LMFDB - Embedded newform 4024.2.a.e.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.582422 q^{3} -3.18976 q^{5} -3.00819 q^{7} -2.66078 q^{9} +O(q^{10})$	$q-0.582422 q^{3} -3.18976 q^{5} -3.00819 q^{7} -2.66078 q^{9} +2.52679 q^{11} +2.23127 q^{13} +1.85779 q^{15} +4.66138 q^{17} -4.31551 q^{19} +1.75203 q^{21} +4.90091 q^{23} +5.17455 q^{25} +3.29697 q^{27} +8.71758 q^{29} -1.05116 q^{31} -1.47166 q^{33} +9.59538 q^{35} -5.94648 q^{37} -1.29954 q^{39} -3.55493 q^{41} +4.35311 q^{43} +8.48726 q^{45} +2.89673 q^{47} +2.04918 q^{49} -2.71489 q^{51} -1.27320 q^{53} -8.05985 q^{55} +2.51345 q^{57} -0.00300337 q^{59} +5.70760 q^{61} +8.00413 q^{63} -7.11720 q^{65} -13.8584 q^{67} -2.85440 q^{69} +2.05176 q^{71} -7.67426 q^{73} -3.01378 q^{75} -7.60106 q^{77} +10.7106 q^{79} +6.06212 q^{81} -0.508840 q^{83} -14.8687 q^{85} -5.07731 q^{87} -7.67083 q^{89} -6.71207 q^{91} +0.612222 q^{93} +13.7654 q^{95} +13.4633 q^{97} -6.72325 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−0.582422	−0.336262	−0.168131	−	0.985765i	$-0.553773\pi$
$3$	−0.582422	−0.336262	−0.168131	+	0.985765i	$0.553773\pi$
$4$	0	0
$5$	−3.18976	−1.42650	−0.713251	−	0.700908i	$-0.752778\pi$
$5$	−3.18976	−1.42650	−0.713251	+	0.700908i	$0.752778\pi$
$6$	0	0
$7$	−3.00819	−1.13699	−0.568494	−	0.822688i	$-0.692474\pi$
$7$	−3.00819	−1.13699	−0.568494	+	0.822688i	$0.692474\pi$
$8$	0	0
$9$	−2.66078	−0.886928
$10$	0	0
$11$	2.52679	0.761857	0.380928	−	0.924605i	$-0.375604\pi$
$11$	2.52679	0.761857	0.380928	+	0.924605i	$0.375604\pi$
$12$	0	0
$13$	2.23127	0.618842	0.309421	−	0.950925i	$-0.399865\pi$
$13$	2.23127	0.618842	0.309421	+	0.950925i	$0.399865\pi$
$14$	0	0
$15$	1.85779	0.479678
$16$	0	0
$17$	4.66138	1.13055	0.565275	−	0.824902i	$-0.308770\pi$
$17$	4.66138	1.13055	0.565275	+	0.824902i	$0.308770\pi$
$18$	0	0
$19$	−4.31551	−0.990045	−0.495022	−	0.868880i	$-0.664840\pi$
$19$	−4.31551	−0.990045	−0.495022	+	0.868880i	$0.664840\pi$
$20$	0	0
$21$	1.75203	0.382325
$22$	0	0
$23$	4.90091	1.02191	0.510955	−	0.859607i	$-0.329292\pi$
$23$	4.90091	1.02191	0.510955	+	0.859607i	$0.329292\pi$
$24$	0	0
$25$	5.17455	1.03491
$26$	0	0
$27$	3.29697	0.634502
$28$	0	0
$29$	8.71758	1.61881	0.809407	−	0.587249i	$-0.199789\pi$
$29$	8.71758	1.61881	0.809407	+	0.587249i	$0.199789\pi$
$30$	0	0
$31$	−1.05116	−0.188795	−0.0943974	−	0.995535i	$-0.530092\pi$
$31$	−1.05116	−0.188795	−0.0943974	+	0.995535i	$0.530092\pi$
$32$	0	0
$33$	−1.47166	−0.256183
$34$	0	0
$35$	9.59538	1.62192
$36$	0	0
$37$	−5.94648	−0.977595	−0.488797	−	0.872397i	$-0.662564\pi$
$37$	−5.94648	−0.977595	−0.488797	+	0.872397i	$0.662564\pi$
$38$	0	0
$39$	−1.29954	−0.208093
$40$	0	0
$41$	−3.55493	−0.555187	−0.277593	−	0.960699i	$-0.589537\pi$
$41$	−3.55493	−0.555187	−0.277593	+	0.960699i	$0.589537\pi$
$42$	0	0
$43$	4.35311	0.663843	0.331921	−	0.943307i	$-0.392303\pi$
$43$	4.35311	0.663843	0.331921	+	0.943307i	$0.392303\pi$
$44$	0	0
$45$	8.48726	1.26521
$46$	0	0
$47$	2.89673	0.422531	0.211266	−	0.977429i	$-0.432241\pi$
$47$	2.89673	0.422531	0.211266	+	0.977429i	$0.432241\pi$
$48$	0	0
$49$	2.04918	0.292740
$50$	0	0
$51$	−2.71489	−0.380161
$52$	0	0
$53$	−1.27320	−0.174888	−0.0874441	−	0.996169i	$-0.527870\pi$
$53$	−1.27320	−0.174888	−0.0874441	+	0.996169i	$0.527870\pi$
$54$	0	0
$55$	−8.05985	−1.08679
$56$	0	0
$57$	2.51345	0.332914
$58$	0	0
$59$	−0.00300337	−0.000391006 0	−0.000195503	−	1.00000i	$-0.500062\pi$
$59$	−0.00300337	−0.000391006 0	−0.000195503		1.00000i	$0.500062\pi$
$60$	0	0
$61$	5.70760	0.730784	0.365392	−	0.930854i	$-0.380935\pi$
$61$	5.70760	0.730784	0.365392	+	0.930854i	$0.380935\pi$
$62$	0	0
$63$	8.00413	1.00843
$64$	0	0
$65$	−7.11720	−0.882780
$66$	0	0
$67$	−13.8584	−1.69307	−0.846536	−	0.532332i	$-0.821316\pi$
$67$	−13.8584	−1.69307	−0.846536	+	0.532332i	$0.821316\pi$
$68$	0	0
$69$	−2.85440	−0.343629
$70$	0	0
$71$	2.05176	0.243499	0.121750	−	0.992561i	$-0.461150\pi$
$71$	2.05176	0.243499	0.121750	+	0.992561i	$0.461150\pi$
$72$	0	0
$73$	−7.67426	−0.898204	−0.449102	−	0.893481i	$-0.648256\pi$
$73$	−7.67426	−0.898204	−0.449102	+	0.893481i	$0.648256\pi$
$74$	0	0
$75$	−3.01378	−0.348001
$76$	0	0
$77$	−7.60106	−0.866221
$78$	0	0
$79$	10.7106	1.20503	0.602516	−	0.798107i	$-0.294165\pi$
$79$	10.7106	1.20503	0.602516	+	0.798107i	$0.294165\pi$
$80$	0	0
$81$	6.06212	0.673569
$82$	0	0
$83$	−0.508840	−0.0558524	−0.0279262	−	0.999610i	$-0.508890\pi$
$83$	−0.508840	−0.0558524	−0.0279262	+	0.999610i	$0.508890\pi$
