LMFDB - Embedded newform 4024.2.a.e.1.13

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			1.13
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)

$f(q)$	$=$	$q-0.782310 q^{3} +2.74195 q^{5} -3.13445 q^{7} -2.38799 q^{9} +O(q^{10})$	$q-0.782310 q^{3} +2.74195 q^{5} -3.13445 q^{7} -2.38799 q^{9} +4.20847 q^{11} +4.79589 q^{13} -2.14505 q^{15} -4.78682 q^{17} -6.84756 q^{19} +2.45211 q^{21} +2.52215 q^{23} +2.51828 q^{25} +4.21508 q^{27} -0.824514 q^{29} -4.65958 q^{31} -3.29233 q^{33} -8.59451 q^{35} -5.55076 q^{37} -3.75187 q^{39} +6.84093 q^{41} -9.50815 q^{43} -6.54775 q^{45} -3.54490 q^{47} +2.82479 q^{49} +3.74478 q^{51} +7.03271 q^{53} +11.5394 q^{55} +5.35692 q^{57} -0.838859 q^{59} -1.86285 q^{61} +7.48504 q^{63} +13.1501 q^{65} +3.63949 q^{67} -1.97310 q^{69} -7.42049 q^{71} +3.97944 q^{73} -1.97008 q^{75} -13.1913 q^{77} -2.08663 q^{79} +3.86647 q^{81} -2.93129 q^{83} -13.1252 q^{85} +0.645025 q^{87} +9.36528 q^{89} -15.0325 q^{91} +3.64523 q^{93} -18.7757 q^{95} -4.28098 q^{97} -10.0498 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.782310	−0.451667	−0.225833	−	0.974166i	$-0.572511\pi$
$3$	−0.782310	−0.451667	−0.225833	+	0.974166i	$0.572511\pi$
$4$	0	0
$5$	2.74195	1.22624	0.613118	−	0.789991i	$-0.289915\pi$
$5$	2.74195	1.22624	0.613118	+	0.789991i	$0.289915\pi$
$6$	0	0
$7$	−3.13445	−1.18471	−0.592356	−	0.805677i	$-0.701802\pi$
$7$	−3.13445	−1.18471	−0.592356	+	0.805677i	$0.701802\pi$
$8$	0	0
$9$	−2.38799	−0.795997
$10$	0	0
$11$	4.20847	1.26890	0.634451	−	0.772963i	$-0.281226\pi$
$11$	4.20847	1.26890	0.634451	+	0.772963i	$0.281226\pi$
$12$	0	0
$13$	4.79589	1.33014	0.665070	−	0.746781i	$-0.268402\pi$
$13$	4.79589	1.33014	0.665070	+	0.746781i	$0.268402\pi$
$14$	0	0
$15$	−2.14505	−0.553851
$16$	0	0
$17$	−4.78682	−1.16097	−0.580487	−	0.814270i	$-0.697138\pi$
$17$	−4.78682	−1.16097	−0.580487	+	0.814270i	$0.697138\pi$
$18$	0	0
$19$	−6.84756	−1.57094	−0.785469	−	0.618901i	$-0.787578\pi$
$19$	−6.84756	−1.57094	−0.785469	+	0.618901i	$0.787578\pi$
$20$	0	0
$21$	2.45211	0.535095
$22$	0	0
$23$	2.52215	0.525904	0.262952	−	0.964809i	$-0.415304\pi$
$23$	2.52215	0.525904	0.262952	+	0.964809i	$0.415304\pi$
$24$	0	0
$25$	2.51828	0.503657
$26$	0	0
$27$	4.21508	0.811193
$28$	0	0
$29$	−0.824514	−0.153108	−0.0765542	−	0.997065i	$-0.524392\pi$
$29$	−0.824514	−0.153108	−0.0765542	+	0.997065i	$0.524392\pi$
$30$	0	0
$31$	−4.65958	−0.836884	−0.418442	−	0.908243i	$-0.637424\pi$
$31$	−4.65958	−0.836884	−0.418442	+	0.908243i	$0.637424\pi$
$32$	0	0
$33$	−3.29233	−0.573122
$34$	0	0
$35$	−8.59451	−1.45274
$36$	0	0
$37$	−5.55076	−0.912540	−0.456270	−	0.889841i	$-0.650815\pi$
$37$	−5.55076	−0.912540	−0.456270	+	0.889841i	$0.650815\pi$
$38$	0	0
$39$	−3.75187	−0.600780
$40$	0	0
$41$	6.84093	1.06837	0.534187	−	0.845366i	$-0.320618\pi$
$41$	6.84093	1.06837	0.534187	+	0.845366i	$0.320618\pi$
$42$	0	0
$43$	−9.50815	−1.44998	−0.724990	−	0.688760i	$-0.758155\pi$
$43$	−9.50815	−1.44998	−0.724990	+	0.688760i	$0.758155\pi$
$44$	0	0
$45$	−6.54775	−0.976081
$46$	0	0
$47$	−3.54490	−0.517077	−0.258539	−	0.966001i	$-0.583241\pi$
$47$	−3.54490	−0.517077	−0.258539	+	0.966001i	$0.583241\pi$
$48$	0	0
$49$	2.82479	0.403541
$50$	0	0
$51$	3.74478	0.524374
$52$	0	0
$53$	7.03271	0.966017	0.483009	−	0.875616i	$-0.339544\pi$
$53$	7.03271	0.966017	0.483009	+	0.875616i	$0.339544\pi$
$54$	0	0
$55$	11.5394	1.55598
$56$	0	0
$57$	5.35692	0.709541
$58$	0	0
$59$	−0.838859	−0.109210	−0.0546051	−	0.998508i	$-0.517390\pi$
$59$	−0.838859	−0.109210	−0.0546051	+	0.998508i	$0.517390\pi$
$60$	0	0
$61$	−1.86285	−0.238514	−0.119257	−	0.992863i	$-0.538051\pi$
$61$	−1.86285	−0.238514	−0.119257	+	0.992863i	$0.538051\pi$
$62$	0	0
$63$	7.48504	0.943027
$64$	0	0
$65$	13.1501	1.63107
$66$	0	0
$67$	3.63949	0.444635	0.222317	−	0.974974i	$-0.428638\pi$
$67$	3.63949	0.444635	0.222317	+	0.974974i	$0.428638\pi$
$68$	0	0
$69$	−1.97310	−0.237533
$70$	0	0
$71$	−7.42049	−0.880651	−0.440325	−	0.897838i	$-0.645137\pi$
$71$	−7.42049	−0.880651	−0.440325	+	0.897838i	$0.645137\pi$
$72$	0	0
$73$	3.97944	0.465758	0.232879	−	0.972506i	$-0.425185\pi$
$73$	3.97944	0.465758	0.232879	+	0.972506i	$0.425185\pi$
$74$	0	0
$75$	−1.97008	−0.227485
$76$	0	0
$77$	−13.1913	−1.50328
$78$	0	0
$79$	−2.08663	−0.234764	−0.117382	−	0.993087i	$-0.537450\pi$
$79$	−2.08663	−0.234764	−0.117382	+	0.993087i	$0.537450\pi$
$80$	0	0
$81$	3.86647	0.429608
$82$	0	0
