LMFDB - Embedded newform 845.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 $ x^9 - 17x^7 - 9x^6 + 59x^5 + 32x^4 - 44x^3 - 23x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.76052$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.20556 q^{2} -0.0130567 q^{3} +2.86449 q^{4} -1.00000 q^{5} +0.0287972 q^{6} -4.60897 q^{7} -1.90669 q^{8} -2.99983 q^{9} +2.20556 q^{10} -2.93232 q^{11} -0.0374007 q^{12} +10.1654 q^{14} +0.0130567 q^{15} -1.52367 q^{16} -3.35206 q^{17} +6.61630 q^{18} +2.46021 q^{19} -2.86449 q^{20} +0.0601778 q^{21} +6.46741 q^{22} -1.58329 q^{23} +0.0248950 q^{24} +1.00000 q^{25} +0.0783378 q^{27} -13.2024 q^{28} +8.26322 q^{29} -0.0287972 q^{30} +9.77959 q^{31} +7.17392 q^{32} +0.0382863 q^{33} +7.39317 q^{34} +4.60897 q^{35} -8.59299 q^{36} -4.12917 q^{37} -5.42613 q^{38} +1.90669 q^{40} -7.26508 q^{41} -0.132726 q^{42} -0.705329 q^{43} -8.39961 q^{44} +2.99983 q^{45} +3.49204 q^{46} -8.57322 q^{47} +0.0198940 q^{48} +14.2426 q^{49} -2.20556 q^{50} +0.0437667 q^{51} -12.4639 q^{53} -0.172779 q^{54} +2.93232 q^{55} +8.78787 q^{56} -0.0321221 q^{57} -18.2250 q^{58} +5.33445 q^{59} +0.0374007 q^{60} +1.92092 q^{61} -21.5695 q^{62} +13.8261 q^{63} -12.7752 q^{64} -0.0844427 q^{66} +7.29793 q^{67} -9.60195 q^{68} +0.0206725 q^{69} -10.1654 q^{70} +6.68083 q^{71} +5.71974 q^{72} +12.4541 q^{73} +9.10713 q^{74} -0.0130567 q^{75} +7.04724 q^{76} +13.5150 q^{77} +0.984840 q^{79} +1.52367 q^{80} +8.99847 q^{81} +16.0236 q^{82} +7.84405 q^{83} +0.172379 q^{84} +3.35206 q^{85} +1.55565 q^{86} -0.107890 q^{87} +5.59102 q^{88} +0.412834 q^{89} -6.61630 q^{90} -4.53532 q^{92} -0.127689 q^{93} +18.9087 q^{94} -2.46021 q^{95} -0.0936675 q^{96} +3.38416 q^{97} -31.4129 q^{98} +8.79646 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9} + 3 q^{10} + 9 q^{11} + 12 q^{12} - 2 q^{14} - 7 q^{15} + 37 q^{16} - q^{17} + 10 q^{18} + 4 q^{19} - 17 q^{20} + q^{21}+ \cdots + q^{99}+O(q^{100}) $ 9 * q - 3 * q^2 + 7 * q^3 + 17 * q^4 - 9 * q^5 + 2 * q^6 - 7 * q^7 - 12 * q^8 + 16 * q^9 + 3 * q^10 + 9 * q^11 + 12 * q^12 - 2 * q^14 - 7 * q^15 + 37 * q^16 - q^17 + 10 * q^18 + 4 * q^19 - 17 * q^20 + q^21 + 12 * q^22 + 14 * q^23 + 35 * q^24 + 9 * q^25 + 22 * q^27 - 18 * q^28 + 12 * q^29 - 2 * q^30 + 7 * q^31 - 22 * q^32 + 8 * q^33 + 30 * q^34 + 7 * q^35 + 3 * q^36 + 5 * q^37 - 47 * q^38 + 12 * q^40 + 10 * q^41 - 11 * q^42 + 39 * q^43 + 25 * q^44 - 16 * q^45 - 6 * q^46 - 36 * q^47 - 3 * q^48 + 16 * q^49 - 3 * q^50 + 43 * q^51 - 8 * q^53 + 2 * q^54 - 9 * q^55 - 29 * q^56 + 32 * q^57 - 21 * q^58 + 21 * q^59 - 12 * q^60 - 3 * q^61 - 10 * q^62 - 35 * q^63 + 34 * q^64 - 49 * q^66 - q^67 - 20 * q^68 - 13 * q^69 + 2 * q^70 + q^71 + 3 * q^72 - 15 * q^74 + 7 * q^75 + 5 * q^76 - 4 * q^77 + 39 * q^79 - 37 * q^80 + 29 * q^81 - 4 * q^82 - 7 * q^83 - 12 * q^84 + q^85 + 24 * q^86 + 16 * q^87 + 42 * q^88 + 19 * q^89 - 10 * q^90 - 27 * q^92 - 31 * q^93 + 16 * q^94 - 4 * q^95 + 7 * q^96 + 34 * q^97 - 48 * q^98 + q^99

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 $

$2$	−2.20556	−1.55957	−0.779783	−	0.626050i	$-0.784671\pi$
$2$	−2.20556	−1.55957	−0.779783	+	0.626050i	$0.784671\pi$
$3$	−0.0130567	−0.00753827	−0.00376913	−	0.999993i	$-0.501200\pi$
$3$	−0.0130567	−0.00753827	−0.00376913	+	0.999993i	$0.501200\pi$
$4$	2.86449	1.43225
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.0287972	0.0117564
$7$	−4.60897	−1.74203	−0.871013	−	0.491259i	$-0.836537\pi$
$7$	−4.60897	−1.74203	−0.871013	+	0.491259i	$0.836537\pi$
$8$	−1.90669	−0.674116
$9$	−2.99983	−0.999943
$10$	2.20556	0.697459
$11$	−2.93232	−0.884128	−0.442064	−	0.896984i	$-0.645754\pi$
$11$	−2.93232	−0.884128	−0.442064	+	0.896984i	$0.645754\pi$
$12$	−0.0374007	−0.0107967
$13$	0	0
$13$	0	0
$14$	10.1654	2.71681
$15$	0.0130567	0.00337122
$16$	−1.52367	−0.380917
$17$	−3.35206	−0.812994	−0.406497	−	0.913652i	$-0.633250\pi$
$17$	−3.35206	−0.812994	−0.406497	+	0.913652i	$0.633250\pi$
$18$	6.61630	1.55948
$19$	2.46021	0.564410	0.282205	−	0.959354i	$-0.408934\pi$
$19$	2.46021	0.564410	0.282205	+	0.959354i	$0.408934\pi$
$20$	−2.86449	−0.640520
$21$	0.0601778	0.0131319
$22$	6.46741	1.37886
$23$	−1.58329	−0.330139	−0.165069	−	0.986282i	$-0.552785\pi$
$23$	−1.58329	−0.330139	−0.165069	+	0.986282i	$0.552785\pi$
$24$	0.0248950	0.00508167
$25$	1.00000	0.200000
$26$	0	0
$27$	0.0783378	0.0150761
$28$	−13.2024	−2.49501
$29$	8.26322	1.53444	0.767221	−	0.641383i	$-0.221639\pi$
$29$	8.26322	1.53444	0.767221	+	0.641383i	$0.221639\pi$
$30$	−0.0287972	−0.00525763
$31$	9.77959	1.75647	0.878233	−	0.478233i	$-0.158723\pi$
$31$	9.77959	1.75647	0.878233	+	0.478233i	$0.158723\pi$
$32$	7.17392	1.26818
$33$	0.0382863	0.00666479
$34$	7.39317	1.26792
$35$	4.60897	0.779058
$36$	−8.59299	−1.43216
$37$	−4.12917	−0.678831	−0.339416	−	0.940637i	$-0.610229\pi$
$37$	−4.12917	−0.678831	−0.339416	+	0.940637i	$0.610229\pi$
$38$	−5.42613	−0.880235
$39$	0	0
$40$	1.90669	0.301474
$41$	−7.26508	−1.13462	−0.567308	−	0.823506i	$-0.692015\pi$
$41$	−7.26508	−1.13462	−0.567308	+	0.823506i	$0.692015\pi$
$42$	−0.132726	−0.0204800
$43$	−0.705329	−0.107562	−0.0537809	−	0.998553i	$-0.517127\pi$
$43$	−0.705329	−0.107562	−0.0537809	+	0.998553i	$0.517127\pi$
$44$	−8.39961	−1.26629
$45$	2.99983	0.447188
$46$	3.49204	0.514873
$47$	−8.57322	−1.25053	−0.625266	−	0.780411i	$-0.715010\pi$
$47$	−8.57322	−1.25053	−0.625266	+	0.780411i	$0.715010\pi$
$48$	0.0198940	0.00287146
$49$	14.2426	2.03466
$50$	−2.20556	−0.311913
$51$	0.0437667	0.00612857
$52$	0	0
$53$	−12.4639	−1.71204	−0.856022	−	0.516940i	$-0.827071\pi$
$53$	−12.4639	−1.71204	−0.856022	+	0.516940i	$0.827071\pi$
$54$	−0.172779	−0.0235122
$55$	2.93232	0.395394
$56$	8.78787	1.17433
$57$	−0.0321221	−0.00425467
$58$	−18.2250	−2.39306
$59$	5.33445	0.694487	0.347243	−	0.937775i	$-0.387118\pi$
$59$	5.33445	0.694487	0.347243	+	0.937775i	$0.387118\pi$
