LMFDB - Embedded newform 2415.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,2,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2415.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:	$19.2838720881$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2508628.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - 2x^{2} + 23x + 8 $ x^5 - 10x^3 - 2x^2 + 23*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.356193$ of defining polynomial
Character	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.356193 q^{2} -1.00000 q^{3} -1.87313 q^{4} -1.00000 q^{5} -0.356193 q^{6} -1.00000 q^{7} -1.37958 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.356193 q^{2} -1.00000 q^{3} -1.87313 q^{4} -1.00000 q^{5} -0.356193 q^{6} -1.00000 q^{7} -1.37958 q^{8} +1.00000 q^{9} -0.356193 q^{10} +1.09197 q^{11} +1.87313 q^{12} +3.51693 q^{13} -0.356193 q^{14} +1.00000 q^{15} +3.25486 q^{16} -2.00000 q^{17} +0.356193 q^{18} -1.23147 q^{19} +1.87313 q^{20} +1.00000 q^{21} +0.388950 q^{22} -1.00000 q^{23} +1.37958 q^{24} +1.00000 q^{25} +1.25271 q^{26} -1.00000 q^{27} +1.87313 q^{28} +0.264228 q^{29} +0.356193 q^{30} +8.90915 q^{31} +3.91852 q^{32} -1.09197 q^{33} -0.712386 q^{34} +1.00000 q^{35} -1.87313 q^{36} -4.49355 q^{37} -0.438641 q^{38} -3.51693 q^{39} +1.37958 q^{40} -1.21669 q^{41} +0.356193 q^{42} +3.39221 q^{43} -2.04539 q^{44} -1.00000 q^{45} -0.356193 q^{46} -6.60890 q^{47} -3.25486 q^{48} +1.00000 q^{49} +0.356193 q^{50} +2.00000 q^{51} -6.58766 q^{52} +4.83822 q^{53} -0.356193 q^{54} -1.09197 q^{55} +1.37958 q^{56} +1.23147 q^{57} +0.0941159 q^{58} +4.56447 q^{59} -1.87313 q^{60} -12.9777 q^{61} +3.17337 q^{62} -1.00000 q^{63} -5.11397 q^{64} -3.51693 q^{65} -0.388950 q^{66} -9.21780 q^{67} +3.74625 q^{68} +1.00000 q^{69} +0.356193 q^{70} -1.48633 q^{71} -1.37958 q^{72} +6.90915 q^{73} -1.60057 q^{74} -1.00000 q^{75} +2.30670 q^{76} -1.09197 q^{77} -1.25271 q^{78} -0.677477 q^{79} -3.25486 q^{80} +1.00000 q^{81} -0.433375 q^{82} -8.08981 q^{83} -1.87313 q^{84} +2.00000 q^{85} +1.20828 q^{86} -0.264228 q^{87} -1.50645 q^{88} -11.2188 q^{89} -0.356193 q^{90} -3.51693 q^{91} +1.87313 q^{92} -8.90915 q^{93} -2.35404 q^{94} +1.23147 q^{95} -3.91852 q^{96} -7.48633 q^{97} +0.356193 q^{98} +1.09197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9	$ 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9} + q^{11} - 10 q^{12} + 5 q^{15} + 8 q^{16} - 10 q^{17} + 3 q^{19} - 10 q^{20} + 5 q^{21} + 12 q^{22} - 5 q^{23} + 6 q^{24} + 5 q^{25} - 14 q^{26} - 5 q^{27} - 10 q^{28} + 4 q^{29} + 2 q^{31} - 12 q^{32} - q^{33} + 5 q^{35} + 10 q^{36} - 4 q^{37} - 24 q^{38} + 6 q^{40} - 9 q^{41} - 8 q^{43} + 2 q^{44} - 5 q^{45} - 11 q^{47} - 8 q^{48} + 5 q^{49} + 10 q^{51} - 22 q^{52} - 19 q^{53} - q^{55} + 6 q^{56} - 3 q^{57} + 8 q^{58} + 5 q^{59} + 10 q^{60} - 17 q^{61} - 24 q^{62} - 5 q^{63} - 8 q^{64} - 12 q^{66} - 2 q^{67} - 20 q^{68} + 5 q^{69} + 10 q^{71} - 6 q^{72} - 8 q^{73} - 34 q^{74} - 5 q^{75} - 8 q^{76} - q^{77} + 14 q^{78} + 24 q^{79} - 8 q^{80} + 5 q^{81} - 8 q^{82} - 24 q^{83} + 10 q^{84} + 10 q^{85} - 10 q^{86} - 4 q^{87} - 26 q^{88} - 4 q^{89} - 10 q^{92} - 2 q^{93} + 2 q^{94} - 3 q^{95} + 12 q^{96} - 20 q^{97} + q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9 + q^11 - 10 * q^12 + 5 * q^15 + 8 * q^16 - 10 * q^17 + 3 * q^19 - 10 * q^20 + 5 * q^21 + 12 * q^22 - 5 * q^23 + 6 * q^24 + 5 * q^25 - 14 * q^26 - 5 * q^27 - 10 * q^28 + 4 * q^29 + 2 * q^31 - 12 * q^32 - q^33 + 5 * q^35 + 10 * q^36 - 4 * q^37 - 24 * q^38 + 6 * q^40 - 9 * q^41 - 8 * q^43 + 2 * q^44 - 5 * q^45 - 11 * q^47 - 8 * q^48 + 5 * q^49 + 10 * q^51 - 22 * q^52 - 19 * q^53 - q^55 + 6 * q^56 - 3 * q^57 + 8 * q^58 + 5 * q^59 + 10 * q^60 - 17 * q^61 - 24 * q^62 - 5 * q^63 - 8 * q^64 - 12 * q^66 - 2 * q^67 - 20 * q^68 + 5 * q^69 + 10 * q^71 - 6 * q^72 - 8 * q^73 - 34 * q^74 - 5 * q^75 - 8 * q^76 - q^77 + 14 * q^78 + 24 * q^79 - 8 * q^80 + 5 * q^81 - 8 * q^82 - 24 * q^83 + 10 * q^84 + 10 * q^85 - 10 * q^86 - 4 * q^87 - 26 * q^88 - 4 * q^89 - 10 * q^92 - 2 * q^93 + 2 * q^94 - 3 * q^95 + 12 * q^96 - 20 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.356193	0.251866	0.125933	−	0.992039i	$-0.459808\pi$
$2$	0.356193	0.251866	0.125933	+	0.992039i	$0.459808\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.87313	−0.936563
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.356193	−0.145415
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.37958	−0.487755
$9$	1.00000	0.333333
$10$	−0.356193	−0.112638
$11$	1.09197	0.329240	0.164620	−	0.986357i	$-0.447360\pi$
$11$	1.09197	0.329240	0.164620	+	0.986357i	$0.447360\pi$
$12$	1.87313	0.540725
$13$	3.51693	0.975422	0.487711	−	0.873005i	$-0.337832\pi$
$13$	3.51693	0.975422	0.487711	+	0.873005i	$0.337832\pi$
$14$	−0.356193	−0.0951965
$15$	1.00000	0.258199
$16$	3.25486	0.813714
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0.356193	0.0839554
$19$	−1.23147	−0.282519	−0.141259	−	0.989973i	$-0.545115\pi$
$19$	−1.23147	−0.282519	−0.141259	+	0.989973i	$0.545115\pi$
$20$	1.87313	0.418844
$21$	1.00000	0.218218
$22$	0.388950	0.0829244
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.37958	0.281606
$25$	1.00000	0.200000
$26$	1.25271	0.245676
$27$	−1.00000	−0.192450
$28$	1.87313	0.353988
$29$	0.264228	0.0490658	0.0245329	−	0.999699i	$-0.492190\pi$
$29$	0.264228	0.0490658	0.0245329	+	0.999699i	$0.492190\pi$
$30$	0.356193	0.0650316
$31$	8.90915	1.60013	0.800065	−	0.599913i	$-0.204798\pi$
$31$	8.90915	1.60013	0.800065	+	0.599913i	$0.204798\pi$
$32$	3.91852	0.692702
$33$	−1.09197	−0.190087
$34$	−0.712386	−0.122173
$35$	1.00000	0.169031
$36$	−1.87313	−0.312188
$37$	−4.49355	−0.738735	−0.369367	−	0.929283i	$-0.620426\pi$
$37$	−4.49355	−0.738735	−0.369367	+	0.929283i	$0.620426\pi$
$38$	−0.438641	−0.0711569
$39$	−3.51693	−0.563160
$40$	1.37958	0.218131
$41$	−1.21669	−0.190015	−0.0950073	−	0.995477i	$-0.530287\pi$
$41$	−1.21669	−0.190015	−0.0950073	+	0.995477i	$0.530287\pi$
$42$	0.356193	0.0549617
$43$	3.39221	0.517307	0.258654	−	0.965970i	$-0.416721\pi$
