LMFDB - Embedded newform 1045.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.506287$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46888 q^{2} -1.50629 q^{3} +0.157597 q^{4} -1.00000 q^{5} +2.21255 q^{6} -2.37015 q^{7} +2.70626 q^{8} -0.731099 q^{9} +O(q^{10})$	\(q-1.46888 q^{2} -1.50629 q^{3} +0.157597 q^{4} -1.00000 q^{5} +2.21255 q^{6} -2.37015 q^{7} +2.70626 q^{8} -0.731099 q^{9} +1.46888 q^{10} +1.00000 q^{11} -0.237386 q^{12} +1.65997 q^{13} +3.48145 q^{14} +1.50629 q^{15} -4.29036 q^{16} +4.24068 q^{17} +1.07389 q^{18} +1.00000 q^{19} -0.157597 q^{20} +3.57012 q^{21} -1.46888 q^{22} +5.50880 q^{23} -4.07641 q^{24} +1.00000 q^{25} -2.43829 q^{26} +5.62011 q^{27} -0.373527 q^{28} -8.58739 q^{29} -2.21255 q^{30} -7.07342 q^{31} +0.889478 q^{32} -1.50629 q^{33} -6.22904 q^{34} +2.37015 q^{35} -0.115219 q^{36} +5.65554 q^{37} -1.46888 q^{38} -2.50039 q^{39} -2.70626 q^{40} +7.82338 q^{41} -5.24406 q^{42} +10.7847 q^{43} +0.157597 q^{44} +0.731099 q^{45} -8.09175 q^{46} -10.4338 q^{47} +6.46251 q^{48} -1.38241 q^{49} -1.46888 q^{50} -6.38769 q^{51} +0.261606 q^{52} +1.30293 q^{53} -8.25524 q^{54} -1.00000 q^{55} -6.41424 q^{56} -1.50629 q^{57} +12.6138 q^{58} -5.42557 q^{59} +0.237386 q^{60} +3.28504 q^{61} +10.3900 q^{62} +1.73281 q^{63} +7.27418 q^{64} -1.65997 q^{65} +2.21255 q^{66} -13.9741 q^{67} +0.668318 q^{68} -8.29784 q^{69} -3.48145 q^{70} -3.87314 q^{71} -1.97854 q^{72} -11.2843 q^{73} -8.30728 q^{74} -1.50629 q^{75} +0.157597 q^{76} -2.37015 q^{77} +3.67276 q^{78} +6.81590 q^{79} +4.29036 q^{80} -6.27220 q^{81} -11.4916 q^{82} -13.9585 q^{83} +0.562639 q^{84} -4.24068 q^{85} -15.8413 q^{86} +12.9351 q^{87} +2.70626 q^{88} -3.25738 q^{89} -1.07389 q^{90} -3.93437 q^{91} +0.868169 q^{92} +10.6546 q^{93} +15.3259 q^{94} -1.00000 q^{95} -1.33981 q^{96} -1.67891 q^{97} +2.03059 q^{98} -0.731099 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) $ 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9	\( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9} + q^{10} + 5 q^{11} + 3 q^{12} - 3 q^{13} + 2 q^{14} + 3 q^{15} - 11 q^{16} - 11 q^{17} + 3 q^{18} + 5 q^{19} - q^{20} - 3 q^{21} - q^{22} + 5 q^{24} + 5 q^{25} - 8 q^{26} - 8 q^{28} - 15 q^{29} + 4 q^{30} - 9 q^{31} - 3 q^{33} - 14 q^{34} - 3 q^{35} - 7 q^{36} + 11 q^{37} - q^{38} - 10 q^{39} - 3 q^{40} - 23 q^{41} - 15 q^{42} + 9 q^{43} + q^{44} + 4 q^{45} + 8 q^{46} - 6 q^{47} + 4 q^{48} - 12 q^{49} - q^{50} - 27 q^{52} - 13 q^{53} + 9 q^{54} - 5 q^{55} - 12 q^{56} - 3 q^{57} + 17 q^{58} - 21 q^{59} - 3 q^{60} - 31 q^{61} - 18 q^{62} + 10 q^{63} - q^{64} + 3 q^{65} - 4 q^{66} + 5 q^{68} - 2 q^{70} - 28 q^{71} - 20 q^{72} - 14 q^{73} + 21 q^{74} - 3 q^{75} + q^{76} + 3 q^{77} + 13 q^{78} + 3 q^{79} + 11 q^{80} - 3 q^{81} - 18 q^{82} - 33 q^{83} + 13 q^{84} + 11 q^{85} - 20 q^{86} + 22 q^{87} + 3 q^{88} - 10 q^{89} - 3 q^{90} + 14 q^{91} + 21 q^{92} + 30 q^{93} + 14 q^{94} - 5 q^{95} - 3 q^{96} - 10 q^{97} + 8 q^{98} - 4 q^{99}+O(q^{100}) \) 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9 + q^10 + 5 * q^11 + 3 * q^12 - 3 * q^13 + 2 * q^14 + 3 * q^15 - 11 * q^16 - 11 * q^17 + 3 * q^18 + 5 * q^19 - q^20 - 3 * q^21 - q^22 + 5 * q^24 + 5 * q^25 - 8 * q^26 - 8 * q^28 - 15 * q^29 + 4 * q^30 - 9 * q^31 - 3 * q^33 - 14 * q^34 - 3 * q^35 - 7 * q^36 + 11 * q^37 - q^38 - 10 * q^39 - 3 * q^40 - 23 * q^41 - 15 * q^42 + 9 * q^43 + q^44 + 4 * q^45 + 8 * q^46 - 6 * q^47 + 4 * q^48 - 12 * q^49 - q^50 - 27 * q^52 - 13 * q^53 + 9 * q^54 - 5 * q^55 - 12 * q^56 - 3 * q^57 + 17 * q^58 - 21 * q^59 - 3 * q^60 - 31 * q^61 - 18 * q^62 + 10 * q^63 - q^64 + 3 * q^65 - 4 * q^66 + 5 * q^68 - 2 * q^70 - 28 * q^71 - 20 * q^72 - 14 * q^73 + 21 * q^74 - 3 * q^75 + q^76 + 3 * q^77 + 13 * q^78 + 3 * q^79 + 11 * q^80 - 3 * q^81 - 18 * q^82 - 33 * q^83 + 13 * q^84 + 11 * q^85 - 20 * q^86 + 22 * q^87 + 3 * q^88 - 10 * q^89 - 3 * q^90 + 14 * q^91 + 21 * q^92 + 30 * q^93 + 14 * q^94 - 5 * q^95 - 3 * q^96 - 10 * q^97 + 8 * 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 $

$2$	−1.46888	−1.03865	−0.519326	−	0.854576i	$-0.673817\pi$
$2$	−1.46888	−1.03865	−0.519326	+	0.854576i	$0.673817\pi$
$3$	−1.50629	−0.869655	−0.434828	−	0.900514i	$-0.643191\pi$
$3$	−1.50629	−0.869655	−0.434828	+	0.900514i	$0.643191\pi$
$4$	0.157597	0.0787984
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.21255	0.903269
$7$	−2.37015	−0.895831	−0.447915	−	0.894076i	$-0.647833\pi$
$7$	−2.37015	−0.895831	−0.447915	+	0.894076i	$0.647833\pi$
$8$	2.70626	0.956808
$9$	−0.731099	−0.243700
$10$	1.46888	0.464499
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.237386	−0.0685274
$13$	1.65997	0.460393	0.230196	−	0.973144i	$-0.426063\pi$
$13$	1.65997	0.460393	0.230196	+	0.973144i	$0.426063\pi$
$14$	3.48145	0.930457
$15$	1.50629	0.388922
$16$	−4.29036	−1.07259
$17$	4.24068	1.02852	0.514258	−	0.857635i	$-0.328067\pi$
$17$	4.24068	1.02852	0.514258	+	0.857635i	$0.328067\pi$
$18$	1.07389	0.253119
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.157597	−0.0352397
$21$	3.57012	0.779064
$22$	−1.46888	−0.313165
$23$	5.50880	1.14866	0.574332	−	0.818622i	$-0.305262\pi$
$23$	5.50880	1.14866	0.574332	+	0.818622i	$0.305262\pi$
$24$	−4.07641	−0.832093
$25$	1.00000	0.200000
$26$	−2.43829	−0.478188
$27$	5.62011	1.08159
$28$	−0.373527	−0.0705900
$29$	−8.58739	−1.59464	−0.797319	−	0.603558i	$-0.793749\pi$
$29$	−8.58739	−1.59464	−0.797319	+	0.603558i	$0.793749\pi$
$30$	−2.21255	−0.403954
$31$	−7.07342	−1.27042	−0.635212	−	0.772338i	$-0.719087\pi$
$31$	−7.07342	−1.27042	−0.635212	+	0.772338i	$0.719087\pi$
$32$	0.889478	0.157239
$33$	−1.50629	−0.262211
$34$	−6.22904	−1.06827
$35$	2.37015	0.400628
$36$	−0.115219	−0.0192031
$37$	5.65554	0.929764	0.464882	−	0.885373i	$-0.346097\pi$
$37$	5.65554	0.929764	0.464882	+	0.885373i	$0.346097\pi$
$38$	−1.46888	−0.238283
$39$	−2.50039	−0.400383
$40$	−2.70626	−0.427898
$41$	7.82338	1.22181	0.610903	−	0.791705i	$-0.290806\pi$
$41$	7.82338	1.22181	0.610903	+	0.791705i	$0.290806\pi$
$42$	−5.24406	−0.809177
$43$	10.7847	1.64464	0.822322	−	0.569022i	$-0.192678\pi$
