LMFDB - Embedded newform 529.4.a.f.1.1

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			1.1
Root			$4.65331$ 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)

$f(q)$	$=$	$q-5.65331 q^{2} +6.30662 q^{3} +23.9599 q^{4} -35.6533 q^{6} -90.2265 q^{8} +12.7735 q^{9} +151.106 q^{12} -12.8397 q^{13} +318.399 q^{16} -72.2126 q^{18} -569.025 q^{24} -125.000 q^{25} +72.5871 q^{26} -89.7212 q^{27} -257.293 q^{29} +196.920 q^{31} -1078.20 q^{32} +306.052 q^{36} -80.9754 q^{39} -478.812 q^{41} +532.544 q^{47} +2008.02 q^{48} -343.000 q^{49} +706.664 q^{50} -307.639 q^{52} +507.222 q^{54} +1454.56 q^{58} -396.000 q^{59} -1113.25 q^{62} +3548.19 q^{64} -396.948 q^{71} -1152.51 q^{72} +413.641 q^{73} -788.328 q^{75} +457.779 q^{78} -910.722 q^{81} +2706.87 q^{82} -1622.65 q^{87} +1241.90 q^{93} -3010.64 q^{94} -6799.78 q^{96} +1939.09 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.65331	−1.99875	−0.999374	−	0.0353837i	$-0.988735\pi$
$2$	−5.65331	−1.99875	−0.999374	+	0.0353837i	$0.988735\pi$
$3$	6.30662	1.21371	0.606855	−	0.794812i	$-0.292431\pi$
$3$	6.30662	1.21371	0.606855	+	0.794812i	$0.292431\pi$
$4$	23.9599	2.99499
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−35.6533	−2.42590
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−90.2265	−3.98749
$9$	12.7735	0.473093
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	151.106	3.63505
$13$	−12.8397	−0.273931	−0.136966	−	0.990576i	$-0.543735\pi$
$13$	−12.8397	−0.273931	−0.136966	+	0.990576i	$0.543735\pi$
$14$	0	0
$15$	0	0
$16$	318.399	4.97498
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−72.2126	−0.945593
$19$	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
$24$	−569.025	−4.83965
$25$	−125.000	−1.00000
$26$	72.5871	0.547519
$27$	−89.7212	−0.639513
$28$	0	0
$29$	−257.293	−1.64752	−0.823760	−	0.566939i	$-0.808127\pi$
$29$	−257.293	−1.64752	−0.823760	+	0.566939i	$0.808127\pi$
$30$	0	0
$31$	196.920	1.14090	0.570449	−	0.821333i	$-0.306769\pi$
$31$	196.920	1.14090	0.570449	+	0.821333i	$0.306769\pi$
$32$	−1078.20	−5.95625
$33$	0	0
$34$	0	0
$35$	0	0
$36$	306.052	1.41691
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−80.9754	−0.332473
$40$	0	0
$41$	−478.812	−1.82385	−0.911925	−	0.410356i	$-0.865404\pi$
$41$	−478.812	−1.82385	−0.911925	+	0.410356i	$0.865404\pi$
$42$	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
$47$	532.544	1.65276	0.826378	−	0.563116i	$-0.190398\pi$
$47$	532.544	1.65276	0.826378	+	0.563116i	$0.190398\pi$
$48$	2008.02	6.03819
$49$	−343.000	−1.00000
$50$	706.664	1.99875
$51$	0	0
$52$	−307.639	−0.820421
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	507.222	1.27822
$55$	0	0
$56$	0	0
$57$	0	0
$58$	1454.56	3.29298
$59$	−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	$0.5\pi$
$62$	−1113.25	−2.28037
$63$	0	0
$64$	3548.19	6.93006
$65$	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
$71$	−396.948	−0.663507	−0.331754	−	0.943366i	$-0.607640\pi$
$71$	−396.948	−0.663507	−0.331754	+	0.943366i	$0.607640\pi$
$72$	−1152.51	−1.88645
$73$	413.641	0.663192	0.331596	−	0.943421i	$-0.392413\pi$
$73$	413.641	0.663192	0.331596	+	0.943421i	$0.392413\pi$
$74$	0	0
$75$	−788.328	−1.21371
$76$	0	0
$77$	0	0
$78$	457.779	0.664530
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−910.722	−1.24928
$82$	2706.87	3.64542
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1622.65	−1.99961
$88$	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
$93$	1241.90	1.38472
$94$	−3010.64	−3.30344
$95$	0	0
$96$	−6799.78	−7.22917
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	1939.09	1.99875
$99$	0	0
$100$	−2994.99	−2.99499
$101$	−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	$0.5\pi$
$104$	1158.49	1.09230
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−2149.71	−1.91534
$109$	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
$116$	−6164.72	−4.93431
