LMFDB - Embedded newform 8024.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11439 q^{3} +4.12614 q^{5} +1.17263 q^{7} +6.69943 q^{9} +O(q^{10})$	$q-3.11439 q^{3} +4.12614 q^{5} +1.17263 q^{7} +6.69943 q^{9} -4.55495 q^{11} -2.99424 q^{13} -12.8504 q^{15} -1.00000 q^{17} -3.45676 q^{19} -3.65201 q^{21} -0.461059 q^{23} +12.0251 q^{25} -11.5215 q^{27} +5.51517 q^{29} +0.588605 q^{31} +14.1859 q^{33} +4.83842 q^{35} +3.02077 q^{37} +9.32522 q^{39} -4.05248 q^{41} +8.04701 q^{43} +27.6428 q^{45} +1.26780 q^{47} -5.62495 q^{49} +3.11439 q^{51} -5.96344 q^{53} -18.7944 q^{55} +10.7657 q^{57} -1.00000 q^{59} +6.85935 q^{61} +7.85592 q^{63} -12.3547 q^{65} -9.06822 q^{67} +1.43592 q^{69} -0.0811455 q^{71} -13.4210 q^{73} -37.4508 q^{75} -5.34125 q^{77} +0.604690 q^{79} +15.7841 q^{81} -1.82936 q^{83} -4.12614 q^{85} -17.1764 q^{87} +13.3031 q^{89} -3.51112 q^{91} -1.83315 q^{93} -14.2631 q^{95} -2.19162 q^{97} -30.5156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	−3.11439	−1.79809	−0.899047	−	0.437851i	$-0.855740\pi$
$3$	−3.11439	−1.79809	−0.899047	+	0.437851i	$0.855740\pi$
$4$	0	0
$5$	4.12614	1.84527	0.922634	−	0.385677i	$-0.126032\pi$
$5$	4.12614	1.84527	0.922634	+	0.385677i	$0.126032\pi$
$6$	0	0
$7$	1.17263	0.443211	0.221605	−	0.975136i	$-0.428870\pi$
$7$	1.17263	0.443211	0.221605	+	0.975136i	$0.428870\pi$
$8$	0	0
$9$	6.69943	2.23314
$10$	0	0
$11$	−4.55495	−1.37337	−0.686685	−	0.726955i	$-0.740935\pi$
$11$	−4.55495	−1.37337	−0.686685	+	0.726955i	$0.740935\pi$
$12$	0	0
$13$	−2.99424	−0.830452	−0.415226	−	0.909718i	$-0.636297\pi$
$13$	−2.99424	−0.830452	−0.415226	+	0.909718i	$0.636297\pi$
$14$	0	0
$15$	−12.8504	−3.31797
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.45676	−0.793035	−0.396518	−	0.918027i	$-0.629781\pi$
$19$	−3.45676	−0.793035	−0.396518	+	0.918027i	$0.629781\pi$
$20$	0	0
$21$	−3.65201	−0.796935
$22$	0	0
$23$	−0.461059	−0.0961375	−0.0480688	−	0.998844i	$-0.515307\pi$
$23$	−0.461059	−0.0961375	−0.0480688	+	0.998844i	$0.515307\pi$
$24$	0	0
$25$	12.0251	2.40501
$26$	0	0
$27$	−11.5215	−2.21731
$28$	0	0
$29$	5.51517	1.02414	0.512071	−	0.858943i	$-0.328878\pi$
$29$	5.51517	1.02414	0.512071	+	0.858943i	$0.328878\pi$
$30$	0	0
$31$	0.588605	0.105717	0.0528583	−	0.998602i	$-0.483167\pi$
$31$	0.588605	0.105717	0.0528583	+	0.998602i	$0.483167\pi$
$32$	0	0
$33$	14.1859	2.46945
$34$	0	0
$35$	4.83842	0.817842
$36$	0	0
$37$	3.02077	0.496611	0.248306	−	0.968682i	$-0.420126\pi$
$37$	3.02077	0.496611	0.248306	+	0.968682i	$0.420126\pi$
$38$	0	0
$39$	9.32522	1.49323
$40$	0	0
$41$	−4.05248	−0.632891	−0.316446	−	0.948611i	$-0.602490\pi$
$41$	−4.05248	−0.632891	−0.316446	+	0.948611i	$0.602490\pi$
$42$	0	0
$43$	8.04701	1.22716	0.613579	−	0.789634i	$-0.289729\pi$
$43$	8.04701	1.22716	0.613579	+	0.789634i	$0.289729\pi$
$44$	0	0
$45$	27.6428	4.12075
$46$	0	0
$47$	1.26780	0.184928	0.0924641	−	0.995716i	$-0.470526\pi$
$47$	1.26780	0.184928	0.0924641	+	0.995716i	$0.470526\pi$
$48$	0	0
$49$	−5.62495	−0.803564
$50$	0	0
$51$	3.11439	0.436102
$52$	0	0
$53$	−5.96344	−0.819142	−0.409571	−	0.912278i	$-0.634322\pi$
$53$	−5.96344	−0.819142	−0.409571	+	0.912278i	$0.634322\pi$
$54$	0	0
$55$	−18.7944	−2.53423
$56$	0	0
$57$	10.7657	1.42595
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	6.85935	0.878250	0.439125	−	0.898426i	$-0.355289\pi$
$61$	6.85935	0.878250	0.439125	+	0.898426i	$0.355289\pi$
$62$	0	0
$63$	7.85592	0.989753
$64$	0	0
$65$	−12.3547	−1.53241
$66$	0	0
$67$	−9.06822	−1.10786	−0.553930	−	0.832563i	$-0.686872\pi$
$67$	−9.06822	−1.10786	−0.553930	+	0.832563i	$0.686872\pi$
$68$	0	0
$69$	1.43592	0.172864
$70$	0	0
$71$	−0.0811455	−0.00963020	−0.00481510	−	0.999988i	$-0.501533\pi$
$71$	−0.0811455	−0.00963020	−0.00481510	+	0.999988i	$0.501533\pi$
$72$	0	0
$73$	−13.4210	−1.57081	−0.785403	−	0.618984i	$-0.787544\pi$
$73$	−13.4210	−1.57081	−0.785403	+	0.618984i	$0.787544\pi$
$74$	0	0
$75$	−37.4508	−4.32444
$76$	0	0
$77$	−5.34125	−0.608692
$78$	0	0
$79$	0.604690	0.0680329	0.0340165	−	0.999421i	$-0.489170\pi$
$79$	0.604690	0.0680329	0.0340165	+	0.999421i	$0.489170\pi$
$80$	0	0
$81$	15.7841	1.75379
$82$	0	0
$83$	−1.82936	−0.200798	−0.100399	−	0.994947i	$-0.532012\pi$
$83$	−1.82936	−0.200798	−0.100399	+	0.994947i	$0.532012\pi$
$84$	0	0
$85$	−4.12614	−0.447543
$86$	0	0
$87$	−17.1764	−1.84150
$88$	0	0
$89$	13.3031	1.41012	0.705061	−	0.709146i	$-0.250919\pi$
