LMFDB - Embedded newform 8045.2.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$1$
Dimension:	$127$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69512 q^{2} -2.02504 q^{3} +5.26369 q^{4} +1.00000 q^{5} +5.45774 q^{6} +1.98108 q^{7} -8.79604 q^{8} +1.10080 q^{9} +O(q^{10})$	\(q-2.69512 q^{2} -2.02504 q^{3} +5.26369 q^{4} +1.00000 q^{5} +5.45774 q^{6} +1.98108 q^{7} -8.79604 q^{8} +1.10080 q^{9} -2.69512 q^{10} +5.26915 q^{11} -10.6592 q^{12} -4.42681 q^{13} -5.33925 q^{14} -2.02504 q^{15} +13.1790 q^{16} -6.55116 q^{17} -2.96679 q^{18} -5.87166 q^{19} +5.26369 q^{20} -4.01177 q^{21} -14.2010 q^{22} -5.90806 q^{23} +17.8124 q^{24} +1.00000 q^{25} +11.9308 q^{26} +3.84596 q^{27} +10.4278 q^{28} +1.10131 q^{29} +5.45774 q^{30} +2.85633 q^{31} -17.9271 q^{32} -10.6703 q^{33} +17.6562 q^{34} +1.98108 q^{35} +5.79427 q^{36} -7.59295 q^{37} +15.8249 q^{38} +8.96449 q^{39} -8.79604 q^{40} -6.53739 q^{41} +10.8122 q^{42} +9.58355 q^{43} +27.7352 q^{44} +1.10080 q^{45} +15.9229 q^{46} +8.18695 q^{47} -26.6881 q^{48} -3.07533 q^{49} -2.69512 q^{50} +13.2664 q^{51} -23.3014 q^{52} +10.6791 q^{53} -10.3653 q^{54} +5.26915 q^{55} -17.4257 q^{56} +11.8904 q^{57} -2.96815 q^{58} +13.1581 q^{59} -10.6592 q^{60} -5.27552 q^{61} -7.69816 q^{62} +2.18077 q^{63} +21.9575 q^{64} -4.42681 q^{65} +28.7577 q^{66} +6.12199 q^{67} -34.4833 q^{68} +11.9641 q^{69} -5.33925 q^{70} +15.2060 q^{71} -9.68269 q^{72} -11.7336 q^{73} +20.4639 q^{74} -2.02504 q^{75} -30.9066 q^{76} +10.4386 q^{77} -24.1604 q^{78} +12.8129 q^{79} +13.1790 q^{80} -11.0906 q^{81} +17.6191 q^{82} -5.89742 q^{83} -21.1167 q^{84} -6.55116 q^{85} -25.8288 q^{86} -2.23019 q^{87} -46.3477 q^{88} +3.13956 q^{89} -2.96679 q^{90} -8.76987 q^{91} -31.0982 q^{92} -5.78419 q^{93} -22.0648 q^{94} -5.87166 q^{95} +36.3031 q^{96} +11.1472 q^{97} +8.28839 q^{98} +5.80028 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) $ 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.69512	−1.90574	−0.952870	−	0.303379i	$-0.901885\pi$
$2$	−2.69512	−1.90574	−0.952870	+	0.303379i	$0.901885\pi$
$3$	−2.02504	−1.16916	−0.584580	−	0.811336i	$-0.698741\pi$
$3$	−2.02504	−1.16916	−0.584580	+	0.811336i	$0.698741\pi$
$4$	5.26369	2.63184
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	5.45774	2.22811
$7$	1.98108	0.748777	0.374389	−	0.927272i	$-0.377853\pi$
$7$	1.98108	0.748777	0.374389	+	0.927272i	$0.377853\pi$
$8$	−8.79604	−3.10987
$9$	1.10080	0.366934
$10$	−2.69512	−0.852273
$11$	5.26915	1.58871	0.794354	−	0.607455i	$-0.207809\pi$
$11$	5.26915	1.58871	0.794354	+	0.607455i	$0.207809\pi$
$12$	−10.6592	−3.07705
$13$	−4.42681	−1.22778	−0.613889	−	0.789393i	$-0.710396\pi$
$13$	−4.42681	−1.22778	−0.613889	+	0.789393i	$0.710396\pi$
$14$	−5.33925	−1.42697
$15$	−2.02504	−0.522864
$16$	13.1790	3.29476
$17$	−6.55116	−1.58889	−0.794445	−	0.607336i	$-0.792238\pi$
$17$	−6.55116	−1.58889	−0.794445	+	0.607336i	$0.792238\pi$
$18$	−2.96679	−0.699280
$19$	−5.87166	−1.34705	−0.673526	−	0.739164i	$-0.735221\pi$
$19$	−5.87166	−1.34705	−0.673526	+	0.739164i	$0.735221\pi$
$20$	5.26369	1.17700
$21$	−4.01177	−0.875440
$22$	−14.2010	−3.02767
$23$	−5.90806	−1.23191	−0.615957	−	0.787779i	$-0.711231\pi$
$23$	−5.90806	−1.23191	−0.615957	+	0.787779i	$0.711231\pi$
$24$	17.8124	3.63593
$25$	1.00000	0.200000
$26$	11.9308	2.33982
$27$	3.84596	0.740156
$28$	10.4278	1.97067
$29$	1.10131	0.204507	0.102254	−	0.994758i	$-0.467395\pi$
$29$	1.10131	0.204507	0.102254	+	0.994758i	$0.467395\pi$
$30$	5.45774	0.996443
$31$	2.85633	0.513012	0.256506	−	0.966543i	$-0.417429\pi$
$31$	2.85633	0.513012	0.256506	+	0.966543i	$0.417429\pi$
$32$	−17.9271	−3.16909
$33$	−10.6703	−1.85745
$34$	17.6562	3.02801
$35$	1.98108	0.334863
$36$	5.79427	0.965712
$37$	−7.59295	−1.24827	−0.624136	−	0.781316i	$-0.714549\pi$
$37$	−7.59295	−1.24827	−0.624136	+	0.781316i	$0.714549\pi$
$38$	15.8249	2.56713
$39$	8.96449	1.43547
$40$	−8.79604	−1.39078
$41$	−6.53739	−1.02097	−0.510484	−	0.859887i	$-0.670534\pi$
$41$	−6.53739	−1.02097	−0.510484	+	0.859887i	$0.670534\pi$
$42$	10.8122	1.66836
$43$	9.58355	1.46148	0.730738	−	0.682657i	$-0.239176\pi$
$43$	9.58355	1.46148	0.730738	+	0.682657i	$0.239176\pi$
$44$	27.7352	4.18123
$45$	1.10080	0.164098
$46$	15.9229	2.34771
$47$	8.18695	1.19419	0.597095	−	0.802171i	$-0.296322\pi$
$47$	8.18695	1.19419	0.597095	+	0.802171i	$0.296322\pi$
$48$	−26.6881	−3.85210
$49$	−3.07533	−0.439333
$50$	−2.69512	−0.381148
$51$	13.2664	1.85767
$52$	−23.3014	−3.23132
$53$	10.6791	1.46688	0.733442	−	0.679752i	$-0.237913\pi$
$53$	10.6791	1.46688	0.733442	+	0.679752i	$0.237913\pi$
$54$	−10.3653	−1.41054
$55$	5.26915	0.710492
$56$	−17.4257	−2.32860
$57$	11.8904	1.57492
$58$	−2.96815	−0.389738
$59$	13.1581	1.71304	0.856522	−	0.516111i	$-0.172621\pi$
$59$	13.1581	1.71304	0.856522	+	0.516111i	$0.172621\pi$
$60$	−10.6592	−1.37610
$61$	−5.27552	−0.675461	−0.337730	−	0.941243i	$-0.609659\pi$
$61$	−5.27552	−0.675461	−0.337730	+	0.941243i	$0.609659\pi$
$62$	−7.69816	−0.977667
$63$	2.18077	0.274752
$64$	21.9575	2.74469
$65$	−4.42681	−0.549079
$66$	28.7577	3.53982
$67$	6.12199	0.747920	0.373960	−	0.927445i	$-0.378000\pi$
$67$	6.12199	0.747920	0.373960	+	0.927445i	$0.378000\pi$
$68$	−34.4833	−4.18171
$69$	11.9641	1.44030
$70$	−5.33925	−0.638163
$71$	15.2060	1.80462	0.902310	−	0.431087i	$-0.141870\pi$
$71$	15.2060	1.80462	0.902310	+	0.431087i	$0.141870\pi$
$72$	−9.68269	−1.14112
$73$	−11.7336	−1.37331	−0.686654	−	0.726984i	$-0.740921\pi$