$117$	−164.009	−0.129595
$118$	2238.71	1.74653
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	−3019.69	−2.21363
$124$	4718.19	3.41698
$125$	0	0
$126$	0	0
$127$	−2325.71	−1.62499	−0.812495	−	0.582968i	$-0.801891\pi$
$127$	−2325.71	−1.62499	−0.812495	+	0.582968i	$0.801891\pi$
$128$	−11433.5	−7.89519
$129$	0	0
$130$	0	0
$131$	−459.352	−0.306364	−0.153182	−	0.988198i	$-0.548952\pi$
$131$	−459.352	−0.306364	−0.153182	+	0.988198i	$0.548952\pi$
$132$	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
$139$	3021.95	1.84401	0.922007	−	0.387172i	$-0.126548\pi$
$139$	3021.95	1.84401	0.922007	+	0.387172i	$0.126548\pi$
$140$	0	0
$141$	3358.55	2.00597
$142$	2244.07	1.32618
$143$	0	0
$144$	4067.07	2.35363
$145$	0	0
$146$	−2338.44	−1.32555
$147$	−2163.17	−1.21371
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	4456.66	2.42590
$151$	2530.68	1.36386	0.681932	−	0.731415i	$-0.261140\pi$
$151$	2530.68	1.36386	0.681932	+	0.731415i	$0.261140\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−1940.17	−0.995754
$157$	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
$162$	5148.60	2.49699
$163$	2496.34	1.19956	0.599781	−	0.800164i	$-0.295254\pi$
$163$	2496.34	1.19956	0.599781	+	0.800164i	$0.295254\pi$
$164$	−11472.3	−5.46242
$165$	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
$169$	−2032.14	−0.924962
$170$	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	$0.780380\pi$
$174$	9173.34	3.99672
$175$	0	0
$176$	0	0
$177$	−2497.42	−1.06055
$178$	0	0
$179$	−2023.41	−0.844898	−0.422449	−	0.906387i	$-0.638829\pi$
$179$	−2023.41	−0.844898	−0.422449	+	0.906387i	$0.638829\pi$
$180$	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
$186$	−7020.85	−2.76771
$187$	0	0
$188$	12759.7	4.94999
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	22377.1	8.41109
$193$	−4934.03	−1.84020	−0.920101	−	0.391681i	$-0.871894\pi$
$193$	−4934.03	−1.84020	−0.920101	+	0.391681i	$0.871894\pi$
$194$	0	0
$195$	0	0
$196$	−8218.26	−2.99499
$197$	2269.22	0.820685	0.410342	−	0.911931i	$-0.365409\pi$
$197$	2269.22	0.820685	0.410342	+	0.911931i	$0.365409\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	11278.3	3.98749
$201$	0	0
$202$	9056.61	3.15456
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−4088.16	−1.36280
$209$	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
$213$	−2503.40	−0.805306
$214$	0	0
$215$	0	0
$216$	8095.23	2.55005
$217$	0	0
$218$	0	0
$219$	2608.68	0.804923
$220$	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
$225$	−1596.69	−0.473093
$226$	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
$232$	23214.6	6.56946
$233$	744.688	0.209383	0.104691	−	0.994505i	$-0.466615\pi$
$233$	744.688	0.209383	0.104691	+	0.994505i	$0.466615\pi$
$234$	927.191	0.259027
$235$	0	0
$236$	−9488.13	−2.61705
$237$	0	0
$238$	0	0
$239$	−3193.91	−0.864422	−0.432211	−	0.901773i	$-0.642266\pi$
$239$	−3193.91	−0.864422	−0.432211	+	0.901773i	$0.642266\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	7524.56	1.99875
$243$	−3321.11	−0.876746
$244$	0	0
$245$	0	0
$246$	17071.2	4.42448
$247$	0	0
$248$	−17767.4	−4.54932
$249$	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
$254$	13148.0	3.24795
$255$	0	0
$256$	36251.4	8.85044
$257$	4904.62	1.19043	0.595217	−	0.803565i	$-0.297066\pi$
$257$	4904.62	1.19043	0.595217	+	0.803565i	$0.297066\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3286.53	−0.779429
$262$	2596.86	0.612345
$263$	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
$269$	−8093.25	−1.83440	−0.917202	−	0.398424i	$-0.869557\pi$
$269$	−8093.25	−1.83440	−0.917202	+	0.398424i	$0.869557\pi$
$270$	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
$277$	−6905.84	−1.49795	−0.748974	−	0.662599i	$-0.769453\pi$
$277$	−6905.84	−1.49795	−0.748974	+	0.662599i	$0.769453\pi$
$278$	−17084.0	−3.68572
$279$	2515.36	0.539751
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	−18987.0	−4.00942
