LMFDB - Embedded newform 1521.4.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.83218$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.03025 q^{2} +8.24289 q^{4} -8.08864 q^{5} +5.95078 q^{7} -0.978887 q^{8} +O(q^{10})$	$q-4.03025 q^{2} +8.24289 q^{4} -8.08864 q^{5} +5.95078 q^{7} -0.978887 q^{8} +32.5992 q^{10} -17.2359 q^{11} -23.9831 q^{14} -61.9979 q^{16} -92.9299 q^{17} +13.3832 q^{19} -66.6738 q^{20} +69.4648 q^{22} -219.710 q^{23} -59.5738 q^{25} +49.0516 q^{28} +199.485 q^{29} +307.777 q^{31} +257.698 q^{32} +374.531 q^{34} -48.1337 q^{35} -333.777 q^{37} -53.9376 q^{38} +7.91787 q^{40} +200.689 q^{41} +116.806 q^{43} -142.073 q^{44} +885.487 q^{46} -338.610 q^{47} -307.588 q^{49} +240.097 q^{50} +26.6215 q^{53} +139.415 q^{55} -5.82514 q^{56} -803.976 q^{58} +280.058 q^{59} -207.084 q^{61} -1240.42 q^{62} -542.603 q^{64} -285.981 q^{67} -766.011 q^{68} +193.991 q^{70} +317.673 q^{71} -63.0668 q^{73} +1345.20 q^{74} +110.316 q^{76} -102.567 q^{77} -623.835 q^{79} +501.479 q^{80} -808.824 q^{82} -659.874 q^{83} +751.677 q^{85} -470.755 q^{86} +16.8720 q^{88} -1273.19 q^{89} -1811.05 q^{92} +1364.68 q^{94} -108.252 q^{95} +603.746 q^{97} +1239.66 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) $ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8	$ 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) $ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.03025	−1.42491	−0.712454	−	0.701719i	$-0.752416\pi$
$2$	−4.03025	−1.42491	−0.712454	+	0.701719i	$0.752416\pi$
$3$	0	0
$3$	0	0
$4$	8.24289	1.03036
$5$	−8.08864	−0.723470	−0.361735	−	0.932281i	$-0.617816\pi$
$5$	−8.08864	−0.723470	−0.361735	+	0.932281i	$0.617816\pi$
$6$	0	0
$7$	5.95078	0.321312	0.160656	−	0.987010i	$-0.448639\pi$
$7$	5.95078	0.321312	0.160656	+	0.987010i	$0.448639\pi$
$8$	−0.978887	−0.0432611
$9$	0	0
$10$	32.5992	1.03088
$11$	−17.2359	−0.472437	−0.236219	−	0.971700i	$-0.575908\pi$
$11$	−17.2359	−0.472437	−0.236219	+	0.971700i	$0.575908\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−23.9831	−0.457840
$15$	0	0
$16$	−61.9979	−0.968718
$17$	−92.9299	−1.32581	−0.662907	−	0.748702i	$-0.730677\pi$
$17$	−92.9299	−1.32581	−0.662907	+	0.748702i	$0.730677\pi$
$18$	0	0
$19$	13.3832	0.161596	0.0807979	−	0.996731i	$-0.474253\pi$
$19$	13.3832	0.161596	0.0807979	+	0.996731i	$0.474253\pi$
$20$	−66.6738	−0.745435
$21$	0	0
$22$	69.4648	0.673179
$23$	−219.710	−1.99186	−0.995930	−	0.0901293i	$-0.971272\pi$
$23$	−219.710	−1.99186	−0.995930	+	0.0901293i	$0.971272\pi$
$24$	0	0
$25$	−59.5738	−0.476591
$26$	0	0
$27$	0	0
$28$	49.0516	0.331067
$29$	199.485	1.27736	0.638681	−	0.769471i	$-0.279480\pi$
$29$	199.485	1.27736	0.638681	+	0.769471i	$0.279480\pi$
$30$	0	0
$31$	307.777	1.78317	0.891586	−	0.452851i	$-0.149593\pi$
$31$	307.777	1.78317	0.891586	+	0.452851i	$0.149593\pi$
$32$	257.698	1.42359
$33$	0	0
$34$	374.531	1.88916
$35$	−48.1337	−0.232460
$36$	0	0
$37$	−333.777	−1.48304	−0.741521	−	0.670930i	$-0.765895\pi$
$37$	−333.777	−1.48304	−0.741521	+	0.670930i	$0.765895\pi$
$38$	−53.9376	−0.230259
$39$	0	0
$40$	7.91787	0.0312981
$41$	200.689	0.764446	0.382223	−	0.924070i	$-0.375159\pi$
$41$	200.689	0.764446	0.382223	+	0.924070i	$0.375159\pi$
$42$	0	0
$43$	116.806	0.414248	0.207124	−	0.978315i	$-0.433590\pi$
$43$	116.806	0.414248	0.207124	+	0.978315i	$0.433590\pi$
$44$	−142.073	−0.486781
$45$	0	0
$46$	885.487	2.83822
$47$	−338.610	−1.05088	−0.525440	−	0.850831i	$-0.676099\pi$
$47$	−338.610	−1.05088	−0.525440	+	0.850831i	$0.676099\pi$
$48$	0	0
$49$	−307.588	−0.896759
$50$	240.097	0.679097
$51$	0	0
$52$	0	0
$53$	26.6215	0.0689951	0.0344975	−	0.999405i	$-0.489017\pi$
$53$	26.6215	0.0689951	0.0344975	+	0.999405i	$0.489017\pi$
$54$	0	0
$55$	139.415	0.341794
$56$	−5.82514	−0.0139003
$57$	0	0
$58$	−803.976	−1.82012
$59$	280.058	0.617973	0.308987	−	0.951066i	$-0.400010\pi$
$59$	280.058	0.617973	0.308987	+	0.951066i	$0.400010\pi$
$60$	0	0
$61$	−207.084	−0.434663	−0.217332	−	0.976098i	$-0.569735\pi$
$61$	−207.084	−0.434663	−0.217332	+	0.976098i	$0.569735\pi$
$62$	−1240.42	−2.54086
$63$	0	0
$64$	−542.603	−1.05977
$65$	0	0
$66$	0	0
$67$	−285.981	−0.521465	−0.260732	−	0.965411i	$-0.583964\pi$
$67$	−285.981	−0.521465	−0.260732	+	0.965411i	$0.583964\pi$
$68$	−766.011	−1.36607
$69$	0	0
$70$	193.991	0.331233
$71$	317.673	0.530998	0.265499	−	0.964111i	$-0.414463\pi$
$71$	317.673	0.530998	0.265499	+	0.964111i	$0.414463\pi$
$72$	0	0
$73$	−63.0668	−0.101115	−0.0505576	−	0.998721i	$-0.516100\pi$
$73$	−63.0668	−0.101115	−0.0505576	+	0.998721i	$0.516100\pi$
$74$	1345.20	2.11320
$75$	0	0
$76$	110.316	0.166502
$77$	−102.567	−0.151800
$78$	0	0
$79$	−623.835	−0.888443	−0.444221	−	0.895917i	$-0.646520\pi$
$79$	−623.835	−0.888443	−0.444221	+	0.895917i	$0.646520\pi$