$83$	−2.93129	−0.321750	−0.160875	−	0.986975i	$-0.551432\pi$
$83$	−2.93129	−0.321750	−0.160875	+	0.986975i	$0.551432\pi$
$84$	0	0
$85$	−13.1252	−1.42363
$86$	0	0
$87$	0.645025	0.0691540
$88$	0	0
$89$	9.36528	0.992718	0.496359	−	0.868117i	$-0.334670\pi$
$89$	9.36528	0.992718	0.496359	+	0.868117i	$0.334670\pi$
$90$	0	0
$91$	−15.0325	−1.57583
$92$	0	0
$93$	3.64523	0.377993
$94$	0	0
$95$	−18.7757	−1.92634
$96$	0	0
$97$	−4.28098	−0.434668	−0.217334	−	0.976097i	$-0.569736\pi$
$97$	−4.28098	−0.434668	−0.217334	+	0.976097i	$0.569736\pi$
$98$	0	0
$99$	−10.0498	−1.01004
$100$	0	0
$101$	−12.8269	−1.27633	−0.638163	−	0.769901i	$-0.720306\pi$
$101$	−12.8269	−1.27633	−0.638163	+	0.769901i	$0.720306\pi$
$102$	0	0
$103$	−19.4646	−1.91791	−0.958954	−	0.283563i	$-0.908484\pi$
$103$	−19.4646	−1.91791	−0.958954	+	0.283563i	$0.908484\pi$
$104$	0	0
$105$	6.72357	0.656153
$106$	0	0
$107$	−0.178210	−0.0172282	−0.00861410	−	0.999963i	$-0.502742\pi$
$107$	−0.178210	−0.0172282	−0.00861410	+	0.999963i	$0.502742\pi$
$108$	0	0
$109$	−0.772846	−0.0740253	−0.0370126	−	0.999315i	$-0.511784\pi$
$109$	−0.772846	−0.0740253	−0.0370126	+	0.999315i	$0.511784\pi$
$110$	0	0
$111$	4.34242	0.412164
$112$	0	0
$113$	8.77320	0.825313	0.412657	−	0.910887i	$-0.364601\pi$
$113$	8.77320	0.825313	0.412657	+	0.910887i	$0.364601\pi$
$114$	0	0
$115$	6.91560	0.644883
$116$	0	0
$117$	−11.4525	−1.05879
$118$	0	0
$119$	15.0040	1.37542
$120$	0	0
$121$	6.71126	0.610115
$122$	0	0
$123$	−5.35173	−0.482549
$124$	0	0
$125$	−6.80474	−0.608634
$126$	0	0
$127$	−18.3686	−1.62995	−0.814974	−	0.579497i	$-0.803249\pi$
$127$	−18.3686	−1.62995	−0.814974	+	0.579497i	$0.803249\pi$
$128$	0	0
$129$	7.43832	0.654908
$130$	0	0
$131$	−0.428029	−0.0373971	−0.0186985	−	0.999825i	$-0.505952\pi$
$131$	−0.428029	−0.0373971	−0.0186985	+	0.999825i	$0.505952\pi$
$132$	0	0
$133$	21.4634	1.86111
$134$	0	0
$135$	11.5575	0.994714
$136$	0	0
$137$	1.88264	0.160845	0.0804224	−	0.996761i	$-0.474373\pi$
$137$	1.88264	0.160845	0.0804224	+	0.996761i	$0.474373\pi$
$138$	0	0
$139$	−9.25944	−0.785375	−0.392688	−	0.919672i	$-0.628455\pi$
$139$	−9.25944	−0.785375	−0.392688	+	0.919672i	$0.628455\pi$
$140$	0	0
$141$	2.77321	0.233547
$142$	0	0
$143$	20.1834	1.68782
$144$	0	0
$145$	−2.26077	−0.187747
$146$	0	0
$147$	−2.20986	−0.182266
$148$	0	0
$149$	1.24700	0.102158	0.0510791	−	0.998695i	$-0.483734\pi$
$149$	1.24700	0.102158	0.0510791	+	0.998695i	$0.483734\pi$
$150$	0	0
$151$	−7.45091	−0.606347	−0.303173	−	0.952935i	$-0.598046\pi$
$151$	−7.45091	−0.606347	−0.303173	+	0.952935i	$0.598046\pi$
$152$	0	0
$153$	11.4309	0.924132
$154$	0	0
$155$	−12.7763	−1.02622
$156$	0	0
$157$	1.31185	0.104697	0.0523484	−	0.998629i	$-0.483329\pi$
$157$	1.31185	0.104697	0.0523484	+	0.998629i	$0.483329\pi$
$158$	0	0
$159$	−5.50176	−0.436318
$160$	0	0
$161$	−7.90555	−0.623044
$162$	0	0
$163$	−13.7568	−1.07752	−0.538760	−	0.842460i	$-0.681107\pi$
$163$	−13.7568	−1.07752	−0.538760	+	0.842460i	$0.681107\pi$
$164$	0	0
$165$	−9.02741	−0.702783
$166$	0	0
$167$	12.7662	0.987876	0.493938	−	0.869497i	$-0.335557\pi$
$167$	12.7662	0.987876	0.493938	+	0.869497i	$0.335557\pi$
$168$	0	0
$169$	10.0005	0.769272
$170$	0	0
$171$	16.3519	1.25046
$172$	0	0
$173$	15.6207	1.18762	0.593810	−	0.804605i	$-0.297623\pi$
$173$	15.6207	1.18762	0.593810	+	0.804605i	$0.297623\pi$
$174$	0	0
$175$	−7.89344	−0.596688
$176$	0	0
$177$	0.656248	0.0493266
$178$	0	0
$179$	−12.2302	−0.914127	−0.457064	−	0.889434i	$-0.651099\pi$
$179$	−12.2302	−0.914127	−0.457064	+	0.889434i	$0.651099\pi$
$180$	0	0
$181$	−23.8693	−1.77419	−0.887096	−	0.461586i	$-0.847281\pi$
$181$	−23.8693	−1.77419	−0.887096	+	0.461586i	$0.847281\pi$
$182$	0	0
$183$	1.45733	0.107729
$184$	0	0
$185$	−15.2199	−1.11899
$186$	0	0
$187$	−20.1452	−1.47316
$188$	0	0
$189$	−13.2120	−0.961029
$190$	0	0
$191$	23.9177	1.73062	0.865311	−	0.501235i	$-0.167121\pi$
$191$	23.9177	1.73062	0.865311	+	0.501235i	$0.167121\pi$
$192$	0	0
$193$	2.76256	0.198854	0.0994268	−	0.995045i	$-0.468299\pi$
$193$	2.76256	0.198854	0.0994268	+	0.995045i	$0.468299\pi$
$194$	0	0
$195$	−10.2874	−0.736699
$196$	0	0
$197$	−6.40833	−0.456574	−0.228287	−	0.973594i	$-0.573313\pi$
$197$	−6.40833	−0.456574	−0.228287	+	0.973594i	$0.573313\pi$
$198$	0	0
$199$	11.0453	0.782978	0.391489	−	0.920183i	$-0.371960\pi$
$199$	11.0453	0.782978	0.391489	+	0.920183i	$0.371960\pi$
$200$	0	0
$201$	−2.84721	−0.200827
$202$	0	0
$203$	2.58440	0.181389
$204$	0	0
$205$	18.7575	1.31008
