LMFDB - Embedded newform 245.4.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.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:	$14.4554679514$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 $ x^5 - x^4 - 37x^3 + 21x^2 + 288*x + 64
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.227497$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.227497 q^{2} +1.80858 q^{3} -7.94824 q^{4} +5.00000 q^{5} -0.411448 q^{6} +3.62818 q^{8} -23.7290 q^{9} +O(q^{10})$	\(q-0.227497 q^{2} +1.80858 q^{3} -7.94824 q^{4} +5.00000 q^{5} -0.411448 q^{6} +3.62818 q^{8} -23.7290 q^{9} -1.13749 q^{10} +17.7019 q^{11} -14.3751 q^{12} +62.3178 q^{13} +9.04291 q^{15} +62.7606 q^{16} -87.1623 q^{17} +5.39829 q^{18} +101.724 q^{19} -39.7412 q^{20} -4.02713 q^{22} +93.6396 q^{23} +6.56187 q^{24} +25.0000 q^{25} -14.1771 q^{26} -91.7477 q^{27} +297.385 q^{29} -2.05724 q^{30} -91.2779 q^{31} -43.3033 q^{32} +32.0153 q^{33} +19.8292 q^{34} +188.604 q^{36} +281.210 q^{37} -23.1418 q^{38} +112.707 q^{39} +18.1409 q^{40} +271.754 q^{41} -7.81066 q^{43} -140.699 q^{44} -118.645 q^{45} -21.3028 q^{46} +92.4524 q^{47} +113.508 q^{48} -5.68743 q^{50} -157.640 q^{51} -495.317 q^{52} +180.304 q^{53} +20.8723 q^{54} +88.5094 q^{55} +183.975 q^{57} -67.6544 q^{58} +99.7011 q^{59} -71.8753 q^{60} -434.218 q^{61} +20.7655 q^{62} -492.233 q^{64} +311.589 q^{65} -7.28340 q^{66} -461.554 q^{67} +692.787 q^{68} +169.355 q^{69} +518.417 q^{71} -86.0932 q^{72} +542.866 q^{73} -63.9744 q^{74} +45.2146 q^{75} -808.524 q^{76} -25.6405 q^{78} +239.701 q^{79} +313.803 q^{80} +474.751 q^{81} -61.8232 q^{82} -299.184 q^{83} -435.812 q^{85} +1.77690 q^{86} +537.846 q^{87} +64.2257 q^{88} -1054.36 q^{89} +26.9914 q^{90} -744.271 q^{92} -165.084 q^{93} -21.0327 q^{94} +508.618 q^{95} -78.3176 q^{96} -288.854 q^{97} -420.049 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 8 q^{3} + 35 q^{4} + 25 q^{5} - 16 q^{6} + 33 q^{8} + 81 q^{9}+O(q^{10}) $ 5 * q + q^2 + 8 * q^3 + 35 * q^4 + 25 * q^5 - 16 * q^6 + 33 * q^8 + 81 * q^9	\( 5 q + q^{2} + 8 q^{3} + 35 q^{4} + 25 q^{5} - 16 q^{6} + 33 q^{8} + 81 q^{9} + 5 q^{10} + 47 q^{11} + 98 q^{12} - q^{13} + 40 q^{15} + 171 q^{16} + 2 q^{17} - 51 q^{18} + 21 q^{19} + 175 q^{20} + 523 q^{22} + 201 q^{23} - 848 q^{24} + 125 q^{25} + 47 q^{26} + 518 q^{27} + 190 q^{29} - 80 q^{30} - 388 q^{31} - 95 q^{32} + 262 q^{33} - 130 q^{34} + 1229 q^{36} - 145 q^{37} - 835 q^{38} + 14 q^{39} + 165 q^{40} + 281 q^{41} + 568 q^{43} + 1091 q^{44} + 405 q^{45} + 337 q^{46} + 473 q^{47} - 70 q^{48} + 25 q^{50} + 732 q^{51} + 379 q^{52} + 351 q^{53} + 774 q^{54} + 235 q^{55} + 954 q^{57} + 1818 q^{58} - 708 q^{59} + 490 q^{60} - 1944 q^{61} - 448 q^{62} - 125 q^{64} - 5 q^{65} - 1482 q^{66} + 1118 q^{67} + 3118 q^{68} - 374 q^{69} + 864 q^{71} - 2219 q^{72} + 1652 q^{73} - 3285 q^{74} + 200 q^{75} - 691 q^{76} - 5574 q^{78} + 218 q^{79} + 855 q^{80} - 455 q^{81} - 1027 q^{82} + 1502 q^{83} + 10 q^{85} - 4264 q^{86} - 390 q^{87} + 2131 q^{88} - 2322 q^{89} - 255 q^{90} - 2957 q^{92} - 2288 q^{93} + 2677 q^{94} + 105 q^{95} - 4592 q^{96} - 598 q^{97} + 35 q^{99}+O(q^{100}) \) 5 * q + q^2 + 8 * q^3 + 35 * q^4 + 25 * q^5 - 16 * q^6 + 33 * q^8 + 81 * q^9 + 5 * q^10 + 47 * q^11 + 98 * q^12 - q^13 + 40 * q^15 + 171 * q^16 + 2 * q^17 - 51 * q^18 + 21 * q^19 + 175 * q^20 + 523 * q^22 + 201 * q^23 - 848 * q^24 + 125 * q^25 + 47 * q^26 + 518 * q^27 + 190 * q^29 - 80 * q^30 - 388 * q^31 - 95 * q^32 + 262 * q^33 - 130 * q^34 + 1229 * q^36 - 145 * q^37 - 835 * q^38 + 14 * q^39 + 165 * q^40 + 281 * q^41 + 568 * q^43 + 1091 * q^44 + 405 * q^45 + 337 * q^46 + 473 * q^47 - 70 * q^48 + 25 * q^50 + 732 * q^51 + 379 * q^52 + 351 * q^53 + 774 * q^54 + 235 * q^55 + 954 * q^57 + 1818 * q^58 - 708 * q^59 + 490 * q^60 - 1944 * q^61 - 448 * q^62 - 125 * q^64 - 5 * q^65 - 1482 * q^66 + 1118 * q^67 + 3118 * q^68 - 374 * q^69 + 864 * q^71 - 2219 * q^72 + 1652 * q^73 - 3285 * q^74 + 200 * q^75 - 691 * q^76 - 5574 * q^78 + 218 * q^79 + 855 * q^80 - 455 * q^81 - 1027 * q^82 + 1502 * q^83 + 10 * q^85 - 4264 * q^86 - 390 * q^87 + 2131 * q^88 - 2322 * q^89 - 255 * q^90 - 2957 * q^92 - 2288 * q^93 + 2677 * q^94 + 105 * q^95 - 4592 * q^96 - 598 * q^97 + 35 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.227497	−0.0804324	−0.0402162	−	0.999191i	$-0.512805\pi$
$2$	−0.227497	−0.0804324	−0.0402162	+	0.999191i	$0.512805\pi$
$3$	1.80858	0.348062	0.174031	−	0.984740i	$-0.444321\pi$
$3$	1.80858	0.348062	0.174031	+	0.984740i	$0.444321\pi$
$4$	−7.94824	−0.993531
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−0.411448	−0.0279955
$7$	0	0
$7$	0	0
$8$	3.62818	0.160345
$9$	−23.7290	−0.878853
$10$	−1.13749	−0.0359705
$11$	17.7019	0.485211	0.242605	−	0.970125i	$-0.421998\pi$
$11$	17.7019	0.485211	0.242605	+	0.970125i	$0.421998\pi$
$12$	−14.3751	−0.345810
$13$	62.3178	1.32953	0.664763	−	0.747054i	$-0.268533\pi$
$13$	62.3178	1.32953	0.664763	+	0.747054i	$0.268533\pi$
$14$	0	0
$15$	9.04291	0.155658
$16$	62.7606	0.980634
$17$	−87.1623	−1.24353	−0.621764	−	0.783205i	$-0.713584\pi$
$17$	−87.1623	−1.24353	−0.621764	+	0.783205i	$0.713584\pi$
$18$	5.39829	0.0706883
$19$	101.724	1.22826	0.614131	−	0.789204i	$-0.289507\pi$
$19$	101.724	1.22826	0.614131	+	0.789204i	$0.289507\pi$
$20$	−39.7412	−0.444320
$21$	0	0
$22$	−4.02713	−0.0390267
$23$	93.6396	0.848922	0.424461	−	0.905446i	$-0.360464\pi$
$23$	93.6396	0.848922	0.424461	+	0.905446i	$0.360464\pi$
$24$	6.56187	0.0558098
$25$	25.0000	0.200000
$26$	−14.1771	−0.106937
$27$	−91.7477	−0.653957
$28$	0	0
$29$	297.385	1.90424	0.952122	−	0.305718i	$-0.0988963\pi$
$29$	297.385	1.90424	0.952122	+	0.305718i	$0.0988963\pi$
$30$	−2.05724	−0.0125200
$31$	−91.2779	−0.528838	−0.264419	−	0.964408i	$-0.585180\pi$
$31$	−91.2779	−0.528838	−0.264419	+	0.964408i	$0.585180\pi$
$32$	−43.3033	−0.239219
$33$	32.0153	0.168883
$34$	19.8292	0.100020
$35$	0	0
$36$	188.604	0.873167
$37$	281.210	1.24948	0.624738	−	0.780835i	$-0.285206\pi$
$37$	281.210	1.24948	0.624738	+	0.780835i	$0.285206\pi$
