LMFDB - Embedded newform 1045.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 28x^4 - 17x^3 - 28x^2 + 6*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.57959$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57959 q^{2} -2.53274 q^{3} +4.65426 q^{4} +1.00000 q^{5} +6.53343 q^{6} +2.87748 q^{7} -6.84689 q^{8} +3.41479 q^{9} +O(q^{10})$	\(q-2.57959 q^{2} -2.53274 q^{3} +4.65426 q^{4} +1.00000 q^{5} +6.53343 q^{6} +2.87748 q^{7} -6.84689 q^{8} +3.41479 q^{9} -2.57959 q^{10} +1.00000 q^{11} -11.7880 q^{12} -3.24663 q^{13} -7.42271 q^{14} -2.53274 q^{15} +8.35361 q^{16} -2.18913 q^{17} -8.80874 q^{18} -1.00000 q^{19} +4.65426 q^{20} -7.28793 q^{21} -2.57959 q^{22} -4.80351 q^{23} +17.3414 q^{24} +1.00000 q^{25} +8.37495 q^{26} -1.05055 q^{27} +13.3926 q^{28} -7.17277 q^{29} +6.53343 q^{30} +4.44632 q^{31} -7.85508 q^{32} -2.53274 q^{33} +5.64705 q^{34} +2.87748 q^{35} +15.8933 q^{36} +3.55048 q^{37} +2.57959 q^{38} +8.22288 q^{39} -6.84689 q^{40} +4.09066 q^{41} +18.7998 q^{42} -4.31844 q^{43} +4.65426 q^{44} +3.41479 q^{45} +12.3911 q^{46} -5.27433 q^{47} -21.1576 q^{48} +1.27991 q^{49} -2.57959 q^{50} +5.54451 q^{51} -15.1107 q^{52} +2.40580 q^{53} +2.70999 q^{54} +1.00000 q^{55} -19.7018 q^{56} +2.53274 q^{57} +18.5028 q^{58} -0.644769 q^{59} -11.7880 q^{60} +14.9985 q^{61} -11.4697 q^{62} +9.82600 q^{63} +3.55562 q^{64} -3.24663 q^{65} +6.53343 q^{66} -0.802158 q^{67} -10.1888 q^{68} +12.1661 q^{69} -7.42271 q^{70} -14.6459 q^{71} -23.3807 q^{72} -5.51740 q^{73} -9.15875 q^{74} -2.53274 q^{75} -4.65426 q^{76} +2.87748 q^{77} -21.2116 q^{78} -4.11472 q^{79} +8.35361 q^{80} -7.58358 q^{81} -10.5522 q^{82} -2.66797 q^{83} -33.9199 q^{84} -2.18913 q^{85} +11.1398 q^{86} +18.1668 q^{87} -6.84689 q^{88} -0.707380 q^{89} -8.80874 q^{90} -9.34212 q^{91} -22.3568 q^{92} -11.2614 q^{93} +13.6056 q^{94} -1.00000 q^{95} +19.8949 q^{96} +15.2075 q^{97} -3.30163 q^{98} +3.41479 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * 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 $

$2$	−2.57959	−1.82404	−0.912021	−	0.410143i	$-0.865479\pi$
$2$	−2.57959	−1.82404	−0.912021	+	0.410143i	$0.865479\pi$
$3$	−2.53274	−1.46228	−0.731140	−	0.682227i	$-0.761011\pi$
$3$	−2.53274	−1.46228	−0.731140	+	0.682227i	$0.761011\pi$
$4$	4.65426	2.32713
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	6.53343	2.66726
$7$	2.87748	1.08759	0.543793	−	0.839219i	$-0.316988\pi$
$7$	2.87748	1.08759	0.543793	+	0.839219i	$0.316988\pi$
$8$	−6.84689	−2.42074
$9$	3.41479	1.13826
$10$	−2.57959	−0.815736
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−11.7880	−3.40292
$13$	−3.24663	−0.900453	−0.450226	−	0.892914i	$-0.648657\pi$
$13$	−3.24663	−0.900453	−0.450226	+	0.892914i	$0.648657\pi$
$14$	−7.42271	−1.98380
$15$	−2.53274	−0.653952
$16$	8.35361	2.08840
$17$	−2.18913	−0.530942	−0.265471	−	0.964119i	$-0.585527\pi$
$17$	−2.18913	−0.530942	−0.265471	+	0.964119i	$0.585527\pi$
$18$	−8.80874	−2.07624
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.65426	1.04072
$21$	−7.28793	−1.59036
$22$	−2.57959	−0.549969
$23$	−4.80351	−1.00160	−0.500801	−	0.865563i	$-0.666961\pi$
$23$	−4.80351	−1.00160	−0.500801	+	0.865563i	$0.666961\pi$
$24$	17.3414	3.53980
$25$	1.00000	0.200000
$26$	8.37495	1.64246
$27$	−1.05055	−0.202179
$28$	13.3926	2.53095
$29$	−7.17277	−1.33195	−0.665975	−	0.745974i	$-0.731984\pi$
$29$	−7.17277	−1.33195	−0.665975	+	0.745974i	$0.731984\pi$
$30$	6.53343	1.19284
$31$	4.44632	0.798582	0.399291	−	0.916824i	$-0.369256\pi$
$31$	4.44632	0.798582	0.399291	+	0.916824i	$0.369256\pi$
$32$	−7.85508	−1.38859
$33$	−2.53274	−0.440894
$34$	5.64705	0.968461
$35$	2.87748	0.486383
$36$	15.8933	2.64889
$37$	3.55048	0.583695	0.291847	−	0.956465i	$-0.405730\pi$
$37$	3.55048	0.583695	0.291847	+	0.956465i	$0.405730\pi$
$38$	2.57959	0.418464
$39$	8.22288	1.31671
$40$	−6.84689	−1.08259
$41$	4.09066	0.638853	0.319427	−	0.947611i	$-0.396510\pi$
$41$	4.09066	0.638853	0.319427	+	0.947611i	$0.396510\pi$
$42$	18.7998	2.90088
$43$	−4.31844	−0.658557	−0.329278	−	0.944233i	$-0.606805\pi$
$43$	−4.31844	−0.658557	−0.329278	+	0.944233i	$0.606805\pi$
$44$	4.65426	0.701656
$45$	3.41479	0.509047
$46$	12.3911	1.82696
$47$	−5.27433	−0.769340	−0.384670	−	0.923054i	$-0.625685\pi$
$47$	−5.27433	−0.769340	−0.384670	+	0.923054i	$0.625685\pi$
$48$	−21.1576	−3.05383