$89$	13.3031	1.41012	0.705061	+	0.709146i	$0.250919\pi$
$90$	0	0
$91$	−3.51112	−0.368065
$92$	0	0
$93$	−1.83315	−0.190088
$94$	0	0
$95$	−14.2631	−1.46336
$96$	0	0
$97$	−2.19162	−0.222525	−0.111262	−	0.993791i	$-0.535489\pi$
$97$	−2.19162	−0.222525	−0.111262	+	0.993791i	$0.535489\pi$
$98$	0	0
$99$	−30.5156	−3.06693
$100$	0	0
$101$	−15.5982	−1.55208	−0.776039	−	0.630685i	$-0.782774\pi$
$101$	−15.5982	−1.55208	−0.776039	+	0.630685i	$0.782774\pi$
$102$	0	0
$103$	−2.38971	−0.235466	−0.117733	−	0.993045i	$-0.537563\pi$
$103$	−2.38971	−0.235466	−0.117733	+	0.993045i	$0.537563\pi$
$104$	0	0
$105$	−15.0687	−1.47056
$106$	0	0
$107$	−2.18994	−0.211710	−0.105855	−	0.994382i	$-0.533758\pi$
$107$	−2.18994	−0.211710	−0.105855	+	0.994382i	$0.533758\pi$
$108$	0	0
$109$	−18.0955	−1.73324	−0.866619	−	0.498971i	$-0.833711\pi$
$109$	−18.0955	−1.73324	−0.866619	+	0.498971i	$0.833711\pi$
$110$	0	0
$111$	−9.40786	−0.892954
$112$	0	0
$113$	10.9603	1.03106	0.515528	−	0.856873i	$-0.327596\pi$
$113$	10.9603	1.03106	0.515528	+	0.856873i	$0.327596\pi$
$114$	0	0
$115$	−1.90240	−0.177399
$116$	0	0
$117$	−20.0597	−1.85452
$118$	0	0
$119$	−1.17263	−0.107494
$120$	0	0
$121$	9.74758	0.886143
$122$	0	0
$123$	12.6210	1.13800
$124$	0	0
$125$	28.9864	2.59263
$126$	0	0
$127$	19.5601	1.73567	0.867837	−	0.496848i	$-0.165510\pi$
$127$	19.5601	1.73567	0.867837	+	0.496848i	$0.165510\pi$
$128$	0	0
$129$	−25.0615	−2.20654
$130$	0	0
$131$	0.264375	0.0230985	0.0115493	−	0.999933i	$-0.496324\pi$
$131$	0.264375	0.0230985	0.0115493	+	0.999933i	$0.496324\pi$
$132$	0	0
$133$	−4.05348	−0.351482
$134$	0	0
$135$	−47.5393	−4.09153
$136$	0	0
$137$	10.7900	0.921849	0.460925	−	0.887439i	$-0.347518\pi$
$137$	10.7900	0.921849	0.460925	+	0.887439i	$0.347518\pi$
$138$	0	0
$139$	−10.8712	−0.922082	−0.461041	−	0.887379i	$-0.652524\pi$
$139$	−10.8712	−0.922082	−0.461041	+	0.887379i	$0.652524\pi$
$140$	0	0
$141$	−3.94844	−0.332518
$142$	0	0
$143$	13.6386	1.14052
$144$	0	0
$145$	22.7564	1.88982
$146$	0	0
$147$	17.5183	1.44488
$148$	0	0
$149$	−17.6464	−1.44565	−0.722823	−	0.691033i	$-0.757156\pi$
$149$	−17.6464	−1.44565	−0.722823	+	0.691033i	$0.757156\pi$
$150$	0	0
$151$	19.6472	1.59887	0.799435	−	0.600752i	$-0.205132\pi$
$151$	19.6472	1.59887	0.799435	+	0.600752i	$0.205132\pi$
$152$	0	0
$153$	−6.69943	−0.541617
$154$	0	0
$155$	2.42867	0.195075
$156$	0	0
$157$	4.34965	0.347140	0.173570	−	0.984822i	$-0.444470\pi$
$157$	4.34965	0.347140	0.173570	+	0.984822i	$0.444470\pi$
$158$	0	0
$159$	18.5725	1.47289
$160$	0	0
$161$	−0.540650	−0.0426092
$162$	0	0
$163$	0.633892	0.0496503	0.0248251	−	0.999692i	$-0.492097\pi$
$163$	0.633892	0.0496503	0.0248251	+	0.999692i	$0.492097\pi$
$164$	0	0
$165$	58.5331	4.55679
$166$	0	0
$167$	−22.0070	−1.70295	−0.851475	−	0.524394i	$-0.824292\pi$
$167$	−22.0070	−1.70295	−0.851475	+	0.524394i	$0.824292\pi$
$168$	0	0
$169$	−4.03455	−0.310350
$170$	0	0
$171$	−23.1583	−1.77096
$172$	0	0
$173$	−6.34228	−0.482195	−0.241097	−	0.970501i	$-0.577507\pi$
$173$	−6.34228	−0.482195	−0.241097	+	0.970501i	$0.577507\pi$
$174$	0	0
$175$	14.1009	1.06593
$176$	0	0
$177$	3.11439	0.234092
$178$	0	0
$179$	−13.7344	−1.02656	−0.513278	−	0.858222i	$-0.671569\pi$
$179$	−13.7344	−1.02656	−0.513278	+	0.858222i	$0.671569\pi$
$180$	0	0
$181$	−16.7361	−1.24398	−0.621991	−	0.783025i	$-0.713676\pi$
$181$	−16.7361	−1.24398	−0.621991	+	0.783025i	$0.713676\pi$
$182$	0	0
$183$	−21.3627	−1.57918
$184$	0	0
$185$	12.4641	0.916381
$186$	0	0
$187$	4.55495	0.333091
$188$	0	0
$189$	−13.5104	−0.982736
$190$	0	0
$191$	16.3773	1.18502	0.592511	−	0.805562i	$-0.298137\pi$
$191$	16.3773	1.18502	0.592511	+	0.805562i	$0.298137\pi$
$192$	0	0
$193$	−23.2060	−1.67041	−0.835203	−	0.549941i	$-0.814650\pi$
$193$	−23.2060	−1.67041	−0.835203	+	0.549941i	$0.814650\pi$
$194$	0	0
$195$	38.4772	2.75541
$196$	0	0
$197$	16.7561	1.19382	0.596910	−	0.802308i	$-0.296395\pi$
$197$	16.7561	1.19382	0.596910	+	0.802308i	$0.296395\pi$
$198$	0	0
$199$	−11.4111	−0.808912	−0.404456	−	0.914558i	$-0.632539\pi$
$199$	−11.4111	−0.808912	−0.404456	+	0.914558i	$0.632539\pi$
$200$	0	0
$201$	28.2420	1.99204
$202$	0	0
$203$	6.46723	0.453911
$204$	0	0
$205$	−16.7211	−1.16785
$206$	0	0
$207$	−3.08884	−0.214689
$208$	0	0
$209$	15.7454	1.08913
$210$	0	0
$211$	15.0814	1.03825	0.519124	−	0.854699i	$-0.326258\pi$
$211$	15.0814	1.03825	0.519124	+	0.854699i	$0.326258\pi$
$212$	0	0
$213$	0.252719	0.0173160
$214$	0	0
$215$	33.2031	2.26443
