LMFDB - Embedded newform 625.2.a.e.1.8

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			1.8
Root			$3.01367$ 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)

$f(q)$	$=$	$q+2.01367 q^{2} -3.02566 q^{3} +2.05487 q^{4} -6.09269 q^{6} -0.369971 q^{7} +0.110485 q^{8} +6.15465 q^{9} +O(q^{10})$	\(q+2.01367 q^{2} -3.02566 q^{3} +2.05487 q^{4} -6.09269 q^{6} -0.369971 q^{7} +0.110485 q^{8} +6.15465 q^{9} -1.74633 q^{11} -6.21734 q^{12} -1.11622 q^{13} -0.745000 q^{14} -3.88725 q^{16} -5.48800 q^{17} +12.3934 q^{18} -3.75219 q^{19} +1.11941 q^{21} -3.51653 q^{22} -7.24619 q^{23} -0.334290 q^{24} -2.24770 q^{26} -9.54490 q^{27} -0.760242 q^{28} +4.19284 q^{29} +0.305684 q^{31} -8.04862 q^{32} +5.28380 q^{33} -11.0510 q^{34} +12.6470 q^{36} +9.21956 q^{37} -7.55568 q^{38} +3.37731 q^{39} -4.18641 q^{41} +2.25412 q^{42} -7.17118 q^{43} -3.58847 q^{44} -14.5914 q^{46} -0.810273 q^{47} +11.7615 q^{48} -6.86312 q^{49} +16.6048 q^{51} -2.29369 q^{52} +3.91508 q^{53} -19.2203 q^{54} -0.0408762 q^{56} +11.3529 q^{57} +8.44299 q^{58} -1.85738 q^{59} +9.68874 q^{61} +0.615546 q^{62} -2.27704 q^{63} -8.43275 q^{64} +10.6398 q^{66} +12.4701 q^{67} -11.2771 q^{68} +21.9245 q^{69} +11.6767 q^{71} +0.679995 q^{72} -3.43137 q^{73} +18.5652 q^{74} -7.71026 q^{76} +0.646091 q^{77} +6.80080 q^{78} +5.69346 q^{79} +10.4157 q^{81} -8.43005 q^{82} -7.13371 q^{83} +2.30024 q^{84} -14.4404 q^{86} -12.6861 q^{87} -0.192943 q^{88} +1.52999 q^{89} +0.412970 q^{91} -14.8900 q^{92} -0.924896 q^{93} -1.63162 q^{94} +24.3524 q^{96} -6.39742 q^{97} -13.8201 q^{98} -10.7480 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.01367	1.42388	0.711940	−	0.702240i	$-0.247817\pi$
$2$	2.01367	1.42388	0.711940	+	0.702240i	$0.247817\pi$
$3$	−3.02566	−1.74687	−0.873434	−	0.486942i	$-0.838112\pi$
$3$	−3.02566	−1.74687	−0.873434	+	0.486942i	$0.838112\pi$
$4$	2.05487	1.02743
$5$	0	0
$5$	0	0
$6$	−6.09269	−2.48733
$7$	−0.369971	−0.139836	−0.0699180	−	0.997553i	$-0.522274\pi$
$7$	−0.369971	−0.139836	−0.0699180	+	0.997553i	$0.522274\pi$
$8$	0.110485	0.0390623
$9$	6.15465	2.05155
$10$	0	0
$11$	−1.74633	−0.526538	−0.263269	−	0.964722i	$-0.584801\pi$
$11$	−1.74633	−0.526538	−0.263269	+	0.964722i	$0.584801\pi$
$12$	−6.21734	−1.79479
$13$	−1.11622	−0.309584	−0.154792	−	0.987947i	$-0.549471\pi$
$13$	−1.11622	−0.309584	−0.154792	+	0.987947i	$0.549471\pi$
$14$	−0.745000	−0.199110
$15$	0	0
$16$	−3.88725	−0.971814
$17$	−5.48800	−1.33103	−0.665517	−	0.746382i	$-0.731789\pi$
$17$	−5.48800	−1.33103	−0.665517	+	0.746382i	$0.731789\pi$
$18$	12.3934	2.92116
$19$	−3.75219	−0.860812	−0.430406	−	0.902635i	$-0.641630\pi$
$19$	−3.75219	−0.860812	−0.430406	+	0.902635i	$0.641630\pi$
$20$	0	0
$21$	1.11941	0.244275
$22$	−3.51653	−0.749726
$23$	−7.24619	−1.51094	−0.755468	−	0.655186i	$-0.772590\pi$
$23$	−7.24619	−1.51094	−0.755468	+	0.655186i	$0.772590\pi$
$24$	−0.334290	−0.0682366
$25$	0	0
$26$	−2.24770	−0.440811
$27$	−9.54490	−1.83692
$28$	−0.760242	−0.143672
$29$	4.19284	0.778590	0.389295	−	0.921113i	$-0.372719\pi$
$29$	4.19284	0.778590	0.389295	+	0.921113i	$0.372719\pi$
$30$	0	0
$31$	0.305684	0.0549024	0.0274512	−	0.999623i	$-0.491261\pi$
$31$	0.305684	0.0549024	0.0274512	+	0.999623i	$0.491261\pi$
$32$	−8.04862	−1.42281
$33$	5.28380	0.919792
$34$	−11.0510	−1.89523
$35$	0	0
$36$	12.6470	2.10783
$37$	9.21956	1.51569	0.757843	−	0.652436i	$-0.226253\pi$
$37$	9.21956	1.51569	0.757843	+	0.652436i	$0.226253\pi$
$38$	−7.55568	−1.22569
$39$	3.37731	0.540803
$40$	0	0
$41$	−4.18641	−0.653807	−0.326904	−	0.945058i	$-0.606005\pi$
$41$	−4.18641	−0.653807	−0.326904	+	0.945058i	$0.606005\pi$
$42$	2.25412	0.347818
$43$	−7.17118	−1.09359	−0.546797	−	0.837265i	$-0.684153\pi$
$43$	−7.17118	−1.09359	−0.546797	+	0.837265i	$0.684153\pi$
$44$	−3.58847	−0.540983
$45$	0	0
$46$	−14.5914	−2.15139
$47$	−0.810273	−0.118190	−0.0590952	−	0.998252i	$-0.518822\pi$
$47$	−0.810273	−0.118190	−0.0590952	+	0.998252i	$0.518822\pi$
$48$	11.7615	1.69763
$49$	−6.86312	−0.980446
$50$	0	0
$51$	16.6048	2.32514
$52$	−2.29369	−0.318077
$53$	3.91508	0.537778	0.268889	−	0.963171i	$-0.413343\pi$
$53$	3.91508	0.537778	0.268889	+	0.963171i	$0.413343\pi$
$54$	−19.2203	−2.61555
$55$	0	0
$56$	−0.0408762	−0.00546231
$57$	11.3529	1.50372
$58$	8.44299	1.10862
$59$	−1.85738	−0.241810	−0.120905	−	0.992664i	$-0.538580\pi$
$59$	−1.85738	−0.241810	−0.120905	+	0.992664i	$0.538580\pi$
$60$	0	0
$61$	9.68874	1.24052	0.620258	−	0.784398i	$-0.287028\pi$
$61$	9.68874	1.24052	0.620258	+	0.784398i	$0.287028\pi$
$62$	0.615546	0.0781744
$63$	−2.27704	−0.286880
$64$	−8.43275	−1.05409
$65$	0	0
$66$	10.6398	1.30967
$67$	12.4701	1.52346	0.761730	−	0.647894i	$-0.224350\pi$
$67$	12.4701	1.52346	0.761730	+	0.647894i	$0.224350\pi$
$68$	−11.2771	−1.36755
$69$	21.9245	2.63941
$70$	0	0
$71$	11.6767	1.38577	0.692885	−	0.721048i	$-0.256339\pi$
$71$	11.6767	1.38577	0.692885	+	0.721048i	$0.256339\pi$
$72$	0.679995	0.0801382
$73$	−3.43137	−0.401611	−0.200806	−	0.979631i	$-0.564356\pi$
$73$	−3.43137	−0.401611	−0.200806	+	0.979631i	$0.564356\pi$
$74$	18.5652	2.15816
$75$	0	0
$76$	−7.71026	−0.884427
$77$	0.646091	0.0736290
$78$	6.80080	0.770039
$79$	5.69346	0.640564	0.320282	−	0.947322i	$-0.396222\pi$
$79$	5.69346	0.640564	0.320282	+	0.947322i	$0.396222\pi$
$80$	0	0
$81$	10.4157	1.15730
