LMFDB - Embedded newform 8007.2.a.f.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.50666 q^{2} -1.00000 q^{3} +0.270014 q^{4} -0.475693 q^{5} +1.50666 q^{6} -0.891979 q^{7} +2.60649 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.50666 q^{2} -1.00000 q^{3} +0.270014 q^{4} -0.475693 q^{5} +1.50666 q^{6} -0.891979 q^{7} +2.60649 q^{8} +1.00000 q^{9} +0.716706 q^{10} -4.53256 q^{11} -0.270014 q^{12} +1.62404 q^{13} +1.34391 q^{14} +0.475693 q^{15} -4.46712 q^{16} -1.00000 q^{17} -1.50666 q^{18} +0.833908 q^{19} -0.128444 q^{20} +0.891979 q^{21} +6.82902 q^{22} -3.25633 q^{23} -2.60649 q^{24} -4.77372 q^{25} -2.44688 q^{26} -1.00000 q^{27} -0.240847 q^{28} +4.74015 q^{29} -0.716706 q^{30} +10.3297 q^{31} +1.51743 q^{32} +4.53256 q^{33} +1.50666 q^{34} +0.424308 q^{35} +0.270014 q^{36} -4.87105 q^{37} -1.25641 q^{38} -1.62404 q^{39} -1.23989 q^{40} +4.26901 q^{41} -1.34391 q^{42} -9.65601 q^{43} -1.22386 q^{44} -0.475693 q^{45} +4.90616 q^{46} +12.1483 q^{47} +4.46712 q^{48} -6.20437 q^{49} +7.19235 q^{50} +1.00000 q^{51} +0.438515 q^{52} +0.534008 q^{53} +1.50666 q^{54} +2.15611 q^{55} -2.32494 q^{56} -0.833908 q^{57} -7.14177 q^{58} +12.6728 q^{59} +0.128444 q^{60} +0.228820 q^{61} -15.5633 q^{62} -0.891979 q^{63} +6.64800 q^{64} -0.772546 q^{65} -6.82902 q^{66} +8.20386 q^{67} -0.270014 q^{68} +3.25633 q^{69} -0.639286 q^{70} -8.50947 q^{71} +2.60649 q^{72} -6.38859 q^{73} +7.33900 q^{74} +4.77372 q^{75} +0.225167 q^{76} +4.04295 q^{77} +2.44688 q^{78} -4.83950 q^{79} +2.12498 q^{80} +1.00000 q^{81} -6.43193 q^{82} -17.0444 q^{83} +0.240847 q^{84} +0.475693 q^{85} +14.5483 q^{86} -4.74015 q^{87} -11.8141 q^{88} -1.87298 q^{89} +0.716706 q^{90} -1.44861 q^{91} -0.879254 q^{92} -10.3297 q^{93} -18.3033 q^{94} -0.396684 q^{95} -1.51743 q^{96} -8.80052 q^{97} +9.34786 q^{98} -4.53256 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	−1.50666	−1.06537	−0.532684	−	0.846315i	$-0.678816\pi$
$2$	−1.50666	−1.06537	−0.532684	+	0.846315i	$0.678816\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.270014	0.135007
$5$	−0.475693	−0.212736	−0.106368	−	0.994327i	$-0.533922\pi$
$5$	−0.475693	−0.212736	−0.106368	+	0.994327i	$0.533922\pi$
$6$	1.50666	0.615090
$7$	−0.891979	−0.337136	−0.168568	−	0.985690i	$-0.553914\pi$
$7$	−0.891979	−0.337136	−0.168568	+	0.985690i	$0.553914\pi$
$8$	2.60649	0.921535
$9$	1.00000	0.333333
$10$	0.716706	0.226642
$11$	−4.53256	−1.36662	−0.683310	−	0.730129i	$-0.739460\pi$
$11$	−4.53256	−1.36662	−0.683310	+	0.730129i	$0.739460\pi$
$12$	−0.270014	−0.0779464
$13$	1.62404	0.450429	0.225214	−	0.974309i	$-0.427692\pi$
$13$	1.62404	0.450429	0.225214	+	0.974309i	$0.427692\pi$
$14$	1.34391	0.359174
$15$	0.475693	0.122823
$16$	−4.46712	−1.11678
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.50666	−0.355122
$19$	0.833908	0.191312	0.0956558	−	0.995414i	$-0.469505\pi$
$19$	0.833908	0.191312	0.0956558	+	0.995414i	$0.469505\pi$
$20$	−0.128444	−0.0287209
$21$	0.891979	0.194646
$22$	6.82902	1.45595
$23$	−3.25633	−0.678991	−0.339495	−	0.940608i	$-0.610256\pi$
$23$	−3.25633	−0.678991	−0.339495	+	0.940608i	$0.610256\pi$
$24$	−2.60649	−0.532048
$25$	−4.77372	−0.954743
$26$	−2.44688	−0.479872
$27$	−1.00000	−0.192450
$28$	−0.240847	−0.0455158
$29$	4.74015	0.880223	0.440111	−	0.897943i	$-0.354939\pi$
$29$	4.74015	0.880223	0.440111	+	0.897943i	$0.354939\pi$
$30$	−0.716706	−0.130852
$31$	10.3297	1.85527	0.927633	−	0.373493i	$-0.121840\pi$
$31$	10.3297	1.85527	0.927633	+	0.373493i	$0.121840\pi$
$32$	1.51743	0.268246
$33$	4.53256	0.789018
$34$	1.50666	0.258389
$35$	0.424308	0.0717211
$36$	0.270014	0.0450024
$37$	−4.87105	−0.800795	−0.400398	−	0.916341i	$-0.631128\pi$
$37$	−4.87105	−0.800795	−0.400398	+	0.916341i	$0.631128\pi$
$38$	−1.25641	−0.203817
$39$	−1.62404	−0.260055
$40$	−1.23989	−0.196044
$41$	4.26901	0.666707	0.333353	−	0.942802i	$-0.391820\pi$
$41$	4.26901	0.666707	0.333353	+	0.942802i	$0.391820\pi$
$42$	−1.34391	−0.207369
$43$	−9.65601	−1.47253	−0.736264	−	0.676695i	$-0.763412\pi$
$43$	−9.65601	−1.47253	−0.736264	+	0.676695i	$0.763412\pi$
$44$	−1.22386	−0.184503
$45$	−0.475693	−0.0709121
$46$	4.90616	0.723374
$47$	12.1483	1.77201	0.886003	−	0.463679i	$-0.153471\pi$
$47$	12.1483	1.77201	0.886003	+	0.463679i	$0.153471\pi$
$48$	4.46712	0.644773
$49$	−6.20437	−0.886339
$50$	7.19235	1.01715
$51$	1.00000	0.140028
$52$	0.438515	0.0608110
$53$	0.534008	0.0733516	0.0366758	−	0.999327i	$-0.488323\pi$
$53$	0.534008	0.0733516	0.0366758	+	0.999327i	$0.488323\pi$
$54$	1.50666	0.205030
$55$	2.15611	0.290729
$56$	−2.32494	−0.310683
$57$	−0.833908	−0.110454
$58$	−7.14177	−0.937761
$59$	12.6728	1.64985	0.824926	−	0.565241i	$-0.191217\pi$
$59$	12.6728	1.64985	0.824926	+	0.565241i	$0.191217\pi$
$60$	0.128444	0.0165820
$61$	0.228820	0.0292974	0.0146487	−	0.999893i	$-0.495337\pi$
$61$	0.228820	0.0292974	0.0146487	+	0.999893i	$0.495337\pi$
$62$	−15.5633	−1.97654
$63$	−0.891979	−0.112379
$64$	6.64800	0.831000
$65$	−0.772546	−0.0958225
$66$	−6.82902	−0.840594
$67$	8.20386	1.00226	0.501130	−	0.865372i	$-0.332918\pi$
$67$	8.20386	1.00226	0.501130	+	0.865372i	$0.332918\pi$
$68$	−0.270014	−0.0327440
$69$	3.25633	0.392016
$70$	−0.639286	−0.0764093
$71$	−8.50947	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$71$	−8.50947	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$72$	2.60649	0.307178
$73$	−6.38859	−0.747728	−0.373864	−	0.927483i	$-0.621967\pi$
$73$	−6.38859	−0.747728	−0.373864	+	0.927483i	$0.621967\pi$
$74$	7.33900	0.853141
$75$	4.77372	0.551221
$76$	0.225167	0.0258284
$77$	4.04295	0.460737
