LMFDB - Embedded newform 531.8.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,8,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.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:	$165.876448532$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-10.5486$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.54862 q^{2} -36.8240 q^{4} -248.981 q^{5} -1453.31 q^{7} -1573.84 q^{8} +O(q^{10})$	\(q+9.54862 q^{2} -36.8240 q^{4} -248.981 q^{5} -1453.31 q^{7} -1573.84 q^{8} -2377.42 q^{10} +6465.27 q^{11} +4142.86 q^{13} -13877.1 q^{14} -10314.5 q^{16} +25815.2 q^{17} +26877.5 q^{19} +9168.46 q^{20} +61734.4 q^{22} -58835.3 q^{23} -16133.6 q^{25} +39558.6 q^{26} +53516.5 q^{28} -13734.2 q^{29} +25833.8 q^{31} +102962. q^{32} +246499. q^{34} +361845. q^{35} +124281. q^{37} +256643. q^{38} +391856. q^{40} +201398. q^{41} +24676.8 q^{43} -238077. q^{44} -561796. q^{46} +412000. q^{47} +1.28856e6 q^{49} -154053. q^{50} -152556. q^{52} +43501.3 q^{53} -1.60973e6 q^{55} +2.28727e6 q^{56} -131143. q^{58} -205379. q^{59} -1.61032e6 q^{61} +246677. q^{62} +2.30341e6 q^{64} -1.03149e6 q^{65} +1.37101e6 q^{67} -950617. q^{68} +3.45512e6 q^{70} -2.87276e6 q^{71} +4.37743e6 q^{73} +1.18671e6 q^{74} -989737. q^{76} -9.39602e6 q^{77} -960831. q^{79} +2.56812e6 q^{80} +1.92307e6 q^{82} -6.21351e6 q^{83} -6.42749e6 q^{85} +235630. q^{86} -1.01753e7 q^{88} -9.66681e6 q^{89} -6.02084e6 q^{91} +2.16655e6 q^{92} +3.93403e6 q^{94} -6.69199e6 q^{95} -1.18859e7 q^{97} +1.23039e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) $ 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * 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 $

$2$	9.54862	0.843986	0.421993	−	0.906599i	$-0.361331\pi$
$2$	9.54862	0.843986	0.421993	+	0.906599i	$0.361331\pi$
$3$	0	0
$3$	0	0
$4$	−36.8240	−0.287687
$5$	−248.981	−0.890781	−0.445390	−	0.895336i	$-0.646935\pi$
$5$	−248.981	−0.890781	−0.445390	+	0.895336i	$0.646935\pi$
$6$	0	0
$7$	−1453.31	−1.60145	−0.800726	−	0.599031i	$-0.795553\pi$
$7$	−1453.31	−1.60145	−0.800726	+	0.599031i	$0.795553\pi$
$8$	−1573.84	−1.08679
$9$	0	0
$10$	−2377.42	−0.751807
$11$	6465.27	1.46458	0.732288	−	0.680995i	$-0.238452\pi$
$11$	6465.27	1.46458	0.732288	+	0.680995i	$0.238452\pi$
$12$	0	0
$13$	4142.86	0.522996	0.261498	−	0.965204i	$-0.415784\pi$
$13$	4142.86	0.522996	0.261498	+	0.965204i	$0.415784\pi$
$14$	−13877.1	−1.35160
$15$	0	0
$16$	−10314.5	−0.629549
$17$	25815.2	1.27440	0.637198	−	0.770700i	$-0.280093\pi$
$17$	25815.2	1.27440	0.637198	+	0.770700i	$0.280093\pi$
$18$	0	0
$19$	26877.5	0.898984	0.449492	−	0.893284i	$-0.351605\pi$
$19$	26877.5	0.898984	0.449492	+	0.893284i	$0.351605\pi$
$20$	9168.46	0.256266
$21$	0	0
$22$	61734.4	1.23608
$23$	−58835.3	−1.00830	−0.504151	−	0.863616i	$-0.668194\pi$
$23$	−58835.3	−1.00830	−0.504151	+	0.863616i	$0.668194\pi$
$24$	0	0
$25$	−16133.6	−0.206510
$26$	39558.6	0.441401
$27$	0	0
$28$	53516.5	0.460717
$29$	−13734.2	−0.104571	−0.0522856	−	0.998632i	$-0.516651\pi$
$29$	−13734.2	−0.104571	−0.0522856	+	0.998632i	$0.516651\pi$
$30$	0	0
$31$	25833.8	0.155748	0.0778741	−	0.996963i	$-0.475187\pi$
$31$	25833.8	0.155748	0.0778741	+	0.996963i	$0.475187\pi$
$32$	102962.	0.555460
$33$	0	0
$34$	246499.	1.07557
$35$	361845.	1.42654
$36$	0	0
$37$	124281.	0.403365	0.201682	−	0.979451i	$-0.435359\pi$
$37$	124281.	0.403365	0.201682	+	0.979451i	$0.435359\pi$
$38$	256643.	0.758730
$39$	0	0
$40$	391856.	0.968092
$41$	201398.	0.456365	0.228182	−	0.973618i	$-0.426722\pi$
$41$	201398.	0.456365	0.228182	+	0.973618i	$0.426722\pi$
$42$	0	0
$43$	24676.8	0.0473314	0.0236657	−	0.999720i	$-0.492466\pi$
$43$	24676.8	0.0473314	0.0236657	+	0.999720i	$0.492466\pi$
$44$	−238077.	−0.421340
$45$	0	0
$46$	−561796.	−0.850993
$47$	412000.	0.578835	0.289418	−	0.957203i	$-0.406538\pi$
$47$	412000.	0.578835	0.289418	+	0.957203i	$0.406538\pi$
$48$	0	0
$49$	1.28856e6	1.56465
$50$	−154053.	−0.174291
$51$	0	0
$52$	−152556.	−0.150459
$53$	43501.3	0.0401362	0.0200681	−	0.999799i	$-0.493612\pi$
$53$	43501.3	0.0401362	0.0200681	+	0.999799i	$0.493612\pi$
$54$	0	0
$55$	−1.60973e6	−1.30462
$56$	2.28727e6	1.74044
$57$	0	0
$58$	−131143.	−0.0882566
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	−1.61032e6	−0.908362	−0.454181	−	0.890909i	$-0.650068\pi$
$61$	−1.61032e6	−0.908362	−0.454181	+	0.890909i	$0.650068\pi$
$62$	246677.	0.131449
$63$	0	0
$64$	2.30341e6	1.09835
$65$	−1.03149e6	−0.465875
$66$	0	0
$67$	1.37101e6	0.556904	0.278452	−	0.960450i	$-0.410179\pi$
$67$	1.37101e6	0.556904	0.278452	+	0.960450i	$0.410179\pi$
$68$	−950617.	−0.366627
$69$	0	0
$70$	3.45512e6	1.20398
$71$	−2.87276e6	−0.952568	−0.476284	−	0.879292i	$-0.658017\pi$
$71$	−2.87276e6	−0.952568	−0.476284	+	0.879292i	$0.658017\pi$
$72$	0	0
$73$	4.37743e6	1.31701	0.658505	−	0.752576i	$-0.271189\pi$
$73$	4.37743e6	1.31701	0.658505	+	0.752576i	$0.271189\pi$
$74$	1.18671e6	0.340434
$75$	0	0
$76$	−989737.	−0.258626
$77$	−9.39602e6	−2.34545
$78$	0	0
$79$	−960831.	−0.219256	−0.109628	−	0.993973i	$-0.534966\pi$
$79$	−960831.	−0.219256	−0.109628	+	0.993973i	$0.534966\pi$
