LMFDB - Embedded newform 624.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.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:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.67544 q^{5} -22.9737 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +5.67544 q^{5} -22.9737 q^{7} +9.00000 q^{9} +32.5964 q^{11} -13.0000 q^{13} -17.0263 q^{15} +107.842 q^{17} -116.921 q^{19} +68.9210 q^{21} +65.1929 q^{23} -92.7893 q^{25} -27.0000 q^{27} +92.5964 q^{29} -51.1843 q^{31} -97.7893 q^{33} -130.386 q^{35} +267.737 q^{37} +39.0000 q^{39} -392.605 q^{41} -317.631 q^{43} +51.0790 q^{45} +114.965 q^{47} +184.789 q^{49} -323.526 q^{51} -618.000 q^{53} +184.999 q^{55} +350.763 q^{57} +600.824 q^{59} -857.263 q^{61} -206.763 q^{63} -73.7808 q^{65} -422.236 q^{67} -195.579 q^{69} +428.596 q^{71} -220.474 q^{73} +278.368 q^{75} -748.860 q^{77} +235.895 q^{79} +81.0000 q^{81} -64.3687 q^{83} +612.051 q^{85} -277.789 q^{87} -1074.97 q^{89} +298.658 q^{91} +153.553 q^{93} -663.579 q^{95} -703.105 q^{97} +293.368 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 24 q^{5} - 8 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 24 * q^5 - 8 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 24 q^{5} - 8 q^{7} + 18 q^{9} - 36 q^{11} - 26 q^{13} - 72 q^{15} - 12 q^{17} - 120 q^{19} + 24 q^{21} - 72 q^{23} + 118 q^{25} - 54 q^{27} + 84 q^{29} - 368 q^{31} + 108 q^{33} + 144 q^{35} + 156 q^{37} + 78 q^{39} - 216 q^{41} - 104 q^{43} + 216 q^{45} + 660 q^{47} + 66 q^{49} + 36 q^{51} - 1236 q^{53} - 1072 q^{55} + 360 q^{57} + 468 q^{59} - 652 q^{61} - 72 q^{63} - 312 q^{65} + 256 q^{67} + 216 q^{69} + 756 q^{71} - 1124 q^{73} - 354 q^{75} - 1776 q^{77} + 320 q^{79} + 162 q^{81} - 660 q^{83} - 1584 q^{85} - 252 q^{87} - 2112 q^{89} + 104 q^{91} + 1104 q^{93} - 720 q^{95} - 116 q^{97} - 324 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 24 * q^5 - 8 * q^7 + 18 * q^9 - 36 * q^11 - 26 * q^13 - 72 * q^15 - 12 * q^17 - 120 * q^19 + 24 * q^21 - 72 * q^23 + 118 * q^25 - 54 * q^27 + 84 * q^29 - 368 * q^31 + 108 * q^33 + 144 * q^35 + 156 * q^37 + 78 * q^39 - 216 * q^41 - 104 * q^43 + 216 * q^45 + 660 * q^47 + 66 * q^49 + 36 * q^51 - 1236 * q^53 - 1072 * q^55 + 360 * q^57 + 468 * q^59 - 652 * q^61 - 72 * q^63 - 312 * q^65 + 256 * q^67 + 216 * q^69 + 756 * q^71 - 1124 * q^73 - 354 * q^75 - 1776 * q^77 + 320 * q^79 + 162 * q^81 - 660 * q^83 - 1584 * q^85 - 252 * q^87 - 2112 * q^89 + 104 * q^91 + 1104 * q^93 - 720 * q^95 - 116 * q^97 - 324 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.67544	0.507627	0.253814	−	0.967253i	$-0.418315\pi$
$5$	5.67544	0.507627	0.253814	+	0.967253i	$0.418315\pi$
$6$	0	0
$7$	−22.9737	−1.24046	−0.620231	−	0.784419i	$-0.712961\pi$
$7$	−22.9737	−1.24046	−0.620231	+	0.784419i	$0.712961\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	32.5964	0.893472	0.446736	−	0.894666i	$-0.352586\pi$
$11$	32.5964	0.893472	0.446736	+	0.894666i	$0.352586\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−17.0263	−0.293079
$16$	0	0
$17$	107.842	1.53856	0.769280	−	0.638912i	$-0.220615\pi$
$17$	107.842	1.53856	0.769280	+	0.638912i	$0.220615\pi$
$18$	0	0
$19$	−116.921	−1.41176	−0.705882	−	0.708329i	$-0.749449\pi$
$19$	−116.921	−1.41176	−0.705882	+	0.708329i	$0.749449\pi$
$20$	0	0
$21$	68.9210	0.716181
$22$	0	0
$23$	65.1929	0.591029	0.295514	−	0.955338i	$-0.404509\pi$
$23$	65.1929	0.591029	0.295514	+	0.955338i	$0.404509\pi$
$24$	0	0
$25$	−92.7893	−0.742315
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	92.5964	0.592922	0.296461	−	0.955045i	$-0.404194\pi$
$29$	92.5964	0.592922	0.296461	+	0.955045i	$0.404194\pi$
$30$	0	0
$31$	−51.1843	−0.296548	−0.148274	−	0.988946i	$-0.547372\pi$
$31$	−51.1843	−0.296548	−0.148274	+	0.988946i	$0.547372\pi$
$32$	0	0
$33$	−97.7893	−0.515847
$34$	0	0
$35$	−130.386	−0.629692
$36$	0	0
$37$	267.737	1.18961	0.594806	−	0.803869i	$-0.297229\pi$
$37$	267.737	1.18961	0.594806	+	0.803869i	$0.297229\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−392.605	−1.49548	−0.747739	−	0.663993i	$-0.768861\pi$
$41$	−392.605	−1.49548	−0.747739	+	0.663993i	$0.768861\pi$
$42$	0	0
$43$	−317.631	−1.12647	−0.563236	−	0.826296i	$-0.690444\pi$
$43$	−317.631	−1.12647	−0.563236	+	0.826296i	$0.690444\pi$
$44$	0	0
$45$	51.0790	0.169209
$46$	0	0
$47$	114.965	0.356795	0.178398	−	0.983958i	$-0.442909\pi$
$47$	114.965	0.356795	0.178398	+	0.983958i	$0.442909\pi$
$48$	0	0
$49$	184.789	0.538744
$50$	0	0
$51$	−323.526	−0.888288
$52$	0	0
$53$	−618.000	−1.60168	−0.800838	−	0.598881i	$-0.795612\pi$
$53$	−618.000	−1.60168	−0.800838	+	0.598881i	$0.795612\pi$
$54$	0	0
$55$	184.999	0.453551
$56$	0	0
$57$	350.763	0.815082
$58$	0	0
$59$	600.824	1.32577	0.662887	−	0.748720i	$-0.269331\pi$
$59$	600.824	1.32577	0.662887	+	0.748720i	$0.269331\pi$
$60$	0	0
$61$	−857.263	−1.79936	−0.899682	−	0.436545i	$-0.856202\pi$