$38$	−23.1418	−0.0987921
$39$	112.707	0.462757
$40$	18.1409	0.0717082
$41$	271.754	1.03514	0.517570	−	0.855641i	$-0.326837\pi$
$41$	271.754	1.03514	0.517570	+	0.855641i	$0.326837\pi$
$42$	0	0
$43$	−7.81066	−0.0277003	−0.0138502	−	0.999904i	$-0.504409\pi$
$43$	−7.81066	−0.0277003	−0.0138502	+	0.999904i	$0.504409\pi$
$44$	−140.699	−0.482072
$45$	−118.645	−0.393035
$46$	−21.3028	−0.0682809
$47$	92.4524	0.286927	0.143464	−	0.989656i	$-0.454176\pi$
$47$	92.4524	0.286927	0.143464	+	0.989656i	$0.454176\pi$
$48$	113.508	0.341321
$49$	0	0
$50$	−5.68743	−0.0160865
$51$	−157.640	−0.432825
$52$	−495.317	−1.32092
$53$	180.304	0.467295	0.233648	−	0.972321i	$-0.424934\pi$
$53$	180.304	0.467295	0.233648	+	0.972321i	$0.424934\pi$
$54$	20.8723	0.0525994
$55$	88.5094	0.216993
$56$	0	0
$57$	183.975	0.427511
$58$	−67.6544	−0.153163
$59$	99.7011	0.220000	0.110000	−	0.993932i	$-0.464915\pi$
$59$	99.7011	0.220000	0.110000	+	0.993932i	$0.464915\pi$
$60$	−71.8753	−0.154651
$61$	−434.218	−0.911408	−0.455704	−	0.890131i	$-0.650612\pi$
$61$	−434.218	−0.911408	−0.455704	+	0.890131i	$0.650612\pi$
$62$	20.7655	0.0425358
$63$	0	0
$64$	−492.233	−0.961393
$65$	311.589	0.594582
$66$	−7.28340	−0.0135837
$67$	−461.554	−0.841609	−0.420805	−	0.907151i	$-0.638252\pi$
$67$	−461.554	−0.841609	−0.420805	+	0.907151i	$0.638252\pi$
$68$	692.787	1.23548
$69$	169.355	0.295478
$70$	0	0
$71$	518.417	0.866546	0.433273	−	0.901263i	$-0.357359\pi$
$71$	518.417	0.866546	0.433273	+	0.901263i	$0.357359\pi$
$72$	−86.0932	−0.140919
$73$	542.866	0.870380	0.435190	−	0.900339i	$-0.356681\pi$
$73$	542.866	0.870380	0.435190	+	0.900339i	$0.356681\pi$
$74$	−63.9744	−0.100498
$75$	45.2146	0.0696124
$76$	−808.524	−1.22032
$77$	0	0
$78$	−25.6405	−0.0372207
$79$	239.701	0.341373	0.170687	−	0.985325i	$-0.445401\pi$
$79$	239.701	0.341373	0.170687	+	0.985325i	$0.445401\pi$
$80$	313.803	0.438553
$81$	474.751	0.651235
$82$	−61.8232	−0.0832589
$83$	−299.184	−0.395660	−0.197830	−	0.980236i	$-0.563389\pi$
$83$	−299.184	−0.395660	−0.197830	+	0.980236i	$0.563389\pi$
$84$	0	0
$85$	−435.812	−0.556122
$86$	1.77690	0.00222801
$87$	537.846	0.662795
$88$	64.2257	0.0778009
$89$	−1054.36	−1.25575	−0.627876	−	0.778314i	$-0.716075\pi$
$89$	−1054.36	−1.25575	−0.627876	+	0.778314i	$0.716075\pi$
$90$	26.9914	0.0316128
$91$	0	0
$92$	−744.271	−0.843430
$93$	−165.084	−0.184069
$94$	−21.0327	−0.0230782
$95$	508.618	0.549296
$96$	−78.3176	−0.0832631
$97$	−288.854	−0.302358	−0.151179	−	0.988506i	$-0.548307\pi$
$97$	−288.854	−0.302358	−0.151179	+	0.988506i	$0.548307\pi$
$98$	0	0
$99$	−420.049	−0.426429
$100$	−198.706	−0.198706
$101$	−354.839	−0.349582	−0.174791	−	0.984606i	$-0.555925\pi$
$101$	−354.839	−0.349582	−0.174791	+	0.984606i	$0.555925\pi$
$102$	35.8627	0.0348131
$103$	1064.15	1.01799	0.508997	−	0.860768i	$-0.330016\pi$
$103$	1064.15	1.01799	0.508997	+	0.860768i	$0.330016\pi$
$104$	226.100	0.213182
$105$	0	0
$106$	−41.0186	−0.0375857
$107$	−1393.71	−1.25920	−0.629601	−	0.776918i	$-0.716782\pi$
$107$	−1393.71	−1.25920	−0.629601	+	0.776918i	$0.716782\pi$
$108$	729.233	0.649727
$109$	−144.484	−0.126964	−0.0634818	−	0.997983i	$-0.520220\pi$
$109$	−144.484	−0.126964	−0.0634818	+	0.997983i	$0.520220\pi$
$110$	−20.1357	−0.0174533
$111$	508.591	0.434895
$112$	0	0
$113$	−2023.91	−1.68489	−0.842447	−	0.538780i	$-0.818886\pi$
$113$	−2023.91	−1.68489	−0.842447	+	0.538780i	$0.818886\pi$
$114$	−41.8539	−0.0343858
$115$	468.198	0.379650
$116$	−2363.69	−1.89193
$117$	−1478.74	−1.16846
$118$	−22.6817	−0.0176951
$119$	0	0
$120$	32.8093	0.0249589
$121$	−1017.64	−0.764570
$122$	98.7834	0.0733068
$123$	491.489	0.360293
$124$	725.499	0.525417
$125$	125.000	0.0894427
$126$	0	0
$127$	1606.56	1.12252	0.561258	−	0.827641i	$-0.310318\pi$
$127$	1606.56	1.12252	0.561258	+	0.827641i	$0.310318\pi$
$128$	458.408	0.316546
$129$	−14.1262	−0.00964143
$130$	−70.8856	−0.0478237
$131$	−2848.32	−1.89969	−0.949843	−	0.312726i	$-0.898758\pi$
$131$	−2848.32	−1.89969	−0.949843	+	0.312726i	$0.898758\pi$
$132$	−254.466	−0.167791
$133$	0	0
$134$	105.002	0.0676927
$135$	−458.738	−0.292459
$136$	−316.241	−0.199393
$137$	841.717	0.524910	0.262455	−	0.964944i	$-0.415468\pi$
$137$	841.717	0.524910	0.262455	+	0.964944i	$0.415468\pi$
$138$	−38.5278	−0.0237660
$139$	−2429.10	−1.48226	−0.741128	−	0.671363i	$-0.765709\pi$
$139$	−2429.10	−1.48226	−0.741128	+	0.671363i	$0.765709\pi$
$140$	0	0
$141$	167.208	0.0998684
$142$	−117.938	−0.0696984
$143$	1103.14	0.645100
$144$	−1489.25	−0.861833
$145$	1486.93	0.851604
$146$	−123.501	−0.0700067
$147$	0	0
$148$	−2235.12	−1.24139
$149$	−577.947	−0.317767	−0.158884	−	0.987297i	$-0.550789\pi$
$149$	−577.947	−0.317767	−0.158884	+	0.987297i	$0.550789\pi$
$150$	−10.2862	−0.00559909
$151$	783.616	0.422316	0.211158	−	0.977452i	$-0.432276\pi$
$151$	783.616	0.422316	0.211158	+	0.977452i	$0.432276\pi$
$152$	369.072	0.196945
$153$	2068.28	1.09288
$154$	0	0
$155$	−456.389	−0.236504
$156$	−895.822	−0.459764
$157$	1372.40	0.697642	0.348821	−	0.937189i	$-0.386582\pi$
$157$	1372.40	0.697642	0.348821	+	0.937189i	$0.386582\pi$
$158$	−54.5314	−0.0274575
$159$	326.095	0.162648
$160$	−216.517	−0.106982
$161$	0	0
$162$	−108.004	−0.0523804
$163$	3754.78	1.80428	0.902139	−	0.431445i	$-0.141996\pi$
$163$	3754.78	1.80428	0.902139	+	0.431445i	$0.141996\pi$
$164$	−2159.96	−1.02844
$165$	160.077	0.0755270
$166$	68.0636	0.0318239
$167$	1945.55	0.901505	0.450752	−	0.892649i	$-0.351156\pi$
$167$	1945.55	0.901505	0.450752	+	0.892649i	$0.351156\pi$
$168$	0	0
$169$	1686.50	0.767639
$170$	99.1459	0.0447303
$171$	−2413.80	−1.07946
$172$	62.0810	0.0275211
$173$	2067.34	0.908538	0.454269	−	0.890864i	$-0.349900\pi$
$173$	2067.34	0.908538	0.454269	+	0.890864i	$0.349900\pi$
$174$	−122.359	−0.0533102
$175$	0	0
$176$	1110.98	0.475814
$177$	180.318	0.0765735
$178$	239.864	0.101003
$179$	2395.92	1.00044	0.500222	−	0.865897i	$-0.333252\pi$
$179$	2395.92	1.00044	0.500222	+	0.865897i	$0.333252\pi$
$180$	943.021	0.390492
$181$	−1319.40	−0.541826	−0.270913	−	0.962604i	$-0.587326\pi$
