LMFDB - Embedded newform 4023.2.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4023,2,Mod(1,4023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4023, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4023 = 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4023.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:	$32.1238167332$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.305611 q^{2} -1.90660 q^{4} +1.63568 q^{5} -3.53407 q^{7} +1.19390 q^{8} +O(q^{10})$	\(q-0.305611 q^{2} -1.90660 q^{4} +1.63568 q^{5} -3.53407 q^{7} +1.19390 q^{8} -0.499884 q^{10} +3.40800 q^{11} -1.61987 q^{13} +1.08005 q^{14} +3.44833 q^{16} +2.63124 q^{17} -2.77340 q^{19} -3.11860 q^{20} -1.04152 q^{22} +4.69302 q^{23} -2.32454 q^{25} +0.495052 q^{26} +6.73806 q^{28} -5.63940 q^{29} -6.83543 q^{31} -3.44165 q^{32} -0.804137 q^{34} -5.78062 q^{35} -4.59407 q^{37} +0.847583 q^{38} +1.95285 q^{40} +9.48524 q^{41} -12.3879 q^{43} -6.49770 q^{44} -1.43424 q^{46} +12.4560 q^{47} +5.48963 q^{49} +0.710404 q^{50} +3.08846 q^{52} +2.81468 q^{53} +5.57441 q^{55} -4.21933 q^{56} +1.72347 q^{58} +11.9950 q^{59} +8.73342 q^{61} +2.08898 q^{62} -5.84486 q^{64} -2.64960 q^{65} +10.9026 q^{67} -5.01672 q^{68} +1.76662 q^{70} -5.04883 q^{71} +9.96534 q^{73} +1.40400 q^{74} +5.28777 q^{76} -12.0441 q^{77} +4.47523 q^{79} +5.64039 q^{80} -2.89880 q^{82} -16.4364 q^{83} +4.30388 q^{85} +3.78587 q^{86} +4.06882 q^{88} -13.3055 q^{89} +5.72475 q^{91} -8.94772 q^{92} -3.80668 q^{94} -4.53641 q^{95} -15.6727 q^{97} -1.67769 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

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 $

$2$	−0.305611	−0.216100	−0.108050	−	0.994145i	$-0.534461\pi$
$2$	−0.305611	−0.216100	−0.108050	+	0.994145i	$0.534461\pi$
$3$	0	0
$3$	0	0
$4$	−1.90660	−0.953301
$5$	1.63568	0.731500	0.365750	−	0.930713i	$-0.380812\pi$
$5$	1.63568	0.731500	0.365750	+	0.930713i	$0.380812\pi$
$6$	0	0
$7$	−3.53407	−1.33575	−0.667876	−	0.744273i	$-0.732796\pi$
$7$	−3.53407	−1.33575	−0.667876	+	0.744273i	$0.732796\pi$
$8$	1.19390	0.422108
$9$	0	0
$10$	−0.499884	−0.158077
$11$	3.40800	1.02755	0.513775	−	0.857925i	$-0.328246\pi$
$11$	3.40800	1.02755	0.513775	+	0.857925i	$0.328246\pi$
$12$	0	0
$13$	−1.61987	−0.449272	−0.224636	−	0.974443i	$-0.572119\pi$
$13$	−1.61987	−0.449272	−0.224636	+	0.974443i	$0.572119\pi$
$14$	1.08005	0.288656
$15$	0	0
$16$	3.44833	0.862083
$17$	2.63124	0.638169	0.319085	−	0.947726i	$-0.396625\pi$
$17$	2.63124	0.638169	0.319085	+	0.947726i	$0.396625\pi$
$18$	0	0
$19$	−2.77340	−0.636262	−0.318131	−	0.948047i	$-0.603055\pi$
$19$	−2.77340	−0.636262	−0.318131	+	0.948047i	$0.603055\pi$
$20$	−3.11860	−0.697340
$21$	0	0
$22$	−1.04152	−0.222054
$23$	4.69302	0.978563	0.489281	−	0.872126i	$-0.337259\pi$
$23$	4.69302	0.978563	0.489281	+	0.872126i	$0.337259\pi$
$24$	0	0
$25$	−2.32454	−0.464907
$26$	0.495052	0.0970877
$27$	0	0
$28$	6.73806	1.27337
$29$	−5.63940	−1.04721	−0.523605	−	0.851961i	$-0.675413\pi$
$29$	−5.63940	−1.04721	−0.523605	+	0.851961i	$0.675413\pi$
$30$	0	0
$31$	−6.83543	−1.22768	−0.613840	−	0.789431i	$-0.710376\pi$
$31$	−6.83543	−1.22768	−0.613840	+	0.789431i	$0.710376\pi$
$32$	−3.44165	−0.608404
$33$	0	0
$34$	−0.804137	−0.137908
$35$	−5.78062	−0.977103
$36$	0	0
$37$	−4.59407	−0.755261	−0.377630	−	0.925956i	$-0.623261\pi$
$37$	−4.59407	−0.755261	−0.377630	+	0.925956i	$0.623261\pi$
$38$	0.847583	0.137496
$39$	0	0
$40$	1.95285	0.308772
$41$	9.48524	1.48135	0.740673	−	0.671866i	$-0.234507\pi$
$41$	9.48524	1.48135	0.740673	+	0.671866i	$0.234507\pi$
$42$	0	0
$43$	−12.3879	−1.88913	−0.944565	−	0.328324i	$-0.893516\pi$
$43$	−12.3879	−1.88913	−0.944565	+	0.328324i	$0.893516\pi$
$44$	−6.49770	−0.979565
$45$	0	0
$46$	−1.43424	−0.211467
$47$	12.4560	1.81689	0.908445	−	0.418005i	$-0.137271\pi$
$47$	12.4560	1.81689	0.908445	+	0.418005i	$0.137271\pi$
$48$	0	0
$49$	5.48963	0.784233
$50$	0.710404	0.100466
$51$	0	0
$52$	3.08846	0.428292
$53$	2.81468	0.386626	0.193313	−	0.981137i	$-0.438077\pi$
$53$	2.81468	0.386626	0.193313	+	0.981137i	$0.438077\pi$
$54$	0	0
$55$	5.57441	0.751654
$56$	−4.21933	−0.563832
$57$	0	0
$58$	1.72347	0.226302
$59$	11.9950	1.56162	0.780809	−	0.624769i	$-0.214807\pi$
$59$	11.9950	1.56162	0.780809	+	0.624769i	$0.214807\pi$
$60$	0	0
$61$	8.73342	1.11820	0.559100	−	0.829100i	$-0.311147\pi$
$61$	8.73342	1.11820	0.559100	+	0.829100i	$0.311147\pi$
$62$	2.08898	0.265301
$63$	0	0
$64$	−5.84486	−0.730607
$65$	−2.64960	−0.328643
$66$	0	0
$67$	10.9026	1.33196	0.665981	−	0.745969i	$-0.268013\pi$
$67$	10.9026	1.33196	0.665981	+	0.745969i	$0.268013\pi$
$68$	−5.01672	−0.608367
$69$	0	0
$70$	1.76662	0.211152
$71$	−5.04883	−0.599186	−0.299593	−	0.954067i	$-0.596851\pi$
$71$	−5.04883	−0.599186	−0.299593	+	0.954067i	$0.596851\pi$
$72$	0	0
$73$	9.96534	1.16636	0.583178	−	0.812345i	$-0.301809\pi$
