LMFDB - Embedded newform 529.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [529,4,Mod(1,529)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(529, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("529.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 529 = 23^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	529.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:	$31.2120103930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{69}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 17 $ x^2 - x - 17
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$-3.65331$ of defining polynomial
Character	$\chi$	$=$	529.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.65331 q^{2} -10.3066 q^{3} -0.959936 q^{4} -27.3467 q^{6} -23.7735 q^{8} +79.2265 q^{9} +9.89370 q^{12} +86.8397 q^{13} -55.3990 q^{16} +210.213 q^{18} +245.025 q^{24} -125.000 q^{25} +230.413 q^{26} -538.279 q^{27} -24.7073 q^{29} +147.080 q^{31} +43.1971 q^{32} -76.0523 q^{36} -895.025 q^{39} +52.8120 q^{41} -580.544 q^{47} +570.977 q^{48} -343.000 q^{49} -331.664 q^{50} -83.3606 q^{52} -1428.22 q^{54} -65.5561 q^{58} -396.000 q^{59} +390.249 q^{62} +557.808 q^{64} -779.052 q^{71} -1883.49 q^{72} +812.359 q^{73} +1288.33 q^{75} -2374.78 q^{78} +3408.72 q^{81} +140.127 q^{82} +254.648 q^{87} -1515.90 q^{93} -1540.36 q^{94} -445.216 q^{96} -910.086 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 4 q^{3} + 23 q^{4} - 63 q^{6} - 114 q^{8} + 92 q^{9} + 161 q^{12} + 74 q^{13} + 263 q^{16} + 138 q^{18} - 324 q^{24} - 250 q^{25} + 303 q^{26} - 628 q^{27} - 282 q^{29} + 344 q^{31} - 1035 q^{32}+ \cdots + 1029 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 - 4 * q^3 + 23 * q^4 - 63 * q^6 - 114 * q^8 + 92 * q^9 + 161 * q^12 + 74 * q^13 + 263 * q^16 + 138 * q^18 - 324 * q^24 - 250 * q^25 + 303 * q^26 - 628 * q^27 - 282 * q^29 + 344 * q^31 - 1035 * q^32 + 230 * q^36 - 976 * q^39 - 426 * q^41 - 48 * q^47 + 2579 * q^48 - 686 * q^49 + 375 * q^50 - 391 * q^52 - 921 * q^54 + 1389 * q^58 - 792 * q^59 - 723 * q^62 + 4106 * q^64 - 1176 * q^71 - 3036 * q^72 + 1226 * q^73 + 500 * q^75 - 1917 * q^78 + 2498 * q^81 + 2847 * q^82 - 1368 * q^87 - 274 * q^93 - 4551 * q^94 - 7245 * q^96 + 1029 * q^98

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 $

$2$	2.65331	0.938087	0.469044	−	0.883175i	$-0.344599\pi$
$2$	2.65331	0.938087	0.469044	+	0.883175i	$0.344599\pi$
$3$	−10.3066	−1.98351	−0.991755	−	0.128146i	$-0.959097\pi$
$3$	−10.3066	−1.98351	−0.991755	+	0.128146i	$0.959097\pi$
$4$	−0.959936	−0.119992
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−27.3467	−1.86071
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−23.7735	−1.05065
$9$	79.2265	2.93431
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	9.89370	0.238005
$13$	86.8397	1.85269	0.926347	−	0.376672i	$-0.122932\pi$
$13$	86.8397	1.85269	0.926347	+	0.376672i	$0.122932\pi$
$14$	0	0
$15$	0	0
$16$	−55.3990	−0.865610
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	210.213	2.75264
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0
$23$	0	0
$24$	245.025	2.08398
$25$	−125.000	−1.00000
$26$	230.413	1.73799
$27$	−538.279	−3.83673
$28$	0	0
$29$	−24.7073	−0.158208	−0.0791039	−	0.996866i	$-0.525206\pi$
$29$	−24.7073	−0.158208	−0.0791039	+	0.996866i	$0.525206\pi$
$30$	0	0
$31$	147.080	0.852141	0.426070	−	0.904690i	$-0.359898\pi$
$31$	147.080	0.852141	0.426070	+	0.904690i	$0.359898\pi$