$61$	−857.263	−1.79936	−0.899682	+	0.436545i	$0.856202\pi$
$62$	0	0
$63$	−206.763	−0.413487
$64$	0	0
$65$	−73.7808	−0.140790
$66$	0	0
$67$	−422.236	−0.769916	−0.384958	−	0.922934i	$-0.625784\pi$
$67$	−422.236	−0.769916	−0.384958	+	0.922934i	$0.625784\pi$
$68$	0	0
$69$	−195.579	−0.341230
$70$	0	0
$71$	428.596	0.716409	0.358205	−	0.933643i	$-0.383389\pi$
$71$	428.596	0.716409	0.358205	+	0.933643i	$0.383389\pi$
$72$	0	0
$73$	−220.474	−0.353487	−0.176743	−	0.984257i	$-0.556556\pi$
$73$	−220.474	−0.353487	−0.176743	+	0.984257i	$0.556556\pi$
$74$	0	0
$75$	278.368	0.428576
$76$	0	0
$77$	−748.860	−1.10832
$78$	0	0
$79$	235.895	0.335952	0.167976	−	0.985791i	$-0.446277\pi$
$79$	235.895	0.335952	0.167976	+	0.985791i	$0.446277\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−64.3687	−0.0851251	−0.0425625	−	0.999094i	$-0.513552\pi$
$83$	−64.3687	−0.0851251	−0.0425625	+	0.999094i	$0.513552\pi$
$84$	0	0
$85$	612.051	0.781015
$86$	0	0
$87$	−277.789	−0.342323
$88$	0	0
$89$	−1074.97	−1.28030	−0.640152	−	0.768249i	$-0.721128\pi$
$89$	−1074.97	−1.28030	−0.640152	+	0.768249i	$0.721128\pi$
$90$	0	0
$91$	298.658	0.344042
$92$	0	0
$93$	153.553	0.171212
$94$	0	0
$95$	−663.579	−0.716650
$96$	0	0
$97$	−703.105	−0.735974	−0.367987	−	0.929831i	$-0.619953\pi$
$97$	−703.105	−0.735974	−0.367987	+	0.929831i	$0.619953\pi$
$98$	0	0
$99$	293.368	0.297824
$100$	0	0
$101$	−246.965	−0.243306	−0.121653	−	0.992573i	$-0.538820\pi$
$101$	−246.965	−0.243306	−0.121653	+	0.992573i	$0.538820\pi$
$102$	0	0
$103$	−1473.84	−1.40992	−0.704961	−	0.709246i	$-0.749036\pi$
$103$	−1473.84	−1.40992	−0.704961	+	0.709246i	$0.749036\pi$
$104$	0	0
$105$	391.157	0.363553
$106$	0	0
$107$	−670.070	−0.605403	−0.302701	−	0.953085i	$-0.597888\pi$
$107$	−670.070	−0.605403	−0.302701	+	0.953085i	$0.597888\pi$
$108$	0	0
$109$	−329.895	−0.289892	−0.144946	−	0.989440i	$-0.546301\pi$
$109$	−329.895	−0.289892	−0.144946	+	0.989440i	$0.546301\pi$
$110$	0	0
$111$	−803.210	−0.686823
$112$	0	0
$113$	−59.5089	−0.0495409	−0.0247705	−	0.999693i	$-0.507885\pi$
$113$	−59.5089	−0.0495409	−0.0247705	+	0.999693i	$0.507885\pi$
$114$	0	0
$115$	369.999	0.300022
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−2477.53	−1.90852
$120$	0	0
$121$	−268.472	−0.201707
$122$	0	0
$123$	1177.81	0.863415
$124$	0	0
$125$	−1236.05	−0.884446
$126$	0	0
$127$	−1981.10	−1.38421	−0.692105	−	0.721797i	$-0.743316\pi$
$127$	−1981.10	−1.38421	−0.692105	+	0.721797i	$0.743316\pi$
$128$	0	0
$129$	952.894	0.650369
$130$	0	0
$131$	−1665.12	−1.11055	−0.555276	−	0.831666i	$-0.687387\pi$
$131$	−1665.12	−1.11055	−0.555276	+	0.831666i	$0.687387\pi$
$132$	0	0
$133$	2686.10	1.75124
$134$	0	0
$135$	−153.237	−0.0976929
$136$	0	0
$137$	−3030.08	−1.88961	−0.944806	−	0.327629i	$-0.893750\pi$
$137$	−3030.08	−1.88961	−0.944806	+	0.327629i	$0.893750\pi$
$138$	0	0
$139$	−37.7879	−0.0230585	−0.0115292	−	0.999934i	$-0.503670\pi$
$139$	−37.7879	−0.0230585	−0.0115292	+	0.999934i	$0.503670\pi$
$140$	0	0
$141$	−344.895	−0.205996
$142$	0	0
$143$	−423.754	−0.247805
$144$	0	0
$145$	525.526	0.300983
$146$	0	0
$147$	−554.368	−0.311044
$148$	0	0
$149$	1877.76	1.03243	0.516215	−	0.856459i	$-0.327340\pi$
$149$	1877.76	1.03243	0.516215	+	0.856459i	$0.327340\pi$
$150$	0	0
$151$	−559.657	−0.301617	−0.150809	−	0.988563i	$-0.548188\pi$
$151$	−559.657	−0.301617	−0.150809	+	0.988563i	$0.548188\pi$
$152$	0	0
$153$	970.578	0.512853
$154$	0	0
$155$	−290.494	−0.150536
$156$	0	0
$157$	−1463.68	−0.744042	−0.372021	−	0.928224i	$-0.621335\pi$
$157$	−1463.68	−0.744042	−0.372021	+	0.928224i	$0.621335\pi$
$158$	0	0
$159$	1854.00	0.924728
$160$	0	0
$161$	−1497.72	−0.733148
$162$	0	0
$163$	2818.39	1.35432	0.677158	−	0.735837i	$-0.263211\pi$
$163$	2818.39	1.35432	0.677158	+	0.735837i	$0.263211\pi$
$164$	0	0
$165$	−554.998	−0.261858
$166$	0	0
$167$	798.051	0.369791	0.184895	−	0.982758i	$-0.440805\pi$
$167$	798.051	0.369791	0.184895	+	0.982758i	$0.440805\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1052.29	−0.470588
$172$	0	0
$173$	4305.28	1.89205	0.946024	−	0.324096i	$-0.105060\pi$
$173$	4305.28	1.89205	0.946024	+	0.324096i	$0.105060\pi$
$174$	0	0
$175$	2131.71	0.920813
$176$	0	0
$177$	−1802.47	−0.765436
$178$	0	0
$179$	−1086.84	−0.453823	−0.226912	−	0.973915i	$-0.572863\pi$
$179$	−1086.84	−0.453823	−0.226912	+	0.973915i	$0.572863\pi$
$180$	0	0
$181$	3358.21	1.37908	0.689541	−	0.724247i	$-0.257812\pi$
$181$	3358.21	1.37908	0.689541	+	0.724247i	$0.257812\pi$
$182$	0	0
$183$	2571.79	1.03886
$184$	0	0
$185$	1519.52	0.603879
$186$	0	0
$187$	3515.27	1.37466
$188$	0	0
$189$	620.289	0.238727
$190$	0	0
$191$	−1242.77	−0.470806	−0.235403	−	0.971898i	$-0.575641\pi$
$191$	−1242.77	−0.470806	−0.235403	+	0.971898i	$0.575641\pi$
$192$	0	0
$193$	379.577	0.141568	0.0707839	−	0.997492i	$-0.477450\pi$
