LMFDB - Embedded newform 625.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.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:	$4.99065012633$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.499011$ of defining polynomial
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.500989 q^{2} -3.09326 q^{3} -1.74901 q^{4} +1.54969 q^{6} -0.0237879 q^{7} +1.87821 q^{8} +6.56824 q^{9} +O(q^{10})$	\(q-0.500989 q^{2} -3.09326 q^{3} -1.74901 q^{4} +1.54969 q^{6} -0.0237879 q^{7} +1.87821 q^{8} +6.56824 q^{9} +3.58329 q^{11} +5.41014 q^{12} -3.77587 q^{13} +0.0119175 q^{14} +2.55705 q^{16} +3.62303 q^{17} -3.29062 q^{18} -2.43084 q^{19} +0.0735820 q^{21} -1.79519 q^{22} -1.71538 q^{23} -5.80980 q^{24} +1.89167 q^{26} -11.0375 q^{27} +0.0416052 q^{28} +3.85734 q^{29} -6.00979 q^{31} -5.03748 q^{32} -11.0840 q^{33} -1.81510 q^{34} -11.4879 q^{36} +0.369309 q^{37} +1.21782 q^{38} +11.6797 q^{39} -7.80900 q^{41} -0.0368638 q^{42} -0.174574 q^{43} -6.26720 q^{44} +0.859388 q^{46} -7.81082 q^{47} -7.90962 q^{48} -6.99943 q^{49} -11.2070 q^{51} +6.60403 q^{52} +8.97184 q^{53} +5.52966 q^{54} -0.0446787 q^{56} +7.51920 q^{57} -1.93249 q^{58} -4.45536 q^{59} +9.21403 q^{61} +3.01084 q^{62} -0.156244 q^{63} -2.59038 q^{64} +5.55298 q^{66} +4.47385 q^{67} -6.33671 q^{68} +5.30612 q^{69} -9.69458 q^{71} +12.3366 q^{72} +3.95387 q^{73} -0.185020 q^{74} +4.25156 q^{76} -0.0852388 q^{77} -5.85142 q^{78} -9.68349 q^{79} +14.4370 q^{81} +3.91223 q^{82} +8.95717 q^{83} -0.128696 q^{84} +0.0874600 q^{86} -11.9317 q^{87} +6.73018 q^{88} -17.0151 q^{89} +0.0898200 q^{91} +3.00022 q^{92} +18.5898 q^{93} +3.91314 q^{94} +15.5822 q^{96} -2.76438 q^{97} +3.50664 q^{98} +23.5359 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9	$ 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9} + q^{11} - 10 q^{12} - 10 q^{13} - 8 q^{14} + 13 q^{16} - 15 q^{17} + 5 q^{18} - 10 q^{19} - 14 q^{21} + 5 q^{22} - 30 q^{23} + 5 q^{24} + 11 q^{26} - 20 q^{27} + 5 q^{28} + 10 q^{29} - 9 q^{31} - 30 q^{32} - 5 q^{33} + 7 q^{34} + 3 q^{36} + 10 q^{37} - 20 q^{38} + 8 q^{39} - 4 q^{41} + 35 q^{42} - 18 q^{44} - 9 q^{46} - 30 q^{47} - 5 q^{48} - 4 q^{49} - 14 q^{51} - 5 q^{52} - 10 q^{53} - 20 q^{54} + 10 q^{57} + 30 q^{58} - 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} - 18 q^{66} - 10 q^{67} - 40 q^{68} + 3 q^{69} - 9 q^{71} + 15 q^{72} - 18 q^{74} - 10 q^{76} - 5 q^{77} + 30 q^{78} - 20 q^{79} + 8 q^{81} + 45 q^{82} - 40 q^{83} - 28 q^{84} - 24 q^{86} - 40 q^{87} + 40 q^{88} - 5 q^{89} + 6 q^{91} - 15 q^{92} + 40 q^{93} + 47 q^{94} + 71 q^{96} + 30 q^{98} - 22 q^{99}+O(q^{100}) $ 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9 + q^11 - 10 * q^12 - 10 * q^13 - 8 * q^14 + 13 * q^16 - 15 * q^17 + 5 * q^18 - 10 * q^19 - 14 * q^21 + 5 * q^22 - 30 * q^23 + 5 * q^24 + 11 * q^26 - 20 * q^27 + 5 * q^28 + 10 * q^29 - 9 * q^31 - 30 * q^32 - 5 * q^33 + 7 * q^34 + 3 * q^36 + 10 * q^37 - 20 * q^38 + 8 * q^39 - 4 * q^41 + 35 * q^42 - 18 * q^44 - 9 * q^46 - 30 * q^47 - 5 * q^48 - 4 * q^49 - 14 * q^51 - 5 * q^52 - 10 * q^53 - 20 * q^54 + 10 * q^57 + 30 * q^58 - 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 - 18 * q^66 - 10 * q^67 - 40 * q^68 + 3 * q^69 - 9 * q^71 + 15 * q^72 - 18 * q^74 - 10 * q^76 - 5 * q^77 + 30 * q^78 - 20 * q^79 + 8 * q^81 + 45 * q^82 - 40 * q^83 - 28 * q^84 - 24 * q^86 - 40 * q^87 + 40 * q^88 - 5 * q^89 + 6 * q^91 - 15 * q^92 + 40 * q^93 + 47 * q^94 + 71 * q^96 + 30 * q^98 - 22 * 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 $

$2$	−0.500989	−0.354253	−0.177126	−	0.984188i	$-0.556680\pi$
$2$	−0.500989	−0.354253	−0.177126	+	0.984188i	$0.556680\pi$
$3$	−3.09326	−1.78589	−0.892946	−	0.450163i	$-0.851366\pi$
$3$	−3.09326	−1.78589	−0.892946	+	0.450163i	$0.851366\pi$
$4$	−1.74901	−0.874505
$5$	0	0
$5$	0	0
$6$	1.54969	0.632658
$7$	−0.0237879	−0.00899097	−0.00449549	−	0.999990i	$-0.501431\pi$
$7$	−0.0237879	−0.00899097	−0.00449549	+	0.999990i	$0.501431\pi$
$8$	1.87821	0.664049
$9$	6.56824	2.18941
$10$	0	0
$11$	3.58329	1.08040	0.540201	−	0.841536i	$-0.318348\pi$
$11$	3.58329	1.08040	0.540201	+	0.841536i	$0.318348\pi$
$12$	5.41014	1.56177
$13$	−3.77587	−1.04724	−0.523619	−	0.851952i	$-0.675418\pi$
$13$	−3.77587	−1.04724	−0.523619	+	0.851952i	$0.675418\pi$
$14$	0.0119175	0.00318508
$15$	0	0
$16$	2.55705	0.639264
$17$	3.62303	0.878713	0.439357	−	0.898313i	$-0.355206\pi$
$17$	3.62303	0.878713	0.439357	+	0.898313i	$0.355206\pi$
$18$	−3.29062	−0.775606
$19$	−2.43084	−0.557672	−0.278836	−	0.960339i	$-0.589949\pi$
$19$	−2.43084	−0.557672	−0.278836	+	0.960339i	$0.589949\pi$
$20$	0	0
$21$	0.0735820	0.0160569
$22$	−1.79519	−0.382736
$23$	−1.71538	−0.357682	−0.178841	−	0.983878i	$-0.557235\pi$
$23$	−1.71538	−0.357682	−0.178841	+	0.983878i	$0.557235\pi$
$24$	−5.80980	−1.18592
$25$	0	0
$26$	1.89167	0.370987
$27$	−11.0375	−2.12416
$28$	0.0416052	0.00786265
$29$	3.85734	0.716290	0.358145	−	0.933666i	$-0.383409\pi$
$29$	3.85734	0.716290	0.358145	+	0.933666i	$0.383409\pi$
$30$	0	0
$31$	−6.00979	−1.07939	−0.539695	−	0.841861i	$-0.681460\pi$
$31$	−6.00979	−1.07939	−0.539695	+	0.841861i	$0.681460\pi$
$32$	−5.03748	−0.890510
$33$	−11.0840	−1.92948
$34$	−1.81510	−0.311287
$35$	0	0
$36$	−11.4879	−1.91465
$37$	0.369309	0.0607139	0.0303570	−	0.999539i	$-0.490336\pi$
$37$	0.369309	0.0607139	0.0303570	+	0.999539i	$0.490336\pi$
$38$	1.21782	0.197557
$39$	11.6797	1.87025
$40$	0	0
$41$	−7.80900	−1.21956	−0.609780	−	0.792570i	$-0.708742\pi$
$41$	−7.80900	−1.21956	−0.609780	+	0.792570i	$0.708742\pi$
$42$	−0.0368638	−0.00568821
$43$	−0.174574	−0.0266224	−0.0133112	−	0.999911i	$-0.504237\pi$
$43$	−0.174574	−0.0266224	−0.0133112	+	0.999911i	$0.504237\pi$
$44$	−6.26720	−0.944817
$45$	0	0
$46$	0.859388	0.126710
$47$	−7.81082	−1.13932	−0.569662	−	0.821879i	$-0.692926\pi$
