LMFDB - Embedded newform 1155.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, 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("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
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			1.1
Root			$-0.652223$ of defining polynomial
Character	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.06644 q^{2} -1.00000 q^{3} +2.27016 q^{4} +1.00000 q^{5} +2.06644 q^{6} +1.00000 q^{7} -0.558268 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.06644 q^{2} -1.00000 q^{3} +2.27016 q^{4} +1.00000 q^{5} +2.06644 q^{6} +1.00000 q^{7} -0.558268 q^{8} +1.00000 q^{9} -2.06644 q^{10} +1.00000 q^{11} -2.27016 q^{12} +0.491831 q^{13} -2.06644 q^{14} -1.00000 q^{15} -3.38669 q^{16} +7.40303 q^{17} -2.06644 q^{18} -0.729840 q^{19} +2.27016 q^{20} -1.00000 q^{21} -2.06644 q^{22} +8.92701 q^{23} +0.558268 q^{24} +1.00000 q^{25} -1.01634 q^{26} -1.00000 q^{27} +2.27016 q^{28} -4.91120 q^{29} +2.06644 q^{30} -7.64104 q^{31} +8.11492 q^{32} -1.00000 q^{33} -15.2979 q^{34} +1.00000 q^{35} +2.27016 q^{36} +7.10072 q^{37} +1.50817 q^{38} -0.491831 q^{39} -0.558268 q^{40} -2.04849 q^{41} +2.06644 q^{42} -9.55172 q^{43} +2.27016 q^{44} +1.00000 q^{45} -18.4471 q^{46} -6.68900 q^{47} +3.38669 q^{48} +1.00000 q^{49} -2.06644 q^{50} -7.40303 q^{51} +1.11654 q^{52} -0.862713 q^{53} +2.06644 q^{54} +1.00000 q^{55} -0.558268 q^{56} +0.729840 q^{57} +10.1487 q^{58} +10.5038 q^{59} -2.27016 q^{60} +4.86271 q^{61} +15.7897 q^{62} +1.00000 q^{63} -9.99559 q^{64} +0.491831 q^{65} +2.06644 q^{66} -3.49236 q^{67} +16.8061 q^{68} -8.92701 q^{69} -2.06644 q^{70} -6.18136 q^{71} -0.558268 q^{72} -0.132873 q^{73} -14.6732 q^{74} -1.00000 q^{75} -1.65685 q^{76} +1.00000 q^{77} +1.01634 q^{78} +9.77391 q^{79} -3.38669 q^{80} +1.00000 q^{81} +4.23307 q^{82} +15.0599 q^{83} -2.27016 q^{84} +7.40303 q^{85} +19.7380 q^{86} +4.91120 q^{87} -0.558268 q^{88} -0.745653 q^{89} -2.06644 q^{90} +0.491831 q^{91} +20.2657 q^{92} +7.64104 q^{93} +13.8224 q^{94} -0.729840 q^{95} -8.11492 q^{96} -3.76199 q^{97} -2.06644 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 6 q^{12} + 4 q^{13} + 2 q^{14} - 4 q^{15} + 6 q^{16} + 6 q^{17} + 2 q^{18} - 6 q^{19} + 6 q^{20} - 4 q^{21} + 2 q^{22} + 10 q^{23} - 6 q^{24} + 4 q^{25} - 4 q^{27} + 6 q^{28} + 6 q^{29} - 2 q^{30} - 8 q^{31} + 14 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{35} + 6 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} + 6 q^{40} - 2 q^{42} + 6 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{46} - 6 q^{48} + 4 q^{49} + 2 q^{50} - 6 q^{51} - 12 q^{52} + 14 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 6 q^{57} + 20 q^{58} + 2 q^{59} - 6 q^{60} + 2 q^{61} + 20 q^{62} + 4 q^{63} - 2 q^{64} + 4 q^{65} - 2 q^{66} - 12 q^{67} + 20 q^{68} - 10 q^{69} + 2 q^{70} + 4 q^{71} + 6 q^{72} + 20 q^{73} - 32 q^{74} - 4 q^{75} + 16 q^{76} + 4 q^{77} - 4 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 14 q^{83} - 6 q^{84} + 6 q^{85} + 40 q^{86} - 6 q^{87} + 6 q^{88} - 6 q^{89} + 2 q^{90} + 4 q^{91} + 40 q^{92} + 8 q^{93} + 4 q^{94} - 6 q^{95} - 14 q^{96} - 14 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 6 * q^12 + 4 * q^13 + 2 * q^14 - 4 * q^15 + 6 * q^16 + 6 * q^17 + 2 * q^18 - 6 * q^19 + 6 * q^20 - 4 * q^21 + 2 * q^22 + 10 * q^23 - 6 * q^24 + 4 * q^25 - 4 * q^27 + 6 * q^28 + 6 * q^29 - 2 * q^30 - 8 * q^31 + 14 * q^32 - 4 * q^33 - 16 * q^34 + 4 * q^35 + 6 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 + 6 * q^40 - 2 * q^42 + 6 * q^43 + 6 * q^44 + 4 * q^45 - 4 * q^46 - 6 * q^48 + 4 * q^49 + 2 * q^50 - 6 * q^51 - 12 * q^52 + 14 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 6 * q^57 + 20 * q^58 + 2 * q^59 - 6 * q^60 + 2 * q^61 + 20 * q^62 + 4 * q^63 - 2 * q^64 + 4 * q^65 - 2 * q^66 - 12 * q^67 + 20 * q^68 - 10 * q^69 + 2 * q^70 + 4 * q^71 + 6 * q^72 + 20 * q^73 - 32 * q^74 - 4 * q^75 + 16 * q^76 + 4 * q^77 - 4 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 14 * q^83 - 6 * q^84 + 6 * q^85 + 40 * q^86 - 6 * q^87 + 6 * q^88 - 6 * q^89 + 2 * q^90 + 4 * q^91 + 40 * q^92 + 8 * q^93 + 4 * q^94 - 6 * q^95 - 14 * q^96 - 14 * q^97 + 2 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.06644	−1.46119	−0.730596	−	0.682810i	$-0.760757\pi$
$2$	−2.06644	−1.46119	−0.730596	+	0.682810i	$0.760757\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.27016	1.13508
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.06644	0.843619
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.558268	−0.197377
$9$	1.00000	0.333333
$10$	−2.06644	−0.653465
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.27016	−0.655339
$13$	0.491831	0.136409	0.0682047	−	0.997671i	$-0.478273\pi$
$13$	0.491831	0.136409	0.0682047	+	0.997671i	$0.478273\pi$
$14$	−2.06644	−0.552278
$15$	−1.00000	−0.258199
$16$	−3.38669	−0.846674
$17$	7.40303	1.79550	0.897750	−	0.440506i	$-0.145201\pi$
$17$	7.40303	1.79550	0.897750	+	0.440506i	$0.145201\pi$
$18$	−2.06644	−0.487064
$19$	−0.729840	−0.167437	−0.0837184	−	0.996489i	$-0.526680\pi$
$19$	−0.729840	−0.167437	−0.0837184	+	0.996489i	$0.526680\pi$
$20$	2.27016	0.507623
$21$	−1.00000	−0.218218
$22$	−2.06644	−0.440566
$23$	8.92701	1.86141	0.930706	−	0.365769i	$-0.119194\pi$
$23$	8.92701	1.86141	0.930706	+	0.365769i	$0.119194\pi$
$24$	0.558268	0.113956
$25$	1.00000	0.200000
$26$	−1.01634	−0.199320
$27$	−1.00000	−0.192450
$28$	2.27016	0.429020
$29$	−4.91120	−0.911987	−0.455994	−	0.889983i	$-0.650716\pi$
$29$	−4.91120	−0.911987	−0.455994	+	0.889983i	$0.650716\pi$
$30$	2.06644	0.377278
$31$	−7.64104	−1.37237	−0.686186	−	0.727426i	$-0.740716\pi$
$31$	−7.64104	−1.37237	−0.686186	+	0.727426i	$0.740716\pi$
$32$	8.11492	1.43453
$33$	−1.00000	−0.174078
$34$	−15.2979	−2.62357
$35$	1.00000	0.169031
$36$	2.27016	0.378360
$37$	7.10072	1.16735	0.583676	−	0.811987i	$-0.301614\pi$
$37$	7.10072	1.16735	0.583676	+	0.811987i	$0.301614\pi$
$38$	1.50817	0.244657
$39$	−0.491831	−0.0787560
$40$	−0.558268	−0.0882699
$41$	−2.04849	−0.319920	−0.159960	−	0.987123i	$-0.551137\pi$
$41$	−2.04849	−0.319920	−0.159960	+	0.987123i	$0.551137\pi$