$181$	−1319.40	−0.541826	−0.270913	+	0.962604i	$0.587326\pi$
$182$	0	0
$183$	−785.319	−0.317227
$184$	339.742	0.136120
$185$	1406.05	0.558782
$186$	37.5561	0.0148051
$187$	−1542.94	−0.603373
$188$	−734.834	−0.285071
$189$	0	0
$190$	−115.709	−0.0441812
$191$	2108.07	0.798611	0.399306	−	0.916818i	$-0.369251\pi$
$191$	2108.07	0.798611	0.399306	+	0.916818i	$0.369251\pi$
$192$	−890.244	−0.334624
$193$	3631.62	1.35445	0.677227	−	0.735774i	$-0.263181\pi$
$193$	3631.62	1.35445	0.677227	+	0.735774i	$0.263181\pi$
$194$	65.7135	0.0243194
$195$	563.534	0.206951
$196$	0	0
$197$	3280.46	1.18641	0.593206	−	0.805051i	$-0.297862\pi$
$197$	3280.46	1.18641	0.593206	+	0.805051i	$0.297862\pi$
$198$	95.5599	0.0342987
$199$	648.733	0.231093	0.115546	−	0.993302i	$-0.463138\pi$
$199$	648.733	0.231093	0.115546	+	0.993302i	$0.463138\pi$
$200$	90.7045	0.0320689
$201$	−834.759	−0.292932
$202$	80.7249	0.0281177
$203$	0	0
$204$	1252.96	0.430024
$205$	1358.77	0.462929
$206$	−242.090	−0.0818798
$207$	−2221.98	−0.746078
$208$	3911.10	1.30378
$209$	1800.70	0.595966
$210$	0	0
$211$	5072.33	1.65495	0.827474	−	0.561504i	$-0.189777\pi$
$211$	5072.33	1.65495	0.827474	+	0.561504i	$0.189777\pi$
$212$	−1433.10	−0.464272
$213$	937.600	0.301612
$214$	317.064	0.101281
$215$	−39.0533	−0.0123880
$216$	−332.877	−0.104858
$217$	0	0
$218$	32.8697	0.0102120
$219$	981.819	0.302946
$220$	−703.495	−0.215589
$221$	−5431.76	−1.65330
$222$	−115.703	−0.0349796
$223$	163.717	0.0491629	0.0245814	−	0.999698i	$-0.492175\pi$
$223$	163.717	0.0491629	0.0245814	+	0.999698i	$0.492175\pi$
$224$	0	0
$225$	−593.226	−0.175771
$226$	460.433	0.135520
$227$	−6326.62	−1.84984	−0.924918	−	0.380166i	$-0.875867\pi$
$227$	−6326.62	−1.84984	−0.924918	+	0.380166i	$0.875867\pi$
$228$	−1462.28	−0.424746
$229$	−3484.47	−1.00550	−0.502752	−	0.864431i	$-0.667679\pi$
$229$	−3484.47	−1.00550	−0.502752	+	0.864431i	$0.667679\pi$
$230$	−106.514	−0.0305361
$231$	0	0
$232$	1078.97	0.305335
$233$	−4496.13	−1.26417	−0.632084	−	0.774900i	$-0.717800\pi$
$233$	−4496.13	−1.26417	−0.632084	+	0.774900i	$0.717800\pi$
$234$	336.409	0.0939819
$235$	462.262	0.128318
$236$	−792.449	−0.218576
$237$	433.519	0.118819
$238$	0	0
$239$	−1529.73	−0.414016	−0.207008	−	0.978339i	$-0.566373\pi$
$239$	−1529.73	−0.414016	−0.207008	+	0.978339i	$0.566373\pi$
$240$	567.538	0.152644
$241$	−4572.40	−1.22213	−0.611067	−	0.791579i	$-0.709259\pi$
$241$	−4572.40	−1.22213	−0.611067	+	0.791579i	$0.709259\pi$
$242$	231.511	0.0614963
$243$	3335.81	0.880627
$244$	3451.27	0.905512
$245$	0	0
$246$	−111.812	−0.0289793
$247$	6339.18	1.63301
$248$	−331.173	−0.0847963
$249$	−541.100	−0.137714
$250$	−28.4372	−0.00719409
$251$	537.143	0.135076	0.0675382	−	0.997717i	$-0.478486\pi$
$251$	537.143	0.135076	0.0675382	+	0.997717i	$0.478486\pi$
$252$	0	0
$253$	1657.60	0.411906
$254$	−365.489	−0.0902866
$255$	−788.201	−0.193565
$256$	3833.58	0.935932
$257$	878.553	0.213240	0.106620	−	0.994300i	$-0.465997\pi$
$257$	878.553	0.213240	0.106620	+	0.994300i	$0.465997\pi$
$258$	3.21368	0.000775484 0
$259$	0	0
$260$	−2476.58	−0.590736
$261$	−7056.67	−1.67355
$262$	647.985	0.152796
$263$	3714.03	0.870786	0.435393	−	0.900241i	$-0.356609\pi$
$263$	3714.03	0.870786	0.435393	+	0.900241i	$0.356609\pi$
$264$	116.157	0.0270795
$265$	901.519	0.208981
$266$	0	0
$267$	−1906.90	−0.437079
$268$	3668.55	0.836165
$269$	154.460	0.0350096	0.0175048	−	0.999847i	$-0.494428\pi$
$269$	154.460	0.0350096	0.0175048	+	0.999847i	$0.494428\pi$
$270$	104.362	0.0235232
$271$	−2406.30	−0.539382	−0.269691	−	0.962947i	$-0.586921\pi$
$271$	−2406.30	−0.539382	−0.269691	+	0.962947i	$0.586921\pi$
$272$	−5470.36	−1.21944
$273$	0	0
$274$	−191.488	−0.0422198
$275$	442.547	0.0970422
$276$	−1346.08	−0.293566
$277$	1449.76	0.314467	0.157234	−	0.987561i	$-0.449742\pi$
$277$	1449.76	0.314467	0.157234	+	0.987561i	$0.449742\pi$
$278$	552.614	0.119222
$279$	2165.94	0.464771
$280$	0	0
$281$	2037.07	0.432460	0.216230	−	0.976342i	$-0.430624\pi$
$281$	2037.07	0.432460	0.216230	+	0.976342i	$0.430624\pi$
$282$	−38.0393	−0.00803266
$283$	7366.22	1.54727	0.773633	−	0.633634i	$-0.218437\pi$
$283$	7366.22	1.54727	0.773633	+	0.633634i	$0.218437\pi$
$284$	−4120.51	−0.860940
$285$	919.877	0.191189
$286$	−250.962	−0.0518870
$287$	0	0
$288$	1027.55	0.210239
$289$	2684.27	0.546361
$290$	−338.272	−0.0684966
$291$	−522.417	−0.105239
$292$	−4314.84	−0.864749
$293$	542.935	0.108255	0.0541273	−	0.998534i	$-0.482762\pi$
$293$	542.935	0.108255	0.0541273	+	0.998534i	$0.482762\pi$
$294$	0	0
$295$	498.506	0.0983868
$296$	1020.28	0.200346
$297$	−1624.11	−0.317307
$298$	131.481	0.0255588
$299$	5835.41	1.12866
$300$	−359.377	−0.0691620
$301$	0	0
$302$	−178.271	−0.0339679
$303$	−641.756	−0.121676
$304$	6384.23	1.20448
$305$	−2171.09	−0.407594
$306$	−470.527	−0.0879028
$307$	−5314.23	−0.987945	−0.493972	−	0.869478i	$-0.664456\pi$
$307$	−5314.23	−0.987945	−0.493972	+	0.869478i	$0.664456\pi$
$308$	0	0
$309$	1924.60	0.354325
$310$	103.827	0.0190226
$311$	8840.00	1.61180	0.805901	−	0.592051i	$-0.201682\pi$
$311$	8840.00	1.61180	0.805901	+	0.592051i	$0.201682\pi$
$312$	408.921	0.0742006
$313$	1896.34	0.342453	0.171226	−	0.985232i	$-0.445227\pi$
$313$	1896.34	0.342453	0.171226	+	0.985232i	$0.445227\pi$
$314$	−312.218	−0.0561130
$315$	0	0
$316$	−1905.20	−0.339165
$317$	−6072.81	−1.07597	−0.537985	−	0.842954i	$-0.680814\pi$
$317$	−6072.81	−1.07597	−0.537985	+	0.842954i	$0.680814\pi$
$318$	−74.1856	−0.0130821
$319$	5264.28	0.923960
$320$	−2461.17	−0.429948
$321$	−2520.63	−0.438280
$322$	0	0
$323$	−8866.46	−1.52738
$324$	−3773.43	−0.647022
$325$	1557.94	0.265905
$326$	−854.203	−0.145122
$327$	−261.311	−0.0441912
$328$	985.971	0.165979
$329$	0	0
$330$	−36.4170	−0.00607482
$331$	3341.78	0.554927	0.277463	−	0.960736i	$-0.410506\pi$
$331$	3341.78	0.554927	0.277463	+	0.960736i	$0.410506\pi$
$332$	2377.99	0.393100
$333$	−6672.83	−1.09810
$334$	−442.608	−0.0725102
$335$	−2307.77	−0.376379
$336$	0	0
$337$	−8568.87	−1.38509	−0.692547	−	0.721373i	$-0.743511\pi$