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−9510.84	−1.98720
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−13772.4	−2.81786
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	9910.81	1.98625
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	12229.1	2.42590
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−18888.3	−3.63505
$301$	0	0
$302$	−14306.7	−2.72602
$303$	−10103.2	−1.91556
$304$	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
$311$	10960.5	1.99843	0.999217	−	0.0395544i	$-0.0125938\pi$
$311$	10960.5	1.99843	0.999217	+	0.0395544i	$0.0125938\pi$
$312$	7306.13	1.32573
$313$	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
$324$	−21820.8	−3.74157
$325$	1604.97	0.273931
$326$	−14112.6	−2.39762
$327$	0	0
$328$	43201.5	7.27258
$329$	0	0
$330$	0	0
$331$	6046.92	1.00414	0.502068	−	0.864828i	$-0.332573\pi$
$331$	6046.92	1.00414	0.502068	+	0.864828i	$0.332573\pi$
$332$	0	0
$333$	0	0
$334$	−10176.0	−1.66708
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	11488.3	1.84877
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	19843.1	3.08316
$347$	9180.00	1.42020	0.710098	−	0.704103i	$-0.248650\pi$
$347$	9180.00	1.42020	0.710098	+	0.704103i	$0.248650\pi$
$348$	−38878.6	−5.98882
$349$	2664.80	0.408721	0.204360	−	0.978896i	$-0.434489\pi$
$349$	2664.80	0.408721	0.204360	+	0.978896i	$0.434489\pi$
$350$	0	0
$351$	1152.00	0.175182
$352$	0	0
$353$	12810.6	1.93156	0.965782	−	0.259355i	$-0.0835101\pi$
$353$	12810.6	1.93156	0.965782	+	0.259355i	$0.0835101\pi$
$354$	14118.7	2.11978
$355$	0	0
$356$	0	0
$357$	0	0
$358$	11439.0	1.68874
$359$	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
$363$	−8394.12	−1.21371
$364$	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
$369$	−6116.11	−0.862850
$370$	0	0
$371$	0	0
$372$	29755.8	4.14723
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−48049.6	−6.59034
$377$	3303.57	0.451307
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−14667.4	−1.97227
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−72106.5	−9.58248
$385$	0	0
$386$	27893.6	3.67810
$387$	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
$392$	30947.7	3.98749
$393$	−2896.96	−0.371838
$394$	−12828.6	−1.64034
$395$	0	0
$396$	0	0
$397$	3404.30	0.430370	0.215185	−	0.976573i	$-0.430965\pi$
$397$	3404.30	0.430370	0.215185	+	0.976573i	$0.430965\pi$
$398$	0	0
$399$	0	0
$400$	−39799.9	−4.97498
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−2528.40	−0.312528
$404$	−38383.8	−4.72690
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16157.1	1.95334	0.976671	−	0.214742i	$-0.0688912\pi$
$409$	16157.1	1.95334	0.976671	+	0.214742i	$0.0688912\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	13843.8	1.63160
$417$	19058.3	2.23810
$418$	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	$0.5\pi$
$422$	13952.4	1.60946
$423$	6802.45	0.781907
$424$	0	0
$425$	0	0
$426$	14152.5	1.60960
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−28567.1	−3.18157
$433$	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
$438$	−14747.7	−1.60884
$439$	18272.0	1.98650	0.993250	−	0.115995i	$-0.0370058\pi$
$439$	18272.0	1.98650	0.993250	+	0.115995i	$0.0370058\pi$
$440$	0	0
$441$	−4381.31	−0.473093
$442$	0	0
$443$	13724.1	1.47191	0.735953	−	0.677033i	$-0.236735\pi$
$443$	13724.1	1.47191	0.735953	+	0.677033i	$0.236735\pi$
$444$	0	0
$445$	0	0
$446$	27362.0	2.90500
$447$	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	$0.918900\pi$
$450$	9026.58	0.945593
$451$	0	0
$452$	0	0
$453$	15960.0	1.65534
$454$	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
$461$	−15365.9	−1.55241	−0.776205	−	0.630480i	$-0.782858\pi$
$461$	−15365.9	−1.55241	−0.776205	+	0.630480i	$0.782858\pi$
$462$	0	0
$463$	−11680.0	−1.17239	−0.586194	−	0.810171i	$-0.699374\pi$
$463$	−11680.0	−1.17239	−0.586194	+	0.810171i	$0.699374\pi$
$464$	−81921.8	−8.19638