$32$	43.1971	0.238633
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−76.0523	−0.352094
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−895.025	−3.67484
$40$	0	0
$41$	52.8120	0.201167	0.100583	−	0.994929i	$-0.467929\pi$
$41$	52.8120	0.201167	0.100583	+	0.994929i	$0.467929\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−580.544	−1.80172	−0.900862	−	0.434106i	$-0.857064\pi$
$47$	−580.544	−1.80172	−0.900862	+	0.434106i	$0.857064\pi$
$48$	570.977	1.71695
$49$	−343.000	−1.00000
$50$	−331.664	−0.938087
$51$	0	0
$52$	−83.3606	−0.222308
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1428.22	−3.59919
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−65.5561	−0.148413
$59$	−396.000	−0.873810	−0.436905	−	0.899508i	$-0.643925\pi$
$59$	−396.000	−0.873810	−0.436905	+	0.899508i	$0.643925\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	390.249	0.799383
$63$	0	0
$64$	557.808	1.08947
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−779.052	−1.30220	−0.651102	−	0.758990i	$-0.725693\pi$
$71$	−779.052	−1.30220	−0.651102	+	0.758990i	$0.725693\pi$
$72$	−1883.49	−3.08294
$73$	812.359	1.30246	0.651229	−	0.758881i	$-0.274254\pi$
$73$	812.359	1.30246	0.651229	+	0.758881i	$0.274254\pi$
$74$	0	0
$75$	1288.33	1.98351
$76$	0	0
$77$	0	0
$78$	−2374.78	−3.44732
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	3408.72	4.67589
$82$	140.127	0.188712
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	254.648	0.313807
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1515.90	−1.69023
$94$	−1540.36	−1.69017
$95$	0	0
$96$	−445.216	−0.473330
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−910.086	−0.938087
$99$	0	0
$100$	119.992	0.119992
$101$	−1602.00	−1.57827	−0.789133	−	0.614222i	$-0.789470\pi$
$101$	−1602.00	−1.57827	−0.789133	+	0.614222i	$0.789470\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−2064.49	−1.94653
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	516.713	0.460377
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	23.7174	0.0189837
$117$	6880.01	5.43638
$118$	−1050.71	−0.819710
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	−544.313	−0.399017
$124$	−141.187	−0.102250
$125$	0	0
$126$	0	0
$127$	−282.285	−0.197234	−0.0986172	−	0.995125i	$-0.531442\pi$
$127$	−282.285	−0.197234	−0.0986172	+	0.995125i	$0.531442\pi$
$128$	1134.46	0.783384
$129$	0	0
$130$	0	0
$131$	−2336.65	−1.55843	−0.779213	−	0.626759i	$-0.784381\pi$
$131$	−2336.65	−1.55843	−0.779213	+	0.626759i	$0.784381\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−2609.95	−1.59261	−0.796305	−	0.604896i	$-0.793215\pi$
$139$	−2609.95	−1.59261	−0.796305	+	0.604896i	$0.793215\pi$
$140$	0	0
$141$	5983.45	3.57374
$142$	−2067.07	−1.22158
$143$	0	0
$144$	−4389.07	−2.53997
$145$	0	0
$146$	2155.44	1.22182
$147$	3535.17	1.98351
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	3418.34	1.86071
$151$	1085.32	0.584917	0.292458	−	0.956278i	$-0.405527\pi$
$151$	1085.32	0.584917	0.292458	+	0.956278i	$0.405527\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	859.166	0.440951
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9044.40	4.38639
$163$	−4132.34	−1.98571	−0.992853	−	0.119344i	$-0.961921\pi$
$163$	−4132.34	−1.98571	−0.992853	+	0.119344i	$0.961921\pi$
$164$	−50.6961	−0.0241384
$165$	0	0
$166$	0	0
$167$	1800.00	0.834061	0.417030	−	0.908892i	$-0.363071\pi$
$167$	1800.00	0.834061	0.417030	+	0.908892i	$0.363071\pi$