$337$	−8568.87	−1.38509	−0.692547	+	0.721373i	$0.743511\pi$
$338$	−383.675	−0.0617431
$339$	−3660.40	−0.586447
$340$	3463.94	0.552525
$341$	−1615.79	−0.256598
$342$	549.133	0.0868237
$343$	0	0
$344$	−28.3385	−0.00444160
$345$	846.775	0.132142
$346$	−470.315	−0.0730760
$347$	−6246.73	−0.966403	−0.483202	−	0.875509i	$-0.660526\pi$
$347$	−6246.73	−0.966403	−0.483202	+	0.875509i	$0.660526\pi$
$348$	−4274.93	−0.658507
$349$	−7091.01	−1.08760	−0.543801	−	0.839214i	$-0.683015\pi$
$349$	−7091.01	−1.08760	−0.543801	+	0.839214i	$0.683015\pi$
$350$	0	0
$351$	−5717.51	−0.869453
$352$	−766.550	−0.116072
$353$	−7289.22	−1.09905	−0.549527	−	0.835476i	$-0.685192\pi$
$353$	−7289.22	−1.09905	−0.549527	+	0.835476i	$0.685192\pi$
$354$	−41.0218	−0.00615899
$355$	2592.09	0.387531
$356$	8380.31	1.24763
$357$	0	0
$358$	−545.065	−0.0804681
$359$	−8668.21	−1.27435	−0.637174	−	0.770720i	$-0.719897\pi$
$359$	−8668.21	−1.27435	−0.637174	+	0.770720i	$0.719897\pi$
$360$	−430.466	−0.0630210
$361$	3488.68	0.508628
$362$	300.161	0.0435804
$363$	−1840.49	−0.266118
$364$	0	0
$365$	2714.33	0.389246
$366$	178.658	0.0255153
$367$	3777.62	0.537302	0.268651	−	0.963238i	$-0.413422\pi$
$367$	3777.62	0.537302	0.268651	+	0.963238i	$0.413422\pi$
$368$	5876.88	0.832482
$369$	−6448.45	−0.909737
$370$	−319.872	−0.0449442
$371$	0	0
$372$	1312.12	0.182878
$373$	−1742.13	−0.241834	−0.120917	−	0.992663i	$-0.538584\pi$
$373$	−1742.13	−0.241834	−0.120917	+	0.992663i	$0.538584\pi$
$374$	351.014	0.0485308
$375$	226.073	0.0311316
$376$	335.434	0.0460072
$377$	18532.4	2.53174
$378$	0	0
$379$	−3471.30	−0.470471	−0.235236	−	0.971938i	$-0.575586\pi$
$379$	−3471.30	−0.470471	−0.235236	+	0.971938i	$0.575586\pi$
$380$	−4042.62	−0.545742
$381$	2905.60	0.390705
$382$	−479.580	−0.0642342
$383$	−14445.1	−1.92718	−0.963591	−	0.267382i	$-0.913841\pi$
$383$	−14445.1	−1.92718	−0.963591	+	0.267382i	$0.913841\pi$
$384$	829.069	0.110178
$385$	0	0
$386$	−826.183	−0.108942
$387$	185.339	0.0243445
$388$	2295.88	0.300402
$389$	−7729.56	−1.00747	−0.503733	−	0.863860i	$-0.668040\pi$
$389$	−7729.56	−1.00747	−0.503733	+	0.863860i	$0.668040\pi$
$390$	−128.202	−0.0166456
$391$	−8161.85	−1.05566
$392$	0	0
$393$	−5151.43	−0.661209
$394$	−746.296	−0.0954260
$395$	1198.51	0.152667
$396$	3338.65	0.423670
$397$	10069.6	1.27300	0.636499	−	0.771278i	$-0.280382\pi$
$397$	10069.6	1.27300	0.636499	+	0.771278i	$0.280382\pi$
$398$	−147.585	−0.0185874
$399$	0	0
$400$	1569.01	0.196127
$401$	1503.17	0.187194	0.0935972	−	0.995610i	$-0.470163\pi$
$401$	1503.17	0.187194	0.0935972	+	0.995610i	$0.470163\pi$
$402$	189.905	0.0235612
$403$	−5688.23	−0.703104
$404$	2820.35	0.347321
$405$	2373.75	0.291241
$406$	0	0
$407$	4977.94	0.606259
$408$	−571.948	−0.0694010
$409$	613.381	0.0741559	0.0370779	−	0.999312i	$-0.488195\pi$
$409$	613.381	0.0741559	0.0370779	+	0.999312i	$0.488195\pi$
$410$	−309.116	−0.0372345
$411$	1522.32	0.182701
$412$	−8458.10	−1.01141
$413$	0	0
$414$	505.494	0.0600089
$415$	−1495.92	−0.176944
$416$	−2698.57	−0.318048
$417$	−4393.23	−0.515917
$418$	−409.654	−0.0479350
$419$	−12334.4	−1.43813	−0.719065	−	0.694943i	$-0.755429\pi$
$419$	−12334.4	−1.43813	−0.719065	+	0.694943i	$0.755429\pi$
$420$	0	0
$421$	−2432.22	−0.281566	−0.140783	−	0.990040i	$-0.544962\pi$
$421$	−2432.22	−0.281566	−0.140783	+	0.990040i	$0.544962\pi$
$422$	−1153.94	−0.133111
$423$	−2193.81	−0.252167
$424$	654.175	0.0749282
$425$	−2179.06	−0.248705
$426$	−213.301	−0.0242594
$427$	0	0
$428$	11077.5	1.25106
$429$	1995.12	0.224535
$430$	8.88452	0.000996394 0
$431$	−1310.48	−0.146458	−0.0732292	−	0.997315i	$-0.523330\pi$
$431$	−1310.48	−0.146458	−0.0732292	+	0.997315i	$0.523330\pi$
$432$	−5758.13	−0.641292
$433$	−4954.86	−0.549920	−0.274960	−	0.961456i	$-0.588665\pi$
$433$	−4954.86	−0.549920	−0.274960	+	0.961456i	$0.588665\pi$
$434$	0	0
$435$	2689.23	0.296411
$436$	1148.39	0.126142
$437$	9525.36	1.04270
$438$	−223.361	−0.0243667
$439$	8195.34	0.890984	0.445492	−	0.895286i	$-0.353029\pi$
$439$	8195.34	0.890984	0.445492	+	0.895286i	$0.353029\pi$
$440$	321.128	0.0347936
$441$	0	0
$442$	1235.71	0.132979
$443$	2956.70	0.317104	0.158552	−	0.987351i	$-0.449317\pi$
$443$	2956.70	0.317104	0.158552	+	0.987351i	$0.449317\pi$
$444$	−4042.41	−0.432081
$445$	−5271.80	−0.561589
$446$	−37.2452	−0.00395429
$447$	−1045.27	−0.110603
$448$	0	0
$449$	−10453.5	−1.09873	−0.549364	−	0.835583i	$-0.685130\pi$
$449$	−10453.5	−1.09873	−0.549364	+	0.835583i	$0.685130\pi$
$450$	134.957	0.0141377
$451$	4810.55	0.502262
$452$	16086.5	1.67399
$453$	1417.23	0.146992
$454$	1439.29	0.148787
$455$	0	0
$456$	667.497	0.0685491
$457$	2245.19	0.229816	0.114908	−	0.993376i	$-0.463343\pi$
$457$	2245.19	0.229816	0.114908	+	0.993376i	$0.463343\pi$
$458$	792.707	0.0808751
$459$	7996.94	0.813214
$460$	−3721.35	−0.377194
$461$	12441.1	1.25692	0.628459	−	0.777842i	$-0.283686\pi$
$461$	12441.1	1.25692	0.628459	+	0.777842i	$0.283686\pi$
$462$	0	0
$463$	17086.0	1.71502	0.857510	−	0.514468i	$-0.172010\pi$
$463$	17086.0	1.71502	0.857510	+	0.514468i	$0.172010\pi$
$464$	18664.1	1.86737
$465$	−825.418	−0.0823179
$466$	1022.86	0.101680
$467$	11709.1	1.16024	0.580121	−	0.814531i	$-0.303005\pi$
$467$	11709.1	1.16024	0.580121	+	0.814531i	$0.303005\pi$
$468$	11753.4	1.16090
$469$	0	0
$470$	−105.163	−0.0103209
$471$	2482.11	0.242823
$472$	361.734	0.0352757
$473$	−138.263	−0.0134405
$474$	−98.6245	−0.00955690
$475$	2543.09	0.245652
$476$	0	0
$477$	−4278.44	−0.410684
$478$	348.009	0.0333003
$479$	−11692.0	−1.11528	−0.557641	−	0.830082i	$-0.688293\pi$
$479$	−11692.0	−1.11528	−0.557641	+	0.830082i	$0.688293\pi$
$480$	−391.588	−0.0372364
$481$	17524.4	1.66121
$482$	1040.21	0.0982992
$483$	0	0
$484$	8088.48	0.759624
$485$	−1444.27	−0.135218
$486$	−758.888	−0.0708310
$487$	7128.84	0.663324	0.331662	−	0.943398i	$-0.392391\pi$
$487$	7128.84	0.663324	0.331662	+	0.943398i	$0.392391\pi$
$488$	−1575.42	−0.146139
$489$	6790.84	0.628001
$490$	0	0
$491$	−4330.34	−0.398015	−0.199008	−	0.979998i	$-0.563772\pi$
