LMFDB - Embedded newform 625.2.a.e.1.2

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			1.2
Root			$-1.47435$ 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)

$f(q)$	$=$	$q-2.47435 q^{2} -2.11675 q^{3} +4.12242 q^{4} +5.23759 q^{6} +0.973070 q^{7} -5.25163 q^{8} +1.48063 q^{9} +O(q^{10})$	\(q-2.47435 q^{2} -2.11675 q^{3} +4.12242 q^{4} +5.23759 q^{6} +0.973070 q^{7} -5.25163 q^{8} +1.48063 q^{9} -5.38225 q^{11} -8.72615 q^{12} +1.99670 q^{13} -2.40772 q^{14} +4.74954 q^{16} +2.04301 q^{17} -3.66361 q^{18} +6.20428 q^{19} -2.05975 q^{21} +13.3176 q^{22} -1.93813 q^{23} +11.1164 q^{24} -4.94054 q^{26} +3.21612 q^{27} +4.01141 q^{28} +4.81812 q^{29} -6.64014 q^{31} -1.24877 q^{32} +11.3929 q^{33} -5.05512 q^{34} +6.10380 q^{36} +0.978913 q^{37} -15.3516 q^{38} -4.22652 q^{39} +2.73319 q^{41} +5.09654 q^{42} -3.99413 q^{43} -22.1879 q^{44} +4.79562 q^{46} -7.21339 q^{47} -10.0536 q^{48} -6.05313 q^{49} -4.32454 q^{51} +8.23125 q^{52} -13.2746 q^{53} -7.95782 q^{54} -5.11020 q^{56} -13.1329 q^{57} -11.9217 q^{58} +6.54024 q^{59} +2.72672 q^{61} +16.4301 q^{62} +1.44076 q^{63} -6.40916 q^{64} -28.1900 q^{66} -9.56957 q^{67} +8.42215 q^{68} +4.10254 q^{69} -5.68520 q^{71} -7.77574 q^{72} +9.35281 q^{73} -2.42218 q^{74} +25.5767 q^{76} -5.23731 q^{77} +10.4579 q^{78} -3.18171 q^{79} -11.2496 q^{81} -6.76289 q^{82} -6.11387 q^{83} -8.49115 q^{84} +9.88290 q^{86} -10.1988 q^{87} +28.2656 q^{88} -3.00743 q^{89} +1.94293 q^{91} -7.98980 q^{92} +14.0555 q^{93} +17.8485 q^{94} +2.64334 q^{96} -5.95526 q^{97} +14.9776 q^{98} -7.96914 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.47435	−1.74963	−0.874816	−	0.484455i	$-0.839018\pi$
$2$	−2.47435	−1.74963	−0.874816	+	0.484455i	$0.839018\pi$
$3$	−2.11675	−1.22211	−0.611053	−	0.791589i	$-0.709254\pi$
$3$	−2.11675	−1.22211	−0.611053	+	0.791589i	$0.709254\pi$
$4$	4.12242	2.06121
$5$	0	0
$5$	0	0
$6$	5.23759	2.13824
$7$	0.973070	0.367786	0.183893	−	0.982946i	$-0.441130\pi$
$7$	0.973070	0.367786	0.183893	+	0.982946i	$0.441130\pi$
$8$	−5.25163	−1.85673
$9$	1.48063	0.493544
$10$	0	0
$11$	−5.38225	−1.62281	−0.811405	−	0.584484i	$-0.801297\pi$
$11$	−5.38225	−1.62281	−0.811405	+	0.584484i	$0.801297\pi$
$12$	−8.72615	−2.51902
$13$	1.99670	0.553785	0.276892	−	0.960901i	$-0.410695\pi$
$13$	1.99670	0.553785	0.276892	+	0.960901i	$0.410695\pi$
$14$	−2.40772	−0.643490
$15$	0	0
$16$	4.74954	1.18738
$17$	2.04301	0.495502	0.247751	−	0.968824i	$-0.420308\pi$
$17$	2.04301	0.495502	0.247751	+	0.968824i	$0.420308\pi$
$18$	−3.66361	−0.863521
$19$	6.20428	1.42336	0.711680	−	0.702504i	$-0.247935\pi$
$19$	6.20428	1.42336	0.711680	+	0.702504i	$0.247935\pi$
$20$	0	0
$21$	−2.05975	−0.449474
$22$	13.3176	2.83932
$23$	−1.93813	−0.404128	−0.202064	−	0.979372i	$-0.564765\pi$
$23$	−1.93813	−0.404128	−0.202064	+	0.979372i	$0.564765\pi$
$24$	11.1164	2.26912
$25$	0	0
$26$	−4.94054	−0.968920
$27$	3.21612	0.618943
$28$	4.01141	0.758085
$29$	4.81812	0.894702	0.447351	−	0.894359i	$-0.352368\pi$
$29$	4.81812	0.894702	0.447351	+	0.894359i	$0.352368\pi$
$30$	0	0
$31$	−6.64014	−1.19260	−0.596302	−	0.802760i	$-0.703364\pi$
$31$	−6.64014	−1.19260	−0.596302	+	0.802760i	$0.703364\pi$
$32$	−1.24877	−0.220754
$33$	11.3929	1.98325
$34$	−5.05512	−0.866947
$35$	0	0
$36$	6.10380	1.01730
$37$	0.978913	0.160932	0.0804662	−	0.996757i	$-0.474359\pi$
$37$	0.978913	0.160932	0.0804662	+	0.996757i	$0.474359\pi$
$38$	−15.3516	−2.49035
$39$	−4.22652	−0.676784
$40$	0	0
$41$	2.73319	0.426853	0.213427	−	0.976959i	$-0.431538\pi$
$41$	2.73319	0.426853	0.213427	+	0.976959i	$0.431538\pi$
$42$	5.09654	0.786413
$43$	−3.99413	−0.609100	−0.304550	−	0.952496i	$-0.598506\pi$
$43$	−3.99413	−0.609100	−0.304550	+	0.952496i	$0.598506\pi$
$44$	−22.1879	−3.34496
$45$	0	0
$46$	4.79562	0.707076
$47$	−7.21339	−1.05218	−0.526091	−	0.850428i	$-0.676343\pi$
$47$	−7.21339	−1.05218	−0.526091	+	0.850428i	$0.676343\pi$
$48$	−10.0536	−1.45111
$49$	−6.05313	−0.864734
$50$	0	0
$51$	−4.32454	−0.605557
$52$	8.23125	1.14147
$53$	−13.2746	−1.82340	−0.911700	−	0.410857i	$-0.865230\pi$
$53$	−13.2746	−1.82340	−0.911700	+	0.410857i	$0.865230\pi$
$54$	−7.95782	−1.08292
$55$	0	0
$56$	−5.11020	−0.682880
$57$	−13.1329	−1.73950
$58$	−11.9217	−1.56540
$59$	6.54024	0.851467	0.425733	−	0.904849i	$-0.360016\pi$
$59$	6.54024	0.851467	0.425733	+	0.904849i	$0.360016\pi$
$60$	0	0
$61$	2.72672	0.349121	0.174560	−	0.984646i	$-0.444150\pi$
$61$	2.72672	0.349121	0.174560	+	0.984646i	$0.444150\pi$
$62$	16.4301	2.08662
$63$	1.44076	0.181519
$64$	−6.40916	−0.801146
$65$	0	0
$66$	−28.1900	−3.46995
$67$	−9.56957	−1.16911	−0.584555	−	0.811354i	$-0.698731\pi$
$67$	−9.56957	−1.16911	−0.584555	+	0.811354i	$0.698731\pi$
$68$	8.42215	1.02134
$69$	4.10254	0.493888
$70$	0	0
$71$	−5.68520	−0.674710	−0.337355	−	0.941378i	$-0.609532\pi$
$71$	−5.68520	−0.674710	−0.337355	+	0.941378i	$0.609532\pi$
$72$	−7.77574	−0.916379
$73$	9.35281	1.09466	0.547332	−	0.836916i	$-0.315644\pi$
$73$	9.35281	1.09466	0.547332	+	0.836916i	$0.315644\pi$
$74$	−2.42218	−0.281572
$75$	0	0
$76$	25.5767	2.93385
$77$	−5.23731	−0.596847
$78$	10.4579	1.18412
$79$	−3.18171	−0.357970	−0.178985	−	0.983852i	$-0.557281\pi$
$79$	−3.18171	−0.357970	−0.178985	+	0.983852i	$0.557281\pi$
$80$	0	0
$81$	−11.2496	−1.24996
$82$	−6.76289	−0.746836