$80$	501.479	0.700838
$81$	0	0
$82$	−808.824	−1.08926
$83$	−659.874	−0.872658	−0.436329	−	0.899787i	$-0.643722\pi$
$83$	−659.874	−0.872658	−0.436329	+	0.899787i	$0.643722\pi$
$84$	0	0
$85$	751.677	0.959186
$86$	−470.755	−0.590265
$87$	0	0
$88$	16.8720	0.0204382
$89$	−1273.19	−1.51638	−0.758191	−	0.652032i	$-0.773917\pi$
$89$	−1273.19	−1.51638	−0.758191	+	0.652032i	$0.773917\pi$
$90$	0	0
$91$	0	0
$92$	−1811.05	−2.05233
$93$	0	0
$94$	1364.68	1.49741
$95$	−108.252	−0.116910
$96$	0	0
$97$	603.746	0.631970	0.315985	−	0.948764i	$-0.397665\pi$
$97$	603.746	0.631970	0.315985	+	0.948764i	$0.397665\pi$
$98$	1239.66	1.27780
$99$	0	0
$100$	−491.060	−0.491060
$101$	740.588	0.729616	0.364808	−	0.931083i	$-0.381135\pi$
$101$	740.588	0.729616	0.364808	+	0.931083i	$0.381135\pi$
$102$	0	0
$103$	−1888.57	−1.80666	−0.903332	−	0.428942i	$-0.858887\pi$
$103$	−1888.57	−1.80666	−0.903332	+	0.428942i	$0.858887\pi$
$104$	0	0
$105$	0	0
$106$	−107.291	−0.0983116
$107$	919.463	0.830728	0.415364	−	0.909655i	$-0.363654\pi$
$107$	919.463	0.830728	0.415364	+	0.909655i	$0.363654\pi$
$108$	0	0
$109$	1570.40	1.37998	0.689988	−	0.723821i	$-0.257616\pi$
$109$	1570.40	1.37998	0.689988	+	0.723821i	$0.257616\pi$
$110$	−561.876	−0.487025
$111$	0	0
$112$	−368.936	−0.311260
$113$	1324.66	1.10277	0.551386	−	0.834250i	$-0.314099\pi$
$113$	1324.66	1.10277	0.551386	+	0.834250i	$0.314099\pi$
$114$	0	0
$115$	1777.16	1.44105
$116$	1644.34	1.31614
$117$	0	0
$118$	−1128.70	−0.880555
$119$	−553.006	−0.426000
$120$	0	0
$121$	−1033.92	−0.776803
$122$	834.601	0.619354
$123$	0	0
$124$	2536.97	1.83731
$125$	1492.95	1.06827
$126$	0	0
$127$	−2350.39	−1.64223	−0.821115	−	0.570763i	$-0.806648\pi$
$127$	−2350.39	−1.64223	−0.821115	+	0.570763i	$0.806648\pi$
$128$	125.240	0.0864824
$129$	0	0
$130$	0	0
$131$	−2308.69	−1.53978	−0.769890	−	0.638177i	$-0.779689\pi$
$131$	−2308.69	−1.53978	−0.769890	+	0.638177i	$0.779689\pi$
$132$	0	0
$133$	79.6405	0.0519226
$134$	1152.57	0.743039
$135$	0	0
$136$	90.9680	0.0573562
$137$	374.912	0.233802	0.116901	−	0.993144i	$-0.462704\pi$
$137$	374.912	0.233802	0.116901	+	0.993144i	$0.462704\pi$
$138$	0	0
$139$	487.711	0.297605	0.148802	−	0.988867i	$-0.452458\pi$
$139$	487.711	0.297605	0.148802	+	0.988867i	$0.452458\pi$
$140$	−396.761	−0.239517
$141$	0	0
$142$	−1280.30	−0.756623
$143$	0	0
$144$	0	0
$145$	−1613.57	−0.924134
$146$	254.175	0.144080
$147$	0	0
$148$	−2751.28	−1.52807
$149$	−1055.91	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$149$	−1055.91	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$150$	0	0
$151$	−888.560	−0.478874	−0.239437	−	0.970912i	$-0.576963\pi$
$151$	−888.560	−0.478874	−0.239437	+	0.970912i	$0.576963\pi$
$152$	−13.1007	−0.00699081
$153$	0	0
$154$	413.370	0.216301
$155$	−2489.50	−1.29007
$156$	0	0
$157$	−3648.56	−1.85469	−0.927346	−	0.374206i	$-0.877915\pi$
$157$	−3648.56	−1.85469	−0.927346	+	0.374206i	$0.877915\pi$
$158$	2514.21	1.26595
$159$	0	0
$160$	−2084.43	−1.02993
$161$	−1307.45	−0.640008
$162$	0	0
$163$	1386.49	0.666249	0.333125	−	0.942883i	$-0.391897\pi$
$163$	1386.49	0.666249	0.333125	+	0.942883i	$0.391897\pi$
$164$	1654.25	0.787655
$165$	0	0
$166$	2659.46	1.24346
$167$	3376.19	1.56442	0.782209	−	0.623017i	$-0.214093\pi$
$167$	3376.19	1.56442	0.782209	+	0.623017i	$0.214093\pi$
$168$	0	0
$169$	0	0
$170$	−3029.44	−1.36675
$171$	0	0
$172$	962.815	0.426825
$173$	−341.817	−0.150219	−0.0751095	−	0.997175i	$-0.523931\pi$
$173$	−341.817	−0.150219	−0.0751095	+	0.997175i	$0.523931\pi$
$174$	0	0
$175$	−354.511	−0.153134
$176$	1068.59	0.457658
$177$	0	0
$178$	5131.28	2.16070
$179$	2885.55	1.20489	0.602446	−	0.798159i	$-0.294193\pi$
$179$	2885.55	1.20489	0.602446	+	0.798159i	$0.294193\pi$
$180$	0	0
$181$	3795.62	1.55871	0.779354	−	0.626584i	$-0.215547\pi$
$181$	3795.62	1.55871	0.779354	+	0.626584i	$0.215547\pi$
$182$	0	0
$183$	0	0
$184$	215.072	0.0861701
$185$	2699.80	1.07294
$186$	0	0
$187$	1601.73	0.626363
$188$	−2791.12	−1.08278
$189$	0	0
$190$	436.282	0.166586
$191$	2805.90	1.06297	0.531486	−	0.847067i	$-0.321634\pi$
$191$	2805.90	1.06297	0.531486	+	0.847067i	$0.321634\pi$
$192$	0	0
$193$	−2485.64	−0.927048	−0.463524	−	0.886084i	$-0.653415\pi$
$193$	−2485.64	−0.927048	−0.463524	+	0.886084i	$0.653415\pi$
$194$	−2433.24	−0.900499
$195$	0	0
$196$	−2535.41	−0.923985
$197$	−2750.70	−0.994818	−0.497409	−	0.867516i	$-0.665715\pi$
$197$	−2750.70	−0.994818	−0.497409	+	0.867516i	$0.665715\pi$
$198$	0	0
$199$	3998.53	1.42436	0.712182	−	0.701994i	$-0.247707\pi$
$199$	3998.53	1.42436	0.712182	+	0.701994i	$0.247707\pi$
$200$	58.3161	0.0206178
$201$	0	0
$202$	−2984.75	−1.03964
$203$	1187.09	0.410432
$204$	0	0
$205$	−1623.30	−0.553054
$206$	7611.41	2.57433
$207$	0	0
$208$	0	0