$491$	−4330.34	−0.398015	−0.199008	+	0.979998i	$0.563772\pi$
$492$	−3906.47	−0.357962
$493$	−25920.8	−2.36798
$494$	−1442.15	−0.131347
$495$	−2100.24	−0.190705
$496$	−5728.65	−0.518597
$497$	0	0
$498$	123.099	0.0110767
$499$	9635.67	0.864433	0.432216	−	0.901770i	$-0.357732\pi$
$499$	9635.67	0.864433	0.432216	+	0.901770i	$0.357732\pi$
$500$	−993.531	−0.0888641
$501$	3518.69	0.313779
$502$	−122.199	−0.0108645
$503$	−18763.3	−1.66325	−0.831626	−	0.555335i	$-0.812590\pi$
$503$	−18763.3	−1.66325	−0.831626	+	0.555335i	$0.812590\pi$
$504$	0	0
$505$	−1774.19	−0.156338
$506$	−377.099	−0.0331306
$507$	3050.18	0.267186
$508$	−12769.4	−1.11525
$509$	−15804.5	−1.37627	−0.688137	−	0.725580i	$-0.741571\pi$
$509$	−15804.5	−1.37627	−0.688137	+	0.725580i	$0.741571\pi$
$510$	179.314	0.0155689
$511$	0	0
$512$	−4539.39	−0.391826
$513$	−9332.90	−0.803231
$514$	−199.869	−0.0171514
$515$	5320.73	0.455261
$516$	112.279	0.00957906
$517$	1636.58	0.139220
$518$	0	0
$519$	3738.96	0.316228
$520$	1130.50	0.0953380
$521$	−9451.64	−0.794787	−0.397393	−	0.917648i	$-0.630085\pi$
$521$	−9451.64	−0.794787	−0.397393	+	0.917648i	$0.630085\pi$
$522$	1605.37	0.134608
$523$	−2704.59	−0.226125	−0.113062	−	0.993588i	$-0.536066\pi$
$523$	−2704.59	−0.226125	−0.113062	+	0.993588i	$0.536066\pi$
$524$	22639.2	1.88740
$525$	0	0
$526$	−844.931	−0.0700394
$527$	7955.99	0.657625
$528$	2009.30	0.165613
$529$	−3398.62	−0.279331
$530$	−205.093	−0.0168088
$531$	−2365.81	−0.193347
$532$	0	0
$533$	16935.1	1.37625
$534$	433.814	0.0351553
$535$	−6968.53	−0.563132
$536$	−1674.60	−0.134947
$537$	4333.22	0.348216
$538$	−35.1392	−0.00281591
$539$	0	0
$540$	3646.16	0.290567
$541$	60.7636	0.00482889	0.00241445	−	0.999997i	$-0.499231\pi$
$541$	60.7636	0.00482889	0.00241445	+	0.999997i	$0.499231\pi$
$542$	547.427	0.0433838
$543$	−2386.25	−0.188589
$544$	3774.42	0.297476
$545$	−722.419	−0.0567799
$546$	0	0
$547$	14267.4	1.11523	0.557614	−	0.830100i	$-0.311717\pi$
$547$	14267.4	1.11523	0.557614	+	0.830100i	$0.311717\pi$
$548$	−6690.17	−0.521515
$549$	10303.6	0.800994
$550$	−100.678	−0.00780534
$551$	30251.1	2.33891
$552$	614.451	0.0473782
$553$	0	0
$554$	−329.816	−0.0252934
$555$	2542.95	0.194491
$556$	19307.1	1.47267
$557$	−19601.8	−1.49112	−0.745559	−	0.666439i	$-0.767818\pi$
$557$	−19601.8	−1.49112	−0.745559	+	0.666439i	$0.767818\pi$
$558$	−492.744	−0.0373827
$559$	−486.743	−0.0368283
$560$	0	0
$561$	−2790.53	−0.210011
$562$	−463.427	−0.0347838
$563$	−7081.18	−0.530082	−0.265041	−	0.964237i	$-0.585385\pi$
$563$	−7081.18	−0.530082	−0.265041	+	0.964237i	$0.585385\pi$
$564$	−1329.01	−0.0992223
$565$	−10119.5	−0.753507
$566$	−1675.79	−0.124450
$567$	0	0
$568$	1880.91	0.138946
$569$	−12034.6	−0.886669	−0.443334	−	0.896356i	$-0.646205\pi$
$569$	−12034.6	−0.886669	−0.443334	+	0.896356i	$0.646205\pi$
$570$	−209.270	−0.0153778
$571$	4179.38	0.306308	0.153154	−	0.988202i	$-0.451057\pi$
$571$	4179.38	0.306308	0.153154	+	0.988202i	$0.451057\pi$
$572$	−8768.04	−0.640927
$573$	3812.62	0.277966
$574$	0	0
$575$	2340.99	0.169784
$576$	11680.2	0.844923
$577$	−16691.6	−1.20430	−0.602151	−	0.798382i	$-0.705689\pi$
$577$	−16691.6	−1.20430	−0.602151	+	0.798382i	$0.705689\pi$
$578$	−610.664	−0.0439451
$579$	6568.09	0.471434
$580$	−11818.5	−0.846095
$581$	0	0
$582$	118.848	0.00846465
$583$	3191.72	0.226737
$584$	1969.62	0.139561
$585$	−7393.70	−0.522550
$586$	−123.516	−0.00870718
$587$	8878.89	0.624311	0.312156	−	0.950031i	$-0.398949\pi$
$587$	8878.89	0.624311	0.312156	+	0.950031i	$0.398949\pi$
$588$	0	0
$589$	−9285.11	−0.649552
$590$	−113.409	−0.00791349
$591$	5932.99	0.412945
$592$	17648.9	1.22528
$593$	−23777.1	−1.64656	−0.823278	−	0.567639i	$-0.807857\pi$
$593$	−23777.1	−1.64656	−0.823278	+	0.567639i	$0.807857\pi$
$594$	369.480	0.0255218
$595$	0	0
$596$	4593.67	0.315711
$597$	1173.29	0.0804346
$598$	−1327.54	−0.0907812
$599$	12709.4	0.866934	0.433467	−	0.901169i	$-0.357290\pi$
$599$	12709.4	0.866934	0.433467	+	0.901169i	$0.357290\pi$
$600$	164.047	0.0111620
$601$	13296.4	0.902446	0.451223	−	0.892411i	$-0.350988\pi$
$601$	13296.4	0.902446	0.451223	+	0.892411i	$0.350988\pi$
$602$	0	0
$603$	10952.2	0.739651
$604$	−6228.37	−0.419584
$605$	−5088.22	−0.341926
$606$	145.998	0.00978672
$607$	5838.48	0.390406	0.195203	−	0.980763i	$-0.437463\pi$
$607$	5838.48	0.390406	0.195203	+	0.980763i	$0.437463\pi$
$608$	−4404.97	−0.293824
$609$	0	0
$610$	493.917	0.0327838
$611$	5761.43	0.381477
$612$	−16439.2	−1.08581
$613$	4270.31	0.281364	0.140682	−	0.990055i	$-0.455070\pi$
$613$	4270.31	0.281364	0.140682	+	0.990055i	$0.455070\pi$
$614$	1208.97	0.0794628
$615$	2457.44	0.161128
$616$	0	0
$617$	−2869.54	−0.187234	−0.0936168	−	0.995608i	$-0.529843\pi$
$617$	−2869.54	−0.187234	−0.0936168	+	0.995608i	$0.529843\pi$
$618$	−437.841	−0.0284992
$619$	6882.06	0.446871	0.223436	−	0.974719i	$-0.428273\pi$
$619$	6882.06	0.446871	0.223436	+	0.974719i	$0.428273\pi$
$620$	3627.49	0.234974
$621$	−8591.22	−0.555159
$622$	−2011.07	−0.129641
$623$	0	0
$624$	7073.54	0.453796
$625$	625.000	0.0400000
$626$	−431.413	−0.0275443
$627$	3256.71	0.207433
$628$	−10908.2	−0.693129
$629$	−24510.9	−1.55376
$630$	0	0
$631$	−28101.9	−1.77293	−0.886467	−	0.462793i	$-0.846847\pi$
$631$	−28101.9	−1.77293	−0.886467	+	0.462793i	$0.846847\pi$
$632$	869.679	0.0547373
$633$	9173.74	0.576024
$634$	1381.55	0.0865429
$635$	8032.82	0.502004
$636$	−2591.88	−0.161595
$637$	0	0
$638$	−1197.61	−0.0743164
$639$	−12301.5	−0.761567
$640$	2292.04	0.141564
$641$	13677.8	0.842809	0.421404	−	0.906873i	$-0.361537\pi$
$641$	13677.8	0.842809	0.421404	+	0.906873i	$0.361537\pi$
$642$	573.437	0.0352520
$643$	17919.1	1.09901	0.549503	−	0.835492i	$-0.314817\pi$
$643$	17919.1	1.09901	0.549503	+	0.835492i	$0.314817\pi$
$644$	0	0
$645$	−70.6311	−0.00431178
$646$	2017.10	0.122851
$647$	31488.2	1.91334	0.956669	−	0.291176i	$-0.0940466\pi$
$647$	31488.2	1.91334	0.956669	+	0.291176i	$0.0940466\pi$
$648$	1722.48	0.104422
$649$	1764.90	0.106746
$650$	−354.428	−0.0213874
$651$	0	0
$652$	−29843.9	−1.79261