$83$	−6.11387	−0.671085	−0.335542	−	0.942025i	$-0.608920\pi$
$83$	−6.11387	−0.671085	−0.335542	+	0.942025i	$0.608920\pi$
$84$	−8.49115	−0.926460
$85$	0	0
$86$	9.88290	1.06570
$87$	−10.1988	−1.09342
$88$	28.2656	3.01312
$89$	−3.00743	−0.318787	−0.159393	−	0.987215i	$-0.550954\pi$
$89$	−3.00743	−0.318787	−0.159393	+	0.987215i	$0.550954\pi$
$90$	0	0
$91$	1.94293	0.203674
$92$	−7.98980	−0.832994
$93$	14.0555	1.45749
$94$	17.8485	1.84093
$95$	0	0
$96$	2.64334	0.269785
$97$	−5.95526	−0.604666	−0.302333	−	0.953202i	$-0.597765\pi$
$97$	−5.95526	−0.604666	−0.302333	+	0.953202i	$0.597765\pi$
$98$	14.9776	1.51297
$99$	−7.96914	−0.800929
$100$	0	0
$101$	−7.77373	−0.773515	−0.386758	−	0.922181i	$-0.626405\pi$
$101$	−7.77373	−0.773515	−0.386758	+	0.922181i	$0.626405\pi$
$102$	10.7004	1.05950
$103$	−4.95056	−0.487793	−0.243897	−	0.969801i	$-0.578426\pi$
$103$	−4.95056	−0.487793	−0.243897	+	0.969801i	$0.578426\pi$
$104$	−10.4859	−1.02823
$105$	0	0
$106$	32.8459	3.19028
$107$	−1.16798	−0.112913	−0.0564567	−	0.998405i	$-0.517980\pi$
$107$	−1.16798	−0.112913	−0.0564567	+	0.998405i	$0.517980\pi$
$108$	13.2582	1.27577
$109$	−17.6879	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$109$	−17.6879	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$110$	0	0
$111$	−2.07212	−0.196676
$112$	4.62163	0.436703
$113$	−5.03643	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$113$	−5.03643	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$114$	32.4955	3.04348
$115$	0	0
$116$	19.8623	1.84417
$117$	2.95638	0.273317
$118$	−16.1829	−1.48975
$119$	1.98799	0.182239
$120$	0	0
$121$	17.9687	1.63351
$122$	−6.74687	−0.610833
$123$	−5.78549	−0.521660
$124$	−27.3735	−2.45821
$125$	0	0
$126$	−3.56495	−0.317591
$127$	−7.79457	−0.691656	−0.345828	−	0.938298i	$-0.612402\pi$
$127$	−7.79457	−0.691656	−0.345828	+	0.938298i	$0.612402\pi$
$128$	18.3561	1.62246
$129$	8.45458	0.744385
$130$	0	0
$131$	−9.79015	−0.855369	−0.427685	−	0.903928i	$-0.640671\pi$
$131$	−9.79015	−0.855369	−0.427685	+	0.903928i	$0.640671\pi$
$132$	46.9663	4.08789
$133$	6.03720	0.523491
$134$	23.6785	2.04551
$135$	0	0
$136$	−10.7291	−0.920015
$137$	3.63326	0.310411	0.155205	−	0.987882i	$-0.450396\pi$
$137$	3.63326	0.310411	0.155205	+	0.987882i	$0.450396\pi$
$138$	−10.1511	−0.864122
$139$	−1.86079	−0.157830	−0.0789149	−	0.996881i	$-0.525146\pi$
$139$	−1.86079	−0.157830	−0.0789149	+	0.996881i	$0.525146\pi$
$140$	0	0
$141$	15.2690	1.28588
$142$	14.0672	1.18049
$143$	−10.7467	−0.898688
$144$	7.03232	0.586027
$145$	0	0
$146$	−23.1421	−1.91526
$147$	12.8130	1.05680
$148$	4.03550	0.331716
$149$	16.8530	1.38066	0.690328	−	0.723497i	$-0.257466\pi$
$149$	16.8530	1.38066	0.690328	+	0.723497i	$0.257466\pi$
$150$	0	0
$151$	−14.3201	−1.16536	−0.582678	−	0.812703i	$-0.697995\pi$
$151$	−14.3201	−1.16536	−0.582678	+	0.812703i	$0.697995\pi$
$152$	−32.5826	−2.64280
$153$	3.02495	0.244552
$154$	12.9590	1.04426
$155$	0	0
$156$	−17.4235	−1.39500
$157$	21.6869	1.73080	0.865400	−	0.501081i	$-0.167064\pi$
$157$	21.6869	1.73080	0.865400	+	0.501081i	$0.167064\pi$
$158$	7.87266	0.626315
$159$	28.0989	2.22839
$160$	0	0
$161$	−1.88594	−0.148633
$162$	27.8355	2.18697
$163$	19.7210	1.54467	0.772334	−	0.635217i	$-0.219089\pi$
$163$	19.7210	1.54467	0.772334	+	0.635217i	$0.219089\pi$
$164$	11.2674	0.879835
$165$	0	0
$166$	15.1279	1.17415
$167$	−6.07947	−0.470444	−0.235222	−	0.971942i	$-0.575582\pi$
$167$	−6.07947	−0.470444	−0.235222	+	0.971942i	$0.575582\pi$
$168$	10.8170	0.834552
$169$	−9.01319	−0.693322
$170$	0	0
$171$	9.18626	0.702491
$172$	−16.4655	−1.25548
$173$	−15.0309	−1.14278	−0.571388	−	0.820680i	$-0.693595\pi$
$173$	−15.0309	−1.14278	−0.571388	+	0.820680i	$0.693595\pi$
$174$	25.2353	1.91308
$175$	0	0
$176$	−25.5632	−1.92690
$177$	−13.8441	−1.04058
$178$	7.44144	0.557759
$179$	−20.0167	−1.49612	−0.748060	−	0.663631i	$-0.769014\pi$
$179$	−20.0167	−1.49612	−0.748060	+	0.663631i	$0.769014\pi$
$180$	0	0
$181$	20.2993	1.50883	0.754417	−	0.656395i	$-0.227920\pi$
$181$	20.2993	1.50883	0.754417	+	0.656395i	$0.227920\pi$
$182$	−4.80749	−0.356355
$183$	−5.77178	−0.426663
$184$	10.1783	0.750357
$185$	0	0
$186$	−34.7783	−2.55007
$187$	−10.9960	−0.804106
$188$	−29.7367	−2.16877
$189$	3.12951	0.227638
$190$	0	0
$191$	−9.47681	−0.685718	−0.342859	−	0.939387i	$-0.611395\pi$
$191$	−9.47681	−0.685718	−0.342859	+	0.939387i	$0.611395\pi$
$192$	13.5666	0.979085
$193$	−2.72478	−0.196134	−0.0980671	−	0.995180i	$-0.531266\pi$
$193$	−2.72478	−0.196134	−0.0980671	+	0.995180i	$0.531266\pi$
$194$	14.7354	1.05794
$195$	0	0
$196$	−24.9536	−1.78240
$197$	−0.780036	−0.0555753	−0.0277876	−	0.999614i	$-0.508846\pi$
$197$	−0.780036	−0.0555753	−0.0277876	+	0.999614i	$0.508846\pi$
$198$	19.7185	1.40133
$199$	−7.42526	−0.526363	−0.263181	−	0.964746i	$-0.584772\pi$
$199$	−7.42526	−0.526363	−0.263181	+	0.964746i	$0.584772\pi$
$200$	0	0
$201$	20.2564	1.42878
$202$	19.2350	1.35337
$203$	4.68836	0.329059
$204$	−17.8276	−1.24818
$205$	0	0
$206$	12.2494	0.853459
$207$	−2.86966	−0.199455
$208$	9.48340	0.657555
$209$	−33.3930	−2.30984
$210$	0	0
$211$	−2.98128	−0.205240	−0.102620	−	0.994721i	$-0.532723\pi$
$211$	−2.98128	−0.205240	−0.102620	+	0.994721i	$0.532723\pi$
$212$	−54.7233	−3.75841
$213$	12.0342	0.824567
$214$	2.89001	0.197557
$215$	0	0