$82$	−8.43005	−0.930943
$83$	−7.13371	−0.783027	−0.391513	−	0.920172i	$-0.628048\pi$
$83$	−7.13371	−0.783027	−0.391513	+	0.920172i	$0.628048\pi$
$84$	2.30024	0.250977
$85$	0	0
$86$	−14.4404	−1.55715
$87$	−12.6861	−1.36009
$88$	−0.192943	−0.0205678
$89$	1.52999	0.162179	0.0810894	−	0.996707i	$-0.474160\pi$
$89$	1.52999	0.162179	0.0810894	+	0.996707i	$0.474160\pi$
$90$	0	0
$91$	0.412970	0.0432911
$92$	−14.8900	−1.55239
$93$	−0.924896	−0.0959072
$94$	−1.63162	−0.168289
$95$	0	0
$96$	24.3524	2.48546
$97$	−6.39742	−0.649559	−0.324780	−	0.945790i	$-0.605290\pi$
$97$	−6.39742	−0.649559	−0.324780	+	0.945790i	$0.605290\pi$
$98$	−13.8201	−1.39604
$99$	−10.7480	−1.08022
$100$	0	0
$101$	−12.2487	−1.21879	−0.609396	−	0.792866i	$-0.708588\pi$
$101$	−12.2487	−1.21879	−0.609396	+	0.792866i	$0.708588\pi$
$102$	33.4367	3.31072
$103$	7.66730	0.755482	0.377741	−	0.925911i	$-0.376701\pi$
$103$	7.66730	0.755482	0.377741	+	0.925911i	$0.376701\pi$
$104$	−0.123326	−0.0120931
$105$	0	0
$106$	7.88369	0.765731
$107$	0.758003	0.0732789	0.0366394	−	0.999329i	$-0.488335\pi$
$107$	0.758003	0.0732789	0.0366394	+	0.999329i	$0.488335\pi$
$108$	−19.6135	−1.88731
$109$	−3.26839	−0.313055	−0.156528	−	0.987674i	$-0.550030\pi$
$109$	−3.26839	−0.313055	−0.156528	+	0.987674i	$0.550030\pi$
$110$	0	0
$111$	−27.8953	−2.64771
$112$	1.43817	0.135895
$113$	−0.847957	−0.0797691	−0.0398846	−	0.999204i	$-0.512699\pi$
$113$	−0.847957	−0.0797691	−0.0398846	+	0.999204i	$0.512699\pi$
$114$	22.8609	2.14112
$115$	0	0
$116$	8.61572	0.799950
$117$	−6.86995	−0.635128
$118$	−3.74015	−0.344309
$119$	2.03040	0.186127
$120$	0	0
$121$	−7.95034	−0.722758
$122$	19.5099	1.76635
$123$	12.6667	1.14212
$124$	0.628139	0.0564086
$125$	0	0
$126$	−4.58521	−0.408483
$127$	−14.3225	−1.27092	−0.635459	−	0.772135i	$-0.719189\pi$
$127$	−14.3225	−1.27092	−0.635459	+	0.772135i	$0.719189\pi$
$128$	−0.883546	−0.0780951
$129$	21.6976	1.91036
$130$	0	0
$131$	−16.6266	−1.45267	−0.726334	−	0.687342i	$-0.758778\pi$
$131$	−16.6266	−1.45267	−0.726334	+	0.687342i	$0.758778\pi$
$132$	10.8575	0.945025
$133$	1.38820	0.120373
$134$	25.1106	2.16923
$135$	0	0
$136$	−0.606340	−0.0519932
$137$	3.07189	0.262449	0.131224	−	0.991353i	$-0.458109\pi$
$137$	3.07189	0.262449	0.131224	+	0.991353i	$0.458109\pi$
$138$	44.1488	3.75820
$139$	−15.7200	−1.33335	−0.666677	−	0.745347i	$-0.732284\pi$
$139$	−15.7200	−1.33335	−0.666677	+	0.745347i	$0.732284\pi$
$140$	0	0
$141$	2.45161	0.206463
$142$	23.5130	1.97317
$143$	1.94929	0.163008
$144$	−23.9247	−1.99372
$145$	0	0
$146$	−6.90964	−0.571846
$147$	20.7655	1.71271
$148$	18.9450	1.55727
$149$	14.8504	1.21660	0.608298	−	0.793709i	$-0.291853\pi$
$149$	14.8504	1.21660	0.608298	+	0.793709i	$0.291853\pi$
$150$	0	0
$151$	−0.712013	−0.0579428	−0.0289714	−	0.999580i	$-0.509223\pi$
$151$	−0.712013	−0.0579428	−0.0289714	+	0.999580i	$0.509223\pi$
$152$	−0.414560	−0.0336253
$153$	−33.7767	−2.73068
$154$	1.30102	0.104839
$155$	0	0
$156$	6.93993	0.555639
$157$	22.0704	1.76141	0.880704	−	0.473667i	$-0.157070\pi$
$157$	22.0704	1.76141	0.880704	+	0.473667i	$0.157070\pi$
$158$	11.4647	0.912086
$159$	−11.8457	−0.939428
$160$	0	0
$161$	2.68088	0.211283
$162$	20.9739	1.64786
$163$	13.1619	1.03092	0.515460	−	0.856914i	$-0.327621\pi$
$163$	13.1619	1.03092	0.515460	+	0.856914i	$0.327621\pi$
$164$	−8.60251	−0.671744
$165$	0	0
$166$	−14.3649	−1.11494
$167$	−10.4081	−0.805403	−0.402702	−	0.915331i	$-0.631929\pi$
$167$	−10.4081	−0.805403	−0.402702	+	0.915331i	$0.631929\pi$
$168$	0.123678	0.00954194
$169$	−11.7540	−0.904158
$170$	0	0
$171$	−23.0934	−1.76600
$172$	−14.7358	−1.12360
$173$	9.19247	0.698890	0.349445	−	0.936957i	$-0.386370\pi$
$173$	9.19247	0.698890	0.349445	+	0.936957i	$0.386370\pi$
$174$	−25.5457	−1.93661
$175$	0	0
$176$	6.78842	0.511697
$177$	5.61981	0.422411
$178$	3.08090	0.230923
$179$	−21.6873	−1.62099	−0.810494	−	0.585747i	$-0.800801\pi$
$179$	−21.6873	−1.62099	−0.810494	+	0.585747i	$0.800801\pi$
$180$	0	0
$181$	10.7680	0.800378	0.400189	−	0.916433i	$-0.368944\pi$
$181$	10.7680	0.800378	0.400189	+	0.916433i	$0.368944\pi$
$182$	0.831586	0.0616413
$183$	−29.3149	−2.16702
$184$	−0.800594	−0.0590206
$185$	0	0
$186$	−1.86244	−0.136560
$187$	9.58385	0.700840
$188$	−1.66500	−0.121433
$189$	3.53134	0.256867
$190$	0	0
$191$	−3.83941	−0.277810	−0.138905	−	0.990306i	$-0.544358\pi$
$191$	−3.83941	−0.277810	−0.138905	+	0.990306i	$0.544358\pi$
$192$	25.5147	1.84136
$193$	10.5334	0.758208	0.379104	−	0.925354i	$-0.376232\pi$
$193$	10.5334	0.758208	0.379104	+	0.925354i	$0.376232\pi$
$194$	−12.8823	−0.924894
$195$	0	0
$196$	−14.1028	−1.00734
$197$	9.65302	0.687749	0.343875	−	0.939016i	$-0.388260\pi$
$197$	9.65302	0.687749	0.343875	+	0.939016i	$0.388260\pi$
$198$	−21.6430	−1.53810
$199$	−15.8462	−1.12331	−0.561654	−	0.827372i	$-0.689835\pi$
$199$	−15.8462	−1.12331	−0.561654	+	0.827372i	$0.689835\pi$
$200$	0	0
$201$	−37.7302	−2.66129
$202$	−24.6649	−1.73541
$203$	−1.55123	−0.108875
$204$	34.1207	2.38893
$205$	0	0
$206$	15.4394	1.07572
$207$	−44.5978	−3.09976
$208$	4.33904	0.300858
$209$	6.55256	0.453250
$210$	0	0
$211$	7.25106	0.499184	0.249592	−	0.968351i	$-0.419704\pi$
$211$	7.25106	0.499184	0.249592	+	0.968351i	$0.419704\pi$
$212$	8.04498	0.552531
$213$	−35.3298	−2.42076
$214$	1.52637	0.104340
$215$	0	0
$216$	−1.05457	−0.0717542