$209$	−230.671	−0.0763438
$210$	0	0
$211$	−1375.80	−0.448882	−0.224441	−	0.974488i	$-0.572056\pi$
$211$	−1375.80	−0.448882	−0.224441	+	0.974488i	$0.572056\pi$
$212$	219.438	0.0710898
$213$	0	0
$214$	−3705.66	−1.18371
$215$	−944.799	−0.299696
$216$	0	0
$217$	1831.51	0.572955
$218$	−6329.12	−1.96634
$219$	0	0
$220$	1149.18	0.352171
$221$	0	0
$222$	0	0
$223$	1694.37	0.508805	0.254403	−	0.967098i	$-0.418121\pi$
$223$	1694.37	0.508805	0.254403	+	0.967098i	$0.418121\pi$
$224$	1533.50	0.457418
$225$	0	0
$226$	−5338.70	−1.57135
$227$	405.400	0.118534	0.0592672	−	0.998242i	$-0.481124\pi$
$227$	405.400	0.118534	0.0592672	+	0.998242i	$0.481124\pi$
$228$	0	0
$229$	2359.54	0.680884	0.340442	−	0.940265i	$-0.389423\pi$
$229$	2359.54	0.680884	0.340442	+	0.940265i	$0.389423\pi$
$230$	−7162.39	−2.05337
$231$	0	0
$232$	−195.274	−0.0552601
$233$	−938.507	−0.263878	−0.131939	−	0.991258i	$-0.542120\pi$
$233$	−938.507	−0.263878	−0.131939	+	0.991258i	$0.542120\pi$
$234$	0	0
$235$	2738.90	0.760280
$236$	2308.49	0.636736
$237$	0	0
$238$	2228.75	0.607010
$239$	−4664.25	−1.26237	−0.631183	−	0.775634i	$-0.717430\pi$
$239$	−4664.25	−1.26237	−0.631183	+	0.775634i	$0.717430\pi$
$240$	0	0
$241$	−3774.94	−1.00898	−0.504492	−	0.863416i	$-0.668320\pi$
$241$	−3774.94	−1.00898	−0.504492	+	0.863416i	$0.668320\pi$
$242$	4166.97	1.10687
$243$	0	0
$244$	−1706.97	−0.447860
$245$	2487.97	0.648778
$246$	0	0
$247$	0	0
$248$	−301.279	−0.0771421
$249$	0	0
$250$	−6016.96	−1.52219
$251$	960.695	0.241588	0.120794	−	0.992678i	$-0.461456\pi$
$251$	960.695	0.241588	0.120794	+	0.992678i	$0.461456\pi$
$252$	0	0
$253$	3786.90	0.941029
$254$	9472.64	2.34003
$255$	0	0
$256$	3836.08	0.936542
$257$	16.7302	0.00406070	0.00203035	−	0.999998i	$-0.499354\pi$
$257$	16.7302	0.00406070	0.00203035	+	0.999998i	$0.499354\pi$
$258$	0	0
$259$	−1986.23	−0.476519
$260$	0	0
$261$	0	0
$262$	9304.58	2.19404
$263$	400.736	0.0939560	0.0469780	−	0.998896i	$-0.485041\pi$
$263$	400.736	0.0939560	0.0469780	+	0.998896i	$0.485041\pi$
$264$	0	0
$265$	−215.332	−0.0499159
$266$	−320.971	−0.0739849
$267$	0	0
$268$	−2357.31	−0.537297
$269$	−2236.97	−0.507027	−0.253514	−	0.967332i	$-0.581586\pi$
$269$	−2236.97	−0.507027	−0.253514	+	0.967332i	$0.581586\pi$
$270$	0	0
$271$	−4018.89	−0.900850	−0.450425	−	0.892814i	$-0.648727\pi$
$271$	−4018.89	−0.900850	−0.450425	+	0.892814i	$0.648727\pi$
$272$	5761.46	1.28434
$273$	0	0
$274$	−1510.99	−0.333147
$275$	1026.81	0.225159
$276$	0	0
$277$	6792.95	1.47346	0.736731	−	0.676186i	$-0.236368\pi$
$277$	6792.95	1.47346	0.736731	+	0.676186i	$0.236368\pi$
$278$	−1965.59	−0.424059
$279$	0	0
$280$	47.1175	0.0100565
$281$	7286.80	1.54695	0.773477	−	0.633824i	$-0.218516\pi$
$281$	7286.80	1.54695	0.773477	+	0.633824i	$0.218516\pi$
$282$	0	0
$283$	−2429.77	−0.510369	−0.255185	−	0.966892i	$-0.582136\pi$
$283$	−2429.77	−0.510369	−0.255185	+	0.966892i	$0.582136\pi$
$284$	2618.54	0.547120
$285$	0	0
$286$	0	0
$287$	1194.25	0.245626
$288$	0	0
$289$	3722.97	0.757780
$290$	6503.07	1.31681
$291$	0	0
$292$	−519.853	−0.104185
$293$	−4211.25	−0.839672	−0.419836	−	0.907600i	$-0.637912\pi$
$293$	−4211.25	−0.839672	−0.419836	+	0.907600i	$0.637912\pi$
$294$	0	0
$295$	−2265.29	−0.447085
$296$	326.730	0.0641580
$297$	0	0
$298$	4255.56	0.827242
$299$	0	0
$300$	0	0
$301$	695.084	0.133103
$302$	3581.12	0.682351
$303$	0	0
$304$	−829.731	−0.156541
$305$	1675.03	0.314466
$306$	0	0
$307$	8212.27	1.52671	0.763353	−	0.645982i	$-0.223552\pi$
$307$	8212.27	1.52671	0.763353	+	0.645982i	$0.223552\pi$
$308$	−845.447	−0.156408
$309$	0	0
$310$	10033.3	1.83823
$311$	4238.55	0.772818	0.386409	−	0.922328i	$-0.373715\pi$
$311$	4238.55	0.772818	0.386409	+	0.922328i	$0.373715\pi$
$312$	0	0
$313$	3807.82	0.687638	0.343819	−	0.939036i	$-0.388279\pi$
$313$	3807.82	0.687638	0.343819	+	0.939036i	$0.388279\pi$
$314$	14704.6	2.64276
$315$	0	0
$316$	−5142.20	−0.915416
$317$	10497.7	1.85996	0.929981	−	0.367608i	$-0.119823\pi$
$317$	10497.7	1.85996	0.929981	+	0.367608i	$0.119823\pi$
$318$	0	0
$319$	−3438.31	−0.603474
$320$	4388.92	0.766713
$321$	0	0
$322$	5269.34	0.911953
$323$	−1243.70	−0.214246
$324$	0	0
$325$	0	0
$326$	−5587.91	−0.949343
$327$	0	0
$328$	−196.452	−0.0330708
$329$	−2014.99	−0.337660
$330$	0	0
$331$	11905.2	1.97695	0.988474	−	0.151389i	$-0.0483746\pi$
$331$	11905.2	1.97695	0.988474	+	0.151389i	$0.0483746\pi$
$332$	−5439.27	−0.899152
$333$	0	0
$334$	−13606.9	−2.22915
$335$	2313.20	0.377264
$336$	0	0
$337$	8981.18	1.45174	0.725869	−	0.687833i	$-0.241438\pi$
$337$	8981.18	1.45174	0.725869	+	0.687833i	$0.241438\pi$
$338$	0	0
$339$	0	0
$340$	6195.99	0.988308
$341$	−5304.80	−0.842437
$342$	0	0
$343$	−3871.51	−0.609451
$344$	−114.339	−0.0179208
$345$	0	0
$346$	1377.61	0.214048