$43$	3.39221	0.517307	0.258654	+	0.965970i	$0.416721\pi$
$44$	−2.04539	−0.308354
$45$	−1.00000	−0.149071
$46$	−0.356193	−0.0525178
$47$	−6.60890	−0.964007	−0.482004	−	0.876169i	$-0.660091\pi$
$47$	−6.60890	−0.964007	−0.482004	+	0.876169i	$0.660091\pi$
$48$	−3.25486	−0.469798
$49$	1.00000	0.142857
$50$	0.356193	0.0503733
$51$	2.00000	0.280056
$52$	−6.58766	−0.913544
$53$	4.83822	0.664580	0.332290	−	0.943177i	$-0.392179\pi$
$53$	4.83822	0.664580	0.332290	+	0.943177i	$0.392179\pi$
$54$	−0.356193	−0.0484717
$55$	−1.09197	−0.147241
$56$	1.37958	0.184354
$57$	1.23147	0.163112
$58$	0.0941159	0.0123580
$59$	4.56447	0.594244	0.297122	−	0.954840i	$-0.403973\pi$
$59$	4.56447	0.594244	0.297122	+	0.954840i	$0.403973\pi$
$60$	−1.87313	−0.241820
$61$	−12.9777	−1.66163	−0.830814	−	0.556551i	$-0.812124\pi$
$61$	−12.9777	−1.66163	−0.830814	+	0.556551i	$0.812124\pi$
$62$	3.17337	0.403019
$63$	−1.00000	−0.125988
$64$	−5.11397	−0.639246
$65$	−3.51693	−0.436222
$66$	−0.388950	−0.0478764
$67$	−9.21780	−1.12613	−0.563067	−	0.826411i	$-0.690379\pi$
$67$	−9.21780	−1.12613	−0.563067	+	0.826411i	$0.690379\pi$
$68$	3.74625	0.454300
$69$	1.00000	0.120386
$70$	0.356193	0.0425732
$71$	−1.48633	−0.176395	−0.0881973	−	0.996103i	$-0.528111\pi$
$71$	−1.48633	−0.176395	−0.0881973	+	0.996103i	$0.528111\pi$
$72$	−1.37958	−0.162585
$73$	6.90915	0.808654	0.404327	−	0.914614i	$-0.367506\pi$
$73$	6.90915	0.808654	0.404327	+	0.914614i	$0.367506\pi$
$74$	−1.60057	−0.186062
$75$	−1.00000	−0.115470
$76$	2.30670	0.264597
$77$	−1.09197	−0.124441
$78$	−1.25271	−0.141841
$79$	−0.677477	−0.0762222	−0.0381111	−	0.999274i	$-0.512134\pi$
$79$	−0.677477	−0.0762222	−0.0381111	+	0.999274i	$0.512134\pi$
$80$	−3.25486	−0.363904
$81$	1.00000	0.111111
$82$	−0.433375	−0.0478583
$83$	−8.08981	−0.887972	−0.443986	−	0.896034i	$-0.646436\pi$
$83$	−8.08981	−0.887972	−0.443986	+	0.896034i	$0.646436\pi$
$84$	−1.87313	−0.204375
$85$	2.00000	0.216930
$86$	1.20828	0.130292
$87$	−0.264228	−0.0283282
$88$	−1.50645	−0.160588
$89$	−11.2188	−1.18919	−0.594593	−	0.804027i	$-0.702687\pi$
$89$	−11.2188	−1.18919	−0.594593	+	0.804027i	$0.702687\pi$
$90$	−0.356193	−0.0375460
$91$	−3.51693	−0.368675
$92$	1.87313	0.195287
$93$	−8.90915	−0.923835
$94$	−2.35404	−0.242801
$95$	1.23147	0.126346
$96$	−3.91852	−0.399932
$97$	−7.48633	−0.760121	−0.380061	−	0.924962i	$-0.624097\pi$
$97$	−7.48633	−0.760121	−0.380061	+	0.924962i	$0.624097\pi$
$98$	0.356193	0.0359809
$99$	1.09197	0.109747
$100$	−1.87313	−0.187313
$101$	−8.52957	−0.848724	−0.424362	−	0.905493i	$-0.639502\pi$
$101$	−8.52957	−0.848724	−0.424362	+	0.905493i	$0.639502\pi$
$102$	0.712386	0.0705367
$103$	−8.26534	−0.814408	−0.407204	−	0.913337i	$-0.633496\pi$
$103$	−8.26534	−0.814408	−0.407204	+	0.913337i	$0.633496\pi$
$104$	−4.85189	−0.475767
$105$	−1.00000	−0.0975900
$106$	1.72334	0.167385
$107$	−2.88250	−0.278662	−0.139331	−	0.990246i	$-0.544495\pi$
$107$	−2.88250	−0.278662	−0.139331	+	0.990246i	$0.544495\pi$
$108$	1.87313	0.180242
$109$	1.07073	0.102557	0.0512786	−	0.998684i	$-0.483670\pi$
$109$	1.07073	0.102557	0.0512786	+	0.998684i	$0.483670\pi$
$110$	−0.388950	−0.0370849
$111$	4.49355	0.426509
$112$	−3.25486	−0.307555
$113$	−6.50833	−0.612252	−0.306126	−	0.951991i	$-0.599033\pi$
$113$	−6.50833	−0.612252	−0.306126	+	0.951991i	$0.599033\pi$
$114$	0.438641	0.0410825
$115$	1.00000	0.0932505
$116$	−0.494932	−0.0459533
$117$	3.51693	0.325141
$118$	1.62583	0.149670
$119$	2.00000	0.183340
$120$	−1.37958	−0.125938
$121$	−9.80761	−0.891601
$122$	−4.62257	−0.418508
$123$	1.21669	0.109705
$124$	−16.6880	−1.49862
$125$	−1.00000	−0.0894427
$126$	−0.356193	−0.0317322
$127$	−12.5253	−1.11144	−0.555719	−	0.831370i	$-0.687557\pi$
$127$	−12.5253	−1.11144	−0.555719	+	0.831370i	$0.687557\pi$
$128$	−9.65859	−0.853707
$129$	−3.39221	−0.298668
$130$	−1.25271	−0.109870
$131$	−14.7579	−1.28941	−0.644703	−	0.764433i	$-0.723019\pi$
$131$	−14.7579	−1.28941	−0.644703	+	0.764433i	$0.723019\pi$
$132$	2.04539	0.178028
$133$	1.23147	0.106782
$134$	−3.28331	−0.283635
$135$	1.00000	0.0860663
$136$	2.75916	0.236596
$137$	1.95572	0.167089	0.0835443	−	0.996504i	$-0.473376\pi$
$137$	1.95572	0.167089	0.0835443	+	0.996504i	$0.473376\pi$
$138$	0.356193	0.0303211
$139$	14.5834	1.23694	0.618472	−	0.785807i	$-0.287752\pi$
$139$	14.5834	1.23694	0.618472	+	0.785807i	$0.287752\pi$
$140$	−1.87313	−0.158308
$141$	6.60890	0.556570
$142$	−0.529419	−0.0444279
$143$	3.84037	0.321148
$144$	3.25486	0.271238
$145$	−0.264228	−0.0219429
$146$	2.46099	0.203673
$147$	−1.00000	−0.0824786
$148$	8.41698	0.691872
$149$	−7.81288	−0.640056	−0.320028	−	0.947408i	$-0.603692\pi$
$149$	−7.81288	−0.640056	−0.320028	+	0.947408i	$0.603692\pi$
$150$	−0.356193	−0.0290830
$151$	8.96294	0.729394	0.364697	−	0.931126i	$-0.381173\pi$
$151$	8.96294	0.729394	0.364697	+	0.931126i	$0.381173\pi$
$152$	1.69891	0.137800
$153$	−2.00000	−0.161690
$154$	−0.388950	−0.0313425
$155$	−8.90915	−0.715600
$156$	6.58766	0.527435
$157$	−9.98633	−0.796996	−0.398498	−	0.917169i	$-0.630468\pi$
$157$	−9.98633	−0.796996	−0.398498	+	0.917169i	$0.630468\pi$
$158$	−0.241313	−0.0191978
$159$	−4.83822	−0.383696
$160$	−3.91852	−0.309786
$161$	1.00000	0.0788110
$162$	0.356193	0.0279851
$163$	8.27360	0.648038	0.324019	−	0.946051i	$-0.394966\pi$
$163$	8.27360	0.648038	0.324019	+	0.946051i	$0.394966\pi$
$164$	2.27901	0.177961
$165$	1.09197	0.0850094
$166$	−2.88153	−0.223650
$167$	−10.4575	−0.809228	−0.404614	−	0.914488i	$-0.632594\pi$
$167$	−10.4575	−0.809228	−0.404614	+	0.914488i	$0.632594\pi$
$168$	−1.37958	−0.106437
$169$	−0.631175	−0.0485520
$170$	0.712386	0.0546375
$171$	−1.23147	−0.0941729
$172$	−6.35404	−0.484491
$173$	5.17337	0.393324	0.196662	−	0.980471i	$-0.436990\pi$
$173$	5.17337	0.393324	0.196662	+	0.980471i	$0.436990\pi$
$174$	−0.0941159	−0.00713491
$175$	−1.00000	−0.0755929
$176$	3.55419	0.267907
$177$	−4.56447	−0.343087
$178$	−3.99604	−0.299516
$179$	18.6531	1.39419	0.697097	−	0.716977i	$-0.254475\pi$
$179$	18.6531	1.39419	0.697097	+	0.716977i	$0.254475\pi$
$180$	1.87313	0.139615
$181$	2.73765	0.203488	0.101744	−	0.994811i	$-0.467558\pi$
