LMFDB - Embedded newform 1840.4.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 123x^{6} + 335x^{5} + 4492x^{4} - 7035x^{3} - 45582x^{2} + 36684x + 124632 $ x^8 - 4x^7 - 123x^6 + 335x^5 + 4492x^4 - 7035x^3 - 45582x^2 + 36684*x + 124632
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.32486$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32486 q^{3} +5.00000 q^{5} -13.2387 q^{7} -25.2447 q^{9} +O(q^{10})$	$q-1.32486 q^{3} +5.00000 q^{5} -13.2387 q^{7} -25.2447 q^{9} -15.3690 q^{11} -80.8192 q^{13} -6.62430 q^{15} -25.5938 q^{17} -151.027 q^{19} +17.5394 q^{21} +23.0000 q^{23} +25.0000 q^{25} +69.2169 q^{27} -271.203 q^{29} +111.777 q^{31} +20.3617 q^{33} -66.1935 q^{35} -355.044 q^{37} +107.074 q^{39} +196.718 q^{41} +248.056 q^{43} -126.224 q^{45} +247.903 q^{47} -167.737 q^{49} +33.9082 q^{51} +238.140 q^{53} -76.8448 q^{55} +200.090 q^{57} -39.8761 q^{59} +643.745 q^{61} +334.208 q^{63} -404.096 q^{65} -349.352 q^{67} -30.4718 q^{69} -355.305 q^{71} +691.084 q^{73} -33.1215 q^{75} +203.465 q^{77} +1032.52 q^{79} +589.905 q^{81} +480.064 q^{83} -127.969 q^{85} +359.306 q^{87} -585.254 q^{89} +1069.94 q^{91} -148.089 q^{93} -755.136 q^{95} -1088.60 q^{97} +387.985 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9}+O(q^{10}) $ 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9	$ 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9} + 35 q^{11} - 54 q^{13} + 60 q^{15} - 11 q^{17} + 107 q^{19} - 148 q^{21} + 184 q^{23} + 200 q^{25} + 405 q^{27} + 37 q^{29} + 290 q^{31} + 289 q^{33} + 155 q^{35} - 172 q^{37} + 311 q^{39} + 308 q^{41} + 538 q^{43} + 310 q^{45} + 1035 q^{47} + 585 q^{49} + 387 q^{51} + 46 q^{53} + 175 q^{55} + 286 q^{57} + 1256 q^{59} + 399 q^{61} + 1234 q^{63} - 270 q^{65} + 1598 q^{67} + 276 q^{69} + 750 q^{71} + 177 q^{73} + 300 q^{75} - 1228 q^{77} + 1292 q^{79} + 380 q^{81} + 2094 q^{83} - 55 q^{85} + 2245 q^{87} + 484 q^{89} + 679 q^{91} - 2111 q^{93} + 535 q^{95} - 2225 q^{97} + 2117 q^{99}+O(q^{100}) $ 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9 + 35 * q^11 - 54 * q^13 + 60 * q^15 - 11 * q^17 + 107 * q^19 - 148 * q^21 + 184 * q^23 + 200 * q^25 + 405 * q^27 + 37 * q^29 + 290 * q^31 + 289 * q^33 + 155 * q^35 - 172 * q^37 + 311 * q^39 + 308 * q^41 + 538 * q^43 + 310 * q^45 + 1035 * q^47 + 585 * q^49 + 387 * q^51 + 46 * q^53 + 175 * q^55 + 286 * q^57 + 1256 * q^59 + 399 * q^61 + 1234 * q^63 - 270 * q^65 + 1598 * q^67 + 276 * q^69 + 750 * q^71 + 177 * q^73 + 300 * q^75 - 1228 * q^77 + 1292 * q^79 + 380 * q^81 + 2094 * q^83 - 55 * q^85 + 2245 * q^87 + 484 * q^89 + 679 * q^91 - 2111 * q^93 + 535 * q^95 - 2225 * q^97 + 2117 * 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
$3$	−1.32486	−0.254969	−0.127485	−	0.991841i	$-0.540690\pi$
$3$	−1.32486	−0.254969	−0.127485	+	0.991841i	$0.540690\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−13.2387	−0.714822	−0.357411	−	0.933947i	$-0.616341\pi$
$7$	−13.2387	−0.714822	−0.357411	+	0.933947i	$0.616341\pi$
$8$	0	0
$9$	−25.2447	−0.934991
$10$	0	0
$11$	−15.3690	−0.421265	−0.210632	−	0.977565i	$-0.567552\pi$
$11$	−15.3690	−0.421265	−0.210632	+	0.977565i	$0.567552\pi$
$12$	0	0
$13$	−80.8192	−1.72425	−0.862124	−	0.506698i	$-0.830866\pi$
$13$	−80.8192	−1.72425	−0.862124	+	0.506698i	$0.830866\pi$
$14$	0	0
$15$	−6.62430	−0.114026
$16$	0	0
$17$	−25.5938	−0.365142	−0.182571	−	0.983193i	$-0.558442\pi$
$17$	−25.5938	−0.365142	−0.182571	+	0.983193i	$0.558442\pi$
$18$	0	0
$19$	−151.027	−1.82358	−0.911789	−	0.410658i	$-0.865299\pi$
$19$	−151.027	−1.82358	−0.911789	+	0.410658i	$0.865299\pi$
$20$	0	0
$21$	17.5394	0.182258
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	69.2169	0.493363
$28$	0	0
$29$	−271.203	−1.73659	−0.868295	−	0.496048i	$-0.834784\pi$
$29$	−271.203	−1.73659	−0.868295	+	0.496048i	$0.834784\pi$
$30$	0	0
$31$	111.777	0.647606	0.323803	−	0.946125i	$-0.395039\pi$
$31$	111.777	0.647606	0.323803	+	0.946125i	$0.395039\pi$
$32$	0	0
$33$	20.3617	0.107410
$34$	0	0
$35$	−66.1935	−0.319678
$36$	0	0
$37$	−355.044	−1.57754	−0.788768	−	0.614691i	$-0.789281\pi$
$37$	−355.044	−1.57754	−0.788768	+	0.614691i	$0.789281\pi$
$38$	0	0
$39$	107.074	0.439630
$40$	0	0
$41$	196.718	0.749323	0.374661	−	0.927162i	$-0.377759\pi$
$41$	196.718	0.749323	0.374661	+	0.927162i	$0.377759\pi$
$42$	0	0
$43$	248.056	0.879724	0.439862	−	0.898065i	$-0.355027\pi$
$43$	248.056	0.879724	0.439862	+	0.898065i	$0.355027\pi$
$44$	0	0
$45$	−126.224	−0.418141
$46$	0	0
$47$	247.903	0.769370	0.384685	−	0.923048i	$-0.374310\pi$
$47$	247.903	0.769370	0.384685	+	0.923048i	$0.374310\pi$
$48$	0	0
$49$	−167.737	−0.489029
$50$	0	0
$51$	33.9082	0.0931000
$52$	0	0
$53$	238.140	0.617190	0.308595	−	0.951194i	$-0.400141\pi$
$53$	238.140	0.617190	0.308595	+	0.951194i	$0.400141\pi$
$54$	0	0
$55$	−76.8448	−0.188395