$47$	−7.81082	−1.13932	−0.569662	+	0.821879i	$0.692926\pi$
$48$	−7.90962	−1.14166
$49$	−6.99943	−0.999919
$50$	0	0
$51$	−11.2070	−1.56929
$52$	6.60403	0.915815
$53$	8.97184	1.23238	0.616189	−	0.787599i	$-0.288676\pi$
$53$	8.97184	1.23238	0.616189	+	0.787599i	$0.288676\pi$
$54$	5.52966	0.752491
$55$	0	0
$56$	−0.0446787	−0.00597045
$57$	7.51920	0.995943
$58$	−1.93249	−0.253748
$59$	−4.45536	−0.580038	−0.290019	−	0.957021i	$-0.593662\pi$
$59$	−4.45536	−0.580038	−0.290019	+	0.957021i	$0.593662\pi$
$60$	0	0
$61$	9.21403	1.17974	0.589868	−	0.807500i	$-0.299180\pi$
$61$	9.21403	1.17974	0.589868	+	0.807500i	$0.299180\pi$
$62$	3.01084	0.382377
$63$	−0.156244	−0.0196849
$64$	−2.59038	−0.323798
$65$	0	0
$66$	5.55298	0.683525
$67$	4.47385	0.546568	0.273284	−	0.961933i	$-0.411890\pi$
$67$	4.47385	0.546568	0.273284	+	0.961933i	$0.411890\pi$
$68$	−6.33671	−0.768439
$69$	5.30612	0.638782
$70$	0	0
$71$	−9.69458	−1.15054	−0.575268	−	0.817965i	$-0.695102\pi$
$71$	−9.69458	−1.15054	−0.575268	+	0.817965i	$0.695102\pi$
$72$	12.3366	1.45388
$73$	3.95387	0.462765	0.231383	−	0.972863i	$-0.425675\pi$
$73$	3.95387	0.462765	0.231383	+	0.972863i	$0.425675\pi$
$74$	−0.185020	−0.0215081
$75$	0	0
$76$	4.25156	0.487687
$77$	−0.0852388	−0.00971386
$78$	−5.85142	−0.662543
$79$	−9.68349	−1.08948	−0.544739	−	0.838606i	$-0.683371\pi$
$79$	−9.68349	−1.08948	−0.544739	+	0.838606i	$0.683371\pi$
$80$	0	0
$81$	14.4370	1.60411
$82$	3.91223	0.432033
$83$	8.95717	0.983177	0.491589	−	0.870828i	$-0.336416\pi$
$83$	8.95717	0.983177	0.491589	+	0.870828i	$0.336416\pi$
$84$	−0.128696	−0.0140418
$85$	0	0
$86$	0.0874600	0.00943105
$87$	−11.9317	−1.27922
$88$	6.73018	0.717440
$89$	−17.0151	−1.80359	−0.901797	−	0.432161i	$-0.857751\pi$
$89$	−17.0151	−1.80359	−0.901797	+	0.432161i	$0.857751\pi$
$90$	0	0
$91$	0.0898200	0.00941569
$92$	3.00022	0.312795
$93$	18.5898	1.92767
$94$	3.91314	0.403609
$95$	0	0
$96$	15.5822	1.59036
$97$	−2.76438	−0.280680	−0.140340	−	0.990103i	$-0.544820\pi$
$97$	−2.76438	−0.280680	−0.140340	+	0.990103i	$0.544820\pi$
$98$	3.50664	0.354224
$99$	23.5359	2.36545
$100$	0	0
$101$	10.3526	1.03012	0.515062	−	0.857153i	$-0.327769\pi$
$101$	10.3526	1.03012	0.515062	+	0.857153i	$0.327769\pi$
$102$	5.61457	0.555925
$103$	−18.2913	−1.80230	−0.901150	−	0.433508i	$-0.857276\pi$
$103$	−18.2913	−1.80230	−0.901150	+	0.433508i	$0.857276\pi$
$104$	−7.09189	−0.695417
$105$	0	0
$106$	−4.49480	−0.436573
$107$	−15.8786	−1.53504	−0.767522	−	0.641023i	$-0.778510\pi$
$107$	−15.8786	−1.53504	−0.767522	+	0.641023i	$0.778510\pi$
$108$	19.3046	1.85759
$109$	5.96109	0.570969	0.285484	−	0.958383i	$-0.407845\pi$
$109$	5.96109	0.570969	0.285484	+	0.958383i	$0.407845\pi$
$110$	0	0
$111$	−1.14237	−0.108429
$112$	−0.0608269	−0.00574760
$113$	−2.42830	−0.228435	−0.114218	−	0.993456i	$-0.536436\pi$
$113$	−2.42830	−0.228435	−0.114218	+	0.993456i	$0.536436\pi$
$114$	−3.76704	−0.352816
$115$	0	0
$116$	−6.74652	−0.626399
$117$	−24.8008	−2.29284
$118$	2.23209	0.205480
$119$	−0.0861842	−0.00790049
$120$	0	0
$121$	1.83995	0.167268
$122$	−4.61613	−0.417925
$123$	24.1552	2.17800
$124$	10.5112	0.943932
$125$	0	0
$126$	0.0782768	0.00697345
$127$	17.1073	1.51803	0.759013	−	0.651075i	$-0.225682\pi$
$127$	17.1073	1.51803	0.759013	+	0.651075i	$0.225682\pi$
$128$	11.3727	1.00522
$129$	0.540004	0.0475447
$130$	0	0
$131$	4.78955	0.418464	0.209232	−	0.977866i	$-0.432904\pi$
$131$	4.78955	0.418464	0.209232	+	0.977866i	$0.432904\pi$
$132$	19.3861	1.68734
$133$	0.0578245	0.00501402
$134$	−2.24135	−0.193623
$135$	0	0
$136$	6.80482	0.583509
$137$	−12.8104	−1.09446	−0.547231	−	0.836982i	$-0.684318\pi$
$137$	−12.8104	−1.09446	−0.547231	+	0.836982i	$0.684318\pi$
$138$	−2.65831	−0.226290
$139$	−7.94020	−0.673479	−0.336739	−	0.941598i	$-0.609324\pi$
$139$	−7.94020	−0.673479	−0.336739	+	0.941598i	$0.609324\pi$
$140$	0	0
$141$	24.1609	2.03471
$142$	4.85688	0.407581
$143$	−13.5300	−1.13144
$144$	16.7953	1.39961
$145$	0	0
$146$	−1.98085	−0.163936
$147$	21.6510	1.78575
$148$	−0.645924	−0.0530946
$149$	−5.62724	−0.461002	−0.230501	−	0.973072i	$-0.574036\pi$
$149$	−5.62724	−0.461002	−0.230501	+	0.973072i	$0.574036\pi$
$150$	0	0
$151$	−7.36960	−0.599730	−0.299865	−	0.953982i	$-0.596942\pi$
$151$	−7.36960	−0.599730	−0.299865	+	0.953982i	$0.596942\pi$
$152$	−4.56563	−0.370322
$153$	23.7969	1.92387
$154$	0.0427037	0.00344116
$155$	0	0
$156$	−20.4280	−1.63555
$157$	−8.63091	−0.688822	−0.344411	−	0.938819i	$-0.611921\pi$
$157$	−8.63091	−0.688822	−0.344411	+	0.938819i	$0.611921\pi$
$158$	4.85133	0.385951
$159$	−27.7522	−2.20089
$160$	0	0
$161$	0.0408053	0.00321591
$162$	−7.23280	−0.568262
$163$	−4.74964	−0.372020	−0.186010	−	0.982548i	$-0.559556\pi$
$163$	−4.74964	−0.372020	−0.186010	+	0.982548i	$0.559556\pi$
$164$	13.6580	1.06651
$165$	0	0
$166$	−4.48745	−0.348293
$167$	−18.5967	−1.43906	−0.719528	−	0.694463i	$-0.755642\pi$
$167$	−18.5967	−1.43906	−0.719528	+	0.694463i	$0.755642\pi$
$168$	0.138203	0.0106626
$169$	1.25720	0.0967079
$170$	0	0
$171$	−15.9663	−1.22097
$172$	0.305332	0.0232814
$173$	−15.7957	−1.20092	−0.600462	−	0.799653i	$-0.705017\pi$
$173$	−15.7957	−1.20092	−0.600462	+	0.799653i	$0.705017\pi$
$174$	5.97767	0.453166
$175$	0	0
$176$	9.16266	0.690661
$177$	13.7816	1.03589
$178$	8.52437	0.638928
$179$	9.36946	0.700306	0.350153	−	0.936692i	$-0.386130\pi$
$179$	9.36946	0.700306	0.350153	+	0.936692i	$0.386130\pi$
$180$	0	0
$181$	−5.84630	−0.434552	−0.217276	−	0.976110i	$-0.569717\pi$
$181$	−5.84630	−0.434552	−0.217276	+	0.976110i	$0.569717\pi$
$182$	−0.0449988	−0.00333554
$183$	−28.5014	−2.10688
$184$	−3.22186	−0.237518
$185$	0	0
$186$	−9.31330	−0.682884
$187$	12.9824	0.949364
$188$	13.6612	0.996345
$189$	0.262558	0.0190983
$190$	0	0
$191$	−21.8947	−1.58425	−0.792123	−	0.610362i	$-0.791024\pi$
