341457	−	0.939898i	$-0.389080\pi$
$863$	20.0618	0.682913	0.341457	+	0.939898i	$0.389080\pi$
$864$	0	0
$865$	−45.2836	−1.53969
$866$	0	0
$867$	−18.5376	−0.629569
$868$	0	0
$869$	−62.7446	−2.12847
$870$	0	0
$871$	−0.0114988	−0.000389621 0
$872$	0	0
$873$	−3.63451	−0.123009
$874$	0	0
$875$	115.145	3.89260
$876$	0	0
$877$	−9.09037	−0.306960	−0.153480	−	0.988152i	$-0.549048\pi$
$877$	−9.09037	−0.306960	−0.153480	+	0.988152i	$0.549048\pi$
$878$	0	0
$879$	−23.7631	−0.801508
$880$	0	0
$881$	39.0237	1.31474	0.657370	−	0.753568i	$-0.271669\pi$
$881$	39.0237	1.31474	0.657370	+	0.753568i	$0.271669\pi$
$882$	0	0
$883$	−16.1429	−0.543251	−0.271626	−	0.962403i	$-0.587561\pi$
$883$	−16.1429	−0.543251	−0.271626	+	0.962403i	$0.587561\pi$
$884$	0	0
$885$	38.9083	1.30789
$886$	0	0
$887$	9.47697	0.318206	0.159103	−	0.987262i	$-0.449140\pi$
$887$	9.47697	0.318206	0.159103	+	0.987262i	$0.449140\pi$
$888$	0	0
$889$	55.0307	1.84567
$890$	0	0
$891$	23.3553	0.782433
$892$	0	0
$893$	−8.34961	−0.279409
$894$	0	0
$895$	−60.6817	−2.02837
$896$	0	0
$897$	−7.22523	−0.241244
$898$	0	0
$899$	46.2700	1.54319
$900$	0	0
$901$	36.2410	1.20736
$902$	0	0
$903$	−2.86334	−0.0952859
$904$	0	0
$905$	1.32642	0.0440916
$906$	0	0
$907$	−26.3810	−0.875968	−0.437984	−	0.898983i	$-0.644307\pi$
$907$	−26.3810	−0.875968	−0.437984	+	0.898983i	$0.644307\pi$
$908$	0	0
$909$	2.11437	0.0701291
$910$	0	0
$911$	10.2778	0.340518	0.170259	−	0.985399i	$-0.445540\pi$
$911$	10.2778	0.340518	0.170259	+	0.985399i	$0.445540\pi$
$912$	0	0
$913$	29.1560	0.964922
$914$	0	0
$915$	30.6177	1.01219
$916$	0	0
$917$	59.3804	1.96091
$918$	0	0
$919$	−57.6282	−1.90098	−0.950489	−	0.310758i	$-0.899417\pi$
$919$	−57.6282	−1.90098	−0.950489	+	0.310758i	$0.899417\pi$
$920$	0	0
$921$	−23.2786	−0.767057
$922$	0	0
$923$	19.6049	0.645302
$924$	0	0
$925$	52.1882	1.71594
$926$	0	0
$927$	9.53822	0.313276
$928$	0	0
$929$	7.09965	0.232932	0.116466	−	0.993195i	$-0.462843\pi$
$929$	7.09965	0.232932	0.116466	+	0.993195i	$0.462843\pi$
$930$	0	0
$931$	9.64757	0.316186
$932$	0	0
$933$	−17.5600	−0.574888
$934$	0	0
$935$	−135.445	−4.42953
$936$	0	0
$937$	−41.9006	−1.36883	−0.684416	−	0.729092i	$-0.739943\pi$
$937$	−41.9006	−1.36883	−0.684416	+	0.729092i	$0.739943\pi$
$938$	0	0
$939$	−39.7584	−1.29747
$940$	0	0
$941$	24.7271	0.806080	0.403040	−	0.915182i	$-0.367954\pi$
$941$	24.7271	0.806080	0.403040	+	0.915182i	$0.367954\pi$
$942$	0	0
$943$	23.8281	0.775951
$944$	0	0
$945$	82.0174	2.66803
$946$	0	0
$947$	43.8837	1.42603	0.713014	−	0.701150i	$-0.247330\pi$
$947$	43.8837	1.42603	0.713014	+	0.701150i	$0.247330\pi$
$948$	0	0
$949$	−0.679217	−0.0220483
$950$	0	0
$951$	−2.22540	−0.0721635
$952$	0	0
$953$	28.2222	0.914207	0.457103	−	0.889414i	$-0.348887\pi$