$73$	9.96534	1.16636	0.583178	+	0.812345i	$0.301809\pi$
$74$	1.40400	0.163212
$75$	0	0
$76$	5.28777	0.606549
$77$	−12.0441	−1.37255
$78$	0	0
$79$	4.47523	0.503503	0.251752	−	0.967792i	$-0.418993\pi$
$79$	4.47523	0.503503	0.251752	+	0.967792i	$0.418993\pi$
$80$	5.64039	0.630614
$81$	0	0
$82$	−2.89880	−0.320119
$83$	−16.4364	−1.80413	−0.902065	−	0.431601i	$-0.857949\pi$
$83$	−16.4364	−1.80413	−0.902065	+	0.431601i	$0.857949\pi$
$84$	0	0
$85$	4.30388	0.466821
$86$	3.78587	0.408241
$87$	0	0
$88$	4.06882	0.433737
$89$	−13.3055	−1.41038	−0.705189	−	0.709019i	$-0.749138\pi$
$89$	−13.3055	−1.41038	−0.705189	+	0.709019i	$0.749138\pi$
$90$	0	0
$91$	5.72475	0.600117
$92$	−8.94772	−0.932865
$93$	0	0
$94$	−3.80668	−0.392629
$95$	−4.53641	−0.465426
$96$	0	0
$97$	−15.6727	−1.59132	−0.795661	−	0.605742i	$-0.792876\pi$
$97$	−15.6727	−1.59132	−0.795661	+	0.605742i	$0.792876\pi$
$98$	−1.67769	−0.169473
$99$	0	0
$100$	4.43196	0.443196
$101$	−6.81158	−0.677777	−0.338889	−	0.940826i	$-0.610051\pi$
$101$	−6.81158	−0.677777	−0.338889	+	0.940826i	$0.610051\pi$
$102$	0	0
$103$	−5.64803	−0.556517	−0.278258	−	0.960506i	$-0.589757\pi$
$103$	−5.64803	−0.556517	−0.278258	+	0.960506i	$0.589757\pi$
$104$	−1.93397	−0.189642
$105$	0	0
$106$	−0.860198	−0.0835498
$107$	0.952294	0.0920617	0.0460309	−	0.998940i	$-0.485343\pi$
$107$	0.952294	0.0920617	0.0460309	+	0.998940i	$0.485343\pi$
$108$	0	0
$109$	−12.1665	−1.16534	−0.582669	−	0.812709i	$-0.697992\pi$
$109$	−12.1665	−1.16534	−0.582669	+	0.812709i	$0.697992\pi$
$110$	−1.70360	−0.162432
$111$	0	0
$112$	−12.1866	−1.15153
$113$	−14.4131	−1.35587	−0.677936	−	0.735121i	$-0.737125\pi$
$113$	−14.4131	−1.35587	−0.677936	+	0.735121i	$0.737125\pi$
$114$	0	0
$115$	7.67630	0.715819
$116$	10.7521	0.998307
$117$	0	0
$118$	−3.66582	−0.337466
$119$	−9.29898	−0.852436
$120$	0	0
$121$	0.614468	0.0558607
$122$	−2.66903	−0.241643
$123$	0	0
$124$	13.0324	1.17035
$125$	−11.9806	−1.07158
$126$	0	0
$127$	8.60930	0.763952	0.381976	−	0.924172i	$-0.375244\pi$
$127$	8.60930	0.763952	0.381976	+	0.924172i	$0.375244\pi$
$128$	8.66956	0.766288
$129$	0	0
$130$	0.809749	0.0710197
$131$	12.4781	1.09021	0.545107	−	0.838366i	$-0.316489\pi$
$131$	12.4781	1.09021	0.545107	+	0.838366i	$0.316489\pi$
$132$	0	0
$133$	9.80139	0.849888
$134$	−3.33195	−0.287837
$135$	0	0
$136$	3.14144	0.269376
$137$	−16.9230	−1.44583	−0.722914	−	0.690938i	$-0.757198\pi$
$137$	−16.9230	−1.44583	−0.722914	+	0.690938i	$0.757198\pi$
$138$	0	0
$139$	−23.0137	−1.95200	−0.976000	−	0.217771i	$-0.930121\pi$
$139$	−23.0137	−1.95200	−0.976000	+	0.217771i	$0.930121\pi$
$140$	11.0213	0.931473
$141$	0	0
$142$	1.54298	0.129484
$143$	−5.52053	−0.461650
$144$	0	0
$145$	−9.22428	−0.766035
$146$	−3.04552	−0.252049
$147$	0	0
$148$	8.75907	0.719991
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	0.110557	0.00899702	0.00449851	−	0.999990i	$-0.498568\pi$
$151$	0.110557	0.00899702	0.00449851	+	0.999990i	$0.498568\pi$
$152$	−3.31117	−0.268571
$153$	0	0
$154$	3.68081	0.296609
$155$	−11.1806	−0.898048
$156$	0	0
$157$	−5.55212	−0.443107	−0.221554	−	0.975148i	$-0.571113\pi$
$157$	−5.55212	−0.443107	−0.221554	+	0.975148i	$0.571113\pi$
$158$	−1.36768	−0.108807
$159$	0	0
$160$	−5.62946	−0.445048
$161$	−16.5855	−1.30712
$162$	0	0
$163$	5.01697	0.392960	0.196480	−	0.980508i	$-0.437049\pi$
$163$	5.01697	0.392960	0.196480	+	0.980508i	$0.437049\pi$
$164$	−18.0846	−1.41217
$165$	0	0
$166$	5.02315	0.389872
$167$	−1.51685	−0.117378	−0.0586888	−	0.998276i	$-0.518692\pi$
$167$	−1.51685	−0.117378	−0.0586888	+	0.998276i	$0.518692\pi$
$168$	0	0
$169$	−10.3760	−0.798154
$170$	−1.31531	−0.100880
$171$	0	0
$172$	23.6187	1.80091
$173$	−4.97703	−0.378397	−0.189198	−	0.981939i	$-0.560589\pi$
$173$	−4.97703	−0.378397	−0.189198	+	0.981939i	$0.560589\pi$
$174$	0	0
$175$	8.21507	0.621001
$176$	11.7519	0.885834
$177$	0	0
$178$	4.06631	0.304782
$179$	11.3993	0.852020	0.426010	−	0.904718i	$-0.359919\pi$
$179$	11.3993	0.852020	0.426010	+	0.904718i	$0.359919\pi$
$180$	0	0
$181$	−6.56349	−0.487860	−0.243930	−	0.969793i	$-0.578437\pi$
$181$	−6.56349	−0.487860	−0.243930	+	0.969793i	$0.578437\pi$
$182$	−1.74955	−0.129685
$183$	0	0
$184$	5.60301	0.413059
$185$	−7.51445	−0.552474
$186$	0	0
$187$	8.96726	0.655751
$188$	−23.7486	−1.73204
$189$	0	0
$190$	1.38638	0.100579
$191$	−8.52455	−0.616815	−0.308407	−	0.951254i	$-0.599796\pi$
$191$	−8.52455	−0.616815	−0.308407	+	0.951254i	$0.599796\pi$
$192$	0	0
$193$	8.44184	0.607657	0.303829	−	0.952727i	$-0.401735\pi$
$193$	8.44184	0.607657	0.303829	+	0.952727i	$0.401735\pi$
$194$	4.78976	0.343884
$195$	0	0
$196$	−10.4665	−0.747610
$197$	15.7509	1.12220	0.561101	−	0.827747i	$-0.310378\pi$
$197$	15.7509	1.12220	0.561101	+	0.827747i	$0.310378\pi$
$198$	0	0
$199$	−12.7755	−0.905629	−0.452814	−	0.891605i	$-0.649580\pi$
$199$	−12.7755	−0.905629	−0.452814	+	0.891605i	$0.649580\pi$
