LMFDB - Embedded newform 8048.2.a.t.1.4

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:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-1.85682 q^{3} -3.94450 q^{5} +1.81629 q^{7} +0.447775 q^{9} +O(q^{10})$	$q-1.85682 q^{3} -3.94450 q^{5} +1.81629 q^{7} +0.447775 q^{9} -2.71418 q^{11} +5.27486 q^{13} +7.32422 q^{15} +4.97041 q^{17} +2.12206 q^{19} -3.37252 q^{21} +9.05426 q^{23} +10.5591 q^{25} +4.73902 q^{27} -2.73663 q^{29} +10.5394 q^{31} +5.03973 q^{33} -7.16435 q^{35} -8.01191 q^{37} -9.79445 q^{39} -1.80308 q^{41} +2.30268 q^{43} -1.76625 q^{45} -7.62130 q^{47} -3.70109 q^{49} -9.22916 q^{51} +1.84529 q^{53} +10.7061 q^{55} -3.94028 q^{57} +9.51643 q^{59} -10.2188 q^{61} +0.813289 q^{63} -20.8067 q^{65} -0.110762 q^{67} -16.8121 q^{69} -9.11297 q^{71} -9.81791 q^{73} -19.6063 q^{75} -4.92973 q^{77} +8.77024 q^{79} -10.1428 q^{81} -9.69425 q^{83} -19.6058 q^{85} +5.08143 q^{87} +8.04688 q^{89} +9.58066 q^{91} -19.5697 q^{93} -8.37047 q^{95} +2.50943 q^{97} -1.21534 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * 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
$3$	−1.85682	−1.07203	−0.536017	−	0.844207i	$-0.680072\pi$
$3$	−1.85682	−1.07203	−0.536017	+	0.844207i	$0.680072\pi$
$4$	0	0
$5$	−3.94450	−1.76403	−0.882017	−	0.471218i	$-0.843815\pi$
$5$	−3.94450	−1.76403	−0.882017	+	0.471218i	$0.843815\pi$
$6$	0	0
$7$	1.81629	0.686493	0.343246	−	0.939245i	$-0.388473\pi$
$7$	1.81629	0.686493	0.343246	+	0.939245i	$0.388473\pi$
$8$	0	0
$9$	0.447775	0.149258
$10$	0	0
$11$	−2.71418	−0.818355	−0.409178	−	0.912455i	$-0.634184\pi$
$11$	−2.71418	−0.818355	−0.409178	+	0.912455i	$0.634184\pi$
$12$	0	0
$13$	5.27486	1.46298	0.731491	−	0.681851i	$-0.238825\pi$
$13$	5.27486	1.46298	0.731491	+	0.681851i	$0.238825\pi$
$14$	0	0
$15$	7.32422	1.89111
$16$	0	0
$17$	4.97041	1.20550	0.602751	−	0.797929i	$-0.294071\pi$
$17$	4.97041	1.20550	0.602751	+	0.797929i	$0.294071\pi$
$18$	0	0
$19$	2.12206	0.486834	0.243417	−	0.969922i	$-0.421732\pi$
$19$	2.12206	0.486834	0.243417	+	0.969922i	$0.421732\pi$
$20$	0	0
$21$	−3.37252	−0.735944
$22$	0	0
$23$	9.05426	1.88794	0.943972	−	0.330026i	$-0.107058\pi$
$23$	9.05426	1.88794	0.943972	+	0.330026i	$0.107058\pi$
$24$	0	0
$25$	10.5591	2.11182
$26$	0	0
$27$	4.73902	0.912025
$28$	0	0
$29$	−2.73663	−0.508180	−0.254090	−	0.967181i	$-0.581776\pi$
$29$	−2.73663	−0.508180	−0.254090	+	0.967181i	$0.581776\pi$
$30$	0	0
$31$	10.5394	1.89293	0.946464	−	0.322810i	$-0.104628\pi$
$31$	10.5394	1.89293	0.946464	+	0.322810i	$0.104628\pi$
$32$	0	0
$33$	5.03973	0.877305
$34$	0	0
$35$	−7.16435	−1.21100
$36$	0	0
$37$	−8.01191	−1.31715	−0.658575	−	0.752515i	$-0.728840\pi$
$37$	−8.01191	−1.31715	−0.658575	+	0.752515i	$0.728840\pi$
$38$	0	0
$39$	−9.79445	−1.56837
$40$	0	0
$41$	−1.80308	−0.281593	−0.140797	−	0.990039i	$-0.544966\pi$
$41$	−1.80308	−0.281593	−0.140797	+	0.990039i	$0.544966\pi$
$42$	0	0
$43$	2.30268	0.351156	0.175578	−	0.984466i	$-0.443821\pi$
$43$	2.30268	0.351156	0.175578	+	0.984466i	$0.443821\pi$
$44$	0	0
$45$	−1.76625	−0.263297
$46$	0	0
$47$	−7.62130	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$47$	−7.62130	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$48$	0	0
$49$	−3.70109	−0.528728
$50$	0	0
$51$	−9.22916	−1.29234
$52$	0	0
$53$	1.84529	0.253469	0.126735	−	0.991937i	$-0.459550\pi$
$53$	1.84529	0.253469	0.126735	+	0.991937i	$0.459550\pi$
$54$	0	0
$55$	10.7061	1.44361
$56$	0	0
$57$	−3.94028	−0.521903
$58$	0	0
$59$	9.51643	1.23893	0.619467	−	0.785023i	$-0.287349\pi$
$59$	9.51643	1.23893	0.619467	+	0.785023i	$0.287349\pi$
$60$	0	0
$61$	−10.2188	−1.30838	−0.654192	−	0.756328i	$-0.726991\pi$
$61$	−10.2188	−1.30838	−0.654192	+	0.756328i	$0.726991\pi$
$62$	0	0
$63$	0.813289	0.102465
$64$	0	0
$65$	−20.8067	−2.58075
$66$	0	0
$67$	−0.110762	−0.0135317	−0.00676584	−	0.999977i	$-0.502154\pi$
$67$	−0.110762	−0.0135317	−0.00676584	+	0.999977i	$0.502154\pi$
$68$	0	0
$69$	−16.8121	−2.02394
$70$	0	0
$71$	−9.11297	−1.08151	−0.540756	−	0.841180i	$-0.681862\pi$
$71$	−9.11297	−1.08151	−0.540756	+	0.841180i	$0.681862\pi$
$72$	0	0
$73$	−9.81791	−1.14910	−0.574550	−	0.818470i	$-0.694823\pi$
$73$	−9.81791	−1.14910	−0.574550	+	0.818470i	$0.694823\pi$
$74$	0	0
$75$	−19.6063	−2.26394
$76$	0	0
$77$	−4.92973	−0.561795
$78$	0	0
$79$	8.77024	0.986730	0.493365	−	0.869823i	$-0.335767\pi$
$79$	8.77024	0.986730	0.493365	+	0.869823i	$0.335767\pi$
$80$	0	0
$81$	−10.1428	−1.12698