$191$	−21.8947	−1.58425	−0.792123	+	0.610362i	$0.791024\pi$
$192$	8.01272	0.578268
$193$	25.2541	1.81783	0.908916	−	0.416980i	$-0.136911\pi$
$193$	25.2541	1.81783	0.908916	+	0.416980i	$0.136911\pi$
$194$	1.38492	0.0994317
$195$	0	0
$196$	12.2421	0.874434
$197$	−14.4538	−1.02979	−0.514897	−	0.857252i	$-0.672170\pi$
$197$	−14.4538	−1.02979	−0.514897	+	0.857252i	$0.672170\pi$
$198$	−11.7912	−0.837966
$199$	−3.77734	−0.267768	−0.133884	−	0.990997i	$-0.542745\pi$
$199$	−3.77734	−0.267768	−0.133884	+	0.990997i	$0.542745\pi$
$200$	0	0
$201$	−13.8388	−0.976112
$202$	−5.18655	−0.364924
$203$	−0.0917579	−0.00644014
$204$	19.6011	1.37235
$205$	0	0
$206$	9.16377	0.638470
$207$	−11.2670	−0.783113
$208$	−9.65511	−0.669461
$209$	−8.71039	−0.602510
$210$	0	0
$211$	−18.9006	−1.30117	−0.650586	−	0.759432i	$-0.725477\pi$
$211$	−18.9006	−1.30117	−0.650586	+	0.759432i	$0.725477\pi$
$212$	−15.6918	−1.07772
$213$	29.9878	2.05473
$214$	7.95502	0.543794
$215$	0	0
$216$	−20.7307	−1.41055
$217$	0.142960	0.00970477
$218$	−2.98644	−0.202267
$219$	−12.2303	−0.826449
$220$	0	0
$221$	−13.6801	−0.920222
$222$	0.572313	0.0384111
$223$	11.3556	0.760426	0.380213	−	0.924899i	$-0.375851\pi$
$223$	11.3556	0.760426	0.380213	+	0.924899i	$0.375851\pi$
$224$	0.119831	0.00800655
$225$	0	0
$226$	1.21655	0.0809239
$227$	−11.1977	−0.743216	−0.371608	−	0.928390i	$-0.621194\pi$
$227$	−11.1977	−0.743216	−0.371608	+	0.928390i	$0.621194\pi$
$228$	−13.1512	−0.870957
$229$	9.64969	0.637669	0.318835	−	0.947810i	$-0.396709\pi$
$229$	9.64969	0.637669	0.318835	+	0.947810i	$0.396709\pi$
$230$	0	0
$231$	0.263665	0.0173479
$232$	7.24491	0.475651
$233$	15.5261	1.01715	0.508573	−	0.861019i	$-0.330173\pi$
$233$	15.5261	1.01715	0.508573	+	0.861019i	$0.330173\pi$
$234$	12.4249	0.812244
$235$	0	0
$236$	7.79246	0.507246
$237$	29.9535	1.94569
$238$	0.0431773	0.00279877
$239$	14.3342	0.927203	0.463601	−	0.886044i	$-0.346557\pi$
$239$	14.3342	0.927203	0.463601	+	0.886044i	$0.346557\pi$
$240$	0	0
$241$	25.3114	1.63045	0.815224	−	0.579145i	$-0.196614\pi$
$241$	25.3114	1.63045	0.815224	+	0.579145i	$0.196614\pi$
$242$	−0.921794	−0.0592552
$243$	−11.5450	−0.740613
$244$	−16.1154	−1.03168
$245$	0	0
$246$	−12.1015	−0.771565
$247$	9.17853	0.584016
$248$	−11.2877	−0.716768
$249$	−27.7068	−1.75585
$250$	0	0
$251$	4.23698	0.267436	0.133718	−	0.991019i	$-0.457308\pi$
$251$	4.23698	0.267436	0.133718	+	0.991019i	$0.457308\pi$
$252$	0.273273	0.0172146
$253$	−6.14671	−0.386440
$254$	−8.57057	−0.537766
$255$	0	0
$256$	−0.516849	−0.0323031
$257$	20.4007	1.27256	0.636281	−	0.771458i	$-0.280472\pi$
$257$	20.4007	1.27256	0.636281	+	0.771458i	$0.280472\pi$
$258$	−0.270536	−0.0168428
$259$	−0.00878507	−0.000545877 0
$260$	0	0
$261$	25.3359	1.56825
$262$	−2.39951	−0.148242
$263$	−28.6678	−1.76773	−0.883865	−	0.467741i	$-0.845068\pi$
$263$	−28.6678	−1.76773	−0.883865	+	0.467741i	$0.845068\pi$
$264$	−20.8182	−1.28127
$265$	0	0
$266$	−0.0289694	−0.00177623
$267$	52.6320	3.22102
$268$	−7.82481	−0.477977
$269$	7.94652	0.484508	0.242254	−	0.970213i	$-0.422113\pi$
$269$	7.94652	0.484508	0.242254	+	0.970213i	$0.422113\pi$
$270$	0	0
$271$	9.55487	0.580417	0.290208	−	0.956963i	$-0.406275\pi$
$271$	9.55487	0.580417	0.290208	+	0.956963i	$0.406275\pi$
$272$	9.26428	0.561729
$273$	−0.277836	−0.0168154
$274$	6.41785	0.387717
$275$	0	0
$276$	−9.28045	−0.558618
$277$	−17.1904	−1.03287	−0.516436	−	0.856326i	$-0.672742\pi$
$277$	−17.1904	−1.03287	−0.516436	+	0.856326i	$0.672742\pi$
$278$	3.97795	0.238582
$279$	−39.4737	−2.36323
$280$	0	0
$281$	27.2182	1.62370	0.811851	−	0.583864i	$-0.198460\pi$
$281$	27.2182	1.62370	0.811851	+	0.583864i	$0.198460\pi$
$282$	−12.1043	−0.720803
$283$	3.47901	0.206806	0.103403	−	0.994640i	$-0.467027\pi$
$283$	3.47901	0.206806	0.103403	+	0.994640i	$0.467027\pi$
$284$	16.9559	1.00615
$285$	0	0
$286$	6.77840	0.400815
$287$	0.185760	0.0109650
$288$	−33.0874	−1.94969
$289$	−3.87366	−0.227863
$290$	0	0
$291$	8.55092	0.501264
$292$	−6.91535	−0.404690
$293$	−23.4941	−1.37254	−0.686271	−	0.727346i	$-0.740754\pi$
$293$	−23.4941	−1.37254	−0.686271	+	0.727346i	$0.740754\pi$
$294$	−10.8469	−0.632607
$295$	0	0
$296$	0.693640	0.0403170
$297$	−39.5504	−2.29495
$298$	2.81919	0.163311
$299$	6.47706	0.374578
$300$	0	0
$301$	0.00415276	0.000239361 0
$302$	3.69209	0.212456
$303$	−32.0233	−1.83969
$304$	−6.21578	−0.356500
$305$	0	0
$306$	−11.9220	−0.681535
$307$	−1.11253	−0.0634952	−0.0317476	−	0.999496i	$-0.510107\pi$
$307$	−1.11253	−0.0634952	−0.0317476	+	0.999496i	$0.510107\pi$
$308$	0.149083	0.00849482
$309$	56.5798	3.21871
$310$	0	0
$311$	−14.7529	−0.836561	−0.418280	−	0.908318i	$-0.637367\pi$
$311$	−14.7529	−0.836561	−0.418280	+	0.908318i	$0.637367\pi$
$312$	21.9370	1.24194
$313$	−4.94834	−0.279697	−0.139848	−	0.990173i	$-0.544661\pi$
$313$	−4.94834	−0.279697	−0.139848	+	0.990173i	$0.544661\pi$
$314$	4.32400	0.244017
$315$	0	0
$316$	16.9365	0.952754
$317$	−22.6855	−1.27414	−0.637072	−	0.770804i	$-0.719855\pi$
$317$	−22.6855	−1.27414	−0.637072	+	0.770804i	$0.719855\pi$
$318$	13.9036	0.779673
$319$	13.8220	0.773881
$320$	0	0
$321$	49.1166	2.74142
$322$	−0.0204430	−0.00113925
$323$	−8.80699	−0.490034
$324$	−25.2505	−1.40281
$325$	0	0
$326$	2.37952	0.131789
$327$	−18.4392	−1.01969
$328$	−14.6670	−0.809848
$329$	0.185803	0.0102436
$330$	0	0
$331$	1.73258	0.0952312	0.0476156	−	0.998866i	$-0.484838\pi$
$331$	1.73258	0.0952312	0.0476156	+	0.998866i	$0.484838\pi$
$332$	−15.6662	−0.859793
$333$	2.42571	0.132928
$334$	9.31676	0.509790
$335$	0	0
$336$	0.188153	0.0102646
$337$	−13.9811	−0.761596	−0.380798	−	0.924658i	$-0.624351\pi$
$337$	−13.9811	−0.761596	−0.380798	+	0.924658i	$0.624351\pi$
$338$	−0.629845	−0.0342590
$339$	7.51136	0.407961
$340$	0	0
$341$	−21.5348	−1.16617
$342$	7.99895	0.432534
$343$	0.333017	0.0179812
$344$	−0.327888	−0.0176785