$216$	0	0
$217$	0.690213	0.0468547
$218$	0	0
$219$	41.7982	2.82446
$220$	0	0
$221$	2.99424	0.201414
$222$	0	0
$223$	−3.30334	−0.221208	−0.110604	−	0.993865i	$-0.535279\pi$
$223$	−3.30334	−0.221208	−0.110604	+	0.993865i	$0.535279\pi$
$224$	0	0
$225$	80.5611	5.37074
$226$	0	0
$227$	15.7175	1.04320	0.521602	−	0.853189i	$-0.325334\pi$
$227$	15.7175	1.04320	0.521602	+	0.853189i	$0.325334\pi$
$228$	0	0
$229$	−5.92162	−0.391311	−0.195656	−	0.980673i	$-0.562684\pi$
$229$	−5.92162	−0.391311	−0.195656	+	0.980673i	$0.562684\pi$
$230$	0	0
$231$	16.6347	1.09449
$232$	0	0
$233$	−4.46527	−0.292530	−0.146265	−	0.989245i	$-0.546725\pi$
$233$	−4.46527	−0.292530	−0.146265	+	0.989245i	$0.546725\pi$
$234$	0	0
$235$	5.23114	0.341242
$236$	0	0
$237$	−1.88324	−0.122330
$238$	0	0
$239$	−3.55971	−0.230259	−0.115129	−	0.993351i	$-0.536728\pi$
$239$	−3.55971	−0.230259	−0.115129	+	0.993351i	$0.536728\pi$
$240$	0	0
$241$	22.7638	1.46635	0.733174	−	0.680041i	$-0.238038\pi$
$241$	22.7638	1.46635	0.733174	+	0.680041i	$0.238038\pi$
$242$	0	0
$243$	−14.5935	−0.936170
$244$	0	0
$245$	−23.2094	−1.48279
$246$	0	0
$247$	10.3504	0.658578
$248$	0	0
$249$	5.69733	0.361054
$250$	0	0
$251$	−5.85116	−0.369322	−0.184661	−	0.982802i	$-0.559119\pi$
$251$	−5.85116	−0.369322	−0.184661	+	0.982802i	$0.559119\pi$
$252$	0	0
$253$	2.10010	0.132032
$254$	0	0
$255$	12.8504	0.804725
$256$	0	0
$257$	−6.93306	−0.432472	−0.216236	−	0.976341i	$-0.569378\pi$
$257$	−6.93306	−0.432472	−0.216236	+	0.976341i	$0.569378\pi$
$258$	0	0
$259$	3.54223	0.220103
$260$	0	0
$261$	36.9485	2.28706
$262$	0	0
$263$	−29.8344	−1.83967	−0.919834	−	0.392309i	$-0.871676\pi$
$263$	−29.8344	−1.83967	−0.919834	+	0.392309i	$0.871676\pi$
$264$	0	0
$265$	−24.6060	−1.51154
$266$	0	0
$267$	−41.4310	−2.53553
$268$	0	0
$269$	−24.2491	−1.47850	−0.739248	−	0.673433i	$-0.764819\pi$
$269$	−24.2491	−1.47850	−0.739248	+	0.673433i	$0.764819\pi$
$270$	0	0
$271$	−24.8194	−1.50767	−0.753835	−	0.657064i	$-0.771798\pi$
$271$	−24.8194	−1.50767	−0.753835	+	0.657064i	$0.771798\pi$
$272$	0	0
$273$	10.9350	0.661816
$274$	0	0
$275$	−54.7736	−3.30297
$276$	0	0
$277$	23.0008	1.38199	0.690993	−	0.722862i	$-0.257174\pi$
$277$	23.0008	1.38199	0.690993	+	0.722862i	$0.257174\pi$
$278$	0	0
$279$	3.94332	0.236080
$280$	0	0
$281$	−31.3246	−1.86867	−0.934336	−	0.356394i	$-0.884006\pi$
$281$	−31.3246	−1.86867	−0.934336	+	0.356394i	$0.884006\pi$
$282$	0	0
$283$	−16.5797	−0.985560	−0.492780	−	0.870154i	$-0.664019\pi$
$283$	−16.5797	−0.985560	−0.492780	+	0.870154i	$0.664019\pi$
$284$	0	0
$285$	44.4209	2.63126
$286$	0	0
$287$	−4.75204	−0.280504
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.82555	0.400121
$292$	0	0
$293$	−15.5910	−0.910836	−0.455418	−	0.890278i	$-0.650510\pi$
$293$	−15.5910	−0.910836	−0.455418	+	0.890278i	$0.650510\pi$
$294$	0	0
$295$	−4.12614	−0.240233
$296$	0	0
$297$	52.4798	3.04519
$298$	0	0
$299$	1.38052	0.0798376
$300$	0	0
$301$	9.43612	0.543889
$302$	0	0
$303$	48.5789	2.79078
$304$	0	0
$305$	28.3027	1.62061
$306$	0	0
$307$	−29.5457	−1.68626	−0.843132	−	0.537707i	$-0.819291\pi$
$307$	−29.5457	−1.68626	−0.843132	+	0.537707i	$0.819291\pi$
$308$	0	0
$309$	7.44251	0.423389
$310$	0	0
$311$	12.4271	0.704678	0.352339	−	0.935873i	$-0.385386\pi$
$311$	12.4271	0.704678	0.352339	+	0.935873i	$0.385386\pi$
$312$	0	0
$313$	−2.14612	−0.121306	−0.0606530	−	0.998159i	$-0.519318\pi$
$313$	−2.14612	−0.121306	−0.0606530	+	0.998159i	$0.519318\pi$
$314$	0	0
$315$	32.4147	1.82636
$316$	0	0
$317$	0.262583	0.0147481	0.00737407	−	0.999973i	$-0.497653\pi$
$317$	0.262583	0.0147481	0.00737407	+	0.999973i	$0.497653\pi$
$318$	0	0
$319$	−25.1214	−1.40653
$320$	0	0
$321$	6.82034	0.380674
$322$	0	0
$323$	3.45676	0.192339
$324$	0	0
$325$	−36.0059	−1.99725
$326$	0	0
$327$	56.3566	3.11652
$328$	0	0
$329$	1.48666	0.0819621
$330$	0	0
$331$	−11.0540	−0.607584	−0.303792	−	0.952738i	$-0.598253\pi$
$331$	−11.0540	−0.607584	−0.303792	+	0.952738i	$0.598253\pi$
$332$	0	0
$333$	20.2374	1.10900
$334$	0	0
$335$	−37.4168	−2.04430
$336$	0	0
$337$	−2.23252	−0.121613	−0.0608065	−	0.998150i	$-0.519367\pi$
$337$	−2.23252	−0.121613	−0.0608065	+	0.998150i	$0.519367\pi$
$338$	0	0
$339$	−34.1346	−1.85394
$340$	0	0
$341$	−2.68107	−0.145188
$342$	0	0
$343$	−14.8043	−0.799359
$344$	0	0
$345$	5.92481	0.318981
$346$	0	0
$347$	−3.64605	−0.195730	−0.0978650	−	0.995200i	$-0.531201\pi$
$347$	−3.64605	−0.195730	−0.0978650	+	0.995200i	$0.531201\pi$