$347$	5712.87	0.883813	0.441906	−	0.897061i	$-0.354302\pi$
$347$	5712.87	0.883813	0.441906	+	0.897061i	$0.354302\pi$
$348$	0	0
$349$	9507.28	1.45820	0.729102	−	0.684406i	$-0.239938\pi$
$349$	9507.28	1.45820	0.729102	+	0.684406i	$0.239938\pi$
$350$	1428.77	0.218202
$351$	0	0
$352$	−4441.65	−0.672559
$353$	3972.96	0.599035	0.299518	−	0.954091i	$-0.403174\pi$
$353$	3972.96	0.599035	0.299518	+	0.954091i	$0.403174\pi$
$354$	0	0
$355$	−2569.55	−0.384162
$356$	−10494.8	−1.56242
$357$	0	0
$358$	−11629.5	−1.71686
$359$	109.866	0.0161519	0.00807594	−	0.999967i	$-0.497429\pi$
$359$	109.866	0.0161519	0.00807594	+	0.999967i	$0.497429\pi$
$360$	0	0
$361$	−6679.89	−0.973887
$362$	−15297.3	−2.22101
$363$	0	0
$364$	0	0
$365$	510.125	0.0731539
$366$	0	0
$367$	7020.02	0.998480	0.499240	−	0.866464i	$-0.333613\pi$
$367$	7020.02	0.998480	0.499240	+	0.866464i	$0.333613\pi$
$368$	13621.6	1.92955
$369$	0	0
$370$	−10880.9	−1.52884
$371$	158.418	0.0221689
$372$	0	0
$373$	2227.18	0.309166	0.154583	−	0.987980i	$-0.450597\pi$
$373$	2227.18	0.309166	0.154583	+	0.987980i	$0.450597\pi$
$374$	−6455.36	−0.892510
$375$	0	0
$376$	331.461	0.0454622
$377$	0	0
$378$	0	0
$379$	−11004.6	−1.49147	−0.745734	−	0.666244i	$-0.767901\pi$
$379$	−11004.6	−1.49147	−0.745734	+	0.666244i	$0.767901\pi$
$380$	−892.309	−0.120459
$381$	0	0
$382$	−11308.5	−1.51464
$383$	4436.85	0.591938	0.295969	−	0.955198i	$-0.404357\pi$
$383$	4436.85	0.591938	0.295969	+	0.955198i	$0.404357\pi$
$384$	0	0
$385$	829.627	0.109823
$386$	10017.7	1.32096
$387$	0	0
$388$	4976.61	0.651157
$389$	2561.58	0.333874	0.166937	−	0.985968i	$-0.446612\pi$
$389$	2561.58	0.333874	0.166937	+	0.985968i	$0.446612\pi$
$390$	0	0
$391$	20417.7	2.64083
$392$	301.094	0.0387948
$393$	0	0
$394$	11086.0	1.41752
$395$	5045.98	0.642762
$396$	0	0
$397$	2206.90	0.278996	0.139498	−	0.990222i	$-0.455451\pi$
$397$	2206.90	0.278996	0.139498	+	0.990222i	$0.455451\pi$
$398$	−16115.1	−2.02959
$399$	0	0
$400$	3693.45	0.461682
$401$	8071.02	1.00511	0.502553	−	0.864546i	$-0.332394\pi$
$401$	8071.02	1.00511	0.502553	+	0.864546i	$0.332394\pi$
$402$	0	0
$403$	0	0
$404$	6104.58	0.751768
$405$	0	0
$406$	−4784.28	−0.584827
$407$	5752.93	0.700644
$408$	0	0
$409$	6298.36	0.761452	0.380726	−	0.924688i	$-0.375674\pi$
$409$	6298.36	0.761452	0.380726	+	0.924688i	$0.375674\pi$
$410$	6542.29	0.788051
$411$	0	0
$412$	−15567.3	−1.86152
$413$	1666.56	0.198562
$414$	0	0
$415$	5337.49	0.631342
$416$	0	0
$417$	0	0
$418$	929.662	0.108783
$419$	1432.01	0.166965	0.0834824	−	0.996509i	$-0.473396\pi$
$419$	1432.01	0.166965	0.0834824	+	0.996509i	$0.473396\pi$
$420$	0	0
$421$	9641.43	1.11614	0.558069	−	0.829794i	$-0.311542\pi$
$421$	9641.43	1.11614	0.558069	+	0.829794i	$0.311542\pi$
$422$	5544.82	0.639615
$423$	0	0
$424$	−26.0594	−0.00298480
$425$	5536.19	0.631870
$426$	0	0
$427$	−1232.31	−0.139662
$428$	7579.03	0.855949
$429$	0	0
$430$	3807.77	0.427040
$431$	−3370.61	−0.376698	−0.188349	−	0.982102i	$-0.560314\pi$
$431$	−3370.61	−0.376698	−0.188349	+	0.982102i	$0.560314\pi$
$432$	0	0
$433$	6248.91	0.693541	0.346770	−	0.937950i	$-0.387278\pi$
$433$	6248.91	0.693541	0.346770	+	0.937950i	$0.387278\pi$
$434$	−7381.45	−0.816407
$435$	0	0
$436$	12944.7	1.42187
$437$	−2940.43	−0.321876
$438$	0	0
$439$	2866.16	0.311604	0.155802	−	0.987788i	$-0.450204\pi$
$439$	2866.16	0.311604	0.155802	+	0.987788i	$0.450204\pi$
$440$	−136.471	−0.0147864
$441$	0	0
$442$	0	0
$443$	−10776.3	−1.15575	−0.577877	−	0.816124i	$-0.696119\pi$
$443$	−10776.3	−1.15575	−0.577877	+	0.816124i	$0.696119\pi$
$444$	0	0
$445$	10298.4	1.09706
$446$	−6828.74	−0.725000
$447$	0	0
$448$	−3228.91	−0.340517
$449$	−13834.9	−1.45414	−0.727069	−	0.686564i	$-0.759118\pi$
$449$	−13834.9	−1.45414	−0.727069	+	0.686564i	$0.759118\pi$
$450$	0	0
$451$	−3459.04	−0.361153
$452$	10919.0	1.13625
$453$	0	0
$454$	−1633.86	−0.168901
$455$	0	0
$456$	0	0
$457$	17233.0	1.76395	0.881976	−	0.471295i	$-0.156213\pi$
$457$	17233.0	1.76395	0.881976	+	0.471295i	$0.156213\pi$
$458$	−9509.51	−0.970197
$459$	0	0
$460$	14648.9	1.48480
$461$	−11485.0	−1.16033	−0.580164	−	0.814500i	$-0.697012\pi$
$461$	−11485.0	−1.16033	−0.580164	+	0.814500i	$0.697012\pi$
$462$	0	0
$463$	1613.61	0.161967	0.0809837	−	0.996715i	$-0.474194\pi$
$463$	1613.61	0.161967	0.0809837	+	0.996715i	$0.474194\pi$
$464$	−12367.7	−1.23740
$465$	0	0
$466$	3782.41	0.376002
$467$	8149.20	0.807495	0.403747	−	0.914871i	$-0.367707\pi$
$467$	8149.20	0.807495	0.403747	+	0.914871i	$0.367707\pi$
$468$	0	0
$469$	−1701.81	−0.167553
$470$	−11038.4	−1.08333
$471$	0	0
$472$	−274.145	−0.0267342
$473$	−2013.24	−0.195706
$474$	0	0
$475$	−797.289	−0.0770150
$476$	−4558.36	−0.438933
$477$	0	0
$478$	18798.1	1.79875
$479$	−1827.62	−0.174334	−0.0871670	−	0.996194i	$-0.527781\pi$