$78$	2.44688	0.277054
$79$	−4.83950	−0.544486	−0.272243	−	0.962228i	$-0.587766\pi$
$79$	−4.83950	−0.544486	−0.272243	+	0.962228i	$0.587766\pi$
$80$	2.12498	0.237580
$81$	1.00000	0.111111
$82$	−6.43193	−0.710288
$83$	−17.0444	−1.87087	−0.935435	−	0.353499i	$-0.884992\pi$
$83$	−17.0444	−1.87087	−0.935435	+	0.353499i	$0.884992\pi$
$84$	0.240847	0.0262786
$85$	0.475693	0.0515961
$86$	14.5483	1.56878
$87$	−4.74015	−0.508197
$88$	−11.8141	−1.25939
$89$	−1.87298	−0.198536	−0.0992679	−	0.995061i	$-0.531650\pi$
$89$	−1.87298	−0.198536	−0.0992679	+	0.995061i	$0.531650\pi$
$90$	0.716706	0.0755474
$91$	−1.44861	−0.151856
$92$	−0.879254	−0.0916685
$93$	−10.3297	−1.07114
$94$	−18.3033	−1.88784
$95$	−0.396684	−0.0406989
$96$	−1.51743	−0.154872
$97$	−8.80052	−0.893558	−0.446779	−	0.894644i	$-0.647429\pi$
$97$	−8.80052	−0.893558	−0.446779	+	0.894644i	$0.647429\pi$
$98$	9.34786	0.944276
$99$	−4.53256	−0.455540
$100$	−1.28897	−0.128897
$101$	12.2767	1.22157	0.610787	−	0.791795i	$-0.290853\pi$
$101$	12.2767	1.22157	0.610787	+	0.791795i	$0.290853\pi$
$102$	−1.50666	−0.149181
$103$	5.16951	0.509367	0.254684	−	0.967024i	$-0.418029\pi$
$103$	5.16951	0.509367	0.254684	+	0.967024i	$0.418029\pi$
$104$	4.23306	0.415086
$105$	−0.424308	−0.0414082
$106$	−0.804566	−0.0781464
$107$	8.96822	0.866991	0.433495	−	0.901156i	$-0.357280\pi$
$107$	8.96822	0.866991	0.433495	+	0.901156i	$0.357280\pi$
$108$	−0.270014	−0.0259821
$109$	5.66265	0.542383	0.271192	−	0.962525i	$-0.412582\pi$
$109$	5.66265	0.542383	0.271192	+	0.962525i	$0.412582\pi$
$110$	−3.24851	−0.309734
$111$	4.87105	0.462339
$112$	3.98458	0.376507
$113$	0.871145	0.0819504	0.0409752	−	0.999160i	$-0.486954\pi$
$113$	0.871145	0.0819504	0.0409752	+	0.999160i	$0.486954\pi$
$114$	1.25641	0.117674
$115$	1.54901	0.144446
$116$	1.27991	0.118836
$117$	1.62404	0.150143
$118$	−19.0935	−1.75770
$119$	0.891979	0.0817676
$120$	1.23989	0.113186
$121$	9.54413	0.867648
$122$	−0.344753	−0.0312125
$123$	−4.26901	−0.384923
$124$	2.78916	0.250474
$125$	4.64929	0.415845
$126$	1.34391	0.119725
$127$	17.9453	1.59239	0.796194	−	0.605041i	$-0.206843\pi$
$127$	17.9453	1.59239	0.796194	+	0.605041i	$0.206843\pi$
$128$	−13.0511	−1.15357
$129$	9.65601	0.850164
$130$	1.16396	0.102086
$131$	−8.71992	−0.761863	−0.380931	−	0.924603i	$-0.624397\pi$
$131$	−8.71992	−0.761863	−0.380931	+	0.924603i	$0.624397\pi$
$132$	1.22386	0.106523
$133$	−0.743829	−0.0644981
$134$	−12.3604	−1.06778
$135$	0.475693	0.0409411
$136$	−2.60649	−0.223505
$137$	1.67736	0.143307	0.0716534	−	0.997430i	$-0.477172\pi$
$137$	1.67736	0.143307	0.0716534	+	0.997430i	$0.477172\pi$
$138$	−4.90616	−0.417640
$139$	6.01855	0.510486	0.255243	−	0.966877i	$-0.417844\pi$
$139$	6.01855	0.510486	0.255243	+	0.966877i	$0.417844\pi$
$140$	0.114569	0.00968286
$141$	−12.1483	−1.02307
$142$	12.8208	1.07590
$143$	−7.36108	−0.615564
$144$	−4.46712	−0.372260
$145$	−2.25485	−0.187255
$146$	9.62542	0.796605
$147$	6.20437	0.511728
$148$	−1.31525	−0.108113
$149$	15.0198	1.23047	0.615236	−	0.788343i	$-0.289061\pi$
$149$	15.0198	1.23047	0.615236	+	0.788343i	$0.289061\pi$
$150$	−7.19235	−0.587253
$151$	−8.21834	−0.668799	−0.334400	−	0.942431i	$-0.608534\pi$
$151$	−8.21834	−0.668799	−0.334400	+	0.942431i	$0.608534\pi$
$152$	2.17358	0.176300
$153$	−1.00000	−0.0808452
$154$	−6.09134	−0.490854
$155$	−4.91375	−0.394682
$156$	−0.438515	−0.0351093
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	7.29146	0.580078
$159$	−0.534008	−0.0423496
$160$	−0.721829	−0.0570656
$161$	2.90457	0.228913
$162$	−1.50666	−0.118374
$163$	11.0945	0.868986	0.434493	−	0.900675i	$-0.356928\pi$
$163$	11.0945	0.868986	0.434493	+	0.900675i	$0.356928\pi$
$164$	1.15269	0.0900101
$165$	−2.15611	−0.167853
$166$	25.6801	1.99316
$167$	−21.9002	−1.69469	−0.847345	−	0.531043i	$-0.821800\pi$
$167$	−21.9002	−1.69469	−0.847345	+	0.531043i	$0.821800\pi$
$168$	2.32494	0.179373
$169$	−10.3625	−0.797114
$170$	−0.716706	−0.0549688
$171$	0.833908	0.0637706
$172$	−2.60726	−0.198802
$173$	4.93490	0.375194	0.187597	−	0.982246i	$-0.439930\pi$
$173$	4.93490	0.375194	0.187597	+	0.982246i	$0.439930\pi$
$174$	7.14177	0.541416
$175$	4.25806	0.321879
$176$	20.2475	1.52621
$177$	−12.6728	−0.952543
$178$	2.82194	0.211514
$179$	11.6311	0.869347	0.434673	−	0.900588i	$-0.356864\pi$
$179$	11.6311	0.869347	0.434673	+	0.900588i	$0.356864\pi$
$180$	−0.128444	−0.00957363
$181$	−5.85579	−0.435258	−0.217629	−	0.976032i	$-0.569832\pi$
$181$	−5.85579	−0.435258	−0.217629	+	0.976032i	$0.569832\pi$
$182$	2.18256	0.161782
$183$	−0.228820	−0.0169149
$184$	−8.48760	−0.625714
$185$	2.31712	0.170358
$186$	15.5633	1.14116
$187$	4.53256	0.331454
$188$	3.28020	0.239233
$189$	0.891979	0.0648819
$190$	0.597667	0.0433593
$191$	10.5939	0.766548	0.383274	−	0.923635i	$-0.374796\pi$
$191$	10.5939	0.766548	0.383274	+	0.923635i	$0.374796\pi$
$192$	−6.64800	−0.479778
$193$	14.9917	1.07913	0.539564	−	0.841944i	$-0.318589\pi$
$193$	14.9917	1.07913	0.539564	+	0.841944i	$0.318589\pi$
$194$	13.2594	0.951967
$195$	0.772546	0.0553231
$196$	−1.67527	−0.119662
$197$	−1.76016	−0.125406	−0.0627032	−	0.998032i	$-0.519972\pi$
$197$	−1.76016	−0.125406	−0.0627032	+	0.998032i	$0.519972\pi$
$198$	6.82902	0.485317
$199$	−5.97216	−0.423355	−0.211678	−	0.977340i	$-0.567893\pi$
$199$	−5.97216	−0.423355	−0.211678	+	0.977340i	$0.567893\pi$
$200$	−12.4427	−0.879829
$201$	−8.20386	−0.578655
$202$	−18.4967	−1.30142
$203$	−4.22811	−0.296755
$204$	0.270014	0.0189048
$205$	−2.03074	−0.141833
$206$	−7.78868	−0.542663
$207$	−3.25633	−0.226330
$208$	−7.25480	−0.503030