$168$	0	0
$169$	5344.14	2.43247
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3510.00	−1.54255	−0.771273	−	0.636505i	$-0.780380\pi$
$173$	−3510.00	−1.54255	−0.771273	+	0.636505i	$0.780380\pi$
$174$	675.662	0.294378
$175$	0	0
$176$	0	0
$177$	4081.42	1.73321
$178$	0	0
$179$	4771.41	1.99236	0.996178	−	0.0873419i	$-0.0278373\pi$
$179$	4771.41	1.99236	0.996178	+	0.0873419i	$0.0278373\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	−4022.15	−1.58558
$187$	0	0
$188$	557.285	0.216192
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−5749.11	−2.16097
$193$	648.026	0.241689	0.120844	−	0.992671i	$-0.461440\pi$
$193$	648.026	0.241689	0.120844	+	0.992671i	$0.461440\pi$
$194$	0	0
$195$	0	0
$196$	329.258	0.119992
$197$	3232.78	1.16917	0.584585	−	0.811333i	$-0.301257\pi$
$197$	3232.78	1.16917	0.584585	+	0.811333i	$0.301257\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	2971.69	1.05065
$201$	0	0
$202$	−4250.61	−1.48055
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−4810.84	−1.60371
$209$	0	0
$210$	0	0
$211$	−2468.00	−0.805233	−0.402616	−	0.915369i	$-0.631899\pi$
$211$	−2468.00	−0.805233	−0.402616	+	0.915369i	$0.631899\pi$
$212$	0	0
$213$	8029.40	2.58294
$214$	0	0
$215$	0	0
$216$	12796.8	4.03107
$217$	0	0
$218$	0	0
$219$	−8372.68	−2.58344
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4840.00	−1.45341	−0.726705	−	0.686950i	$-0.758949\pi$
$223$	−4840.00	−1.45341	−0.726705	+	0.686950i	$0.758949\pi$
$224$	0	0
$225$	−9903.31	−2.93431
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	587.378	0.166221
$233$	−6498.69	−1.82722	−0.913612	−	0.406587i	$-0.866719\pi$
$233$	−6498.69	−1.82722	−0.913612	+	0.406587i	$0.866719\pi$
$234$	18254.8	5.09980
$235$	0	0
$236$	380.135	0.104850
$237$	0	0
$238$	0	0
$239$	−4174.09	−1.12970	−0.564852	−	0.825192i	$-0.691067\pi$
$239$	−4174.09	−1.12970	−0.564852	+	0.825192i	$0.691067\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−3531.56	−0.938087
$243$	−20598.9	−5.43794
$244$	0	0
$245$	0	0
$246$	−1444.23	−0.374313
$247$	0	0
$248$	−3496.61	−0.895302
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	−748.991	−0.185023
$255$	0	0
$256$	−1452.38	−0.354586
$257$	−8186.62	−1.98703	−0.993516	−	0.113691i	$-0.963733\pi$
$257$	−8186.62	−1.98703	−0.993516	+	0.113691i	$0.963733\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1957.47	−0.464231
$262$	−6199.86	−1.46194
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7091.25	1.60729	0.803646	−	0.595108i	$-0.202891\pi$
$269$	7091.25	1.60729	0.803646	+	0.595108i	$0.202891\pi$
$270$	0	0
$271$	−8912.00	−1.99766	−0.998829	−	0.0483752i	$-0.984596\pi$
$271$	−8912.00	−1.99766	−0.998829	+	0.0483752i	$0.984596\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8743.84	1.89663	0.948315	−	0.317331i	$-0.102787\pi$
$277$	8743.84	1.89663	0.948315	+	0.317331i	$0.102787\pi$
$278$	−6925.00	−1.49401
$279$	11652.6	2.50045
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	15876.0	3.35248
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	747.840	0.156254
$285$	0	0
$286$	0	0
$287$	0	0
$288$	3422.36	0.700223
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	−779.812	−0.156284
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	9379.91	1.86071
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−1236.71	−0.238005
$301$	0	0
$302$	2879.70	0.548703
$303$	16511.2	3.13051
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10420.0	1.93714	0.968568	−	0.248749i	$-0.0800193\pi$