$73$	−11.7336	−1.37331	−0.686654	+	0.726984i	$0.740921\pi$
$74$	20.4639	2.37888
$75$	−2.02504	−0.233832
$76$	−30.9066	−3.54523
$77$	10.4386	1.18959
$78$	−24.1604	−2.73563
$79$	12.8129	1.44156	0.720782	−	0.693162i	$-0.243783\pi$
$79$	12.8129	1.44156	0.720782	+	0.693162i	$0.243783\pi$
$80$	13.1790	1.47346
$81$	−11.0906	−1.23229
$82$	17.6191	1.94570
$83$	−5.89742	−0.647326	−0.323663	−	0.946172i	$-0.604914\pi$
$83$	−5.89742	−0.647326	−0.323663	+	0.946172i	$0.604914\pi$
$84$	−21.1167	−2.30402
$85$	−6.55116	−0.710573
$86$	−25.8288	−2.78519
$87$	−2.23019	−0.239102
$88$	−46.3477	−4.94068
$89$	3.13956	0.332793	0.166396	−	0.986059i	$-0.446787\pi$
$89$	3.13956	0.332793	0.166396	+	0.986059i	$0.446787\pi$
$90$	−2.96679	−0.312727
$91$	−8.76987	−0.919332
$92$	−31.0982	−3.24221
$93$	−5.78419	−0.599792
$94$	−22.0648	−2.27581
$95$	−5.87166	−0.602420
$96$	36.3031	3.70517
$97$	11.1472	1.13183	0.565914	−	0.824464i	$-0.308523\pi$
$97$	11.1472	1.13183	0.565914	+	0.824464i	$0.308523\pi$
$98$	8.28839	0.837254
$99$	5.80028	0.582951
$100$	5.26369	0.526369
$101$	−7.61304	−0.757526	−0.378763	−	0.925494i	$-0.623650\pi$
$101$	−7.61304	−0.757526	−0.378763	+	0.925494i	$0.623650\pi$
$102$	−35.7545	−3.54023
$103$	16.8696	1.66221	0.831106	−	0.556114i	$-0.187708\pi$
$103$	16.8696	1.66221	0.831106	+	0.556114i	$0.187708\pi$
$104$	38.9385	3.81823
$105$	−4.01177	−0.391509
$106$	−28.7814	−2.79550
$107$	9.99908	0.966648	0.483324	−	0.875442i	$-0.339429\pi$
$107$	9.99908	0.966648	0.483324	+	0.875442i	$0.339429\pi$
$108$	20.2439	1.94797
$109$	8.91104	0.853523	0.426761	−	0.904364i	$-0.359654\pi$
$109$	8.91104	0.853523	0.426761	+	0.904364i	$0.359654\pi$
$110$	−14.2010	−1.35401
$111$	15.3760	1.45943
$112$	26.1087	2.46704
$113$	−9.91682	−0.932896	−0.466448	−	0.884549i	$-0.654466\pi$
$113$	−9.91682	−0.932896	−0.466448	+	0.884549i	$0.654466\pi$
$114$	−32.0460	−3.00138
$115$	−5.90806	−0.550929
$116$	5.79693	0.538231
$117$	−4.87304	−0.450513
$118$	−35.4628	−3.26462
$119$	−12.9784	−1.18972
$120$	17.8124	1.62604
$121$	16.7639	1.52400
$122$	14.2182	1.28725
$123$	13.2385	1.19367
$124$	15.0348	1.35017
$125$	1.00000	0.0894427
$126$	−5.87745	−0.523605
$127$	−15.3909	−1.36572	−0.682861	−	0.730548i	$-0.739265\pi$
$127$	−15.3909	−1.36572	−0.682861	+	0.730548i	$0.739265\pi$
$128$	−23.3242	−2.06158
$129$	−19.4071	−1.70870
$130$	11.9308	1.04640
$131$	−3.06019	−0.267370	−0.133685	−	0.991024i	$-0.542681\pi$
$131$	−3.06019	−0.267370	−0.133685	+	0.991024i	$0.542681\pi$
$132$	−56.1649	−4.88853
$133$	−11.6322	−1.00864
$134$	−16.4995	−1.42534
$135$	3.84596	0.331008
$136$	57.6243	4.94124
$137$	5.71277	0.488075	0.244037	−	0.969766i	$-0.421528\pi$
$137$	5.71277	0.488075	0.244037	+	0.969766i	$0.421528\pi$
$138$	−32.2446	−2.74485
$139$	−21.3359	−1.80969	−0.904846	−	0.425740i	$-0.860014\pi$
$139$	−21.3359	−1.80969	−0.904846	+	0.425740i	$0.860014\pi$
$140$	10.4278	0.881308
$141$	−16.5789	−1.39620
$142$	−40.9821	−3.43914
$143$	−23.3256	−1.95058
$144$	14.5075	1.20896
$145$	1.10131	0.0914584
$146$	31.6234	2.61717
$147$	6.22767	0.513650
$148$	−39.9669	−3.28526
$149$	1.16286	0.0952649	0.0476325	−	0.998865i	$-0.484832\pi$
$149$	1.16286	0.0952649	0.0476325	+	0.998865i	$0.484832\pi$
$150$	5.45774	0.445623
$151$	−16.6160	−1.35219	−0.676095	−	0.736815i	$-0.736329\pi$
$151$	−16.6160	−1.35219	−0.676095	+	0.736815i	$0.736329\pi$
$152$	51.6474	4.18916
$153$	−7.21152	−0.583017
$154$	−28.1333	−2.26705
$155$	2.85633	0.229426
$156$	47.1863	3.77793
$157$	11.9405	0.952957	0.476478	−	0.879186i	$-0.341913\pi$
$157$	11.9405	0.952957	0.476478	+	0.879186i	$0.341913\pi$
$158$	−34.5323	−2.74725
$159$	−21.6256	−1.71502
$160$	−17.9271	−1.41726
$161$	−11.7043	−0.922430
$162$	29.8906	2.34843
$163$	−7.72598	−0.605146	−0.302573	−	0.953126i	$-0.597846\pi$
$163$	−7.72598	−0.605146	−0.302573	+	0.953126i	$0.597846\pi$
$164$	−34.4108	−2.68703
$165$	−10.6703	−0.830679
$166$	15.8943	1.23363
$167$	23.6708	1.83170	0.915851	−	0.401518i	$-0.131517\pi$
$167$	23.6708	1.83170	0.915851	+	0.401518i	$0.131517\pi$
$168$	35.2877	2.72251
$169$	6.59668	0.507437
$170$	17.6562	1.35417
$171$	−6.46353	−0.494278
$172$	50.4448	3.84638
$173$	9.87103	0.750481	0.375240	−	0.926928i	$-0.377560\pi$
$173$	9.87103	0.750481	0.375240	+	0.926928i	$0.377560\pi$
$174$	6.01064	0.455665
$175$	1.98108	0.149755
$176$	69.4424	5.23442
$177$	−26.6458	−2.00282
$178$	−8.46151	−0.634217
$179$	19.6343	1.46754	0.733769	−	0.679399i	$-0.237759\pi$
$179$	19.6343	1.46754	0.733769	+	0.679399i	$0.237759\pi$
$180$	5.79427	0.431880
$181$	−12.6231	−0.938266	−0.469133	−	0.883128i	$-0.655433\pi$
$181$	−12.6231	−0.938266	−0.469133	+	0.883128i	$0.655433\pi$
$182$	23.6359	1.75201
$183$	10.6832	0.789722
$184$	51.9675	3.83110
$185$	−7.59295	−0.558244
$186$	15.5891	1.14305
$187$	−34.5191	−2.52428
$188$	43.0936	3.14292
$189$	7.61915	0.554212
$190$	15.8249	1.14806
$191$	−9.40000	−0.680160	−0.340080	−	0.940397i	$-0.610454\pi$
$191$	−9.40000	−0.680160	−0.340080	+	0.940397i	$0.610454\pi$
$192$	−44.4650	−3.20898
$193$	−17.6990	−1.27400	−0.637000	−	0.770864i	$-0.719825\pi$
$193$	−17.6990	−1.27400	−0.637000	+	0.770864i	$0.719825\pi$
$194$	−30.0431	−2.15697
$195$	8.96449	0.641960
$196$	−16.1876	−1.15625
$197$	−14.9556	−1.06554	−0.532770	−	0.846260i	$-0.678849\pi$
$197$	−14.9556	−1.06554	−0.532770	+	0.846260i	$0.678849\pi$
$198$	−15.6325	−1.11095
$199$	−12.3320	−0.874193	−0.437096	−	0.899415i	$-0.643993\pi$
$199$	−12.3320	−0.874193	−0.437096	+	0.899415i	$0.643993\pi$
$200$	−8.79604	−0.621974
$201$	−12.3973	−0.874438
$202$	20.5181	1.44365
