LMFDB - Embedded newform 625.2.a.g.1.3

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:	$0$
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			1.3
Root			$1.32675$ 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)

$f(q)$	$=$	$q-0.326751 q^{2} -1.71538 q^{3} -1.89323 q^{4} +0.560502 q^{6} +3.42409 q^{7} +1.27212 q^{8} -0.0574791 q^{9} +O(q^{10})$	\(q-0.326751 q^{2} -1.71538 q^{3} -1.89323 q^{4} +0.560502 q^{6} +3.42409 q^{7} +1.27212 q^{8} -0.0574791 q^{9} -5.34111 q^{11} +3.24761 q^{12} -3.52114 q^{13} -1.11883 q^{14} +3.37080 q^{16} +2.55787 q^{17} +0.0187814 q^{18} -2.02579 q^{19} -5.87362 q^{21} +1.74522 q^{22} +7.57082 q^{23} -2.18216 q^{24} +1.15054 q^{26} +5.24473 q^{27} -6.48261 q^{28} +4.74270 q^{29} +1.62421 q^{31} -3.64565 q^{32} +9.16203 q^{33} -0.835788 q^{34} +0.108821 q^{36} -0.0134290 q^{37} +0.661929 q^{38} +6.04008 q^{39} +9.67740 q^{41} +1.91921 q^{42} +2.32645 q^{43} +10.1120 q^{44} -2.47377 q^{46} +6.94647 q^{47} -5.78220 q^{48} +4.72443 q^{49} -4.38772 q^{51} +6.66634 q^{52} +1.72246 q^{53} -1.71372 q^{54} +4.35585 q^{56} +3.47500 q^{57} -1.54968 q^{58} -0.0221830 q^{59} +3.91768 q^{61} -0.530712 q^{62} -0.196814 q^{63} -5.55038 q^{64} -2.99370 q^{66} +4.11832 q^{67} -4.84265 q^{68} -12.9868 q^{69} +2.33894 q^{71} -0.0731202 q^{72} +1.51373 q^{73} +0.00438793 q^{74} +3.83530 q^{76} -18.2885 q^{77} -1.97360 q^{78} +0.426831 q^{79} -8.82426 q^{81} -3.16210 q^{82} -6.04187 q^{83} +11.1201 q^{84} -0.760170 q^{86} -8.13552 q^{87} -6.79453 q^{88} -6.09362 q^{89} -12.0567 q^{91} -14.3333 q^{92} -2.78613 q^{93} -2.26977 q^{94} +6.25367 q^{96} -16.0018 q^{97} -1.54371 q^{98} +0.307002 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.326751	−0.231048	−0.115524	−	0.993305i	$-0.536855\pi$
$2$	−0.326751	−0.231048	−0.115524	+	0.993305i	$0.536855\pi$
$3$	−1.71538	−0.990374	−0.495187	−	0.868786i	$-0.664900\pi$
$3$	−1.71538	−0.990374	−0.495187	+	0.868786i	$0.664900\pi$
$4$	−1.89323	−0.946617
$5$	0	0
$5$	0	0
$6$	0.560502	0.228824
$7$	3.42409	1.29419	0.647093	−	0.762411i	$-0.275984\pi$
$7$	3.42409	1.29419	0.647093	+	0.762411i	$0.275984\pi$
$8$	1.27212	0.449762
$9$	−0.0574791	−0.0191597
$10$	0	0
$11$	−5.34111	−1.61041	−0.805203	−	0.592999i	$-0.797944\pi$
$11$	−5.34111	−1.61041	−0.805203	+	0.592999i	$0.797944\pi$
$12$	3.24761	0.937505
$13$	−3.52114	−0.976588	−0.488294	−	0.872679i	$-0.662381\pi$
$13$	−3.52114	−0.976588	−0.488294	+	0.872679i	$0.662381\pi$
$14$	−1.11883	−0.299019
$15$	0	0
$16$	3.37080	0.842700
$17$	2.55787	0.620375	0.310188	−	0.950675i	$-0.399608\pi$
$17$	2.55787	0.620375	0.310188	+	0.950675i	$0.399608\pi$
$18$	0.0187814	0.00442681
$19$	−2.02579	−0.464748	−0.232374	−	0.972626i	$-0.574649\pi$
$19$	−2.02579	−0.464748	−0.232374	+	0.972626i	$0.574649\pi$
$20$	0	0
$21$	−5.87362	−1.28173
$22$	1.74522	0.372081
$23$	7.57082	1.57863	0.789313	−	0.613991i	$-0.210437\pi$
$23$	7.57082	1.57863	0.789313	+	0.613991i	$0.210437\pi$
$24$	−2.18216	−0.445432
$25$	0	0
$26$	1.15054	0.225639
$27$	5.24473	1.00935
$28$	−6.48261	−1.22510
$29$	4.74270	0.880697	0.440348	−	0.897827i	$-0.354855\pi$
$29$	4.74270	0.880697	0.440348	+	0.897827i	$0.354855\pi$
$30$	0	0
$31$	1.62421	0.291716	0.145858	−	0.989306i	$-0.453406\pi$
$31$	1.62421	0.291716	0.145858	+	0.989306i	$0.453406\pi$
$32$	−3.64565	−0.644466
$33$	9.16203	1.59490
$34$	−0.835788	−0.143336
$35$	0	0
$36$	0.108821	0.0181369
$37$	−0.0134290	−0.00220771	−0.00110385	−	0.999999i	$-0.500351\pi$
$37$	−0.0134290	−0.00220771	−0.00110385	+	0.999999i	$0.500351\pi$
$38$	0.661929	0.107379
$39$	6.04008	0.967187
$40$	0	0
$41$	9.67740	1.51136	0.755678	−	0.654943i	$-0.227307\pi$
$41$	9.67740	1.51136	0.755678	+	0.654943i	$0.227307\pi$
$42$	1.91921	0.296141
$43$	2.32645	0.354780	0.177390	−	0.984141i	$-0.443235\pi$
$43$	2.32645	0.354780	0.177390	+	0.984141i	$0.443235\pi$
$44$	10.1120	1.52444
$45$	0	0
$46$	−2.47377	−0.364738
$47$	6.94647	1.01325	0.506624	−	0.862167i	$-0.330893\pi$
$47$	6.94647	1.01325	0.506624	+	0.862167i	$0.330893\pi$
$48$	−5.78220	−0.834588
$49$	4.72443	0.674918
$50$	0	0
$51$	−4.38772	−0.614403
$52$	6.66634	0.924454
$53$	1.72246	0.236598	0.118299	−	0.992978i	$-0.462256\pi$
$53$	1.72246	0.236598	0.118299	+	0.992978i	$0.462256\pi$
$54$	−1.71372	−0.233208
$55$	0	0
$56$	4.35585	0.582075
$57$	3.47500	0.460275
$58$	−1.54968	−0.203483
$59$	−0.0221830	−0.00288798	−0.00144399	−	0.999999i	$-0.500460\pi$
$59$	−0.0221830	−0.00288798	−0.00144399	+	0.999999i	$0.500460\pi$
$60$	0	0
$61$	3.91768	0.501607	0.250804	−	0.968038i	$-0.419305\pi$
$61$	3.91768	0.501607	0.250804	+	0.968038i	$0.419305\pi$
$62$	−0.530712	−0.0674004
$63$	−0.196814	−0.0247962
$64$	−5.55038	−0.693798
$65$	0	0
$66$	−2.99370	−0.368499
$67$	4.11832	0.503133	0.251566	−	0.967840i	$-0.419054\pi$
$67$	4.11832	0.503133	0.251566	+	0.967840i	$0.419054\pi$
$68$	−4.84265	−0.587258
$69$	−12.9868	−1.56343
$70$	0	0
$71$	2.33894	0.277581	0.138790	−	0.990322i	$-0.455679\pi$
$71$	2.33894	0.277581	0.138790	+	0.990322i	$0.455679\pi$
$72$	−0.0731202	−0.00861730
$73$	1.51373	0.177169	0.0885843	−	0.996069i	$-0.471766\pi$
$73$	1.51373	0.177169	0.0885843	+	0.996069i	$0.471766\pi$
$74$	0.00438793	0.000510086 0
$75$	0	0
$76$	3.83530	0.439939
$77$	−18.2885	−2.08417
$78$	−1.97360	−0.223467
$79$	0.426831	0.0480222	0.0240111	−	0.999712i	$-0.492356\pi$
$79$	0.426831	0.0480222	0.0240111	+	0.999712i	$0.492356\pi$