$209$	−3.77974	−0.261450
$210$	0.639286	0.0441150
$211$	8.82507	0.607543	0.303771	−	0.952745i	$-0.401754\pi$
$211$	8.82507	0.607543	0.303771	+	0.952745i	$0.401754\pi$
$212$	0.144190	0.00990298
$213$	8.50947	0.583059
$214$	−13.5120	−0.923664
$215$	4.59329	0.313260
$216$	−2.60649	−0.177349
$217$	−9.21386	−0.625478
$218$	−8.53166	−0.577837
$219$	6.38859	0.431701
$220$	0.582179	0.0392505
$221$	−1.62404	−0.109245
$222$	−7.33900	−0.492561
$223$	11.5235	0.771669	0.385835	−	0.922568i	$-0.373914\pi$
$223$	11.5235	0.771669	0.385835	+	0.922568i	$0.373914\pi$
$224$	−1.35351	−0.0904354
$225$	−4.77372	−0.318248
$226$	−1.31252	−0.0873072
$227$	5.12093	0.339888	0.169944	−	0.985454i	$-0.445641\pi$
$227$	5.12093	0.339888	0.169944	+	0.985454i	$0.445641\pi$
$228$	−0.225167	−0.0149120
$229$	−20.4102	−1.34875	−0.674373	−	0.738391i	$-0.735586\pi$
$229$	−20.4102	−1.34875	−0.674373	+	0.738391i	$0.735586\pi$
$230$	−2.33383	−0.153888
$231$	−4.04295	−0.266007
$232$	12.3552	0.811156
$233$	−5.32447	−0.348818	−0.174409	−	0.984673i	$-0.555801\pi$
$233$	−5.32447	−0.348818	−0.174409	+	0.984673i	$0.555801\pi$
$234$	−2.44688	−0.159957
$235$	−5.77884	−0.376970
$236$	3.42182	0.222742
$237$	4.83950	0.314359
$238$	−1.34391	−0.0871125
$239$	−12.0876	−0.781881	−0.390940	−	0.920416i	$-0.627850\pi$
$239$	−12.0876	−0.781881	−0.390940	+	0.920416i	$0.627850\pi$
$240$	−2.12498	−0.137167
$241$	19.7085	1.26954	0.634770	−	0.772701i	$-0.281095\pi$
$241$	19.7085	1.26954	0.634770	+	0.772701i	$0.281095\pi$
$242$	−14.3797	−0.924363
$243$	−1.00000	−0.0641500
$244$	0.0617846	0.00395535
$245$	2.95138	0.188556
$246$	6.43193	0.410085
$247$	1.35430	0.0861722
$248$	26.9243	1.70969
$249$	17.0444	1.08015
$250$	−7.00488	−0.443027
$251$	1.59923	0.100942	0.0504712	−	0.998726i	$-0.483928\pi$
$251$	1.59923	0.100942	0.0504712	+	0.998726i	$0.483928\pi$
$252$	−0.240847	−0.0151719
$253$	14.7595	0.927922
$254$	−27.0374	−1.69648
$255$	−0.475693	−0.0297890
$256$	6.36754	0.397971
$257$	23.6021	1.47226	0.736128	−	0.676842i	$-0.236652\pi$
$257$	23.6021	1.47226	0.736128	+	0.676842i	$0.236652\pi$
$258$	−14.5483	−0.905737
$259$	4.34487	0.269977
$260$	−0.208598	−0.0129367
$261$	4.74015	0.293408
$262$	13.1379	0.811663
$263$	29.6514	1.82839	0.914193	−	0.405279i	$-0.132826\pi$
$263$	29.6514	1.82839	0.914193	+	0.405279i	$0.132826\pi$
$264$	11.8141	0.727108
$265$	−0.254023	−0.0156045
$266$	1.12069	0.0687142
$267$	1.87298	0.114625
$268$	2.21516	0.135312
$269$	−10.9975	−0.670528	−0.335264	−	0.942124i	$-0.608825\pi$
$269$	−10.9975	−0.670528	−0.335264	+	0.942124i	$0.608825\pi$
$270$	−0.716706	−0.0436173
$271$	12.6363	0.767601	0.383800	−	0.923416i	$-0.374615\pi$
$271$	12.6363	0.767601	0.383800	+	0.923416i	$0.374615\pi$
$272$	4.46712	0.270859
$273$	1.44861	0.0876740
$274$	−2.52721	−0.152674
$275$	21.6372	1.30477
$276$	0.879254	0.0529249
$277$	−31.7187	−1.90579	−0.952896	−	0.303297i	$-0.901913\pi$
$277$	−31.7187	−1.90579	−0.952896	+	0.303297i	$0.901913\pi$
$278$	−9.06788	−0.543855
$279$	10.3297	0.618422
$280$	1.10596	0.0660935
$281$	24.2286	1.44536	0.722679	−	0.691184i	$-0.242910\pi$
$281$	24.2286	1.44536	0.722679	+	0.691184i	$0.242910\pi$
$282$	18.3033	1.08994
$283$	−26.5012	−1.57533	−0.787667	−	0.616101i	$-0.788711\pi$
$283$	−26.5012	−1.57533	−0.787667	+	0.616101i	$0.788711\pi$
$284$	−2.29768	−0.136342
$285$	0.396684	0.0234975
$286$	11.0906	0.655802
$287$	−3.80787	−0.224771
$288$	1.51743	0.0894153
$289$	1.00000	0.0588235
$290$	3.39729	0.199496
$291$	8.80052	0.515896
$292$	−1.72501	−0.100949
$293$	14.4985	0.847012	0.423506	−	0.905893i	$-0.360799\pi$
$293$	14.4985	0.847012	0.423506	+	0.905893i	$0.360799\pi$
$294$	−9.34786	−0.545178
$295$	−6.02834	−0.350983
$296$	−12.6964	−0.737961
$297$	4.53256	0.263006
$298$	−22.6297	−1.31090
$299$	−5.28841	−0.305837
$300$	1.28897	0.0744188
$301$	8.61296	0.496443
$302$	12.3822	0.712517
$303$	−12.2767	−0.705276
$304$	−3.72517	−0.213653
$305$	−0.108848	−0.00623261
$306$	1.50666	0.0861298
$307$	9.63388	0.549834	0.274917	−	0.961468i	$-0.411350\pi$
$307$	9.63388	0.549834	0.274917	+	0.961468i	$0.411350\pi$
$308$	1.09165	0.0622027
$309$	−5.16951	−0.294083
$310$	7.40334	0.420482
$311$	0.353212	0.0200288	0.0100144	−	0.999950i	$-0.496812\pi$
$311$	0.353212	0.0200288	0.0100144	+	0.999950i	$0.496812\pi$
$312$	−4.23306	−0.239650
$313$	−1.87195	−0.105809	−0.0529045	−	0.998600i	$-0.516848\pi$
$313$	−1.87195	−0.105809	−0.0529045	+	0.998600i	$0.516848\pi$
$314$	1.50666	0.0850256
$315$	0.424308	0.0239070
$316$	−1.30673	−0.0735095
$317$	−23.7694	−1.33502	−0.667512	−	0.744599i	$-0.732641\pi$
$317$	−23.7694	−1.33502	−0.667512	+	0.744599i	$0.732641\pi$
$318$	0.804566	0.0451178
$319$	−21.4850	−1.20293
$320$	−3.16240	−0.176784
$321$	−8.96822	−0.500557
$322$	−4.37620	−0.243876
$323$	−0.833908	−0.0463999
$324$	0.270014	0.0150008
$325$	−7.75272	−0.430044
$326$	−16.7156	−0.925789
$327$	−5.66265	−0.313145
$328$	11.1271	0.614394
$329$	−10.8360	−0.597408
$330$	3.24851	0.178825
$331$	−19.4227	−1.06757	−0.533785	−	0.845620i	$-0.679231\pi$
$331$	−19.4227	−1.06757	−0.533785	+	0.845620i	$0.679231\pi$
$332$	−4.60224	−0.252581
$333$	−4.87105	−0.266932
$334$	32.9961	1.80547
$335$	−3.90252	−0.213217
$336$	−3.98458	−0.217377
$337$	24.0399	1.30954	0.654769	−	0.755829i	$-0.272766\pi$
$337$	24.0399	1.30954	0.654769	+	0.755829i	$0.272766\pi$
$338$	15.6127	0.849219
$339$	−0.871145	−0.0473141
$340$	0.128444	0.00696584
$341$	−46.8199	−2.53544
$342$	−1.25641	−0.0679391
$343$	11.7780	0.635954
$344$	−25.1683	−1.35699
$345$	−1.54901	−0.0833959
$346$	−7.43520	−0.399719