$203$	2.18177	0.153130
$204$	69.8301	4.88909
$205$	−6.53739	−0.456591
$206$	−45.4657	−3.16774
$207$	−6.50359	−0.452031
$208$	−58.3412	−4.04523
$209$	−30.9387	−2.14007
$210$	10.8122	0.746114
$211$	−4.85671	−0.334350	−0.167175	−	0.985927i	$-0.553464\pi$
$211$	−4.85671	−0.334350	−0.167175	+	0.985927i	$0.553464\pi$
$212$	56.2113	3.86061
$213$	−30.7928	−2.10989
$214$	−26.9488	−1.84218
$215$	9.58355	0.653592
$216$	−33.8292	−2.30179
$217$	5.65861	0.384131
$218$	−24.0164	−1.62659
$219$	23.7610	1.60562
$220$	27.7352	1.86990
$221$	29.0008	1.95080
$222$	−41.4403	−2.78129
$223$	−2.70604	−0.181210	−0.0906048	−	0.995887i	$-0.528880\pi$
$223$	−2.70604	−0.181210	−0.0906048	+	0.995887i	$0.528880\pi$
$224$	−35.5149	−2.37294
$225$	1.10080	0.0733867
$226$	26.7270	1.77786
$227$	−13.9737	−0.927465	−0.463732	−	0.885975i	$-0.653490\pi$
$227$	−13.9737	−0.927465	−0.463732	+	0.885975i	$0.653490\pi$
$228$	62.5872	4.14494
$229$	0.827790	0.0547019	0.0273510	−	0.999626i	$-0.491293\pi$
$229$	0.827790	0.0547019	0.0273510	+	0.999626i	$0.491293\pi$
$230$	15.9229	1.04993
$231$	−21.1386	−1.39082
$232$	−9.68713	−0.635991
$233$	−0.360224	−0.0235991	−0.0117995	−	0.999930i	$-0.503756\pi$
$233$	−0.360224	−0.0235991	−0.0117995	+	0.999930i	$0.503756\pi$
$234$	13.1334	0.858560
$235$	8.18695	0.534058
$236$	69.2604	4.50847
$237$	−25.9467	−1.68542
$238$	34.9783	2.26731
$239$	17.5374	1.13440	0.567201	−	0.823579i	$-0.308026\pi$
$239$	17.5374	1.13440	0.567201	+	0.823579i	$0.308026\pi$
$240$	−26.6881	−1.72271
$241$	−0.00727693	−0.000468748 0	−0.000234374	−	1.00000i	$-0.500075\pi$
$241$	−0.00727693	−0.000468748 0	−0.000234374		1.00000i	$0.500075\pi$
$242$	−45.1809	−2.90434
$243$	10.9211	0.700592
$244$	−27.7687	−1.77771
$245$	−3.07533	−0.196475
$246$	−35.6794	−2.27483
$247$	25.9928	1.65388
$248$	−25.1244	−1.59540
$249$	11.9425	0.756827
$250$	−2.69512	−0.170455
$251$	30.1400	1.90242	0.951210	−	0.308544i	$-0.0998418\pi$
$251$	30.1400	1.90242	0.951210	+	0.308544i	$0.0998418\pi$
$252$	11.4789	0.723103
$253$	−31.1304	−1.95715
$254$	41.4804	2.60271
$255$	13.2664	0.830773
$256$	18.9464	1.18415
$257$	5.72393	0.357049	0.178524	−	0.983935i	$-0.442868\pi$
$257$	5.72393	0.357049	0.178524	+	0.983935i	$0.442868\pi$
$258$	52.3045	3.25634
$259$	−15.0422	−0.934678
$260$	−23.3014	−1.44509
$261$	1.21232	0.0750406
$262$	8.24760	0.509538
$263$	−13.2419	−0.816530	−0.408265	−	0.912863i	$-0.633866\pi$
$263$	−13.2419	−0.816530	−0.408265	+	0.912863i	$0.633866\pi$
$264$	93.8561	5.77644
$265$	10.6791	0.656010
$266$	31.3503	1.92221
$267$	−6.35775	−0.389088
$268$	32.2242	1.96841
$269$	−15.6367	−0.953385	−0.476693	−	0.879070i	$-0.658164\pi$
$269$	−15.6367	−0.953385	−0.476693	+	0.879070i	$0.658164\pi$
$270$	−10.3653	−0.630814
$271$	−25.2668	−1.53485	−0.767423	−	0.641141i	$-0.778462\pi$
$271$	−25.2668	−1.53485	−0.767423	+	0.641141i	$0.778462\pi$
$272$	−86.3380	−5.23501
$273$	17.7594	1.07485
$274$	−15.3966	−0.930144
$275$	5.26915	0.317742
$276$	62.9751	3.79066
$277$	23.1226	1.38930	0.694652	−	0.719346i	$-0.255558\pi$
$277$	23.1226	1.38930	0.694652	+	0.719346i	$0.255558\pi$
$278$	57.5030	3.44880
$279$	3.14425	0.188241
$280$	−17.4257	−1.04138
$281$	−30.2264	−1.80315	−0.901577	−	0.432620i	$-0.857589\pi$
$281$	−30.2264	−1.80315	−0.901577	+	0.432620i	$0.857589\pi$
$282$	44.6822	2.66079
$283$	−14.1279	−0.839815	−0.419907	−	0.907567i	$-0.637937\pi$
$283$	−14.1279	−0.839815	−0.419907	+	0.907567i	$0.637937\pi$
$284$	80.0397	4.74948
$285$	11.8904	0.704325
$286$	62.8652	3.71730
$287$	−12.9511	−0.764478
$288$	−19.7341	−1.16284
$289$	25.9177	1.52457
$290$	−2.96815	−0.174296
$291$	−22.5736	−1.32329
$292$	−61.7618	−3.61434
$293$	−7.17970	−0.419442	−0.209721	−	0.977761i	$-0.567256\pi$
$293$	−7.17970	−0.419442	−0.209721	+	0.977761i	$0.567256\pi$
$294$	−16.7843	−0.978883
$295$	13.1581	0.766097
$296$	66.7879	3.88197
$297$	20.2649	1.17589
$298$	−3.13404	−0.181550
$299$	26.1539	1.51252
$300$	−10.6592	−0.615409
$301$	18.9858	1.09432
$302$	44.7821	2.57692
$303$	15.4167	0.885668
$304$	−77.3829	−4.43821
$305$	−5.27552	−0.302075
$306$	19.4359	1.11108
$307$	7.04655	0.402168	0.201084	−	0.979574i	$-0.435554\pi$
$307$	7.04655	0.402168	0.201084	+	0.979574i	$0.435554\pi$
$308$	54.9456	3.13081
$309$	−34.1617	−1.94339
$310$	−7.69816	−0.437226
$311$	−33.2281	−1.88419	−0.942096	−	0.335343i	$-0.891148\pi$
$311$	−33.2281	−1.88419	−0.942096	+	0.335343i	$0.891148\pi$
$312$	−78.8521	−4.46412
$313$	29.9172	1.69102	0.845510	−	0.533960i	$-0.179297\pi$
$313$	29.9172	1.69102	0.845510	+	0.533960i	$0.179297\pi$
$314$	−32.1812	−1.81609
$315$	2.18077	0.122873
$316$	67.4431	3.79397
$317$	−5.20921	−0.292578	−0.146289	−	0.989242i	$-0.546733\pi$
$317$	−5.20921	−0.292578	−0.146289	+	0.989242i	$0.546733\pi$
$318$	58.2836	3.26838
$319$	5.80294	0.324902
$320$	21.9575	1.22746
$321$	−20.2486	−1.13017
$322$	31.5446	1.75791
$323$	38.4662	2.14032
$324$	−58.3777	−3.24320
$325$	−4.42681	−0.245555
$326$	20.8225	1.15325
$327$	−18.0452	−0.997904
$328$	57.5031	3.17508
$329$	16.2190	0.894182
$330$	28.7577	1.58306
$331$	20.8958	1.14854	0.574269	−	0.818667i	$-0.305286\pi$
$331$	20.8958	1.14854	0.574269	+	0.818667i	$0.305286\pi$
$332$	−31.0422	−1.70366
$333$	−8.35832	−0.458033
$334$	−63.7957	−3.49075
$335$	6.12199	0.334480
$336$	−52.8713	−2.88437
$337$	−8.76570	−0.477498	−0.238749	−	0.971081i	$-0.576737\pi$
$337$	−8.76570	−0.477498	−0.238749	+	0.971081i	$0.576737\pi$
$338$	−17.7789	−0.967043
$339$	20.0820	1.09070
$340$	−34.4833	−1.87012
$341$	15.0504	0.815026
$342$	17.4200	0.941966