$80$	0	0
$81$	−8.82426	−0.980473
$82$	−3.16210	−0.349196
$83$	−6.04187	−0.663181	−0.331591	−	0.943423i	$-0.607585\pi$
$83$	−6.04187	−0.663181	−0.331591	+	0.943423i	$0.607585\pi$
$84$	11.1201	1.21331
$85$	0	0
$86$	−0.760170	−0.0819712
$87$	−8.13552	−0.872219
$88$	−6.79453	−0.724299
$89$	−6.09362	−0.645922	−0.322961	−	0.946412i	$-0.604678\pi$
$89$	−6.09362	−0.645922	−0.322961	+	0.946412i	$0.604678\pi$
$90$	0	0
$91$	−12.0567	−1.26389
$92$	−14.3333	−1.49435
$93$	−2.78613	−0.288908
$94$	−2.26977	−0.234109
$95$	0	0
$96$	6.25367	0.638262
$97$	−16.0018	−1.62474	−0.812370	−	0.583143i	$-0.801823\pi$
$97$	−16.0018	−1.62474	−0.812370	+	0.583143i	$0.801823\pi$
$98$	−1.54371	−0.155938
$99$	0.307002	0.0308549
$100$	0	0
$101$	1.44418	0.143701	0.0718505	−	0.997415i	$-0.477110\pi$
$101$	1.44418	0.143701	0.0718505	+	0.997415i	$0.477110\pi$
$102$	1.43369	0.141957
$103$	14.6657	1.44506	0.722529	−	0.691341i	$-0.242980\pi$
$103$	14.6657	1.44506	0.722529	+	0.691341i	$0.242980\pi$
$104$	−4.47930	−0.439232
$105$	0	0
$106$	−0.562816	−0.0546655
$107$	12.2169	1.18106	0.590528	−	0.807017i	$-0.298919\pi$
$107$	12.2169	1.18106	0.590528	+	0.807017i	$0.298919\pi$
$108$	−9.92950	−0.955467
$109$	−15.3516	−1.47041	−0.735207	−	0.677843i	$-0.762915\pi$
$109$	−15.3516	−1.47041	−0.735207	+	0.677843i	$0.762915\pi$
$110$	0	0
$111$	0.0230357	0.00218646
$112$	11.5419	1.09061
$113$	18.5544	1.74545	0.872727	−	0.488209i	$-0.162350\pi$
$113$	18.5544	1.74545	0.872727	+	0.488209i	$0.162350\pi$
$114$	−1.13546	−0.106345
$115$	0	0
$116$	−8.97904	−0.833683
$117$	0.202392	0.0187111
$118$	0.00724831	0.000667261 0
$119$	8.75840	0.802881
$120$	0	0
$121$	17.5275	1.59341
$122$	−1.28011	−0.115895
$123$	−16.6004	−1.49681
$124$	−3.07500	−0.276144
$125$	0	0
$126$	0.0643092	0.00572911
$127$	0.677902	0.0601541	0.0300771	−	0.999548i	$-0.490425\pi$
$127$	0.677902	0.0601541	0.0300771	+	0.999548i	$0.490425\pi$
$128$	9.10489	0.804766
$129$	−3.99074	−0.351365
$130$	0	0
$131$	7.05058	0.616012	0.308006	−	0.951384i	$-0.400338\pi$
$131$	7.05058	0.616012	0.308006	+	0.951384i	$0.400338\pi$
$132$	−17.3459	−1.50976
$133$	−6.93650	−0.601471
$134$	−1.34567	−0.116248
$135$	0	0
$136$	3.25392	0.279021
$137$	−10.9186	−0.932837	−0.466418	−	0.884564i	$-0.654456\pi$
$137$	−10.9186	−0.932837	−0.466418	+	0.884564i	$0.654456\pi$
$138$	4.24346	0.361227
$139$	19.5102	1.65483	0.827416	−	0.561589i	$-0.189810\pi$
$139$	19.5102	1.65483	0.827416	+	0.561589i	$0.189810\pi$
$140$	0	0
$141$	−11.9158	−1.00349
$142$	−0.764251	−0.0641345
$143$	18.8068	1.57270
$144$	−0.193751	−0.0161459
$145$	0	0
$146$	−0.494613	−0.0409345
$147$	−8.10418	−0.668421
$148$	0.0254242	0.00208985
$149$	−12.7945	−1.04817	−0.524085	−	0.851666i	$-0.675593\pi$
$149$	−12.7945	−1.04817	−0.524085	+	0.851666i	$0.675593\pi$
$150$	0	0
$151$	2.15617	0.175466	0.0877331	−	0.996144i	$-0.472038\pi$
$151$	2.15617	0.175466	0.0877331	+	0.996144i	$0.472038\pi$
$152$	−2.57705	−0.209026
$153$	−0.147024	−0.0118862
$154$	5.97578	0.481542
$155$	0	0
$156$	−11.4353	−0.915556
$157$	−15.7474	−1.25678	−0.628389	−	0.777899i	$-0.716285\pi$
$157$	−15.7474	−1.25678	−0.628389	+	0.777899i	$0.716285\pi$
$158$	−0.139467	−0.0110954
$159$	−2.95467	−0.234321
$160$	0	0
$161$	25.9232	2.04304
$162$	2.88334	0.226536
$163$	7.39219	0.579001	0.289500	−	0.957178i	$-0.406511\pi$
$163$	7.39219	0.579001	0.289500	+	0.957178i	$0.406511\pi$
$164$	−18.3216	−1.43068
$165$	0	0
$166$	1.97419	0.153227
$167$	−10.8165	−0.837007	−0.418504	−	0.908215i	$-0.637445\pi$
$167$	−10.8165	−0.837007	−0.418504	+	0.908215i	$0.637445\pi$
$168$	−7.47194	−0.576472
$169$	−0.601591	−0.0462762
$170$	0	0
$171$	0.116441	0.00890444
$172$	−4.40451	−0.335841
$173$	14.9983	1.14030	0.570149	−	0.821541i	$-0.306885\pi$
$173$	14.9983	1.14030	0.570149	+	0.821541i	$0.306885\pi$
$174$	2.65829	0.201524
$175$	0	0
$176$	−18.0038	−1.35709
$177$	0.0380522	0.00286018
$178$	1.99110	0.149239
$179$	7.39841	0.552983	0.276492	−	0.961016i	$-0.410828\pi$
$179$	7.39841	0.552983	0.276492	+	0.961016i	$0.410828\pi$
$180$	0	0
$181$	−10.9177	−0.811503	−0.405752	−	0.913983i	$-0.632990\pi$
$181$	−10.9177	−0.811503	−0.405752	+	0.913983i	$0.632990\pi$
$182$	3.93954	0.292018
$183$	−6.72030	−0.496779
$184$	9.63098	0.710006
$185$	0	0
$186$	0.910371	0.0667516
$187$	−13.6619	−0.999056
$188$	−13.1513	−0.959157
$189$	17.9585	1.30629
$190$	0	0
$191$	−1.75142	−0.126728	−0.0633642	−	0.997990i	$-0.520183\pi$
$191$	−1.75142	−0.126728	−0.0633642	+	0.997990i	$0.520183\pi$
$192$	9.52100	0.687119
$193$	−9.53146	−0.686089	−0.343045	−	0.939319i	$-0.611458\pi$
$193$	−9.53146	−0.686089	−0.343045	+	0.939319i	$0.611458\pi$
$194$	5.22862	0.375393
$195$	0	0
$196$	−8.94444	−0.638889
$197$	23.9169	1.70401	0.852004	−	0.523535i	$-0.175387\pi$
$197$	23.9169	1.70401	0.852004	+	0.523535i	$0.175387\pi$
$198$	−0.100313	−0.00712896
$199$	23.8281	1.68913	0.844566	−	0.535451i	$-0.179858\pi$
$199$	23.8281	1.68913	0.844566	+	0.535451i	$0.179858\pi$
$200$	0	0
$201$	−7.06448	−0.498290
$202$	−0.471886	−0.0332018
$203$	16.2394	1.13979
$204$	8.30697	0.581605
$205$	0	0
$206$	−4.79204	−0.333878
$207$	−0.435164	−0.0302460
$208$	−11.8691	−0.822971
$209$	10.8200	0.748434
$210$	0	0
$211$	−15.3923	−1.05965	−0.529826	−	0.848107i	$-0.677743\pi$
$211$	−15.3923	−1.05965	−0.529826	+	0.848107i	$0.677743\pi$
$212$	−3.26102	−0.223968
$213$	−4.01216	−0.274909
$214$	−3.99190	−0.272881
$215$	0	0