$216$	−16.8899	−1.14921
$217$	−6.46132	−0.438623
$218$	43.7662	2.96422
$219$	−19.7976	−1.33779
$220$	0	0
$221$	4.07927	0.274402
$222$	5.12715	0.344111
$223$	−24.8720	−1.66556	−0.832778	−	0.553607i	$-0.813251\pi$
$223$	−24.8720	−1.66556	−0.832778	+	0.553607i	$0.813251\pi$
$224$	−1.21514	−0.0811903
$225$	0	0
$226$	12.4619	0.828953
$227$	17.5594	1.16546	0.582729	−	0.812667i	$-0.301985\pi$
$227$	17.5594	1.16546	0.582729	+	0.812667i	$0.301985\pi$
$228$	−54.1394	−3.58547
$229$	7.01849	0.463795	0.231897	−	0.972740i	$-0.425507\pi$
$229$	7.01849	0.463795	0.231897	+	0.972740i	$0.425507\pi$
$230$	0	0
$231$	11.0861	0.729410
$232$	−25.3030	−1.66122
$233$	8.69126	0.569383	0.284692	−	0.958619i	$-0.408109\pi$
$233$	8.69126	0.569383	0.284692	+	0.958619i	$0.408109\pi$
$234$	−7.31513	−0.478205
$235$	0	0
$236$	26.9617	1.75505
$237$	6.73488	0.437477
$238$	−4.91899	−0.318851
$239$	21.3430	1.38057	0.690283	−	0.723540i	$-0.257486\pi$
$239$	21.3430	1.38057	0.690283	+	0.723540i	$0.257486\pi$
$240$	0	0
$241$	−21.3897	−1.37783	−0.688916	−	0.724841i	$-0.741913\pi$
$241$	−21.3897	−1.37783	−0.688916	+	0.724841i	$0.741913\pi$
$242$	−44.4608	−2.85805
$243$	14.1643	0.908639
$244$	11.2407	0.719612
$245$	0	0
$246$	14.3153	0.912713
$247$	12.3881	0.788235
$248$	34.8716	2.21435
$249$	12.9415	0.820137
$250$	0	0
$251$	14.9016	0.940580	0.470290	−	0.882512i	$-0.344149\pi$
$251$	14.9016	0.940580	0.470290	+	0.882512i	$0.344149\pi$
$252$	5.93942	0.374148
$253$	10.4315	0.655823
$254$	19.2865	1.21014
$255$	0	0
$256$	−32.6011	−2.03757
$257$	−19.0689	−1.18948	−0.594741	−	0.803917i	$-0.702746\pi$
$257$	−19.0689	−1.18948	−0.594741	+	0.803917i	$0.702746\pi$
$258$	−20.9196	−1.30240
$259$	0.952551	0.0591887
$260$	0	0
$261$	7.13386	0.441575
$262$	24.2243	1.49658
$263$	6.78341	0.418283	0.209142	−	0.977885i	$-0.432933\pi$
$263$	6.78341	0.418283	0.209142	+	0.977885i	$0.432933\pi$
$264$	−59.8312	−3.68236
$265$	0	0
$266$	−14.9382	−0.915917
$267$	6.36597	0.389591
$268$	−39.4498	−2.40978
$269$	0.494663	0.0301601	0.0150801	−	0.999886i	$-0.495200\pi$
$269$	0.494663	0.0301601	0.0150801	+	0.999886i	$0.495200\pi$
$270$	0	0
$271$	9.82610	0.596893	0.298446	−	0.954426i	$-0.403532\pi$
$271$	9.82610	0.596893	0.298446	+	0.954426i	$0.403532\pi$
$272$	9.70334	0.588352
$273$	−4.11270	−0.248912
$274$	−8.98998	−0.543104
$275$	0	0
$276$	16.9124	1.01801
$277$	−25.1399	−1.51051	−0.755255	−	0.655431i	$-0.772487\pi$
$277$	−25.1399	−1.51051	−0.755255	+	0.655431i	$0.772487\pi$
$278$	4.60424	0.276144
$279$	−9.83161	−0.588603
$280$	0	0
$281$	−7.77050	−0.463549	−0.231775	−	0.972770i	$-0.574453\pi$
$281$	−7.77050	−0.463549	−0.231775	+	0.972770i	$0.574453\pi$
$282$	−37.7808	−2.24981
$283$	−28.9617	−1.72160	−0.860798	−	0.508946i	$-0.830035\pi$
$283$	−28.9617	−1.72160	−0.860798	+	0.508946i	$0.830035\pi$
$284$	−23.4368	−1.39072
$285$	0	0
$286$	26.5912	1.57237
$287$	2.65959	0.156991
$288$	−1.84898	−0.108952
$289$	−12.8261	−0.754477
$290$	0	0
$291$	12.6058	0.738966
$292$	38.5562	2.25633
$293$	0.154329	0.00901602	0.00450801	−	0.999990i	$-0.498565\pi$
$293$	0.154329	0.00901602	0.00450801	+	0.999990i	$0.498565\pi$
$294$	−31.7038	−1.84901
$295$	0	0
$296$	−5.14089	−0.298808
$297$	−17.3100	−1.00443
$298$	−41.7004	−2.41564
$299$	−3.86987	−0.223800
$300$	0	0
$301$	−3.88657	−0.224018
$302$	35.4330	2.03894
$303$	16.4551	0.945318
$304$	29.4675	1.69007
$305$	0	0
$306$	−7.48478	−0.427877
$307$	14.4875	0.826845	0.413423	−	0.910539i	$-0.364333\pi$
$307$	14.4875	0.826845	0.413423	+	0.910539i	$0.364333\pi$
$308$	−21.5904	−1.23023
$309$	10.4791	0.596135
$310$	0	0
$311$	18.5385	1.05122	0.525610	−	0.850726i	$-0.323837\pi$
$311$	18.5385	1.05122	0.525610	+	0.850726i	$0.323837\pi$
$312$	22.1961	1.25661
$313$	−6.85703	−0.387582	−0.193791	−	0.981043i	$-0.562078\pi$
$313$	−6.85703	−0.387582	−0.193791	+	0.981043i	$0.562078\pi$
$314$	−53.6610	−3.02826
$315$	0	0
$316$	−13.1163	−0.737852
$317$	27.2810	1.53225	0.766127	−	0.642689i	$-0.222181\pi$
$317$	27.2810	1.53225	0.766127	+	0.642689i	$0.222181\pi$
$318$	−69.5266	−3.89886
$319$	−25.9323	−1.45193
$320$	0	0
$321$	2.47233	0.137992
$322$	4.66647	0.260052
$323$	12.6754	0.705278
$324$	−46.3757	−2.57643
$325$	0	0
$326$	−48.7967	−2.70260
$327$	37.4410	2.07049
$328$	−14.3537	−0.792551
$329$	−7.01914	−0.386978
$330$	0	0
$331$	−13.0429	−0.716901	−0.358450	−	0.933549i	$-0.616695\pi$
$331$	−13.0429	−0.716901	−0.358450	+	0.933549i	$0.616695\pi$
$332$	−25.2040	−1.38325
$333$	1.44941	0.0794273
$334$	15.0428	0.823103
$335$	0	0
$336$	−9.78284	−0.533698
$337$	19.6507	1.07044	0.535222	−	0.844712i	$-0.320228\pi$
$337$	19.6507	1.07044	0.535222	+	0.844712i	$0.320228\pi$
$338$	22.3018	1.21306
$339$	10.6609	0.579018
$340$	0	0
$341$	35.7389	1.93537
$342$	−22.7301	−1.22910
$343$	−12.7016	−0.685823
$344$	20.9757	1.13093
$345$	0	0
$346$	37.1917	1.99944
$347$	−4.38943	−0.235637	−0.117818	−	0.993035i	$-0.537590\pi$
$347$	−4.38943	−0.235637	−0.117818	+	0.993035i	$0.537590\pi$
$348$	−42.0436	−2.25377
$349$	27.1955	1.45574	0.727872	−	0.685713i	$-0.240510\pi$
$349$	27.1955	1.45574	0.727872	+	0.685713i	$0.240510\pi$
$350$	0	0
$351$	6.42163	0.342761
$352$	6.72122	0.358242
$353$	15.9317	0.847960	0.423980	−	0.905672i	$-0.360633\pi$
$353$	15.9317	0.847960	0.423980	+	0.905672i	$0.360633\pi$
$354$	34.2551	1.82064
$355$	0	0