$49$	1.27991	0.182844
$50$	−2.57959	−0.364808
$51$	5.54451	0.776386
$52$	−15.1107	−2.09547
$53$	2.40580	0.330462	0.165231	−	0.986255i	$-0.447163\pi$
$53$	2.40580	0.330462	0.165231	+	0.986255i	$0.447163\pi$
$54$	2.70999	0.368783
$55$	1.00000	0.134840
$56$	−19.7018	−2.63276
$57$	2.53274	0.335470
$58$	18.5028	2.42953
$59$	−0.644769	−0.0839418	−0.0419709	−	0.999119i	$-0.513364\pi$
$59$	−0.644769	−0.0839418	−0.0419709	+	0.999119i	$0.513364\pi$
$60$	−11.7880	−1.52183
$61$	14.9985	1.92035	0.960177	−	0.279391i	$-0.0901325\pi$
$61$	14.9985	1.92035	0.960177	+	0.279391i	$0.0901325\pi$
$62$	−11.4697	−1.45665
$63$	9.82600	1.23796
$64$	3.55562	0.444452
$65$	−3.24663	−0.402695
$66$	6.53343	0.804209
$67$	−0.802158	−0.0979992	−0.0489996	−	0.998799i	$-0.515603\pi$
$67$	−0.802158	−0.0979992	−0.0489996	+	0.998799i	$0.515603\pi$
$68$	−10.1888	−1.23557
$69$	12.1661	1.46462
$70$	−7.42271	−0.887184
$71$	−14.6459	−1.73815	−0.869073	−	0.494683i	$-0.835284\pi$
$71$	−14.6459	−1.73815	−0.869073	+	0.494683i	$0.835284\pi$
$72$	−23.3807	−2.75544
$73$	−5.51740	−0.645762	−0.322881	−	0.946439i	$-0.604651\pi$
$73$	−5.51740	−0.645762	−0.322881	+	0.946439i	$0.604651\pi$
$74$	−9.15875	−1.06468
$75$	−2.53274	−0.292456
$76$	−4.65426	−0.533880
$77$	2.87748	0.327920
$78$	−21.2116	−2.40174
$79$	−4.11472	−0.462942	−0.231471	−	0.972842i	$-0.574354\pi$
$79$	−4.11472	−0.462942	−0.231471	+	0.972842i	$0.574354\pi$
$80$	8.35361	0.933962
$81$	−7.58358	−0.842620
$82$	−10.5522	−1.16530
$83$	−2.66797	−0.292848	−0.146424	−	0.989222i	$-0.546776\pi$
$83$	−2.66797	−0.292848	−0.146424	+	0.989222i	$0.546776\pi$
$84$	−33.9199	−3.70096
$85$	−2.18913	−0.237445
$86$	11.1398	1.20123
$87$	18.1668	1.94768
$88$	−6.84689	−0.729881
$89$	−0.707380	−0.0749822	−0.0374911	−	0.999297i	$-0.511937\pi$
$89$	−0.707380	−0.0749822	−0.0374911	+	0.999297i	$0.511937\pi$
$90$	−8.80874	−0.928523
$91$	−9.34212	−0.979320
$92$	−22.3568	−2.33086
$93$	−11.2614	−1.16775
$94$	13.6056	1.40331
$95$	−1.00000	−0.102598
$96$	19.8949	2.03051
$97$	15.2075	1.54409	0.772044	−	0.635569i	$-0.219234\pi$
$97$	15.2075	1.54409	0.772044	+	0.635569i	$0.219234\pi$
$98$	−3.30163	−0.333515
$99$	3.41479	0.343199
$100$	4.65426	0.465426
$101$	−19.6872	−1.95895	−0.979474	−	0.201571i	$-0.935395\pi$
$101$	−19.6872	−1.95895	−0.979474	+	0.201571i	$0.935395\pi$
$102$	−14.3025	−1.41616
$103$	−3.23128	−0.318387	−0.159194	−	0.987247i	$-0.550889\pi$
$103$	−3.23128	−0.318387	−0.159194	+	0.987247i	$0.550889\pi$
$104$	22.2293	2.17976
$105$	−7.28793	−0.711229
$106$	−6.20596	−0.602776
$107$	−8.40655	−0.812692	−0.406346	−	0.913719i	$-0.633197\pi$
$107$	−8.40655	−0.812692	−0.406346	+	0.913719i	$0.633197\pi$
$108$	−4.88955	−0.470497
$109$	−7.18301	−0.688008	−0.344004	−	0.938968i	$-0.611783\pi$
$109$	−7.18301	−0.688008	−0.344004	+	0.938968i	$0.611783\pi$
$110$	−2.57959	−0.245954
$111$	−8.99244	−0.853525
$112$	24.0374	2.27132
$113$	16.8313	1.58335	0.791676	−	0.610942i	$-0.209209\pi$
$113$	16.8313	1.58335	0.791676	+	0.610942i	$0.209209\pi$
$114$	−6.53343	−0.611912
$115$	−4.80351	−0.447930
$116$	−33.3839	−3.09962
$117$	−11.0866	−1.02495
$118$	1.66324	0.153113
$119$	−6.29919	−0.577445
$120$	17.3414	1.58305
$121$	1.00000	0.0909091
$122$	−38.6898	−3.50281
$123$	−10.3606	−0.934183
$124$	20.6943	1.85840
$125$	1.00000	0.0894427
$126$	−25.3470	−2.25809
$127$	0.143673	0.0127489	0.00637446	−	0.999980i	$-0.497971\pi$
$127$	0.143673	0.0127489	0.00637446	+	0.999980i	$0.497971\pi$
$128$	6.53814	0.577895
$129$	10.9375	0.962994
$130$	8.37495	0.734532
$131$	−22.7411	−1.98690	−0.993448	−	0.114283i	$-0.963543\pi$
$131$	−22.7411	−1.98690	−0.993448	+	0.114283i	$0.963543\pi$
$132$	−11.7880	−1.02602
$133$	−2.87748	−0.249509
$134$	2.06923	0.178755
$135$	−1.05055	−0.0904173
$136$	14.9887	1.28527
$137$	14.4751	1.23669	0.618346	−	0.785906i	$-0.287803\pi$
$137$	14.4751	1.23669	0.618346	+	0.785906i	$0.287803\pi$
$138$	−31.3834	−2.67153
$139$	−14.2226	−1.20634	−0.603172	−	0.797611i	$-0.706097\pi$
$139$	−14.2226	−1.20634	−0.603172	+	0.797611i	$0.706097\pi$
$140$	13.3926	1.13188
$141$	13.3585	1.12499
$142$	37.7803	3.17045
$143$	−3.24663	−0.271497
$144$	28.5258	2.37715
$145$	−7.17277	−0.595666
$146$	14.2326	1.17790
$147$	−3.24168	−0.267369
$148$	16.5248	1.35833
$149$	21.1287	1.73093	0.865465	−	0.500969i	$-0.167023\pi$
$149$	21.1287	1.73093	0.865465	+	0.500969i	$0.167023\pi$
$150$	6.53343	0.533452