$200$	−2.77527	−0.196241
$201$	0	0
$202$	2.08170	0.146468
$203$	19.9300	1.39881
$204$	0	0
$205$	15.5149	1.08361
$206$	1.72610	0.120263
$207$	0	0
$208$	−5.58587	−0.387310
$209$	−9.45176	−0.653792
$210$	0	0
$211$	−14.6958	−1.01170	−0.505850	−	0.862622i	$-0.668821\pi$
$211$	−14.6958	−1.01170	−0.505850	+	0.862622i	$0.668821\pi$
$212$	−5.36647	−0.368571
$213$	0	0
$214$	−0.291032	−0.0198945
$215$	−20.2626	−1.38190
$216$	0	0
$217$	24.1569	1.63988
$218$	3.71822	0.251829
$219$	0	0
$220$	−10.6282	−0.716552
$221$	−4.26228	−0.286712
$222$	0	0
$223$	−29.2278	−1.95724	−0.978618	−	0.205685i	$-0.934058\pi$
$223$	−29.2278	−1.95724	−0.978618	+	0.205685i	$0.934058\pi$
$224$	12.1630	0.812677
$225$	0	0
$226$	4.40481	0.293004
$227$	−24.8935	−1.65224	−0.826120	−	0.563494i	$-0.809457\pi$
$227$	−24.8935	−1.65224	−0.826120	+	0.563494i	$0.809457\pi$
$228$	0	0
$229$	0.102112	0.00674772	0.00337386	−	0.999994i	$-0.498926\pi$
$229$	0.102112	0.00674772	0.00337386	+	0.999994i	$0.498926\pi$
$230$	−2.34597	−0.154688
$231$	0	0
$232$	−6.73289	−0.442036
$233$	−28.2129	−1.84829	−0.924143	−	0.382046i	$-0.875220\pi$
$233$	−28.2129	−1.84829	−0.924143	+	0.382046i	$0.875220\pi$
$234$	0	0
$235$	20.3740	1.32906
$236$	−22.8697	−1.48869
$237$	0	0
$238$	2.84187	0.184211
$239$	7.85767	0.508271	0.254135	−	0.967169i	$-0.418209\pi$
$239$	7.85767	0.508271	0.254135	+	0.967169i	$0.418209\pi$
$240$	0	0
$241$	−21.2146	−1.36656	−0.683278	−	0.730158i	$-0.739446\pi$
$241$	−21.2146	−1.36656	−0.683278	+	0.730158i	$0.739446\pi$
$242$	−0.187788	−0.0120715
$243$	0	0
$244$	−16.6512	−1.06598
$245$	8.97931	0.573667
$246$	0	0
$247$	4.49256	0.285855
$248$	−8.16083	−0.518213
$249$	0	0
$250$	3.66142	0.231568
$251$	−7.56275	−0.477356	−0.238678	−	0.971099i	$-0.576714\pi$
$251$	−7.56275	−0.477356	−0.238678	+	0.971099i	$0.576714\pi$
$252$	0	0
$253$	15.9938	1.00552
$254$	−2.63110	−0.165090
$255$	0	0
$256$	9.04020	0.565013
$257$	29.0286	1.81075	0.905377	−	0.424609i	$-0.139588\pi$
$257$	29.0286	1.81075	0.905377	+	0.424609i	$0.139588\pi$
$258$	0	0
$259$	16.2358	1.00884
$260$	5.05174	0.313296
$261$	0	0
$262$	−3.81344	−0.235595
$263$	28.5695	1.76167	0.880835	−	0.473423i	$-0.156982\pi$
$263$	28.5695	1.76167	0.880835	+	0.473423i	$0.156982\pi$
$264$	0	0
$265$	4.60393	0.282817
$266$	−2.99542	−0.183661
$267$	0	0
$268$	−20.7869	−1.26976
$269$	−14.1136	−0.860523	−0.430262	−	0.902704i	$-0.641579\pi$
$269$	−14.1136	−0.860523	−0.430262	+	0.902704i	$0.641579\pi$
$270$	0	0
$271$	−1.60205	−0.0973179	−0.0486589	−	0.998815i	$-0.515495\pi$
$271$	−1.60205	−0.0973179	−0.0486589	+	0.998815i	$0.515495\pi$
$272$	9.07339	0.550155
$273$	0	0
$274$	5.17186	0.312443
$275$	−7.92202	−0.477716
$276$	0	0
$277$	−31.2828	−1.87960	−0.939801	−	0.341722i	$-0.888990\pi$
$277$	−31.2828	−1.87960	−0.939801	+	0.341722i	$0.888990\pi$
$278$	7.03326	0.421827
$279$	0	0
$280$	−6.90149	−0.412443
$281$	26.1541	1.56022	0.780111	−	0.625641i	$-0.215162\pi$
$281$	26.1541	1.56022	0.780111	+	0.625641i	$0.215162\pi$
$282$	0	0
$283$	−19.0406	−1.13185	−0.565924	−	0.824457i	$-0.691481\pi$
$283$	−19.0406	−1.13185	−0.565924	+	0.824457i	$0.691481\pi$
$284$	9.62611	0.571205
$285$	0	0
$286$	1.68714	0.0997626
$287$	−33.5215	−1.97871
$288$	0	0
$289$	−10.0766	−0.592740
$290$	2.81905	0.165540
$291$	0	0
$292$	−18.9999	−1.11189
$293$	9.64954	0.563732	0.281866	−	0.959454i	$-0.409047\pi$
$293$	9.64954	0.563732	0.281866	+	0.959454i	$0.409047\pi$
$294$	0	0
$295$	19.6201	1.14232
$296$	−5.48487	−0.318802
$297$	0	0
$298$	0.305611	0.0177036
$299$	−7.60211	−0.439641
$300$	0	0
$301$	43.7795	2.52341
$302$	−0.0337875	−0.00194425
$303$	0	0
$304$	−9.56362	−0.548511
$305$	14.2851	0.817964
$306$	0	0
$307$	17.8112	1.01654	0.508270	−	0.861198i	$-0.330285\pi$
$307$	17.8112	1.01654	0.508270	+	0.861198i	$0.330285\pi$
$308$	22.9633	1.30846
$309$	0	0
$310$	3.41692	0.194068
$311$	24.1349	1.36857	0.684283	−	0.729216i	$-0.260115\pi$
$311$	24.1349	1.36857	0.684283	+	0.729216i	$0.260115\pi$
$312$	0	0
$313$	−16.1597	−0.913399	−0.456699	−	0.889621i	$-0.650969\pi$
$313$	−16.1597	−0.913399	−0.456699	+	0.889621i	$0.650969\pi$
$314$	1.69679	0.0957554
$315$	0	0
$316$	−8.53249	−0.479990
$317$	−4.30766	−0.241942	−0.120971	−	0.992656i	$-0.538601\pi$
$317$	−4.30766	−0.241942	−0.120971	+	0.992656i	$0.538601\pi$
$318$	0	0
$319$	−19.2191	−1.07606
$320$	−9.56035	−0.534440
$321$	0	0
$322$	5.06870	0.282468
$323$	−7.29748	−0.406043
$324$	0	0
$325$	3.76546	0.208870
$326$	−1.53324	−0.0849185
$327$	0	0
$328$	11.3244	0.625288
$329$	−44.0202	−2.42691
$330$	0	0
$331$	8.16566	0.448825	0.224413	−	0.974494i	$-0.427954\pi$
$331$	8.16566	0.448825	0.224413	+	0.974494i	$0.427954\pi$
$332$	31.3377	1.71988
$333$	0	0
$334$	0.463567	0.0253653
$335$	17.8332	0.974331
$336$	0	0
$337$	1.74800	0.0952198	0.0476099	−	0.998866i	$-0.484840\pi$
$337$	1.74800	0.0952198	0.0476099	+	0.998866i	$0.484840\pi$