$348$	0	0
$349$	0.937629	0.0501901	0.0250951	−	0.999685i	$-0.492011\pi$
$349$	0.937629	0.0501901	0.0250951	+	0.999685i	$0.492011\pi$
$350$	0	0
$351$	34.4981	1.84137
$352$	0	0
$353$	24.0257	1.27876	0.639379	−	0.768892i	$-0.279192\pi$
$353$	24.0257	1.27876	0.639379	+	0.768892i	$0.279192\pi$
$354$	0	0
$355$	−0.334818	−0.0177703
$356$	0	0
$357$	3.65201	0.193285
$358$	0	0
$359$	27.6941	1.46164	0.730820	−	0.682570i	$-0.239138\pi$
$359$	27.6941	1.46164	0.730820	+	0.682570i	$0.239138\pi$
$360$	0	0
$361$	−7.05081	−0.371095
$362$	0	0
$363$	−30.3578	−1.59337
$364$	0	0
$365$	−55.3769	−2.89856
$366$	0	0
$367$	9.07744	0.473838	0.236919	−	0.971529i	$-0.423862\pi$
$367$	9.07744	0.473838	0.236919	+	0.971529i	$0.423862\pi$
$368$	0	0
$369$	−27.1493	−1.41334
$370$	0	0
$371$	−6.99288	−0.363052
$372$	0	0
$373$	−20.6744	−1.07048	−0.535240	−	0.844700i	$-0.679779\pi$
$373$	−20.6744	−1.07048	−0.535240	+	0.844700i	$0.679779\pi$
$374$	0	0
$375$	−90.2751	−4.66179
$376$	0	0
$377$	−16.5137	−0.850501
$378$	0	0
$379$	24.1185	1.23889	0.619443	−	0.785042i	$-0.287359\pi$
$379$	24.1185	1.23889	0.619443	+	0.785042i	$0.287359\pi$
$380$	0	0
$381$	−60.9177	−3.12091
$382$	0	0
$383$	27.8429	1.42270	0.711352	−	0.702836i	$-0.248083\pi$
$383$	27.8429	1.42270	0.711352	+	0.702836i	$0.248083\pi$
$384$	0	0
$385$	−22.0388	−1.12320
$386$	0	0
$387$	53.9104	2.74042
$388$	0	0
$389$	14.0303	0.711364	0.355682	−	0.934607i	$-0.384249\pi$
$389$	14.0303	0.711364	0.355682	+	0.934607i	$0.384249\pi$
$390$	0	0
$391$	0.461059	0.0233168
$392$	0	0
$393$	−0.823366	−0.0415333
$394$	0	0
$395$	2.49504	0.125539
$396$	0	0
$397$	21.3589	1.07197	0.535986	−	0.844227i	$-0.319940\pi$
$397$	21.3589	1.07197	0.535986	+	0.844227i	$0.319940\pi$
$398$	0	0
$399$	12.6241	0.631997
$400$	0	0
$401$	25.8587	1.29132	0.645661	−	0.763624i	$-0.276582\pi$
$401$	25.8587	1.29132	0.645661	+	0.763624i	$0.276582\pi$
$402$	0	0
$403$	−1.76242	−0.0877925
$404$	0	0
$405$	65.1275	3.23621
$406$	0	0
$407$	−13.7595	−0.682031
$408$	0	0
$409$	−2.18497	−0.108040	−0.0540199	−	0.998540i	$-0.517203\pi$
$409$	−2.18497	−0.108040	−0.0540199	+	0.998540i	$0.517203\pi$
$410$	0	0
$411$	−33.6042	−1.65757
$412$	0	0
$413$	−1.17263	−0.0577011
$414$	0	0
$415$	−7.54819	−0.370526
$416$	0	0
$417$	33.8571	1.65799
$418$	0	0
$419$	−19.8867	−0.971528	−0.485764	−	0.874090i	$-0.661459\pi$
$419$	−19.8867	−0.971528	−0.485764	+	0.874090i	$0.661459\pi$
$420$	0	0
$421$	−5.67589	−0.276626	−0.138313	−	0.990389i	$-0.544168\pi$
$421$	−5.67589	−0.276626	−0.138313	+	0.990389i	$0.544168\pi$
$422$	0	0
$423$	8.49357	0.412971
$424$	0	0
$425$	−12.0251	−0.583301
$426$	0	0
$427$	8.04345	0.389250
$428$	0	0
$429$	−42.4759	−2.05076
$430$	0	0
$431$	−2.12421	−0.102320	−0.0511599	−	0.998690i	$-0.516292\pi$
$431$	−2.12421	−0.102320	−0.0511599	+	0.998690i	$0.516292\pi$
$432$	0	0
$433$	24.0825	1.15733	0.578667	−	0.815564i	$-0.303573\pi$
$433$	24.0825	1.15733	0.578667	+	0.815564i	$0.303573\pi$
$434$	0	0
$435$	−70.8724	−3.39807
$436$	0	0
$437$	1.59377	0.0762404
$438$	0	0
$439$	−32.7727	−1.56416	−0.782078	−	0.623180i	$-0.785840\pi$
$439$	−32.7727	−1.56416	−0.782078	+	0.623180i	$0.785840\pi$
$440$	0	0
$441$	−37.6840	−1.79448
$442$	0	0
$443$	−19.3652	−0.920067	−0.460033	−	0.887902i	$-0.652163\pi$
$443$	−19.3652	−0.920067	−0.460033	+	0.887902i	$0.652163\pi$
$444$	0	0
$445$	54.8904	2.60205
$446$	0	0
$447$	54.9577	2.59941
$448$	0	0
$449$	−24.7554	−1.16828	−0.584140	−	0.811653i	$-0.698568\pi$
$449$	−24.7554	−1.16828	−0.584140	+	0.811653i	$0.698568\pi$
$450$	0	0
$451$	18.4589	0.869194
$452$	0	0
$453$	−61.1892	−2.87492
$454$	0	0
$455$	−14.4874	−0.679179
$456$	0	0
$457$	13.6010	0.636230	0.318115	−	0.948052i	$-0.396950\pi$
$457$	13.6010	0.636230	0.318115	+	0.948052i	$0.396950\pi$
$458$	0	0
$459$	11.5215	0.537777
$460$	0	0
$461$	−18.6549	−0.868844	−0.434422	−	0.900709i	$-0.643047\pi$
$461$	−18.6549	−0.868844	−0.434422	+	0.900709i	$0.643047\pi$
$462$	0	0
$463$	22.5423	1.04763	0.523814	−	0.851833i	$-0.324509\pi$
$463$	22.5423	1.04763	0.523814	+	0.851833i	$0.324509\pi$
$464$	0	0
$465$	−7.56383	−0.350764
$466$	0	0
$467$	24.5657	1.13676	0.568382	−	0.822765i	$-0.307570\pi$
$467$	24.5657	1.13676	0.568382	+	0.822765i	$0.307570\pi$
$468$	0	0
$469$	−10.6336	−0.491015
$470$	0	0
$471$	−13.5465	−0.624190
$472$	0	0
$473$	−36.6537	−1.68534
$474$	0	0
$475$	−41.5678	−1.90726
$476$	0	0
$477$	−39.9517	−1.82926
$478$	0	0