$56$	0	0
$57$	200.090	0.464957
$58$	0	0
$59$	−39.8761	−0.0879902	−0.0439951	−	0.999032i	$-0.514009\pi$
$59$	−39.8761	−0.0879902	−0.0439951	+	0.999032i	$0.514009\pi$
$60$	0	0
$61$	643.745	1.35120	0.675599	−	0.737269i	$-0.263885\pi$
$61$	643.745	1.35120	0.675599	+	0.737269i	$0.263885\pi$
$62$	0	0
$63$	334.208	0.668352
$64$	0	0
$65$	−404.096	−0.771107
$66$	0	0
$67$	−349.352	−0.637017	−0.318508	−	0.947920i	$-0.603182\pi$
$67$	−349.352	−0.637017	−0.318508	+	0.947920i	$0.603182\pi$
$68$	0	0
$69$	−30.4718	−0.0531648
$70$	0	0
$71$	−355.305	−0.593901	−0.296951	−	0.954893i	$-0.595970\pi$
$71$	−355.305	−0.593901	−0.296951	+	0.954893i	$0.595970\pi$
$72$	0	0
$73$	691.084	1.10802	0.554009	−	0.832511i	$-0.313097\pi$
$73$	691.084	1.10802	0.554009	+	0.832511i	$0.313097\pi$
$74$	0	0
$75$	−33.1215	−0.0509939
$76$	0	0
$77$	203.465	0.301130
$78$	0	0
$79$	1032.52	1.47047	0.735236	−	0.677811i	$-0.237071\pi$
$79$	1032.52	1.47047	0.735236	+	0.677811i	$0.237071\pi$
$80$	0	0
$81$	589.905	0.809198
$82$	0	0
$83$	480.064	0.634865	0.317433	−	0.948281i	$-0.397179\pi$
$83$	480.064	0.634865	0.317433	+	0.948281i	$0.397179\pi$
$84$	0	0
$85$	−127.969	−0.163297
$86$	0	0
$87$	359.306	0.442777
$88$	0	0
$89$	−585.254	−0.697043	−0.348522	−	0.937301i	$-0.613316\pi$
$89$	−585.254	−0.697043	−0.348522	+	0.937301i	$0.613316\pi$
$90$	0	0
$91$	1069.94	1.23253
$92$	0	0
$93$	−148.089	−0.165120
$94$	0	0
$95$	−755.136	−0.815529
$96$	0	0
$97$	−1088.60	−1.13949	−0.569744	−	0.821822i	$-0.692958\pi$
$97$	−1088.60	−1.13949	−0.569744	+	0.821822i	$0.692958\pi$
$98$	0	0
$99$	387.985	0.393879
$100$	0	0
$101$	−799.197	−0.787357	−0.393679	−	0.919248i	$-0.628798\pi$
$101$	−799.197	−0.787357	−0.393679	+	0.919248i	$0.628798\pi$
$102$	0	0
$103$	1200.08	1.14804	0.574019	−	0.818842i	$-0.305384\pi$
$103$	1200.08	1.14804	0.574019	+	0.818842i	$0.305384\pi$
$104$	0	0
$105$	87.6971	0.0815082
$106$	0	0
$107$	−564.273	−0.509816	−0.254908	−	0.966965i	$-0.582045\pi$
$107$	−564.273	−0.509816	−0.254908	+	0.966965i	$0.582045\pi$
$108$	0	0
$109$	−311.208	−0.273471	−0.136736	−	0.990608i	$-0.543661\pi$
$109$	−311.208	−0.273471	−0.136736	+	0.990608i	$0.543661\pi$
$110$	0	0
$111$	470.383	0.402223
$112$	0	0
$113$	−1946.07	−1.62010	−0.810050	−	0.586361i	$-0.800560\pi$
$113$	−1946.07	−1.62010	−0.810050	+	0.586361i	$0.800560\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	2040.26	1.61216
$118$	0	0
$119$	338.829	0.261012
$120$	0	0
$121$	−1094.80	−0.822536
$122$	0	0
$123$	−260.624	−0.191054
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2132.04	−1.48967	−0.744835	−	0.667248i	$-0.767472\pi$
$127$	−2132.04	−1.48967	−0.744835	+	0.667248i	$0.767472\pi$
$128$	0	0
$129$	−328.639	−0.224303
$130$	0	0
$131$	927.294	0.618458	0.309229	−	0.950988i	$-0.399929\pi$
$131$	927.294	0.618458	0.309229	+	0.950988i	$0.399929\pi$
$132$	0	0
$133$	1999.40	1.30354
$134$	0	0
$135$	346.085	0.220639
$136$	0	0
$137$	1028.94	0.641668	0.320834	−	0.947135i	$-0.396037\pi$
$137$	1028.94	0.641668	0.320834	+	0.947135i	$0.396037\pi$
$138$	0	0
$139$	2943.50	1.79615	0.898074	−	0.439845i	$-0.144967\pi$
$139$	2943.50	1.79615	0.898074	+	0.439845i	$0.144967\pi$
$140$	0	0
$141$	−328.437	−0.196166
$142$	0	0
$143$	1242.11	0.726365
$144$	0	0
$145$	−1356.01	−0.776627
$146$	0	0
$147$	222.228	0.124687
$148$	0	0
$149$	−1422.75	−0.782257	−0.391129	−	0.920336i	$-0.627915\pi$
$149$	−1422.75	−0.782257	−0.391129	+	0.920336i	$0.627915\pi$
$150$	0	0
$151$	−679.457	−0.366182	−0.183091	−	0.983096i	$-0.558610\pi$
$151$	−679.457	−0.366182	−0.183091	+	0.983096i	$0.558610\pi$
$152$	0	0
$153$	646.110	0.341405
$154$	0	0
$155$	558.886	0.289618
$156$	0	0
$157$	−1342.43	−0.682405	−0.341203	−	0.939990i	$-0.610834\pi$
$157$	−1342.43	−0.682405	−0.341203	+	0.939990i	$0.610834\pi$
$158$	0	0
$159$	−315.502	−0.157365
$160$	0	0
$161$	−304.490	−0.149051
$162$	0	0
$163$	−1573.64	−0.756176	−0.378088	−	0.925770i	$-0.623418\pi$
$163$	−1573.64	−0.756176	−0.378088	+	0.925770i	$0.623418\pi$
$164$	0	0
$165$	101.809	0.0480350
$166$	0	0
$167$	2742.77	1.27091	0.635454	−	0.772139i	$-0.280813\pi$
$167$	2742.77	1.27091	0.635454	+	0.772139i	$0.280813\pi$
$168$	0	0
$169$	4334.75	1.97303
$170$	0	0
$171$	3812.64	1.70503
$172$	0	0
$173$	−2693.84	−1.18386	−0.591932	−	0.805988i	$-0.701635\pi$
$173$	−2693.84	−1.18386	−0.591932	+	0.805988i	$0.701635\pi$
$174$	0	0
$175$	−330.967	−0.142964
$176$	0	0
$177$	52.8302	0.0224348
$178$	0	0
$179$	−756.139	−0.315735	−0.157867	−	0.987460i	$-0.550462\pi$
$179$	−756.139	−0.315735	−0.157867	+	0.987460i	$0.550462\pi$
$180$	0	0
$181$	−121.589	−0.0499315	−0.0249658	−	0.999688i	$-0.507948\pi$
$181$	−121.589	−0.0499315	−0.0249658	+	0.999688i	$0.507948\pi$
$182$	0	0
$183$	−852.872	−0.344514
$184$	0	0
$185$	−1775.22	−0.705495
$186$	0	0
$187$	393.350	0.153822
$188$	0	0
$189$	−916.342	−0.352667
$190$	0	0