$953$	28.2222	0.914207	0.457103	+	0.889414i	$0.348887\pi$
$954$	0	0
$955$	20.5599	0.665303
$956$	0	0
$957$	−79.2735	−2.56255
$958$	0	0
$959$	−49.9272	−1.61223
$960$	0	0
$961$	−10.3984	−0.335433
$962$	0	0
$963$	−6.06590	−0.195471
$964$	0	0
$965$	−14.3795	−0.462892
$966$	0	0
$967$	12.8404	0.412920	0.206460	−	0.978455i	$-0.433806\pi$
$967$	12.8404	0.412920	0.206460	+	0.978455i	$0.433806\pi$
$968$	0	0
$969$	−14.9890	−0.481517
$970$	0	0
$971$	−23.7708	−0.762842	−0.381421	−	0.924401i	$-0.624565\pi$
$971$	−23.7708	−0.762842	−0.381421	+	0.924401i	$0.624565\pi$
$972$	0	0
$973$	66.5966	2.13499
$974$	0	0
$975$	−24.5655	−0.786726
$976$	0	0
$977$	34.1275	1.09184	0.545918	−	0.837838i	$-0.316181\pi$
$977$	34.1275	1.09184	0.545918	+	0.837838i	$0.316181\pi$
$978$	0	0
$979$	−26.8096	−0.856840
$980$	0	0
$981$	−1.40674	−0.0449136
$982$	0	0
$983$	48.9637	1.56170	0.780850	−	0.624718i	$-0.214786\pi$
$983$	48.9637	1.56170	0.780850	+	0.624718i	$0.214786\pi$
$984$	0	0
$985$	−21.6699	−0.690460
$986$	0	0
$987$	19.2826	0.613773
$988$	0	0
$989$	2.34851	0.0746783
$990$	0	0
$991$	10.7179	0.340465	0.170233	−	0.985404i	$-0.445548\pi$
$991$	10.7179	0.340465	0.170233	+	0.985404i	$0.445548\pi$
$992$	0	0
$993$	−37.3463	−1.18515
$994$	0	0
$995$	82.7551	2.62351
$996$	0	0
$997$	−41.3877	−1.31076	−0.655381	−	0.755298i	$-0.727492\pi$
$997$	−41.3877	−1.31076	−0.655381	+	0.755298i	$0.727492\pi$
$998$	0	0
$999$	22.7828	0.720815

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.10		26
4.3	odd	2		503.2.a.f.1.5	✓	26
12.11	even	2		4527.2.a.o.1.22		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.5	✓	26	4.3	odd	2
4527.2.a.o.1.22		26	12.11	even	2
8048.2.a.u.1.10		26	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)

\(f(q)\)	\(=\)	\(q-1.34769 q^{3} +4.23270 q^{5} +3.43664 q^{7} -1.18373 q^{9} +O(q^{10})\)	\(q-1.34769 q^{3} +4.23270 q^{5} +3.43664 q^{7} -1.18373 q^{9} -5.77016 q^{11} +1.41129 q^{13} -5.70437 q^{15} +5.54572 q^{17} +2.00551 q^{19} -4.63154 q^{21} +3.79879 q^{23} +12.9157 q^{25} +5.63837 q^{27} -10.1941 q^{29} -4.53890 q^{31} +7.77639 q^{33} +14.5463 q^{35} +4.04066 q^{37} -1.90198 q^{39} +6.27255 q^{41} +0.618226 q^{43} -5.01037 q^{45} -4.16333 q^{47} +4.81053 q^{49} -7.47392 q^{51} +6.53494 q^{53} -24.4234 q^{55} -2.70281 q^{57} -6.82079 q^{59} -5.36740 q^{61} -4.06805 q^{63} +5.97356 q^{65} -0.00814770 q^{67} -5.11960 q^{69} +13.8914 q^{71} -0.481274 q^{73} -17.4064 q^{75} -19.8300 q^{77} +10.8740 q^{79} -4.04760 q^{81} -5.05289 q^{83} +23.4734 q^{85} +13.7385 q^{87} +4.64626 q^{89} +4.85010 q^{91} +6.11703 q^{93} +8.48873 q^{95} +3.07039 q^{97} +6.83030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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