$338$	3.17103	0.172481
$339$	0	0
$340$	−8.20578	−0.445021
$341$	−23.2951	−1.26150
$342$	0	0
$343$	5.33774	0.288211
$344$	−14.7899	−0.797417
$345$	0	0
$346$	1.52104	0.0817714
$347$	−13.7440	−0.737815	−0.368908	−	0.929466i	$-0.620268\pi$
$347$	−13.7440	−0.737815	−0.368908	+	0.929466i	$0.620268\pi$
$348$	0	0
$349$	27.5051	1.47231	0.736157	−	0.676811i	$-0.236638\pi$
$349$	27.5051	1.47231	0.736157	+	0.676811i	$0.236638\pi$
$350$	−2.51062	−0.134198
$351$	0	0
$352$	−11.7292	−0.625166
$353$	−28.3401	−1.50839	−0.754195	−	0.656650i	$-0.771973\pi$
$353$	−28.3401	−1.50839	−0.754195	+	0.656650i	$0.771973\pi$
$354$	0	0
$355$	−8.25830	−0.438305
$356$	25.3683	1.34451
$357$	0	0
$358$	−3.48374	−0.184121
$359$	−12.6727	−0.668841	−0.334420	−	0.942424i	$-0.608541\pi$
$359$	−12.6727	−0.668841	−0.334420	+	0.942424i	$0.608541\pi$
$360$	0	0
$361$	−11.3082	−0.595170
$362$	2.00588	0.105427
$363$	0	0
$364$	−10.9148	−0.572092
$365$	16.3002	0.853189
$366$	0	0
$367$	−0.224210	−0.0117037	−0.00585183	−	0.999983i	$-0.501863\pi$
$367$	−0.224210	−0.0117037	−0.00585183	+	0.999983i	$0.501863\pi$
$368$	16.1831	0.843603
$369$	0	0
$370$	2.29650	0.119389
$371$	−9.94726	−0.516436
$372$	0	0
$373$	20.4341	1.05804	0.529018	−	0.848610i	$-0.322560\pi$
$373$	20.4341	1.05804	0.529018	+	0.848610i	$0.322560\pi$
$374$	−2.74050	−0.141708
$375$	0	0
$376$	14.8712	0.766924
$377$	9.13513	0.470483
$378$	0	0
$379$	12.7493	0.654886	0.327443	−	0.944871i	$-0.393813\pi$
$379$	12.7493	0.654886	0.327443	+	0.944871i	$0.393813\pi$
$380$	8.64913	0.443691
$381$	0	0
$382$	2.60520	0.133294
$383$	−18.9807	−0.969867	−0.484933	−	0.874551i	$-0.661156\pi$
$383$	−18.9807	−0.969867	−0.484933	+	0.874551i	$0.661156\pi$
$384$	0	0
$385$	−19.7004	−1.00402
$386$	−2.57992	−0.131315
$387$	0	0
$388$	29.8816	1.51701
$389$	−5.72304	−0.290170	−0.145085	−	0.989419i	$-0.546345\pi$
$389$	−5.72304	−0.290170	−0.145085	+	0.989419i	$0.546345\pi$
$390$	0	0
$391$	12.3485	0.624489
$392$	6.55408	0.331031
$393$	0	0
$394$	−4.81364	−0.242508
$395$	7.32007	0.368313
$396$	0	0
$397$	−2.01956	−0.101359	−0.0506795	−	0.998715i	$-0.516139\pi$
$397$	−2.01956	−0.101359	−0.0506795	+	0.998715i	$0.516139\pi$
$398$	3.90433	0.195706
$399$	0	0
$400$	−8.01577	−0.400789
$401$	31.8242	1.58923	0.794613	−	0.607116i	$-0.207674\pi$
$401$	31.8242	1.58923	0.794613	+	0.607116i	$0.207674\pi$
$402$	0	0
$403$	11.0725	0.551563
$404$	12.9870	0.646126
$405$	0	0
$406$	−6.09084	−0.302283
$407$	−15.6566	−0.776069
$408$	0	0
$409$	−31.7115	−1.56803	−0.784017	−	0.620739i	$-0.786833\pi$
$409$	−31.7115	−1.56803	−0.784017	+	0.620739i	$0.786833\pi$
$410$	−4.74152	−0.234167
$411$	0	0
$412$	10.7685	0.530528
$413$	−42.3912	−2.08594
$414$	0	0
$415$	−26.8848	−1.31972
$416$	5.57505	0.273339
$417$	0	0
$418$	2.88856	0.141284
$419$	−11.2750	−0.550819	−0.275410	−	0.961327i	$-0.588814\pi$
$419$	−11.2750	−0.550819	−0.275410	+	0.961327i	$0.588814\pi$
$420$	0	0
$421$	−11.4940	−0.560184	−0.280092	−	0.959973i	$-0.590365\pi$
$421$	−11.4940	−0.560184	−0.280092	+	0.959973i	$0.590365\pi$
$422$	4.49120	0.218628
$423$	0	0
$424$	3.36045	0.163198
$425$	−6.11641	−0.296689
$426$	0	0
$427$	−30.8645	−1.49364
$428$	−1.81565	−0.0877625
$429$	0	0
$430$	6.19249	0.298628
$431$	9.26265	0.446166	0.223083	−	0.974799i	$-0.428388\pi$
$431$	9.26265	0.446166	0.223083	+	0.974799i	$0.428388\pi$
$432$	0	0
$433$	−30.5177	−1.46659	−0.733293	−	0.679913i	$-0.762018\pi$
$433$	−30.5177	−1.46659	−0.733293	+	0.679913i	$0.762018\pi$
$434$	−7.38261	−0.354377
$435$	0	0
$436$	23.1966	1.11092
$437$	−13.0156	−0.622622
$438$	0	0
$439$	0.692973	0.0330738	0.0165369	−	0.999863i	$-0.494736\pi$
$439$	0.692973	0.0330738	0.0165369	+	0.999863i	$0.494736\pi$
$440$	6.65530	0.317279
$441$	0	0
$442$	1.30260	0.0619584
$443$	−8.80697	−0.418432	−0.209216	−	0.977870i	$-0.567091\pi$
$443$	−8.80697	−0.418432	−0.209216	+	0.977870i	$0.567091\pi$
$444$	0	0
$445$	−21.7636	−1.03169
$446$	8.93234	0.422959
$447$	0	0
$448$	20.6561	0.975910
$449$	−15.2076	−0.717693	−0.358846	−	0.933397i	$-0.616830\pi$
$449$	−15.2076	−0.717693	−0.358846	+	0.933397i	$0.616830\pi$
$450$	0	0
$451$	32.3257	1.52216
$452$	27.4801	1.29255
$453$	0	0
$454$	7.60774	0.357049
$455$	9.36388	0.438986
$456$	0	0
$457$	16.2345	0.759418	0.379709	−	0.925106i	$-0.376024\pi$
$457$	16.2345	0.759418	0.379709	+	0.925106i	$0.376024\pi$
$458$	−0.0312065	−0.00145818
$459$	0	0
$460$	−14.6357	−0.682391
$461$	34.7376	1.61789	0.808945	−	0.587884i	$-0.200039\pi$
$461$	34.7376	1.61789	0.808945	+	0.587884i	$0.200039\pi$
$462$	0	0
$463$	−31.4505	−1.46163	−0.730814	−	0.682577i	$-0.760859\pi$
$463$	−31.4505	−1.46163	−0.730814	+	0.682577i	$0.760859\pi$
$464$	−19.4465	−0.902783
$465$	0	0
$466$	8.62217	0.399415
$467$	16.9108	0.782540	0.391270	−	0.920276i	$-0.372036\pi$
$467$	16.9108	0.782540	0.391270	+	0.920276i	$0.372036\pi$
$468$	0	0
$469$	−38.5305	−1.77917