$216$	6.67192	0.453967
$217$	5.56144	0.377535
$218$	5.01614	0.339736
$219$	−2.59662	−0.175463
$220$	0	0
$221$	−9.00662	−0.605851
$222$	−0.00752696	−0.000505176 0
$223$	−19.5753	−1.31086	−0.655429	−	0.755257i	$-0.727512\pi$
$223$	−19.5753	−1.31086	−0.655429	+	0.755257i	$0.727512\pi$
$224$	−12.4831	−0.834059
$225$	0	0
$226$	−6.06268	−0.403283
$227$	1.34207	0.0890760	0.0445380	−	0.999008i	$-0.485818\pi$
$227$	1.34207	0.0890760	0.0445380	+	0.999008i	$0.485818\pi$
$228$	−6.57898	−0.435704
$229$	−22.3702	−1.47827	−0.739133	−	0.673560i	$-0.764764\pi$
$229$	−22.3702	−1.47827	−0.739133	+	0.673560i	$0.764764\pi$
$230$	0	0
$231$	31.3717	2.06410
$232$	6.03327	0.396104
$233$	18.3651	1.20314	0.601568	−	0.798822i	$-0.294543\pi$
$233$	18.3651	1.20314	0.601568	+	0.798822i	$0.294543\pi$
$234$	−0.0661317	−0.00432317
$235$	0	0
$236$	0.0419976	0.00273381
$237$	−0.732176	−0.0475600
$238$	−2.86182	−0.185504
$239$	−12.0037	−0.776458	−0.388229	−	0.921563i	$-0.626913\pi$
$239$	−12.0037	−0.776458	−0.388229	+	0.921563i	$0.626913\pi$
$240$	0	0
$241$	10.1170	0.651692	0.325846	−	0.945423i	$-0.394351\pi$
$241$	10.1170	0.651692	0.325846	+	0.945423i	$0.394351\pi$
$242$	−5.72713	−0.368154
$243$	−0.597258	−0.0383141
$244$	−7.41708	−0.474830
$245$	0	0
$246$	5.42420	0.345834
$247$	7.13309	0.453867
$248$	2.06618	0.131203
$249$	10.3641	0.656797
$250$	0	0
$251$	16.7258	1.05573	0.527863	−	0.849330i	$-0.322994\pi$
$251$	16.7258	1.05573	0.527863	+	0.849330i	$0.322994\pi$
$252$	0.372615	0.0234725
$253$	−40.4366	−2.54223
$254$	−0.221505	−0.0138985
$255$	0	0
$256$	8.12573	0.507858
$257$	15.0170	0.936732	0.468366	−	0.883535i	$-0.344843\pi$
$257$	15.0170	0.936732	0.468366	+	0.883535i	$0.344843\pi$
$258$	1.30398	0.0811822
$259$	−0.0459821	−0.00285719
$260$	0	0
$261$	−0.272606	−0.0168739
$262$	−2.30379	−0.142328
$263$	6.17182	0.380571	0.190285	−	0.981729i	$-0.439059\pi$
$263$	6.17182	0.380571	0.190285	+	0.981729i	$0.439059\pi$
$264$	11.6552	0.717327
$265$	0	0
$266$	2.26651	0.138969
$267$	10.4529	0.639704
$268$	−7.79694	−0.476274
$269$	11.1052	0.677098	0.338549	−	0.940949i	$-0.390064\pi$
$269$	11.1052	0.677098	0.338549	+	0.940949i	$0.390064\pi$
$270$	0	0
$271$	−1.16149	−0.0705554	−0.0352777	−	0.999378i	$-0.511232\pi$
$271$	−1.16149	−0.0705554	−0.0352777	+	0.999378i	$0.511232\pi$
$272$	8.62208	0.522790
$273$	20.6818	1.25172
$274$	3.56766	0.215530
$275$	0	0
$276$	24.5871	1.47997
$277$	−2.17486	−0.130675	−0.0653374	−	0.997863i	$-0.520812\pi$
$277$	−2.17486	−0.130675	−0.0653374	+	0.997863i	$0.520812\pi$
$278$	−6.37497	−0.382346
$279$	−0.0933580	−0.00558920
$280$	0	0
$281$	−24.1177	−1.43874	−0.719370	−	0.694627i	$-0.755569\pi$
$281$	−24.1177	−1.43874	−0.719370	+	0.694627i	$0.755569\pi$
$282$	3.89351	0.231855
$283$	16.2144	0.963845	0.481923	−	0.876214i	$-0.339939\pi$
$283$	16.2144	0.963845	0.481923	+	0.876214i	$0.339939\pi$
$284$	−4.42816	−0.262763
$285$	0	0
$286$	−6.14514	−0.363370
$287$	33.1363	1.95598
$288$	0.209549	0.0123478
$289$	−10.4573	−0.615135
$290$	0	0
$291$	27.4492	1.60910
$292$	−2.86584	−0.167711
$293$	3.48929	0.203846	0.101923	−	0.994792i	$-0.467500\pi$
$293$	3.48929	0.203846	0.101923	+	0.994792i	$0.467500\pi$
$294$	2.64805	0.154437
$295$	0	0
$296$	−0.0170832	−0.000992943 0
$297$	−28.0127	−1.62546
$298$	4.18063	0.242178
$299$	−26.6579	−1.54167
$300$	0	0
$301$	7.96599	0.459152
$302$	−0.704529	−0.0405411
$303$	−2.47731	−0.142318
$304$	−6.82854	−0.391644
$305$	0	0
$306$	0.0480403	0.00274628
$307$	8.92690	0.509485	0.254742	−	0.967009i	$-0.418009\pi$
$307$	8.92690	0.509485	0.254742	+	0.967009i	$0.418009\pi$
$308$	34.6244	1.97291
$309$	−25.1573	−1.43115
$310$	0	0
$311$	−27.1101	−1.53727	−0.768635	−	0.639687i	$-0.779064\pi$
$311$	−27.1101	−1.53727	−0.768635	+	0.639687i	$0.779064\pi$
$312$	7.68370	0.435004
$313$	−20.1073	−1.13653	−0.568267	−	0.822844i	$-0.692386\pi$
$313$	−20.1073	−1.13653	−0.568267	+	0.822844i	$0.692386\pi$
$314$	5.14548	0.290376
$315$	0	0
$316$	−0.808091	−0.0454587
$317$	9.67993	0.543679	0.271840	−	0.962343i	$-0.412368\pi$
$317$	9.67993	0.543679	0.271840	+	0.962343i	$0.412368\pi$
$318$	0.965442	0.0541393
$319$	−25.3313	−1.41828
$320$	0	0
$321$	−20.9567	−1.16969
$322$	−8.47044	−0.472039
$323$	−5.18171	−0.288318
$324$	16.7064	0.928132
$325$	0	0
$326$	−2.41540	−0.133777
$327$	26.3337	1.45626
$328$	12.3108	0.679750
$329$	23.7854	1.31133
$330$	0	0
$331$	19.6759	1.08148	0.540742	−	0.841189i	$-0.318144\pi$
$331$	19.6759	1.08148	0.540742	+	0.841189i	$0.318144\pi$
$332$	11.4387	0.627778
$333$	0.000771885 0	4.22990e−5 0
$334$	3.53431	0.193389
$335$	0	0
$336$	−19.7988	−1.08011
$337$	18.8552	1.02711	0.513554	−	0.858057i	$-0.328329\pi$
$337$	18.8552	1.02711	0.513554	+	0.858057i	$0.328329\pi$
$338$	0.196571	0.0106920
$339$	−31.8279	−1.72865
$340$	0	0
$341$	−8.67508	−0.469782
$342$	−0.0380471	−0.00205735
$343$	−7.79178	−0.420717
$344$	2.95952	0.159567
$345$	0	0
$346$	−4.90071	−0.263464
$347$	25.9753	1.39442	0.697212	−	0.716865i	$-0.254423\pi$
$347$	25.9753	1.39442	0.697212	+	0.716865i	$0.254423\pi$
$348$	15.4024	0.825657
$349$	−26.1490	−1.39972	−0.699861	−	0.714279i	$-0.746755\pi$
$349$	−26.1490	−1.39972	−0.699861	+	0.714279i	$0.746755\pi$
$350$	0	0
$351$	−18.4674	−0.985718
$352$	19.4718	1.03785
$353$	−5.32892	−0.283630	−0.141815	−	0.989893i	$-0.545294\pi$
$353$	−5.32892	−0.283630	−0.141815	+	0.989893i	$0.545294\pi$
$354$	−0.0124336	−0.000660838 0