$151$	−18.8390	−1.53310	−0.766549	−	0.642186i	$-0.778028\pi$
$151$	−18.8390	−1.53310	−0.766549	+	0.642186i	$0.778028\pi$
$152$	6.84689	0.555356
$153$	−7.47542	−0.604352
$154$	−7.42271	−0.598139
$155$	4.44632	0.357137
$156$	38.2714	3.06416
$157$	−6.41658	−0.512099	−0.256050	−	0.966664i	$-0.582421\pi$
$157$	−6.41658	−0.512099	−0.256050	+	0.966664i	$0.582421\pi$
$158$	10.6143	0.844425
$159$	−6.09327	−0.483228
$160$	−7.85508	−0.620998
$161$	−13.8220	−1.08933
$162$	19.5625	1.53698
$163$	−2.05394	−0.160877	−0.0804383	−	0.996760i	$-0.525632\pi$
$163$	−2.05394	−0.160877	−0.0804383	+	0.996760i	$0.525632\pi$
$164$	19.0390	1.48669
$165$	−2.53274	−0.197174
$166$	6.88226	0.534167
$167$	11.0855	0.857823	0.428912	−	0.903346i	$-0.358897\pi$
$167$	11.0855	0.857823	0.428912	+	0.903346i	$0.358897\pi$
$168$	49.8996	3.84984
$169$	−2.45940	−0.189185
$170$	5.64705	0.433109
$171$	−3.41479	−0.261135
$172$	−20.0992	−1.53255
$173$	−23.7772	−1.80775	−0.903875	−	0.427796i	$-0.859290\pi$
$173$	−23.7772	−1.80775	−0.903875	+	0.427796i	$0.859290\pi$
$174$	−46.8627	−3.55266
$175$	2.87748	0.217517
$176$	8.35361	0.629677
$177$	1.63303	0.122746
$178$	1.82475	0.136771
$179$	−10.3606	−0.774385	−0.387192	−	0.921999i	$-0.626555\pi$
$179$	−10.3606	−0.774385	−0.387192	+	0.921999i	$0.626555\pi$
$180$	15.8933	1.18462
$181$	14.9465	1.11096	0.555480	−	0.831530i	$-0.312534\pi$
$181$	14.9465	1.11096	0.555480	+	0.831530i	$0.312534\pi$
$182$	24.0988	1.78632
$183$	−37.9872	−2.80810
$184$	32.8891	2.42462
$185$	3.55048	0.261036
$186$	29.0497	2.13003
$187$	−2.18913	−0.160085
$188$	−24.5481	−1.79035
$189$	−3.02295	−0.219887
$190$	2.57959	0.187143
$191$	9.79290	0.708590	0.354295	−	0.935134i	$-0.384721\pi$
$191$	9.79290	0.708590	0.354295	+	0.935134i	$0.384721\pi$
$192$	−9.00546	−0.649913
$193$	−4.90663	−0.353187	−0.176593	−	0.984284i	$-0.556508\pi$
$193$	−4.90663	−0.353187	−0.176593	+	0.984284i	$0.556508\pi$
$194$	−39.2291	−2.81648
$195$	8.22288	0.588852
$196$	5.95702	0.425501
$197$	0.934222	0.0665606	0.0332803	−	0.999446i	$-0.489405\pi$
$197$	0.934222	0.0665606	0.0332803	+	0.999446i	$0.489405\pi$
$198$	−8.80874	−0.626010
$199$	4.45374	0.315717	0.157858	−	0.987462i	$-0.449541\pi$
$199$	4.45374	0.315717	0.157858	+	0.987462i	$0.449541\pi$
$200$	−6.84689	−0.484148
$201$	2.03166	0.143302
$202$	50.7848	3.57320
$203$	−20.6395	−1.44861
$204$	25.8056	1.80675
$205$	4.09066	0.285704
$206$	8.33536	0.580752
$207$	−16.4030	−1.14009
$208$	−27.1211	−1.88051
$209$	−1.00000	−0.0691714
$210$	18.7998	1.29731
$211$	−25.9060	−1.78344	−0.891722	−	0.452584i	$-0.850502\pi$
$211$	−25.9060	−1.78344	−0.891722	+	0.452584i	$0.850502\pi$
$212$	11.1972	0.769028
$213$	37.0943	2.54166
$214$	21.6854	1.48238
$215$	−4.31844	−0.294515
$216$	7.19303	0.489423
$217$	12.7942	0.868527
$218$	18.5292	1.25496
$219$	13.9741	0.944286
$220$	4.65426	0.313790
$221$	7.10729	0.478088
$222$	23.1968	1.55687
$223$	6.74565	0.451722	0.225861	−	0.974160i	$-0.427480\pi$
$223$	6.74565	0.451722	0.225861	+	0.974160i	$0.427480\pi$
$224$	−22.6028	−1.51022
$225$	3.41479	0.227653
$226$	−43.4177	−2.88810
$227$	−8.82323	−0.585619	−0.292809	−	0.956171i	$-0.594590\pi$
$227$	−8.82323	−0.585619	−0.292809	+	0.956171i	$0.594590\pi$
$228$	11.7880	0.780682
$229$	−10.2524	−0.677500	−0.338750	−	0.940876i	$-0.610004\pi$
$229$	−10.2524	−0.677500	−0.338750	+	0.940876i	$0.610004\pi$
$230$	12.3911	0.817043
$231$	−7.28793	−0.479510
$232$	49.1111	3.22430
$233$	−24.6627	−1.61570	−0.807852	−	0.589385i	$-0.799370\pi$
$233$	−24.6627	−1.61570	−0.807852	+	0.589385i	$0.799370\pi$
$234$	28.5987	1.86956
$235$	−5.27433	−0.344059
$236$	−3.00092	−0.195343
$237$	10.4215	0.676950
$238$	16.2493	1.05328
$239$	23.7512	1.53634	0.768168	−	0.640248i	$-0.221169\pi$
$239$	23.7512	1.53634	0.768168	+	0.640248i	$0.221169\pi$
$240$	−21.1576	−1.36571
$241$	−1.03358	−0.0665791	−0.0332895	−	0.999446i	$-0.510598\pi$
$241$	−1.03358	−0.0665791	−0.0332895	+	0.999446i	$0.510598\pi$
$242$	−2.57959	−0.165822
$243$	22.3589	1.43433
$244$	69.8067	4.46891
$245$	1.27991	0.0817703
$246$	26.7260	1.70399
$247$	3.24663	0.206578
$248$	−30.4434	−1.93316
$249$	6.75729	0.428226
$250$	−2.57959	−0.163147
$251$	−28.3593	−1.79002	−0.895011	−	0.446045i	$-0.852832\pi$
$251$	−28.3593	−1.79002	−0.895011	+	0.446045i	$0.852832\pi$
$252$	45.7327	2.88089
$253$	−4.80351	−0.301994
$254$	−0.370617	−0.0232546