$356$	−12.3979	−0.657087
$357$	−4.20808	−0.222715
$358$	49.5284	2.61766
$359$	−5.01508	−0.264686	−0.132343	−	0.991204i	$-0.542250\pi$
$359$	−5.01508	−0.264686	−0.132343	+	0.991204i	$0.542250\pi$
$360$	0	0
$361$	19.4931	1.02595
$362$	−50.2276	−2.63991
$363$	−38.0352	−1.99633
$364$	8.00958	0.419816
$365$	0	0
$366$	14.2814	0.746502
$367$	−28.3669	−1.48074	−0.740370	−	0.672200i	$-0.765349\pi$
$367$	−28.3669	−1.48074	−0.740370	+	0.672200i	$0.765349\pi$
$368$	−9.20522	−0.479855
$369$	4.04686	0.210671
$370$	0	0
$371$	−12.9171	−0.670621
$372$	57.9428	3.00420
$373$	32.7218	1.69427	0.847135	−	0.531377i	$-0.178325\pi$
$373$	32.7218	1.69427	0.847135	+	0.531377i	$0.178325\pi$
$374$	27.2080	1.40689
$375$	0	0
$376$	37.8821	1.95362
$377$	9.62033	0.495472
$378$	−7.74352	−0.398283
$379$	4.48483	0.230370	0.115185	−	0.993344i	$-0.463254\pi$
$379$	4.48483	0.230370	0.115185	+	0.993344i	$0.463254\pi$
$380$	0	0
$381$	16.4992	0.845278
$382$	23.4490	1.19975
$383$	−16.2839	−0.832069	−0.416034	−	0.909349i	$-0.636580\pi$
$383$	−16.2839	−0.832069	−0.416034	+	0.909349i	$0.636580\pi$
$384$	−38.8553	−1.98282
$385$	0	0
$386$	6.74208	0.343163
$387$	−5.91385	−0.300618
$388$	−24.5501	−1.24634
$389$	21.4899	1.08958	0.544791	−	0.838572i	$-0.316609\pi$
$389$	21.4899	1.08958	0.544791	+	0.838572i	$0.316609\pi$
$390$	0	0
$391$	−3.95962	−0.200246
$392$	31.7888	1.60558
$393$	20.7233	1.04535
$394$	1.93008	0.0972363
$395$	0	0
$396$	−32.8522	−1.65088
$397$	20.8870	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$397$	20.8870	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$398$	18.3727	0.920941
$399$	−12.7792	−0.639762
$400$	0	0
$401$	−0.633423	−0.0316316	−0.0158158	−	0.999875i	$-0.505035\pi$
$401$	−0.633423	−0.0316316	−0.0158158	+	0.999875i	$0.505035\pi$
$402$	−50.1215	−2.49983
$403$	−13.2584	−0.660447
$404$	−32.0466	−1.59438
$405$	0	0
$406$	−11.6007	−0.575732
$407$	−5.26876	−0.261163
$408$	22.7109	1.12436
$409$	−4.29822	−0.212534	−0.106267	−	0.994338i	$-0.533890\pi$
$409$	−4.29822	−0.212534	−0.106267	+	0.994338i	$0.533890\pi$
$410$	0	0
$411$	−7.69071	−0.379355
$412$	−20.4083	−1.00545
$413$	6.36411	0.313158
$414$	7.10055	0.348973
$415$	0	0
$416$	−2.49343	−0.122250
$417$	3.93882	0.192885
$418$	82.6261	4.04137
$419$	33.6215	1.64252	0.821258	−	0.570556i	$-0.193272\pi$
$419$	33.6215	1.64252	0.821258	+	0.570556i	$0.193272\pi$
$420$	0	0
$421$	−21.3768	−1.04184	−0.520920	−	0.853605i	$-0.674411\pi$
$421$	−21.3768	−1.04184	−0.520920	+	0.853605i	$0.674411\pi$
$422$	7.37674	0.359094
$423$	−10.6804	−0.519298
$424$	69.7130	3.38556
$425$	0	0
$426$	−29.7768	−1.44269
$427$	2.65329	0.128402
$428$	−4.81493	−0.232738
$429$	22.7482	1.09829
$430$	0	0
$431$	26.0905	1.25673	0.628367	−	0.777917i	$-0.283723\pi$
$431$	26.0905	1.25673	0.628367	+	0.777917i	$0.283723\pi$
$432$	15.2751	0.734923
$433$	−11.7676	−0.565513	−0.282757	−	0.959192i	$-0.591249\pi$
$433$	−11.7676	−0.565513	−0.282757	+	0.959192i	$0.591249\pi$
$434$	15.9876	0.767429
$435$	0	0
$436$	−72.9172	−3.49210
$437$	−12.0247	−0.575220
$438$	48.9862	2.34065
$439$	4.92572	0.235092	0.117546	−	0.993067i	$-0.462497\pi$
$439$	4.92572	0.235092	0.117546	+	0.993067i	$0.462497\pi$
$440$	0	0
$441$	−8.96247	−0.426784
$442$	−10.0936	−0.480102
$443$	−9.35961	−0.444689	−0.222344	−	0.974968i	$-0.571371\pi$
$443$	−9.35961	−0.444689	−0.222344	+	0.974968i	$0.571371\pi$
$444$	−8.54214	−0.405392
$445$	0	0
$446$	61.5422	2.91411
$447$	−35.6737	−1.68731
$448$	−6.23657	−0.294650
$449$	7.41602	0.349983	0.174992	−	0.984570i	$-0.444010\pi$
$449$	7.41602	0.349983	0.174992	+	0.984570i	$0.444010\pi$
$450$	0	0
$451$	−14.7107	−0.692702
$452$	−20.7623	−0.976576
$453$	30.3121	1.42419
$454$	−43.4481	−2.03912
$455$	0	0
$456$	68.9692	3.22978
$457$	−8.79714	−0.411513	−0.205756	−	0.978603i	$-0.565965\pi$
$457$	−8.79714	−0.411513	−0.205756	+	0.978603i	$0.565965\pi$
$458$	−17.3662	−0.811471
$459$	6.57056	0.306688
$460$	0	0
$461$	−42.1526	−1.96324	−0.981621	−	0.190838i	$-0.938879\pi$
$461$	−42.1526	−1.96324	−0.981621	+	0.190838i	$0.938879\pi$
$462$	−27.4309	−1.27620
$463$	−23.0565	−1.07153	−0.535763	−	0.844368i	$-0.679976\pi$
$463$	−23.0565	−1.07153	−0.535763	+	0.844368i	$0.679976\pi$
$464$	22.8838	1.06235
$465$	0	0
$466$	−21.5053	−0.996212
$467$	−4.37964	−0.202666	−0.101333	−	0.994853i	$-0.532311\pi$
$467$	−4.37964	−0.202666	−0.101333	+	0.994853i	$0.532311\pi$
$468$	12.1875	0.563365
$469$	−9.31186	−0.429982
$470$	0	0
$471$	−45.9057	−2.11522
$472$	−34.3469	−1.58095
$473$	21.4974	0.988453
$474$	−16.6645	−0.765424
$475$	0	0
$476$	8.19534	0.375633
$477$	−19.6547	−0.899929
$478$	−52.8102	−2.41548
$479$	14.1913	0.648417	0.324209	−	0.945986i	$-0.394902\pi$
$479$	14.1913	0.648417	0.324209	+	0.945986i	$0.394902\pi$
$480$	0	0
$481$	1.95460	0.0891219
$482$	52.9257	2.41070
$483$	3.99206	0.181645
$484$	74.0744	3.36702
$485$	0	0
$486$	−35.0475	−1.58978
$487$	−22.0694	−1.00006	−0.500029	−	0.866009i	$-0.666677\pi$
$487$	−22.0694	−1.00006	−0.500029	+	0.866009i	$0.666677\pi$
$488$	−14.3197	−0.648223
$489$	−41.7444	−1.88775
$490$	0	0
$491$	−7.95704	−0.359096	−0.179548	−	0.983749i	$-0.557464\pi$
$491$	−7.95704	−0.359096	−0.179548	+	0.983749i	$0.557464\pi$
$492$	−23.8502	−1.07525
$493$	9.84345	0.443327
$494$	−30.6525	−1.37912
$495$	0	0
$496$	−31.5376	−1.41608
$497$	−5.53210	−0.248149