$42$	2.06644	0.318858
$43$	−9.55172	−1.45662	−0.728312	−	0.685246i	$-0.759695\pi$
$43$	−9.55172	−1.45662	−0.728312	+	0.685246i	$0.759695\pi$
$44$	2.27016	0.342239
$45$	1.00000	0.149071
$46$	−18.4471	−2.71988
$47$	−6.68900	−0.975692	−0.487846	−	0.872930i	$-0.662217\pi$
$47$	−6.68900	−0.975692	−0.487846	+	0.872930i	$0.662217\pi$
$48$	3.38669	0.488827
$49$	1.00000	0.142857
$50$	−2.06644	−0.292238
$51$	−7.40303	−1.03663
$52$	1.11654	0.154836
$53$	−0.862713	−0.118503	−0.0592514	−	0.998243i	$-0.518871\pi$
$53$	−0.862713	−0.118503	−0.0592514	+	0.998243i	$0.518871\pi$
$54$	2.06644	0.281206
$55$	1.00000	0.134840
$56$	−0.558268	−0.0746016
$57$	0.729840	0.0966697
$58$	10.1487	1.33259
$59$	10.5038	1.36747	0.683736	−	0.729729i	$-0.260354\pi$
$59$	10.5038	1.36747	0.683736	+	0.729729i	$0.260354\pi$
$60$	−2.27016	−0.293076
$61$	4.86271	0.622607	0.311303	−	0.950311i	$-0.399234\pi$
$61$	4.86271	0.622607	0.311303	+	0.950311i	$0.399234\pi$
$62$	15.7897	2.00530
$63$	1.00000	0.125988
$64$	−9.99559	−1.24945
$65$	0.491831	0.0610041
$66$	2.06644	0.254361
$67$	−3.49236	−0.426659	−0.213330	−	0.976980i	$-0.568431\pi$
$67$	−3.49236	−0.426659	−0.213330	+	0.976980i	$0.568431\pi$
$68$	16.8061	2.03803
$69$	−8.92701	−1.07469
$70$	−2.06644	−0.246986
$71$	−6.18136	−0.733593	−0.366796	−	0.930301i	$-0.619545\pi$
$71$	−6.18136	−0.733593	−0.366796	+	0.930301i	$0.619545\pi$
$72$	−0.558268	−0.0657925
$73$	−0.132873	−0.0155516	−0.00777580	−	0.999970i	$-0.502475\pi$
$73$	−0.132873	−0.0155516	−0.00777580	+	0.999970i	$0.502475\pi$
$74$	−14.6732	−1.70572
$75$	−1.00000	−0.115470
$76$	−1.65685	−0.190054
$77$	1.00000	0.113961
$78$	1.01634	0.115078
$79$	9.77391	1.09965	0.549826	−	0.835279i	$-0.314694\pi$
$79$	9.77391	1.09965	0.549826	+	0.835279i	$0.314694\pi$
$80$	−3.38669	−0.378644
$81$	1.00000	0.111111
$82$	4.23307	0.467464
$83$	15.0599	1.65304	0.826519	−	0.562909i	$-0.190318\pi$
$83$	15.0599	1.65304	0.826519	+	0.562909i	$0.190318\pi$
$84$	−2.27016	−0.247695
$85$	7.40303	0.802972
$86$	19.7380	2.12841
$87$	4.91120	0.526536
$88$	−0.558268	−0.0595115
$89$	−0.745653	−0.0790391	−0.0395195	−	0.999219i	$-0.512583\pi$
$89$	−0.745653	−0.0790391	−0.0395195	+	0.999219i	$0.512583\pi$
$90$	−2.06644	−0.217822
$91$	0.491831	0.0515579
$92$	20.2657	2.11285
$93$	7.64104	0.792339
$94$	13.8224	1.42567
$95$	−0.729840	−0.0748800
$96$	−8.11492	−0.828226
$97$	−3.76199	−0.381972	−0.190986	−	0.981593i	$-0.561169\pi$
$97$	−3.76199	−0.381972	−0.190986	+	0.981593i	$0.561169\pi$
$98$	−2.06644	−0.208742
$99$	1.00000	0.100504
$100$	2.27016	0.227016
$101$	9.68953	0.964144	0.482072	−	0.876132i	$-0.339884\pi$
$101$	9.68953	0.964144	0.482072	+	0.876132i	$0.339884\pi$
$102$	15.2979	1.51472
$103$	−12.3023	−1.21218	−0.606091	−	0.795395i	$-0.707263\pi$
$103$	−12.3023	−1.21218	−0.606091	+	0.795395i	$0.707263\pi$
$104$	−0.274573	−0.0269241
$105$	−1.00000	−0.0975900
$106$	1.78274	0.173155
$107$	15.2336	1.47269	0.736344	−	0.676608i	$-0.236551\pi$
$107$	15.2336	1.47269	0.736344	+	0.676608i	$0.236551\pi$
$108$	−2.27016	−0.218446
$109$	18.5114	1.77307	0.886536	−	0.462660i	$-0.153105\pi$
$109$	18.5114	1.77307	0.886536	+	0.462660i	$0.153105\pi$
$110$	−2.06644	−0.197027
$111$	−7.10072	−0.673971
$112$	−3.38669	−0.320013
$113$	−12.7494	−1.19936	−0.599682	−	0.800238i	$-0.704706\pi$
$113$	−12.7494	−1.19936	−0.599682	+	0.800238i	$0.704706\pi$
$114$	−1.50817	−0.141253
$115$	8.92701	0.832448
$116$	−11.1492	−1.03518
$117$	0.491831	0.0454698
$118$	−21.7053	−1.99814
$119$	7.40303	0.678635
$120$	0.558268	0.0509626
$121$	1.00000	0.0909091
$122$	−10.0485	−0.909747
$123$	2.04849	0.184706
$124$	−17.3464	−1.55775
$125$	1.00000	0.0894427
$126$	−2.06644	−0.184093
$127$	14.5038	1.28700	0.643500	−	0.765446i	$-0.277482\pi$
$127$	14.5038	1.28700	0.643500	+	0.765446i	$0.277482\pi$
$128$	4.42539	0.391153
$129$	9.55172	0.840982
$130$	−1.01634	−0.0891387
$131$	−1.43960	−0.125778	−0.0628891	−	0.998021i	$-0.520031\pi$
$131$	−1.43960	−0.125778	−0.0628891	+	0.998021i	$0.520031\pi$
$132$	−2.27016	−0.197592
$133$	−0.729840	−0.0632852
$134$	7.21673	0.623431
$135$	−1.00000	−0.0860663
$136$	−4.13287	−0.354391
$137$	13.7581	1.17543	0.587717	−	0.809067i	$-0.300027\pi$
$137$	13.7581	1.17543	0.587717	+	0.809067i	$0.300027\pi$
$138$	18.4471	1.57032
$139$	0.197173	0.0167240	0.00836201	−	0.999965i	$-0.497338\pi$
$139$	0.197173	0.0167240	0.00836201	+	0.999965i	$0.497338\pi$
$140$	2.27016	0.191864
$141$	6.68900	0.563316
$142$	12.7734	1.07192
$143$	0.491831	0.0411290
$144$	−3.38669	−0.282225
$145$	−4.91120	−0.407853
$146$	0.274573	0.0227239
$147$	−1.00000	−0.0824786
$148$	16.1198	1.32504
$149$	18.9389	1.55154	0.775769	−	0.631017i	$-0.217362\pi$
$149$	18.9389	1.55154	0.775769	+	0.631017i	$0.217362\pi$
$150$	2.06644	0.168724
$151$	−7.93143	−0.645450	−0.322725	−	0.946493i	$-0.604599\pi$
$151$	−7.93143	−0.645450	−0.322725	+	0.946493i	$0.604599\pi$
$152$	0.407446	0.0330483
$153$	7.40303	0.598500
$154$	−2.06644	−0.166518
$155$	−7.64104	−0.613743
$156$	−1.11654	−0.0893944
$157$	−3.28597	−0.262249	−0.131125	−	0.991366i	$-0.541859\pi$
$157$	−3.28597	−0.262249	−0.131125	+	0.991366i	$0.541859\pi$
$158$	−20.1972	−1.60680
$159$	0.862713	0.0684176
$160$	8.11492	0.641541
$161$	8.92701	0.703547
$162$	−2.06644	−0.162355
$163$	18.9231	1.48217	0.741087	−	0.671409i	$-0.234311\pi$
$163$	18.9231	1.48217	0.741087	+	0.671409i	$0.234311\pi$
$164$	−4.65039	−0.363135
$165$	−1.00000	−0.0778499
$166$	−31.1203	−2.41540
$167$	0.883465	0.0683646	0.0341823	−	0.999416i	$-0.489117\pi$
$167$	0.883465	0.0683646	0.0341823	+	0.999416i	$0.489117\pi$
$168$	0.558268	0.0430713
$169$	−12.7581	−0.981392
$170$	−15.2979	−1.17330
$171$	−0.729840	−0.0558123
$172$	−21.6839	−1.65338
$173$	9.62096	0.731468	0.365734	−	0.930719i	$-0.380818\pi$
$173$	9.62096	0.731468	0.365734	+	0.930719i	$0.380818\pi$
$174$	−10.1487	−0.769370
$175$	1.00000	0.0755929
$176$	−3.38669	−0.255282
$177$	−10.5038	−0.789511
$178$	1.54084	0.115491
$179$	1.13235	0.0846356	0.0423178	−	0.999104i	$-0.486526\pi$
$179$	1.13235	0.0846356	0.0423178	+	0.999104i	$0.486526\pi$
$180$	2.27016	0.169208
$181$	3.08064	0.228982	0.114491	−	0.993424i	$-0.463476\pi$