$465$	0	0
$466$	−4209.95	−0.418503
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−3929.63	−0.388135
$469$	0	0
$470$	0	0
$471$	0	0
$472$	35729.7	3.48431
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	18056.2	1.72776
$479$	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
$484$	−31890.7	−2.99499
$485$	0	0
$486$	18775.3	1.75239
$487$	2335.12	0.217278	0.108639	−	0.994081i	$-0.465351\pi$
$487$	2335.12	0.217278	0.108639	+	0.994081i	$0.465351\pi$
$488$	0	0
$489$	15743.5	1.45592
$490$	0	0
$491$	16210.8	1.48998	0.744992	−	0.667073i	$-0.232453\pi$
$491$	16210.8	1.48998	0.744992	+	0.667073i	$0.232453\pi$
$492$	−72351.5	−6.62979
$493$	0	0
$494$	0	0
$495$	0	0
$496$	62699.1	5.67595
$497$	0	0
$498$	0	0
$499$	21525.2	1.93106	0.965530	−	0.260292i	$-0.0838188\pi$
$499$	21525.2	1.93106	0.965530	+	0.260292i	$0.0838188\pi$
$500$	0	0
$501$	11351.9	1.01231
$502$	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
$507$	−12815.9	−1.12264
$508$	−55724.0	−4.86683
$509$	14675.7	1.27797	0.638986	−	0.769218i	$-0.279354\pi$
$509$	14675.7	1.27797	0.638986	+	0.769218i	$0.279354\pi$
$510$	0	0
$511$	0	0
$512$	−113473.	−9.79459
$513$	0	0
$514$	−27727.3	−2.37938
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−22136.2	−1.87220
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	18579.8	1.55788
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−11006.0	−0.917559
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−5058.31	−0.413393
$532$	0	0
$533$	6147.82	0.499609
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12760.9	−1.02546
$538$	45753.7	3.66651
$539$	0	0
$540$	0	0
$541$	−21313.0	−1.69375	−0.846873	−	0.531796i	$-0.821517\pi$
$541$	−21313.0	−1.69375	−0.846873	+	0.531796i	$0.821517\pi$
$542$	50382.3	3.99282
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1569.66	−0.122694	−0.0613471	−	0.998116i	$-0.519540\pi$
$547$	−1569.66	−0.122694	−0.0613471	+	0.998116i	$0.519540\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	39040.9	2.99402
$555$	0	0
$556$	72405.6	5.52281
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	−14220.1	−1.07883
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	80470.7	6.00785
$565$	0	0
$566$	0	0
$567$	0	0
$568$	35815.2	2.64573
$569$	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
$576$	45322.9	3.27856
$577$	25741.2	1.85723	0.928615	−	0.371044i	$-0.121000\pi$
$577$	25741.2	1.85723	0.928615	+	0.371044i	$0.121000\pi$
$578$	27774.7	1.99875
$579$	−31117.0	−2.23347
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−37321.4	−2.64447
$585$	0	0
$586$	0	0
$587$	23488.7	1.65159	0.825794	−	0.563972i	$-0.190728\pi$
$587$	23488.7	1.65159	0.825794	+	0.563972i	$0.190728\pi$
$588$	−51829.5	−3.63505
$589$	0	0
$590$	0	0
$591$	14311.1	0.996074
$592$	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	$0.188339\pi$
$600$	71128.1	4.83965
$601$	−25680.9	−1.74301	−0.871504	−	0.490388i	$-0.836855\pi$
$601$	−25680.9	−1.74301	−0.871504	+	0.490388i	$0.836855\pi$
$602$	0	0
$603$	0	0
$604$	60634.8	4.08476
$605$	0	0
$606$	57116.6	3.82872
$607$	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
$611$	−6837.73	−0.452741
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−58907.5	−3.87185
$615$	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
$622$	−61963.2	−3.99437
$623$	0	0
$624$	−25782.5	−1.65405
$625$	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
$633$	−15564.7	−0.977319
$634$	−8649.57	−0.541827
$635$	0	0
$636$	0	0
$637$	4404.03	0.273931
$638$	0	0
$639$	−5070.41	−0.313901
$640$	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
$647$	207.659	0.0126181	0.00630906	−	0.999980i	$-0.497992\pi$
$647$	207.659	0.0126181	0.00630906	+	0.999980i	$0.497992\pi$
$648$	82171.3	4.98147
$649$	0	0
$650$	−9073.38	−0.547519
$651$	0	0
$652$	59812.2	3.59268
$653$	−10700.5	−0.641261	−0.320631	−	0.947204i	$-0.603895\pi$
$653$	−10700.5	−0.641261	−0.320631	+	0.947204i	$0.603895\pi$