$191$	136.915	0.0518683	0.0259342	−	0.999664i	$-0.491744\pi$
$191$	136.915	0.0518683	0.0259342	+	0.999664i	$0.491744\pi$
$192$	0	0
$193$	405.019	0.151056	0.0755282	−	0.997144i	$-0.475936\pi$
$193$	405.019	0.151056	0.0755282	+	0.997144i	$0.475936\pi$
$194$	0	0
$195$	535.371	0.196609
$196$	0	0
$197$	−1280.45	−0.463089	−0.231545	−	0.972824i	$-0.574378\pi$
$197$	−1280.45	−0.463089	−0.231545	+	0.972824i	$0.574378\pi$
$198$	0	0
$199$	322.354	0.114830	0.0574148	−	0.998350i	$-0.481714\pi$
$199$	322.354	0.114830	0.0574148	+	0.998350i	$0.481714\pi$
$200$	0	0
$201$	462.842	0.162420
$202$	0	0
$203$	3590.37	1.24135
$204$	0	0
$205$	983.591	0.335107
$206$	0	0
$207$	−580.629	−0.194959
$208$	0	0
$209$	2321.13	0.768210
$210$	0	0
$211$	1405.71	0.458640	0.229320	−	0.973351i	$-0.426350\pi$
$211$	1405.71	0.458640	0.229320	+	0.973351i	$0.426350\pi$
$212$	0	0
$213$	470.730	0.151427
$214$	0	0
$215$	1240.28	0.393424
$216$	0	0
$217$	−1479.78	−0.462923
$218$	0	0
$219$	−915.589	−0.282510
$220$	0	0
$221$	2068.47	0.629596
$222$	0	0
$223$	−3389.58	−1.01786	−0.508930	−	0.860808i	$-0.669959\pi$
$223$	−3389.58	−1.01786	−0.508930	+	0.860808i	$0.669959\pi$
$224$	0	0
$225$	−631.119	−0.186998
$226$	0	0
$227$	−1998.96	−0.584474	−0.292237	−	0.956346i	$-0.594400\pi$
$227$	−1998.96	−0.584474	−0.292237	+	0.956346i	$0.594400\pi$
$228$	0	0
$229$	4477.72	1.29212	0.646062	−	0.763285i	$-0.276415\pi$
$229$	4477.72	1.29212	0.646062	+	0.763285i	$0.276415\pi$
$230$	0	0
$231$	−269.562	−0.0767788
$232$	0	0
$233$	−4039.58	−1.13580	−0.567901	−	0.823097i	$-0.692244\pi$
$233$	−4039.58	−1.13580	−0.567901	+	0.823097i	$0.692244\pi$
$234$	0	0
$235$	1239.52	0.344073
$236$	0	0
$237$	−1367.94	−0.374925
$238$	0	0
$239$	−6285.11	−1.70104	−0.850522	−	0.525939i	$-0.823714\pi$
$239$	−6285.11	−1.70104	−0.850522	+	0.525939i	$0.823714\pi$
$240$	0	0
$241$	792.699	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$241$	792.699	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$242$	0	0
$243$	−2650.40	−0.699684
$244$	0	0
$245$	−838.684	−0.218700
$246$	0	0
$247$	12205.9	3.14430
$248$	0	0
$249$	−636.017	−0.161871
$250$	0	0
$251$	−3842.86	−0.966372	−0.483186	−	0.875518i	$-0.660520\pi$
$251$	−3842.86	−0.966372	−0.483186	+	0.875518i	$0.660520\pi$
$252$	0	0
$253$	−353.486	−0.0878398
$254$	0	0
$255$	169.541	0.0416356
$256$	0	0
$257$	3660.31	0.888420	0.444210	−	0.895923i	$-0.353484\pi$
$257$	3660.31	0.888420	0.444210	+	0.895923i	$0.353484\pi$
$258$	0	0
$259$	4700.32	1.12766
$260$	0	0
$261$	6846.45	1.62370
$262$	0	0
$263$	5194.77	1.21796	0.608980	−	0.793185i	$-0.291579\pi$
$263$	5194.77	1.21796	0.608980	+	0.793185i	$0.291579\pi$
$264$	0	0
$265$	1190.70	0.276016
$266$	0	0
$267$	775.380	0.177725
$268$	0	0
$269$	5839.98	1.32368	0.661840	−	0.749645i	$-0.269776\pi$
$269$	5839.98	1.32368	0.661840	+	0.749645i	$0.269776\pi$
$270$	0	0
$271$	4825.09	1.08156	0.540781	−	0.841163i	$-0.318129\pi$
$271$	4825.09	1.08156	0.540781	+	0.841163i	$0.318129\pi$
$272$	0	0
$273$	−1417.52	−0.314258
$274$	0	0
$275$	−384.224	−0.0842530
$276$	0	0
$277$	3814.60	0.827426	0.413713	−	0.910407i	$-0.364232\pi$
$277$	3814.60	0.827426	0.413713	+	0.910407i	$0.364232\pi$
$278$	0	0
$279$	−2821.79	−0.605505
$280$	0	0
$281$	4821.32	1.02354	0.511771	−	0.859122i	$-0.328989\pi$
$281$	4821.32	1.02354	0.511771	+	0.859122i	$0.328989\pi$
$282$	0	0
$283$	4897.54	1.02872	0.514361	−	0.857574i	$-0.328029\pi$
$283$	4897.54	1.02872	0.514361	+	0.857574i	$0.328029\pi$
$284$	0	0
$285$	1000.45	0.207935
$286$	0	0
$287$	−2604.29	−0.535633
$288$	0	0
$289$	−4257.96	−0.866671
$290$	0	0
$291$	1442.24	0.290535
$292$	0	0
$293$	634.063	0.126424	0.0632122	−	0.998000i	$-0.479865\pi$
$293$	634.063	0.126424	0.0632122	+	0.998000i	$0.479865\pi$
$294$	0	0
$295$	−199.380	−0.0393504
$296$	0	0
$297$	−1063.79	−0.207837
$298$	0	0
$299$	−1858.84	−0.359530
$300$	0	0
$301$	−3283.93	−0.628846
$302$	0	0
$303$	1058.82	0.200752
$304$	0	0
$305$	3218.73	0.604274
$306$	0	0
$307$	−316.952	−0.0589232	−0.0294616	−	0.999566i	$-0.509379\pi$
$307$	−316.952	−0.0589232	−0.0294616	+	0.999566i	$0.509379\pi$
$308$	0	0
$309$	−1589.94	−0.292714
$310$	0	0
$311$	−1919.14	−0.349918	−0.174959	−	0.984576i	$-0.555979\pi$
$311$	−1919.14	−0.349918	−0.174959	+	0.984576i	$0.555979\pi$
$312$	0	0
$313$	−1095.62	−0.197853	−0.0989265	−	0.995095i	$-0.531541\pi$
$313$	−1095.62	−0.197853	−0.0989265	+	0.995095i	$0.531541\pi$
$314$	0	0
$315$	1671.04	0.298896
$316$	0	0
$317$	−579.206	−0.102623	−0.0513114	−	0.998683i	$-0.516340\pi$
$317$	−579.206	−0.102623	−0.0513114	+	0.998683i	$0.516340\pi$
$318$	0	0
$319$	4168.11	0.731565
$320$	0	0
$321$	747.583	0.129988
$322$	0	0
$323$	3865.36	0.665866
$324$	0	0
$325$	−2020.48	−0.344850
$326$	0	0
$327$	412.307	0.0697267
$328$	0	0
$329$	−3281.92	−0.549963
$330$	0	0
$331$	4092.53	0.679595	0.339797	−	0.940499i	$-0.389641\pi$