$255$	5.54451	0.347210
$256$	−23.9769	−1.49856
$257$	−1.33012	−0.0829709	−0.0414854	−	0.999139i	$-0.513209\pi$
$257$	−1.33012	−0.0829709	−0.0414854	+	0.999139i	$0.513209\pi$
$258$	−28.2142	−1.75654
$259$	10.2164	0.634818
$260$	−15.1107	−0.937123
$261$	−24.4935	−1.51611
$262$	58.6625	3.62418
$263$	20.2489	1.24860	0.624301	−	0.781184i	$-0.285384\pi$
$263$	20.2489	1.24860	0.624301	+	0.781184i	$0.285384\pi$
$264$	17.3414	1.06729
$265$	2.40580	0.147787
$266$	7.42271	0.455116
$267$	1.79161	0.109645
$268$	−3.73345	−0.228057
$269$	8.88752	0.541882	0.270941	−	0.962596i	$-0.412665\pi$
$269$	8.88752	0.541882	0.270941	+	0.962596i	$0.412665\pi$
$270$	2.70999	0.164925
$271$	22.4396	1.36311	0.681553	−	0.731768i	$-0.261305\pi$
$271$	22.4396	1.36311	0.681553	+	0.731768i	$0.261305\pi$
$272$	−18.2871	−1.10882
$273$	23.6612	1.43204
$274$	−37.3398	−2.25578
$275$	1.00000	0.0603023
$276$	56.6240	3.40836
$277$	−11.4303	−0.686782	−0.343391	−	0.939193i	$-0.611576\pi$
$277$	−11.4303	−0.686782	−0.343391	+	0.939193i	$0.611576\pi$
$278$	36.6884	2.20042
$279$	15.1832	0.908996
$280$	−19.7018	−1.17741
$281$	11.1655	0.666078	0.333039	−	0.942913i	$-0.391926\pi$
$281$	11.1655	0.666078	0.333039	+	0.942913i	$0.391926\pi$
$282$	−34.4594	−2.05203
$283$	−29.8642	−1.77524	−0.887620	−	0.460577i	$-0.847643\pi$
$283$	−29.8642	−1.77524	−0.887620	+	0.460577i	$0.847643\pi$
$284$	−68.1657	−4.04489
$285$	2.53274	0.150027
$286$	8.37495	0.495221
$287$	11.7708	0.694808
$288$	−26.8234	−1.58059
$289$	−12.2077	−0.718100
$290$	18.5028	1.08652
$291$	−38.5167	−2.25789
$292$	−25.6794	−1.50277
$293$	−20.0619	−1.17203	−0.586015	−	0.810300i	$-0.699304\pi$
$293$	−20.0619	−1.17203	−0.586015	+	0.810300i	$0.699304\pi$
$294$	8.36218	0.487692
$295$	−0.644769	−0.0375399
$296$	−24.3097	−1.41297
$297$	−1.05055	−0.0609593
$298$	−54.5033	−3.15729
$299$	15.5952	0.901895
$300$	−11.7880	−0.680583
$301$	−12.4262	−0.716237
$302$	48.5968	2.79643
$303$	49.8626	2.86453
$304$	−8.35361	−0.479113
$305$	14.9985	0.858809
$306$	19.2835	1.10236
$307$	−27.4343	−1.56576	−0.782878	−	0.622175i	$-0.786249\pi$
$307$	−27.4343	−1.56576	−0.782878	+	0.622175i	$0.786249\pi$
$308$	13.3926	0.763111
$309$	8.18400	0.465572
$310$	−11.4697	−0.651432
$311$	−26.3891	−1.49639	−0.748195	−	0.663479i	$-0.769079\pi$
$311$	−26.3891	−1.49639	−0.748195	+	0.663479i	$0.769079\pi$
$312$	−56.3011	−3.18742
$313$	−11.2050	−0.633343	−0.316671	−	0.948535i	$-0.602565\pi$
$313$	−11.2050	−0.633343	−0.316671	+	0.948535i	$0.602565\pi$
$314$	16.5521	0.934090
$315$	9.82600	0.553632
$316$	−19.1510	−1.07733
$317$	−26.8283	−1.50683	−0.753414	−	0.657546i	$-0.771594\pi$
$317$	−26.8283	−1.50683	−0.753414	+	0.657546i	$0.771594\pi$
$318$	15.7181	0.881428
$319$	−7.17277	−0.401598
$320$	3.55562	0.198765
$321$	21.2916	1.18838
$322$	35.6551	1.98698
$323$	2.18913	0.121806
$324$	−35.2960	−1.96089
$325$	−3.24663	−0.180091
$326$	5.29830	0.293446
$327$	18.1927	1.00606
$328$	−28.0083	−1.54650
$329$	−15.1768	−0.836724
$330$	6.53343	0.359653
$331$	3.32900	0.182979	0.0914893	−	0.995806i	$-0.470837\pi$
$331$	3.32900	0.182979	0.0914893	+	0.995806i	$0.470837\pi$
$332$	−12.4174	−0.681495
$333$	12.1241	0.664398
$334$	−28.5960	−1.56471
$335$	−0.802158	−0.0438266
$336$	−60.8805	−3.32130
$337$	7.13553	0.388697	0.194348	−	0.980933i	$-0.437741\pi$
$337$	7.13553	0.388697	0.194348	+	0.980933i	$0.437741\pi$
$338$	6.34424	0.345081
$339$	−42.6292	−2.31530
$340$	−10.1888	−0.552564
$341$	4.44632	0.240782
$342$	8.80874	0.476322
$343$	−16.4595	−0.888728
$344$	29.5679	1.59419
$345$	12.1661	0.654999
$346$	61.3354	3.29741
$347$	−6.96374	−0.373833	−0.186916	−	0.982376i	$-0.559849\pi$
$347$	−6.96374	−0.373833	−0.186916	+	0.982376i	$0.559849\pi$
$348$	84.5529	4.53251
$349$	−24.9796	−1.33712	−0.668562	−	0.743656i	$-0.733090\pi$
$349$	−24.9796	−1.33712	−0.668562	+	0.743656i	$0.733090\pi$
$350$	−7.42271	−0.396761
$351$	3.41076	0.182053
$352$	−7.85508	−0.418677
$353$	15.5105	0.825541	0.412770	−	0.910835i	$-0.364561\pi$
$353$	15.5105	0.825541	0.412770	+	0.910835i	$0.364561\pi$
$354$	−4.21255	−0.223895
$355$	−14.6459	−0.777323
$356$	−3.29233	−0.174493
$357$	15.9542	0.844387
$358$	26.7259	1.41251
$359$	15.2641	0.805610	0.402805	−	0.915286i	$-0.368035\pi$
$359$	15.2641	0.805610	0.402805	+	0.915286i	$0.368035\pi$
$360$	−23.3807	−1.23227
$361$	1.00000	0.0526316
$362$	−38.5556	−2.02644
$363$	−2.53274	−0.132935