$80$	2.56812e6	0.560790
$81$	0	0
$82$	1.92307e6	0.385166
$83$	−6.21351e6	−1.19279	−0.596395	−	0.802691i	$-0.703401\pi$
$83$	−6.21351e6	−1.19279	−0.596395	+	0.802691i	$0.703401\pi$
$84$	0	0
$85$	−6.42749e6	−1.13521
$86$	235630.	0.0399471
$87$	0	0
$88$	−1.01753e7	−1.59169
$89$	−9.66681e6	−1.45351	−0.726755	−	0.686896i	$-0.758973\pi$
$89$	−9.66681e6	−1.45351	−0.726755	+	0.686896i	$0.758973\pi$
$90$	0	0
$91$	−6.02084e6	−0.837553
$92$	2.16655e6	0.290075
$93$	0	0
$94$	3.93403e6	0.488529
$95$	−6.69199e6	−0.800798
$96$	0	0
$97$	−1.18859e7	−1.32231	−0.661153	−	0.750251i	$-0.729933\pi$
$97$	−1.18859e7	−1.32231	−0.661153	+	0.750251i	$0.729933\pi$
$98$	1.23039e7	1.32054
$99$	0	0
$100$	594101.	0.0594101
$101$	−1.70394e7	−1.64562	−0.822808	−	0.568319i	$-0.807594\pi$
$101$	−1.70394e7	−1.64562	−0.822808	+	0.568319i	$0.807594\pi$
$102$	0	0
$103$	1.76135e7	1.58824	0.794118	−	0.607764i	$-0.207933\pi$
$103$	1.76135e7	1.58824	0.794118	+	0.607764i	$0.207933\pi$
$104$	−6.52020e6	−0.568387
$105$	0	0
$106$	415377.	0.0338744
$107$	−2.01177e7	−1.58758	−0.793789	−	0.608193i	$-0.791895\pi$
$107$	−2.01177e7	−1.58758	−0.793789	+	0.608193i	$0.791895\pi$
$108$	0	0
$109$	1.35078e7	0.999063	0.499531	−	0.866296i	$-0.333506\pi$
$109$	1.35078e7	0.999063	0.499531	+	0.866296i	$0.333506\pi$
$110$	−1.53707e7	−1.10108
$111$	0	0
$112$	1.49902e7	1.00819
$113$	477253.	0.0311153	0.0155577	−	0.999879i	$-0.495048\pi$
$113$	477253.	0.0311153	0.0155577	+	0.999879i	$0.495048\pi$
$114$	0	0
$115$	1.46489e7	0.898176
$116$	505749.	0.0300838
$117$	0	0
$118$	−1.96109e6	−0.109878
$119$	−3.75174e7	−2.04088
$120$	0	0
$121$	2.23125e7	1.14499
$122$	−1.53764e7	−0.766645
$123$	0	0
$124$	−951304.	−0.0448068
$125$	2.34686e7	1.07474
$126$	0	0
$127$	2.32024e7	1.00512	0.502562	−	0.864541i	$-0.332391\pi$
$127$	2.32024e7	1.00512	0.502562	+	0.864541i	$0.332391\pi$
$128$	8.81518e6	0.371532
$129$	0	0
$130$	−9.84932e6	−0.393192
$131$	4.19666e6	0.163100	0.0815501	−	0.996669i	$-0.474013\pi$
$131$	4.19666e6	0.163100	0.0815501	+	0.996669i	$0.474013\pi$
$132$	0	0
$133$	−3.90613e7	−1.43968
$134$	1.30913e7	0.470019
$135$	0	0
$136$	−4.06290e7	−1.38500
$137$	2.12221e7	0.705126	0.352563	−	0.935788i	$-0.385310\pi$
$137$	2.12221e7	0.705126	0.352563	+	0.935788i	$0.385310\pi$
$138$	0	0
$139$	5.22180e7	1.64918	0.824591	−	0.565729i	$-0.191405\pi$
$139$	5.22180e7	1.64918	0.824591	+	0.565729i	$0.191405\pi$
$140$	−1.33246e7	−0.410398
$141$	0	0
$142$	−2.74309e7	−0.803954
$143$	2.67847e7	0.765968
$144$	0	0
$145$	3.41956e6	0.0931500
$146$	4.17984e7	1.11154
$147$	0	0
$148$	−4.57651e6	−0.116043
$149$	−3.35373e7	−0.830569	−0.415285	−	0.909692i	$-0.636318\pi$
$149$	−3.35373e7	−0.830569	−0.415285	+	0.909692i	$0.636318\pi$
$150$	0	0
$151$	−2.21646e7	−0.523890	−0.261945	−	0.965083i	$-0.584364\pi$
$151$	−2.21646e7	−0.523890	−0.261945	+	0.965083i	$0.584364\pi$
$152$	−4.23010e7	−0.977007
$153$	0	0
$154$	−8.97189e7	−1.97953
$155$	−6.43213e6	−0.138738
$156$	0	0
$157$	−5.80763e7	−1.19771	−0.598853	−	0.800859i	$-0.704377\pi$
$157$	−5.80763e7	−1.19771	−0.598853	+	0.800859i	$0.704377\pi$
$158$	−9.17461e6	−0.185049
$159$	0	0
$160$	−2.56356e7	−0.494793
$161$	8.55057e7	1.61475
$162$	0	0
$163$	−7.64837e7	−1.38329	−0.691643	−	0.722239i	$-0.743113\pi$
$163$	−7.64837e7	−1.38329	−0.691643	+	0.722239i	$0.743113\pi$
$164$	−7.41628e6	−0.131290
$165$	0	0
$166$	−5.93304e7	−1.00670
$167$	−7.60086e7	−1.26286	−0.631430	−	0.775433i	$-0.717532\pi$
$167$	−7.60086e7	−1.26286	−0.631430	+	0.775433i	$0.717532\pi$
$168$	0	0
$169$	−4.55852e7	−0.726475
$170$	−6.13736e7	−0.958099
$171$	0	0
$172$	−908698.	−0.0136166
$173$	9.21166e7	1.35262	0.676311	−	0.736616i	$-0.263578\pi$
$173$	9.21166e7	1.35262	0.676311	+	0.736616i	$0.263578\pi$
$174$	0	0
$175$	2.34470e7	0.330715
$176$	−6.66862e7	−0.922023
$177$	0	0
$178$	−9.23047e7	−1.22674
$179$	−5.97291e7	−0.778396	−0.389198	−	0.921154i	$-0.627248\pi$
$179$	−5.97291e7	−0.778396	−0.389198	+	0.921154i	$0.627248\pi$
$180$	0	0
$181$	−9.97102e7	−1.24987	−0.624934	−	0.780677i	$-0.714874\pi$
$181$	−9.97102e7	−1.24987	−0.624934	+	0.780677i	$0.714874\pi$
$182$	−5.74907e7	−0.706883
$183$	0	0
$184$	9.25974e7	1.09581
$185$	−3.09435e7	−0.359309
$186$	0	0
$187$	1.66902e8	1.86645
$188$	−1.51715e7	−0.166523
$189$	0	0
$190$	−6.38993e7	−0.675862
$191$	2.03999e6	0.0211841	0.0105921	−	0.999944i	$-0.496628\pi$
$191$	2.03999e6	0.0211841	0.0105921	+	0.999944i	$0.496628\pi$
$192$	0	0
$193$	1.32752e8	1.32920	0.664602	−	0.747198i	$-0.268601\pi$
$193$	1.32752e8	1.32920	0.664602	+	0.747198i	$0.268601\pi$
$194$	−1.13494e8	−1.11601
$195$	0	0
$196$	−4.74497e7	−0.450130
$197$	1.07186e8	0.998867	0.499434	−	0.866352i	$-0.333542\pi$
$197$	1.07186e8	0.998867	0.499434	+	0.866352i	$0.333542\pi$
$198$	0	0
$199$	−1.08221e8	−0.973474	−0.486737	−	0.873549i	$-0.661813\pi$
$199$	−1.08221e8	−0.973474	−0.486737	+	0.873549i	$0.661813\pi$
$200$	2.53916e7	0.224433
$201$	0	0
$202$	−1.62702e8	−1.38888
$203$	1.99601e7	0.167466
$204$	0	0
$205$	−5.01443e7	−0.406521
$206$	1.68184e8	1.34045
$207$	0	0
$208$	−4.27316e7	−0.329252