$479$	−1.75100	−0.0800050	−0.0400025	−	0.999200i	$-0.512737\pi$
$479$	−1.75100	−0.0800050	−0.0400025	+	0.999200i	$0.512737\pi$
$480$	0	0
$481$	−9.04490	−0.412412
$482$	0	0
$483$	1.68379	0.0766153
$484$	0	0
$485$	−9.04292	−0.410618
$486$	0	0
$487$	13.1349	0.595200	0.297600	−	0.954691i	$-0.403814\pi$
$487$	13.1349	0.595200	0.297600	+	0.954691i	$0.403814\pi$
$488$	0	0
$489$	−1.97419	−0.0892759
$490$	0	0
$491$	−3.28449	−0.148227	−0.0741135	−	0.997250i	$-0.523613\pi$
$491$	−3.28449	−0.148227	−0.0741135	+	0.997250i	$0.523613\pi$
$492$	0	0
$493$	−5.51517	−0.248391
$494$	0	0
$495$	−125.912	−5.65931
$496$	0	0
$497$	−0.0951532	−0.00426821
$498$	0	0
$499$	−43.7636	−1.95913	−0.979564	−	0.201134i	$-0.935537\pi$
$499$	−43.7636	−1.95913	−0.979564	+	0.201134i	$0.935537\pi$
$500$	0	0
$501$	68.5383	3.06207
$502$	0	0
$503$	9.74232	0.434389	0.217194	−	0.976128i	$-0.430309\pi$
$503$	9.74232	0.434389	0.217194	+	0.976128i	$0.430309\pi$
$504$	0	0
$505$	−64.3604	−2.86400
$506$	0	0
$507$	12.5652	0.558038
$508$	0	0
$509$	−29.9256	−1.32643	−0.663214	−	0.748430i	$-0.730808\pi$
$509$	−29.9256	−1.32643	−0.663214	+	0.748430i	$0.730808\pi$
$510$	0	0
$511$	−15.7378	−0.696198
$512$	0	0
$513$	39.8270	1.75841
$514$	0	0
$515$	−9.86031	−0.434497
$516$	0	0
$517$	−5.77478	−0.253975
$518$	0	0
$519$	19.7523	0.867032
$520$	0	0
$521$	−28.4691	−1.24725	−0.623627	−	0.781722i	$-0.714341\pi$
$521$	−28.4691	−1.24725	−0.623627	+	0.781722i	$0.714341\pi$
$522$	0	0
$523$	−36.9677	−1.61649	−0.808243	−	0.588849i	$-0.799581\pi$
$523$	−36.9677	−1.61649	−0.808243	+	0.588849i	$0.799581\pi$
$524$	0	0
$525$	−43.9157	−1.91664
$526$	0	0
$527$	−0.588605	−0.0256400
$528$	0	0
$529$	−22.7874	−0.990758
$530$	0	0
$531$	−6.69943	−0.290731
$532$	0	0
$533$	12.1341	0.525586
$534$	0	0
$535$	−9.03602	−0.390661
$536$	0	0
$537$	42.7742	1.84584
$538$	0	0
$539$	25.6214	1.10359
$540$	0	0
$541$	26.4418	1.13682	0.568412	−	0.822744i	$-0.307558\pi$
$541$	26.4418	1.13682	0.568412	+	0.822744i	$0.307558\pi$
$542$	0	0
$543$	52.1226	2.23680
$544$	0	0
$545$	−74.6648	−3.19829
$546$	0	0
$547$	−26.5755	−1.13629	−0.568143	−	0.822930i	$-0.692338\pi$
$547$	−26.5755	−1.13629	−0.568143	+	0.822930i	$0.692338\pi$
$548$	0	0
$549$	45.9538	1.96126
$550$	0	0
$551$	−19.0646	−0.812181
$552$	0	0
$553$	0.709074	0.0301529
$554$	0	0
$555$	−38.8182	−1.64774
$556$	0	0
$557$	−17.3239	−0.734039	−0.367019	−	0.930213i	$-0.619622\pi$
$557$	−17.3239	−0.734039	−0.367019	+	0.930213i	$0.619622\pi$
$558$	0	0
$559$	−24.0946	−1.01909
$560$	0	0
$561$	−14.1859	−0.598929
$562$	0	0
$563$	−13.0724	−0.550938	−0.275469	−	0.961310i	$-0.588833\pi$
$563$	−13.0724	−0.550938	−0.275469	+	0.961310i	$0.588833\pi$
$564$	0	0
$565$	45.2237	1.90257
$566$	0	0
$567$	18.5088	0.777299
$568$	0	0
$569$	−9.49116	−0.397890	−0.198945	−	0.980011i	$-0.563752\pi$
$569$	−9.49116	−0.397890	−0.198945	+	0.980011i	$0.563752\pi$
$570$	0	0
$571$	21.5508	0.901872	0.450936	−	0.892556i	$-0.351090\pi$
$571$	21.5508	0.901872	0.450936	+	0.892556i	$0.351090\pi$
$572$	0	0
$573$	−51.0054	−2.13078
$574$	0	0
$575$	−5.54427	−0.231212
$576$	0	0
$577$	3.36917	0.140261	0.0701303	−	0.997538i	$-0.477659\pi$
$577$	3.36917	0.140261	0.0701303	+	0.997538i	$0.477659\pi$
$578$	0	0
$579$	72.2727	3.00355
$580$	0	0
$581$	−2.14515	−0.0889958
$582$	0	0
$583$	27.1632	1.12498
$584$	0	0
$585$	−82.7692	−3.42208
$586$	0	0
$587$	−17.7700	−0.733447	−0.366723	−	0.930330i	$-0.619520\pi$
$587$	−17.7700	−0.733447	−0.366723	+	0.930330i	$0.619520\pi$
$588$	0	0
$589$	−2.03467	−0.0838370
$590$	0	0
$591$	−52.1849	−2.14660
$592$	0	0
$593$	−23.7553	−0.975513	−0.487757	−	0.872980i	$-0.662185\pi$
$593$	−23.7553	−0.975513	−0.487757	+	0.872980i	$0.662185\pi$
$594$	0	0
$595$	−4.83842	−0.198356
$596$	0	0
$597$	35.5386	1.45450
$598$	0	0
$599$	3.74074	0.152842	0.0764212	−	0.997076i	$-0.475651\pi$
$599$	3.74074	0.152842	0.0764212	+	0.997076i	$0.475651\pi$
$600$	0	0
$601$	−8.54129	−0.348407	−0.174203	−	0.984710i	$-0.555735\pi$
$601$	−8.54129	−0.348407	−0.174203	+	0.984710i	$0.555735\pi$
$602$	0	0
$603$	−60.7520	−2.47401
$604$	0	0
$605$	40.2199	1.63517
$606$	0	0
$607$	−18.8510	−0.765139	−0.382569	−	0.923927i	$-0.624961\pi$
$607$	−18.8510	−0.765139	−0.382569	+	0.923927i	$0.624961\pi$
$608$	0	0
$609$	−20.1415	−0.816174
$610$	0	0
$611$	−3.79610	−0.153574
$612$	0	0
$613$	49.3590	1.99359	0.996796	−	0.0799889i	$-0.0254885\pi$
$613$	49.3590	1.99359	0.996796	+	0.0799889i	$0.0254885\pi$
$614$	0	0