$217$	−0.113094	−0.00767733
$218$	−6.58147	−0.445753
$219$	10.3822	0.701562
$220$	0	0
$221$	6.12583	0.412068
$222$	−56.1719	−3.77001
$223$	−22.8653	−1.53118	−0.765588	−	0.643331i	$-0.777552\pi$
$223$	−22.8653	−1.53118	−0.765588	+	0.643331i	$0.777552\pi$
$224$	2.97776	0.198960
$225$	0	0
$226$	−1.70751	−0.113582
$227$	0.665418	0.0441654	0.0220827	−	0.999756i	$-0.492970\pi$
$227$	0.665418	0.0441654	0.0220827	+	0.999756i	$0.492970\pi$
$228$	23.3286	1.54498
$229$	−23.4732	−1.55115	−0.775576	−	0.631254i	$-0.782541\pi$
$229$	−23.4732	−1.55115	−0.775576	+	0.631254i	$0.782541\pi$
$230$	0	0
$231$	−1.95486	−0.128620
$232$	0.463245	0.0304135
$233$	−17.8293	−1.16803	−0.584017	−	0.811742i	$-0.698520\pi$
$233$	−17.8293	−1.16803	−0.584017	+	0.811742i	$0.698520\pi$
$234$	−13.8338	−0.904345
$235$	0	0
$236$	−3.81667	−0.248444
$237$	−17.2265	−1.11898
$238$	4.08856	0.265022
$239$	−23.8706	−1.54406	−0.772030	−	0.635587i	$-0.780758\pi$
$239$	−23.8706	−1.54406	−0.772030	+	0.635587i	$0.780758\pi$
$240$	0	0
$241$	−4.21325	−0.271399	−0.135700	−	0.990750i	$-0.543328\pi$
$241$	−4.21325	−0.271399	−0.135700	+	0.990750i	$0.543328\pi$
$242$	−16.0094	−1.02912
$243$	−2.87982	−0.184741
$244$	19.9091	1.27455
$245$	0	0
$246$	25.5065	1.62623
$247$	4.18828	0.266494
$248$	0.0337734	0.00214461
$249$	21.5842	1.36784
$250$	0	0
$251$	19.5741	1.23551	0.617755	−	0.786371i	$-0.288042\pi$
$251$	19.5741	1.23551	0.617755	+	0.786371i	$0.288042\pi$
$252$	−4.67902	−0.294751
$253$	12.6542	0.795565
$254$	−28.8408	−1.80963
$255$	0	0
$256$	15.0863	0.942896
$257$	18.4169	1.14881	0.574407	−	0.818570i	$-0.305233\pi$
$257$	18.4169	1.14881	0.574407	+	0.818570i	$0.305233\pi$
$258$	43.6918	2.72013
$259$	−3.41097	−0.211948
$260$	0	0
$261$	25.8054	1.59732
$262$	−33.4804	−2.06843
$263$	4.32450	0.266660	0.133330	−	0.991072i	$-0.457433\pi$
$263$	4.32450	0.266660	0.133330	+	0.991072i	$0.457433\pi$
$264$	0.583780	0.0359292
$265$	0	0
$266$	2.79538	0.171396
$267$	−4.62924	−0.283305
$268$	25.6243	1.56526
$269$	4.73895	0.288939	0.144469	−	0.989509i	$-0.453852\pi$
$269$	4.73895	0.288939	0.144469	+	0.989509i	$0.453852\pi$
$270$	0	0
$271$	9.96528	0.605348	0.302674	−	0.953094i	$-0.402121\pi$
$271$	9.96528	0.605348	0.302674	+	0.953094i	$0.402121\pi$
$272$	21.3332	1.29352
$273$	−1.24951	−0.0756238
$274$	6.18577	0.373696
$275$	0	0
$276$	45.0520	2.71181
$277$	17.6092	1.05804	0.529018	−	0.848611i	$-0.322560\pi$
$277$	17.6092	1.05804	0.529018	+	0.848611i	$0.322560\pi$
$278$	−31.6549	−1.89854
$279$	1.88137	0.112635
$280$	0	0
$281$	−25.4964	−1.52099	−0.760494	−	0.649345i	$-0.775043\pi$
$281$	−25.4964	−1.52099	−0.760494	+	0.649345i	$0.775043\pi$
$282$	4.93674	0.293979
$283$	16.1004	0.957072	0.478536	−	0.878068i	$-0.341168\pi$
$283$	16.1004	0.957072	0.478536	+	0.878068i	$0.341168\pi$
$284$	23.9941	1.42379
$285$	0	0
$286$	3.92523	0.232104
$287$	1.54885	0.0914258
$288$	−49.5364	−2.91896
$289$	13.1181	0.771654
$290$	0	0
$291$	19.3564	1.13469
$292$	−7.05101	−0.412629
$293$	−24.9049	−1.45496	−0.727481	−	0.686128i	$-0.759309\pi$
$293$	−24.9049	−1.45496	−0.727481	+	0.686128i	$0.759309\pi$
$294$	41.8149	2.43869
$295$	0	0
$296$	1.01862	0.0592062
$297$	16.6685	0.967207
$298$	29.9039	1.73229
$299$	8.08836	0.467762
$300$	0	0
$301$	2.65313	0.152924
$302$	−1.43376	−0.0825035
$303$	37.0605	2.12907
$304$	14.5857	0.836549
$305$	0	0
$306$	−68.0151	−3.88817
$307$	−1.74743	−0.0997311	−0.0498655	−	0.998756i	$-0.515879\pi$
$307$	−1.74743	−0.0997311	−0.0498655	+	0.998756i	$0.515879\pi$
$308$	1.32763	0.0756489
$309$	−23.1987	−1.31973
$310$	0	0
$311$	18.3262	1.03919	0.519593	−	0.854414i	$-0.326084\pi$
$311$	18.3262	1.03919	0.519593	+	0.854414i	$0.326084\pi$
$312$	0.373142	0.0211250
$313$	−3.30758	−0.186955	−0.0934777	−	0.995621i	$-0.529798\pi$
$313$	−3.30758	−0.186955	−0.0934777	+	0.995621i	$0.529798\pi$
$314$	44.4425	2.50803
$315$	0	0
$316$	11.6993	0.658137
$317$	0.999043	0.0561118	0.0280559	−	0.999606i	$-0.491068\pi$
$317$	0.999043	0.0561118	0.0280559	+	0.999606i	$0.491068\pi$
$318$	−23.8534	−1.33763
$319$	−7.32207	−0.409957
$320$	0	0
$321$	−2.29346	−0.128009
$322$	5.39842	0.300842
$323$	20.5920	1.14577
$324$	21.4030	1.18905
$325$	0	0
$326$	26.5037	1.46790
$327$	9.88906	0.546866
$328$	−0.462534	−0.0255392
$329$	0.299778	0.0165273
$330$	0	0
$331$	−14.2009	−0.780555	−0.390277	−	0.920697i	$-0.627621\pi$
$331$	−14.2009	−0.780555	−0.390277	+	0.920697i	$0.627621\pi$
$332$	−14.6588	−0.804508
$333$	56.7432	3.10951
$334$	−20.9585	−1.14680
$335$	0	0
$336$	−4.35143	−0.237390
$337$	−13.1530	−0.716489	−0.358245	−	0.933628i	$-0.616625\pi$
$337$	−13.1530	−0.716489	−0.358245	+	0.933628i	$0.616625\pi$
$338$	−23.6688	−1.28741
$339$	2.56563	0.139346
$340$	0	0
$341$	−0.533824	−0.0289082
$342$	−46.5025	−2.51457
$343$	5.12896	0.276938
$344$	−0.792306	−0.0427183
$345$	0	0
$346$	18.5106	0.995136
$347$	−13.5474	−0.727265	−0.363633	−	0.931542i	$-0.618464\pi$
$347$	−13.5474	−0.727265	−0.363633	+	0.931542i	$0.618464\pi$
$348$	−26.0683	−1.39741
$349$	−32.0976	−1.71814	−0.859072	−	0.511854i	$-0.828959\pi$
$349$	−32.0976	−1.71814	−0.859072	+	0.511854i	$0.828959\pi$
$350$	0	0
$351$	10.6542	0.568681
$352$	14.0555	0.749162
$353$	−18.3122	−0.974662	−0.487331	−	0.873217i	$-0.662029\pi$
$353$	−18.3122	−0.974662	−0.487331	+	0.873217i	$0.662029\pi$
$354$	11.3164	0.601462
$355$	0	0
$356$	3.14393	0.166628