$498$	−32.0220	−1.43494
$499$	−4.09384	−0.183266	−0.0916328	−	0.995793i	$-0.529209\pi$
$499$	−4.09384	−0.183266	−0.0916328	+	0.995793i	$0.529209\pi$
$500$	0	0
$501$	12.8687	0.574932
$502$	−36.8718	−1.64567
$503$	−26.9554	−1.20188	−0.600941	−	0.799293i	$-0.705208\pi$
$503$	−26.9554	−1.20188	−0.600941	+	0.799293i	$0.705208\pi$
$504$	−7.56633	−0.337031
$505$	0	0
$506$	−25.8112	−1.14745
$507$	19.0787	0.847314
$508$	−32.1325	−1.42565
$509$	33.8837	1.50187	0.750935	−	0.660376i	$-0.229603\pi$
$509$	33.8837	1.50187	0.750935	+	0.660376i	$0.229603\pi$
$510$	0	0
$511$	9.10093	0.402602
$512$	43.9545	1.94253
$513$	19.9537	0.880978
$514$	47.1831	2.08116
$515$	0	0
$516$	34.8534	1.53433
$517$	38.8243	1.70749
$518$	−2.35695	−0.103558
$519$	31.8166	1.39659
$520$	0	0
$521$	−26.6469	−1.16742	−0.583711	−	0.811962i	$-0.698400\pi$
$521$	−26.6469	−1.16742	−0.583711	+	0.811962i	$0.698400\pi$
$522$	−17.6517	−0.772594
$523$	1.80935	0.0791172	0.0395586	−	0.999217i	$-0.487405\pi$
$523$	1.80935	0.0791172	0.0395586	+	0.999217i	$0.487405\pi$
$524$	−40.3592	−1.76310
$525$	0	0
$526$	−16.7846	−0.731842
$527$	−13.5659	−0.590938
$528$	54.1109	2.35488
$529$	−19.2437	−0.836680
$530$	0	0
$531$	9.68370	0.420237
$532$	24.8879	1.07903
$533$	5.45737	0.236385
$534$	−15.7517	−0.681641
$535$	0	0
$536$	50.2558	2.17072
$537$	42.3704	1.82842
$538$	−1.22397	−0.0527691
$539$	32.5795	1.40330
$540$	0	0
$541$	−33.5233	−1.44128	−0.720640	−	0.693309i	$-0.756152\pi$
$541$	−33.5233	−1.44128	−0.720640	+	0.693309i	$0.756152\pi$
$542$	−24.3132	−1.04434
$543$	−42.9685	−1.84396
$544$	−2.55126	−0.109384
$545$	0	0
$546$	10.1763	0.435504
$547$	27.5146	1.17644	0.588219	−	0.808702i	$-0.299829\pi$
$547$	27.5146	1.17644	0.588219	+	0.808702i	$0.299829\pi$
$548$	14.9779	0.639822
$549$	4.03727	0.172306
$550$	0	0
$551$	29.8929	1.27348
$552$	−21.5450	−0.917017
$553$	−3.09602	−0.131656
$554$	62.2050	2.64284
$555$	0	0
$556$	−7.67095	−0.325321
$557$	−2.16761	−0.0918448	−0.0459224	−	0.998945i	$-0.514623\pi$
$557$	−2.16761	−0.0918448	−0.0459224	+	0.998945i	$0.514623\pi$
$558$	24.3269	1.02984
$559$	−7.97509	−0.337310
$560$	0	0
$561$	23.2758	0.982704
$562$	19.2270	0.811040
$563$	−33.4836	−1.41116	−0.705582	−	0.708628i	$-0.749314\pi$
$563$	−33.4836	−1.41116	−0.705582	+	0.708628i	$0.749314\pi$
$564$	62.9451	2.65047
$565$	0	0
$566$	71.6616	3.01216
$567$	−10.9467	−0.459717
$568$	29.8566	1.25275
$569$	17.2323	0.722415	0.361208	−	0.932485i	$-0.382365\pi$
$569$	17.2323	0.722415	0.361208	+	0.932485i	$0.382365\pi$
$570$	0	0
$571$	42.7928	1.79082	0.895411	−	0.445240i	$-0.146882\pi$
$571$	42.7928	1.79082	0.895411	+	0.445240i	$0.146882\pi$
$572$	−44.3027	−1.85239
$573$	20.0600	0.838020
$574$	−6.58076	−0.274676
$575$	0	0
$576$	−9.48962	−0.395401
$577$	31.8385	1.32546	0.662728	−	0.748860i	$-0.269399\pi$
$577$	31.8385	1.32546	0.662728	+	0.748860i	$0.269399\pi$
$578$	31.7363	1.32006
$579$	5.76769	0.239697
$580$	0	0
$581$	−5.94923	−0.246816
$582$	−31.1912	−1.29292
$583$	71.4470	2.95903
$584$	−49.1175	−2.03250
$585$	0	0
$586$	−0.381866	−0.0157747
$587$	24.8027	1.02372	0.511858	−	0.859070i	$-0.328957\pi$
$587$	24.8027	1.02372	0.511858	+	0.859070i	$0.328957\pi$
$588$	52.8205	2.17828
$589$	−41.1973	−1.69750
$590$	0	0
$591$	1.65114	0.0679189
$592$	4.64938	0.191089
$593$	−17.8431	−0.732728	−0.366364	−	0.930472i	$-0.619397\pi$
$593$	−17.8431	−0.732728	−0.366364	+	0.930472i	$0.619397\pi$
$594$	42.8310	1.75738
$595$	0	0
$596$	69.4754	2.84582
$597$	15.7174	0.643271
$598$	9.57541	0.391568
$599$	−28.1028	−1.14825	−0.574126	−	0.818767i	$-0.694658\pi$
$599$	−28.1028	−1.14825	−0.574126	+	0.818767i	$0.694658\pi$
$600$	0	0
$601$	−26.6296	−1.08624	−0.543121	−	0.839654i	$-0.682758\pi$
$601$	−26.6296	−1.08624	−0.543121	+	0.839654i	$0.682758\pi$
$602$	9.61675	0.391949
$603$	−14.1690	−0.577007
$604$	−59.0336	−2.40204
$605$	0	0
$606$	−40.7156	−1.65396
$607$	19.2757	0.782375	0.391187	−	0.920311i	$-0.372064\pi$
$607$	19.2757	0.782375	0.391187	+	0.920311i	$0.372064\pi$
$608$	−7.74774	−0.314213
$609$	−9.92410	−0.402145
$610$	0	0
$611$	−14.4030	−0.582682
$612$	12.4701	0.504074
$613$	0.465680	0.0188086	0.00940432	−	0.999956i	$-0.497006\pi$
$613$	0.465680	0.0188086	0.00940432	+	0.999956i	$0.497006\pi$
$614$	−35.8472	−1.44668
$615$	0	0
$616$	27.5044	1.10818
$617$	44.2765	1.78251	0.891253	−	0.453507i	$-0.149827\pi$
$617$	44.2765	1.78251	0.891253	+	0.453507i	$0.149827\pi$
$618$	−25.9290	−1.04302
$619$	−36.1620	−1.45347	−0.726736	−	0.686917i	$-0.758964\pi$
$619$	−36.1620	−1.45347	−0.726736	+	0.686917i	$0.758964\pi$
$620$	0	0
$621$	−6.23326	−0.250132
$622$	−45.8707	−1.83925
$623$	−2.92644	−0.117245
$624$	−20.0740	−0.803603
$625$	0	0
$626$	16.9667	0.678126
$627$	70.6847	2.82287
$628$	89.4025	3.56755
$629$	1.99993	0.0797424
$630$	0	0
$631$	−0.199431	−0.00793921	−0.00396961	−	0.999992i	$-0.501264\pi$
$631$	−0.199431	−0.00793921	−0.00396961	+	0.999992i	$0.501264\pi$
$632$	16.7091	0.664654
$633$	6.31062	0.250825
$634$	−67.5028	−2.68088
$635$	0	0
$636$	115.836	4.59318
$637$	−12.0863	−0.478876
$638$	64.1657	2.54035
$639$	−8.41770	−0.332999
$640$	0	0
$641$	−30.9078	−1.22079	−0.610393	−	0.792099i	$-0.708988\pi$
$641$	−30.9078	−1.22079	−0.610393	+	0.792099i	$0.708988\pi$
$642$	−6.11742	−0.241436
$643$	22.2489	0.877412	0.438706	−	0.898631i	$-0.355437\pi$