$82$	0	0
$83$	−9.69425	−1.06408	−0.532041	−	0.846719i	$-0.678575\pi$
$83$	−9.69425	−1.06408	−0.532041	+	0.846719i	$0.678575\pi$
$84$	0	0
$85$	−19.6058	−2.12655
$86$	0	0
$87$	5.08143	0.544787
$88$	0	0
$89$	8.04688	0.852968	0.426484	−	0.904495i	$-0.359752\pi$
$89$	8.04688	0.852968	0.426484	+	0.904495i	$0.359752\pi$
$90$	0	0
$91$	9.58066	1.00433
$92$	0	0
$93$	−19.5697	−2.02928
$94$	0	0
$95$	−8.37047	−0.858792
$96$	0	0
$97$	2.50943	0.254794	0.127397	−	0.991852i	$-0.459338\pi$
$97$	2.50943	0.254794	0.127397	+	0.991852i	$0.459338\pi$
$98$	0	0
$99$	−1.21534	−0.122146
$100$	0	0
$101$	−0.603573	−0.0600577	−0.0300289	−	0.999549i	$-0.509560\pi$
$101$	−0.603573	−0.0600577	−0.0300289	+	0.999549i	$0.509560\pi$
$102$	0	0
$103$	15.0372	1.48166	0.740830	−	0.671693i	$-0.234433\pi$
$103$	15.0372	1.48166	0.740830	+	0.671693i	$0.234433\pi$
$104$	0	0
$105$	13.3029	1.29823
$106$	0	0
$107$	3.40595	0.329266	0.164633	−	0.986355i	$-0.447356\pi$
$107$	3.40595	0.329266	0.164633	+	0.986355i	$0.447356\pi$
$108$	0	0
$109$	16.1087	1.54293	0.771466	−	0.636271i	$-0.219524\pi$
$109$	16.1087	1.54293	0.771466	+	0.636271i	$0.219524\pi$
$110$	0	0
$111$	14.8767	1.41203
$112$	0	0
$113$	11.5573	1.08721	0.543607	−	0.839340i	$-0.317058\pi$
$113$	11.5573	1.08721	0.543607	+	0.839340i	$0.317058\pi$
$114$	0	0
$115$	−35.7145	−3.33040
$116$	0	0
$117$	2.36195	0.218362
$118$	0	0
$119$	9.02771	0.827569
$120$	0	0
$121$	−3.63325	−0.330295
$122$	0	0
$123$	3.34799	0.301878
$124$	0	0
$125$	−21.9278	−1.96128
$126$	0	0
$127$	11.8100	1.04796	0.523982	−	0.851729i	$-0.324446\pi$
$127$	11.8100	1.04796	0.523982	+	0.851729i	$0.324446\pi$
$128$	0	0
$129$	−4.27566	−0.376451
$130$	0	0
$131$	16.0194	1.39962	0.699812	−	0.714327i	$-0.253267\pi$
$131$	16.0194	1.39962	0.699812	+	0.714327i	$0.253267\pi$
$132$	0	0
$133$	3.85428	0.334208
$134$	0	0
$135$	−18.6931	−1.60884
$136$	0	0
$137$	−17.1525	−1.46544	−0.732718	−	0.680532i	$-0.761749\pi$
$137$	−17.1525	−1.46544	−0.732718	+	0.680532i	$0.761749\pi$
$138$	0	0
$139$	7.09606	0.601880	0.300940	−	0.953643i	$-0.402700\pi$
$139$	7.09606	0.601880	0.300940	+	0.953643i	$0.402700\pi$
$140$	0	0
$141$	14.1514	1.19176
$142$	0	0
$143$	−14.3169	−1.19724
$144$	0	0
$145$	10.7947	0.896447
$146$	0	0
$147$	6.87226	0.566814
$148$	0	0
$149$	−1.77562	−0.145464	−0.0727322	−	0.997352i	$-0.523172\pi$
$149$	−1.77562	−0.145464	−0.0727322	+	0.997352i	$0.523172\pi$
$150$	0	0
$151$	−3.65506	−0.297445	−0.148722	−	0.988879i	$-0.547516\pi$
$151$	−3.65506	−0.297445	−0.148722	+	0.988879i	$0.547516\pi$
$152$	0	0
$153$	2.22563	0.179931
$154$	0	0
$155$	−41.5726	−3.33919
$156$	0	0
$157$	6.14391	0.490337	0.245169	−	0.969480i	$-0.421157\pi$
$157$	6.14391	0.490337	0.245169	+	0.969480i	$0.421157\pi$
$158$	0	0
$159$	−3.42636	−0.271728
$160$	0	0
$161$	16.4452	1.29606
$162$	0	0
$163$	−17.3428	−1.35840	−0.679198	−	0.733955i	$-0.737672\pi$
$163$	−17.3428	−1.35840	−0.679198	+	0.733955i	$0.737672\pi$
$164$	0	0
$165$	−19.8792	−1.54760
$166$	0	0
$167$	−10.9456	−0.846999	−0.423499	−	0.905896i	$-0.639198\pi$
$167$	−10.9456	−0.846999	−0.423499	+	0.905896i	$0.639198\pi$
$168$	0	0
$169$	14.8241	1.14032
$170$	0	0
$171$	0.950206	0.0726641
$172$	0	0
$173$	17.2125	1.30864	0.654321	−	0.756217i	$-0.272955\pi$
$173$	17.2125	1.30864	0.654321	+	0.756217i	$0.272955\pi$
$174$	0	0
$175$	19.1783	1.44975
$176$	0	0
$177$	−17.6703	−1.32818
$178$	0	0
$179$	−14.2541	−1.06540	−0.532701	−	0.846304i	$-0.678823\pi$
$179$	−14.2541	−1.06540	−0.532701	+	0.846304i	$0.678823\pi$
$180$	0	0
$181$	3.97051	0.295126	0.147563	−	0.989053i	$-0.452857\pi$
$181$	3.97051	0.295126	0.147563	+	0.989053i	$0.452857\pi$
$182$	0	0
$183$	18.9745	1.40263
$184$	0	0
$185$	31.6030	2.32350
$186$	0	0
$187$	−13.4906	−0.986529
$188$	0	0
$189$	8.60743	0.626098
$190$	0	0
$191$	19.7564	1.42952	0.714762	−	0.699368i	$-0.246535\pi$
$191$	19.7564	1.42952	0.714762	+	0.699368i	$0.246535\pi$
$192$	0	0
$193$	14.1509	1.01860	0.509302	−	0.860588i	$-0.329903\pi$
$193$	14.1509	1.01860	0.509302	+	0.860588i	$0.329903\pi$
$194$	0	0
$195$	38.6342	2.76665
$196$	0	0
$197$	−8.30175	−0.591475	−0.295738	−	0.955269i	$-0.595565\pi$
$197$	−8.30175	−0.591475	−0.295738	+	0.955269i	$0.595565\pi$
$198$	0	0
$199$	8.38049	0.594077	0.297038	−	0.954866i	$-0.404001\pi$
$199$	8.38049	0.594077	0.297038	+	0.954866i	$0.404001\pi$
$200$	0	0
$201$	0.205664	0.0145064
$202$	0	0
$203$	−4.97052	−0.348862
$204$	0	0
$205$	7.11224	0.496740
$206$	0	0
$207$	4.05427	0.281791