$84$	0	0
$85$	−14.8687	−1.61273
$86$	0	0
$87$	−5.07731	−0.544345
$88$	0	0
$89$	−7.67083	−0.813106	−0.406553	−	0.913627i	$-0.633269\pi$
$89$	−7.67083	−0.813106	−0.406553	+	0.913627i	$0.633269\pi$
$90$	0	0
$91$	−6.71207	−0.703616
$92$	0	0
$93$	0.612222	0.0634845
$94$	0	0
$95$	13.7654	1.41230
$96$	0	0
$97$	13.4633	1.36699	0.683497	−	0.729954i	$-0.260458\pi$
$97$	13.4633	1.36699	0.683497	+	0.729954i	$0.260458\pi$
$98$	0	0
$99$	−6.72325	−0.675712
$100$	0	0
$101$	−3.61557	−0.359763	−0.179882	−	0.983688i	$-0.557571\pi$
$101$	−3.61557	−0.359763	−0.179882	+	0.983688i	$0.557571\pi$
$102$	0	0
$103$	−8.69287	−0.856534	−0.428267	−	0.903652i	$-0.640876\pi$
$103$	−8.69287	−0.856534	−0.428267	+	0.903652i	$0.640876\pi$
$104$	0	0
$105$	−5.58857	−0.545388
$106$	0	0
$107$	−17.6501	−1.70630	−0.853150	−	0.521665i	$-0.825311\pi$
$107$	−17.6501	−1.70630	−0.853150	+	0.521665i	$0.825311\pi$
$108$	0	0
$109$	−2.89913	−0.277686	−0.138843	−	0.990314i	$-0.544338\pi$
$109$	−2.89913	−0.277686	−0.138843	+	0.990314i	$0.544338\pi$
$110$	0	0
$111$	3.46336	0.328728
$112$	0	0
$113$	−2.14609	−0.201887	−0.100944	−	0.994892i	$-0.532186\pi$
$113$	−2.14609	−0.201887	−0.100944	+	0.994892i	$0.532186\pi$
$114$	0	0
$115$	−15.6327	−1.45776
$116$	0	0
$117$	−5.93692	−0.548869
$118$	0	0
$119$	−14.0223	−1.28542
$120$	0	0
$121$	−4.61532	−0.419575
$122$	0	0
$123$	2.07047	0.186688
$124$	0	0
$125$	−0.556783	−0.0498002
$126$	0	0
$127$	0.0409748	0.00363592	0.00181796	−	0.999998i	$-0.499421\pi$
$127$	0.0409748	0.00363592	0.00181796	+	0.999998i	$0.499421\pi$
$128$	0	0
$129$	−2.53535	−0.223225
$130$	0	0
$131$	9.71411	0.848726	0.424363	−	0.905492i	$-0.360498\pi$
$131$	9.71411	0.848726	0.424363	+	0.905492i	$0.360498\pi$
$132$	0	0
$133$	12.9818	1.12567
$134$	0	0
$135$	−10.5165	−0.905119
$136$	0	0
$137$	1.00835	0.0861489	0.0430745	−	0.999072i	$-0.486285\pi$
$137$	1.00835	0.0861489	0.0430745	+	0.999072i	$0.486285\pi$
$138$	0	0
$139$	2.41995	0.205258	0.102629	−	0.994720i	$-0.467275\pi$
$139$	2.41995	0.205258	0.102629	+	0.994720i	$0.467275\pi$
$140$	0	0
$141$	−1.68712	−0.142081
$142$	0	0
$143$	5.63795	0.471469
$144$	0	0
$145$	−27.8070	−2.30924
$146$	0	0
$147$	−1.19349	−0.0984372
$148$	0	0
$149$	−12.3931	−1.01528	−0.507639	−	0.861570i	$-0.669482\pi$
$149$	−12.3931	−1.01528	−0.507639	+	0.861570i	$0.669482\pi$
$150$	0	0
$151$	−16.4456	−1.33833	−0.669163	−	0.743115i	$-0.733347\pi$
$151$	−16.4456	−1.33833	−0.669163	+	0.743115i	$0.733347\pi$
$152$	0	0
$153$	−12.4029	−1.00272
$154$	0	0
$155$	3.35296	0.269316
$156$	0	0
$157$	13.7202	1.09499	0.547494	−	0.836810i	$-0.315582\pi$
$157$	13.7202	1.09499	0.547494	+	0.836810i	$0.315582\pi$
$158$	0	0
$159$	0.741543	0.0588082
$160$	0	0
$161$	−14.7428	−1.16190
$162$	0	0
$163$	−7.52511	−0.589412	−0.294706	−	0.955588i	$-0.595222\pi$
$163$	−7.52511	−0.589412	−0.294706	+	0.955588i	$0.595222\pi$
$164$	0	0
$165$	4.69424	0.365446
$166$	0	0
$167$	−14.3791	−1.11269	−0.556346	−	0.830951i	$-0.687797\pi$
$167$	−14.3791	−1.11269	−0.556346	+	0.830951i	$0.687797\pi$
$168$	0	0
$169$	−8.02145	−0.617034
$170$	0	0
$171$	11.4826	0.878099
$172$	0	0
$173$	7.55769	0.574601	0.287300	−	0.957841i	$-0.407242\pi$
$173$	7.55769	0.574601	0.287300	+	0.957841i	$0.407242\pi$
$174$	0	0
$175$	−15.5660	−1.17668
$176$	0	0
$177$	0.00174923	0.000131480 0
$178$	0	0
$179$	−2.87815	−0.215123	−0.107562	−	0.994198i	$-0.534304\pi$
$179$	−2.87815	−0.215123	−0.107562	+	0.994198i	$0.534304\pi$
$180$	0	0
$181$	25.9448	1.92846	0.964231	−	0.265063i	$-0.0853929\pi$
$181$	25.9448	1.92846	0.964231	+	0.265063i	$0.0853929\pi$
$182$	0	0
$183$	−3.32424	−0.245735
$184$	0	0
$185$	18.9678	1.39454
$186$	0	0
$187$	11.7783	0.861317
$188$	0	0
$189$	−9.91789	−0.721420
$190$	0	0
$191$	−24.8829	−1.80046	−0.900232	−	0.435410i	$-0.856603\pi$
$191$	−24.8829	−1.80046	−0.900232	+	0.435410i	$0.856603\pi$
$192$	0	0
$193$	−0.757929	−0.0545569	−0.0272785	−	0.999628i	$-0.508684\pi$
$193$	−0.757929	−0.0545569	−0.0272785	+	0.999628i	$0.508684\pi$
$194$	0	0
$195$	4.14522	0.296845
$196$	0	0
$197$	0.768600	0.0547605	0.0273802	−	0.999625i	$-0.491284\pi$
$197$	0.768600	0.0547605	0.0273802	+	0.999625i	$0.491284\pi$
$198$	0	0
$199$	−8.12600	−0.576037	−0.288018	−	0.957625i	$-0.592996\pi$
$199$	−8.12600	−0.576037	−0.288018	+	0.957625i	$0.592996\pi$
$200$	0	0
$201$	8.07144	0.569315
$202$	0	0
$203$	−26.2241	−1.84057
$204$	0	0
$205$	11.3394	0.791976
$206$	0	0
$207$	−13.0403	−0.906361
$208$	0	0
$209$	−10.9044	−0.754272
$210$	0	0