$357$	−6.14332	−0.325139
$358$	−43.6711	−2.30809
$359$	−14.4364	−0.761927	−0.380963	−	0.924590i	$-0.624408\pi$
$359$	−14.4364	−0.761927	−0.380963	+	0.924590i	$0.624408\pi$
$360$	0	0
$361$	−4.92106	−0.259003
$362$	21.6832	1.13964
$363$	24.0551	1.26256
$364$	0.848599	0.0444787
$365$	0	0
$366$	−59.0305	−3.08557
$367$	−10.4130	−0.543555	−0.271777	−	0.962360i	$-0.587611\pi$
$367$	−10.4130	−0.543555	−0.271777	+	0.962360i	$0.587611\pi$
$368$	28.1678	1.46835
$369$	−25.7659	−1.34132
$370$	0	0
$371$	−1.44847	−0.0752008
$372$	−1.90054	−0.0985383
$373$	−10.0225	−0.518944	−0.259472	−	0.965751i	$-0.583549\pi$
$373$	−10.0225	−0.518944	−0.259472	+	0.965751i	$0.583549\pi$
$374$	19.2987	0.997912
$375$	0	0
$376$	−0.0895228	−0.00461679
$377$	−4.68014	−0.241039
$378$	7.11096	0.365748
$379$	14.2995	0.734516	0.367258	−	0.930119i	$-0.380297\pi$
$379$	14.2995	0.734516	0.367258	+	0.930119i	$0.380297\pi$
$380$	0	0
$381$	43.3351	2.22013
$382$	−7.73131	−0.395568
$383$	−5.03705	−0.257381	−0.128690	−	0.991685i	$-0.541077\pi$
$383$	−5.03705	−0.257381	−0.128690	+	0.991685i	$0.541077\pi$
$384$	2.67331	0.136422
$385$	0	0
$386$	21.2107	1.07960
$387$	−44.1361	−2.24356
$388$	−13.1458	−0.667379
$389$	10.1326	0.513743	0.256872	−	0.966446i	$-0.417308\pi$
$389$	10.1326	0.513743	0.256872	+	0.966446i	$0.417308\pi$
$390$	0	0
$391$	39.7671	2.01111
$392$	−0.758270	−0.0382984
$393$	50.3064	2.53762
$394$	19.4380	0.979272
$395$	0	0
$396$	−22.0858	−1.10985
$397$	−19.6679	−0.987104	−0.493552	−	0.869716i	$-0.664302\pi$
$397$	−19.6679	−0.987104	−0.493552	+	0.869716i	$0.664302\pi$
$398$	−31.9091	−1.59946
$399$	−4.20024	−0.210275
$400$	0	0
$401$	−23.0931	−1.15321	−0.576606	−	0.817022i	$-0.695623\pi$
$401$	−23.0931	−1.15321	−0.576606	+	0.817022i	$0.695623\pi$
$402$	−75.9762	−3.78935
$403$	−0.341211	−0.0169969
$404$	−25.1695	−1.25223
$405$	0	0
$406$	−3.12366	−0.155025
$407$	−16.1004	−0.798066
$408$	1.83458	0.0908253
$409$	38.6338	1.91032	0.955159	−	0.296093i	$-0.0956838\pi$
$409$	38.6338	1.91032	0.955159	+	0.296093i	$0.0956838\pi$
$410$	0	0
$411$	−9.29450	−0.458464
$412$	15.7553	0.776207
$413$	0.687177	0.0338138
$414$	−89.8052	−4.41368
$415$	0	0
$416$	8.98405	0.440479
$417$	47.5635	2.32919
$418$	13.1947	0.645373
$419$	−10.7891	−0.527082	−0.263541	−	0.964648i	$-0.584890\pi$
$419$	−10.7891	−0.527082	−0.263541	+	0.964648i	$0.584890\pi$
$420$	0	0
$421$	32.4550	1.58176	0.790880	−	0.611971i	$-0.209623\pi$
$421$	32.4550	1.58176	0.790880	+	0.611971i	$0.209623\pi$
$422$	14.6013	0.710778
$423$	−4.98694	−0.242473
$424$	0.432557	0.0210068
$425$	0	0
$426$	−71.1426	−3.44687
$427$	−3.58456	−0.173469
$428$	1.55760	0.0752892
$429$	−5.89790	−0.284753
$430$	0	0
$431$	13.7920	0.664339	0.332170	−	0.943220i	$-0.392219\pi$
$431$	13.7920	0.664339	0.332170	+	0.943220i	$0.392219\pi$
$432$	37.1035	1.78514
$433$	21.1120	1.01458	0.507290	−	0.861776i	$-0.330647\pi$
$433$	21.1120	1.01458	0.507290	+	0.861776i	$0.330647\pi$
$434$	−0.227734	−0.0109316
$435$	0	0
$436$	−6.71612	−0.321644
$437$	27.1891	1.30063
$438$	20.9063	0.998940
$439$	36.3457	1.73468	0.867342	−	0.497713i	$-0.165827\pi$
$439$	36.3457	1.73468	0.867342	+	0.497713i	$0.165827\pi$
$440$	0	0
$441$	−42.2401	−2.01143
$442$	12.3354	0.586735
$443$	−6.38810	−0.303508	−0.151754	−	0.988418i	$-0.548492\pi$
$443$	−6.38810	−0.303508	−0.151754	+	0.988418i	$0.548492\pi$
$444$	−57.3212	−2.72034
$445$	0	0
$446$	−46.0432	−2.18021
$447$	−44.9324	−2.12523
$448$	3.11988	0.147400
$449$	−35.1628	−1.65943	−0.829717	−	0.558185i	$-0.811498\pi$
$449$	−35.1628	−1.65943	−0.829717	+	0.558185i	$0.811498\pi$
$450$	0	0
$451$	7.31084	0.344254
$452$	−1.74244	−0.0819575
$453$	2.15431	0.101218
$454$	1.33993	0.0628862
$455$	0	0
$456$	1.25432	0.0587389
$457$	−22.2994	−1.04312	−0.521561	−	0.853214i	$-0.674650\pi$
$457$	−22.2994	−1.04312	−0.521561	+	0.853214i	$0.674650\pi$
$458$	−47.2673	−2.20865
$459$	52.3824	2.44500
$460$	0	0
$461$	−1.88541	−0.0878122	−0.0439061	−	0.999036i	$-0.513980\pi$
$461$	−1.88541	−0.0878122	−0.0439061	+	0.999036i	$0.513980\pi$
$462$	−3.93644	−0.183140
$463$	12.9152	0.600218	0.300109	−	0.953905i	$-0.402977\pi$
$463$	12.9152	0.600218	0.300109	+	0.953905i	$0.402977\pi$
$464$	−16.2986	−0.756645
$465$	0	0
$466$	−35.9023	−1.66314
$467$	2.61816	0.121154	0.0605769	−	0.998164i	$-0.480706\pi$
$467$	2.61816	0.121154	0.0605769	+	0.998164i	$0.480706\pi$
$468$	−14.1168	−0.652551
$469$	−4.61357	−0.213035
$470$	0	0
$471$	−66.7776	−3.07695
$472$	−0.205212	−0.00944566
$473$	12.5232	0.575819
$474$	−34.6885	−1.59329
$475$	0	0
$476$	4.17221	0.191233
$477$	24.0960	1.10328
$478$	−48.0675	−2.19855
$479$	6.73316	0.307646	0.153823	−	0.988098i	$-0.450841\pi$
$479$	6.73316	0.307646	0.153823	+	0.988098i	$0.450841\pi$
$480$	0	0
$481$	−10.2911	−0.469233
$482$	−8.48409	−0.386440
$483$	−8.11146	−0.369084
$484$	−16.3369	−0.742586
$485$	0	0
$486$	−5.79902	−0.263049
$487$	5.55795	0.251855	0.125927	−	0.992039i	$-0.459809\pi$
$487$	5.55795	0.251855	0.125927	+	0.992039i	$0.459809\pi$
$488$	1.07046	0.0484574
$489$	−39.8235	−1.80088
$490$	0	0
$491$	−5.55199	−0.250558	−0.125279	−	0.992122i	$-0.539983\pi$
$491$	−5.55199	−0.250558	−0.125279	+	0.992122i	$0.539983\pi$
$492$	26.0283	1.17345
$493$	−23.0103	−1.03633
$494$	8.43381	0.379455
$495$	0	0
$496$	−1.18827	−0.0533549
$497$	−4.32005	−0.193781
$498$	43.4635	1.94765