$208$	0	0
$209$	−5.75965	−0.398403
$210$	0	0
$211$	5.73796	0.395018	0.197509	−	0.980301i	$-0.436715\pi$
$211$	5.73796	0.395018	0.197509	+	0.980301i	$0.436715\pi$
$212$	0	0
$213$	16.9211	1.15942
$214$	0	0
$215$	−9.08292	−0.619450
$216$	0	0
$217$	19.1426	1.29948
$218$	0	0
$219$	18.2301	1.23187
$220$	0	0
$221$	26.2182	1.76363
$222$	0	0
$223$	−18.2148	−1.21975	−0.609877	−	0.792496i	$-0.708781\pi$
$223$	−18.2148	−1.21975	−0.609877	+	0.792496i	$0.708781\pi$
$224$	0	0
$225$	4.72809	0.315206
$226$	0	0
$227$	−14.8215	−0.983736	−0.491868	−	0.870670i	$-0.663686\pi$
$227$	−14.8215	−0.983736	−0.491868	+	0.870670i	$0.663686\pi$
$228$	0	0
$229$	19.5441	1.29151	0.645757	−	0.763543i	$-0.276542\pi$
$229$	19.5441	1.29151	0.645757	+	0.763543i	$0.276542\pi$
$230$	0	0
$231$	9.15361	0.602264
$232$	0	0
$233$	−9.96151	−0.652600	−0.326300	−	0.945266i	$-0.605802\pi$
$233$	−9.96151	−0.652600	−0.326300	+	0.945266i	$0.605802\pi$
$234$	0	0
$235$	30.0622	1.96104
$236$	0	0
$237$	−16.2848	−1.05781
$238$	0	0
$239$	−0.0513772	−0.00332332	−0.00166166	−	0.999999i	$-0.500529\pi$
$239$	−0.0513772	−0.00332332	−0.00166166	+	0.999999i	$0.500529\pi$
$240$	0	0
$241$	11.0663	0.712841	0.356421	−	0.934326i	$-0.383997\pi$
$241$	11.0663	0.712841	0.356421	+	0.934326i	$0.383997\pi$
$242$	0	0
$243$	4.61633	0.296137
$244$	0	0
$245$	14.5990	0.932693
$246$	0	0
$247$	11.1936	0.712230
$248$	0	0
$249$	18.0005	1.14073
$250$	0	0
$251$	0.610374	0.0385264	0.0192632	−	0.999814i	$-0.493868\pi$
$251$	0.610374	0.0385264	0.0192632	+	0.999814i	$0.493868\pi$
$252$	0	0
$253$	−24.5749	−1.54501
$254$	0	0
$255$	36.4044	2.27973
$256$	0	0
$257$	−7.62796	−0.475819	−0.237909	−	0.971287i	$-0.576462\pi$
$257$	−7.62796	−0.475819	−0.237909	+	0.971287i	$0.576462\pi$
$258$	0	0
$259$	−14.5519	−0.904214
$260$	0	0
$261$	−1.22540	−0.0758502
$262$	0	0
$263$	−16.9652	−1.04612	−0.523059	−	0.852297i	$-0.675209\pi$
$263$	−16.9652	−1.04612	−0.523059	+	0.852297i	$0.675209\pi$
$264$	0	0
$265$	−7.27873	−0.447129
$266$	0	0
$267$	−14.9416	−0.914411
$268$	0	0
$269$	17.6936	1.07880	0.539398	−	0.842051i	$-0.318652\pi$
$269$	17.6936	1.07880	0.539398	+	0.842051i	$0.318652\pi$
$270$	0	0
$271$	21.9554	1.33370	0.666849	−	0.745193i	$-0.267643\pi$
$271$	21.9554	1.33370	0.666849	+	0.745193i	$0.267643\pi$
$272$	0	0
$273$	−17.7896	−1.07667
$274$	0	0
$275$	−28.6592	−1.72822
$276$	0	0
$277$	−5.55843	−0.333973	−0.166987	−	0.985959i	$-0.553404\pi$
$277$	−5.55843	−0.333973	−0.166987	+	0.985959i	$0.553404\pi$
$278$	0	0
$279$	4.71927	0.282535
$280$	0	0
$281$	−32.1458	−1.91766	−0.958828	−	0.283987i	$-0.908343\pi$
$281$	−32.1458	−1.91766	−0.958828	+	0.283987i	$0.908343\pi$
$282$	0	0
$283$	29.3997	1.74763	0.873816	−	0.486256i	$-0.161638\pi$
$283$	29.3997	1.74763	0.873816	+	0.486256i	$0.161638\pi$
$284$	0	0
$285$	15.5424	0.920655
$286$	0	0
$287$	−3.27491	−0.193312
$288$	0	0
$289$	7.70502	0.453237
$290$	0	0
$291$	−4.65956	−0.273148
$292$	0	0
$293$	−31.3039	−1.82879	−0.914396	−	0.404820i	$-0.867334\pi$
$293$	−31.3039	−1.82879	−0.914396	+	0.404820i	$0.867334\pi$
$294$	0	0
$295$	−37.5376	−2.18552
$296$	0	0
$297$	−12.8625	−0.746360
$298$	0	0
$299$	47.7599	2.76203
$300$	0	0
$301$	4.18233	0.241066
$302$	0	0
$303$	1.12073	0.0643840
$304$	0	0
$305$	40.3081	2.30803
$306$	0	0
$307$	4.44264	0.253555	0.126778	−	0.991931i	$-0.459537\pi$
$307$	4.44264	0.253555	0.126778	+	0.991931i	$0.459537\pi$
$308$	0	0
$309$	−27.9213	−1.58839
$310$	0	0
$311$	2.74656	0.155743	0.0778714	−	0.996963i	$-0.475188\pi$
$311$	2.74656	0.155743	0.0778714	+	0.996963i	$0.475188\pi$
$312$	0	0
$313$	−21.1389	−1.19484	−0.597420	−	0.801928i	$-0.703808\pi$
$313$	−21.1389	−1.19484	−0.597420	+	0.801928i	$0.703808\pi$
$314$	0	0
$315$	−3.20802	−0.180751
$316$	0	0
$317$	3.95470	0.222118	0.111059	−	0.993814i	$-0.464576\pi$
$317$	3.95470	0.222118	0.111059	+	0.993814i	$0.464576\pi$
$318$	0	0
$319$	7.42771	0.415872
$320$	0	0
$321$	−6.32423	−0.352984
$322$	0	0
$323$	10.5475	0.586880
$324$	0	0
$325$	55.6976	3.08955
$326$	0	0
$327$	−29.9109	−1.65408
$328$	0	0
$329$	−13.8425	−0.763161
$330$	0	0
$331$	−19.6232	−1.07859	−0.539294	−	0.842117i	$-0.681309\pi$
$331$	−19.6232	−1.07859	−0.539294	+	0.842117i	$0.681309\pi$
$332$	0	0
$333$	−3.58753	−0.196596
$334$	0	0
$335$	0.436899	0.0238704
$336$	0	0
$337$	−15.0816	−0.821546	−0.410773	−	0.911738i	$-0.634741\pi$
$337$	−15.0816	−0.821546	−0.410773	+	0.911738i	$0.634741\pi$
$338$	0	0
$339$	−21.4597	−1.16553
$340$	0	0