$654$	0	0
$655$	0	0
$656$	−152453.	−9.07363
$657$	5283.65	0.313751
$658$	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	$0.5\pi$
$662$	−34185.1	−2.00701
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	43127.9	2.49801
$669$	−30524.1	−1.76402
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−25066.4	−1.43572	−0.717859	−	0.696188i	$-0.754878\pi$
$673$	−25066.4	−1.43572	−0.717859	+	0.696188i	$0.754878\pi$
$674$	0	0
$675$	11215.1	0.639513
$676$	−48690.0	−2.77025
$677$	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
$683$	−21910.1	−1.22748	−0.613738	−	0.789510i	$-0.710335\pi$
$683$	−21910.1	−1.22748	−0.613738	+	0.789510i	$0.710335\pi$
$684$	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	$0.446937\pi$
$692$	−84099.4	−4.61991
$693$	0	0
$694$	−51897.4	−2.83861
$695$	0	0
$696$	146406.	7.97342
$697$	0	0
$698$	−15065.0	−0.816930
$699$	4696.47	0.254130
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−6512.60	−0.350145
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−72422.6	−3.86071
$707$	0	0
$708$	−59838.1	−3.17635
$709$	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
$716$	−48480.8	−2.53046
$717$	−20142.8	−1.04916
$718$	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
$722$	38776.1	1.99875
$723$	0	0
$724$	0	0
$725$	32161.6	1.64752
$726$	47454.6	2.42590
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	3644.50	0.185160
$730$	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
$738$	34576.3	1.72462
$739$	−37876.3	−1.88539	−0.942695	−	0.333656i	$-0.891718\pi$
$739$	−37876.3	−1.88539	−0.942695	+	0.333656i	$0.891718\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−112052.	−5.52155
$745$	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	$0.5\pi$
$752$	169561.	8.22243
$753$	0	0
$754$	−18676.1	−0.902048
$755$	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
$761$	−6372.08	−0.303532	−0.151766	−	0.988416i	$-0.548496\pi$
$761$	−6372.08	−0.303532	−0.151766	+	0.988416i	$0.548496\pi$
$762$	82919.4	3.94207
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5084.54	0.239364
$768$	228624.	10.7419
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	30931.6	1.44484
$772$	−118219.	−5.51139
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−24615.0	−1.14090
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	23084.6	1.05361
$784$	−109211.	−4.97498
$785$	0	0
$786$	16377.4	0.743209
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	54370.3	2.45794
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−19245.6	−0.860201
$795$	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
$800$	134775.	5.95625
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	14293.8	0.624664
$807$	−51041.1	−2.22643
$808$	144543.	6.29332
$809$	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
$811$	−41578.7	−1.80028	−0.900139	−	0.435602i	$-0.856536\pi$
$811$	−41578.7	−1.80028	−0.900139	+	0.435602i	$0.856536\pi$
$812$	0	0
$813$	−56204.6	−2.42458
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−91341.1	−3.90424
$819$	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
$823$	−46745.0	−1.97986	−0.989931	−	0.141548i	$-0.954792\pi$
$823$	−46745.0	−1.97986	−0.989931	+	0.141548i	$0.954792\pi$
$824$	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
$831$	−43552.5	−1.81808
$832$	−45557.9	−1.89836
$833$	0	0
$834$	−107742.	−4.47340
$835$	0	0
$836$	0	0
$837$	−17667.9	−0.729619
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	41810.6	1.71432
$842$	0	0
$843$	0	0
$844$	−59133.1	−2.41167
$845$	0	0
$846$	−38456.4	−1.56283
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−59981.3	−2.41188
$853$	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
$857$	50108.5	1.99728	0.998642	−	0.0520955i	$-0.0165900\pi$
$857$	50108.5	1.99728	0.998642	+	0.0520955i	$0.0165900\pi$
$858$	0	0
$859$	−24600.8	−0.977148	−0.488574	−	0.872523i	$-0.662483\pi$