$193$	379.577	0.141568	0.0707839	+	0.997492i	$0.477450\pi$
$194$	0	0
$195$	221.342	0.0812854
$196$	0	0
$197$	4342.55	1.57053	0.785264	−	0.619161i	$-0.212527\pi$
$197$	4342.55	1.57053	0.785264	+	0.619161i	$0.212527\pi$
$198$	0	0
$199$	−318.369	−0.113410	−0.0567049	−	0.998391i	$-0.518059\pi$
$199$	−318.369	−0.113410	−0.0567049	+	0.998391i	$0.518059\pi$
$200$	0	0
$201$	1266.71	0.444511
$202$	0	0
$203$	−2127.28	−0.735496
$204$	0	0
$205$	−2228.21	−0.759145
$206$	0	0
$207$	586.736	0.197010
$208$	0	0
$209$	−3811.21	−1.26137
$210$	0	0
$211$	4944.21	1.61314	0.806572	−	0.591136i	$-0.201320\pi$
$211$	4944.21	1.61314	0.806572	+	0.591136i	$0.201320\pi$
$212$	0	0
$213$	−1285.79	−0.413619
$214$	0	0
$215$	−1802.70	−0.571828
$216$	0	0
$217$	1175.89	0.367856
$218$	0	0
$219$	661.422	0.204086
$220$	0	0
$221$	−1401.95	−0.426720
$222$	0	0
$223$	1335.45	0.401023	0.200511	−	0.979691i	$-0.435740\pi$
$223$	1335.45	0.401023	0.200511	+	0.979691i	$0.435740\pi$
$224$	0	0
$225$	−835.104	−0.247438
$226$	0	0
$227$	2381.68	0.696378	0.348189	−	0.937424i	$-0.386797\pi$
$227$	2381.68	0.696378	0.348189	+	0.937424i	$0.386797\pi$
$228$	0	0
$229$	−5005.89	−1.44454	−0.722268	−	0.691614i	$-0.756900\pi$
$229$	−5005.89	−1.44454	−0.722268	+	0.691614i	$0.756900\pi$
$230$	0	0
$231$	2246.58	0.639888
$232$	0	0
$233$	943.666	0.265329	0.132664	−	0.991161i	$-0.457647\pi$
$233$	943.666	0.265329	0.132664	+	0.991161i	$0.457647\pi$
$234$	0	0
$235$	652.478	0.181119
$236$	0	0
$237$	−707.684	−0.193962
$238$	0	0
$239$	4493.15	1.21606	0.608029	−	0.793915i	$-0.291960\pi$
$239$	4493.15	1.21606	0.608029	+	0.793915i	$0.291960\pi$
$240$	0	0
$241$	−2780.90	−0.743291	−0.371646	−	0.928375i	$-0.621206\pi$
$241$	−2780.90	−0.743291	−0.371646	+	0.928375i	$0.621206\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1048.76	0.273481
$246$	0	0
$247$	1519.97	0.391553
$248$	0	0
$249$	193.106	0.0491470
$250$	0	0
$251$	−1999.16	−0.502732	−0.251366	−	0.967892i	$-0.580880\pi$
$251$	−1999.16	−0.502732	−0.251366	+	0.967892i	$0.580880\pi$
$252$	0	0
$253$	2125.06	0.528068
$254$	0	0
$255$	−1836.15	−0.450919
$256$	0	0
$257$	−6149.42	−1.49257	−0.746285	−	0.665627i	$-0.768164\pi$
$257$	−6149.42	−1.49257	−0.746285	+	0.665627i	$0.768164\pi$
$258$	0	0
$259$	−6150.89	−1.47567
$260$	0	0
$261$	833.368	0.197641
$262$	0	0
$263$	−3362.31	−0.788324	−0.394162	−	0.919041i	$-0.628965\pi$
$263$	−3362.31	−0.788324	−0.394162	+	0.919041i	$0.628965\pi$
$264$	0	0
$265$	−3507.42	−0.813054
$266$	0	0
$267$	3224.92	0.739183
$268$	0	0
$269$	1972.32	0.447042	0.223521	−	0.974699i	$-0.428245\pi$
$269$	1972.32	0.447042	0.223521	+	0.974699i	$0.428245\pi$
$270$	0	0
$271$	−3662.55	−0.820975	−0.410487	−	0.911866i	$-0.634641\pi$
$271$	−3662.55	−0.820975	−0.410487	+	0.911866i	$0.634641\pi$
$272$	0	0
$273$	−895.973	−0.198633
$274$	0	0
$275$	−3024.60	−0.663238
$276$	0	0
$277$	8367.26	1.81495	0.907473	−	0.420111i	$-0.138009\pi$
$277$	8367.26	1.81495	0.907473	+	0.420111i	$0.138009\pi$
$278$	0	0
$279$	−460.659	−0.0988492
$280$	0	0
$281$	934.689	0.198430	0.0992151	−	0.995066i	$-0.468367\pi$
$281$	934.689	0.198430	0.0992151	+	0.995066i	$0.468367\pi$
$282$	0	0
$283$	5297.94	1.11283	0.556413	−	0.830906i	$-0.312177\pi$
$283$	5297.94	1.11283	0.556413	+	0.830906i	$0.312177\pi$
$284$	0	0
$285$	1990.74	0.413758
$286$	0	0
$287$	9019.58	1.85508
$288$	0	0
$289$	6716.90	1.36717
$290$	0	0
$291$	2109.31	0.424915
$292$	0	0
$293$	5212.53	1.03932	0.519658	−	0.854375i	$-0.326060\pi$
$293$	5212.53	1.03932	0.519658	+	0.854375i	$0.326060\pi$
$294$	0	0
$295$	3409.94	0.672999
$296$	0	0
$297$	−880.104	−0.171949
$298$	0	0
$299$	−847.508	−0.163922
$300$	0	0
$301$	7297.16	1.39735
$302$	0	0
$303$	740.895	0.140473
$304$	0	0
$305$	−4865.35	−0.913406
$306$	0	0
$307$	5296.34	0.984619	0.492309	−	0.870420i	$-0.336153\pi$
$307$	5296.34	0.984619	0.492309	+	0.870420i	$0.336153\pi$
$308$	0	0
$309$	4421.53	0.814019
$310$	0	0
$311$	−9716.63	−1.77164	−0.885819	−	0.464031i	$-0.846403\pi$
$311$	−9716.63	−1.77164	−0.885819	+	0.464031i	$0.846403\pi$
$312$	0	0
$313$	−5412.10	−0.977348	−0.488674	−	0.872466i	$-0.662519\pi$
$313$	−5412.10	−0.977348	−0.488674	+	0.872466i	$0.662519\pi$
$314$	0	0
$315$	−1173.47	−0.209897
$316$	0	0
$317$	−734.293	−0.130101	−0.0650505	−	0.997882i	$-0.520721\pi$
$317$	−734.293	−0.130101	−0.0650505	+	0.997882i	$0.520721\pi$
$318$	0	0
$319$	3018.31	0.529759
$320$	0	0
$321$	2010.21	0.349529
$322$	0	0
$323$	−12609.0	−2.17208
$324$	0	0
$325$	1206.26	0.205881
$326$	0	0
$327$	989.684	0.167369
$328$	0	0
$329$	−2641.17	−0.442591
$330$	0	0
$331$	−11844.4	−1.96686	−0.983428	−	0.181301i	$-0.941969\pi$
$331$	−11844.4	−1.96686	−0.983428	+	0.181301i	$0.941969\pi$
$332$	0	0
$333$	2409.63	0.396537
$334$	0	0
$335$	−2396.38	−0.390830
$336$	0	0
$337$	−394.103	−0.0637037	−0.0318518	−	0.999493i	$-0.510140\pi$
$337$	−394.103	−0.0637037	−0.0318518	+	0.999493i	$0.510140\pi$