$307$	10420.0	1.93714	0.968568	+	0.248749i	$0.0800193\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5104.51	−0.930707	−0.465354	−	0.885125i	$-0.654073\pi$
$311$	−5104.51	−0.930707	−0.465354	+	0.885125i	$0.654073\pi$
$312$	21277.9	3.86097
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1530.00	0.271083	0.135542	−	0.990772i	$-0.456723\pi$
$317$	1530.00	0.271083	0.135542	+	0.990772i	$0.456723\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−3272.15	−0.561069
$325$	−10855.0	−1.85269
$326$	−10964.4	−1.86277
$327$	0	0
$328$	−1255.53	−0.211356
$329$	0	0
$330$	0	0
$331$	5997.08	0.995859	0.497930	−	0.867218i	$-0.334094\pi$
$331$	5997.08	0.995859	0.497930	+	0.867218i	$0.334094\pi$
$332$	0	0
$333$	0	0
$334$	4775.96	0.782422
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	14179.7	2.28187
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−9313.12	−1.44704
$347$	9180.00	1.42020	0.710098	−	0.704103i	$-0.248650\pi$
$347$	9180.00	1.42020	0.710098	+	0.704103i	$0.248650\pi$
$348$	−244.446	−0.0376543
$349$	−12386.8	−1.89986	−0.949929	−	0.312467i	$-0.898845\pi$
$349$	−12386.8	−1.89986	−0.949929	+	0.312467i	$0.898845\pi$
$350$	0	0
$351$	−46744.0	−7.10829
$352$	0	0
$353$	−9384.65	−1.41500	−0.707499	−	0.706714i	$-0.750177\pi$
$353$	−9384.65	−1.41500	−0.707499	+	0.706714i	$0.750177\pi$
$354$	10829.3	1.62590
$355$	0	0
$356$	0	0
$357$	0	0
$358$	12660.0	1.86900
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	13718.1	1.98351
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	4184.11	0.590287
$370$	0	0
$371$	0	0
$372$	1455.17	0.202814
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	13801.6	1.89298
$377$	−2145.57	−0.293110
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	2909.41	0.391217
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−11692.5	−1.55385
$385$	0	0
$386$	1719.41	0.226725
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	8154.31	1.05065
$393$	24083.0	3.09116
$394$	8577.58	1.09678
$395$	0	0
$396$	0	0
$397$	11677.7	1.47629	0.738145	−	0.674642i	$-0.235702\pi$
$397$	11677.7	1.47629	0.738145	+	0.674642i	$0.235702\pi$
$398$	0	0
$399$	0	0
$400$	6924.88	0.865610
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	12772.4	1.57876
$404$	1537.82	0.189379
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11155.1	−1.34862	−0.674308	−	0.738451i	$-0.735558\pi$
$409$	−11155.1	−1.34862	−0.674308	+	0.738451i	$0.735558\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	3751.23	0.442113
$417$	26899.7	3.15896
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	−6548.37	−0.755379
$423$	−45994.5	−5.28682
$424$	0	0
$425$	0	0
$426$	21304.5	2.42302
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	29820.1	3.32111
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−22215.3	−2.42349
$439$	−10984.0	−1.19416	−0.597080	−	0.802182i	$-0.703672\pi$
$439$	−10984.0	−1.19416	−0.597080	+	0.802182i	$0.703672\pi$
$440$	0	0
$441$	−27174.7	−2.93431
$442$	0	0
$443$	4071.85	0.436703	0.218352	−	0.975870i	$-0.429932\pi$
$443$	4071.85	0.436703	0.218352	+	0.975870i	$0.429932\pi$
$444$	0	0
$445$	0	0
$446$	−12842.0	−1.36343
$447$	0	0
$448$	0	0
$449$	−18414.0	−1.93544	−0.967718	−	0.252037i	$-0.918900\pi$
$449$	−18414.0	−1.93544	−0.967718	+	0.252037i	$0.918900\pi$
$450$	−26276.6	−2.75264
$451$	0	0
$452$	0	0
$453$	−11186.0	−1.16019
