LMFDB - Embedded newform 8045.2.a.b.1.12

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:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.37056 q^{2} -2.51302 q^{3} +3.61953 q^{4} -1.00000 q^{5} +5.95725 q^{6} +3.42592 q^{7} -3.83919 q^{8} +3.31526 q^{9} +O(q^{10})$	\(q-2.37056 q^{2} -2.51302 q^{3} +3.61953 q^{4} -1.00000 q^{5} +5.95725 q^{6} +3.42592 q^{7} -3.83919 q^{8} +3.31526 q^{9} +2.37056 q^{10} +0.370238 q^{11} -9.09595 q^{12} -1.94964 q^{13} -8.12134 q^{14} +2.51302 q^{15} +1.86195 q^{16} +5.91964 q^{17} -7.85900 q^{18} +5.75226 q^{19} -3.61953 q^{20} -8.60940 q^{21} -0.877670 q^{22} +2.52664 q^{23} +9.64796 q^{24} +1.00000 q^{25} +4.62173 q^{26} -0.792245 q^{27} +12.4002 q^{28} -1.35344 q^{29} -5.95725 q^{30} -8.55839 q^{31} +3.26452 q^{32} -0.930415 q^{33} -14.0328 q^{34} -3.42592 q^{35} +11.9997 q^{36} +1.49853 q^{37} -13.6361 q^{38} +4.89948 q^{39} +3.83919 q^{40} +3.75696 q^{41} +20.4091 q^{42} -3.28400 q^{43} +1.34009 q^{44} -3.31526 q^{45} -5.98955 q^{46} -1.86407 q^{47} -4.67912 q^{48} +4.73693 q^{49} -2.37056 q^{50} -14.8761 q^{51} -7.05679 q^{52} +8.76522 q^{53} +1.87806 q^{54} -0.370238 q^{55} -13.1528 q^{56} -14.4555 q^{57} +3.20840 q^{58} -1.19747 q^{59} +9.09595 q^{60} -14.0369 q^{61} +20.2881 q^{62} +11.3578 q^{63} -11.4626 q^{64} +1.94964 q^{65} +2.20560 q^{66} +4.36382 q^{67} +21.4263 q^{68} -6.34950 q^{69} +8.12134 q^{70} +0.344492 q^{71} -12.7279 q^{72} -1.97808 q^{73} -3.55236 q^{74} -2.51302 q^{75} +20.8205 q^{76} +1.26841 q^{77} -11.6145 q^{78} +4.84546 q^{79} -1.86195 q^{80} -7.95484 q^{81} -8.90608 q^{82} -6.28468 q^{83} -31.1620 q^{84} -5.91964 q^{85} +7.78491 q^{86} +3.40121 q^{87} -1.42142 q^{88} +3.88786 q^{89} +7.85900 q^{90} -6.67932 q^{91} +9.14527 q^{92} +21.5074 q^{93} +4.41888 q^{94} -5.75226 q^{95} -8.20381 q^{96} -9.64645 q^{97} -11.2292 q^{98} +1.22743 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) $ 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * 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 $

$2$	−2.37056	−1.67624	−0.838118	−	0.545489i	$-0.816344\pi$
$2$	−2.37056	−1.67624	−0.838118	+	0.545489i	$0.816344\pi$
$3$	−2.51302	−1.45089	−0.725446	−	0.688279i	$-0.758366\pi$
$3$	−2.51302	−1.45089	−0.725446	+	0.688279i	$0.758366\pi$
$4$	3.61953	1.80977
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	5.95725	2.43204
$7$	3.42592	1.29488	0.647438	−	0.762118i	$-0.275840\pi$
$7$	3.42592	1.29488	0.647438	+	0.762118i	$0.275840\pi$
$8$	−3.83919	−1.35736
$9$	3.31526	1.10509
$10$	2.37056	0.749635
$11$	0.370238	0.111631	0.0558155	−	0.998441i	$-0.482224\pi$
$11$	0.370238	0.111631	0.0558155	+	0.998441i	$0.482224\pi$
$12$	−9.09595	−2.62577
$13$	−1.94964	−0.540733	−0.270367	−	0.962757i	$-0.587145\pi$
$13$	−1.94964	−0.540733	−0.270367	+	0.962757i	$0.587145\pi$
$14$	−8.12134	−2.17052
$15$	2.51302	0.648858
$16$	1.86195	0.465488
$17$	5.91964	1.43572	0.717861	−	0.696186i	$-0.245121\pi$
$17$	5.91964	1.43572	0.717861	+	0.696186i	$0.245121\pi$
$18$	−7.85900	−1.85238
$19$	5.75226	1.31966	0.659830	−	0.751415i	$-0.270628\pi$
$19$	5.75226	1.31966	0.659830	+	0.751415i	$0.270628\pi$
$20$	−3.61953	−0.809352
$21$	−8.60940	−1.87872
$22$	−0.877670	−0.187120
$23$	2.52664	0.526842	0.263421	−	0.964681i	$-0.415149\pi$
$23$	2.52664	0.526842	0.263421	+	0.964681i	$0.415149\pi$
$24$	9.64796	1.96938
$25$	1.00000	0.200000
$26$	4.62173	0.906396
$27$	−0.792245	−0.152468
$28$	12.4002	2.34342
$29$	−1.35344	−0.251327	−0.125663	−	0.992073i	$-0.540106\pi$
$29$	−1.35344	−0.251327	−0.125663	+	0.992073i	$0.540106\pi$
$30$	−5.95725	−1.08764
$31$	−8.55839	−1.53713	−0.768566	−	0.639770i	$-0.779030\pi$
$31$	−8.55839	−1.53713	−0.768566	+	0.639770i	$0.779030\pi$
$32$	3.26452	0.577092
$33$	−0.930415	−0.161964
$34$	−14.0328	−2.40661
$35$	−3.42592	−0.579086
$36$	11.9997	1.99995
$37$	1.49853	0.246357	0.123179	−	0.992385i	$-0.460691\pi$
$37$	1.49853	0.246357	0.123179	+	0.992385i	$0.460691\pi$
$38$	−13.6361	−2.21206
$39$	4.89948	0.784545
$40$	3.83919	0.607030
$41$	3.75696	0.586738	0.293369	−	0.955999i	$-0.405223\pi$
$41$	3.75696	0.586738	0.293369	+	0.955999i	$0.405223\pi$
$42$	20.4091	3.14919
$43$	−3.28400	−0.500806	−0.250403	−	0.968142i	$-0.580563\pi$
$43$	−3.28400	−0.500806	−0.250403	+	0.968142i	$0.580563\pi$
$44$	1.34009	0.202026
$45$	−3.31526	−0.494209
$46$	−5.98955	−0.883111
$47$	−1.86407	−0.271903	−0.135951	−	0.990716i	$-0.543409\pi$
$47$	−1.86407	−0.271903	−0.135951	+	0.990716i	$0.543409\pi$
$48$	−4.67912	−0.675372
$49$	4.73693	0.676705
$50$	−2.37056	−0.335247
$51$	−14.8761	−2.08308
$52$	−7.05679	−0.978601
$53$	8.76522	1.20400	0.601998	−	0.798498i	$-0.294371\pi$
$53$	8.76522	1.20400	0.601998	+	0.798498i	$0.294371\pi$
$54$	1.87806	0.255572
$55$	−0.370238	−0.0499229
$56$	−13.1528	−1.75761
$57$	−14.4555	−1.91468
$58$	3.20840	0.421283
$59$	−1.19747	−0.155897	−0.0779487	−	0.996957i	$-0.524837\pi$
$59$	−1.19747	−0.155897	−0.0779487	+	0.996957i	$0.524837\pi$
$60$	9.09595	1.17428
$61$	−14.0369	−1.79724	−0.898618	−	0.438732i	$-0.855428\pi$
$61$	−14.0369	−1.79724	−0.898618	+	0.438732i	$0.855428\pi$
$62$	20.2881	2.57660
$63$	11.3578	1.43095
$64$	−11.4626	−1.43283
$65$	1.94964	0.241823
$66$	2.20560	0.271491
$67$	4.36382	0.533126	0.266563	−	0.963818i	$-0.414112\pi$
$67$	4.36382	0.533126	0.266563	+	0.963818i	$0.414112\pi$
$68$	21.4263	2.59832
$69$	−6.34950	−0.764390
$70$	8.12134	0.970685
$71$	0.344492	0.0408836	0.0204418	−	0.999791i	$-0.493493\pi$
$71$	0.344492	0.0408836	0.0204418	+	0.999791i	$0.493493\pi$
$72$	−12.7279	−1.50000