$343$	−19.9600	−1.07774
$344$	−84.2973	−4.54501
$345$	11.9641	0.644124
$346$	−26.6037	−1.43022
$347$	12.0126	0.644872	0.322436	−	0.946591i	$-0.395498\pi$
$347$	12.0126	0.644872	0.322436	+	0.946591i	$0.395498\pi$
$348$	−11.7390	−0.629278
$349$	−28.0643	−1.50225	−0.751124	−	0.660161i	$-0.770488\pi$
$349$	−28.0643	−1.50225	−0.751124	+	0.660161i	$0.770488\pi$
$350$	−5.33925	−0.285395
$351$	−17.0254	−0.908746
$352$	−94.4604	−5.03475
$353$	−20.9692	−1.11608	−0.558039	−	0.829815i	$-0.688446\pi$
$353$	−20.9692	−1.11608	−0.558039	+	0.829815i	$0.688446\pi$
$354$	71.8137	3.81686
$355$	15.2060	0.807051
$356$	16.5257	0.875859
$357$	26.2818	1.39098
$358$	−52.9169	−2.79675
$359$	5.19846	0.274364	0.137182	−	0.990546i	$-0.456195\pi$
$359$	5.19846	0.274364	0.137182	+	0.990546i	$0.456195\pi$
$360$	−9.68269	−0.510323
$361$	15.4764	0.814548
$362$	34.0207	1.78809
$363$	−33.9477	−1.78179
$364$	−46.1618	−2.41954
$365$	−11.7336	−0.614162
$366$	−28.7924	−1.50500
$367$	4.70051	0.245365	0.122682	−	0.992446i	$-0.460850\pi$
$367$	4.70051	0.245365	0.122682	+	0.992446i	$0.460850\pi$
$368$	−77.8625	−4.05887
$369$	−7.19636	−0.374628
$370$	20.4639	1.06387
$371$	21.1561	1.09837
$372$	−30.4462	−1.57856
$373$	−22.8622	−1.18376	−0.591880	−	0.806026i	$-0.701614\pi$
$373$	−22.8622	−1.18376	−0.591880	+	0.806026i	$0.701614\pi$
$374$	93.0331	4.81063
$375$	−2.02504	−0.104573
$376$	−72.0128	−3.71377
$377$	−4.87527	−0.251089
$378$	−20.5346	−1.05618
$379$	−6.55915	−0.336921	−0.168460	−	0.985708i	$-0.553880\pi$
$379$	−6.55915	−0.336921	−0.168460	+	0.985708i	$0.553880\pi$
$380$	−30.9066	−1.58548
$381$	31.1673	1.59675
$382$	25.3341	1.29621
$383$	−14.7300	−0.752670	−0.376335	−	0.926484i	$-0.622816\pi$
$383$	−14.7300	−0.752670	−0.376335	+	0.926484i	$0.622816\pi$
$384$	47.2324	2.41032
$385$	10.4386	0.532000
$386$	47.7009	2.42791
$387$	10.5496	0.536265
$388$	58.6755	2.97880
$389$	−10.8894	−0.552112	−0.276056	−	0.961142i	$-0.589028\pi$
$389$	−10.8894	−0.552112	−0.276056	+	0.961142i	$0.589028\pi$
$390$	−24.1604	−1.22341
$391$	38.7046	1.95738
$392$	27.0507	1.36627
$393$	6.19703	0.312599
$394$	40.3071	2.03064
$395$	12.8129	0.644687
$396$	30.5309	1.53424
$397$	−29.1829	−1.46465	−0.732323	−	0.680958i	$-0.761564\pi$
$397$	−29.1829	−1.46465	−0.732323	+	0.680958i	$0.761564\pi$
$398$	33.2363	1.66598
$399$	23.5558	1.17926
$400$	13.1790	0.658952
$401$	7.11714	0.355413	0.177707	−	0.984084i	$-0.443132\pi$
$401$	7.11714	0.355413	0.177707	+	0.984084i	$0.443132\pi$
$402$	33.4122	1.66645
$403$	−12.6444	−0.629864
$404$	−40.0727	−1.99369
$405$	−11.0906	−0.551098
$406$	−5.88014	−0.291827
$407$	−40.0084	−1.98314
$408$	−116.692	−5.77710
$409$	18.8583	0.932481	0.466241	−	0.884658i	$-0.345608\pi$
$409$	18.8583	0.932481	0.466241	+	0.884658i	$0.345608\pi$
$410$	17.6191	0.870144
$411$	−11.5686	−0.570637
$412$	88.7964	4.37468
$413$	26.0673	1.28269
$414$	17.5280	0.861453
$415$	−5.89742	−0.289493
$416$	79.3598	3.89093
$417$	43.2062	2.11582
$418$	83.3835	4.07842
$419$	−14.7887	−0.722475	−0.361237	−	0.932474i	$-0.617646\pi$
$419$	−14.7887	−0.722475	−0.361237	+	0.932474i	$0.617646\pi$
$420$	−21.1167	−1.03039
$421$	−9.44097	−0.460125	−0.230062	−	0.973176i	$-0.573893\pi$
$421$	−9.44097	−0.460125	−0.230062	+	0.973176i	$0.573893\pi$
$422$	13.0894	0.637183
$423$	9.01220	0.438188
$424$	−93.9336	−4.56182
$425$	−6.55116	−0.317778
$426$	82.9904	4.02090
$427$	−10.4512	−0.505770
$428$	52.6320	2.54407
$429$	47.2353	2.28054
$430$	−25.8288	−1.24558
$431$	4.96522	0.239166	0.119583	−	0.992824i	$-0.461844\pi$
$431$	4.96522	0.239166	0.119583	+	0.992824i	$0.461844\pi$
$432$	50.6861	2.43864
$433$	−22.3959	−1.07628	−0.538139	−	0.842856i	$-0.680873\pi$
$433$	−22.3959	−1.07628	−0.538139	+	0.842856i	$0.680873\pi$
$434$	−15.2507	−0.732055
$435$	−2.23019	−0.106929
$436$	46.9049	2.24634
$437$	34.6901	1.65945
$438$	−64.0387	−3.05989
$439$	−16.7511	−0.799484	−0.399742	−	0.916628i	$-0.630900\pi$
$439$	−16.7511	−0.799484	−0.399742	+	0.916628i	$0.630900\pi$
$440$	−46.3477	−2.20954
$441$	−3.38532	−0.161206
$442$	−78.1607	−3.71772
$443$	13.4592	0.639466	0.319733	−	0.947508i	$-0.396407\pi$
$443$	13.4592	0.639466	0.319733	+	0.947508i	$0.396407\pi$
$444$	80.9347	3.84099
$445$	3.13956	0.148830
$446$	7.29310	0.345338
$447$	−2.35484	−0.111380
$448$	43.4996	2.05516
$449$	2.61108	0.123224	0.0616122	−	0.998100i	$-0.480376\pi$
$449$	2.61108	0.123224	0.0616122	+	0.998100i	$0.480376\pi$
$450$	−2.96679	−0.139856
$451$	−34.4465	−1.62202
$452$	−52.1990	−2.45524
$453$	33.6481	1.58092
$454$	37.6607	1.76751
$455$	−8.76987	−0.411138
$456$	−104.588	−4.89779
$457$	−4.42540	−0.207012	−0.103506	−	0.994629i	$-0.533006\pi$
$457$	−4.42540	−0.207012	−0.103506	+	0.994629i	$0.533006\pi$
$458$	−2.23100	−0.104248
$459$	−25.1955	−1.17603
$460$	−31.0982	−1.44996
$461$	27.7534	1.29260	0.646302	−	0.763082i	$-0.276315\pi$
$461$	27.7534	1.29260	0.646302	+	0.763082i	$0.276315\pi$
$462$	56.9712	2.65054
$463$	−6.34361	−0.294812	−0.147406	−	0.989076i	$-0.547092\pi$
$463$	−6.34361	−0.294812	−0.147406	+	0.989076i	$0.547092\pi$
$464$	14.5141	0.673802
$465$	−5.78419	−0.268235
$466$	0.970848	0.0449737
$467$	−24.8058	−1.14788	−0.573938	−	0.818898i	$-0.694585\pi$
$467$	−24.8058	−1.14788	−0.573938	+	0.818898i	$0.694585\pi$
$468$	−25.6502	−1.18568
$469$	12.1281	0.560026
$470$	−22.0648	−1.01777
$471$	−24.1801	−1.11416
$472$	−115.740	−5.32735
$473$	50.4971	2.32186
$474$	69.9295	3.21197
$475$	−5.87166	−0.269410
$476$	−68.3141	−3.13117
$477$	11.7555	0.538249
$478$	−47.2655	−2.16187