$345$	0	0
$346$	7.91347	0.425431
$347$	−5.08173	−0.272802	−0.136401	−	0.990654i	$-0.543553\pi$
$347$	−5.08173	−0.272802	−0.136401	+	0.990654i	$0.543553\pi$
$348$	20.8687	1.11868
$349$	−8.88643	−0.475680	−0.237840	−	0.971304i	$-0.576439\pi$
$349$	−8.88643	−0.475680	−0.237840	+	0.971304i	$0.576439\pi$
$350$	0	0
$351$	41.6761	2.22450
$352$	−18.0508	−0.962108
$353$	−9.41440	−0.501078	−0.250539	−	0.968107i	$-0.580608\pi$
$353$	−9.41440	−0.501078	−0.250539	+	0.968107i	$0.580608\pi$
$354$	−6.90442	−0.366966
$355$	0	0
$356$	29.7595	1.57725
$357$	0.266590	0.0141094
$358$	−4.69400	−0.248086
$359$	9.11498	0.481070	0.240535	−	0.970640i	$-0.422677\pi$
$359$	9.11498	0.481070	0.240535	+	0.970640i	$0.422677\pi$
$360$	0	0
$361$	−13.0910	−0.689002
$362$	2.92893	0.153941
$363$	−5.69143	−0.298723
$364$	−0.157096	−0.00823407
$365$	0	0
$366$	14.2789	0.746369
$367$	17.8784	0.933246	0.466623	−	0.884456i	$-0.345471\pi$
$367$	17.8784	0.933246	0.466623	+	0.884456i	$0.345471\pi$
$368$	−4.38633	−0.228653
$369$	−51.2914	−2.67012
$370$	0	0
$371$	−0.213421	−0.0110803
$372$	−32.5138	−1.68576
$373$	3.07395	0.159163	0.0795814	−	0.996828i	$-0.474642\pi$
$373$	3.07395	0.159163	0.0795814	+	0.996828i	$0.474642\pi$
$374$	−6.50402	−0.336315
$375$	0	0
$376$	−14.6704	−0.756567
$377$	−14.5648	−0.750126
$378$	−0.131539	−0.00676563
$379$	25.5866	1.31430	0.657149	−	0.753761i	$-0.271762\pi$
$379$	25.5866	1.31430	0.657149	+	0.753761i	$0.271762\pi$
$380$	0	0
$381$	−52.9173	−2.71103
$382$	10.9690	0.561224
$383$	2.22529	0.113707	0.0568535	−	0.998383i	$-0.481893\pi$
$383$	2.22529	0.113707	0.0568535	+	0.998383i	$0.481893\pi$
$384$	−35.1788	−1.79521
$385$	0	0
$386$	−12.6520	−0.643972
$387$	−1.14665	−0.0582873
$388$	4.83492	0.245456
$389$	4.64413	0.235467	0.117733	−	0.993045i	$-0.462437\pi$
$389$	4.64413	0.235467	0.117733	+	0.993045i	$0.462437\pi$
$390$	0	0
$391$	−6.21488	−0.314300
$392$	−13.1464	−0.663995
$393$	−14.8153	−0.747333
$394$	7.24122	0.364807
$395$	0	0
$396$	−41.1645	−2.06859
$397$	37.9452	1.90442	0.952208	−	0.305452i	$-0.0988074\pi$
$397$	37.9452	1.90442	0.952208	+	0.305452i	$0.0988074\pi$
$398$	1.89241	0.0948577
$399$	−0.178866	−0.00895449
$400$	0	0
$401$	−16.7187	−0.834890	−0.417445	−	0.908702i	$-0.637074\pi$
$401$	−16.7187	−0.834890	−0.417445	+	0.908702i	$0.637074\pi$
$402$	6.93308	0.345791
$403$	22.6922	1.13038
$404$	−18.1068	−0.900848
$405$	0	0
$406$	0.0459697	0.00228144
$407$	1.32334	0.0655955
$408$	−21.0491	−1.04208
$409$	0.473132	0.0233949	0.0116974	−	0.999932i	$-0.496277\pi$
$409$	0.473132	0.0233949	0.0116974	+	0.999932i	$0.496277\pi$
$410$	0	0
$411$	39.6257	1.95459
$412$	31.9917	1.57612
$413$	0.105983	0.00521511
$414$	5.64467	0.277420
$415$	0	0
$416$	19.0209	0.932576
$417$	24.5611	1.20276
$418$	4.36381	0.213441
$419$	24.2348	1.18395	0.591975	−	0.805957i	$-0.298349\pi$
$419$	24.2348	1.18395	0.591975	+	0.805957i	$0.298349\pi$
$420$	0	0
$421$	−31.3482	−1.52782	−0.763908	−	0.645325i	$-0.776722\pi$
$421$	−31.3482	−1.52782	−0.763908	+	0.645325i	$0.776722\pi$
$422$	9.46901	0.460944
$423$	−51.3033	−2.49445
$424$	16.8510	0.818359
$425$	0	0
$426$	−15.0236	−0.727895
$427$	−0.219182	−0.0106070
$428$	27.7719	1.34240
$429$	41.8519	2.02063
$430$	0	0
$431$	−2.00066	−0.0963686	−0.0481843	−	0.998838i	$-0.515343\pi$
$431$	−2.00066	−0.0963686	−0.0481843	+	0.998838i	$0.515343\pi$
$432$	−28.2234	−1.35790
$433$	−7.07253	−0.339884	−0.169942	−	0.985454i	$-0.554358\pi$
$433$	−7.07253	−0.339884	−0.169942	+	0.985454i	$0.554358\pi$
$434$	−0.0716215	−0.00343794
$435$	0	0
$436$	−10.4260	−0.499315
$437$	4.16981	0.199469
$438$	6.12726	0.292772
$439$	12.9620	0.618644	0.309322	−	0.950957i	$-0.399898\pi$
$439$	12.9620	0.618644	0.309322	+	0.950957i	$0.399898\pi$
$440$	0	0
$441$	−45.9739	−2.18924
$442$	6.85358	0.325991
$443$	−21.8687	−1.03901	−0.519506	−	0.854467i	$-0.673884\pi$
$443$	−21.8687	−1.03901	−0.519506	+	0.854467i	$0.673884\pi$
$444$	1.99801	0.0948213
$445$	0	0
$446$	−5.68902	−0.269383
$447$	17.4065	0.823299
$448$	0.0616197	0.00291126
$449$	0.399626	0.0188595	0.00942976	−	0.999956i	$-0.496998\pi$
$449$	0.399626	0.0188595	0.00942976	+	0.999956i	$0.496998\pi$
$450$	0	0
$451$	−27.9819	−1.31762
$452$	4.24712	0.199768
$453$	22.7961	1.07105
$454$	5.60992	0.263287
$455$	0	0
$456$	14.1227	0.661355
$457$	35.2247	1.64774	0.823872	−	0.566777i	$-0.191810\pi$
$457$	35.2247	1.64774	0.823872	+	0.566777i	$0.191810\pi$
$458$	−4.83439	−0.225896
$459$	−39.9891	−1.86653
$460$	0	0
$461$	−15.9615	−0.743402	−0.371701	−	0.928353i	$-0.621225\pi$
$461$	−15.9615	−0.743402	−0.371701	+	0.928353i	$0.621225\pi$
$462$	−0.132094	−0.00614555
$463$	30.4831	1.41667	0.708336	−	0.705875i	$-0.249446\pi$
$463$	30.4831	1.41667	0.708336	+	0.705875i	$0.249446\pi$
$464$	9.86342	0.457898
$465$	0	0
$466$	−7.77839	−0.360327
$467$	−6.64657	−0.307567	−0.153783	−	0.988105i	$-0.549146\pi$
$467$	−6.64657	−0.307567	−0.153783	+	0.988105i	$0.549146\pi$
$468$	43.3769	2.00510
$469$	−0.106424	−0.00491418
$470$	0	0
$471$	26.6976	1.23016
$472$	−8.36811	−0.385174
$473$	−0.625551	−0.0287628
$474$	−15.0064	−0.689267
$475$	0	0
$476$	0.150737	0.00690902
$477$	58.9292	2.69818
$478$	−7.18129	−0.328464
$479$	22.2408	1.01621	0.508104	−	0.861296i	$-0.330347\pi$
$479$	22.2408	1.01621	0.508104	+	0.861296i	$0.330347\pi$
$480$	0	0
$481$	−1.39446	−0.0635820
$482$	−12.6807	−0.577591
$483$	−0.126221	−0.00574327
$484$	−3.21809	−0.146277
$485$	0	0
$486$	5.78393	0.262364
$487$	−10.1851	−0.461531	−0.230765	−	0.973009i	$-0.574123\pi$
$487$	−10.1851	−0.461531	−0.230765	+	0.973009i	$0.574123\pi$
$488$	17.3059	0.783402
$489$	14.6918	0.664388
$490$	0	0
$491$	6.64004	0.299661	0.149830	−	0.988712i	$-0.452127\pi$
$491$	6.64004	0.299661	0.149830	+	0.988712i	$0.452127\pi$
$492$	−42.2478	−1.90468
$493$	13.9752	0.629413
$494$	−4.59834	−0.206889
$495$	0	0