$181$	3.08064	0.228982	0.114491	+	0.993424i	$0.463476\pi$
$182$	−1.01634	−0.0753360
$183$	−4.86271	−0.359462
$184$	−4.98366	−0.367400
$185$	7.10072	0.522056
$186$	−15.7897	−1.15776
$187$	7.40303	0.541363
$188$	−15.1851	−1.10749
$189$	−1.00000	−0.0727393
$190$	1.50817	0.109414
$191$	−17.3464	−1.25514	−0.627570	−	0.778560i	$-0.715950\pi$
$191$	−17.3464	−1.25514	−0.627570	+	0.778560i	$0.715950\pi$
$192$	9.99559	0.721369
$193$	12.6405	0.909884	0.454942	−	0.890521i	$-0.349660\pi$
$193$	12.6405	0.909884	0.454942	+	0.890521i	$0.349660\pi$
$194$	7.77391	0.558135
$195$	−0.491831	−0.0352208
$196$	2.27016	0.162154
$197$	16.7091	1.19047	0.595237	−	0.803550i	$-0.297058\pi$
$197$	16.7091	1.19047	0.595237	+	0.803550i	$0.297058\pi$
$198$	−2.06644	−0.146855
$199$	−1.96732	−0.139460	−0.0697300	−	0.997566i	$-0.522214\pi$
$199$	−1.96732	−0.139460	−0.0697300	+	0.997566i	$0.522214\pi$
$200$	−0.558268	−0.0394755
$201$	3.49236	0.246332
$202$	−20.0228	−1.40880
$203$	−4.91120	−0.344699
$204$	−16.8061	−1.17666
$205$	−2.04849	−0.143073
$206$	25.4219	1.77123
$207$	8.92701	0.620470
$208$	−1.66568	−0.115494
$209$	−0.729840	−0.0504841
$210$	2.06644	0.142598
$211$	−22.6644	−1.56028	−0.780140	−	0.625605i	$-0.784852\pi$
$211$	−22.6644	−1.56028	−0.780140	+	0.625605i	$0.784852\pi$
$212$	−1.95850	−0.134510
$213$	6.18136	0.423540
$214$	−31.4793	−2.15188
$215$	−9.55172	−0.651422
$216$	0.558268	0.0379853
$217$	−7.64104	−0.518708
$218$	−38.2526	−2.59080
$219$	0.132873	0.00897872
$220$	2.27016	0.153054
$221$	3.64104	0.244923
$222$	14.6732	0.984800
$223$	−13.7620	−0.921571	−0.460786	−	0.887512i	$-0.652432\pi$
$223$	−13.7620	−0.921571	−0.460786	+	0.887512i	$0.652432\pi$
$224$	8.11492	0.542201
$225$	1.00000	0.0666667
$226$	26.3459	1.75250
$227$	2.18952	0.145324	0.0726618	−	0.997357i	$-0.476851\pi$
$227$	2.18952	0.145324	0.0726618	+	0.997357i	$0.476851\pi$
$228$	1.65685	0.109728
$229$	14.1487	0.934971	0.467486	−	0.884001i	$-0.345160\pi$
$229$	14.1487	0.934971	0.467486	+	0.884001i	$0.345160\pi$
$230$	−18.4471	−1.21637
$231$	−1.00000	−0.0657952
$232$	2.74176	0.180006
$233$	−18.0038	−1.17947	−0.589733	−	0.807598i	$-0.700767\pi$
$233$	−18.0038	−1.17947	−0.589733	+	0.807598i	$0.700767\pi$
$234$	−1.01634	−0.0664401
$235$	−6.68900	−0.436343
$236$	23.8452	1.55219
$237$	−9.77391	−0.634884
$238$	−15.2979	−0.991615
$239$	0.270685	0.0175091	0.00875457	−	0.999962i	$-0.497213\pi$
$239$	0.270685	0.0175091	0.00875457	+	0.999962i	$0.497213\pi$
$240$	3.38669	0.218610
$241$	−18.4760	−1.19014	−0.595072	−	0.803672i	$-0.702877\pi$
$241$	−18.4760	−1.19014	−0.595072	+	0.803672i	$0.702877\pi$
$242$	−2.06644	−0.132836
$243$	−1.00000	−0.0641500
$244$	11.0391	0.706708
$245$	1.00000	0.0638877
$246$	−4.23307	−0.269891
$247$	−0.358958	−0.0228400
$248$	4.26575	0.270875
$249$	−15.0599	−0.954382
$250$	−2.06644	−0.130693
$251$	−22.4988	−1.42011	−0.710056	−	0.704145i	$-0.751331\pi$
$251$	−22.4988	−1.42011	−0.710056	+	0.704145i	$0.751331\pi$
$252$	2.27016	0.143007
$253$	8.92701	0.561237
$254$	−29.9711	−1.88055
$255$	−7.40303	−0.463596
$256$	10.8464	0.677898
$257$	28.0027	1.74676	0.873381	−	0.487038i	$-0.161923\pi$
$257$	28.0027	1.74676	0.873381	+	0.487038i	$0.161923\pi$
$258$	−19.7380	−1.22884
$259$	7.10072	0.441217
$260$	1.11654	0.0692446
$261$	−4.91120	−0.303996
$262$	2.97483	0.183786
$263$	−28.1240	−1.73420	−0.867101	−	0.498132i	$-0.834020\pi$
$263$	−28.1240	−1.73420	−0.867101	+	0.498132i	$0.834020\pi$
$264$	0.558268	0.0343590
$265$	−0.862713	−0.0529961
$266$	1.50817	0.0924718
$267$	0.745653	0.0456332
$268$	−7.92821	−0.484292
$269$	−26.9624	−1.64393	−0.821963	−	0.569540i	$-0.807121\pi$
$269$	−26.9624	−1.64393	−0.821963	+	0.569540i	$0.807121\pi$
$270$	2.06644	0.125759
$271$	19.6645	1.19453	0.597267	−	0.802043i	$-0.296253\pi$
$271$	19.6645	1.19453	0.597267	+	0.802043i	$0.296253\pi$
$272$	−25.0718	−1.52020
$273$	−0.491831	−0.0297670
$274$	−28.4302	−1.71753
$275$	1.00000	0.0603023
$276$	−20.2657	−1.21985
$277$	22.1655	1.33180	0.665899	−	0.746042i	$-0.268048\pi$
$277$	22.1655	1.33180	0.665899	+	0.746042i	$0.268048\pi$
$278$	−0.407446	−0.0244370
$279$	−7.64104	−0.457457
$280$	−0.558268	−0.0333629
$281$	−2.10020	−0.125287	−0.0626436	−	0.998036i	$-0.519953\pi$
$281$	−2.10020	−0.125287	−0.0626436	+	0.998036i	$0.519953\pi$
$282$	−13.8224	−0.823112
$283$	3.64104	0.216437	0.108219	−	0.994127i	$-0.465485\pi$
$283$	3.64104	0.216437	0.108219	+	0.994127i	$0.465485\pi$
$284$	−14.0327	−0.832686
$285$	0.729840	0.0432320
$286$	−1.01634	−0.0600973
$287$	−2.04849	−0.120918
$288$	8.11492	0.478177
$289$	37.8049	2.22382
$290$	10.1487	0.595951
$291$	3.76199	0.220532
$292$	−0.301643	−0.0176523
$293$	−22.0304	−1.28703	−0.643516	−	0.765432i	$-0.722525\pi$
$293$	−22.0304	−1.28703	−0.643516	+	0.765432i	$0.722525\pi$
$294$	2.06644	0.120517
$295$	10.5038	0.611552
$296$	−3.96410	−0.230409
$297$	−1.00000	−0.0580259
$298$	−39.1361	−2.26709
$299$	4.39058	0.253914
$300$	−2.27016	−0.131068
$301$	−9.55172	−0.550552
$302$	16.3898	0.943126
$303$	−9.68953	−0.556649
$304$	2.47175	0.141764
$305$	4.86271	0.278438
$306$	−15.2979	−0.874523
$307$	−3.39538	−0.193785	−0.0968923	−	0.995295i	$-0.530890\pi$
$307$	−3.39538	−0.193785	−0.0968923	+	0.995295i	$0.530890\pi$
$308$	2.27016	0.129354
$309$	12.3023	0.699854
$310$	15.7897	0.896796
$311$	25.6438	1.45412	0.727062	−	0.686572i	$-0.240885\pi$
$311$	25.6438	1.45412	0.727062	+	0.686572i	$0.240885\pi$
$312$	0.274573	0.0155447
$313$	2.17371	0.122865	0.0614326	−	0.998111i	$-0.480433\pi$
$313$	2.17371	0.122865	0.0614326	+	0.998111i	$0.480433\pi$
$314$	6.79025	0.383196
$315$	1.00000	0.0563436
$316$	22.1883	1.24819
$317$	6.60462	0.370952	0.185476	−	0.982649i	$-0.440617\pi$
$317$	6.60462	0.370952	0.185476	+	0.982649i	$0.440617\pi$
$318$	−1.78274	−0.0999712
$319$	−4.91120	−0.274974
$320$	−9.99559	−0.558770
$321$	−15.2336	−0.850256
$322$	−18.4471	−1.02802
$323$	−5.40303	−0.300633
$324$	2.27016	0.126120
$325$	0.491831	0.0272819
$326$	−39.1034	−2.16574
$327$	−18.5114	−1.02368
$328$	1.14360	0.0631450
$329$	−6.68900	−0.368777
$330$	2.06644	0.113754
$331$	−21.5316	−1.18349	−0.591743	−	0.806127i	$-0.701560\pi$