$499$	19.2580	0.862107	0.431054	−	0.902326i	$-0.358142\pi$
$499$	19.2580	0.862107	0.431054	+	0.902326i	$0.358142\pi$
$500$	0	0
$501$	31.4914	1.40693
$502$	39.4159	1.75922
$503$	−31.1565	−1.38920	−0.694600	−	0.719396i	$-0.744419\pi$
$503$	−31.1565	−1.38920	−0.694600	+	0.719396i	$0.744419\pi$
$504$	−0.251579	−0.0112062
$505$	0	0
$506$	25.4814	1.13279
$507$	35.5638	1.57944
$508$	−29.4309	−1.30578
$509$	−21.2250	−0.940782	−0.470391	−	0.882458i	$-0.655887\pi$
$509$	−21.2250	−0.940782	−0.470391	+	0.882458i	$0.655887\pi$
$510$	0	0
$511$	1.26951	0.0561597
$512$	32.1460	1.42067
$513$	35.8143	1.58124
$514$	37.0855	1.63577
$515$	0	0
$516$	44.5856	1.96277
$517$	1.41500	0.0622317
$518$	−6.86858	−0.301788
$519$	−27.8133	−1.22087
$520$	0	0
$521$	24.0095	1.05188	0.525938	−	0.850523i	$-0.323714\pi$
$521$	24.0095	1.05188	0.525938	+	0.850523i	$0.323714\pi$
$522$	51.9636	2.27439
$523$	−22.8190	−0.997804	−0.498902	−	0.866658i	$-0.666263\pi$
$523$	−22.8190	−0.997804	−0.498902	+	0.866658i	$0.666263\pi$
$524$	−34.1654	−1.49252
$525$	0	0
$526$	8.70813	0.379692
$527$	−1.67759	−0.0730770
$528$	−20.5395	−0.893867
$529$	29.5073	1.28293
$530$	0	0
$531$	−11.4315	−0.496086
$532$	2.85257	0.123675
$533$	4.67296	0.202408
$534$	−9.32177	−0.403392
$535$	0	0
$536$	1.37775	0.0595098
$537$	65.6186	2.83165
$538$	9.54268	0.411414
$539$	11.9853	0.516242
$540$	0	0
$541$	27.5072	1.18263	0.591314	−	0.806442i	$-0.298609\pi$
$541$	27.5072	1.18263	0.591314	+	0.806442i	$0.298609\pi$
$542$	20.0668	0.861943
$543$	−32.5803	−1.39816
$544$	44.1708	1.89381
$545$	0	0
$546$	−2.51610	−0.107679
$547$	−0.243224	−0.0103995	−0.00519975	−	0.999986i	$-0.501655\pi$
$547$	−0.243224	−0.0103995	−0.00519975	+	0.999986i	$0.501655\pi$
$548$	6.31232	0.269649
$549$	59.6308	2.54498
$550$	0	0
$551$	−15.7323	−0.670220
$552$	2.42233	0.103101
$553$	−2.10642	−0.0895739
$554$	35.4592	1.50652
$555$	0	0
$556$	−32.3025	−1.36993
$557$	−27.7280	−1.17487	−0.587436	−	0.809271i	$-0.699862\pi$
$557$	−27.7280	−1.17487	−0.587436	+	0.809271i	$0.699862\pi$
$558$	3.78847	0.160379
$559$	8.00463	0.338560
$560$	0	0
$561$	−28.9975	−1.22428
$562$	−51.3414	−2.16570
$563$	12.4049	0.522805	0.261403	−	0.965230i	$-0.415815\pi$
$563$	12.4049	0.522805	0.261403	+	0.965230i	$0.415815\pi$
$564$	5.03774	0.212127
$565$	0	0
$566$	32.4210	1.36276
$567$	−3.85353	−0.161833
$568$	1.29010	0.0541313
$569$	−26.6074	−1.11544	−0.557720	−	0.830029i	$-0.688324\pi$
$569$	−26.6074	−1.11544	−0.557720	+	0.830029i	$0.688324\pi$
$570$	0	0
$571$	12.9219	0.540765	0.270382	−	0.962753i	$-0.412850\pi$
$571$	12.9219	0.540765	0.270382	+	0.962753i	$0.412850\pi$
$572$	4.00553	0.167480
$573$	11.6168	0.485298
$574$	3.11888	0.130179
$575$	0	0
$576$	−51.9006	−2.16253
$577$	28.3432	1.17994	0.589971	−	0.807425i	$-0.299139\pi$
$577$	28.3432	1.17994	0.589971	+	0.807425i	$0.299139\pi$
$578$	26.4156	1.09874
$579$	−31.8704	−1.32449
$580$	0	0
$581$	2.63927	0.109495
$582$	38.9775	1.61567
$583$	−6.83702	−0.283160
$584$	−0.379114	−0.0156878
$585$	0	0
$586$	−50.1503	−2.07169
$587$	−8.38885	−0.346245	−0.173123	−	0.984900i	$-0.555386\pi$
$587$	−8.38885	−0.346245	−0.173123	+	0.984900i	$0.555386\pi$
$588$	42.6704	1.75970
$589$	−1.14698	−0.0472606
$590$	0	0
$591$	−29.2068	−1.20141
$592$	−35.8388	−1.47297
$593$	30.9031	1.26904	0.634518	−	0.772908i	$-0.281199\pi$
$593$	30.9031	1.26904	0.634518	+	0.772908i	$0.281199\pi$
$594$	33.5649	1.37719
$595$	0	0
$596$	30.5157	1.24997
$597$	47.9454	1.96227
$598$	16.2873	0.666037
$599$	32.6384	1.33357	0.666784	−	0.745251i	$-0.267671\pi$
$599$	32.6384	1.33357	0.666784	+	0.745251i	$0.267671\pi$
$600$	0	0
$601$	16.9351	0.690796	0.345398	−	0.938456i	$-0.387744\pi$
$601$	16.9351	0.690796	0.345398	+	0.938456i	$0.387744\pi$
$602$	5.34253	0.217745
$603$	76.7488	3.12545
$604$	−1.46309	−0.0595324
$605$	0	0
$606$	74.6276	3.03154
$607$	36.3044	1.47355	0.736775	−	0.676138i	$-0.236348\pi$
$607$	36.3044	1.47355	0.736775	+	0.676138i	$0.236348\pi$
$608$	30.2000	1.22477
$609$	4.69350	0.190190
$610$	0	0
$611$	0.904445	0.0365899
$612$	−69.4066	−2.80560
$613$	−30.7941	−1.24376	−0.621881	−	0.783112i	$-0.713631\pi$
$613$	−30.7941	−1.24376	−0.621881	+	0.783112i	$0.713631\pi$
$614$	−3.51875	−0.142005
$615$	0	0
$616$	0.0713833	0.00287611
$617$	−35.3238	−1.42208	−0.711042	−	0.703150i	$-0.751776\pi$
$617$	−35.3238	−1.42208	−0.711042	+	0.703150i	$0.751776\pi$
$618$	−46.7145	−1.87913
$619$	3.95098	0.158803	0.0794017	−	0.996843i	$-0.474699\pi$
$619$	3.95098	0.158803	0.0794017	+	0.996843i	$0.474699\pi$
$620$	0	0
$621$	69.1642	2.77546
$622$	36.9030	1.47967
$623$	−0.566053	−0.0226784
$624$	−13.1285	−0.525560
$625$	0	0
$626$	−6.66037	−0.266202
$627$	−19.8258	−0.791768
$628$	45.3517	1.80973
$629$	−50.5969	−2.01743
$630$	0	0
$631$	13.5268	0.538494	0.269247	−	0.963071i	$-0.413225\pi$
$631$	13.5268	0.538494	0.269247	+	0.963071i	$0.413225\pi$
$632$	0.629040	0.0250219
$633$	−21.9393	−0.872008
$634$	2.01174	0.0798965
$635$	0	0
$636$	−24.3414	−0.965200
$637$	7.66077	0.303531
$638$	−14.7442	−0.583730
$639$	71.8660	2.84298
$640$	0	0
$641$	−36.1269	−1.42693	−0.713464	−	0.700692i	$-0.752875\pi$
$641$	−36.1269	−1.42693	−0.713464	+	0.700692i	$0.752875\pi$
$642$	−4.61828	−0.182269
$643$	7.35135	0.289909	0.144954	−	0.989438i	$-0.453696\pi$
$643$	7.35135	0.289909	0.144954	+	0.989438i	$0.453696\pi$
$644$	5.50886	0.217080