$60$	0.0374007	0.00482841
$61$	1.92092	0.245949	0.122974	−	0.992410i	$-0.460757\pi$
$61$	1.92092	0.245949	0.122974	+	0.992410i	$0.460757\pi$
$62$	−21.5695	−2.73932
$63$	13.8261	1.74193
$64$	−12.7752	−1.59690
$65$	0	0
$66$	−0.0844427	−0.0103942
$67$	7.29793	0.891584	0.445792	−	0.895137i	$-0.352922\pi$
$67$	7.29793	0.891584	0.445792	+	0.895137i	$0.352922\pi$
$68$	−9.60195	−1.16441
$69$	0.0206725	0.00248867
$70$	−10.1654	−1.21499
$71$	6.68083	0.792868	0.396434	−	0.918063i	$-0.370248\pi$
$71$	6.68083	0.792868	0.396434	+	0.918063i	$0.370248\pi$
$72$	5.71974	0.674078
$73$	12.4541	1.45765	0.728824	−	0.684702i	$-0.240067\pi$
$73$	12.4541	1.45765	0.728824	+	0.684702i	$0.240067\pi$
$74$	9.10713	1.05868
$75$	−0.0130567	−0.00150765
$76$	7.04724	0.808374
$77$	13.5150	1.54017
$78$	0	0
$79$	0.984840	0.110803	0.0554016	−	0.998464i	$-0.482356\pi$
$79$	0.984840	0.110803	0.0554016	+	0.998464i	$0.482356\pi$
$80$	1.52367	0.170351
$81$	8.99847	0.999830
$82$	16.0236	1.76951
$83$	7.84405	0.860997	0.430498	−	0.902591i	$-0.358338\pi$
$83$	7.84405	0.860997	0.430498	+	0.902591i	$0.358338\pi$
$84$	0.172379	0.0188081
$85$	3.35206	0.363582
$86$	1.55565	0.167750
$87$	−0.107890	−0.0115670
$88$	5.59102	0.596005
$89$	0.412834	0.0437604	0.0218802	−	0.999761i	$-0.493035\pi$
$89$	0.412834	0.0437604	0.0218802	+	0.999761i	$0.493035\pi$
$90$	−6.61630	−0.697419
$91$	0	0
$92$	−4.53532	−0.472840
$93$	−0.127689	−0.0132407
$94$	18.9087	1.95029
$95$	−2.46021	−0.252412
$96$	−0.0936675	−0.00955989
$97$	3.38416	0.343609	0.171805	−	0.985131i	$-0.445040\pi$
$97$	3.38416	0.343609	0.171805	+	0.985131i	$0.445040\pi$
$98$	−31.4129	−3.17318
$99$	8.79646	0.884077
$100$	2.86449	0.286449
$101$	−10.5432	−1.04909	−0.524545	−	0.851383i	$-0.675765\pi$
$101$	−10.5432	−1.04909	−0.524545	+	0.851383i	$0.675765\pi$
$102$	−0.0965301	−0.00955790
$103$	14.1152	1.39081	0.695404	−	0.718619i	$-0.255226\pi$
$103$	14.1152	1.39081	0.695404	+	0.718619i	$0.255226\pi$
$104$	0	0
$105$	−0.0601778	−0.00587275
$106$	27.4898	2.67004
$107$	−3.97165	−0.383954	−0.191977	−	0.981399i	$-0.561490\pi$
$107$	−3.97165	−0.383954	−0.191977	+	0.981399i	$0.561490\pi$
$108$	0.224398	0.0215927
$109$	−10.1737	−0.974460	−0.487230	−	0.873274i	$-0.661993\pi$
$109$	−10.1737	−0.974460	−0.487230	+	0.873274i	$0.661993\pi$
$110$	−6.46741	−0.616643
$111$	0.0539132	0.00511721
$112$	7.02254	0.663568
$113$	−2.59984	−0.244572	−0.122286	−	0.992495i	$-0.539023\pi$
$113$	−2.59984	−0.244572	−0.122286	+	0.992495i	$0.539023\pi$
$114$	0.0708472	0.00663545
$115$	1.58329	0.147642
$116$	23.6699	2.19770
$117$	0	0
$118$	−11.7655	−1.08310
$119$	15.4495	1.41626
$120$	−0.0248950	−0.00227259
$121$	−2.40150	−0.218318
$122$	−4.23670	−0.383573
$123$	0.0948578	0.00855304
$124$	28.0135	2.51569
$125$	−1.00000	−0.0894427
$126$	−30.4943	−2.71665
$127$	19.9719	1.77222	0.886109	−	0.463476i	$-0.153398\pi$
$127$	19.9719	1.77222	0.886109	+	0.463476i	$0.153398\pi$
$128$	13.8286	1.22228
$129$	0.00920925	0.000810829 0
$130$	0	0
$131$	−11.4400	−0.999514	−0.499757	−	0.866166i	$-0.666577\pi$
$131$	−11.4400	−0.999514	−0.499757	+	0.866166i	$0.666577\pi$
$132$	0.109671	0.00954562
$133$	−11.3390	−0.983217
$134$	−16.0960	−1.39048
$135$	−0.0783378	−0.00674224
$136$	6.39133	0.548052
$137$	−3.12520	−0.267004	−0.133502	−	0.991049i	$-0.542622\pi$
$137$	−3.12520	−0.267004	−0.133502	+	0.991049i	$0.542622\pi$
$138$	−0.0455944	−0.00388125
$139$	6.09526	0.516993	0.258497	−	0.966012i	$-0.416773\pi$
$139$	6.09526	0.516993	0.258497	+	0.966012i	$0.416773\pi$
$140$	13.2024	1.11580
$141$	0.111938	0.00942685
$142$	−14.7350	−1.23653
$143$	0	0
$144$	4.57075	0.380896
$145$	−8.26322	−0.686223
$146$	−27.4684	−2.27330
$147$	−0.185961	−0.0153378
$148$	−11.8280	−0.972253
$149$	14.6196	1.19769	0.598844	−	0.800866i	$-0.295627\pi$
$149$	14.6196	1.19769	0.598844	+	0.800866i	$0.295627\pi$
$150$	0.0287972	0.00235129
$151$	−12.2510	−0.996973	−0.498487	−	0.866897i	$-0.666111\pi$
$151$	−12.2510	−0.996973	−0.498487	+	0.866897i	$0.666111\pi$
$152$	−4.69085	−0.380478
$153$	10.0556	0.812948
$154$	−29.8081	−2.40200
$155$	−9.77959	−0.785515
$156$	0	0
$157$	7.13991	0.569827	0.284914	−	0.958553i	$-0.408035\pi$
$157$	7.13991	0.569827	0.284914	+	0.958553i	$0.408035\pi$
$158$	−2.17212	−0.172805
$159$	0.162736	0.0129058
$160$	−7.17392	−0.567148
$161$	7.29733	0.575110
$162$	−19.8467	−1.55930
$163$	−2.39002	−0.187201	−0.0936004	−	0.995610i	$-0.529838\pi$
$163$	−2.39002	−0.187201	−0.0936004	+	0.995610i	$0.529838\pi$
$164$	−20.8108	−1.62505
$165$	−0.0382863	−0.00298059
$166$	−17.3005	−1.34278
$167$	7.43841	0.575601	0.287801	−	0.957690i	$-0.407076\pi$
$167$	7.43841	0.575601	0.287801	+	0.957690i	$0.407076\pi$
$168$	−0.114740	−0.00885240
$169$	0	0
$170$	−7.39317	−0.567030
$171$	−7.38020	−0.564378
$172$	−2.02041	−0.154055
$173$	2.54119	0.193203	0.0966015	−	0.995323i	$-0.469203\pi$
$173$	2.54119	0.193203	0.0966015	+	0.995323i	$0.469203\pi$
$174$	0.237958	0.0180396
$175$	−4.60897	−0.348405
$176$	4.46788	0.336779
$177$	−0.0696502	−0.00523523
$178$	−0.910531	−0.0682472
$179$	−10.6913	−0.799105	−0.399553	−	0.916710i	$-0.630835\pi$
$179$	−10.6913	−0.799105	−0.399553	+	0.916710i	$0.630835\pi$
$180$	8.59299	0.640484
$181$	−11.7908	−0.876402	−0.438201	−	0.898877i	$-0.644384\pi$
$181$	−11.7908	−0.876402	−0.438201	+	0.898877i	$0.644384\pi$
$182$	0	0
$183$	−0.0250808	−0.00185403
$184$	3.01884	0.222552
$185$	4.12917	0.303583
$186$	0.281625	0.0206498
$187$	9.82931	0.718790
$188$	−24.5579	−1.79107
$189$	−0.361056	−0.0262630
$190$	5.42613	0.393653
$191$	13.4746	0.974990	0.487495	−	0.873126i	$-0.337911\pi$
$191$	13.4746	0.974990	0.487495	+	0.873126i	$0.337911\pi$
$192$	0.166801	0.0120378
$193$	−6.98737	−0.502962	−0.251481	−	0.967862i	$-0.580918\pi$
$193$	−6.98737	−0.502962	−0.251481	+	0.967862i	$0.580918\pi$
$194$	−7.46397	−0.535882
$195$	0	0
$196$	40.7978	2.91413