$73$	−1.97808	−0.231517	−0.115758	−	0.993277i	$-0.536930\pi$
$73$	−1.97808	−0.231517	−0.115758	+	0.993277i	$0.536930\pi$
$74$	−3.55236	−0.412953
$75$	−2.51302	−0.290178
$76$	20.8205	2.38828
$77$	1.26841	0.144548
$78$	−11.6145	−1.31508
$79$	4.84546	0.545156	0.272578	−	0.962134i	$-0.412124\pi$
$79$	4.84546	0.545156	0.272578	+	0.962134i	$0.412124\pi$
$80$	−1.86195	−0.208173
$81$	−7.95484	−0.883872
$82$	−8.90608	−0.983512
$83$	−6.28468	−0.689833	−0.344916	−	0.938633i	$-0.612093\pi$
$83$	−6.28468	−0.689833	−0.344916	+	0.938633i	$0.612093\pi$
$84$	−31.1620	−3.40005
$85$	−5.91964	−0.642075
$86$	7.78491	0.839468
$87$	3.40121	0.364648
$88$	−1.42142	−0.151523
$89$	3.88786	0.412113	0.206056	−	0.978540i	$-0.433937\pi$
$89$	3.88786	0.412113	0.206056	+	0.978540i	$0.433937\pi$
$90$	7.85900	0.828411
$91$	−6.67932	−0.700183
$92$	9.14527	0.953461
$93$	21.5074	2.23021
$94$	4.41888	0.455773
$95$	−5.75226	−0.590170
$96$	−8.20381	−0.837298
$97$	−9.64645	−0.979449	−0.489725	−	0.871877i	$-0.662903\pi$
$97$	−9.64645	−0.979449	−0.489725	+	0.871877i	$0.662903\pi$
$98$	−11.2292	−1.13432
$99$	1.22743	0.123362
$100$	3.61953	0.361953
$101$	−13.4091	−1.33425	−0.667126	−	0.744945i	$-0.732476\pi$
$101$	−13.4091	−1.33425	−0.667126	+	0.744945i	$0.732476\pi$
$102$	35.2647	3.49173
$103$	0.234502	0.0231062	0.0115531	−	0.999933i	$-0.496322\pi$
$103$	0.234502	0.0231062	0.0115531	+	0.999933i	$0.496322\pi$
$104$	7.48505	0.733969
$105$	8.60940	0.840191
$106$	−20.7784	−2.01818
$107$	5.81850	0.562496	0.281248	−	0.959635i	$-0.409252\pi$
$107$	5.81850	0.562496	0.281248	+	0.959635i	$0.409252\pi$
$108$	−2.86756	−0.275931
$109$	−10.4288	−0.998903	−0.499451	−	0.866342i	$-0.666465\pi$
$109$	−10.4288	−0.998903	−0.499451	+	0.866342i	$0.666465\pi$
$110$	0.877670	0.0836826
$111$	−3.76584	−0.357438
$112$	6.37890	0.602749
$113$	−9.22531	−0.867844	−0.433922	−	0.900950i	$-0.642871\pi$
$113$	−9.22531	−0.867844	−0.433922	+	0.900950i	$0.642871\pi$
$114$	34.2676	3.20946
$115$	−2.52664	−0.235611
$116$	−4.89881	−0.454843
$117$	−6.46356	−0.597556
$118$	2.83867	0.261321
$119$	20.2802	1.85908
$120$	−9.64796	−0.880734
$121$	−10.8629	−0.987539
$122$	33.2752	3.01259
$123$	−9.44130	−0.851294
$124$	−30.9774	−2.78185
$125$	−1.00000	−0.0894427
$126$	−26.9243	−2.39861
$127$	−21.5941	−1.91616	−0.958082	−	0.286494i	$-0.907510\pi$
$127$	−21.5941	−1.91616	−0.958082	+	0.286494i	$0.907510\pi$
$128$	20.6438	1.82467
$129$	8.25275	0.726614
$130$	−4.62173	−0.405353
$131$	−21.2593	−1.85743	−0.928717	−	0.370789i	$-0.879087\pi$
$131$	−21.2593	−1.85743	−0.928717	+	0.370789i	$0.879087\pi$
$132$	−3.36767	−0.293118
$133$	19.7068	1.70880
$134$	−10.3447	−0.893645
$135$	0.792245	0.0681856
$136$	−22.7266	−1.94879
$137$	11.4407	0.977448	0.488724	−	0.872439i	$-0.337463\pi$
$137$	11.4407	0.977448	0.488724	+	0.872439i	$0.337463\pi$
$138$	15.0518	1.28130
$139$	−22.9490	−1.94651	−0.973254	−	0.229731i	$-0.926215\pi$
$139$	−22.9490	−1.94651	−0.973254	+	0.229731i	$0.926215\pi$
$140$	−12.4002	−1.04801
$141$	4.68444	0.394501
$142$	−0.816636	−0.0685306
$143$	−0.721832	−0.0603626
$144$	6.17285	0.514404
$145$	1.35344	0.112397
$146$	4.68915	0.388077
$147$	−11.9040	−0.981825
$148$	5.42399	0.445849
$149$	−17.3773	−1.42361	−0.711803	−	0.702379i	$-0.752121\pi$
$149$	−17.3773	−1.42361	−0.711803	+	0.702379i	$0.752121\pi$
$150$	5.95725	0.486407
$151$	3.27668	0.266653	0.133326	−	0.991072i	$-0.457434\pi$
$151$	3.27668	0.266653	0.133326	+	0.991072i	$0.457434\pi$
$152$	−22.0840	−1.79125
$153$	19.6251	1.58660
$154$	−3.00683	−0.242297
$155$	8.55839	0.687427
$156$	17.7338	1.41984
$157$	−0.946342	−0.0755263	−0.0377632	−	0.999287i	$-0.512023\pi$
$157$	−0.946342	−0.0755263	−0.0377632	+	0.999287i	$0.512023\pi$
$158$	−11.4864	−0.913811
$159$	−22.0272	−1.74687
$160$	−3.26452	−0.258083
$161$	8.65609	0.682195
$162$	18.8574	1.48158
$163$	−6.75167	−0.528832	−0.264416	−	0.964409i	$-0.585179\pi$
$163$	−6.75167	−0.528832	−0.264416	+	0.964409i	$0.585179\pi$
$164$	13.5984	1.06186
$165$	0.930415	0.0724327
$166$	14.8982	1.15632
$167$	−8.71831	−0.674643	−0.337322	−	0.941389i	$-0.609521\pi$
$167$	−8.71831	−0.674643	−0.337322	+	0.941389i	$0.609521\pi$
$168$	33.0531	2.55010
$169$	−9.19890	−0.707608
$170$	14.0328	1.07627
$171$	19.0702	1.45834
$172$	−11.8866	−0.906341
$173$	−21.7531	−1.65385	−0.826927	−	0.562309i	$-0.809913\pi$
$173$	−21.7531	−1.65385	−0.826927	+	0.562309i	$0.809913\pi$
$174$	−8.06276	−0.611236
$175$	3.42592	0.258975
$176$	0.689366	0.0519629
$177$	3.00927	0.226190
$178$	−9.21640	−0.690798
$179$	22.8080	1.70475	0.852376	−	0.522929i	$-0.175161\pi$
$179$	22.8080	1.70475	0.852376	+	0.522929i	$0.175161\pi$
$180$	−11.9997	−0.894403
$181$	−0.280696	−0.0208639	−0.0104320	−	0.999946i	$-0.503321\pi$
$181$	−0.280696	−0.0208639	−0.0104320	+	0.999946i	$0.503321\pi$
$182$	15.8337	1.17367
$183$	35.2749	2.60759
$184$	−9.70027	−0.715114
$185$	−1.49853	−0.110174
$186$	−50.9845	−3.73836
$187$	2.19168	0.160271
$188$	−6.74707	−0.492081
$189$	−2.71417	−0.197427
$190$	13.6361	0.989263
$191$	4.55381	0.329502	0.164751	−	0.986335i	$-0.447318\pi$
$191$	4.55381	0.329502	0.164751	+	0.986335i	$0.447318\pi$
$192$	28.8058	2.07888
$193$	3.41188	0.245592	0.122796	−	0.992432i	$-0.460814\pi$
$193$	3.41188	0.245592	0.122796	+	0.992432i	$0.460814\pi$
$194$	22.8675	1.64179
$195$	−4.89948	−0.350859
$196$	17.1455	1.22468
$197$	4.46638	0.318217	0.159108	−	0.987261i	$-0.449138\pi$
$197$	4.46638	0.318217	0.159108	+	0.987261i	$0.449138\pi$
$198$	−2.90970	−0.206783
$199$	−2.72287	−0.193019	−0.0965096	−	0.995332i	$-0.530768\pi$
$199$	−2.72287	−0.193019	−0.0965096	+	0.995332i	$0.530768\pi$