$181$	2.73765	0.203488	0.101744	+	0.994811i	$0.467558\pi$
$182$	−1.25271	−0.0928568
$183$	12.9777	0.959341
$184$	1.37958	0.101704
$185$	4.49355	0.330372
$186$	−3.17337	−0.232683
$187$	−2.18393	−0.159705
$188$	12.3793	0.902854
$189$	1.00000	0.0727393
$190$	0.438641	0.0318223
$191$	19.0106	1.37556	0.687778	−	0.725921i	$-0.258586\pi$
$191$	19.0106	1.37556	0.687778	+	0.725921i	$0.258586\pi$
$192$	5.11397	0.369069
$193$	0.825389	0.0594128	0.0297064	−	0.999559i	$-0.490543\pi$
$193$	0.825389	0.0594128	0.0297064	+	0.999559i	$0.490543\pi$
$194$	−2.66658	−0.191449
$195$	3.51693	0.251853
$196$	−1.87313	−0.133795
$197$	4.95357	0.352927	0.176464	−	0.984307i	$-0.443534\pi$
$197$	4.95357	0.352927	0.176464	+	0.984307i	$0.443534\pi$
$198$	0.388950	0.0276415
$199$	−7.36902	−0.522376	−0.261188	−	0.965288i	$-0.584114\pi$
$199$	−7.36902	−0.522376	−0.261188	+	0.965288i	$0.584114\pi$
$200$	−1.37958	−0.0975510
$201$	9.21780	0.650173
$202$	−3.03817	−0.213765
$203$	−0.264228	−0.0185451
$204$	−3.74625	−0.262290
$205$	1.21669	0.0849772
$206$	−2.94405	−0.205122
$207$	−1.00000	−0.0695048
$208$	11.4471	0.793715
$209$	−1.34472	−0.0930164
$210$	−0.356193	−0.0245796
$211$	15.8172	1.08890	0.544450	−	0.838793i	$-0.316738\pi$
$211$	15.8172	1.08890	0.544450	+	0.838793i	$0.316738\pi$
$212$	−9.06260	−0.622422
$213$	1.48633	0.101841
$214$	−1.02672	−0.0701854
$215$	−3.39221	−0.231347
$216$	1.37958	0.0938685
$217$	−8.90915	−0.604792
$218$	0.381386	0.0258307
$219$	−6.90915	−0.466877
$220$	2.04539	0.137900
$221$	−7.03387	−0.473149
$222$	1.60057	0.107423
$223$	3.25036	0.217660	0.108830	−	0.994060i	$-0.465290\pi$
$223$	3.25036	0.217660	0.108830	+	0.994060i	$0.465290\pi$
$224$	−3.91852	−0.261817
$225$	1.00000	0.0666667
$226$	−2.31822	−0.154206
$227$	−22.0677	−1.46469	−0.732344	−	0.680935i	$-0.761573\pi$
$227$	−22.0677	−1.46469	−0.732344	+	0.680935i	$0.761573\pi$
$228$	−2.30670	−0.152765
$229$	−3.18316	−0.210349	−0.105175	−	0.994454i	$-0.533540\pi$
$229$	−3.18316	−0.210349	−0.105175	+	0.994454i	$0.533540\pi$
$230$	0.356193	0.0234867
$231$	1.09197	0.0718460
$232$	−0.364523	−0.0239321
$233$	−18.3320	−1.20097	−0.600483	−	0.799637i	$-0.705025\pi$
$233$	−18.3320	−1.20097	−0.600483	+	0.799637i	$0.705025\pi$
$234$	1.25271	0.0818920
$235$	6.60890	0.431117
$236$	−8.54984	−0.556547
$237$	0.677477	0.0440069
$238$	0.712386	0.0461771
$239$	−21.5498	−1.39394	−0.696969	−	0.717101i	$-0.745469\pi$
$239$	−21.5498	−1.39394	−0.696969	+	0.717101i	$0.745469\pi$
$240$	3.25486	0.210100
$241$	2.71350	0.174792	0.0873958	−	0.996174i	$-0.472146\pi$
$241$	2.71350	0.174792	0.0873958	+	0.996174i	$0.472146\pi$
$242$	−3.49340	−0.224564
$243$	−1.00000	−0.0641500
$244$	24.3089	1.55622
$245$	−1.00000	−0.0638877
$246$	0.433375	0.0276310
$247$	−4.33100	−0.275575
$248$	−12.2909	−0.780471
$249$	8.08981	0.512671
$250$	−0.356193	−0.0225276
$251$	−0.528455	−0.0333558	−0.0166779	−	0.999861i	$-0.505309\pi$
$251$	−0.528455	−0.0333558	−0.0166779	+	0.999861i	$0.505309\pi$
$252$	1.87313	0.117996
$253$	−1.09197	−0.0686513
$254$	−4.46141	−0.279934
$255$	−2.00000	−0.125245
$256$	6.78762	0.424226
$257$	0.611248	0.0381286	0.0190643	−	0.999818i	$-0.493931\pi$
$257$	0.611248	0.0381286	0.0190643	+	0.999818i	$0.493931\pi$
$258$	−1.20828	−0.0752243
$259$	4.49355	0.279215
$260$	6.58766	0.408550
$261$	0.264228	0.0163553
$262$	−5.25666	−0.324758
$263$	24.9839	1.54057	0.770287	−	0.637698i	$-0.220113\pi$
$263$	24.9839	1.54057	0.770287	+	0.637698i	$0.220113\pi$
$264$	1.50645	0.0927158
$265$	−4.83822	−0.297209
$266$	0.438641	0.0268948
$267$	11.2188	0.686577
$268$	17.2661	1.05470
$269$	−10.3935	−0.633704	−0.316852	−	0.948475i	$-0.602626\pi$
$269$	−10.3935	−0.633704	−0.316852	+	0.948475i	$0.602626\pi$
$270$	0.356193	0.0216772
$271$	29.5655	1.79598	0.897989	−	0.440018i	$-0.145028\pi$
$271$	29.5655	1.79598	0.897989	+	0.440018i	$0.145028\pi$
$272$	−6.50971	−0.394709
$273$	3.51693	0.212855
$274$	0.696614	0.0420840
$275$	1.09197	0.0658480
$276$	−1.87313	−0.112749
$277$	16.8086	1.00993	0.504965	−	0.863140i	$-0.331505\pi$
$277$	16.8086	1.00993	0.504965	+	0.863140i	$0.331505\pi$
$278$	5.19449	0.311545
$279$	8.90915	0.533377
$280$	−1.37958	−0.0824457
$281$	−15.9440	−0.951139	−0.475570	−	0.879678i	$-0.657758\pi$
$281$	−15.9440	−0.951139	−0.475570	+	0.879678i	$0.657758\pi$
$282$	2.35404	0.140181
$283$	−7.16524	−0.425929	−0.212965	−	0.977060i	$-0.568312\pi$
$283$	−7.16524	−0.425929	−0.212965	+	0.977060i	$0.568312\pi$
$284$	2.78408	0.165205
$285$	−1.23147	−0.0729460
$286$	1.36791	0.0808863
$287$	1.21669	0.0718188
$288$	3.91852	0.230901
$289$	−13.0000	−0.764706
$290$	−0.0941159	−0.00552668
$291$	7.48633	0.438856
$292$	−12.9417	−0.757356
$293$	−25.5645	−1.49350	−0.746748	−	0.665107i	$-0.768386\pi$
$293$	−25.5645	−1.49350	−0.746748	+	0.665107i	$0.768386\pi$
$294$	−0.356193	−0.0207736
$295$	−4.56447	−0.265754
$296$	6.19921	0.360322
$297$	−1.09197	−0.0633622
$298$	−2.78289	−0.161209
$299$	−3.51693	−0.203390
$300$	1.87313	0.108145
$301$	−3.39221	−0.195524
$302$	3.19253	0.183710
$303$	8.52957	0.490011
$304$	−4.00826	−0.229889
$305$	12.9777	0.743102
$306$	−0.712386	−0.0407244
$307$	9.50633	0.542555	0.271277	−	0.962501i	$-0.412554\pi$
$307$	9.50633	0.542555	0.271277	+	0.962501i	$0.412554\pi$
$308$	2.04539	0.116547
$309$	8.26534	0.470199
$310$	−3.17337	−0.180235
$311$	−9.25382	−0.524736	−0.262368	−	0.964968i	$-0.584503\pi$
$311$	−9.25382	−0.524736	−0.262368	+	0.964968i	$0.584503\pi$
$312$	4.85189	0.274684
$313$	5.06850	0.286489	0.143244	−	0.989687i	$-0.454247\pi$
$313$	5.06850	0.286489	0.143244	+	0.989687i	$0.454247\pi$
$314$	−3.55706	−0.200736
$315$	1.00000	0.0563436
$316$	1.26900	0.0713869
$317$	−12.8164	−0.719842	−0.359921	−	0.932983i	$-0.617196\pi$
$317$	−12.8164	−0.719842	−0.359921	+	0.932983i	$0.617196\pi$
$318$	−1.72334	−0.0966400
$319$	0.288527	0.0161544
$320$	5.11397	0.285879
$321$	2.88250	0.160885
$322$	0.356193	0.0198498
$323$	2.46294	0.137042
$324$	−1.87313	−0.104063
$325$	3.51693	0.195084
$326$	2.94700	0.163219
$327$	−1.07073	−0.0592115
$328$	1.67852	0.0926806
$329$	6.60890	0.364360
$330$	0.388950	0.0214110