$331$	4092.53	0.679595	0.339797	+	0.940499i	$0.389641\pi$
$332$	0	0
$333$	8962.99	1.47498
$334$	0	0
$335$	−1746.76	−0.284883
$336$	0	0
$337$	4344.10	0.702190	0.351095	−	0.936340i	$-0.385809\pi$
$337$	4344.10	0.702190	0.351095	+	0.936340i	$0.385809\pi$
$338$	0	0
$339$	2578.28	0.413076
$340$	0	0
$341$	−1717.90	−0.272814
$342$	0	0
$343$	6761.49	1.06439
$344$	0	0
$345$	−152.359	−0.0237760
$346$	0	0
$347$	−10371.1	−1.60447	−0.802236	−	0.597007i	$-0.796356\pi$
$347$	−10371.1	−1.60447	−0.802236	+	0.597007i	$0.796356\pi$
$348$	0	0
$349$	1055.46	0.161884	0.0809418	−	0.996719i	$-0.474207\pi$
$349$	1055.46	0.161884	0.0809418	+	0.996719i	$0.474207\pi$
$350$	0	0
$351$	−5594.06	−0.850680
$352$	0	0
$353$	−4673.92	−0.704724	−0.352362	−	0.935864i	$-0.614621\pi$
$353$	−4673.92	−0.704724	−0.352362	+	0.935864i	$0.614621\pi$
$354$	0	0
$355$	−1776.53	−0.265601
$356$	0	0
$357$	−448.901	−0.0665500
$358$	0	0
$359$	3013.12	0.442970	0.221485	−	0.975164i	$-0.428910\pi$
$359$	3013.12	0.442970	0.221485	+	0.975164i	$0.428910\pi$
$360$	0	0
$361$	15950.2	2.32544
$362$	0	0
$363$	1450.45	0.209721
$364$	0	0
$365$	3455.42	0.495521
$366$	0	0
$367$	−565.884	−0.0804874	−0.0402437	−	0.999190i	$-0.512813\pi$
$367$	−565.884	−0.0804874	−0.0402437	+	0.999190i	$0.512813\pi$
$368$	0	0
$369$	−4966.10	−0.700610
$370$	0	0
$371$	−3152.67	−0.441181
$372$	0	0
$373$	−1612.64	−0.223858	−0.111929	−	0.993716i	$-0.535703\pi$
$373$	−1612.64	−0.223858	−0.111929	+	0.993716i	$0.535703\pi$
$374$	0	0
$375$	−165.607	−0.0228051
$376$	0	0
$377$	21918.4	2.99431
$378$	0	0
$379$	−9322.19	−1.26345	−0.631727	−	0.775191i	$-0.717654\pi$
$379$	−9322.19	−1.26345	−0.631727	+	0.775191i	$0.717654\pi$
$380$	0	0
$381$	2824.66	0.379820
$382$	0	0
$383$	7212.65	0.962270	0.481135	−	0.876647i	$-0.340225\pi$
$383$	7212.65	0.962270	0.481135	+	0.876647i	$0.340225\pi$
$384$	0	0
$385$	1017.32	0.134669
$386$	0	0
$387$	−6262.10	−0.822533
$388$	0	0
$389$	−1764.20	−0.229944	−0.114972	−	0.993369i	$-0.536678\pi$
$389$	−1764.20	−0.229944	−0.114972	+	0.993369i	$0.536678\pi$
$390$	0	0
$391$	−588.658	−0.0761374
$392$	0	0
$393$	−1228.53	−0.157688
$394$	0	0
$395$	5162.59	0.657615
$396$	0	0
$397$	355.515	0.0449440	0.0224720	−	0.999747i	$-0.492846\pi$
$397$	355.515	0.0449440	0.0224720	+	0.999747i	$0.492846\pi$
$398$	0	0
$399$	−2648.93	−0.332361
$400$	0	0
$401$	13076.5	1.62846	0.814228	−	0.580545i	$-0.197161\pi$
$401$	13076.5	1.62846	0.814228	+	0.580545i	$0.197161\pi$
$402$	0	0
$403$	−9033.74	−1.11663
$404$	0	0
$405$	2949.53	0.361884
$406$	0	0
$407$	5456.65	0.664560
$408$	0	0
$409$	4527.39	0.547348	0.273674	−	0.961823i	$-0.411761\pi$
$409$	4527.39	0.547348	0.273674	+	0.961823i	$0.411761\pi$
$410$	0	0
$411$	−1363.20	−0.163606
$412$	0	0
$413$	527.907	0.0628974
$414$	0	0
$415$	2400.32	0.283920
$416$	0	0
$417$	−3899.72	−0.457962
$418$	0	0
$419$	−7612.72	−0.887603	−0.443802	−	0.896125i	$-0.646371\pi$
$419$	−7612.72	−0.887603	−0.443802	+	0.896125i	$0.646371\pi$
$420$	0	0
$421$	−9316.02	−1.07847	−0.539234	−	0.842156i	$-0.681286\pi$
$421$	−9316.02	−1.07847	−0.539234	+	0.842156i	$0.681286\pi$
$422$	0	0
$423$	−6258.26	−0.719354
$424$	0	0
$425$	−639.846	−0.0730284
$426$	0	0
$427$	−8522.35	−0.965867
$428$	0	0
$429$	−1645.62	−0.185201
$430$	0	0
$431$	−12010.0	−1.34223	−0.671114	−	0.741354i	$-0.734184\pi$
$431$	−12010.0	−1.34223	−0.671114	+	0.741354i	$0.734184\pi$
$432$	0	0
$433$	−6252.22	−0.693908	−0.346954	−	0.937882i	$-0.612784\pi$
$433$	−6252.22	−0.693908	−0.346954	+	0.937882i	$0.612784\pi$
$434$	0	0
$435$	1796.53	0.198016
$436$	0	0
$437$	−3473.62	−0.380242
$438$	0	0
$439$	8678.65	0.943530	0.471765	−	0.881724i	$-0.343617\pi$
$439$	8678.65	0.943530	0.471765	+	0.881724i	$0.343617\pi$
$440$	0	0
$441$	4234.48	0.457237
$442$	0	0
$443$	15502.9	1.66267	0.831336	−	0.555770i	$-0.187576\pi$
$443$	15502.9	1.66267	0.831336	+	0.555770i	$0.187576\pi$
$444$	0	0
$445$	−2926.27	−0.311727
$446$	0	0
$447$	1884.95	0.199452
$448$	0	0
$449$	18480.7	1.94245	0.971225	−	0.238164i	$-0.0765456\pi$
$449$	18480.7	1.94245	0.971225	+	0.238164i	$0.0765456\pi$
$450$	0	0
$451$	−3023.35	−0.315663
$452$	0	0
$453$	900.185	0.0933651
$454$	0	0
$455$	5349.71	0.551205
$456$	0	0
$457$	896.208	0.0917348	0.0458674	−	0.998948i	$-0.485395\pi$
$457$	896.208	0.0917348	0.0458674	+	0.998948i	$0.485395\pi$
$458$	0	0
$459$	−1771.53	−0.180148
$460$	0	0
$461$	−4385.54	−0.443070	−0.221535	−	0.975152i	$-0.571107\pi$
$461$	−4385.54	−0.443070	−0.221535	+	0.975152i	$0.571107\pi$
$462$	0	0
$463$	3270.08	0.328236	0.164118	−	0.986441i	$-0.447522\pi$
$463$	3270.08	0.328236	0.164118	+	0.986441i	$0.447522\pi$
$464$	0	0
$465$	−740.445	−0.0738437
$466$	0	0
$467$	−5285.33	−0.523717	−0.261858	−	0.965106i	$-0.584335\pi$
$467$	−5285.33	−0.523717	−0.261858	+	0.965106i	$0.584335\pi$
$468$	0	0
$469$	4624.97	0.455354
$470$	0	0