$43$	10.7847	1.64464	0.822322	+	0.569022i	$0.192678\pi$
$44$	0.157597	0.0237586
$45$	0.731099	0.108986
$46$	−8.09175	−1.19306
$47$	−10.4338	−1.52192	−0.760960	−	0.648799i	$-0.775272\pi$
$47$	−10.4338	−1.52192	−0.760960	+	0.648799i	$0.775272\pi$
$48$	6.46251	0.932783
$49$	−1.38241	−0.197487
$50$	−1.46888	−0.207730
$51$	−6.38769	−0.894455
$52$	0.261606	0.0362782
$53$	1.30293	0.178971	0.0894857	−	0.995988i	$-0.471478\pi$
$53$	1.30293	0.178971	0.0894857	+	0.995988i	$0.471478\pi$
$54$	−8.25524	−1.12340
$55$	−1.00000	−0.134840
$56$	−6.41424	−0.857138
$57$	−1.50629	−0.199513
$58$	12.6138	1.65627
$59$	−5.42557	−0.706349	−0.353174	−	0.935557i	$-0.614898\pi$
$59$	−5.42557	−0.706349	−0.353174	+	0.935557i	$0.614898\pi$
$60$	0.237386	0.0306464
$61$	3.28504	0.420607	0.210303	−	0.977636i	$-0.432555\pi$
$61$	3.28504	0.420607	0.210303	+	0.977636i	$0.432555\pi$
$62$	10.3900	1.31953
$63$	1.73281	0.218314
$64$	7.27418	0.909273
$65$	−1.65997	−0.205894
$66$	2.21255	0.272346
$67$	−13.9741	−1.70721	−0.853603	−	0.520925i	$-0.825587\pi$
$67$	−13.9741	−1.70721	−0.853603	+	0.520925i	$0.825587\pi$
$68$	0.668318	0.0810454
$69$	−8.29784	−0.998942
$70$	−3.48145	−0.416113
$71$	−3.87314	−0.459657	−0.229828	−	0.973231i	$-0.573816\pi$
$71$	−3.87314	−0.459657	−0.229828	+	0.973231i	$0.573816\pi$
$72$	−1.97854	−0.233174
$73$	−11.2843	−1.32072	−0.660361	−	0.750948i	$-0.729597\pi$
$73$	−11.2843	−1.32072	−0.660361	+	0.750948i	$0.729597\pi$
$74$	−8.30728	−0.965702
$75$	−1.50629	−0.173931
$76$	0.157597	0.0180776
$77$	−2.37015	−0.270103
$78$	3.67276	0.415859
$79$	6.81590	0.766848	0.383424	−	0.923572i	$-0.374745\pi$
$79$	6.81590	0.766848	0.383424	+	0.923572i	$0.374745\pi$
$80$	4.29036	0.479676
$81$	−6.27220	−0.696911
$82$	−11.4916	−1.26903
$83$	−13.9585	−1.53214	−0.766070	−	0.642757i	$-0.777791\pi$
$83$	−13.9585	−1.53214	−0.766070	+	0.642757i	$0.777791\pi$
$84$	0.562639	0.0613890
$85$	−4.24068	−0.459967
$86$	−15.8413	−1.70821
$87$	12.9351	1.38679
$88$	2.70626	0.288488
$89$	−3.25738	−0.345282	−0.172641	−	0.984985i	$-0.555230\pi$
$89$	−3.25738	−0.345282	−0.172641	+	0.984985i	$0.555230\pi$
$90$	−1.07389	−0.113198
$91$	−3.93437	−0.412434
$92$	0.868169	0.0905129
$93$	10.6546	1.10483
$94$	15.3259	1.58075
$95$	−1.00000	−0.102598
$96$	−1.33981	−0.136744
$97$	−1.67891	−0.170468	−0.0852338	−	0.996361i	$-0.527164\pi$
$97$	−1.67891	−0.170468	−0.0852338	+	0.996361i	$0.527164\pi$
$98$	2.03059	0.205120
$99$	−0.731099	−0.0734782
$100$	0.157597	0.0157597
$101$	−6.91537	−0.688105	−0.344052	−	0.938950i	$-0.611800\pi$
$101$	−6.91537	−0.688105	−0.344052	+	0.938950i	$0.611800\pi$
$102$	9.38272	0.929028
$103$	11.3483	1.11818	0.559090	−	0.829107i	$-0.311150\pi$
$103$	11.3483	1.11818	0.559090	+	0.829107i	$0.311150\pi$
$104$	4.49231	0.440508
$105$	−3.57012	−0.348408
$106$	−1.91384	−0.185889
$107$	−10.2046	−0.986515	−0.493257	−	0.869883i	$-0.664194\pi$
$107$	−10.2046	−0.986515	−0.493257	+	0.869883i	$0.664194\pi$
$108$	0.885710	0.0852275
$109$	2.11487	0.202568	0.101284	−	0.994858i	$-0.467705\pi$
$109$	2.11487	0.202568	0.101284	+	0.994858i	$0.467705\pi$
$110$	1.46888	0.140052
$111$	−8.51886	−0.808575
$112$	10.1688	0.960859
$113$	6.93753	0.652628	0.326314	−	0.945261i	$-0.394193\pi$
$113$	6.93753	0.652628	0.326314	+	0.945261i	$0.394193\pi$
$114$	2.21255	0.207224
$115$	−5.50880	−0.513698
$116$	−1.35334	−0.125655
$117$	−1.21360	−0.112198
$118$	7.96949	0.733651
$119$	−10.0510	−0.921377
$120$	4.07641	0.372123
$121$	1.00000	0.0909091
$122$	−4.82532	−0.436864
$123$	−11.7843	−1.06255
$124$	−1.11475	−0.100107
$125$	−1.00000	−0.0894427
$126$	−2.54528	−0.226752
$127$	0.182433	0.0161884	0.00809418	−	0.999967i	$-0.497424\pi$
$127$	0.182433	0.0161884	0.00809418	+	0.999967i	$0.497424\pi$
$128$	−12.4638	−1.10166
$129$	−16.2448	−1.43027
$130$	2.43829	0.213852
$131$	−18.8389	−1.64596	−0.822980	−	0.568071i	$-0.807690\pi$
$131$	−18.8389	−1.64596	−0.822980	+	0.568071i	$0.807690\pi$
$132$	−0.237386	−0.0206618
$133$	−2.37015	−0.205518
$134$	20.5262	1.77319
$135$	−5.62011	−0.483702
$136$	11.4764	0.984093
$137$	−19.3608	−1.65410	−0.827052	−	0.562126i	$-0.809984\pi$
$137$	−19.3608	−1.65410	−0.827052	+	0.562126i	$0.809984\pi$
$138$	12.1885	1.03755
$139$	−15.1666	−1.28642	−0.643209	−	0.765691i	$-0.722397\pi$
$139$	−15.1666	−1.28642	−0.643209	+	0.765691i	$0.722397\pi$
$140$	0.373527	0.0315688
$141$	15.7162	1.32355
$142$	5.68916	0.477423
$143$	1.65997	0.138814
$144$	3.13667	0.261390
$145$	8.58739	0.713144
$146$	16.5752	1.37177
$147$	2.08230	0.171746
$148$	0.891294	0.0732639
$149$	−1.39216	−0.114050	−0.0570249	−	0.998373i	$-0.518161\pi$
$149$	−1.39216	−0.114050	−0.0570249	+	0.998373i	$0.518161\pi$
$150$	2.21255	0.180654
$151$	−0.180055	−0.0146527	−0.00732635	−	0.999973i	$-0.502332\pi$
$151$	−0.180055	−0.0146527	−0.00732635	+	0.999973i	$0.502332\pi$
$152$	2.70626	0.219507
$153$	−3.10036	−0.250649
$154$	3.48145	0.280543
$155$	7.07342	0.568151
$156$	−0.394054	−0.0315495
$157$	23.2337	1.85425	0.927126	−	0.374751i	$-0.122272\pi$
$157$	23.2337	1.85425	0.927126	+	0.374751i	$0.122272\pi$
$158$	−10.0117	−0.796489
$159$	−1.96259	−0.155643
$160$	−0.889478	−0.0703194
$161$	−13.0567	−1.02901
$162$	9.21308	0.723848
$163$	−6.61201	−0.517892	−0.258946	−	0.965892i	$-0.583375\pi$
$163$	−6.61201	−0.517892	−0.258946	+	0.965892i	$0.583375\pi$
$164$	1.23294	0.0962764
$165$	1.50629	0.117264
$166$	20.5033	1.59136
$167$	−2.70433	−0.209267	−0.104634	−	0.994511i	$-0.533367\pi$
$167$	−2.70433	−0.209267	−0.104634	+	0.994511i	$0.533367\pi$
$168$	9.66168	0.745415
$169$	−10.2445	−0.788038
$170$	6.22904	0.477745
$171$	−0.731099	−0.0559085
$172$	1.69963	0.129595
$173$	19.7234	1.49954	0.749770	−	0.661698i	$-0.230164\pi$
$173$	19.7234	1.49954	0.749770	+	0.661698i	$0.230164\pi$
$174$	−19.0000	−1.44039
$175$	−2.37015	−0.179166
$176$	−4.29036	−0.323398
$177$	8.17247	0.614280
$178$	4.78469	0.358627
$179$	12.9360	0.966879	0.483439	−	0.875378i	$-0.339387\pi$
$179$	12.9360	0.966879	0.483439	+	0.875378i	$0.339387\pi$
$180$	0.115219	0.00858790
$181$	16.6367	1.23660	0.618299	−	0.785943i	$-0.287822\pi$