$200$	−3.83919	−0.271472
$201$	−10.9664	−0.773508
$202$	31.7869	2.23652
$203$	−4.63677	−0.325437
$204$	−53.8447	−3.76988
$205$	−3.75696	−0.262397
$206$	−0.555900	−0.0387314
$207$	8.37648	0.582205
$208$	−3.63014	−0.251705
$209$	2.12971	0.147315
$210$	−20.4091	−1.40836
$211$	8.89809	0.612570	0.306285	−	0.951940i	$-0.400914\pi$
$211$	8.89809	0.612570	0.306285	+	0.951940i	$0.400914\pi$
$212$	31.7260	2.17895
$213$	−0.865713	−0.0593177
$214$	−13.7931	−0.942876
$215$	3.28400	0.223967
$216$	3.04158	0.206953
$217$	−29.3204	−1.99040
$218$	24.7222	1.67440
$219$	4.97095	0.335906
$220$	−1.34009	−0.0903488
$221$	−11.5412	−0.776343
$222$	8.92714	0.599150
$223$	22.6156	1.51445	0.757227	−	0.653151i	$-0.226553\pi$
$223$	22.6156	1.51445	0.757227	+	0.653151i	$0.226553\pi$
$224$	11.1840	0.747263
$225$	3.31526	0.221017
$226$	21.8691	1.45471
$227$	−23.9709	−1.59101	−0.795504	−	0.605949i	$-0.792794\pi$
$227$	−23.9709	−1.59101	−0.795504	+	0.605949i	$0.792794\pi$
$228$	−52.3223	−3.46513
$229$	13.0423	0.861859	0.430930	−	0.902386i	$-0.358186\pi$
$229$	13.0423	0.861859	0.430930	+	0.902386i	$0.358186\pi$
$230$	5.98955	0.394939
$231$	−3.18753	−0.209724
$232$	5.19610	0.341141
$233$	18.5845	1.21751	0.608754	−	0.793359i	$-0.291670\pi$
$233$	18.5845	1.21751	0.608754	+	0.793359i	$0.291670\pi$
$234$	15.3222	1.00165
$235$	1.86407	0.121599
$236$	−4.33429	−0.282138
$237$	−12.1767	−0.790963
$238$	−48.0753	−3.11626
$239$	9.52080	0.615849	0.307925	−	0.951411i	$-0.400366\pi$
$239$	9.52080	0.615849	0.307925	+	0.951411i	$0.400366\pi$
$240$	4.67912	0.302036
$241$	−12.2541	−0.789354	−0.394677	−	0.918820i	$-0.629143\pi$
$241$	−12.2541	−0.789354	−0.394677	+	0.918820i	$0.629143\pi$
$242$	25.7512	1.65535
$243$	22.3674	1.43487
$244$	−50.8069	−3.25258
$245$	−4.73693	−0.302632
$246$	22.3811	1.42697
$247$	−11.2148	−0.713584
$248$	32.8573	2.08644
$249$	15.7935	1.00087
$250$	2.37056	0.149927
$251$	16.1863	1.02167	0.510835	−	0.859679i	$-0.329336\pi$
$251$	16.1863	1.02167	0.510835	+	0.859679i	$0.329336\pi$
$252$	41.1100	2.58968
$253$	0.935460	0.0588119
$254$	51.1899	3.21194
$255$	14.8761	0.931580
$256$	−26.0119	−1.62575
$257$	19.1440	1.19417	0.597085	−	0.802178i	$-0.296326\pi$
$257$	19.1440	1.19417	0.597085	+	0.802178i	$0.296326\pi$
$258$	−19.5636	−1.21798
$259$	5.13386	0.319002
$260$	7.05679	0.437644
$261$	−4.48699	−0.277738
$262$	50.3964	3.11350
$263$	−8.76520	−0.540485	−0.270243	−	0.962792i	$-0.587104\pi$
$263$	−8.76520	−0.540485	−0.270243	+	0.962792i	$0.587104\pi$
$264$	3.57204	0.219844
$265$	−8.76522	−0.538443
$266$	−46.7160	−2.86434
$267$	−9.77027	−0.597931
$268$	15.7950	0.964833
$269$	23.1670	1.41252	0.706259	−	0.707953i	$-0.250381\pi$
$269$	23.1670	1.41252	0.706259	+	0.707953i	$0.250381\pi$
$270$	−1.87806	−0.114295
$271$	−17.2427	−1.04742	−0.523710	−	0.851897i	$-0.675452\pi$
$271$	−17.2427	−1.04742	−0.523710	+	0.851897i	$0.675452\pi$
$272$	11.0221	0.668311
$273$	16.7852	1.01589
$274$	−27.1209	−1.63843
$275$	0.370238	0.0223262
$276$	−22.9822	−1.38337
$277$	7.31216	0.439345	0.219673	−	0.975574i	$-0.429501\pi$
$277$	7.31216	0.439345	0.219673	+	0.975574i	$0.429501\pi$
$278$	54.4019	3.26281
$279$	−28.3733	−1.69866
$280$	13.1528	0.786028
$281$	10.1150	0.603409	0.301705	−	0.953401i	$-0.402444\pi$
$281$	10.1150	0.603409	0.301705	+	0.953401i	$0.402444\pi$
$282$	−11.1047	−0.661277
$283$	12.9484	0.769705	0.384853	−	0.922978i	$-0.374252\pi$
$283$	12.9484	0.769705	0.384853	+	0.922978i	$0.374252\pi$
$284$	1.24690	0.0739898
$285$	14.4555	0.856272
$286$	1.71114	0.101182
$287$	12.8710	0.759754
$288$	10.8227	0.637736
$289$	18.0421	1.06130
$290$	−3.20840	−0.188404
$291$	24.2417	1.42107
$292$	−7.15973	−0.418991
$293$	18.9734	1.10844	0.554219	−	0.832371i	$-0.313017\pi$
$293$	18.9734	1.10844	0.554219	+	0.832371i	$0.313017\pi$
$294$	28.2191	1.64577
$295$	1.19747	0.0697195
$296$	−5.75316	−0.334396
$297$	−0.293319	−0.0170201
$298$	41.1939	2.38630
$299$	−4.92605	−0.284881
$300$	−9.09595	−0.525155
$301$	−11.2507	−0.648481
$302$	−7.76756	−0.446973
$303$	33.6972	1.93585
$304$	10.7104	0.614285
$305$	14.0369	0.803749
$306$	−46.5224	−2.65951
$307$	7.82938	0.446847	0.223423	−	0.974722i	$-0.428277\pi$
$307$	7.82938	0.446847	0.223423	+	0.974722i	$0.428277\pi$
$308$	4.59104	0.261599
$309$	−0.589308	−0.0335245
$310$	−20.2881	−1.15229
$311$	3.86523	0.219177	0.109589	−	0.993977i	$-0.465047\pi$
$311$	3.86523	0.219177	0.109589	+	0.993977i	$0.465047\pi$
$312$	−18.8101	−1.06491
$313$	−9.44463	−0.533842	−0.266921	−	0.963718i	$-0.586006\pi$
$313$	−9.44463	−0.533842	−0.266921	+	0.963718i	$0.586006\pi$
$314$	2.24336	0.126600
$315$	−11.3578	−0.639940
$316$	17.5383	0.986606
$317$	1.45014	0.0814480	0.0407240	−	0.999170i	$-0.487034\pi$
$317$	1.45014	0.0814480	0.0407240	+	0.999170i	$0.487034\pi$
$318$	52.2166	2.92816
$319$	−0.501094	−0.0280559
$320$	11.4626	0.640781
$321$	−14.6220	−0.816120
$322$	−20.5197	−1.14352
$323$	34.0513	1.89466
$324$	−28.7928	−1.59960
$325$	−1.94964	−0.108147
$326$	16.0052	0.886447
$327$	26.2079	1.44930
$328$	−14.4237	−0.796415
$329$	−6.38616	−0.352080
$330$	−2.20560	−0.121414
$331$	−4.55136	−0.250165	−0.125083	−	0.992146i	$-0.539920\pi$
$331$	−4.55136	−0.250165	−0.125083	+	0.992146i	$0.539920\pi$
$332$	−22.7476	−1.24844
$333$	4.96802	0.272246
$334$	20.6672	1.13086
$335$	−4.36382	−0.238421
$336$	−16.0303	−0.874524
$337$	9.63459	0.524829	0.262415	−	0.964955i	$-0.415481\pi$
$337$	9.63459	0.524829	0.262415	+	0.964955i	$0.415481\pi$
$338$	21.8065	1.18612
$339$	23.1834	1.25915
$340$	−21.4263	−1.16201
$341$	−3.16864	−0.171592