$355$	0	0
$356$	11.5366	0.611441
$357$	−15.0240	−0.795152
$358$	−2.41744	−0.127766
$359$	9.89929	0.522465	0.261232	−	0.965276i	$-0.415871\pi$
$359$	9.89929	0.522465	0.261232	+	0.965276i	$0.415871\pi$
$360$	0	0
$361$	−14.8962	−0.784009
$362$	3.56736	0.187496
$363$	−30.0663	−1.57807
$364$	22.8262	1.19642
$365$	0	0
$366$	2.19587	0.114780
$367$	16.4392	0.858118	0.429059	−	0.903276i	$-0.358845\pi$
$367$	16.4392	0.858118	0.429059	+	0.903276i	$0.358845\pi$
$368$	25.5197	1.33031
$369$	−0.556248	−0.0289571
$370$	0	0
$371$	5.89787	0.306202
$372$	5.27479	0.273485
$373$	22.9933	1.19055	0.595273	−	0.803524i	$-0.297044\pi$
$373$	22.9933	1.19055	0.595273	+	0.803524i	$0.297044\pi$
$374$	4.46404	0.230830
$375$	0	0
$376$	8.83674	0.455720
$377$	−16.6997	−0.860078
$378$	−5.86795	−0.301815
$379$	16.4246	0.843674	0.421837	−	0.906672i	$-0.361385\pi$
$379$	16.4246	0.843674	0.421837	+	0.906672i	$0.361385\pi$
$380$	0	0
$381$	−1.16286	−0.0595751
$382$	0.572279	0.0292803
$383$	4.70503	0.240416	0.120208	−	0.992749i	$-0.461644\pi$
$383$	4.70503	0.240416	0.120208	+	0.992749i	$0.461644\pi$
$384$	−15.6183	−0.797020
$385$	0	0
$386$	3.11441	0.158520
$387$	−0.133722	−0.00679748
$388$	30.2952	1.53801
$389$	3.53865	0.179417	0.0897083	−	0.995968i	$-0.471407\pi$
$389$	3.53865	0.179417	0.0897083	+	0.995968i	$0.471407\pi$
$390$	0	0
$391$	19.3652	0.979340
$392$	6.01003	0.303552
$393$	−12.0944	−0.610082
$394$	−7.81487	−0.393707
$395$	0	0
$396$	−0.581227	−0.0292078
$397$	−8.76374	−0.439839	−0.219920	−	0.975518i	$-0.570580\pi$
$397$	−8.76374	−0.439839	−0.219920	+	0.975518i	$0.570580\pi$
$398$	−7.78587	−0.390270
$399$	11.8987	0.595681
$400$	0	0
$401$	22.7677	1.13697	0.568483	−	0.822695i	$-0.307531\pi$
$401$	22.7677	1.13697	0.568483	+	0.822695i	$0.307531\pi$
$402$	2.30833	0.115129
$403$	−5.71906	−0.284887
$404$	−2.73416	−0.136030
$405$	0	0
$406$	−5.30626	−0.263345
$407$	0.0717256	0.00355531
$408$	−5.58170	−0.276335
$409$	28.0426	1.38662	0.693309	−	0.720640i	$-0.256152\pi$
$409$	28.0426	1.38662	0.693309	+	0.720640i	$0.256152\pi$
$410$	0	0
$411$	18.7295	0.923857
$412$	−27.7657	−1.36792
$413$	−0.0759566	−0.00373758
$414$	0.142190	0.00698827
$415$	0	0
$416$	12.8368	0.629378
$417$	−33.4673	−1.63890
$418$	−3.53544	−0.172924
$419$	34.0901	1.66541	0.832705	−	0.553717i	$-0.186791\pi$
$419$	34.0901	1.66541	0.832705	+	0.553717i	$0.186791\pi$
$420$	0	0
$421$	−32.1390	−1.56636	−0.783180	−	0.621796i	$-0.786403\pi$
$421$	−32.1390	−1.56636	−0.783180	+	0.621796i	$0.786403\pi$
$422$	5.02946	0.244830
$423$	−0.399277	−0.0194135
$424$	2.19117	0.106413
$425$	0	0
$426$	1.31098	0.0635171
$427$	13.4145	0.649173
$428$	−23.1295	−1.11801
$429$	−32.2608	−1.55756
$430$	0	0
$431$	−17.7549	−0.855222	−0.427611	−	0.903963i	$-0.640645\pi$
$431$	−17.7549	−0.855222	−0.427611	+	0.903963i	$0.640645\pi$
$432$	17.6789	0.850579
$433$	3.06764	0.147421	0.0737107	−	0.997280i	$-0.476516\pi$
$433$	3.06764	0.147421	0.0737107	+	0.997280i	$0.476516\pi$
$434$	−1.81721	−0.0872287
$435$	0	0
$436$	29.0641	1.39192
$437$	−15.3369	−0.733664
$438$	0.848448	0.0405404
$439$	14.7475	0.703861	0.351931	−	0.936026i	$-0.385525\pi$
$439$	14.7475	0.703861	0.351931	+	0.936026i	$0.385525\pi$
$440$	0	0
$441$	−0.271556	−0.0129312
$442$	2.94292	0.139981
$443$	−10.1857	−0.483935	−0.241968	−	0.970284i	$-0.577793\pi$
$443$	−10.1857	−0.483935	−0.241968	+	0.970284i	$0.577793\pi$
$444$	−0.0436121	−0.00206974
$445$	0	0
$446$	6.39624	0.302871
$447$	21.9475	1.03808
$448$	−19.0050	−0.897904
$449$	18.9484	0.894230	0.447115	−	0.894477i	$-0.352451\pi$
$449$	18.9484	0.894230	0.447115	+	0.894477i	$0.352451\pi$
$450$	0	0
$451$	−51.6881	−2.43390
$452$	−35.1279	−1.65228
$453$	−3.69864	−0.173777
$454$	−0.438522	−0.0205808
$455$	0	0
$456$	4.42061	0.207014
$457$	−30.4392	−1.42389	−0.711943	−	0.702237i	$-0.752185\pi$
$457$	−30.4392	−1.42389	−0.711943	+	0.702237i	$0.752185\pi$
$458$	7.30949	0.341550
$459$	13.4154	0.626175
$460$	0	0
$461$	2.36972	0.110369	0.0551844	−	0.998476i	$-0.482425\pi$
$461$	2.36972	0.110369	0.0551844	+	0.998476i	$0.482425\pi$
$462$	−10.2507	−0.476907
$463$	0.320982	0.0149173	0.00745865	−	0.999972i	$-0.497626\pi$
$463$	0.320982	0.0149173	0.00745865	+	0.999972i	$0.497626\pi$
$464$	15.9867	0.742164
$465$	0	0
$466$	−6.00081	−0.277982
$467$	12.8071	0.592644	0.296322	−	0.955088i	$-0.404240\pi$
$467$	12.8071	0.592644	0.296322	+	0.955088i	$0.404240\pi$
$468$	−0.383175	−0.0177123
$469$	14.1015	0.651148
$470$	0	0
$471$	27.0127	1.24468
$472$	−0.0282194	−0.00129890
$473$	−12.4258	−0.571341
$474$	0.239239	0.0109886
$475$	0	0
$476$	−16.5817	−0.760021
$477$	−0.0990055	−0.00453315
$478$	3.92224	0.179399
$479$	−20.3446	−0.929569	−0.464784	−	0.885424i	$-0.653868\pi$
$479$	−20.3446	−0.929569	−0.464784	+	0.885424i	$0.653868\pi$
$480$	0	0
$481$	0.0472852	0.00215602
$482$	−3.30573	−0.150572
$483$	−44.4681	−2.02337
$484$	−33.1837	−1.50835
$485$	0	0
$486$	0.195155	0.00885239
$487$	15.4919	0.702003	0.351002	−	0.936375i	$-0.385841\pi$
$487$	15.4919	0.702003	0.351002	+	0.936375i	$0.385841\pi$
$488$	4.98375	0.225604
$489$	−12.6804	−0.573427
$490$	0	0
$491$	−27.5085	−1.24144	−0.620722	−	0.784031i	$-0.713160\pi$
$491$	−27.5085	−1.24144	−0.620722	+	0.784031i	$0.713160\pi$
$492$	31.4284	1.41690
$493$	12.1312	0.546362
$494$	−2.33074	−0.104865
$495$	0	0
$496$	5.47488	0.245829
$497$	8.00875	0.359241
$498$	−3.38648	−0.151752