$181$	16.6367	1.23660	0.618299	+	0.785943i	$0.287822\pi$
$182$	5.77910	0.428376
$183$	−4.94822	−0.365783
$184$	14.9083	1.09905
$185$	−5.65554	−0.415803
$186$	−15.6503	−1.14754
$187$	4.24068	0.310109
$188$	−1.64433	−0.119925
$189$	−13.3205	−0.968922
$190$	1.46888	0.106563
$191$	−19.8500	−1.43629	−0.718147	−	0.695891i	$-0.755010\pi$
$191$	−19.8500	−1.43629	−0.718147	+	0.695891i	$0.755010\pi$
$192$	−10.9570	−0.790754
$193$	8.74942	0.629797	0.314899	−	0.949125i	$-0.398029\pi$
$193$	8.74942	0.629797	0.314899	+	0.949125i	$0.398029\pi$
$194$	2.46611	0.177056
$195$	2.50039	0.179057
$196$	−0.217863	−0.0155616
$197$	21.3374	1.52023	0.760113	−	0.649791i	$-0.225143\pi$
$197$	21.3374	1.52023	0.760113	+	0.649791i	$0.225143\pi$
$198$	1.07389	0.0763183
$199$	−7.15659	−0.507317	−0.253659	−	0.967294i	$-0.581634\pi$
$199$	−7.15659	−0.507317	−0.253659	+	0.967294i	$0.581634\pi$
$200$	2.70626	0.191362
$201$	21.0490	1.48468
$202$	10.1578	0.714702
$203$	20.3534	1.42853
$204$	−1.00668	−0.0704816
$205$	−7.82338	−0.546408
$206$	−16.6692	−1.16140
$207$	−4.02748	−0.279929
$208$	−7.12186	−0.493812
$209$	1.00000	0.0691714
$210$	5.24406	0.361875
$211$	−7.22711	−0.497534	−0.248767	−	0.968563i	$-0.580025\pi$
$211$	−7.22711	−0.497534	−0.248767	+	0.968563i	$0.580025\pi$
$212$	0.205338	0.0141027
$213$	5.83406	0.399743
$214$	14.9893	1.02465
$215$	−10.7847	−0.735507
$216$	15.2095	1.03487
$217$	16.7650	1.13809
$218$	−3.10648	−0.210398
$219$	16.9973	1.14857
$220$	−0.157597	−0.0106252
$221$	7.03941	0.473522
$222$	12.5132	0.839828
$223$	21.7964	1.45960	0.729799	−	0.683662i	$-0.239614\pi$
$223$	21.7964	1.45960	0.729799	+	0.683662i	$0.239614\pi$
$224$	−2.10819	−0.140860
$225$	−0.731099	−0.0487399
$226$	−10.1904	−0.677854
$227$	−22.7886	−1.51253	−0.756265	−	0.654265i	$-0.772978\pi$
$227$	−22.7886	−1.51253	−0.756265	+	0.654265i	$0.772978\pi$
$228$	−0.237386	−0.0157213
$229$	10.1912	0.673455	0.336728	−	0.941602i	$-0.390680\pi$
$229$	10.1912	0.673455	0.336728	+	0.941602i	$0.390680\pi$
$230$	8.09175	0.533554
$231$	3.57012	0.234897
$232$	−23.2397	−1.52576
$233$	−5.99398	−0.392679	−0.196339	−	0.980536i	$-0.562905\pi$
$233$	−5.99398	−0.392679	−0.196339	+	0.980536i	$0.562905\pi$
$234$	1.78263	0.116534
$235$	10.4338	0.680623
$236$	−0.855052	−0.0556591
$237$	−10.2667	−0.666894
$238$	14.7637	0.956990
$239$	−2.37974	−0.153932	−0.0769661	−	0.997034i	$-0.524523\pi$
$239$	−2.37974	−0.153932	−0.0769661	+	0.997034i	$0.524523\pi$
$240$	−6.46251	−0.417153
$241$	−26.4711	−1.70516	−0.852578	−	0.522600i	$-0.824962\pi$
$241$	−26.4711	−1.70516	−0.852578	+	0.522600i	$0.824962\pi$
$242$	−1.46888	−0.0944229
$243$	−7.41259	−0.475518
$244$	0.517712	0.0331431
$245$	1.38241	0.0883188
$246$	17.3096	1.10362
$247$	1.65997	0.105621
$248$	−19.1425	−1.21555
$249$	21.0255	1.33243
$250$	1.46888	0.0928999
$251$	2.71782	0.171547	0.0857737	−	0.996315i	$-0.472664\pi$
$251$	2.71782	0.171547	0.0857737	+	0.996315i	$0.472664\pi$
$252$	0.273085	0.0172028
$253$	5.50880	0.346335
$254$	−0.267972	−0.0168141
$255$	6.38769	0.400012
$256$	3.75945	0.234966
$257$	4.53933	0.283155	0.141578	−	0.989927i	$-0.454783\pi$
$257$	4.53933	0.283155	0.141578	+	0.989927i	$0.454783\pi$
$258$	23.8616	1.48556
$259$	−13.4044	−0.832912
$260$	−0.261606	−0.0162241
$261$	6.27823	0.388613
$262$	27.6720	1.70958
$263$	−9.63745	−0.594270	−0.297135	−	0.954835i	$-0.596031\pi$
$263$	−9.63745	−0.594270	−0.297135	+	0.954835i	$0.596031\pi$
$264$	−4.07641	−0.250886
$265$	−1.30293	−0.0800384
$266$	3.48145	0.213461
$267$	4.90655	0.300276
$268$	−2.20227	−0.134525
$269$	−1.53070	−0.0933282	−0.0466641	−	0.998911i	$-0.514859\pi$
$269$	−1.53070	−0.0933282	−0.0466641	+	0.998911i	$0.514859\pi$
$270$	8.25524	0.502398
$271$	19.4804	1.18335	0.591675	−	0.806177i	$-0.298467\pi$
$271$	19.4804	1.18335	0.591675	+	0.806177i	$0.298467\pi$
$272$	−18.1940	−1.10318
$273$	5.92629	0.358676
$274$	28.4386	1.71804
$275$	1.00000	0.0603023
$276$	−1.30771	−0.0787150
$277$	−20.7875	−1.24900	−0.624499	−	0.781026i	$-0.714697\pi$
$277$	−20.7875	−1.24900	−0.624499	+	0.781026i	$0.714697\pi$
$278$	22.2779	1.33614
$279$	5.17137	0.309602
$280$	6.41424	0.383324
$281$	12.5809	0.750514	0.375257	−	0.926921i	$-0.377555\pi$
$281$	12.5809	0.750514	0.375257	+	0.926921i	$0.377555\pi$
$282$	−23.0852	−1.37470
$283$	−2.21849	−0.131875	−0.0659377	−	0.997824i	$-0.521004\pi$
$283$	−2.21849	−0.131875	−0.0659377	+	0.997824i	$0.521004\pi$
$284$	−0.610394	−0.0362202
$285$	1.50629	0.0892248
$286$	−2.43829	−0.144179
$287$	−18.5425	−1.09453
$288$	−0.650296	−0.0383191
$289$	0.983395	0.0578467
$290$	−12.6138	−0.740709
$291$	2.52892	0.148248
$292$	−1.77836	−0.104071
$293$	−27.7472	−1.62101	−0.810506	−	0.585731i	$-0.800808\pi$
$293$	−27.7472	−1.62101	−0.810506	+	0.585731i	$0.800808\pi$
$294$	−3.05865	−0.178384
$295$	5.42557	0.315889
$296$	15.3054	0.889606
$297$	5.62011	0.326112
$298$	2.04490	0.118458
$299$	9.14444	0.528837
$300$	−0.237386	−0.0137055
$301$	−25.5612	−1.47332
$302$	0.264479	0.0152191
$303$	10.4165	0.598414
$304$	−4.29036	−0.246069
$305$	−3.28504	−0.188101
$306$	4.55404	0.260337
$307$	8.75808	0.499850	0.249925	−	0.968265i	$-0.419594\pi$
$307$	8.75808	0.499850	0.249925	+	0.968265i	$0.419594\pi$
$308$	−0.373527	−0.0212837
$309$	−17.0938	−0.972432
$310$	−10.3900	−0.590111
$311$	14.4078	0.816993	0.408497	−	0.912760i	$-0.366053\pi$
$311$	14.4078	0.816993	0.408497	+	0.912760i	$0.366053\pi$
$312$	−6.76671	−0.383090
$313$	17.4993	0.989120	0.494560	−	0.869143i	$-0.335329\pi$
$313$	17.4993	0.989120	0.494560	+	0.869143i	$0.335329\pi$
$314$	−34.1274	−1.92592
$315$	−1.73281	−0.0976328
$316$	1.07416	0.0604264
$317$	−17.5241	−0.984253	−0.492126	−	0.870524i	$-0.663780\pi$
$317$	−17.5241	−0.984253	−0.492126	+	0.870524i	$0.663780\pi$
$318$	2.88280	0.161659
$319$	−8.58739	−0.480802
$320$	−7.27418	−0.406639
$321$	15.3710	0.857928
$322$	19.1786	1.06878
$323$	4.24068	0.235958
$324$	−0.988478	−0.0549154
$325$	1.65997	0.0920786
$326$	9.71222	0.537910
$327$	−3.18560	−0.176164
$328$	21.1721	1.16903
$329$	24.7295	1.36338
$330$	−2.21255	−0.121797
$331$	16.9716	0.932844	0.466422	−	0.884562i	$-0.345543\pi$