$347$	27.7183	1.48800	0.743998	−	0.668182i	$-0.232927\pi$
$347$	27.7183	1.48800	0.743998	+	0.668182i	$0.232927\pi$
$348$	−1.27991	−0.0686102
$349$	−23.0288	−1.23270	−0.616352	−	0.787471i	$-0.711390\pi$
$349$	−23.0288	−1.23270	−0.616352	+	0.787471i	$0.711390\pi$
$350$	−6.41543	−0.342919
$351$	−1.62404	−0.0866850
$352$	−6.87783	−0.366590
$353$	34.6520	1.84434	0.922170	−	0.386785i	$-0.126414\pi$
$353$	34.6520	1.84434	0.922170	+	0.386785i	$0.126414\pi$
$354$	19.0935	1.01481
$355$	4.04789	0.214840
$356$	−0.505732	−0.0268037
$357$	−0.891979	−0.0472085
$358$	−17.5240	−0.926173
$359$	−27.3200	−1.44190	−0.720948	−	0.692989i	$-0.756294\pi$
$359$	−27.3200	−1.44190	−0.720948	+	0.692989i	$0.756294\pi$
$360$	−1.23989	−0.0653480
$361$	−18.3046	−0.963400
$362$	8.82267	0.463709
$363$	−9.54413	−0.500937
$364$	−0.391146	−0.0205016
$365$	3.03901	0.159069
$366$	0.344753	0.0180205
$367$	−1.74124	−0.0908922	−0.0454461	−	0.998967i	$-0.514471\pi$
$367$	−1.74124	−0.0908922	−0.0454461	+	0.998967i	$0.514471\pi$
$368$	14.5464	0.758283
$369$	4.26901	0.222236
$370$	−3.49111	−0.181494
$371$	−0.476324	−0.0247295
$372$	−2.78916	−0.144611
$373$	−27.3740	−1.41737	−0.708687	−	0.705523i	$-0.750712\pi$
$373$	−27.3740	−1.41737	−0.708687	+	0.705523i	$0.750712\pi$
$374$	−6.82902	−0.353120
$375$	−4.64929	−0.240088
$376$	31.6644	1.63297
$377$	7.69820	0.396478
$378$	−1.34391	−0.0691231
$379$	32.0482	1.64621	0.823103	−	0.567892i	$-0.192241\pi$
$379$	32.0482	1.64621	0.823103	+	0.567892i	$0.192241\pi$
$380$	−0.107110	−0.00549464
$381$	−17.9453	−0.919366
$382$	−15.9614	−0.816655
$383$	−12.0255	−0.614475	−0.307238	−	0.951633i	$-0.599405\pi$
$383$	−12.0255	−0.614475	−0.307238	+	0.951633i	$0.599405\pi$
$384$	13.0511	0.666011
$385$	−1.92320	−0.0980155
$386$	−22.5874	−1.14967
$387$	−9.65601	−0.490843
$388$	−2.37627	−0.120637
$389$	5.51186	0.279462	0.139731	−	0.990189i	$-0.455376\pi$
$389$	5.51186	0.279462	0.139731	+	0.990189i	$0.455376\pi$
$390$	−1.16396	−0.0589395
$391$	3.25633	0.164679
$392$	−16.1717	−0.816792
$393$	8.71992	0.439862
$394$	2.65196	0.133604
$395$	2.30212	0.115832
$396$	−1.22386	−0.0615011
$397$	−30.9226	−1.55196	−0.775980	−	0.630757i	$-0.782744\pi$
$397$	−30.9226	−1.55196	−0.775980	+	0.630757i	$0.782744\pi$
$398$	8.99800	0.451029
$399$	0.743829	0.0372380
$400$	21.3248	1.06624
$401$	−32.0355	−1.59978	−0.799889	−	0.600149i	$-0.795108\pi$
$401$	−32.0355	−1.59978	−0.799889	+	0.600149i	$0.795108\pi$
$402$	12.3604	0.616480
$403$	16.7759	0.835665
$404$	3.31487	0.164921
$405$	−0.475693	−0.0236374
$406$	6.37031	0.316153
$407$	22.0783	1.09438
$408$	2.60649	0.129041
$409$	11.3502	0.561231	0.280616	−	0.959820i	$-0.409461\pi$
$409$	11.3502	0.561231	0.280616	+	0.959820i	$0.409461\pi$
$410$	3.05962	0.151104
$411$	−1.67736	−0.0827383
$412$	1.39584	0.0687682
$413$	−11.3038	−0.556225
$414$	4.90616	0.241125
$415$	8.10791	0.398002
$416$	2.46437	0.120826
$417$	−6.01855	−0.294729
$418$	5.69477	0.278540
$419$	−9.03760	−0.441516	−0.220758	−	0.975329i	$-0.570853\pi$
$419$	−9.03760	−0.441516	−0.220758	+	0.975329i	$0.570853\pi$
$420$	−0.114569	−0.00559040
$421$	−26.6087	−1.29683	−0.648415	−	0.761287i	$-0.724568\pi$
$421$	−26.6087	−1.29683	−0.648415	+	0.761287i	$0.724568\pi$
$422$	−13.2963	−0.647256
$423$	12.1483	0.590669
$424$	1.39189	0.0675960
$425$	4.77372	0.231559
$426$	−12.8208	−0.621172
$427$	−0.204103	−0.00987721
$428$	2.42155	0.117050
$429$	7.36108	0.355396
$430$	−6.92052	−0.333737
$431$	2.18676	0.105332	0.0526662	−	0.998612i	$-0.483228\pi$
$431$	2.18676	0.105332	0.0526662	+	0.998612i	$0.483228\pi$
$432$	4.46712	0.214924
$433$	−25.5406	−1.22740	−0.613702	−	0.789538i	$-0.710320\pi$
$433$	−25.5406	−1.22740	−0.613702	+	0.789538i	$0.710320\pi$
$434$	13.8821	0.666363
$435$	2.25485	0.108112
$436$	1.52899	0.0732256
$437$	−2.71548	−0.129899
$438$	−9.62542	−0.459920
$439$	−38.2033	−1.82334	−0.911672	−	0.410919i	$-0.865208\pi$
$439$	−38.2033	−1.82334	−0.911672	+	0.410919i	$0.865208\pi$
$440$	5.61988	0.267917
$441$	−6.20437	−0.295446
$442$	2.44688	0.116386
$443$	2.13639	0.101503	0.0507515	−	0.998711i	$-0.483838\pi$
$443$	2.13639	0.101503	0.0507515	+	0.998711i	$0.483838\pi$
$444$	1.31525	0.0624191
$445$	0.890965	0.0422358
$446$	−17.3619	−0.822111
$447$	−15.0198	−0.710413
$448$	−5.92988	−0.280160
$449$	23.2737	1.09835	0.549176	−	0.835707i	$-0.314942\pi$
$449$	23.2737	1.09835	0.549176	+	0.835707i	$0.314942\pi$
$450$	7.19235	0.339051
$451$	−19.3495	−0.911134
$452$	0.235221	0.0110639
$453$	8.21834	0.386131
$454$	−7.71548	−0.362105
$455$	0.689094	0.0323052
$456$	−2.17358	−0.101787
$457$	−36.0245	−1.68515	−0.842577	−	0.538576i	$-0.818962\pi$
$457$	−36.0245	−1.68515	−0.842577	+	0.538576i	$0.818962\pi$
$458$	30.7512	1.43691
$459$	1.00000	0.0466760
$460$	0.418255	0.0195012
$461$	−38.6054	−1.79803	−0.899016	−	0.437916i	$-0.855717\pi$
$461$	−38.6054	−1.79803	−0.899016	+	0.437916i	$0.855717\pi$
$462$	6.09134	0.283395
$463$	−40.6770	−1.89042	−0.945210	−	0.326464i	$-0.894143\pi$
$463$	−40.6770	−1.89042	−0.945210	+	0.326464i	$0.894143\pi$
$464$	−21.1748	−0.983016
$465$	4.91375	0.227870
$466$	8.02215	0.371619
$467$	−1.34897	−0.0624228	−0.0312114	−	0.999513i	$-0.509937\pi$
$467$	−1.34897	−0.0624228	−0.0312114	+	0.999513i	$0.509937\pi$
$468$	0.438515	0.0202703
$469$	−7.31767	−0.337899
$470$	8.70673	0.401612
$471$	1.00000	0.0460776
$472$	33.0315	1.52040
$473$	43.7665	2.01238
$474$	−7.29146	−0.334908
$475$	−3.98084	−0.182654
$476$	0.240847	0.0110392
$477$	0.534008	0.0244505
$478$	18.2118	0.832990
$479$	21.1557	0.966630	0.483315	−	0.875447i	$-0.339433\pi$
$479$	21.1557	0.966630	0.483315	+	0.875447i	$0.339433\pi$