$206$	0	0
$207$	−6.02286	−0.418618
$208$	0	0
$209$	−28.8178	−1.99337
$210$	0	0
$211$	−18.6613	−1.28470	−0.642348	−	0.766413i	$-0.722040\pi$
$211$	−18.6613	−1.28470	−0.642348	+	0.766413i	$0.722040\pi$
$212$	0	0
$213$	5.80513	0.397761
$214$	0	0
$215$	−26.0709	−1.77802
$216$	0	0
$217$	14.6052	0.991466
$218$	0	0
$219$	−3.11315	−0.210367
$220$	0	0
$221$	−22.9570	−1.54426
$222$	0	0
$223$	0.873691	0.0585067	0.0292534	−	0.999572i	$-0.490687\pi$
$223$	0.873691	0.0585067	0.0292534	+	0.999572i	$0.490687\pi$
$224$	0	0
$225$	−6.01364	−0.400909
$226$	0	0
$227$	2.85679	0.189612	0.0948059	−	0.995496i	$-0.469777\pi$
$227$	2.85679	0.189612	0.0948059	+	0.995496i	$0.469777\pi$
$228$	0	0
$229$	−9.68538	−0.640028	−0.320014	−	0.947413i	$-0.603688\pi$
$229$	−9.68538	−0.640028	−0.320014	+	0.947413i	$0.603688\pi$
$230$	0	0
$231$	10.3197	0.678984
$232$	0	0
$233$	1.70544	0.111727	0.0558637	−	0.998438i	$-0.482209\pi$
$233$	1.70544	0.111727	0.0558637	+	0.998438i	$0.482209\pi$
$234$	0	0
$235$	−9.71994	−0.634059
$236$	0	0
$237$	1.63239	0.106035
$238$	0	0
$239$	−14.0402	−0.908186	−0.454093	−	0.890954i	$-0.650037\pi$
$239$	−14.0402	−0.908186	−0.454093	+	0.890954i	$0.650037\pi$
$240$	0	0
$241$	5.22606	0.336640	0.168320	−	0.985732i	$-0.446166\pi$
$241$	5.22606	0.336640	0.168320	+	0.985732i	$0.446166\pi$
$242$	0	0
$243$	−15.6700	−1.00523
$244$	0	0
$245$	7.74542	0.494837
$246$	0	0
$247$	−32.8401	−2.08957
$248$	0	0
$249$	2.29318	0.145324
$250$	0	0
$251$	15.8111	0.997985	0.498993	−	0.866606i	$-0.333703\pi$
$251$	15.8111	0.997985	0.498993	+	0.866606i	$0.333703\pi$
$252$	0	0
$253$	10.6144	0.667321
$254$	0	0
$255$	10.2680	0.643006
$256$	0	0
$257$	−18.0283	−1.12457	−0.562286	−	0.826943i	$-0.690078\pi$
$257$	−18.0283	−1.12457	−0.562286	+	0.826943i	$0.690078\pi$
$258$	0	0
$259$	17.3986	1.08110
$260$	0	0
$261$	1.96893	0.121874
$262$	0	0
$263$	11.0511	0.681437	0.340719	−	0.940165i	$-0.389330\pi$
$263$	11.0511	0.681437	0.340719	+	0.940165i	$0.389330\pi$
$264$	0	0
$265$	19.2833	1.18457
$266$	0	0
$267$	−7.32656	−0.448378
$268$	0	0
$269$	23.6994	1.44498	0.722489	−	0.691382i	$-0.242998\pi$
$269$	23.6994	1.44498	0.722489	+	0.691382i	$0.242998\pi$
$270$	0	0
$271$	−26.2018	−1.59165	−0.795823	−	0.605530i	$-0.792961\pi$
$271$	−26.2018	−1.59165	−0.795823	+	0.605530i	$0.792961\pi$
$272$	0	0
$273$	11.7601	0.711751
$274$	0	0
$275$	10.5981	0.639091
$276$	0	0
$277$	−13.5784	−0.815845	−0.407922	−	0.913017i	$-0.633747\pi$
$277$	−13.5784	−0.815845	−0.407922	+	0.913017i	$0.633747\pi$
$278$	0	0
$279$	11.1270	0.666157
$280$	0	0
$281$	−14.0210	−0.836424	−0.418212	−	0.908350i	$-0.637343\pi$
$281$	−14.0210	−0.836424	−0.418212	+	0.908350i	$0.637343\pi$
$282$	0	0
$283$	−1.00737	−0.0598821	−0.0299411	−	0.999552i	$-0.509532\pi$
$283$	−1.00737	−0.0598821	−0.0299411	+	0.999552i	$0.509532\pi$
$284$	0	0
$285$	14.6884	0.870065
$286$	0	0
$287$	−21.4426	−1.26572
$288$	0	0
$289$	5.91362	0.347860
$290$	0	0
$291$	3.34906	0.196325
$292$	0	0
$293$	−15.7631	−0.920889	−0.460445	−	0.887688i	$-0.652310\pi$
$293$	−15.7631	−0.920889	−0.460445	+	0.887688i	$0.652310\pi$
$294$	0	0
$295$	−2.30011	−0.133918
$296$	0	0
$297$	17.7391	1.02932
$298$	0	0
$299$	12.0959	0.699526
$300$	0	0
$301$	29.8028	1.71781
$302$	0	0
$303$	10.0346	0.576475
$304$	0	0
$305$	−5.10785	−0.292474
$306$	0	0
$307$	9.15464	0.522483	0.261241	−	0.965274i	$-0.415868\pi$
$307$	9.15464	0.522483	0.261241	+	0.965274i	$0.415868\pi$
$308$	0	0
$309$	15.2274	0.866255
$310$	0	0
$311$	18.8592	1.06940	0.534702	−	0.845040i	$-0.320424\pi$
$311$	18.8592	1.06940	0.534702	+	0.845040i	$0.320424\pi$
$312$	0	0
$313$	−9.93305	−0.561449	−0.280725	−	0.959788i	$-0.590575\pi$
$313$	−9.93305	−0.561449	−0.280725	+	0.959788i	$0.590575\pi$
$314$	0	0
$315$	20.5236	1.15637
$316$	0	0
$317$	−15.9605	−0.896428	−0.448214	−	0.893926i	$-0.647940\pi$
$317$	−15.9605	−0.896428	−0.448214	+	0.893926i	$0.647940\pi$
$318$	0	0
$319$	−3.46994	−0.194280
$320$	0	0
$321$	0.139415	0.00778141
$322$	0	0
$323$	32.7780	1.82382
$324$	0	0
$325$	12.0774	0.669934
$326$	0	0
$327$	0.604606	0.0334348
$328$	0	0
$329$	11.1113	0.612587
$330$	0	0
$331$	−19.3622	−1.06424	−0.532122	−	0.846667i	$-0.678605\pi$
$331$	−19.3622	−1.06424	−0.532122	+	0.846667i	$0.678605\pi$
$332$	0	0
$333$	13.2552	0.726379
$334$	0	0
$335$	9.97930	0.545228
$336$	0	0
$337$	−25.8009	−1.40546	−0.702732	−	0.711455i	$-0.748036\pi$
$337$	−25.8009	−1.40546	−0.702732	+	0.711455i	$0.748036\pi$
$338$	0	0