$499$	4.91044	0.219821	0.109911	−	0.993941i	$-0.464943\pi$
$499$	4.91044	0.219821	0.109911	+	0.993941i	$0.464943\pi$
$500$	0	0
$501$	18.5544	0.828950
$502$	−5.46518	−0.243923
$503$	41.0454	1.83012	0.915062	−	0.403314i	$-0.132142\pi$
$503$	41.0454	1.83012	0.915062	+	0.403314i	$0.132142\pi$
$504$	−0.250371	−0.0111524
$505$	0	0
$506$	13.2127	0.587377
$507$	1.03196	0.0458308
$508$	−1.28343	−0.0569429
$509$	41.4404	1.83681	0.918407	−	0.395637i	$-0.129476\pi$
$509$	41.4404	1.83681	0.918407	+	0.395637i	$0.129476\pi$
$510$	0	0
$511$	5.18315	0.229289
$512$	−20.8649	−0.922106
$513$	−10.6247	−0.469093
$514$	−4.90681	−0.216430
$515$	0	0
$516$	7.55540	0.332608
$517$	−37.1019	−1.63174
$518$	0.0150247	0.000660147 0
$519$	−25.7277	−1.12932
$520$	0	0
$521$	−1.62447	−0.0711691	−0.0355846	−	0.999367i	$-0.511329\pi$
$521$	−1.62447	−0.0711691	−0.0355846	+	0.999367i	$0.511329\pi$
$522$	0.0890743	0.00389868
$523$	−19.0009	−0.830853	−0.415427	−	0.909627i	$-0.636368\pi$
$523$	−19.0009	−0.830853	−0.415427	+	0.909627i	$0.636368\pi$
$524$	−13.3484	−0.583128
$525$	0	0
$526$	−2.01665	−0.0879301
$527$	4.15451	0.180973
$528$	30.8834	1.34403
$529$	34.3174	1.49206
$530$	0	0
$531$	0.00127506	5.53328e−5 0
$532$	13.1324	0.569362
$533$	−34.0755	−1.47597
$534$	−3.41548	−0.147802
$535$	0	0
$536$	5.23899	0.226290
$537$	−12.6911	−0.547660
$538$	−3.62865	−0.156442
$539$	−25.2337	−1.08689
$540$	0	0
$541$	−33.5195	−1.44112	−0.720558	−	0.693394i	$-0.756114\pi$
$541$	−33.5195	−1.44112	−0.720558	+	0.693394i	$0.756114\pi$
$542$	0.379518	0.0163017
$543$	18.7279	0.803691
$544$	−9.32511	−0.399811
$545$	0	0
$546$	−6.75780	−0.289207
$547$	−17.4258	−0.745072	−0.372536	−	0.928018i	$-0.621512\pi$
$547$	−17.4258	−0.745072	−0.372536	+	0.928018i	$0.621512\pi$
$548$	20.6714	0.883039
$549$	−0.225185	−0.00961065
$550$	0	0
$551$	−9.60771	−0.409302
$552$	−16.5208	−0.703171
$553$	1.46151	0.0621497
$554$	0.710638	0.0301921
$555$	0	0
$556$	−36.9373	−1.56649
$557$	−16.2929	−0.690351	−0.345175	−	0.938538i	$-0.612181\pi$
$557$	−16.2929	−0.690351	−0.345175	+	0.938538i	$0.612181\pi$
$558$	0.0305048	0.00129137
$559$	−8.19175	−0.346474
$560$	0	0
$561$	23.4353	0.989439
$562$	7.88048	0.332418
$563$	23.0999	0.973545	0.486773	−	0.873529i	$-0.338174\pi$
$563$	23.0999	0.973545	0.486773	+	0.873529i	$0.338174\pi$
$564$	22.5594	0.949924
$565$	0	0
$566$	−5.29807	−0.222694
$567$	−30.2151	−1.26891
$568$	2.97541	0.124845
$569$	−9.28334	−0.389178	−0.194589	−	0.980885i	$-0.562337\pi$
$569$	−9.28334	−0.389178	−0.194589	+	0.980885i	$0.562337\pi$
$570$	0	0
$571$	22.2143	0.929640	0.464820	−	0.885405i	$-0.346119\pi$
$571$	22.2143	0.929640	0.464820	+	0.885405i	$0.346119\pi$
$572$	−35.6057	−1.48875
$573$	3.00435	0.125509
$574$	−10.8273	−0.451924
$575$	0	0
$576$	0.319031	0.0132930
$577$	14.6093	0.608192	0.304096	−	0.952641i	$-0.401646\pi$
$577$	14.6093	0.608192	0.304096	+	0.952641i	$0.401646\pi$
$578$	3.41693	0.142126
$579$	16.3500	0.679485
$580$	0	0
$581$	−20.6879	−0.858280
$582$	−8.96905	−0.371779
$583$	−9.19986	−0.381019
$584$	1.92564	0.0796837
$585$	0	0
$586$	−1.14013	−0.0470983
$587$	23.3435	0.963489	0.481744	−	0.876312i	$-0.340003\pi$
$587$	23.3435	0.963489	0.481744	+	0.876312i	$0.340003\pi$
$588$	15.3431	0.632739
$589$	−3.29030	−0.135575
$590$	0	0
$591$	−41.0265	−1.68760
$592$	−0.0452664	−0.00186044
$593$	−41.9815	−1.72397	−0.861986	−	0.506932i	$-0.830780\pi$
$593$	−41.9815	−1.72397	−0.861986	+	0.506932i	$0.830780\pi$
$594$	9.15319	0.375560
$595$	0	0
$596$	24.2231	0.992215
$597$	−40.8743	−1.67287
$598$	8.71050	0.356199
$599$	−25.4160	−1.03847	−0.519236	−	0.854631i	$-0.673783\pi$
$599$	−25.4160	−1.03847	−0.519236	+	0.854631i	$0.673783\pi$
$600$	0	0
$601$	37.1379	1.51489	0.757444	−	0.652900i	$-0.226448\pi$
$601$	37.1379	1.51489	0.757444	+	0.652900i	$0.226448\pi$
$602$	−2.60289	−0.106086
$603$	−0.236717	−0.00963988
$604$	−4.08213	−0.166099
$605$	0	0
$606$	0.809463	0.0328822
$607$	−18.9242	−0.768109	−0.384054	−	0.923310i	$-0.625473\pi$
$607$	−18.9242	−0.768109	−0.384054	+	0.923310i	$0.625473\pi$
$608$	7.38532	0.299514
$609$	−27.8568	−1.12881
$610$	0	0
$611$	−24.4595	−0.989525
$612$	0.278351	0.0112517
$613$	9.20317	0.371713	0.185856	−	0.982577i	$-0.440494\pi$
$613$	9.20317	0.371713	0.185856	+	0.982577i	$0.440494\pi$
$614$	−2.91687	−0.117715
$615$	0	0
$616$	−23.2651	−0.937378
$617$	−32.7375	−1.31796	−0.658981	−	0.752159i	$-0.729012\pi$
$617$	−32.7375	−1.31796	−0.658981	+	0.752159i	$0.729012\pi$
$618$	8.22017	0.330664
$619$	36.3952	1.46285	0.731424	−	0.681923i	$-0.238856\pi$
$619$	36.3952	1.46285	0.731424	+	0.681923i	$0.238856\pi$
$620$	0	0
$621$	39.7069	1.59338
$622$	8.85824	0.355183
$623$	−20.8651	−0.835943
$624$	20.3599	0.815049
$625$	0	0
$626$	6.57010	0.262594
$627$	−18.5604	−0.741229
$628$	29.8135	1.18969
$629$	−0.0343496	−0.00136961
$630$	0	0
$631$	12.1083	0.482024	0.241012	−	0.970522i	$-0.422521\pi$
$631$	12.1083	0.482024	0.241012	+	0.970522i	$0.422521\pi$
$632$	0.542980	0.0215986
$633$	26.4037	1.04945
$634$	−3.16293	−0.125616
$635$	0	0
$636$	5.59388	0.221812
$637$	−16.6354	−0.659117
$638$	8.27703	0.327691
$639$	−0.134440	−0.00531837
$640$	0	0
$641$	9.75177	0.385172	0.192586	−	0.981280i	$-0.438313\pi$
$641$	9.75177	0.385172	0.192586	+	0.981280i	$0.438313\pi$
$642$	6.84762	0.270254
$643$	6.77862	0.267323	0.133661	−	0.991027i	$-0.457327\pi$
$643$	6.77862	0.267323	0.133661	+	0.991027i	$0.457327\pi$