$341$	−28.6057	−1.54909
$342$	0	0
$343$	−19.4363	−1.04946
$344$	0	0
$345$	66.3154	3.57030
$346$	0	0
$347$	21.5127	1.15486	0.577430	−	0.816440i	$-0.304055\pi$
$347$	21.5127	1.15486	0.577430	+	0.816440i	$0.304055\pi$
$348$	0	0
$349$	−11.1162	−0.595037	−0.297519	−	0.954716i	$-0.596159\pi$
$349$	−11.1162	−0.595037	−0.297519	+	0.954716i	$0.596159\pi$
$350$	0	0
$351$	24.9976	1.33428
$352$	0	0
$353$	13.7137	0.729909	0.364955	−	0.931025i	$-0.381085\pi$
$353$	13.7137	0.729909	0.364955	+	0.931025i	$0.381085\pi$
$354$	0	0
$355$	35.9461	1.90782
$356$	0	0
$357$	−16.7628	−0.887183
$358$	0	0
$359$	−19.9773	−1.05436	−0.527182	−	0.849753i	$-0.676751\pi$
$359$	−19.9773	−1.05436	−0.527182	+	0.849753i	$0.676751\pi$
$360$	0	0
$361$	−14.4969	−0.762992
$362$	0	0
$363$	6.74628	0.354088
$364$	0	0
$365$	38.7267	2.02705
$366$	0	0
$367$	−2.64315	−0.137972	−0.0689858	−	0.997618i	$-0.521976\pi$
$367$	−2.64315	−0.137972	−0.0689858	+	0.997618i	$0.521976\pi$
$368$	0	0
$369$	−0.807373	−0.0420302
$370$	0	0
$371$	3.35157	0.174005
$372$	0	0
$373$	33.4238	1.73062	0.865310	−	0.501236i	$-0.167121\pi$
$373$	33.4238	1.73062	0.865310	+	0.501236i	$0.167121\pi$
$374$	0	0
$375$	40.7159	2.10256
$376$	0	0
$377$	−14.4354	−0.743459
$378$	0	0
$379$	3.12671	0.160608	0.0803041	−	0.996770i	$-0.474411\pi$
$379$	3.12671	0.160608	0.0803041	+	0.996770i	$0.474411\pi$
$380$	0	0
$381$	−21.9289	−1.12345
$382$	0	0
$383$	−24.2996	−1.24165	−0.620826	−	0.783948i	$-0.713203\pi$
$383$	−24.2996	−1.24165	−0.620826	+	0.783948i	$0.713203\pi$
$384$	0	0
$385$	19.4453	0.991025
$386$	0	0
$387$	1.03108	0.0524129
$388$	0	0
$389$	17.1964	0.871890	0.435945	−	0.899973i	$-0.356414\pi$
$389$	17.1964	0.871890	0.435945	+	0.899973i	$0.356414\pi$
$390$	0	0
$391$	45.0034	2.27592
$392$	0	0
$393$	−29.7452	−1.50045
$394$	0	0
$395$	−34.5942	−1.74062
$396$	0	0
$397$	−27.4852	−1.37944	−0.689722	−	0.724075i	$-0.742267\pi$
$397$	−27.4852	−1.37944	−0.689722	+	0.724075i	$0.742267\pi$
$398$	0	0
$399$	−7.15669	−0.358283
$400$	0	0
$401$	29.2028	1.45832	0.729160	−	0.684343i	$-0.239911\pi$
$401$	29.2028	1.45832	0.729160	+	0.684343i	$0.239911\pi$
$402$	0	0
$403$	55.5937	2.76932
$404$	0	0
$405$	40.0084	1.98803
$406$	0	0
$407$	21.7457	1.07790
$408$	0	0
$409$	−6.14208	−0.303706	−0.151853	−	0.988403i	$-0.548524\pi$
$409$	−6.14208	−0.303706	−0.151853	+	0.988403i	$0.548524\pi$
$410$	0	0
$411$	31.8491	1.57100
$412$	0	0
$413$	17.2846	0.850519
$414$	0	0
$415$	38.2390	1.87708
$416$	0	0
$417$	−13.1761	−0.645236
$418$	0	0
$419$	−26.3111	−1.28538	−0.642692	−	0.766125i	$-0.722182\pi$
$419$	−26.3111	−1.28538	−0.642692	+	0.766125i	$0.722182\pi$
$420$	0	0
$421$	−36.3892	−1.77350	−0.886750	−	0.462250i	$-0.847042\pi$
$421$	−36.3892	−1.77350	−0.886750	+	0.462250i	$0.847042\pi$
$422$	0	0
$423$	−3.41263	−0.165928
$424$	0	0
$425$	52.4830	2.54580
$426$	0	0
$427$	−18.5603	−0.898197
$428$	0	0
$429$	26.5839	1.28348
$430$	0	0
$431$	−6.02116	−0.290029	−0.145015	−	0.989430i	$-0.546323\pi$
$431$	−6.02116	−0.290029	−0.145015	+	0.989430i	$0.546323\pi$
$432$	0	0
$433$	32.3038	1.55242	0.776212	−	0.630472i	$-0.217139\pi$
$433$	32.3038	1.55242	0.776212	+	0.630472i	$0.217139\pi$
$434$	0	0
$435$	−20.0437	−0.961023
$436$	0	0
$437$	19.2137	0.919116
$438$	0	0
$439$	0.683681	0.0326303	0.0163152	−	0.999867i	$-0.494806\pi$
$439$	0.683681	0.0326303	0.0163152	+	0.999867i	$0.494806\pi$
$440$	0	0
$441$	−1.65726	−0.0789170
$442$	0	0
$443$	13.3451	0.634044	0.317022	−	0.948418i	$-0.397317\pi$
$443$	13.3451	0.634044	0.317022	+	0.948418i	$0.397317\pi$
$444$	0	0
$445$	−31.7409	−1.50466
$446$	0	0
$447$	3.29700	0.155943
$448$	0	0
$449$	37.5727	1.77317	0.886584	−	0.462568i	$-0.153072\pi$
$449$	37.5727	1.77317	0.886584	+	0.462568i	$0.153072\pi$
$450$	0	0
$451$	4.89387	0.230443
$452$	0	0
$453$	6.78679	0.318871
$454$	0	0
$455$	−37.7909	−1.77167
$456$	0	0
$457$	41.1984	1.92718	0.963590	−	0.267385i	$-0.0861594\pi$
$457$	41.1984	1.92718	0.963590	+	0.267385i	$0.0861594\pi$
$458$	0	0
$459$	23.5549	1.09945
$460$	0	0
$461$	−28.5726	−1.33076	−0.665379	−	0.746506i	$-0.731730\pi$
$461$	−28.5726	−1.33076	−0.665379	+	0.746506i	$0.731730\pi$
$462$	0	0
$463$	27.7473	1.28953	0.644763	−	0.764383i	$-0.276956\pi$
$463$	27.7473	1.28953	0.644763	+	0.764383i	$0.276956\pi$
$464$	0	0
$465$	77.1927	3.57973
$466$	0	0
$467$	26.2547	1.21492	0.607462	−	0.794348i	$-0.292188\pi$
$467$	26.2547	1.21492	0.607462	+	0.794348i	$0.292188\pi$
$468$	0	0
$469$	−0.201175	−0.00928941
$470$	0	0