$331$	16.2146	0.891235	0.445618	−	0.895223i	$-0.352984\pi$
$331$	16.2146	0.891235	0.445618	+	0.895223i	$0.352984\pi$
$332$	15.1532	0.831643
$333$	−4.49355	−0.246245
$334$	−3.72490	−0.203817
$335$	9.21780	0.503622
$336$	3.25486	0.177567
$337$	2.43142	0.132448	0.0662240	−	0.997805i	$-0.478905\pi$
$337$	2.43142	0.132448	0.0662240	+	0.997805i	$0.478905\pi$
$338$	−0.224820	−0.0122286
$339$	6.50833	0.353484
$340$	−3.74625	−0.203169
$341$	9.72848	0.526827
$342$	−0.438641	−0.0237190
$343$	−1.00000	−0.0539949
$344$	−4.67983	−0.252319
$345$	−1.00000	−0.0538382
$346$	1.84272	0.0990651
$347$	−7.35926	−0.395066	−0.197533	−	0.980296i	$-0.563293\pi$
$347$	−7.35926	−0.395066	−0.197533	+	0.980296i	$0.563293\pi$
$348$	0.494932	0.0265311
$349$	28.0370	1.50079	0.750393	−	0.660992i	$-0.229864\pi$
$349$	28.0370	1.50079	0.750393	+	0.660992i	$0.229864\pi$
$350$	−0.356193	−0.0190393
$351$	−3.51693	−0.187720
$352$	4.27888	0.228065
$353$	−8.28435	−0.440932	−0.220466	−	0.975395i	$-0.570758\pi$
$353$	−8.28435	−0.440932	−0.220466	+	0.975395i	$0.570758\pi$
$354$	−1.62583	−0.0864120
$355$	1.48633	0.0788861
$356$	21.0142	1.11375
$357$	−2.00000	−0.105851
$358$	6.64408	0.351151
$359$	−10.0845	−0.532238	−0.266119	−	0.963940i	$-0.585741\pi$
$359$	−10.0845	−0.532238	−0.266119	+	0.963940i	$0.585741\pi$
$360$	1.37958	0.0727102
$361$	−17.4835	−0.920183
$362$	0.975131	0.0512517
$363$	9.80761	0.514766
$364$	6.58766	0.345287
$365$	−6.90915	−0.361641
$366$	4.62257	0.241626
$367$	−15.5677	−0.812629	−0.406315	−	0.913733i	$-0.633186\pi$
$367$	−15.5677	−0.812629	−0.406315	+	0.913733i	$0.633186\pi$
$368$	−3.25486	−0.169671
$369$	−1.21669	−0.0633382
$370$	1.60057	0.0832096
$371$	−4.83822	−0.251188
$372$	16.6880	0.865230
$373$	−20.2980	−1.05099	−0.525496	−	0.850796i	$-0.676120\pi$
$373$	−20.2980	−1.05099	−0.525496	+	0.850796i	$0.676120\pi$
$374$	−0.777900	−0.0402243
$375$	1.00000	0.0516398
$376$	9.11750	0.470199
$377$	0.929271	0.0478599
$378$	0.356193	0.0183206
$379$	−11.0033	−0.565200	−0.282600	−	0.959238i	$-0.591197\pi$
$379$	−11.0033	−0.565200	−0.282600	+	0.959238i	$0.591197\pi$
$380$	−2.30670	−0.118331
$381$	12.5253	0.641689
$382$	6.77142	0.346456
$383$	−6.48072	−0.331149	−0.165575	−	0.986197i	$-0.552948\pi$
$383$	−6.48072	−0.331149	−0.165575	+	0.986197i	$0.552948\pi$
$384$	9.65859	0.492888
$385$	1.09197	0.0556517
$386$	0.293998	0.0149641
$387$	3.39221	0.172436
$388$	14.0228	0.711902
$389$	5.93018	0.300672	0.150336	−	0.988635i	$-0.451964\pi$
$389$	5.93018	0.300672	0.150336	+	0.988635i	$0.451964\pi$
$390$	1.25271	0.0634333
$391$	2.00000	0.101144
$392$	−1.37958	−0.0696793
$393$	14.7579	0.744439
$394$	1.76443	0.0888905
$395$	0.677477	0.0340876
$396$	−2.04539	−0.102785
$397$	−36.3657	−1.82514	−0.912571	−	0.408918i	$-0.865906\pi$
$397$	−36.3657	−1.82514	−0.912571	+	0.408918i	$0.865906\pi$
$398$	−2.62479	−0.131569
$399$	−1.23147	−0.0616506
$400$	3.25486	0.162743
$401$	17.8987	0.893820	0.446910	−	0.894579i	$-0.352524\pi$
$401$	17.8987	0.893820	0.446910	+	0.894579i	$0.352524\pi$
$402$	3.28331	0.163757
$403$	31.3329	1.56080
$404$	15.9770	0.794883
$405$	−1.00000	−0.0496904
$406$	−0.0941159	−0.00467090
$407$	−4.90680	−0.243221
$408$	−2.75916	−0.136599
$409$	−21.8555	−1.08069	−0.540344	−	0.841444i	$-0.681706\pi$
$409$	−21.8555	−1.08069	−0.540344	+	0.841444i	$0.681706\pi$
$410$	0.433375	0.0214029
$411$	−1.95572	−0.0964686
$412$	15.4820	0.762745
$413$	−4.56447	−0.224603
$414$	−0.356193	−0.0175059
$415$	8.08981	0.397113
$416$	13.7812	0.675677
$417$	−14.5834	−0.714150
$418$	−0.478980	−0.0234277
$419$	−12.8963	−0.630026	−0.315013	−	0.949087i	$-0.602009\pi$
$419$	−12.8963	−0.630026	−0.315013	+	0.949087i	$0.602009\pi$
$420$	1.87313	0.0913992
$421$	37.1081	1.80854	0.904270	−	0.426962i	$-0.140416\pi$
$421$	37.1081	1.80854	0.904270	+	0.426962i	$0.140416\pi$
$422$	5.63396	0.274257
$423$	−6.60890	−0.321336
$424$	−6.67471	−0.324152
$425$	−2.00000	−0.0970143
$426$	0.529419	0.0256504
$427$	12.9777	0.628036
$428$	5.39928	0.260984
$429$	−3.84037	−0.185415
$430$	−1.20828	−0.0582685
$431$	19.3184	0.930533	0.465267	−	0.885171i	$-0.345958\pi$
$431$	19.3184	0.930533	0.465267	+	0.885171i	$0.345958\pi$
$432$	−3.25486	−0.156599
$433$	23.1128	1.11073	0.555365	−	0.831607i	$-0.312579\pi$
$433$	23.1128	1.11073	0.555365	+	0.831607i	$0.312579\pi$
$434$	−3.17337	−0.152327
$435$	0.264228	0.0126687
$436$	−2.00561	−0.0960514
$437$	1.23147	0.0589092
$438$	−2.46099	−0.117591
$439$	1.52645	0.0728535	0.0364268	−	0.999336i	$-0.488402\pi$
$439$	1.52645	0.0728535	0.0364268	+	0.999336i	$0.488402\pi$
$440$	1.50645	0.0718173
$441$	1.00000	0.0476190
$442$	−2.50541	−0.119170
$443$	−22.3553	−1.06213	−0.531066	−	0.847330i	$-0.678208\pi$
$443$	−22.3553	−1.06213	−0.531066	+	0.847330i	$0.678208\pi$
$444$	−8.41698	−0.399452
$445$	11.2188	0.531820
$446$	1.15775	0.0548212
$447$	7.81288	0.369536
$448$	5.11397	0.241612
$449$	−1.62195	−0.0765446	−0.0382723	−	0.999267i	$-0.512185\pi$
$449$	−1.62195	−0.0765446	−0.0382723	+	0.999267i	$0.512185\pi$
$450$	0.356193	0.0167911
$451$	−1.32858	−0.0625604
$452$	12.1909	0.573413
$453$	−8.96294	−0.421116
$454$	−7.86037	−0.368905
$455$	3.51693	0.164876
$456$	−1.69891	−0.0795588
$457$	12.7499	0.596416	0.298208	−	0.954501i	$-0.403611\pi$
$457$	12.7499	0.596416	0.298208	+	0.954501i	$0.403611\pi$
$458$	−1.13382	−0.0529799
$459$	2.00000	0.0933520
$460$	−1.87313	−0.0873350
$461$	−35.6504	−1.66040	−0.830202	−	0.557462i	$-0.811775\pi$
$461$	−35.6504	−1.66040	−0.830202	+	0.557462i	$0.811775\pi$
$462$	0.388950	0.0180956
$463$	23.7000	1.10143	0.550717	−	0.834692i	$-0.314354\pi$
$463$	23.7000	1.10143	0.550717	+	0.834692i	$0.314354\pi$
$464$	0.860023	0.0399256
$465$	8.90915	0.413152
$466$	−6.52971	−0.302483
$467$	−30.7333	−1.42217	−0.711085	−	0.703106i	$-0.751796\pi$
$467$	−30.7333	−1.42217	−0.711085	+	0.703106i	$0.751796\pi$
$468$	−6.58766	−0.304515
$469$	9.21780	0.425638
$470$	2.35404	0.108584
$471$	9.98633	0.460146
$472$	−6.29706	−0.289845
$473$	3.70418	0.170318
$474$	0.241313	0.0110839
$475$	−1.23147	−0.0565037
$476$	−3.74625	−0.171709
$477$	4.83822	0.221527
$478$	−7.67587	−0.351086