$331$	16.9716	0.932844	0.466422	+	0.884562i	$0.345543\pi$
$332$	−2.19981	−0.120730
$333$	−4.13475	−0.226583
$334$	3.97232	0.217356
$335$	13.9741	0.763485
$336$	−15.3171	−0.835616
$337$	−7.28651	−0.396921	−0.198461	−	0.980109i	$-0.563594\pi$
$337$	−7.28651	−0.396921	−0.198461	+	0.980109i	$0.563594\pi$
$338$	15.0479	0.818498
$339$	−10.4499	−0.567561
$340$	−0.668318	−0.0362446
$341$	−7.07342	−0.383047
$342$	1.07389	0.0580695
$343$	19.8675	1.07275
$344$	29.1861	1.57361
$345$	8.29784	0.446741
$346$	−28.9712	−1.55750
$347$	−36.1538	−1.94084	−0.970420	−	0.241424i	$-0.922386\pi$
$347$	−36.1538	−1.94084	−0.970420	+	0.241424i	$0.922386\pi$
$348$	2.03853	0.109276
$349$	−24.1766	−1.29414	−0.647071	−	0.762430i	$-0.724006\pi$
$349$	−24.1766	−1.29414	−0.647071	+	0.762430i	$0.724006\pi$
$350$	3.48145	0.186091
$351$	9.32921	0.497956
$352$	0.889478	0.0474093
$353$	30.1314	1.60373	0.801865	−	0.597505i	$-0.203841\pi$
$353$	30.1314	1.60373	0.801865	+	0.597505i	$0.203841\pi$
$354$	−12.0043	−0.638023
$355$	3.87314	0.205565
$356$	−0.513352	−0.0272076
$357$	15.1398	0.801281
$358$	−19.0013	−1.00425
$359$	−21.4837	−1.13386	−0.566932	−	0.823764i	$-0.691870\pi$
$359$	−21.4837	−1.13386	−0.566932	+	0.823764i	$0.691870\pi$
$360$	1.97854	0.104278
$361$	1.00000	0.0526316
$362$	−24.4373	−1.28439
$363$	−1.50629	−0.0790596
$364$	−0.620044	−0.0324991
$365$	11.2843	0.590645
$366$	7.26832	0.379921
$367$	−23.3008	−1.21629	−0.608146	−	0.793825i	$-0.708086\pi$
$367$	−23.3008	−1.21629	−0.608146	+	0.793825i	$0.708086\pi$
$368$	−23.6347	−1.23205
$369$	−5.71966	−0.297754
$370$	8.30728	0.431875
$371$	−3.08814	−0.160328
$372$	1.67913	0.0870589
$373$	5.55402	0.287576	0.143788	−	0.989609i	$-0.454072\pi$
$373$	5.55402	0.287576	0.143788	+	0.989609i	$0.454072\pi$
$374$	−6.22904	−0.322096
$375$	1.50629	0.0777843
$376$	−28.2365	−1.45619
$377$	−14.2548	−0.734160
$378$	19.5661	1.00637
$379$	−17.2424	−0.885685	−0.442842	−	0.896599i	$-0.646030\pi$
$379$	−17.2424	−0.885685	−0.442842	+	0.896599i	$0.646030\pi$
$380$	−0.157597	−0.00808454
$381$	−0.274797	−0.0140783
$382$	29.1572	1.49181
$383$	3.88761	0.198647	0.0993237	−	0.995055i	$-0.468332\pi$
$383$	3.88761	0.198647	0.0993237	+	0.995055i	$0.468332\pi$
$384$	18.7741	0.958062
$385$	2.37015	0.120794
$386$	−12.8518	−0.654140
$387$	−7.88464	−0.400799
$388$	−0.264591	−0.0134326
$389$	−7.93411	−0.402275	−0.201138	−	0.979563i	$-0.564464\pi$
$389$	−7.93411	−0.402275	−0.201138	+	0.979563i	$0.564464\pi$
$390$	−3.67276	−0.185978
$391$	23.3611	1.18142
$392$	−3.74116	−0.188957
$393$	28.3767	1.43142
$394$	−31.3420	−1.57899
$395$	−6.81590	−0.342945
$396$	−0.115219	−0.00578996
$397$	−13.1297	−0.658963	−0.329482	−	0.944162i	$-0.606874\pi$
$397$	−13.1297	−0.658963	−0.329482	+	0.944162i	$0.606874\pi$
$398$	10.5121	0.526926
$399$	3.57012	0.178730
$400$	−4.29036	−0.214518
$401$	−26.9636	−1.34650	−0.673249	−	0.739416i	$-0.735102\pi$
$401$	−26.9636	−1.34650	−0.673249	+	0.739416i	$0.735102\pi$
$402$	−30.9183	−1.54207
$403$	−11.7417	−0.584894
$404$	−1.08984	−0.0542215
$405$	6.27220	0.311668
$406$	−29.8966	−1.48374
$407$	5.65554	0.280335
$408$	−17.2868	−0.855822
$409$	0.624583	0.0308836	0.0154418	−	0.999881i	$-0.495085\pi$
$409$	0.624583	0.0308836	0.0154418	+	0.999881i	$0.495085\pi$
$410$	11.4916	0.567528
$411$	29.1629	1.43850
$412$	1.78845	0.0881108
$413$	12.8594	0.632769
$414$	5.91587	0.290749
$415$	13.9585	0.685194
$416$	1.47651	0.0723917
$417$	22.8453	1.11874
$418$	−1.46888	−0.0718451
$419$	3.31831	0.162110	0.0810551	−	0.996710i	$-0.474171\pi$
$419$	3.31831	0.162110	0.0810551	+	0.996710i	$0.474171\pi$
$420$	−0.562639	−0.0274540
$421$	12.1529	0.592298	0.296149	−	0.955142i	$-0.404298\pi$
$421$	12.1529	0.592298	0.296149	+	0.955142i	$0.404298\pi$
$422$	10.6157	0.516765
$423$	7.62811	0.370891
$424$	3.52607	0.171241
$425$	4.24068	0.205703
$426$	−8.56950	−0.415194
$427$	−7.78603	−0.376792
$428$	−1.60821	−0.0777357
$429$	−2.50039	−0.120720
$430$	15.8413	0.763936
$431$	−16.4877	−0.794182	−0.397091	−	0.917779i	$-0.629980\pi$
$431$	−16.4877	−0.794182	−0.397091	+	0.917779i	$0.629980\pi$
$432$	−24.1123	−1.16010
$433$	−15.6967	−0.754336	−0.377168	−	0.926145i	$-0.623102\pi$
$433$	−15.6967	−0.754336	−0.377168	+	0.926145i	$0.623102\pi$
$434$	−24.6258	−1.18207
$435$	−12.9351	−0.620190
$436$	0.333297	0.0159620
$437$	5.50880	0.263522
$438$	−24.9670	−1.19297
$439$	−9.97406	−0.476036	−0.238018	−	0.971261i	$-0.576498\pi$
$439$	−9.97406	−0.476036	−0.238018	+	0.971261i	$0.576498\pi$
$440$	−2.70626	−0.129016
$441$	1.01068	0.0481275
$442$	−10.3400	−0.491824
$443$	−40.8203	−1.93943	−0.969717	−	0.244233i	$-0.921464\pi$
$443$	−40.8203	−1.93943	−0.969717	+	0.244233i	$0.921464\pi$
$444$	−1.34254	−0.0637144
$445$	3.25738	0.154415
$446$	−32.0163	−1.51601
$447$	2.09699	0.0991841
$448$	−17.2409	−0.814554
$449$	38.2592	1.80556	0.902782	−	0.430099i	$-0.141521\pi$
$449$	38.2592	1.80556	0.902782	+	0.430099i	$0.141521\pi$
$450$	1.07389	0.0506238
$451$	7.82338	0.368389
$452$	1.09333	0.0514260
$453$	0.271215	0.0127428
$454$	33.4736	1.57099
$455$	3.93437	0.184446
$456$	−4.07641	−0.190895
$457$	8.81696	0.412440	0.206220	−	0.978506i	$-0.433884\pi$
$457$	8.81696	0.412440	0.206220	+	0.978506i	$0.433884\pi$
$458$	−14.9697	−0.699486
$459$	23.8331	1.11243
$460$	−0.868169	−0.0404786
$461$	−15.1755	−0.706795	−0.353398	−	0.935473i	$-0.614974\pi$
$461$	−15.1755	−0.706795	−0.353398	+	0.935473i	$0.614974\pi$
$462$	−5.24406	−0.243976
$463$	−20.0721	−0.932828	−0.466414	−	0.884566i	$-0.654454\pi$
$463$	−20.0721	−0.932828	−0.466414	+	0.884566i	$0.654454\pi$
$464$	36.8430	1.71039
$465$	−10.6546	−0.494096
$466$	8.80441	0.407856
$467$	−13.3824	−0.619265	−0.309633	−	0.950856i	$-0.600206\pi$
$467$	−13.3824	−0.619265	−0.309633	+	0.950856i	$0.600206\pi$
$468$	−0.191260	−0.00884098
$469$	33.1206	1.52937
$470$	−15.3259	−0.706931
$471$	−34.9966	−1.61256
$472$	−14.6830	−0.675840
$473$	10.7847	0.495879
$474$	15.0805	0.692671
$475$	1.00000	0.0458831
$476$	−1.58401	−0.0726030
$477$	−0.952571	−0.0436152
$478$	3.49554	0.159882
$479$	−33.6330	−1.53673	−0.768365	−	0.640012i	$-0.778930\pi$