$209$	1.73771e8	1.31663
$210$	0	0
$211$	1.65214e8	1.21076	0.605380	−	0.795937i	$-0.293021\pi$
$211$	1.65214e8	1.21076	0.605380	+	0.795937i	$0.293021\pi$
$212$	−1.60189e6	−0.0115467
$213$	0	0
$214$	−1.92096e8	−1.33989
$215$	−6.14406e6	−0.0421619
$216$	0	0
$217$	−3.75445e7	−0.249423
$218$	1.28981e8	0.843195
$219$	0	0
$220$	5.92765e7	0.375321
$221$	1.06949e8	0.666504
$222$	0	0
$223$	1.30090e7	0.0785558	0.0392779	−	0.999228i	$-0.487494\pi$
$223$	1.30090e7	0.0785558	0.0392779	+	0.999228i	$0.487494\pi$
$224$	−1.49635e8	−0.889542
$225$	0	0
$226$	4.55711e6	0.0262609
$227$	−3.00513e8	−1.70519	−0.852595	−	0.522572i	$-0.824973\pi$
$227$	−3.00513e8	−1.70519	−0.852595	+	0.522572i	$0.824973\pi$
$228$	0	0
$229$	4.51247e7	0.248308	0.124154	−	0.992263i	$-0.460378\pi$
$229$	4.51247e7	0.248308	0.124154	+	0.992263i	$0.460378\pi$
$230$	1.39876e8	0.758048
$231$	0	0
$232$	2.16155e7	0.113647
$233$	−2.62782e8	−1.36097	−0.680486	−	0.732761i	$-0.738231\pi$
$233$	−2.62782e8	−1.36097	−0.680486	+	0.732761i	$0.738231\pi$
$234$	0	0
$235$	−1.02580e8	−0.515615
$236$	7.56287e6	0.0374537
$237$	0	0
$238$	−3.58239e8	−1.72248
$239$	3.84804e8	1.82325	0.911626	−	0.411022i	$-0.134828\pi$
$239$	3.84804e8	1.82325	0.911626	+	0.411022i	$0.134828\pi$
$240$	0	0
$241$	1.89745e8	0.873194	0.436597	−	0.899657i	$-0.356184\pi$
$241$	1.89745e8	0.873194	0.436597	+	0.899657i	$0.356184\pi$
$242$	2.13054e8	0.966352
$243$	0	0
$244$	5.92985e7	0.261324
$245$	−3.20826e8	−1.39376
$246$	0	0
$247$	1.11350e8	0.470165
$248$	−4.06583e7	−0.169266
$249$	0	0
$250$	2.24092e8	0.907062
$251$	1.89901e8	0.758002	0.379001	−	0.925396i	$-0.376268\pi$
$251$	1.89901e8	0.758002	0.379001	+	0.925396i	$0.376268\pi$
$252$	0	0
$253$	−3.80386e8	−1.47674
$254$	2.21550e8	0.848310
$255$	0	0
$256$	−2.10663e8	−0.784781
$257$	−3.23214e8	−1.18775	−0.593874	−	0.804558i	$-0.702402\pi$
$257$	−3.23214e8	−1.18775	−0.593874	+	0.804558i	$0.702402\pi$
$258$	0	0
$259$	−1.80618e8	−0.645969
$260$	3.79836e7	0.134026
$261$	0	0
$262$	4.00723e7	0.137654
$263$	9.96217e7	0.337683	0.168841	−	0.985643i	$-0.445997\pi$
$263$	9.96217e7	0.337683	0.168841	+	0.985643i	$0.445997\pi$
$264$	0	0
$265$	−1.08310e7	−0.0357526
$266$	−3.72981e8	−1.21507
$267$	0	0
$268$	−5.04861e7	−0.160214
$269$	−5.17058e8	−1.61960	−0.809798	−	0.586709i	$-0.800423\pi$
$269$	−5.17058e8	−1.61960	−0.809798	+	0.586709i	$0.800423\pi$
$270$	0	0
$271$	9.10242e6	0.0277821	0.0138910	−	0.999904i	$-0.495578\pi$
$271$	9.10242e6	0.0277821	0.0138910	+	0.999904i	$0.495578\pi$
$272$	−2.66272e8	−0.802294
$273$	0	0
$274$	2.02642e8	0.595117
$275$	−1.04308e8	−0.302449
$276$	0	0
$277$	1.04754e8	0.296135	0.148068	−	0.988977i	$-0.452695\pi$
$277$	1.04754e8	0.296135	0.148068	+	0.988977i	$0.452695\pi$
$278$	4.98610e8	1.39189
$279$	0	0
$280$	−5.69487e8	−1.55035
$281$	2.20844e8	0.593765	0.296882	−	0.954914i	$-0.404053\pi$
$281$	2.20844e8	0.593765	0.296882	+	0.954914i	$0.404053\pi$
$282$	0	0
$283$	4.56240e8	1.19658	0.598289	−	0.801280i	$-0.295847\pi$
$283$	4.56240e8	1.19658	0.598289	+	0.801280i	$0.295847\pi$
$284$	1.05787e8	0.274041
$285$	0	0
$286$	2.55757e8	0.646466
$287$	−2.92693e8	−0.730846
$288$	0	0
$289$	2.56086e8	0.624084
$290$	3.26521e7	0.0786173
$291$	0	0
$292$	−1.61194e8	−0.378887
$293$	−1.06837e8	−0.248133	−0.124067	−	0.992274i	$-0.539594\pi$
$293$	−1.06837e8	−0.248133	−0.124067	+	0.992274i	$0.539594\pi$
$294$	0	0
$295$	5.11354e7	0.115970
$296$	−1.95598e8	−0.438373
$297$	0	0
$298$	−3.20235e8	−0.700989
$299$	−2.43746e8	−0.527338
$300$	0	0
$301$	−3.58630e7	−0.0757991
$302$	−2.11641e8	−0.442156
$303$	0	0
$304$	−2.77229e8	−0.565954
$305$	4.00940e8	0.809151
$306$	0	0
$307$	−4.91926e8	−0.970322	−0.485161	−	0.874425i	$-0.661239\pi$
$307$	−4.91926e8	−0.970322	−0.485161	+	0.874425i	$0.661239\pi$
$308$	3.45998e8	0.674756
$309$	0	0
$310$	−6.14179e7	−0.117093
$311$	34302.1	6.46634e−5 0	3.23317e−5	−	1.00000i	$-0.499990\pi$
$311$	34302.1	6.46634e−5 0	3.23317e−5		1.00000i	$0.499990\pi$
$312$	0	0
$313$	−2.14984e8	−0.396279	−0.198139	−	0.980174i	$-0.563490\pi$
$313$	−2.14984e8	−0.396279	−0.198139	+	0.980174i	$0.563490\pi$
$314$	−5.54548e8	−1.01085
$315$	0	0
$316$	3.53816e7	0.0630772
$317$	4.71181e8	0.830769	0.415385	−	0.909646i	$-0.363647\pi$
$317$	4.71181e8	0.830769	0.415385	+	0.909646i	$0.363647\pi$
$318$	0	0
$319$	−8.87956e7	−0.153152
$320$	−5.73504e8	−0.978388
$321$	0	0
$322$	8.16461e8	1.36282
$323$	6.93849e8	1.14566
$324$	0	0
$325$	−6.68390e7	−0.108004
$326$	−7.30313e8	−1.16747
$327$	0	0
$328$	−3.16969e8	−0.495973
$329$	−5.98762e8	−0.926977
$330$	0	0
$331$	1.20626e9	1.82829	0.914143	−	0.405391i	$-0.132865\pi$
$331$	1.20626e9	1.82829	0.914143	+	0.405391i	$0.132865\pi$
$332$	2.28806e8	0.343150
$333$	0	0
$334$	−7.25777e8	−1.06584
$335$	−3.41356e8	−0.496079
$336$	0	0
$337$	−5.37145e8	−0.764517	−0.382258	−	0.924055i	$-0.624854\pi$
$337$	−5.37145e8	−0.764517	−0.382258	+	0.924055i	$0.624854\pi$
$338$	−4.35276e8	−0.613135
$339$	0	0
$340$	2.36686e8	0.326584
$341$	1.67023e8	0.228105
$342$	0	0
$343$	−6.75807e8	−0.904259
$344$	−3.88374e7	−0.0514394
$345$	0	0
$346$	8.79586e8	1.14159