$338$	0	0
$339$	178.527	0.0286025
$340$	0	0
$341$	−1668.43	−0.264957
$342$	0	0
$343$	3634.68	0.572170
$344$	0	0
$345$	−1110.00	−0.173218
$346$	0	0
$347$	369.369	0.0571435	0.0285717	−	0.999592i	$-0.490904\pi$
$347$	369.369	0.0571435	0.0285717	+	0.999592i	$0.490904\pi$
$348$	0	0
$349$	−779.523	−0.119561	−0.0597807	−	0.998212i	$-0.519040\pi$
$349$	−779.523	−0.119561	−0.0597807	+	0.998212i	$0.519040\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	11043.3	1.66509	0.832545	−	0.553958i	$-0.186883\pi$
$353$	11043.3	1.66509	0.832545	+	0.553958i	$0.186883\pi$
$354$	0	0
$355$	2432.48	0.363669
$356$	0	0
$357$	7432.58	1.10189
$358$	0	0
$359$	4373.42	0.642953	0.321477	−	0.946918i	$-0.395821\pi$
$359$	4373.42	0.642953	0.321477	+	0.946918i	$0.395821\pi$
$360$	0	0
$361$	6811.52	0.993078
$362$	0	0
$363$	805.416	0.116456
$364$	0	0
$365$	−1251.29	−0.179439
$366$	0	0
$367$	−1471.37	−0.209277	−0.104639	−	0.994510i	$-0.533369\pi$
$367$	−1471.37	−0.209277	−0.104639	+	0.994510i	$0.533369\pi$
$368$	0	0
$369$	−3533.44	−0.498493
$370$	0	0
$371$	14197.7	1.98682
$372$	0	0
$373$	6819.26	0.946617	0.473308	−	0.880897i	$-0.343060\pi$
$373$	6819.26	0.946617	0.473308	+	0.880897i	$0.343060\pi$
$374$	0	0
$375$	3708.15	0.510635
$376$	0	0
$377$	−1203.75	−0.164447
$378$	0	0
$379$	−12907.5	−1.74938	−0.874692	−	0.484680i	$-0.838936\pi$
$379$	−12907.5	−1.74938	−0.874692	+	0.484680i	$0.838936\pi$
$380$	0	0
$381$	5943.31	0.799174
$382$	0	0
$383$	13211.1	1.76255	0.881274	−	0.472606i	$-0.156687\pi$
$383$	13211.1	1.76255	0.881274	+	0.472606i	$0.156687\pi$
$384$	0	0
$385$	−4250.11	−0.562612
$386$	0	0
$387$	−2858.68	−0.375491
$388$	0	0
$389$	−5715.52	−0.744958	−0.372479	−	0.928041i	$-0.621492\pi$
$389$	−5715.52	−0.744958	−0.372479	+	0.928041i	$0.621492\pi$
$390$	0	0
$391$	7030.53	0.909333
$392$	0	0
$393$	4995.37	0.641178
$394$	0	0
$395$	1338.81	0.170538
$396$	0	0
$397$	6447.73	0.815120	0.407560	−	0.913179i	$-0.366380\pi$
$397$	6447.73	0.815120	0.407560	+	0.913179i	$0.366380\pi$
$398$	0	0
$399$	−8058.31	−1.01108
$400$	0	0
$401$	−9866.44	−1.22869	−0.614347	−	0.789036i	$-0.710581\pi$
$401$	−9866.44	−1.22869	−0.614347	+	0.789036i	$0.710581\pi$
$402$	0	0
$403$	665.396	0.0822475
$404$	0	0
$405$	459.711	0.0564030
$406$	0	0
$407$	8727.26	1.06289
$408$	0	0
$409$	−898.259	−0.108597	−0.0542984	−	0.998525i	$-0.517292\pi$
$409$	−898.259	−0.108597	−0.0542984	+	0.998525i	$0.517292\pi$
$410$	0	0
$411$	9090.23	1.09097
$412$	0	0
$413$	−13803.1	−1.64457
$414$	0	0
$415$	−365.321	−0.0432118
$416$	0	0
$417$	113.364	0.0133128
$418$	0	0
$419$	−16305.5	−1.90113	−0.950566	−	0.310524i	$-0.899496\pi$
$419$	−16305.5	−1.90113	−0.950566	+	0.310524i	$0.899496\pi$
$420$	0	0
$421$	−7803.99	−0.903428	−0.451714	−	0.892163i	$-0.649187\pi$
$421$	−7803.99	−0.903428	−0.451714	+	0.892163i	$0.649187\pi$
$422$	0	0
$423$	1034.69	0.118932
$424$	0	0
$425$	−10006.6	−1.14210
$426$	0	0
$427$	19694.5	2.23204
$428$	0	0
$429$	1271.26	0.143070
$430$	0	0
$431$	−7404.68	−0.827543	−0.413771	−	0.910381i	$-0.635789\pi$
$431$	−7404.68	−0.827543	−0.413771	+	0.910381i	$0.635789\pi$
$432$	0	0
$433$	6269.05	0.695777	0.347889	−	0.937536i	$-0.386899\pi$
$433$	6269.05	0.695777	0.347889	+	0.937536i	$0.386899\pi$
$434$	0	0
$435$	−1576.58	−0.173773
$436$	0	0
$437$	−7622.42	−0.834393
$438$	0	0
$439$	−10573.0	−1.14948	−0.574740	−	0.818336i	$-0.694897\pi$
$439$	−10573.0	−1.14948	−0.574740	+	0.818336i	$0.694897\pi$
$440$	0	0
$441$	1663.10	0.179581
$442$	0	0
$443$	3593.58	0.385409	0.192704	−	0.981257i	$-0.438274\pi$
$443$	3593.58	0.385409	0.192704	+	0.981257i	$0.438274\pi$
$444$	0	0
$445$	−6100.95	−0.649917
$446$	0	0
$447$	−5633.28	−0.596074
$448$	0	0
$449$	6617.79	0.695574	0.347787	−	0.937574i	$-0.386933\pi$
$449$	6617.79	0.695574	0.347787	+	0.937574i	$0.386933\pi$
$450$	0	0
$451$	−12797.5	−1.33617
$452$	0	0
$453$	1678.97	0.174139
$454$	0	0
$455$	1695.02	0.174645
$456$	0	0
$457$	−14995.6	−1.53494	−0.767468	−	0.641088i	$-0.778484\pi$
$457$	−14995.6	−1.53494	−0.767468	+	0.641088i	$0.778484\pi$
$458$	0	0
$459$	−2911.73	−0.296096
$460$	0	0
$461$	−5981.77	−0.604336	−0.302168	−	0.953255i	$-0.597710\pi$
$461$	−5981.77	−0.604336	−0.302168	+	0.953255i	$0.597710\pi$
$462$	0	0
$463$	−6186.02	−0.620926	−0.310463	−	0.950585i	$-0.600484\pi$
$463$	−6186.02	−0.620926	−0.310463	+	0.950585i	$0.600484\pi$
$464$	0	0
$465$	871.482	0.0869118
$466$	0	0
$467$	−9009.53	−0.892744	−0.446372	−	0.894848i	$-0.647284\pi$
$467$	−9009.53	−0.892744	−0.446372	+	0.894848i	$0.647284\pi$
$468$	0	0
$469$	9700.32	0.955051
$470$	0	0
$471$	4391.05	0.429573
$472$	0	0
$473$	−10353.7	−1.00647
$474$	0	0
$475$	10849.0	1.04797
$476$	0	0
$477$	−5562.00	−0.533892
$478$	0	0
$479$	−10666.3	−1.01745	−0.508723	−	0.860930i	$-0.669882\pi$
$479$	−10666.3	−1.01745	−0.508723	+	0.860930i	$0.669882\pi$
$480$	0	0
$481$	−3480.58	−0.329939
$482$	0	0