$197$	−1.43795	−0.102450	−0.0512248	−	0.998687i	$-0.516313\pi$
$197$	−1.43795	−0.102450	−0.0512248	+	0.998687i	$0.516313\pi$
$198$	−19.4011	−1.37878
$199$	−2.42852	−0.172153	−0.0860765	−	0.996289i	$-0.527433\pi$
$199$	−2.42852	−0.172153	−0.0860765	+	0.996289i	$0.527433\pi$
$200$	−1.90669	−0.134823
$201$	−0.0952866	−0.00672100
$202$	23.2537	1.63613
$203$	−38.0849	−2.67304
$204$	0.125369	0.00877761
$205$	7.26508	0.507415
$206$	−31.1318	−2.16906
$207$	4.74960	0.330120
$208$	0	0
$209$	−7.21411	−0.499011
$210$	0.132726	0.00915894
$211$	18.2640	1.25735	0.628674	−	0.777669i	$-0.283598\pi$
$211$	18.2640	1.25735	0.628674	+	0.777669i	$0.283598\pi$
$212$	−35.7026	−2.45207
$213$	−0.0872293	−0.00597685
$214$	8.75970	0.598801
$215$	0.705329	0.0481031
$216$	−0.149366	−0.0101631
$217$	−45.0738	−3.05981
$218$	22.4386	1.51973
$219$	−0.162610	−0.0109881
$220$	8.39961	0.566301
$221$	0	0
$222$	−0.118909	−0.00798063
$223$	−15.5841	−1.04359	−0.521793	−	0.853072i	$-0.674737\pi$
$223$	−15.5841	−1.04359	−0.521793	+	0.853072i	$0.674737\pi$
$224$	−33.0644	−2.20921
$225$	−2.99983	−0.199989
$226$	5.73409	0.381426
$227$	4.03380	0.267733	0.133866	−	0.990999i	$-0.457261\pi$
$227$	4.03380	0.267733	0.133866	+	0.990999i	$0.457261\pi$
$228$	−0.0920135	−0.00609374
$229$	−22.5147	−1.48781	−0.743906	−	0.668284i	$-0.767029\pi$
$229$	−22.5147	−1.48781	−0.743906	+	0.668284i	$0.767029\pi$
$230$	−3.49204	−0.230258
$231$	−0.176460	−0.0116102
$232$	−15.7554	−1.03439
$233$	−7.69759	−0.504286	−0.252143	−	0.967690i	$-0.581135\pi$
$233$	−7.69759	−0.504286	−0.252143	+	0.967690i	$0.581135\pi$
$234$	0	0
$235$	8.57322	0.559255
$236$	15.2805	0.994676
$237$	−0.0128587	−0.000835264 0
$238$	−34.0749	−2.20875
$239$	−18.7203	−1.21092	−0.605459	−	0.795877i	$-0.707010\pi$
$239$	−18.7203	−1.21092	−0.605459	+	0.795877i	$0.707010\pi$
$240$	−0.0198940	−0.00128415
$241$	22.6183	1.45697	0.728486	−	0.685061i	$-0.240225\pi$
$241$	22.6183	1.45697	0.728486	+	0.685061i	$0.240225\pi$
$242$	5.29665	0.340482
$243$	−0.352503	−0.0226131
$244$	5.50246	0.352259
$245$	−14.2426	−0.909926
$246$	−0.209214	−0.0133390
$247$	0	0
$248$	−18.6466	−1.18406
$249$	−0.102417	−0.00649042
$250$	2.20556	0.139492
$251$	1.78412	0.112612	0.0563062	−	0.998414i	$-0.482068\pi$
$251$	1.78412	0.112612	0.0563062	+	0.998414i	$0.482068\pi$
$252$	39.6048	2.49487
$253$	4.64271	0.291885
$254$	−44.0492	−2.76389
$255$	−0.0437667	−0.00274078
$256$	−4.94936	−0.309335
$257$	−4.59847	−0.286845	−0.143422	−	0.989662i	$-0.545811\pi$
$257$	−4.59847	−0.286845	−0.143422	+	0.989662i	$0.545811\pi$
$258$	−0.0203115	−0.00126454
$259$	19.0312	1.18254
$260$	0	0
$261$	−24.7883	−1.53435
$262$	25.2315	1.55881
$263$	18.8127	1.16004	0.580022	−	0.814601i	$-0.303044\pi$
$263$	18.8127	1.16004	0.580022	+	0.814601i	$0.303044\pi$
$264$	−0.0730001	−0.00449284
$265$	12.4639	0.765649
$266$	25.0089	1.53339
$267$	−0.00539024	−0.000329877 0
$268$	20.9049	1.27697
$269$	1.85412	0.113048	0.0565238	−	0.998401i	$-0.481998\pi$
$269$	1.85412	0.113048	0.0565238	+	0.998401i	$0.481998\pi$
$270$	0.172779	0.0105150
$271$	−0.149698	−0.00909352	−0.00454676	−	0.999990i	$-0.501447\pi$
$271$	−0.149698	−0.00909352	−0.00454676	+	0.999990i	$0.501447\pi$
$272$	5.10743	0.309683
$273$	0	0
$274$	6.89281	0.416410
$275$	−2.93232	−0.176826
$276$	0.0592161	0.00356439
$277$	22.0184	1.32296	0.661478	−	0.749965i	$-0.269929\pi$
$277$	22.0184	1.32296	0.661478	+	0.749965i	$0.269929\pi$
$278$	−13.4435	−0.806285
$279$	−29.3371	−1.75637
$280$	−8.78787	−0.525176
$281$	0.438029	0.0261306	0.0130653	−	0.999915i	$-0.495841\pi$
$281$	0.438029	0.0261306	0.0130653	+	0.999915i	$0.495841\pi$
$282$	−0.246885	−0.0147018
$283$	13.8290	0.822049	0.411025	−	0.911624i	$-0.365171\pi$
$283$	13.8290	0.822049	0.411025	+	0.911624i	$0.365171\pi$
$284$	19.1372	1.13558
$285$	0.0321221	0.00190275
$286$	0	0
$287$	33.4845	1.97653
$288$	−21.5205	−1.26811
$289$	−5.76370	−0.339041
$290$	18.2250	1.07021
$291$	−0.0441858	−0.00259022
$292$	35.6748	2.08771
$293$	−14.5656	−0.850929	−0.425464	−	0.904975i	$-0.639889\pi$
$293$	−14.5656	−0.850929	−0.425464	+	0.904975i	$0.639889\pi$
$294$	0.410148	0.0239203
$295$	−5.33445	−0.310584
$296$	7.87304	0.457611
$297$	−0.229711	−0.0133292
$298$	−32.2445	−1.86787
$299$	0	0
$300$	−0.0374007	−0.00215933
$301$	3.25084	0.187375
$302$	27.0203	1.55485
$303$	0.137659	0.00790833
$304$	−3.74854	−0.214994
$305$	−1.92092	−0.109992
$306$	−22.1782	−1.26785
$307$	9.05062	0.516546	0.258273	−	0.966072i	$-0.416847\pi$
$307$	9.05062	0.516546	0.258273	+	0.966072i	$0.416847\pi$
$308$	38.7135	2.20591
$309$	−0.184297	−0.0104843
$310$	21.5695	1.22506
$311$	7.28812	0.413272	0.206636	−	0.978418i	$-0.433748\pi$
$311$	7.28812	0.413272	0.206636	+	0.978418i	$0.433748\pi$
$312$	0	0
$313$	12.9725	0.733250	0.366625	−	0.930369i	$-0.380513\pi$
$313$	12.9725	0.733250	0.366625	+	0.930369i	$0.380513\pi$
$314$	−15.7475	−0.888683
$315$	−13.8261	−0.779014
$316$	2.82107	0.158697
$317$	−15.5048	−0.870838	−0.435419	−	0.900228i	$-0.643400\pi$
$317$	−15.5048	−0.870838	−0.435419	+	0.900228i	$0.643400\pi$
$318$	−0.358925	−0.0201275
$319$	−24.2304	−1.35664
$320$	12.7752	0.714154
$321$	0.0518565	0.00289434
$322$	−16.0947	−0.896922
$323$	−8.24676	−0.458862
$324$	25.7760	1.43200
$325$	0	0
$326$	5.27133	0.291952
$327$	0.132834	0.00734574
$328$	13.8523	0.764863
$329$	39.5137	2.17846
$330$	0.0844427	0.00464842
$331$	−26.0853	−1.43378	−0.716890	−	0.697186i	$-0.754435\pi$
$331$	−26.0853	−1.43378	−0.716890	+	0.697186i	$0.754435\pi$
$332$	22.4692	1.23316
$333$	12.3868	0.678793
$334$	−16.4058	−0.897688
$335$	−7.29793	−0.398728
$336$	−0.0916910	−0.00500215
$337$	−14.4785	−0.788697	−0.394348	−	0.918961i	$-0.629030\pi$
$337$	−14.4785	−0.788697	−0.394348	+	0.918961i	$0.629030\pi$
$338$	0	0
$339$	0.0339452	0.00184365
$340$	9.60195	0.520739
$341$	−28.6769	−1.55294
$342$	16.2775	0.880185
$343$	−33.3809	−1.80240
$344$	1.34484	0.0725091