$347$	1.07234e9	1.37778	0.688888	−	0.724867i	$-0.258099\pi$
$347$	1.07234e9	1.37778	0.688888	+	0.724867i	$0.258099\pi$
$348$	0	0
$349$	−6.30212e8	−0.793592	−0.396796	−	0.917907i	$-0.629878\pi$
$349$	−6.30212e8	−0.793592	−0.396796	+	0.917907i	$0.629878\pi$
$350$	2.23886e8	0.279119
$351$	0	0
$352$	6.65678e8	0.813513
$353$	1.30728e9	1.58182	0.790908	−	0.611936i	$-0.209609\pi$
$353$	1.30728e9	1.58182	0.790908	+	0.611936i	$0.209609\pi$
$354$	0	0
$355$	7.15263e8	0.848529
$356$	3.55970e8	0.418156
$357$	0	0
$358$	−5.70330e8	−0.656956
$359$	−1.57238e9	−1.79360	−0.896802	−	0.442432i	$-0.854116\pi$
$359$	−1.57238e9	−1.79360	−0.896802	+	0.442432i	$0.854116\pi$
$360$	0	0
$361$	−1.71470e8	−0.191828
$362$	−9.52094e8	−1.05487
$363$	0	0
$364$	2.21711e8	0.240953
$365$	−1.08990e9	−1.17317
$366$	0	0
$367$	−1.48548e9	−1.56868	−0.784342	−	0.620329i	$-0.786999\pi$
$367$	−1.48548e9	−1.56868	−0.784342	+	0.620329i	$0.786999\pi$
$368$	6.06858e8	0.634775
$369$	0	0
$370$	−2.95468e8	−0.303252
$371$	−6.32207e7	−0.0642763
$372$	0	0
$373$	−9.92307e8	−0.990069	−0.495034	−	0.868873i	$-0.664845\pi$
$373$	−9.92307e8	−0.990069	−0.495034	+	0.868873i	$0.664845\pi$
$374$	1.59368e9	1.57526
$375$	0	0
$376$	−6.48422e8	−0.629072
$377$	−5.68990e7	−0.0546903
$378$	0	0
$379$	−1.37321e9	−1.29569	−0.647844	−	0.761773i	$-0.724329\pi$
$379$	−1.37321e9	−1.29569	−0.647844	+	0.761773i	$0.724329\pi$
$380$	2.46426e8	0.230379
$381$	0	0
$382$	1.94790e7	0.0178791
$383$	−1.58828e9	−1.44455	−0.722275	−	0.691606i	$-0.756903\pi$
$383$	−1.58828e9	−1.44455	−0.722275	+	0.691606i	$0.756903\pi$
$384$	0	0
$385$	2.33943e9	2.08928
$386$	1.26760e9	1.12183
$387$	0	0
$388$	4.37687e8	0.380410
$389$	1.66679e8	0.143568	0.0717841	−	0.997420i	$-0.477131\pi$
$389$	1.66679e8	0.143568	0.0717841	+	0.997420i	$0.477131\pi$
$390$	0	0
$391$	−1.51884e9	−1.28497
$392$	−2.02798e9	−1.70045
$393$	0	0
$394$	1.02348e9	0.843030
$395$	2.39229e8	0.195309
$396$	0	0
$397$	−1.07153e9	−0.859482	−0.429741	−	0.902952i	$-0.641395\pi$
$397$	−1.07153e9	−0.859482	−0.429741	+	0.902952i	$0.641395\pi$
$398$	−1.03336e9	−0.821599
$399$	0	0
$400$	1.66410e8	0.130008
$401$	−1.56895e8	−0.121508	−0.0607538	−	0.998153i	$-0.519350\pi$
$401$	−1.56895e8	−0.121508	−0.0607538	+	0.998153i	$0.519350\pi$
$402$	0	0
$403$	1.07026e8	0.0814557
$404$	6.27457e8	0.473423
$405$	0	0
$406$	1.90591e8	0.141339
$407$	8.03508e8	0.590758
$408$	0	0
$409$	−1.06873e9	−0.772392	−0.386196	−	0.922417i	$-0.626211\pi$
$409$	−1.06873e9	−0.772392	−0.386196	+	0.922417i	$0.626211\pi$
$410$	−4.78808e8	−0.343098
$411$	0	0
$412$	−6.48598e8	−0.456915
$413$	2.98479e8	0.208491
$414$	0	0
$415$	1.54705e9	1.06251
$416$	4.26557e8	0.290503
$417$	0	0
$418$	1.65927e9	1.11122
$419$	−9.24350e7	−0.0613885	−0.0306943	−	0.999529i	$-0.509772\pi$
$419$	−9.24350e7	−0.0613885	−0.0306943	+	0.999529i	$0.509772\pi$
$420$	0	0
$421$	8.98575e8	0.586904	0.293452	−	0.955974i	$-0.405196\pi$
$421$	8.98575e8	0.586904	0.293452	+	0.955974i	$0.405196\pi$
$422$	1.57756e9	1.02186
$423$	0	0
$424$	−6.84641e7	−0.0436197
$425$	−4.16491e8	−0.263175
$426$	0	0
$427$	2.34029e9	1.45470
$428$	7.40814e8	0.456726
$429$	0	0
$430$	−5.86672e7	−0.0355841
$431$	7.64033e8	0.459665	0.229833	−	0.973230i	$-0.426182\pi$
$431$	7.64033e8	0.459665	0.229833	+	0.973230i	$0.426182\pi$
$432$	0	0
$433$	1.90959e8	0.113040	0.0565199	−	0.998401i	$-0.482000\pi$
$433$	1.90959e8	0.113040	0.0565199	+	0.998401i	$0.482000\pi$
$434$	−3.58498e8	−0.210510
$435$	0	0
$436$	−4.97412e8	−0.287417
$437$	−1.58135e9	−0.906447
$438$	0	0
$439$	2.64998e9	1.49492	0.747458	−	0.664309i	$-0.231274\pi$
$439$	2.64998e9	1.49492	0.747458	+	0.664309i	$0.231274\pi$
$440$	2.53345e9	1.41785
$441$	0	0
$442$	1.02121e9	0.562520
$443$	−3.33143e9	−1.82061	−0.910307	−	0.413934i	$-0.864154\pi$
$443$	−3.33143e9	−1.82061	−0.910307	+	0.413934i	$0.864154\pi$
$444$	0	0
$445$	2.40685e9	1.29476
$446$	1.24218e8	0.0663001
$447$	0	0
$448$	−3.34755e9	−1.75895
$449$	1.02214e8	0.0532902	0.0266451	−	0.999645i	$-0.491518\pi$
$449$	1.02214e8	0.0532902	0.0266451	+	0.999645i	$0.491518\pi$
$450$	0	0
$451$	1.30209e9	0.668381
$452$	−1.75743e7	−0.00895148
$453$	0	0
$454$	−2.86948e9	−1.43916
$455$	1.49907e9	0.746076
$456$	0	0
$457$	−2.28925e9	−1.12198	−0.560992	−	0.827821i	$-0.689580\pi$
$457$	−2.28925e9	−1.12198	−0.560992	+	0.827821i	$0.689580\pi$
$458$	4.30879e8	0.209568
$459$	0	0
$460$	−5.39429e8	−0.258394
$461$	−6.39371e8	−0.303948	−0.151974	−	0.988384i	$-0.548563\pi$
$461$	−6.39371e8	−0.303948	−0.151974	+	0.988384i	$0.548563\pi$
$462$	0	0
$463$	−1.36469e9	−0.638998	−0.319499	−	0.947587i	$-0.603515\pi$
$463$	−1.36469e9	−0.638998	−0.319499	+	0.947587i	$0.603515\pi$
$464$	1.41662e8	0.0658327
$465$	0	0
$466$	−2.50920e9	−1.14864
$467$	−2.57730e9	−1.17100	−0.585498	−	0.810674i	$-0.699101\pi$
$467$	−2.57730e9	−1.17100	−0.585498	+	0.810674i	$0.699101\pi$
$468$	0	0
$469$	−1.99250e9	−0.891855
$470$	−9.79498e8	−0.435172
$471$	0	0
$472$	3.23234e8	0.141488
$473$	1.59542e8	0.0693205
$474$	0	0
$475$	−4.33630e8	−0.185649
$476$	1.38154e9	0.587136
$477$	0	0
$478$	3.67434e9	1.53880