$479$	−33.6330	−1.53673	−0.768365	+	0.640012i	$0.778930\pi$
$480$	1.33981	0.0611537
$481$	9.38802	0.428057
$482$	38.8828	1.77106
$483$	19.6671	0.894883
$484$	0.157597	0.00716349
$485$	1.67891	0.0762354
$486$	10.8882	0.493897
$487$	22.2554	1.00849	0.504244	−	0.863561i	$-0.331771\pi$
$487$	22.2554	1.00849	0.504244	+	0.863561i	$0.331771\pi$
$488$	8.89019	0.402440
$489$	9.95958	0.450388
$490$	−2.03059	−0.0917325
$491$	−0.633170	−0.0285746	−0.0142873	−	0.999898i	$-0.504548\pi$
$491$	−0.633170	−0.0285746	−0.0142873	+	0.999898i	$0.504548\pi$
$492$	−1.85716	−0.0837272
$493$	−36.4164	−1.64011
$494$	−2.43829	−0.109704
$495$	0.731099	0.0328604
$496$	30.3475	1.36264
$497$	9.17990	0.411775
$498$	−30.8838	−1.38394
$499$	−32.3433	−1.44788	−0.723942	−	0.689861i	$-0.757672\pi$
$499$	−32.3433	−1.44788	−0.723942	+	0.689861i	$0.757672\pi$
$500$	−0.157597	−0.00704794
$501$	4.07350	0.181990
$502$	−3.99215	−0.178178
$503$	−33.0415	−1.47325	−0.736625	−	0.676302i	$-0.763581\pi$
$503$	−33.0415	−1.47325	−0.736625	+	0.676302i	$0.763581\pi$
$504$	4.68944	0.208884
$505$	6.91537	0.307730
$506$	−8.09175	−0.359722
$507$	15.4312	0.685322
$508$	0.0287509	0.00127562
$509$	−28.2619	−1.25269	−0.626343	−	0.779548i	$-0.715449\pi$
$509$	−28.2619	−1.25269	−0.626343	+	0.779548i	$0.715449\pi$
$510$	−9.38272	−0.415474
$511$	26.7453	1.18314
$512$	19.4055	0.857609
$513$	5.62011	0.248134
$514$	−6.66771	−0.294100
$515$	−11.3483	−0.500066
$516$	−2.56012	−0.112703
$517$	−10.4338	−0.458876
$518$	19.6895	0.865106
$519$	−29.7091	−1.30408
$520$	−4.49231	−0.197001
$521$	−8.11601	−0.355569	−0.177784	−	0.984069i	$-0.556893\pi$
$521$	−8.11601	−0.355569	−0.177784	+	0.984069i	$0.556893\pi$
$522$	−9.22194	−0.403633
$523$	−12.4906	−0.546175	−0.273088	−	0.961989i	$-0.588045\pi$
$523$	−12.4906	−0.546175	−0.273088	+	0.961989i	$0.588045\pi$
$524$	−2.96894	−0.129699
$525$	3.57012	0.155813
$526$	14.1562	0.617240
$527$	−29.9961	−1.30665
$528$	6.46251	0.281245
$529$	7.34690	0.319430
$530$	1.91384	0.0831321
$531$	3.96663	0.172137
$532$	−0.373527	−0.0161945
$533$	12.9866	0.562511
$534$	−7.20711	−0.311882
$535$	10.2046	0.441183
$536$	−37.8175	−1.63347
$537$	−19.4853	−0.840851
$538$	2.24840	0.0969355
$539$	−1.38241	−0.0595445
$540$	−0.885710	−0.0381149
$541$	−36.0202	−1.54863	−0.774315	−	0.632801i	$-0.781905\pi$
$541$	−36.0202	−1.54863	−0.774315	+	0.632801i	$0.781905\pi$
$542$	−28.6143	−1.22909
$543$	−25.0597	−1.07541
$544$	3.77200	0.161723
$545$	−2.11487	−0.0905912
$546$	−8.70499	−0.372539
$547$	24.1206	1.03132	0.515662	−	0.856792i	$-0.327546\pi$
$547$	24.1206	1.03132	0.515662	+	0.856792i	$0.327546\pi$
$548$	−3.05120	−0.130341
$549$	−2.40169	−0.102502
$550$	−1.46888	−0.0626331
$551$	−8.58739	−0.365835
$552$	−22.4561	−0.955796
$553$	−16.1547	−0.686966
$554$	30.5342	1.29727
$555$	8.51886	0.361606
$556$	−2.39021	−0.101368
$557$	17.1586	0.727035	0.363517	−	0.931587i	$-0.381576\pi$
$557$	17.1586	0.727035	0.363517	+	0.931587i	$0.381576\pi$
$558$	−7.59610	−0.321569
$559$	17.9022	0.757182
$560$	−10.1688	−0.429709
$561$	−6.38769	−0.269688
$562$	−18.4798	−0.779523
$563$	−44.9936	−1.89626	−0.948128	−	0.317890i	$-0.897026\pi$
$563$	−44.9936	−1.89626	−0.948128	+	0.317890i	$0.897026\pi$
$564$	2.47683	0.104293
$565$	−6.93753	−0.291864
$566$	3.25868	0.136973
$567$	14.8660	0.624314
$568$	−10.4817	−0.439803
$569$	−9.46330	−0.396722	−0.198361	−	0.980129i	$-0.563562\pi$
$569$	−9.46330	−0.396722	−0.198361	+	0.980129i	$0.563562\pi$
$570$	−2.21255	−0.0926735
$571$	9.56809	0.400412	0.200206	−	0.979754i	$-0.435839\pi$
$571$	9.56809	0.400412	0.200206	+	0.979754i	$0.435839\pi$
$572$	0.261606	0.0109383
$573$	29.8998	1.24908
$574$	27.2367	1.13684
$575$	5.50880	0.229733
$576$	−5.31814	−0.221589
$577$	29.1262	1.21254	0.606270	−	0.795259i	$-0.292665\pi$
$577$	29.1262	1.21254	0.606270	+	0.795259i	$0.292665\pi$
$578$	−1.44448	−0.0600826
$579$	−13.1791	−0.547707
$580$	1.35334	0.0561946
$581$	33.0836	1.37254
$582$	−3.71467	−0.153978
$583$	1.30293	0.0539619
$584$	−30.5382	−1.26368
$585$	1.21360	0.0501763
$586$	40.7573	1.68367
$587$	18.2859	0.754740	0.377370	−	0.926063i	$-0.376829\pi$
$587$	18.2859	0.754740	0.377370	+	0.926063i	$0.376829\pi$
$588$	0.328164	0.0135333
$589$	−7.07342	−0.291455
$590$	−7.96949	−0.328099
$591$	−32.1402	−1.32207
$592$	−24.2643	−0.997255
$593$	−29.4744	−1.21037	−0.605184	−	0.796085i	$-0.706901\pi$
$593$	−29.4744	−1.21037	−0.605184	+	0.796085i	$0.706901\pi$
$594$	−8.25524	−0.338717
$595$	10.0510	0.412052
$596$	−0.219399	−0.00898694
$597$	10.7799	0.441191
$598$	−13.4321	−0.549278
$599$	29.2797	1.19634	0.598169	−	0.801370i	$-0.295895\pi$
$599$	29.2797	1.19634	0.598169	+	0.801370i	$0.295895\pi$
$600$	−4.07641	−0.166419
$601$	−27.3933	−1.11740	−0.558698	−	0.829371i	$-0.688699\pi$
$601$	−27.3933	−1.11740	−0.558698	+	0.829371i	$0.688699\pi$
$602$	37.5462	1.53027
$603$	10.2164	0.416045
$604$	−0.0283762	−0.00115461
$605$	−1.00000	−0.0406558
$606$	−15.3006	−0.621544
$607$	−40.2259	−1.63272	−0.816360	−	0.577544i	$-0.804011\pi$
$607$	−40.2259	−1.63272	−0.816360	+	0.577544i	$0.804011\pi$
$608$	0.889478	0.0360731
$609$	−30.6580	−1.24233
$610$	4.82532	0.195372
$611$	−17.3197	−0.700681
$612$	−0.488606	−0.0197507
$613$	−17.9992	−0.726980	−0.363490	−	0.931598i	$-0.618415\pi$
$613$	−17.9992	−0.726980	−0.363490	+	0.931598i	$0.618415\pi$
$614$	−12.8645	−0.519171
$615$	11.7843	0.475187
$616$	−6.41424	−0.258437
$617$	42.5251	1.71199	0.855997	−	0.516981i	$-0.172944\pi$
$617$	42.5251	1.71199	0.855997	+	0.516981i	$0.172944\pi$
$618$	25.1087	1.01002
$619$	4.71299	0.189431	0.0947155	−	0.995504i	$-0.469806\pi$
$619$	4.71299	0.189431	0.0947155	+	0.995504i	$0.469806\pi$
$620$	1.11475	0.0447694
$621$	30.9601	1.24238
$622$	−21.1633	−0.848572
$623$	7.72046	0.309314
$624$	10.7276	0.429447
$625$	1.00000	0.0400000
$626$	−25.7043	−1.02735
$627$	−1.50629	−0.0601553
$628$	3.66156	0.146112
$629$	23.9833	0.956278
$630$	2.54528	0.101407
$631$	44.8168	1.78413	0.892065	−	0.451907i	$-0.149256\pi$
$631$	44.8168	1.78413	0.892065	+	0.451907i	$0.149256\pi$
$632$	18.4456	0.733727
$633$	10.8861	0.432684
$634$	25.7408	1.02230
$635$	−0.182433	−0.00723965