$645$	0	0
$646$	41.4655	1.63144
$647$	−43.7004	−1.71804	−0.859020	−	0.511943i	$-0.828926\pi$
$647$	−43.7004	−1.71804	−0.859020	+	0.511943i	$0.828926\pi$
$648$	1.15078	0.0452069
$649$	3.24359	0.127322
$650$	0	0
$651$	0.342185	0.0134113
$652$	27.0459	1.05920
$653$	35.8445	1.40270	0.701352	−	0.712815i	$-0.252580\pi$
$653$	35.8445	1.40270	0.701352	+	0.712815i	$0.252580\pi$
$654$	19.9133	0.778672
$655$	0	0
$656$	16.2736	0.635379
$657$	−21.1189	−0.823925
$658$	0.603653	0.0235329
$659$	0.0683150	0.00266117	0.00133059	−	0.999999i	$-0.499576\pi$
$659$	0.0683150	0.00266117	0.00133059	+	0.999999i	$0.499576\pi$
$660$	0	0
$661$	−26.3573	−1.02518	−0.512590	−	0.858634i	$-0.671314\pi$
$661$	−26.3573	−1.02518	−0.512590	+	0.858634i	$0.671314\pi$
$662$	−28.5960	−1.11142
$663$	−18.5347	−0.719828
$664$	−0.788166	−0.0305868
$665$	0	0
$666$	114.262	4.42756
$667$	−30.3821	−1.17640
$668$	−21.3873	−0.827498
$669$	69.1828	2.67476
$670$	0	0
$671$	−16.9197	−0.653179
$672$	−9.00970	−0.347557
$673$	5.07703	0.195705	0.0978525	−	0.995201i	$-0.468803\pi$
$673$	5.07703	0.195705	0.0978525	+	0.995201i	$0.468803\pi$
$674$	−26.4858	−1.02019
$675$	0	0
$676$	−24.1530	−0.928962
$677$	−38.5508	−1.48163	−0.740814	−	0.671710i	$-0.765560\pi$
$677$	−38.5508	−1.48163	−0.740814	+	0.671710i	$0.765560\pi$
$678$	5.16634	0.198412
$679$	2.36686	0.0908318
$680$	0	0
$681$	−2.01333	−0.0771511
$682$	−1.07494	−0.0411618
$683$	−36.6133	−1.40097	−0.700484	−	0.713668i	$-0.747032\pi$
$683$	−36.6133	−1.40097	−0.700484	+	0.713668i	$0.747032\pi$
$684$	−47.4539	−1.81445
$685$	0	0
$686$	10.3280	0.394326
$687$	71.0220	2.70966
$688$	27.8762	1.06277
$689$	−4.37010	−0.166488
$690$	0	0
$691$	29.3768	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$691$	29.3768	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$692$	18.8893	0.718063
$693$	3.97647	0.151053
$694$	−27.2801	−1.03554
$695$	0	0
$696$	−1.40162	−0.0531284
$697$	22.9750	0.870240
$698$	−64.6339	−2.44643
$699$	53.9454	2.04040
$700$	0	0
$701$	−0.566147	−0.0213831	−0.0106915	−	0.999943i	$-0.503403\pi$
$701$	−0.566147	−0.0213831	−0.0106915	+	0.999943i	$0.503403\pi$
$702$	21.4541	0.809733
$703$	−34.5936	−1.30472
$704$	14.7264	0.555020
$705$	0	0
$706$	−36.8748	−1.38780
$707$	4.53167	0.170431
$708$	11.5480	0.433999
$709$	0.657257	0.0246838	0.0123419	−	0.999924i	$-0.496071\pi$
$709$	0.657257	0.0246838	0.0123419	+	0.999924i	$0.496071\pi$
$710$	0	0
$711$	35.0412	1.31415
$712$	0.169041	0.00633507
$713$	−2.21504	−0.0829540
$714$	−12.3706	−0.462959
$715$	0	0
$716$	−44.5646	−1.66546
$717$	72.2244	2.69727
$718$	−29.0702	−1.08489
$719$	35.1052	1.30920	0.654601	−	0.755975i	$-0.272837\pi$
$719$	35.1052	1.30920	0.654601	+	0.755975i	$0.272837\pi$
$720$	0	0
$721$	−2.83668	−0.105644
$722$	−9.90939	−0.368789
$723$	12.7479	0.474099
$724$	22.1268	0.822336
$725$	0	0
$726$	48.4389	1.79774
$727$	41.2180	1.52869	0.764346	−	0.644807i	$-0.223062\pi$
$727$	41.2180	1.52869	0.764346	+	0.644807i	$0.223062\pi$
$728$	0.0456269	0.00169105
$729$	−22.5338	−0.834587
$730$	0	0
$731$	39.3554	1.45561
$732$	−60.2382	−2.22647
$733$	40.1082	1.48143	0.740715	−	0.671820i	$-0.234487\pi$
$733$	40.1082	1.48143	0.740715	+	0.671820i	$0.234487\pi$
$734$	−20.9684	−0.773957
$735$	0	0
$736$	58.3218	2.14977
$737$	−21.7768	−0.802160
$738$	−51.8840	−1.90987
$739$	−28.5434	−1.04999	−0.524993	−	0.851106i	$-0.675932\pi$
$739$	−28.5434	−1.04999	−0.524993	+	0.851106i	$0.675932\pi$
$740$	0	0
$741$	−12.6723	−0.465530
$742$	−2.91674	−0.107077
$743$	−29.6851	−1.08904	−0.544520	−	0.838748i	$-0.683288\pi$
$743$	−29.6851	−1.08904	−0.544520	+	0.838748i	$0.683288\pi$
$744$	−0.102187	−0.00374635
$745$	0	0
$746$	−20.1820	−0.738914
$747$	−43.9055	−1.60642
$748$	19.6935	0.720067
$749$	−0.280439	−0.0102470
$750$	0	0
$751$	45.2113	1.64978	0.824892	−	0.565290i	$-0.191236\pi$
$751$	45.2113	1.64978	0.824892	+	0.565290i	$0.191236\pi$
$752$	3.14974	0.114859
$753$	−59.2248	−2.15827
$754$	−9.42425	−0.343211
$755$	0	0
$756$	7.25644	0.263914
$757$	−5.69813	−0.207102	−0.103551	−	0.994624i	$-0.533020\pi$
$757$	−5.69813	−0.207102	−0.103551	+	0.994624i	$0.533020\pi$
$758$	28.7945	1.04586
$759$	−38.2875	−1.38975
$760$	0	0
$761$	−41.6303	−1.50910	−0.754548	−	0.656245i	$-0.772144\pi$
$761$	−41.6303	−1.50910	−0.754548	+	0.656245i	$0.772144\pi$
$762$	87.2627	3.16119
$763$	1.20921	0.0437764
$764$	−7.88948	−0.285431
$765$	0	0
$766$	−10.1429	−0.366480
$767$	2.07325	0.0748607
$768$	−45.6462	−1.64712
$769$	−15.9261	−0.574308	−0.287154	−	0.957884i	$-0.592709\pi$
$769$	−15.9261	−0.574308	−0.287154	+	0.957884i	$0.592709\pi$
$770$	0	0
$771$	−55.7233	−2.00683
$772$	21.6447	0.779009
$773$	47.8320	1.72040	0.860198	−	0.509959i	$-0.170340\pi$
$773$	47.8320	1.72040	0.860198	+	0.509959i	$0.170340\pi$
$774$	−88.8755	−3.19456
$775$	0	0
$776$	−0.706817	−0.0253732
$777$	10.3205	0.370245
$778$	20.4037	0.731509
$779$	15.7082	0.562805
$780$	0	0
$781$	−20.3914	−0.729661
$782$	80.0778	2.86358
$783$	−40.0202	−1.43021
$784$	26.6787	0.952811
$785$	0	0
$786$	101.300	3.61327
$787$	−40.5561	−1.44567	−0.722834	−	0.691021i	$-0.757161\pi$
$787$	−40.5561	−1.44567	−0.722834	+	0.691021i	$0.757161\pi$
$788$	19.8357	0.706617
$789$	−13.0845	−0.465821
$790$	0	0
$791$	0.313720	0.0111546
$792$	−1.18749	−0.0421958
$793$	−10.8148	−0.384044
$794$	−39.6047	−1.40552
$795$	0	0