$479$	−2.51029e9	−1.04364	−0.521818	−	0.853057i	$-0.674746\pi$
$479$	−2.51029e9	−1.04364	−0.521818	+	0.853057i	$0.674746\pi$
$480$	0	0
$481$	5.14877e8	0.210958
$482$	1.81180e9	0.736964
$483$	0	0
$484$	−8.21635e8	−0.329398
$485$	2.95937e9	1.17788
$486$	0	0
$487$	−1.03970e9	−0.407905	−0.203952	−	0.978981i	$-0.565379\pi$
$487$	−1.03970e9	−0.407905	−0.203952	+	0.978981i	$0.565379\pi$
$488$	2.53439e9	0.987199
$489$	0	0
$490$	−3.06344e9	−1.17631
$491$	4.29594e9	1.63785	0.818923	−	0.573903i	$-0.194571\pi$
$491$	4.29594e9	1.63785	0.818923	+	0.573903i	$0.194571\pi$
$492$	0	0
$493$	−3.54552e8	−0.133265
$494$	1.06324e9	0.396813
$495$	0	0
$496$	−2.66464e8	−0.0980511
$497$	4.17501e9	1.52549
$498$	0	0
$499$	−2.36884e9	−0.853461	−0.426730	−	0.904379i	$-0.640335\pi$
$499$	−2.36884e9	−0.853461	−0.426730	+	0.904379i	$0.640335\pi$
$500$	−8.64206e8	−0.309188
$501$	0	0
$502$	1.81330e9	0.639743
$503$	−3.86205e9	−1.35310	−0.676551	−	0.736396i	$-0.736526\pi$
$503$	−3.86205e9	−1.35310	−0.676551	+	0.736396i	$0.736526\pi$
$504$	0	0
$505$	4.24247e9	1.46588
$506$	−3.63216e9	−1.24634
$507$	0	0
$508$	−8.54403e8	−0.289161
$509$	2.82144e8	0.0948327	0.0474163	−	0.998875i	$-0.484901\pi$
$509$	2.82144e8	0.0948327	0.0474163	+	0.998875i	$0.484901\pi$
$510$	0	0
$511$	−6.36175e9	−2.10913
$512$	−3.13988e9	−1.03388
$513$	0	0
$514$	−3.08625e9	−1.00244
$515$	−4.38542e9	−1.41477
$516$	0	0
$517$	2.66369e9	0.847748
$518$	−1.72465e9	−0.545189
$519$	0	0
$520$	1.62340e9	0.506308
$521$	−1.57821e9	−0.488915	−0.244457	−	0.969660i	$-0.578610\pi$
$521$	−1.57821e9	−0.488915	−0.244457	+	0.969660i	$0.578610\pi$
$522$	0	0
$523$	−2.13247e9	−0.651818	−0.325909	−	0.945401i	$-0.605670\pi$
$523$	−2.13247e9	−0.651818	−0.325909	+	0.945401i	$0.605670\pi$
$524$	−1.54538e8	−0.0469218
$525$	0	0
$526$	9.51249e8	0.285000
$527$	6.66906e8	0.198485
$528$	0	0
$529$	5.67661e7	0.0166722
$530$	−1.03421e8	−0.0301747
$531$	0	0
$532$	1.43839e9	0.414177
$533$	8.34364e8	0.238677
$534$	0	0
$535$	5.00892e9	1.41418
$536$	−2.15776e9	−0.605237
$537$	0	0
$538$	−4.93719e9	−1.36692
$539$	8.33086e9	2.29155
$540$	0	0
$541$	1.40969e9	0.382766	0.191383	−	0.981515i	$-0.438703\pi$
$541$	1.40969e9	0.382766	0.191383	+	0.981515i	$0.438703\pi$
$542$	8.69155e7	0.0234477
$543$	0	0
$544$	2.65799e9	0.707875
$545$	−3.36319e9	−0.889946
$546$	0	0
$547$	2.03003e9	0.530331	0.265166	−	0.964203i	$-0.414573\pi$
$547$	2.03003e9	0.530331	0.265166	+	0.964203i	$0.414573\pi$
$548$	−7.81482e8	−0.202856
$549$	0	0
$550$	−9.95995e8	−0.255263
$551$	−3.69143e8	−0.0940078
$552$	0	0
$553$	1.39638e9	0.351129
$554$	1.00025e9	0.249934
$555$	0	0
$556$	−1.92287e9	−0.474448
$557$	1.08167e9	0.265216	0.132608	−	0.991169i	$-0.457665\pi$
$557$	1.08167e9	0.265216	0.132608	+	0.991169i	$0.457665\pi$
$558$	0	0
$559$	1.02233e8	0.0247542
$560$	−3.73227e9	−0.898079
$561$	0	0
$562$	2.10876e9	0.501129
$563$	−3.83004e8	−0.0904533	−0.0452266	−	0.998977i	$-0.514401\pi$
$563$	−3.83004e8	−0.0904533	−0.0452266	+	0.998977i	$0.514401\pi$
$564$	0	0
$565$	−1.18827e8	−0.0277169
$566$	4.35646e9	1.00990
$567$	0	0
$568$	4.52127e9	1.03524
$569$	−2.51189e9	−0.571620	−0.285810	−	0.958286i	$-0.592263\pi$
$569$	−2.51189e9	−0.571620	−0.285810	+	0.958286i	$0.592263\pi$
$570$	0	0
$571$	3.94620e9	0.887060	0.443530	−	0.896259i	$-0.353726\pi$
$571$	3.94620e9	0.887060	0.443530	+	0.896259i	$0.353726\pi$
$572$	−9.86318e8	−0.220359
$573$	0	0
$574$	−2.79482e9	−0.616824
$575$	9.49223e8	0.208224
$576$	0	0
$577$	3.82097e9	0.828053	0.414027	−	0.910265i	$-0.364122\pi$
$577$	3.82097e9	0.828053	0.414027	+	0.910265i	$0.364122\pi$
$578$	2.44526e9	0.526718
$579$	0	0
$580$	−1.25922e8	−0.0267980
$581$	9.03014e9	1.91020
$582$	0	0
$583$	2.81248e8	0.0587826
$584$	−6.88938e9	−1.43131
$585$	0	0
$586$	−1.02014e9	−0.209421
$587$	5.26096e9	1.07357	0.536786	−	0.843718i	$-0.319638\pi$
$587$	5.26096e9	1.07357	0.536786	+	0.843718i	$0.319638\pi$
$588$	0	0
$589$	6.94350e8	0.140015
$590$	4.88273e8	0.0978769
$591$	0	0
$592$	−1.28190e9	−0.253938
$593$	−5.47065e9	−1.07733	−0.538664	−	0.842521i	$-0.681071\pi$
$593$	−5.47065e9	−1.07733	−0.538664	+	0.842521i	$0.681071\pi$
$594$	0	0
$595$	9.34111e9	1.81798
$596$	1.23498e9	0.238944
$597$	0	0
$598$	−2.32744e9	−0.445066
$599$	6.15736e9	1.17058	0.585289	−	0.810825i	$-0.300981\pi$
$599$	6.15736e9	1.17058	0.585289	+	0.810825i	$0.300981\pi$
$600$	0	0
$601$	−7.30212e9	−1.37211	−0.686054	−	0.727550i	$-0.740659\pi$
$601$	−7.30212e9	−1.37211	−0.686054	+	0.727550i	$0.740659\pi$
$602$	−3.42442e8	−0.0639734
$603$	0	0
$604$	8.16187e8	0.150716
$605$	−5.55539e9	−1.01993
$606$	0	0
$607$	−6.94268e9	−1.25999	−0.629995	−	0.776599i	$-0.716943\pi$
$607$	−6.94268e9	−1.25999	−0.629995	+	0.776599i	$0.716943\pi$
$608$	2.76737e9	0.499349
$609$	0	0
$610$	3.82842e9	0.682913
$611$	1.70686e9	0.302728
$612$	0	0
$613$	3.42370e9	0.600323	0.300161	−	0.953888i	$-0.402960\pi$
$613$	3.42370e9	0.600323	0.300161	+	0.953888i	$0.402960\pi$
$614$	−4.69722e9	−0.818938
$615$	0	0
$616$	1.47878e10	2.54901
$617$	2.18204e9	0.373995	0.186997	−	0.982360i	$-0.440124\pi$