$615$	52.0761	2.09991
$616$	0	0
$617$	−27.3583	−1.10141	−0.550703	−	0.834702i	$-0.685640\pi$
$617$	−27.3583	−1.10141	−0.550703	+	0.834702i	$0.685640\pi$
$618$	0	0
$619$	23.3006	0.936531	0.468266	−	0.883588i	$-0.344879\pi$
$619$	23.3006	0.936531	0.468266	+	0.883588i	$0.344879\pi$
$620$	0	0
$621$	5.31209	0.213167
$622$	0	0
$623$	15.5995	0.624981
$624$	0	0
$625$	59.4769	2.37907
$626$	0	0
$627$	−49.0373	−1.95836
$628$	0	0
$629$	−3.02077	−0.120446
$630$	0	0
$631$	−32.3083	−1.28617	−0.643086	−	0.765794i	$-0.722346\pi$
$631$	−32.3083	−1.28617	−0.643086	+	0.765794i	$0.722346\pi$
$632$	0	0
$633$	−46.9694	−1.86687
$634$	0	0
$635$	80.7076	3.20278
$636$	0	0
$637$	16.8424	0.667321
$638$	0	0
$639$	−0.543629	−0.0215056
$640$	0	0
$641$	16.4350	0.649143	0.324571	−	0.945861i	$-0.394780\pi$
$641$	16.4350	0.649143	0.324571	+	0.945861i	$0.394780\pi$
$642$	0	0
$643$	−38.7344	−1.52754	−0.763768	−	0.645490i	$-0.776653\pi$
$643$	−38.7344	−1.52754	−0.763768	+	0.645490i	$0.776653\pi$
$644$	0	0
$645$	−103.407	−4.07167
$646$	0	0
$647$	44.7163	1.75798	0.878989	−	0.476843i	$-0.158219\pi$
$647$	44.7163	1.75798	0.878989	+	0.476843i	$0.158219\pi$
$648$	0	0
$649$	4.55495	0.178797
$650$	0	0
$651$	−2.14959	−0.0842492
$652$	0	0
$653$	5.26890	0.206188	0.103094	−	0.994672i	$-0.467126\pi$
$653$	5.26890	0.206188	0.103094	+	0.994672i	$0.467126\pi$
$654$	0	0
$655$	1.09085	0.0426229
$656$	0	0
$657$	−89.9130	−3.50784
$658$	0	0
$659$	−29.3417	−1.14299	−0.571495	−	0.820605i	$-0.693636\pi$
$659$	−29.3417	−1.14299	−0.571495	+	0.820605i	$0.693636\pi$
$660$	0	0
$661$	26.2952	1.02276	0.511382	−	0.859353i	$-0.329134\pi$
$661$	26.2952	1.02276	0.511382	+	0.859353i	$0.329134\pi$
$662$	0	0
$663$	−9.32522	−0.362162
$664$	0	0
$665$	−16.7253	−0.648578
$666$	0	0
$667$	−2.54282	−0.0984585
$668$	0	0
$669$	10.2879	0.397753
$670$	0	0
$671$	−31.2440	−1.20616
$672$	0	0
$673$	−3.51890	−0.135644	−0.0678218	−	0.997697i	$-0.521605\pi$
$673$	−3.51890	−0.135644	−0.0678218	+	0.997697i	$0.521605\pi$
$674$	0	0
$675$	−138.547	−5.33266
$676$	0	0
$677$	−25.5634	−0.982483	−0.491241	−	0.871024i	$-0.663457\pi$
$677$	−25.5634	−0.982483	−0.491241	+	0.871024i	$0.663457\pi$
$678$	0	0
$679$	−2.56994	−0.0986254
$680$	0	0
$681$	−48.9503	−1.87578
$682$	0	0
$683$	−4.34941	−0.166426	−0.0832128	−	0.996532i	$-0.526518\pi$
$683$	−4.34941	−0.166426	−0.0832128	+	0.996532i	$0.526518\pi$
$684$	0	0
$685$	44.5210	1.70106
$686$	0	0
$687$	18.4422	0.703615
$688$	0	0
$689$	17.8560	0.680258
$690$	0	0
$691$	48.4384	1.84269	0.921343	−	0.388752i	$-0.127094\pi$
$691$	48.4384	1.84269	0.921343	+	0.388752i	$0.127094\pi$
$692$	0	0
$693$	−35.7834	−1.35930
$694$	0	0
$695$	−44.8561	−1.70149
$696$	0	0
$697$	4.05248	0.153499
$698$	0	0
$699$	13.9066	0.525996
$700$	0	0
$701$	21.1163	0.797553	0.398777	−	0.917048i	$-0.369435\pi$
$701$	21.1163	0.797553	0.398777	+	0.917048i	$0.369435\pi$
$702$	0	0
$703$	−10.4421	−0.393830
$704$	0	0
$705$	−16.2918	−0.613586
$706$	0	0
$707$	−18.2908	−0.687897
$708$	0	0
$709$	29.3265	1.10138	0.550690	−	0.834710i	$-0.314365\pi$
$709$	29.3265	1.10138	0.550690	+	0.834710i	$0.314365\pi$
$710$	0	0
$711$	4.05108	0.151927
$712$	0	0
$713$	−0.271382	−0.0101633
$714$	0	0
$715$	56.2748	2.10456
$716$	0	0
$717$	11.0863	0.414027
$718$	0	0
$719$	34.8870	1.30107	0.650533	−	0.759478i	$-0.274545\pi$
$719$	34.8870	1.30107	0.650533	+	0.759478i	$0.274545\pi$
$720$	0	0
$721$	−2.80224	−0.104361
$722$	0	0
$723$	−70.8955	−2.63663
$724$	0	0
$725$	66.3203	2.46308
$726$	0	0
$727$	4.49022	0.166533	0.0832665	−	0.996527i	$-0.473465\pi$
$727$	4.49022	0.166533	0.0832665	+	0.996527i	$0.473465\pi$
$728$	0	0
$729$	−1.90262	−0.0704674
$730$	0	0
$731$	−8.04701	−0.297629
$732$	0	0
$733$	−25.4027	−0.938271	−0.469135	−	0.883126i	$-0.655434\pi$
$733$	−25.4027	−0.938271	−0.469135	+	0.883126i	$0.655434\pi$
$734$	0	0
$735$	72.2830	2.66620
$736$	0	0
$737$	41.3053	1.52150
$738$	0	0
$739$	−33.2763	−1.22409	−0.612045	−	0.790823i	$-0.709653\pi$
$739$	−33.2763	−1.22409	−0.612045	+	0.790823i	$0.709653\pi$
$740$	0	0
$741$	−32.2351	−1.18418
$742$	0	0
$743$	−10.8147	−0.396752	−0.198376	−	0.980126i	$-0.563567\pi$
$743$	−10.8147	−0.396752	−0.198376	+	0.980126i	$0.563567\pi$
$744$	0	0
$745$	−72.8115	−2.66761
$746$	0	0
$747$	−12.2557	−0.448411
$748$	0	0
$749$	−2.56798	−0.0938320
$750$	0	0
$751$	−4.67715	−0.170672	−0.0853358	−	0.996352i	$-0.527196\pi$
$751$	−4.67715	−0.170672	−0.0853358	+	0.996352i	$0.527196\pi$