$479$	−1827.62	−0.174334	−0.0871670	+	0.996194i	$0.527781\pi$
$480$	0	0
$481$	0	0
$482$	15213.9	1.43771
$483$	0	0
$484$	−8522.52	−0.800387
$485$	−4883.49	−0.457212
$486$	0	0
$487$	−7838.95	−0.729398	−0.364699	−	0.931125i	$-0.618828\pi$
$487$	−7838.95	−0.729398	−0.364699	+	0.931125i	$0.618828\pi$
$488$	202.712	0.0188040
$489$	0	0
$490$	−10027.1	−0.924449
$491$	−17196.1	−1.58055	−0.790273	−	0.612755i	$-0.790061\pi$
$491$	−17196.1	−1.58055	−0.790273	+	0.612755i	$0.790061\pi$
$492$	0	0
$493$	−18538.2	−1.69354
$494$	0	0
$495$	0	0
$496$	−19081.5	−1.72739
$497$	1890.40	0.170616
$498$	0	0
$499$	5355.28	0.480431	0.240216	−	0.970720i	$-0.422782\pi$
$499$	5355.28	0.480431	0.240216	+	0.970720i	$0.422782\pi$
$500$	12306.2	1.10070
$501$	0	0
$502$	−3871.84	−0.344240
$503$	−2979.80	−0.264141	−0.132071	−	0.991240i	$-0.542163\pi$
$503$	−2979.80	−0.264141	−0.132071	+	0.991240i	$0.542163\pi$
$504$	0	0
$505$	−5990.35	−0.527856
$506$	−15262.1	−1.34088
$507$	0	0
$508$	−19374.0	−1.69209
$509$	−15353.7	−1.33701	−0.668507	−	0.743706i	$-0.733066\pi$
$509$	−15353.7	−1.33701	−0.668507	+	0.743706i	$0.733066\pi$
$510$	0	0
$511$	−375.297	−0.0324895
$512$	−16462.3	−1.42097
$513$	0	0
$514$	−67.4267	−0.00578611
$515$	15276.0	1.30707
$516$	0	0
$517$	5836.24	0.496475
$518$	8005.00	0.678995
$519$	0	0
$520$	0	0
$521$	−433.801	−0.0364783	−0.0182391	−	0.999834i	$-0.505806\pi$
$521$	−433.801	−0.0364783	−0.0182391	+	0.999834i	$0.505806\pi$
$522$	0	0
$523$	18900.3	1.58021	0.790106	−	0.612970i	$-0.210026\pi$
$523$	18900.3	1.58021	0.790106	+	0.612970i	$0.210026\pi$
$524$	−19030.3	−1.58653
$525$	0	0
$526$	−1615.06	−0.133879
$527$	−28601.7	−2.36415
$528$	0	0
$529$	36105.7	2.96751
$530$	867.839	0.0711255
$531$	0	0
$532$	656.468	0.0534990
$533$	0	0
$534$	0	0
$535$	−7437.21	−0.601007
$536$	279.943	0.0225592
$537$	0	0
$538$	9015.53	0.722467
$539$	5301.55	0.423662
$540$	0	0
$541$	13247.5	1.05278	0.526391	−	0.850243i	$-0.323545\pi$
$541$	13247.5	1.05278	0.526391	+	0.850243i	$0.323545\pi$
$542$	16197.1	1.28363
$543$	0	0
$544$	−23947.9	−1.88742
$545$	−12702.4	−0.998372
$546$	0	0
$547$	−5543.52	−0.433316	−0.216658	−	0.976248i	$-0.569516\pi$
$547$	−5543.52	−0.433316	−0.216658	+	0.976248i	$0.569516\pi$
$548$	3090.36	0.240901
$549$	0	0
$550$	−4138.28	−0.320831
$551$	2669.76	0.206416
$552$	0	0
$553$	−3712.31	−0.285467
$554$	−27377.3	−2.09955
$555$	0	0
$556$	4020.14	0.306640
$557$	−6733.19	−0.512198	−0.256099	−	0.966651i	$-0.582437\pi$
$557$	−6733.19	−0.512198	−0.256099	+	0.966651i	$0.582437\pi$
$558$	0	0
$559$	0	0
$560$	2984.19	0.225188
$561$	0	0
$562$	−29367.6	−2.20427
$563$	−24282.4	−1.81773	−0.908863	−	0.417096i	$-0.863048\pi$
$563$	−24282.4	−1.81773	−0.908863	+	0.417096i	$0.863048\pi$
$564$	0	0
$565$	−10714.7	−0.797823
$566$	9792.55	0.727229
$567$	0	0
$568$	−310.966	−0.0229716
$569$	19876.2	1.46441	0.732207	−	0.681082i	$-0.238490\pi$
$569$	19876.2	1.46441	0.732207	+	0.681082i	$0.238490\pi$
$570$	0	0
$571$	225.014	0.0164913	0.00824567	−	0.999966i	$-0.497375\pi$
$571$	225.014	0.0164913	0.00824567	+	0.999966i	$0.497375\pi$
$572$	0	0
$573$	0	0
$574$	−4813.14	−0.349994
$575$	13089.0	0.949302
$576$	0	0
$577$	−12258.1	−0.884424	−0.442212	−	0.896910i	$-0.645806\pi$
$577$	−12258.1	−0.884424	−0.442212	+	0.896910i	$0.645806\pi$
$578$	−15004.5	−1.07977
$579$	0	0
$580$	−13300.4	−0.952191
$581$	−3926.77	−0.280395
$582$	0	0
$583$	−458.844	−0.0325958
$584$	61.7353	0.00437436
$585$	0	0
$586$	16972.4	1.19645
$587$	−10282.3	−0.722992	−0.361496	−	0.932374i	$-0.617734\pi$
$587$	−10282.3	−0.722992	−0.361496	+	0.932374i	$0.617734\pi$
$588$	0	0
$589$	4119.04	0.288153
$590$	9129.67	0.637055
$591$	0	0
$592$	20693.5	1.43665
$593$	−12355.1	−0.855586	−0.427793	−	0.903877i	$-0.640709\pi$
$593$	−12355.1	−0.855586	−0.427793	+	0.903877i	$0.640709\pi$
$594$	0	0
$595$	4473.07	0.308198
$596$	−8703.71	−0.598184
$597$	0	0
$598$	0	0
$599$	−13035.2	−0.889157	−0.444578	−	0.895740i	$-0.646646\pi$
$599$	−13035.2	−0.889157	−0.444578	+	0.895740i	$0.646646\pi$
$600$	0	0
$601$	−5153.36	−0.349767	−0.174884	−	0.984589i	$-0.555955\pi$
$601$	−5153.36	−0.349767	−0.174884	+	0.984589i	$0.555955\pi$
$602$	−2801.36	−0.189659
$603$	0	0
$604$	−7324.30	−0.493413
$605$	8363.05	0.561994
$606$	0	0
$607$	−1406.55	−0.0940530	−0.0470265	−	0.998894i	$-0.514975\pi$
$607$	−1406.55	−0.0940530	−0.0470265	+	0.998894i	$0.514975\pi$
$608$	3448.83	0.230047
$609$	0	0
$610$	−6750.79	−0.448085
$611$	0	0
$612$	0	0
$613$	−15984.3	−1.05318	−0.526590	−	0.850120i	$-0.676530\pi$
$613$	−15984.3	−1.05318	−0.526590	+	0.850120i	$0.676530\pi$
$614$	−33097.5	−2.17541
$615$	0	0
$616$	100.401	0.00656703
$617$	10324.4	0.673652	0.336826	−	0.941567i	$-0.390647\pi$
$617$	10324.4	0.673652	0.336826	+	0.941567i	$0.390647\pi$
$618$	0	0