$471$	1778.53	0.173992
$472$	0	0
$473$	−3812.35	−0.370597
$474$	0	0
$475$	−3775.68	−0.364716
$476$	0	0
$477$	−6011.79	−0.577067
$478$	0	0
$479$	−863.567	−0.0823745	−0.0411873	−	0.999151i	$-0.513114\pi$
$479$	−863.567	−0.0823745	−0.0411873	+	0.999151i	$0.513114\pi$
$480$	0	0
$481$	28694.4	2.72006
$482$	0	0
$483$	403.406	0.0380034
$484$	0	0
$485$	−5442.99	−0.509595
$486$	0	0
$487$	−13312.4	−1.23869	−0.619345	−	0.785119i	$-0.712602\pi$
$487$	−13312.4	−1.23869	−0.619345	+	0.785119i	$0.712602\pi$
$488$	0	0
$489$	2084.85	0.192802
$490$	0	0
$491$	6004.86	0.551926	0.275963	−	0.961168i	$-0.411003\pi$
$491$	6004.86	0.551926	0.275963	+	0.961168i	$0.411003\pi$
$492$	0	0
$493$	6941.12	0.634102
$494$	0	0
$495$	1939.93	0.176148
$496$	0	0
$497$	4703.78	0.424534
$498$	0	0
$499$	8570.42	0.768867	0.384433	−	0.923153i	$-0.374397\pi$
$499$	8570.42	0.768867	0.384433	+	0.923153i	$0.374397\pi$
$500$	0	0
$501$	−3633.78	−0.324042
$502$	0	0
$503$	−12905.5	−1.14399	−0.571994	−	0.820258i	$-0.693830\pi$
$503$	−12905.5	−1.14399	−0.571994	+	0.820258i	$0.693830\pi$
$504$	0	0
$505$	−3995.99	−0.352117
$506$	0	0
$507$	−5742.93	−0.503062
$508$	0	0
$509$	6693.92	0.582913	0.291457	−	0.956584i	$-0.405860\pi$
$509$	6693.92	0.582913	0.291457	+	0.956584i	$0.405860\pi$
$510$	0	0
$511$	−9149.05	−0.792036
$512$	0	0
$513$	−10453.6	−0.899687
$514$	0	0
$515$	6000.42	0.513418
$516$	0	0
$517$	−3810.01	−0.324109
$518$	0	0
$519$	3568.95	0.301849
$520$	0	0
$521$	−5655.04	−0.475532	−0.237766	−	0.971323i	$-0.576415\pi$
$521$	−5655.04	−0.475532	−0.237766	+	0.971323i	$0.576415\pi$
$522$	0	0
$523$	−5714.02	−0.477738	−0.238869	−	0.971052i	$-0.576777\pi$
$523$	−5714.02	−0.477738	−0.238869	+	0.971052i	$0.576777\pi$
$524$	0	0
$525$	438.485	0.0364516
$526$	0	0
$527$	−2860.81	−0.236468
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	1006.66	0.0822700
$532$	0	0
$533$	−15898.6	−1.29202
$534$	0	0
$535$	−2821.37	−0.227997
$536$	0	0
$537$	1001.78	0.0805026
$538$	0	0
$539$	2577.94	0.206011
$540$	0	0
$541$	−16901.3	−1.34315	−0.671575	−	0.740937i	$-0.734382\pi$
$541$	−16901.3	−1.34315	−0.671575	+	0.740937i	$0.734382\pi$
$542$	0	0
$543$	161.088	0.0127310
$544$	0	0
$545$	−1556.04	−0.122300
$546$	0	0
$547$	8224.09	0.642846	0.321423	−	0.946936i	$-0.395839\pi$
$547$	8224.09	0.642846	0.321423	+	0.946936i	$0.395839\pi$
$548$	0	0
$549$	−16251.2	−1.26336
$550$	0	0
$551$	40959.0	3.16681
$552$	0	0
$553$	−13669.2	−1.05113
$554$	0	0
$555$	2351.91	0.179880
$556$	0	0
$557$	2574.81	0.195868	0.0979339	−	0.995193i	$-0.468777\pi$
$557$	2574.81	0.195868	0.0979339	+	0.995193i	$0.468777\pi$
$558$	0	0
$559$	−20047.7	−1.51686
$560$	0	0
$561$	−521.134	−0.0392198
$562$	0	0
$563$	−12445.2	−0.931622	−0.465811	−	0.884884i	$-0.654237\pi$
$563$	−12445.2	−0.931622	−0.465811	+	0.884884i	$0.654237\pi$
$564$	0	0
$565$	−9730.37	−0.724531
$566$	0	0
$567$	−7809.58	−0.578433
$568$	0	0
$569$	12680.4	0.934252	0.467126	−	0.884191i	$-0.345289\pi$
$569$	12680.4	0.934252	0.467126	+	0.884191i	$0.345289\pi$
$570$	0	0
$571$	−20714.2	−1.51814	−0.759072	−	0.651006i	$-0.774347\pi$
$571$	−20714.2	−1.51814	−0.759072	+	0.651006i	$0.774347\pi$
$572$	0	0
$573$	−181.394	−0.0132248
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−17705.2	−1.27743	−0.638713	−	0.769445i	$-0.720533\pi$
$577$	−17705.2	−1.27743	−0.638713	+	0.769445i	$0.720533\pi$
$578$	0	0
$579$	−536.593	−0.0385147
$580$	0	0
$581$	−6355.42	−0.453816
$582$	0	0
$583$	−3659.97	−0.260001
$584$	0	0
$585$	10201.3	0.720978
$586$	0	0
$587$	−23495.1	−1.65204	−0.826018	−	0.563644i	$-0.809399\pi$
$587$	−23495.1	−1.65204	−0.826018	+	0.563644i	$0.809399\pi$
$588$	0	0
$589$	−16881.4	−1.18096
$590$	0	0
$591$	1696.42	0.118074
$592$	0	0
$593$	26347.1	1.82453	0.912265	−	0.409600i	$-0.134332\pi$
$593$	26347.1	1.82453	0.912265	+	0.409600i	$0.134332\pi$
$594$	0	0
$595$	1694.15	0.116728
$596$	0	0
$597$	−427.074	−0.0292780
$598$	0	0
$599$	−4155.08	−0.283426	−0.141713	−	0.989908i	$-0.545261\pi$
$599$	−4155.08	−0.283426	−0.141713	+	0.989908i	$0.545261\pi$
$600$	0	0
$601$	19177.1	1.30158	0.650791	−	0.759257i	$-0.274437\pi$
$601$	19177.1	1.30158	0.650791	+	0.759257i	$0.274437\pi$
$602$	0	0
$603$	8819.30	0.595605
$604$	0	0
$605$	−5473.98	−0.367849
$606$	0	0
$607$	−23874.4	−1.59643	−0.798214	−	0.602373i	$-0.794222\pi$
$607$	−23874.4	−1.59643	−0.798214	+	0.602373i	$0.794222\pi$
$608$	0	0
$609$	−4756.74	−0.316507
$610$	0	0
$611$	−20035.4	−1.32659
$612$	0	0
$613$	−8555.23	−0.563691	−0.281846	−	0.959460i	$-0.590947\pi$
$613$	−8555.23	−0.563691	−0.281846	+	0.959460i	$0.590947\pi$
$614$	0	0
$615$	−1303.12	−0.0854421
$616$	0	0
$617$	−20759.6	−1.35454	−0.677269	−	0.735735i	$-0.736837\pi$
$617$	−20759.6	−1.35454	−0.677269	+	0.735735i	$0.736837\pi$
$618$	0	0
$619$	13007.8	0.844634	0.422317	−	0.906448i	$-0.361217\pi$