$653$	−1110.46	−0.0665477	−0.0332739	−	0.999446i	$-0.510593\pi$
$653$	−1110.46	−0.0665477	−0.0332739	+	0.999446i	$0.510593\pi$
$654$	59.4475	0.00355441
$655$	−14241.6	−0.849566
$656$	17055.4	1.01509
$657$	−12881.7	−0.764936
$658$	0	0
$659$	12793.3	0.756230	0.378115	−	0.925759i	$-0.376572\pi$
$659$	12793.3	0.756230	0.378115	+	0.925759i	$0.376572\pi$
$660$	−1272.33	−0.0750384
$661$	2039.60	0.120017	0.0600086	−	0.998198i	$-0.480887\pi$
$661$	2039.60	0.120017	0.0600086	+	0.998198i	$0.480887\pi$
$662$	−760.246	−0.0446341
$663$	−9823.79	−0.575452
$664$	−1085.50	−0.0634419
$665$	0	0
$666$	1518.05	0.0883232
$667$	27847.1	1.61656
$668$	−15463.7	−0.895672
$669$	296.096	0.0171117
$670$	525.012	0.0302731
$671$	−7686.48	−0.442225
$672$	0	0
$673$	−4039.94	−0.231394	−0.115697	−	0.993285i	$-0.536910\pi$
$673$	−4039.94	−0.231394	−0.115697	+	0.993285i	$0.536910\pi$
$674$	1949.40	0.111406
$675$	−2293.69	−0.130791
$676$	−13404.7	−0.762673
$677$	−1121.55	−0.0636701	−0.0318351	−	0.999493i	$-0.510135\pi$
$677$	−1121.55	−0.0636701	−0.0318351	+	0.999493i	$0.510135\pi$
$678$	832.731	0.0471694
$679$	0	0
$680$	−1581.20	−0.0891712
$681$	−11442.2	−0.643858
$682$	367.588	0.0206388
$683$	−10232.0	−0.573232	−0.286616	−	0.958046i	$-0.592530\pi$
$683$	−10232.0	−0.573232	−0.286616	+	0.958046i	$0.592530\pi$
$684$	19185.5	1.07248
$685$	4208.59	0.234747
$686$	0	0
$687$	−6301.95	−0.349977
$688$	−490.201	−0.0271639
$689$	11236.1	0.621281
$690$	−192.639	−0.0106285
$691$	−22514.6	−1.23950	−0.619750	−	0.784799i	$-0.712766\pi$
$691$	−22514.6	−1.23950	−0.619750	+	0.784799i	$0.712766\pi$
$692$	−16431.7	−0.902661
$693$	0	0
$694$	1421.11	0.0777302
$695$	−12145.5	−0.662885
$696$	1951.40	0.106276
$697$	−23686.7	−1.28723
$698$	1613.19	0.0874785
$699$	−8131.62	−0.440009
$700$	0	0
$701$	−19172.5	−1.03300	−0.516501	−	0.856287i	$-0.672766\pi$
$701$	−19172.5	−1.03300	−0.516501	+	0.856287i	$0.672766\pi$
$702$	1300.72	0.0699322
$703$	28605.6	1.53468
$704$	−8713.45	−0.466478
$705$	836.039	0.0446625
$706$	1658.28	0.0883996
$707$	0	0
$708$	−1433.21	−0.0760781
$709$	−26333.2	−1.39487	−0.697437	−	0.716646i	$-0.745676\pi$
$709$	−26333.2	−1.39487	−0.697437	+	0.716646i	$0.745676\pi$
$710$	−589.692	−0.0311701
$711$	−5687.88	−0.300017
$712$	−3825.41	−0.201353
$713$	−8547.23	−0.448943
$714$	0	0
$715$	5515.71	0.288498
$716$	−19043.3	−0.993971
$717$	−2766.64	−0.144103
$718$	1971.99	0.102499
$719$	−17047.7	−0.884246	−0.442123	−	0.896955i	$-0.645774\pi$
$719$	−17047.7	−0.884246	−0.442123	+	0.896955i	$0.645774\pi$
$720$	−7446.24	−0.385423
$721$	0	0
$722$	−793.666	−0.0409102
$723$	−8269.57	−0.425378
$724$	10486.9	0.538321
$725$	7434.64	0.380849
$726$	418.707	0.0214045
$727$	16021.2	0.817325	0.408662	−	0.912686i	$-0.365995\pi$
$727$	16021.2	0.817325	0.408662	+	0.912686i	$0.365995\pi$
$728$	0	0
$729$	−6785.17	−0.344722
$730$	−617.503	−0.0313080
$731$	680.795	0.0344461
$732$	6241.91	0.315174
$733$	−7544.16	−0.380150	−0.190075	−	0.981770i	$-0.560873\pi$
$733$	−7544.16	−0.380150	−0.190075	+	0.981770i	$0.560873\pi$
$734$	−859.397	−0.0432165
$735$	0	0
$736$	−4054.91	−0.203079
$737$	−8170.38	−0.408358
$738$	1467.00	0.0731723
$739$	17543.9	0.873291	0.436645	−	0.899634i	$-0.356166\pi$
$739$	17543.9	0.873291	0.436645	+	0.899634i	$0.356166\pi$
$740$	−11175.6	−0.555167
$741$	11464.9	0.568387
$742$	0	0
$743$	−26595.9	−1.31320	−0.656600	−	0.754239i	$-0.728006\pi$
$743$	−26595.9	−1.31320	−0.656600	+	0.754239i	$0.728006\pi$
$744$	−598.953	−0.0295144
$745$	−2889.74	−0.142110
$746$	396.330	0.0194513
$747$	7099.36	0.347727
$748$	12263.6	0.599470
$749$	0	0
$750$	−51.4310	−0.00250399
$751$	9616.06	0.467237	0.233619	−	0.972328i	$-0.424943\pi$
$751$	9616.06	0.467237	0.233619	+	0.972328i	$0.424943\pi$
$752$	5802.37	0.281370
$753$	971.468	0.0470149
$754$	−4216.07	−0.203634
$755$	3918.08	0.188866
$756$	0	0
$757$	8519.83	0.409060	0.204530	−	0.978860i	$-0.434433\pi$
$757$	8519.83	0.409060	0.204530	+	0.978860i	$0.434433\pi$
$758$	789.711	0.0378411
$759$	2997.90	0.143369
$760$	1845.36	0.0880765
$761$	4228.45	0.201421	0.100710	−	0.994916i	$-0.467888\pi$
$761$	4228.45	0.201421	0.100710	+	0.994916i	$0.467888\pi$
$762$	−661.017	−0.0314253
$763$	0	0
$764$	−16755.5	−0.793444
$765$	10341.4	0.488750
$766$	3286.22	0.155008
$767$	6213.15	0.292495
$768$	6933.34	0.325762
$769$	25293.4	1.18609	0.593046	−	0.805169i	$-0.297925\pi$
$769$	25293.4	1.18609	0.593046	+	0.805169i	$0.297925\pi$
$770$	0	0
$771$	1588.94	0.0742207
$772$	−28865.0	−1.34569
$773$	−15296.8	−0.711756	−0.355878	−	0.934533i	$-0.615818\pi$
$773$	−15296.8	−0.711756	−0.355878	+	0.934533i	$0.615818\pi$
$774$	−42.1642	−0.00195809
$775$	−2281.95	−0.105768
$776$	−1048.02	−0.0484814
$777$	0	0
$778$	1758.45	0.0810329
$779$	27643.7	1.27142
$780$	−4479.11	−0.205613
$781$	9176.96	0.420458
$782$	1856.80	0.0849092
$783$	−27284.4	−1.24529
$784$	0	0
$785$	6862.02	0.311995
$786$	1171.94	0.0531826
$787$	−16789.8	−0.760474	−0.380237	−	0.924889i	$-0.624158\pi$
$787$	−16789.8	−0.760474	−0.380237	+	0.924889i	$0.624158\pi$
$788$	−26073.9	−1.17874
$789$	6717.12	0.303087
$790$	−272.657	−0.0122794
$791$	0	0
$792$	−1524.01	−0.0683755
$793$	−27059.5	−1.21174
$794$	−2290.81	−0.102390
$795$	1630.47	0.0727382
$796$	−5156.29	−0.229598
$797$	18124.8	0.805537	0.402768	−	0.915302i	$-0.368048\pi$
$797$	18124.8	0.805537	0.402768	+	0.915302i	$0.368048\pi$
$798$	0	0
$799$	−8058.37	−0.356802
$800$	−1082.58	−0.0478439
$801$	25018.9	1.10362
$802$	−341.968	−0.0150565
$803$	9609.76	0.422318
$804$	6634.87	0.291037
$805$	0	0
$806$	1294.06	0.0565524
$807$	279.353	0.0121855
$808$	−1287.42	−0.0560536
$809$	−10777.9	−0.468395	−0.234197	−	0.972189i	$-0.575246\pi$
$809$	−10777.9	−0.468395	−0.234197	+	0.972189i	$0.575246\pi$
$810$	−540.022	−0.0234252
$811$	−11986.8	−0.519007	−0.259503	−	0.965742i	$-0.583559\pi$
$811$	−11986.8	−0.519007	−0.259503	+	0.965742i	$0.583559\pi$
$812$	0	0
$813$	−4352.00	−0.187738
$814$	−1132.47	−0.0487629
$815$	18773.9	0.806898
$816$	−9893.59	−0.424442
$817$	−794.528	−0.0340233
$818$	−139.543	−0.00596454