$483$	4493.16	0.423283
$484$	0	0
$485$	−3990.43	−0.373600
$486$	0	0
$487$	3171.12	0.295066	0.147533	−	0.989057i	$-0.452867\pi$
$487$	3171.12	0.295066	0.147533	+	0.989057i	$0.452867\pi$
$488$	0	0
$489$	−8455.18	−0.781915
$490$	0	0
$491$	12570.6	1.15540	0.577701	−	0.816248i	$-0.303950\pi$
$491$	12570.6	1.15540	0.577701	+	0.816248i	$0.303950\pi$
$492$	0	0
$493$	9985.79	0.912246
$494$	0	0
$495$	1664.99	0.151184
$496$	0	0
$497$	−9846.43	−0.888678
$498$	0	0
$499$	−4547.81	−0.407992	−0.203996	−	0.978972i	$-0.565393\pi$
$499$	−4547.81	−0.407992	−0.203996	+	0.978972i	$0.565393\pi$
$500$	0	0
$501$	−2394.15	−0.213499
$502$	0	0
$503$	22192.6	1.96723	0.983616	−	0.180274i	$-0.0576984\pi$
$503$	22192.6	1.96723	0.983616	+	0.180274i	$0.0576984\pi$
$504$	0	0
$505$	−1401.64	−0.123509
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	12027.0	1.04732	0.523662	−	0.851926i	$-0.324565\pi$
$509$	12027.0	1.04732	0.523662	+	0.851926i	$0.324565\pi$
$510$	0	0
$511$	5065.10	0.438487
$512$	0	0
$513$	3156.87	0.271694
$514$	0	0
$515$	−8364.71	−0.715715
$516$	0	0
$517$	3747.45	0.318787
$518$	0	0
$519$	−12915.8	−1.09237
$520$	0	0
$521$	9841.05	0.827532	0.413766	−	0.910383i	$-0.364213\pi$
$521$	9841.05	0.827532	0.413766	+	0.910383i	$0.364213\pi$
$522$	0	0
$523$	−8447.46	−0.706275	−0.353137	−	0.935571i	$-0.614885\pi$
$523$	−8447.46	−0.706275	−0.353137	+	0.935571i	$0.614885\pi$
$524$	0	0
$525$	−6395.13	−0.531631
$526$	0	0
$527$	−5519.82	−0.456257
$528$	0	0
$529$	−7916.89	−0.650685
$530$	0	0
$531$	5407.42	0.441925
$532$	0	0
$533$	5103.86	0.414771
$534$	0	0
$535$	−3802.94	−0.307319
$536$	0	0
$537$	3260.52	0.262015
$538$	0	0
$539$	6023.47	0.481353
$540$	0	0
$541$	287.472	0.0228454	0.0114227	−	0.999935i	$-0.496364\pi$
$541$	287.472	0.0228454	0.0114227	+	0.999935i	$0.496364\pi$
$542$	0	0
$543$	−10074.6	−0.796213
$544$	0	0
$545$	−1872.30	−0.147157
$546$	0	0
$547$	14498.4	1.13329	0.566643	−	0.823964i	$-0.308242\pi$
$547$	14498.4	1.13329	0.566643	+	0.823964i	$0.308242\pi$
$548$	0	0
$549$	−7715.36	−0.599788
$550$	0	0
$551$	−10826.5	−0.837065
$552$	0	0
$553$	−5419.37	−0.416736
$554$	0	0
$555$	−4558.57	−0.348650
$556$	0	0
$557$	7846.18	0.596864	0.298432	−	0.954431i	$-0.403536\pi$
$557$	7846.18	0.596864	0.298432	+	0.954431i	$0.403536\pi$
$558$	0	0
$559$	4129.21	0.312427
$560$	0	0
$561$	−10545.8	−0.793661
$562$	0	0
$563$	−24877.5	−1.86228	−0.931140	−	0.364663i	$-0.881184\pi$
$563$	−24877.5	−1.86228	−0.931140	+	0.364663i	$0.881184\pi$
$564$	0	0
$565$	−337.739	−0.0251483
$566$	0	0
$567$	−1860.87	−0.137829
$568$	0	0
$569$	950.720	0.0700462	0.0350231	−	0.999387i	$-0.488850\pi$
$569$	950.720	0.0700462	0.0350231	+	0.999387i	$0.488850\pi$
$570$	0	0
$571$	15853.2	1.16188	0.580942	−	0.813945i	$-0.302684\pi$
$571$	15853.2	1.16188	0.580942	+	0.813945i	$0.302684\pi$
$572$	0	0
$573$	3728.32	0.271820
$574$	0	0
$575$	−6049.20	−0.438729
$576$	0	0
$577$	4575.80	0.330144	0.165072	−	0.986282i	$-0.447214\pi$
$577$	4575.80	0.330144	0.165072	+	0.986282i	$0.447214\pi$
$578$	0	0
$579$	−1138.73	−0.0817342
$580$	0	0
$581$	1478.78	0.105594
$582$	0	0
$583$	−20144.6	−1.43105
$584$	0	0
$585$	−664.027	−0.0469302
$586$	0	0
$587$	8391.13	0.590015	0.295008	−	0.955495i	$-0.404678\pi$
$587$	8391.13	0.590015	0.295008	+	0.955495i	$0.404678\pi$
$588$	0	0
$589$	5984.52	0.418655
$590$	0	0
$591$	−13027.7	−0.906745
$592$	0	0
$593$	14064.6	0.973967	0.486984	−	0.873411i	$-0.338097\pi$
$593$	14064.6	0.973967	0.486984	+	0.873411i	$0.338097\pi$
$594$	0	0
$595$	−14061.1	−0.968819
$596$	0	0
$597$	955.106	0.0654772
$598$	0	0
$599$	−5955.20	−0.406215	−0.203108	−	0.979156i	$-0.565104\pi$
$599$	−5955.20	−0.406215	−0.203108	+	0.979156i	$0.565104\pi$
$600$	0	0
$601$	−29271.3	−1.98669	−0.993344	−	0.115186i	$-0.963254\pi$
$601$	−29271.3	−1.98669	−0.993344	+	0.115186i	$0.963254\pi$
$602$	0	0
$603$	−3800.13	−0.256639
$604$	0	0
$605$	−1523.70	−0.102392
$606$	0	0
$607$	−14184.9	−0.948516	−0.474258	−	0.880386i	$-0.657284\pi$
$607$	−14184.9	−0.948516	−0.474258	+	0.880386i	$0.657284\pi$
$608$	0	0
$609$	6381.84	0.424639
$610$	0	0
$611$	−1494.55	−0.0989573
$612$	0	0
$613$	−1539.42	−0.101430	−0.0507148	−	0.998713i	$-0.516150\pi$
$613$	−1539.42	−0.101430	−0.0507148	+	0.998713i	$0.516150\pi$
$614$	0	0
$615$	6684.62	0.438293
$616$	0	0
$617$	−26387.4	−1.72175	−0.860873	−	0.508820i	$-0.830082\pi$
$617$	−26387.4	−1.72175	−0.860873	+	0.508820i	$0.830082\pi$
$618$	0	0
$619$	−22476.7	−1.45948	−0.729738	−	0.683727i	$-0.760358\pi$
$619$	−22476.7	−1.45948	−0.729738	+	0.683727i	$0.760358\pi$
$620$	0	0
$621$	−1760.21	−0.113743
$622$	0	0
$623$	24696.1	1.58817
$624$	0	0
$625$	4583.53	0.293346
$626$	0	0
$627$	11433.6	0.728254
$628$	0	0
$629$	28873.3	1.83029
$630$	0	0
$631$	−12025.1	−0.758657	−0.379329	−	0.925262i	$-0.623845\pi$
$631$	−12025.1	−0.758657	−0.379329	+	0.925262i	$0.623845\pi$
$632$	0	0