$364$	−43.4806	−2.27900
$365$	−5.51740	−0.288794
$366$	97.9913	5.12209
$367$	−21.5170	−1.12318	−0.561589	−	0.827416i	$-0.689810\pi$
$367$	−21.5170	−1.12318	−0.561589	+	0.827416i	$0.689810\pi$
$368$	−40.1267	−2.09175
$369$	13.9687	0.727183
$370$	−9.15875	−0.476141
$371$	6.92264	0.359406
$372$	−52.4134	−2.71751
$373$	10.6278	0.550287	0.275143	−	0.961403i	$-0.411275\pi$
$373$	10.6278	0.550287	0.275143	+	0.961403i	$0.411275\pi$
$374$	5.64705	0.292002
$375$	−2.53274	−0.130790
$376$	36.1127	1.86237
$377$	23.2873	1.19936
$378$	7.79796	0.401084
$379$	34.4023	1.76713	0.883563	−	0.468312i	$-0.155138\pi$
$379$	34.4023	1.76713	0.883563	+	0.468312i	$0.155138\pi$
$380$	−4.65426	−0.238758
$381$	−0.363887	−0.0186425
$382$	−25.2616	−1.29250
$383$	−29.1210	−1.48801	−0.744006	−	0.668173i	$-0.767077\pi$
$383$	−29.1210	−1.48801	−0.744006	+	0.668173i	$0.767077\pi$
$384$	−16.5594	−0.845045
$385$	2.87748	0.146650
$386$	12.6571	0.644227
$387$	−14.7466	−0.749611
$388$	70.7797	3.59329
$389$	2.33965	0.118625	0.0593125	−	0.998239i	$-0.481109\pi$
$389$	2.33965	0.118625	0.0593125	+	0.998239i	$0.481109\pi$
$390$	−21.2116	−1.07409
$391$	10.5155	0.531792
$392$	−8.76338	−0.442618
$393$	57.5973	2.90540
$394$	−2.40991	−0.121409
$395$	−4.11472	−0.207034
$396$	15.8933	0.798669
$397$	30.7176	1.54167	0.770837	−	0.637033i	$-0.219838\pi$
$397$	30.7176	1.54167	0.770837	+	0.637033i	$0.219838\pi$
$398$	−11.4888	−0.575881
$399$	7.28793	0.364853
$400$	8.35361	0.417681
$401$	36.7578	1.83560	0.917798	−	0.397048i	$-0.129965\pi$
$401$	36.7578	1.83560	0.917798	+	0.397048i	$0.129965\pi$
$402$	−5.24084	−0.261389
$403$	−14.4355	−0.719085
$404$	−91.6293	−4.55873
$405$	−7.58358	−0.376831
$406$	53.2414	2.64233
$407$	3.55048	0.175991
$408$	−37.9626	−1.87943
$409$	−34.6219	−1.71194	−0.855971	−	0.517024i	$-0.827040\pi$
$409$	−34.6219	−1.71194	−0.855971	+	0.517024i	$0.827040\pi$
$410$	−10.5522	−0.521136
$411$	−36.6617	−1.80839
$412$	−15.0392	−0.740929
$413$	−1.85531	−0.0912939
$414$	42.3129	2.07956
$415$	−2.66797	−0.130966
$416$	25.5025	1.25036
$417$	36.0222	1.76401
$418$	2.57959	0.126172
$419$	−5.15373	−0.251776	−0.125888	−	0.992044i	$-0.540178\pi$
$419$	−5.15373	−0.251776	−0.125888	+	0.992044i	$0.540178\pi$
$420$	−33.9199	−1.65512
$421$	−9.69018	−0.472270	−0.236135	−	0.971720i	$-0.575881\pi$
$421$	−9.69018	−0.472270	−0.236135	+	0.971720i	$0.575881\pi$
$422$	66.8268	3.25308
$423$	−18.0107	−0.875711
$424$	−16.4722	−0.799962
$425$	−2.18913	−0.106188
$426$	−95.6878	−4.63609
$427$	43.1578	2.08855
$428$	−39.1263	−1.89124
$429$	8.22288	0.397004
$430$	11.1398	0.537209
$431$	0.196679	0.00947368	0.00473684	−	0.999989i	$-0.498492\pi$
$431$	0.196679	0.00947368	0.00473684	+	0.999989i	$0.498492\pi$
$432$	−8.77592	−0.422232
$433$	−19.3120	−0.928075	−0.464038	−	0.885816i	$-0.653600\pi$
$433$	−19.3120	−0.928075	−0.464038	+	0.885816i	$0.653600\pi$
$434$	−33.0037	−1.58423
$435$	18.1668	0.871030
$436$	−33.4316	−1.60108
$437$	4.80351	0.229783
$438$	−36.0475	−1.72242
$439$	13.9280	0.664747	0.332373	−	0.943148i	$-0.392151\pi$
$439$	13.9280	0.664747	0.332373	+	0.943148i	$0.392151\pi$
$440$	−6.84689	−0.326413
$441$	4.37061	0.208124
$442$	−18.3339	−0.872053
$443$	29.4381	1.39864	0.699322	−	0.714807i	$-0.253485\pi$
$443$	29.4381	1.39864	0.699322	+	0.714807i	$0.253485\pi$
$444$	−41.8532	−1.98626
$445$	−0.707380	−0.0335330
$446$	−17.4010	−0.823960
$447$	−53.5136	−2.53110
$448$	10.2312	0.483380
$449$	7.30837	0.344903	0.172452	−	0.985018i	$-0.444831\pi$
$449$	7.30837	0.344903	0.172452	+	0.985018i	$0.444831\pi$
$450$	−8.80874	−0.415248
$451$	4.09066	0.192622
$452$	78.3370	3.68466
$453$	47.7144	2.24182
$454$	22.7603	1.06819
$455$	−9.34212	−0.437965
$456$	−17.3414	−0.812086
$457$	12.2572	0.573366	0.286683	−	0.958026i	$-0.407447\pi$
$457$	12.2572	0.573366	0.286683	+	0.958026i	$0.407447\pi$
$458$	26.4470	1.23579
$459$	2.29980	0.107345
$460$	−22.3568	−1.04239
$461$	−8.26361	−0.384875	−0.192437	−	0.981309i	$-0.561639\pi$
$461$	−8.26361	−0.384875	−0.192437	+	0.981309i	$0.561639\pi$
$462$	18.7998	0.874647
$463$	6.86256	0.318930	0.159465	−	0.987204i	$-0.449023\pi$
$463$	6.86256	0.318930	0.159465	+	0.987204i	$0.449023\pi$
$464$	−59.9185	−2.78165
$465$	−11.2614	−0.522234
$466$	63.6194	2.94711
$467$	26.8861	1.24414	0.622071	−	0.782961i	$-0.286291\pi$
$467$	26.8861	1.24414	0.622071	+	0.782961i	$0.286291\pi$