$470$	−6.22653	−0.287209
$471$	0	0
$472$	14.3209	0.659172
$473$	−42.2178	−1.94118
$474$	0	0
$475$	6.44687	0.295803
$476$	17.7294	0.812628
$477$	0	0
$478$	−2.40139	−0.109837
$479$	15.0101	0.685827	0.342914	−	0.939367i	$-0.388586\pi$
$479$	15.0101	0.685827	0.342914	+	0.939367i	$0.388586\pi$
$480$	0	0
$481$	7.44182	0.339318
$482$	6.48344	0.295313
$483$	0	0
$484$	−1.17154	−0.0532520
$485$	−25.6356	−1.16405
$486$	0	0
$487$	−17.5594	−0.795692	−0.397846	−	0.917452i	$-0.630242\pi$
$487$	−17.5594	−0.795692	−0.397846	+	0.917452i	$0.630242\pi$
$488$	10.4268	0.472001
$489$	0	0
$490$	−2.74418	−0.123969
$491$	42.0390	1.89719	0.948597	−	0.316488i	$-0.102504\pi$
$491$	42.0390	1.89719	0.948597	+	0.316488i	$0.102504\pi$
$492$	0	0
$493$	−14.8386	−0.668298
$494$	−1.37298	−0.0617732
$495$	0	0
$496$	−23.5708	−1.05836
$497$	17.8429	0.800364
$498$	0	0
$499$	21.7477	0.973563	0.486781	−	0.873524i	$-0.338171\pi$
$499$	21.7477	0.973563	0.486781	+	0.873524i	$0.338171\pi$
$500$	22.8423	1.02154
$501$	0	0
$502$	2.31126	0.103157
$503$	12.1713	0.542693	0.271347	−	0.962482i	$-0.412531\pi$
$503$	12.1713	0.542693	0.271347	+	0.962482i	$0.412531\pi$
$504$	0	0
$505$	−11.1416	−0.495794
$506$	−4.88789	−0.217293
$507$	0	0
$508$	−16.4145	−0.728276
$509$	22.2123	0.984543	0.492271	−	0.870442i	$-0.336167\pi$
$509$	22.2123	0.984543	0.492271	+	0.870442i	$0.336167\pi$
$510$	0	0
$511$	−35.2182	−1.55796
$512$	−20.1019	−0.888387
$513$	0	0
$514$	−8.87147	−0.391304
$515$	−9.23840	−0.407092
$516$	0	0
$517$	42.4499	1.86695
$518$	−4.96183	−0.218010
$519$	0	0
$520$	−3.16337	−0.138723
$521$	21.6867	0.950110	0.475055	−	0.879956i	$-0.342428\pi$
$521$	21.6867	0.950110	0.475055	+	0.879956i	$0.342428\pi$
$522$	0	0
$523$	−36.9180	−1.61431	−0.807156	−	0.590339i	$-0.798994\pi$
$523$	−36.9180	−1.61431	−0.807156	+	0.590339i	$0.798994\pi$
$524$	−23.7907	−1.03930
$525$	0	0
$526$	−8.73116	−0.380697
$527$	−17.9856	−0.783467
$528$	0	0
$529$	−0.975551	−0.0424152
$530$	−1.40701	−0.0611167
$531$	0	0
$532$	−18.6873	−0.810199
$533$	−15.3649	−0.665528
$534$	0	0
$535$	1.55765	0.0673432
$536$	13.0166	0.562232
$537$	0	0
$538$	4.31329	0.185959
$539$	18.7087	0.805840
$540$	0	0
$541$	22.5914	0.971281	0.485641	−	0.874159i	$-0.338586\pi$
$541$	22.5914	0.971281	0.485641	+	0.874159i	$0.338586\pi$
$542$	0.489606	0.0210304
$543$	0	0
$544$	−9.05581	−0.388265
$545$	−19.9005	−0.852445
$546$	0	0
$547$	−2.27839	−0.0974169	−0.0487084	−	0.998813i	$-0.515511\pi$
$547$	−2.27839	−0.0974169	−0.0487084	+	0.998813i	$0.515511\pi$
$548$	32.2654	1.37831
$549$	0	0
$550$	2.42106	0.103234
$551$	15.6403	0.666301
$552$	0	0
$553$	−15.8158	−0.672555
$554$	9.56038	0.406182
$555$	0	0
$556$	43.8780	1.86084
$557$	−14.7170	−0.623578	−0.311789	−	0.950151i	$-0.600928\pi$
$557$	−14.7170	−0.623578	−0.311789	+	0.950151i	$0.600928\pi$
$558$	0	0
$559$	20.0668	0.848734
$560$	−19.9335	−0.842344
$561$	0	0
$562$	−7.99299	−0.337164
$563$	31.0953	1.31051	0.655256	−	0.755407i	$-0.272561\pi$
$563$	31.0953	1.31051	0.655256	+	0.755407i	$0.272561\pi$
$564$	0	0
$565$	−23.5753	−0.991821
$566$	5.81904	0.244592
$567$	0	0
$568$	−6.02781	−0.252921
$569$	−1.46472	−0.0614043	−0.0307021	−	0.999529i	$-0.509774\pi$
$569$	−1.46472	−0.0614043	−0.0307021	+	0.999529i	$0.509774\pi$
$570$	0	0
$571$	27.0412	1.13164	0.565821	−	0.824528i	$-0.308560\pi$
$571$	27.0412	1.13164	0.565821	+	0.824528i	$0.308560\pi$
$572$	10.5255	0.440092
$573$	0	0
$574$	10.2445	0.427599
$575$	−10.9091	−0.454941
$576$	0	0
$577$	23.9350	0.996426	0.498213	−	0.867055i	$-0.333990\pi$
$577$	23.9350	0.996426	0.498213	+	0.867055i	$0.333990\pi$
$578$	3.07952	0.128091
$579$	0	0
$580$	17.5870	0.730262
$581$	58.0874	2.40987
$582$	0	0
$583$	9.59243	0.397278
$584$	11.8976	0.492328
$585$	0	0
$586$	−2.94901	−0.121822
$587$	−39.8918	−1.64651	−0.823256	−	0.567670i	$-0.807845\pi$
$587$	−39.8918	−1.64651	−0.823256	+	0.567670i	$0.807845\pi$
$588$	0	0
$589$	18.9574	0.781126
$590$	−5.99612	−0.246856
$591$	0	0
$592$	−15.8419	−0.651098
$593$	−34.8118	−1.42955	−0.714774	−	0.699356i	$-0.753470\pi$
$593$	−34.8118	−1.42955	−0.714774	+	0.699356i	$0.753470\pi$
$594$	0	0
$595$	−15.2102	−0.623557
$596$	1.90660	0.0780974
$597$	0	0
$598$	2.32329	0.0950064
$599$	10.1936	0.416498	0.208249	−	0.978076i	$-0.433224\pi$
$599$	10.1936	0.416498	0.208249	+	0.978076i	$0.433224\pi$
$600$	0	0
$601$	−10.8285	−0.441702	−0.220851	−	0.975308i	$-0.570883\pi$
$601$	−10.8285	−0.441702	−0.220851	+	0.975308i	$0.570883\pi$
$602$	−13.3795	−0.545308
$603$	0	0
$604$	−0.210789	−0.00857687
$605$	1.00508	0.0408621
$606$	0	0
$607$	19.9228	0.808642	0.404321	−	0.914617i	$-0.367508\pi$
$607$	19.9228	0.808642	0.404321	+	0.914617i	$0.367508\pi$
$608$	9.54509	0.387105
$609$	0	0
$610$	−4.36570	−0.176762
$611$	−20.1771	−0.816278
$612$	0	0
$613$	11.2848	0.455787	0.227893	−	0.973686i	$-0.426816\pi$
$613$	11.2848	0.455787	0.227893	+	0.973686i	$0.426816\pi$