$633$	−14832.6	−0.931349
$634$	0	0
$635$	−11243.6	−0.702662
$636$	0	0
$637$	−2402.26	−0.149421
$638$	0	0
$639$	3857.37	0.238803
$640$	0	0
$641$	30396.6	1.87300	0.936500	−	0.350667i	$-0.114045\pi$
$641$	30396.6	1.87300	0.936500	+	0.350667i	$0.114045\pi$
$642$	0	0
$643$	−946.866	−0.0580727	−0.0290363	−	0.999578i	$-0.509244\pi$
$643$	−946.866	−0.0580727	−0.0290363	+	0.999578i	$0.509244\pi$
$644$	0	0
$645$	5408.10	0.330145
$646$	0	0
$647$	6154.71	0.373982	0.186991	−	0.982362i	$-0.440126\pi$
$647$	6154.71	0.373982	0.186991	+	0.982362i	$0.440126\pi$
$648$	0	0
$649$	19584.7	1.18454
$650$	0	0
$651$	−3527.68	−0.212382
$652$	0	0
$653$	25100.8	1.50424	0.752122	−	0.659024i	$-0.229031\pi$
$653$	25100.8	1.50424	0.752122	+	0.659024i	$0.229031\pi$
$654$	0	0
$655$	−9450.31	−0.563747
$656$	0	0
$657$	−1984.27	−0.117829
$658$	0	0
$659$	−29931.0	−1.76927	−0.884634	−	0.466286i	$-0.845592\pi$
$659$	−29931.0	−1.76927	−0.884634	+	0.466286i	$0.845592\pi$
$660$	0	0
$661$	−21629.4	−1.27275	−0.636374	−	0.771381i	$-0.719567\pi$
$661$	−21629.4	−1.27275	−0.636374	+	0.771381i	$0.719567\pi$
$662$	0	0
$663$	4205.84	0.246367
$664$	0	0
$665$	15244.8	0.888976
$666$	0	0
$667$	6036.63	0.350434
$668$	0	0
$669$	−4006.34	−0.231531
$670$	0	0
$671$	−27943.7	−1.60768
$672$	0	0
$673$	−23196.7	−1.32863	−0.664315	−	0.747453i	$-0.731277\pi$
$673$	−23196.7	−1.32863	−0.664315	+	0.747453i	$0.731277\pi$
$674$	0	0
$675$	2505.31	0.142859
$676$	0	0
$677$	−12108.8	−0.687415	−0.343707	−	0.939077i	$-0.611683\pi$
$677$	−12108.8	−0.687415	−0.343707	+	0.939077i	$0.611683\pi$
$678$	0	0
$679$	16152.9	0.912947
$680$	0	0
$681$	−7145.05	−0.402054
$682$	0	0
$683$	18673.1	1.04613	0.523065	−	0.852293i	$-0.324788\pi$
$683$	18673.1	1.04613	0.523065	+	0.852293i	$0.324788\pi$
$684$	0	0
$685$	−17197.0	−0.959219
$686$	0	0
$687$	15017.7	0.834003
$688$	0	0
$689$	8034.00	0.444225
$690$	0	0
$691$	23165.4	1.27533	0.637666	−	0.770313i	$-0.279900\pi$
$691$	23165.4	1.27533	0.637666	+	0.770313i	$0.279900\pi$
$692$	0	0
$693$	−6739.74	−0.369439
$694$	0	0
$695$	−214.463	−0.0117051
$696$	0	0
$697$	−42339.3	−2.30088
$698$	0	0
$699$	−2831.00	−0.153188
$700$	0	0
$701$	−24631.3	−1.32712	−0.663561	−	0.748123i	$-0.730956\pi$
$701$	−24631.3	−1.32712	−0.663561	+	0.748123i	$0.730956\pi$
$702$	0	0
$703$	−31304.0	−1.67945
$704$	0	0
$705$	−1957.43	−0.104569
$706$	0	0
$707$	5673.69	0.301812
$708$	0	0
$709$	12590.4	0.666916	0.333458	−	0.942765i	$-0.391785\pi$
$709$	12590.4	0.666916	0.333458	+	0.942765i	$0.391785\pi$
$710$	0	0
$711$	2123.05	0.111984
$712$	0	0
$713$	−3336.85	−0.175268
$714$	0	0
$715$	−2404.99	−0.125792
$716$	0	0
$717$	−13479.5	−0.702092
$718$	0	0
$719$	25388.9	1.31689	0.658446	−	0.752628i	$-0.271214\pi$
$719$	25388.9	1.31689	0.658446	+	0.752628i	$0.271214\pi$
$720$	0	0
$721$	33859.6	1.74895
$722$	0	0
$723$	8342.69	0.429139
$724$	0	0
$725$	−8591.96	−0.440134
$726$	0	0
$727$	12928.4	0.659542	0.329771	−	0.944061i	$-0.393028\pi$
$727$	12928.4	0.659542	0.329771	+	0.944061i	$0.393028\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−34254.0	−1.73315
$732$	0	0
$733$	−1139.67	−0.0574277	−0.0287139	−	0.999588i	$-0.509141\pi$
$733$	−1139.67	−0.0574277	−0.0287139	+	0.999588i	$0.509141\pi$
$734$	0	0
$735$	−3146.28	−0.157895
$736$	0	0
$737$	−13763.4	−0.687899
$738$	0	0
$739$	32440.8	1.61482	0.807412	−	0.589987i	$-0.200867\pi$
$739$	32440.8	1.61482	0.807412	+	0.589987i	$0.200867\pi$
$740$	0	0
$741$	−4559.92	−0.226063
$742$	0	0
$743$	30710.3	1.51635	0.758177	−	0.652049i	$-0.226090\pi$
$743$	30710.3	1.51635	0.758177	+	0.652049i	$0.226090\pi$
$744$	0	0
$745$	10657.1	0.524090
$746$	0	0
$747$	−579.318	−0.0283750
$748$	0	0
$749$	15394.0	0.750979
$750$	0	0
$751$	7540.98	0.366410	0.183205	−	0.983075i	$-0.441353\pi$
$751$	7540.98	0.366410	0.183205	+	0.983075i	$0.441353\pi$
$752$	0	0
$753$	5997.47	0.290252
$754$	0	0
$755$	−3176.30	−0.153109
$756$	0	0
$757$	−14925.3	−0.716603	−0.358301	−	0.933606i	$-0.616644\pi$
$757$	−14925.3	−0.716603	−0.358301	+	0.933606i	$0.616644\pi$
$758$	0	0
$759$	−6375.17	−0.304880
$760$	0	0
$761$	30708.3	1.46278	0.731391	−	0.681959i	$-0.238872\pi$
$761$	30708.3	1.46278	0.731391	+	0.681959i	$0.238872\pi$
$762$	0	0
$763$	7578.89	0.359599
$764$	0	0
$765$	5508.46	0.260338
$766$	0	0
$767$	−7810.71	−0.367703
$768$	0	0
$769$	−27546.2	−1.29173	−0.645866	−	0.763451i	$-0.723504\pi$
$769$	−27546.2	−1.29173	−0.645866	+	0.763451i	$0.723504\pi$
$770$	0	0
$771$	18448.3	0.861735
$772$	0	0
$773$	−8345.20	−0.388300	−0.194150	−	0.980972i	$-0.562195\pi$
$773$	−8345.20	−0.388300	−0.194150	+	0.980972i	$0.562195\pi$
$774$	0	0
$775$	4749.36	0.220132
$776$	0	0
$777$	18452.7	0.851977
$778$	0	0
$779$	45903.8	2.11126
$780$	0	0
$781$	13970.7	0.640092
$782$	0	0
$783$	−2500.10	−0.114108
$784$	0	0
$785$	−8307.05	−0.377696
$786$	0	0
$787$	3237.72	0.146648	0.0733241	−	0.997308i	$-0.476639\pi$