$211$	−22.7624	−1.56703	−0.783513	−	0.621375i	$-0.786574\pi$
$211$	−22.7624	−1.56703	−0.783513	+	0.621375i	$0.786574\pi$
$212$	0	0
$213$	−1.19499	−0.0818794
$214$	0	0
$215$	−13.8854	−0.946974
$216$	0	0
$217$	3.16210	0.214657
$218$	0	0
$219$	4.46966	0.302032
$220$	0	0
$221$	10.4008	0.699632
$222$	0	0
$223$	−6.71230	−0.449489	−0.224745	−	0.974418i	$-0.572155\pi$
$223$	−6.71230	−0.449489	−0.224745	+	0.974418i	$0.572155\pi$
$224$	0	0
$225$	−13.7684	−0.917891
$226$	0	0
$227$	−15.5457	−1.03181	−0.515903	−	0.856647i	$-0.672543\pi$
$227$	−15.5457	−1.03181	−0.515903	+	0.856647i	$0.672543\pi$
$228$	0	0
$229$	18.3849	1.21491	0.607455	−	0.794354i	$-0.292190\pi$
$229$	18.3849	1.21491	0.607455	+	0.794354i	$0.292190\pi$
$230$	0	0
$231$	4.42703	0.291277
$232$	0	0
$233$	−4.20286	−0.275339	−0.137669	−	0.990478i	$-0.543961\pi$
$233$	−4.20286	−0.275339	−0.137669	+	0.990478i	$0.543961\pi$
$234$	0	0
$235$	−9.23987	−0.602742
$236$	0	0
$237$	−6.23807	−0.405206
$238$	0	0
$239$	2.73687	0.177033	0.0885166	−	0.996075i	$-0.471787\pi$
$239$	2.73687	0.177033	0.0885166	+	0.996075i	$0.471787\pi$
$240$	0	0
$241$	19.6636	1.26664	0.633321	−	0.773889i	$-0.281691\pi$
$241$	19.6636	1.26664	0.633321	+	0.773889i	$0.281691\pi$
$242$	0	0
$243$	−13.4216	−0.860997
$244$	0	0
$245$	−6.53638	−0.417594
$246$	0	0
$247$	−9.62905	−0.612682
$248$	0	0
$249$	0.296360	0.0187810
$250$	0	0
$251$	−15.3180	−0.966862	−0.483431	−	0.875383i	$-0.660610\pi$
$251$	−15.3180	−0.966862	−0.483431	+	0.875383i	$0.660610\pi$
$252$	0	0
$253$	12.3836	0.778549
$254$	0	0
$255$	8.65985	0.542301
$256$	0	0
$257$	−9.79154	−0.610779	−0.305390	−	0.952227i	$-0.598787\pi$
$257$	−9.79154	−0.610779	−0.305390	+	0.952227i	$0.598787\pi$
$258$	0	0
$259$	17.8881	1.11151
$260$	0	0
$261$	−23.1956	−1.43577
$262$	0	0
$263$	13.3562	0.823580	0.411790	−	0.911279i	$-0.364904\pi$
$263$	13.3562	0.823580	0.411790	+	0.911279i	$0.364904\pi$
$264$	0	0
$265$	4.06121	0.249478
$266$	0	0
$267$	4.46766	0.273417
$268$	0	0
$269$	27.0075	1.64668	0.823338	−	0.567551i	$-0.192109\pi$
$269$	27.0075	1.64668	0.823338	+	0.567551i	$0.192109\pi$
$270$	0	0
$271$	8.75940	0.532096	0.266048	−	0.963960i	$-0.414282\pi$
$271$	8.75940	0.532096	0.266048	+	0.963960i	$0.414282\pi$
$272$	0	0
$273$	3.90926	0.236599
$274$	0	0
$275$	13.0750	0.788453
$276$	0	0
$277$	15.8058	0.949676	0.474838	−	0.880073i	$-0.342507\pi$
$277$	15.8058	0.949676	0.474838	+	0.880073i	$0.342507\pi$
$278$	0	0
$279$	2.79692	0.167447
$280$	0	0
$281$	5.54498	0.330786	0.165393	−	0.986228i	$-0.447111\pi$
$281$	5.54498	0.330786	0.165393	+	0.986228i	$0.447111\pi$
$282$	0	0
$283$	20.6049	1.22483	0.612417	−	0.790535i	$-0.290197\pi$
$283$	20.6049	1.22483	0.612417	+	0.790535i	$0.290197\pi$
$284$	0	0
$285$	−8.01729	−0.474903
$286$	0	0
$287$	10.6939	0.631240
$288$	0	0
$289$	4.72845	0.278144
$290$	0	0
$291$	−7.84134	−0.459668
$292$	0	0
$293$	−5.17567	−0.302366	−0.151183	−	0.988506i	$-0.548308\pi$
$293$	−5.17567	−0.302366	−0.151183	+	0.988506i	$0.548308\pi$
$294$	0	0
$295$	0.00958003	0.000557771 0
$296$	0	0
$297$	8.33075	0.483399
$298$	0	0
$299$	10.9352	0.632401
$300$	0	0
$301$	−13.0950	−0.754781
$302$	0	0
$303$	2.10579	0.120975
$304$	0	0
$305$	−18.2059	−1.04247
$306$	0	0
$307$	−12.8287	−0.732175	−0.366088	−	0.930580i	$-0.619303\pi$
$307$	−12.8287	−0.732175	−0.366088	+	0.930580i	$0.619303\pi$
$308$	0	0
$309$	5.06292	0.288020
$310$	0	0
$311$	−2.63295	−0.149301	−0.0746505	−	0.997210i	$-0.523784\pi$
$311$	−2.63295	−0.149301	−0.0746505	+	0.997210i	$0.523784\pi$
$312$	0	0
$313$	25.5870	1.44626	0.723131	−	0.690711i	$-0.242702\pi$
$313$	25.5870	1.44626	0.723131	+	0.690711i	$0.242702\pi$
$314$	0	0
$315$	−25.5312	−1.43852
$316$	0	0
$317$	−34.3875	−1.93139	−0.965696	−	0.259675i	$-0.916384\pi$
$317$	−34.3875	−1.93139	−0.965696	+	0.259675i	$0.916384\pi$
$318$	0	0
$319$	22.0275	1.23330
$320$	0	0
$321$	10.2798	0.573763
$322$	0	0
$323$	−20.1162	−1.11930
$324$	0	0
$325$	11.5458	0.640446
$326$	0	0
$327$	1.68852	0.0933753
$328$	0	0
$329$	−8.71390	−0.480413
$330$	0	0
$331$	1.17717	0.0647029	0.0323514	−	0.999477i	$-0.489700\pi$
$331$	1.17717	0.0647029	0.0323514	+	0.999477i	$0.489700\pi$
$332$	0	0
$333$	15.8223	0.867056
$334$	0	0
$335$	44.2049	2.41517
$336$	0	0
$337$	−7.00207	−0.381427	−0.190714	−	0.981646i	$-0.561080\pi$
$337$	−7.00207	−0.381427	−0.190714	+	0.981646i	$0.561080\pi$
$338$	0	0
$339$	1.24993	0.0678869
$340$	0	0
$341$	−2.65608	−0.143835
$342$	0	0
$343$	14.8930	0.804146
$344$	0	0