$643$	22.2489	0.877412	0.438706	+	0.898631i	$0.355437\pi$
$644$	−7.77463	−0.306363
$645$	0	0
$646$	−31.3634	−1.23398
$647$	35.0696	1.37873	0.689364	−	0.724415i	$-0.257890\pi$
$647$	35.0696	1.37873	0.689364	+	0.724415i	$0.257890\pi$
$648$	59.0789	2.32084
$649$	−35.2012	−1.38177
$650$	0	0
$651$	13.6770	0.536044
$652$	81.2983	3.18389
$653$	4.00012	0.156537	0.0782683	−	0.996932i	$-0.475061\pi$
$653$	4.00012	0.156537	0.0782683	+	0.996932i	$0.475061\pi$
$654$	−92.6422	−3.62260
$655$	0	0
$656$	12.9814	0.506839
$657$	13.8481	0.540265
$658$	17.3678	0.677068
$659$	10.0349	0.390904	0.195452	−	0.980713i	$-0.437383\pi$
$659$	10.0349	0.390904	0.195452	+	0.980713i	$0.437383\pi$
$660$	0	0
$661$	−48.5672	−1.88905	−0.944523	−	0.328445i	$-0.893475\pi$
$661$	−48.5672	−1.88905	−0.944523	+	0.328445i	$0.893475\pi$
$662$	32.2727	1.25431
$663$	−8.63481	−0.335348
$664$	32.1078	1.24602
$665$	0	0
$666$	−3.58636	−0.138968
$667$	−9.33814	−0.361574
$668$	−25.0622	−0.969684
$669$	52.6479	2.03549
$670$	0	0
$671$	−14.6759	−0.566557
$672$	2.57216	0.0992232
$673$	−5.06306	−0.195167	−0.0975834	−	0.995227i	$-0.531111\pi$
$673$	−5.06306	−0.195167	−0.0975834	+	0.995227i	$0.531111\pi$
$674$	−48.6229	−1.87288
$675$	0	0
$676$	−37.1562	−1.42908
$677$	−33.6669	−1.29393	−0.646963	−	0.762521i	$-0.723961\pi$
$677$	−33.6669	−1.29393	−0.646963	+	0.762521i	$0.723961\pi$
$678$	−26.3787	−1.01307
$679$	−5.79489	−0.222387
$680$	0	0
$681$	−37.1689	−1.42431
$682$	−88.4307	−3.38619
$683$	−32.4523	−1.24175	−0.620877	−	0.783908i	$-0.713223\pi$
$683$	−32.4523	−1.24175	−0.620877	+	0.783908i	$0.713223\pi$
$684$	37.8697	1.44798
$685$	0	0
$686$	31.4283	1.19994
$687$	−14.8564	−0.566807
$688$	−18.9703	−0.723235
$689$	−26.5053	−1.00977
$690$	0	0
$691$	−30.7811	−1.17097	−0.585484	−	0.810684i	$-0.699095\pi$
$691$	−30.7811	−1.17097	−0.585484	+	0.810684i	$0.699095\pi$
$692$	−61.9637	−2.35550
$693$	−7.75453	−0.294570
$694$	10.8610	0.412278
$695$	0	0
$696$	53.5601	2.03019
$697$	5.58394	0.211507
$698$	−67.2914	−2.54702
$699$	−18.3972	−0.695847
$700$	0	0
$701$	−13.1755	−0.497633	−0.248817	−	0.968551i	$-0.580042\pi$
$701$	−13.1755	−0.497633	−0.248817	+	0.968551i	$0.580042\pi$
$702$	−15.8894	−0.599706
$703$	6.07345	0.229065
$704$	34.4958	1.30011
$705$	0	0
$706$	−39.4207	−1.48362
$707$	−7.56439	−0.284488
$708$	−57.0711	−2.14486
$709$	−10.6307	−0.399246	−0.199623	−	0.979873i	$-0.563972\pi$
$709$	−10.6307	−0.399246	−0.199623	+	0.979873i	$0.563972\pi$
$710$	0	0
$711$	−4.71094	−0.176674
$712$	15.7939	0.591901
$713$	12.8695	0.481965
$714$	10.4123	0.389670
$715$	0	0
$716$	−82.5174	−3.08382
$717$	−45.1779	−1.68720
$718$	12.4091	0.463102
$719$	−15.3464	−0.572324	−0.286162	−	0.958181i	$-0.592380\pi$
$719$	−15.3464	−0.572324	−0.286162	+	0.958181i	$0.592380\pi$
$720$	0	0
$721$	−4.81724	−0.179404
$722$	−48.2328	−1.79504
$723$	45.2767	1.68386
$724$	83.6823	3.11003
$725$	0	0
$726$	94.1124	3.49284
$727$	−23.8156	−0.883272	−0.441636	−	0.897194i	$-0.645602\pi$
$727$	−23.8156	−0.883272	−0.441636	+	0.897194i	$0.645602\pi$
$728$	−10.2035	−0.378168
$729$	3.76661	0.139504
$730$	0	0
$731$	−8.16005	−0.301810
$732$	−23.7937	−0.879442
$733$	4.09180	0.151134	0.0755670	−	0.997141i	$-0.475923\pi$
$733$	4.09180	0.151134	0.0755670	+	0.997141i	$0.475923\pi$
$734$	70.1897	2.59075
$735$	0	0
$736$	2.42029	0.0892130
$737$	51.5059	1.89724
$738$	−10.0134	−0.368597
$739$	2.45909	0.0904590	0.0452295	−	0.998977i	$-0.485598\pi$
$739$	2.45909	0.0904590	0.0452295	+	0.998977i	$0.485598\pi$
$740$	0	0
$741$	−26.2225	−0.963307
$742$	31.9614	1.17334
$743$	11.1921	0.410598	0.205299	−	0.978699i	$-0.434183\pi$
$743$	11.1921	0.410598	0.205299	+	0.978699i	$0.434183\pi$
$744$	−73.8144	−2.70617
$745$	0	0
$746$	−80.9653	−2.96435
$747$	−9.05240	−0.331210
$748$	−45.3301	−1.65743
$749$	−1.13653	−0.0415279
$750$	0	0
$751$	−10.3616	−0.378101	−0.189051	−	0.981967i	$-0.560541\pi$
$751$	−10.3616	−0.378101	−0.189051	+	0.981967i	$0.560541\pi$
$752$	−34.2603	−1.24934
$753$	−31.5430	−1.14949
$754$	−23.8041	−0.866894
$755$	0	0
$756$	12.9012	0.469211
$757$	5.84722	0.212521	0.106260	−	0.994338i	$-0.466112\pi$
$757$	5.84722	0.212521	0.106260	+	0.994338i	$0.466112\pi$
$758$	−11.0971	−0.403063
$759$	−22.0809	−0.801486
$760$	0	0
$761$	41.6931	1.51137	0.755687	−	0.654932i	$-0.227303\pi$
$761$	41.6931	1.51137	0.755687	+	0.654932i	$0.227303\pi$
$762$	−40.8248	−1.47893
$763$	−17.2116	−0.623102
$764$	−39.0674	−1.41341
$765$	0	0
$766$	40.2921	1.45581
$767$	13.0589	0.471530
$768$	69.0084	2.49013
$769$	−17.3076	−0.624129	−0.312065	−	0.950061i	$-0.601021\pi$
$769$	−17.3076	−0.624129	−0.312065	+	0.950061i	$0.601021\pi$
$770$	0	0
$771$	40.3640	1.45367
$772$	−11.2327	−0.404274
$773$	−17.8695	−0.642723	−0.321361	−	0.946957i	$-0.604140\pi$
$773$	−17.8695	−0.642723	−0.321361	+	0.946957i	$0.604140\pi$
$774$	14.6329	0.525970
$775$	0	0
$776$	31.2748	1.12270
$777$	−2.01631	−0.0723348
$778$	−53.1736	−1.90637
$779$	16.9575	0.607565
$780$	0	0
$781$	30.5992	1.09493
$782$	9.79749	0.350358
$783$	15.4956	0.553769
$784$	−28.7496	−1.02677
$785$	0	0
$786$	−51.2768	−1.82898
$787$	−9.91104	−0.353290	−0.176645	−	0.984275i	$-0.556525\pi$
$787$	−9.91104	−0.353290	−0.176645	+	0.984275i	$0.556525\pi$
$788$	−3.21564	−0.114552
$789$	−14.3588	−0.511187
$790$	0	0
$791$	−4.90080	−0.174252
$792$	41.8510	1.48711