$614$	−5.44331	−0.219674
$615$	0	0
$616$	−14.3795	−0.579366
$617$	−38.0300	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$617$	−38.0300	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$618$	0	0
$619$	28.0309	1.12666	0.563328	−	0.826233i	$-0.309521\pi$
$619$	28.0309	1.12666	0.563328	+	0.826233i	$0.309521\pi$
$620$	21.3170	0.856110
$621$	0	0
$622$	−7.37591	−0.295747
$623$	47.0225	1.88392
$624$	0	0
$625$	−7.97386	−0.318954
$626$	4.93858	0.197385
$627$	0	0
$628$	10.5857	0.422414
$629$	−12.0881	−0.481984
$630$	0	0
$631$	−3.58906	−0.142878	−0.0714390	−	0.997445i	$-0.522759\pi$
$631$	−3.58906	−0.142878	−0.0714390	+	0.997445i	$0.522759\pi$
$632$	5.34299	0.212533
$633$	0	0
$634$	1.31647	0.0522836
$635$	14.0821	0.558831
$636$	0	0
$637$	−8.89252	−0.352334
$638$	5.87357	0.232537
$639$	0	0
$640$	14.1807	0.560540
$641$	−17.6313	−0.696393	−0.348197	−	0.937422i	$-0.613206\pi$
$641$	−17.6313	−0.696393	−0.348197	+	0.937422i	$0.613206\pi$
$642$	0	0
$643$	9.61672	0.379246	0.189623	−	0.981857i	$-0.439273\pi$
$643$	9.61672	0.379246	0.189623	+	0.981857i	$0.439273\pi$
$644$	31.6219	1.24608
$645$	0	0
$646$	2.23019	0.0877458
$647$	43.9383	1.72739	0.863697	−	0.504012i	$-0.168143\pi$
$647$	43.9383	1.72739	0.863697	+	0.504012i	$0.168143\pi$
$648$	0	0
$649$	40.8790	1.60464
$650$	−1.15077	−0.0451368
$651$	0	0
$652$	−9.56537	−0.374609
$653$	7.96035	0.311513	0.155756	−	0.987796i	$-0.450219\pi$
$653$	7.96035	0.311513	0.155756	+	0.987796i	$0.450219\pi$
$654$	0	0
$655$	20.4102	0.797492
$656$	32.7083	1.27704
$657$	0	0
$658$	13.4531	0.524456
$659$	8.96854	0.349365	0.174682	−	0.984625i	$-0.444110\pi$
$659$	8.96854	0.349365	0.174682	+	0.984625i	$0.444110\pi$
$660$	0	0
$661$	−37.2703	−1.44965	−0.724824	−	0.688934i	$-0.758079\pi$
$661$	−37.2703	−1.44965	−0.724824	+	0.688934i	$0.758079\pi$
$662$	−2.49552	−0.0969910
$663$	0	0
$664$	−19.6234	−0.761538
$665$	16.0320	0.621694
$666$	0	0
$667$	−26.4658	−1.02476
$668$	2.89203	0.111896
$669$	0	0
$670$	−5.45002	−0.210553
$671$	29.7635	1.14901
$672$	0	0
$673$	7.24605	0.279315	0.139657	−	0.990200i	$-0.455400\pi$
$673$	7.24605	0.279315	0.139657	+	0.990200i	$0.455400\pi$
$674$	−0.534210	−0.0205770
$675$	0	0
$676$	19.7829	0.760881
$677$	−21.9263	−0.842695	−0.421347	−	0.906899i	$-0.638443\pi$
$677$	−21.9263	−0.842695	−0.421347	+	0.906899i	$0.638443\pi$
$678$	0	0
$679$	55.3884	2.12561
$680$	5.13841	0.197049
$681$	0	0
$682$	7.11926	0.272611
$683$	−13.2757	−0.507982	−0.253991	−	0.967207i	$-0.581743\pi$
$683$	−13.2757	−0.507982	−0.253991	+	0.967207i	$0.581743\pi$
$684$	0	0
$685$	−27.6807	−1.05762
$686$	−1.63127	−0.0622823
$687$	0	0
$688$	−42.7175	−1.62859
$689$	−4.55943	−0.173700
$690$	0	0
$691$	−37.1778	−1.41431	−0.707156	−	0.707058i	$-0.750022\pi$
$691$	−37.1778	−1.41431	−0.707156	+	0.707058i	$0.750022\pi$
$692$	9.48921	0.360726
$693$	0	0
$694$	4.20032	0.159442
$695$	−37.6432	−1.42789
$696$	0	0
$697$	24.9579	0.945349
$698$	−8.40587	−0.318167
$699$	0	0
$700$	−15.6629	−0.592000
$701$	21.0544	0.795212	0.397606	−	0.917556i	$-0.369841\pi$
$701$	21.0544	0.795212	0.397606	+	0.917556i	$0.369841\pi$
$702$	0	0
$703$	12.7412	0.480544
$704$	−19.9193	−0.750736
$705$	0	0
$706$	8.66105	0.325963
$707$	24.0726	0.905342
$708$	0	0
$709$	31.4549	1.18131	0.590656	−	0.806924i	$-0.298869\pi$
$709$	31.4549	1.18131	0.590656	+	0.806924i	$0.298869\pi$
$710$	2.52383	0.0947176
$711$	0	0
$712$	−15.8854	−0.595332
$713$	−32.0788	−1.20136
$714$	0	0
$715$	−9.02985	−0.337697
$716$	−21.7338	−0.812232
$717$	0	0
$718$	3.87293	0.144536
$719$	−30.6402	−1.14269	−0.571344	−	0.820711i	$-0.693578\pi$
$719$	−30.6402	−1.14269	−0.571344	+	0.820711i	$0.693578\pi$
$720$	0	0
$721$	19.9605	0.743368
$722$	3.45593	0.128616
$723$	0	0
$724$	12.5140	0.465078
$725$	13.1090	0.486856
$726$	0	0
$727$	2.27198	0.0842631	0.0421315	−	0.999112i	$-0.486585\pi$
$727$	2.27198	0.0842631	0.0421315	+	0.999112i	$0.486585\pi$
$728$	6.83479	0.253314
$729$	0	0
$730$	−4.98151	−0.184374
$731$	−32.5954	−1.20558
$732$	0	0
$733$	26.6681	0.985007	0.492504	−	0.870310i	$-0.336082\pi$
$733$	26.6681	0.985007	0.492504	+	0.870310i	$0.336082\pi$
$734$	0.0685211	0.00252916
$735$	0	0
$736$	−16.1518	−0.595362
$737$	37.1560	1.36866
$738$	0	0
$739$	−4.14116	−0.152335	−0.0761675	−	0.997095i	$-0.524268\pi$
$739$	−4.14116	−0.152335	−0.0761675	+	0.997095i	$0.524268\pi$
$740$	14.3271	0.526674
$741$	0	0
$742$	3.04000	0.111602
$743$	26.3954	0.968354	0.484177	−	0.874970i	$-0.339119\pi$
$743$	26.3954	0.968354	0.484177	+	0.874970i	$0.339119\pi$
$744$	0	0
$745$	−1.63568	−0.0599269
$746$	−6.24489	−0.228642
$747$	0	0
$748$	−17.0970	−0.625128
$749$	−3.36547	−0.122972
$750$	0	0
$751$	−48.9987	−1.78799	−0.893994	−	0.448080i	$-0.852108\pi$
$751$	−48.9987	−1.78799	−0.893994	+	0.448080i	$0.852108\pi$
$752$	42.9523	1.56631
$753$	0	0
$754$	−2.79180	−0.101671
$755$	0.180837	0.00658132