$345$	9.10484	0.490188
$346$	0	0
$347$	9.80693	0.526464	0.263232	−	0.964733i	$-0.415212\pi$
$347$	9.80693	0.526464	0.263232	+	0.964733i	$0.415212\pi$
$348$	0	0
$349$	6.78721	0.363311	0.181656	−	0.983362i	$-0.441854\pi$
$349$	6.78721	0.363311	0.181656	+	0.983362i	$0.441854\pi$
$350$	0	0
$351$	7.35642	0.392656
$352$	0	0
$353$	−9.07998	−0.483279	−0.241639	−	0.970366i	$-0.577685\pi$
$353$	−9.07998	−0.483279	−0.241639	+	0.970366i	$0.577685\pi$
$354$	0	0
$355$	−6.54462	−0.347352
$356$	0	0
$357$	8.16690	0.432238
$358$	0	0
$359$	0.430390	0.0227151	0.0113575	−	0.999936i	$-0.496385\pi$
$359$	0.430390	0.0227151	0.0113575	+	0.999936i	$0.496385\pi$
$360$	0	0
$361$	−0.376411	−0.0198111
$362$	0	0
$363$	2.68807	0.141087
$364$	0	0
$365$	24.4790	1.28129
$366$	0	0
$367$	−24.8771	−1.29857	−0.649287	−	0.760544i	$-0.724932\pi$
$367$	−24.8771	−1.29857	−0.649287	+	0.760544i	$0.724932\pi$
$368$	0	0
$369$	9.45890	0.492411
$370$	0	0
$371$	3.83004	0.198846
$372$	0	0
$373$	19.1319	0.990610	0.495305	−	0.868719i	$-0.335056\pi$
$373$	19.1319	0.990610	0.495305	+	0.868719i	$0.335056\pi$
$374$	0	0
$375$	0.324283	0.0167459
$376$	0	0
$377$	19.4512	1.00179
$378$	0	0
$379$	2.14993	0.110434	0.0552172	−	0.998474i	$-0.482415\pi$
$379$	2.14993	0.110434	0.0552172	+	0.998474i	$0.482415\pi$
$380$	0	0
$381$	−0.0238646	−0.00122262
$382$	0	0
$383$	4.43686	0.226713	0.113356	−	0.993554i	$-0.463840\pi$
$383$	4.43686	0.226713	0.113356	+	0.993554i	$0.463840\pi$
$384$	0	0
$385$	24.2455	1.23567
$386$	0	0
$387$	−11.5827	−0.588781
$388$	0	0
$389$	−1.84046	−0.0933151	−0.0466576	−	0.998911i	$-0.514857\pi$
$389$	−1.84046	−0.0933151	−0.0466576	+	0.998911i	$0.514857\pi$
$390$	0	0
$391$	22.8450	1.15532
$392$	0	0
$393$	−5.65771	−0.285394
$394$	0	0
$395$	−34.1641	−1.71898
$396$	0	0
$397$	−22.0415	−1.10623	−0.553116	−	0.833104i	$-0.686561\pi$
$397$	−22.0415	−1.10623	−0.553116	+	0.833104i	$0.686561\pi$
$398$	0	0
$399$	−7.56091	−0.378519
$400$	0	0
$401$	−39.0667	−1.95090	−0.975450	−	0.220221i	$-0.929322\pi$
$401$	−39.0667	−1.95090	−0.975450	+	0.220221i	$0.929322\pi$
$402$	0	0
$403$	−2.34543	−0.116834
$404$	0	0
$405$	−19.3367	−0.960849
$406$	0	0
$407$	−15.0255	−0.744787
$408$	0	0
$409$	−32.7868	−1.62120	−0.810601	−	0.585599i	$-0.800859\pi$
$409$	−32.7868	−1.62120	−0.810601	+	0.585599i	$0.800859\pi$
$410$	0	0
$411$	−0.587284	−0.0289686
$412$	0	0
$413$	0.00903470	0.000444569 0
$414$	0	0
$415$	1.62308	0.0796737
$416$	0	0
$417$	−1.40944	−0.0690204
$418$	0	0
$419$	−2.58084	−0.126082	−0.0630412	−	0.998011i	$-0.520080\pi$
$419$	−2.58084	−0.126082	−0.0630412	+	0.998011i	$0.520080\pi$
$420$	0	0
$421$	32.1994	1.56930	0.784651	−	0.619938i	$-0.212842\pi$
$421$	32.1994	1.56930	0.784651	+	0.619938i	$0.212842\pi$
$422$	0	0
$423$	−7.70757	−0.374755
$424$	0	0
$425$	24.1206	1.17002
$426$	0	0
$427$	−17.1695	−0.830892
$428$	0	0
$429$	−3.28367	−0.158537
$430$	0	0
$431$	−20.4995	−0.987426	−0.493713	−	0.869625i	$-0.664361\pi$
$431$	−20.4995	−0.987426	−0.493713	+	0.869625i	$0.664361\pi$
$432$	0	0
$433$	8.16873	0.392564	0.196282	−	0.980547i	$-0.437113\pi$
$433$	8.16873	0.392564	0.196282	+	0.980547i	$0.437113\pi$
$434$	0	0
$435$	16.1954	0.776510
$436$	0	0
$437$	−21.1499	−1.01174
$438$	0	0
$439$	−17.0604	−0.814249	−0.407125	−	0.913373i	$-0.633469\pi$
$439$	−17.0604	−0.814249	−0.407125	+	0.913373i	$0.633469\pi$
$440$	0	0
$441$	−5.45242	−0.259639
$442$	0	0
$443$	14.6672	0.696857	0.348429	−	0.937335i	$-0.386715\pi$
$443$	14.6672	0.696857	0.348429	+	0.937335i	$0.386715\pi$
$444$	0	0
$445$	24.4681	1.15990
$446$	0	0
$447$	7.21799	0.341399
$448$	0	0
$449$	−21.2796	−1.00425	−0.502124	−	0.864795i	$-0.667448\pi$
$449$	−21.2796	−1.00425	−0.502124	+	0.864795i	$0.667448\pi$
$450$	0	0
$451$	−8.98257	−0.422973
$452$	0	0
$453$	9.57830	0.450028
$454$	0	0
$455$	21.4099	1.00371
$456$	0	0
$457$	−23.0703	−1.07918	−0.539592	−	0.841927i	$-0.681422\pi$
$457$	−23.0703	−1.07918	−0.539592	+	0.841927i	$0.681422\pi$
$458$	0	0
$459$	15.3684	0.717336
$460$	0	0
$461$	0.559977	0.0260807	0.0130404	−	0.999915i	$-0.495849\pi$
$461$	0.559977	0.0260807	0.0130404	+	0.999915i	$0.495849\pi$
$462$	0	0
$463$	−23.1038	−1.07372	−0.536862	−	0.843670i	$-0.680390\pi$
$463$	−23.1038	−1.07372	−0.536862	+	0.843670i	$0.680390\pi$
$464$	0	0
$465$	−1.95284	−0.0905608
$466$	0	0
$467$	−41.3409	−1.91303	−0.956515	−	0.291685i	$-0.905784\pi$
$467$	−41.3409	−1.91303	−0.956515	+	0.291685i	$0.905784\pi$
$468$	0	0
$469$	41.6886	1.92500
$470$	0	0
$471$	−7.99093	−0.368203