$787$	3237.72	0.146648	0.0733241	+	0.997308i	$0.476639\pi$
$788$	0	0
$789$	10086.9	0.455139
$790$	0	0
$791$	1367.14	0.0614536
$792$	0	0
$793$	11144.4	0.499054
$794$	0	0
$795$	10522.3	0.469417
$796$	0	0
$797$	−12384.7	−0.550424	−0.275212	−	0.961384i	$-0.588748\pi$
$797$	−12384.7	−0.550424	−0.275212	+	0.961384i	$0.588748\pi$
$798$	0	0
$799$	12398.1	0.548951
$800$	0	0
$801$	−9674.76	−0.426768
$802$	0	0
$803$	−7186.67	−0.315831
$804$	0	0
$805$	−8500.22	−0.372166
$806$	0	0
$807$	−5916.95	−0.258100
$808$	0	0
$809$	17730.5	0.770547	0.385273	−	0.922802i	$-0.374107\pi$
$809$	17730.5	0.770547	0.385273	+	0.922802i	$0.374107\pi$
$810$	0	0
$811$	−28706.9	−1.24295	−0.621477	−	0.783433i	$-0.713467\pi$
$811$	−28706.9	−1.24295	−0.621477	+	0.783433i	$0.713467\pi$
$812$	0	0
$813$	10987.7	0.473990
$814$	0	0
$815$	15995.6	0.687488
$816$	0	0
$817$	37137.8	1.59031
$818$	0	0
$819$	2687.92	0.114681
$820$	0	0
$821$	37863.1	1.60954	0.804769	−	0.593588i	$-0.202289\pi$
$821$	37863.1	1.60954	0.804769	+	0.593588i	$0.202289\pi$
$822$	0	0
$823$	44615.9	1.88969	0.944844	−	0.327519i	$-0.106213\pi$
$823$	44615.9	1.88969	0.944844	+	0.327519i	$0.106213\pi$
$824$	0	0
$825$	9073.81	0.382920
$826$	0	0
$827$	−4034.61	−0.169646	−0.0848229	−	0.996396i	$-0.527032\pi$
$827$	−4034.61	−0.169646	−0.0848229	+	0.996396i	$0.527032\pi$
$828$	0	0
$829$	37488.6	1.57061	0.785303	−	0.619111i	$-0.212507\pi$
$829$	37488.6	1.57061	0.785303	+	0.619111i	$0.212507\pi$
$830$	0	0
$831$	−25101.8	−1.04786
$832$	0	0
$833$	19928.0	0.828891
$834$	0	0
$835$	4529.30	0.187716
$836$	0	0
$837$	1381.98	0.0570706
$838$	0	0
$839$	31830.3	1.30978	0.654890	−	0.755724i	$-0.272715\pi$
$839$	31830.3	1.30978	0.654890	+	0.755724i	$0.272715\pi$
$840$	0	0
$841$	−15814.9	−0.648444
$842$	0	0
$843$	−2804.07	−0.114564
$844$	0	0
$845$	959.150	0.0390482
$846$	0	0
$847$	6167.78	0.250210
$848$	0	0
$849$	−15893.8	−0.642491
$850$	0	0
$851$	17454.5	0.703095
$852$	0	0
$853$	−19108.8	−0.767025	−0.383513	−	0.923536i	$-0.625286\pi$
$853$	−19108.8	−0.767025	−0.383513	+	0.923536i	$0.625286\pi$
$854$	0	0
$855$	−5972.21	−0.238883
$856$	0	0
$857$	−42574.5	−1.69699	−0.848493	−	0.529206i	$-0.822490\pi$
$857$	−42574.5	−1.69699	−0.848493	+	0.529206i	$0.822490\pi$
$858$	0	0
$859$	12988.3	0.515897	0.257948	−	0.966159i	$-0.416954\pi$
$859$	12988.3	0.515897	0.257948	+	0.966159i	$0.416954\pi$
$860$	0	0
$861$	−27058.7	−1.07103
$862$	0	0
$863$	−4189.79	−0.165263	−0.0826316	−	0.996580i	$-0.526332\pi$
$863$	−4189.79	−0.165263	−0.0826316	+	0.996580i	$0.526332\pi$
$864$	0	0
$865$	24434.4	0.960455
$866$	0	0
$867$	−20150.7	−0.789335
$868$	0	0
$869$	7689.33	0.300164
$870$	0	0
$871$	5489.07	0.213536
$872$	0	0
$873$	−6327.94	−0.245325
$874$	0	0
$875$	28396.6	1.09712
$876$	0	0
$877$	20491.8	0.789008	0.394504	−	0.918894i	$-0.370916\pi$
$877$	20491.8	0.789008	0.394504	+	0.918894i	$0.370916\pi$
$878$	0	0
$879$	−15637.6	−0.600049
$880$	0	0
$881$	−5798.27	−0.221735	−0.110868	−	0.993835i	$-0.535363\pi$
$881$	−5798.27	−0.221735	−0.110868	+	0.993835i	$0.535363\pi$
$882$	0	0
$883$	−41600.3	−1.58546	−0.792731	−	0.609572i	$-0.791341\pi$
$883$	−41600.3	−1.58546	−0.792731	+	0.609572i	$0.791341\pi$
$884$	0	0
$885$	−10229.8	−0.388556
$886$	0	0
$887$	20971.6	0.793866	0.396933	−	0.917848i	$-0.370075\pi$
$887$	20971.6	0.793866	0.396933	+	0.917848i	$0.370075\pi$
$888$	0	0
$889$	45513.2	1.71706
$890$	0	0
$891$	2640.31	0.0992747
$892$	0	0
$893$	−13441.8	−0.503711
$894$	0	0
$895$	−6168.31	−0.230373
$896$	0	0
$897$	2542.52	0.0946403
$898$	0	0
$899$	−4739.49	−0.175830
$900$	0	0
$901$	−66646.4	−2.46428
$902$	0	0
$903$	−21891.5	−0.806758
$904$	0	0
$905$	19059.3	0.700060
$906$	0	0
$907$	−3476.48	−0.127271	−0.0636354	−	0.997973i	$-0.520269\pi$
$907$	−3476.48	−0.127271	−0.0636354	+	0.997973i	$0.520269\pi$
$908$	0	0
$909$	−2222.69	−0.0811021
$910$	0	0
$911$	−33785.7	−1.22873	−0.614363	−	0.789024i	$-0.710587\pi$
$911$	−33785.7	−1.22873	−0.614363	+	0.789024i	$0.710587\pi$
$912$	0	0
$913$	−2098.19	−0.0760569
$914$	0	0
$915$	14596.0	0.527355
$916$	0	0
$917$	38254.0	1.37760
$918$	0	0
$919$	−47481.8	−1.70433	−0.852165	−	0.523273i	$-0.824711\pi$
$919$	−47481.8	−1.70433	−0.852165	+	0.523273i	$0.824711\pi$
$920$	0	0
$921$	−15889.0	−0.568470
$922$	0	0
$923$	−5571.75	−0.198696
$924$	0	0
$925$	−24843.1	−0.883066
$926$	0	0
$927$	−13264.6	−0.469974
$928$	0	0
$929$	32226.3	1.13812	0.569059	−	0.822297i	$-0.307308\pi$
$929$	32226.3	1.13812	0.569059	+	0.822297i	$0.307308\pi$
$930$	0	0
$931$	−21605.8	−0.760580
$932$	0	0
$933$	29149.9	1.02286
$934$	0	0
$935$	19950.7	0.697815
$936$	0	0
$937$	19088.0	0.665505	0.332753	−	0.943014i	$-0.392023\pi$
$937$	19088.0	0.665505	0.332753	+	0.943014i	$0.392023\pi$
$938$	0	0
$939$	16236.3	0.564272
$940$	0	0
$941$	49937.4	1.72998	0.864990	−	0.501789i	$-0.167325\pi$
$941$	49937.4	1.72998	0.864990	+	0.501789i	$0.167325\pi$
$942$	0	0