$619$	13007.8	0.844634	0.422317	+	0.906448i	$0.361217\pi$
$620$	0	0
$621$	1591.99	0.102873
$622$	0	0
$623$	7748.01	0.498262
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−3075.17	−0.195870
$628$	0	0
$629$	9086.93	0.576025
$630$	0	0
$631$	−22464.0	−1.41724	−0.708619	−	0.705591i	$-0.750682\pi$
$631$	−22464.0	−1.41724	−0.708619	+	0.705591i	$0.750682\pi$
$632$	0	0
$633$	−1862.37	−0.116939
$634$	0	0
$635$	−10660.2	−0.666201
$636$	0	0
$637$	13556.4	0.843207
$638$	0	0
$639$	8969.60	0.555292
$640$	0	0
$641$	30843.6	1.90054	0.950271	−	0.311423i	$-0.100806\pi$
$641$	30843.6	1.90054	0.950271	+	0.311423i	$0.100806\pi$
$642$	0	0
$643$	12342.2	0.756964	0.378482	−	0.925609i	$-0.376446\pi$
$643$	12342.2	0.756964	0.378482	+	0.925609i	$0.376446\pi$
$644$	0	0
$645$	−1643.19	−0.100311
$646$	0	0
$647$	−7767.97	−0.472010	−0.236005	−	0.971752i	$-0.575838\pi$
$647$	−7767.97	−0.472010	−0.236005	+	0.971752i	$0.575838\pi$
$648$	0	0
$649$	612.853	0.0370672
$650$	0	0
$651$	1960.51	0.118031
$652$	0	0
$653$	−5503.77	−0.329830	−0.164915	−	0.986308i	$-0.552735\pi$
$653$	−5503.77	−0.329830	−0.164915	+	0.986308i	$0.552735\pi$
$654$	0	0
$655$	4636.47	0.276583
$656$	0	0
$657$	−17446.2	−1.03599
$658$	0	0
$659$	9838.41	0.581563	0.290782	−	0.956789i	$-0.406085\pi$
$659$	9838.41	0.581563	0.290782	+	0.956789i	$0.406085\pi$
$660$	0	0
$661$	28999.5	1.70643	0.853215	−	0.521559i	$-0.174649\pi$
$661$	28999.5	1.70643	0.853215	+	0.521559i	$0.174649\pi$
$662$	0	0
$663$	−2740.44	−0.160528
$664$	0	0
$665$	9997.01	0.582959
$666$	0	0
$667$	−6237.67	−0.362104
$668$	0	0
$669$	4490.71	0.259523
$670$	0	0
$671$	−9893.69	−0.569213
$672$	0	0
$673$	4734.80	0.271193	0.135597	−	0.990764i	$-0.456705\pi$
$673$	4734.80	0.271193	0.135597	+	0.990764i	$0.456705\pi$
$674$	0	0
$675$	1730.42	0.0986726
$676$	0	0
$677$	−32417.9	−1.84036	−0.920179	−	0.391497i	$-0.871957\pi$
$677$	−32417.9	−1.84036	−0.920179	+	0.391497i	$0.871957\pi$
$678$	0	0
$679$	14411.6	0.814532
$680$	0	0
$681$	2648.34	0.149023
$682$	0	0
$683$	30414.5	1.70392	0.851960	−	0.523607i	$-0.175414\pi$
$683$	30414.5	1.70392	0.851960	+	0.523607i	$0.175414\pi$
$684$	0	0
$685$	5144.71	0.286963
$686$	0	0
$687$	−5932.35	−0.329452
$688$	0	0
$689$	−19246.3	−1.06419
$690$	0	0
$691$	9356.67	0.515115	0.257558	−	0.966263i	$-0.417082\pi$
$691$	9356.67	0.515115	0.257558	+	0.966263i	$0.417082\pi$
$692$	0	0
$693$	−5136.42	−0.281553
$694$	0	0
$695$	14717.5	0.803262
$696$	0	0
$697$	−5034.77	−0.273609
$698$	0	0
$699$	5351.88	0.289594
$700$	0	0
$701$	−16569.8	−0.892769	−0.446384	−	0.894841i	$-0.647289\pi$
$701$	−16569.8	−0.892769	−0.446384	+	0.894841i	$0.647289\pi$
$702$	0	0
$703$	53621.2	2.87676
$704$	0	0
$705$	−1642.18	−0.0877280
$706$	0	0
$707$	10580.3	0.562821
$708$	0	0
$709$	10170.1	0.538712	0.269356	−	0.963041i	$-0.413189\pi$
$709$	10170.1	0.538712	0.269356	+	0.963041i	$0.413189\pi$
$710$	0	0
$711$	−26065.7	−1.37488
$712$	0	0
$713$	2570.88	0.135035
$714$	0	0
$715$	6210.53	0.324840
$716$	0	0
$717$	8326.88	0.433714
$718$	0	0
$719$	−16607.2	−0.861398	−0.430699	−	0.902496i	$-0.641733\pi$
$719$	−16607.2	−0.861398	−0.430699	+	0.902496i	$0.641733\pi$
$720$	0	0
$721$	−15887.6	−0.820643
$722$	0	0
$723$	−1050.21	−0.0540220
$724$	0	0
$725$	−6780.07	−0.347318
$726$	0	0
$727$	32179.2	1.64162	0.820812	−	0.571199i	$-0.193521\pi$
$727$	32179.2	1.64162	0.820812	+	0.571199i	$0.193521\pi$
$728$	0	0
$729$	−12416.0	−0.630800
$730$	0	0
$731$	−6348.69	−0.321224
$732$	0	0
$733$	−13693.8	−0.690031	−0.345016	−	0.938597i	$-0.612126\pi$
$733$	−13693.8	−0.690031	−0.345016	+	0.938597i	$0.612126\pi$
$734$	0	0
$735$	1111.14	0.0557619
$736$	0	0
$737$	5369.17	0.268353
$738$	0	0
$739$	29132.6	1.45015	0.725074	−	0.688671i	$-0.241806\pi$
$739$	29132.6	1.45015	0.725074	+	0.688671i	$0.241806\pi$
$740$	0	0
$741$	−16171.1	−0.801700
$742$	0	0
$743$	35966.3	1.77587	0.887937	−	0.459964i	$-0.152138\pi$
$743$	35966.3	1.77587	0.887937	+	0.459964i	$0.152138\pi$
$744$	0	0
$745$	−7113.76	−0.349836
$746$	0	0
$747$	−12119.1	−0.593593
$748$	0	0
$749$	7470.24	0.364428
$750$	0	0
$751$	4200.62	0.204105	0.102052	−	0.994779i	$-0.467459\pi$
$751$	4200.62	0.204105	0.102052	+	0.994779i	$0.467459\pi$
$752$	0	0
$753$	5091.25	0.246395
$754$	0	0
$755$	−3397.28	−0.163761
$756$	0	0
$757$	−3869.05	−0.185764	−0.0928818	−	0.995677i	$-0.529608\pi$
$757$	−3869.05	−0.185764	−0.0928818	+	0.995677i	$0.529608\pi$
$758$	0	0
$759$	468.319	0.0223965
$760$	0	0
$761$	−16200.5	−0.771704	−0.385852	−	0.922561i	$-0.626093\pi$
$761$	−16200.5	−0.771704	−0.385852	+	0.922561i	$0.626093\pi$
$762$	0	0
$763$	4119.99	0.195483
$764$	0	0
$765$	3230.55	0.152681
$766$	0	0
$767$	3222.75	0.151717
$768$	0	0
$769$	−32488.0	−1.52347	−0.761734	−	0.647890i	$-0.775652\pi$
$769$	−32488.0	−1.52347	−0.761734	+	0.647890i	$0.775652\pi$
$770$	0	0
$771$	−4849.40	−0.226520
$772$	0	0