$345$	−0.0206725	−0.00111297
$346$	−5.60474	−0.301313
$347$	6.81412	0.365801	0.182901	−	0.983131i	$-0.441451\pi$
$347$	6.81412	0.365801	0.182901	+	0.983131i	$0.441451\pi$
$348$	−0.309050	−0.0165668
$349$	33.7357	1.80583	0.902915	−	0.429818i	$-0.141422\pi$
$349$	33.7357	1.80583	0.902915	+	0.429818i	$0.141422\pi$
$350$	10.1654	0.543361
$351$	0	0
$352$	−21.0362	−1.12123
$353$	18.7215	0.996447	0.498223	−	0.867049i	$-0.333986\pi$
$353$	18.7215	0.996447	0.498223	+	0.867049i	$0.333986\pi$
$354$	0.153618	0.00816468
$355$	−6.68083	−0.354581
$356$	1.18256	0.0626756
$357$	−0.201719	−0.0106761
$358$	23.5803	1.24626
$359$	27.9262	1.47389	0.736945	−	0.675952i	$-0.236268\pi$
$359$	27.9262	1.47389	0.736945	+	0.675952i	$0.236268\pi$
$360$	−5.71974	−0.301457
$361$	−12.9474	−0.681441
$362$	26.0053	1.36681
$363$	0.0313556	0.00164574
$364$	0	0
$365$	−12.4541	−0.651880
$366$	0.0553172	0.00289148
$367$	2.50988	0.131015	0.0655074	−	0.997852i	$-0.479133\pi$
$367$	2.50988	0.131015	0.0655074	+	0.997852i	$0.479133\pi$
$368$	2.41241	0.125755
$369$	21.7940	1.13455
$370$	−9.10713	−0.473457
$371$	57.4456	2.98243
$372$	−0.365763	−0.0189640
$373$	22.1779	1.14833	0.574165	−	0.818739i	$-0.305327\pi$
$373$	22.1779	1.14833	0.574165	+	0.818739i	$0.305327\pi$
$374$	−21.6791	−1.12100
$375$	0.0130567	0.000674243 0
$376$	16.3465	0.843005
$377$	0	0
$378$	0.796331	0.0409589
$379$	1.42116	0.0730001	0.0365001	−	0.999334i	$-0.488379\pi$
$379$	1.42116	0.0730001	0.0365001	+	0.999334i	$0.488379\pi$
$380$	−7.04724	−0.361516
$381$	−0.260766	−0.0133595
$382$	−29.7191	−1.52056
$383$	−21.0733	−1.07679	−0.538397	−	0.842691i	$-0.680970\pi$
$383$	−21.0733	−1.07679	−0.538397	+	0.842691i	$0.680970\pi$
$384$	−0.180555	−0.00921390
$385$	−13.5150	−0.688787
$386$	15.4111	0.784402
$387$	2.11587	0.107556
$388$	9.69390	0.492133
$389$	13.4713	0.683021	0.341511	−	0.939878i	$-0.389061\pi$
$389$	13.4713	0.683021	0.341511	+	0.939878i	$0.389061\pi$
$390$	0	0
$391$	5.30728	0.268401
$392$	−27.1562	−1.37160
$393$	0.149368	0.00753461
$394$	3.17148	0.159777
$395$	−0.984840	−0.0495527
$396$	25.1974	1.26622
$397$	−27.1859	−1.36442	−0.682210	−	0.731156i	$-0.738981\pi$
$397$	−27.1859	−1.36442	−0.682210	+	0.731156i	$0.738981\pi$
$398$	5.35624	0.268484
$399$	0.148050	0.00741176
$400$	−1.52367	−0.0761834
$401$	3.27520	0.163556	0.0817779	−	0.996651i	$-0.473940\pi$
$401$	3.27520	0.163556	0.0817779	+	0.996651i	$0.473940\pi$
$402$	0.210160	0.0104818
$403$	0	0
$404$	−30.2010	−1.50256
$405$	−8.99847	−0.447137
$406$	83.9986	4.16878
$407$	12.1080	0.600173
$408$	−0.0834495	−0.00413137
$409$	19.1677	0.947782	0.473891	−	0.880583i	$-0.342849\pi$
$409$	19.1677	0.947782	0.473891	+	0.880583i	$0.342849\pi$
$410$	−16.0236	−0.791348
$411$	0.0408047	0.00201275
$412$	40.4327	1.99198
$413$	−24.5863	−1.20981
$414$	−10.4755	−0.514844
$415$	−7.84405	−0.385049
$416$	0	0
$417$	−0.0795837	−0.00389723
$418$	15.9112	0.778240
$419$	0.392463	0.0191731	0.00958653	−	0.999954i	$-0.496948\pi$
$419$	0.392463	0.0191731	0.00958653	+	0.999954i	$0.496948\pi$
$420$	−0.172379	−0.00841122
$421$	−35.1816	−1.71465	−0.857323	−	0.514779i	$-0.827874\pi$
$421$	−35.1816	−1.71465	−0.857323	+	0.514779i	$0.827874\pi$
$422$	−40.2824	−1.96092
$423$	25.7182	1.25046
$424$	23.7647	1.15412
$425$	−3.35206	−0.162599
$426$	0.192389	0.00932130
$427$	−8.85346	−0.428449
$428$	−11.3768	−0.549916
$429$	0	0
$430$	−1.55565	−0.0750199
$431$	−2.27454	−0.109561	−0.0547803	−	0.998498i	$-0.517446\pi$
$431$	−2.27454	−0.109561	−0.0547803	+	0.998498i	$0.517446\pi$
$432$	−0.119361	−0.00574275
$433$	11.8827	0.571048	0.285524	−	0.958372i	$-0.407832\pi$
$433$	11.8827	0.571048	0.285524	+	0.958372i	$0.407832\pi$
$434$	99.4130	4.77197
$435$	0.107890	0.00517293
$436$	−29.1424	−1.39567
$437$	−3.89522	−0.186334
$438$	0.358645	0.0171367
$439$	−35.7273	−1.70517	−0.852585	−	0.522588i	$-0.824967\pi$
$439$	−35.7273	−1.70517	−0.852585	+	0.522588i	$0.824967\pi$
$440$	−5.59102	−0.266541
$441$	−42.7254	−2.03454
$442$	0	0
$443$	4.98904	0.237036	0.118518	−	0.992952i	$-0.462186\pi$
$443$	4.98904	0.237036	0.118518	+	0.992952i	$0.462186\pi$
$444$	0.154434	0.00732911
$445$	−0.412834	−0.0195702
$446$	34.3716	1.62754
$447$	−0.190884	−0.00902849
$448$	58.8804	2.78184
$449$	23.4886	1.10850	0.554248	−	0.832352i	$-0.313006\pi$
$449$	23.4886	1.10850	0.554248	+	0.832352i	$0.313006\pi$
$450$	6.61630	0.311895
$451$	21.3035	1.00314
$452$	−7.44721	−0.350287
$453$	0.159957	0.00751545
$454$	−8.89679	−0.417547
$455$	0	0
$456$	0.0612468	0.00286815
$457$	−40.2510	−1.88286	−0.941430	−	0.337208i	$-0.890518\pi$
$457$	−40.2510	−1.88286	−0.941430	+	0.337208i	$0.890518\pi$
$458$	49.6575	2.32034
$459$	−0.262593	−0.0122568
$460$	4.53532	0.211460
$461$	31.2140	1.45378	0.726890	−	0.686754i	$-0.240965\pi$
$461$	31.2140	1.45378	0.726890	+	0.686754i	$0.240965\pi$
$462$	0.389194	0.0181069
$463$	17.7491	0.824872	0.412436	−	0.910987i	$-0.364678\pi$
$463$	17.7491	0.824872	0.412436	+	0.910987i	$0.364678\pi$
$464$	−12.5904	−0.584495
$465$	0.127689	0.00592142
$466$	16.9775	0.786467
$467$	26.6645	1.23389	0.616944	−	0.787007i	$-0.288370\pi$
$467$	26.6645	1.23389	0.616944	+	0.787007i	$0.288370\pi$
$468$	0	0
$469$	−33.6359	−1.55316
$470$	−18.9087	−0.872196
$471$	−0.0932234	−0.00429551
$472$	−10.1711	−0.468165
$473$	2.06825	0.0950983
$474$	0.0283607	0.00130265
$475$	2.46021	0.112882
$476$	44.2551	2.02843
$477$	37.3895	1.71195
$478$	41.2888	1.88851
$479$	20.4630	0.934980	0.467490	−	0.883998i	$-0.345158\pi$
$479$	20.4630	0.934980	0.467490	+	0.883998i	$0.345158\pi$
$480$	0.0936675	0.00427531
$481$	0	0
$482$	−49.8860	−2.27224
$483$	−0.0952788	−0.00433534
$484$	−6.87908	−0.312686
$485$	−3.38416	−0.153667
$486$	0.777467	0.0352666
$487$	8.11789	0.367857	0.183928	−	0.982940i	$-0.441119\pi$
$487$	8.11789	0.367857	0.183928	+	0.982940i	$0.441119\pi$
$488$	−3.66260	−0.165798
$489$	0.0312057	0.00141117
$490$	31.4129	1.41909