$619$	−7423.00	−0.481996	−0.240998	−	0.970526i	$-0.577475\pi$
$619$	−7423.00	−0.481996	−0.240998	+	0.970526i	$0.577475\pi$
$620$	−20520.6	−1.32924
$621$	0	0
$622$	−17082.4	−1.10119
$623$	−7576.48	−0.487232
$624$	0	0
$625$	−4629.23	−0.296271
$626$	−15346.4	−0.979820
$627$	0	0
$628$	−30074.6	−1.91100
$629$	31017.8	1.96624
$630$	0	0
$631$	1190.30	0.0750954	0.0375477	−	0.999295i	$-0.488045\pi$
$631$	1190.30	0.0750954	0.0375477	+	0.999295i	$0.488045\pi$
$632$	610.665	0.0384350
$633$	0	0
$634$	−42308.2	−2.65027
$635$	19011.5	1.18810
$636$	0	0
$637$	0	0
$638$	13857.2	0.859894
$639$	0	0
$640$	−1013.02	−0.0625674
$641$	−705.064	−0.0434451	−0.0217226	−	0.999764i	$-0.506915\pi$
$641$	−705.064	−0.0434451	−0.0217226	+	0.999764i	$0.506915\pi$
$642$	0	0
$643$	14641.4	0.897976	0.448988	−	0.893538i	$-0.351785\pi$
$643$	14641.4	0.897976	0.448988	+	0.893538i	$0.351785\pi$
$644$	−10777.1	−0.659439
$645$	0	0
$646$	5012.42	0.305280
$647$	−1520.83	−0.0924111	−0.0462056	−	0.998932i	$-0.514713\pi$
$647$	−1520.83	−0.0924111	−0.0462056	+	0.998932i	$0.514713\pi$
$648$	0	0
$649$	−4827.04	−0.291954
$650$	0	0
$651$	0	0
$652$	11428.7	0.686477
$653$	22054.4	1.32168	0.660838	−	0.750529i	$-0.270201\pi$
$653$	22054.4	1.32168	0.660838	+	0.750529i	$0.270201\pi$
$654$	0	0
$655$	18674.2	1.11398
$656$	−12442.3	−0.740532
$657$	0	0
$658$	8120.92	0.481134
$659$	12652.3	0.747895	0.373947	−	0.927450i	$-0.378004\pi$
$659$	12652.3	0.747895	0.373947	+	0.927450i	$0.378004\pi$
$660$	0	0
$661$	−10893.2	−0.640994	−0.320497	−	0.947250i	$-0.603850\pi$
$661$	−10893.2	−0.640994	−0.320497	+	0.947250i	$0.603850\pi$
$662$	−47981.0	−2.81697
$663$	0	0
$664$	645.942	0.0377522
$665$	−644.184	−0.0375645
$666$	0	0
$667$	−43829.0	−2.54433
$668$	27829.6	1.61191
$669$	0	0
$670$	−9322.76	−0.537567
$671$	3569.28	0.205351
$672$	0	0
$673$	−9019.75	−0.516621	−0.258310	−	0.966062i	$-0.583166\pi$
$673$	−9019.75	−0.516621	−0.258310	+	0.966062i	$0.583166\pi$
$674$	−36196.4	−2.06859
$675$	0	0
$676$	0	0
$677$	24923.6	1.41491	0.707454	−	0.706760i	$-0.249844\pi$
$677$	24923.6	1.41491	0.707454	+	0.706760i	$0.249844\pi$
$678$	0	0
$679$	3592.76	0.203060
$680$	−735.807	−0.0414955
$681$	0	0
$682$	21379.7	1.20039
$683$	24634.2	1.38009	0.690044	−	0.723767i	$-0.257591\pi$
$683$	24634.2	1.38009	0.690044	+	0.723767i	$0.257591\pi$
$684$	0	0
$685$	−3032.53	−0.169149
$686$	15603.1	0.868411
$687$	0	0
$688$	−7241.70	−0.401290
$689$	0	0
$690$	0	0
$691$	−10340.2	−0.569264	−0.284632	−	0.958637i	$-0.591871\pi$
$691$	−10340.2	−0.569264	−0.284632	+	0.958637i	$0.591871\pi$
$692$	−2817.56	−0.154780
$693$	0	0
$694$	−23024.3	−1.25935
$695$	−3944.92	−0.215308
$696$	0	0
$697$	−18650.0	−1.01351
$698$	−38316.7	−2.07780
$699$	0	0
$700$	−2922.19	−0.157784
$701$	20833.9	1.12252	0.561258	−	0.827641i	$-0.310317\pi$
$701$	20833.9	1.12252	0.561258	+	0.827641i	$0.310317\pi$
$702$	0	0
$703$	−4467.00	−0.239653
$704$	9352.23	0.500676
$705$	0	0
$706$	−16012.0	−0.853569
$707$	4407.07	0.234434
$708$	0	0
$709$	9189.56	0.486772	0.243386	−	0.969930i	$-0.421742\pi$
$709$	9189.56	0.486772	0.243386	+	0.969930i	$0.421742\pi$
$710$	10355.9	0.547395
$711$	0	0
$712$	1246.31	0.0656004
$713$	−67621.8	−3.55183
$714$	0	0
$715$	0	0
$716$	23785.2	1.24147
$717$	0	0
$718$	−442.788	−0.0230149
$719$	−7503.22	−0.389183	−0.194592	−	0.980884i	$-0.562338\pi$
$719$	−7503.22	−0.389183	−0.194592	+	0.980884i	$0.562338\pi$
$720$	0	0
$721$	−11238.5	−0.580503
$722$	26921.6	1.38770
$723$	0	0
$724$	31286.8	1.60603
$725$	−11884.1	−0.608779
$726$	0	0
$727$	−20727.2	−1.05740	−0.528700	−	0.848809i	$-0.677320\pi$
$727$	−20727.2	−1.05740	−0.528700	+	0.848809i	$0.677320\pi$
$728$	0	0
$729$	0	0
$730$	−2055.93	−0.104237
$731$	−10854.7	−0.549216
$732$	0	0
$733$	17361.3	0.874836	0.437418	−	0.899258i	$-0.355893\pi$
$733$	17361.3	0.874836	0.437418	+	0.899258i	$0.355893\pi$
$734$	−28292.4	−1.42274
$735$	0	0
$736$	−56618.9	−2.83560
$737$	4929.13	0.246359
$738$	0	0
$739$	18093.8	0.900663	0.450331	−	0.892861i	$-0.351306\pi$
$739$	18093.8	0.900663	0.450331	+	0.892861i	$0.351306\pi$
$740$	22254.1	1.10551
$741$	0	0
$742$	−638.466	−0.0315887
$743$	14875.3	0.734482	0.367241	−	0.930126i	$-0.380302\pi$
$743$	14875.3	0.734482	0.367241	+	0.930126i	$0.380302\pi$
$744$	0	0
$745$	8540.85	0.420017
$746$	−8976.07	−0.440533
$747$	0	0
$748$	13202.9	0.645380
$749$	5471.52	0.266923
$750$	0	0
$751$	−17750.9	−0.862502	−0.431251	−	0.902232i	$-0.641928\pi$
$751$	−17750.9	−0.862502	−0.431251	+	0.902232i	$0.641928\pi$
$752$	20993.1	1.01801
$753$	0	0
$754$	0	0
$755$	7187.25	0.346451
$756$	0	0
$757$	−10220.3	−0.490702	−0.245351	−	0.969434i	$-0.578903\pi$
$757$	−10220.3	−0.490702	−0.245351	+	0.969434i	$0.578903\pi$
$758$	44351.1	2.12520
$759$	0	0
$760$	105.967	0.00505765
$761$	14112.0	0.672221	0.336110	−	0.941823i	$-0.390888\pi$