$496$	−15.3674	−0.690015
$497$	0.230614	0.0103444
$498$	13.8808	0.622015
$499$	1.08397	0.0485253	0.0242626	−	0.999706i	$-0.492276\pi$
$499$	1.08397	0.0485253	0.0242626	+	0.999706i	$0.492276\pi$
$500$	0	0
$501$	57.5244	2.57000
$502$	−2.12268	−0.0947399
$503$	1.98603	0.0885525	0.0442762	−	0.999019i	$-0.485902\pi$
$503$	1.98603	0.0885525	0.0442762	+	0.999019i	$0.485902\pi$
$504$	−0.293460	−0.0130718
$505$	0	0
$506$	3.07944	0.136898
$507$	−3.88885	−0.172710
$508$	−29.9208	−1.32752
$509$	2.30915	0.102351	0.0511757	−	0.998690i	$-0.483703\pi$
$509$	2.30915	0.102351	0.0511757	+	0.998690i	$0.483703\pi$
$510$	0	0
$511$	−0.0940541	−0.00416071
$512$	−22.4865	−0.993773
$513$	26.8303	1.18459
$514$	−10.2205	−0.450809
$515$	0	0
$516$	−0.944472	−0.0415780
$517$	−27.9884	−1.23093
$518$	0.00440122	0.000193379 0
$519$	48.8601	2.14472
$520$	0	0
$521$	−11.0447	−0.483879	−0.241940	−	0.970291i	$-0.577784\pi$
$521$	−11.0447	−0.483879	−0.241940	+	0.970291i	$0.577784\pi$
$522$	−12.6930	−0.555559
$523$	−26.0025	−1.13701	−0.568505	−	0.822680i	$-0.692478\pi$
$523$	−26.0025	−1.13701	−0.568505	+	0.822680i	$0.692478\pi$
$524$	−8.37696	−0.365949
$525$	0	0
$526$	14.3623	0.626224
$527$	−21.7736	−0.948475
$528$	−28.3425	−1.23345
$529$	−20.0575	−0.872064
$530$	0	0
$531$	−29.2638	−1.26994
$532$	−0.101136	−0.00438478
$533$	29.4858	1.27717
$534$	−26.3681	−1.14106
$535$	0	0
$536$	8.40286	0.362948
$537$	−28.9821	−1.25067
$538$	−3.98112	−0.171638
$539$	−25.0810	−1.08031
$540$	0	0
$541$	−0.0182391	−0.000784162 0	−0.000392081	−	1.00000i	$-0.500125\pi$
$541$	−0.0182391	−0.000784162 0	−0.000392081		1.00000i	$0.500125\pi$
$542$	−4.78689	−0.205614
$543$	18.0841	0.776063
$544$	−18.2510	−0.782503
$545$	0	0
$546$	0.139193	0.00595691
$547$	9.96048	0.425879	0.212940	−	0.977065i	$-0.431696\pi$
$547$	9.96048	0.425879	0.212940	+	0.977065i	$0.431696\pi$
$548$	22.4054	0.957113
$549$	60.5199	2.58293
$550$	0	0
$551$	−9.37656	−0.399455
$552$	9.96602	0.424182
$553$	0.230350	0.00979547
$554$	8.61222	0.365898
$555$	0	0
$556$	13.8875	0.588960
$557$	34.9291	1.47999	0.739996	−	0.672611i	$-0.234827\pi$
$557$	34.9291	1.47999	0.739996	+	0.672611i	$0.234827\pi$
$558$	19.7759	0.837181
$559$	0.659171	0.0278800
$560$	0	0
$561$	−40.1577	−1.69546
$562$	−13.6360	−0.575201
$563$	−15.7939	−0.665634	−0.332817	−	0.942991i	$-0.607999\pi$
$563$	−15.7939	−0.665634	−0.332817	+	0.942991i	$0.607999\pi$
$564$	−42.2576	−1.77937
$565$	0	0
$566$	−1.74295	−0.0732616
$567$	−0.343426	−0.0144225
$568$	−18.2085	−0.764012
$569$	−14.5605	−0.610408	−0.305204	−	0.952287i	$-0.598725\pi$
$569$	−14.5605	−0.610408	−0.305204	+	0.952287i	$0.598725\pi$
$570$	0	0
$571$	27.6138	1.15560	0.577802	−	0.816177i	$-0.303911\pi$
$571$	27.6138	1.15560	0.577802	+	0.816177i	$0.303911\pi$
$572$	23.6642	0.989448
$573$	67.7259	2.82929
$574$	−0.0930636	−0.00388440
$575$	0	0
$576$	−17.0142	−0.708927
$577$	27.1520	1.13035	0.565176	−	0.824970i	$-0.308808\pi$
$577$	27.1520	1.13035	0.565176	+	0.824970i	$0.308808\pi$
$578$	1.94066	0.0807210
$579$	−78.1175	−3.24645
$580$	0	0
$581$	−0.213072	−0.00883972
$582$	−4.28392	−0.177574
$583$	32.1487	1.33146
$584$	7.42621	0.307299
$585$	0	0
$586$	11.7703	0.486227
$587$	10.7037	0.441787	0.220894	−	0.975298i	$-0.429103\pi$
$587$	10.7037	0.441787	0.220894	+	0.975298i	$0.429103\pi$
$588$	−37.8679	−1.56165
$589$	14.6088	0.601946
$590$	0	0
$591$	44.7094	1.83910
$592$	0.944342	0.0388122
$593$	3.84629	0.157948	0.0789740	−	0.996877i	$-0.474836\pi$
$593$	3.84629	0.157948	0.0789740	+	0.996877i	$0.474836\pi$
$594$	19.8143	0.812993
$595$	0	0
$596$	9.84210	0.403148
$597$	11.6843	0.478206
$598$	−3.24494	−0.132695
$599$	46.1423	1.88532	0.942662	−	0.333750i	$-0.108314\pi$
$599$	46.1423	1.88532	0.942662	+	0.333750i	$0.108314\pi$
$600$	0	0
$601$	−13.3119	−0.543005	−0.271503	−	0.962438i	$-0.587521\pi$
$601$	−13.3119	−0.543005	−0.271503	+	0.962438i	$0.587521\pi$
$602$	−0.00208049	−8.47943e−5 0
$603$	29.3853	1.19666
$604$	12.8895	0.524466
$605$	0	0
$606$	16.0433	0.651716
$607$	−34.0838	−1.38342	−0.691709	−	0.722176i	$-0.743142\pi$
$607$	−34.0838	−1.38342	−0.691709	+	0.722176i	$0.743142\pi$
$608$	12.2453	0.496613
$609$	0.283831	0.0115014
$610$	0	0
$611$	29.4926	1.19314
$612$	−41.6210	−1.68243
$613$	−25.4809	−1.02916	−0.514582	−	0.857441i	$-0.672053\pi$
$613$	−25.4809	−1.02916	−0.514582	+	0.857441i	$0.672053\pi$
$614$	0.557363	0.0224933
$615$	0	0
$616$	−0.160097	−0.00645048
$617$	26.9453	1.08478	0.542389	−	0.840127i	$-0.317520\pi$
$617$	26.9453	1.08478	0.542389	+	0.840127i	$0.317520\pi$
$618$	−28.3459	−1.14024
$619$	39.3289	1.58076	0.790381	−	0.612615i	$-0.209882\pi$
$619$	39.3289	1.58076	0.790381	+	0.612615i	$0.209882\pi$
$620$	0	0
$621$	18.9335	0.759775
$622$	7.39105	0.296354
$623$	0.404752	0.0162161
$624$	29.8657	1.19559
$625$	0	0
$626$	2.47906	0.0990833
$627$	26.9435	1.07602
$628$	15.0956	0.602378
$629$	1.33802	0.0533502
$630$	0	0
$631$	11.2443	0.447630	0.223815	−	0.974632i	$-0.428149\pi$
$631$	11.2443	0.447630	0.223815	+	0.974632i	$0.428149\pi$
$632$	−18.1877	−0.723467
$633$	58.4645	2.32375
$634$	11.3652	0.451369
$635$	0	0
$636$	48.5389	1.92469
$637$	26.4290	1.04715
$638$	−6.92465	−0.274150
$639$	−63.6763	−2.51900
$640$	0	0
$641$	17.4773	0.690311	0.345155	−	0.938546i	$-0.387826\pi$
$641$	17.4773	0.690311	0.345155	+	0.938546i	$0.387826\pi$
$642$	−24.6069	−0.971157
$643$	8.72320	0.344009	0.172005	−	0.985096i	$-0.444976\pi$
$643$	8.72320	0.344009	0.172005	+	0.985096i	$0.444976\pi$
$644$	−0.0713689	−0.00281233
$645$	0	0
$646$	4.41221	0.173596
$647$	−25.0654	−0.985421	−0.492710	−	0.870193i	$-0.663994\pi$
$647$	−25.0654	−0.985421	−0.492710	+	0.870193i	$0.663994\pi$
$648$	27.1158	1.06521
$649$	−15.9648	−0.626674
$650$	0	0
$651$	−0.442212	−0.0173317
$652$	8.30716	0.325334
$653$	−29.9777	−1.17312	−0.586559	−	0.809906i	$-0.699518\pi$