$472$	0	0
$473$	10.9994	0.505753
$474$	0	0
$475$	−22.3308	−1.02461
$476$	0	0
$477$	3.38772	0.155113
$478$	0	0
$479$	−19.6439	−0.897552	−0.448776	−	0.893644i	$-0.648140\pi$
$479$	−19.6439	−0.897552	−0.448776	+	0.893644i	$0.648140\pi$
$480$	0	0
$481$	−13.2682	−0.604977
$482$	0	0
$483$	8.58656	0.390702
$484$	0	0
$485$	−42.9447	−1.95002
$486$	0	0
$487$	−16.2834	−0.737871	−0.368935	−	0.929455i	$-0.620278\pi$
$487$	−16.2834	−0.737871	−0.368935	+	0.929455i	$0.620278\pi$
$488$	0	0
$489$	4.38279	0.198197
$490$	0	0
$491$	17.5310	0.791164	0.395582	−	0.918431i	$-0.370543\pi$
$491$	17.5310	0.791164	0.395582	+	0.918431i	$0.370543\pi$
$492$	0	0
$493$	40.6359	1.83015
$494$	0	0
$495$	21.4455	0.963905
$496$	0	0
$497$	−6.17207	−0.276855
$498$	0	0
$499$	−9.28253	−0.415543	−0.207772	−	0.978177i	$-0.566621\pi$
$499$	−9.28253	−0.415543	−0.207772	+	0.978177i	$0.566621\pi$
$500$	0	0
$501$	8.37474	0.374156
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	11.5328	0.513203
$506$	0	0
$507$	4.67187	0.207485
$508$	0	0
$509$	24.1334	1.06970	0.534848	−	0.844948i	$-0.320369\pi$
$509$	24.1334	1.06970	0.534848	+	0.844948i	$0.320369\pi$
$510$	0	0
$511$	23.0856	1.02125
$512$	0	0
$513$	−14.2281	−0.628185
$514$	0	0
$515$	27.7282	1.22185
$516$	0	0
$517$	7.31943	0.321908
$518$	0	0
$519$	−4.40177	−0.193216
$520$	0	0
$521$	−20.6519	−0.904775	−0.452388	−	0.891821i	$-0.649428\pi$
$521$	−20.6519	−0.904775	−0.452388	+	0.891821i	$0.649428\pi$
$522$	0	0
$523$	−35.1612	−1.53749	−0.768745	−	0.639556i	$-0.779118\pi$
$523$	−35.1612	−1.53749	−0.768745	+	0.639556i	$0.779118\pi$
$524$	0	0
$525$	9.06600	0.395673
$526$	0	0
$527$	−4.89988	−0.213442
$528$	0	0
$529$	1.01891	0.0443006
$530$	0	0
$531$	0.00799133	0.000346794 0
$532$	0	0
$533$	−7.93200	−0.343573
$534$	0	0
$535$	56.2996	2.43404
$536$	0	0
$537$	1.67630	0.0723376
$538$	0	0
$539$	5.17785	0.223026
$540$	0	0
$541$	1.90490	0.0818980	0.0409490	−	0.999161i	$-0.486962\pi$
$541$	1.90490	0.0818980	0.0409490	+	0.999161i	$0.486962\pi$
$542$	0	0
$543$	−15.1108	−0.648468
$544$	0	0
$545$	9.24752	0.396120
$546$	0	0
$547$	37.1196	1.58712	0.793559	−	0.608493i	$-0.208226\pi$
$547$	37.1196	1.58712	0.793559	+	0.608493i	$0.208226\pi$
$548$	0	0
$549$	−15.1867	−0.648153
$550$	0	0
$551$	−37.6207	−1.60270
$552$	0	0
$553$	−32.2194	−1.37011
$554$	0	0
$555$	−11.0473	−0.468931
$556$	0	0
$557$	26.6711	1.13009	0.565045	−	0.825060i	$-0.308859\pi$
$557$	26.6711	1.13009	0.565045	+	0.825060i	$0.308859\pi$
$558$	0	0
$559$	9.71295	0.410814
$560$	0	0
$561$	−6.85997	−0.289628
$562$	0	0
$563$	15.4899	0.652823	0.326411	−	0.945228i	$-0.394161\pi$
$563$	15.4899	0.652823	0.326411	+	0.945228i	$0.394161\pi$
$564$	0	0
$565$	6.84550	0.287993
$566$	0	0
$567$	−18.2360	−0.765840
$568$	0	0
$569$	−23.3797	−0.980128	−0.490064	−	0.871686i	$-0.663027\pi$
$569$	−23.3797	−0.980128	−0.490064	+	0.871686i	$0.663027\pi$
$570$	0	0
$571$	−42.9335	−1.79671	−0.898356	−	0.439267i	$-0.855238\pi$
$571$	−42.9335	−1.79671	−0.898356	+	0.439267i	$0.855238\pi$
$572$	0	0
$573$	14.4924	0.605427
$574$	0	0
$575$	25.3600	1.05759
$576$	0	0
$577$	31.5253	1.31241	0.656207	−	0.754581i	$-0.272160\pi$
$577$	31.5253	1.31241	0.656207	+	0.754581i	$0.272160\pi$
$578$	0	0
$579$	0.441435	0.0183454
$580$	0	0
$581$	1.53068	0.0635035
$582$	0	0
$583$	−3.21712	−0.133240
$584$	0	0
$585$	18.9373	0.782963
$586$	0	0
$587$	−12.9156	−0.533084	−0.266542	−	0.963823i	$-0.585881\pi$
$587$	−12.9156	−0.533084	−0.266542	+	0.963823i	$0.585881\pi$
$588$	0	0
$589$	4.53631	0.186915
$590$	0	0
$591$	−0.447650	−0.0184138
$592$	0	0
$593$	11.2717	0.462873	0.231436	−	0.972850i	$-0.425657\pi$
$593$	11.2717	0.462873	0.231436	+	0.972850i	$0.425657\pi$
$594$	0	0
$595$	44.7277	1.83366
$596$	0	0
$597$	4.73277	0.193699
$598$	0	0
$599$	−9.10043	−0.371833	−0.185917	−	0.982566i	$-0.559525\pi$
$599$	−9.10043	−0.371833	−0.185917	+	0.982566i	$0.559525\pi$
$600$	0	0
$601$	−5.72284	−0.233440	−0.116720	−	0.993165i	$-0.537238\pi$
$601$	−5.72284	−0.233440	−0.116720	+	0.993165i	$0.537238\pi$
$602$	0	0
$603$	36.8742	1.50163
$604$	0	0
$605$	14.7218	0.598524
$606$	0	0
$607$	34.4033	1.39639	0.698194	−	0.715909i	$-0.253987\pi$
$607$	34.4033	1.39639	0.698194	+	0.715909i	$0.253987\pi$
$608$	0	0
$609$	15.2735	0.618913
$610$	0	0
$611$	6.46338	0.261480
$612$	0	0
$613$	10.8110	0.436653	0.218326	−	0.975876i	$-0.429940\pi$
$613$	10.8110	0.436653	0.218326	+	0.975876i	$0.429940\pi$
$614$	0	0
$615$	−6.60430	−0.266311
$616$	0	0