$339$	−6.86336	−0.372767
$340$	0	0
$341$	−19.6097	−1.06193
$342$	0	0
$343$	13.0870	0.706632
$344$	0	0
$345$	−5.41014	−0.291272
$346$	0	0
$347$	−10.3617	−0.556244	−0.278122	−	0.960546i	$-0.589712\pi$
$347$	−10.3617	−0.556244	−0.278122	+	0.960546i	$0.589712\pi$
$348$	0	0
$349$	−30.6268	−1.63942	−0.819708	−	0.572782i	$-0.805864\pi$
$349$	−30.6268	−1.63942	−0.819708	+	0.572782i	$0.805864\pi$
$350$	0	0
$351$	20.2150	1.07900
$352$	0	0
$353$	19.3678	1.03085	0.515423	−	0.856936i	$-0.327635\pi$
$353$	19.3678	1.03085	0.515423	+	0.856936i	$0.327635\pi$
$354$	0	0
$355$	−20.3466	−1.07989
$356$	0	0
$357$	−11.7378	−0.621231
$358$	0	0
$359$	7.32531	0.386615	0.193308	−	0.981138i	$-0.438078\pi$
$359$	7.32531	0.386615	0.193308	+	0.981138i	$0.438078\pi$
$360$	0	0
$361$	27.8891	1.46785
$362$	0	0
$363$	−5.25029	−0.275569
$364$	0	0
$365$	10.9114	0.571129
$366$	0	0
$367$	−10.8217	−0.564890	−0.282445	−	0.959284i	$-0.591145\pi$
$367$	−10.8217	−0.564890	−0.282445	+	0.959284i	$0.591145\pi$
$368$	0	0
$369$	−16.3361	−0.850423
$370$	0	0
$371$	−22.0437	−1.14445
$372$	0	0
$373$	7.66951	0.397112	0.198556	−	0.980090i	$-0.436375\pi$
$373$	7.66951	0.397112	0.198556	+	0.980090i	$0.436375\pi$
$374$	0	0
$375$	5.32342	0.274900
$376$	0	0
$377$	−3.95427	−0.203655
$378$	0	0
$379$	31.2594	1.60569	0.802844	−	0.596188i	$-0.203319\pi$
$379$	31.2594	1.60569	0.802844	+	0.596188i	$0.203319\pi$
$380$	0	0
$381$	14.3699	0.736194
$382$	0	0
$383$	27.6800	1.41438	0.707191	−	0.707022i	$-0.249962\pi$
$383$	27.6800	1.41438	0.707191	+	0.707022i	$0.249962\pi$
$384$	0	0
$385$	−36.1698	−1.84338
$386$	0	0
$387$	22.7054	1.15418
$388$	0	0
$389$	−10.8563	−0.550436	−0.275218	−	0.961382i	$-0.588750\pi$
$389$	−10.8563	−0.550436	−0.275218	+	0.961382i	$0.588750\pi$
$390$	0	0
$391$	−12.0731	−0.610561
$392$	0	0
$393$	0.334851	0.0168910
$394$	0	0
$395$	−5.72144	−0.287877
$396$	0	0
$397$	28.0305	1.40681	0.703406	−	0.710789i	$-0.251662\pi$
$397$	28.0305	1.40681	0.703406	+	0.710789i	$0.251662\pi$
$398$	0	0
$399$	−16.7910	−0.840601
$400$	0	0
$401$	−14.1461	−0.706423	−0.353212	−	0.935543i	$-0.614910\pi$
$401$	−14.1461	−0.706423	−0.353212	+	0.935543i	$0.614910\pi$
$402$	0	0
$403$	−22.3468	−1.11317
$404$	0	0
$405$	10.6017	0.526801
$406$	0	0
$407$	−23.3603	−1.15792
$408$	0	0
$409$	11.6422	0.575668	0.287834	−	0.957680i	$-0.407065\pi$
$409$	11.6422	0.575668	0.287834	+	0.957680i	$0.407065\pi$
$410$	0	0
$411$	−1.47281	−0.0726483
$412$	0	0
$413$	2.62936	0.129383
$414$	0	0
$415$	−8.03744	−0.394542
$416$	0	0
$417$	7.24375	0.354728
$418$	0	0
$419$	−10.3865	−0.507415	−0.253708	−	0.967281i	$-0.581650\pi$
$419$	−10.3865	−0.507415	−0.253708	+	0.967281i	$0.581650\pi$
$420$	0	0
$421$	−33.7537	−1.64505	−0.822527	−	0.568726i	$-0.807436\pi$
$421$	−33.7537	−1.64505	−0.822527	+	0.568726i	$0.807436\pi$
$422$	0	0
$423$	8.46519	0.411592
$424$	0	0
$425$	−12.0546	−0.584732
$426$	0	0
$427$	5.83902	0.282570
$428$	0	0
$429$	−15.7897	−0.762332
$430$	0	0
$431$	33.8760	1.63175	0.815876	−	0.578227i	$-0.196255\pi$
$431$	33.8760	1.63175	0.815876	+	0.578227i	$0.196255\pi$
$432$	0	0
$433$	−25.1478	−1.20853	−0.604263	−	0.796785i	$-0.706532\pi$
$433$	−25.1478	−1.20853	−0.604263	+	0.796785i	$0.706532\pi$
$434$	0	0
$435$	1.76863	0.0847992
$436$	0	0
$437$	−17.2706	−0.826163
$438$	0	0
$439$	−33.5536	−1.60143	−0.800714	−	0.599047i	$-0.795546\pi$
$439$	−33.5536	−1.60143	−0.800714	+	0.599047i	$0.795546\pi$
$440$	0	0
$441$	−6.74556	−0.321217
$442$	0	0
$443$	19.4957	0.926266	0.463133	−	0.886289i	$-0.346725\pi$
$443$	19.4957	0.926266	0.463133	+	0.886289i	$0.346725\pi$
$444$	0	0
$445$	25.6791	1.21731
$446$	0	0
$447$	−0.975540	−0.0461415
$448$	0	0
$449$	−27.3024	−1.28848	−0.644240	−	0.764823i	$-0.722826\pi$
$449$	−27.3024	−1.28848	−0.644240	+	0.764823i	$0.722826\pi$
$450$	0	0
$451$	28.7899	1.35566
$452$	0	0
$453$	5.82892	0.273867
$454$	0	0
$455$	−41.2183	−1.93234
$456$	0	0
$457$	−38.1038	−1.78242	−0.891211	−	0.453589i	$-0.850143\pi$
$457$	−38.1038	−1.78242	−0.891211	+	0.453589i	$0.850143\pi$
$458$	0	0
$459$	−20.1768	−0.941773
$460$	0	0
$461$	40.3315	1.87843	0.939213	−	0.343334i	$-0.111556\pi$
$461$	40.3315	1.87843	0.939213	+	0.343334i	$0.111556\pi$
$462$	0	0
$463$	2.16542	0.100636	0.0503178	−	0.998733i	$-0.483977\pi$
$463$	2.16542	0.100636	0.0503178	+	0.998733i	$0.483977\pi$
$464$	0	0
$465$	9.99504	0.463509
$466$	0	0
$467$	5.71207	0.264323	0.132162	−	0.991228i	$-0.457808\pi$
$467$	5.71207	0.264323	0.132162	+	0.991228i	$0.457808\pi$