$331$	−21.5316	−1.18349	−0.591743	+	0.806127i	$0.701560\pi$
$332$	34.1883	1.87633
$333$	7.10072	0.389117
$334$	−1.82562	−0.0998937
$335$	−3.49236	−0.190808
$336$	3.38669	0.184759
$337$	2.31541	0.126128	0.0630642	−	0.998009i	$-0.479913\pi$
$337$	2.31541	0.126128	0.0630642	+	0.998009i	$0.479913\pi$
$338$	26.3638	1.43400
$339$	12.7494	0.692453
$340$	16.8061	0.911437
$341$	−7.64104	−0.413786
$342$	1.50817	0.0815524
$343$	1.00000	0.0539949
$344$	5.33241	0.287505
$345$	−8.92701	−0.480614
$346$	−19.8811	−1.06881
$347$	18.9063	1.01494	0.507471	−	0.861669i	$-0.330581\pi$
$347$	18.9063	1.01494	0.507471	+	0.861669i	$0.330581\pi$
$348$	11.1492	0.597660
$349$	−12.9913	−0.695409	−0.347705	−	0.937604i	$-0.613039\pi$
$349$	−12.9913	−0.695409	−0.347705	+	0.937604i	$0.613039\pi$
$350$	−2.06644	−0.110456
$351$	−0.491831	−0.0262520
$352$	8.11492	0.432527
$353$	−0.201445	−0.0107218	−0.00536092	−	0.999986i	$-0.501706\pi$
$353$	−0.201445	−0.0107218	−0.00536092	+	0.999986i	$0.501706\pi$
$354$	21.7053	1.15363
$355$	−6.18136	−0.328073
$356$	−1.69275	−0.0897156
$357$	−7.40303	−0.391810
$358$	−2.33992	−0.123669
$359$	−10.4384	−0.550918	−0.275459	−	0.961313i	$-0.588830\pi$
$359$	−10.4384	−0.550918	−0.275459	+	0.961313i	$0.588830\pi$
$360$	−0.558268	−0.0294233
$361$	−18.4673	−0.971965
$362$	−6.36594	−0.334587
$363$	−1.00000	−0.0524864
$364$	1.11654	0.0585223
$365$	−0.132873	−0.00695488
$366$	10.0485	0.525243
$367$	2.53538	0.132346	0.0661729	−	0.997808i	$-0.478921\pi$
$367$	2.53538	0.132346	0.0661729	+	0.997808i	$0.478921\pi$
$368$	−30.2331	−1.57601
$369$	−2.04849	−0.106640
$370$	−14.6732	−0.762823
$371$	−0.862713	−0.0447898
$372$	17.3464	0.899368
$373$	31.9830	1.65602	0.828009	−	0.560715i	$-0.189474\pi$
$373$	31.9830	1.65602	0.828009	+	0.560715i	$0.189474\pi$
$374$	−15.2979	−0.791035
$375$	−1.00000	−0.0516398
$376$	3.73425	0.192579
$377$	−2.41548	−0.124404
$378$	2.06644	0.106286
$379$	23.8301	1.22407	0.612034	−	0.790832i	$-0.290352\pi$
$379$	23.8301	1.22407	0.612034	+	0.790832i	$0.290352\pi$
$380$	−1.65685	−0.0849948
$381$	−14.5038	−0.743050
$382$	35.8452	1.83400
$383$	−15.5679	−0.795483	−0.397742	−	0.917497i	$-0.630206\pi$
$383$	−15.5679	−0.795483	−0.397742	+	0.917497i	$0.630206\pi$
$384$	−4.42539	−0.225832
$385$	1.00000	0.0509647
$386$	−26.1208	−1.32951
$387$	−9.55172	−0.485541
$388$	−8.54032	−0.433569
$389$	30.4798	1.54539	0.772693	−	0.634780i	$-0.218909\pi$
$389$	30.4798	1.54539	0.772693	+	0.634780i	$0.218909\pi$
$390$	1.01634	0.0514643
$391$	66.0870	3.34216
$392$	−0.558268	−0.0281968
$393$	1.43960	0.0726180
$394$	−34.5283	−1.73951
$395$	9.77391	0.491779
$396$	2.27016	0.114080
$397$	−39.5390	−1.98441	−0.992203	−	0.124634i	$-0.960224\pi$
$397$	−39.5390	−1.98441	−0.992203	+	0.124634i	$0.960224\pi$
$398$	4.06535	0.203778
$399$	0.729840	0.0365377
$400$	−3.38669	−0.169335
$401$	4.52023	0.225730	0.112865	−	0.993610i	$-0.463997\pi$
$401$	4.52023	0.225730	0.112865	+	0.993610i	$0.463997\pi$
$402$	−7.21673	−0.359938
$403$	−3.75810	−0.187204
$404$	21.9968	1.09438
$405$	1.00000	0.0496904
$406$	10.1487	0.503671
$407$	7.10072	0.351970
$408$	4.13287	0.204608
$409$	−2.87037	−0.141930	−0.0709652	−	0.997479i	$-0.522608\pi$
$409$	−2.87037	−0.141930	−0.0709652	+	0.997479i	$0.522608\pi$
$410$	4.23307	0.209056
$411$	−13.7581	−0.678637
$412$	−27.9282	−1.37592
$413$	10.5038	0.516856
$414$	−18.4471	−0.906626
$415$	15.0599	0.739261
$416$	3.99117	0.195683
$417$	−0.197173	−0.00965562
$418$	1.50817	0.0737670
$419$	−30.2619	−1.47839	−0.739194	−	0.673492i	$-0.764793\pi$
$419$	−30.2619	−1.47839	−0.739194	+	0.673492i	$0.764793\pi$
$420$	−2.27016	−0.110772
$421$	27.3061	1.33082	0.665408	−	0.746480i	$-0.268258\pi$
$421$	27.3061	1.33082	0.665408	+	0.746480i	$0.268258\pi$
$422$	46.8345	2.27987
$423$	−6.68900	−0.325231
$424$	0.481625	0.0233898
$425$	7.40303	0.359100
$426$	−12.7734	−0.618873
$427$	4.86271	0.235323
$428$	34.5827	1.67162
$429$	−0.491831	−0.0237458
$430$	19.7380	0.951852
$431$	−8.09698	−0.390018	−0.195009	−	0.980801i	$-0.562474\pi$
$431$	−8.09698	−0.390018	−0.195009	+	0.980801i	$0.562474\pi$
$432$	3.38669	0.162942
$433$	−7.66995	−0.368594	−0.184297	−	0.982871i	$-0.559001\pi$
$433$	−7.66995	−0.368594	−0.184297	+	0.982871i	$0.559001\pi$
$434$	15.7897	0.757931
$435$	4.91120	0.235474
$436$	42.0238	2.01258
$437$	−6.51530	−0.311669
$438$	−0.274573	−0.0131196
$439$	−6.39979	−0.305446	−0.152723	−	0.988269i	$-0.548804\pi$
$439$	−6.39979	−0.305446	−0.152723	+	0.988269i	$0.548804\pi$
$440$	−0.558268	−0.0266144
$441$	1.00000	0.0476190
$442$	−7.52398	−0.357879
$443$	−28.4182	−1.35019	−0.675095	−	0.737731i	$-0.735897\pi$
$443$	−28.4182	−1.35019	−0.675095	+	0.737731i	$0.735897\pi$
$444$	−16.1198	−0.765011
$445$	−0.745653	−0.0353473
$446$	28.4383	1.34659
$447$	−18.9389	−0.895781
$448$	−9.99559	−0.472247
$449$	−16.2368	−0.766264	−0.383132	−	0.923694i	$-0.625154\pi$
$449$	−16.2368	−0.766264	−0.383132	+	0.923694i	$0.625154\pi$
$450$	−2.06644	−0.0974127
$451$	−2.04849	−0.0964595
$452$	−28.9432	−1.36137
$453$	7.93143	0.372651
$454$	−4.52451	−0.212346
$455$	0.491831	0.0230574
$456$	−0.407446	−0.0190804
$457$	10.0320	0.469278	0.234639	−	0.972083i	$-0.424609\pi$
$457$	10.0320	0.469278	0.234639	+	0.972083i	$0.424609\pi$
$458$	−29.2374	−1.36617
$459$	−7.40303	−0.345544
$460$	20.2657	0.944895
$461$	23.8824	1.11232	0.556158	−	0.831077i	$-0.312275\pi$
$461$	23.8824	1.11232	0.556158	+	0.831077i	$0.312275\pi$
$462$	2.06644	0.0961393
$463$	−2.50869	−0.116589	−0.0582945	−	0.998299i	$-0.518566\pi$
$463$	−2.50869	−0.116589	−0.0582945	+	0.998299i	$0.518566\pi$
$464$	16.6327	0.772156
$465$	7.64104	0.354345
$466$	37.2036	1.72343
$467$	21.4434	0.992280	0.496140	−	0.868242i	$-0.334750\pi$
$467$	21.4434	0.992280	0.496140	+	0.868242i	$0.334750\pi$
$468$	1.11654	0.0516119
$469$	−3.49236	−0.161262
$470$	13.8224	0.637580
$471$	3.28597	0.151410
$472$	−5.86391	−0.269908
$473$	−9.55172	−0.439188
$474$	20.1972	0.927687
$475$	−0.729840	−0.0334874
$476$	16.8061	0.770305
$477$	−0.862713	−0.0395009
$478$	−0.559353	−0.0255842
$479$	−24.2695	−1.10890	−0.554451	−	0.832216i	$-0.687072\pi$
$479$	−24.2695	−1.10890	−0.554451	+	0.832216i	$0.687072\pi$