$756$	0	0
$757$	−39.0094	−1.41782	−0.708911	−	0.705298i	$-0.750813\pi$
$757$	−39.0094	−1.41782	−0.708911	+	0.705298i	$0.750813\pi$
$758$	−3.89632	−0.141521
$759$	0	0
$760$	−5.41603	−0.196460
$761$	−20.3545	−0.737851	−0.368926	−	0.929459i	$-0.620274\pi$
$761$	−20.3545	−0.737851	−0.368926	+	0.929459i	$0.620274\pi$
$762$	0	0
$763$	42.9972	1.55660
$764$	16.2529	0.588010
$765$	0	0
$766$	5.80071	0.209588
$767$	−19.4304	−0.701592
$768$	0	0
$769$	12.3911	0.446834	0.223417	−	0.974723i	$-0.428279\pi$
$769$	12.3911	0.446834	0.223417	+	0.974723i	$0.428279\pi$
$770$	6.02065	0.216969
$771$	0	0
$772$	−16.0952	−0.579280
$773$	24.4773	0.880387	0.440193	−	0.897903i	$-0.354910\pi$
$773$	24.4773	0.880387	0.440193	+	0.897903i	$0.354910\pi$
$774$	0	0
$775$	15.8892	0.570757
$776$	−18.7117	−0.671710
$777$	0	0
$778$	1.74903	0.0627056
$779$	−26.3064	−0.942524
$780$	0	0
$781$	−17.2064	−0.615694
$782$	−3.77383	−0.134952
$783$	0	0
$784$	18.9301	0.676074
$785$	−9.08151	−0.324133
$786$	0	0
$787$	23.5061	0.837904	0.418952	−	0.908008i	$-0.362398\pi$
$787$	23.5061	0.837904	0.418952	+	0.908008i	$0.362398\pi$
$788$	−30.0306	−1.06980
$789$	0	0
$790$	−2.23710	−0.0795923
$791$	50.9369	1.81111
$792$	0	0
$793$	−14.1471	−0.502376
$794$	0.617201	0.0219037
$795$	0	0
$796$	24.3577	0.863337
$797$	−25.5059	−0.903465	−0.451732	−	0.892153i	$-0.649194\pi$
$797$	−25.5059	−0.903465	−0.451732	+	0.892153i	$0.649194\pi$
$798$	0	0
$799$	32.7746	1.15948
$800$	8.00025	0.282851
$801$	0	0
$802$	−9.72585	−0.343432
$803$	33.9619	1.19849
$804$	0	0
$805$	−27.1286	−0.956157
$806$	−3.38389	−0.119193
$807$	0	0
$808$	−8.13235	−0.286095
$809$	−12.4633	−0.438187	−0.219093	−	0.975704i	$-0.570310\pi$
$809$	−12.4633	−0.438187	−0.219093	+	0.975704i	$0.570310\pi$
$810$	0	0
$811$	12.1894	0.428026	0.214013	−	0.976831i	$-0.431346\pi$
$811$	12.1894	0.428026	0.214013	+	0.976831i	$0.431346\pi$
$812$	−37.9986	−1.33349
$813$	0	0
$814$	4.78484	0.167708
$815$	8.20618	0.287450
$816$	0	0
$817$	34.3565	1.20198
$818$	9.69141	0.338852
$819$	0	0
$820$	−29.5807	−1.03300
$821$	2.08885	0.0729013	0.0364507	−	0.999335i	$-0.488395\pi$
$821$	2.08885	0.0729013	0.0364507	+	0.999335i	$0.488395\pi$
$822$	0	0
$823$	29.9137	1.04272	0.521362	−	0.853335i	$-0.325424\pi$
$823$	29.9137	1.04272	0.521362	+	0.853335i	$0.325424\pi$
$824$	−6.74319	−0.234910
$825$	0	0
$826$	12.9552	0.450770
$827$	38.5243	1.33962	0.669810	−	0.742532i	$-0.266375\pi$
$827$	38.5243	1.33962	0.669810	+	0.742532i	$0.266375\pi$
$828$	0	0
$829$	−5.01966	−0.174340	−0.0871699	−	0.996193i	$-0.527782\pi$
$829$	−5.01966	−0.174340	−0.0871699	+	0.996193i	$0.527782\pi$
$830$	8.21629	0.285192
$831$	0	0
$832$	9.46794	0.328242
$833$	14.4445	0.500474
$834$	0	0
$835$	−2.48109	−0.0858617
$836$	18.0207	0.623260
$837$	0	0
$838$	3.44577	0.119032
$839$	−20.2606	−0.699474	−0.349737	−	0.936848i	$-0.613729\pi$
$839$	−20.2606	−0.699474	−0.349737	+	0.936848i	$0.613729\pi$
$840$	0	0
$841$	2.80286	0.0966502
$842$	3.51270	0.121056
$843$	0	0
$844$	28.0190	0.964454
$845$	−16.9719	−0.583850
$846$	0	0
$847$	−2.17157	−0.0746160
$848$	9.70595	0.333304
$849$	0	0
$850$	1.86924	0.0641145
$851$	−21.5601	−0.739070
$852$	0	0
$853$	−38.8874	−1.33148	−0.665740	−	0.746184i	$-0.731884\pi$
$853$	−38.8874	−1.33148	−0.665740	+	0.746184i	$0.731884\pi$
$854$	9.43254	0.322775
$855$	0	0
$856$	1.13695	0.0388600
$857$	−10.8572	−0.370874	−0.185437	−	0.982656i	$-0.559370\pi$
$857$	−10.8572	−0.370874	−0.185437	+	0.982656i	$0.559370\pi$
$858$	0	0
$859$	−20.9307	−0.714145	−0.357073	−	0.934077i	$-0.616225\pi$
$859$	−20.9307	−0.714145	−0.357073	+	0.934077i	$0.616225\pi$
$860$	38.6328	1.31737
$861$	0	0
$862$	−2.83077	−0.0964165
$863$	30.8139	1.04892	0.524459	−	0.851435i	$-0.324267\pi$
$863$	30.8139	1.04892	0.524459	+	0.851435i	$0.324267\pi$
$864$	0	0
$865$	−8.14085	−0.276797
$866$	9.32655	0.316929
$867$	0	0
$868$	−46.0575	−1.56329
$869$	15.2516	0.517375
$870$	0	0
$871$	−17.6608	−0.598414
$872$	−14.5256	−0.491899
$873$	0	0
$874$	3.97773	0.134549
$875$	42.3404	1.43137
$876$	0	0
$877$	42.4847	1.43461	0.717303	−	0.696762i	$-0.245376\pi$
$877$	42.4847	1.43461	0.717303	+	0.696762i	$0.245376\pi$
$878$	−0.211781	−0.00714725
$879$	0	0
$880$	19.2224	0.647988
$881$	43.4583	1.46415	0.732073	−	0.681226i	$-0.238553\pi$
$881$	43.4583	1.46415	0.732073	+	0.681226i	$0.238553\pi$
$882$	0	0
$883$	−13.0935	−0.440632	−0.220316	−	0.975429i	$-0.570709\pi$
$883$	−13.0935	−0.440632	−0.220316	+	0.975429i	$0.570709\pi$
$884$	8.12647	0.273323
$885$	0	0
$886$	2.69151	0.0904230
$887$	−45.3324	−1.52211	−0.761057	−	0.648685i	$-0.775319\pi$
$887$	−45.3324	−1.52211	−0.761057	+	0.648685i	$0.775319\pi$
$888$	0	0
$889$	−30.4258	−1.02045
$890$	6.65119	0.222949
$891$	0	0
$892$	55.7257	1.86584
$893$	−34.5454	−1.15602
$894$	0	0
$895$	18.6456	0.623253
$896$	−30.6388	−1.02357
$897$	0	0
$898$	4.64763	0.155093
$899$	38.5477	1.28564