$491$	3.26996	0.147571	0.0737857	−	0.997274i	$-0.476492\pi$
$491$	3.26996	0.147571	0.0737857	+	0.997274i	$0.476492\pi$
$492$	0.271719	0.0122501
$493$	−27.6988	−1.24749
$494$	0	0
$495$	−8.79646	−0.395371
$496$	−14.9008	−0.669068
$497$	−30.7917	−1.38120
$498$	0.225887	0.0101222
$499$	8.89340	0.398123	0.199062	−	0.979987i	$-0.436211\pi$
$499$	8.89340	0.398123	0.199062	+	0.979987i	$0.436211\pi$
$500$	−2.86449	−0.128104
$501$	−0.0971208	−0.00433904
$502$	−3.93498	−0.175627
$503$	−23.7974	−1.06107	−0.530537	−	0.847662i	$-0.678010\pi$
$503$	−23.7974	−1.06107	−0.530537	+	0.847662i	$0.678010\pi$
$504$	−26.3621	−1.17426
$505$	10.5432	0.469168
$506$	−10.2398	−0.455213
$507$	0	0
$508$	57.2093	2.53825
$509$	−1.26770	−0.0561898	−0.0280949	−	0.999605i	$-0.508944\pi$
$509$	−1.26770	−0.0561898	−0.0280949	+	0.999605i	$0.508944\pi$
$510$	0.0965301	0.00427442
$511$	−57.4008	−2.53926
$512$	−16.7410	−0.739855
$513$	0.192727	0.00850911
$514$	10.1422	0.447354
$515$	−14.1152	−0.621988
$516$	0.0263798	0.00116131
$517$	25.1394	1.10563
$518$	−41.9745	−1.84425
$519$	−0.0331794	−0.00145642
$520$	0	0
$521$	14.3916	0.630506	0.315253	−	0.949008i	$-0.397911\pi$
$521$	14.3916	0.630506	0.315253	+	0.949008i	$0.397911\pi$
$522$	54.6720	2.39293
$523$	19.0185	0.831622	0.415811	−	0.909451i	$-0.363498\pi$
$523$	19.0185	0.831622	0.415811	+	0.909451i	$0.363498\pi$
$524$	−32.7697	−1.43155
$525$	0.0601778	0.00262637
$526$	−41.4926	−1.80916
$527$	−32.7818	−1.42800
$528$	−0.0583357	−0.00253873
$529$	−20.4932	−0.891008
$530$	−27.4898	−1.19408
$531$	−16.0024	−0.694447
$532$	−32.4805	−1.40821
$533$	0	0
$534$	0.0118885	0.000514465 0
$535$	3.97165	0.171709
$536$	−13.9149	−0.601031
$537$	0.139593	0.00602387
$538$	−4.08937	−0.176305
$539$	−41.7638	−1.79890
$540$	−0.224398	−0.00965655
$541$	−13.8969	−0.597475	−0.298738	−	0.954335i	$-0.596566\pi$
$541$	−13.8969	−0.597475	−0.298738	+	0.954335i	$0.596566\pi$
$542$	0.330168	0.0141819
$543$	0.153948	0.00660656
$544$	−24.0474	−1.03102
$545$	10.1737	0.435792
$546$	0	0
$547$	37.0840	1.58560	0.792798	−	0.609484i	$-0.208623\pi$
$547$	37.0840	1.58560	0.792798	+	0.609484i	$0.208623\pi$
$548$	−8.95211	−0.382415
$549$	−5.76243	−0.245935
$550$	6.46741	0.275771
$551$	20.3292	0.866054
$552$	−0.0394160	−0.00167766
$553$	−4.53910	−0.193022
$554$	−48.5628	−2.06324
$555$	−0.0539132	−0.00228849
$556$	17.4598	0.740461
$557$	−31.8065	−1.34768	−0.673842	−	0.738875i	$-0.735357\pi$
$557$	−31.8065	−1.34768	−0.673842	+	0.738875i	$0.735357\pi$
$558$	64.7047	2.73917
$559$	0	0
$560$	−7.02254	−0.296757
$561$	−0.128338	−0.00541843
$562$	−0.966100	−0.0407525
$563$	26.3595	1.11092	0.555461	−	0.831543i	$-0.312542\pi$
$563$	26.3595	1.11092	0.555461	+	0.831543i	$0.312542\pi$
$564$	0.320645	0.0135016
$565$	2.59984	0.109376
$566$	−30.5007	−1.28204
$567$	−41.4736	−1.74173
$568$	−12.7383	−0.534485
$569$	17.7846	0.745569	0.372785	−	0.927918i	$-0.378403\pi$
$569$	17.7846	0.745569	0.372785	+	0.927918i	$0.378403\pi$
$570$	−0.0708472	−0.00296746
$571$	19.6399	0.821903	0.410952	−	0.911657i	$-0.365196\pi$
$571$	19.6399	0.821903	0.410952	+	0.911657i	$0.365196\pi$
$572$	0	0
$573$	−0.175934	−0.00734974
$574$	−73.8522	−3.08253
$575$	−1.58329	−0.0660277
$576$	38.3233	1.59681
$577$	−18.1454	−0.755403	−0.377701	−	0.925927i	$-0.623285\pi$
$577$	−18.1454	−0.755403	−0.377701	+	0.925927i	$0.623285\pi$
$578$	12.7122	0.528757
$579$	0.0912317	0.00379146
$580$	−23.6699	−0.982840
$581$	−36.1530	−1.49988
$582$	0.0974545	0.00403962
$583$	36.5480	1.51366
$584$	−23.7462	−0.982624
$585$	0	0
$586$	32.1252	1.32708
$587$	0.423843	0.0174939	0.00874694	−	0.999962i	$-0.497216\pi$
$587$	0.423843	0.0174939	0.00874694	+	0.999962i	$0.497216\pi$
$588$	−0.532683	−0.0219675
$589$	24.0598	0.991367
$590$	11.7655	0.484376
$591$	0.0187748	0.000772293 0
$592$	6.29149	0.258578
$593$	47.5064	1.95085	0.975426	−	0.220327i	$-0.0707125\pi$
$593$	47.5064	1.95085	0.975426	+	0.220327i	$0.0707125\pi$
$594$	0.506642	0.0207878
$595$	−15.4495	−0.633369
$596$	41.8778	1.71538
$597$	0.0317083	0.00129774
$598$	0	0
$599$	25.7871	1.05363	0.526816	−	0.849979i	$-0.323386\pi$
$599$	25.7871	1.05363	0.526816	+	0.849979i	$0.323386\pi$
$600$	0.0248950	0.00101633
$601$	−12.6776	−0.517130	−0.258565	−	0.965994i	$-0.583250\pi$
$601$	−12.6776	−0.517130	−0.258565	+	0.965994i	$0.583250\pi$
$602$	−7.16992	−0.292224
$603$	−21.8925	−0.891533
$604$	−35.0929	−1.42791
$605$	2.40150	0.0976349
$606$	−0.303616	−0.0123336
$607$	−4.40448	−0.178772	−0.0893862	−	0.995997i	$-0.528491\pi$
$607$	−4.40448	−0.178772	−0.0893862	+	0.995997i	$0.528491\pi$
$608$	17.6493	0.715775
$609$	0.497262	0.0201501
$610$	4.23670	0.171539
$611$	0	0
$612$	28.8042	1.16434
$613$	33.3599	1.34739	0.673696	−	0.739008i	$-0.264706\pi$
$613$	33.3599	1.34739	0.673696	+	0.739008i	$0.264706\pi$
$614$	−19.9617	−0.805588
$615$	−0.0948578	−0.00382503
$616$	−25.7688	−1.03826
$617$	32.5535	1.31055	0.655277	−	0.755389i	$-0.272552\pi$
$617$	32.5535	1.31055	0.655277	+	0.755389i	$0.272552\pi$
$618$	0.406477	0.0163509
$619$	35.5813	1.43013	0.715067	−	0.699056i	$-0.246396\pi$
$619$	35.5813	1.43013	0.715067	+	0.699056i	$0.246396\pi$
$620$	−28.0135	−1.12505
$621$	−0.124031	−0.00497721
$622$	−16.0744	−0.644524
$623$	−1.90274	−0.0762317
$624$	0	0
$625$	1.00000	0.0400000
$626$	−28.6116	−1.14355
$627$	0.0941922	0.00376168
$628$	20.4522	0.816133
$629$	13.8412	0.551886
$630$	30.4943	1.21492
$631$	−25.4117	−1.01162	−0.505812	−	0.862644i	$-0.668807\pi$
$631$	−25.4117	−1.01162	−0.505812	+	0.862644i	$0.668807\pi$
$632$	−1.87778	−0.0746942
$633$	−0.238467	−0.00947822
$634$	34.1968	1.35813
$635$	−19.9719	−0.792560
$636$	0.466157	0.0184843
$637$	0	0
$638$	53.4416	2.11577
$639$	−20.0413	−0.792823
$640$	−13.8286	−0.546622
$641$	−10.1479	−0.400817	−0.200408	−	0.979712i	$-0.564227\pi$
$641$	−10.1479	−0.400817	−0.200408	+	0.979712i	$0.564227\pi$
$642$	−0.114372	−0.00451392