$342$	−45.2070	−2.44452
$343$	−7.75309	−0.418627
$344$	12.6079	0.679773
$345$	6.34950	0.341846
$346$	51.5668	2.77225
$347$	−8.84242	−0.474686	−0.237343	−	0.971426i	$-0.576276\pi$
$347$	−8.84242	−0.474686	−0.237343	+	0.971426i	$0.576276\pi$
$348$	12.3108	0.659928
$349$	−24.7464	−1.32464	−0.662322	−	0.749220i	$-0.730429\pi$
$349$	−24.7464	−1.32464	−0.662322	+	0.749220i	$0.730429\pi$
$350$	−8.12134	−0.434104
$351$	1.54459	0.0824443
$352$	1.20865	0.0644213
$353$	−27.2374	−1.44970	−0.724849	−	0.688908i	$-0.758091\pi$
$353$	−27.2374	−1.44970	−0.724849	+	0.688908i	$0.758091\pi$
$354$	−7.13363	−0.379148
$355$	−0.344492	−0.0182837
$356$	14.0723	0.745828
$357$	−50.9645	−2.69733
$358$	−54.0677	−2.85757
$359$	4.23070	0.223288	0.111644	−	0.993748i	$-0.464388\pi$
$359$	4.23070	0.223288	0.111644	+	0.993748i	$0.464388\pi$
$360$	12.7279	0.670820
$361$	14.0885	0.741501
$362$	0.665405	0.0349729
$363$	27.2987	1.43281
$364$	−24.1760	−1.26717
$365$	1.97808	0.103537
$366$	−83.6211	−4.37094
$367$	−12.1857	−0.636088	−0.318044	−	0.948076i	$-0.603026\pi$
$367$	−12.1857	−0.636088	−0.318044	+	0.948076i	$0.603026\pi$
$368$	4.70449	0.245239
$369$	12.4553	0.648396
$370$	3.55236	0.184678
$371$	30.0290	1.55903
$372$	77.8467	4.03616
$373$	−6.89227	−0.356868	−0.178434	−	0.983952i	$-0.557103\pi$
$373$	−6.89227	−0.356868	−0.178434	+	0.983952i	$0.557103\pi$
$374$	−5.19549	−0.268652
$375$	2.51302	0.129772
$376$	7.15653	0.369070
$377$	2.63872	0.135901
$378$	6.43409	0.330934
$379$	2.39095	0.122815	0.0614074	−	0.998113i	$-0.480441\pi$
$379$	2.39095	0.122815	0.0614074	+	0.998113i	$0.480441\pi$
$380$	−20.8205	−1.06807
$381$	54.2663	2.78015
$382$	−10.7951	−0.552323
$383$	9.82565	0.502067	0.251034	−	0.967978i	$-0.419230\pi$
$383$	9.82565	0.502067	0.251034	+	0.967978i	$0.419230\pi$
$384$	−51.8782	−2.64740
$385$	−1.26841	−0.0646440
$386$	−8.08804	−0.411671
$387$	−10.8873	−0.553433
$388$	−34.9157	−1.77257
$389$	−28.8174	−1.46110	−0.730551	−	0.682858i	$-0.760737\pi$
$389$	−28.8174	−1.46110	−0.730551	+	0.682858i	$0.760737\pi$
$390$	11.6145	0.588123
$391$	14.9568	0.756399
$392$	−18.1860	−0.918532
$393$	53.4250	2.69493
$394$	−10.5878	−0.533406
$395$	−4.84546	−0.243801
$396$	4.44274	0.223256
$397$	19.9158	0.999544	0.499772	−	0.866157i	$-0.333417\pi$
$397$	19.9158	0.999544	0.499772	+	0.866157i	$0.333417\pi$
$398$	6.45472	0.323546
$399$	−49.5235	−2.47928
$400$	1.86195	0.0930976
$401$	−11.9161	−0.595060	−0.297530	−	0.954712i	$-0.596163\pi$
$401$	−11.9161	−0.595060	−0.297530	+	0.954712i	$0.596163\pi$
$402$	25.9964	1.29658
$403$	16.6858	0.831179
$404$	−48.5346	−2.41468
$405$	7.95484	0.395279
$406$	10.9917	0.545510
$407$	0.554815	0.0275011
$408$	57.1124	2.82748
$409$	7.95173	0.393188	0.196594	−	0.980485i	$-0.437012\pi$
$409$	7.95173	0.393188	0.196594	+	0.980485i	$0.437012\pi$
$410$	8.90608	0.439840
$411$	−28.7508	−1.41817
$412$	0.848788	0.0418168
$413$	−4.10244	−0.201868
$414$	−19.8569	−0.975913
$415$	6.28468	0.308503
$416$	−6.36465	−0.312053
$417$	57.6712	2.82417
$418$	−5.04859	−0.246935
$419$	−13.0326	−0.636683	−0.318342	−	0.947976i	$-0.603126\pi$
$419$	−13.0326	−0.636683	−0.318342	+	0.947976i	$0.603126\pi$
$420$	31.1620	1.52055
$421$	−13.2376	−0.645162	−0.322581	−	0.946542i	$-0.604550\pi$
$421$	−13.2376	−0.645162	−0.322581	+	0.946542i	$0.604550\pi$
$422$	−21.0934	−1.02681
$423$	−6.17988	−0.300476
$424$	−33.6514	−1.63426
$425$	5.91964	0.287145
$426$	2.05222	0.0994304
$427$	−48.0892	−2.32720
$428$	21.0603	1.01799
$429$	1.81398	0.0875796
$430$	−7.78491	−0.375422
$431$	15.9241	0.767039	0.383519	−	0.923533i	$-0.374712\pi$
$431$	15.9241	0.767039	0.383519	+	0.923533i	$0.374712\pi$
$432$	−1.47512	−0.0709718
$433$	−1.90909	−0.0917448	−0.0458724	−	0.998947i	$-0.514607\pi$
$433$	−1.90909	−0.0917448	−0.0458724	+	0.998947i	$0.514607\pi$
$434$	69.5056	3.33637
$435$	−3.40121	−0.163076
$436$	−37.7476	−1.80778
$437$	14.5339	0.695252
$438$	−11.7839	−0.563057
$439$	28.5572	1.36296	0.681481	−	0.731836i	$-0.261336\pi$
$439$	28.5572	1.36296	0.681481	+	0.731836i	$0.261336\pi$
$440$	1.42142	0.0677633
$441$	15.7041	0.747817
$442$	27.3590	1.30133
$443$	−29.7858	−1.41517	−0.707583	−	0.706631i	$-0.750214\pi$
$443$	−29.7858	−1.41517	−0.707583	+	0.706631i	$0.750214\pi$
$444$	−13.6306	−0.646879
$445$	−3.88786	−0.184302
$446$	−53.6116	−2.53858
$447$	43.6695	2.06550
$448$	−39.2701	−1.85534
$449$	16.6280	0.784725	0.392363	−	0.919811i	$-0.371658\pi$
$449$	16.6280	0.784725	0.392363	+	0.919811i	$0.371658\pi$
$450$	−7.85900	−0.370477
$451$	1.39097	0.0654982
$452$	−33.3913	−1.57059
$453$	−8.23436	−0.386884
$454$	56.8244	2.66690
$455$	6.67932	0.313131
$456$	55.4976	2.59891
$457$	32.6766	1.52854	0.764272	−	0.644894i	$-0.223098\pi$
$457$	32.6766	1.52854	0.764272	+	0.644894i	$0.223098\pi$
$458$	−30.9175	−1.44468
$459$	−4.68980	−0.218901
$460$	−9.14527	−0.426401
$461$	−19.0662	−0.887999	−0.444000	−	0.896027i	$-0.646441\pi$
$461$	−19.0662	−0.887999	−0.444000	+	0.896027i	$0.646441\pi$
$462$	7.55621	0.351547
$463$	−11.0280	−0.512516	−0.256258	−	0.966608i	$-0.582490\pi$
$463$	−11.0280	−0.512516	−0.256258	+	0.966608i	$0.582490\pi$
$464$	−2.52003	−0.116990
$465$	−21.5074	−0.997381
$466$	−44.0555	−2.04083
$467$	−20.4601	−0.946783	−0.473391	−	0.880852i	$-0.656970\pi$
$467$	−20.4601	−0.946783	−0.473391	+	0.880852i	$0.656970\pi$
$468$	−23.3951	−1.08144
$469$	14.9501	0.690332
$470$	−4.41888	−0.203828
$471$	2.37817	0.109580
$472$	4.59732	0.211609
$473$	−1.21586	−0.0559054
$474$	28.8656	1.32584
$475$	5.75226	0.263932
$476$	73.4049	3.36451
$477$	29.0590	1.33052