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18491.9	1.86823	0.934115	−	0.356973i	$-0.116191\pi$
$461$	18491.9	1.86823	0.934115	+	0.356973i	$0.116191\pi$
$462$	0	0
$463$	−11680.0	−1.17239	−0.586194	−	0.810171i	$-0.699374\pi$
$463$	−11680.0	−1.17239	−0.586194	+	0.810171i	$0.699374\pi$
$464$	1368.76	0.136946
$465$	0	0
$466$	−17243.0	−1.71410
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−6604.37	−0.652323
$469$	0	0
$470$	0	0
$471$	0	0
$472$	9414.31	0.918069
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−11075.2	−1.05976
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1277.67	0.119992
$485$	0	0
$486$	−54655.3	−5.10126
$487$	17336.9	1.61316	0.806580	−	0.591125i	$-0.201316\pi$
$487$	17336.9	1.61316	0.806580	+	0.591125i	$0.201316\pi$
$488$	0	0
$489$	42590.5	3.93867
$490$	0	0
$491$	4465.22	0.410412	0.205206	−	0.978719i	$-0.434214\pi$
$491$	4465.22	0.410412	0.205206	+	0.978719i	$0.434214\pi$
$492$	522.506	0.0478788
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−8148.10	−0.737622
$497$	0	0
$498$	0	0
$499$	−5737.17	−0.514691	−0.257346	−	0.966319i	$-0.582848\pi$
$499$	−5737.17	−0.514691	−0.257346	+	0.966319i	$0.582848\pi$
$500$	0	0
$501$	−18551.9	−1.65437
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−55080.1	−4.82483
$508$	270.976	0.0236665
$509$	−22637.7	−1.97131	−0.985656	−	0.168769i	$-0.946021\pi$
$509$	−22637.7	−1.97131	−0.985656	+	0.168769i	$0.946021\pi$
$510$	0	0
$511$	0	0
$512$	−12929.3	−1.11602
$513$	0	0
$514$	−21721.7	−1.86401
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	36176.2	3.05966
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	−5193.78	−0.435490
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	2243.03	0.186999
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−31373.7	−2.56403
$532$	0	0
$533$	4586.18	0.372701
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−49177.1	−3.95186
$538$	18815.3	1.50778
$539$	0	0
$540$	0	0
$541$	22247.0	1.76797	0.883985	−	0.467515i	$-0.154851\pi$
$541$	22247.0	1.76797	0.883985	+	0.467515i	$0.154851\pi$
$542$	−23646.3	−1.87398
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22901.7	1.79014	0.895068	−	0.445930i	$-0.147127\pi$
$547$	22901.7	1.79014	0.895068	+	0.445930i	$0.147127\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	23200.1	1.77920
$555$	0	0
$556$	2505.38	0.191100
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	30918.1	2.34564
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−5743.72	−0.428820
$565$	0	0
$566$	0	0
$567$	0	0
$568$	18520.8	1.36816
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	44193.1	3.19684
$577$	−3963.24	−0.285948	−0.142974	−	0.989726i	$-0.545667\pi$
$577$	−3963.24	−0.285948	−0.142974	+	0.989726i	$0.545667\pi$
$578$	−13035.7	−0.938087
$579$	−6678.96	−0.479392
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−19312.6	−1.36843
$585$	0	0
$586$	0	0
$587$	−25636.7	−1.80262	−0.901311	−	0.433172i	$-0.857394\pi$
$587$	−25636.7	−1.80262	−0.901311	+	0.433172i	$0.857394\pi$
$588$	−3393.54	−0.238005
$589$	0	0
$590$	0	0
$591$	−33319.1	−2.31906
$592$	0	0
$593$	26370.0	1.82611	0.913057	−	0.407831i	$-0.133715\pi$
$593$	26370.0	1.82611	0.913057	+	0.407831i	$0.133715\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24336.0	1.66000	0.830002	−	0.557761i	$-0.188339\pi$
$599$	24336.0	1.66000	0.830002	+	0.557761i	$0.188339\pi$
$600$	−30628.1	−2.08398
$601$	25354.9	1.72088	0.860441	−	0.509550i	$-0.170188\pi$
$601$	25354.9	1.72088	0.860441	+	0.509550i	$0.170188\pi$