$643$	−1.85439	−0.0731298	−0.0365649	−	0.999331i	$-0.511642\pi$
$643$	−1.85439	−0.0731298	−0.0365649	+	0.999331i	$0.511642\pi$
$644$	20.9032	0.823700
$645$	−0.00920925	−0.000362614 0
$646$	18.1887	0.715626
$647$	−29.6034	−1.16383	−0.581914	−	0.813250i	$-0.697696\pi$
$647$	−29.6034	−1.16383	−0.581914	+	0.813250i	$0.697696\pi$
$648$	−17.1573	−0.674001
$649$	−15.6423	−0.614015
$650$	0	0
$651$	0.588514	0.0230657
$652$	−6.84619	−0.268118
$653$	6.78882	0.265667	0.132834	−	0.991138i	$-0.457592\pi$
$653$	6.78882	0.265667	0.132834	+	0.991138i	$0.457592\pi$
$654$	−0.292973	−0.0114562
$655$	11.4400	0.446996
$656$	11.0696	0.432195
$657$	−37.3603	−1.45756
$658$	−87.1498	−3.39745
$659$	7.94959	0.309672	0.154836	−	0.987940i	$-0.450515\pi$
$659$	7.94959	0.309672	0.154836	+	0.987940i	$0.450515\pi$
$660$	−0.109671	−0.00426893
$661$	−40.3832	−1.57072	−0.785361	−	0.619038i	$-0.787523\pi$
$661$	−40.3832	−1.57072	−0.785361	+	0.619038i	$0.787523\pi$
$662$	57.5328	2.23607
$663$	0	0
$664$	−14.9562	−0.580412
$665$	11.3390	0.439708
$666$	−27.3198	−1.05862
$667$	−13.0831	−0.506579
$668$	21.3073	0.824402
$669$	0.203476	0.00786684
$670$	16.0960	0.621843
$671$	−5.63275	−0.217450
$672$	0.431710	0.0166536
$673$	9.63920	0.371564	0.185782	−	0.982591i	$-0.440518\pi$
$673$	9.63920	0.371564	0.185782	+	0.982591i	$0.440518\pi$
$674$	31.9333	1.23002
$675$	0.0783378	0.00301522
$676$	0	0
$677$	51.1565	1.96610	0.983051	−	0.183330i	$-0.0586878\pi$
$677$	51.1565	1.96610	0.983051	+	0.183330i	$0.0586878\pi$
$678$	−0.0748681	−0.00287529
$679$	−15.5975	−0.598577
$680$	−6.39133	−0.245097
$681$	−0.0526680	−0.00201824
$682$	63.2485	2.42191
$683$	8.34502	0.319313	0.159657	−	0.987173i	$-0.448961\pi$
$683$	8.34502	0.319313	0.159657	+	0.987173i	$0.448961\pi$
$684$	−21.1405	−0.808328
$685$	3.12520	0.119408
$686$	73.6236	2.81096
$687$	0.293967	0.0112155
$688$	1.07469	0.0409721
$689$	0	0
$690$	0.0455944	0.00173575
$691$	36.1243	1.37423	0.687117	−	0.726547i	$-0.258876\pi$
$691$	36.1243	1.37423	0.687117	+	0.726547i	$0.258876\pi$
$692$	7.27921	0.276714
$693$	−40.5426	−1.54009
$694$	−15.0289	−0.570491
$695$	−6.09526	−0.231206
$696$	0.205713	0.00779753
$697$	24.3530	0.922435
$698$	−74.4061	−2.81631
$699$	0.100505	0.00380144
$700$	−13.2024	−0.499002
$701$	−45.9823	−1.73673	−0.868364	−	0.495928i	$-0.834828\pi$
$701$	−45.9823	−1.73673	−0.868364	+	0.495928i	$0.834828\pi$
$702$	0	0
$703$	−10.1586	−0.383139
$704$	37.4609	1.41186
$705$	−0.111938	−0.00421582
$706$	−41.2915	−1.55402
$707$	48.5934	1.82754
$708$	−0.199512	−0.00749813
$709$	−38.0949	−1.43069	−0.715343	−	0.698774i	$-0.753729\pi$
$709$	−38.0949	−1.43069	−0.715343	+	0.698774i	$0.753729\pi$
$710$	14.7350	0.552993
$711$	−2.95435	−0.110797
$712$	−0.787147	−0.0294996
$713$	−15.4839	−0.579877
$714$	0.444904	0.0166501
$715$	0	0
$716$	−30.6251	−1.14452
$717$	0.244425	0.00912822
$718$	−61.5930	−2.29863
$719$	−27.8672	−1.03927	−0.519635	−	0.854388i	$-0.673932\pi$
$719$	−27.8672	−1.03927	−0.519635	+	0.854388i	$0.673932\pi$
$720$	−4.57075	−0.170342
$721$	−65.0563	−2.42282
$722$	28.5562	1.06275
$723$	−0.295319	−0.0109830
$724$	−33.7746	−1.25522
$725$	8.26322	0.306888
$726$	−0.0691566	−0.00256664
$727$	−13.7750	−0.510885	−0.255443	−	0.966824i	$-0.582221\pi$
$727$	−13.7750	−0.510885	−0.255443	+	0.966824i	$0.582221\pi$
$728$	0	0
$729$	−26.9908	−0.999659
$730$	27.4684	1.01665
$731$	2.36431	0.0874470
$732$	−0.0718438	−0.00265542
$733$	6.58392	0.243183	0.121591	−	0.992580i	$-0.461200\pi$
$733$	6.58392	0.243183	0.121591	+	0.992580i	$0.461200\pi$
$734$	−5.53570	−0.204326
$735$	0.185961	0.00685927
$736$	−11.3584	−0.418676
$737$	−21.3999	−0.788274
$738$	−48.0680	−1.76941
$739$	−5.47501	−0.201401	−0.100701	−	0.994917i	$-0.532108\pi$
$739$	−5.47501	−0.201401	−0.100701	+	0.994917i	$0.532108\pi$
$740$	11.8280	0.434805
$741$	0	0
$742$	−126.700	−4.65129
$743$	26.6714	0.978479	0.489239	−	0.872150i	$-0.337274\pi$
$743$	26.6714	0.978479	0.489239	+	0.872150i	$0.337274\pi$
$744$	0.243463	0.00892578
$745$	−14.6196	−0.535622
$746$	−48.9148	−1.79090
$747$	−23.5308	−0.860948
$748$	28.1560	1.02948
$749$	18.3052	0.668857
$750$	−0.0287972	−0.00105153
$751$	21.8067	0.795740	0.397870	−	0.917442i	$-0.369750\pi$
$751$	21.8067	0.795740	0.397870	+	0.917442i	$0.369750\pi$
$752$	13.0627	0.476349
$753$	−0.0232946	−0.000848903 0
$754$	0	0
$755$	12.2510	0.445860
$756$	−1.03424	−0.0376151
$757$	−8.33422	−0.302912	−0.151456	−	0.988464i	$-0.548396\pi$
$757$	−8.33422	−0.302912	−0.151456	+	0.988464i	$0.548396\pi$
$758$	−3.13446	−0.113849
$759$	−0.0606183	−0.00220031
$760$	4.69085	0.170155
$761$	7.33096	0.265747	0.132874	−	0.991133i	$-0.457580\pi$
$761$	7.33096	0.265747	0.132874	+	0.991133i	$0.457580\pi$
$762$	0.575135	0.0208350
$763$	46.8901	1.69753
$764$	38.5980	1.39643
$765$	−10.0556	−0.363561
$766$	46.4783	1.67933
$767$	0	0
$768$	0.0646221	0.00233185
$769$	22.3346	0.805407	0.402703	−	0.915331i	$-0.368071\pi$
$769$	22.3346	0.805407	0.402703	+	0.915331i	$0.368071\pi$
$770$	29.8081	1.07421
$771$	0.0600407	0.00216231
$772$	−20.0153	−0.720365
$773$	27.1591	0.976844	0.488422	−	0.872608i	$-0.337573\pi$
$773$	27.1591	0.976844	0.488422	+	0.872608i	$0.337573\pi$
$774$	−4.66667	−0.167740
$775$	9.77959	0.351293
$776$	−6.45254	−0.231633
$777$	−0.248484	−0.00891432
$778$	−29.7117	−1.06522
$779$	−17.8736	−0.640388
$780$	0	0
$781$	−19.5903	−0.700997
$782$	−11.7055	−0.418589
$783$	0.647322	0.0231334
$784$	−21.7010	−0.775036
$785$	−7.13991	−0.254834
$786$	−0.329440	−0.0117507
$787$	−11.5806	−0.412805	−0.206403	−	0.978467i	$-0.566176\pi$
$787$	−11.5806	−0.412805	−0.206403	+	0.978467i	$0.566176\pi$
$788$	−4.11899	−0.146733
$789$	−0.245632	−0.00874472
$790$	2.17212	0.0772807
$791$	11.9826	0.426051
$792$	−16.7721	−0.595971
$793$	0	0
$794$	59.9601	2.12790
$795$	−0.162736	−0.00577167
$796$	−6.95647	−0.246566
$797$	−47.7004	−1.68963	−0.844817	−	0.535055i	$-0.820291\pi$