$653$	−29.9777	−1.17312	−0.586559	+	0.809906i	$0.699518\pi$
$654$	9.23784	0.361228
$655$	0	0
$656$	−19.9680	−0.779621
$657$	25.9699	1.01318
$658$	−0.0930852	−0.00362884
$659$	−4.36601	−0.170076	−0.0850379	−	0.996378i	$-0.527101\pi$
$659$	−4.36601	−0.170076	−0.0850379	+	0.996378i	$0.527101\pi$
$660$	0	0
$661$	−19.4529	−0.756628	−0.378314	−	0.925677i	$-0.623496\pi$
$661$	−19.4529	−0.756628	−0.378314	+	0.925677i	$0.623496\pi$
$662$	−0.868004	−0.0337359
$663$	42.3160	1.64342
$664$	16.8235	0.652878
$665$	0	0
$666$	−1.21525	−0.0470901
$667$	−6.61681	−0.256204
$668$	32.5258	1.25846
$669$	−35.1257	−1.35804
$670$	0	0
$671$	33.0165	1.27459
$672$	−0.370668	−0.0142988
$673$	36.2275	1.39647	0.698234	−	0.715870i	$-0.253970\pi$
$673$	36.2275	1.39647	0.698234	+	0.715870i	$0.253970\pi$
$674$	7.00436	0.269798
$675$	0	0
$676$	−2.19886	−0.0845715
$677$	−2.04046	−0.0784214	−0.0392107	−	0.999231i	$-0.512484\pi$
$677$	−2.04046	−0.0784214	−0.0392107	+	0.999231i	$0.512484\pi$
$678$	−3.76311	−0.144521
$679$	0.0657586	0.00252359
$680$	0	0
$681$	34.6373	1.32730
$682$	10.7887	0.413121
$683$	−15.9105	−0.608799	−0.304400	−	0.952544i	$-0.598456\pi$
$683$	−15.9105	−0.608799	−0.304400	+	0.952544i	$0.598456\pi$
$684$	27.9252	1.06775
$685$	0	0
$686$	−0.166838	−0.00636990
$687$	−29.8490	−1.13881
$688$	−0.446396	−0.0170187
$689$	−33.8765	−1.29059
$690$	0	0
$691$	−3.90166	−0.148426	−0.0742130	−	0.997242i	$-0.523644\pi$
$691$	−3.90166	−0.148426	−0.0742130	+	0.997242i	$0.523644\pi$
$692$	27.6268	1.05021
$693$	−0.559869	−0.0212677
$694$	2.54589	0.0966408
$695$	0	0
$696$	−22.4104	−0.849462
$697$	−28.2922	−1.07164
$698$	4.45201	0.168511
$699$	−48.0261	−1.81651
$700$	0	0
$701$	30.3587	1.14663	0.573316	−	0.819335i	$-0.305657\pi$
$701$	30.3587	1.14663	0.573316	+	0.819335i	$0.305657\pi$
$702$	−20.8793	−0.788037
$703$	−0.897729	−0.0338585
$704$	−9.28208	−0.349832
$705$	0	0
$706$	4.71651	0.177508
$707$	−0.246267	−0.00926181
$708$	−24.1041	−0.905887
$709$	−48.5677	−1.82400	−0.911999	−	0.410193i	$-0.865461\pi$
$709$	−48.5677	−1.82400	−0.911999	+	0.410193i	$0.865461\pi$
$710$	0	0
$711$	−63.6035	−2.38532
$712$	−31.9579	−1.19767
$713$	10.3091	0.386078
$714$	−0.133559	−0.00499831
$715$	0	0
$716$	−16.3873	−0.612421
$717$	−44.3394	−1.65588
$718$	−4.56651	−0.170421
$719$	4.76238	0.177607	0.0888033	−	0.996049i	$-0.471696\pi$
$719$	4.76238	0.177607	0.0888033	+	0.996049i	$0.471696\pi$
$720$	0	0
$721$	0.435112	0.0162044
$722$	6.55847	0.244081
$723$	−78.2946	−2.91181
$724$	10.2252	0.380018
$725$	0	0
$726$	2.85135	0.105823
$727$	3.38691	0.125614	0.0628068	−	0.998026i	$-0.479995\pi$
$727$	3.38691	0.125614	0.0628068	+	0.998026i	$0.479995\pi$
$728$	0.168701	0.00625248
$729$	−7.59939	−0.281459
$730$	0	0
$731$	−0.632488	−0.0233934
$732$	49.8491	1.84248
$733$	−15.4009	−0.568847	−0.284423	−	0.958699i	$-0.591802\pi$
$733$	−15.4009	−0.568847	−0.284423	+	0.958699i	$0.591802\pi$
$734$	−8.95690	−0.330605
$735$	0	0
$736$	8.64121	0.318519
$737$	16.0311	0.590513
$738$	25.6964	0.945899
$739$	9.98465	0.367291	0.183646	−	0.982993i	$-0.441210\pi$
$739$	9.98465	0.367291	0.183646	+	0.982993i	$0.441210\pi$
$740$	0	0
$741$	−28.3915	−1.04299
$742$	0.106922	0.00392522
$743$	41.4419	1.52036	0.760178	−	0.649715i	$-0.225112\pi$
$743$	41.4419	1.52036	0.760178	+	0.649715i	$0.225112\pi$
$744$	34.9157	1.28007
$745$	0	0
$746$	−1.54001	−0.0563839
$747$	58.8328	2.15258
$748$	−22.7063	−0.830223
$749$	0.377719	0.0138015
$750$	0	0
$751$	21.1036	0.770082	0.385041	−	0.922900i	$-0.374187\pi$
$751$	21.1036	0.770082	0.385041	+	0.922900i	$0.374187\pi$
$752$	−19.9727	−0.728329
$753$	−13.1061	−0.477612
$754$	7.29682	0.265734
$755$	0	0
$756$	−0.459217	−0.0167015
$757$	−40.7168	−1.47988	−0.739938	−	0.672675i	$-0.765145\pi$
$757$	−40.7168	−1.47988	−0.739938	+	0.672675i	$0.765145\pi$
$758$	−12.8186	−0.465594
$759$	19.0133	0.690141
$760$	0	0
$761$	−5.07664	−0.184028	−0.0920140	−	0.995758i	$-0.529330\pi$
$761$	−5.07664	−0.184028	−0.0920140	+	0.995758i	$0.529330\pi$
$762$	26.5110	0.960392
$763$	−0.141802	−0.00513357
$764$	38.2941	1.38543
$765$	0	0
$766$	−1.11485	−0.0402810
$767$	16.8229	0.607438
$768$	1.59875	0.0576899
$769$	−46.7687	−1.68652	−0.843261	−	0.537505i	$-0.819367\pi$
$769$	−46.7687	−1.68652	−0.843261	+	0.537505i	$0.819367\pi$
$770$	0	0
$771$	−63.1046	−2.27266
$772$	−44.1697	−1.58970
$773$	−19.0938	−0.686756	−0.343378	−	0.939197i	$-0.611571\pi$
$773$	−19.0938	−0.686756	−0.343378	+	0.939197i	$0.611571\pi$
$774$	0.574458	0.0206485
$775$	0	0
$776$	−5.19209	−0.186385
$777$	0.0271745	0.000974878 0
$778$	−2.32666	−0.0834147
$779$	18.9824	0.680115
$780$	0	0
$781$	−34.7385	−1.24304
$782$	3.11359	0.111342
$783$	−42.5753	−1.52152
$784$	−17.8979	−0.639212
$785$	0	0
$786$	7.42230	0.264745
$787$	4.02513	0.143481	0.0717403	−	0.997423i	$-0.477145\pi$
$787$	4.02513	0.143481	0.0717403	+	0.997423i	$0.477145\pi$
$788$	25.2799	0.900559
$789$	88.6768	3.15698
$790$	0	0
$791$	0.0577642	0.00205386
$792$	44.2054	1.57077
$793$	−34.7910	−1.23546
$794$	−19.0101	−0.674645
$795$	0	0
$796$	6.60660	0.234165
$797$	15.9847	0.566205	0.283103	−	0.959090i	$-0.408636\pi$
$797$	15.9847	0.566205	0.283103	+	0.959090i	$0.408636\pi$
$798$	0.0896099	0.00317216
$799$	−28.2988	−1.00114
$800$	0	0
$801$	−111.759	−3.94881
$802$	8.37587	0.295762
$803$	14.1678	0.499972
$804$	24.2042	0.853615
$805$	0	0
$806$	−11.3685	−0.400440
$807$	−24.5806	−0.865279
$808$	19.4444	0.684052
$809$	−29.2015	−1.02667	−0.513335	−	0.858188i	$-0.671590\pi$
$809$	−29.2015	−1.02667	−0.513335	+	0.858188i	$0.671590\pi$
$810$	0	0
$811$	7.11719	0.249918	0.124959	−	0.992162i	$-0.460120\pi$
$811$	7.11719	0.249918	0.124959	+	0.992162i	$0.460120\pi$
$812$	0.160485	0.00563194
$813$	−29.5557	−1.03656
$814$	−0.662978	−0.0232374
$815$	0	0
$816$	−28.6568	−1.00319
$817$	0.424362	0.0148466
$818$	−0.237034	−0.00828770