$480$	0.721829	0.0329468
$481$	−7.91079	−0.360701
$482$	−29.6940	−1.35253
$483$	−2.90457	−0.132163
$484$	2.57705	0.117139
$485$	4.18634	0.190092
$486$	1.50666	0.0683433
$487$	−8.44142	−0.382517	−0.191259	−	0.981540i	$-0.561257\pi$
$487$	−8.44142	−0.382517	−0.191259	+	0.981540i	$0.561257\pi$
$488$	0.596418	0.0269986
$489$	−11.0945	−0.501709
$490$	−4.44671	−0.200882
$491$	32.0169	1.44490	0.722451	−	0.691423i	$-0.243016\pi$
$491$	32.0169	1.44490	0.722451	+	0.691423i	$0.243016\pi$
$492$	−1.15269	−0.0519674
$493$	−4.74015	−0.213485
$494$	−2.04047	−0.0918051
$495$	2.15611	0.0969098
$496$	−46.1439	−2.07192
$497$	7.59027	0.340470
$498$	−25.6801	−1.15075
$499$	4.86839	0.217939	0.108970	−	0.994045i	$-0.465245\pi$
$499$	4.86839	0.217939	0.108970	+	0.994045i	$0.465245\pi$
$500$	1.25537	0.0561420
$501$	21.9002	0.978430
$502$	−2.40949	−0.107541
$503$	12.6016	0.561878	0.280939	−	0.959726i	$-0.409354\pi$
$503$	12.6016	0.561878	0.280939	+	0.959726i	$0.409354\pi$
$504$	−2.32494	−0.103561
$505$	−5.83992	−0.259873
$506$	−22.2375	−0.988577
$507$	10.3625	0.460214
$508$	4.84549	0.214984
$509$	40.0736	1.77623	0.888116	−	0.459620i	$-0.152014\pi$
$509$	40.0736	1.77623	0.888116	+	0.459620i	$0.152014\pi$
$510$	0.716706	0.0317363
$511$	5.69849	0.252086
$512$	16.5085	0.729580
$513$	−0.833908	−0.0368179
$514$	−35.5602	−1.56849
$515$	−2.45910	−0.108361
$516$	2.60726	0.114778
$517$	−55.0628	−2.42166
$518$	−6.54623	−0.287625
$519$	−4.93490	−0.216618
$520$	−2.01364	−0.0883038
$521$	−12.2478	−0.536584	−0.268292	−	0.963338i	$-0.586459\pi$
$521$	−12.2478	−0.536584	−0.268292	+	0.963338i	$0.586459\pi$
$522$	−7.14177	−0.312587
$523$	−16.8777	−0.738010	−0.369005	−	0.929427i	$-0.620302\pi$
$523$	−16.8777	−0.738010	−0.369005	+	0.929427i	$0.620302\pi$
$524$	−2.35450	−0.102857
$525$	−4.25806	−0.185837
$526$	−44.6745	−1.94790
$527$	−10.3297	−0.449968
$528$	−20.2475	−0.881160
$529$	−12.3963	−0.538971
$530$	0.382726	0.0166246
$531$	12.6728	0.549951
$532$	−0.200844	−0.00870770
$533$	6.93305	0.300304
$534$	−2.82194	−0.122117
$535$	−4.26612	−0.184440
$536$	21.3833	0.923618
$537$	−11.6311	−0.501917
$538$	16.5694	0.714358
$539$	28.1217	1.21129
$540$	0.128444	0.00552734
$541$	23.2815	1.00095	0.500474	−	0.865752i	$-0.333159\pi$
$541$	23.2815	1.00095	0.500474	+	0.865752i	$0.333159\pi$
$542$	−19.0386	−0.817777
$543$	5.85579	0.251296
$544$	−1.51743	−0.0650592
$545$	−2.69368	−0.115385
$546$	−2.18256	−0.0934050
$547$	−8.58081	−0.366889	−0.183445	−	0.983030i	$-0.558725\pi$
$547$	−8.58081	−0.366889	−0.183445	+	0.983030i	$0.558725\pi$
$548$	0.452912	0.0193474
$549$	0.228820	0.00976579
$550$	−32.5998	−1.39006
$551$	3.95285	0.168397
$552$	8.48760	0.361256
$553$	4.31673	0.183566
$554$	47.7892	2.03037
$555$	−2.31712	−0.0983563
$556$	1.62509	0.0689193
$557$	11.1953	0.474362	0.237181	−	0.971466i	$-0.423777\pi$
$557$	11.1953	0.474362	0.237181	+	0.971466i	$0.423777\pi$
$558$	−15.5633	−0.658846
$559$	−15.6818	−0.663268
$560$	−1.89543	−0.0800967
$561$	−4.53256	−0.191365
$562$	−36.5042	−1.53984
$563$	6.52292	0.274908	0.137454	−	0.990508i	$-0.456108\pi$
$563$	6.52292	0.274908	0.137454	+	0.990508i	$0.456108\pi$
$564$	−3.28020	−0.138121
$565$	−0.414397	−0.0174338
$566$	39.9283	1.67831
$567$	−0.891979	−0.0374596
$568$	−22.1799	−0.930647
$569$	−12.3512	−0.517788	−0.258894	−	0.965906i	$-0.583358\pi$
$569$	−12.3512	−0.517788	−0.258894	+	0.965906i	$0.583358\pi$
$570$	−0.597667	−0.0250335
$571$	20.5319	0.859234	0.429617	−	0.903011i	$-0.358649\pi$
$571$	20.5319	0.859234	0.429617	+	0.903011i	$0.358649\pi$
$572$	−1.98759	−0.0831055
$573$	−10.5939	−0.442567
$574$	5.73715	0.239464
$575$	15.5448	0.648262
$576$	6.64800	0.277000
$577$	−26.7263	−1.11263	−0.556316	−	0.830971i	$-0.687786\pi$
$577$	−26.7263	−1.11263	−0.556316	+	0.830971i	$0.687786\pi$
$578$	−1.50666	−0.0626687
$579$	−14.9917	−0.623035
$580$	−0.608842	−0.0252808
$581$	15.2033	0.630738
$582$	−13.2594	−0.549618
$583$	−2.42042	−0.100244
$584$	−16.6518	−0.689058
$585$	−0.772546	−0.0319408
$586$	−21.8443	−0.902379
$587$	−35.6223	−1.47029	−0.735145	−	0.677910i	$-0.762886\pi$
$587$	−35.6223	−1.47029	−0.735145	+	0.677910i	$0.762886\pi$
$588$	1.67527	0.0690869
$589$	8.61401	0.354934
$590$	9.08263	0.373926
$591$	1.76016	0.0724034
$592$	21.7596	0.894312
$593$	−31.0176	−1.27374	−0.636871	−	0.770971i	$-0.719772\pi$
$593$	−31.0176	−1.27374	−0.636871	+	0.770971i	$0.719772\pi$
$594$	−6.82902	−0.280198
$595$	−0.424308	−0.0173949
$596$	4.05556	0.166122
$597$	5.97216	0.244424
$598$	7.96782	0.325829
$599$	40.6580	1.66124	0.830621	−	0.556838i	$-0.187985\pi$
$599$	40.6580	1.66124	0.830621	+	0.556838i	$0.187985\pi$
$600$	12.4427	0.507970
$601$	−17.3442	−0.707484	−0.353742	−	0.935343i	$-0.615091\pi$
$601$	−17.3442	−0.707484	−0.353742	+	0.935343i	$0.615091\pi$
$602$	−12.9768	−0.528894
$603$	8.20386	0.334087
$604$	−2.21907	−0.0902926
$605$	−4.54007	−0.184580
$606$	18.4967	0.751377
$607$	−17.0943	−0.693835	−0.346917	−	0.937896i	$-0.612772\pi$
$607$	−17.0943	−0.693835	−0.346917	+	0.937896i	$0.612772\pi$
$608$	1.26540	0.0513185
$609$	4.22811	0.171332
$610$	0.163996	0.00664002
$611$	19.7293	0.798163
$612$	−0.270014	−0.0109147
$613$	13.7027	0.553448	0.276724	−	0.960950i	$-0.410751\pi$
$613$	13.7027	0.553448	0.276724	+	0.960950i	$0.410751\pi$
$614$	−14.5149	−0.585775
$615$	2.03074	0.0818872
$616$	10.5379	0.424585
$617$	−36.3555	−1.46362	−0.731808	−	0.681511i	$-0.761323\pi$
$617$	−36.3555	−1.46362	−0.731808	+	0.681511i	$0.761323\pi$
$618$	7.78868	0.313307
$619$	41.6344	1.67343	0.836714	−	0.547640i	$-0.184474\pi$