$943$	−25595.1	−0.883870
$944$	0	0
$945$	3520.42	0.121184
$946$	0	0
$947$	8750.88	0.300280	0.150140	−	0.988665i	$-0.452028\pi$
$947$	8750.88	0.300280	0.150140	+	0.988665i	$0.452028\pi$
$948$	0	0
$949$	2866.16	0.0980396
$950$	0	0
$951$	2202.88	0.0751138
$952$	0	0
$953$	−16754.9	−0.569512	−0.284756	−	0.958600i	$-0.591913\pi$
$953$	−16754.9	−0.569512	−0.284756	+	0.958600i	$0.591913\pi$
$954$	0	0
$955$	−7053.29	−0.238994
$956$	0	0
$957$	−9054.94	−0.305857
$958$	0	0
$959$	69612.0	2.34399
$960$	0	0
$961$	−27171.2	−0.912059
$962$	0	0
$963$	−6030.63	−0.201801
$964$	0	0
$965$	2154.27	0.0718636
$966$	0	0
$967$	27044.5	0.899371	0.449686	−	0.893187i	$-0.351536\pi$
$967$	27044.5	0.899371	0.449686	+	0.893187i	$0.351536\pi$
$968$	0	0
$969$	37827.0	1.25405
$970$	0	0
$971$	−19022.0	−0.628677	−0.314339	−	0.949311i	$-0.601783\pi$
$971$	−19022.0	−0.628677	−0.314339	+	0.949311i	$0.601783\pi$
$972$	0	0
$973$	868.128	0.0286032
$974$	0	0
$975$	−3618.78	−0.118865
$976$	0	0
$977$	13969.3	0.457440	0.228720	−	0.973492i	$-0.426546\pi$
$977$	13969.3	0.457440	0.228720	+	0.973492i	$0.426546\pi$
$978$	0	0
$979$	−35040.3	−1.14392
$980$	0	0
$981$	−2969.05	−0.0966305
$982$	0	0
$983$	−17008.5	−0.551869	−0.275934	−	0.961177i	$-0.588987\pi$
$983$	−17008.5	−0.551869	−0.275934	+	0.961177i	$0.588987\pi$
$984$	0	0
$985$	24645.9	0.797243
$986$	0	0
$987$	7923.51	0.255530
$988$	0	0
$989$	−20707.3	−0.665777
$990$	0	0
$991$	−22607.7	−0.724679	−0.362339	−	0.932046i	$-0.618022\pi$
$991$	−22607.7	−0.724679	−0.362339	+	0.932046i	$0.618022\pi$
$992$	0	0
$993$	35533.3	1.13556
$994$	0	0
$995$	−1806.88	−0.0575699
$996$	0	0
$997$	33660.7	1.06925	0.534626	−	0.845088i	$-0.320452\pi$
$997$	33660.7	1.06925	0.534626	+	0.845088i	$0.320452\pi$
$998$	0	0
$999$	−7228.89	−0.228941

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.m.1.1		2
3.2	odd	2		1872.4.a.s.1.2		2
4.3	odd	2		156.4.a.d.1.1	✓	2
8.3	odd	2		2496.4.a.t.1.2		2
8.5	even	2		2496.4.a.bb.1.2		2
12.11	even	2		468.4.a.d.1.2		2
52.31	even	4		2028.4.b.f.337.2		4
52.47	even	4		2028.4.b.f.337.3		4
52.51	odd	2		2028.4.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.a.d.1.1	✓	2	4.3	odd	2
468.4.a.d.1.2		2	12.11	even	2
624.4.a.m.1.1		2	1.1	even	1	trivial
1872.4.a.s.1.2		2	3.2	odd	2
2028.4.a.e.1.2		2	52.51	odd	2
2028.4.b.f.337.2		4	52.31	even	4
2028.4.b.f.337.3		4	52.47	even	4
2496.4.a.t.1.2		2	8.3	odd	2
2496.4.a.bb.1.2		2	8.5	even	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.67544 q^{5} -22.9737 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +5.67544 q^{5} -22.9737 q^{7} +9.00000 q^{9} +32.5964 q^{11} -13.0000 q^{13} -17.0263 q^{15} +107.842 q^{17} -116.921 q^{19} +68.9210 q^{21} +65.1929 q^{23} -92.7893 q^{25} -27.0000 q^{27} +92.5964 q^{29} -51.1843 q^{31} -97.7893 q^{33} -130.386 q^{35} +267.737 q^{37} +39.0000 q^{39} -392.605 q^{41} -317.631 q^{43} +51.0790 q^{45} +114.965 q^{47} +184.789 q^{49} -323.526 q^{51} -618.000 q^{53} +184.999 q^{55} +350.763 q^{57} +600.824 q^{59} -857.263 q^{61} -206.763 q^{63} -73.7808 q^{65} -422.236 q^{67} -195.579 q^{69} +428.596 q^{71} -220.474 q^{73} +278.368 q^{75} -748.860 q^{77} +235.895 q^{79} +81.0000 q^{81} -64.3687 q^{83} +612.051 q^{85} -277.789 q^{87} -1074.97 q^{89} +298.658 q^{91} +153.553 q^{93} -663.579 q^{95} -703.105 q^{97} +293.368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 24 q^{5} - 8 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 24 * q^5 - 8 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 24 q^{5} - 8 q^{7} + 18 q^{9} - 36 q^{11} - 26 q^{13} - 72 q^{15} - 12 q^{17} - 120 q^{19} + 24 q^{21} - 72 q^{23} + 118 q^{25} - 54 q^{27} + 84 q^{29} - 368 q^{31} + 108 q^{33} + 144 q^{35} + 156 q^{37} + 78 q^{39} - 216 q^{41} - 104 q^{43} + 216 q^{45} + 660 q^{47} + 66 q^{49} + 36 q^{51} - 1236 q^{53} - 1072 q^{55} + 360 q^{57} + 468 q^{59} - 652 q^{61} - 72 q^{63} - 312 q^{65} + 256 q^{67} + 216 q^{69} + 756 q^{71} - 1124 q^{73} - 354 q^{75} - 1776 q^{77} + 320 q^{79} + 162 q^{81} - 660 q^{83} - 1584 q^{85} - 252 q^{87} - 2112 q^{89} + 104 q^{91} + 1104 q^{93} - 720 q^{95} - 116 q^{97} - 324 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 24 * q^5 - 8 * q^7 + 18 * q^9 - 36 * q^11 - 26 * q^13 - 72 * q^15 - 12 * q^17 - 120 * q^19 + 24 * q^21 - 72 * q^23 + 118 * q^25 - 54 * q^27 + 84 * q^29 - 368 * q^31 + 108 * q^33 + 144 * q^35 + 156 * q^37 + 78 * q^39 - 216 * q^41 - 104 * q^43 + 216 * q^45 + 660 * q^47 + 66 * q^49 + 36 * q^51 - 1236 * q^53 - 1072 * q^55 + 360 * q^57 + 468 * q^59 - 652 * q^61 - 72 * q^63 - 312 * q^65 + 256 * q^67 + 216 * q^69 + 756 * q^71 - 1124 * q^73 - 354 * q^75 - 1776 * q^77 + 320 * q^79 + 162 * q^81 - 660 * q^83 - 1584 * q^85 - 252 * q^87 - 2112 * q^89 + 104 * q^91 + 1104 * q^93 - 720 * q^95 - 116 * q^97 - 324 * q^99

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