$819$	0.589959	0.0206148
$820$	0	0
$821$	−32.6032	−1.13786	−0.568929	−	0.822387i	$-0.692642\pi$
$821$	−32.6032	−1.13786	−0.568929	+	0.822387i	$0.692642\pi$
$822$	−19.8521	−0.692420
$823$	−27.1361	−0.945905	−0.472953	−	0.881088i	$-0.656812\pi$
$823$	−27.1361	−0.945905	−0.472953	+	0.881088i	$0.656812\pi$
$824$	−34.3551	−1.19681
$825$	0	0
$826$	−0.0530966	−0.00184747
$827$	54.8232	1.90639	0.953195	−	0.302357i	$-0.0977736\pi$
$827$	54.8232	1.90639	0.953195	+	0.302357i	$0.0977736\pi$
$828$	19.7062	0.684836
$829$	0.471969	0.0163922	0.00819608	−	0.999966i	$-0.497391\pi$
$829$	0.471969	0.0163922	0.00819608	+	0.999966i	$0.497391\pi$
$830$	0	0
$831$	53.1744	1.84460
$832$	9.78095	0.339093
$833$	−25.3591	−0.878642
$834$	−12.3048	−0.426082
$835$	0	0
$836$	15.2346	0.526898
$837$	66.3329	2.29280
$838$	−12.1414	−0.419418
$839$	−38.4737	−1.32826	−0.664129	−	0.747618i	$-0.731198\pi$
$839$	−38.4737	−1.32826	−0.664129	+	0.747618i	$0.731198\pi$
$840$	0	0
$841$	−14.1209	−0.486929
$842$	15.7051	0.541233
$843$	−84.1929	−2.89976
$844$	33.0574	1.13788
$845$	0	0
$846$	25.7024	0.883667
$847$	−0.0437684	−0.00150390
$848$	22.9415	0.787814
$849$	−10.7615	−0.369333
$850$	0	0
$851$	−0.633505	−0.0217163
$852$	−52.4490	−1.79687
$853$	35.8541	1.22762	0.613811	−	0.789453i	$-0.289636\pi$
$853$	35.8541	1.22762	0.613811	+	0.789453i	$0.289636\pi$
$854$	0.109808	0.00375755
$855$	0	0
$856$	−29.8234	−1.01934
$857$	−0.386299	−0.0131957	−0.00659786	−	0.999978i	$-0.502100\pi$
$857$	−0.386299	−0.0131957	−0.00659786	+	0.999978i	$0.502100\pi$
$858$	−20.9673	−0.715813
$859$	22.2889	0.760487	0.380244	−	0.924886i	$-0.375840\pi$
$859$	22.2889	0.760487	0.380244	+	0.924886i	$0.375840\pi$
$860$	0	0
$861$	−0.574602	−0.0195824
$862$	1.00231	0.0341389
$863$	−9.29808	−0.316510	−0.158255	−	0.987398i	$-0.550587\pi$
$863$	−9.29808	−0.316510	−0.158255	+	0.987398i	$0.550587\pi$
$864$	55.6011	1.89159
$865$	0	0
$866$	3.54326	0.120405
$867$	11.9822	0.406938
$868$	−0.250039	−0.00848686
$869$	−34.6987	−1.17707
$870$	0	0
$871$	−16.8927	−0.572387
$872$	11.1962	0.379151
$873$	−18.1571	−0.614524
$874$	−2.08903	−0.0706626
$875$	0	0
$876$	21.3910	0.722734
$877$	−29.1786	−0.985292	−0.492646	−	0.870230i	$-0.663970\pi$
$877$	−29.1786	−0.985292	−0.492646	+	0.870230i	$0.663970\pi$
$878$	−6.49384	−0.219156
$879$	72.6734	2.45121
$880$	0	0
$881$	−40.0737	−1.35012	−0.675059	−	0.737764i	$-0.735882\pi$
$881$	−40.0737	−1.35012	−0.675059	+	0.737764i	$0.735882\pi$
$882$	23.0325	0.775543
$883$	−30.8339	−1.03764	−0.518822	−	0.854882i	$-0.673629\pi$
$883$	−30.8339	−1.03764	−0.518822	+	0.854882i	$0.673629\pi$
$884$	23.9266	0.804739
$885$	0	0
$886$	10.9560	0.368073
$887$	38.5528	1.29448	0.647239	−	0.762287i	$-0.275924\pi$
$887$	38.5528	1.29448	0.647239	+	0.762287i	$0.275924\pi$
$888$	−2.14561	−0.0720019
$889$	−0.406946	−0.0136485
$890$	0	0
$891$	51.7320	1.73309
$892$	−19.8610	−0.664996
$893$	18.9868	0.635370
$894$	−8.72047	−0.291656
$895$	0	0
$896$	−0.270533	−0.00903787
$897$	−20.0352	−0.668956
$898$	−0.200208	−0.00668104
$899$	−23.1818	−0.773156
$900$	0	0
$901$	32.5052	1.08291
$902$	14.0186	0.466769
$903$	−0.0128455	−0.000427473 0
$904$	−4.56087	−0.151692
$905$	0	0
$906$	−11.4206	−0.379424
$907$	4.77670	0.158608	0.0793038	−	0.996850i	$-0.474730\pi$
$907$	4.77670	0.158608	0.0793038	+	0.996850i	$0.474730\pi$
$908$	19.5849	0.649946
$909$	67.9984	2.25537
$910$	0	0
$911$	−11.5332	−0.382111	−0.191056	−	0.981579i	$-0.561191\pi$
$911$	−11.5332	−0.382111	−0.191056	+	0.981579i	$0.561191\pi$
$912$	19.2270	0.636670
$913$	32.0961	1.06223
$914$	−17.6472	−0.583718
$915$	0	0
$916$	−16.8774	−0.557645
$917$	−0.113933	−0.00376240
$918$	20.0341	0.661224
$919$	−41.3268	−1.36325	−0.681623	−	0.731704i	$-0.738726\pi$
$919$	−41.3268	−1.36325	−0.681623	+	0.731704i	$0.738726\pi$
$920$	0	0
$921$	3.44133	0.113396
$922$	7.99655	0.263352
$923$	36.6055	1.20488
$924$	−0.461153	−0.0151708
$925$	0	0
$926$	−15.2717	−0.501860
$927$	−120.142	−3.94598
$928$	−19.4313	−0.637863
$929$	−13.4174	−0.440212	−0.220106	−	0.975476i	$-0.570640\pi$
$929$	−13.4174	−0.440212	−0.220106	+	0.975476i	$0.570640\pi$
$930$	0	0
$931$	17.0145	0.557627
$932$	−27.1552	−0.889499
$933$	45.6345	1.49401
$934$	3.32986	0.108956
$935$	0	0
$936$	−46.5812	−1.52256
$937$	−41.9929	−1.37185	−0.685923	−	0.727674i	$-0.740602\pi$
$937$	−41.9929	−1.37185	−0.685923	+	0.727674i	$0.740602\pi$
$938$	0.0533170	0.00174086
$939$	15.3065	0.499508
$940$	0	0
$941$	40.0565	1.30580	0.652902	−	0.757443i	$-0.273551\pi$
$941$	40.0565	1.30580	0.652902	+	0.757443i	$0.273551\pi$
$942$	−13.3752	−0.435789
$943$	13.3954	0.436215
$944$	−11.3926	−0.370797
$945$	0	0
$946$	0.313394	0.0101893
$947$	−43.9751	−1.42900	−0.714499	−	0.699637i	$-0.753345\pi$
$947$	−43.9751	−1.42900	−0.714499	+	0.699637i	$0.753345\pi$
$948$	−52.3890	−1.70152
$949$	−14.9293	−0.484625
$950$	0	0
$951$	70.1721	2.27549
$952$	−0.161872	−0.00524631
$953$	−40.6171	−1.31572	−0.657859	−	0.753141i	$-0.728538\pi$
$953$	−40.6171	−1.31572	−0.657859	+	0.753141i	$0.728538\pi$
$954$	−29.5229	−0.955839
$955$	0	0
$956$	−25.0707	−0.810843
$957$	−42.7548	−1.38207
$958$	−11.1424	−0.359994
$959$	0.304731	0.00984028
$960$	0	0
$961$	5.11756	0.165083
$962$	0.698610	0.0225241
$963$	−104.295	−3.36084
$964$	−44.2698	−1.42584
$965$	0	0
$966$	0.0632355	0.00203457
$967$	16.0002	0.514531	0.257265	−	0.966341i	$-0.417179\pi$
$967$	16.0002	0.514531	0.257265	+	0.966341i	$0.417179\pi$
$968$	3.45581	0.111074
$969$	27.2423	0.875148
$970$	0	0
$971$	55.5201	1.78173	0.890863	−	0.454272i	$-0.150101\pi$
$971$	55.5201	1.78173	0.890863	+	0.454272i	$0.150101\pi$
$972$	20.1923	0.647670
$973$	0.188880	0.00605523
$974$	5.10263	0.163499
$975$	0	0
$976$	23.5608	0.754162
$977$	−6.32539	−0.202367	−0.101184	−	0.994868i	$-0.532263\pi$
$977$	−6.32539	−0.202367	−0.101184	+	0.994868i	$0.532263\pi$