$761$	14112.0	0.672221	0.336110	+	0.941823i	$0.390888\pi$
$762$	0	0
$763$	9345.13	0.443403
$764$	23128.7	1.09525
$765$	0	0
$766$	−17881.6	−0.843457
$767$	0	0
$768$	0	0
$769$	−22736.7	−1.06620	−0.533100	−	0.846052i	$-0.678973\pi$
$769$	−22736.7	−1.06620	−0.533100	+	0.846052i	$0.678973\pi$
$770$	−3343.60	−0.156487
$771$	0	0
$772$	−20488.8	−0.955194
$773$	343.173	0.0159678	0.00798388	−	0.999968i	$-0.497459\pi$
$773$	343.173	0.0159678	0.00798388	+	0.999968i	$0.497459\pi$
$774$	0	0
$775$	−18335.4	−0.849844
$776$	−590.999	−0.0273397
$777$	0	0
$778$	−10323.8	−0.475740
$779$	2685.86	0.123531
$780$	0	0
$781$	−5475.37	−0.250863
$782$	−82288.3	−3.76294
$783$	0	0
$784$	19069.8	0.868706
$785$	29511.9	1.34181
$786$	0	0
$787$	−19086.4	−0.864496	−0.432248	−	0.901755i	$-0.642279\pi$
$787$	−19086.4	−0.864496	−0.432248	+	0.901755i	$0.642279\pi$
$788$	−22673.7	−1.02502
$789$	0	0
$790$	−20336.6	−0.915876
$791$	7882.74	0.354334
$792$	0	0
$793$	0	0
$794$	−8894.36	−0.397543
$795$	0	0
$796$	32959.5	1.46761
$797$	−12031.1	−0.534709	−0.267354	−	0.963598i	$-0.586149\pi$
$797$	−12031.1	−0.534709	−0.267354	+	0.963598i	$0.586149\pi$
$798$	0	0
$799$	31467.0	1.39327
$800$	−15352.1	−0.678471
$801$	0	0
$802$	−32528.2	−1.43218
$803$	1087.01	0.0477706
$804$	0	0
$805$	10575.5	0.463027
$806$	0	0
$807$	0	0
$808$	−724.952	−0.0315640
$809$	−10175.4	−0.442210	−0.221105	−	0.975250i	$-0.570966\pi$
$809$	−10175.4	−0.442210	−0.221105	+	0.975250i	$0.570966\pi$
$810$	0	0
$811$	26754.1	1.15840	0.579201	−	0.815185i	$-0.303365\pi$
$811$	26754.1	1.15840	0.579201	+	0.815185i	$0.303365\pi$
$812$	9785.08	0.422893
$813$	0	0
$814$	−23185.7	−0.998353
$815$	−11214.9	−0.482011
$816$	0	0
$817$	1563.23	0.0669408
$818$	−25383.9	−1.08500
$819$	0	0
$820$	−13380.7	−0.569845
$821$	15777.5	0.670693	0.335347	−	0.942095i	$-0.391147\pi$
$821$	15777.5	0.670693	0.335347	+	0.942095i	$0.391147\pi$
$822$	0	0
$823$	−13863.9	−0.587197	−0.293599	−	0.955929i	$-0.594853\pi$
$823$	−13863.9	−0.587197	−0.293599	+	0.955929i	$0.594853\pi$
$824$	1848.70	0.0781583
$825$	0	0
$826$	−6716.66	−0.282933
$827$	26835.0	1.12835	0.564175	−	0.825655i	$-0.309194\pi$
$827$	26835.0	1.12835	0.564175	+	0.825655i	$0.309194\pi$
$828$	0	0
$829$	625.251	0.0261953	0.0130976	−	0.999914i	$-0.495831\pi$
$829$	625.251	0.0261953	0.0130976	+	0.999914i	$0.495831\pi$
$830$	−21511.4	−0.899604
$831$	0	0
$832$	0	0
$833$	28584.2	1.18893
$834$	0	0
$835$	−27308.8	−1.13181
$836$	−1901.40	−0.0786617
$837$	0	0
$838$	−5771.35	−0.237909
$839$	27307.4	1.12367	0.561833	−	0.827251i	$-0.310096\pi$
$839$	27307.4	1.12367	0.561833	+	0.827251i	$0.310096\pi$
$840$	0	0
$841$	15405.4	0.631656
$842$	−38857.3	−1.59039
$843$	0	0
$844$	−11340.6	−0.462510
$845$	0	0
$846$	0	0
$847$	−6152.66	−0.249596
$848$	−1650.48	−0.0668368
$849$	0	0
$850$	−22312.2	−0.900356
$851$	73334.2	2.95401
$852$	0	0
$853$	44801.9	1.79834	0.899172	−	0.437595i	$-0.144170\pi$
$853$	44801.9	1.79834	0.899172	+	0.437595i	$0.144170\pi$
$854$	4966.53	0.199006
$855$	0	0
$856$	−900.051	−0.0359382
$857$	25167.1	1.00314	0.501571	−	0.865116i	$-0.332756\pi$
$857$	25167.1	1.00314	0.501571	+	0.865116i	$0.332756\pi$
$858$	0	0
$859$	4059.10	0.161228	0.0806138	−	0.996745i	$-0.474312\pi$
$859$	4059.10	0.161228	0.0806138	+	0.996745i	$0.474312\pi$
$860$	−7787.87	−0.308795
$861$	0	0
$862$	13584.4	0.536759
$863$	−818.924	−0.0323019	−0.0161509	−	0.999870i	$-0.505141\pi$
$863$	−818.924	−0.0323019	−0.0161509	+	0.999870i	$0.505141\pi$
$864$	0	0
$865$	2764.84	0.108679
$866$	−25184.6	−0.988231
$867$	0	0
$868$	15096.9	0.590350
$869$	10752.3	0.419733
$870$	0	0
$871$	0	0
$872$	−1537.25	−0.0596993
$873$	0	0
$874$	11850.7	0.458644
$875$	8884.23	0.343248
$876$	0	0
$877$	190.251	0.00732533	0.00366267	−	0.999993i	$-0.498834\pi$
$877$	190.251	0.00732533	0.00366267	+	0.999993i	$0.498834\pi$
$878$	−11551.3	−0.444007
$879$	0	0
$880$	−8643.43	−0.331102
$881$	19803.4	0.757315	0.378657	−	0.925537i	$-0.376386\pi$
$881$	19803.4	0.757315	0.378657	+	0.925537i	$0.376386\pi$
$882$	0	0
$883$	19652.1	0.748974	0.374487	−	0.927232i	$-0.377819\pi$
$883$	19652.1	0.748974	0.374487	+	0.927232i	$0.377819\pi$
$884$	0	0
$885$	0	0
$886$	43431.2	1.64684
$887$	26295.0	0.995379	0.497689	−	0.867355i	$-0.334182\pi$
$887$	26295.0	0.995379	0.497689	+	0.867355i	$0.334182\pi$
$888$	0	0
$889$	−13986.6	−0.527668
$890$	−41505.1	−1.56321
$891$	0	0
$892$	13966.5	0.524253
$893$	−4531.69	−0.169818
$894$	0	0
$895$	−23340.2	−0.871704
$896$	745.275	0.0277878
$897$	0	0
$898$	55758.0	2.07201
$899$	61397.0	2.27776
$900$	0	0
$901$	−2473.93	−0.0914746
$902$	13940.8	0.514609
$903$	0	0
$904$	−1296.69	−0.0477072
$905$	−30701.4	−1.12768
$906$	0	0
$907$	42417.3	1.55286	0.776429	−	0.630204i	$-0.217029\pi$