$797$	−47.7004	−1.68963	−0.844817	+	0.535055i	$0.820291\pi$
$798$	−0.326532	−0.0115591
$799$	28.7379	1.01668
$800$	7.17392	0.253636
$801$	−1.23843	−0.0437579
$802$	−7.22365	−0.255076
$803$	−36.5195	−1.28875
$804$	−0.272948	−0.00962612
$805$	−7.29733	−0.257197
$806$	0	0
$807$	−0.0242086	−0.000852184 0
$808$	20.1027	0.707209
$809$	36.3910	1.27944	0.639720	−	0.768608i	$-0.279050\pi$
$809$	36.3910	1.27944	0.639720	+	0.768608i	$0.279050\pi$
$810$	19.8467	0.697340
$811$	36.8088	1.29253	0.646266	−	0.763112i	$-0.276330\pi$
$811$	36.8088	1.29253	0.646266	+	0.763112i	$0.276330\pi$
$812$	−109.094	−3.82845
$813$	0.00195456	6.85494e−5 0
$814$	−26.7050	−0.936010
$815$	2.39002	0.0837188
$816$	−0.0666860	−0.00233448
$817$	−1.73526	−0.0607089
$818$	−42.2755	−1.47813
$819$	0	0
$820$	20.8108	0.726744
$821$	−33.6668	−1.17498	−0.587491	−	0.809231i	$-0.699884\pi$
$821$	−33.6668	−1.17498	−0.587491	+	0.809231i	$0.699884\pi$
$822$	−0.0899971	−0.00313901
$823$	31.8809	1.11130	0.555649	−	0.831417i	$-0.312470\pi$
$823$	31.8809	1.11130	0.555649	+	0.831417i	$0.312470\pi$
$824$	−26.9132	−0.937566
$825$	0.0382863	0.00133296
$826$	54.2266	1.88678
$827$	18.4625	0.642003	0.321002	−	0.947079i	$-0.395981\pi$
$827$	18.4625	0.642003	0.321002	+	0.947079i	$0.395981\pi$
$828$	13.6052	0.472813
$829$	−34.9663	−1.21443	−0.607214	−	0.794538i	$-0.707713\pi$
$829$	−34.9663	−1.21443	−0.607214	+	0.794538i	$0.707713\pi$
$830$	17.3005	0.600510
$831$	−0.287487	−0.00997280
$832$	0	0
$833$	−47.7420	−1.65416
$834$	0.175527	0.00607799
$835$	−7.43841	−0.257417
$836$	−20.6648	−0.714706
$837$	0.766111	0.0264807
$838$	−0.865600	−0.0299017
$839$	28.7234	0.991641	0.495820	−	0.868425i	$-0.334867\pi$
$839$	28.7234	0.991641	0.495820	+	0.868425i	$0.334867\pi$
$840$	0.114740	0.00395892
$841$	39.2808	1.35451
$842$	77.5951	2.67410
$843$	−0.00571920	−0.000196980 0
$844$	52.3172	1.80083
$845$	0	0
$846$	−56.7230	−1.95018
$847$	11.0684	0.380316
$848$	18.9908	0.652147
$849$	−0.180561	−0.00619683
$850$	7.39317	0.253584
$851$	6.53767	0.224108
$852$	−0.249868	−0.00856032
$853$	13.7453	0.470629	0.235315	−	0.971919i	$-0.424388\pi$
$853$	13.7453	0.470629	0.235315	+	0.971919i	$0.424388\pi$
$854$	19.5268	0.668195
$855$	7.38020	0.252398
$856$	7.57269	0.258829
$857$	−12.6495	−0.432099	−0.216049	−	0.976382i	$-0.569317\pi$
$857$	−12.6495	−0.432099	−0.216049	+	0.976382i	$0.569317\pi$
$858$	0	0
$859$	−11.0929	−0.378485	−0.189243	−	0.981930i	$-0.560603\pi$
$859$	−11.0929	−0.378485	−0.189243	+	0.981930i	$0.560603\pi$
$860$	2.02041	0.0688954
$861$	−0.437196	−0.0148996
$862$	5.01663	0.170867
$863$	33.0295	1.12434	0.562169	−	0.827022i	$-0.309967\pi$
$863$	33.0295	1.12434	0.562169	+	0.827022i	$0.309967\pi$
$864$	0.561989	0.0191192
$865$	−2.54119	−0.0864030
$866$	−26.2081	−0.890587
$867$	0.0752546	0.00255578
$868$	−129.114	−4.38240
$869$	−2.88787	−0.0979641
$870$	−0.237958	−0.00806753
$871$	0	0
$872$	19.3980	0.656899
$873$	−10.1519	−0.343590
$874$	8.59114	0.290600
$875$	4.60897	0.155812
$876$	−0.465794	−0.0157377
$877$	−15.4397	−0.521362	−0.260681	−	0.965425i	$-0.583947\pi$
$877$	−15.4397	−0.521362	−0.260681	+	0.965425i	$0.583947\pi$
$878$	78.7987	2.65933
$879$	0.190178	0.00641453
$880$	−4.46788	−0.150612
$881$	−16.6260	−0.560143	−0.280072	−	0.959979i	$-0.590358\pi$
$881$	−16.6260	−0.560143	−0.280072	+	0.959979i	$0.590358\pi$
$882$	94.2333	3.17300
$883$	57.2845	1.92778	0.963888	−	0.266309i	$-0.0858041\pi$
$883$	57.2845	1.92778	0.963888	+	0.266309i	$0.0858041\pi$
$884$	0	0
$885$	0.0696502	0.00234126
$886$	−11.0036	−0.369674
$887$	18.4657	0.620018	0.310009	−	0.950734i	$-0.399668\pi$
$887$	18.4657	0.620018	0.310009	+	0.950734i	$0.399668\pi$
$888$	−0.102796	−0.00344960
$889$	−92.0498	−3.08725
$890$	0.910531	0.0305211
$891$	−26.3864	−0.883977
$892$	−44.6405	−1.49467
$893$	−21.0919	−0.705813
$894$	0.421005	0.0140805
$895$	10.6913	0.357371
$896$	−63.7354	−2.12925
$897$	0	0
$898$	−51.8055	−1.72877
$899$	80.8109	2.69519
$900$	−8.59299	−0.286433
$901$	41.7796	1.39188
$902$	−46.9862	−1.56447
$903$	−0.0424451	−0.00141249
$904$	4.95708	0.164870
$905$	11.7908	0.391939
$906$	−0.352795	−0.0117208
$907$	44.9489	1.49250	0.746251	−	0.665664i	$-0.231852\pi$
$907$	44.9489	1.49250	0.746251	+	0.665664i	$0.231852\pi$
$908$	11.5548	0.383459
$909$	31.6279	1.04903
$910$	0	0
$911$	−44.3728	−1.47014	−0.735069	−	0.677992i	$-0.762850\pi$
$911$	−44.3728	−1.47014	−0.735069	+	0.677992i	$0.762850\pi$
$912$	0.0489434	0.00162068
$913$	−23.0013	−0.761231
$914$	88.7759	2.93645
$915$	0.0250808	0.000829146 0
$916$	−64.4931	−2.13091
$917$	52.7264	1.74118
$918$	0.579164	0.0191153
$919$	25.1680	0.830217	0.415108	−	0.909772i	$-0.363744\pi$
$919$	25.1680	0.830217	0.415108	+	0.909772i	$0.363744\pi$
$920$	−3.01884	−0.0995282
$921$	−0.118171	−0.00389386
$922$	−68.8443	−2.26726
$923$	0	0
$924$	−0.505469	−0.0166287
$925$	−4.12917	−0.135766
$926$	−39.1468	−1.28644
$927$	−42.3430	−1.39073
$928$	59.2797	1.94595
$929$	37.5601	1.23231	0.616153	−	0.787626i	$-0.288690\pi$
$929$	37.5601	1.23231	0.616153	+	0.787626i	$0.288690\pi$
$930$	−0.281625	−0.00923485
$931$	35.0397	1.14838
$932$	−22.0497	−0.722262
$933$	−0.0951586	−0.00311535
$934$	−58.8102	−1.92433
$935$	−9.82931	−0.321453
$936$	0	0
$937$	−17.1212	−0.559326	−0.279663	−	0.960098i	$-0.590223\pi$
$937$	−17.1212	−0.559326	−0.279663	+	0.960098i	$0.590223\pi$
$938$	74.1860	2.42226
$939$	−0.169378	−0.00552743
$940$	24.5579	0.800991
$941$	−0.208568	−0.00679913	−0.00339956	−	0.999994i	$-0.501082\pi$
$941$	−0.208568	−0.00679913	−0.00339956	+	0.999994i	$0.501082\pi$
$942$	0.205610	0.00669913
$943$	11.5027	0.374580
$944$	−8.12794	−0.264542
$945$	0.361056	0.0117452
$946$	−4.56165	−0.148312
$947$	0.414059	0.0134551	0.00672755	−	0.999977i	$-0.497859\pi$
$947$	0.414059	0.0134551	0.00672755	+	0.999977i	$0.497859\pi$
$948$	−0.0368337	−0.00119630
$949$	0	0