$480$	−8.11492	−0.370394
$481$	3.49236	0.159238
$482$	38.1795	1.73903
$483$	−8.92701	−0.406193
$484$	2.27016	0.103189
$485$	−3.76199	−0.170823
$486$	2.06644	0.0937355
$487$	−18.7091	−0.847790	−0.423895	−	0.905711i	$-0.639337\pi$
$487$	−18.7091	−0.847790	−0.423895	+	0.905711i	$0.639337\pi$
$488$	−2.71470	−0.122888
$489$	−18.9231	−0.855733
$490$	−2.06644	−0.0933521
$491$	27.3425	1.23395	0.616975	−	0.786983i	$-0.288358\pi$
$491$	27.3425	1.23395	0.616975	+	0.786983i	$0.288358\pi$
$492$	4.65039	0.209656
$493$	−36.3578	−1.63747
$494$	0.741764	0.0333736
$495$	1.00000	0.0449467
$496$	25.8779	1.16195
$497$	−6.18136	−0.277272
$498$	31.1203	1.39453
$499$	14.4105	0.645104	0.322552	−	0.946552i	$-0.395459\pi$
$499$	14.4105	0.645104	0.322552	+	0.946552i	$0.395459\pi$
$500$	2.27016	0.101525
$501$	−0.883465	−0.0394703
$502$	46.4924	2.07506
$503$	36.4063	1.62328	0.811638	−	0.584161i	$-0.198576\pi$
$503$	36.4063	1.62328	0.811638	+	0.584161i	$0.198576\pi$
$504$	−0.558268	−0.0248672
$505$	9.68953	0.431178
$506$	−18.4471	−0.820074
$507$	12.7581	0.566607
$508$	32.9258	1.46085
$509$	−0.503755	−0.0223286	−0.0111643	−	0.999938i	$-0.503554\pi$
$509$	−0.503755	−0.0223286	−0.0111643	+	0.999938i	$0.503554\pi$
$510$	15.2979	0.677402
$511$	−0.132873	−0.00587795
$512$	−31.2641	−1.38169
$513$	0.729840	0.0322232
$514$	−57.8658	−2.55235
$515$	−12.3023	−0.542105
$516$	21.6839	0.954582
$517$	−6.68900	−0.294182
$518$	−14.6732	−0.644703
$519$	−9.62096	−0.422313
$520$	−0.274573	−0.0120408
$521$	38.5102	1.68716	0.843582	−	0.537000i	$-0.180443\pi$
$521$	38.5102	1.68716	0.843582	+	0.537000i	$0.180443\pi$
$522$	10.1487	0.444196
$523$	−32.9575	−1.44113	−0.720565	−	0.693388i	$-0.756117\pi$
$523$	−32.9575	−1.44113	−0.720565	+	0.693388i	$0.756117\pi$
$524$	−3.26811	−0.142768
$525$	−1.00000	−0.0436436
$526$	58.1166	2.53400
$527$	−56.5669	−2.46409
$528$	3.38669	0.147387
$529$	56.6916	2.46485
$530$	1.78274	0.0774374
$531$	10.5038	0.455824
$532$	−1.65685	−0.0718337
$533$	−1.00751	−0.0436401
$534$	−1.54084	−0.0666789
$535$	15.2336	0.658606
$536$	1.94967	0.0842129
$537$	−1.13235	−0.0488644
$538$	55.7161	2.40209
$539$	1.00000	0.0430730
$540$	−2.27016	−0.0976921
$541$	−17.3906	−0.747680	−0.373840	−	0.927493i	$-0.621959\pi$
$541$	−17.3906	−0.747680	−0.373840	+	0.927493i	$0.621959\pi$
$542$	−40.6355	−1.74544
$543$	−3.08064	−0.132203
$544$	60.0750	2.57570
$545$	18.5114	0.792942
$546$	1.01634	0.0434952
$547$	16.0920	0.688046	0.344023	−	0.938961i	$-0.388210\pi$
$547$	16.0920	0.688046	0.344023	+	0.938961i	$0.388210\pi$
$548$	31.2331	1.33421
$549$	4.86271	0.207536
$550$	−2.06644	−0.0881131
$551$	3.58439	0.152700
$552$	4.98366	0.212119
$553$	9.77391	0.415629
$554$	−45.8037	−1.94601
$555$	−7.10072	−0.301409
$556$	0.447615	0.0189831
$557$	6.29737	0.266828	0.133414	−	0.991060i	$-0.457406\pi$
$557$	6.29737	0.266828	0.133414	+	0.991060i	$0.457406\pi$
$558$	15.7897	0.668432
$559$	−4.69783	−0.198697
$560$	−3.38669	−0.143114
$561$	−7.40303	−0.312556
$562$	4.33992	0.183069
$563$	10.8192	0.455973	0.227987	−	0.973664i	$-0.426786\pi$
$563$	10.8192	0.455973	0.227987	+	0.973664i	$0.426786\pi$
$564$	15.1851	0.639408
$565$	−12.7494	−0.536372
$566$	−7.52398	−0.316257
$567$	1.00000	0.0419961
$568$	3.45085	0.144795
$569$	−14.4362	−0.605198	−0.302599	−	0.953118i	$-0.597854\pi$
$569$	−14.4362	−0.605198	−0.302599	+	0.953118i	$0.597854\pi$
$570$	−1.50817	−0.0631702
$571$	−29.2162	−1.22266	−0.611330	−	0.791375i	$-0.709365\pi$
$571$	−29.2162	−1.22266	−0.611330	+	0.791375i	$0.709365\pi$
$572$	1.11654	0.0466847
$573$	17.3464	0.724656
$574$	4.23307	0.176685
$575$	8.92701	0.372282
$576$	−9.99559	−0.416483
$577$	17.7734	0.739917	0.369958	−	0.929048i	$-0.379372\pi$
$577$	17.7734	0.739917	0.369958	+	0.929048i	$0.379372\pi$
$578$	−78.1214	−3.24942
$579$	−12.6405	−0.525322
$580$	−11.1492	−0.462946
$581$	15.0599	0.624789
$582$	−7.77391	−0.322239
$583$	−0.862713	−0.0357299
$584$	0.0741786	0.00306953
$585$	0.491831	0.0203347
$586$	45.5245	1.88060
$587$	−38.8415	−1.60316	−0.801579	−	0.597889i	$-0.796006\pi$
$587$	−38.8415	−1.60316	−0.801579	+	0.597889i	$0.796006\pi$
$588$	−2.27016	−0.0936198
$589$	5.57674	0.229786
$590$	−21.7053	−0.893595
$591$	−16.7091	−0.687320
$592$	−24.0480	−0.988366
$593$	−12.9748	−0.532813	−0.266406	−	0.963861i	$-0.585836\pi$
$593$	−12.9748	−0.532813	−0.266406	+	0.963861i	$0.585836\pi$
$594$	2.06644	0.0847869
$595$	7.40303	0.303495
$596$	42.9944	1.76112
$597$	1.96732	0.0805172
$598$	−9.07286	−0.371017
$599$	40.7156	1.66359	0.831796	−	0.555081i	$-0.187313\pi$
$599$	40.7156	1.66359	0.831796	+	0.555081i	$0.187313\pi$
$600$	0.558268	0.0227912
$601$	−23.8791	−0.974047	−0.487023	−	0.873389i	$-0.661917\pi$
$601$	−23.8791	−0.974047	−0.487023	+	0.873389i	$0.661917\pi$
$602$	19.7380	0.804462
$603$	−3.49236	−0.142220
$604$	−18.0056	−0.732638
$605$	1.00000	0.0406558
$606$	20.0228	0.813371
$607$	−46.5478	−1.88932	−0.944659	−	0.328053i	$-0.893608\pi$
$607$	−46.5478	−1.88932	−0.944659	+	0.328053i	$0.893608\pi$
$608$	−5.92260	−0.240193
$609$	4.91120	0.199012
$610$	−10.0485	−0.406851
$611$	−3.28986	−0.133094
$612$	16.8061	0.679345
$613$	28.8661	1.16589	0.582945	−	0.812511i	$-0.301900\pi$
$613$	28.8661	1.16589	0.582945	+	0.812511i	$0.301900\pi$
$614$	7.01634	0.283156
$615$	2.04849	0.0826030
$616$	−0.558268	−0.0224932
$617$	25.9269	1.04378	0.521888	−	0.853014i	$-0.325228\pi$
$617$	25.9269	1.04378	0.521888	+	0.853014i	$0.325228\pi$
$618$	−25.4219	−1.02262
$619$	−8.77339	−0.352632	−0.176316	−	0.984334i	$-0.556418\pi$
$619$	−8.77339	−0.352632	−0.176316	+	0.984334i	$0.556418\pi$
$620$	−17.3464	−0.696648
$621$	−8.92701	−0.358229
$622$	−52.9912	−2.12475
$623$	−0.745653	−0.0298740
$624$	1.66568	0.0666806
$625$	1.00000	0.0400000
$626$	−4.49183	−0.179530
$627$	0.729840	0.0291470
$628$	−7.45968	−0.297674
$629$	52.5669	2.09598
$630$	−2.06644	−0.0823288
$631$	−22.0478	−0.877710	−0.438855	−	0.898558i	$-0.644616\pi$
$631$	−22.0478	−0.877710	−0.438855	+	0.898558i	$0.644616\pi$
$632$	−5.45646	−0.217046
$633$	22.6644	0.900828
$634$	−13.6480	−0.542032
$635$	14.5038	0.575564
$636$	1.95850	0.0776595
$637$	0.491831	0.0194871