$602$	0	0
$603$	0	0
$604$	−1041.84	−0.0701853
$605$	0	0
$606$	43809.4	2.93669
$607$	8840.00	0.591111	0.295556	−	0.955326i	$-0.404495\pi$
$607$	8840.00	0.591111	0.295556	+	0.955326i	$0.404495\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−50414.3	−3.33804
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	27647.5	1.81720
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	−13543.8	−0.873085
$623$	0	0
$624$	49583.5	3.18098
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	25436.7	1.59719
$634$	4059.57	0.254300
$635$	0	0
$636$	0	0
$637$	−29786.0	−1.85269
$638$	0	0
$639$	−61721.6	−3.82108
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28400.3	1.72571	0.862854	−	0.505454i	$-0.168675\pi$
$647$	28400.3	1.72571	0.862854	+	0.505454i	$0.168675\pi$
$648$	−81037.3	−4.91272
$649$	0	0
$650$	−28801.6	−1.73799
$651$	0	0
$652$	3966.78	0.238269
$653$	32726.5	1.96124	0.980618	−	0.195928i	$-0.0627719\pi$
$653$	32726.5	1.96124	0.980618	+	0.195928i	$0.0627719\pi$
$654$	0	0
$655$	0	0
$656$	−2925.73	−0.174132
$657$	64360.4	3.82182
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	15912.1	0.934203
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1727.88	−0.100081
$669$	49884.1	2.88285
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−8519.60	−0.487974	−0.243987	−	0.969778i	$-0.578455\pi$
$673$	−8519.60	−0.487974	−0.243987	+	0.969778i	$0.578455\pi$
$674$	0	0
$675$	67284.9	3.83673
$676$	−5130.03	−0.291877
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13453.9	−0.753734	−0.376867	−	0.926267i	$-0.622999\pi$
$683$	−13453.9	−0.753734	−0.376867	+	0.926267i	$0.622999\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6028.00	0.331861	0.165930	−	0.986137i	$-0.446937\pi$
$691$	6028.00	0.331861	0.165930	+	0.986137i	$0.446937\pi$
$692$	3369.37	0.185093
$693$	0	0
$694$	24357.4	1.33227
$695$	0	0
$696$	−6053.89	−0.329701
$697$	0	0
$698$	−32866.0	−1.78223
$699$	66979.5	3.62432
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−124026.	−6.66820
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−24900.4	−1.32739
$707$	0	0
$708$	−3917.90	−0.207972
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4580.25	−0.239067
$717$	43020.8	2.24078
$718$	0	0
$719$	−37944.0	−1.96811	−0.984056	−	0.177859i	$-0.943083\pi$
$719$	−37944.0	−1.96811	−0.984056	+	0.177859i	$0.943083\pi$
$720$	0	0
$721$	0	0
$722$	−18199.1	−0.938087
$723$	0	0
$724$	0	0
$725$	3088.41	0.158208
$726$	36398.4	1.86071
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	120269.	6.11032
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	11101.7	0.553741
$739$	7328.32	0.364786	0.182393	−	0.983226i	$-0.441616\pi$
$739$	7328.32	0.364786	0.182393	+	0.983226i	$0.441616\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	36038.2	1.77584
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	32161.6	1.55959
$753$	0	0
$754$	−5692.87	−0.274963
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32753.9	−1.56022	−0.780111	−	0.625641i	$-0.784837\pi$
$761$	−32753.9	−1.56022	−0.780111	+	0.625641i	$0.784837\pi$
$762$	7719.57	0.366995
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−34388.5	−1.61890
$768$	14969.2	0.703324
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	84376.4	3.94130
$772$	−622.063	−0.0290007
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−18385.0	−0.852141
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	13299.4	0.607001