$468$	0	0
$469$	−11.4078	−0.526764
$470$	0	0
$471$	−1.02627	−0.0472881
$472$	0	0
$473$	−40.0148	−1.83988
$474$	0	0
$475$	−17.2441	−0.791214
$476$	0	0
$477$	−16.7940	−0.768947
$478$	0	0
$479$	−26.1923	−1.19676	−0.598378	−	0.801214i	$-0.704188\pi$
$479$	−26.1923	−1.19676	−0.598378	+	0.801214i	$0.704188\pi$
$480$	0	0
$481$	−26.6208	−1.21381
$482$	0	0
$483$	6.18459	0.281409
$484$	0	0
$485$	−11.7382	−0.533006
$486$	0	0
$487$	23.5525	1.06726	0.533632	−	0.845717i	$-0.320827\pi$
$487$	23.5525	1.06726	0.533632	+	0.845717i	$0.320827\pi$
$488$	0	0
$489$	10.7621	0.486680
$490$	0	0
$491$	−18.8746	−0.851797	−0.425898	−	0.904771i	$-0.640042\pi$
$491$	−18.8746	−0.851797	−0.425898	+	0.904771i	$0.640042\pi$
$492$	0	0
$493$	3.94680	0.177755
$494$	0	0
$495$	−27.5560	−1.23855
$496$	0	0
$497$	23.2592	1.04332
$498$	0	0
$499$	3.09166	0.138402	0.0692008	−	0.997603i	$-0.477955\pi$
$499$	3.09166	0.138402	0.0692008	+	0.997603i	$0.477955\pi$
$500$	0	0
$501$	−9.98711	−0.446191
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−35.1708	−1.56508
$506$	0	0
$507$	−7.82352	−0.347455
$508$	0	0
$509$	−34.1838	−1.51517	−0.757586	−	0.652735i	$-0.773621\pi$
$509$	−34.1838	−1.51517	−0.757586	+	0.652735i	$0.773621\pi$
$510$	0	0
$511$	−12.4733	−0.551788
$512$	0	0
$513$	−28.8630	−1.27433
$514$	0	0
$515$	−53.3710	−2.35181
$516$	0	0
$517$	−14.9186	−0.656121
$518$	0	0
$519$	−12.2202	−0.536409
$520$	0	0
$521$	0.854474	0.0374352	0.0187176	−	0.999825i	$-0.494042\pi$
$521$	0.854474	0.0374352	0.0187176	+	0.999825i	$0.494042\pi$
$522$	0	0
$523$	13.1703	0.575897	0.287948	−	0.957646i	$-0.407027\pi$
$523$	13.1703	0.575897	0.287948	+	0.957646i	$0.407027\pi$
$524$	0	0
$525$	6.17512	0.269504
$526$	0	0
$527$	22.3045	0.971601
$528$	0	0
$529$	−16.6388	−0.723425
$530$	0	0
$531$	2.00319	0.0869309
$532$	0	0
$533$	32.8083	1.42109
$534$	0	0
$535$	−0.488643	−0.0211259
$536$	0	0
$537$	9.56780	0.412881
$538$	0	0
$539$	11.8880	0.512054
$540$	0	0
$541$	17.1478	0.737243	0.368621	−	0.929580i	$-0.379830\pi$
$541$	17.1478	0.737243	0.368621	+	0.929580i	$0.379830\pi$
$542$	0	0
$543$	18.6732	0.801344
$544$	0	0
$545$	−2.11911	−0.0907725
$546$	0	0
$547$	15.6202	0.667869	0.333935	−	0.942596i	$-0.391623\pi$
$547$	15.6202	0.667869	0.333935	+	0.942596i	$0.391623\pi$
$548$	0	0
$549$	4.44847	0.189856
$550$	0	0
$551$	5.64591	0.240524
$552$	0	0
$553$	6.54045	0.278128
$554$	0	0
$555$	11.9067	0.505411
$556$	0	0
$557$	25.5458	1.08241	0.541206	−	0.840890i	$-0.317968\pi$
$557$	25.5458	1.08241	0.541206	+	0.840890i	$0.317968\pi$
$558$	0	0
$559$	−45.6000	−1.92867
$560$	0	0
$561$	15.7598	0.665379
$562$	0	0
$563$	−33.3755	−1.40661	−0.703305	−	0.710889i	$-0.748293\pi$
$563$	−33.3755	−1.40661	−0.703305	+	0.710889i	$0.748293\pi$
$564$	0	0
$565$	24.0557	1.01203
$566$	0	0
$567$	−12.1193	−0.508962
$568$	0	0
$569$	24.1877	1.01400	0.507001	−	0.861945i	$-0.330754\pi$
$569$	24.1877	1.01400	0.507001	+	0.861945i	$0.330754\pi$
$570$	0	0
$571$	−19.9168	−0.833490	−0.416745	−	0.909023i	$-0.636829\pi$
$571$	−19.9168	−0.833490	−0.416745	+	0.909023i	$0.636829\pi$
$572$	0	0
$573$	−18.7110	−0.781665
$574$	0	0
$575$	6.35148	0.264875
$576$	0	0
$577$	8.75746	0.364578	0.182289	−	0.983245i	$-0.441649\pi$
$577$	8.75746	0.364578	0.182289	+	0.983245i	$0.441649\pi$
$578$	0	0
$579$	−2.16118	−0.0898156
$580$	0	0
$581$	9.18797	0.381181
$582$	0	0
$583$	29.5970	1.22578
$584$	0	0
$585$	−31.4023	−1.29832
$586$	0	0
$587$	30.4852	1.25826	0.629129	−	0.777301i	$-0.283412\pi$
$587$	30.4852	1.25826	0.629129	+	0.777301i	$0.283412\pi$
$588$	0	0
$589$	31.9067	1.31469
$590$	0	0
$591$	5.01330	0.206220
$592$	0	0
$593$	32.4939	1.33437	0.667183	−	0.744894i	$-0.267500\pi$
$593$	32.4939	1.33437	0.667183	+	0.744894i	$0.267500\pi$
$594$	0	0
$595$	41.1403	1.68659
$596$	0	0
$597$	−8.64083	−0.353646
$598$	0	0
$599$	28.3811	1.15962	0.579811	−	0.814751i	$-0.303126\pi$
$599$	28.3811	1.15962	0.579811	+	0.814751i	$0.303126\pi$
$600$	0	0
$601$	35.8320	1.46162	0.730809	−	0.682582i	$-0.239143\pi$
$601$	35.8320	1.46162	0.730809	+	0.682582i	$0.239143\pi$
$602$	0	0
$603$	−8.69107	−0.353928
$604$	0	0
$605$	18.4019	0.748145
$606$	0	0
$607$	13.1031	0.531840	0.265920	−	0.963995i	$-0.414324\pi$
$607$	13.1031	0.531840	0.265920	+	0.963995i	$0.414324\pi$
$608$	0	0
$609$	−2.02180	−0.0819275
$610$	0	0
$611$	−17.0009	−0.687785
$612$	0	0
$613$	40.3520	1.62980	0.814902	−	0.579599i	$-0.196791\pi$
$613$	40.3520	1.62980	0.814902	+	0.579599i	$0.196791\pi$