$479$	19.0796	0.871768	0.435884	−	0.900003i	$-0.356436\pi$
$479$	19.0796	0.871768	0.435884	+	0.900003i	$0.356436\pi$
$480$	36.3031	1.65700
$481$	33.6126	1.53260
$482$	0.0196122	0.000893312 0
$483$	23.7018	1.07847
$484$	88.2402	4.01092
$485$	11.1472	0.506169
$486$	−29.4338	−1.33515
$487$	−9.43906	−0.427725	−0.213862	−	0.976864i	$-0.568604\pi$
$487$	−9.43906	−0.427725	−0.213862	+	0.976864i	$0.568604\pi$
$488$	46.4037	2.10060
$489$	15.6455	0.707512
$490$	8.28839	0.374431
$491$	3.21648	0.145158	0.0725788	−	0.997363i	$-0.476877\pi$
$491$	3.21648	0.145158	0.0725788	+	0.997363i	$0.476877\pi$
$492$	69.6833	3.14157
$493$	−7.21483	−0.324940
$494$	−70.0537	−3.15186
$495$	5.80028	0.260703
$496$	37.6437	1.69025
$497$	30.1243	1.35126
$498$	−32.1866	−1.44232
$499$	20.0313	0.896722	0.448361	−	0.893853i	$-0.352008\pi$
$499$	20.0313	0.896722	0.448361	+	0.893853i	$0.352008\pi$
$500$	5.26369	0.235399
$501$	−47.9344	−2.14155
$502$	−81.2310	−3.62552
$503$	19.2196	0.856958	0.428479	−	0.903552i	$-0.359050\pi$
$503$	19.2196	0.856958	0.428479	+	0.903552i	$0.359050\pi$
$504$	−19.1822	−0.854442
$505$	−7.61304	−0.338776
$506$	83.9004	3.72983
$507$	−13.3586	−0.593275
$508$	−81.0130	−3.59437
$509$	8.10294	0.359157	0.179578	−	0.983744i	$-0.442527\pi$
$509$	8.10294	0.359157	0.179578	+	0.983744i	$0.442527\pi$
$510$	−35.7545	−1.58324
$511$	−23.2451	−1.02830
$512$	−4.41459	−0.195099
$513$	−22.5822	−0.997028
$514$	−15.4267	−0.680442
$515$	16.8696	0.743364
$516$	−102.153	−4.49703
$517$	43.1383	1.89722
$518$	40.5406	1.78125
$519$	−19.9893	−0.877432
$520$	38.9385	1.70756
$521$	−19.4267	−0.851099	−0.425549	−	0.904935i	$-0.639919\pi$
$521$	−19.4267	−0.851099	−0.425549	+	0.904935i	$0.639919\pi$
$522$	−3.26734	−0.143008
$523$	−15.9294	−0.696545	−0.348273	−	0.937393i	$-0.613232\pi$
$523$	−15.9294	−0.696545	−0.348273	+	0.937393i	$0.613232\pi$
$524$	−16.1079	−0.703677
$525$	−4.01177	−0.175088
$526$	35.6885	1.55609
$527$	−18.7123	−0.815119
$528$	−140.624	−6.11987
$529$	11.9051	0.517614
$530$	−28.7814	−1.25019
$531$	14.4845	0.628573
$532$	−61.2284	−2.65459
$533$	28.9398	1.25352
$534$	17.1349	0.741501
$535$	9.99908	0.432298
$536$	−53.8493	−2.32593
$537$	−39.7604	−1.71579
$538$	42.1428	1.81690
$539$	−16.2044	−0.697971
$540$	20.2439	0.871161
$541$	12.4791	0.536519	0.268259	−	0.963347i	$-0.413552\pi$
$541$	12.4791	0.536519	0.268259	+	0.963347i	$0.413552\pi$
$542$	68.0971	2.92502
$543$	25.5623	1.09698
$544$	117.443	5.03533
$545$	8.91104	0.381707
$546$	−47.8637	−2.04838
$547$	−2.15079	−0.0919612	−0.0459806	−	0.998942i	$-0.514641\pi$
$547$	−2.15079	−0.0919612	−0.0459806	+	0.998942i	$0.514641\pi$
$548$	30.0702	1.28454
$549$	−5.80729	−0.247849
$550$	−14.2010	−0.605533
$551$	−6.46649	−0.275482
$552$	−105.236	−4.47916
$553$	25.3834	1.07941
$554$	−62.3184	−2.64765
$555$	15.3760	0.652677
$556$	−112.306	−4.76283
$557$	9.00569	0.381583	0.190792	−	0.981631i	$-0.438895\pi$
$557$	9.00569	0.381583	0.190792	+	0.981631i	$0.438895\pi$
$558$	−8.47414	−0.358739
$559$	−42.4246	−1.79437
$560$	26.1087	1.10329
$561$	69.9026	2.95129
$562$	81.4637	3.43634
$563$	36.2762	1.52886	0.764430	−	0.644707i	$-0.223020\pi$
$563$	36.2762	1.52886	0.764430	+	0.644707i	$0.223020\pi$
$564$	−87.2663	−3.67457
$565$	−9.91682	−0.417204
$566$	38.0763	1.60047
$567$	−21.9714	−0.922713
$568$	−133.753	−5.61214
$569$	−18.0947	−0.758570	−0.379285	−	0.925280i	$-0.623830\pi$
$569$	−18.0947	−0.758570	−0.379285	+	0.925280i	$0.623830\pi$
$570$	−32.0460	−1.34226
$571$	14.2031	0.594382	0.297191	−	0.954818i	$-0.403950\pi$
$571$	14.2031	0.594382	0.297191	+	0.954818i	$0.403950\pi$
$572$	−122.778	−5.13362
$573$	19.0354	0.795215
$574$	34.9047	1.45690
$575$	−5.90806	−0.246383
$576$	24.1709	1.00712
$577$	−5.74622	−0.239218	−0.119609	−	0.992821i	$-0.538164\pi$
$577$	−5.74622	−0.239218	−0.119609	+	0.992821i	$0.538164\pi$
$578$	−69.8515	−2.90544
$579$	35.8412	1.48951
$580$	5.79693	0.240704
$581$	−11.6832	−0.484703
$582$	60.8386	2.52184
$583$	56.2697	2.33045
$584$	103.209	4.27081
$585$	−4.87304	−0.201475
$586$	19.3502	0.799348
$587$	−8.52826	−0.351999	−0.176000	−	0.984390i	$-0.556316\pi$
$587$	−8.52826	−0.351999	−0.176000	+	0.984390i	$0.556316\pi$
$588$	32.7805	1.35185
$589$	−16.7714	−0.691053
$590$	−35.4628	−1.45998
$591$	30.2857	1.24579
$592$	−100.068	−4.11276
$593$	−33.8458	−1.38988	−0.694941	−	0.719067i	$-0.744569\pi$
$593$	−33.8458	−1.38988	−0.694941	+	0.719067i	$0.744569\pi$
$594$	−54.6165	−2.24094
$595$	−12.9784	−0.532061
$596$	6.12092	0.250722
$597$	24.9729	1.02207
$598$	−70.4879	−2.88246
$599$	−7.51801	−0.307177	−0.153589	−	0.988135i	$-0.549083\pi$
$599$	−7.51801	−0.307177	−0.153589	+	0.988135i	$0.549083\pi$
$600$	17.8124	0.727187
$601$	−39.3434	−1.60485	−0.802424	−	0.596754i	$-0.796457\pi$
$601$	−39.3434	−1.60485	−0.802424	+	0.596754i	$0.796457\pi$
$602$	−51.1690	−2.08549
$603$	6.73909	0.274437
$604$	−87.4614	−3.55875
$605$	16.7639	0.681551
$606$	−41.5500	−1.68785
$607$	5.67961	0.230528	0.115264	−	0.993335i	$-0.463229\pi$
$607$	5.67961	0.230528	0.115264	+	0.993335i	$0.463229\pi$
$608$	105.262	4.26892
$609$	−4.41818	−0.179034
$610$	14.2182	0.575677
$611$	−36.2421	−1.46620
$612$	−37.9592	−1.53441
$613$	−12.8967	−0.520891	−0.260446	−	0.965489i	$-0.583869\pi$
$613$	−12.8967	−0.520891	−0.260446	+	0.965489i	$0.583869\pi$
$614$	−18.9913	−0.766428
$615$	13.2385	0.533828
$616$	−91.8184	−3.69947
$617$	26.9199	1.08376	0.541878	−	0.840457i	$-0.317714\pi$
$617$	26.9199	1.08376	0.541878	+	0.840457i	$0.317714\pi$
$618$	92.0700	3.70360