$796$	−32.5619	−1.15413
$797$	−27.8316	−0.985847	−0.492924	−	0.870073i	$-0.664072\pi$
$797$	−27.8316	−0.985847	−0.492924	+	0.870073i	$0.664072\pi$
$798$	−8.45789	−0.299406
$799$	4.44678	0.157316
$800$	0	0
$801$	9.41656	0.332718
$802$	−46.5018	−1.64204
$803$	5.99230	0.211464
$804$	−77.5306	−2.73429
$805$	0	0
$806$	−0.687086	−0.0242016
$807$	−14.3385	−0.504738
$808$	−1.35330	−0.0476088
$809$	−8.24706	−0.289951	−0.144976	−	0.989435i	$-0.546310\pi$
$809$	−8.24706	−0.289951	−0.144976	+	0.989435i	$0.546310\pi$
$810$	0	0
$811$	−18.3560	−0.644567	−0.322283	−	0.946643i	$-0.604450\pi$
$811$	−18.3560	−0.644567	−0.322283	+	0.946643i	$0.604450\pi$
$812$	−3.18757	−0.111862
$813$	−30.1516	−1.05746
$814$	−32.4209	−1.13635
$815$	0	0
$816$	−64.5472	−2.25961
$817$	26.9076	0.941379
$818$	77.7957	2.72006
$819$	2.54169	0.0888137
$820$	0	0
$821$	−15.8635	−0.553640	−0.276820	−	0.960922i	$-0.589281\pi$
$821$	−15.8635	−0.553640	−0.276820	+	0.960922i	$0.589281\pi$
$822$	−18.7161	−0.652797
$823$	32.4705	1.13185	0.565925	−	0.824457i	$-0.308519\pi$
$823$	32.4705	1.13185	0.565925	+	0.824457i	$0.308519\pi$
$824$	0.847120	0.0295108
$825$	0	0
$826$	1.38375	0.0481468
$827$	−28.7946	−1.00129	−0.500643	−	0.865654i	$-0.666903\pi$
$827$	−28.7946	−1.00129	−0.500643	+	0.865654i	$0.666903\pi$
$828$	−91.6425	−3.18480
$829$	−8.59224	−0.298421	−0.149210	−	0.988805i	$-0.547673\pi$
$829$	−8.59224	−0.298421	−0.149210	+	0.988805i	$0.547673\pi$
$830$	0	0
$831$	−53.2796	−1.84825
$832$	9.41283	0.326331
$833$	37.6648	1.30501
$834$	95.7772	3.31649
$835$	0	0
$836$	13.4646	0.465684
$837$	−2.91772	−0.100851
$838$	−21.7257	−0.750501
$839$	−47.0541	−1.62449	−0.812244	−	0.583318i	$-0.801754\pi$
$839$	−47.0541	−1.62449	−0.812244	+	0.583318i	$0.801754\pi$
$840$	0	0
$841$	−11.4201	−0.393797
$842$	65.3537	2.25224
$843$	77.1436	2.65697
$844$	14.9000	0.512878
$845$	0	0
$846$	−10.0421	−0.345253
$847$	2.94140	0.101068
$848$	−15.2189	−0.522620
$849$	−48.7145	−1.67188
$850$	0	0
$851$	−66.8067	−2.29011
$852$	−72.5981	−2.48717
$853$	18.6891	0.639904	0.319952	−	0.947434i	$-0.396333\pi$
$853$	18.6891	0.639904	0.319952	+	0.947434i	$0.396333\pi$
$854$	−7.21811	−0.246999
$855$	0	0
$856$	0.0837478	0.00286244
$857$	−3.06228	−0.104606	−0.0523028	−	0.998631i	$-0.516656\pi$
$857$	−3.06228	−0.104606	−0.0523028	+	0.998631i	$0.516656\pi$
$858$	−11.8764	−0.405454
$859$	16.9664	0.578885	0.289442	−	0.957195i	$-0.406530\pi$
$859$	16.9664	0.578885	0.289442	+	0.957195i	$0.406530\pi$
$860$	0	0
$861$	−4.68630	−0.159709
$862$	27.7726	0.945939
$863$	21.8969	0.745380	0.372690	−	0.927956i	$-0.378435\pi$
$863$	21.8969	0.745380	0.372690	+	0.927956i	$0.378435\pi$
$864$	76.8233	2.61358
$865$	0	0
$866$	42.5127	1.44464
$867$	−39.6910	−1.34798
$868$	−0.232393	−0.00788795
$869$	−9.94264	−0.337281
$870$	0	0
$871$	−13.9194	−0.471640
$872$	−0.361108	−0.0122287
$873$	−39.3738	−1.33260
$874$	54.7499	1.85194
$875$	0	0
$876$	21.3340	0.720808
$877$	−46.8320	−1.58141	−0.790703	−	0.612200i	$-0.790285\pi$
$877$	−46.8320	−1.58141	−0.790703	+	0.612200i	$0.790285\pi$
$878$	73.1882	2.46998
$879$	75.3540	2.54163
$880$	0	0
$881$	0.0281377	0.000947982 0	0.000473991	−	1.00000i	$-0.499849\pi$
$881$	0.0281377	0.000947982 0	0.000473991		1.00000i	$0.499849\pi$
$882$	−85.0576	−2.86404
$883$	29.2717	0.985070	0.492535	−	0.870293i	$-0.336070\pi$
$883$	29.2717	0.985070	0.492535	+	0.870293i	$0.336070\pi$
$884$	12.5878	0.423372
$885$	0	0
$886$	−12.8635	−0.432159
$887$	−19.8797	−0.667494	−0.333747	−	0.942663i	$-0.608313\pi$
$887$	−19.8797	−0.667494	−0.333747	+	0.942663i	$0.608313\pi$
$888$	−3.08201	−0.103425
$889$	5.29892	0.177720
$890$	0	0
$891$	−18.1893	−0.609365
$892$	−46.9852	−1.57318
$893$	3.04030	0.101740
$894$	−90.4791	−3.02607
$895$	0	0
$896$	0.326887	0.0109205
$897$	−24.4727	−0.817119
$898$	−70.8062	−2.36283
$899$	1.28168	0.0427465
$900$	0	0
$901$	−21.4860	−0.715801
$902$	14.7216	0.490177
$903$	−8.02748	−0.267138
$904$	−0.0936864	−0.00311596
$905$	0	0
$906$	4.33807	0.144123
$907$	14.2958	0.474685	0.237343	−	0.971426i	$-0.423724\pi$
$907$	14.2958	0.474685	0.237343	+	0.971426i	$0.423724\pi$
$908$	1.36735	0.0453770
$909$	−75.3865	−2.50041
$910$	0	0
$911$	19.4005	0.642766	0.321383	−	0.946949i	$-0.395852\pi$
$911$	19.4005	0.642766	0.321383	+	0.946949i	$0.395852\pi$
$912$	−44.1315	−1.46134
$913$	12.4578	0.412293
$914$	−44.9036	−1.48528
$915$	0	0
$916$	−48.2343	−1.59371
$917$	6.15135	0.203135
$918$	105.481	3.48139
$919$	4.29914	0.141816	0.0709078	−	0.997483i	$-0.477410\pi$
$919$	4.29914	0.141816	0.0709078	+	0.997483i	$0.477410\pi$
$920$	0	0
$921$	5.28714	0.174217
$922$	−3.79659	−0.125034
$923$	−13.0338	−0.429013
$924$	−4.01697	−0.132149
$925$	0	0
$926$	26.0069	0.854639
$927$	47.1895	1.54991
$928$	−33.7465	−1.10778
$929$	43.0317	1.41182	0.705912	−	0.708300i	$-0.250538\pi$
$929$	43.0317	1.41182	0.705912	+	0.708300i	$0.250538\pi$
$930$	0	0
$931$	25.7517	0.843979
$932$	−36.6368	−1.20008
$933$	−55.4490	−1.81532
$934$	5.27211	0.172509
$935$	0	0
$936$	−0.759025	−0.0248095
$937$	−16.9808	−0.554740	−0.277370	−	0.960763i	$-0.589463\pi$
$937$	−16.9808	−0.554740	−0.277370	+	0.960763i	$0.589463\pi$
$938$	−9.29020	−0.303336
$939$	10.0076	0.326586
$940$	0	0
$941$	−18.0448	−0.588242	−0.294121	−	0.955768i	$-0.595027\pi$
$941$	−18.0448	−0.588242	−0.294121	+	0.955768i	$0.595027\pi$
$942$	−134.468	−4.38120