$478$	−22.5696	−1.03231
$479$	6.82394	0.311794	0.155897	−	0.987773i	$-0.450173\pi$
$479$	6.82394	0.311794	0.155897	+	0.987773i	$0.450173\pi$
$480$	8.20381	0.374451
$481$	−2.92160	−0.133214
$482$	29.0489	1.32314
$483$	−21.7529	−0.989791
$484$	−39.3187	−1.78721
$485$	9.64645	0.438023
$486$	−53.0232	−2.40518
$487$	−12.4376	−0.563602	−0.281801	−	0.959473i	$-0.590932\pi$
$487$	−12.4376	−0.563602	−0.281801	+	0.959473i	$0.590932\pi$
$488$	53.8902	2.43950
$489$	16.9671	0.767277
$490$	11.2292	0.507282
$491$	−42.1512	−1.90226	−0.951130	−	0.308792i	$-0.900075\pi$
$491$	−42.1512	−1.90226	−0.951130	+	0.308792i	$0.900075\pi$
$492$	−34.1731	−1.54064
$493$	−8.01185	−0.360836
$494$	26.5854	1.19613
$495$	−1.22743	−0.0551691
$496$	−15.9353	−0.715517
$497$	1.18020	0.0529392
$498$	−37.4394	−1.67770
$499$	−13.3751	−0.598750	−0.299375	−	0.954136i	$-0.596778\pi$
$499$	−13.3751	−0.598750	−0.299375	+	0.954136i	$0.596778\pi$
$500$	−3.61953	−0.161870
$501$	21.9093	0.978834
$502$	−38.3705	−1.71256
$503$	39.9589	1.78168	0.890839	−	0.454320i	$-0.150118\pi$
$503$	39.9589	1.78168	0.890839	+	0.454320i	$0.150118\pi$
$504$	−43.6048	−1.94231
$505$	13.4091	0.596696
$506$	−2.21756	−0.0985826
$507$	23.1170	1.02666
$508$	−78.1605	−3.46781
$509$	−4.15940	−0.184362	−0.0921810	−	0.995742i	$-0.529384\pi$
$509$	−4.15940	−0.184362	−0.0921810	+	0.995742i	$0.529384\pi$
$510$	−35.2647	−1.56155
$511$	−6.77675	−0.299786
$512$	20.3752	0.900464
$513$	−4.55720	−0.201205
$514$	−45.3819	−2.00171
$515$	−0.234502	−0.0103334
$516$	29.8711	1.31500
$517$	−0.690151	−0.0303528
$518$	−12.1701	−0.534723
$519$	54.6658	2.39956
$520$	−7.48505	−0.328241
$521$	2.92480	0.128138	0.0640690	−	0.997945i	$-0.479592\pi$
$521$	2.92480	0.128138	0.0640690	+	0.997945i	$0.479592\pi$
$522$	10.6367	0.465554
$523$	−42.4665	−1.85693	−0.928464	−	0.371422i	$-0.878870\pi$
$523$	−42.4665	−1.85693	−0.928464	+	0.371422i	$0.878870\pi$
$524$	−76.9487	−3.36152
$525$	−8.60940	−0.375745
$526$	20.7784	0.905981
$527$	−50.6626	−2.20690
$528$	−1.73239	−0.0753925
$529$	−16.6161	−0.722438
$530$	20.7784	0.902558
$531$	−3.96992	−0.172280
$532$	71.3294	3.09252
$533$	−7.32472	−0.317269
$534$	23.1610	1.00227
$535$	−5.81850	−0.251556
$536$	−16.7536	−0.723643
$537$	−57.3170	−2.47341
$538$	−54.9187	−2.36771
$539$	1.75379	0.0755412
$540$	2.86756	0.123400
$541$	−20.1492	−0.866283	−0.433141	−	0.901326i	$-0.642595\pi$
$541$	−20.1492	−0.866283	−0.433141	+	0.901326i	$0.642595\pi$
$542$	40.8748	1.75572
$543$	0.705393	0.0302713
$544$	19.3248	0.828544
$545$	10.4288	0.446723
$546$	−39.7903	−1.70287
$547$	13.6079	0.581834	0.290917	−	0.956748i	$-0.406040\pi$
$547$	13.6079	0.581834	0.290917	+	0.956748i	$0.406040\pi$
$548$	41.4101	1.76895
$549$	−46.5358	−1.98610
$550$	−0.877670	−0.0374240
$551$	−7.78532	−0.331666
$552$	24.3770	1.03755
$553$	16.6001	0.705910
$554$	−17.3339	−0.736446
$555$	3.76584	0.159851
$556$	−83.0646	−3.52273
$557$	−22.2634	−0.943331	−0.471666	−	0.881777i	$-0.656347\pi$
$557$	−22.2634	−0.943331	−0.471666	+	0.881777i	$0.656347\pi$
$558$	67.2604	2.84736
$559$	6.40262	0.270802
$560$	−6.37890	−0.269558
$561$	−5.50772	−0.232536
$562$	−23.9781	−1.01146
$563$	−45.7223	−1.92696	−0.963482	−	0.267774i	$-0.913712\pi$
$563$	−45.7223	−1.92696	−0.963482	+	0.267774i	$0.913712\pi$
$564$	16.9555	0.713955
$565$	9.22531	0.388112
$566$	−30.6950	−1.29021
$567$	−27.2527	−1.14450
$568$	−1.32257	−0.0554938
$569$	6.12232	0.256661	0.128331	−	0.991731i	$-0.459038\pi$
$569$	6.12232	0.256661	0.128331	+	0.991731i	$0.459038\pi$
$570$	−34.2676	−1.43531
$571$	−5.93514	−0.248378	−0.124189	−	0.992259i	$-0.539633\pi$
$571$	−5.93514	−0.248378	−0.124189	+	0.992259i	$0.539633\pi$
$572$	−2.61269	−0.109242
$573$	−11.4438	−0.478072
$574$	−30.5115	−1.27353
$575$	2.52664	0.105368
$576$	−38.0016	−1.58340
$577$	−37.8002	−1.57364	−0.786822	−	0.617180i	$-0.788275\pi$
$577$	−37.8002	−1.57364	−0.786822	+	0.617180i	$0.788275\pi$
$578$	−42.7698	−1.77899
$579$	−8.57411	−0.356328
$580$	4.89881	0.203412
$581$	−21.5308	−0.893248
$582$	−57.4663	−2.38206
$583$	3.24522	0.134403
$584$	7.59423	0.314252
$585$	6.46356	0.267235
$586$	−44.9775	−1.85800
$587$	16.1743	0.667584	0.333792	−	0.942647i	$-0.391672\pi$
$587$	16.1743	0.667584	0.333792	+	0.942647i	$0.391672\pi$
$588$	−43.0869	−1.77687
$589$	−49.2301	−2.02849
$590$	−2.83867	−0.116866
$591$	−11.2241	−0.461698
$592$	2.79020	0.114676
$593$	20.1976	0.829417	0.414709	−	0.909954i	$-0.363883\pi$
$593$	20.1976	0.829417	0.414709	+	0.909954i	$0.363883\pi$
$594$	0.695330	0.0285297
$595$	−20.2802	−0.831407
$596$	−62.8978	−2.57639
$597$	6.84262	0.280050
$598$	11.6775	0.477528
$599$	−24.3832	−0.996269	−0.498134	−	0.867100i	$-0.665981\pi$
$599$	−24.3832	−0.996269	−0.498134	+	0.867100i	$0.665981\pi$
$600$	9.64796	0.393876
$601$	−2.65553	−0.108321	−0.0541606	−	0.998532i	$-0.517248\pi$
$601$	−2.65553	−0.108321	−0.0541606	+	0.998532i	$0.517248\pi$
$602$	26.6705	1.08701
$603$	14.4672	0.589150
$604$	11.8601	0.482579
$605$	10.8629	0.441641
$606$	−79.8811	−3.24495
$607$	−13.6750	−0.555052	−0.277526	−	0.960718i	$-0.589514\pi$
$607$	−13.6750	−0.555052	−0.277526	+	0.960718i	$0.589514\pi$
$608$	18.7784	0.761565
$609$	11.6523	0.472174
$610$	−33.2752	−1.34727
$611$	3.63427	0.147027
$612$	71.0337	2.87137
$613$	13.2648	0.535761	0.267881	−	0.963452i	$-0.413677\pi$
$613$	13.2648	0.535761	0.267881	+	0.963452i	$0.413677\pi$
$614$	−18.5600	−0.749020
$615$	9.44130	0.380710
$616$	−4.86966	−0.196204
$617$	29.6762	1.19472	0.597359	−	0.801974i	$-0.296217\pi$
$617$	29.6762	1.19472	0.597359	+	0.801974i	$0.296217\pi$