$644$	−49.0787	−1.93397
$645$	0	0
$646$	1.69313	0.0666153
$647$	−40.1188	−1.57723	−0.788616	−	0.614886i	$-0.789202\pi$
$647$	−40.1188	−1.57723	−0.788616	+	0.614886i	$0.789202\pi$
$648$	−11.2255	−0.440979
$649$	0.118482	0.00465082
$650$	0	0
$651$	−9.53997	−0.373901
$652$	−13.9951	−0.548092
$653$	6.14210	0.240359	0.120179	−	0.992752i	$-0.461653\pi$
$653$	6.14210	0.240359	0.120179	+	0.992752i	$0.461653\pi$
$654$	−8.60458	−0.336466
$655$	0	0
$656$	32.6206	1.27362
$657$	−0.0870078	−0.00339450
$658$	−7.77190	−0.302980
$659$	−44.1645	−1.72040	−0.860202	−	0.509954i	$-0.829662\pi$
$659$	−44.1645	−1.72040	−0.860202	+	0.509954i	$0.829662\pi$
$660$	0	0
$661$	27.7447	1.07915	0.539573	−	0.841939i	$-0.318586\pi$
$661$	27.7447	1.07915	0.539573	+	0.841939i	$0.318586\pi$
$662$	−6.42911	−0.249875
$663$	15.4498	0.600019
$664$	−7.68597	−0.298274
$665$	0	0
$666$	−0.000252214 0	−9.77310e−6 0
$667$	35.9061	1.39029
$668$	20.4782	0.792325
$669$	33.5790	1.29824
$670$	0	0
$671$	−20.9248	−0.807792
$672$	21.4131	0.826030
$673$	−41.8324	−1.61252	−0.806259	−	0.591562i	$-0.798511\pi$
$673$	−41.8324	−1.61252	−0.806259	+	0.591562i	$0.798511\pi$
$674$	−6.16096	−0.237311
$675$	0	0
$676$	1.13895	0.0438059
$677$	22.6274	0.869643	0.434821	−	0.900517i	$-0.356812\pi$
$677$	22.6274	0.869643	0.434821	+	0.900517i	$0.356812\pi$
$678$	10.3998	0.399401
$679$	−54.7918	−2.10272
$680$	0	0
$681$	−2.30215	−0.0882186
$682$	2.83459	0.108542
$683$	−49.0024	−1.87502	−0.937512	−	0.347953i	$-0.886877\pi$
$683$	−49.0024	−1.87502	−0.937512	+	0.347953i	$0.886877\pi$
$684$	−0.220449	−0.00842909
$685$	0	0
$686$	2.54597	0.0972057
$687$	38.3734	1.46404
$688$	7.84200	0.298973
$689$	−6.06502	−0.231059
$690$	0	0
$691$	36.2583	1.37933	0.689665	−	0.724128i	$-0.257758\pi$
$691$	36.2583	1.37933	0.689665	+	0.724128i	$0.257758\pi$
$692$	−28.3953	−1.07943
$693$	1.05121	0.0399320
$694$	−8.48744	−0.322179
$695$	0	0
$696$	−10.3493	−0.392291
$697$	24.7536	0.937608
$698$	8.54421	0.323403
$699$	−31.5030	−1.19155
$700$	0	0
$701$	−8.32362	−0.314379	−0.157189	−	0.987568i	$-0.550243\pi$
$701$	−8.32362	−0.314379	−0.157189	+	0.987568i	$0.550243\pi$
$702$	6.03425	0.227748
$703$	0.0272043	0.00102603
$704$	29.6452	1.11730
$705$	0	0
$706$	1.74123	0.0655321
$707$	4.94500	0.185976
$708$	−0.0720417	−0.00270749
$709$	37.3097	1.40119	0.700597	−	0.713557i	$-0.252917\pi$
$709$	37.3097	1.40119	0.700597	+	0.713557i	$0.252917\pi$
$710$	0	0
$711$	−0.0245339	−0.000920092 0
$712$	−7.75180	−0.290511
$713$	12.2966	0.460511
$714$	4.90910	0.183718
$715$	0	0
$716$	−14.0069	−0.523463
$717$	20.5910	0.768983
$718$	−3.23460	−0.120714
$719$	−7.66524	−0.285865	−0.142933	−	0.989732i	$-0.545653\pi$
$719$	−7.66524	−0.285865	−0.142933	+	0.989732i	$0.545653\pi$
$720$	0	0
$721$	50.2169	1.87017
$722$	4.86734	0.181144
$723$	−17.3544	−0.645419
$724$	20.6697	0.768183
$725$	0	0
$726$	9.82419	0.364610
$727$	−45.9089	−1.70267	−0.851334	−	0.524624i	$-0.824206\pi$
$727$	−45.9089	−1.70267	−0.851334	+	0.524624i	$0.824206\pi$
$728$	−15.3376	−0.568448
$729$	27.4973	1.01842
$730$	0	0
$731$	5.95076	0.220097
$732$	12.7231	0.470259
$733$	23.7131	0.875864	0.437932	−	0.899008i	$-0.355711\pi$
$733$	23.7131	0.875864	0.437932	+	0.899008i	$0.355711\pi$
$734$	−5.37152	−0.198266
$735$	0	0
$736$	−27.6006	−1.01737
$737$	−21.9964	−0.810249
$738$	0.181755	0.00669048
$739$	−3.53683	−0.130104	−0.0650522	−	0.997882i	$-0.520721\pi$
$739$	−3.53683	−0.130104	−0.0650522	+	0.997882i	$0.520721\pi$
$740$	0	0
$741$	−12.2359	−0.449498
$742$	−1.92714	−0.0707474
$743$	−15.7201	−0.576715	−0.288358	−	0.957523i	$-0.593109\pi$
$743$	−15.7201	−0.576715	−0.288358	+	0.957523i	$0.593109\pi$
$744$	−3.54429	−0.129940
$745$	0	0
$746$	−7.51307	−0.275073
$747$	0.347281	0.0127063
$748$	25.8651	0.945724
$749$	41.8320	1.52851
$750$	0	0
$751$	−14.1856	−0.517642	−0.258821	−	0.965925i	$-0.583334\pi$
$751$	−14.1856	−0.517642	−0.258821	+	0.965925i	$0.583334\pi$
$752$	23.4152	0.853864
$753$	−28.6911	−1.04556
$754$	5.45664	0.198719
$755$	0	0
$756$	−33.9996	−1.23655
$757$	12.7388	0.463000	0.231500	−	0.972835i	$-0.425637\pi$
$757$	12.7388	0.463000	0.231500	+	0.972835i	$0.425637\pi$
$758$	−5.36675	−0.194929
$759$	69.3641	2.51776
$760$	0	0
$761$	32.2299	1.16833	0.584167	−	0.811633i	$-0.301421\pi$
$761$	32.2299	1.16833	0.584167	+	0.811633i	$0.301421\pi$
$762$	0.379965	0.0137647
$763$	−52.5652	−1.90299
$764$	3.31585	0.119963
$765$	0	0
$766$	−1.53737	−0.0555476
$767$	0.0781093	0.00282036
$768$	−13.9387	−0.502969
$769$	−11.3687	−0.409965	−0.204983	−	0.978766i	$-0.565714\pi$
$769$	−11.3687	−0.409965	−0.204983	+	0.978766i	$0.565714\pi$
$770$	0	0
$771$	−25.7597	−0.927715
$772$	18.0453	0.649464
$773$	27.1994	0.978295	0.489147	−	0.872201i	$-0.337308\pi$
$773$	27.1994	0.978295	0.489147	+	0.872201i	$0.337308\pi$
$774$	0.0436939	0.00157054
$775$	0	0
$776$	−20.3562	−0.730746
$777$	0.0788766	0.00282968
$778$	−1.15626	−0.0414538
$779$	−19.6044	−0.702400
$780$	0	0
$781$	−12.4925	−0.447018
$782$	−6.32760	−0.226275
$783$	24.8742	0.888931
$784$	15.9251	0.568754
$785$	0	0
$786$	3.95186	0.140958
$787$	−25.7266	−0.917054	−0.458527	−	0.888680i	$-0.651623\pi$
$787$	−25.7266	−0.917054	−0.458527	+	0.888680i	$0.651623\pi$
$788$	−45.2803	−1.61304
$789$	−10.5870	−0.376908
$790$	0	0
$791$	63.5321	2.25894
$792$	0.390543	0.0138774
$793$	−13.7947	−0.489864
$794$	2.86356	0.101624
$795$	0	0
$796$	−45.1122	−1.59896