$619$	9.43571	0.379253	0.189627	−	0.981856i	$-0.439272\pi$
$619$	9.43571	0.379253	0.189627	+	0.981856i	$0.439272\pi$
$620$	15.0348	0.603813
$621$	−22.7222	−0.911809
$622$	89.5538	3.59078
$623$	6.21972	0.249188
$624$	118.143	4.72952
$625$	1.00000	0.0400000
$626$	−80.6305	−3.22264
$627$	62.6521	2.50209
$628$	62.8511	2.50803
$629$	49.7426	1.98337
$630$	−5.87745	−0.234163
$631$	−0.569141	−0.0226572	−0.0113286	−	0.999936i	$-0.503606\pi$
$631$	−0.569141	−0.0226572	−0.0113286	+	0.999936i	$0.503606\pi$
$632$	−112.703	−4.48308
$633$	9.83504	0.390908
$634$	14.0395	0.557578
$635$	−15.3909	−0.610770
$636$	−113.830	−4.51367
$637$	13.6139	0.539403
$638$	−15.6396	−0.619180
$639$	16.7388	0.662176
$640$	−23.3242	−0.921969
$641$	−46.2209	−1.82561	−0.912807	−	0.408391i	$-0.866090\pi$
$641$	−46.2209	−1.82561	−0.912807	+	0.408391i	$0.866090\pi$
$642$	54.5724	2.15380
$643$	−28.6827	−1.13113	−0.565567	−	0.824702i	$-0.691343\pi$
$643$	−28.6827	−1.13113	−0.565567	+	0.824702i	$0.691343\pi$
$644$	−61.6079	−2.42769
$645$	−19.4071	−0.764154
$646$	−103.671	−4.07889
$647$	−5.64862	−0.222070	−0.111035	−	0.993816i	$-0.535417\pi$
$647$	−5.64862	−0.222070	−0.111035	+	0.993816i	$0.535417\pi$
$648$	97.5538	3.83227
$649$	69.3322	2.72153
$650$	11.9308	0.467965
$651$	−11.4589	−0.449111
$652$	−40.6672	−1.59265
$653$	−24.5471	−0.960603	−0.480301	−	0.877104i	$-0.659473\pi$
$653$	−24.5471	−0.960603	−0.480301	+	0.877104i	$0.659473\pi$
$654$	48.6342	1.90175
$655$	−3.06019	−0.119572
$656$	−86.1565	−3.36385
$657$	−12.9163	−0.503913
$658$	−43.7122	−1.70408
$659$	−9.30688	−0.362544	−0.181272	−	0.983433i	$-0.558022\pi$
$659$	−9.30688	−0.362544	−0.181272	+	0.983433i	$0.558022\pi$
$660$	−56.1649	−2.18622
$661$	22.8710	0.889580	0.444790	−	0.895635i	$-0.353278\pi$
$661$	22.8710	0.889580	0.444790	+	0.895635i	$0.353278\pi$
$662$	−56.3168	−2.18881
$663$	−58.7278	−2.28080
$664$	51.8739	2.01310
$665$	−11.6322	−0.451078
$666$	22.5267	0.872892
$667$	−6.50657	−0.251936
$668$	124.596	4.82076
$669$	5.47984	0.211863
$670$	−16.4995	−0.637432
$671$	−27.7975	−1.07311
$672$	71.9192	2.77434
$673$	−37.0459	−1.42802	−0.714008	−	0.700138i	$-0.753122\pi$
$673$	−37.0459	−1.42802	−0.714008	+	0.700138i	$0.753122\pi$
$674$	23.6246	0.909988
$675$	3.84596	0.148031
$676$	34.7229	1.33550
$677$	6.20479	0.238470	0.119235	−	0.992866i	$-0.461956\pi$
$677$	6.20479	0.238470	0.119235	+	0.992866i	$0.461956\pi$
$678$	−54.1234	−2.07860
$679$	22.0835	0.847487
$680$	57.6243	2.20979
$681$	28.2973	1.08435
$682$	−40.5627	−1.55323
$683$	2.59240	0.0991954	0.0495977	−	0.998769i	$-0.484206\pi$
$683$	2.59240	0.0991954	0.0495977	+	0.998769i	$0.484206\pi$
$684$	−34.0220	−1.30086
$685$	5.71277	0.218274
$686$	53.7947	2.05389
$687$	−1.67631	−0.0639552
$688$	126.302	4.81522
$689$	−47.2743	−1.80101
$690$	−32.2446	−1.22753
$691$	−22.8019	−0.867425	−0.433713	−	0.901051i	$-0.642797\pi$
$691$	−22.8019	−0.867425	−0.433713	+	0.901051i	$0.642797\pi$
$692$	51.9581	1.97515
$693$	11.4908	0.436500
$694$	−32.3756	−1.22896
$695$	−21.3359	−0.809319
$696$	19.6169	0.743575
$697$	42.8275	1.62221
$698$	75.6368	2.86290
$699$	0.729469	0.0275911
$700$	10.4278	0.394133
$701$	−18.5519	−0.700695	−0.350348	−	0.936620i	$-0.613937\pi$
$701$	−18.5519	−0.700695	−0.350348	+	0.936620i	$0.613937\pi$
$702$	45.8854	1.73183
$703$	44.5832	1.68149
$704$	115.698	4.36052
$705$	−16.5789	−0.624398
$706$	56.5146	2.12695
$707$	−15.0820	−0.567218
$708$	−140.255	−5.27111
$709$	−14.9104	−0.559972	−0.279986	−	0.960004i	$-0.590330\pi$
$709$	−14.9104	−0.559972	−0.279986	+	0.960004i	$0.590330\pi$
$710$	−40.9821	−1.53803
$711$	14.1044	0.528958
$712$	−27.6157	−1.03494
$713$	−16.8753	−0.631987
$714$	−70.8326	−2.65084
$715$	−23.3256	−0.872326
$716$	103.349	3.86233
$717$	−35.5141	−1.32630
$718$	−14.0105	−0.522867
$719$	9.16597	0.341833	0.170916	−	0.985286i	$-0.445327\pi$
$719$	9.16597	0.341833	0.170916	+	0.985286i	$0.445327\pi$
$720$	14.5075	0.540663
$721$	33.4200	1.24463
$722$	−41.7108	−1.55232
$723$	0.0147361	0.000548041 0
$724$	−66.4439	−2.46937
$725$	1.10131	0.0409014
$726$	91.4933	3.39563
$727$	3.38696	0.125615	0.0628076	−	0.998026i	$-0.479995\pi$
$727$	3.38696	0.125615	0.0628076	+	0.998026i	$0.479995\pi$
$728$	77.1401	2.85900
$729$	11.1561	0.413190
$730$	31.6234	1.17043
$731$	−62.7834	−2.32213
$732$	56.2328	2.07842
$733$	15.3114	0.565540	0.282770	−	0.959188i	$-0.408747\pi$
$733$	15.3114	0.565540	0.282770	+	0.959188i	$0.408747\pi$
$734$	−12.6685	−0.467602
$735$	6.22767	0.229711
$736$	105.914	3.90404
$737$	32.2577	1.18823
$738$	19.3951	0.713943
$739$	−4.04075	−0.148641	−0.0743207	−	0.997234i	$-0.523679\pi$
$739$	−4.04075	−0.148641	−0.0743207	+	0.997234i	$0.523679\pi$
$740$	−39.9669	−1.46921
$741$	−52.6365	−1.93365
$742$	−57.0183	−2.09321
$743$	−14.4971	−0.531847	−0.265924	−	0.963994i	$-0.585677\pi$
$743$	−14.4971	−0.531847	−0.265924	+	0.963994i	$0.585677\pi$
$744$	50.8780	1.86528
$745$	1.16286	0.0426038
$746$	61.6164	2.25594
$747$	−6.49188	−0.237526
$748$	−181.698	−6.64352
$749$	19.8090	0.723804
$750$	5.45774	0.199289
$751$	−40.8721	−1.49144	−0.745722	−	0.666258i	$-0.767895\pi$
$751$	−40.8721	−1.49144	−0.745722	+	0.666258i	$0.767895\pi$
$752$	107.896	3.93457
$753$	−61.0348	−2.22423
$754$	13.1395	0.478511
$755$	−16.6160	−0.604717
$756$	40.1048	1.45860
$757$	−51.8701	−1.88525	−0.942626	−	0.333850i	$-0.891652\pi$
$757$	−51.8701	−1.88525	−0.942626	+	0.333850i	$0.891652\pi$
$758$	17.6777	0.642084
$759$	63.0405	2.28822
$760$	51.6474	1.87345
$761$	51.3725	1.86225	0.931127	−	0.364696i	$-0.118827\pi$