$619$	41.6344	1.67343	0.836714	+	0.547640i	$0.184474\pi$
$620$	−1.32678	−0.0532849
$621$	3.25633	0.130672
$622$	−0.532169	−0.0213380
$623$	1.67066	0.0669337
$624$	7.25480	0.290424
$625$	21.6570	0.866278
$626$	2.82039	0.112726
$627$	3.77974	0.150948
$628$	−0.270014	−0.0107747
$629$	4.87105	0.194221
$630$	−0.639286	−0.0254698
$631$	−16.5375	−0.658348	−0.329174	−	0.944269i	$-0.606770\pi$
$631$	−16.5375	−0.658348	−0.329174	+	0.944269i	$0.606770\pi$
$632$	−12.6141	−0.501763
$633$	−8.82507	−0.350765
$634$	35.8123	1.42229
$635$	−8.53645	−0.338759
$636$	−0.144190	−0.00571749
$637$	−10.0762	−0.399232
$638$	32.3705	1.28156
$639$	−8.50947	−0.336629
$640$	6.20832	0.245405
$641$	−15.5384	−0.613730	−0.306865	−	0.951753i	$-0.599280\pi$
$641$	−15.5384	−0.613730	−0.306865	+	0.951753i	$0.599280\pi$
$642$	13.5120	0.533277
$643$	−41.0163	−1.61753	−0.808763	−	0.588135i	$-0.799862\pi$
$643$	−41.0163	−1.61753	−0.808763	+	0.588135i	$0.799862\pi$
$644$	0.784276	0.0309048
$645$	−4.59329	−0.180861
$646$	1.25641	0.0494329
$647$	−17.9079	−0.704034	−0.352017	−	0.935994i	$-0.614504\pi$
$647$	−17.9079	−0.704034	−0.352017	+	0.935994i	$0.614504\pi$
$648$	2.60649	0.102393
$649$	−57.4401	−2.25472
$650$	11.6807	0.458154
$651$	9.21386	0.361120
$652$	2.99566	0.117319
$653$	6.04928	0.236727	0.118363	−	0.992970i	$-0.462235\pi$
$653$	6.04928	0.236727	0.118363	+	0.992970i	$0.462235\pi$
$654$	8.53166	0.333614
$655$	4.14800	0.162076
$656$	−19.0702	−0.744565
$657$	−6.38859	−0.249243
$658$	16.3261	0.636459
$659$	22.8914	0.891722	0.445861	−	0.895102i	$-0.352898\pi$
$659$	22.8914	0.891722	0.445861	+	0.895102i	$0.352898\pi$
$660$	−0.582179	−0.0226613
$661$	−43.3445	−1.68590	−0.842952	−	0.537988i	$-0.819185\pi$
$661$	−43.3445	−1.68590	−0.842952	+	0.537988i	$0.819185\pi$
$662$	29.2634	1.13735
$663$	1.62404	0.0630726
$664$	−44.4262	−1.72407
$665$	0.353834	0.0137211
$666$	7.33900	0.284380
$667$	−15.4355	−0.597663
$668$	−5.91337	−0.228795
$669$	−11.5235	−0.445523
$670$	5.87975	0.227155
$671$	−1.03714	−0.0400384
$672$	1.35351	0.0522129
$673$	32.4736	1.25176	0.625882	−	0.779918i	$-0.284739\pi$
$673$	32.4736	1.25176	0.625882	+	0.779918i	$0.284739\pi$
$674$	−36.2199	−1.39514
$675$	4.77372	0.183740
$676$	−2.79802	−0.107616
$677$	1.41176	0.0542582	0.0271291	−	0.999632i	$-0.491363\pi$
$677$	1.41176	0.0542582	0.0271291	+	0.999632i	$0.491363\pi$
$678$	1.31252	0.0504069
$679$	7.84988	0.301251
$680$	1.23989	0.0475476
$681$	−5.12093	−0.196234
$682$	70.5416	2.70118
$683$	−23.4829	−0.898547	−0.449274	−	0.893394i	$-0.648317\pi$
$683$	−23.4829	−0.898547	−0.449274	+	0.893394i	$0.648317\pi$
$684$	0.225167	0.00860947
$685$	−0.797910	−0.0304866
$686$	−17.7454	−0.677524
$687$	20.4102	0.778699
$688$	43.1346	1.64449
$689$	0.867251	0.0330396
$690$	2.33383	0.0888473
$691$	42.1455	1.60329	0.801645	−	0.597800i	$-0.203958\pi$
$691$	42.1455	1.60329	0.801645	+	0.597800i	$0.203958\pi$
$692$	1.33249	0.0506538
$693$	4.04295	0.153579
$694$	−41.7619	−1.58526
$695$	−2.86298	−0.108599
$696$	−12.3552	−0.468321
$697$	−4.26901	−0.161700
$698$	34.6965	1.31328
$699$	5.32447	0.201390
$700$	1.14973	0.0434559
$701$	−19.6202	−0.741047	−0.370523	−	0.928823i	$-0.620822\pi$
$701$	−19.6202	−0.741047	−0.370523	+	0.928823i	$0.620822\pi$
$702$	2.44688	0.0923514
$703$	−4.06201	−0.153201
$704$	−30.1325	−1.13566
$705$	5.77884	0.217644
$706$	−52.2087	−1.96490
$707$	−10.9505	−0.411837
$708$	−3.42182	−0.128600
$709$	−15.3331	−0.575846	−0.287923	−	0.957654i	$-0.592965\pi$
$709$	−15.3331	−0.575846	−0.287923	+	0.957654i	$0.592965\pi$
$710$	−6.09878	−0.228883
$711$	−4.83950	−0.181495
$712$	−4.88192	−0.182958
$713$	−33.6368	−1.25971
$714$	1.34391	0.0502944
$715$	3.50161	0.130953
$716$	3.14055	0.117368
$717$	12.0876	0.451419
$718$	41.1619	1.53615
$719$	−23.0278	−0.858793	−0.429396	−	0.903116i	$-0.641274\pi$
$719$	−23.0278	−0.858793	−0.429396	+	0.903116i	$0.641274\pi$
$720$	2.12498	0.0791932
$721$	−4.61110	−0.171726
$722$	27.5787	1.02637
$723$	−19.7085	−0.732969
$724$	−1.58115	−0.0587629
$725$	−22.6281	−0.840387
$726$	14.3797	0.533681
$727$	32.1931	1.19398	0.596989	−	0.802250i	$-0.296364\pi$
$727$	32.1931	1.19398	0.596989	+	0.802250i	$0.296364\pi$
$728$	−3.77580	−0.139941
$729$	1.00000	0.0370370
$730$	−4.57874	−0.169467
$731$	9.65601	0.357140
$732$	−0.0617846	−0.00228362
$733$	−6.04021	−0.223100	−0.111550	−	0.993759i	$-0.535582\pi$
$733$	−6.04021	−0.223100	−0.111550	+	0.993759i	$0.535582\pi$
$734$	2.62346	0.0968336
$735$	−2.95138	−0.108863
$736$	−4.94124	−0.182136
$737$	−37.1845	−1.36971
$738$	−6.43193	−0.236763
$739$	−31.3883	−1.15464	−0.577319	−	0.816519i	$-0.695901\pi$
$739$	−31.3883	−1.15464	−0.577319	+	0.816519i	$0.695901\pi$
$740$	0.625656	0.0229996
$741$	−1.35430	−0.0497516
$742$	0.717656	0.0263460
$743$	29.7628	1.09189	0.545946	−	0.837820i	$-0.316170\pi$
$743$	29.7628	1.09189	0.545946	+	0.837820i	$0.316170\pi$
$744$	−26.9243	−0.987091
$745$	−7.14482	−0.261766
$746$	41.2433	1.51002
$747$	−17.0444	−0.623623
$748$	1.22386	0.0447486
$749$	−7.99947	−0.292294
$750$	7.00488	0.255782
$751$	−36.6038	−1.33569	−0.667846	−	0.744300i	$-0.732783\pi$
$751$	−36.6038	−1.33569	−0.667846	+	0.744300i	$0.732783\pi$
$752$	−54.2678	−1.97894
$753$	−1.59923	−0.0582792
$754$	−11.5985	−0.422394
$755$	3.90941	0.142278
$756$	0.240847	0.00875952
$757$	−44.9897	−1.63518	−0.817588	−	0.575803i	$-0.804690\pi$
$757$	−44.9897	−1.63518	−0.817588	+	0.575803i	$0.804690\pi$
$758$	−48.2856	−1.75381
$759$	−14.7595	−0.535736
$760$	−1.03395	−0.0375055
$761$	−39.2938	−1.42440	−0.712200	−	0.701977i	$-0.752301\pi$
$761$	−39.2938	−1.42440	−0.712200	+	0.701977i	$0.752301\pi$