$471$	−11.4081	−0.525658
$472$	0	0
$473$	−6.24988	−0.287370
$474$	0	0
$475$	22.4070	1.02810
$476$	0	0
$477$	0.826273	0.0378324
$478$	0	0
$479$	−16.6815	−0.762197	−0.381099	−	0.924534i	$-0.624454\pi$
$479$	−16.6815	−0.762197	−0.381099	+	0.924534i	$0.624454\pi$
$480$	0	0
$481$	−42.2617	−1.92697
$482$	0	0
$483$	−30.5357	−1.38942
$484$	0	0
$485$	−9.89846	−0.449466
$486$	0	0
$487$	−6.69307	−0.303292	−0.151646	−	0.988435i	$-0.548457\pi$
$487$	−6.69307	−0.303292	−0.151646	+	0.988435i	$0.548457\pi$
$488$	0	0
$489$	32.2025	1.45625
$490$	0	0
$491$	−4.11207	−0.185575	−0.0927876	−	0.995686i	$-0.529578\pi$
$491$	−4.11207	−0.185575	−0.0927876	+	0.995686i	$0.529578\pi$
$492$	0	0
$493$	−13.6022	−0.612613
$494$	0	0
$495$	4.79391	0.215470
$496$	0	0
$497$	−16.5518	−0.742450
$498$	0	0
$499$	34.8606	1.56057	0.780287	−	0.625422i	$-0.215073\pi$
$499$	34.8606	1.56057	0.780287	+	0.625422i	$0.215073\pi$
$500$	0	0
$501$	20.3241	0.908012
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	2.38079	0.105944
$506$	0	0
$507$	−27.5257	−1.22246
$508$	0	0
$509$	−23.1500	−1.02611	−0.513053	−	0.858357i	$-0.671486\pi$
$509$	−23.1500	−1.02611	−0.513053	+	0.858357i	$0.671486\pi$
$510$	0	0
$511$	−17.8322	−0.788848
$512$	0	0
$513$	10.0565	0.444005
$514$	0	0
$515$	−59.3142	−2.61370
$516$	0	0
$517$	20.6855	0.909749
$518$	0	0
$519$	−31.9605	−1.40291
$520$	0	0
$521$	12.4515	0.545512	0.272756	−	0.962083i	$-0.412065\pi$
$521$	12.4515	0.545512	0.272756	+	0.962083i	$0.412065\pi$
$522$	0	0
$523$	−32.8249	−1.43533	−0.717666	−	0.696388i	$-0.754789\pi$
$523$	−32.8249	−1.43533	−0.717666	+	0.696388i	$0.754789\pi$
$524$	0	0
$525$	−35.6107	−1.55418
$526$	0	0
$527$	52.3851	2.28193
$528$	0	0
$529$	58.9796	2.56433
$530$	0	0
$531$	4.26122	0.184921
$532$	0	0
$533$	−9.51098	−0.411966
$534$	0	0
$535$	−13.4348	−0.580836
$536$	0	0
$537$	26.4673	1.14215
$538$	0	0
$539$	10.0454	0.432687
$540$	0	0
$541$	−9.88043	−0.424793	−0.212396	−	0.977184i	$-0.568127\pi$
$541$	−9.88043	−0.424793	−0.212396	+	0.977184i	$0.568127\pi$
$542$	0	0
$543$	−7.37252	−0.316385
$544$	0	0
$545$	−63.5407	−2.72178
$546$	0	0
$547$	23.5992	1.00903	0.504516	−	0.863403i	$-0.331671\pi$
$547$	23.5992	1.00903	0.504516	+	0.863403i	$0.331671\pi$
$548$	0	0
$549$	−4.57573	−0.195287
$550$	0	0
$551$	−5.80731	−0.247400
$552$	0	0
$553$	15.9293	0.677383
$554$	0	0
$555$	−58.6810	−2.49087
$556$	0	0
$557$	−16.7564	−0.709993	−0.354997	−	0.934868i	$-0.615518\pi$
$557$	−16.7564	−0.709993	−0.354997	+	0.934868i	$0.615518\pi$
$558$	0	0
$559$	12.1463	0.513734
$560$	0	0
$561$	25.0496	1.05759
$562$	0	0
$563$	−7.78871	−0.328255	−0.164127	−	0.986439i	$-0.552481\pi$
$563$	−7.78871	−0.328255	−0.164127	+	0.986439i	$0.552481\pi$
$564$	0	0
$565$	−45.5876	−1.91788
$566$	0	0
$567$	−18.4223	−0.773664
$568$	0	0
$569$	18.3575	0.769586	0.384793	−	0.923003i	$-0.374273\pi$
$569$	18.3575	0.769586	0.384793	+	0.923003i	$0.374273\pi$
$570$	0	0
$571$	32.0415	1.34089	0.670447	−	0.741958i	$-0.266103\pi$
$571$	32.0415	1.34089	0.670447	+	0.741958i	$0.266103\pi$
$572$	0	0
$573$	−36.6841	−1.53250
$574$	0	0
$575$	95.6047	3.98699
$576$	0	0
$577$	2.34196	0.0974970	0.0487485	−	0.998811i	$-0.484477\pi$
$577$	2.34196	0.0974970	0.0487485	+	0.998811i	$0.484477\pi$
$578$	0	0
$579$	−26.2757	−1.09198
$580$	0	0
$581$	−17.6076	−0.730485
$582$	0	0
$583$	−5.00843	−0.207428
$584$	0	0
$585$	−9.31671	−0.385199
$586$	0	0
$587$	38.5078	1.58939	0.794694	−	0.607011i	$-0.207632\pi$
$587$	38.5078	1.58939	0.794694	+	0.607011i	$0.207632\pi$
$588$	0	0
$589$	22.3652	0.921542
$590$	0	0
$591$	15.4148	0.634082
$592$	0	0
$593$	41.0143	1.68426	0.842128	−	0.539278i	$-0.181303\pi$
$593$	41.0143	1.68426	0.842128	+	0.539278i	$0.181303\pi$
$594$	0	0
$595$	−35.6098	−1.45986
$596$	0	0
$597$	−15.5610	−0.636871
$598$	0	0
$599$	−12.5079	−0.511061	−0.255530	−	0.966801i	$-0.582250\pi$
$599$	−12.5079	−0.511061	−0.255530	+	0.966801i	$0.582250\pi$
$600$	0	0
$601$	27.5176	1.12247	0.561234	−	0.827657i	$-0.310327\pi$
$601$	27.5176	1.12247	0.561234	+	0.827657i	$0.310327\pi$
$602$	0	0
$603$	−0.0495963	−0.00201972
$604$	0	0
$605$	14.3313	0.582652
$606$	0	0
$607$	−11.6359	−0.472287	−0.236144	−	0.971718i	$-0.575884\pi$
$607$	−11.6359	−0.472287	−0.236144	+	0.971718i	$0.575884\pi$
$608$	0	0
$609$	9.22935	0.373992
$610$	0	0
$611$	−40.2013	−1.62637
$612$	0	0
$613$	5.13331	0.207332	0.103666	−	0.994612i	$-0.466943\pi$
$613$	5.13331	0.207332	0.103666	+	0.994612i	$0.466943\pi$
$614$	0	0