$614$	0	0
$615$	−14.6742	−0.591720
$616$	0	0
$617$	2.32963	0.0937873	0.0468937	−	0.998900i	$-0.485068\pi$
$617$	2.32963	0.0937873	0.0468937	+	0.998900i	$0.485068\pi$
$618$	0	0
$619$	10.7165	0.430734	0.215367	−	0.976533i	$-0.430905\pi$
$619$	10.7165	0.430734	0.215367	+	0.976533i	$0.430905\pi$
$620$	0	0
$621$	10.6311	0.426609
$622$	0	0
$623$	−29.3550	−1.17608
$624$	0	0
$625$	−31.2497	−1.24999
$626$	0	0
$627$	22.5445	0.900339
$628$	0	0
$629$	26.5705	1.05944
$630$	0	0
$631$	−47.9839	−1.91021	−0.955104	−	0.296272i	$-0.904257\pi$
$631$	−47.9839	−1.91021	−0.955104	+	0.296272i	$0.904257\pi$
$632$	0	0
$633$	14.5989	0.580255
$634$	0	0
$635$	−50.3657	−1.99870
$636$	0	0
$637$	13.5474	0.536766
$638$	0	0
$639$	17.7201	0.700995
$640$	0	0
$641$	13.3959	0.529104	0.264552	−	0.964371i	$-0.414776\pi$
$641$	13.3959	0.529104	0.264552	+	0.964371i	$0.414776\pi$
$642$	0	0
$643$	28.3774	1.11910	0.559548	−	0.828798i	$-0.310975\pi$
$643$	28.3774	1.11910	0.559548	+	0.828798i	$0.310975\pi$
$644$	0	0
$645$	20.3955	0.803072
$646$	0	0
$647$	−41.6159	−1.63609	−0.818045	−	0.575154i	$-0.804942\pi$
$647$	−41.6159	−1.63609	−0.818045	+	0.575154i	$0.804942\pi$
$648$	0	0
$649$	−3.53032	−0.138577
$650$	0	0
$651$	−11.4258	−0.447813
$652$	0	0
$653$	27.3149	1.06891	0.534457	−	0.845196i	$-0.320516\pi$
$653$	27.3149	1.06891	0.534457	+	0.845196i	$0.320516\pi$
$654$	0	0
$655$	−1.17363	−0.0458576
$656$	0	0
$657$	−9.50286	−0.370742
$658$	0	0
$659$	49.5904	1.93177	0.965884	−	0.258975i	$-0.0833848\pi$
$659$	49.5904	1.93177	0.965884	+	0.258975i	$0.0833848\pi$
$660$	0	0
$661$	16.8535	0.655527	0.327764	−	0.944760i	$-0.393705\pi$
$661$	16.8535	0.655527	0.327764	+	0.944760i	$0.393705\pi$
$662$	0	0
$663$	17.9595	0.697490
$664$	0	0
$665$	58.8514	2.28216
$666$	0	0
$667$	−2.07954	−0.0805203
$668$	0	0
$669$	−0.683498	−0.0264255
$670$	0	0
$671$	−7.83977	−0.302651
$672$	0	0
$673$	12.3614	0.476498	0.238249	−	0.971204i	$-0.423427\pi$
$673$	12.3614	0.476498	0.238249	+	0.971204i	$0.423427\pi$
$674$	0	0
$675$	10.6148	0.408563
$676$	0	0
$677$	−10.6556	−0.409530	−0.204765	−	0.978811i	$-0.565643\pi$
$677$	−10.6556	−0.409530	−0.204765	+	0.978811i	$0.565643\pi$
$678$	0	0
$679$	13.4185	0.514956
$680$	0	0
$681$	−2.23490	−0.0856414
$682$	0	0
$683$	18.8908	0.722834	0.361417	−	0.932404i	$-0.382293\pi$
$683$	18.8908	0.722834	0.361417	+	0.932404i	$0.382293\pi$
$684$	0	0
$685$	5.16210	0.197234
$686$	0	0
$687$	7.57697	0.289080
$688$	0	0
$689$	33.7281	1.28494
$690$	0	0
$691$	2.56497	0.0975760	0.0487880	−	0.998809i	$-0.484464\pi$
$691$	2.56497	0.0975760	0.0487880	+	0.998809i	$0.484464\pi$
$692$	0	0
$693$	31.5006	1.19661
$694$	0	0
$695$	−25.3889	−0.963056
$696$	0	0
$697$	−32.7463	−1.24035
$698$	0	0
$699$	−1.33419	−0.0504636
$700$	0	0
$701$	−5.46188	−0.206292	−0.103146	−	0.994666i	$-0.532891\pi$
$701$	−5.46188	−0.206292	−0.103146	+	0.994666i	$0.532891\pi$
$702$	0	0
$703$	38.0092	1.43354
$704$	0	0
$705$	7.60401	0.286383
$706$	0	0
$707$	40.2054	1.51208
$708$	0	0
$709$	24.0617	0.903654	0.451827	−	0.892106i	$-0.350772\pi$
$709$	24.0617	0.903654	0.451827	+	0.892106i	$0.350772\pi$
$710$	0	0
$711$	4.98286	0.186872
$712$	0	0
$713$	−11.7521	−0.440121
$714$	0	0
$715$	55.3418	2.06966
$716$	0	0
$717$	10.9838	0.410198
$718$	0	0
$719$	37.1305	1.38473	0.692367	−	0.721545i	$-0.256568\pi$
$719$	37.1305	1.38473	0.692367	+	0.721545i	$0.256568\pi$
$720$	0	0
$721$	61.0110	2.27217
$722$	0	0
$723$	−4.08840	−0.152049
$724$	0	0
$725$	−2.07636	−0.0771140
$726$	0	0
$727$	25.0661	0.929649	0.464825	−	0.885403i	$-0.346117\pi$
$727$	25.0661	0.929649	0.464825	+	0.885403i	$0.346117\pi$
$728$	0	0
$729$	0.659399	0.0244222
$730$	0	0
$731$	45.5138	1.68339
$732$	0	0
$733$	19.9521	0.736948	0.368474	−	0.929638i	$-0.379880\pi$
$733$	19.9521	0.736948	0.368474	+	0.929638i	$0.379880\pi$
$734$	0	0
$735$	−6.05932	−0.223501
$736$	0	0
$737$	15.3167	0.564198
$738$	0	0
$739$	1.65665	0.0609410	0.0304705	−	0.999536i	$-0.490299\pi$
$739$	1.65665	0.0609410	0.0304705	+	0.999536i	$0.490299\pi$
$740$	0	0
$741$	25.6912	0.943789
$742$	0	0
$743$	−21.0417	−0.771947	−0.385973	−	0.922510i	$-0.626134\pi$
$743$	−21.0417	−0.771947	−0.385973	+	0.922510i	$0.626134\pi$
$744$	0	0
$745$	3.41921	0.125270
$746$	0	0
$747$	6.99988	0.256112
$748$	0	0
$749$	0.558591	0.0204105
$750$	0	0
$751$	12.1436	0.443128	0.221564	−	0.975146i	$-0.428884\pi$
$751$	12.1436	0.443128	0.221564	+	0.975146i	$0.428884\pi$
$752$	0	0
$753$	−12.3692	−0.450757