$784$	19001.9	0.865610
$785$	0	0
$786$	63899.6	2.89978
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−3103.27	−0.140291
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	30984.6	1.38489
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−5399.64	−0.238633
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	33889.2	1.48101
$807$	−73086.9	−3.18808
$808$	38085.2	1.65821
$809$	27846.0	1.21015	0.605076	−	0.796168i	$-0.293143\pi$
$809$	27846.0	1.21015	0.605076	+	0.796168i	$0.293143\pi$
$810$	0	0
$811$	38214.7	1.65462	0.827312	−	0.561743i	$-0.189869\pi$
$811$	38214.7	1.65462	0.827312	+	0.561743i	$0.189869\pi$
$812$	0	0
$813$	91852.6	3.96238
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−29597.9	−1.26512
$819$	0	0
$820$	0	0
$821$	24462.0	1.03987	0.519933	−	0.854207i	$-0.325957\pi$
$821$	24462.0	1.03987	0.519933	+	0.854207i	$0.325957\pi$
$822$	0	0
$823$	29161.0	1.23510	0.617550	−	0.786532i	$-0.288125\pi$
$823$	29161.0	1.23510	0.617550	+	0.786532i	$0.288125\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−4966.00	−0.208053	−0.104027	−	0.994575i	$-0.533173\pi$
$829$	−4966.00	−0.208053	−0.104027	+	0.994575i	$0.533173\pi$
$830$	0	0
$831$	−90119.5	−3.76199
$832$	48439.9	2.01845
$833$	0	0
$834$	71373.4	2.96338
$835$	0	0
$836$	0	0
$837$	−79170.1	−3.26944
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−23778.6	−0.974970
$842$	0	0
$843$	0	0
$844$	2369.12	0.0966215
$845$	0	0
$846$	−122038.	−4.95950
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−7707.71	−0.309932
$853$	24590.0	0.987041	0.493520	−	0.869734i	$-0.335710\pi$
$853$	24590.0	0.987041	0.493520	+	0.869734i	$0.335710\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22790.5	−0.908410	−0.454205	−	0.890897i	$-0.650077\pi$
$857$	−22790.5	−0.908410	−0.454205	+	0.890897i	$0.650077\pi$
$858$	0	0
$859$	−25747.2	−1.02268	−0.511340	−	0.859379i	$-0.670851\pi$
$859$	−25747.2	−1.02268	−0.511340	+	0.859379i	$0.670851\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46290.8	−1.82591	−0.912954	−	0.408063i	$-0.866205\pi$
$863$	−46290.8	−1.82591	−0.912954	+	0.408063i	$0.866205\pi$
$864$	−23252.1	−0.915570
$865$	0	0
$866$	0	0
$867$	50636.4	1.98351
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	8037.23	0.309992
$877$	34090.0	1.31259	0.656293	−	0.754506i	$-0.272124\pi$
$877$	34090.0	1.31259	0.656293	+	0.754506i	$0.272124\pi$
$878$	−29143.9	−1.12023
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−72102.9	−2.75264
$883$	2860.00	0.109000	0.0544998	−	0.998514i	$-0.482644\pi$
$883$	2860.00	0.109000	0.0544998	+	0.998514i	$0.482644\pi$
$884$	0	0
$885$	0	0
$886$	10803.9	0.409666
$887$	−44547.4	−1.68631	−0.843153	−	0.537673i	$-0.819303\pi$
$887$	−44547.4	−1.68631	−0.843153	+	0.537673i	$0.819303\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	4646.09	0.174398
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−48858.1	−1.81561
$899$	−3633.95	−0.134815
$900$	9506.54	0.352094
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−29680.0	−1.08836
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−126921.	−4.63113
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−107395.	−3.84233
$922$	49064.8	1.75256
$923$	−67652.7	−2.41258
$924$	0	0
$925$	0	0
$926$	−30990.7	−1.09980
$927$	0	0
$928$	−1067.28	−0.0377535
$929$	−839.150	−0.0296357	−0.0148179	−	0.999890i	$-0.504717\pi$
$929$	−839.150	−0.0296357	−0.0148179	+	0.999890i	$0.504717\pi$