$615$	−13.2061	−0.532523
$616$	0	0
$617$	1.96505	0.0791098	0.0395549	−	0.999217i	$-0.487406\pi$
$617$	1.96505	0.0791098	0.0395549	+	0.999217i	$0.487406\pi$
$618$	0	0
$619$	14.2750	0.573759	0.286879	−	0.957967i	$-0.407382\pi$
$619$	14.2750	0.573759	0.286879	+	0.957967i	$0.407382\pi$
$620$	0	0
$621$	42.9083	1.72185
$622$	0	0
$623$	14.6155	0.585556
$624$	0	0
$625$	33.6988	1.34795
$626$	0	0
$627$	10.6946	0.427102
$628$	0	0
$629$	−39.8225	−1.58783
$630$	0	0
$631$	−15.8374	−0.630478	−0.315239	−	0.949012i	$-0.602085\pi$
$631$	−15.8374	−0.630478	−0.315239	+	0.949012i	$0.602085\pi$
$632$	0	0
$633$	−10.6544	−0.423473
$634$	0	0
$635$	−46.5844	−1.84864
$636$	0	0
$637$	−19.5227	−0.773519
$638$	0	0
$639$	−4.08056	−0.161425
$640$	0	0
$641$	−0.559972	−0.0221175	−0.0110588	−	0.999939i	$-0.503520\pi$
$641$	−0.559972	−0.0221175	−0.0110588	+	0.999939i	$0.503520\pi$
$642$	0	0
$643$	12.9759	0.511718	0.255859	−	0.966714i	$-0.417642\pi$
$643$	12.9759	0.511718	0.255859	+	0.966714i	$0.417642\pi$
$644$	0	0
$645$	16.8653	0.664072
$646$	0	0
$647$	15.1782	0.596717	0.298359	−	0.954454i	$-0.403561\pi$
$647$	15.1782	0.596717	0.298359	+	0.954454i	$0.403561\pi$
$648$	0	0
$649$	−25.8293	−1.01389
$650$	0	0
$651$	−35.5442	−1.39309
$652$	0	0
$653$	16.1007	0.630071	0.315035	−	0.949080i	$-0.397984\pi$
$653$	16.1007	0.630071	0.315035	+	0.949080i	$0.397984\pi$
$654$	0	0
$655$	−63.1886	−2.46898
$656$	0	0
$657$	−4.39621	−0.171513
$658$	0	0
$659$	−15.9865	−0.622747	−0.311374	−	0.950288i	$-0.600789\pi$
$659$	−15.9865	−0.622747	−0.311374	+	0.950288i	$0.600789\pi$
$660$	0	0
$661$	35.5386	1.38229	0.691146	−	0.722715i	$-0.257106\pi$
$661$	35.5386	1.38229	0.691146	+	0.722715i	$0.257106\pi$
$662$	0	0
$663$	−48.6825	−1.89067
$664$	0	0
$665$	−15.2032	−0.589555
$666$	0	0
$667$	−24.7782	−0.959416
$668$	0	0
$669$	33.8216	1.30762
$670$	0	0
$671$	27.7357	1.07072
$672$	0	0
$673$	−6.58630	−0.253883	−0.126942	−	0.991910i	$-0.540516\pi$
$673$	−6.58630	−0.253883	−0.126942	+	0.991910i	$0.540516\pi$
$674$	0	0
$675$	50.0397	1.92603
$676$	0	0
$677$	−45.9657	−1.76661	−0.883303	−	0.468803i	$-0.844685\pi$
$677$	−45.9657	−1.76661	−0.883303	+	0.468803i	$0.844685\pi$
$678$	0	0
$679$	4.55786	0.174915
$680$	0	0
$681$	27.5208	1.05460
$682$	0	0
$683$	1.79922	0.0688452	0.0344226	−	0.999407i	$-0.489041\pi$
$683$	1.79922	0.0688452	0.0344226	+	0.999407i	$0.489041\pi$
$684$	0	0
$685$	67.6580	2.58508
$686$	0	0
$687$	−36.2899	−1.38455
$688$	0	0
$689$	9.73361	0.370821
$690$	0	0
$691$	−18.2317	−0.693567	−0.346783	−	0.937945i	$-0.612726\pi$
$691$	−18.2317	−0.693567	−0.346783	+	0.937945i	$0.612726\pi$
$692$	0	0
$693$	−2.20741	−0.0838526
$694$	0	0
$695$	−27.9904	−1.06174
$696$	0	0
$697$	−8.96204	−0.339462
$698$	0	0
$699$	18.4967	0.699610
$700$	0	0
$701$	−9.05386	−0.341960	−0.170980	−	0.985275i	$-0.554693\pi$
$701$	−9.05386	−0.341960	−0.170980	+	0.985275i	$0.554693\pi$
$702$	0	0
$703$	−17.0018	−0.641234
$704$	0	0
$705$	−55.8201	−2.10231
$706$	0	0
$707$	−1.09626	−0.0412292
$708$	0	0
$709$	6.55608	0.246219	0.123109	−	0.992393i	$-0.460713\pi$
$709$	6.55608	0.246219	0.123109	+	0.992393i	$0.460713\pi$
$710$	0	0
$711$	3.92710	0.147278
$712$	0	0
$713$	95.4262	3.57374
$714$	0	0
$715$	56.4730	2.11197
$716$	0	0
$717$	0.0953982	0.00356271
$718$	0	0
$719$	−8.20776	−0.306098	−0.153049	−	0.988219i	$-0.548909\pi$
$719$	−8.20776	−0.306098	−0.153049	+	0.988219i	$0.548909\pi$
$720$	0	0
$721$	27.3119	1.01715
$722$	0	0
$723$	−20.5481	−0.764191
$724$	0	0
$725$	−28.8963	−1.07318
$726$	0	0
$727$	31.2111	1.15756	0.578778	−	0.815485i	$-0.303530\pi$
$727$	31.2111	1.15756	0.578778	+	0.815485i	$0.303530\pi$
$728$	0	0
$729$	21.8568	0.809511
$730$	0	0
$731$	11.4453	0.423319
$732$	0	0
$733$	2.96047	0.109347	0.0546737	−	0.998504i	$-0.482588\pi$
$733$	2.96047	0.109347	0.0546737	+	0.998504i	$0.482588\pi$
$734$	0	0
$735$	−27.1076	−0.999880
$736$	0	0
$737$	0.300627	0.0110737
$738$	0	0
$739$	51.3789	1.89000	0.945001	−	0.327066i	$-0.106060\pi$
$739$	51.3789	1.89000	0.945001	+	0.327066i	$0.106060\pi$
$740$	0	0
$741$	−20.7844	−0.763535
$742$	0	0
$743$	−53.2529	−1.95366	−0.976830	−	0.214015i	$-0.931346\pi$
$743$	−53.2529	−1.95366	−0.976830	+	0.214015i	$0.931346\pi$
$744$	0	0
$745$	7.00393	0.256604
$746$	0	0
$747$	−4.34084	−0.158823
$748$	0	0
$749$	6.18619	0.226038
$750$	0	0
$751$	1.58365	0.0577881	0.0288940	−	0.999582i	$-0.490801\pi$
$751$	1.58365	0.0577881	0.0288940	+	0.999582i	$0.490801\pi$
$752$	0	0
$753$	−1.13335	−0.0413017