$907$	42417.3	1.55286	0.776429	+	0.630204i	$0.217029\pi$
$908$	3341.66	0.122133
$909$	0	0
$910$	0	0
$911$	872.245	0.0317220	0.0158610	−	0.999874i	$-0.494951\pi$
$911$	872.245	0.0317220	0.0158610	+	0.999874i	$0.494951\pi$
$912$	0	0
$913$	11373.5	0.412276
$914$	−69453.2	−2.51347
$915$	0	0
$916$	19449.4	0.701557
$917$	−13738.5	−0.494749
$918$	0	0
$919$	−17181.4	−0.616716	−0.308358	−	0.951270i	$-0.599779\pi$
$919$	−17181.4	−0.616716	−0.308358	+	0.951270i	$0.599779\pi$
$920$	−1739.64	−0.0623415
$921$	0	0
$922$	46287.5	1.65336
$923$	0	0
$924$	0	0
$925$	19884.4	0.706804
$926$	−6503.25	−0.230788
$927$	0	0
$928$	51407.0	1.81845
$929$	56042.9	1.97923	0.989617	−	0.143728i	$-0.0459091\pi$
$929$	56042.9	1.97923	0.989617	+	0.143728i	$0.0459091\pi$
$930$	0	0
$931$	−4116.52	−0.144912
$932$	−7736.00	−0.271890
$933$	0	0
$934$	−32843.3	−1.15060
$935$	−12955.8	−0.453155
$936$	0	0
$937$	−36672.5	−1.27859	−0.639295	−	0.768961i	$-0.720774\pi$
$937$	−36672.5	−1.27859	−0.639295	+	0.768961i	$0.720774\pi$
$938$	6858.71	0.238747
$939$	0	0
$940$	22576.4	0.783363
$941$	−21069.0	−0.729895	−0.364947	−	0.931028i	$-0.618913\pi$
$941$	−21069.0	−0.729895	−0.364947	+	0.931028i	$0.618913\pi$
$942$	0	0
$943$	−44093.4	−1.52267
$944$	−17363.0	−0.598642
$945$	0	0
$946$	8113.87	0.278863
$947$	−1838.70	−0.0630938	−0.0315469	−	0.999502i	$-0.510043\pi$
$947$	−1838.70	−0.0630938	−0.0315469	+	0.999502i	$0.510043\pi$
$948$	0	0
$949$	0	0
$950$	3213.27	0.109739
$951$	0	0
$952$	541.330	0.0184292
$953$	−13599.8	−0.462266	−0.231133	−	0.972922i	$-0.574243\pi$
$953$	−13599.8	−0.462266	−0.231133	+	0.972922i	$0.574243\pi$
$954$	0	0
$955$	−22695.9	−0.769029
$956$	−38446.9	−1.30069
$957$	0	0
$958$	7365.75	0.248410
$959$	2231.02	0.0751235
$960$	0	0
$961$	64935.6	2.17971
$962$	0	0
$963$	0	0
$964$	−31116.4	−1.03962
$965$	20105.5	0.670692
$966$	0	0
$967$	−2081.30	−0.0692141	−0.0346070	−	0.999401i	$-0.511018\pi$
$967$	−2081.30	−0.0692141	−0.0346070	+	0.999401i	$0.511018\pi$
$968$	1012.10	0.0336054
$969$	0	0
$970$	19681.6	0.651484
$971$	−1636.62	−0.0540904	−0.0270452	−	0.999634i	$-0.508610\pi$
$971$	−1636.62	−0.0540904	−0.0270452	+	0.999634i	$0.508610\pi$
$972$	0	0
$973$	2902.26	0.0956240
$974$	31592.9	1.03932
$975$	0	0
$976$	12838.8	0.421066
$977$	−29387.1	−0.962311	−0.481156	−	0.876635i	$-0.659783\pi$
$977$	−29387.1	−0.962311	−0.481156	+	0.876635i	$0.659783\pi$
$978$	0	0
$979$	21944.6	0.716396
$980$	20508.1	0.668476
$981$	0	0
$982$	69304.4	2.25213
$983$	24084.4	0.781457	0.390728	−	0.920506i	$-0.372223\pi$
$983$	24084.4	0.781457	0.390728	+	0.920506i	$0.372223\pi$
$984$	0	0
$985$	22249.4	0.719722
$986$	74713.4	2.41314
$987$	0	0
$988$	0	0
$989$	−25663.4	−0.825125
$990$	0	0
$991$	1413.43	0.0453068	0.0226534	−	0.999743i	$-0.492789\pi$
$991$	1413.43	0.0453068	0.0226534	+	0.999743i	$0.492789\pi$
$992$	79313.5	2.53851
$993$	0	0
$994$	−7618.79	−0.243112
$995$	−32342.7	−1.03049
$996$	0	0
$997$	−33357.4	−1.05962	−0.529809	−	0.848117i	$-0.677737\pi$
$997$	−33357.4	−1.05962	−0.529809	+	0.848117i	$0.677737\pi$
$998$	−21583.1	−0.684570
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bf.1.3		9
3.2	odd	2		507.4.a.p.1.7	yes	9
13.12	even	2		1521.4.a.bi.1.7		9
39.5	even	4		507.4.b.k.337.4		18
39.8	even	4		507.4.b.k.337.15		18
39.38	odd	2		507.4.a.o.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.3	✓	9	39.38	odd	2
507.4.a.p.1.7	yes	9	3.2	odd	2
507.4.b.k.337.4		18	39.5	even	4
507.4.b.k.337.15		18	39.8	even	4
1521.4.a.bf.1.3		9	1.1	even	1	trivial
1521.4.a.bi.1.7		9	13.12	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-4.03025 q^{2} +8.24289 q^{4} -8.08864 q^{5} +5.95078 q^{7} -0.978887 q^{8} +O(q^{10})\)	\(q-4.03025 q^{2} +8.24289 q^{4} -8.08864 q^{5} +5.95078 q^{7} -0.978887 q^{8} +32.5992 q^{10} -17.2359 q^{11} -23.9831 q^{14} -61.9979 q^{16} -92.9299 q^{17} +13.3832 q^{19} -66.6738 q^{20} +69.4648 q^{22} -219.710 q^{23} -59.5738 q^{25} +49.0516 q^{28} +199.485 q^{29} +307.777 q^{31} +257.698 q^{32} +374.531 q^{34} -48.1337 q^{35} -333.777 q^{37} -53.9376 q^{38} +7.91787 q^{40} +200.689 q^{41} +116.806 q^{43} -142.073 q^{44} +885.487 q^{46} -338.610 q^{47} -307.588 q^{49} +240.097 q^{50} +26.6215 q^{53} +139.415 q^{55} -5.82514 q^{56} -803.976 q^{58} +280.058 q^{59} -207.084 q^{61} -1240.42 q^{62} -542.603 q^{64} -285.981 q^{67} -766.011 q^{68} +193.991 q^{70} +317.673 q^{71} -63.0668 q^{73} +1345.20 q^{74} +110.316 q^{76} -102.567 q^{77} -623.835 q^{79} +501.479 q^{80} -808.824 q^{82} -659.874 q^{83} +751.677 q^{85} -470.755 q^{86} +16.8720 q^{88} -1273.19 q^{89} -1811.05 q^{92} +1364.68 q^{94} -108.252 q^{95} +603.746 q^{97} +1239.66 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

Introduction

L-functions

Modular forms

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