$638$	10.1487	0.401790
$639$	−6.18136	−0.244531
$640$	4.42539	0.174929
$641$	−15.2092	−0.600729	−0.300364	−	0.953825i	$-0.597108\pi$
$641$	−15.2092	−0.600729	−0.300364	+	0.953825i	$0.597108\pi$
$642$	31.4793	1.24239
$643$	17.5517	0.692172	0.346086	−	0.938203i	$-0.387511\pi$
$643$	17.5517	0.692172	0.346086	+	0.938203i	$0.387511\pi$
$644$	20.2657	0.798582
$645$	9.55172	0.376099
$646$	11.1650	0.439282
$647$	−20.9837	−0.824953	−0.412476	−	0.910968i	$-0.635336\pi$
$647$	−20.9837	−0.824953	−0.412476	+	0.910968i	$0.635336\pi$
$648$	−0.558268	−0.0219308
$649$	10.5038	0.412308
$650$	−1.01634	−0.0398640
$651$	7.64104	0.299476
$652$	42.9585	1.68239
$653$	13.3942	0.524156	0.262078	−	0.965047i	$-0.415592\pi$
$653$	13.3942	0.524156	0.262078	+	0.965047i	$0.415592\pi$
$654$	38.2526	1.49580
$655$	−1.43960	−0.0562497
$656$	6.93760	0.270868
$657$	−0.132873	−0.00518386
$658$	13.8224	0.538853
$659$	−24.9308	−0.971165	−0.485583	−	0.874191i	$-0.661392\pi$
$659$	−24.9308	−0.971165	−0.485583	+	0.874191i	$0.661392\pi$
$660$	−2.27016	−0.0883658
$661$	−23.4634	−0.912622	−0.456311	−	0.889820i	$-0.650830\pi$
$661$	−23.4634	−0.912622	−0.456311	+	0.889820i	$0.650830\pi$
$662$	44.4938	1.72930
$663$	−3.64104	−0.141406
$664$	−8.40745	−0.326272
$665$	−0.729840	−0.0283020
$666$	−14.6732	−0.568575
$667$	−43.8424	−1.69758
$668$	2.00561	0.0775992
$669$	13.7620	0.532069
$670$	7.21673	0.278807
$671$	4.86271	0.187723
$672$	−8.11492	−0.313040
$673$	6.28378	0.242222	0.121111	−	0.992639i	$-0.461354\pi$
$673$	6.28378	0.242222	0.121111	+	0.992639i	$0.461354\pi$
$674$	−4.78465	−0.184298
$675$	−1.00000	−0.0384900
$676$	−28.9629	−1.11396
$677$	1.10461	0.0424537	0.0212268	−	0.999775i	$-0.493243\pi$
$677$	1.10461	0.0424537	0.0212268	+	0.999775i	$0.493243\pi$
$678$	−26.3459	−1.01181
$679$	−3.76199	−0.144372
$680$	−4.13287	−0.158488
$681$	−2.18952	−0.0839026
$682$	15.7897	0.604620
$683$	44.7254	1.71137	0.855686	−	0.517496i	$-0.173136\pi$
$683$	44.7254	1.71137	0.855686	+	0.517496i	$0.173136\pi$
$684$	−1.65685	−0.0633514
$685$	13.7581	0.525670
$686$	−2.06644	−0.0788969
$687$	−14.1487	−0.539806
$688$	32.3488	1.23328
$689$	−0.424309	−0.0161649
$690$	18.4471	0.702269
$691$	−17.1045	−0.650685	−0.325343	−	0.945596i	$-0.605480\pi$
$691$	−17.1045	−0.650685	−0.325343	+	0.945596i	$0.605480\pi$
$692$	21.8411	0.830274
$693$	1.00000	0.0379869
$694$	−39.0686	−1.48302
$695$	0.197173	0.00747921
$696$	−2.74176	−0.103926
$697$	−15.1650	−0.574416
$698$	26.8457	1.01613
$699$	18.0038	0.680965
$700$	2.27016	0.0858040
$701$	6.74992	0.254941	0.127471	−	0.991842i	$-0.459314\pi$
$701$	6.74992	0.254941	0.127471	+	0.991842i	$0.459314\pi$
$702$	1.01634	0.0383592
$703$	−5.18239	−0.195458
$704$	−9.99559	−0.376723
$705$	6.68900	0.251922
$706$	0.416274	0.0156667
$707$	9.68953	0.364412
$708$	−23.8452	−0.896157
$709$	−15.3061	−0.574831	−0.287416	−	0.957806i	$-0.592796\pi$
$709$	−15.3061	−0.574831	−0.287416	+	0.957806i	$0.592796\pi$
$710$	12.7734	0.479377
$711$	9.77391	0.366550
$712$	0.416274	0.0156005
$713$	−68.2117	−2.55455
$714$	15.2979	0.572509
$715$	0.491831	0.0183934
$716$	2.57061	0.0960682
$717$	−0.270685	−0.0101089
$718$	21.5703	0.804996
$719$	−17.0757	−0.636816	−0.318408	−	0.947954i	$-0.603148\pi$
$719$	−17.0757	−0.636816	−0.318408	+	0.947954i	$0.603148\pi$
$720$	−3.38669	−0.126215
$721$	−12.3023	−0.458162
$722$	38.1616	1.42023
$723$	18.4760	0.687131
$724$	6.99354	0.259913
$725$	−4.91120	−0.182397
$726$	2.06644	0.0766926
$727$	−45.0342	−1.67023	−0.835113	−	0.550079i	$-0.814598\pi$
$727$	−45.0342	−1.67023	−0.835113	+	0.550079i	$0.814598\pi$
$728$	−0.274573	−0.0101764
$729$	1.00000	0.0370370
$730$	0.274573	0.0101624
$731$	−70.7117	−2.61537
$732$	−11.0391	−0.408018
$733$	−0.174376	−0.00644072	−0.00322036	−	0.999995i	$-0.501025\pi$
$733$	−0.174376	−0.00644072	−0.00322036	+	0.999995i	$0.501025\pi$
$734$	−5.23920	−0.193382
$735$	−1.00000	−0.0368856
$736$	72.4420	2.67025
$737$	−3.49236	−0.128643
$738$	4.23307	0.155821
$739$	−23.8736	−0.878205	−0.439102	−	0.898437i	$-0.644703\pi$
$739$	−23.8736	−0.878205	−0.439102	+	0.898437i	$0.644703\pi$
$740$	16.1198	0.592575
$741$	0.358958	0.0131867
$742$	1.78274	0.0654465
$743$	39.9454	1.46545	0.732727	−	0.680522i	$-0.238247\pi$
$743$	39.9454	1.46545	0.732727	+	0.680522i	$0.238247\pi$
$744$	−4.26575	−0.156390
$745$	18.9389	0.693869
$746$	−66.0909	−2.41976
$747$	15.0599	0.551012
$748$	16.8061	0.614491
$749$	15.2336	0.556623
$750$	2.06644	0.0754556
$751$	14.2418	0.519690	0.259845	−	0.965650i	$-0.416329\pi$
$751$	14.2418	0.519690	0.259845	+	0.965650i	$0.416329\pi$
$752$	22.6536	0.826092
$753$	22.4988	0.819902
$754$	4.99144	0.181777
$755$	−7.93143	−0.288654
$756$	−2.27016	−0.0825649
$757$	14.0970	0.512363	0.256182	−	0.966629i	$-0.417535\pi$
$757$	14.0970	0.512363	0.256182	+	0.966629i	$0.417535\pi$
$758$	−49.2433	−1.78860
$759$	−8.92701	−0.324030
$760$	0.407446	0.0147796
$761$	−37.5980	−1.36293	−0.681463	−	0.731853i	$-0.738656\pi$
$761$	−37.5980	−1.36293	−0.681463	+	0.731853i	$0.738656\pi$
$762$	29.9711	1.08574
$763$	18.5114	0.670158
$764$	−39.3791	−1.42468
$765$	7.40303	0.267657
$766$	32.1701	1.16235
$767$	5.16607	0.186536
$768$	−10.8464	−0.391385
$769$	15.5970	0.562442	0.281221	−	0.959643i	$-0.409261\pi$
$769$	15.5970	0.562442	0.281221	+	0.959643i	$0.409261\pi$
$770$	−2.06644	−0.0744692
$771$	−28.0027	−1.00849
$772$	28.6960	1.03279
$773$	−33.8136	−1.21619	−0.608095	−	0.793864i	$-0.708066\pi$
$773$	−33.8136	−1.21619	−0.608095	+	0.793864i	$0.708066\pi$
$774$	19.7380	0.709468
$775$	−7.64104	−0.274474
$776$	2.10020	0.0753927
$777$	−7.10072	−0.254737
$778$	−62.9845	−2.25810
$779$	1.49507	0.0535664
$780$	−1.11654	−0.0399784
$781$	−6.18136	−0.221186
$782$	−136.565	−4.88354
$783$	4.91120	0.175512
$784$	−3.38669	−0.120953
$785$	−3.28597	−0.117281
$786$	−2.97483	−0.106109
$787$	−28.1371	−1.00298	−0.501490	−	0.865163i	$-0.667215\pi$
$787$	−28.1371	−1.00298	−0.501490	+	0.865163i	$0.667215\pi$
$788$	37.9323	1.35128
$789$	28.1240	1.00124
$790$	−20.1972	−0.718583
$791$	−12.7494	−0.453317
$792$	−0.558268	−0.0198372
$793$	2.39163	0.0849294
$794$	81.7048	2.89960
$795$	0.862713	0.0305973