$468$	−51.5997	−2.38520
$469$	−2.30819	−0.106583
$470$	13.6056	0.627579
$471$	16.2516	0.748832
$472$	4.41466	0.203201
$473$	−4.31844	−0.198562
$474$	−26.8832	−1.23479
$475$	−1.00000	−0.0458831
$476$	−29.3180	−1.34379
$477$	8.21530	0.376153
$478$	−61.2682	−2.80234
$479$	−15.7331	−0.718862	−0.359431	−	0.933172i	$-0.617029\pi$
$479$	−15.7331	−0.718862	−0.359431	+	0.933172i	$0.617029\pi$
$480$	19.8949	0.908074
$481$	−11.5271	−0.525589
$482$	2.66622	0.121443
$483$	35.0076	1.59290
$484$	4.65426	0.211557
$485$	15.2075	0.690537
$486$	−57.6768	−2.61627
$487$	−35.9154	−1.62748	−0.813742	−	0.581226i	$-0.802573\pi$
$487$	−35.9154	−1.62748	−0.813742	+	0.581226i	$0.802573\pi$
$488$	−102.693	−4.64868
$489$	5.20209	0.235247
$490$	−3.30163	−0.149152
$491$	26.6321	1.20189	0.600945	−	0.799290i	$-0.294791\pi$
$491$	26.6321	1.20189	0.600945	+	0.799290i	$0.294791\pi$
$492$	−48.2209	−2.17396
$493$	15.7021	0.707188
$494$	−8.37495	−0.376807
$495$	3.41479	0.153483
$496$	37.1428	1.66776
$497$	−42.1433	−1.89038
$498$	−17.4310	−0.781102
$499$	−9.83268	−0.440171	−0.220086	−	0.975481i	$-0.570634\pi$
$499$	−9.83268	−0.440171	−0.220086	+	0.975481i	$0.570634\pi$
$500$	4.65426	0.208145
$501$	−28.0768	−1.25438
$502$	73.1552	3.26507
$503$	5.17947	0.230941	0.115471	−	0.993311i	$-0.463162\pi$
$503$	5.17947	0.230941	0.115471	+	0.993311i	$0.463162\pi$
$504$	−67.2775	−2.99678
$505$	−19.6872	−0.876068
$506$	12.3911	0.550850
$507$	6.22904	0.276641
$508$	0.668691	0.0296684
$509$	−28.2015	−1.25001	−0.625005	−	0.780621i	$-0.714903\pi$
$509$	−28.2015	−1.25001	−0.625005	+	0.780621i	$0.714903\pi$
$510$	−14.3025	−0.633327
$511$	−15.8762	−0.702322
$512$	48.7742	2.15554
$513$	1.05055	0.0463831
$514$	3.43117	0.151342
$515$	−3.23128	−0.142387
$516$	50.9060	2.24101
$517$	−5.27433	−0.231965
$518$	−26.3542	−1.15794
$519$	60.2217	2.64344
$520$	22.2293	0.974819
$521$	−2.47096	−0.108255	−0.0541273	−	0.998534i	$-0.517238\pi$
$521$	−2.47096	−0.108255	−0.0541273	+	0.998534i	$0.517238\pi$
$522$	63.1830	2.76545
$523$	−33.9213	−1.48327	−0.741637	−	0.670802i	$-0.765950\pi$
$523$	−33.9213	−1.48327	−0.741637	+	0.670802i	$0.765950\pi$
$524$	−105.843	−4.62377
$525$	−7.28793	−0.318071
$526$	−52.2339	−2.27750
$527$	−9.73357	−0.424001
$528$	−21.1576	−0.920764
$529$	0.0737212	0.00320527
$530$	−6.20596	−0.269570
$531$	−2.20175	−0.0955478
$532$	−13.3926	−0.580641
$533$	−13.2808	−0.575257
$534$	−4.62162	−0.199997
$535$	−8.40655	−0.363447
$536$	5.49228	0.237231
$537$	26.2406	1.13237
$538$	−22.9261	−0.988415
$539$	1.27991	0.0551295
$540$	−4.88955	−0.210413
$541$	41.6591	1.79106	0.895532	−	0.444998i	$-0.146795\pi$
$541$	41.6591	1.79106	0.895532	+	0.444998i	$0.146795\pi$
$542$	−57.8848	−2.48636
$543$	−37.8555	−1.62454
$544$	17.1958	0.737263
$545$	−7.18301	−0.307687
$546$	−61.0360	−2.61210
$547$	26.9367	1.15173	0.575864	−	0.817545i	$-0.304666\pi$
$547$	26.9367	1.15173	0.575864	+	0.817545i	$0.304666\pi$
$548$	67.3709	2.87794
$549$	51.2165	2.18587
$550$	−2.57959	−0.109994
$551$	7.17277	0.305570
$552$	−83.2997	−3.54547
$553$	−11.8400	−0.503489
$554$	29.4855	1.25272
$555$	−8.99244	−0.381708
$556$	−66.1957	−2.80732
$557$	−31.8497	−1.34952	−0.674758	−	0.738039i	$-0.735752\pi$
$557$	−31.8497	−1.34952	−0.674758	+	0.738039i	$0.735752\pi$
$558$	−39.1664	−1.65805
$559$	14.0204	0.592999
$560$	24.0374	1.01576
$561$	5.54451	0.234089
$562$	−28.8024	−1.21495
$563$	16.8415	0.709786	0.354893	−	0.934907i	$-0.384517\pi$
$563$	16.8415	0.709786	0.354893	+	0.934907i	$0.384517\pi$
$564$	62.1740	2.61800
$565$	16.8313	0.708096
$566$	77.0371	3.23811
$567$	−21.8216	−0.916422
$568$	100.279	4.20760
$569$	26.8314	1.12483	0.562416	−	0.826854i	$-0.309872\pi$
$569$	26.8314	1.12483	0.562416	+	0.826854i	$0.309872\pi$
$570$	−6.53343	−0.273655
$571$	−27.7315	−1.16053	−0.580263	−	0.814429i	$-0.697050\pi$
$571$	−27.7315	−1.16053	−0.580263	+	0.814429i	$0.697050\pi$
$572$	−15.1107	−0.631808
$573$	−24.8029	−1.03616
$574$	−30.3638	−1.26736
$575$	−4.80351	−0.200320
$576$	12.1417	0.505903
$577$	18.6576	0.776727	0.388363	−	0.921506i	$-0.373041\pi$
$577$	18.6576	0.776727	0.388363	+	0.921506i	$0.373041\pi$
$578$	31.4908	1.30985
$579$	12.4272	0.516458
$580$	−33.3839	−1.38619
$581$	−7.67704	−0.318497
$582$	99.3571	4.11849
$583$	2.40580	0.0996380
$584$	37.7770	1.56322
$585$	−11.0866	−0.458372
$586$	51.7514	2.13783
$587$	20.8036	0.858655	0.429328	−	0.903149i	$-0.358751\pi$