$479$	−12.3935	−0.566274	−0.283137	−	0.959079i	$-0.591375\pi$
$479$	−12.3935	−0.566274	−0.283137	+	0.959079i	$0.591375\pi$
$480$	3.91852	0.178855
$481$	−15.8035	−0.720578
$482$	0.966528	0.0440241
$483$	−1.00000	−0.0455016
$484$	18.3709	0.835041
$485$	7.48633	0.339937
$486$	−0.356193	−0.0161572
$487$	7.70099	0.348965	0.174483	−	0.984660i	$-0.444175\pi$
$487$	7.70099	0.348965	0.174483	+	0.984660i	$0.444175\pi$
$488$	17.9038	0.810467
$489$	−8.27360	−0.374145
$490$	−0.356193	−0.0160911
$491$	18.2181	0.822173	0.411087	−	0.911596i	$-0.365149\pi$
$491$	18.2181	0.822173	0.411087	+	0.911596i	$0.365149\pi$
$492$	−2.27901	−0.102746
$493$	−0.528455	−0.0238004
$494$	−1.54267	−0.0694080
$495$	−1.09197	−0.0490802
$496$	28.9980	1.30205
$497$	1.48633	0.0666709
$498$	2.88153	0.129125
$499$	5.51832	0.247034	0.123517	−	0.992342i	$-0.460583\pi$
$499$	5.51832	0.247034	0.123517	+	0.992342i	$0.460583\pi$
$500$	1.87313	0.0837688
$501$	10.4575	0.467208
$502$	−0.188232	−0.00840120
$503$	−29.8598	−1.33138	−0.665691	−	0.746227i	$-0.731863\pi$
$503$	−29.8598	−1.33138	−0.665691	+	0.746227i	$0.731863\pi$
$504$	1.37958	0.0614514
$505$	8.52957	0.379561
$506$	−0.388950	−0.0172909
$507$	0.631175	0.0280315
$508$	23.4614	1.04093
$509$	−8.50391	−0.376929	−0.188465	−	0.982080i	$-0.560351\pi$
$509$	−8.50391	−0.376929	−0.188465	+	0.982080i	$0.560351\pi$
$510$	−0.712386	−0.0315450
$511$	−6.90915	−0.305643
$512$	21.7349	0.960555
$513$	1.23147	0.0543707
$514$	0.217722	0.00960331
$515$	8.26534	0.364214
$516$	6.35404	0.279721
$517$	−7.21669	−0.317390
$518$	1.60057	0.0703250
$519$	−5.17337	−0.227086
$520$	4.85189	0.212769
$521$	10.0520	0.440388	0.220194	−	0.975456i	$-0.429331\pi$
$521$	10.0520	0.440388	0.220194	+	0.975456i	$0.429331\pi$
$522$	0.0941159	0.00411934
$523$	−12.6484	−0.553075	−0.276537	−	0.961003i	$-0.589187\pi$
$523$	−12.6484	−0.553075	−0.276537	+	0.961003i	$0.589187\pi$
$524$	27.6435	1.20761
$525$	1.00000	0.0436436
$526$	8.89909	0.388019
$527$	−17.8183	−0.776177
$528$	−3.55419	−0.154676
$529$	1.00000	0.0434783
$530$	−1.72334	−0.0748570
$531$	4.56447	0.198081
$532$	−2.30670	−0.100008
$533$	−4.27901	−0.185345
$534$	3.99604	0.172926
$535$	2.88250	0.124621
$536$	12.7167	0.549277
$537$	−18.6531	−0.804939
$538$	−3.70210	−0.159609
$539$	1.09197	0.0470343
$540$	−1.87313	−0.0806065
$541$	−31.9370	−1.37308	−0.686538	−	0.727093i	$-0.740871\pi$
$541$	−31.9370	−1.37308	−0.686538	+	0.727093i	$0.740871\pi$
$542$	10.5310	0.452346
$543$	−2.73765	−0.117484
$544$	−7.83703	−0.336010
$545$	−1.07073	−0.0458650
$546$	1.25271	0.0536109
$547$	31.4935	1.34656	0.673282	−	0.739386i	$-0.264884\pi$
$547$	31.4935	1.34656	0.673282	+	0.739386i	$0.264884\pi$
$548$	−3.66331	−0.156489
$549$	−12.9777	−0.553876
$550$	0.388950	0.0165849
$551$	−0.325388	−0.0138620
$552$	−1.37958	−0.0587188
$553$	0.677477	0.0288093
$554$	5.98709	0.254367
$555$	−4.49355	−0.190740
$556$	−27.3165	−1.15848
$557$	11.7403	0.497454	0.248727	−	0.968574i	$-0.419988\pi$
$557$	11.7403	0.497454	0.248727	+	0.968574i	$0.419988\pi$
$558$	3.17337	0.134340
$559$	11.9302	0.504593
$560$	3.25486	0.137543
$561$	2.18393	0.0922056
$562$	−5.67914	−0.239560
$563$	−17.5155	−0.738192	−0.369096	−	0.929391i	$-0.620333\pi$
$563$	−17.5155	−0.738192	−0.369096	+	0.929391i	$0.620333\pi$
$564$	−12.3793	−0.521263
$565$	6.50833	0.273807
$566$	−2.55221	−0.107277
$567$	−1.00000	−0.0419961
$568$	2.05051	0.0860374
$569$	4.07315	0.170755	0.0853776	−	0.996349i	$-0.472790\pi$
$569$	4.07315	0.170755	0.0853776	+	0.996349i	$0.472790\pi$
$570$	−0.438641	−0.0183726
$571$	−2.24020	−0.0937493	−0.0468746	−	0.998901i	$-0.514926\pi$
$571$	−2.24020	−0.0937493	−0.0468746	+	0.998901i	$0.514926\pi$
$572$	−7.19350	−0.300775
$573$	−19.0106	−0.794177
$574$	0.433375	0.0180887
$575$	−1.00000	−0.0417029
$576$	−5.11397	−0.213082
$577$	−38.0313	−1.58326	−0.791632	−	0.610998i	$-0.790768\pi$
$577$	−38.0313	−1.58326	−0.791632	+	0.610998i	$0.790768\pi$
$578$	−4.63051	−0.192604
$579$	−0.825389	−0.0343020
$580$	0.494932	0.0205509
$581$	8.08981	0.335622
$582$	2.66658	0.110533
$583$	5.28317	0.218806
$584$	−9.53172	−0.394425
$585$	−3.51693	−0.145407
$586$	−9.10591	−0.376161
$587$	−32.4152	−1.33792	−0.668959	−	0.743299i	$-0.733260\pi$
$587$	−32.4152	−1.33792	−0.668959	+	0.743299i	$0.733260\pi$
$588$	1.87313	0.0772464
$589$	−10.9713	−0.452067
$590$	−1.62583	−0.0669345
$591$	−4.95357	−0.203763
$592$	−14.6259	−0.601119
$593$	−2.36069	−0.0969420	−0.0484710	−	0.998825i	$-0.515435\pi$
$593$	−2.36069	−0.0969420	−0.0484710	+	0.998825i	$0.515435\pi$
$594$	−0.388950	−0.0159588
$595$	−2.00000	−0.0819920
$596$	14.6345	0.599453
$597$	7.36902	0.301594
$598$	−1.25271	−0.0512270
$599$	−6.19911	−0.253289	−0.126644	−	0.991948i	$-0.540421\pi$
$599$	−6.19911	−0.253289	−0.126644	+	0.991948i	$0.540421\pi$
$600$	1.37958	0.0563211
$601$	−5.16151	−0.210542	−0.105271	−	0.994444i	$-0.533571\pi$
$601$	−5.16151	−0.210542	−0.105271	+	0.994444i	$0.533571\pi$
$602$	−1.20828	−0.0492459
$603$	−9.21780	−0.375378
$604$	−16.7887	−0.683124
$605$	9.80761	0.398736
$606$	3.03817	0.123417
$607$	−28.9807	−1.17629	−0.588145	−	0.808755i	$-0.700142\pi$
$607$	−28.9807	−1.17629	−0.588145	+	0.808755i	$0.700142\pi$
$608$	−4.82554	−0.195701
$609$	0.264228	0.0107070
$610$	4.62257	0.187162
$611$	−23.2431	−0.940314
$612$	3.74625	0.151433
$613$	−0.893449	−0.0360861	−0.0180430	−	0.999837i	$-0.505744\pi$
$613$	−0.893449	−0.0360861	−0.0180430	+	0.999837i	$0.505744\pi$
$614$	3.38608	0.136651
$615$	−1.21669	−0.0490616
$616$	1.50645	0.0606967
$617$	25.7548	1.03685	0.518424	−	0.855124i	$-0.326519\pi$
$617$	25.7548	1.03685	0.518424	+	0.855124i	$0.326519\pi$
$618$	2.94405	0.118427
$619$	31.9239	1.28313	0.641564	−	0.767070i	$-0.278286\pi$
$619$	31.9239	1.28313	0.641564	+	0.767070i	$0.278286\pi$
$620$	16.6880	0.670205
$621$	1.00000	0.0401286
$622$	−3.29614	−0.132163
$623$	11.2188	0.449470
$624$	−11.4471	−0.458251
$625$	1.00000	0.0400000
$626$	1.80536	0.0721568
$627$	1.34472	0.0537030
$628$	18.7057	0.746437
$629$	8.98709	0.358339
$630$	0.356193	0.0141911
$631$	−12.6956	−0.505404	−0.252702	−	0.967544i	$-0.581319\pi$