$900$	0	0
$901$	7.40609	0.246733
$902$	−9.87910	−0.328938
$903$	0	0
$904$	−17.2078	−0.572324
$905$	−10.7358	−0.356870
$906$	0	0
$907$	−31.1130	−1.03309	−0.516545	−	0.856260i	$-0.672782\pi$
$907$	−31.1130	−1.03309	−0.516545	+	0.856260i	$0.672782\pi$
$908$	47.4620	1.57508
$909$	0	0
$910$	−2.86171	−0.0948647
$911$	47.7208	1.58106	0.790530	−	0.612424i	$-0.209805\pi$
$911$	47.7208	1.58106	0.790530	+	0.612424i	$0.209805\pi$
$912$	0	0
$913$	−56.0153	−1.85383
$914$	−4.96145	−0.164110
$915$	0	0
$916$	−0.194686	−0.00643261
$917$	−44.0983	−1.45626
$918$	0	0
$919$	44.9434	1.48255	0.741273	−	0.671203i	$-0.234222\pi$
$919$	44.9434	1.48255	0.741273	+	0.671203i	$0.234222\pi$
$920$	9.16475	0.302153
$921$	0	0
$922$	−10.6162	−0.349626
$923$	8.17848	0.269198
$924$	0	0
$925$	10.6791	0.351126
$926$	9.61162	0.315858
$927$	0	0
$928$	19.4089	0.637127
$929$	−32.6595	−1.07152	−0.535761	−	0.844370i	$-0.679975\pi$
$929$	−32.6595	−1.07152	−0.535761	+	0.844370i	$0.679975\pi$
$930$	0	0
$931$	−15.2250	−0.498978
$932$	53.7907	1.76197
$933$	0	0
$934$	−5.16815	−0.169107
$935$	14.6676	0.479682
$936$	0	0
$937$	17.5359	0.572873	0.286437	−	0.958099i	$-0.407529\pi$
$937$	17.5359	0.572873	0.286437	+	0.958099i	$0.407529\pi$
$938$	11.7753	0.384479
$939$	0	0
$940$	−38.8452	−1.26699
$941$	14.9992	0.488961	0.244480	−	0.969654i	$-0.421383\pi$
$941$	14.9992	0.488961	0.244480	+	0.969654i	$0.421383\pi$
$942$	0	0
$943$	44.5144	1.44959
$944$	41.3628	1.34625
$945$	0	0
$946$	12.9022	0.419488
$947$	4.99323	0.162258	0.0811292	−	0.996704i	$-0.474147\pi$
$947$	4.99323	0.162258	0.0811292	+	0.996704i	$0.474147\pi$
$948$	0	0
$949$	−16.1426	−0.524011
$950$	−1.97024	−0.0639229
$951$	0	0
$952$	−11.1021	−0.359820
$953$	24.5163	0.794161	0.397081	−	0.917784i	$-0.370023\pi$
$953$	24.5163	0.794161	0.397081	+	0.917784i	$0.370023\pi$
$954$	0	0
$955$	−13.9435	−0.451200
$956$	−14.9815	−0.484535
$957$	0	0
$958$	−4.58724	−0.148207
$959$	59.8070	1.93127
$960$	0	0
$961$	15.7231	0.507197
$962$	−2.27431	−0.0733266
$963$	0	0
$964$	40.4479	1.30274
$965$	13.8082	0.444502
$966$	0	0
$967$	24.3735	0.783800	0.391900	−	0.920008i	$-0.371818\pi$
$967$	24.3735	0.783800	0.391900	+	0.920008i	$0.371818\pi$
$968$	0.733614	0.0235792
$969$	0	0
$970$	7.83453	0.251552
$971$	−6.16139	−0.197728	−0.0988642	−	0.995101i	$-0.531521\pi$
$971$	−6.16139	−0.197728	−0.0988642	+	0.995101i	$0.531521\pi$
$972$	0	0
$973$	81.3321	2.60739
$974$	5.36635	0.171949
$975$	0	0
$976$	30.1157	0.963982
$977$	20.1046	0.643203	0.321601	−	0.946875i	$-0.395779\pi$
$977$	20.1046	0.643203	0.321601	+	0.946875i	$0.395779\pi$
$978$	0	0
$979$	−45.3451	−1.44924
$980$	−17.1200	−0.546877
$981$	0	0
$982$	−12.8476	−0.409983
$983$	42.4939	1.35534	0.677672	−	0.735364i	$-0.262989\pi$
$983$	42.4939	1.35534	0.677672	+	0.735364i	$0.262989\pi$
$984$	0	0
$985$	25.7634	0.820891
$986$	4.53485	0.144419
$987$	0	0
$988$	−8.56553	−0.272506
$989$	−58.1365	−1.84863
$990$	0	0
$991$	−1.68049	−0.0533824	−0.0266912	−	0.999644i	$-0.508497\pi$
$991$	−1.68049	−0.0533824	−0.0266912	+	0.999644i	$0.508497\pi$
$992$	23.5252	0.746925
$993$	0	0
$994$	−5.45300	−0.172959
$995$	−20.8966	−0.662468
$996$	0	0
$997$	1.88157	0.0595898	0.0297949	−	0.999556i	$-0.490515\pi$
$997$	1.88157	0.0595898	0.0297949	+	0.999556i	$0.490515\pi$
$998$	−6.64636	−0.210387
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.13	✓	25
3.2	odd	2		4023.2.a.f.1.13	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.13	✓	25	1.1	even	1	trivial
4023.2.a.f.1.13	yes	25	3.2	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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q-0.305611 q^{2} -1.90660 q^{4} +1.63568 q^{5} -3.53407 q^{7} +1.19390 q^{8} +O(q^{10})\)	\(q-0.305611 q^{2} -1.90660 q^{4} +1.63568 q^{5} -3.53407 q^{7} +1.19390 q^{8} -0.499884 q^{10} +3.40800 q^{11} -1.61987 q^{13} +1.08005 q^{14} +3.44833 q^{16} +2.63124 q^{17} -2.77340 q^{19} -3.11860 q^{20} -1.04152 q^{22} +4.69302 q^{23} -2.32454 q^{25} +0.495052 q^{26} +6.73806 q^{28} -5.63940 q^{29} -6.83543 q^{31} -3.44165 q^{32} -0.804137 q^{34} -5.78062 q^{35} -4.59407 q^{37} +0.847583 q^{38} +1.95285 q^{40} +9.48524 q^{41} -12.3879 q^{43} -6.49770 q^{44} -1.43424 q^{46} +12.4560 q^{47} +5.48963 q^{49} +0.710404 q^{50} +3.08846 q^{52} +2.81468 q^{53} +5.57441 q^{55} -4.21933 q^{56} +1.72347 q^{58} +11.9950 q^{59} +8.73342 q^{61} +2.08898 q^{62} -5.84486 q^{64} -2.64960 q^{65} +10.9026 q^{67} -5.01672 q^{68} +1.76662 q^{70} -5.04883 q^{71} +9.96534 q^{73} +1.40400 q^{74} +5.28777 q^{76} -12.0441 q^{77} +4.47523 q^{79} +5.64039 q^{80} -2.89880 q^{82} -16.4364 q^{83} +4.30388 q^{85} +3.78587 q^{86} +4.06882 q^{88} -13.3055 q^{89} +5.72475 q^{91} -8.94772 q^{92} -3.80668 q^{94} -4.53641 q^{95} -15.6727 q^{97} -1.67769 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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