$617$	2.18204e9	0.373995	0.186997	+	0.982360i	$0.440124\pi$
$618$	0	0
$619$	1.11884e10	1.89605	0.948025	−	0.318197i	$-0.103077\pi$
$619$	1.11884e10	1.89605	0.948025	+	0.318197i	$0.103077\pi$
$620$	2.36856e8	0.0399130
$621$	0	0
$622$	327537.	5.45751e−5 0
$623$	1.40488e10	2.32773
$624$	0	0
$625$	−4.58279e9	−0.750844
$626$	−2.05280e9	−0.334454
$627$	0	0
$628$	2.13860e9	0.344564
$629$	3.20833e9	0.514046
$630$	0	0
$631$	4.03660e9	0.639606	0.319803	−	0.947484i	$-0.396383\pi$
$631$	4.03660e9	0.639606	0.319803	+	0.947484i	$0.396383\pi$
$632$	1.51220e9	0.238286
$633$	0	0
$634$	4.49913e9	0.701158
$635$	−5.77694e9	−0.895344
$636$	0	0
$637$	5.33831e9	0.818305
$638$	−8.47875e8	−0.129259
$639$	0	0
$640$	−2.19481e9	−0.330954
$641$	−2.82614e9	−0.423830	−0.211915	−	0.977288i	$-0.567970\pi$
$641$	−2.82614e9	−0.423830	−0.211915	+	0.977288i	$0.567970\pi$
$642$	0	0
$643$	7.92165e9	1.17511	0.587553	−	0.809186i	$-0.300091\pi$
$643$	7.92165e9	1.17511	0.587553	+	0.809186i	$0.300091\pi$
$644$	−3.14866e9	−0.464542
$645$	0	0
$646$	6.62530e9	0.966922
$647$	6.23017e9	0.904347	0.452173	−	0.891930i	$-0.350649\pi$
$647$	6.23017e9	0.904347	0.452173	+	0.891930i	$0.350649\pi$
$648$	0	0
$649$	−1.32783e9	−0.190672
$650$	−6.38220e8	−0.0911536
$651$	0	0
$652$	2.81643e9	0.397954
$653$	−8.41425e9	−1.18255	−0.591275	−	0.806470i	$-0.701375\pi$
$653$	−8.41425e9	−1.18255	−0.591275	+	0.806470i	$0.701375\pi$
$654$	0	0
$655$	−1.04489e9	−0.145286
$656$	−2.07733e9	−0.287304
$657$	0	0
$658$	−5.71735e9	−0.782356
$659$	−7.96272e9	−1.08383	−0.541917	−	0.840432i	$-0.682301\pi$
$659$	−7.96272e9	−1.08383	−0.541917	+	0.840432i	$0.682301\pi$
$660$	0	0
$661$	1.16395e10	1.56758	0.783790	−	0.621026i	$-0.213284\pi$
$661$	1.16395e10	1.56758	0.783790	+	0.621026i	$0.213284\pi$
$662$	1.15181e10	1.54305
$663$	0	0
$664$	9.77908e9	1.29631
$665$	9.72551e9	1.28244
$666$	0	0
$667$	8.08058e8	0.105439
$668$	2.79894e9	0.363308
$669$	0	0
$670$	−3.25948e9	−0.418684
$671$	−1.04112e10	−1.33037
$672$	0	0
$673$	6.37828e8	0.0806586	0.0403293	−	0.999186i	$-0.487159\pi$
$673$	6.37828e8	0.0806586	0.0403293	+	0.999186i	$0.487159\pi$
$674$	−5.12899e9	−0.645242
$675$	0	0
$676$	1.67863e9	0.208998
$677$	8.92413e9	1.10536	0.552682	−	0.833392i	$-0.313604\pi$
$677$	8.92413e9	1.10536	0.552682	+	0.833392i	$0.313604\pi$
$678$	0	0
$679$	1.72739e10	2.11761
$680$	1.01158e10	1.23373
$681$	0	0
$682$	1.59484e9	0.192518
$683$	−2.24023e9	−0.269042	−0.134521	−	0.990911i	$-0.542950\pi$
$683$	−2.24023e9	−0.269042	−0.134521	+	0.990911i	$0.542950\pi$
$684$	0	0
$685$	−5.28390e9	−0.628113
$686$	−6.45302e9	−0.763183
$687$	0	0
$688$	−2.54530e8	−0.0297975
$689$	1.80220e8	0.0209911
$690$	0	0
$691$	−1.59406e10	−1.83794	−0.918972	−	0.394322i	$-0.870979\pi$
$691$	−1.59406e10	−1.83794	−0.918972	+	0.394322i	$0.870979\pi$
$692$	−3.39210e9	−0.389132
$693$	0	0
$694$	1.02394e10	1.16282
$695$	−1.30013e10	−1.46906
$696$	0	0
$697$	5.19913e9	0.581589
$698$	−6.01765e9	−0.669781
$699$	0	0
$700$	−8.63411e8	−0.0951425
$701$	−1.79817e10	−1.97160	−0.985800	−	0.167926i	$-0.946293\pi$
$701$	−1.79817e10	−1.97160	−0.985800	+	0.167926i	$0.946293\pi$
$702$	0	0
$703$	3.34036e9	0.362618
$704$	1.48921e10	1.60862
$705$	0	0
$706$	1.24827e10	1.33503
$707$	2.47634e10	2.63538
$708$	0	0
$709$	−5.70361e9	−0.601019	−0.300510	−	0.953779i	$-0.597157\pi$
$709$	−5.70361e9	−0.601019	−0.300510	+	0.953779i	$0.597157\pi$
$710$	6.82977e9	0.716147
$711$	0	0
$712$	1.52140e10	1.57966
$713$	−1.51994e9	−0.157041
$714$	0	0
$715$	−6.66887e9	−0.682309
$716$	2.19946e9	0.223935
$717$	0	0
$718$	−1.50140e10	−1.51378
$719$	−1.38876e10	−1.39340	−0.696702	−	0.717360i	$-0.745350\pi$
$719$	−1.38876e10	−1.39340	−0.696702	+	0.717360i	$0.745350\pi$
$720$	0	0
$721$	−2.55978e10	−2.54348
$722$	−1.63730e9	−0.161900
$723$	0	0
$724$	3.67172e9	0.359571
$725$	2.21582e8	0.0215949
$726$	0	0
$727$	−7.90006e8	−0.0762535	−0.0381268	−	0.999273i	$-0.512139\pi$
$727$	−7.90006e8	−0.0762535	−0.0381268	+	0.999273i	$0.512139\pi$
$728$	9.47584e9	0.910244
$729$	0	0
$730$	−1.04070e10	−0.990138
$731$	6.37037e8	0.0603190
$732$	0	0
$733$	5.65590e8	0.0530442	0.0265221	−	0.999648i	$-0.491557\pi$
$733$	5.65590e8	0.0530442	0.0265221	+	0.999648i	$0.491557\pi$
$734$	−1.41843e10	−1.32395
$735$	0	0
$736$	−6.05780e9	−0.560071
$737$	8.86397e9	0.815628
$738$	0	0
$739$	−1.70229e10	−1.55159	−0.775796	−	0.630984i	$-0.782651\pi$
$739$	−1.70229e10	−1.55159	−0.775796	+	0.630984i	$0.782651\pi$
$740$	1.13946e9	0.103369
$741$	0	0
$742$	−6.03670e8	−0.0542483
$743$	−1.64681e7	−0.00147294	−0.000736468	−	1.00000i	$-0.500234\pi$
$743$	−1.64681e7	−0.00147294	−0.000736468		1.00000i	$0.500234\pi$
$744$	0	0
$745$	8.35014e9	0.739855
$746$	−9.47516e9	−0.835604
$747$	0	0
$748$	−6.14600e9	−0.536954
$749$	2.92372e10	2.54243
$750$	0	0
$751$	−9.72464e9	−0.837787	−0.418893	−	0.908035i	$-0.637582\pi$
$751$	−9.72464e9	−0.837787	−0.418893	+	0.908035i	$0.637582\pi$
$752$	−4.24959e9	−0.364405
$753$	0	0
$754$	−5.43307e8	−0.0461578
$755$	5.51856e9	0.466671
$756$	0	0
$757$	3.21906e9	0.269708	0.134854	−	0.990866i	$-0.456944\pi$
$757$	3.21906e9	0.269708	0.134854	+	0.990866i	$0.456944\pi$