$754$	0	0
$755$	14.4174	0.524703
$756$	0	0
$757$	19.5748	0.711460	0.355730	−	0.934589i	$-0.384232\pi$
$757$	19.5748	0.711460	0.355730	+	0.934589i	$0.384232\pi$
$758$	0	0
$759$	45.6311	1.65630
$760$	0	0
$761$	−40.6377	−1.47312	−0.736558	−	0.676374i	$-0.763550\pi$
$761$	−40.6377	−1.47312	−0.736558	+	0.676374i	$0.763550\pi$
$762$	0	0
$763$	29.2580	1.05921
$764$	0	0
$765$	−8.77899	−0.317405
$766$	0	0
$767$	50.1978	1.81254
$768$	0	0
$769$	13.2485	0.477753	0.238877	−	0.971050i	$-0.423221\pi$
$769$	13.2485	0.477753	0.238877	+	0.971050i	$0.423221\pi$
$770$	0	0
$771$	14.1637	0.510094
$772$	0	0
$773$	−12.0416	−0.433105	−0.216553	−	0.976271i	$-0.569481\pi$
$773$	−12.0416	−0.433105	−0.216553	+	0.976271i	$0.569481\pi$
$774$	0	0
$775$	111.286	3.99751
$776$	0	0
$777$	27.0203	0.969349
$778$	0	0
$779$	−3.82624	−0.137089
$780$	0	0
$781$	24.7342	0.885060
$782$	0	0
$783$	−12.9690	−0.463473
$784$	0	0
$785$	−24.2346	−0.864971
$786$	0	0
$787$	−12.1289	−0.432349	−0.216174	−	0.976355i	$-0.569358\pi$
$787$	−12.1289	−0.432349	−0.216174	+	0.976355i	$0.569358\pi$
$788$	0	0
$789$	31.5013	1.12147
$790$	0	0
$791$	20.9913	0.746365
$792$	0	0
$793$	−53.9028	−1.91414
$794$	0	0
$795$	13.5153	0.479337
$796$	0	0
$797$	−14.4578	−0.512122	−0.256061	−	0.966661i	$-0.582425\pi$
$797$	−14.4578	−0.512122	−0.256061	+	0.966661i	$0.582425\pi$
$798$	0	0
$799$	−37.8810	−1.34013
$800$	0	0
$801$	3.60319	0.127313
$802$	0	0
$803$	26.6475	0.940371
$804$	0	0
$805$	−64.8679	−2.28629
$806$	0	0
$807$	−32.8537	−1.15651
$808$	0	0
$809$	−27.0911	−0.952473	−0.476237	−	0.879317i	$-0.657999\pi$
$809$	−27.0911	−0.952473	−0.476237	+	0.879317i	$0.657999\pi$
$810$	0	0
$811$	−49.0420	−1.72210	−0.861049	−	0.508521i	$-0.830192\pi$
$811$	−49.0420	−1.72210	−0.861049	+	0.508521i	$0.830192\pi$
$812$	0	0
$813$	−40.7673	−1.42977
$814$	0	0
$815$	68.4088	2.39626
$816$	0	0
$817$	4.88643	0.170955
$818$	0	0
$819$	4.28998	0.149904
$820$	0	0
$821$	32.0162	1.11737	0.558686	−	0.829379i	$-0.311306\pi$
$821$	32.0162	1.11737	0.558686	+	0.829379i	$0.311306\pi$
$822$	0	0
$823$	33.4062	1.16447	0.582233	−	0.813022i	$-0.302179\pi$
$823$	33.4062	1.16447	0.582233	+	0.813022i	$0.302179\pi$
$824$	0	0
$825$	53.2149	1.85271
$826$	0	0
$827$	13.8118	0.480282	0.240141	−	0.970738i	$-0.422806\pi$
$827$	13.8118	0.480282	0.240141	+	0.970738i	$0.422806\pi$
$828$	0	0
$829$	−21.2551	−0.738221	−0.369111	−	0.929385i	$-0.620338\pi$
$829$	−21.2551	−0.738221	−0.369111	+	0.929385i	$0.620338\pi$
$830$	0	0
$831$	10.3210	0.358031
$832$	0	0
$833$	−18.3960	−0.637383
$834$	0	0
$835$	43.1751	1.49413
$836$	0	0
$837$	49.9463	1.72640
$838$	0	0
$839$	−1.11482	−0.0384878	−0.0192439	−	0.999815i	$-0.506126\pi$
$839$	−1.11482	−0.0384878	−0.0192439	+	0.999815i	$0.506126\pi$
$840$	0	0
$841$	−21.5108	−0.741753
$842$	0	0
$843$	59.6889	2.05579
$844$	0	0
$845$	−58.4737	−2.01156
$846$	0	0
$847$	−6.59902	−0.226745
$848$	0	0
$849$	−54.5900	−1.87352
$850$	0	0
$851$	−72.5419	−2.48670
$852$	0	0
$853$	−13.5270	−0.463157	−0.231578	−	0.972816i	$-0.574389\pi$
$853$	−13.5270	−0.463157	−0.231578	+	0.972816i	$0.574389\pi$
$854$	0	0
$855$	−3.74809	−0.128182
$856$	0	0
$857$	−30.4430	−1.03991	−0.519957	−	0.854192i	$-0.674052\pi$
$857$	−30.4430	−1.03991	−0.519957	+	0.854192i	$0.674052\pi$
$858$	0	0
$859$	−2.53240	−0.0864043	−0.0432021	−	0.999066i	$-0.513756\pi$
$859$	−2.53240	−0.0864043	−0.0432021	+	0.999066i	$0.513756\pi$
$860$	0	0
$861$	6.08092	0.207237
$862$	0	0
$863$	1.37810	0.0469111	0.0234555	−	0.999725i	$-0.492533\pi$
$863$	1.37810	0.0469111	0.0234555	+	0.999725i	$0.492533\pi$
$864$	0	0
$865$	−67.8947	−2.30849
$866$	0	0
$867$	−14.3068	−0.485885
$868$	0	0
$869$	−23.8040	−0.807495
$870$	0	0
$871$	−0.584252	−0.0197966
$872$	0	0
$873$	1.12366	0.0380302
$874$	0	0
$875$	−39.8272	−1.34641
$876$	0	0
$877$	−15.7869	−0.533085	−0.266543	−	0.963823i	$-0.585881\pi$
$877$	−15.7869	−0.533085	−0.266543	+	0.963823i	$0.585881\pi$
$878$	0	0
$879$	58.1256	1.96053
$880$	0	0
$881$	1.68417	0.0567411	0.0283705	−	0.999597i	$-0.490968\pi$
$881$	1.68417	0.0567411	0.0283705	+	0.999597i	$0.490968\pi$
$882$	0	0
$883$	6.73744	0.226733	0.113367	−	0.993553i	$-0.463837\pi$
$883$	6.73744	0.226733	0.113367	+	0.993553i	$0.463837\pi$
$884$	0	0
$885$	69.7005	2.34296
$886$	0	0
$887$	43.1540	1.44897	0.724484	−	0.689292i	$-0.242078\pi$
$887$	43.1540	1.44897	0.724484	+	0.689292i	$0.242078\pi$
$888$	0	0
$889$	21.4503	0.719420
$890$	0	0
$891$	27.5294	0.922270