$636$	−0.309298	−0.0122644
$637$	−2.29476	−0.0909215
$638$	12.6138	0.499386
$639$	2.83164	0.112018
$640$	12.4638	0.492676
$641$	−16.8943	−0.667286	−0.333643	−	0.942699i	$-0.608278\pi$
$641$	−16.8943	−0.667286	−0.333643	+	0.942699i	$0.608278\pi$
$642$	−22.5782	−0.891089
$643$	−16.4846	−0.650088	−0.325044	−	0.945699i	$-0.605379\pi$
$643$	−16.4846	−0.650088	−0.325044	+	0.945699i	$0.605379\pi$
$644$	−2.05769	−0.0810843
$645$	16.2448	0.639638
$646$	−6.22904	−0.245078
$647$	−9.12229	−0.358634	−0.179317	−	0.983791i	$-0.557389\pi$
$647$	−9.12229	−0.358634	−0.179317	+	0.983791i	$0.557389\pi$
$648$	−16.9742	−0.666810
$649$	−5.42557	−0.212972
$650$	−2.43829	−0.0956376
$651$	−25.2530	−0.989742
$652$	−1.04203	−0.0408091
$653$	42.5343	1.66449	0.832247	−	0.554405i	$-0.187054\pi$
$653$	42.5343	1.66449	0.832247	+	0.554405i	$0.187054\pi$
$654$	4.67926	0.182973
$655$	18.8389	0.736095
$656$	−33.5651	−1.31050
$657$	8.24991	0.321860
$658$	−36.3246	−1.41608
$659$	43.4399	1.69218	0.846089	−	0.533042i	$-0.178951\pi$
$659$	43.4399	1.69218	0.846089	+	0.533042i	$0.178951\pi$
$660$	0.237386	0.00924024
$661$	15.4534	0.601066	0.300533	−	0.953771i	$-0.402835\pi$
$661$	15.4534	0.601066	0.300533	+	0.953771i	$0.402835\pi$
$662$	−24.9292	−0.968901
$663$	−10.6034	−0.411801
$664$	−37.7753	−1.46596
$665$	2.37015	0.0919103
$666$	6.07344	0.235341
$667$	−47.3062	−1.83170
$668$	−0.426193	−0.0164899
$669$	−32.8317	−1.26935
$670$	−20.5262	−0.792996
$671$	3.28504	0.126818
$672$	3.17554	0.122499
$673$	35.3735	1.36355	0.681774	−	0.731563i	$-0.261209\pi$
$673$	35.3735	1.36355	0.681774	+	0.731563i	$0.261209\pi$
$674$	10.7030	0.412263
$675$	5.62011	0.216318
$676$	−1.61450	−0.0620961
$677$	37.7812	1.45205	0.726024	−	0.687670i	$-0.241366\pi$
$677$	37.7812	1.45205	0.726024	+	0.687670i	$0.241366\pi$
$678$	15.3496	0.589499
$679$	3.97926	0.152710
$680$	−11.4764	−0.440100
$681$	34.3261	1.31538
$682$	10.3900	0.397853
$683$	−25.4882	−0.975279	−0.487640	−	0.873045i	$-0.662142\pi$
$683$	−25.4882	−0.975279	−0.487640	+	0.873045i	$0.662142\pi$
$684$	−0.115219	−0.00440550
$685$	19.3608	0.739738
$686$	−29.1829	−1.11421
$687$	−15.3509	−0.585674
$688$	−46.2700	−1.76403
$689$	2.16283	0.0823971
$690$	−12.1885	−0.464008
$691$	−32.0636	−1.21976	−0.609878	−	0.792495i	$-0.708782\pi$
$691$	−32.0636	−1.21976	−0.609878	+	0.792495i	$0.708782\pi$
$692$	3.10834	0.118161
$693$	1.73281	0.0658240
$694$	53.1055	2.01586
$695$	15.1666	0.575304
$696$	35.0057	1.32689
$697$	33.1765	1.25665
$698$	35.5124	1.34416
$699$	9.02865	0.341495
$700$	−0.373527	−0.0141180
$701$	18.0731	0.682612	0.341306	−	0.939952i	$-0.389131\pi$
$701$	18.0731	0.682612	0.341306	+	0.939952i	$0.389131\pi$
$702$	−13.7034	−0.517203
$703$	5.65554	0.213303
$704$	7.27418	0.274156
$705$	−15.7162	−0.591908
$706$	−44.2592	−1.66572
$707$	16.3904	0.616426
$708$	1.28795	0.0484043
$709$	−33.7607	−1.26791	−0.633954	−	0.773371i	$-0.718569\pi$
$709$	−33.7607	−1.26791	−0.633954	+	0.773371i	$0.718569\pi$
$710$	−5.68916	−0.213510
$711$	−4.98309	−0.186881
$712$	−8.81532	−0.330368
$713$	−38.9661	−1.45929
$714$	−22.2384	−0.832252
$715$	−1.65997	−0.0620794
$716$	2.03866	0.0761885
$717$	3.58457	0.133868
$718$	31.5568	1.17769
$719$	33.6320	1.25426	0.627130	−	0.778914i	$-0.284229\pi$
$719$	33.6320	1.25426	0.627130	+	0.778914i	$0.284229\pi$
$720$	−3.13667	−0.116897
$721$	−26.8971	−1.00170
$722$	−1.46888	−0.0546659
$723$	39.8732	1.48290
$724$	2.62189	0.0974419
$725$	−8.58739	−0.318928
$726$	2.21255	0.0821154
$727$	47.2729	1.75326	0.876628	−	0.481169i	$-0.159788\pi$
$727$	47.2729	1.75326	0.876628	+	0.481169i	$0.159788\pi$
$728$	−10.6474	−0.394620
$729$	29.9821	1.11045
$730$	−16.5752	−0.613475
$731$	45.7343	1.69154
$732$	−0.779823	−0.0288231
$733$	−8.02118	−0.296269	−0.148135	−	0.988967i	$-0.547327\pi$
$733$	−8.02118	−0.296269	−0.148135	+	0.988967i	$0.547327\pi$
$734$	34.2260	1.26330
$735$	−2.08230	−0.0768069
$736$	4.89996	0.180615
$737$	−13.9741	−0.514742
$738$	8.40147	0.309263
$739$	−35.8378	−1.31832	−0.659158	−	0.752005i	$-0.729087\pi$
$739$	−35.8378	−1.31832	−0.659158	+	0.752005i	$0.729087\pi$
$740$	−0.891294	−0.0327646
$741$	−2.50039	−0.0918542
$742$	4.53609	0.166525
$743$	−17.7526	−0.651279	−0.325640	−	0.945494i	$-0.605580\pi$
$743$	−17.7526	−0.651279	−0.325640	+	0.945494i	$0.605580\pi$
$744$	28.8342	1.05711
$745$	1.39216	0.0510046
$746$	−8.15816	−0.298692
$747$	10.2050	0.373382
$748$	0.668318	0.0244361
$749$	24.1864	0.883750
$750$	−2.21255	−0.0807909
$751$	40.0829	1.46264	0.731322	−	0.682032i	$-0.238904\pi$
$751$	40.0829	1.46264	0.731322	+	0.682032i	$0.238904\pi$
$752$	44.7645	1.63240
$753$	−4.09382	−0.149187
$754$	20.9386	0.762537
$755$	0.180055	0.00655289
$756$	−2.09926	−0.0763495
$757$	−21.7750	−0.791427	−0.395713	−	0.918374i	$-0.629503\pi$
$757$	−21.7750	−0.791427	−0.395713	+	0.918374i	$0.629503\pi$
$758$	25.3270	0.919919
$759$	−8.29784	−0.301192
$760$	−2.70626	−0.0981664
$761$	6.14066	0.222599	0.111299	−	0.993787i	$-0.464499\pi$
$761$	6.14066	0.222599	0.111299	+	0.993787i	$0.464499\pi$
$762$	0.403643	0.0146224
$763$	−5.01255	−0.181467
$764$	−3.12829	−0.113178
$765$	3.10036	0.112094
$766$	−5.71041	−0.206326
$767$	−9.00628	−0.325198
$768$	−5.66282	−0.204339
$769$	−13.8426	−0.499177	−0.249588	−	0.968352i	$-0.580295\pi$
$769$	−13.8426	−0.499177	−0.249588	+	0.968352i	$0.580295\pi$
$770$	−3.48145	−0.125463
$771$	−6.83753	−0.246248
$772$	1.37888	0.0496270
$773$	−31.8405	−1.14522	−0.572611	−	0.819827i	$-0.694069\pi$
$773$	−31.8405	−1.14522	−0.572611	+	0.819827i	$0.694069\pi$
$774$	11.5816	0.416291
$775$	−7.07342	−0.254085
$776$	−4.54357	−0.163105
$777$	20.1909	0.724346
$778$	11.6542	0.417824
$779$	7.82338	0.280302
$780$	0.394054	0.0141094
$781$	−3.87314	−0.138592
$782$	−34.3145	−1.22709
$783$	−48.2621	−1.72475
$784$	5.93102	0.211822
$785$	−23.2337	−0.829246
$786$	−41.6819	−1.48674
$787$	12.6125	0.449588	0.224794	−	0.974406i	$-0.427829\pi$
$787$	12.6125	0.449588	0.224794	+	0.974406i	$0.427829\pi$
$788$	3.36270	0.119791
$789$	14.5168	0.516810
$790$	10.0117	0.356201
$791$	−16.4430	−0.584644
$792$	−1.97854	−0.0703045
$793$	5.45307	0.193644