$758$	−1.31123e10	−1.09354
$759$	0	0
$760$	1.05321e10	0.870299
$761$	1.87932e10	1.54580	0.772902	−	0.634526i	$-0.218805\pi$
$761$	1.87932e10	1.54580	0.772902	+	0.634526i	$0.218805\pi$
$762$	0	0
$763$	−1.96310e10	−1.59995
$764$	−7.51204e7	−0.00609440
$765$	0	0
$766$	−1.51659e10	−1.21918
$767$	−8.50856e8	−0.0680883
$768$	0	0
$769$	4.53024e9	0.359235	0.179617	−	0.983737i	$-0.442514\pi$
$769$	4.53024e9	0.359235	0.179617	+	0.983737i	$0.442514\pi$
$770$	2.23383e10	1.76333
$771$	0	0
$772$	−4.88846e9	−0.382395
$773$	−2.02996e10	−1.58074	−0.790369	−	0.612631i	$-0.790111\pi$
$773$	−2.02996e10	−1.58074	−0.790369	+	0.612631i	$0.790111\pi$
$774$	0	0
$775$	−4.16792e8	−0.0321635
$776$	1.87066e10	1.43707
$777$	0	0
$778$	1.59156e9	0.121170
$779$	5.41309e9	0.410265
$780$	0	0
$781$	−1.85732e10	−1.39511
$782$	−1.45029e10	−1.08450
$783$	0	0
$784$	−1.32909e10	−0.985024
$785$	1.44599e10	1.06689
$786$	0	0
$787$	1.91261e10	1.39867	0.699333	−	0.714796i	$-0.253480\pi$
$787$	1.91261e10	1.39867	0.699333	+	0.714796i	$0.253480\pi$
$788$	−3.94702e9	−0.287361
$789$	0	0
$790$	2.28430e9	0.164838
$791$	−6.93595e8	−0.0498297
$792$	0	0
$793$	−6.67134e9	−0.475069
$794$	−1.02316e10	−0.725391
$795$	0	0
$796$	3.98511e9	0.280056
$797$	−1.30414e10	−0.912470	−0.456235	−	0.889859i	$-0.650802\pi$
$797$	−1.30414e10	−0.912470	−0.456235	+	0.889859i	$0.650802\pi$
$798$	0	0
$799$	1.06359e10	0.737665
$800$	−1.66114e9	−0.114708
$801$	0	0
$802$	−1.49813e9	−0.102551
$803$	2.83013e10	1.92886
$804$	0	0
$805$	−2.12893e10	−1.43839
$806$	1.02195e9	0.0687475
$807$	0	0
$808$	2.68172e10	1.78844
$809$	−5.94145e9	−0.394523	−0.197262	−	0.980351i	$-0.563205\pi$
$809$	−5.94145e9	−0.394523	−0.197262	+	0.980351i	$0.563205\pi$
$810$	0	0
$811$	5.51763e9	0.363228	0.181614	−	0.983370i	$-0.441868\pi$
$811$	5.51763e9	0.363228	0.181614	+	0.983370i	$0.441868\pi$
$812$	−7.35009e8	−0.0481777
$813$	0	0
$814$	7.67239e9	0.498592
$815$	1.90430e10	1.23221
$816$	0	0
$817$	6.63252e8	0.0425502
$818$	−1.02049e10	−0.651889
$819$	0	0
$820$	1.84651e9	0.116951
$821$	−2.25214e10	−1.42035	−0.710173	−	0.704027i	$-0.751383\pi$
$821$	−2.25214e10	−1.42035	−0.710173	+	0.704027i	$0.751383\pi$
$822$	0	0
$823$	−1.14413e10	−0.715446	−0.357723	−	0.933828i	$-0.616447\pi$
$823$	−1.14413e10	−0.715446	−0.357723	+	0.933828i	$0.616447\pi$
$824$	−2.77208e10	−1.72608
$825$	0	0
$826$	2.85006e9	0.175964
$827$	9.39865e9	0.577825	0.288912	−	0.957356i	$-0.406706\pi$
$827$	9.39865e9	0.577825	0.288912	+	0.957356i	$0.406706\pi$
$828$	0	0
$829$	−1.97222e10	−1.20230	−0.601152	−	0.799135i	$-0.705291\pi$
$829$	−1.97222e10	−1.20230	−0.601152	+	0.799135i	$0.705291\pi$
$830$	1.47721e10	0.896747
$831$	0	0
$832$	9.54268e9	0.574432
$833$	3.32643e10	1.99398
$834$	0	0
$835$	1.89247e10	1.12493
$836$	−6.39892e9	−0.378778
$837$	0	0
$838$	−8.82626e8	−0.0518111
$839$	−1.88016e10	−1.09908	−0.549539	−	0.835468i	$-0.685196\pi$
$839$	−1.88016e10	−1.09908	−0.549539	+	0.835468i	$0.685196\pi$
$840$	0	0
$841$	−1.70612e10	−0.989065
$842$	8.58014e9	0.495339
$843$	0	0
$844$	−6.08383e9	−0.348320
$845$	1.13499e10	0.647130
$846$	0	0
$847$	−3.24269e10	−1.83364
$848$	−4.48695e8	−0.0252677
$849$	0	0
$850$	−3.97691e9	−0.222116
$851$	−7.31209e9	−0.406713
$852$	0	0
$853$	1.36908e10	0.755276	0.377638	−	0.925953i	$-0.376736\pi$
$853$	1.36908e10	0.755276	0.377638	+	0.925953i	$0.376736\pi$
$854$	2.23466e10	1.22775
$855$	0	0
$856$	3.16621e10	1.72537
$857$	1.50548e10	0.817038	0.408519	−	0.912750i	$-0.366045\pi$
$857$	1.50548e10	0.817038	0.408519	+	0.912750i	$0.366045\pi$
$858$	0	0
$859$	−3.48451e10	−1.87571	−0.937856	−	0.347025i	$-0.887192\pi$
$859$	−3.48451e10	−1.87571	−0.937856	+	0.347025i	$0.887192\pi$
$860$	2.26248e8	0.0121294
$861$	0	0
$862$	7.29546e9	0.387951
$863$	−2.13694e10	−1.13176	−0.565879	−	0.824488i	$-0.691463\pi$
$863$	−2.13694e10	−1.13176	−0.565879	+	0.824488i	$0.691463\pi$
$864$	0	0
$865$	−2.29353e10	−1.20489
$866$	1.82339e9	0.0954041
$867$	0	0
$868$	1.38254e9	0.0717559
$869$	−6.21203e9	−0.321118
$870$	0	0
$871$	5.67992e9	0.291258
$872$	−2.12592e10	−1.08577
$873$	0	0
$874$	−1.50997e10	−0.765029
$875$	−3.41070e10	−1.72114
$876$	0	0
$877$	6.17620e9	0.309188	0.154594	−	0.987978i	$-0.450593\pi$
$877$	6.17620e9	0.309188	0.154594	+	0.987978i	$0.450593\pi$
$878$	2.53036e10	1.26169
$879$	0	0
$880$	1.66036e10	0.821320
$881$	−2.32664e10	−1.14634	−0.573170	−	0.819437i	$-0.694286\pi$
$881$	−2.32664e10	−1.14634	−0.573170	+	0.819437i	$0.694286\pi$
$882$	0	0
$883$	3.59095e10	1.75528	0.877640	−	0.479320i	$-0.159117\pi$
$883$	3.59095e10	1.75528	0.877640	+	0.479320i	$0.159117\pi$
$884$	−3.93827e9	−0.191744
$885$	0	0
$886$	−3.18106e10	−1.53657
$887$	4.49012e9	0.216035	0.108018	−	0.994149i	$-0.465550\pi$
$887$	4.49012e9	0.216035	0.108018	+	0.994149i	$0.465550\pi$
$888$	0	0
$889$	−3.37201e10	−1.60966
$890$	2.29821e10	1.09276
$891$	0	0
$892$	−4.79045e8	−0.0225995
$893$	1.10736e10	0.520363
$894$	0	0
$895$	1.48714e10	0.693380
$896$	−1.28112e10	−0.594991
$897$	0	0
$898$	9.76000e8	0.0449762
$899$	−3.54808e8	−0.0162868
$900$	0	0