$943$	30.3355	0.987860
$944$	7.22011	0.234995
$945$	0	0
$946$	25.2176	0.819896
$947$	−7.20872	−0.234252	−0.117126	−	0.993117i	$-0.537368\pi$
$947$	−7.20872	−0.234252	−0.117126	+	0.993117i	$0.537368\pi$
$948$	−35.3981	−1.14968
$949$	3.83017	0.124333
$950$	0	0
$951$	−3.02277	−0.0980200
$952$	0.224329	0.00727053
$953$	−13.8000	−0.447026	−0.223513	−	0.974701i	$-0.571753\pi$
$953$	−13.8000	−0.447026	−0.223513	+	0.974701i	$0.571753\pi$
$954$	48.5213	1.57094
$955$	0	0
$956$	−49.0509	−1.58642
$957$	22.1541	0.716141
$958$	13.5584	0.438051
$959$	−1.13651	−0.0366998
$960$	0	0
$961$	−30.9066	−0.996986
$962$	−20.7228	−0.668131
$963$	4.66524	0.150335
$964$	−8.65767	−0.278845
$965$	0	0
$966$	−16.3338	−0.525531
$967$	−14.8956	−0.479010	−0.239505	−	0.970895i	$-0.576985\pi$
$967$	−14.8956	−0.479010	−0.239505	+	0.970895i	$0.576985\pi$
$968$	−0.878391	−0.0282326
$969$	−62.3045	−2.00151
$970$	0	0
$971$	39.7938	1.27704	0.638522	−	0.769604i	$-0.279546\pi$
$971$	39.7938	1.27704	0.638522	+	0.769604i	$0.279546\pi$
$972$	−5.91766	−0.189809
$973$	5.81595	0.186451
$974$	11.1919	0.358611
$975$	0	0
$976$	−37.6626	−1.20555
$977$	−30.9627	−0.990584	−0.495292	−	0.868726i	$-0.664939\pi$
$977$	−30.9627	−0.990584	−0.495292	+	0.868726i	$0.664939\pi$
$978$	−80.1914	−2.56424
$979$	−2.67187	−0.0853933
$980$	0	0
$981$	−20.1158	−0.642248
$982$	−11.1799	−0.356764
$983$	27.7549	0.885243	0.442621	−	0.896709i	$-0.354049\pi$
$983$	27.7549	0.885243	0.442621	+	0.896709i	$0.354049\pi$
$984$	1.39947	0.0446136
$985$	0	0
$986$	−46.3351	−1.47561
$987$	−0.907027	−0.0288710
$988$	8.60636	0.273805
$989$	51.9637	1.65235
$990$	0	0
$991$	15.0180	0.477062	0.238531	−	0.971135i	$-0.423334\pi$
$991$	15.0180	0.477062	0.238531	+	0.971135i	$0.423334\pi$
$992$	−2.46033	−0.0781156
$993$	42.9673	1.36353
$994$	−8.69915	−0.275920
$995$	0	0
$996$	44.3527	1.40537
$997$	17.9138	0.567335	0.283668	−	0.958923i	$-0.408449\pi$
$997$	17.9138	0.567335	0.283668	+	0.958923i	$0.408449\pi$
$998$	38.7793	1.22754
$999$	−87.9999	−2.78419

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.e.1.8	✓	8
3.2	odd	2		5625.2.a.be.1.1		8
4.3	odd	2		10000.2.a.bn.1.7		8
5.2	odd	4		625.2.b.d.624.13		16
5.3	odd	4		625.2.b.d.624.4		16
5.4	even	2		625.2.a.g.1.1	yes	8
15.14	odd	2		5625.2.a.s.1.8		8
20.19	odd	2		10000.2.a.be.1.2		8
25.2	odd	20		625.2.e.j.124.7		32
25.3	odd	20		625.2.e.k.249.2		32
25.4	even	10		625.2.d.n.376.4		16
25.6	even	5		625.2.d.p.251.1		16
25.8	odd	20		625.2.e.k.374.7		32
25.9	even	10		625.2.d.m.126.1		16
25.11	even	5		625.2.d.q.501.4		16
25.12	odd	20		625.2.e.j.499.2		32
25.13	odd	20		625.2.e.j.499.7		32
25.14	even	10		625.2.d.m.501.1		16
25.16	even	5		625.2.d.q.126.4		16
25.17	odd	20		625.2.e.k.374.2		32
25.19	even	10		625.2.d.n.251.4		16
25.21	even	5		625.2.d.p.376.1		16
25.22	odd	20		625.2.e.k.249.7		32
25.23	odd	20		625.2.e.j.124.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.8	✓	8	1.1	even	1	trivial
625.2.a.g.1.1	yes	8	5.4	even	2
625.2.b.d.624.4		16	5.3	odd	4
625.2.b.d.624.13		16	5.2	odd	4
625.2.d.m.126.1		16	25.9	even	10
625.2.d.m.501.1		16	25.14	even	10
625.2.d.n.251.4		16	25.19	even	10
625.2.d.n.376.4		16	25.4	even	10
625.2.d.p.251.1		16	25.6	even	5
625.2.d.p.376.1		16	25.21	even	5
625.2.d.q.126.4		16	25.16	even	5
625.2.d.q.501.4		16	25.11	even	5
625.2.e.j.124.2		32	25.23	odd	20
625.2.e.j.124.7		32	25.2	odd	20
625.2.e.j.499.2		32	25.12	odd	20
625.2.e.j.499.7		32	25.13	odd	20
625.2.e.k.249.2		32	25.3	odd	20
625.2.e.k.249.7		32	25.22	odd	20
625.2.e.k.374.2		32	25.17	odd	20
625.2.e.k.374.7		32	25.8	odd	20
5625.2.a.s.1.8		8	15.14	odd	2
5625.2.a.be.1.1		8	3.2	odd	2
10000.2.a.be.1.2		8	20.19	odd	2
10000.2.a.bn.1.7		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)

\(f(q)\)	\(=\)	\(q+2.01367 q^{2} -3.02566 q^{3} +2.05487 q^{4} -6.09269 q^{6} -0.369971 q^{7} +0.110485 q^{8} +6.15465 q^{9} +O(q^{10})\)	\(q+2.01367 q^{2} -3.02566 q^{3} +2.05487 q^{4} -6.09269 q^{6} -0.369971 q^{7} +0.110485 q^{8} +6.15465 q^{9} -1.74633 q^{11} -6.21734 q^{12} -1.11622 q^{13} -0.745000 q^{14} -3.88725 q^{16} -5.48800 q^{17} +12.3934 q^{18} -3.75219 q^{19} +1.11941 q^{21} -3.51653 q^{22} -7.24619 q^{23} -0.334290 q^{24} -2.24770 q^{26} -9.54490 q^{27} -0.760242 q^{28} +4.19284 q^{29} +0.305684 q^{31} -8.04862 q^{32} +5.28380 q^{33} -11.0510 q^{34} +12.6470 q^{36} +9.21956 q^{37} -7.55568 q^{38} +3.37731 q^{39} -4.18641 q^{41} +2.25412 q^{42} -7.17118 q^{43} -3.58847 q^{44} -14.5914 q^{46} -0.810273 q^{47} +11.7615 q^{48} -6.86312 q^{49} +16.6048 q^{51} -2.29369 q^{52} +3.91508 q^{53} -19.2203 q^{54} -0.0408762 q^{56} +11.3529 q^{57} +8.44299 q^{58} -1.85738 q^{59} +9.68874 q^{61} +0.615546 q^{62} -2.27704 q^{63} -8.43275 q^{64} +10.6398 q^{66} +12.4701 q^{67} -11.2771 q^{68} +21.9245 q^{69} +11.6767 q^{71} +0.679995 q^{72} -3.43137 q^{73} +18.5652 q^{74} -7.71026 q^{76} +0.646091 q^{77} +6.80080 q^{78} +5.69346 q^{79} +10.4157 q^{81} -8.43005 q^{82} -7.13371 q^{83} +2.30024 q^{84} -14.4404 q^{86} -12.6861 q^{87} -0.192943 q^{88} +1.52999 q^{89} +0.412970 q^{91} -14.8900 q^{92} -0.924896 q^{93} -1.63162 q^{94} +24.3524 q^{96} -6.39742 q^{97} -13.8201 q^{98} -10.7480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) 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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