$793$	5.44444	0.193338
$794$	−51.6819	−1.83412
$795$	0	0
$796$	−30.6101	−1.08495
$797$	−39.7590	−1.40834	−0.704168	−	0.710034i	$-0.748680\pi$
$797$	−39.7590	−1.40834	−0.704168	+	0.710034i	$0.748680\pi$
$798$	31.6204	1.11935
$799$	−14.7370	−0.521359
$800$	0	0
$801$	−4.45290	−0.157335
$802$	1.56731	0.0553437
$803$	−50.3392	−1.77643
$804$	83.5055	2.94501
$805$	0	0
$806$	32.8059	1.15554
$807$	−1.04708	−0.0368589
$808$	40.8248	1.43621
$809$	−5.05110	−0.177587	−0.0887935	−	0.996050i	$-0.528301\pi$
$809$	−5.05110	−0.177587	−0.0887935	+	0.996050i	$0.528301\pi$
$810$	0	0
$811$	11.7743	0.413452	0.206726	−	0.978399i	$-0.433719\pi$
$811$	11.7743	0.413452	0.206726	+	0.978399i	$0.433719\pi$
$812$	19.3274	0.678260
$813$	−20.7994	−0.729467
$814$	13.0368	0.456939
$815$	0	0
$816$	−20.5396	−0.719028
$817$	−24.7807	−0.866968
$818$	10.6353	0.371855
$819$	2.87676	0.100522
$820$	0	0
$821$	5.25401	0.183366	0.0916832	−	0.995788i	$-0.470775\pi$
$821$	5.25401	0.183366	0.0916832	+	0.995788i	$0.470775\pi$
$822$	19.0295	0.663732
$823$	37.0042	1.28989	0.644943	−	0.764230i	$-0.276881\pi$
$823$	37.0042	1.28989	0.644943	+	0.764230i	$0.276881\pi$
$824$	25.9985	0.905701
$825$	0	0
$826$	−15.7471	−0.547910
$827$	−28.1147	−0.977645	−0.488822	−	0.872383i	$-0.662573\pi$
$827$	−28.1147	−0.977645	−0.488822	+	0.872383i	$0.662573\pi$
$828$	−11.8300	−0.411119
$829$	−14.4532	−0.501979	−0.250990	−	0.967990i	$-0.580756\pi$
$829$	−14.4532	−0.501979	−0.250990	+	0.967990i	$0.580756\pi$
$830$	0	0
$831$	53.2149	1.84600
$832$	−12.7972	−0.443662
$833$	−12.3666	−0.428477
$834$	−9.74604	−0.337478
$835$	0	0
$836$	−137.660	−4.76108
$837$	−21.3555	−0.738154
$838$	−83.1914	−2.87380
$839$	16.3694	0.565134	0.282567	−	0.959248i	$-0.408814\pi$
$839$	16.3694	0.565134	0.282567	+	0.959248i	$0.408814\pi$
$840$	0	0
$841$	−5.78575	−0.199509
$842$	52.8937	1.82284
$843$	16.4482	0.566506
$844$	−12.2901	−0.423043
$845$	0	0
$846$	26.4271	0.908581
$847$	17.4848	0.600784
$848$	−63.0480	−2.16508
$849$	61.3048	2.10397
$850$	0	0
$851$	−1.89726	−0.0650373
$852$	49.6099	1.69961
$853$	46.8216	1.60314	0.801571	−	0.597900i	$-0.203998\pi$
$853$	46.8216	1.60314	0.801571	+	0.597900i	$0.203998\pi$
$854$	−6.56517	−0.224656
$855$	0	0
$856$	6.13382	0.209650
$857$	−28.9569	−0.989148	−0.494574	−	0.869136i	$-0.664676\pi$
$857$	−28.9569	−0.989148	−0.494574	+	0.869136i	$0.664676\pi$
$858$	−56.2870	−1.92161
$859$	24.4996	0.835917	0.417959	−	0.908466i	$-0.362746\pi$
$859$	24.4996	0.835917	0.417959	+	0.908466i	$0.362746\pi$
$860$	0	0
$861$	−5.62968	−0.191859
$862$	−64.5571	−2.19882
$863$	−10.4696	−0.356391	−0.178195	−	0.983995i	$-0.557026\pi$
$863$	−10.4696	−0.356391	−0.178195	+	0.983995i	$0.557026\pi$
$864$	−4.01621	−0.136634
$865$	0	0
$866$	29.1171	0.989440
$867$	27.1497	0.922052
$868$	−26.6363	−0.904096
$869$	17.1247	0.580917
$870$	0	0
$871$	−19.1076	−0.647435
$872$	92.8905	3.14567
$873$	−8.81756	−0.298429
$874$	29.7534	1.00642
$875$	0	0
$876$	−81.6139	−2.75748
$877$	−39.9121	−1.34774	−0.673868	−	0.738852i	$-0.735368\pi$
$877$	−39.9121	−1.34774	−0.673868	+	0.738852i	$0.735368\pi$
$878$	−12.1880	−0.411324
$879$	−0.326677	−0.0110185
$880$	0	0
$881$	−7.88965	−0.265809	−0.132904	−	0.991129i	$-0.542430\pi$
$881$	−7.88965	−0.265809	−0.132904	+	0.991129i	$0.542430\pi$
$882$	22.1763	0.746716
$883$	15.1304	0.509178	0.254589	−	0.967049i	$-0.418060\pi$
$883$	15.1304	0.509178	0.254589	+	0.967049i	$0.418060\pi$
$884$	16.8165	0.565600
$885$	0	0
$886$	23.1590	0.778042
$887$	−47.6886	−1.60123	−0.800614	−	0.599180i	$-0.795493\pi$
$887$	−47.6886	−1.60123	−0.800614	+	0.599180i	$0.795493\pi$
$888$	10.8820	0.365175
$889$	−7.58466	−0.254381
$890$	0	0
$891$	60.5483	2.02845
$892$	−102.533	−3.43306
$893$	−44.7539	−1.49763
$894$	88.2693	2.95217
$895$	0	0
$896$	17.8618	0.596719
$897$	8.19154	0.273508
$898$	−18.3498	−0.612342
$899$	−31.9930	−1.06703
$900$	0	0
$901$	−27.1200	−0.903499
$902$	36.3996	1.21197
$903$	8.22690	0.273774
$904$	26.4494	0.879696
$905$	0	0
$906$	−75.0029	−2.49180
$907$	38.9321	1.29272	0.646359	−	0.763033i	$-0.276291\pi$
$907$	38.9321	1.29272	0.646359	+	0.763033i	$0.276291\pi$
$908$	72.3873	2.40226
$909$	−11.5100	−0.381764
$910$	0	0
$911$	37.6740	1.24819	0.624097	−	0.781347i	$-0.285467\pi$
$911$	37.6740	1.24819	0.624097	+	0.781347i	$0.285467\pi$
$912$	−62.3752	−2.06545
$913$	32.9064	1.08904
$914$	21.7672	0.719996
$915$	0	0
$916$	28.9332	0.955980
$917$	−9.52650	−0.314593
$918$	−16.2579	−0.536590
$919$	14.4646	0.477145	0.238572	−	0.971125i	$-0.423321\pi$
$919$	14.4646	0.477145	0.238572	+	0.971125i	$0.423321\pi$
$920$	0	0
$921$	−30.6664	−1.01049
$922$	104.300	3.43495
$923$	−11.3516	−0.373644
$924$	45.7015	1.50347
$925$	0	0
$926$	57.0499	1.87478
$927$	−7.32997	−0.240748
$928$	−6.01674	−0.197509
$929$	−40.8990	−1.34185	−0.670927	−	0.741524i	$-0.734103\pi$
$929$	−40.8990	−1.34185	−0.670927	+	0.741524i	$0.734103\pi$
$930$	0	0
$931$	−37.5553	−1.23083
$932$	35.8291	1.17362
$933$	−39.2413	−1.28470
$934$	10.8368	0.354590
$935$	0	0
$936$	−15.5258	−0.507477
$937$	−33.2563	−1.08644	−0.543218	−	0.839592i	$-0.682794\pi$
$937$	−33.2563	−1.08644	−0.543218	+	0.839592i	$0.682794\pi$
$938$	23.0408	0.752310
$939$	14.5146	0.473666
$940$	0	0
$941$	39.1577	1.27650	0.638252	−	0.769827i	$-0.279658\pi$
$941$	39.1577	1.27650	0.638252	+	0.769827i	$0.279658\pi$