$796$	−4.46614	−0.158298
$797$	−31.3780	−1.11147	−0.555733	−	0.831361i	$-0.687562\pi$
$797$	−31.3780	−1.11147	−0.555733	+	0.831361i	$0.687562\pi$
$798$	−1.50817	−0.0533886
$799$	−49.5189	−1.75185
$800$	8.11492	0.286906
$801$	−0.745653	−0.0263464
$802$	−9.34078	−0.329834
$803$	−0.132873	−0.00468898
$804$	7.92821	0.279606
$805$	8.92701	0.314636
$806$	7.76588	0.273541
$807$	26.9624	0.949122
$808$	−5.40935	−0.190300
$809$	−21.9954	−0.773319	−0.386659	−	0.922223i	$-0.626371\pi$
$809$	−21.9954	−0.773319	−0.386659	+	0.922223i	$0.626371\pi$
$810$	−2.06644	−0.0726072
$811$	12.7134	0.446426	0.223213	−	0.974770i	$-0.428345\pi$
$811$	12.7134	0.446426	0.223213	+	0.974770i	$0.428345\pi$
$812$	−11.1492	−0.391261
$813$	−19.6645	−0.689664
$814$	−14.6732	−0.514295
$815$	18.9231	0.662848
$816$	25.0718	0.877689
$817$	6.97123	0.243892
$818$	5.93143	0.207388
$819$	0.491831	0.0171860
$820$	−4.65039	−0.162399
$821$	−15.5332	−0.542112	−0.271056	−	0.962564i	$-0.587373\pi$
$821$	−15.5332	−0.542112	−0.271056	+	0.962564i	$0.587373\pi$
$822$	28.4302	0.991619
$823$	−23.9384	−0.834441	−0.417220	−	0.908805i	$-0.636996\pi$
$823$	−23.9384	−0.834441	−0.417220	+	0.908805i	$0.636996\pi$
$824$	6.86798	0.239257
$825$	−1.00000	−0.0348155
$826$	−21.7053	−0.755225
$827$	39.5031	1.37366	0.686829	−	0.726819i	$-0.259002\pi$
$827$	39.5031	1.37366	0.686829	+	0.726819i	$0.259002\pi$
$828$	20.2657	0.704283
$829$	−30.3627	−1.05454	−0.527270	−	0.849698i	$-0.676785\pi$
$829$	−30.3627	−1.05454	−0.527270	+	0.849698i	$0.676785\pi$
$830$	−31.1203	−1.08020
$831$	−22.1655	−0.768914
$832$	−4.91614	−0.170437
$833$	7.40303	0.256500
$834$	0.407446	0.0141087
$835$	0.883465	0.0305736
$836$	−1.65685	−0.0573035
$837$	7.64104	0.264113
$838$	62.5342	2.16021
$839$	39.3425	1.35825	0.679127	−	0.734021i	$-0.262359\pi$
$839$	39.3425	1.35825	0.679127	+	0.734021i	$0.262359\pi$
$840$	0.558268	0.0192621
$841$	−4.88010	−0.168279
$842$	−56.4262	−1.94458
$843$	2.10020	0.0723346
$844$	−51.4517	−1.77104
$845$	−12.7581	−0.438892
$846$	13.8224	0.475224
$847$	1.00000	0.0343604
$848$	2.92175	0.100333
$849$	−3.64104	−0.124960
$850$	−15.2979	−0.524714
$851$	63.3882	2.17292
$852$	14.0327	0.480752
$853$	6.63780	0.227274	0.113637	−	0.993522i	$-0.463750\pi$
$853$	6.63780	0.227274	0.113637	+	0.993522i	$0.463750\pi$
$854$	−10.0485	−0.343852
$855$	−0.729840	−0.0249600
$856$	−8.50442	−0.290675
$857$	−11.2178	−0.383192	−0.191596	−	0.981474i	$-0.561366\pi$
$857$	−11.2178	−0.383192	−0.191596	+	0.981474i	$0.561366\pi$
$858$	1.01634	0.0346972
$859$	10.8261	0.369383	0.184692	−	0.982797i	$-0.440871\pi$
$859$	10.8261	0.369383	0.184692	+	0.982797i	$0.440871\pi$
$860$	−21.6839	−0.739416
$861$	2.04849	0.0698123
$862$	16.7319	0.569890
$863$	14.0240	0.477382	0.238691	−	0.971096i	$-0.423282\pi$
$863$	14.0240	0.477382	0.238691	+	0.971096i	$0.423282\pi$
$864$	−8.11492	−0.276075
$865$	9.62096	0.327122
$866$	15.8495	0.538587
$867$	−37.8049	−1.28392
$868$	−17.3464	−0.588775
$869$	9.77391	0.331557
$870$	−10.1487	−0.344073
$871$	−1.71765	−0.0582003
$872$	−10.3343	−0.349964
$873$	−3.76199	−0.127324
$874$	13.4634	0.455408
$875$	1.00000	0.0338062
$876$	0.301643	0.0101916
$877$	−5.14532	−0.173745	−0.0868726	−	0.996219i	$-0.527687\pi$
$877$	−5.14532	−0.173745	−0.0868726	+	0.996219i	$0.527687\pi$
$878$	13.2248	0.446314
$879$	22.0304	0.743069
$880$	−3.38669	−0.114165
$881$	6.67358	0.224838	0.112419	−	0.993661i	$-0.464140\pi$
$881$	6.67358	0.224838	0.112419	+	0.993661i	$0.464140\pi$
$882$	−2.06644	−0.0695805
$883$	−33.8463	−1.13902	−0.569509	−	0.821985i	$-0.692866\pi$
$883$	−33.8463	−1.13902	−0.569509	+	0.821985i	$0.692866\pi$
$884$	8.26575	0.278007
$885$	−10.5038	−0.353080
$886$	58.7244	1.97288
$887$	−21.5447	−0.723401	−0.361701	−	0.932294i	$-0.617804\pi$
$887$	−21.5447	−0.723401	−0.361701	+	0.932294i	$0.617804\pi$
$888$	3.96410	0.133027
$889$	14.5038	0.486440
$890$	1.54084	0.0516492
$891$	1.00000	0.0335013
$892$	−31.2419	−1.04606
$893$	4.88191	0.163367
$894$	39.1361	1.30891
$895$	1.13235	0.0378502
$896$	4.42539	0.147842
$897$	−4.39058	−0.146597
$898$	33.5524	1.11966
$899$	37.5267	1.25159
$900$	2.27016	0.0756720
$901$	−6.38669	−0.212772
$902$	4.23307	0.140946
$903$	9.55172	0.317861
$904$	7.11759	0.236727
$905$	3.08064	0.102404
$906$	−16.3898	−0.544514
$907$	−9.76317	−0.324181	−0.162090	−	0.986776i	$-0.551824\pi$
$907$	−9.76317	−0.324181	−0.162090	+	0.986776i	$0.551824\pi$
$908$	4.97056	0.164954
$909$	9.68953	0.321381
$910$	−1.01634	−0.0336913
$911$	−49.9180	−1.65386	−0.826929	−	0.562306i	$-0.809914\pi$
$911$	−49.9180	−1.65386	−0.826929	+	0.562306i	$0.809914\pi$
$912$	−2.47175	−0.0818477
$913$	15.0599	0.498410
$914$	−20.7305	−0.685704
$915$	−4.86271	−0.160756
$916$	32.1198	1.06127
$917$	−1.43960	−0.0475397
$918$	15.2979	0.504906
$919$	−33.6722	−1.11074	−0.555371	−	0.831603i	$-0.687424\pi$
$919$	−33.6722	−1.11074	−0.555371	+	0.831603i	$0.687424\pi$
$920$	−4.98366	−0.164306
$921$	3.39538	0.111882
$922$	−49.3515	−1.62531
$923$	−3.04019	−0.100069
$924$	−2.27016	−0.0746828
$925$	7.10072	0.233470
$926$	5.18406	0.170359
$927$	−12.3023	−0.404061
$928$	−39.8540	−1.30827
$929$	29.9922	0.984013	0.492006	−	0.870592i	$-0.336264\pi$
$929$	29.9922	0.984013	0.492006	+	0.870592i	$0.336264\pi$
$930$	−15.7897	−0.517766
$931$	−0.729840	−0.0239196
$932$	−40.8714	−1.33879
$933$	−25.6438	−0.839539
$934$	−44.3113	−1.44991
$935$	7.40303	0.242105
$936$	−0.274573	−0.00897471
$937$	19.1924	0.626988	0.313494	−	0.949590i	$-0.398501\pi$
$937$	19.1924	0.626988	0.313494	+	0.949590i	$0.398501\pi$
$938$	7.21673	0.235635
$939$	−2.17371	−0.0709363
$940$	−15.1851	−0.495284
$941$	−19.8372	−0.646673	−0.323337	−	0.946284i	$-0.604805\pi$
$941$	−19.8372	−0.646673	−0.323337	+	0.946284i	$0.604805\pi$
$942$	−6.79025	−0.221238
$943$	−18.2869	−0.595503
$944$	−35.5730	−1.15780
$945$	−1.00000	−0.0325300
$946$	19.7380	0.641738
$947$	19.6514	0.638585	0.319292	−	0.947656i	$-0.396555\pi$
$947$	19.6514	0.638585	0.319292	+	0.947656i	$0.396555\pi$
$948$	−22.1883	−0.720644
$949$	−0.0653510	−0.00212138
$950$	1.50817	0.0489315
$951$	−6.60462	−0.214169
$952$	−4.13287	−0.133947
$953$	28.0150	0.907496	0.453748	−	0.891130i	$-0.350087\pi$