$859$	−24600.8	−0.977148	−0.488574	+	0.872523i	$0.662483\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5226.84	0.206169	0.103084	−	0.994673i	$-0.467129\pi$
$863$	5226.84	0.206169	0.103084	+	0.994673i	$0.467129\pi$
$864$	96737.1	3.80910
$865$	0	0
$866$	0	0
$867$	−30984.4	−1.21371
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	62503.8	2.41074
$877$	34090.0	1.31259	0.656293	−	0.754506i	$-0.272124\pi$
$877$	34090.0	1.31259	0.656293	+	0.754506i	$0.272124\pi$
$878$	−103297.	−3.97051
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	24768.9	0.945593
$883$	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
$886$	−77586.9	−2.94197
$887$	46875.4	1.77443	0.887215	−	0.461355i	$-0.152637\pi$
$887$	46875.4	1.77443	0.887215	+	0.461355i	$0.152637\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−115966.	−4.35295
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	104100.	3.86845
$899$	−50666.1	−1.87965
$900$	−38256.5	−1.41691
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−90227.0	−3.30860
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−20463.2	−0.746667
$910$	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
$921$	65715.0	2.35112
$922$	86868.2	3.10288
$923$	5096.71	0.181755
$924$	0	0
$925$	0	0
$926$	66030.7	2.34331
$927$	0	0
$928$	277412.	9.81304
$929$	−48618.9	−1.71704	−0.858521	−	0.512778i	$-0.828616\pi$
$929$	−48618.9	−1.71704	−0.858521	+	0.512778i	$0.828616\pi$
$930$	0	0
$931$	0	0
$932$	17842.7	0.627099
$933$	69123.8	2.42552
$934$	0	0
$935$	0	0
$936$	14797.9	0.516757
$937$	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
$944$	−126086.	−4.34719
$945$	0	0
$946$	0	0
$947$	57261.1	1.96487	0.982437	−	0.186594i	$-0.0597450\pi$
$947$	57261.1	1.96487	0.982437	+	0.186594i	$0.0597450\pi$
$948$	0	0
$949$	−5311.04	−0.181669
$950$	0	0
$951$	9649.13	0.329016
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	−76525.9	−2.58894
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8986.44	0.301649
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−55815.2	−1.85615	−0.928075	−	0.372394i	$-0.878537\pi$
$967$	−55815.2	−1.85615	−0.928075	+	0.372394i	$0.878537\pi$
$968$	120091.	3.98749
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−79573.6	−2.62585
$973$	0	0
$974$	−13201.2	−0.434284
$975$	10121.9	0.332473
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	−89002.9	−2.91002
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−91644.6	−2.97810
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	272456.	8.82680
$985$	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	$0.546735\pi$
$992$	−212318.	−6.79548
$993$	38135.6	1.21873
$994$	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	$0.905209\pi$
$998$	−121688.	−3.85970
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-5.65331 q^{2} +6.30662 q^{3} +23.9599 q^{4} -35.6533 q^{6} -90.2265 q^{8} +12.7735 q^{9} +151.106 q^{12} -12.8397 q^{13} +318.399 q^{16} -72.2126 q^{18} -569.025 q^{24} -125.000 q^{25} +72.5871 q^{26} -89.7212 q^{27} -257.293 q^{29} +196.920 q^{31} -1078.20 q^{32} +306.052 q^{36} -80.9754 q^{39} -478.812 q^{41} +532.544 q^{47} +2008.02 q^{48} -343.000 q^{49} +706.664 q^{50} -307.639 q^{52} +507.222 q^{54} +1454.56 q^{58} -396.000 q^{59} -1113.25 q^{62} +3548.19 q^{64} -396.948 q^{71} -1152.51 q^{72} +413.641 q^{73} -788.328 q^{75} +457.779 q^{78} -910.722 q^{81} +2706.87 q^{82} -1622.65 q^{87} +1241.90 q^{93} -3010.64 q^{94} -6799.78 q^{96} +1939.09 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 4 q^{3} + 23 q^{4} - 63 q^{6} - 114 q^{8} + 92 q^{9} + 161 q^{12} + 74 q^{13} + 263 q^{16} + 138 q^{18} - 324 q^{24} - 250 q^{25} + 303 q^{26} - 628 q^{27} - 282 q^{29} + 344 q^{31} - 1035 q^{32}+ \cdots + 1029 q^{98}+O(q^{100}) \) 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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