$950$	−5.42613	−0.176047
$951$	0.202441	0.00656461
$952$	−29.4575	−0.954722
$953$	−33.8593	−1.09681	−0.548404	−	0.836213i	$-0.684765\pi$
$953$	−33.8593	−1.09681	−0.548404	+	0.836213i	$0.684765\pi$
$954$	−82.4647	−2.66989
$955$	−13.4746	−0.436029
$956$	−53.6243	−1.73433
$957$	0.316368	0.0102267
$958$	−45.1324	−1.45816
$959$	14.4039	0.465128
$960$	−0.166801	−0.00538348
$961$	64.6403	2.08517
$962$	0	0
$963$	11.9143	0.383932
$964$	64.7899	2.08674
$965$	6.98737	0.224931
$966$	0.210143	0.00676124
$967$	13.2125	0.424885	0.212442	−	0.977174i	$-0.431858\pi$
$967$	13.2125	0.424885	0.212442	+	0.977174i	$0.431858\pi$
$968$	4.57892	0.147172
$969$	0.107675	0.00345902
$970$	7.46397	0.239654
$971$	−17.6019	−0.564872	−0.282436	−	0.959286i	$-0.591142\pi$
$971$	−17.6019	−0.564872	−0.282436	+	0.959286i	$0.591142\pi$
$972$	−1.00974	−0.0323875
$973$	−28.0929	−0.900616
$974$	−17.9045	−0.573697
$975$	0	0
$976$	−2.92685	−0.0936861
$977$	−11.7812	−0.376913	−0.188457	−	0.982082i	$-0.560348\pi$
$977$	−11.7812	−0.376913	−0.188457	+	0.982082i	$0.560348\pi$
$978$	−0.0688260	−0.00220081
$979$	−1.21056	−0.0386897
$980$	−40.7978	−1.30324
$981$	30.5192	0.974404
$982$	−7.21210	−0.230147
$983$	−45.4985	−1.45118	−0.725589	−	0.688128i	$-0.758433\pi$
$983$	−45.4985	−1.45118	−0.725589	+	0.688128i	$0.758433\pi$
$984$	−0.180864	−0.00576574
$985$	1.43795	0.0458169
$986$	61.0914	1.94555
$987$	−0.515917	−0.0164218
$988$	0	0
$989$	1.11674	0.0355103
$990$	19.4011	0.616608
$991$	33.2685	1.05681	0.528405	−	0.848993i	$-0.322790\pi$
$991$	33.2685	1.05681	0.528405	+	0.848993i	$0.322790\pi$
$992$	70.1580	2.22752
$993$	0.340588	0.0108082
$994$	67.9130	2.15407
$995$	2.42852	0.0769892
$996$	−0.293373	−0.00929589
$997$	−39.7870	−1.26007	−0.630033	−	0.776568i	$-0.716959\pi$
$997$	−39.7870	−1.26007	−0.630033	+	0.776568i	$0.716959\pi$
$998$	−19.6149	−0.620899
$999$	−0.323470	−0.0102341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.n.1.3	✓	9
3.2	odd	2		7605.2.a.cs.1.7		9
5.4	even	2		4225.2.a.bt.1.7		9
13.2	odd	12		845.2.m.j.316.5		36
13.3	even	3		845.2.e.p.191.7		18
13.4	even	6		845.2.e.o.146.3		18
13.5	odd	4		845.2.c.h.506.14		18
13.6	odd	12		845.2.m.j.361.14		36
13.7	odd	12		845.2.m.j.361.5		36
13.8	odd	4		845.2.c.h.506.5		18
13.9	even	3		845.2.e.p.146.7		18
13.10	even	6		845.2.e.o.191.3		18
13.11	odd	12		845.2.m.j.316.14		36
13.12	even	2		845.2.a.o.1.7	yes	9
39.38	odd	2		7605.2.a.cp.1.3		9
65.64	even	2		4225.2.a.bs.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.n.1.3	✓	9	1.1	even	1	trivial
845.2.a.o.1.7	yes	9	13.12	even	2
845.2.c.h.506.5		18	13.8	odd	4
845.2.c.h.506.14		18	13.5	odd	4
845.2.e.o.146.3		18	13.4	even	6
845.2.e.o.191.3		18	13.10	even	6
845.2.e.p.146.7		18	13.9	even	3
845.2.e.p.191.7		18	13.3	even	3
845.2.m.j.316.5		36	13.2	odd	12
845.2.m.j.316.14		36	13.11	odd	12
845.2.m.j.361.5		36	13.7	odd	12
845.2.m.j.361.14		36	13.6	odd	12
4225.2.a.bs.1.3		9	65.64	even	2
4225.2.a.bt.1.7		9	5.4	even	2
7605.2.a.cp.1.3		9	39.38	odd	2
7605.2.a.cs.1.7		9	3.2	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20556 q^{2} -0.0130567 q^{3} +2.86449 q^{4} -1.00000 q^{5} +0.0287972 q^{6} -4.60897 q^{7} -1.90669 q^{8} -2.99983 q^{9} +2.20556 q^{10} -2.93232 q^{11} -0.0374007 q^{12} +10.1654 q^{14} +0.0130567 q^{15} -1.52367 q^{16} -3.35206 q^{17} +6.61630 q^{18} +2.46021 q^{19} -2.86449 q^{20} +0.0601778 q^{21} +6.46741 q^{22} -1.58329 q^{23} +0.0248950 q^{24} +1.00000 q^{25} +0.0783378 q^{27} -13.2024 q^{28} +8.26322 q^{29} -0.0287972 q^{30} +9.77959 q^{31} +7.17392 q^{32} +0.0382863 q^{33} +7.39317 q^{34} +4.60897 q^{35} -8.59299 q^{36} -4.12917 q^{37} -5.42613 q^{38} +1.90669 q^{40} -7.26508 q^{41} -0.132726 q^{42} -0.705329 q^{43} -8.39961 q^{44} +2.99983 q^{45} +3.49204 q^{46} -8.57322 q^{47} +0.0198940 q^{48} +14.2426 q^{49} -2.20556 q^{50} +0.0437667 q^{51} -12.4639 q^{53} -0.172779 q^{54} +2.93232 q^{55} +8.78787 q^{56} -0.0321221 q^{57} -18.2250 q^{58} +5.33445 q^{59} +0.0374007 q^{60} +1.92092 q^{61} -21.5695 q^{62} +13.8261 q^{63} -12.7752 q^{64} -0.0844427 q^{66} +7.29793 q^{67} -9.60195 q^{68} +0.0206725 q^{69} -10.1654 q^{70} +6.68083 q^{71} +5.71974 q^{72} +12.4541 q^{73} +9.10713 q^{74} -0.0130567 q^{75} +7.04724 q^{76} +13.5150 q^{77} +0.984840 q^{79} +1.52367 q^{80} +8.99847 q^{81} +16.0236 q^{82} +7.84405 q^{83} +0.172379 q^{84} +3.35206 q^{85} +1.55565 q^{86} -0.107890 q^{87} +5.59102 q^{88} +0.412834 q^{89} -6.61630 q^{90} -4.53532 q^{92} -0.127689 q^{93} +18.9087 q^{94} -2.46021 q^{95} -0.0936675 q^{96} +3.38416 q^{97} -31.4129 q^{98} +8.79646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9} + 3 q^{10} + 9 q^{11} + 12 q^{12} - 2 q^{14} - 7 q^{15} + 37 q^{16} - q^{17} + 10 q^{18} + 4 q^{19} - 17 q^{20} + q^{21}+ \cdots + q^{99}+O(q^{100}) \) 9 * q - 3 * q^2 + 7 * q^3 + 17 * q^4 - 9 * q^5 + 2 * q^6 - 7 * q^7 - 12 * q^8 + 16 * q^9 + 3 * q^10 + 9 * q^11 + 12 * q^12 - 2 * q^14 - 7 * q^15 + 37 * q^16 - q^17 + 10 * q^18 + 4 * q^19 - 17 * q^20 + q^21 + 12 * q^22 + 14 * q^23 + 35 * q^24 + 9 * q^25 + 22 * q^27 - 18 * q^28 + 12 * q^29 - 2 * q^30 + 7 * q^31 - 22 * q^32 + 8 * q^33 + 30 * q^34 + 7 * q^35 + 3 * q^36 + 5 * q^37 - 47 * q^38 + 12 * q^40 + 10 * q^41 - 11 * q^42 + 39 * q^43 + 25 * q^44 - 16 * q^45 - 6 * q^46 - 36 * q^47 - 3 * q^48 + 16 * q^49 - 3 * q^50 + 43 * q^51 - 8 * q^53 + 2 * q^54 - 9 * q^55 - 29 * q^56 + 32 * q^57 - 21 * q^58 + 21 * q^59 - 12 * q^60 - 3 * q^61 - 10 * q^62 - 35 * q^63 + 34 * q^64 - 49 * q^66 - q^67 - 20 * q^68 - 13 * q^69 + 2 * q^70 + q^71 + 3 * q^72 - 15 * q^74 + 7 * q^75 + 5 * q^76 - 4 * q^77 + 39 * q^79 - 37 * q^80 + 29 * q^81 - 4 * q^82 - 7 * q^83 - 12 * q^84 + q^85 + 24 * q^86 + 16 * q^87 + 42 * q^88 + 19 * q^89 - 10 * q^90 - 27 * q^92 - 31 * q^93 + 16 * q^94 - 4 * q^95 + 7 * q^96 + 34 * q^97 - 48 * q^98 + q^99

Introduction

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