$794$	19.2860	0.684433
$795$	1.96259	0.0696058
$796$	−1.12786	−0.0399758
$797$	−11.4262	−0.404738	−0.202369	−	0.979309i	$-0.564864\pi$
$797$	−11.4262	−0.404738	−0.202369	+	0.979309i	$0.564864\pi$
$798$	−5.24406	−0.185638
$799$	−44.2463	−1.56532
$800$	0.889478	0.0314478
$801$	2.38147	0.0841450
$802$	39.6062	1.39854
$803$	−11.2843	−0.398213
$804$	3.31725	0.116990
$805$	13.0567	0.460187
$806$	17.2471	0.607502
$807$	2.30567	0.0811634
$808$	−18.7148	−0.658384
$809$	1.67518	0.0588963	0.0294481	−	0.999566i	$-0.490625\pi$
$809$	1.67518	0.0588963	0.0294481	+	0.999566i	$0.490625\pi$
$810$	−9.21308	−0.323715
$811$	−16.9393	−0.594819	−0.297409	−	0.954750i	$-0.596123\pi$
$811$	−16.9393	−0.594819	−0.297409	+	0.954750i	$0.596123\pi$
$812$	3.20763	0.112566
$813$	−29.3431	−1.02911
$814$	−8.30728	−0.291170
$815$	6.61201	0.231609
$816$	27.4055	0.959383
$817$	10.7847	0.377307
$818$	−0.917435	−0.0320774
$819$	2.87641	0.100510
$820$	−1.23294	−0.0430561
$821$	38.3477	1.33835	0.669173	−	0.743107i	$-0.266649\pi$
$821$	38.3477	1.33835	0.669173	+	0.743107i	$0.266649\pi$
$822$	−42.8367	−1.49410
$823$	44.7069	1.55838	0.779192	−	0.626785i	$-0.215629\pi$
$823$	44.7069	1.55838	0.779192	+	0.626785i	$0.215629\pi$
$824$	30.7115	1.06988
$825$	−1.50629	−0.0524422
$826$	−18.8888	−0.657227
$827$	44.2586	1.53902	0.769511	−	0.638634i	$-0.220500\pi$
$827$	44.2586	1.53902	0.769511	+	0.638634i	$0.220500\pi$
$828$	−0.634717	−0.0220580
$829$	18.1734	0.631190	0.315595	−	0.948894i	$-0.397796\pi$
$829$	18.1734	0.631190	0.315595	+	0.948894i	$0.397796\pi$
$830$	−20.5033	−0.711678
$831$	31.3119	1.08620
$832$	12.0749	0.418623
$833$	−5.86236	−0.203119
$834$	−33.5569	−1.16198
$835$	2.70433	0.0935872
$836$	0.157597	0.00545060
$837$	−39.7534	−1.37408
$838$	−4.87419	−0.168376
$839$	−45.9108	−1.58502	−0.792508	−	0.609861i	$-0.791225\pi$
$839$	−45.9108	−1.58502	−0.792508	+	0.609861i	$0.791225\pi$
$840$	−9.66168	−0.333360
$841$	44.7433	1.54287
$842$	−17.8512	−0.615191
$843$	−18.9505	−0.652688
$844$	−1.13897	−0.0392049
$845$	10.2445	0.352422
$846$	−11.2047	−0.385227
$847$	−2.37015	−0.0814392
$848$	−5.59004	−0.191963
$849$	3.34168	0.114686
$850$	−6.22904	−0.213654
$851$	31.1552	1.06799
$852$	0.919428	0.0314991
$853$	13.2281	0.452921	0.226460	−	0.974020i	$-0.427285\pi$
$853$	13.2281	0.452921	0.226460	+	0.974020i	$0.427285\pi$
$854$	11.4367	0.391356
$855$	0.731099	0.0250030
$856$	−27.6163	−0.943905
$857$	21.6852	0.740751	0.370376	−	0.928882i	$-0.379229\pi$
$857$	21.6852	0.740751	0.370376	+	0.928882i	$0.379229\pi$
$858$	3.67276	0.125386
$859$	−32.6556	−1.11420	−0.557098	−	0.830447i	$-0.688085\pi$
$859$	−32.6556	−1.11420	−0.557098	+	0.830447i	$0.688085\pi$
$860$	−1.69963	−0.0579568
$861$	27.9304	0.951866
$862$	24.2183	0.824879
$863$	10.8412	0.369038	0.184519	−	0.982829i	$-0.440927\pi$
$863$	10.8412	0.369038	0.184519	+	0.982829i	$0.440927\pi$
$864$	4.99896	0.170068
$865$	−19.7234	−0.670615
$866$	23.0565	0.783493
$867$	−1.48127	−0.0503067
$868$	2.64212	0.0896793
$869$	6.81590	0.231213
$870$	19.0000	0.644161
$871$	−23.1965	−0.785985
$872$	5.72340	0.193819
$873$	1.22745	0.0415429
$874$	−8.09175	−0.273707
$875$	2.37015	0.0801256
$876$	2.67872	0.0905057
$877$	19.7918	0.668323	0.334161	−	0.942516i	$-0.391547\pi$
$877$	19.7918	0.668323	0.334161	+	0.942516i	$0.391547\pi$
$878$	14.6507	0.494436
$879$	41.7953	1.40972
$880$	4.29036	0.144628
$881$	−47.4168	−1.59751	−0.798757	−	0.601654i	$-0.794509\pi$
$881$	−47.4168	−1.59751	−0.798757	+	0.601654i	$0.794509\pi$
$882$	−1.48456	−0.0499877
$883$	−50.8156	−1.71008	−0.855040	−	0.518561i	$-0.826468\pi$
$883$	−50.8156	−1.71008	−0.855040	+	0.518561i	$0.826468\pi$
$884$	1.10939	0.0373127
$885$	−8.17247	−0.274714
$886$	59.9600	2.01440
$887$	33.6405	1.12954	0.564769	−	0.825249i	$-0.308965\pi$
$887$	33.6405	1.12954	0.564769	+	0.825249i	$0.308965\pi$
$888$	−23.0543	−0.773651
$889$	−0.432394	−0.0145020
$890$	−4.78469	−0.160383
$891$	−6.27220	−0.210127
$892$	3.43505	0.115014
$893$	−10.4338	−0.349152
$894$	−3.08021	−0.103018
$895$	−12.9360	−0.432401
$896$	29.5411	0.986898
$897$	−13.7742	−0.459906
$898$	−56.1980	−1.87535
$899$	60.7423	2.02587
$900$	−0.115219	−0.00384063
$901$	5.52532	0.184075
$902$	−11.4916	−0.382628
$903$	38.5025	1.28128
$904$	18.7748	0.624440
$905$	−16.6367	−0.553023
$906$	−0.398382	−0.0132353
$907$	5.82074	0.193274	0.0966372	−	0.995320i	$-0.469191\pi$
$907$	5.82074	0.193274	0.0966372	+	0.995320i	$0.469191\pi$
$908$	−3.59140	−0.119185
$909$	5.05582	0.167691
$910$	−5.77910	−0.191575
$911$	3.08428	0.102187	0.0510934	−	0.998694i	$-0.483729\pi$
$911$	3.08428	0.102187	0.0510934	+	0.998694i	$0.483729\pi$
$912$	6.46251	0.213995
$913$	−13.9585	−0.461958
$914$	−12.9510	−0.428382
$915$	4.94822	0.163583
$916$	1.60610	0.0530672
$917$	44.6509	1.47450
$918$	−35.0079	−1.15543
$919$	4.33367	0.142954	0.0714772	−	0.997442i	$-0.477229\pi$
$919$	4.33367	0.142954	0.0714772	+	0.997442i	$0.477229\pi$
$920$	−14.9083	−0.491511
$921$	−13.1922	−0.434697
$922$	22.2910	0.734115
$923$	−6.42929	−0.211623
$924$	0.562639	0.0185095
$925$	5.65554	0.185953
$926$	29.4834	0.968884
$927$	−8.29672	−0.272500
$928$	−7.63830	−0.250739
$929$	15.8629	0.520444	0.260222	−	0.965549i	$-0.416204\pi$
$929$	15.8629	0.520444	0.260222	+	0.965549i	$0.416204\pi$
$930$	15.6503	0.513193
$931$	−1.38241	−0.0453066
$932$	−0.944631	−0.0309424
$933$	−21.7023	−0.710503
$934$	19.6571	0.643201
$935$	−4.24068	−0.138685
$936$	−3.28432	−0.107351
$937$	−4.74377	−0.154972	−0.0774860	−	0.996993i	$-0.524689\pi$
$937$	−4.74377	−0.154972	−0.0774860	+	0.996993i	$0.524689\pi$
$938$	−48.6501	−1.58848
$939$	−26.3590	−0.860194
$940$	1.64433	0.0536320
$941$	12.2501	0.399341	0.199670	−	0.979863i	$-0.436013\pi$
$941$	12.2501	0.399341	0.199670	+	0.979863i	$0.436013\pi$
$942$	51.4057	1.67489
$943$	43.0974	1.40345
$944$	23.2776	0.757622
$945$	13.3205	0.433315
$946$	−15.8413	−0.515046
$947$	−15.9323	−0.517730	−0.258865	−	0.965914i	$-0.583348\pi$
$947$	−15.9323	−0.517730	−0.258865	+	0.965914i	$0.583348\pi$
$948$	−1.61800	−0.0525501
$949$	−18.7315	−0.608051
$950$	−1.46888	−0.0476566