$930$	0	0
$931$	0	0
$932$	6238.32	0.219252
$933$	52610.2	1.84607
$934$	0	0
$935$	0	0
$936$	−163562.	−5.71174
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	21938.0	0.756379
$945$	0	0
$946$	0	0
$947$	−38049.1	−1.30563	−0.652814	−	0.757518i	$-0.726412\pi$
$947$	−38049.1	−1.30563	−0.652814	+	0.757518i	$0.726412\pi$
$948$	0	0
$949$	70545.0	2.41305
$950$	0	0
$951$	−15769.1	−0.537696
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	4006.86	0.135556
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8158.44	−0.273856
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	47303.2	1.57308	0.786540	−	0.617539i	$-0.211870\pi$
$967$	47303.2	1.57308	0.786540	+	0.617539i	$0.211870\pi$
$968$	31642.5	1.05065
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	19773.6	0.652509
$973$	0	0
$974$	46000.2	1.51329
$975$	111878.	3.67484
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	113006.	3.69482
$979$	0	0
$980$	0	0
$981$	0	0
$982$	11847.6	0.385003
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	12940.2	0.419227
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−9128.00	−0.292594	−0.146297	−	0.989241i	$-0.546735\pi$
$991$	−9128.00	−0.292594	−0.146297	+	0.989241i	$0.546735\pi$
$992$	6353.44	0.203349
$993$	−61809.6	−1.97530
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−60190.0	−1.91197	−0.955986	−	0.293412i	$-0.905209\pi$
$997$	−60190.0	−1.91197	−0.955986	+	0.293412i	$0.905209\pi$
$998$	−15222.5	−0.482825
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	529.4.a.f.1.2	✓	2
23.22	odd	2	CM	529.4.a.f.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
529.4.a.f.1.2	✓	2	1.1	even	1	trivial
529.4.a.f.1.2	✓	2	23.22	odd	2	CM

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 529 = 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	529.a (trivial)

\(f(q)\)	\(=\)	\(q+2.65331 q^{2} -10.3066 q^{3} -0.959936 q^{4} -27.3467 q^{6} -23.7735 q^{8} +79.2265 q^{9} +9.89370 q^{12} +86.8397 q^{13} -55.3990 q^{16} +210.213 q^{18} +245.025 q^{24} -125.000 q^{25} +230.413 q^{26} -538.279 q^{27} -24.7073 q^{29} +147.080 q^{31} +43.1971 q^{32} -76.0523 q^{36} -895.025 q^{39} +52.8120 q^{41} -580.544 q^{47} +570.977 q^{48} -343.000 q^{49} -331.664 q^{50} -83.3606 q^{52} -1428.22 q^{54} -65.5561 q^{58} -396.000 q^{59} +390.249 q^{62} +557.808 q^{64} -779.052 q^{71} -1883.49 q^{72} +812.359 q^{73} +1288.33 q^{75} -2374.78 q^{78} +3408.72 q^{81} +140.127 q^{82} +254.648 q^{87} -1515.90 q^{93} -1540.36 q^{94} -445.216 q^{96} -910.086 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 4 q^{3} + 23 q^{4} - 63 q^{6} - 114 q^{8} + 92 q^{9} + 161 q^{12} + 74 q^{13} + 263 q^{16} + 138 q^{18} - 324 q^{24} - 250 q^{25} + 303 q^{26} - 628 q^{27} - 282 q^{29} + 344 q^{31} - 1035 q^{32}+ \cdots + 1029 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 - 4 * q^3 + 23 * q^4 - 63 * q^6 - 114 * q^8 + 92 * q^9 + 161 * q^12 + 74 * q^13 + 263 * q^16 + 138 * q^18 - 324 * q^24 - 250 * q^25 + 303 * q^26 - 628 * q^27 - 282 * q^29 + 344 * q^31 - 1035 * q^32 + 230 * q^36 - 976 * q^39 - 426 * q^41 - 48 * q^47 + 2579 * q^48 - 686 * q^49 + 375 * q^50 - 391 * q^52 - 921 * q^54 + 1389 * q^58 - 792 * q^59 - 723 * q^62 + 4106 * q^64 - 1176 * q^71 - 3036 * q^72 + 1226 * q^73 + 500 * q^75 - 1917 * q^78 + 2498 * q^81 + 2847 * q^82 - 1368 * q^87 - 274 * q^93 - 4551 * q^94 - 7245 * q^96 + 1029 * q^98

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