$752$	0	0
$753$	18.2228	0.664075
$754$	0	0
$755$	81.0674	2.95034
$756$	0	0
$757$	3.85501	0.140113	0.0700564	−	0.997543i	$-0.477682\pi$
$757$	3.85501	0.140113	0.0700564	+	0.997543i	$0.477682\pi$
$758$	0	0
$759$	−6.54054	−0.237407
$760$	0	0
$761$	−17.4761	−0.633509	−0.316755	−	0.948507i	$-0.602593\pi$
$761$	−17.4761	−0.633509	−0.316755	+	0.948507i	$0.602593\pi$
$762$	0	0
$763$	−21.2193	−0.768189
$764$	0	0
$765$	−27.6428	−0.999429
$766$	0	0
$767$	2.99424	0.108116
$768$	0	0
$769$	−33.4052	−1.20462	−0.602311	−	0.798262i	$-0.705753\pi$
$769$	−33.4052	−1.20462	−0.602311	+	0.798262i	$0.705753\pi$
$770$	0	0
$771$	21.5923	0.777626
$772$	0	0
$773$	5.47206	0.196816	0.0984081	−	0.995146i	$-0.468625\pi$
$773$	5.47206	0.196816	0.0984081	+	0.995146i	$0.468625\pi$
$774$	0	0
$775$	7.07802	0.254250
$776$	0	0
$777$	−11.0319	−0.395767
$778$	0	0
$779$	14.0085	0.501905
$780$	0	0
$781$	0.369614	0.0132258
$782$	0	0
$783$	−63.5430	−2.27084
$784$	0	0
$785$	17.9473	0.640566
$786$	0	0
$787$	24.8078	0.884303	0.442151	−	0.896940i	$-0.354215\pi$
$787$	24.8078	0.884303	0.442151	+	0.896940i	$0.354215\pi$
$788$	0	0
$789$	92.9160	3.30790
$790$	0	0
$791$	12.8523	0.456975
$792$	0	0
$793$	−20.5385	−0.729344
$794$	0	0
$795$	76.6328	2.71788
$796$	0	0
$797$	7.73963	0.274152	0.137076	−	0.990561i	$-0.456230\pi$
$797$	7.73963	0.274152	0.137076	+	0.990561i	$0.456230\pi$
$798$	0	0
$799$	−1.26780	−0.0448517
$800$	0	0
$801$	89.1231	3.14901
$802$	0	0
$803$	61.1319	2.15730
$804$	0	0
$805$	−2.23080	−0.0786253
$806$	0	0
$807$	75.5213	2.65848
$808$	0	0
$809$	20.4809	0.720069	0.360034	−	0.932939i	$-0.382765\pi$
$809$	20.4809	0.720069	0.360034	+	0.932939i	$0.382765\pi$
$810$	0	0
$811$	29.8759	1.04909	0.524543	−	0.851384i	$-0.324236\pi$
$811$	29.8759	1.04909	0.524543	+	0.851384i	$0.324236\pi$
$812$	0	0
$813$	77.2972	2.71093
$814$	0	0
$815$	2.61553	0.0916180
$816$	0	0
$817$	−27.8166	−0.973179
$818$	0	0
$819$	−23.5225	−0.821943
$820$	0	0
$821$	−14.5822	−0.508921	−0.254461	−	0.967083i	$-0.581898\pi$
$821$	−14.5822	−0.508921	−0.254461	+	0.967083i	$0.581898\pi$
$822$	0	0
$823$	48.8653	1.70334	0.851668	−	0.524082i	$-0.175591\pi$
$823$	48.8653	1.70334	0.851668	+	0.524082i	$0.175591\pi$
$824$	0	0
$825$	170.586	5.93906
$826$	0	0
$827$	−20.7292	−0.720823	−0.360412	−	0.932793i	$-0.617364\pi$
$827$	−20.7292	−0.720823	−0.360412	+	0.932793i	$0.617364\pi$
$828$	0	0
$829$	4.78801	0.166295	0.0831473	−	0.996537i	$-0.473503\pi$
$829$	4.78801	0.166295	0.0831473	+	0.996537i	$0.473503\pi$
$830$	0	0
$831$	−71.6335	−2.48494
$832$	0	0
$833$	5.62495	0.194893
$834$	0	0
$835$	−90.8040	−3.14240
$836$	0	0
$837$	−6.78161	−0.234407
$838$	0	0
$839$	6.32786	0.218462	0.109231	−	0.994016i	$-0.465161\pi$
$839$	6.32786	0.218462	0.109231	+	0.994016i	$0.465161\pi$
$840$	0	0
$841$	1.41715	0.0488673
$842$	0	0
$843$	97.5572	3.36005
$844$	0	0
$845$	−16.6471	−0.572679
$846$	0	0
$847$	11.4303	0.392748
$848$	0	0
$849$	51.6356	1.77213
$850$	0	0
$851$	−1.39275	−0.0477430
$852$	0	0
$853$	−24.2047	−0.828752	−0.414376	−	0.910106i	$-0.636000\pi$
$853$	−24.2047	−0.828752	−0.414376	+	0.910106i	$0.636000\pi$
$854$	0	0
$855$	−95.5546	−3.26790
$856$	0	0
$857$	−7.42641	−0.253681	−0.126841	−	0.991923i	$-0.540484\pi$
$857$	−7.42641	−0.253681	−0.126841	+	0.991923i	$0.540484\pi$
$858$	0	0
$859$	−35.5180	−1.21186	−0.605930	−	0.795518i	$-0.707199\pi$
$859$	−35.5180	−1.21186	−0.605930	+	0.795518i	$0.707199\pi$
$860$	0	0
$861$	14.7997	0.504373
$862$	0	0
$863$	−29.1511	−0.992315	−0.496157	−	0.868233i	$-0.665256\pi$
$863$	−29.1511	−0.992315	−0.496157	+	0.868233i	$0.665256\pi$
$864$	0	0
$865$	−26.1692	−0.889778
$866$	0	0
$867$	−3.11439	−0.105770
$868$	0	0
$869$	−2.75433	−0.0934343
$870$	0	0
$871$	27.1524	0.920024
$872$	0	0
$873$	−14.6826	−0.496930
$874$	0	0
$875$	33.9902	1.14908
$876$	0	0
$877$	−4.55632	−0.153856	−0.0769279	−	0.997037i	$-0.524511\pi$
$877$	−4.55632	−0.153856	−0.0769279	+	0.997037i	$0.524511\pi$
$878$	0	0
$879$	48.5565	1.63777
$880$	0	0
$881$	46.3799	1.56258	0.781289	−	0.624169i	$-0.214563\pi$
$881$	46.3799	1.56258	0.781289	+	0.624169i	$0.214563\pi$
$882$	0	0
$883$	−37.9096	−1.27576	−0.637880	−	0.770136i	$-0.720188\pi$
$883$	−37.9096	−1.27576	−0.637880	+	0.770136i	$0.720188\pi$
$884$	0	0
$885$	12.8504	0.431962
$886$	0	0
$887$	−35.4535	−1.19041	−0.595207	−	0.803573i	$-0.702930\pi$
$887$	−35.4535	−1.19041	−0.595207	+	0.803573i	$0.702930\pi$
$888$	0	0
$889$	22.9366	0.769269
$890$	0	0