$762$	27.0374	0.979462
$763$	−5.05096	−0.182857
$764$	2.86050	0.103489
$765$	0.475693	0.0171987
$766$	18.1183	0.654642
$767$	20.5811	0.743141
$768$	−6.36754	−0.229769
$769$	−45.9204	−1.65593	−0.827967	−	0.560777i	$-0.810503\pi$
$769$	−45.9204	−1.65593	−0.827967	+	0.560777i	$0.810503\pi$
$770$	2.89761	0.104422
$771$	−23.6021	−0.850008
$772$	4.04798	0.145690
$773$	−16.7502	−0.602465	−0.301232	−	0.953551i	$-0.597398\pi$
$773$	−16.7502	−0.602465	−0.301232	+	0.953551i	$0.597398\pi$
$774$	14.5483	0.522927
$775$	−49.3110	−1.77130
$776$	−22.9385	−0.823445
$777$	−4.34487	−0.155871
$778$	−8.30448	−0.297730
$779$	3.55996	0.127549
$780$	0.208598	0.00746901
$781$	38.5697	1.38013
$782$	−4.90616	−0.175444
$783$	−4.74015	−0.169399
$784$	27.7157	0.989846
$785$	0.475693	0.0169782
$786$	−13.1379	−0.468614
$787$	−22.4933	−0.801800	−0.400900	−	0.916122i	$-0.631302\pi$
$787$	−22.4933	−0.801800	−0.400900	+	0.916122i	$0.631302\pi$
$788$	−0.475269	−0.0169307
$789$	−29.6514	−1.05562
$790$	−3.46850	−0.123404
$791$	−0.777043	−0.0276285
$792$	−11.8141	−0.419796
$793$	0.371613	0.0131964
$794$	46.5897	1.65341
$795$	0.254023	0.00900928
$796$	−1.61257	−0.0571559
$797$	−7.24694	−0.256700	−0.128350	−	0.991729i	$-0.540968\pi$
$797$	−7.24694	−0.256700	−0.128350	+	0.991729i	$0.540968\pi$
$798$	−1.12069	−0.0396722
$799$	−12.1483	−0.429775
$800$	−7.24377	−0.256106
$801$	−1.87298	−0.0661786
$802$	48.2665	1.70435
$803$	28.9567	1.02186
$804$	−2.21516	−0.0781226
$805$	−1.38168	−0.0486980
$806$	−25.2754	−0.890290
$807$	10.9975	0.387129
$808$	31.9990	1.12572
$809$	−25.7925	−0.906817	−0.453408	−	0.891303i	$-0.649792\pi$
$809$	−25.7925	−0.906817	−0.453408	+	0.891303i	$0.649792\pi$
$810$	0.716706	0.0251825
$811$	13.4381	0.471874	0.235937	−	0.971768i	$-0.424184\pi$
$811$	13.4381	0.471874	0.235937	+	0.971768i	$0.424184\pi$
$812$	−1.14165	−0.0400640
$813$	−12.6363	−0.443175
$814$	−33.2645	−1.16592
$815$	−5.27756	−0.184865
$816$	−4.46712	−0.156380
$817$	−8.05222	−0.281712
$818$	−17.1009	−0.597917
$819$	−1.44861	−0.0506186
$820$	−0.548327	−0.0191484
$821$	34.9444	1.21957	0.609784	−	0.792568i	$-0.291256\pi$
$821$	34.9444	1.21957	0.609784	+	0.792568i	$0.291256\pi$
$822$	2.52721	0.0881466
$823$	30.7470	1.07177	0.535886	−	0.844290i	$-0.319978\pi$
$823$	30.7470	1.07177	0.535886	+	0.844290i	$0.319978\pi$
$824$	13.4743	0.469400
$825$	−21.6372	−0.753310
$826$	17.0310	0.592584
$827$	−29.6281	−1.03027	−0.515136	−	0.857109i	$-0.672258\pi$
$827$	−29.6281	−1.03027	−0.515136	+	0.857109i	$0.672258\pi$
$828$	−0.879254	−0.0305562
$829$	16.4807	0.572398	0.286199	−	0.958170i	$-0.407608\pi$
$829$	16.4807	0.572398	0.286199	+	0.958170i	$0.407608\pi$
$830$	−12.2158	−0.424018
$831$	31.7187	1.10031
$832$	10.7966	0.374306
$833$	6.20437	0.214969
$834$	9.06788	0.313995
$835$	10.4178	0.360522
$836$	−1.02058	−0.0352976
$837$	−10.3297	−0.357046
$838$	13.6166	0.470376
$839$	−27.4110	−0.946334	−0.473167	−	0.880973i	$-0.656889\pi$
$839$	−27.4110	−0.946334	−0.473167	+	0.880973i	$0.656889\pi$
$840$	−1.10596	−0.0381591
$841$	−6.53102	−0.225207
$842$	40.0902	1.38160
$843$	−24.2286	−0.834478
$844$	2.38289	0.0820225
$845$	4.92936	0.169575
$846$	−18.3033	−0.629279
$847$	−8.51316	−0.292516
$848$	−2.38548	−0.0819176
$849$	26.5012	0.909520
$850$	−7.19235	−0.246696
$851$	15.8617	0.543733
$852$	2.29768	0.0787171
$853$	−37.3203	−1.27782	−0.638912	−	0.769280i	$-0.720615\pi$
$853$	−37.3203	−1.27782	−0.638912	+	0.769280i	$0.720615\pi$
$854$	0.307512	0.0105229
$855$	−0.396684	−0.0135663
$856$	23.3756	0.798962
$857$	−42.9183	−1.46606	−0.733031	−	0.680196i	$-0.761895\pi$
$857$	−42.9183	−1.46606	−0.733031	+	0.680196i	$0.761895\pi$
$858$	−11.0906	−0.378627
$859$	−40.9598	−1.39753	−0.698765	−	0.715351i	$-0.746267\pi$
$859$	−40.9598	−1.39753	−0.698765	+	0.715351i	$0.746267\pi$
$860$	1.24025	0.0422923
$861$	3.80787	0.129772
$862$	−3.29469	−0.112218
$863$	−44.9940	−1.53161	−0.765807	−	0.643071i	$-0.777660\pi$
$863$	−44.9940	−1.53161	−0.765807	+	0.643071i	$0.777660\pi$
$864$	−1.51743	−0.0516239
$865$	−2.34750	−0.0798173
$866$	38.4809	1.30764
$867$	−1.00000	−0.0339618
$868$	−2.48787	−0.0844439
$869$	21.9353	0.744105
$870$	−3.39729	−0.115179
$871$	13.3234	0.451447
$872$	14.7597	0.499825
$873$	−8.80052	−0.297853
$874$	4.09129	0.138390
$875$	−4.14707	−0.140196
$876$	1.72501	0.0582827
$877$	−26.9678	−0.910639	−0.455319	−	0.890328i	$-0.650475\pi$
$877$	−26.9678	−0.910639	−0.455319	+	0.890328i	$0.650475\pi$
$878$	57.5593	1.94253
$879$	−14.4985	−0.489023
$880$	−9.63159	−0.324681
$881$	−2.92987	−0.0987099	−0.0493550	−	0.998781i	$-0.515717\pi$
$881$	−2.92987	−0.0987099	−0.0493550	+	0.998781i	$0.515717\pi$
$882$	9.34786	0.314759
$883$	49.1624	1.65445	0.827223	−	0.561874i	$-0.189920\pi$
$883$	49.1624	1.65445	0.827223	+	0.561874i	$0.189920\pi$
$884$	−0.438515	−0.0147488
$885$	6.02834	0.202640
$886$	−3.21881	−0.108138
$887$	−51.1999	−1.71912	−0.859562	−	0.511032i	$-0.829263\pi$
$887$	−51.1999	−1.71912	−0.859562	+	0.511032i	$0.829263\pi$
$888$	12.6964	0.426062
$889$	−16.0068	−0.536852
$890$	−1.34238	−0.0449966
$891$	−4.53256	−0.151847
$892$	3.11150	0.104181
$893$	10.1305	0.339006
$894$	22.6297	0.756850
$895$	−5.53281	−0.184942
$896$	11.6413	0.388909
$897$	5.28841	0.176575
$898$	−35.0654	−1.17015
$899$	48.9642	1.63305
$900$	−1.28897	−0.0429657
$901$	−0.534008	−0.0177904
$902$	29.1531	0.970693
$903$	−8.61296	−0.286621
$904$	2.27063	0.0755201
$905$	2.78556	0.0925951
$906$	−12.3822	−0.411372
$907$	43.6283	1.44866	0.724328	−	0.689456i	$-0.242150\pi$
$907$	43.6283	1.44866	0.724328	+	0.689456i	$0.242150\pi$