$587$	20.8036	0.858655	0.429328	+	0.903149i	$0.358751\pi$
$588$	−15.0876	−0.622202
$589$	−4.44632	−0.183207
$590$	1.66324	0.0684744
$591$	−2.36615	−0.0973302
$592$	29.6593	1.21899
$593$	36.0273	1.47946	0.739732	−	0.672902i	$-0.234952\pi$
$593$	36.0273	1.47946	0.739732	+	0.672902i	$0.234952\pi$
$594$	2.70999	0.111192
$595$	−6.29919	−0.258241
$596$	98.3384	4.02810
$597$	−11.2802	−0.461667
$598$	−40.2292	−1.64509
$599$	−2.00165	−0.0817850	−0.0408925	−	0.999164i	$-0.513020\pi$
$599$	−2.00165	−0.0817850	−0.0408925	+	0.999164i	$0.513020\pi$
$600$	17.3414	0.707960
$601$	−9.98124	−0.407143	−0.203572	−	0.979060i	$-0.565255\pi$
$601$	−9.98124	−0.407143	−0.203572	+	0.979060i	$0.565255\pi$
$602$	32.0546	1.30645
$603$	−2.73920	−0.111549
$604$	−87.6817	−3.56772
$605$	1.00000	0.0406558
$606$	−128.625	−5.22502
$607$	−5.36715	−0.217846	−0.108923	−	0.994050i	$-0.534740\pi$
$607$	−5.36715	−0.217846	−0.108923	+	0.994050i	$0.534740\pi$
$608$	7.85508	0.318565
$609$	52.2746	2.11827
$610$	−38.6898	−1.56650
$611$	17.1238	0.692754
$612$	−34.7925	−1.40641
$613$	34.3276	1.38648	0.693239	−	0.720708i	$-0.256183\pi$
$613$	34.3276	1.38648	0.693239	+	0.720708i	$0.256183\pi$
$614$	70.7690	2.85600
$615$	−10.3606	−0.417779
$616$	−19.7018	−0.793808
$617$	14.7316	0.593074	0.296537	−	0.955021i	$-0.404168\pi$
$617$	14.7316	0.593074	0.296537	+	0.955021i	$0.404168\pi$
$618$	−21.1113	−0.849222
$619$	−10.9410	−0.439756	−0.219878	−	0.975527i	$-0.570566\pi$
$619$	−10.9410	−0.439756	−0.219878	+	0.975527i	$0.570566\pi$
$620$	20.6943	0.831104
$621$	5.04635	0.202503
$622$	68.0730	2.72948
$623$	−2.03547	−0.0815496
$624$	68.6907	2.74983
$625$	1.00000	0.0400000
$626$	28.9042	1.15524
$627$	2.53274	0.101148
$628$	−29.8644	−1.19172
$629$	−7.77246	−0.309908
$630$	−25.3470	−1.00985
$631$	8.87490	0.353304	0.176652	−	0.984273i	$-0.443473\pi$
$631$	8.87490	0.353304	0.176652	+	0.984273i	$0.443473\pi$
$632$	28.1730	1.12066
$633$	65.6133	2.60789
$634$	69.2059	2.74852
$635$	0.143673	0.00570149
$636$	−28.3597	−1.12453
$637$	−4.15538	−0.164642
$638$	18.5028	0.732531
$639$	−50.0126	−1.97847
$640$	6.53814	0.258443
$641$	−6.96188	−0.274978	−0.137489	−	0.990503i	$-0.543903\pi$
$641$	−6.96188	−0.274978	−0.137489	+	0.990503i	$0.543903\pi$
$642$	−54.9236	−2.16766
$643$	18.6796	0.736651	0.368325	−	0.929697i	$-0.379931\pi$
$643$	18.6796	0.736651	0.368325	+	0.929697i	$0.379931\pi$
$644$	−64.3313	−2.53501
$645$	10.9375	0.430664
$646$	−5.64705	−0.222180
$647$	−37.0659	−1.45721	−0.728605	−	0.684935i	$-0.759831\pi$
$647$	−37.0659	−1.45721	−0.728605	+	0.684935i	$0.759831\pi$
$648$	51.9239	2.03977
$649$	−0.644769	−0.0253094
$650$	8.37495	0.328493
$651$	−32.4044	−1.27003
$652$	−9.55955	−0.374381
$653$	−28.6203	−1.12000	−0.560000	−	0.828493i	$-0.689199\pi$
$653$	−28.6203	−1.12000	−0.560000	+	0.828493i	$0.689199\pi$
$654$	−46.9297	−1.83510
$655$	−22.7411	−0.888567
$656$	34.1718	1.33418
$657$	−18.8407	−0.735047
$658$	39.1498	1.52622
$659$	40.9443	1.59496	0.797481	−	0.603344i	$-0.206165\pi$
$659$	40.9443	1.59496	0.797481	+	0.603344i	$0.206165\pi$
$660$	−11.7880	−0.458849
$661$	−25.0123	−0.972865	−0.486432	−	0.873718i	$-0.661702\pi$
$661$	−25.0123	−0.972865	−0.486432	+	0.873718i	$0.661702\pi$
$662$	−8.58745	−0.333761
$663$	−18.0010	−0.699099
$664$	18.2673	0.708909
$665$	−2.87748	−0.111584
$666$	−31.2752	−1.21189
$667$	34.4545	1.33408
$668$	51.5949	1.99627
$669$	−17.0850	−0.660544
$670$	2.06923	0.0799415
$671$	14.9985	0.579009
$672$	57.2472	2.20836
$673$	25.8017	0.994583	0.497291	−	0.867584i	$-0.334328\pi$
$673$	25.8017	0.994583	0.497291	+	0.867584i	$0.334328\pi$
$674$	−18.4067	−0.708999
$675$	−1.05055	−0.0404358
$676$	−11.4467	−0.440258
$677$	−5.31731	−0.204361	−0.102180	−	0.994766i	$-0.532582\pi$
$677$	−5.31731	−0.204361	−0.102180	+	0.994766i	$0.532582\pi$
$678$	109.966	4.22321
$679$	43.7593	1.67933
$680$	14.9887	0.574792
$681$	22.3470	0.856338
$682$	−11.4697	−0.439196
$683$	25.7322	0.984614	0.492307	−	0.870422i	$-0.336154\pi$
$683$	25.7322	0.984614	0.492307	+	0.870422i	$0.336154\pi$
$684$	−15.8933	−0.607696
$685$	14.4751	0.553066
$686$	42.4586	1.62108
$687$	25.9668	0.990695
$688$	−36.0746	−1.37533
$689$	−7.81074	−0.297565
$690$	−31.3834	−1.19475
$691$	−11.4339	−0.434965	−0.217483	−	0.976064i	$-0.569785\pi$
$691$	−11.4339	−0.434965	−0.217483	+	0.976064i	$0.569785\pi$