$891$	−71.8959	−2.40860
$892$	0	0
$893$	−4.38249	−0.146655
$894$	0	0
$895$	−56.6700	−1.89427
$896$	0	0
$897$	−4.29948	−0.143555
$898$	0	0
$899$	3.24626	0.108269
$900$	0	0
$901$	5.96344	0.198671
$902$	0	0
$903$	−29.3878	−0.977964
$904$	0	0
$905$	−69.0554	−2.29548
$906$	0	0
$907$	−18.4105	−0.611312	−0.305656	−	0.952142i	$-0.598876\pi$
$907$	−18.4105	−0.611312	−0.305656	+	0.952142i	$0.598876\pi$
$908$	0	0
$909$	−104.499	−3.46601
$910$	0	0
$911$	−4.31180	−0.142856	−0.0714281	−	0.997446i	$-0.522756\pi$
$911$	−4.31180	−0.142856	−0.0714281	+	0.997446i	$0.522756\pi$
$912$	0	0
$913$	8.33263	0.275770
$914$	0	0
$915$	−88.1456	−2.91400
$916$	0	0
$917$	0.310012	0.0102375
$918$	0	0
$919$	11.7584	0.387873	0.193936	−	0.981014i	$-0.437874\pi$
$919$	11.7584	0.387873	0.193936	+	0.981014i	$0.437874\pi$
$920$	0	0
$921$	92.0169	3.03206
$922$	0	0
$923$	0.242969	0.00799742
$924$	0	0
$925$	36.3249	1.19436
$926$	0	0
$927$	−16.0097	−0.525829
$928$	0	0
$929$	−44.8861	−1.47267	−0.736333	−	0.676619i	$-0.763444\pi$
$929$	−44.8861	−1.47267	−0.736333	+	0.676619i	$0.763444\pi$
$930$	0	0
$931$	19.4441	0.637255
$932$	0	0
$933$	−38.7029	−1.26708
$934$	0	0
$935$	18.7944	0.614642
$936$	0	0
$937$	18.6852	0.610420	0.305210	−	0.952285i	$-0.401273\pi$
$937$	18.6852	0.610420	0.305210	+	0.952285i	$0.401273\pi$
$938$	0	0
$939$	6.68387	0.218120
$940$	0	0
$941$	−20.7971	−0.677966	−0.338983	−	0.940793i	$-0.610083\pi$
$941$	−20.7971	−0.677966	−0.338983	+	0.940793i	$0.610083\pi$
$942$	0	0
$943$	1.86843	0.0608446
$944$	0	0
$945$	−55.7458	−1.81341
$946$	0	0
$947$	21.0334	0.683495	0.341747	−	0.939792i	$-0.388981\pi$
$947$	21.0334	0.683495	0.341747	+	0.939792i	$0.388981\pi$
$948$	0	0
$949$	40.1856	1.30448
$950$	0	0
$951$	−0.817786	−0.0265185
$952$	0	0
$953$	21.1419	0.684855	0.342427	−	0.939544i	$-0.388751\pi$
$953$	21.1419	0.684855	0.342427	+	0.939544i	$0.388751\pi$
$954$	0	0
$955$	67.5752	2.18668
$956$	0	0
$957$	78.2377	2.52907
$958$	0	0
$959$	12.6526	0.408573
$960$	0	0
$961$	−30.6535	−0.988824
$962$	0	0
$963$	−14.6714	−0.472779
$964$	0	0
$965$	−95.7514	−3.08235
$966$	0	0
$967$	−2.17163	−0.0698350	−0.0349175	−	0.999390i	$-0.511117\pi$
$967$	−2.17163	−0.0698350	−0.0349175	+	0.999390i	$0.511117\pi$
$968$	0	0
$969$	−10.7657	−0.345844
$970$	0	0
$971$	13.0442	0.418608	0.209304	−	0.977851i	$-0.432880\pi$
$971$	13.0442	0.418608	0.209304	+	0.977851i	$0.432880\pi$
$972$	0	0
$973$	−12.7478	−0.408677
$974$	0	0
$975$	112.136	3.59124
$976$	0	0
$977$	−18.8436	−0.602862	−0.301431	−	0.953488i	$-0.597464\pi$
$977$	−18.8436	−0.602862	−0.301431	+	0.953488i	$0.597464\pi$
$978$	0	0
$979$	−60.5948	−1.93662
$980$	0	0
$981$	−121.230	−3.87057
$982$	0	0
$983$	−0.300627	−0.00958853	−0.00479426	−	0.999989i	$-0.501526\pi$
$983$	−0.300627	−0.00958853	−0.00479426	+	0.999989i	$0.501526\pi$
$984$	0	0
$985$	69.1379	2.20292
$986$	0	0
$987$	−4.63004	−0.147376
$988$	0	0
$989$	−3.71015	−0.117976
$990$	0	0
$991$	−21.4890	−0.682622	−0.341311	−	0.939950i	$-0.610871\pi$
$991$	−21.4890	−0.682622	−0.341311	+	0.939950i	$0.610871\pi$
$992$	0	0
$993$	34.4265	1.09249
$994$	0	0
$995$	−47.0838	−1.49266
$996$	0	0
$997$	36.3345	1.15072	0.575362	−	0.817899i	$-0.304861\pi$
$997$	36.3345	1.15072	0.575362	+	0.817899i	$0.304861\pi$
$998$	0	0
$999$	−34.8037	−1.10114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.2	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.2	✓	23	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.11439 q^{3} +4.12614 q^{5} +1.17263 q^{7} +6.69943 q^{9} +O(q^{10})\)	\(q-3.11439 q^{3} +4.12614 q^{5} +1.17263 q^{7} +6.69943 q^{9} -4.55495 q^{11} -2.99424 q^{13} -12.8504 q^{15} -1.00000 q^{17} -3.45676 q^{19} -3.65201 q^{21} -0.461059 q^{23} +12.0251 q^{25} -11.5215 q^{27} +5.51517 q^{29} +0.588605 q^{31} +14.1859 q^{33} +4.83842 q^{35} +3.02077 q^{37} +9.32522 q^{39} -4.05248 q^{41} +8.04701 q^{43} +27.6428 q^{45} +1.26780 q^{47} -5.62495 q^{49} +3.11439 q^{51} -5.96344 q^{53} -18.7944 q^{55} +10.7657 q^{57} -1.00000 q^{59} +6.85935 q^{61} +7.85592 q^{63} -12.3547 q^{65} -9.06822 q^{67} +1.43592 q^{69} -0.0811455 q^{71} -13.4210 q^{73} -37.4508 q^{75} -5.34125 q^{77} +0.604690 q^{79} +15.7841 q^{81} -1.82936 q^{83} -4.12614 q^{85} -17.1764 q^{87} +13.3031 q^{89} -3.51112 q^{91} -1.83315 q^{93} -14.2631 q^{95} -2.19162 q^{97} -30.5156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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