$631$	−12.6956	−0.505404	−0.252702	+	0.967544i	$0.581319\pi$
$632$	0.934634	0.0371777
$633$	−15.8172	−0.628676
$634$	−4.56511	−0.181304
$635$	12.5253	0.497050
$636$	9.06260	0.359355
$637$	3.51693	0.139346
$638$	0.102771	0.00406876
$639$	−1.48633	−0.0587982
$640$	9.65859	0.381789
$641$	23.2227	0.917240	0.458620	−	0.888632i	$-0.348344\pi$
$641$	23.2227	0.917240	0.458620	+	0.888632i	$0.348344\pi$
$642$	1.02672	0.0405216
$643$	32.5209	1.28250	0.641250	−	0.767332i	$-0.278416\pi$
$643$	32.5209	1.28250	0.641250	+	0.767332i	$0.278416\pi$
$644$	−1.87313	−0.0738115
$645$	3.39221	0.133568
$646$	0.877282	0.0345162
$647$	−25.9604	−1.02061	−0.510305	−	0.859993i	$-0.670468\pi$
$647$	−25.9604	−1.02061	−0.510305	+	0.859993i	$0.670468\pi$
$648$	−1.37958	−0.0541950
$649$	4.98425	0.195649
$650$	1.25271	0.0491352
$651$	8.90915	0.349177
$652$	−15.4975	−0.606929
$653$	−42.2683	−1.65409	−0.827043	−	0.562139i	$-0.809979\pi$
$653$	−42.2683	−1.65409	−0.827043	+	0.562139i	$0.809979\pi$
$654$	−0.381386	−0.0149134
$655$	14.7579	0.576640
$656$	−3.96014	−0.154618
$657$	6.90915	0.269551
$658$	2.35404	0.0917701
$659$	13.2862	0.517558	0.258779	−	0.965937i	$-0.416680\pi$
$659$	13.2862	0.517558	0.258779	+	0.965937i	$0.416680\pi$
$660$	−2.04539	−0.0796167
$661$	8.04136	0.312773	0.156386	−	0.987696i	$-0.450016\pi$
$661$	8.04136	0.312773	0.156386	+	0.987696i	$0.450016\pi$
$662$	5.77553	0.224472
$663$	7.03387	0.273173
$664$	11.1605	0.433113
$665$	−1.23147	−0.0477544
$666$	−1.60057	−0.0620208
$667$	−0.264228	−0.0102309
$668$	19.5883	0.757893
$669$	−3.25036	−0.125666
$670$	3.28331	0.126845
$671$	−14.1712	−0.547074
$672$	3.91852	0.151160
$673$	−13.6753	−0.527145	−0.263573	−	0.964640i	$-0.584901\pi$
$673$	−13.6753	−0.527145	−0.263573	+	0.964640i	$0.584901\pi$
$674$	0.866055	0.0333592
$675$	−1.00000	−0.0384900
$676$	1.18227	0.0454720
$677$	−16.9006	−0.649543	−0.324772	−	0.945792i	$-0.605287\pi$
$677$	−16.9006	−0.649543	−0.324772	+	0.945792i	$0.605287\pi$
$678$	2.31822	0.0890307
$679$	7.48633	0.287299
$680$	−2.75916	−0.105809
$681$	22.0677	0.845637
$682$	3.46521	0.132690
$683$	−48.5492	−1.85768	−0.928841	−	0.370479i	$-0.879194\pi$
$683$	−48.5492	−1.85768	−0.928841	+	0.370479i	$0.879194\pi$
$684$	2.30670	0.0881989
$685$	−1.95572	−0.0747243
$686$	−0.356193	−0.0135995
$687$	3.18316	0.121445
$688$	11.0412	0.420940
$689$	17.0157	0.648246
$690$	−0.356193	−0.0135600
$691$	−3.33770	−0.126972	−0.0634861	−	0.997983i	$-0.520222\pi$
$691$	−3.33770	−0.126972	−0.0634861	+	0.997983i	$0.520222\pi$
$692$	−9.69038	−0.368373
$693$	−1.09197	−0.0414803
$694$	−2.62131	−0.0995037
$695$	−14.5834	−0.553178
$696$	0.364523	0.0138172
$697$	2.43338	0.0921707
$698$	9.98658	0.377998
$699$	18.3320	0.693379
$700$	1.87313	0.0707975
$701$	32.3860	1.22320	0.611602	−	0.791166i	$-0.290526\pi$
$701$	32.3860	1.22320	0.611602	+	0.791166i	$0.290526\pi$
$702$	−1.25271	−0.0472804
$703$	5.53367	0.208706
$704$	−5.58427	−0.210465
$705$	−6.60890	−0.248906
$706$	−2.95083	−0.111056
$707$	8.52957	0.320787
$708$	8.54984	0.321323
$709$	38.2114	1.43506	0.717530	−	0.696528i	$-0.245273\pi$
$709$	38.2114	1.43506	0.717530	+	0.696528i	$0.245273\pi$
$710$	0.529419	0.0198687
$711$	−0.677477	−0.0254074
$712$	15.4772	0.580032
$713$	−8.90915	−0.333650
$714$	−0.712386	−0.0266604
$715$	−3.84037	−0.143622
$716$	−34.9395	−1.30575
$717$	21.5498	0.804791
$718$	−3.59202	−0.134053
$719$	36.7613	1.37096	0.685482	−	0.728090i	$-0.259592\pi$
$719$	36.7613	1.37096	0.685482	+	0.728090i	$0.259592\pi$
$720$	−3.25486	−0.121301
$721$	8.26534	0.307817
$722$	−6.22749	−0.231763
$723$	−2.71350	−0.100916
$724$	−5.12796	−0.190579
$725$	0.264228	0.00981317
$726$	3.49340	0.129652
$727$	32.6354	1.21038	0.605190	−	0.796081i	$-0.293097\pi$
$727$	32.6354	1.21038	0.605190	+	0.796081i	$0.293097\pi$
$728$	4.85189	0.179823
$729$	1.00000	0.0370370
$730$	−2.46099	−0.0910852
$731$	−6.78442	−0.250931
$732$	−24.3089	−0.898484
$733$	8.74660	0.323063	0.161531	−	0.986868i	$-0.448357\pi$
$733$	8.74660	0.323063	0.161531	+	0.986868i	$0.448357\pi$
$734$	−5.54511	−0.204674
$735$	1.00000	0.0368856
$736$	−3.91852	−0.144438
$737$	−10.0655	−0.370768
$738$	−0.433375	−0.0159528
$739$	3.60257	0.132523	0.0662614	−	0.997802i	$-0.478893\pi$
$739$	3.60257	0.132523	0.0662614	+	0.997802i	$0.478893\pi$
$740$	−8.41698	−0.309414
$741$	4.33100	0.159103
$742$	−1.72334	−0.0632657
$743$	−21.5028	−0.788863	−0.394431	−	0.918925i	$-0.629058\pi$
$743$	−21.5028	−0.788863	−0.394431	+	0.918925i	$0.629058\pi$
$744$	12.2909	0.450605
$745$	7.81288	0.286242
$746$	−7.23001	−0.264709
$747$	−8.08981	−0.295991
$748$	4.09078	0.149574
$749$	2.88250	0.105324
$750$	0.356193	0.0130063
$751$	12.2789	0.448063	0.224031	−	0.974582i	$-0.428078\pi$
$751$	12.2789	0.448063	0.224031	+	0.974582i	$0.428078\pi$
$752$	−21.5110	−0.784426
$753$	0.528455	0.0192580
$754$	0.331000	0.0120543
$755$	−8.96294	−0.326195
$756$	−1.87313	−0.0681250
$757$	−28.2463	−1.02663	−0.513315	−	0.858200i	$-0.671583\pi$
$757$	−28.2463	−1.02663	−0.513315	+	0.858200i	$0.671583\pi$
$758$	−3.91928	−0.142355
$759$	1.09197	0.0396358
$760$	−1.69891	−0.0616260
$761$	29.2264	1.05946	0.529729	−	0.848167i	$-0.322294\pi$
$761$	29.2264	1.05946	0.529729	+	0.848167i	$0.322294\pi$
$762$	4.46141	0.161620
$763$	−1.07073	−0.0387630
$764$	−35.6092	−1.28829
$765$	2.00000	0.0723102
$766$	−2.30838	−0.0834053
$767$	16.0530	0.579639
$768$	−6.78762	−0.244927
$769$	−38.3770	−1.38391	−0.691955	−	0.721941i	$-0.743250\pi$
$769$	−38.3770	−1.38391	−0.691955	+	0.721941i	$0.743250\pi$
$770$	0.388950	0.0140168
$771$	−0.611248	−0.0220135
$772$	−1.54606	−0.0556439
$773$	−30.6491	−1.10237	−0.551186	−	0.834383i	$-0.685824\pi$
$773$	−30.6491	−1.10237	−0.551186	+	0.834383i	$0.685824\pi$
$774$	1.20828	0.0434308
$775$	8.90915	0.320026
$776$	10.3280	0.370753
$777$	−4.49355	−0.161205
$778$	2.11229	0.0757292
$779$	1.49831	0.0536827
$780$	−6.58766	−0.235876
$781$	−1.62302	−0.0580761
$782$	0.712386	0.0254749
$783$	−0.264228	−0.00944272
$784$	3.25486	0.116245
$785$	9.98633	0.356427
$786$	5.25666	0.187499
$787$	−12.4659	−0.444362	−0.222181	−	0.975005i	$-0.571318\pi$
$787$	−12.4659	−0.444362	−0.222181	+	0.975005i	$0.571318\pi$