$618$	1.39699	0.0561950
$619$	−23.9740	−0.963596	−0.481798	−	0.876282i	$-0.660016\pi$
$619$	−23.9740	−0.963596	−0.481798	+	0.876282i	$0.660016\pi$
$620$	30.9774	1.24408
$621$	−2.00172	−0.0803263
$622$	−9.16275	−0.367393
$623$	13.3195	0.533635
$624$	9.12260	0.365196
$625$	1.00000	0.0400000
$626$	22.3890	0.894845
$627$	−5.35199	−0.213738
$628$	−3.42532	−0.136685
$629$	8.87078	0.353701
$630$	26.9243	1.07269
$631$	17.5211	0.697505	0.348753	−	0.937215i	$-0.386605\pi$
$631$	17.5211	0.697505	0.348753	+	0.937215i	$0.386605\pi$
$632$	−18.6026	−0.739973
$633$	−22.3611	−0.888772
$634$	−3.43764	−0.136526
$635$	21.5941	0.856935
$636$	−79.7280	−3.16142
$637$	−9.23532	−0.365917
$638$	1.18787	0.0470283
$639$	1.14208	0.0451799
$640$	−20.6438	−0.816017
$641$	−0.264889	−0.0104625	−0.00523125	−	0.999986i	$-0.501665\pi$
$641$	−0.264889	−0.0104625	−0.00523125	+	0.999986i	$0.501665\pi$
$642$	34.6623	1.36801
$643$	5.70138	0.224840	0.112420	−	0.993661i	$-0.464140\pi$
$643$	5.70138	0.224840	0.112420	+	0.993661i	$0.464140\pi$
$644$	31.3310	1.23461
$645$	−8.25275	−0.324952
$646$	−80.7205	−3.17590
$647$	1.58573	0.0623414	0.0311707	−	0.999514i	$-0.490076\pi$
$647$	1.58573	0.0623414	0.0311707	+	0.999514i	$0.490076\pi$
$648$	30.5402	1.19973
$649$	−0.443350	−0.0174030
$650$	4.62173	0.181279
$651$	73.6826	2.88785
$652$	−24.4379	−0.957062
$653$	−20.1690	−0.789272	−0.394636	−	0.918838i	$-0.629129\pi$
$653$	−20.1690	−0.789272	−0.394636	+	0.918838i	$0.629129\pi$
$654$	−62.1272	−2.42937
$655$	21.2593	0.830670
$656$	6.99528	0.273120
$657$	−6.55784	−0.255846
$658$	15.1387	0.590170
$659$	−16.5005	−0.642770	−0.321385	−	0.946949i	$-0.604148\pi$
$659$	−16.5005	−0.642770	−0.321385	+	0.946949i	$0.604148\pi$
$660$	3.36767	0.131086
$661$	−6.90302	−0.268496	−0.134248	−	0.990948i	$-0.542862\pi$
$661$	−6.90302	−0.268496	−0.134248	+	0.990948i	$0.542862\pi$
$662$	10.7893	0.419336
$663$	29.0031	1.12639
$664$	24.1281	0.936351
$665$	−19.7068	−0.764197
$666$	−11.7770	−0.456349
$667$	−3.41965	−0.132410
$668$	−31.5562	−1.22095
$669$	−56.8335	−2.19731
$670$	10.3447	0.399650
$671$	−5.19698	−0.200627
$672$	−28.1056	−1.08420
$673$	−50.8767	−1.96115	−0.980576	−	0.196138i	$-0.937160\pi$
$673$	−50.8767	−1.96115	−0.980576	+	0.196138i	$0.937160\pi$
$674$	−22.8393	−0.879738
$675$	−0.792245	−0.0304935
$676$	−33.2957	−1.28060
$677$	38.3112	1.47242	0.736210	−	0.676753i	$-0.236613\pi$
$677$	38.3112	1.47242	0.736210	+	0.676753i	$0.236613\pi$
$678$	−54.9574	−2.11063
$679$	−33.0480	−1.26827
$680$	22.7266	0.871526
$681$	60.2394	2.30838
$682$	7.51145	0.287628
$683$	2.42660	0.0928512	0.0464256	−	0.998922i	$-0.485217\pi$
$683$	2.42660	0.0928512	0.0464256	+	0.998922i	$0.485217\pi$
$684$	69.0253	2.63925
$685$	−11.4407	−0.437128
$686$	18.3791	0.701718
$687$	−32.7755	−1.25046
$688$	−6.11465	−0.233119
$689$	−17.0890	−0.651041
$690$	−15.0518	−0.573014
$691$	26.2652	0.999174	0.499587	−	0.866264i	$-0.333485\pi$
$691$	26.2652	0.999174	0.499587	+	0.866264i	$0.333485\pi$
$692$	−78.7359	−2.99309
$693$	4.20509	0.159738
$694$	20.9614	0.795685
$695$	22.9490	0.870505
$696$	−13.0579	−0.494958
$697$	22.2398	0.842394
$698$	58.6627	2.22041
$699$	−46.7030	−1.76647
$700$	12.4002	0.468685
$701$	29.9898	1.13270	0.566349	−	0.824166i	$-0.308356\pi$
$701$	29.9898	1.13270	0.566349	+	0.824166i	$0.308356\pi$
$702$	−3.66154	−0.138196
$703$	8.61996	0.325108
$704$	−4.24391	−0.159948
$705$	−4.68444	−0.176426
$706$	64.5677	2.43004
$707$	−45.9384	−1.72769
$708$	10.8921	0.409352
$709$	−10.1930	−0.382807	−0.191403	−	0.981511i	$-0.561304\pi$
$709$	−10.1930	−0.382807	−0.191403	+	0.981511i	$0.561304\pi$
$710$	0.816636	0.0306478
$711$	16.0639	0.602444
$712$	−14.9263	−0.559385
$713$	−21.6240	−0.809826
$714$	120.814	4.52136
$715$	0.721832	0.0269950
$716$	82.5544	3.08520
$717$	−23.9259	−0.893530
$718$	−10.0291	−0.374283
$719$	−15.9625	−0.595300	−0.297650	−	0.954675i	$-0.596203\pi$
$719$	−15.9625	−0.595300	−0.297650	+	0.954675i	$0.596203\pi$
$720$	−6.17285	−0.230048
$721$	0.803385	0.0299196
$722$	−33.3976	−1.24293
$723$	30.7947	1.14527
$724$	−1.01599	−0.0377589
$725$	−1.35344	−0.0502654
$726$	−64.7131	−2.40173
$727$	36.8282	1.36588	0.682941	−	0.730473i	$-0.260701\pi$
$727$	36.8282	1.36588	0.682941	+	0.730473i	$0.260701\pi$
$728$	25.6432	0.950399
$729$	−32.3451	−1.19797
$730$	−4.68915	−0.173553
$731$	−19.4401	−0.719018
$732$	127.679	4.71914
$733$	28.5043	1.05283	0.526414	−	0.850228i	$-0.323536\pi$
$733$	28.5043	1.05283	0.526414	+	0.850228i	$0.323536\pi$
$734$	28.8869	1.06623
$735$	11.9040	0.439086
$736$	8.24829	0.304036
$737$	1.61565	0.0595134
$738$	−29.5259	−1.08686
$739$	−34.8728	−1.28282	−0.641409	−	0.767199i	$-0.721650\pi$
$739$	−34.8728	−1.28282	−0.641409	+	0.767199i	$0.721650\pi$
$740$	−5.42399	−0.199390
$741$	28.1831	1.03533
$742$	−71.1853	−2.61330
$743$	41.0543	1.50614	0.753068	−	0.657943i	$-0.228573\pi$
$743$	41.0543	1.50614	0.753068	+	0.657943i	$0.228573\pi$
$744$	−82.5710	−3.02720
$745$	17.3773	0.636656
$746$	16.3385	0.598195
$747$	−20.8353	−0.762324
$748$	7.93284	0.290053
$749$	19.9337	0.728363
$750$	−5.95725	−0.217528
$751$	45.5207	1.66107	0.830537	−	0.556963i	$-0.188034\pi$
$751$	45.5207	1.66107	0.830537	+	0.556963i	$0.188034\pi$
$752$	−3.47081	−0.126567
$753$	−40.6765	−1.48233
$754$	−6.25522	−0.227802
$755$	−3.27668	−0.119251
$756$	−9.82402	−0.357296
$757$	−19.8188	−0.720328	−0.360164	−	0.932889i	$-0.617279\pi$
$757$	−19.8188	−0.720328	−0.360164	+	0.932889i	$0.617279\pi$
$758$	−5.66787	−0.205866
$759$	−2.35083	−0.0853297
$760$	22.0840	0.801072
$761$	19.2599	0.698170	0.349085	−	0.937091i	$-0.386492\pi$