$819$	0	0
$820$	−10799.8	−0.459934
$821$	10740.4	0.456570	0.228285	−	0.973594i	$-0.426688\pi$
$821$	10740.4	0.456570	0.228285	+	0.973594i	$0.426688\pi$
$822$	−346.322	−0.0146951
$823$	32400.9	1.37233	0.686163	−	0.727448i	$-0.259294\pi$
$823$	32400.9	1.37233	0.686163	+	0.727448i	$0.259294\pi$
$824$	3860.92	0.163230
$825$	800.383	0.0337767
$826$	0	0
$827$	3825.30	0.160845	0.0804225	−	0.996761i	$-0.474373\pi$
$827$	3825.30	0.160845	0.0804225	+	0.996761i	$0.474373\pi$
$828$	17660.8	0.741251
$829$	−31836.3	−1.33380	−0.666900	−	0.745147i	$-0.732379\pi$
$829$	−31836.3	−1.33380	−0.666900	+	0.745147i	$0.732379\pi$
$830$	340.318	0.0142321
$831$	2622.01	0.109454
$832$	−30674.9	−1.27820
$833$	0	0
$834$	999.448	0.0414965
$835$	9727.76	0.403165
$836$	−14312.4	−0.592111
$837$	8374.53	0.345838
$838$	2806.05	0.115672
$839$	−6354.36	−0.261474	−0.130737	−	0.991417i	$-0.541734\pi$
$839$	−6354.36	−0.261474	−0.130737	+	0.991417i	$0.541734\pi$
$840$	0	0
$841$	64049.1	2.62615
$842$	553.324	0.0226470
$843$	3684.20	0.150523
$844$	−40316.2	−1.64424
$845$	8432.52	0.343299
$846$	499.085	0.0202824
$847$	0	0
$848$	11316.0	0.458245
$849$	13322.4	0.538544
$850$	495.730	0.0200040
$851$	26332.4	1.06071
$852$	−7452.28	−0.299661
$853$	−16620.2	−0.667135	−0.333568	−	0.942726i	$-0.608253\pi$
$853$	−16620.2	−0.667135	−0.333568	+	0.942726i	$0.608253\pi$
$854$	0	0
$855$	−12069.0	−0.482750
$856$	−5056.62	−0.201906
$857$	34440.5	1.37277	0.686385	−	0.727238i	$-0.259196\pi$
$857$	34440.5	1.37277	0.686385	+	0.727238i	$0.259196\pi$
$858$	−453.885	−0.0180599
$859$	−29192.5	−1.15953	−0.579765	−	0.814784i	$-0.696856\pi$
$859$	−29192.5	−1.15953	−0.579765	+	0.814784i	$0.696856\pi$
$860$	310.405	0.0123078
$861$	0	0
$862$	298.130	0.0117800
$863$	26062.4	1.02801	0.514006	−	0.857786i	$-0.328161\pi$
$863$	26062.4	1.02801	0.514006	+	0.857786i	$0.328161\pi$
$864$	3972.98	0.156439
$865$	10336.7	0.406311
$866$	1127.22	0.0442314
$867$	4854.72	0.190167
$868$	0	0
$869$	4243.16	0.165638
$870$	−611.793	−0.0238411
$871$	−28763.0	−1.11894
$872$	−524.214	−0.0203579
$873$	6854.23	0.265728
$874$	−2166.99	−0.0838668
$875$	0	0
$876$	−7803.74	−0.300986
$877$	−29048.5	−1.11847	−0.559236	−	0.829009i	$-0.688905\pi$
$877$	−29048.5	−1.11847	−0.559236	+	0.829009i	$0.688905\pi$
$878$	−1864.42	−0.0716640
$879$	981.943	0.0376793
$880$	5554.90	0.212791
$881$	−24440.9	−0.934660	−0.467330	−	0.884083i	$-0.654784\pi$
$881$	−24440.9	−0.934660	−0.467330	+	0.884083i	$0.654784\pi$
$882$	0	0
$883$	38951.9	1.48453	0.742264	−	0.670108i	$-0.233752\pi$
$883$	38951.9	1.48453	0.742264	+	0.670108i	$0.233752\pi$
$884$	43173.0	1.64261
$885$	901.589	0.0342447
$886$	−672.642	−0.0255055
$887$	−3143.83	−0.119007	−0.0595036	−	0.998228i	$-0.518952\pi$
$887$	−3143.83	−0.119007	−0.0595036	+	0.998228i	$0.518952\pi$
$888$	1845.26	0.0697330
$889$	0	0
$890$	1199.32	0.0451700
$891$	8403.98	0.315986
$892$	−1301.26	−0.0488448
$893$	9404.59	0.352422
$894$	237.795	0.00889604
$895$	11979.6	0.447412
$896$	0	0
$897$	10553.8	0.392845
$898$	2378.13	0.0883734
$899$	−27144.7	−1.00704
$900$	4715.10	0.174633
$901$	−15715.7	−0.581094
$902$	−1094.39	−0.0403981
$903$	0	0
$904$	−7343.10	−0.270163
$905$	−6597.02	−0.242312
$906$	−322.417	−0.0118229
$907$	−12839.5	−0.470044	−0.235022	−	0.971990i	$-0.575516\pi$
$907$	−12839.5	−0.470044	−0.235022	+	0.971990i	$0.575516\pi$
$908$	50285.6	1.83787
$909$	8419.98	0.307231
$910$	0	0
$911$	17451.4	0.634678	0.317339	−	0.948312i	$-0.397211\pi$
$911$	17451.4	0.634678	0.317339	+	0.948312i	$0.397211\pi$
$912$	11546.4	0.419232
$913$	−5296.13	−0.191978
$914$	−510.775	−0.0184846
$915$	−3926.60	−0.141868
$916$	27695.4	0.998998
$917$	0	0
$918$	−1819.28	−0.0654088
$919$	−7937.66	−0.284918	−0.142459	−	0.989801i	$-0.545501\pi$
$919$	−7937.66	−0.284918	−0.142459	+	0.989801i	$0.545501\pi$
$920$	1698.71	0.0608747
$921$	−9611.22	−0.343866
$922$	−2830.31	−0.101097
$923$	32306.6	1.15210
$924$	0	0
$925$	7030.24	0.249895
$926$	−3887.02	−0.137943
$927$	−25251.2	−0.894668
$928$	−12877.8	−0.455532
$929$	1781.40	0.0629127	0.0314563	−	0.999505i	$-0.489985\pi$
$929$	1781.40	0.0629127	0.0314563	+	0.999505i	$0.489985\pi$
$930$	187.780	0.00662103
$931$	0	0
$932$	35736.3	1.25599
$933$	15987.9	0.561007
$934$	−2663.79	−0.0933210
$935$	−7714.69	−0.269837
$936$	−5365.14	−0.187356
$937$	46363.0	1.61645	0.808225	−	0.588874i	$-0.200429\pi$
$937$	46363.0	1.61645	0.808225	+	0.588874i	$0.200429\pi$
$938$	0	0
$939$	3429.70	0.119195
$940$	−3674.17	−0.127488
$941$	−16535.3	−0.572833	−0.286416	−	0.958105i	$-0.592464\pi$
$941$	−16535.3	−0.572833	−0.286416	+	0.958105i	$0.592464\pi$
$942$	−564.673	−0.0195308
$943$	25446.9	0.878754
$944$	6257.30	0.215739
$945$	0	0
$946$	31.4545	0.00108105
$947$	−2734.31	−0.0938258	−0.0469129	−	0.998899i	$-0.514938\pi$
$947$	−2734.31	−0.0938258	−0.0469129	+	0.998899i	$0.514938\pi$
$948$	−3445.72	−0.118050
$949$	33830.2	1.15719
$950$	−578.546	−0.0197584
$951$	−10983.2	−0.374505
$952$	0	0
$953$	−35524.6	−1.20751	−0.603754	−	0.797171i	$-0.706329\pi$
$953$	−35524.6	−1.20751	−0.603754	+	0.797171i	$0.706329\pi$
$954$	973.332	0.0330323
$955$	10540.4	0.357150
$956$	12158.7	0.411338
$957$	9520.89	0.321595
$958$	2659.89	0.0897048
$959$	0	0
$960$	−4451.22	−0.149649
$961$	−21459.4	−0.720330
$962$	−3986.74	−0.133615
$963$	33071.3	1.10665
$964$	36342.6	1.21423
$965$	18158.1	0.605730
$966$	0	0
$967$	9328.92	0.310236	0.155118	−	0.987896i	$-0.450424\pi$
$967$	9328.92	0.310236	0.155118	+	0.987896i	$0.450424\pi$
$968$	−3692.19	−0.122595
$969$	−16035.7	−0.531622
$970$	328.568	0.0108760
$971$	−49138.1	−1.62401	−0.812007	−	0.583648i	$-0.801625\pi$
$971$	−49138.1	−1.62401	−0.812007	+	0.583648i	$0.801625\pi$
$972$	−26513.9	−0.874930
$973$	0	0
$974$	−1621.79	−0.0533527
$975$	2817.67	0.0925515
$976$	−27251.8	−0.893758
$977$	14771.3	0.483702	0.241851	−	0.970313i	$-0.422245\pi$
$977$	14771.3	0.483702	0.241851	+	0.970313i	$0.422245\pi$
$978$	−1544.90	−0.0505116
$979$	−18664.1	−0.609304
$980$	0	0
$981$	3428.46	0.111582
$982$	985.141	0.0320133