$754$	0	0
$755$	−20.4300	−0.743524
$756$	0	0
$757$	25.3830	0.922559	0.461280	−	0.887255i	$-0.347391\pi$
$757$	25.3830	0.922559	0.461280	+	0.887255i	$0.347391\pi$
$758$	0	0
$759$	−8.30375	−0.301407
$760$	0	0
$761$	30.0640	1.08982	0.544909	−	0.838495i	$-0.316564\pi$
$761$	30.0640	1.08982	0.544909	+	0.838495i	$0.316564\pi$
$762$	0	0
$763$	2.42245	0.0876986
$764$	0	0
$765$	31.3429	1.13320
$766$	0	0
$767$	−4.02307	−0.145265
$768$	0	0
$769$	−51.9179	−1.87221	−0.936103	−	0.351725i	$-0.885595\pi$
$769$	−51.9179	−1.87221	−0.936103	+	0.351725i	$0.885595\pi$
$770$	0	0
$771$	14.1037	0.507932
$772$	0	0
$773$	−46.7248	−1.68057	−0.840287	−	0.542141i	$-0.817614\pi$
$773$	−46.7248	−1.68057	−0.840287	+	0.542141i	$0.817614\pi$
$774$	0	0
$775$	−11.7341	−0.421502
$776$	0	0
$777$	−13.6111	−0.488296
$778$	0	0
$779$	−46.8437	−1.67835
$780$	0	0
$781$	−31.2290	−1.11746
$782$	0	0
$783$	−3.47539	−0.124200
$784$	0	0
$785$	3.59702	0.128383
$786$	0	0
$787$	7.09460	0.252895	0.126448	−	0.991973i	$-0.459642\pi$
$787$	7.09460	0.252895	0.126448	+	0.991973i	$0.459642\pi$
$788$	0	0
$789$	−8.64535	−0.307783
$790$	0	0
$791$	−27.4992	−0.977758
$792$	0	0
$793$	−8.93403	−0.317257
$794$	0	0
$795$	−15.0855	−0.535029
$796$	0	0
$797$	−30.2747	−1.07239	−0.536193	−	0.844096i	$-0.680138\pi$
$797$	−30.2747	−1.07239	−0.536193	+	0.844096i	$0.680138\pi$
$798$	0	0
$799$	16.9688	0.600313
$800$	0	0
$801$	−22.3642	−0.790201
$802$	0	0
$803$	16.7474	0.591001
$804$	0	0
$805$	−21.6766	−0.764000
$806$	0	0
$807$	−18.5403	−0.652649
$808$	0	0
$809$	26.5275	0.932657	0.466328	−	0.884612i	$-0.345577\pi$
$809$	26.5275	0.932657	0.466328	+	0.884612i	$0.345577\pi$
$810$	0	0
$811$	12.5506	0.440711	0.220355	−	0.975420i	$-0.429278\pi$
$811$	12.5506	0.440711	0.220355	+	0.975420i	$0.429278\pi$
$812$	0	0
$813$	20.4979	0.718894
$814$	0	0
$815$	−37.7206	−1.32129
$816$	0	0
$817$	65.1076	2.27783
$818$	0	0
$819$	35.8974	1.25436
$820$	0	0
$821$	6.90554	0.241005	0.120503	−	0.992713i	$-0.461549\pi$
$821$	6.90554	0.241005	0.120503	+	0.992713i	$0.461549\pi$
$822$	0	0
$823$	29.7448	1.03684	0.518420	−	0.855126i	$-0.326520\pi$
$823$	29.7448	1.03684	0.518420	+	0.855126i	$0.326520\pi$
$824$	0	0
$825$	−8.29103	−0.288656
$826$	0	0
$827$	−0.0703256	−0.00244546	−0.00122273	−	0.999999i	$-0.500389\pi$
$827$	−0.0703256	−0.00244546	−0.00122273	+	0.999999i	$0.500389\pi$
$828$	0	0
$829$	56.8579	1.97476	0.987379	−	0.158376i	$-0.0506259\pi$
$829$	56.8579	1.97476	0.987379	+	0.158376i	$0.0506259\pi$
$830$	0	0
$831$	10.6225	0.368490
$832$	0	0
$833$	−13.5217	−0.468500
$834$	0	0
$835$	35.0042	1.21137
$836$	0	0
$837$	−19.6405	−0.678874
$838$	0	0
$839$	−44.4956	−1.53616	−0.768080	−	0.640354i	$-0.778788\pi$
$839$	−44.4956	−1.53616	−0.768080	+	0.640354i	$0.778788\pi$
$840$	0	0
$841$	−28.3202	−0.976558
$842$	0	0
$843$	10.9688	0.377785
$844$	0	0
$845$	27.4210	0.943309
$846$	0	0
$847$	−21.0361	−0.722810
$848$	0	0
$849$	0.788078	0.0270468
$850$	0	0
$851$	−13.9998	−0.479908
$852$	0	0
$853$	44.7271	1.53143	0.765714	−	0.643182i	$-0.222386\pi$
$853$	44.7271	1.53143	0.765714	+	0.643182i	$0.222386\pi$
$854$	0	0
$855$	44.8361	1.53336
$856$	0	0
$857$	−23.9161	−0.816957	−0.408479	−	0.912768i	$-0.633941\pi$
$857$	−23.9161	−0.816957	−0.408479	+	0.912768i	$0.633941\pi$
$858$	0	0
$859$	−48.8257	−1.66591	−0.832956	−	0.553339i	$-0.813353\pi$
$859$	−48.8257	−1.66591	−0.832956	+	0.553339i	$0.813353\pi$
$860$	0	0
$861$	16.7747	0.571682
$862$	0	0
$863$	−19.2384	−0.654882	−0.327441	−	0.944872i	$-0.606186\pi$
$863$	−19.2384	−0.654882	−0.327441	+	0.944872i	$0.606186\pi$
$864$	0	0
$865$	42.8312	1.45630
$866$	0	0
$867$	−4.62629	−0.157117
$868$	0	0
$869$	−8.78154	−0.297893
$870$	0	0
$871$	17.4546	0.591426
$872$	0	0
$873$	10.2230	0.345994
$874$	0	0
$875$	21.3291	0.721056
$876$	0	0
$877$	51.6079	1.74268	0.871338	−	0.490684i	$-0.163253\pi$
$877$	51.6079	1.74268	0.871338	+	0.490684i	$0.163253\pi$
$878$	0	0
$879$	12.3316	0.415935
$880$	0	0
$881$	53.5766	1.80504	0.902521	−	0.430645i	$-0.141714\pi$
$881$	53.5766	1.80504	0.902521	+	0.430645i	$0.141714\pi$
$882$	0	0
$883$	3.04666	0.102528	0.0512641	−	0.998685i	$-0.483675\pi$
$883$	3.04666	0.102528	0.0512641	+	0.998685i	$0.483675\pi$
$884$	0	0
$885$	1.79940	0.0604861
$886$	0	0
$887$	−15.5393	−0.521759	−0.260879	−	0.965371i	$-0.584012\pi$
$887$	−15.5393	−0.521759	−0.260879	+	0.965371i	$0.584012\pi$
$888$	0	0
$889$	57.5754	1.93102
$890$	0	0
$891$	16.2720	0.545131
$892$	0	0
$893$	24.2739	0.812296