$908$	1.38272	0.0458872
$909$	12.2767	0.407191
$910$	−1.03823	−0.0344169
$911$	−28.4003	−0.940942	−0.470471	−	0.882415i	$-0.655916\pi$
$911$	−28.4003	−0.940942	−0.470471	+	0.882415i	$0.655916\pi$
$912$	3.72517	0.123353
$913$	77.2550	2.55677
$914$	54.2765	1.79531
$915$	0.108848	0.00359840
$916$	−5.51105	−0.182090
$917$	7.77798	0.256852
$918$	−1.50666	−0.0497271
$919$	−35.1193	−1.15848	−0.579240	−	0.815157i	$-0.696650\pi$
$919$	−35.1193	−1.15848	−0.579240	+	0.815157i	$0.696650\pi$
$920$	4.03749	0.133112
$921$	−9.63388	−0.317447
$922$	58.1650	1.91556
$923$	−13.8197	−0.454883
$924$	−1.09165	−0.0359128
$925$	23.2530	0.764554
$926$	61.2862	2.01399
$927$	5.16951	0.169789
$928$	7.19283	0.236116
$929$	−2.24445	−0.0736381	−0.0368191	−	0.999322i	$-0.511723\pi$
$929$	−2.24445	−0.0736381	−0.0368191	+	0.999322i	$0.511723\pi$
$930$	−7.40334	−0.242765
$931$	−5.17388	−0.169567
$932$	−1.43768	−0.0470928
$933$	−0.353212	−0.0115636
$934$	2.03243	0.0665032
$935$	−2.15611	−0.0705122
$936$	4.23306	0.138362
$937$	28.5036	0.931173	0.465586	−	0.885002i	$-0.345843\pi$
$937$	28.5036	0.931173	0.465586	+	0.885002i	$0.345843\pi$
$938$	11.0252	0.359986
$939$	1.87195	0.0610889
$940$	−1.56037	−0.0508936
$941$	17.7126	0.577413	0.288706	−	0.957418i	$-0.406775\pi$
$941$	17.7126	0.577413	0.288706	+	0.957418i	$0.406775\pi$
$942$	−1.50666	−0.0490895
$943$	−13.9013	−0.452688
$944$	−56.6107	−1.84252
$945$	−0.424308	−0.0138027
$946$	−65.9410	−2.14393
$947$	27.3314	0.888152	0.444076	−	0.895989i	$-0.353532\pi$
$947$	27.3314	0.888152	0.444076	+	0.895989i	$0.353532\pi$
$948$	1.30673	0.0424407
$949$	−10.3754	−0.336798
$950$	5.99776	0.194593
$951$	23.7694	0.770776
$952$	2.32494	0.0753517
$953$	1.40686	0.0455726	0.0227863	−	0.999740i	$-0.492746\pi$
$953$	1.40686	0.0455726	0.0227863	+	0.999740i	$0.492746\pi$
$954$	−0.804566	−0.0260488
$955$	−5.03944	−0.163073
$956$	−3.26382	−0.105559
$957$	21.4850	0.694512
$958$	−31.8744	−1.02982
$959$	−1.49617	−0.0483140
$960$	3.16240	0.102066
$961$	75.7023	2.44201
$962$	11.9188	0.384279
$963$	8.96822	0.288997
$964$	5.32159	0.171397
$965$	−7.13146	−0.229570
$966$	4.37620	0.140802
$967$	57.5695	1.85131	0.925655	−	0.378369i	$-0.123515\pi$
$967$	57.5695	1.85131	0.925655	+	0.378369i	$0.123515\pi$
$968$	24.8767	0.799568
$969$	0.833908	0.0267890
$970$	−6.30738	−0.202518
$971$	21.1275	0.678012	0.339006	−	0.940784i	$-0.389909\pi$
$971$	21.1275	0.678012	0.339006	+	0.940784i	$0.389909\pi$
$972$	−0.270014	−0.00866071
$973$	−5.36842	−0.172104
$974$	12.7183	0.407521
$975$	7.75272	0.248286
$976$	−1.02217	−0.0327187
$977$	0.245514	0.00785468	0.00392734	−	0.999992i	$-0.498750\pi$
$977$	0.245514	0.00785468	0.00392734	+	0.999992i	$0.498750\pi$
$978$	16.7156	0.534504
$979$	8.48942	0.271323
$980$	0.796913	0.0254564
$981$	5.66265	0.180794
$982$	−48.2384	−1.53935
$983$	−14.9535	−0.476941	−0.238471	−	0.971150i	$-0.576646\pi$
$983$	−14.9535	−0.476941	−0.238471	+	0.971150i	$0.576646\pi$
$984$	−11.1271	−0.354720
$985$	0.837296	0.0266785
$986$	7.14177	0.227440
$987$	10.8360	0.344914
$988$	0.365681	0.0116339
$989$	31.4431	0.999833
$990$	−3.24851	−0.103245
$991$	−14.0396	−0.445982	−0.222991	−	0.974821i	$-0.571582\pi$
$991$	−14.0396	−0.445982	−0.222991	+	0.974821i	$0.571582\pi$
$992$	15.6745	0.497667
$993$	19.4227	0.616362
$994$	−11.4359	−0.362726
$995$	2.84091	0.0900630
$996$	4.60224	0.145827
$997$	−27.1275	−0.859135	−0.429568	−	0.903035i	$-0.641334\pi$
$997$	−27.1275	−0.859135	−0.429568	+	0.903035i	$0.641334\pi$
$998$	−7.33500	−0.232185
$999$	4.87105	0.154113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.13	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.13	✓	48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-1.50666 q^{2} -1.00000 q^{3} +0.270014 q^{4} -0.475693 q^{5} +1.50666 q^{6} -0.891979 q^{7} +2.60649 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.50666 q^{2} -1.00000 q^{3} +0.270014 q^{4} -0.475693 q^{5} +1.50666 q^{6} -0.891979 q^{7} +2.60649 q^{8} +1.00000 q^{9} +0.716706 q^{10} -4.53256 q^{11} -0.270014 q^{12} +1.62404 q^{13} +1.34391 q^{14} +0.475693 q^{15} -4.46712 q^{16} -1.00000 q^{17} -1.50666 q^{18} +0.833908 q^{19} -0.128444 q^{20} +0.891979 q^{21} +6.82902 q^{22} -3.25633 q^{23} -2.60649 q^{24} -4.77372 q^{25} -2.44688 q^{26} -1.00000 q^{27} -0.240847 q^{28} +4.74015 q^{29} -0.716706 q^{30} +10.3297 q^{31} +1.51743 q^{32} +4.53256 q^{33} +1.50666 q^{34} +0.424308 q^{35} +0.270014 q^{36} -4.87105 q^{37} -1.25641 q^{38} -1.62404 q^{39} -1.23989 q^{40} +4.26901 q^{41} -1.34391 q^{42} -9.65601 q^{43} -1.22386 q^{44} -0.475693 q^{45} +4.90616 q^{46} +12.1483 q^{47} +4.46712 q^{48} -6.20437 q^{49} +7.19235 q^{50} +1.00000 q^{51} +0.438515 q^{52} +0.534008 q^{53} +1.50666 q^{54} +2.15611 q^{55} -2.32494 q^{56} -0.833908 q^{57} -7.14177 q^{58} +12.6728 q^{59} +0.128444 q^{60} +0.228820 q^{61} -15.5633 q^{62} -0.891979 q^{63} +6.64800 q^{64} -0.772546 q^{65} -6.82902 q^{66} +8.20386 q^{67} -0.270014 q^{68} +3.25633 q^{69} -0.639286 q^{70} -8.50947 q^{71} +2.60649 q^{72} -6.38859 q^{73} +7.33900 q^{74} +4.77372 q^{75} +0.225167 q^{76} +4.04295 q^{77} +2.44688 q^{78} -4.83950 q^{79} +2.12498 q^{80} +1.00000 q^{81} -6.43193 q^{82} -17.0444 q^{83} +0.240847 q^{84} +0.475693 q^{85} +14.5483 q^{86} -4.74015 q^{87} -11.8141 q^{88} -1.87298 q^{89} +0.716706 q^{90} -1.44861 q^{91} -0.879254 q^{92} -10.3297 q^{93} -18.3033 q^{94} -0.396684 q^{95} -1.51743 q^{96} -8.80052 q^{97} +9.34786 q^{98} -4.53256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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