$773$	−1876.44	−0.0873103	−0.0436552	−	0.999047i	$-0.513900\pi$
$773$	−1876.44	−0.0873103	−0.0436552	+	0.999047i	$0.513900\pi$
$774$	0	0
$775$	2794.43	0.129521
$776$	0	0
$777$	−6227.26	−0.287518
$778$	0	0
$779$	−29709.8	−1.36645
$780$	0	0
$781$	5460.67	0.250190
$782$	0	0
$783$	−18771.8	−0.856770
$784$	0	0
$785$	−6712.15	−0.305181
$786$	0	0
$787$	34613.9	1.56779	0.783897	−	0.620891i	$-0.213229\pi$
$787$	34613.9	1.56779	0.783897	+	0.620891i	$0.213229\pi$
$788$	0	0
$789$	−6882.35	−0.310542
$790$	0	0
$791$	25763.5	1.15808
$792$	0	0
$793$	−52027.0	−2.32980
$794$	0	0
$795$	−1577.51	−0.0703756
$796$	0	0
$797$	−36433.4	−1.61924	−0.809622	−	0.586952i	$-0.800328\pi$
$797$	−36433.4	−1.61924	−0.809622	+	0.586952i	$0.800328\pi$
$798$	0	0
$799$	−6344.80	−0.280930
$800$	0	0
$801$	14774.6	0.651729
$802$	0	0
$803$	−10621.2	−0.466769
$804$	0	0
$805$	−1522.45	−0.0666575
$806$	0	0
$807$	−7737.15	−0.337498
$808$	0	0
$809$	14416.6	0.626528	0.313264	−	0.949666i	$-0.398577\pi$
$809$	14416.6	0.626528	0.313264	+	0.949666i	$0.398577\pi$
$810$	0	0
$811$	−31008.8	−1.34262	−0.671310	−	0.741177i	$-0.734268\pi$
$811$	−31008.8	−1.34262	−0.671310	+	0.741177i	$0.734268\pi$
$812$	0	0
$813$	−6392.57	−0.275765
$814$	0	0
$815$	−7868.18	−0.338172
$816$	0	0
$817$	−37463.1	−1.60425
$818$	0	0
$819$	−27010.4	−1.15240
$820$	0	0
$821$	−7880.13	−0.334980	−0.167490	−	0.985874i	$-0.553566\pi$
$821$	−7880.13	−0.334980	−0.167490	+	0.985874i	$0.553566\pi$
$822$	0	0
$823$	−28691.6	−1.21522	−0.607610	−	0.794236i	$-0.707871\pi$
$823$	−28691.6	−1.21522	−0.607610	+	0.794236i	$0.707871\pi$
$824$	0	0
$825$	509.043	0.0214819
$826$	0	0
$827$	−21182.0	−0.890653	−0.445327	−	0.895368i	$-0.646912\pi$
$827$	−21182.0	−0.890653	−0.445327	+	0.895368i	$0.646912\pi$
$828$	0	0
$829$	−6621.96	−0.277431	−0.138716	−	0.990332i	$-0.544297\pi$
$829$	−6621.96	−0.277431	−0.138716	+	0.990332i	$0.544297\pi$
$830$	0	0
$831$	−5053.81	−0.210968
$832$	0	0
$833$	4293.03	0.178565
$834$	0	0
$835$	13713.8	0.568367
$836$	0	0
$837$	7736.87	0.319505
$838$	0	0
$839$	−15760.4	−0.648520	−0.324260	−	0.945968i	$-0.605115\pi$
$839$	−15760.4	−0.648520	−0.324260	+	0.945968i	$0.605115\pi$
$840$	0	0
$841$	49162.0	2.01575
$842$	0	0
$843$	−6387.56	−0.260972
$844$	0	0
$845$	21673.7	0.882366
$846$	0	0
$847$	14493.7	0.587967
$848$	0	0
$849$	−6488.55	−0.262293
$850$	0	0
$851$	−8166.00	−0.328939
$852$	0	0
$853$	−36388.9	−1.46065	−0.730324	−	0.683101i	$-0.760631\pi$
$853$	−36388.9	−1.46065	−0.730324	+	0.683101i	$0.760631\pi$
$854$	0	0
$855$	19063.2	0.762512
$856$	0	0
$857$	13705.5	0.546289	0.273145	−	0.961973i	$-0.411936\pi$
$857$	13705.5	0.546289	0.273145	+	0.961973i	$0.411936\pi$
$858$	0	0
$859$	11868.6	0.471420	0.235710	−	0.971823i	$-0.424258\pi$
$859$	11868.6	0.471420	0.235710	+	0.971823i	$0.424258\pi$
$860$	0	0
$861$	3450.32	0.136570
$862$	0	0
$863$	1617.09	0.0637851	0.0318926	−	0.999491i	$-0.489847\pi$
$863$	1617.09	0.0637851	0.0318926	+	0.999491i	$0.489847\pi$
$864$	0	0
$865$	−13469.2	−0.529440
$866$	0	0
$867$	5641.19	0.220975
$868$	0	0
$869$	−15868.7	−0.619458
$870$	0	0
$871$	28234.4	1.09838
$872$	0	0
$873$	27481.4	1.06541
$874$	0	0
$875$	−1654.84	−0.0639357
$876$	0	0
$877$	34187.2	1.31633	0.658164	−	0.752875i	$-0.271333\pi$
$877$	34187.2	1.31633	0.658164	+	0.752875i	$0.271333\pi$
$878$	0	0
$879$	−840.045	−0.0322344
$880$	0	0
$881$	44843.7	1.71489	0.857447	−	0.514572i	$-0.172049\pi$
$881$	44843.7	1.71489	0.857447	+	0.514572i	$0.172049\pi$
$882$	0	0
$883$	−4278.18	−0.163049	−0.0815244	−	0.996671i	$-0.525979\pi$
$883$	−4278.18	−0.163049	−0.0815244	+	0.996671i	$0.525979\pi$
$884$	0	0
$885$	264.151	0.0100331
$886$	0	0
$887$	−41274.8	−1.56243	−0.781213	−	0.624265i	$-0.785399\pi$
$887$	−41274.8	−1.56243	−0.781213	+	0.624265i	$0.785399\pi$
$888$	0	0
$889$	28225.5	1.06485
$890$	0	0
$891$	−9066.23	−0.340887
$892$	0	0
$893$	−37440.1	−1.40301
$894$	0	0
$895$	−3780.70	−0.141201
$896$	0	0
$897$	2462.70	0.0916692
$898$	0	0
$899$	−30314.3	−1.12463
$900$	0	0
$901$	−6094.92	−0.225362
$902$	0	0
$903$	4350.75	0.160337
$904$	0	0
$905$	−607.943	−0.0223300
$906$	0	0
$907$	4746.93	0.173781	0.0868905	−	0.996218i	$-0.472307\pi$
$907$	4746.93	0.173781	0.0868905	+	0.996218i	$0.472307\pi$
$908$	0	0
$909$	20175.5	0.736172
$910$	0	0
$911$	−21904.2	−0.796619	−0.398310	−	0.917251i	$-0.630403\pi$
$911$	−21904.2	−0.796619	−0.398310	+	0.917251i	$0.630403\pi$
$912$	0	0
$913$	−7378.08	−0.267447
$914$	0	0
$915$	−4264.36	−0.154071
$916$	0	0
$917$	−12276.2	−0.442088
$918$	0	0
$919$	−27533.7	−0.988305	−0.494152	−	0.869375i	$-0.664522\pi$
$919$	−27533.7	−0.988305	−0.494152	+	0.869375i	$0.664522\pi$
$920$	0	0
$921$	419.917	0.0150236
$922$	0	0
$923$	28715.5	1.02403
$924$	0	0
$925$	−8876.09	−0.315507
$926$	0	0
$927$	−30295.8	−1.07340
$928$	0	0
$929$	−43283.9	−1.52863	−0.764315	−	0.644843i	$-0.776923\pi$