$788$	−9.27867	−0.330539
$789$	−24.9839	−0.889451
$790$	0.241313	0.00858551
$791$	6.50833	0.231410
$792$	−1.50645	−0.0535295
$793$	−45.6418	−1.62079
$794$	−12.9532	−0.459692
$795$	4.83822	0.171594
$796$	13.8031	0.489238
$797$	−25.2618	−0.894819	−0.447409	−	0.894329i	$-0.647653\pi$
$797$	−25.2618	−0.894819	−0.447409	+	0.894329i	$0.647653\pi$
$798$	−0.438641	−0.0155277
$799$	13.2178	0.467612
$800$	3.91852	0.138540
$801$	−11.2188	−0.396395
$802$	6.37540	0.225123
$803$	7.54455	0.266241
$804$	−17.2661	−0.608929
$805$	−1.00000	−0.0352454
$806$	11.1605	0.393113
$807$	10.3935	0.365869
$808$	11.7672	0.413969
$809$	28.0653	0.986722	0.493361	−	0.869825i	$-0.335768\pi$
$809$	28.0653	0.986722	0.493361	+	0.869825i	$0.335768\pi$
$810$	−0.356193	−0.0125153
$811$	36.4890	1.28130	0.640652	−	0.767832i	$-0.278664\pi$
$811$	36.4890	1.28130	0.640652	+	0.767832i	$0.278664\pi$
$812$	0.494932	0.0173687
$813$	−29.5655	−1.03691
$814$	−1.74777	−0.0612591
$815$	−8.27360	−0.289811
$816$	6.50971	0.227886
$817$	−4.17741	−0.146149
$818$	−7.78479	−0.272189
$819$	−3.51693	−0.122892
$820$	−2.27901	−0.0795865
$821$	3.33478	0.116385	0.0581924	−	0.998305i	$-0.481466\pi$
$821$	3.33478	0.116385	0.0581924	+	0.998305i	$0.481466\pi$
$822$	−0.696614	−0.0242972
$823$	−9.25582	−0.322638	−0.161319	−	0.986902i	$-0.551575\pi$
$823$	−9.25582	−0.322638	−0.161319	+	0.986902i	$0.551575\pi$
$824$	11.4027	0.397232
$825$	−1.09197	−0.0380173
$826$	−1.62583	−0.0565699
$827$	46.8837	1.63031	0.815154	−	0.579245i	$-0.196652\pi$
$827$	46.8837	1.63031	0.815154	+	0.579245i	$0.196652\pi$
$828$	1.87313	0.0650957
$829$	−3.73674	−0.129782	−0.0648911	−	0.997892i	$-0.520670\pi$
$829$	−3.73674	−0.129782	−0.0648911	+	0.997892i	$0.520670\pi$
$830$	2.88153	0.100019
$831$	−16.8086	−0.583083
$832$	−17.9855	−0.623535
$833$	−2.00000	−0.0692959
$834$	−5.19449	−0.179870
$835$	10.4575	0.361898
$836$	2.51884	0.0871158
$837$	−8.90915	−0.307945
$838$	−4.59357	−0.158682
$839$	−26.3428	−0.909456	−0.454728	−	0.890630i	$-0.650263\pi$
$839$	−26.3428	−0.909456	−0.454728	+	0.890630i	$0.650263\pi$
$840$	1.37958	0.0476000
$841$	−28.9302	−0.997593
$842$	13.2176	0.455510
$843$	15.9440	0.549141
$844$	−29.6276	−1.01982
$845$	0.631175	0.0217131
$846$	−2.35404	−0.0809336
$847$	9.80761	0.336994
$848$	15.7477	0.540779
$849$	7.16524	0.245910
$850$	−0.712386	−0.0244346
$851$	4.49355	0.154037
$852$	−2.78408	−0.0953810
$853$	22.1519	0.758467	0.379234	−	0.925301i	$-0.376188\pi$
$853$	22.1519	0.758467	0.379234	+	0.925301i	$0.376188\pi$
$854$	4.62257	0.158181
$855$	1.23147	0.0421154
$856$	3.97663	0.135919
$857$	6.15443	0.210231	0.105116	−	0.994460i	$-0.466479\pi$
$857$	6.15443	0.210231	0.105116	+	0.994460i	$0.466479\pi$
$858$	−1.36791	−0.0466997
$859$	54.7948	1.86957	0.934787	−	0.355209i	$-0.115590\pi$
$859$	54.7948	1.86957	0.934787	+	0.355209i	$0.115590\pi$
$860$	6.35404	0.216671
$861$	−1.21669	−0.0414646
$862$	6.88106	0.234370
$863$	−42.6125	−1.45054	−0.725272	−	0.688462i	$-0.758286\pi$
$863$	−42.6125	−1.45054	−0.725272	+	0.688462i	$0.758286\pi$
$864$	−3.91852	−0.133311
$865$	−5.17337	−0.175900
$866$	8.23261	0.279756
$867$	13.0000	0.441503
$868$	16.6880	0.566426
$869$	−0.739782	−0.0250954
$870$	0.0941159	0.00319083
$871$	−32.4184	−1.09846
$872$	−1.47716	−0.0500228
$873$	−7.48633	−0.253374
$874$	0.438641	0.0148372
$875$	1.00000	0.0338062
$876$	12.9417	0.437260
$877$	0.0709268	0.00239503	0.00119751	−	0.999999i	$-0.499619\pi$
$877$	0.0709268	0.00239503	0.00119751	+	0.999999i	$0.499619\pi$
$878$	0.543711	0.0183494
$879$	25.5645	0.862271
$880$	−3.55419	−0.119812
$881$	1.70418	0.0574152	0.0287076	−	0.999588i	$-0.490861\pi$
$881$	1.70418	0.0574152	0.0287076	+	0.999588i	$0.490861\pi$
$882$	0.356193	0.0119936
$883$	−12.3247	−0.414758	−0.207379	−	0.978261i	$-0.566493\pi$
$883$	−12.3247	−0.414758	−0.207379	+	0.978261i	$0.566493\pi$
$884$	13.1753	0.443134
$885$	4.56447	0.153433
$886$	−7.96280	−0.267515
$887$	15.2062	0.510574	0.255287	−	0.966865i	$-0.417830\pi$
$887$	15.2062	0.510574	0.255287	+	0.966865i	$0.417830\pi$
$888$	−6.19921	−0.208032
$889$	12.5253	0.420084
$890$	3.99604	0.133948
$891$	1.09197	0.0365822
$892$	−6.08833	−0.203852
$893$	8.13866	0.272350
$894$	2.78289	0.0930738
$895$	−18.6531	−0.623503
$896$	9.65859	0.322671
$897$	3.51693	0.117427
$898$	−0.577727	−0.0192790
$899$	2.35404	0.0785117
$900$	−1.87313	−0.0624376
$901$	−9.67644	−0.322369
$902$	−0.473231	−0.0157569
$903$	3.39221	0.112886
$904$	8.97876	0.298629
$905$	−2.73765	−0.0910025
$906$	−3.19253	−0.106065
$907$	−28.7614	−0.955006	−0.477503	−	0.878630i	$-0.658458\pi$
$907$	−28.7614	−0.955006	−0.477503	+	0.878630i	$0.658458\pi$
$908$	41.3357	1.37177
$909$	−8.52957	−0.282908
$910$	1.25271	0.0415268
$911$	−9.24806	−0.306402	−0.153201	−	0.988195i	$-0.548958\pi$
$911$	−9.24806	−0.306402	−0.153201	+	0.988195i	$0.548958\pi$
$912$	4.00826	0.132727
$913$	−8.83380	−0.292356
$914$	4.54143	0.150217
$915$	−12.9777	−0.429030
$916$	5.96247	0.197006
$917$	14.7579	0.487349
$918$	0.712386	0.0235122
$919$	45.6051	1.50437	0.752186	−	0.658951i	$-0.228999\pi$
$919$	45.6051	1.50437	0.752186	+	0.658951i	$0.228999\pi$
$920$	−1.37958	−0.0454834
$921$	−9.50633	−0.313244
$922$	−12.6984	−0.418200
$923$	−5.22732	−0.172059
$924$	−2.04539	−0.0672884
$925$	−4.49355	−0.147747
$926$	8.44178	0.277414
$927$	−8.26534	−0.271469
$928$	1.03538	0.0339880
$929$	−44.1170	−1.44743	−0.723715	−	0.690099i	$-0.757567\pi$
$929$	−44.1170	−1.44743	−0.723715	+	0.690099i	$0.757567\pi$
$930$	3.17337	0.104059
$931$	−1.23147	−0.0403598
$932$	34.3381	1.12478
$933$	9.25382	0.302956
$934$	−10.9470	−0.358197
$935$	2.18393	0.0714222
$936$	−4.85189	−0.158589
$937$	−28.2445	−0.922708	−0.461354	−	0.887216i	$-0.652636\pi$
$937$	−28.2445	−0.922708	−0.461354	+	0.887216i	$0.652636\pi$
$938$	3.28331	0.107204
$939$	−5.06850	−0.165404
$940$	−12.3793	−0.403768
$941$	−27.3830	−0.892661	−0.446330	−	0.894868i	$-0.647269\pi$
$941$	−27.3830	−0.892661	−0.446330	+	0.894868i	$0.647269\pi$
$942$	3.55706	0.115895
$943$	1.21669	0.0396208
$944$	14.8567	0.483545
$945$	−1.00000	−0.0325300
$946$	1.31940	0.0428974