$978$	−7.36046	−0.235362
$979$	−60.9699	−1.94861
$980$	0	0
$981$	39.1539	1.25009
$982$	−3.32659	−0.106156
$983$	−48.6422	−1.55144	−0.775722	−	0.631074i	$-0.782614\pi$
$983$	−48.6422	−1.55144	−0.775722	+	0.631074i	$0.782614\pi$
$984$	45.3687	1.44630
$985$	0	0
$986$	−7.00145	−0.222972
$987$	−0.574736	−0.0182940
$988$	−16.0533	−0.510725
$989$	0.299462	0.00952234
$990$	0	0
$991$	−44.1763	−1.40331	−0.701653	−	0.712519i	$-0.747554\pi$
$991$	−44.1763	−1.40331	−0.701653	+	0.712519i	$0.747554\pi$
$992$	30.2742	0.961207
$993$	−5.35931	−0.170073
$994$	−0.115535	−0.00366455
$995$	0	0
$996$	48.4595	1.53550
$997$	−40.2262	−1.27398	−0.636988	−	0.770873i	$-0.719820\pi$
$997$	−40.2262	−1.27398	−0.636988	+	0.770873i	$0.719820\pi$
$998$	−0.543059	−0.0171902
$999$	−4.07623	−0.128966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.e.1.5	✓	8
3.2	odd	2		5625.2.a.be.1.4		8
4.3	odd	2		10000.2.a.bn.1.8		8
5.2	odd	4		625.2.b.d.624.7		16
5.3	odd	4		625.2.b.d.624.10		16
5.4	even	2		625.2.a.g.1.4	yes	8
15.14	odd	2		5625.2.a.s.1.5		8
20.19	odd	2		10000.2.a.be.1.1		8
25.2	odd	20		625.2.e.k.124.4		32
25.3	odd	20		625.2.e.j.249.5		32
25.4	even	10		625.2.d.m.376.3		16
25.6	even	5		625.2.d.q.251.2		16
25.8	odd	20		625.2.e.j.374.4		32
25.9	even	10		625.2.d.n.126.2		16
25.11	even	5		625.2.d.p.501.3		16
25.12	odd	20		625.2.e.k.499.5		32
25.13	odd	20		625.2.e.k.499.4		32
25.14	even	10		625.2.d.n.501.2		16
25.16	even	5		625.2.d.p.126.3		16
25.17	odd	20		625.2.e.j.374.5		32
25.19	even	10		625.2.d.m.251.3		16
25.21	even	5		625.2.d.q.376.2		16
25.22	odd	20		625.2.e.j.249.4		32
25.23	odd	20		625.2.e.k.124.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.5	✓	8	1.1	even	1	trivial
625.2.a.g.1.4	yes	8	5.4	even	2
625.2.b.d.624.7		16	5.2	odd	4
625.2.b.d.624.10		16	5.3	odd	4
625.2.d.m.251.3		16	25.19	even	10
625.2.d.m.376.3		16	25.4	even	10
625.2.d.n.126.2		16	25.9	even	10
625.2.d.n.501.2		16	25.14	even	10
625.2.d.p.126.3		16	25.16	even	5
625.2.d.p.501.3		16	25.11	even	5
625.2.d.q.251.2		16	25.6	even	5
625.2.d.q.376.2		16	25.21	even	5
625.2.e.j.249.4		32	25.22	odd	20
625.2.e.j.249.5		32	25.3	odd	20
625.2.e.j.374.4		32	25.8	odd	20
625.2.e.j.374.5		32	25.17	odd	20
625.2.e.k.124.4		32	25.2	odd	20
625.2.e.k.124.5		32	25.23	odd	20
625.2.e.k.499.4		32	25.13	odd	20
625.2.e.k.499.5		32	25.12	odd	20
5625.2.a.s.1.5		8	15.14	odd	2
5625.2.a.be.1.4		8	3.2	odd	2
10000.2.a.be.1.1		8	20.19	odd	2
10000.2.a.bn.1.8		8	4.3	odd	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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.500989 q^{2} -3.09326 q^{3} -1.74901 q^{4} +1.54969 q^{6} -0.0237879 q^{7} +1.87821 q^{8} +6.56824 q^{9} +O(q^{10})\)	\(q-0.500989 q^{2} -3.09326 q^{3} -1.74901 q^{4} +1.54969 q^{6} -0.0237879 q^{7} +1.87821 q^{8} +6.56824 q^{9} +3.58329 q^{11} +5.41014 q^{12} -3.77587 q^{13} +0.0119175 q^{14} +2.55705 q^{16} +3.62303 q^{17} -3.29062 q^{18} -2.43084 q^{19} +0.0735820 q^{21} -1.79519 q^{22} -1.71538 q^{23} -5.80980 q^{24} +1.89167 q^{26} -11.0375 q^{27} +0.0416052 q^{28} +3.85734 q^{29} -6.00979 q^{31} -5.03748 q^{32} -11.0840 q^{33} -1.81510 q^{34} -11.4879 q^{36} +0.369309 q^{37} +1.21782 q^{38} +11.6797 q^{39} -7.80900 q^{41} -0.0368638 q^{42} -0.174574 q^{43} -6.26720 q^{44} +0.859388 q^{46} -7.81082 q^{47} -7.90962 q^{48} -6.99943 q^{49} -11.2070 q^{51} +6.60403 q^{52} +8.97184 q^{53} +5.52966 q^{54} -0.0446787 q^{56} +7.51920 q^{57} -1.93249 q^{58} -4.45536 q^{59} +9.21403 q^{61} +3.01084 q^{62} -0.156244 q^{63} -2.59038 q^{64} +5.55298 q^{66} +4.47385 q^{67} -6.33671 q^{68} +5.30612 q^{69} -9.69458 q^{71} +12.3366 q^{72} +3.95387 q^{73} -0.185020 q^{74} +4.25156 q^{76} -0.0852388 q^{77} -5.85142 q^{78} -9.68349 q^{79} +14.4370 q^{81} +3.91223 q^{82} +8.95717 q^{83} -0.128696 q^{84} +0.0874600 q^{86} -11.9317 q^{87} +6.73018 q^{88} -17.0151 q^{89} +0.0898200 q^{91} +3.00022 q^{92} +18.5898 q^{93} +3.91314 q^{94} +15.5822 q^{96} -2.76438 q^{97} +3.50664 q^{98} +23.5359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9} + q^{11} - 10 q^{12} - 10 q^{13} - 8 q^{14} + 13 q^{16} - 15 q^{17} + 5 q^{18} - 10 q^{19} - 14 q^{21} + 5 q^{22} - 30 q^{23} + 5 q^{24} + 11 q^{26} - 20 q^{27} + 5 q^{28} + 10 q^{29} - 9 q^{31} - 30 q^{32} - 5 q^{33} + 7 q^{34} + 3 q^{36} + 10 q^{37} - 20 q^{38} + 8 q^{39} - 4 q^{41} + 35 q^{42} - 18 q^{44} - 9 q^{46} - 30 q^{47} - 5 q^{48} - 4 q^{49} - 14 q^{51} - 5 q^{52} - 10 q^{53} - 20 q^{54} + 10 q^{57} + 30 q^{58} - 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} - 18 q^{66} - 10 q^{67} - 40 q^{68} + 3 q^{69} - 9 q^{71} + 15 q^{72} - 18 q^{74} - 10 q^{76} - 5 q^{77} + 30 q^{78} - 20 q^{79} + 8 q^{81} + 45 q^{82} - 40 q^{83} - 28 q^{84} - 24 q^{86} - 40 q^{87} + 40 q^{88} - 5 q^{89} + 6 q^{91} - 15 q^{92} + 40 q^{93} + 47 q^{94} + 71 q^{96} + 30 q^{98} - 22 q^{99}+O(q^{100}) \) 8 * q - 5 * q^2 - 5 * q^3 + 11 * q^4 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 9 * q^9 + q^11 - 10 * q^12 - 10 * q^13 - 8 * q^14 + 13 * q^16 - 15 * q^17 + 5 * q^18 - 10 * q^19 - 14 * q^21 + 5 * q^22 - 30 * q^23 + 5 * q^24 + 11 * q^26 - 20 * q^27 + 5 * q^28 + 10 * q^29 - 9 * q^31 - 30 * q^32 - 5 * q^33 + 7 * q^34 + 3 * q^36 + 10 * q^37 - 20 * q^38 + 8 * q^39 - 4 * q^41 + 35 * q^42 - 18 * q^44 - 9 * q^46 - 30 * q^47 - 5 * q^48 - 4 * q^49 - 14 * q^51 - 5 * q^52 - 10 * q^53 - 20 * q^54 + 10 * q^57 + 30 * q^58 - 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 - 18 * q^66 - 10 * q^67 - 40 * q^68 + 3 * q^69 - 9 * q^71 + 15 * q^72 - 18 * q^74 - 10 * q^76 - 5 * q^77 + 30 * q^78 - 20 * q^79 + 8 * q^81 + 45 * q^82 - 40 * q^83 - 28 * q^84 - 24 * q^86 - 40 * q^87 + 40 * q^88 - 5 * q^89 + 6 * q^91 - 15 * q^92 + 40 * q^93 + 47 * q^94 + 71 * q^96 + 30 * q^98 - 22 * q^99

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