$761$	51.3725	1.86225	0.931127	+	0.364696i	$0.118827\pi$
$762$	−83.9996	−3.04299
$763$	17.6535	0.639099
$764$	−49.4787	−1.79007
$765$	−7.21152	−0.260733
$766$	39.6993	1.43439
$767$	−58.2486	−2.10324
$768$	−38.3673	−1.38446
$769$	−30.1655	−1.08780	−0.543898	−	0.839152i	$-0.683052\pi$
$769$	−30.1655	−1.08780	−0.543898	+	0.839152i	$0.683052\pi$
$770$	−28.1333	−1.01385
$771$	−11.5912	−0.417447
$772$	−93.1619	−3.35297
$773$	−46.1286	−1.65913	−0.829565	−	0.558411i	$-0.811411\pi$
$773$	−46.1286	−1.65913	−0.829565	+	0.558411i	$0.811411\pi$
$774$	−28.4324	−1.02198
$775$	2.85633	0.102602
$776$	−98.0514	−3.51984
$777$	30.4612	1.09279
$778$	29.3481	1.05218
$779$	38.3853	1.37530
$780$	47.1863	1.68954
$781$	80.1227	2.86702
$782$	−104.314	−3.73025
$783$	4.23558	0.151367
$784$	−40.5299	−1.44750
$785$	11.9405	0.426175
$786$	−16.7017	−0.595732
$787$	−32.9700	−1.17525	−0.587627	−	0.809132i	$-0.699938\pi$
$787$	−32.9700	−1.17525	−0.587627	+	0.809132i	$0.699938\pi$
$788$	−78.7215	−2.80434
$789$	26.8154	0.954654
$790$	−34.5323	−1.22861
$791$	−19.6460	−0.698531
$792$	−51.0196	−1.81290
$793$	23.3537	0.829316
$794$	78.6514	2.79123
$795$	−21.6256	−0.766981
$796$	−64.9119	−2.30074
$797$	−22.1508	−0.784623	−0.392311	−	0.919832i	$-0.628324\pi$
$797$	−22.1508	−0.784623	−0.392311	+	0.919832i	$0.628324\pi$
$798$	−63.4857	−2.24737
$799$	−53.6340	−1.89744
$800$	−17.9271	−0.633817
$801$	3.45603	0.122113
$802$	−19.1816	−0.677325
$803$	−61.8259	−2.18179
$804$	−65.2555	−2.30138
$805$	−11.7043	−0.412523
$806$	34.0783	1.20036
$807$	31.6650	1.11466
$808$	66.9646	2.35581
$809$	−5.00771	−0.176062	−0.0880308	−	0.996118i	$-0.528057\pi$
$809$	−5.00771	−0.176062	−0.0880308	+	0.996118i	$0.528057\pi$
$810$	29.8906	1.05025
$811$	−54.9425	−1.92929	−0.964646	−	0.263550i	$-0.915107\pi$
$811$	−54.9425	−1.92929	−0.964646	+	0.263550i	$0.915107\pi$
$812$	11.4842	0.403015
$813$	51.1663	1.79448
$814$	107.827	3.77935
$815$	−7.72598	−0.270629
$816$	174.838	6.12056
$817$	−56.2713	−1.96868
$818$	−50.8254	−1.77707
$819$	−9.65387	−0.337334
$820$	−34.4108	−1.20168
$821$	46.8901	1.63648	0.818238	−	0.574880i	$-0.194951\pi$
$821$	46.8901	1.63648	0.818238	+	0.574880i	$0.194951\pi$
$822$	31.1788	1.08749
$823$	−39.5902	−1.38003	−0.690014	−	0.723796i	$-0.742396\pi$
$823$	−39.5902	−1.38003	−0.690014	+	0.723796i	$0.742396\pi$
$824$	−148.386	−5.16926
$825$	−10.6703	−0.371491
$826$	−70.2546	−2.44447
$827$	4.51447	0.156983	0.0784917	−	0.996915i	$-0.474990\pi$
$827$	4.51447	0.156983	0.0784917	+	0.996915i	$0.474990\pi$
$828$	−34.2329	−1.18968
$829$	−51.4263	−1.78611	−0.893054	−	0.449949i	$-0.851442\pi$
$829$	−51.4263	−1.78611	−0.893054	+	0.449949i	$0.851442\pi$
$830$	15.8943	0.551698
$831$	−46.8243	−1.62432
$832$	−97.2020	−3.36987
$833$	20.1470	0.698051
$834$	−116.446	−4.03220
$835$	23.6708	0.819162
$836$	−162.852	−5.63234
$837$	10.9853	0.379708
$838$	39.8573	1.37685
$839$	−26.2945	−0.907787	−0.453893	−	0.891056i	$-0.649965\pi$
$839$	−26.2945	−0.907787	−0.453893	+	0.891056i	$0.649965\pi$
$840$	35.2877	1.21754
$841$	−27.7871	−0.958177
$842$	25.4446	0.876878
$843$	61.2097	2.10817
$844$	−25.5642	−0.879956
$845$	6.59668	0.226933
$846$	−24.2890	−0.835072
$847$	33.2107	1.14113
$848$	140.740	4.83303
$849$	28.6095	0.981877
$850$	17.6562	0.605602
$851$	44.8596	1.53777
$852$	−162.084	−5.55290
$853$	−54.5855	−1.86897	−0.934487	−	0.355998i	$-0.884141\pi$
$853$	−54.5855	−1.86897	−0.934487	+	0.355998i	$0.884141\pi$
$854$	28.1673	0.963866
$855$	−6.46353	−0.221048
$856$	−87.9523	−3.00615
$857$	16.5579	0.565607	0.282804	−	0.959178i	$-0.408736\pi$
$857$	16.5579	0.565607	0.282804	+	0.959178i	$0.408736\pi$
$858$	−127.305	−4.34611
$859$	3.10364	0.105895	0.0529474	−	0.998597i	$-0.483138\pi$
$859$	3.10364	0.105895	0.0529474	+	0.998597i	$0.483138\pi$
$860$	50.4448	1.72015
$861$	26.2265	0.893797
$862$	−13.3819	−0.455788
$863$	−34.3725	−1.17005	−0.585027	−	0.811014i	$-0.698916\pi$
$863$	−34.3725	−1.17005	−0.585027	+	0.811014i	$0.698916\pi$
$864$	−68.9468	−2.34562
$865$	9.87103	0.335625
$866$	60.3597	2.05111
$867$	−52.4845	−1.78247
$868$	29.7852	1.01097
$869$	67.5131	2.29022
$870$	6.01064	0.203780
$871$	−27.1009	−0.918279
$872$	−78.3819	−2.65435
$873$	12.2709	0.415306
$874$	−93.4941	−3.16249
$875$	1.98108	0.0669727
$876$	125.070	4.22573
$877$	−5.56230	−0.187825	−0.0939127	−	0.995580i	$-0.529937\pi$
$877$	−5.56230	−0.187825	−0.0939127	+	0.995580i	$0.529937\pi$
$878$	45.1461	1.52361
$879$	14.5392	0.490395
$880$	69.4424	2.34090
$881$	9.70053	0.326819	0.163410	−	0.986558i	$-0.447751\pi$
$881$	9.70053	0.326819	0.163410	+	0.986558i	$0.447751\pi$
$882$	9.12386	0.307216
$883$	50.3978	1.69602	0.848011	−	0.529979i	$-0.177800\pi$
$883$	50.3978	1.69602	0.848011	+	0.529979i	$0.177800\pi$
$884$	152.651	5.13421
$885$	−26.6458	−0.895689
$886$	−36.2742	−1.21866
$887$	13.7753	0.462529	0.231265	−	0.972891i	$-0.425714\pi$
$887$	13.7753	0.462529	0.231265	+	0.972891i	$0.425714\pi$
$888$	−135.248	−4.53864
$889$	−30.4906	−1.02262
$890$	−8.46151	−0.283630
$891$	−58.4383	−1.95776
$892$	−14.2437	−0.476915
$893$	−48.0710	−1.60863
$894$	6.34657	0.212261
$895$	19.6343	0.656303
$896$	−46.2070	−1.54367
$897$	−52.9627	−1.76837
$898$	−7.03718	−0.234834
$899$	3.14569	0.104915
$900$	5.79427	0.193142
$901$	−69.9603	−2.33072
$902$	92.8375	3.09115
$903$	−38.4470	−1.27944
$904$	87.2288	2.90119
$905$	−12.6231	−0.419605
$906$	−90.6857	−3.01283
$907$	5.56873	0.184907	0.0924533	−	0.995717i	$-0.470529\pi$
$907$	5.56873	0.184907	0.0924533	+	0.995717i	$0.470529\pi$