$761$	19.2599	0.698170	0.349085	+	0.937091i	$0.386492\pi$
$762$	−128.641	−4.66018
$763$	−35.7284	−1.29346
$764$	16.4827	0.596322
$765$	−19.6251	−0.709547
$766$	−23.2922	−0.841583
$767$	2.33464	0.0842989
$768$	65.3684	2.35878
$769$	26.6747	0.961914	0.480957	−	0.876744i	$-0.340289\pi$
$769$	26.6747	0.961914	0.480957	+	0.876744i	$0.340289\pi$
$770$	3.00683	0.108359
$771$	−48.1092	−1.73261
$772$	12.3494	0.444465
$773$	18.8509	0.678021	0.339011	−	0.940783i	$-0.389908\pi$
$773$	18.8509	0.678021	0.339011	+	0.940783i	$0.389908\pi$
$774$	25.8090	0.927684
$775$	−8.55839	−0.307427
$776$	37.0346	1.32946
$777$	−12.9015	−0.462838
$778$	68.3134	2.44915
$779$	21.6110	0.774295
$780$	−17.7338	−0.634973
$781$	0.127544	0.00456388
$782$	−35.4560	−1.26790
$783$	1.07225	0.0383192
$784$	8.81994	0.314998
$785$	0.946342	0.0337764
$786$	−126.647	−4.51735
$787$	34.0938	1.21531	0.607656	−	0.794201i	$-0.292110\pi$
$787$	34.0938	1.21531	0.607656	+	0.794201i	$0.292110\pi$
$788$	16.1662	0.575898
$789$	22.0271	0.784185
$790$	11.4864	0.408668
$791$	−31.6052	−1.12375
$792$	−4.71236	−0.167446
$793$	27.3668	0.971825
$794$	−47.2114	−1.67547
$795$	22.0272	0.781223
$796$	−9.85552	−0.349320
$797$	16.3336	0.578566	0.289283	−	0.957244i	$-0.406583\pi$
$797$	16.3336	0.578566	0.289283	+	0.957244i	$0.406583\pi$
$798$	117.398	4.15585
$799$	−11.0346	−0.390377
$800$	3.26452	0.115418
$801$	12.8893	0.455420
$802$	28.2477	0.997461
$803$	−0.732361	−0.0258445
$804$	−39.6931	−1.39987
$805$	−8.65609	−0.305087
$806$	−39.5546	−1.39325
$807$	−58.2192	−2.04941
$808$	51.4800	1.81106
$809$	18.5735	0.653008	0.326504	−	0.945196i	$-0.394129\pi$
$809$	18.5735	0.653008	0.326504	+	0.945196i	$0.394129\pi$
$810$	−18.8574	−0.662581
$811$	55.2711	1.94083	0.970416	−	0.241440i	$-0.0776196\pi$
$811$	55.2711	1.94083	0.970416	+	0.241440i	$0.0776196\pi$
$812$	−16.7829	−0.588965
$813$	43.3312	1.51969
$814$	−1.31522	−0.0460984
$815$	6.75167	0.236501
$816$	−27.6987	−0.969647
$817$	−18.8904	−0.660893
$818$	−18.8500	−0.659076
$819$	−22.1436	−0.773762
$820$	−13.5984	−0.474878
$821$	3.21404	0.112171	0.0560853	−	0.998426i	$-0.482138\pi$
$821$	3.21404	0.112171	0.0560853	+	0.998426i	$0.482138\pi$
$822$	68.1553	2.37719
$823$	26.7318	0.931812	0.465906	−	0.884834i	$-0.345728\pi$
$823$	26.7318	0.931812	0.465906	+	0.884834i	$0.345728\pi$
$824$	−0.900298	−0.0313634
$825$	−0.930415	−0.0323929
$826$	9.72506	0.338378
$827$	−14.6083	−0.507982	−0.253991	−	0.967207i	$-0.581743\pi$
$827$	−14.6083	−0.507982	−0.253991	+	0.967207i	$0.581743\pi$
$828$	30.3189	1.05366
$829$	5.46813	0.189916	0.0949579	−	0.995481i	$-0.469728\pi$
$829$	5.46813	0.189916	0.0949579	+	0.995481i	$0.469728\pi$
$830$	−14.8982	−0.517123
$831$	−18.3756	−0.637442
$832$	22.3480	0.774779
$833$	28.0409	0.971560
$834$	−136.713	−4.73398
$835$	8.71831	0.301710
$836$	7.70854	0.266606
$837$	6.78034	0.234363
$838$	30.8945	1.06723
$839$	−3.61886	−0.124937	−0.0624684	−	0.998047i	$-0.519897\pi$
$839$	−3.61886	−0.124937	−0.0624684	+	0.998047i	$0.519897\pi$
$840$	−33.0531	−1.14044
$841$	−27.1682	−0.936835
$842$	31.3805	1.08144
$843$	−25.4191	−0.875481
$844$	32.2069	1.10861
$845$	9.19890	0.316452
$846$	14.6497	0.503668
$847$	−37.2155	−1.27874
$848$	16.3204	0.560446
$849$	−32.5397	−1.11676
$850$	−14.0328	−0.481322
$851$	3.78626	0.129791
$852$	−3.13348	−0.107351
$853$	−17.5765	−0.601808	−0.300904	−	0.953654i	$-0.597288\pi$
$853$	−17.5765	−0.601808	−0.300904	+	0.953654i	$0.597288\pi$
$854$	113.998	3.90093
$855$	−19.0702	−0.652188
$856$	−22.3383	−0.763509
$857$	−51.4716	−1.75824	−0.879118	−	0.476604i	$-0.841868\pi$
$857$	−51.4716	−1.75824	−0.879118	+	0.476604i	$0.841868\pi$
$858$	−4.30013	−0.146804
$859$	−17.6332	−0.601638	−0.300819	−	0.953681i	$-0.597260\pi$
$859$	−17.6332	−0.601638	−0.300819	+	0.953681i	$0.597260\pi$
$860$	11.8866	0.405328
$861$	−32.3452	−1.10232
$862$	−37.7490	−1.28574
$863$	4.62276	0.157361	0.0786804	−	0.996900i	$-0.474929\pi$
$863$	4.62276	0.157361	0.0786804	+	0.996900i	$0.474929\pi$
$864$	−2.58630	−0.0879878
$865$	21.7531	0.739626
$866$	4.52559	0.153786
$867$	−45.3401	−1.53983
$868$	−106.126	−3.60215
$869$	1.79397	0.0608564
$870$	8.06276	0.273353
$871$	−8.50789	−0.288279
$872$	40.0384	1.35587
$873$	−31.9805	−1.08237
$874$	−34.4535	−1.16541
$875$	−3.42592	−0.115817
$876$	17.9925	0.607911
$877$	−29.0171	−0.979838	−0.489919	−	0.871768i	$-0.662974\pi$
$877$	−29.0171	−0.979838	−0.489919	+	0.871768i	$0.662974\pi$
$878$	−67.6965	−2.28465
$879$	−47.6805	−1.60822
$880$	−0.689366	−0.0232385
$881$	−30.4760	−1.02676	−0.513380	−	0.858161i	$-0.671607\pi$
$881$	−30.4760	−1.02676	−0.513380	+	0.858161i	$0.671607\pi$
$882$	−37.2276	−1.25352
$883$	6.52528	0.219593	0.109797	−	0.993954i	$-0.464980\pi$
$883$	6.52528	0.219593	0.109797	+	0.993954i	$0.464980\pi$
$884$	−41.7736	−1.40500
$885$	−3.00927	−0.101155
$886$	70.6089	2.37215
$887$	39.7867	1.33591	0.667954	−	0.744203i	$-0.267170\pi$
$887$	39.7867	1.33591	0.667954	+	0.744203i	$0.267170\pi$
$888$	14.4578	0.485172
$889$	−73.9796	−2.48120
$890$	9.21640	0.308934
$891$	−2.94519	−0.0986675
$892$	81.8580	2.74081
$893$	−10.7226	−0.358819
$894$	−103.521	−3.46226
$895$	−22.8080	−0.762388
$896$	70.7239	2.36272
$897$	12.3793	0.413331
$898$	−39.4177	−1.31538
$899$	11.5832	0.386323
$900$	11.9997	0.399989
$901$	51.8869	1.72860
$902$	−3.29737	−0.109790
$903$	28.2733	0.940876
$904$	35.4177	1.17798
$905$	0.280696	0.00933064
$906$	19.5200	0.648509
$907$	−28.4354	−0.944180	−0.472090	−	0.881550i	$-0.656500\pi$
$907$	−28.4354	−0.944180	−0.472090	+	0.881550i	$0.656500\pi$