$953$	28.0150	0.907496	0.453748	+	0.891130i	$0.350087\pi$
$954$	1.78274	0.0577184
$955$	−17.3464	−0.561316
$956$	0.614498	0.0198743
$957$	4.91120	0.158757
$958$	50.1514	1.62032
$959$	13.7581	0.444272
$960$	9.99559	0.322606
$961$	27.3855	0.883404
$962$	−7.21673	−0.232677
$963$	15.2336	0.490896
$964$	−41.9435	−1.35091
$965$	12.6405	0.406913
$966$	18.4471	0.593526
$967$	−23.7130	−0.762558	−0.381279	−	0.924460i	$-0.624516\pi$
$967$	−23.7130	−0.762558	−0.381279	+	0.924460i	$0.624516\pi$
$968$	−0.558268	−0.0179434
$969$	5.40303	0.173570
$970$	7.77391	0.249605
$971$	−48.7368	−1.56404	−0.782020	−	0.623254i	$-0.785810\pi$
$971$	−48.7368	−1.56404	−0.782020	+	0.623254i	$0.785810\pi$
$972$	−2.27016	−0.0728154
$973$	0.197173	0.00632109
$974$	38.6611	1.23878
$975$	−0.491831	−0.0157512
$976$	−16.4685	−0.527145
$977$	15.7341	0.503379	0.251690	−	0.967808i	$-0.419014\pi$
$977$	15.7341	0.503379	0.251690	+	0.967808i	$0.419014\pi$
$978$	39.1034	1.25039
$979$	−0.745653	−0.0238312
$980$	2.27016	0.0725176
$981$	18.5114	0.591024
$982$	−56.5015	−1.80304
$983$	−15.5154	−0.494866	−0.247433	−	0.968905i	$-0.579587\pi$
$983$	−15.5154	−0.494866	−0.247433	+	0.968905i	$0.579587\pi$
$984$	−1.14360	−0.0364568
$985$	16.7091	0.532396
$986$	75.1310	2.39266
$987$	6.68900	0.212913
$988$	−0.814892	−0.0259252
$989$	−85.2683	−2.71137
$990$	−2.06644	−0.0656757
$991$	47.5150	1.50936	0.754682	−	0.656090i	$-0.227791\pi$
$991$	47.5150	1.50936	0.754682	+	0.656090i	$0.227791\pi$
$992$	−62.0065	−1.96871
$993$	21.5316	0.683286
$994$	12.7734	0.405147
$995$	−1.96732	−0.0623684
$996$	−34.1883	−1.08330
$997$	−53.8637	−1.70588	−0.852941	−	0.522008i	$-0.825183\pi$
$997$	−53.8637	−1.70588	−0.852941	+	0.522008i	$0.825183\pi$
$998$	−29.7785	−0.942621
$999$	−7.10072	−0.224657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.v.1.1	✓	4
3.2	odd	2		3465.2.a.bj.1.4		4
5.4	even	2		5775.2.a.by.1.4		4
7.6	odd	2		8085.2.a.bq.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.v.1.1	✓	4	1.1	even	1	trivial
3465.2.a.bj.1.4		4	3.2	odd	2
5775.2.a.by.1.4		4	5.4	even	2
8085.2.a.bq.1.1		4	7.6	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:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

\(f(q)\)	\(=\)	\(q-2.06644 q^{2} -1.00000 q^{3} +2.27016 q^{4} +1.00000 q^{5} +2.06644 q^{6} +1.00000 q^{7} -0.558268 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.06644 q^{2} -1.00000 q^{3} +2.27016 q^{4} +1.00000 q^{5} +2.06644 q^{6} +1.00000 q^{7} -0.558268 q^{8} +1.00000 q^{9} -2.06644 q^{10} +1.00000 q^{11} -2.27016 q^{12} +0.491831 q^{13} -2.06644 q^{14} -1.00000 q^{15} -3.38669 q^{16} +7.40303 q^{17} -2.06644 q^{18} -0.729840 q^{19} +2.27016 q^{20} -1.00000 q^{21} -2.06644 q^{22} +8.92701 q^{23} +0.558268 q^{24} +1.00000 q^{25} -1.01634 q^{26} -1.00000 q^{27} +2.27016 q^{28} -4.91120 q^{29} +2.06644 q^{30} -7.64104 q^{31} +8.11492 q^{32} -1.00000 q^{33} -15.2979 q^{34} +1.00000 q^{35} +2.27016 q^{36} +7.10072 q^{37} +1.50817 q^{38} -0.491831 q^{39} -0.558268 q^{40} -2.04849 q^{41} +2.06644 q^{42} -9.55172 q^{43} +2.27016 q^{44} +1.00000 q^{45} -18.4471 q^{46} -6.68900 q^{47} +3.38669 q^{48} +1.00000 q^{49} -2.06644 q^{50} -7.40303 q^{51} +1.11654 q^{52} -0.862713 q^{53} +2.06644 q^{54} +1.00000 q^{55} -0.558268 q^{56} +0.729840 q^{57} +10.1487 q^{58} +10.5038 q^{59} -2.27016 q^{60} +4.86271 q^{61} +15.7897 q^{62} +1.00000 q^{63} -9.99559 q^{64} +0.491831 q^{65} +2.06644 q^{66} -3.49236 q^{67} +16.8061 q^{68} -8.92701 q^{69} -2.06644 q^{70} -6.18136 q^{71} -0.558268 q^{72} -0.132873 q^{73} -14.6732 q^{74} -1.00000 q^{75} -1.65685 q^{76} +1.00000 q^{77} +1.01634 q^{78} +9.77391 q^{79} -3.38669 q^{80} +1.00000 q^{81} +4.23307 q^{82} +15.0599 q^{83} -2.27016 q^{84} +7.40303 q^{85} +19.7380 q^{86} +4.91120 q^{87} -0.558268 q^{88} -0.745653 q^{89} -2.06644 q^{90} +0.491831 q^{91} +20.2657 q^{92} +7.64104 q^{93} +13.8224 q^{94} -0.729840 q^{95} -8.11492 q^{96} -3.76199 q^{97} -2.06644 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 6 q^{12} + 4 q^{13} + 2 q^{14} - 4 q^{15} + 6 q^{16} + 6 q^{17} + 2 q^{18} - 6 q^{19} + 6 q^{20} - 4 q^{21} + 2 q^{22} + 10 q^{23} - 6 q^{24} + 4 q^{25} - 4 q^{27} + 6 q^{28} + 6 q^{29} - 2 q^{30} - 8 q^{31} + 14 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{35} + 6 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} + 6 q^{40} - 2 q^{42} + 6 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{46} - 6 q^{48} + 4 q^{49} + 2 q^{50} - 6 q^{51} - 12 q^{52} + 14 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 6 q^{57} + 20 q^{58} + 2 q^{59} - 6 q^{60} + 2 q^{61} + 20 q^{62} + 4 q^{63} - 2 q^{64} + 4 q^{65} - 2 q^{66} - 12 q^{67} + 20 q^{68} - 10 q^{69} + 2 q^{70} + 4 q^{71} + 6 q^{72} + 20 q^{73} - 32 q^{74} - 4 q^{75} + 16 q^{76} + 4 q^{77} - 4 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 14 q^{83} - 6 q^{84} + 6 q^{85} + 40 q^{86} - 6 q^{87} + 6 q^{88} - 6 q^{89} + 2 q^{90} + 4 q^{91} + 40 q^{92} + 8 q^{93} + 4 q^{94} - 6 q^{95} - 14 q^{96} - 14 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 6 * q^12 + 4 * q^13 + 2 * q^14 - 4 * q^15 + 6 * q^16 + 6 * q^17 + 2 * q^18 - 6 * q^19 + 6 * q^20 - 4 * q^21 + 2 * q^22 + 10 * q^23 - 6 * q^24 + 4 * q^25 - 4 * q^27 + 6 * q^28 + 6 * q^29 - 2 * q^30 - 8 * q^31 + 14 * q^32 - 4 * q^33 - 16 * q^34 + 4 * q^35 + 6 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 + 6 * q^40 - 2 * q^42 + 6 * q^43 + 6 * q^44 + 4 * q^45 - 4 * q^46 - 6 * q^48 + 4 * q^49 + 2 * q^50 - 6 * q^51 - 12 * q^52 + 14 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 6 * q^57 + 20 * q^58 + 2 * q^59 - 6 * q^60 + 2 * q^61 + 20 * q^62 + 4 * q^63 - 2 * q^64 + 4 * q^65 - 2 * q^66 - 12 * q^67 + 20 * q^68 - 10 * q^69 + 2 * q^70 + 4 * q^71 + 6 * q^72 + 20 * q^73 - 32 * q^74 - 4 * q^75 + 16 * q^76 + 4 * q^77 - 4 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 14 * q^83 - 6 * q^84 + 6 * q^85 + 40 * q^86 - 6 * q^87 + 6 * q^88 - 6 * q^89 + 2 * q^90 + 4 * q^91 + 40 * q^92 + 8 * q^93 + 4 * q^94 - 6 * q^95 - 14 * q^96 - 14 * q^97 + 2 * q^98 + 4 * q^99

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