$951$	26.3964	0.855961
$952$	−27.2007	−0.881581
$953$	6.43381	0.208412	0.104206	−	0.994556i	$-0.466770\pi$
$953$	6.43381	0.208412	0.104206	+	0.994556i	$0.466770\pi$
$954$	1.39921	0.0453011
$955$	19.8500	0.642331
$956$	−0.375039	−0.0121296
$957$	12.9351	0.418132
$958$	49.4027	1.59613
$959$	45.8879	1.48180
$960$	10.9570	0.353636
$961$	19.0333	0.613978
$962$	−13.7898	−0.444602
$963$	7.46056	0.240413
$964$	−4.17177	−0.134364
$965$	−8.74942	−0.281654
$966$	−28.8885	−0.929473
$967$	38.3648	1.23373	0.616864	−	0.787069i	$-0.288403\pi$
$967$	38.3648	1.23373	0.616864	+	0.787069i	$0.288403\pi$
$968$	2.70626	0.0869826
$969$	−6.38769	−0.205202
$970$	−2.46611	−0.0791821
$971$	−47.2502	−1.51633	−0.758165	−	0.652062i	$-0.773904\pi$
$971$	−47.2502	−1.51633	−0.758165	+	0.652062i	$0.773904\pi$
$972$	−1.16820	−0.0374700
$973$	35.9472	1.15241
$974$	−32.6904	−1.04747
$975$	−2.50039	−0.0800766
$976$	−14.0940	−0.451138
$977$	5.33212	0.170590	0.0852948	−	0.996356i	$-0.472817\pi$
$977$	5.33212	0.170590	0.0852948	+	0.996356i	$0.472817\pi$
$978$	−14.6294	−0.467796
$979$	−3.25738	−0.104106
$980$	0.217863	0.00695938
$981$	−1.54618	−0.0493657
$982$	0.930048	0.0296790
$983$	2.78056	0.0886861	0.0443430	−	0.999016i	$-0.485881\pi$
$983$	2.78056	0.0886861	0.0443430	+	0.999016i	$0.485881\pi$
$984$	−31.8913	−1.01666
$985$	−21.3374	−0.679866
$986$	53.4912	1.70351
$987$	−37.2498	−1.18567
$988$	0.261606	0.00832279
$989$	59.4105	1.88914
$990$	−1.07389	−0.0341306
$991$	21.2957	0.676480	0.338240	−	0.941060i	$-0.390168\pi$
$991$	21.2957	0.676480	0.338240	+	0.941060i	$0.390168\pi$
$992$	−6.29166	−0.199760
$993$	−25.5641	−0.811253
$994$	−13.4841	−0.427691
$995$	7.15659	0.226879
$996$	3.31354	0.104994
$997$	−30.4272	−0.963638	−0.481819	−	0.876271i	$-0.660024\pi$
$997$	−30.4272	−0.963638	−0.481819	+	0.876271i	$0.660024\pi$
$998$	47.5082	1.50385
$999$	31.7847	1.00562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.e.1.2	✓	5
3.2	odd	2		9405.2.a.u.1.4		5
5.4	even	2		5225.2.a.i.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.e.1.2	✓	5	1.1	even	1	trivial
5225.2.a.i.1.4		5	5.4	even	2
9405.2.a.u.1.4		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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46888 q^{2} -1.50629 q^{3} +0.157597 q^{4} -1.00000 q^{5} +2.21255 q^{6} -2.37015 q^{7} +2.70626 q^{8} -0.731099 q^{9} +O(q^{10})\)	\(q-1.46888 q^{2} -1.50629 q^{3} +0.157597 q^{4} -1.00000 q^{5} +2.21255 q^{6} -2.37015 q^{7} +2.70626 q^{8} -0.731099 q^{9} +1.46888 q^{10} +1.00000 q^{11} -0.237386 q^{12} +1.65997 q^{13} +3.48145 q^{14} +1.50629 q^{15} -4.29036 q^{16} +4.24068 q^{17} +1.07389 q^{18} +1.00000 q^{19} -0.157597 q^{20} +3.57012 q^{21} -1.46888 q^{22} +5.50880 q^{23} -4.07641 q^{24} +1.00000 q^{25} -2.43829 q^{26} +5.62011 q^{27} -0.373527 q^{28} -8.58739 q^{29} -2.21255 q^{30} -7.07342 q^{31} +0.889478 q^{32} -1.50629 q^{33} -6.22904 q^{34} +2.37015 q^{35} -0.115219 q^{36} +5.65554 q^{37} -1.46888 q^{38} -2.50039 q^{39} -2.70626 q^{40} +7.82338 q^{41} -5.24406 q^{42} +10.7847 q^{43} +0.157597 q^{44} +0.731099 q^{45} -8.09175 q^{46} -10.4338 q^{47} +6.46251 q^{48} -1.38241 q^{49} -1.46888 q^{50} -6.38769 q^{51} +0.261606 q^{52} +1.30293 q^{53} -8.25524 q^{54} -1.00000 q^{55} -6.41424 q^{56} -1.50629 q^{57} +12.6138 q^{58} -5.42557 q^{59} +0.237386 q^{60} +3.28504 q^{61} +10.3900 q^{62} +1.73281 q^{63} +7.27418 q^{64} -1.65997 q^{65} +2.21255 q^{66} -13.9741 q^{67} +0.668318 q^{68} -8.29784 q^{69} -3.48145 q^{70} -3.87314 q^{71} -1.97854 q^{72} -11.2843 q^{73} -8.30728 q^{74} -1.50629 q^{75} +0.157597 q^{76} -2.37015 q^{77} +3.67276 q^{78} +6.81590 q^{79} +4.29036 q^{80} -6.27220 q^{81} -11.4916 q^{82} -13.9585 q^{83} +0.562639 q^{84} -4.24068 q^{85} -15.8413 q^{86} +12.9351 q^{87} +2.70626 q^{88} -3.25738 q^{89} -1.07389 q^{90} -3.93437 q^{91} +0.868169 q^{92} +10.6546 q^{93} +15.3259 q^{94} -1.00000 q^{95} -1.33981 q^{96} -1.67891 q^{97} +2.03059 q^{98} -0.731099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9	\( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9} + q^{10} + 5 q^{11} + 3 q^{12} - 3 q^{13} + 2 q^{14} + 3 q^{15} - 11 q^{16} - 11 q^{17} + 3 q^{18} + 5 q^{19} - q^{20} - 3 q^{21} - q^{22} + 5 q^{24} + 5 q^{25} - 8 q^{26} - 8 q^{28} - 15 q^{29} + 4 q^{30} - 9 q^{31} - 3 q^{33} - 14 q^{34} - 3 q^{35} - 7 q^{36} + 11 q^{37} - q^{38} - 10 q^{39} - 3 q^{40} - 23 q^{41} - 15 q^{42} + 9 q^{43} + q^{44} + 4 q^{45} + 8 q^{46} - 6 q^{47} + 4 q^{48} - 12 q^{49} - q^{50} - 27 q^{52} - 13 q^{53} + 9 q^{54} - 5 q^{55} - 12 q^{56} - 3 q^{57} + 17 q^{58} - 21 q^{59} - 3 q^{60} - 31 q^{61} - 18 q^{62} + 10 q^{63} - q^{64} + 3 q^{65} - 4 q^{66} + 5 q^{68} - 2 q^{70} - 28 q^{71} - 20 q^{72} - 14 q^{73} + 21 q^{74} - 3 q^{75} + q^{76} + 3 q^{77} + 13 q^{78} + 3 q^{79} + 11 q^{80} - 3 q^{81} - 18 q^{82} - 33 q^{83} + 13 q^{84} + 11 q^{85} - 20 q^{86} + 22 q^{87} + 3 q^{88} - 10 q^{89} - 3 q^{90} + 14 q^{91} + 21 q^{92} + 30 q^{93} + 14 q^{94} - 5 q^{95} - 3 q^{96} - 10 q^{97} + 8 q^{98} - 4 q^{99}+O(q^{100}) \) 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9 + q^10 + 5 * q^11 + 3 * q^12 - 3 * q^13 + 2 * q^14 + 3 * q^15 - 11 * q^16 - 11 * q^17 + 3 * q^18 + 5 * q^19 - q^20 - 3 * q^21 - q^22 + 5 * q^24 + 5 * q^25 - 8 * q^26 - 8 * q^28 - 15 * q^29 + 4 * q^30 - 9 * q^31 - 3 * q^33 - 14 * q^34 - 3 * q^35 - 7 * q^36 + 11 * q^37 - q^38 - 10 * q^39 - 3 * q^40 - 23 * q^41 - 15 * q^42 + 9 * q^43 + q^44 + 4 * q^45 + 8 * q^46 - 6 * q^47 + 4 * q^48 - 12 * q^49 - q^50 - 27 * q^52 - 13 * q^53 + 9 * q^54 - 5 * q^55 - 12 * q^56 - 3 * q^57 + 17 * q^58 - 21 * q^59 - 3 * q^60 - 31 * q^61 - 18 * q^62 + 10 * q^63 - q^64 + 3 * q^65 - 4 * q^66 + 5 * q^68 - 2 * q^70 - 28 * q^71 - 20 * q^72 - 14 * q^73 + 21 * q^74 - 3 * q^75 + q^76 + 3 * q^77 + 13 * q^78 + 3 * q^79 + 11 * q^80 - 3 * q^81 - 18 * q^82 - 33 * q^83 + 13 * q^84 + 11 * q^85 - 20 * q^86 + 22 * q^87 + 3 * q^88 - 10 * q^89 - 3 * q^90 + 14 * q^91 + 21 * q^92 + 30 * q^93 + 14 * q^94 - 5 * q^95 - 3 * q^96 - 10 * q^97 + 8 * q^98 - 4 * q^99

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