$617$	−15.8098	−0.636478	−0.318239	−	0.948010i	$-0.603092\pi$
$617$	−15.8098	−0.636478	−0.318239	+	0.948010i	$0.603092\pi$
$618$	0	0
$619$	−31.4072	−1.26236	−0.631182	−	0.775635i	$-0.717430\pi$
$619$	−31.4072	−1.26236	−0.631182	+	0.775635i	$0.717430\pi$
$620$	0	0
$621$	16.1581	0.648404
$622$	0	0
$623$	23.0753	0.924492
$624$	0	0
$625$	−24.0968	−0.963871
$626$	0	0
$627$	6.35096	0.253633
$628$	0	0
$629$	−27.7188	−1.10522
$630$	0	0
$631$	−23.5995	−0.939480	−0.469740	−	0.882805i	$-0.655652\pi$
$631$	−23.5995	−0.939480	−0.469740	+	0.882805i	$0.655652\pi$
$632$	0	0
$633$	13.2573	0.526931
$634$	0	0
$635$	−0.130700	−0.00518666
$636$	0	0
$637$	4.57227	0.181160
$638$	0	0
$639$	−5.45929	−0.215966
$640$	0	0
$641$	33.7609	1.33348	0.666738	−	0.745292i	$-0.267690\pi$
$641$	33.7609	1.33348	0.666738	+	0.745292i	$0.267690\pi$
$642$	0	0
$643$	34.5312	1.36178	0.680889	−	0.732386i	$-0.261594\pi$
$643$	34.5312	1.36178	0.680889	+	0.732386i	$0.261594\pi$
$644$	0	0
$645$	8.08714	0.318431
$646$	0	0
$647$	4.49031	0.176532	0.0882662	−	0.996097i	$-0.471867\pi$
$647$	4.49031	0.176532	0.0882662	+	0.996097i	$0.471867\pi$
$648$	0	0
$649$	−0.00758890	−0.000297890 0
$650$	0	0
$651$	−1.84168	−0.0721810
$652$	0	0
$653$	14.7592	0.577570	0.288785	−	0.957394i	$-0.406749\pi$
$653$	14.7592	0.577570	0.288785	+	0.957394i	$0.406749\pi$
$654$	0	0
$655$	−30.9856	−1.21071
$656$	0	0
$657$	20.4195	0.796642
$658$	0	0
$659$	−19.6523	−0.765543	−0.382772	−	0.923843i	$-0.625030\pi$
$659$	−19.6523	−0.765543	−0.382772	+	0.923843i	$0.625030\pi$
$660$	0	0
$661$	4.39338	0.170883	0.0854413	−	0.996343i	$-0.472770\pi$
$661$	4.39338	0.170883	0.0854413	+	0.996343i	$0.472770\pi$
$662$	0	0
$663$	−6.05765	−0.235260
$664$	0	0
$665$	−41.4089	−1.60577
$666$	0	0
$667$	42.7240	1.65428
$668$	0	0
$669$	3.90940	0.151146
$670$	0	0
$671$	14.4219	0.556752
$672$	0	0
$673$	0.0346316	0.00133495	0.000667475	−	1.00000i	$-0.499788\pi$
$673$	0.0346316	0.00133495	0.000667475		1.00000i	$0.499788\pi$
$674$	0	0
$675$	17.0603	0.656653
$676$	0	0
$677$	−36.9750	−1.42107	−0.710533	−	0.703664i	$-0.751546\pi$
$677$	−36.9750	−1.42107	−0.710533	+	0.703664i	$0.751546\pi$
$678$	0	0
$679$	−40.5002	−1.55425
$680$	0	0
$681$	9.05418	0.346957
$682$	0	0
$683$	45.9788	1.75933	0.879666	−	0.475592i	$-0.157766\pi$
$683$	45.9788	1.75933	0.879666	+	0.475592i	$0.157766\pi$
$684$	0	0
$685$	−3.21638	−0.122892
$686$	0	0
$687$	−10.7078	−0.408528
$688$	0	0
$689$	−2.84086	−0.108228
$690$	0	0
$691$	−42.3567	−1.61133	−0.805663	−	0.592375i	$-0.798191\pi$
$691$	−42.3567	−1.61133	−0.805663	+	0.592375i	$0.798191\pi$
$692$	0	0
$693$	20.2248	0.768276
$694$	0	0
$695$	−7.71907	−0.292801
$696$	0	0
$697$	−16.5709	−0.627667
$698$	0	0
$699$	2.44784	0.0925858
$700$	0	0
$701$	−39.2931	−1.48408	−0.742040	−	0.670355i	$-0.766142\pi$
$701$	−39.2931	−1.48408	−0.742040	+	0.670355i	$0.766142\pi$
$702$	0	0
$703$	25.6621	0.967863
$704$	0	0
$705$	5.38150	0.202679
$706$	0	0
$707$	10.8763	0.409046
$708$	0	0
$709$	−1.82289	−0.0684602	−0.0342301	−	0.999414i	$-0.510898\pi$
$709$	−1.82289	−0.0684602	−0.0342301	+	0.999414i	$0.510898\pi$
$710$	0	0
$711$	−28.4985	−1.06878
$712$	0	0
$713$	−5.15166	−0.192931
$714$	0	0
$715$	−17.9837	−0.672552
$716$	0	0
$717$	−1.59401	−0.0595295
$718$	0	0
$719$	−36.6790	−1.36790	−0.683949	−	0.729530i	$-0.739739\pi$
$719$	−36.6790	−1.36790	−0.683949	+	0.729530i	$0.739739\pi$
$720$	0	0
$721$	26.1498	0.973869
$722$	0	0
$723$	−11.4525	−0.425924
$724$	0	0
$725$	45.1096	1.67533
$726$	0	0
$727$	−37.1179	−1.37663	−0.688313	−	0.725414i	$-0.741648\pi$
$727$	−37.1179	−1.37663	−0.688313	+	0.725414i	$0.741648\pi$
$728$	0	0
$729$	−10.3693	−0.384049
$730$	0	0
$731$	20.2915	0.750508
$732$	0	0
$733$	−1.90063	−0.0702013	−0.0351006	−	0.999384i	$-0.511175\pi$
$733$	−1.90063	−0.0702013	−0.0351006	+	0.999384i	$0.511175\pi$
$734$	0	0
$735$	3.80694	0.140421
$736$	0	0
$737$	−35.0173	−1.28988
$738$	0	0
$739$	27.6254	1.01622	0.508108	−	0.861293i	$-0.330345\pi$
$739$	27.6254	1.01622	0.508108	+	0.861293i	$0.330345\pi$
$740$	0	0
$741$	5.60817	0.206021
$742$	0	0
$743$	22.4166	0.822386	0.411193	−	0.911548i	$-0.365112\pi$
$743$	22.4166	0.822386	0.411193	+	0.911548i	$0.365112\pi$
$744$	0	0
$745$	39.5308	1.44830
$746$	0	0
$747$	1.35391	0.0495371
$748$	0	0
$749$	53.0948	1.94004
$750$	0	0
$751$	−14.6077	−0.533042	−0.266521	−	0.963829i	$-0.585874\pi$
$751$	−14.6077	−0.533042	−0.266521	+	0.963829i	$0.585874\pi$
$752$	0	0
$753$	8.92153	0.325119
$754$	0	0
$755$	52.4576	1.90913
$756$	0	0