$901$	1.12299e9	0.0511494
$902$	1.24332e10	0.564105
$903$	0	0
$904$	−7.51120e8	−0.0338158
$905$	2.48259e10	1.11336
$906$	0	0
$907$	4.38937e10	1.95333	0.976667	−	0.214758i	$-0.0688963\pi$
$907$	4.38937e10	1.95333	0.976667	+	0.214758i	$0.0688963\pi$
$908$	1.10661e10	0.490561
$909$	0	0
$910$	1.43141e10	0.629678
$911$	−1.57039e10	−0.688165	−0.344082	−	0.938939i	$-0.611810\pi$
$911$	−1.57039e10	−0.688165	−0.344082	+	0.938939i	$0.611810\pi$
$912$	0	0
$913$	−4.01720e10	−1.74693
$914$	−2.18592e10	−0.946939
$915$	0	0
$916$	−1.66167e9	−0.0714350
$917$	−6.09903e9	−0.261197
$918$	0	0
$919$	−2.38584e10	−1.01400	−0.506998	−	0.861947i	$-0.669245\pi$
$919$	−2.38584e10	−1.01400	−0.506998	+	0.861947i	$0.669245\pi$
$920$	−2.30550e10	−0.976129
$921$	0	0
$922$	−6.10511e9	−0.256528
$923$	−1.19015e10	−0.498189
$924$	0	0
$925$	−2.00509e9	−0.0832986
$926$	−1.30309e10	−0.539305
$927$	0	0
$928$	−1.41411e9	−0.0580850
$929$	−6.67984e9	−0.273345	−0.136672	−	0.990616i	$-0.543641\pi$
$929$	−6.67984e9	−0.273345	−0.136672	+	0.990616i	$0.543641\pi$
$930$	0	0
$931$	3.46332e10	1.40659
$932$	9.67666e9	0.391534
$933$	0	0
$934$	−2.46096e10	−0.988304
$935$	−4.15554e10	−1.66260
$936$	0	0
$937$	2.07249e10	0.823009	0.411504	−	0.911408i	$-0.365003\pi$
$937$	2.07249e10	0.823009	0.411504	+	0.911408i	$0.365003\pi$
$938$	−1.90256e10	−0.752713
$939$	0	0
$940$	3.77741e9	0.148336
$941$	−1.64161e10	−0.642255	−0.321127	−	0.947036i	$-0.604062\pi$
$941$	−1.64161e10	−0.642255	−0.321127	+	0.947036i	$0.604062\pi$
$942$	0	0
$943$	−1.18493e10	−0.460153
$944$	2.11839e9	0.0819603
$945$	0	0
$946$	1.52341e9	0.0585056
$947$	−3.58035e10	−1.36994	−0.684968	−	0.728573i	$-0.740184\pi$
$947$	−3.58035e10	−1.36994	−0.684968	+	0.728573i	$0.740184\pi$
$948$	0	0
$949$	1.81351e10	0.688791
$950$	−4.14057e9	−0.156685
$951$	0	0
$952$	5.90464e10	2.21801
$953$	1.53516e10	0.574550	0.287275	−	0.957848i	$-0.407251\pi$
$953$	1.53516e10	0.574550	0.287275	+	0.957848i	$0.407251\pi$
$954$	0	0
$955$	−5.07917e8	−0.0188704
$956$	−1.41700e10	−0.524526
$957$	0	0
$958$	−2.39698e10	−0.880814
$959$	−3.08422e10	−1.12923
$960$	0	0
$961$	−2.68452e10	−0.975742
$962$	4.91637e9	0.178046
$963$	0	0
$964$	−6.98716e9	−0.251207
$965$	−3.30528e10	−1.18403
$966$	0	0
$967$	−3.79457e10	−1.34949	−0.674745	−	0.738051i	$-0.735747\pi$
$967$	−3.79457e10	−1.34949	−0.674745	+	0.738051i	$0.735747\pi$
$968$	−3.51164e10	−1.24436
$969$	0	0
$970$	2.82579e10	0.994119
$971$	1.75229e10	0.614242	0.307121	−	0.951670i	$-0.400634\pi$
$971$	1.75229e10	0.614242	0.307121	+	0.951670i	$0.400634\pi$
$972$	0	0
$973$	−7.58888e10	−2.64109
$974$	−9.92774e9	−0.344266
$975$	0	0
$976$	1.66097e10	0.571858
$977$	2.92529e10	1.00355	0.501773	−	0.864999i	$-0.332681\pi$
$977$	2.92529e10	1.00355	0.501773	+	0.864999i	$0.332681\pi$
$978$	0	0
$979$	−6.24985e10	−2.12878
$980$	1.18141e10	0.400967
$981$	0	0
$982$	4.10203e10	1.38232
$983$	1.43406e10	0.481536	0.240768	−	0.970583i	$-0.422601\pi$
$983$	1.43406e10	0.481536	0.240768	+	0.970583i	$0.422601\pi$
$984$	0	0
$985$	−2.66873e10	−0.889772
$986$	−3.38548e9	−0.112474
$987$	0	0
$988$	−4.10034e9	−0.135260
$989$	−1.45187e9	−0.0477244
$990$	0	0
$991$	−3.77304e10	−1.23150	−0.615749	−	0.787942i	$-0.711146\pi$
$991$	−3.77304e10	−1.23150	−0.615749	+	0.787942i	$0.711146\pi$
$992$	2.65991e9	0.0865118
$993$	0	0
$994$	3.98655e10	1.28749
$995$	2.69449e10	0.867152
$996$	0	0
$997$	4.59214e10	1.46751	0.733757	−	0.679412i	$-0.237765\pi$
$997$	4.59214e10	1.46751	0.733757	+	0.679412i	$0.237765\pi$
$998$	−2.26191e10	−0.720309
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.13		18
3.2	odd	2		177.8.a.d.1.6	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.6	✓	18	3.2	odd	2
531.8.a.e.1.13		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+9.54862 q^{2} -36.8240 q^{4} -248.981 q^{5} -1453.31 q^{7} -1573.84 q^{8} +O(q^{10})\)	\(q+9.54862 q^{2} -36.8240 q^{4} -248.981 q^{5} -1453.31 q^{7} -1573.84 q^{8} -2377.42 q^{10} +6465.27 q^{11} +4142.86 q^{13} -13877.1 q^{14} -10314.5 q^{16} +25815.2 q^{17} +26877.5 q^{19} +9168.46 q^{20} +61734.4 q^{22} -58835.3 q^{23} -16133.6 q^{25} +39558.6 q^{26} +53516.5 q^{28} -13734.2 q^{29} +25833.8 q^{31} +102962. q^{32} +246499. q^{34} +361845. q^{35} +124281. q^{37} +256643. q^{38} +391856. q^{40} +201398. q^{41} +24676.8 q^{43} -238077. q^{44} -561796. q^{46} +412000. q^{47} +1.28856e6 q^{49} -154053. q^{50} -152556. q^{52} +43501.3 q^{53} -1.60973e6 q^{55} +2.28727e6 q^{56} -131143. q^{58} -205379. q^{59} -1.61032e6 q^{61} +246677. q^{62} +2.30341e6 q^{64} -1.03149e6 q^{65} +1.37101e6 q^{67} -950617. q^{68} +3.45512e6 q^{70} -2.87276e6 q^{71} +4.37743e6 q^{73} +1.18671e6 q^{74} -989737. q^{76} -9.39602e6 q^{77} -960831. q^{79} +2.56812e6 q^{80} +1.92307e6 q^{82} -6.21351e6 q^{83} -6.42749e6 q^{85} +235630. q^{86} -1.01753e7 q^{88} -9.66681e6 q^{89} -6.02084e6 q^{91} +2.16655e6 q^{92} +3.93403e6 q^{94} -6.69199e6 q^{95} -1.18859e7 q^{97} +1.23039e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * q^98

Introduction

L-functions

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