$983$	20651.0	0.670054	0.335027	−	0.942208i	$-0.391254\pi$
$983$	20651.0	0.670054	0.335027	+	0.942208i	$0.391254\pi$
$984$	1783.21	0.0577710
$985$	16402.3	0.530580
$986$	5896.91	0.190462
$987$	0	0
$988$	−50385.4	−1.62244
$989$	−731.387	−0.0235154
$990$	477.799	0.0153389
$991$	−45941.4	−1.47263	−0.736315	−	0.676639i	$-0.763436\pi$
$991$	−45941.4	−1.47263	−0.736315	+	0.676639i	$0.763436\pi$
$992$	3952.63	0.126508
$993$	6043.88	0.193149
$994$	0	0
$995$	3243.67	0.103348
$996$	4300.79	0.136823
$997$	−9025.07	−0.286687	−0.143344	−	0.989673i	$-0.545785\pi$
$997$	−9025.07	−0.286687	−0.143344	+	0.989673i	$0.545785\pi$
$998$	−2192.09	−0.0695284
$999$	−25800.3	−0.817103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.4.a.n.1.3		5
3.2	odd	2		2205.4.a.bt.1.3		5
5.4	even	2		1225.4.a.bf.1.3		5
7.2	even	3		245.4.e.o.116.3		10
7.3	odd	6		35.4.e.c.16.3	yes	10
7.4	even	3		245.4.e.o.226.3		10
7.5	odd	6		35.4.e.c.11.3	✓	10
7.6	odd	2		245.4.a.m.1.3		5
21.5	even	6		315.4.j.g.46.3		10
21.17	even	6		315.4.j.g.226.3		10
21.20	even	2		2205.4.a.bu.1.3		5
28.3	even	6		560.4.q.n.401.3		10
28.19	even	6		560.4.q.n.81.3		10
35.3	even	12		175.4.k.d.149.5		20
35.12	even	12		175.4.k.d.74.5		20
35.17	even	12		175.4.k.d.149.6		20
35.19	odd	6		175.4.e.d.151.3		10
35.24	odd	6		175.4.e.d.51.3		10
35.33	even	12		175.4.k.d.74.6		20
35.34	odd	2		1225.4.a.bg.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.e.c.11.3	✓	10	7.5	odd	6
35.4.e.c.16.3	yes	10	7.3	odd	6
175.4.e.d.51.3		10	35.24	odd	6
175.4.e.d.151.3		10	35.19	odd	6
175.4.k.d.74.5		20	35.12	even	12
175.4.k.d.74.6		20	35.33	even	12
175.4.k.d.149.5		20	35.3	even	12
175.4.k.d.149.6		20	35.17	even	12
245.4.a.m.1.3		5	7.6	odd	2
245.4.a.n.1.3		5	1.1	even	1	trivial
245.4.e.o.116.3		10	7.2	even	3
245.4.e.o.226.3		10	7.4	even	3
315.4.j.g.46.3		10	21.5	even	6
315.4.j.g.226.3		10	21.17	even	6
560.4.q.n.81.3		10	28.19	even	6
560.4.q.n.401.3		10	28.3	even	6
1225.4.a.bf.1.3		5	5.4	even	2
1225.4.a.bg.1.3		5	35.34	odd	2
2205.4.a.bt.1.3		5	3.2	odd	2
2205.4.a.bu.1.3		5	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q-0.227497 q^{2} +1.80858 q^{3} -7.94824 q^{4} +5.00000 q^{5} -0.411448 q^{6} +3.62818 q^{8} -23.7290 q^{9} +O(q^{10})\)	\(q-0.227497 q^{2} +1.80858 q^{3} -7.94824 q^{4} +5.00000 q^{5} -0.411448 q^{6} +3.62818 q^{8} -23.7290 q^{9} -1.13749 q^{10} +17.7019 q^{11} -14.3751 q^{12} +62.3178 q^{13} +9.04291 q^{15} +62.7606 q^{16} -87.1623 q^{17} +5.39829 q^{18} +101.724 q^{19} -39.7412 q^{20} -4.02713 q^{22} +93.6396 q^{23} +6.56187 q^{24} +25.0000 q^{25} -14.1771 q^{26} -91.7477 q^{27} +297.385 q^{29} -2.05724 q^{30} -91.2779 q^{31} -43.3033 q^{32} +32.0153 q^{33} +19.8292 q^{34} +188.604 q^{36} +281.210 q^{37} -23.1418 q^{38} +112.707 q^{39} +18.1409 q^{40} +271.754 q^{41} -7.81066 q^{43} -140.699 q^{44} -118.645 q^{45} -21.3028 q^{46} +92.4524 q^{47} +113.508 q^{48} -5.68743 q^{50} -157.640 q^{51} -495.317 q^{52} +180.304 q^{53} +20.8723 q^{54} +88.5094 q^{55} +183.975 q^{57} -67.6544 q^{58} +99.7011 q^{59} -71.8753 q^{60} -434.218 q^{61} +20.7655 q^{62} -492.233 q^{64} +311.589 q^{65} -7.28340 q^{66} -461.554 q^{67} +692.787 q^{68} +169.355 q^{69} +518.417 q^{71} -86.0932 q^{72} +542.866 q^{73} -63.9744 q^{74} +45.2146 q^{75} -808.524 q^{76} -25.6405 q^{78} +239.701 q^{79} +313.803 q^{80} +474.751 q^{81} -61.8232 q^{82} -299.184 q^{83} -435.812 q^{85} +1.77690 q^{86} +537.846 q^{87} +64.2257 q^{88} -1054.36 q^{89} +26.9914 q^{90} -744.271 q^{92} -165.084 q^{93} -21.0327 q^{94} +508.618 q^{95} -78.3176 q^{96} -288.854 q^{97} -420.049 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 8 q^{3} + 35 q^{4} + 25 q^{5} - 16 q^{6} + 33 q^{8} + 81 q^{9}+O(q^{10}) \) 5 * q + q^2 + 8 * q^3 + 35 * q^4 + 25 * q^5 - 16 * q^6 + 33 * q^8 + 81 * q^9	\( 5 q + q^{2} + 8 q^{3} + 35 q^{4} + 25 q^{5} - 16 q^{6} + 33 q^{8} + 81 q^{9} + 5 q^{10} + 47 q^{11} + 98 q^{12} - q^{13} + 40 q^{15} + 171 q^{16} + 2 q^{17} - 51 q^{18} + 21 q^{19} + 175 q^{20} + 523 q^{22} + 201 q^{23} - 848 q^{24} + 125 q^{25} + 47 q^{26} + 518 q^{27} + 190 q^{29} - 80 q^{30} - 388 q^{31} - 95 q^{32} + 262 q^{33} - 130 q^{34} + 1229 q^{36} - 145 q^{37} - 835 q^{38} + 14 q^{39} + 165 q^{40} + 281 q^{41} + 568 q^{43} + 1091 q^{44} + 405 q^{45} + 337 q^{46} + 473 q^{47} - 70 q^{48} + 25 q^{50} + 732 q^{51} + 379 q^{52} + 351 q^{53} + 774 q^{54} + 235 q^{55} + 954 q^{57} + 1818 q^{58} - 708 q^{59} + 490 q^{60} - 1944 q^{61} - 448 q^{62} - 125 q^{64} - 5 q^{65} - 1482 q^{66} + 1118 q^{67} + 3118 q^{68} - 374 q^{69} + 864 q^{71} - 2219 q^{72} + 1652 q^{73} - 3285 q^{74} + 200 q^{75} - 691 q^{76} - 5574 q^{78} + 218 q^{79} + 855 q^{80} - 455 q^{81} - 1027 q^{82} + 1502 q^{83} + 10 q^{85} - 4264 q^{86} - 390 q^{87} + 2131 q^{88} - 2322 q^{89} - 255 q^{90} - 2957 q^{92} - 2288 q^{93} + 2677 q^{94} + 105 q^{95} - 4592 q^{96} - 598 q^{97} + 35 q^{99}+O(q^{100}) \) 5 * q + q^2 + 8 * q^3 + 35 * q^4 + 25 * q^5 - 16 * q^6 + 33 * q^8 + 81 * q^9 + 5 * q^10 + 47 * q^11 + 98 * q^12 - q^13 + 40 * q^15 + 171 * q^16 + 2 * q^17 - 51 * q^18 + 21 * q^19 + 175 * q^20 + 523 * q^22 + 201 * q^23 - 848 * q^24 + 125 * q^25 + 47 * q^26 + 518 * q^27 + 190 * q^29 - 80 * q^30 - 388 * q^31 - 95 * q^32 + 262 * q^33 - 130 * q^34 + 1229 * q^36 - 145 * q^37 - 835 * q^38 + 14 * q^39 + 165 * q^40 + 281 * q^41 + 568 * q^43 + 1091 * q^44 + 405 * q^45 + 337 * q^46 + 473 * q^47 - 70 * q^48 + 25 * q^50 + 732 * q^51 + 379 * q^52 + 351 * q^53 + 774 * q^54 + 235 * q^55 + 954 * q^57 + 1818 * q^58 - 708 * q^59 + 490 * q^60 - 1944 * q^61 - 448 * q^62 - 125 * q^64 - 5 * q^65 - 1482 * q^66 + 1118 * q^67 + 3118 * q^68 - 374 * q^69 + 864 * q^71 - 2219 * q^72 + 1652 * q^73 - 3285 * q^74 + 200 * q^75 - 691 * q^76 - 5574 * q^78 + 218 * q^79 + 855 * q^80 - 455 * q^81 - 1027 * q^82 + 1502 * q^83 + 10 * q^85 - 4264 * q^86 - 390 * q^87 + 2131 * q^88 - 2322 * q^89 - 255 * q^90 - 2957 * q^92 - 2288 * q^93 + 2677 * q^94 + 105 * q^95 - 4592 * q^96 - 598 * q^97 + 35 * q^99

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