$929$	−43283.9	−1.52863	−0.764315	+	0.644843i	$0.776923\pi$
$930$	0	0
$931$	25332.8	0.891783
$932$	0	0
$933$	2542.59	0.0892183
$934$	0	0
$935$	1966.75	0.0687911
$936$	0	0
$937$	−45113.4	−1.57288	−0.786441	−	0.617666i	$-0.788079\pi$
$937$	−45113.4	−1.57288	−0.786441	+	0.617666i	$0.788079\pi$
$938$	0	0
$939$	1451.54	0.0504464
$940$	0	0
$941$	47251.6	1.63694	0.818469	−	0.574551i	$-0.194823\pi$
$941$	47251.6	1.63694	0.818469	+	0.574551i	$0.194823\pi$
$942$	0	0
$943$	4524.52	0.156245
$944$	0	0
$945$	−4581.71	−0.157718
$946$	0	0
$947$	21772.1	0.747094	0.373547	−	0.927611i	$-0.378142\pi$
$947$	21772.1	0.747094	0.373547	+	0.927611i	$0.378142\pi$
$948$	0	0
$949$	−55852.9	−1.91050
$950$	0	0
$951$	767.366	0.0261657
$952$	0	0
$953$	8971.24	0.304939	0.152470	−	0.988308i	$-0.451277\pi$
$953$	8971.24	0.304939	0.152470	+	0.988308i	$0.451277\pi$
$954$	0	0
$955$	684.577	0.0231962
$956$	0	0
$957$	−5522.15	−0.186526
$958$	0	0
$959$	−13621.9	−0.458679
$960$	0	0
$961$	−17296.9	−0.580607
$962$	0	0
$963$	14244.9	0.476673
$964$	0	0
$965$	2025.09	0.0675545
$966$	0	0
$967$	12271.1	0.408079	0.204039	−	0.978963i	$-0.434593\pi$
$967$	12271.1	0.408079	0.204039	+	0.978963i	$0.434593\pi$
$968$	0	0
$969$	−5121.06	−0.169775
$970$	0	0
$971$	40051.1	1.32369	0.661844	−	0.749641i	$-0.269774\pi$
$971$	40051.1	1.32369	0.661844	+	0.749641i	$0.269774\pi$
$972$	0	0
$973$	−38968.1	−1.28393
$974$	0	0
$975$	2676.85	0.0879260
$976$	0	0
$977$	24519.1	0.802904	0.401452	−	0.915880i	$-0.368506\pi$
$977$	24519.1	0.802904	0.401452	+	0.915880i	$0.368506\pi$
$978$	0	0
$979$	8994.75	0.293640
$980$	0	0
$981$	7856.37	0.255693
$982$	0	0
$983$	−22521.5	−0.730747	−0.365374	−	0.930861i	$-0.619059\pi$
$983$	−22521.5	−0.730747	−0.365374	+	0.930861i	$0.619059\pi$
$984$	0	0
$985$	−6402.27	−0.207100
$986$	0	0
$987$	4348.08	0.140224
$988$	0	0
$989$	5705.28	0.183435
$990$	0	0
$991$	56656.4	1.81609	0.908047	−	0.418868i	$-0.137573\pi$
$991$	56656.4	1.81609	0.908047	+	0.418868i	$0.137573\pi$
$992$	0	0
$993$	−5422.03	−0.173276
$994$	0	0
$995$	1611.77	0.0513534
$996$	0	0
$997$	19626.5	0.623447	0.311723	−	0.950173i	$-0.399094\pi$
$997$	19626.5	0.623447	0.311723	+	0.950173i	$0.399094\pi$
$998$	0	0
$999$	−24575.0	−0.778298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.x.1.3		8
4.3	odd	2		920.4.a.c.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.c.1.6	✓	8	4.3	odd	2
1840.4.a.x.1.3		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32486 q^{3} +5.00000 q^{5} -13.2387 q^{7} -25.2447 q^{9} +O(q^{10})\)	\(q-1.32486 q^{3} +5.00000 q^{5} -13.2387 q^{7} -25.2447 q^{9} -15.3690 q^{11} -80.8192 q^{13} -6.62430 q^{15} -25.5938 q^{17} -151.027 q^{19} +17.5394 q^{21} +23.0000 q^{23} +25.0000 q^{25} +69.2169 q^{27} -271.203 q^{29} +111.777 q^{31} +20.3617 q^{33} -66.1935 q^{35} -355.044 q^{37} +107.074 q^{39} +196.718 q^{41} +248.056 q^{43} -126.224 q^{45} +247.903 q^{47} -167.737 q^{49} +33.9082 q^{51} +238.140 q^{53} -76.8448 q^{55} +200.090 q^{57} -39.8761 q^{59} +643.745 q^{61} +334.208 q^{63} -404.096 q^{65} -349.352 q^{67} -30.4718 q^{69} -355.305 q^{71} +691.084 q^{73} -33.1215 q^{75} +203.465 q^{77} +1032.52 q^{79} +589.905 q^{81} +480.064 q^{83} -127.969 q^{85} +359.306 q^{87} -585.254 q^{89} +1069.94 q^{91} -148.089 q^{93} -755.136 q^{95} -1088.60 q^{97} +387.985 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9}+O(q^{10}) \) 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9	\( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9} + 35 q^{11} - 54 q^{13} + 60 q^{15} - 11 q^{17} + 107 q^{19} - 148 q^{21} + 184 q^{23} + 200 q^{25} + 405 q^{27} + 37 q^{29} + 290 q^{31} + 289 q^{33} + 155 q^{35} - 172 q^{37} + 311 q^{39} + 308 q^{41} + 538 q^{43} + 310 q^{45} + 1035 q^{47} + 585 q^{49} + 387 q^{51} + 46 q^{53} + 175 q^{55} + 286 q^{57} + 1256 q^{59} + 399 q^{61} + 1234 q^{63} - 270 q^{65} + 1598 q^{67} + 276 q^{69} + 750 q^{71} + 177 q^{73} + 300 q^{75} - 1228 q^{77} + 1292 q^{79} + 380 q^{81} + 2094 q^{83} - 55 q^{85} + 2245 q^{87} + 484 q^{89} + 679 q^{91} - 2111 q^{93} + 535 q^{95} - 2225 q^{97} + 2117 q^{99}+O(q^{100}) \) 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9 + 35 * q^11 - 54 * q^13 + 60 * q^15 - 11 * q^17 + 107 * q^19 - 148 * q^21 + 184 * q^23 + 200 * q^25 + 405 * q^27 + 37 * q^29 + 290 * q^31 + 289 * q^33 + 155 * q^35 - 172 * q^37 + 311 * q^39 + 308 * q^41 + 538 * q^43 + 310 * q^45 + 1035 * q^47 + 585 * q^49 + 387 * q^51 + 46 * q^53 + 175 * q^55 + 286 * q^57 + 1256 * q^59 + 399 * q^61 + 1234 * q^63 - 270 * q^65 + 1598 * q^67 + 276 * q^69 + 750 * q^71 + 177 * q^73 + 300 * q^75 - 1228 * q^77 + 1292 * q^79 + 380 * q^81 + 2094 * q^83 - 55 * q^85 + 2245 * q^87 + 484 * q^89 + 679 * q^91 - 2111 * q^93 + 535 * q^95 - 2225 * q^97 + 2117 * q^99

Introduction

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