$908$	−73.5530	−2.44094
$909$	−8.38044	−0.277962
$910$	23.6359	0.783521
$911$	−32.1948	−1.06666	−0.533331	−	0.845907i	$-0.679060\pi$
$911$	−32.1948	−1.06666	−0.533331	+	0.845907i	$0.679060\pi$
$912$	156.704	5.18898
$913$	−31.0744	−1.02841
$914$	11.9270	0.394510
$915$	10.6832	0.353174
$916$	4.35723	0.143967
$917$	−6.06248	−0.200201
$918$	67.9050	2.24120
$919$	−0.0305947	−0.00100923	−0.000504614	−	1.00000i	$-0.500161\pi$
$919$	−0.0305947	−0.00100923	−0.000504614		1.00000i	$0.500161\pi$
$920$	51.9675	1.71332
$921$	−14.2696	−0.470198
$922$	−74.7988	−2.46337
$923$	−67.3142	−2.21567
$924$	−111.267	−3.66042
$925$	−7.59295	−0.249655
$926$	17.0968	0.561836
$927$	18.5701	0.609921
$928$	−19.7432	−0.648101
$929$	−16.3810	−0.537442	−0.268721	−	0.963218i	$-0.586601\pi$
$929$	−16.3810	−0.537442	−0.268721	+	0.963218i	$0.586601\pi$
$930$	15.5891	0.511187
$931$	18.0573	0.591804
$932$	−1.89611	−0.0621090
$933$	67.2883	2.20292
$934$	66.8548	2.18755
$935$	−34.5191	−1.12889
$936$	42.8635	1.40104
$937$	−49.4991	−1.61706	−0.808532	−	0.588452i	$-0.799738\pi$
$937$	−49.4991	−1.61706	−0.808532	+	0.588452i	$0.799738\pi$
$938$	−32.6868	−1.06726
$939$	−60.5836	−1.97707
$940$	43.0936	1.40556
$941$	−11.2207	−0.365784	−0.182892	−	0.983133i	$-0.558546\pi$
$941$	−11.2207	−0.365784	−0.182892	+	0.983133i	$0.558546\pi$
$942$	65.1682	2.12330
$943$	38.6233	1.25775
$944$	173.412	5.64407
$945$	7.61915	0.247851
$946$	−136.096	−4.42486
$947$	30.8103	1.00120	0.500600	−	0.865679i	$-0.333113\pi$
$947$	30.8103	1.00120	0.500600	+	0.865679i	$0.333113\pi$
$948$	−136.575	−4.43576
$949$	51.9423	1.68612
$950$	15.8249	0.513426
$951$	10.5489	0.342071
$952$	114.158	3.69989
$953$	−2.26126	−0.0732493	−0.0366247	−	0.999329i	$-0.511661\pi$
$953$	−2.26126	−0.0732493	−0.0366247	+	0.999329i	$0.511661\pi$
$954$	−31.6826	−1.02576
$955$	−9.40000	−0.304177
$956$	92.3116	2.98557
$957$	−11.7512	−0.379863
$958$	−51.4218	−1.66136
$959$	11.3174	0.365459
$960$	−44.4650	−1.43510
$961$	−22.8414	−0.736819
$962$	−90.5900	−2.92074
$963$	11.0070	0.354695
$964$	−0.0383035	−0.00123367
$965$	−17.6990	−0.569750
$966$	−63.8792	−2.05528
$967$	19.4170	0.624407	0.312204	−	0.950015i	$-0.398933\pi$
$967$	19.4170	0.624407	0.312204	+	0.950015i	$0.398933\pi$
$968$	−147.456	−4.73943
$969$	−77.8957	−2.50237
$970$	−30.0431	−0.964627
$971$	−39.6830	−1.27349	−0.636744	−	0.771075i	$-0.719719\pi$
$971$	−39.6830	−1.27349	−0.636744	+	0.771075i	$0.719719\pi$
$972$	57.4855	1.84385
$973$	−42.2682	−1.35506
$974$	25.4394	0.815132
$975$	8.96449	0.287093
$976$	−69.5263	−2.22548
$977$	27.8937	0.892397	0.446198	−	0.894934i	$-0.352778\pi$
$977$	27.8937	0.892397	0.446198	+	0.894934i	$0.352778\pi$
$978$	−42.1664	−1.34833
$979$	16.5428	0.528711
$980$	−16.1876	−0.517093
$981$	9.80928	0.313186
$982$	−8.66880	−0.276633
$983$	39.4242	1.25744	0.628718	−	0.777633i	$-0.283580\pi$
$983$	39.4242	1.25744	0.628718	+	0.777633i	$0.283580\pi$
$984$	−116.446	−3.71217
$985$	−14.9556	−0.476524
$986$	19.4449	0.619250
$987$	−32.8442	−1.04544
$988$	136.818	4.35275
$989$	−56.6201	−1.80042
$990$	−15.6325	−0.496833
$991$	3.11757	0.0990329	0.0495164	−	0.998773i	$-0.484232\pi$
$991$	3.11757	0.0990329	0.0495164	+	0.998773i	$0.484232\pi$
$992$	−51.2056	−1.62578
$993$	−42.3149	−1.34282
$994$	−81.1887	−2.57515
$995$	−12.3320	−0.390951
$996$	62.8618	1.99185
$997$	−17.3278	−0.548778	−0.274389	−	0.961619i	$-0.588476\pi$
$997$	−17.3278	−0.548778	−0.274389	+	0.961619i	$0.588476\pi$
$998$	−53.9867	−1.70892
$999$	−29.2022	−0.923916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.c.1.6	✓	127

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.c.1.6	✓	127	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69512 q^{2} -2.02504 q^{3} +5.26369 q^{4} +1.00000 q^{5} +5.45774 q^{6} +1.98108 q^{7} -8.79604 q^{8} +1.10080 q^{9} +O(q^{10})\)	\(q-2.69512 q^{2} -2.02504 q^{3} +5.26369 q^{4} +1.00000 q^{5} +5.45774 q^{6} +1.98108 q^{7} -8.79604 q^{8} +1.10080 q^{9} -2.69512 q^{10} +5.26915 q^{11} -10.6592 q^{12} -4.42681 q^{13} -5.33925 q^{14} -2.02504 q^{15} +13.1790 q^{16} -6.55116 q^{17} -2.96679 q^{18} -5.87166 q^{19} +5.26369 q^{20} -4.01177 q^{21} -14.2010 q^{22} -5.90806 q^{23} +17.8124 q^{24} +1.00000 q^{25} +11.9308 q^{26} +3.84596 q^{27} +10.4278 q^{28} +1.10131 q^{29} +5.45774 q^{30} +2.85633 q^{31} -17.9271 q^{32} -10.6703 q^{33} +17.6562 q^{34} +1.98108 q^{35} +5.79427 q^{36} -7.59295 q^{37} +15.8249 q^{38} +8.96449 q^{39} -8.79604 q^{40} -6.53739 q^{41} +10.8122 q^{42} +9.58355 q^{43} +27.7352 q^{44} +1.10080 q^{45} +15.9229 q^{46} +8.18695 q^{47} -26.6881 q^{48} -3.07533 q^{49} -2.69512 q^{50} +13.2664 q^{51} -23.3014 q^{52} +10.6791 q^{53} -10.3653 q^{54} +5.26915 q^{55} -17.4257 q^{56} +11.8904 q^{57} -2.96815 q^{58} +13.1581 q^{59} -10.6592 q^{60} -5.27552 q^{61} -7.69816 q^{62} +2.18077 q^{63} +21.9575 q^{64} -4.42681 q^{65} +28.7577 q^{66} +6.12199 q^{67} -34.4833 q^{68} +11.9641 q^{69} -5.33925 q^{70} +15.2060 q^{71} -9.68269 q^{72} -11.7336 q^{73} +20.4639 q^{74} -2.02504 q^{75} -30.9066 q^{76} +10.4386 q^{77} -24.1604 q^{78} +12.8129 q^{79} +13.1790 q^{80} -11.0906 q^{81} +17.6191 q^{82} -5.89742 q^{83} -21.1167 q^{84} -6.55116 q^{85} -25.8288 q^{86} -2.23019 q^{87} -46.3477 q^{88} +3.13956 q^{89} -2.96679 q^{90} -8.76987 q^{91} -31.0982 q^{92} -5.78419 q^{93} -22.0648 q^{94} -5.87166 q^{95} +36.3031 q^{96} +11.1472 q^{97} +8.28839 q^{98} +5.80028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * q^99

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