$757$	−49.2376	−1.78957	−0.894785	−	0.446497i	$-0.852671\pi$
$757$	−49.2376	−1.78957	−0.894785	+	0.446497i	$0.852671\pi$
$758$	0	0
$759$	−7.21247	−0.261796
$760$	0	0
$761$	−42.9688	−1.55762	−0.778809	−	0.627261i	$-0.784176\pi$
$761$	−42.9688	−1.55762	−0.778809	+	0.627261i	$0.784176\pi$
$762$	0	0
$763$	8.72112	0.315726
$764$	0	0
$765$	39.5623	1.43038
$766$	0	0
$767$	−0.00670133	−0.000241971 0
$768$	0	0
$769$	41.9922	1.51428	0.757138	−	0.653255i	$-0.226597\pi$
$769$	41.9922	1.51428	0.757138	+	0.653255i	$0.226597\pi$
$770$	0	0
$771$	5.70281	0.205382
$772$	0	0
$773$	2.42501	0.0872216	0.0436108	−	0.999049i	$-0.486114\pi$
$773$	2.42501	0.0872216	0.0436108	+	0.999049i	$0.486114\pi$
$774$	0	0
$775$	−5.43931	−0.195386
$776$	0	0
$777$	−10.4184	−0.373759
$778$	0	0
$779$	15.3413	0.549660
$780$	0	0
$781$	5.18437	0.185511
$782$	0	0
$783$	28.7416	1.02714
$784$	0	0
$785$	−43.7640	−1.56200
$786$	0	0
$787$	22.2904	0.794567	0.397284	−	0.917696i	$-0.369953\pi$
$787$	22.2904	0.794567	0.397284	+	0.917696i	$0.369953\pi$
$788$	0	0
$789$	−7.77896	−0.276938
$790$	0	0
$791$	6.45583	0.229543
$792$	0	0
$793$	12.7352	0.452240
$794$	0	0
$795$	−2.36534	−0.0838900
$796$	0	0
$797$	29.9053	1.05930	0.529650	−	0.848216i	$-0.322323\pi$
$797$	29.9053	1.05930	0.529650	+	0.848216i	$0.322323\pi$
$798$	0	0
$799$	13.5028	0.477693
$800$	0	0
$801$	20.4104	0.721167
$802$	0	0
$803$	−19.3912	−0.684302
$804$	0	0
$805$	47.0261	1.65745
$806$	0	0
$807$	−15.7298	−0.553714
$808$	0	0
$809$	−16.2950	−0.572903	−0.286451	−	0.958095i	$-0.592476\pi$
$809$	−16.2950	−0.572903	−0.286451	+	0.958095i	$0.592476\pi$
$810$	0	0
$811$	−20.4980	−0.719781	−0.359890	−	0.932995i	$-0.617186\pi$
$811$	−20.4980	−0.719781	−0.359890	+	0.932995i	$0.617186\pi$
$812$	0	0
$813$	−5.10167	−0.178923
$814$	0	0
$815$	24.0033	0.840798
$816$	0	0
$817$	−18.7859	−0.657234
$818$	0	0
$819$	17.8594	0.624057
$820$	0	0
$821$	6.48507	0.226331	0.113165	−	0.993576i	$-0.463901\pi$
$821$	6.48507	0.226331	0.113165	+	0.993576i	$0.463901\pi$
$822$	0	0
$823$	−17.4490	−0.608235	−0.304117	−	0.952635i	$-0.598361\pi$
$823$	−17.4490	−0.608235	−0.304117	+	0.952635i	$0.598361\pi$
$824$	0	0
$825$	−7.61519	−0.265127
$826$	0	0
$827$	−16.9685	−0.590053	−0.295026	−	0.955489i	$-0.595328\pi$
$827$	−16.9685	−0.590053	−0.295026	+	0.955489i	$0.595328\pi$
$828$	0	0
$829$	−7.05878	−0.245162	−0.122581	−	0.992459i	$-0.539117\pi$
$829$	−7.05878	−0.245162	−0.122581	+	0.992459i	$0.539117\pi$
$830$	0	0
$831$	−9.20563	−0.319340
$832$	0	0
$833$	9.55200	0.330957
$834$	0	0
$835$	45.8660	1.58726
$836$	0	0
$837$	−3.46566	−0.119791
$838$	0	0
$839$	−18.1789	−0.627605	−0.313802	−	0.949488i	$-0.601603\pi$
$839$	−18.1789	−0.627605	−0.313802	+	0.949488i	$0.601603\pi$
$840$	0	0
$841$	46.9961	1.62056
$842$	0	0
$843$	−3.22952	−0.111231
$844$	0	0
$845$	25.5865	0.880201
$846$	0	0
$847$	13.8837	0.477051
$848$	0	0
$849$	−12.0008	−0.411865
$850$	0	0
$851$	−29.1431	−0.999014
$852$	0	0
$853$	4.13293	0.141509	0.0707543	−	0.997494i	$-0.477459\pi$
$853$	4.13293	0.141509	0.0707543	+	0.997494i	$0.477459\pi$
$854$	0	0
$855$	−36.6268	−1.25261
$856$	0	0
$857$	56.1670	1.91863	0.959314	−	0.282341i	$-0.0911109\pi$
$857$	56.1670	1.91863	0.959314	+	0.282341i	$0.0911109\pi$
$858$	0	0
$859$	8.74064	0.298227	0.149113	−	0.988820i	$-0.452358\pi$
$859$	8.74064	0.298227	0.149113	+	0.988820i	$0.452358\pi$
$860$	0	0
$861$	−6.22836	−0.212262
$862$	0	0
$863$	−6.53979	−0.222617	−0.111308	−	0.993786i	$-0.535504\pi$
$863$	−6.53979	−0.222617	−0.111308	+	0.993786i	$0.535504\pi$
$864$	0	0
$865$	−24.1072	−0.819670
$866$	0	0
$867$	−2.75396	−0.0935293
$868$	0	0
$869$	27.0634	0.918062
$870$	0	0
$871$	−30.9218	−1.04774
$872$	0	0
$873$	−35.8230	−1.21242
$874$	0	0
$875$	1.67491	0.0566222
$876$	0	0
$877$	15.0109	0.506884	0.253442	−	0.967351i	$-0.418437\pi$
$877$	15.0109	0.506884	0.253442	+	0.967351i	$0.418437\pi$
$878$	0	0
$879$	3.01442	0.101674
$880$	0	0
$881$	−21.0005	−0.707526	−0.353763	−	0.935335i	$-0.615098\pi$
$881$	−21.0005	−0.707526	−0.353763	+	0.935335i	$0.615098\pi$
$882$	0	0
$883$	27.9658	0.941125	0.470562	−	0.882367i	$-0.344051\pi$
$883$	27.9658	0.941125	0.470562	+	0.882367i	$0.344051\pi$
$884$	0	0
$885$	−0.00557962	−0.000187557 0
$886$	0	0
$887$	31.0929	1.04400	0.521998	−	0.852946i	$-0.325187\pi$
$887$	31.0929	1.04400	0.521998	+	0.852946i	$0.325187\pi$
$888$	0	0
$889$	−0.123260	−0.00413400
$890$	0	0
$891$	15.3177	0.513163
$892$	0	0
$893$	−12.5009	−0.418325
$894$	0	0
$895$	9.18060	0.306874