$908$	−86.7636	−2.87935
$909$	−44.4545	−1.47446
$910$	−15.8337	−0.524882
$911$	−21.0331	−0.696858	−0.348429	−	0.937335i	$-0.613285\pi$
$911$	−21.0331	−0.696858	−0.348429	+	0.937335i	$0.613285\pi$
$912$	−26.9155	−0.891261
$913$	−2.32683	−0.0770068
$914$	−77.4616	−2.56220
$915$	−35.2749	−1.16615
$916$	47.2070	1.55976
$917$	−72.8327	−2.40515
$918$	11.1174	0.366930
$919$	−2.46226	−0.0812224	−0.0406112	−	0.999175i	$-0.512930\pi$
$919$	−2.46226	−0.0812224	−0.0406112	+	0.999175i	$0.512930\pi$
$920$	9.70027	0.319809
$921$	−19.6754	−0.648326
$922$	45.1974	1.48850
$923$	−0.671635	−0.0221071
$924$	−11.5374	−0.379551
$925$	1.49853	0.0492715
$926$	26.1425	0.859097
$927$	0.777434	0.0255343
$928$	−4.41833	−0.145039
$929$	10.6740	0.350201	0.175100	−	0.984551i	$-0.443975\pi$
$929$	10.6740	0.350201	0.175100	+	0.984551i	$0.443975\pi$
$930$	50.9845	1.67185
$931$	27.2481	0.893020
$932$	67.2670	2.20340
$933$	−9.71340	−0.318002
$934$	48.5019	1.58703
$935$	−2.19168	−0.0716754
$936$	24.8148	0.811099
$937$	−25.6731	−0.838703	−0.419351	−	0.907824i	$-0.637742\pi$
$937$	−25.6731	−0.838703	−0.419351	+	0.907824i	$0.637742\pi$
$938$	−35.4401	−1.15716
$939$	23.7345	0.774547
$940$	6.74707	0.220065
$941$	19.5429	0.637080	0.318540	−	0.947909i	$-0.396807\pi$
$941$	19.5429	0.637080	0.318540	+	0.947909i	$0.396807\pi$
$942$	−5.63759	−0.183683
$943$	9.49250	0.309118
$944$	−2.22963	−0.0725684
$945$	2.71417	0.0882919
$946$	2.88227	0.0937107
$947$	−21.3475	−0.693700	−0.346850	−	0.937921i	$-0.612749\pi$
$947$	−21.3475	−0.693700	−0.346850	+	0.937921i	$0.612749\pi$
$948$	−44.0740	−1.43146
$949$	3.85655	0.125189
$950$	−13.6361	−0.442412
$951$	−3.64423	−0.118172
$952$	−77.8596	−2.52344
$953$	−46.6927	−1.51253	−0.756263	−	0.654268i	$-0.772977\pi$
$953$	−46.6927	−1.51253	−0.756263	+	0.654268i	$0.772977\pi$
$954$	−68.8859	−2.23026
$955$	−4.55381	−0.147358
$956$	34.4608	1.11454
$957$	1.25926	0.0407060
$958$	−16.1765	−0.522640
$959$	39.1950	1.26567
$960$	−28.8058	−0.929704
$961$	42.2461	1.36278
$962$	6.92582	0.223298
$963$	19.2898	0.621606
$964$	−44.3540	−1.42855
$965$	−3.41188	−0.109832
$966$	51.5664	1.65912
$967$	−18.0180	−0.579420	−0.289710	−	0.957115i	$-0.593559\pi$
$967$	−18.0180	−0.579420	−0.289710	+	0.957115i	$0.593559\pi$
$968$	41.7048	1.34044
$969$	−85.5715	−2.74895
$970$	−22.8675	−0.734230
$971$	−33.0388	−1.06027	−0.530133	−	0.847915i	$-0.677858\pi$
$971$	−33.0388	−1.06027	−0.530133	+	0.847915i	$0.677858\pi$
$972$	80.9595	2.59678
$973$	−78.6214	−2.52049
$974$	29.4841	0.944730
$975$	4.89948	0.156909
$976$	−26.1360	−0.836592
$977$	9.16103	0.293087	0.146544	−	0.989204i	$-0.453185\pi$
$977$	9.16103	0.293087	0.146544	+	0.989204i	$0.453185\pi$
$978$	−40.2214	−1.28614
$979$	1.43944	0.0460046
$980$	−17.1455	−0.547692
$981$	−34.5743	−1.10387
$982$	99.9219	3.18863
$983$	−32.6515	−1.04142	−0.520710	−	0.853733i	$-0.674333\pi$
$983$	−32.6515	−1.04142	−0.520710	+	0.853733i	$0.674333\pi$
$984$	36.2470	1.15551
$985$	−4.46638	−0.142311
$986$	18.9925	0.604846
$987$	16.0485	0.510831
$988$	−40.5925	−1.29142
$989$	−8.29751	−0.263845
$990$	2.90970	0.0924764
$991$	59.9890	1.90561	0.952807	−	0.303577i	$-0.0981810\pi$
$991$	59.9890	1.90561	0.952807	+	0.303577i	$0.0981810\pi$
$992$	−27.9391	−0.887067
$993$	11.4377	0.362963
$994$	−2.79773	−0.0887386
$995$	2.72287	0.0863208
$996$	57.1651	1.81135
$997$	40.5874	1.28542	0.642708	−	0.766111i	$-0.277811\pi$
$997$	40.5874	1.28542	0.642708	+	0.766111i	$0.277811\pi$
$998$	31.7063	1.00365
$999$	−1.18721	−0.0375615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.b.1.12	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.b.1.12	✓	126	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)

\(f(q)\)	\(=\)	\(q-2.37056 q^{2} -2.51302 q^{3} +3.61953 q^{4} -1.00000 q^{5} +5.95725 q^{6} +3.42592 q^{7} -3.83919 q^{8} +3.31526 q^{9} +O(q^{10})\)	\(q-2.37056 q^{2} -2.51302 q^{3} +3.61953 q^{4} -1.00000 q^{5} +5.95725 q^{6} +3.42592 q^{7} -3.83919 q^{8} +3.31526 q^{9} +2.37056 q^{10} +0.370238 q^{11} -9.09595 q^{12} -1.94964 q^{13} -8.12134 q^{14} +2.51302 q^{15} +1.86195 q^{16} +5.91964 q^{17} -7.85900 q^{18} +5.75226 q^{19} -3.61953 q^{20} -8.60940 q^{21} -0.877670 q^{22} +2.52664 q^{23} +9.64796 q^{24} +1.00000 q^{25} +4.62173 q^{26} -0.792245 q^{27} +12.4002 q^{28} -1.35344 q^{29} -5.95725 q^{30} -8.55839 q^{31} +3.26452 q^{32} -0.930415 q^{33} -14.0328 q^{34} -3.42592 q^{35} +11.9997 q^{36} +1.49853 q^{37} -13.6361 q^{38} +4.89948 q^{39} +3.83919 q^{40} +3.75696 q^{41} +20.4091 q^{42} -3.28400 q^{43} +1.34009 q^{44} -3.31526 q^{45} -5.98955 q^{46} -1.86407 q^{47} -4.67912 q^{48} +4.73693 q^{49} -2.37056 q^{50} -14.8761 q^{51} -7.05679 q^{52} +8.76522 q^{53} +1.87806 q^{54} -0.370238 q^{55} -13.1528 q^{56} -14.4555 q^{57} +3.20840 q^{58} -1.19747 q^{59} +9.09595 q^{60} -14.0369 q^{61} +20.2881 q^{62} +11.3578 q^{63} -11.4626 q^{64} +1.94964 q^{65} +2.20560 q^{66} +4.36382 q^{67} +21.4263 q^{68} -6.34950 q^{69} +8.12134 q^{70} +0.344492 q^{71} -12.7279 q^{72} -1.97808 q^{73} -3.55236 q^{74} -2.51302 q^{75} +20.8205 q^{76} +1.26841 q^{77} -11.6145 q^{78} +4.84546 q^{79} -1.86195 q^{80} -7.95484 q^{81} -8.90608 q^{82} -6.28468 q^{83} -31.1620 q^{84} -5.91964 q^{85} +7.78491 q^{86} +3.40121 q^{87} -1.42142 q^{88} +3.88786 q^{89} +7.85900 q^{90} -6.67932 q^{91} +9.14527 q^{92} +21.5074 q^{93} +4.41888 q^{94} -5.75226 q^{95} -8.20381 q^{96} -9.64645 q^{97} -11.2292 q^{98} +1.22743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

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