$942$	113.587	3.70086
$943$	−5.29728	−0.172503
$944$	31.0631	1.01102
$945$	0	0
$946$	−53.1923	−1.72943
$947$	28.5840	0.928854	0.464427	−	0.885611i	$-0.346260\pi$
$947$	28.5840	0.928854	0.464427	+	0.885611i	$0.346260\pi$
$948$	27.7640	0.901734
$949$	18.6747	0.606208
$950$	0	0
$951$	−57.7471	−1.87258
$952$	−10.4402	−0.338368
$953$	43.3568	1.40446	0.702232	−	0.711948i	$-0.252187\pi$
$953$	43.3568	1.40446	0.702232	+	0.711948i	$0.252187\pi$
$954$	48.6328	1.57454
$955$	0	0
$956$	87.9850	2.84564
$957$	54.8923	1.77442
$958$	−35.1143	−1.13449
$959$	3.53542	0.114165
$960$	0	0
$961$	13.0915	0.422306
$962$	−4.83636	−0.155931
$963$	−1.72936	−0.0557277
$964$	−88.1775	−2.84001
$965$	0	0
$966$	−9.87776	−0.317812
$967$	−32.2551	−1.03726	−0.518628	−	0.855000i	$-0.673557\pi$
$967$	−32.2551	−1.03726	−0.518628	+	0.855000i	$0.673557\pi$
$968$	−94.3647	−3.03300
$969$	−26.8306	−0.861925
$970$	0	0
$971$	10.9579	0.351654	0.175827	−	0.984421i	$-0.443740\pi$
$971$	10.9579	0.351654	0.175827	+	0.984421i	$0.443740\pi$
$972$	58.3912	1.87290
$973$	−1.81068	−0.0580476
$974$	54.6074	1.74973
$975$	0	0
$976$	12.9507	0.414540
$977$	20.1536	0.644770	0.322385	−	0.946609i	$-0.395516\pi$
$977$	20.1536	0.644770	0.322385	+	0.946609i	$0.395516\pi$
$978$	103.290	3.30286
$979$	16.1867	0.517330
$980$	0	0
$981$	−26.1894	−0.836162
$982$	19.6885	0.628286
$983$	16.0511	0.511951	0.255975	−	0.966683i	$-0.417603\pi$
$983$	16.0511	0.511951	0.255975	+	0.966683i	$0.417603\pi$
$984$	30.3832	0.968582
$985$	0	0
$986$	−24.3562	−0.775659
$987$	14.8578	0.472928
$988$	51.0689	1.62472
$989$	7.74115	0.246154
$990$	0	0
$991$	20.5569	0.653010	0.326505	−	0.945195i	$-0.394129\pi$
$991$	20.5569	0.653010	0.326505	+	0.945195i	$0.394129\pi$
$992$	8.29204	0.263272
$993$	27.6085	0.876129
$994$	13.6884	0.434169
$995$	0	0
$996$	53.3506	1.69048
$997$	−21.7296	−0.688183	−0.344092	−	0.938936i	$-0.611813\pi$
$997$	−21.7296	−0.688183	−0.344092	+	0.938936i	$0.611813\pi$
$998$	10.1296	0.320647
$999$	3.14830	0.0996079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.e.1.2	✓	8
3.2	odd	2		5625.2.a.be.1.7		8
4.3	odd	2		10000.2.a.bn.1.6		8
5.2	odd	4		625.2.b.d.624.2		16
5.3	odd	4		625.2.b.d.624.15		16
5.4	even	2		625.2.a.g.1.7	yes	8
15.14	odd	2		5625.2.a.s.1.2		8
20.19	odd	2		10000.2.a.be.1.3		8
25.2	odd	20		625.2.e.k.124.1		32
25.3	odd	20		625.2.e.j.249.8		32
25.4	even	10		625.2.d.m.376.1		16
25.6	even	5		625.2.d.q.251.4		16
25.8	odd	20		625.2.e.j.374.1		32
25.9	even	10		625.2.d.n.126.4		16
25.11	even	5		625.2.d.p.501.1		16
25.12	odd	20		625.2.e.k.499.8		32
25.13	odd	20		625.2.e.k.499.1		32
25.14	even	10		625.2.d.n.501.4		16
25.16	even	5		625.2.d.p.126.1		16
25.17	odd	20		625.2.e.j.374.8		32
25.19	even	10		625.2.d.m.251.1		16
25.21	even	5		625.2.d.q.376.4		16
25.22	odd	20		625.2.e.j.249.1		32
25.23	odd	20		625.2.e.k.124.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.2	✓	8	1.1	even	1	trivial
625.2.a.g.1.7	yes	8	5.4	even	2
625.2.b.d.624.2		16	5.2	odd	4
625.2.b.d.624.15		16	5.3	odd	4
625.2.d.m.251.1		16	25.19	even	10
625.2.d.m.376.1		16	25.4	even	10
625.2.d.n.126.4		16	25.9	even	10
625.2.d.n.501.4		16	25.14	even	10
625.2.d.p.126.1		16	25.16	even	5
625.2.d.p.501.1		16	25.11	even	5
625.2.d.q.251.4		16	25.6	even	5
625.2.d.q.376.4		16	25.21	even	5
625.2.e.j.249.1		32	25.22	odd	20
625.2.e.j.249.8		32	25.3	odd	20
625.2.e.j.374.1		32	25.8	odd	20
625.2.e.j.374.8		32	25.17	odd	20
625.2.e.k.124.1		32	25.2	odd	20
625.2.e.k.124.8		32	25.23	odd	20
625.2.e.k.499.1		32	25.13	odd	20
625.2.e.k.499.8		32	25.12	odd	20
5625.2.a.s.1.2		8	15.14	odd	2
5625.2.a.be.1.7		8	3.2	odd	2
10000.2.a.be.1.3		8	20.19	odd	2
10000.2.a.bn.1.6		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)

\(f(q)\)	\(=\)	\(q-2.47435 q^{2} -2.11675 q^{3} +4.12242 q^{4} +5.23759 q^{6} +0.973070 q^{7} -5.25163 q^{8} +1.48063 q^{9} +O(q^{10})\)	\(q-2.47435 q^{2} -2.11675 q^{3} +4.12242 q^{4} +5.23759 q^{6} +0.973070 q^{7} -5.25163 q^{8} +1.48063 q^{9} -5.38225 q^{11} -8.72615 q^{12} +1.99670 q^{13} -2.40772 q^{14} +4.74954 q^{16} +2.04301 q^{17} -3.66361 q^{18} +6.20428 q^{19} -2.05975 q^{21} +13.3176 q^{22} -1.93813 q^{23} +11.1164 q^{24} -4.94054 q^{26} +3.21612 q^{27} +4.01141 q^{28} +4.81812 q^{29} -6.64014 q^{31} -1.24877 q^{32} +11.3929 q^{33} -5.05512 q^{34} +6.10380 q^{36} +0.978913 q^{37} -15.3516 q^{38} -4.22652 q^{39} +2.73319 q^{41} +5.09654 q^{42} -3.99413 q^{43} -22.1879 q^{44} +4.79562 q^{46} -7.21339 q^{47} -10.0536 q^{48} -6.05313 q^{49} -4.32454 q^{51} +8.23125 q^{52} -13.2746 q^{53} -7.95782 q^{54} -5.11020 q^{56} -13.1329 q^{57} -11.9217 q^{58} +6.54024 q^{59} +2.72672 q^{61} +16.4301 q^{62} +1.44076 q^{63} -6.40916 q^{64} -28.1900 q^{66} -9.56957 q^{67} +8.42215 q^{68} +4.10254 q^{69} -5.68520 q^{71} -7.77574 q^{72} +9.35281 q^{73} -2.42218 q^{74} +25.5767 q^{76} -5.23731 q^{77} +10.4579 q^{78} -3.18171 q^{79} -11.2496 q^{81} -6.76289 q^{82} -6.11387 q^{83} -8.49115 q^{84} +9.88290 q^{86} -10.1988 q^{87} +28.2656 q^{88} -3.00743 q^{89} +1.94293 q^{91} -7.98980 q^{92} +14.0555 q^{93} +17.8485 q^{94} +2.64334 q^{96} -5.95526 q^{97} +14.9776 q^{98} -7.96914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) 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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