$894$	0	0
$895$	−33.5345	−1.12094
$896$	0	0
$897$	−9.46277	−0.315953
$898$	0	0
$899$	3.84188	0.128134
$900$	0	0
$901$	−33.6643	−1.12152
$902$	0	0
$903$	−23.3151	−0.775877
$904$	0	0
$905$	−65.4484	−2.17558
$906$	0	0
$907$	−7.69810	−0.255611	−0.127806	−	0.991799i	$-0.540793\pi$
$907$	−7.69810	−0.255611	−0.127806	+	0.991799i	$0.540793\pi$
$908$	0	0
$909$	30.6306	1.01595
$910$	0	0
$911$	23.8300	0.789524	0.394762	−	0.918783i	$-0.370827\pi$
$911$	23.8300	0.789524	0.394762	+	0.918783i	$0.370827\pi$
$912$	0	0
$913$	−12.3362	−0.408270
$914$	0	0
$915$	3.99592	0.132101
$916$	0	0
$917$	1.34164	0.0443047
$918$	0	0
$919$	−10.2678	−0.338702	−0.169351	−	0.985556i	$-0.554167\pi$
$919$	−10.2678	−0.338702	−0.169351	+	0.985556i	$0.554167\pi$
$920$	0	0
$921$	−7.16177	−0.235988
$922$	0	0
$923$	−35.5879	−1.17139
$924$	0	0
$925$	−13.9784	−0.459607
$926$	0	0
$927$	46.4814	1.52665
$928$	0	0
$929$	27.2402	0.893722	0.446861	−	0.894603i	$-0.352542\pi$
$929$	27.2402	0.893722	0.446861	+	0.894603i	$0.352542\pi$
$930$	0	0
$931$	−19.3429	−0.633938
$932$	0	0
$933$	−14.7537	−0.483015
$934$	0	0
$935$	−55.2371	−1.80645
$936$	0	0
$937$	−16.9940	−0.555169	−0.277584	−	0.960701i	$-0.589534\pi$
$937$	−16.9940	−0.555169	−0.277584	+	0.960701i	$0.589534\pi$
$938$	0	0
$939$	7.77073	0.253588
$940$	0	0
$941$	−53.7017	−1.75062	−0.875312	−	0.483558i	$-0.839344\pi$
$941$	−53.7017	−1.75062	−0.875312	+	0.483558i	$0.839344\pi$
$942$	0	0
$943$	17.2538	0.561862
$944$	0	0
$945$	−36.2265	−1.17845
$946$	0	0
$947$	33.3047	1.08226	0.541129	−	0.840940i	$-0.317997\pi$
$947$	33.3047	1.08226	0.541129	+	0.840940i	$0.317997\pi$
$948$	0	0
$949$	19.0849	0.619523
$950$	0	0
$951$	12.4860	0.404887
$952$	0	0
$953$	25.8323	0.836791	0.418395	−	0.908265i	$-0.362593\pi$
$953$	25.8323	0.836791	0.418395	+	0.908265i	$0.362593\pi$
$954$	0	0
$955$	65.5810	2.12215
$956$	0	0
$957$	2.71457	0.0877497
$958$	0	0
$959$	−5.90105	−0.190555
$960$	0	0
$961$	−9.28836	−0.299624
$962$	0	0
$963$	0.425564	0.0137136
$964$	0	0
$965$	7.57480	0.243842
$966$	0	0
$967$	3.73502	0.120110	0.0600550	−	0.998195i	$-0.480872\pi$
$967$	3.73502	0.120110	0.0600550	+	0.998195i	$0.480872\pi$
$968$	0	0
$969$	−25.6426	−0.823759
$970$	0	0
$971$	−6.61697	−0.212349	−0.106174	−	0.994348i	$-0.533860\pi$
$971$	−6.61697	−0.212349	−0.106174	+	0.994348i	$0.533860\pi$
$972$	0	0
$973$	29.0233	0.930443
$974$	0	0
$975$	−9.44827	−0.302587
$976$	0	0
$977$	21.0556	0.673629	0.336814	−	0.941571i	$-0.390651\pi$
$977$	21.0556	0.673629	0.336814	+	0.941571i	$0.390651\pi$
$978$	0	0
$979$	39.4136	1.25966
$980$	0	0
$981$	1.84555	0.0589239
$982$	0	0
$983$	35.9454	1.14648	0.573240	−	0.819387i	$-0.305686\pi$
$983$	35.9454	1.14648	0.573240	+	0.819387i	$0.305686\pi$
$984$	0	0
$985$	−17.5713	−0.559868
$986$	0	0
$987$	−8.69250	−0.276685
$988$	0	0
$989$	−23.9809	−0.762550
$990$	0	0
$991$	8.89637	0.282602	0.141301	−	0.989967i	$-0.454871\pi$
$991$	8.89637	0.282602	0.141301	+	0.989967i	$0.454871\pi$
$992$	0	0
$993$	15.1473	0.480684
$994$	0	0
$995$	30.2856	0.960117
$996$	0	0
$997$	−10.9190	−0.345809	−0.172905	−	0.984939i	$-0.555315\pi$
$997$	−10.9190	−0.345809	−0.172905	+	0.984939i	$0.555315\pi$
$998$	0	0
$999$	−23.3969	−0.740246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.13	✓	29
4.3	odd	2		8048.2.a.w.1.17		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.13	✓	29	1.1	even	1	trivial
8048.2.a.w.1.17		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)

\(f(q)\)	\(=\)	\(q-0.782310 q^{3} +2.74195 q^{5} -3.13445 q^{7} -2.38799 q^{9} +O(q^{10})\)	\(q-0.782310 q^{3} +2.74195 q^{5} -3.13445 q^{7} -2.38799 q^{9} +4.20847 q^{11} +4.79589 q^{13} -2.14505 q^{15} -4.78682 q^{17} -6.84756 q^{19} +2.45211 q^{21} +2.52215 q^{23} +2.51828 q^{25} +4.21508 q^{27} -0.824514 q^{29} -4.65958 q^{31} -3.29233 q^{33} -8.59451 q^{35} -5.55076 q^{37} -3.75187 q^{39} +6.84093 q^{41} -9.50815 q^{43} -6.54775 q^{45} -3.54490 q^{47} +2.82479 q^{49} +3.74478 q^{51} +7.03271 q^{53} +11.5394 q^{55} +5.35692 q^{57} -0.838859 q^{59} -1.86285 q^{61} +7.48504 q^{63} +13.1501 q^{65} +3.63949 q^{67} -1.97310 q^{69} -7.42049 q^{71} +3.97944 q^{73} -1.97008 q^{75} -13.1913 q^{77} -2.08663 q^{79} +3.86647 q^{81} -2.93129 q^{83} -13.1252 q^{85} +0.645025 q^{87} +9.36528 q^{89} -15.0325 q^{91} +3.64523 q^{93} -18.7757 q^{95} -4.28098 q^{97} -10.0498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

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