$947$	14.8542	0.482696	0.241348	−	0.970439i	$-0.422410\pi$
$947$	14.8542	0.482696	0.241348	+	0.970439i	$0.422410\pi$
$948$	−1.26900	−0.0412152
$949$	24.2990	0.788779
$950$	−0.438641	−0.0142314
$951$	12.8164	0.415601
$952$	−2.75916	−0.0894249
$953$	−24.6405	−0.798183	−0.399092	−	0.916911i	$-0.630674\pi$
$953$	−24.6405	−0.798183	−0.399092	+	0.916911i	$0.630674\pi$
$954$	1.72334	0.0557951
$955$	−19.0106	−0.615167
$956$	40.3654	1.30551
$957$	−0.288527	−0.00932676
$958$	−4.41448	−0.142625
$959$	−1.95572	−0.0631535
$960$	−5.11397	−0.165053
$961$	48.3729	1.56042
$962$	−5.62910	−0.181489
$963$	−2.88250	−0.0928872
$964$	−5.08272	−0.163703
$965$	−0.825389	−0.0265702
$966$	−0.356193	−0.0114603
$967$	39.8115	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$967$	39.8115	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$968$	13.5304	0.434883
$969$	−2.46294	−0.0791210
$970$	2.66658	0.0856186
$971$	20.1631	0.647065	0.323532	−	0.946217i	$-0.395130\pi$
$971$	20.1631	0.647065	0.323532	+	0.946217i	$0.395130\pi$
$972$	1.87313	0.0600806
$973$	−14.5834	−0.467521
$974$	2.74304	0.0878926
$975$	−3.51693	−0.112632
$976$	−42.2406	−1.35209
$977$	−23.7435	−0.759623	−0.379811	−	0.925064i	$-0.624011\pi$
$977$	−23.7435	−0.759623	−0.379811	+	0.925064i	$0.624011\pi$
$978$	−2.94700	−0.0942345
$979$	−12.2505	−0.391528
$980$	1.87313	0.0598348
$981$	1.07073	0.0341857
$982$	6.48917	0.207078
$983$	13.6609	0.435714	0.217857	−	0.975981i	$-0.430093\pi$
$983$	13.6609	0.435714	0.217857	+	0.975981i	$0.430093\pi$
$984$	−1.67852	−0.0535092
$985$	−4.95357	−0.157834
$986$	−0.188232	−0.00599452
$987$	−6.60890	−0.210364
$988$	8.11251	0.258093
$989$	−3.39221	−0.107866
$990$	−0.388950	−0.0123616
$991$	−34.9954	−1.11167	−0.555833	−	0.831294i	$-0.687601\pi$
$991$	−34.9954	−1.11167	−0.555833	+	0.831294i	$0.687601\pi$
$992$	34.9106	1.10841
$993$	−16.2146	−0.514555
$994$	0.529419	0.0167922
$995$	7.36902	0.233614
$996$	−15.1532	−0.480149
$997$	24.2339	0.767496	0.383748	−	0.923438i	$-0.374633\pi$
$997$	24.2339	0.767496	0.383748	+	0.923438i	$0.374633\pi$
$998$	1.96559	0.0622195
$999$	4.49355	0.142170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.2.a.o.1.3	✓	5
3.2	odd	2		7245.2.a.bg.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.o.1.3	✓	5	1.1	even	1	trivial
7245.2.a.bg.1.3		5	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:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

\(f(q)\)	\(=\)	\(q+0.356193 q^{2} -1.00000 q^{3} -1.87313 q^{4} -1.00000 q^{5} -0.356193 q^{6} -1.00000 q^{7} -1.37958 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.356193 q^{2} -1.00000 q^{3} -1.87313 q^{4} -1.00000 q^{5} -0.356193 q^{6} -1.00000 q^{7} -1.37958 q^{8} +1.00000 q^{9} -0.356193 q^{10} +1.09197 q^{11} +1.87313 q^{12} +3.51693 q^{13} -0.356193 q^{14} +1.00000 q^{15} +3.25486 q^{16} -2.00000 q^{17} +0.356193 q^{18} -1.23147 q^{19} +1.87313 q^{20} +1.00000 q^{21} +0.388950 q^{22} -1.00000 q^{23} +1.37958 q^{24} +1.00000 q^{25} +1.25271 q^{26} -1.00000 q^{27} +1.87313 q^{28} +0.264228 q^{29} +0.356193 q^{30} +8.90915 q^{31} +3.91852 q^{32} -1.09197 q^{33} -0.712386 q^{34} +1.00000 q^{35} -1.87313 q^{36} -4.49355 q^{37} -0.438641 q^{38} -3.51693 q^{39} +1.37958 q^{40} -1.21669 q^{41} +0.356193 q^{42} +3.39221 q^{43} -2.04539 q^{44} -1.00000 q^{45} -0.356193 q^{46} -6.60890 q^{47} -3.25486 q^{48} +1.00000 q^{49} +0.356193 q^{50} +2.00000 q^{51} -6.58766 q^{52} +4.83822 q^{53} -0.356193 q^{54} -1.09197 q^{55} +1.37958 q^{56} +1.23147 q^{57} +0.0941159 q^{58} +4.56447 q^{59} -1.87313 q^{60} -12.9777 q^{61} +3.17337 q^{62} -1.00000 q^{63} -5.11397 q^{64} -3.51693 q^{65} -0.388950 q^{66} -9.21780 q^{67} +3.74625 q^{68} +1.00000 q^{69} +0.356193 q^{70} -1.48633 q^{71} -1.37958 q^{72} +6.90915 q^{73} -1.60057 q^{74} -1.00000 q^{75} +2.30670 q^{76} -1.09197 q^{77} -1.25271 q^{78} -0.677477 q^{79} -3.25486 q^{80} +1.00000 q^{81} -0.433375 q^{82} -8.08981 q^{83} -1.87313 q^{84} +2.00000 q^{85} +1.20828 q^{86} -0.264228 q^{87} -1.50645 q^{88} -11.2188 q^{89} -0.356193 q^{90} -3.51693 q^{91} +1.87313 q^{92} -8.90915 q^{93} -2.35404 q^{94} +1.23147 q^{95} -3.91852 q^{96} -7.48633 q^{97} +0.356193 q^{98} +1.09197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9} + q^{11} - 10 q^{12} + 5 q^{15} + 8 q^{16} - 10 q^{17} + 3 q^{19} - 10 q^{20} + 5 q^{21} + 12 q^{22} - 5 q^{23} + 6 q^{24} + 5 q^{25} - 14 q^{26} - 5 q^{27} - 10 q^{28} + 4 q^{29} + 2 q^{31} - 12 q^{32} - q^{33} + 5 q^{35} + 10 q^{36} - 4 q^{37} - 24 q^{38} + 6 q^{40} - 9 q^{41} - 8 q^{43} + 2 q^{44} - 5 q^{45} - 11 q^{47} - 8 q^{48} + 5 q^{49} + 10 q^{51} - 22 q^{52} - 19 q^{53} - q^{55} + 6 q^{56} - 3 q^{57} + 8 q^{58} + 5 q^{59} + 10 q^{60} - 17 q^{61} - 24 q^{62} - 5 q^{63} - 8 q^{64} - 12 q^{66} - 2 q^{67} - 20 q^{68} + 5 q^{69} + 10 q^{71} - 6 q^{72} - 8 q^{73} - 34 q^{74} - 5 q^{75} - 8 q^{76} - q^{77} + 14 q^{78} + 24 q^{79} - 8 q^{80} + 5 q^{81} - 8 q^{82} - 24 q^{83} + 10 q^{84} + 10 q^{85} - 10 q^{86} - 4 q^{87} - 26 q^{88} - 4 q^{89} - 10 q^{92} - 2 q^{93} + 2 q^{94} - 3 q^{95} + 12 q^{96} - 20 q^{97} + q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 10 * q^4 - 5 * q^5 - 5 * q^7 - 6 * q^8 + 5 * q^9 + q^11 - 10 * q^12 + 5 * q^15 + 8 * q^16 - 10 * q^17 + 3 * q^19 - 10 * q^20 + 5 * q^21 + 12 * q^22 - 5 * q^23 + 6 * q^24 + 5 * q^25 - 14 * q^26 - 5 * q^27 - 10 * q^28 + 4 * q^29 + 2 * q^31 - 12 * q^32 - q^33 + 5 * q^35 + 10 * q^36 - 4 * q^37 - 24 * q^38 + 6 * q^40 - 9 * q^41 - 8 * q^43 + 2 * q^44 - 5 * q^45 - 11 * q^47 - 8 * q^48 + 5 * q^49 + 10 * q^51 - 22 * q^52 - 19 * q^53 - q^55 + 6 * q^56 - 3 * q^57 + 8 * q^58 + 5 * q^59 + 10 * q^60 - 17 * q^61 - 24 * q^62 - 5 * q^63 - 8 * q^64 - 12 * q^66 - 2 * q^67 - 20 * q^68 + 5 * q^69 + 10 * q^71 - 6 * q^72 - 8 * q^73 - 34 * q^74 - 5 * q^75 - 8 * q^76 - q^77 + 14 * q^78 + 24 * q^79 - 8 * q^80 + 5 * q^81 - 8 * q^82 - 24 * q^83 + 10 * q^84 + 10 * q^85 - 10 * q^86 - 4 * q^87 - 26 * q^88 - 4 * q^89 - 10 * q^92 - 2 * q^93 + 2 * q^94 - 3 * q^95 + 12 * q^96 - 20 * q^97 + q^99

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