$797$	17.6713	0.625951	0.312975	−	0.949761i	$-0.398674\pi$
$797$	17.6713	0.625951	0.312975	+	0.949761i	$0.398674\pi$
$798$	−3.88792	−0.137631
$799$	17.7682	0.628593
$800$	0	0
$801$	0.350256	0.0123757
$802$	−7.43938	−0.262694
$803$	−8.08500	−0.285314
$804$	13.3747	0.471689
$805$	0	0
$806$	1.86871	0.0658224
$807$	−19.0497	−0.670580
$808$	1.83716	0.0646312
$809$	−40.8576	−1.43648	−0.718238	−	0.695798i	$-0.755051\pi$
$809$	−40.8576	−1.43648	−0.718238	+	0.695798i	$0.755051\pi$
$810$	0	0
$811$	16.6214	0.583656	0.291828	−	0.956471i	$-0.405737\pi$
$811$	16.6214	0.583656	0.291828	+	0.956471i	$0.405737\pi$
$812$	−30.7451	−1.07894
$813$	1.99239	0.0698763
$814$	−0.0234364	−0.000821447 0
$815$	0	0
$816$	−14.7901	−0.517758
$817$	−4.71290	−0.164884
$818$	−9.16296	−0.320375
$819$	0.693009	0.0242157
$820$	0	0
$821$	−43.8983	−1.53206	−0.766031	−	0.642803i	$-0.777771\pi$
$821$	−43.8983	−1.53206	−0.766031	+	0.642803i	$0.777771\pi$
$822$	−6.11988	−0.213455
$823$	26.1962	0.913143	0.456571	−	0.889687i	$-0.349077\pi$
$823$	26.1962	0.913143	0.456571	+	0.889687i	$0.349077\pi$
$824$	18.6565	0.649932
$825$	0	0
$826$	0.0248189	0.000863561 0
$827$	22.3827	0.778321	0.389161	−	0.921170i	$-0.372765\pi$
$827$	22.3827	0.778321	0.389161	+	0.921170i	$0.372765\pi$
$828$	0.823867	0.0286314
$829$	0.841282	0.0292189	0.0146095	−	0.999893i	$-0.495349\pi$
$829$	0.841282	0.0292189	0.0146095	+	0.999893i	$0.495349\pi$
$830$	0	0
$831$	3.73071	0.129417
$832$	19.5437	0.677554
$833$	12.0845	0.418702
$834$	10.9355	0.378665
$835$	0	0
$836$	−20.4848	−0.708480
$837$	8.51853	0.294444
$838$	−11.1390	−0.384790
$839$	35.5528	1.22742	0.613710	−	0.789532i	$-0.289677\pi$
$839$	35.5528	1.22742	0.613710	+	0.789532i	$0.289677\pi$
$840$	0	0
$841$	−6.50681	−0.224373
$842$	10.5015	0.361904
$843$	41.3709	1.42489
$844$	29.1413	1.00308
$845$	0	0
$846$	0.130464	0.00448545
$847$	60.0158	2.06217
$848$	5.80608	0.199381
$849$	−27.8138	−0.954567
$850$	0	0
$851$	−0.101668	−0.00348515
$852$	7.59596	0.260233
$853$	−19.5406	−0.669058	−0.334529	−	0.942386i	$-0.608577\pi$
$853$	−19.5406	−0.669058	−0.334529	+	0.942386i	$0.608577\pi$
$854$	−4.38320	−0.149990
$855$	0	0
$856$	15.5414	0.531194
$857$	−0.570622	−0.0194921	−0.00974604	−	0.999953i	$-0.503102\pi$
$857$	−0.570622	−0.0194921	−0.00974604	+	0.999953i	$0.503102\pi$
$858$	10.5412	0.359872
$859$	−20.0983	−0.685744	−0.342872	−	0.939382i	$-0.611400\pi$
$859$	−20.0983	−0.685744	−0.342872	+	0.939382i	$0.611400\pi$
$860$	0	0
$861$	−56.8413	−1.93715
$862$	5.80142	0.197597
$863$	13.9645	0.475358	0.237679	−	0.971344i	$-0.423613\pi$
$863$	13.9645	0.475358	0.237679	+	0.971344i	$0.423613\pi$
$864$	−19.1205	−0.650491
$865$	0	0
$866$	−1.00236	−0.0340614
$867$	17.9382	0.609213
$868$	−10.5291	−0.357381
$869$	−2.27975	−0.0773353
$870$	0	0
$871$	−14.5012	−0.491353
$872$	−19.5290	−0.661336
$873$	0.919771	0.0311295
$874$	5.01135	0.169511
$875$	0	0
$876$	4.91601	0.166096
$877$	23.4512	0.791891	0.395946	−	0.918274i	$-0.370417\pi$
$877$	23.4512	0.791891	0.395946	+	0.918274i	$0.370417\pi$
$878$	−4.81877	−0.162626
$879$	−5.98545	−0.201884
$880$	0	0
$881$	44.4714	1.49828	0.749139	−	0.662412i	$-0.230467\pi$
$881$	44.4714	1.49828	0.749139	+	0.662412i	$0.230467\pi$
$882$	0.0887311	0.00298773
$883$	−0.758169	−0.0255144	−0.0127572	−	0.999919i	$-0.504061\pi$
$883$	−0.758169	−0.0255144	−0.0127572	+	0.999919i	$0.504061\pi$
$884$	17.0516	0.573509
$885$	0	0
$886$	3.32818	0.111812
$887$	55.4695	1.86248	0.931242	−	0.364402i	$-0.118727\pi$
$887$	55.4695	1.86248	0.931242	+	0.364402i	$0.118727\pi$
$888$	0.0293042	0.000983385 0
$889$	2.32120	0.0778506
$890$	0	0
$891$	47.1314	1.57896
$892$	37.0606	1.24088
$893$	−14.0721	−0.470905
$894$	−7.17136	−0.239846
$895$	0	0
$896$	31.1760	1.04152
$897$	45.7284	1.52683
$898$	−6.19141	−0.206610
$899$	7.70312	0.256914
$900$	0	0
$901$	4.40584	0.146780
$902$	16.8891	0.562347
$903$	−13.6647	−0.454732
$904$	23.6034	0.785038
$905$	0	0
$906$	1.20853	0.0401509
$907$	15.8193	0.525271	0.262636	−	0.964895i	$-0.415408\pi$
$907$	15.8193	0.525271	0.262636	+	0.964895i	$0.415408\pi$
$908$	−2.54084	−0.0843209
$909$	−0.0830100	−0.00275327
$910$	0	0
$911$	−8.02411	−0.265851	−0.132925	−	0.991126i	$-0.542437\pi$
$911$	−8.02411	−0.265851	−0.132925	+	0.991126i	$0.542437\pi$
$912$	11.7135	0.387873
$913$	32.2703	1.06799
$914$	9.94605	0.328986
$915$	0	0
$916$	42.3520	1.39935
$917$	24.1419	0.797235
$918$	−4.38348	−0.144676
$919$	22.8402	0.753428	0.376714	−	0.926330i	$-0.377054\pi$
$919$	22.8402	0.753428	0.376714	+	0.926330i	$0.377054\pi$
$920$	0	0
$921$	−15.3130	−0.504581
$922$	−0.774308	−0.0255005
$923$	−8.23572	−0.271082
$924$	−59.3939	−1.95392
$925$	0	0
$926$	−0.104881	−0.00344661
$927$	−0.842973	−0.0276869
$928$	−17.2902	−0.567579
$929$	−39.7278	−1.30343	−0.651713	−	0.758465i	$-0.725950\pi$
$929$	−39.7278	−1.30343	−0.651713	+	0.758465i	$0.725950\pi$
$930$	0	0
$931$	−9.57070	−0.313667
$932$	−34.7694	−1.13891
$933$	46.5040	1.52247
$934$	−4.18475	−0.136929
$935$	0	0
$936$	0.257466	0.00841555
$937$	30.4367	0.994325	0.497162	−	0.867658i	$-0.334375\pi$
$937$	30.4367	0.994325	0.497162	+	0.867658i	$0.334375\pi$
$938$	−4.60769	−0.150446
$939$	34.4917	1.12559
$940$	0	0
$941$	25.6117	0.834916	0.417458	−	0.908696i	$-0.362921\pi$
$941$	25.6117	0.834916	0.417458	+	0.908696i	$0.362921\pi$
$942$	−8.82643	−0.287581
$943$	73.2659	2.38587
$944$	−0.0747744	−0.00243370
$945$	0	0