$892$	0	0
$893$	−16.1729	−0.541204
$894$	0	0
$895$	56.2253	1.87940
$896$	0	0
$897$	−88.6815	−2.96099
$898$	0	0
$899$	−28.8424	−0.961948
$900$	0	0
$901$	9.17183	0.305558
$902$	0	0
$903$	−7.76584	−0.258431
$904$	0	0
$905$	−15.6617	−0.520612
$906$	0	0
$907$	42.1502	1.39958	0.699788	−	0.714351i	$-0.253278\pi$
$907$	42.1502	1.39958	0.699788	+	0.714351i	$0.253278\pi$
$908$	0	0
$909$	−0.270265	−0.00896412
$910$	0	0
$911$	6.88011	0.227948	0.113974	−	0.993484i	$-0.463642\pi$
$911$	6.88011	0.227948	0.113974	+	0.993484i	$0.463642\pi$
$912$	0	0
$913$	26.3119	0.870797
$914$	0	0
$915$	−74.8448	−2.47429
$916$	0	0
$917$	29.0959	0.960832
$918$	0	0
$919$	−35.9448	−1.18571	−0.592854	−	0.805310i	$-0.701999\pi$
$919$	−35.9448	−1.18571	−0.592854	+	0.805310i	$0.701999\pi$
$920$	0	0
$921$	−8.24919	−0.271820
$922$	0	0
$923$	−48.0696	−1.58223
$924$	0	0
$925$	−84.5984	−2.78158
$926$	0	0
$927$	6.73328	0.221150
$928$	0	0
$929$	30.7846	1.01001	0.505005	−	0.863116i	$-0.331491\pi$
$929$	30.7846	1.01001	0.505005	+	0.863116i	$0.331491\pi$
$930$	0	0
$931$	−7.85395	−0.257403
$932$	0	0
$933$	−5.09985	−0.166962
$934$	0	0
$935$	53.2136	1.74027
$936$	0	0
$937$	40.9876	1.33901	0.669503	−	0.742809i	$-0.266507\pi$
$937$	40.9876	1.33901	0.669503	+	0.742809i	$0.266507\pi$
$938$	0	0
$939$	39.2511	1.28091
$940$	0	0
$941$	−13.8412	−0.451209	−0.225604	−	0.974219i	$-0.572436\pi$
$941$	−13.8412	−0.451209	−0.225604	+	0.974219i	$0.572436\pi$
$942$	0	0
$943$	−16.3255	−0.531633
$944$	0	0
$945$	−33.9520	−1.10446
$946$	0	0
$947$	29.6528	0.963587	0.481794	−	0.876285i	$-0.339985\pi$
$947$	29.6528	0.963587	0.481794	+	0.876285i	$0.339985\pi$
$948$	0	0
$949$	−51.7880	−1.68111
$950$	0	0
$951$	−7.34317	−0.238118
$952$	0	0
$953$	−1.08472	−0.0351374	−0.0175687	−	0.999846i	$-0.505593\pi$
$953$	−1.08472	−0.0351374	−0.0175687	+	0.999846i	$0.505593\pi$
$954$	0	0
$955$	−77.9292	−2.52173
$956$	0	0
$957$	−13.7919	−0.445829
$958$	0	0
$959$	−31.1539	−1.00601
$960$	0	0
$961$	80.0784	2.58317
$962$	0	0
$963$	1.52510	0.0491457
$964$	0	0
$965$	−55.8183	−1.79685
$966$	0	0
$967$	−3.26690	−0.105056	−0.0525282	−	0.998619i	$-0.516728\pi$
$967$	−3.26690	−0.105056	−0.0525282	+	0.998619i	$0.516728\pi$
$968$	0	0
$969$	−19.5848	−0.629156
$970$	0	0
$971$	9.76414	0.313346	0.156673	−	0.987651i	$-0.449923\pi$
$971$	9.76414	0.313346	0.156673	+	0.987651i	$0.449923\pi$
$972$	0	0
$973$	12.8885	0.413186
$974$	0	0
$975$	−103.420	−3.31210
$976$	0	0
$977$	51.3149	1.64171	0.820855	−	0.571136i	$-0.193497\pi$
$977$	51.3149	1.64171	0.820855	+	0.571136i	$0.193497\pi$
$978$	0	0
$979$	−21.8407	−0.698030
$980$	0	0
$981$	7.21306	0.230295
$982$	0	0
$983$	−58.2718	−1.85858	−0.929290	−	0.369351i	$-0.879580\pi$
$983$	−58.2718	−1.85858	−0.929290	+	0.369351i	$0.879580\pi$
$984$	0	0
$985$	32.7463	1.04338
$986$	0	0
$987$	25.7030	0.818135
$988$	0	0
$989$	20.8491	0.662962
$990$	0	0
$991$	58.2487	1.85033	0.925166	−	0.379563i	$-0.123926\pi$
$991$	58.2487	1.85033	0.925166	+	0.379563i	$0.123926\pi$
$992$	0	0
$993$	36.4367	1.15628
$994$	0	0
$995$	−33.0568	−1.04797
$996$	0	0
$997$	38.9963	1.23502	0.617512	−	0.786562i	$-0.288141\pi$
$997$	38.9963	1.23502	0.617512	+	0.786562i	$0.288141\pi$
$998$	0	0
$999$	−37.9686	−1.20127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.4		21
4.3	odd	2		2012.2.a.a.1.18	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.18	✓	21	4.3	odd	2
8048.2.a.t.1.4		21	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)

\(f(q)\)	\(=\)	\(q-1.85682 q^{3} -3.94450 q^{5} +1.81629 q^{7} +0.447775 q^{9} +O(q^{10})\)	\(q-1.85682 q^{3} -3.94450 q^{5} +1.81629 q^{7} +0.447775 q^{9} -2.71418 q^{11} +5.27486 q^{13} +7.32422 q^{15} +4.97041 q^{17} +2.12206 q^{19} -3.37252 q^{21} +9.05426 q^{23} +10.5591 q^{25} +4.73902 q^{27} -2.73663 q^{29} +10.5394 q^{31} +5.03973 q^{33} -7.16435 q^{35} -8.01191 q^{37} -9.79445 q^{39} -1.80308 q^{41} +2.30268 q^{43} -1.76625 q^{45} -7.62130 q^{47} -3.70109 q^{49} -9.22916 q^{51} +1.84529 q^{53} +10.7061 q^{55} -3.94028 q^{57} +9.51643 q^{59} -10.2188 q^{61} +0.813289 q^{63} -20.8067 q^{65} -0.110762 q^{67} -16.8121 q^{69} -9.11297 q^{71} -9.81791 q^{73} -19.6063 q^{75} -4.92973 q^{77} +8.77024 q^{79} -10.1428 q^{81} -9.69425 q^{83} -19.6058 q^{85} +5.08143 q^{87} +8.04688 q^{89} +9.58066 q^{91} -19.5697 q^{93} -8.37047 q^{95} +2.50943 q^{97} -1.21534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

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