$896$	0	0
$897$	−6.36893	−0.212652
$898$	0	0
$899$	−9.16361	−0.305624
$900$	0	0
$901$	−5.93489	−0.197720
$902$	0	0
$903$	7.62679	0.253804
$904$	0	0
$905$	−82.7576	−2.75096
$906$	0	0
$907$	−12.0020	−0.398521	−0.199260	−	0.979947i	$-0.563854\pi$
$907$	−12.0020	−0.398521	−0.199260	+	0.979947i	$0.563854\pi$
$908$	0	0
$909$	9.62026	0.319084
$910$	0	0
$911$	−26.1594	−0.866699	−0.433350	−	0.901226i	$-0.642668\pi$
$911$	−26.1594	−0.866699	−0.433350	+	0.901226i	$0.642668\pi$
$912$	0	0
$913$	−1.28573	−0.0425515
$914$	0	0
$915$	10.6035	0.350541
$916$	0	0
$917$	−29.2218	−0.964990
$918$	0	0
$919$	33.1149	1.09236	0.546179	−	0.837668i	$-0.316082\pi$
$919$	33.1149	1.09236	0.546179	+	0.837668i	$0.316082\pi$
$920$	0	0
$921$	7.47175	0.246203
$922$	0	0
$923$	4.57802	0.150687
$924$	0	0
$925$	−30.7704	−1.01172
$926$	0	0
$927$	23.1299	0.759684
$928$	0	0
$929$	−55.3936	−1.81740	−0.908702	−	0.417445i	$-0.862926\pi$
$929$	−55.3936	−1.81740	−0.908702	+	0.417445i	$0.862926\pi$
$930$	0	0
$931$	−8.84324	−0.289826
$932$	0	0
$933$	1.53349	0.0502042
$934$	0	0
$935$	−37.5700	−1.22867
$936$	0	0
$937$	26.3384	0.860437	0.430218	−	0.902725i	$-0.358437\pi$
$937$	26.3384	0.860437	0.430218	+	0.902725i	$0.358437\pi$
$938$	0	0
$939$	−14.9024	−0.486323
$940$	0	0
$941$	14.0644	0.458486	0.229243	−	0.973369i	$-0.426375\pi$
$941$	14.0644	0.458486	0.229243	+	0.973369i	$0.426375\pi$
$942$	0	0
$943$	−17.4224	−0.567351
$944$	0	0
$945$	31.6357	1.02911
$946$	0	0
$947$	21.5787	0.701215	0.350607	−	0.936523i	$-0.385975\pi$
$947$	21.5787	0.701215	0.350607	+	0.936523i	$0.385975\pi$
$948$	0	0
$949$	−17.1233	−0.555846
$950$	0	0
$951$	20.0280	0.649453
$952$	0	0
$953$	16.2914	0.527730	0.263865	−	0.964560i	$-0.415003\pi$
$953$	16.2914	0.527730	0.263865	+	0.964560i	$0.415003\pi$
$954$	0	0
$955$	79.3705	2.56837
$956$	0	0
$957$	−12.8293	−0.414713
$958$	0	0
$959$	−3.03330	−0.0979502
$960$	0	0
$961$	−29.8951	−0.964357
$962$	0	0
$963$	46.9631	1.51337
$964$	0	0
$965$	2.41761	0.0778257
$966$	0	0
$967$	−25.2548	−0.812138	−0.406069	−	0.913842i	$-0.633101\pi$
$967$	−25.2548	−0.812138	−0.406069	+	0.913842i	$0.633101\pi$
$968$	0	0
$969$	11.7161	0.376376
$970$	0	0
$971$	−20.2860	−0.651009	−0.325504	−	0.945541i	$-0.605534\pi$
$971$	−20.2860	−0.651009	−0.325504	+	0.945541i	$0.605534\pi$
$972$	0	0
$973$	−7.27967	−0.233376
$974$	0	0
$975$	−6.72454	−0.215358
$976$	0	0
$977$	28.8847	0.924103	0.462052	−	0.886853i	$-0.347113\pi$
$977$	28.8847	0.924103	0.462052	+	0.886853i	$0.347113\pi$
$978$	0	0
$979$	−19.3826	−0.619470
$980$	0	0
$981$	7.71396	0.246288
$982$	0	0
$983$	−4.34364	−0.138541	−0.0692703	−	0.997598i	$-0.522067\pi$
$983$	−4.34364	−0.138541	−0.0692703	+	0.997598i	$0.522067\pi$
$984$	0	0
$985$	−2.45165	−0.0781160
$986$	0	0
$987$	5.07517	0.161544
$988$	0	0
$989$	21.3342	0.678388
$990$	0	0
$991$	35.2068	1.11838	0.559191	−	0.829039i	$-0.311112\pi$
$991$	35.2068	1.11838	0.559191	+	0.829039i	$0.311112\pi$
$992$	0	0
$993$	−0.685607	−0.0217571
$994$	0	0
$995$	25.9200	0.821718
$996$	0	0
$997$	−49.1895	−1.55785	−0.778924	−	0.627118i	$-0.784234\pi$
$997$	−49.1895	−1.55785	−0.778924	+	0.627118i	$0.784234\pi$
$998$	0	0
$999$	−19.6053	−0.620285

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.14	✓	29
4.3	odd	2		8048.2.a.w.1.16		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.14	✓	29	1.1	even	1	trivial
8048.2.a.w.1.16		29	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.582422 q^{3} -3.18976 q^{5} -3.00819 q^{7} -2.66078 q^{9} +O(q^{10})\)	\(q-0.582422 q^{3} -3.18976 q^{5} -3.00819 q^{7} -2.66078 q^{9} +2.52679 q^{11} +2.23127 q^{13} +1.85779 q^{15} +4.66138 q^{17} -4.31551 q^{19} +1.75203 q^{21} +4.90091 q^{23} +5.17455 q^{25} +3.29697 q^{27} +8.71758 q^{29} -1.05116 q^{31} -1.47166 q^{33} +9.59538 q^{35} -5.94648 q^{37} -1.29954 q^{39} -3.55493 q^{41} +4.35311 q^{43} +8.48726 q^{45} +2.89673 q^{47} +2.04918 q^{49} -2.71489 q^{51} -1.27320 q^{53} -8.05985 q^{55} +2.51345 q^{57} -0.00300337 q^{59} +5.70760 q^{61} +8.00413 q^{63} -7.11720 q^{65} -13.8584 q^{67} -2.85440 q^{69} +2.05176 q^{71} -7.67426 q^{73} -3.01378 q^{75} -7.60106 q^{77} +10.7106 q^{79} +6.06212 q^{81} -0.508840 q^{83} -14.8687 q^{85} -5.07731 q^{87} -7.67083 q^{89} -6.71207 q^{91} +0.612222 q^{93} +13.7654 q^{95} +13.4633 q^{97} -6.72325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

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Modular forms

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