$946$	4.06016	0.132007
$947$	−58.0741	−1.88716	−0.943578	−	0.331151i	$-0.892563\pi$
$947$	−58.0741	−1.88716	−0.943578	+	0.331151i	$0.892563\pi$
$948$	1.38618	0.0450211
$949$	−5.33005	−0.173021
$950$	0	0
$951$	−16.6047	−0.538446
$952$	11.1417	0.361105
$953$	−45.6216	−1.47783	−0.738915	−	0.673799i	$-0.764661\pi$
$953$	−45.6216	−1.47783	−0.738915	+	0.673799i	$0.764661\pi$
$954$	0.0323502	0.00104738
$955$	0	0
$956$	22.7259	0.735008
$957$	43.4527	1.40463
$958$	6.64762	0.214775
$959$	−37.3862	−1.20726
$960$	0	0
$961$	−28.3620	−0.914902
$962$	−0.0154505	−0.000498144 0
$963$	−0.702219	−0.0226287
$964$	−19.1538	−0.616903
$965$	0	0
$966$	14.5300	0.467495
$967$	−42.4449	−1.36494	−0.682468	−	0.730916i	$-0.739093\pi$
$967$	−42.4449	−1.36494	−0.682468	+	0.730916i	$0.739093\pi$
$968$	22.2971	0.716655
$969$	8.88860	0.285543
$970$	0	0
$971$	−14.4332	−0.463182	−0.231591	−	0.972813i	$-0.574393\pi$
$971$	−14.4332	−0.463182	−0.231591	+	0.972813i	$0.574393\pi$
$972$	1.13075	0.0362688
$973$	66.8047	2.14166
$974$	−5.06198	−0.162196
$975$	0	0
$976$	13.2057	0.422705
$977$	16.5748	0.530276	0.265138	−	0.964211i	$-0.414583\pi$
$977$	16.5748	0.530276	0.265138	+	0.964211i	$0.414583\pi$
$978$	4.14333	0.132489
$979$	32.5467	1.04020
$980$	0	0
$981$	0.882394	0.0281727
$982$	8.98845	0.286833
$983$	33.3174	1.06266	0.531330	−	0.847165i	$-0.321692\pi$
$983$	33.3174	1.06266	0.531330	+	0.847165i	$0.321692\pi$
$984$	−21.1177	−0.673207
$985$	0	0
$986$	−3.96389	−0.126236
$987$	−40.8009	−1.29871
$988$	−13.5046	−0.429639
$989$	17.6131	0.560065
$990$	0	0
$991$	−24.4962	−0.778146	−0.389073	−	0.921207i	$-0.627205\pi$
$991$	−24.4962	−0.778146	−0.389073	+	0.921207i	$0.627205\pi$
$992$	−5.92129	−0.188001
$993$	−33.7516	−1.07107
$994$	−2.61687	−0.0830020
$995$	0	0
$996$	−19.6216	−0.621735
$997$	−59.8575	−1.89571	−0.947854	−	0.318706i	$-0.896752\pi$
$997$	−59.8575	−1.89571	−0.947854	+	0.318706i	$0.896752\pi$
$998$	−1.60449	−0.0507893
$999$	−0.0704313	−0.00222835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.g.1.3	yes	8
3.2	odd	2		5625.2.a.s.1.6		8
4.3	odd	2		10000.2.a.be.1.7		8
5.2	odd	4		625.2.b.d.624.8		16
5.3	odd	4		625.2.b.d.624.9		16
5.4	even	2		625.2.a.e.1.6	✓	8
15.14	odd	2		5625.2.a.be.1.3		8
20.19	odd	2		10000.2.a.bn.1.2		8
25.2	odd	20		625.2.e.j.124.4		32
25.3	odd	20		625.2.e.k.249.5		32
25.4	even	10		625.2.d.p.376.2		16
25.6	even	5		625.2.d.n.251.3		16
25.8	odd	20		625.2.e.k.374.4		32
25.9	even	10		625.2.d.q.126.3		16
25.11	even	5		625.2.d.m.501.2		16
25.12	odd	20		625.2.e.j.499.5		32
25.13	odd	20		625.2.e.j.499.4		32
25.14	even	10		625.2.d.q.501.3		16
25.16	even	5		625.2.d.m.126.2		16
25.17	odd	20		625.2.e.k.374.5		32
25.19	even	10		625.2.d.p.251.2		16
25.21	even	5		625.2.d.n.376.3		16
25.22	odd	20		625.2.e.k.249.4		32
25.23	odd	20		625.2.e.j.124.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.6	✓	8	5.4	even	2
625.2.a.g.1.3	yes	8	1.1	even	1	trivial
625.2.b.d.624.8		16	5.2	odd	4
625.2.b.d.624.9		16	5.3	odd	4
625.2.d.m.126.2		16	25.16	even	5
625.2.d.m.501.2		16	25.11	even	5
625.2.d.n.251.3		16	25.6	even	5
625.2.d.n.376.3		16	25.21	even	5
625.2.d.p.251.2		16	25.19	even	10
625.2.d.p.376.2		16	25.4	even	10
625.2.d.q.126.3		16	25.9	even	10
625.2.d.q.501.3		16	25.14	even	10
625.2.e.j.124.4		32	25.2	odd	20
625.2.e.j.124.5		32	25.23	odd	20
625.2.e.j.499.4		32	25.13	odd	20
625.2.e.j.499.5		32	25.12	odd	20
625.2.e.k.249.4		32	25.22	odd	20
625.2.e.k.249.5		32	25.3	odd	20
625.2.e.k.374.4		32	25.8	odd	20
625.2.e.k.374.5		32	25.17	odd	20
5625.2.a.s.1.6		8	3.2	odd	2
5625.2.a.be.1.3		8	15.14	odd	2
10000.2.a.be.1.7		8	4.3	odd	2
10000.2.a.bn.1.2		8	20.19	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)

\(f(q)\)	\(=\)	\(q-0.326751 q^{2} -1.71538 q^{3} -1.89323 q^{4} +0.560502 q^{6} +3.42409 q^{7} +1.27212 q^{8} -0.0574791 q^{9} +O(q^{10})\)	\(q-0.326751 q^{2} -1.71538 q^{3} -1.89323 q^{4} +0.560502 q^{6} +3.42409 q^{7} +1.27212 q^{8} -0.0574791 q^{9} -5.34111 q^{11} +3.24761 q^{12} -3.52114 q^{13} -1.11883 q^{14} +3.37080 q^{16} +2.55787 q^{17} +0.0187814 q^{18} -2.02579 q^{19} -5.87362 q^{21} +1.74522 q^{22} +7.57082 q^{23} -2.18216 q^{24} +1.15054 q^{26} +5.24473 q^{27} -6.48261 q^{28} +4.74270 q^{29} +1.62421 q^{31} -3.64565 q^{32} +9.16203 q^{33} -0.835788 q^{34} +0.108821 q^{36} -0.0134290 q^{37} +0.661929 q^{38} +6.04008 q^{39} +9.67740 q^{41} +1.91921 q^{42} +2.32645 q^{43} +10.1120 q^{44} -2.47377 q^{46} +6.94647 q^{47} -5.78220 q^{48} +4.72443 q^{49} -4.38772 q^{51} +6.66634 q^{52} +1.72246 q^{53} -1.71372 q^{54} +4.35585 q^{56} +3.47500 q^{57} -1.54968 q^{58} -0.0221830 q^{59} +3.91768 q^{61} -0.530712 q^{62} -0.196814 q^{63} -5.55038 q^{64} -2.99370 q^{66} +4.11832 q^{67} -4.84265 q^{68} -12.9868 q^{69} +2.33894 q^{71} -0.0731202 q^{72} +1.51373 q^{73} +0.00438793 q^{74} +3.83530 q^{76} -18.2885 q^{77} -1.97360 q^{78} +0.426831 q^{79} -8.82426 q^{81} -3.16210 q^{82} -6.04187 q^{83} +11.1201 q^{84} -0.760170 q^{86} -8.13552 q^{87} -6.79453 q^{88} -6.09362 q^{89} -12.0567 q^{91} -14.3333 q^{92} -2.78613 q^{93} -2.26977 q^{94} +6.25367 q^{96} -16.0018 q^{97} -1.54371 q^{98} +0.307002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 5 q^{3} + 11 q^{4} - 4 q^{6} + 10 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10}) \) 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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