$692$	−110.665	−4.20687
$693$	9.82600	0.373259
$694$	17.9635	0.681887
$695$	−14.2226	−0.539494
$696$	−124.386	−4.71483
$697$	−8.95499	−0.339194
$698$	64.4369	2.43897
$699$	62.4642	2.36261
$700$	13.3926	0.506191
$701$	20.5265	0.775273	0.387637	−	0.921812i	$-0.373292\pi$
$701$	20.5265	0.775273	0.387637	+	0.921812i	$0.373292\pi$
$702$	−8.79834	−0.332072
$703$	−3.55048	−0.133909
$704$	3.55562	0.134007
$705$	13.3585	0.503111
$706$	−40.0107	−1.50582
$707$	−56.6495	−2.13052
$708$	7.60057	0.285647
$709$	16.2980	0.612084	0.306042	−	0.952018i	$-0.400995\pi$
$709$	16.2980	0.612084	0.306042	+	0.952018i	$0.400995\pi$
$710$	37.7803	1.41787
$711$	−14.0509	−0.526949
$712$	4.84335	0.181512
$713$	−21.3579	−0.799861
$714$	−41.1553	−1.54020
$715$	−3.24663	−0.121417
$716$	−48.2207	−1.80209
$717$	−60.1556	−2.24655
$718$	−39.3751	−1.46947
$719$	29.3047	1.09288	0.546440	−	0.837498i	$-0.315983\pi$
$719$	29.3047	1.09288	0.546440	+	0.837498i	$0.315983\pi$
$720$	28.5258	1.06309
$721$	−9.29795	−0.346274
$722$	−2.57959	−0.0960022
$723$	2.61781	0.0973572
$724$	69.5647	2.58535
$725$	−7.17277	−0.266390
$726$	6.53343	0.242478
$727$	−24.2144	−0.898063	−0.449032	−	0.893516i	$-0.648231\pi$
$727$	−24.2144	−0.898063	−0.449032	+	0.893516i	$0.648231\pi$
$728$	63.9644	2.37068
$729$	−33.8787	−1.25477
$730$	14.2326	0.526772
$731$	9.45364	0.349655
$732$	−176.802	−6.53481
$733$	−8.54037	−0.315446	−0.157723	−	0.987483i	$-0.550415\pi$
$733$	−8.54037	−0.315446	−0.157723	+	0.987483i	$0.550415\pi$
$734$	55.5049	2.04872
$735$	−3.24168	−0.119571
$736$	37.7320	1.39082
$737$	−0.802158	−0.0295479
$738$	−36.0335	−1.32641
$739$	10.6881	0.393167	0.196583	−	0.980487i	$-0.437015\pi$
$739$	10.6881	0.393167	0.196583	+	0.980487i	$0.437015\pi$
$740$	16.5248	0.607465
$741$	−8.22288	−0.302075
$742$	−17.8576	−0.655571
$743$	−9.69343	−0.355617	−0.177809	−	0.984065i	$-0.556901\pi$
$743$	−9.69343	−0.355617	−0.177809	+	0.984065i	$0.556901\pi$
$744$	77.1054	2.82682
$745$	21.1287	0.774095
$746$	−27.4153	−1.00375
$747$	−9.11056	−0.333338
$748$	−10.1888	−0.372539
$749$	−24.1897	−0.883872
$750$	6.53343	0.238567
$751$	49.2682	1.79782	0.898911	−	0.438131i	$-0.144359\pi$
$751$	49.2682	1.79782	0.898911	+	0.438131i	$0.144359\pi$
$752$	−44.0597	−1.60669
$753$	71.8268	2.61751
$754$	−60.0716	−2.18768
$755$	−18.8390	−0.685622
$756$	−14.0696	−0.511706
$757$	43.3829	1.57678	0.788389	−	0.615177i	$-0.210916\pi$
$757$	43.3829	1.57678	0.788389	+	0.615177i	$0.210916\pi$
$758$	−88.7436	−3.22331
$759$	12.1661	0.441600
$760$	6.84689	0.248363
$761$	3.69301	0.133872	0.0669358	−	0.997757i	$-0.478678\pi$
$761$	3.69301	0.133872	0.0669358	+	0.997757i	$0.478678\pi$
$762$	0.938677	0.0340047
$763$	−20.6690	−0.748268
$764$	45.5787	1.64898
$765$	−7.47542	−0.270274
$766$	75.1200	2.71420
$767$	2.09333	0.0755856
$768$	60.7274	2.19131
$769$	−29.8952	−1.07805	−0.539024	−	0.842290i	$-0.681207\pi$
$769$	−29.8952	−1.07805	−0.539024	+	0.842290i	$0.681207\pi$
$770$	−7.42271	−0.267496
$771$	3.36886	0.121327
$772$	−22.8367	−0.821911
$773$	19.2250	0.691477	0.345738	−	0.938331i	$-0.387628\pi$
$773$	19.2250	0.691477	0.345738	+	0.938331i	$0.387628\pi$
$774$	38.0400	1.36732
$775$	4.44632	0.159716
$776$	−104.124	−3.73784
$777$	−25.8756	−0.928282
$778$	−6.03533	−0.216377
$779$	−4.09066	−0.146563
$780$	38.2714	1.37034
$781$	−14.6459	−0.524071
$782$	−27.1257	−0.970012
$783$	7.53538	0.269292
$784$	10.6918	0.381852
$785$	−6.41658	−0.229018
$786$	−148.577	−5.29957
$787$	−36.7718	−1.31077	−0.655387	−	0.755294i	$-0.727494\pi$
$787$	−36.7718	−1.31077	−0.655387	+	0.755294i	$0.727494\pi$
$788$	4.34811	0.154895
$789$	−51.2854	−1.82581
$790$	10.6143	0.377638
$791$	48.4316	1.72203
$792$	−23.3807	−0.830796
$793$	−48.6944	−1.72919
$794$	−79.2387	−2.81208
$795$	−6.09327	−0.216106
$796$	20.7288	0.734714
$797$	16.5973	0.587905	0.293953	−	0.955820i	$-0.405029\pi$
$797$	16.5973	0.587905	0.293953	+	0.955820i	$0.405029\pi$
$798$	−18.7998	−0.665507
$799$	11.5462	0.408475
$800$	−7.85508	−0.277719
$801$	−2.41555	−0.0853494
$802$	−94.8198	−3.34820
$803$	−5.51740	−0.194705
$804$	9.45587	0.333483
$805$	−13.8220	−0.487162
$806$	37.2377	1.31164
$807$	−22.5098	−0.792383
$808$	134.796	4.74210
$809$	49.5534	1.74221	0.871103	−	0.491100i	$-0.163405\pi$
$809$	49.5534	1.74221	0.871103	+	0.491100i	$0.163405\pi$
$810$	19.5625	0.687356
$811$	39.0493	1.37121	0.685603	−	0.727975i	$-0.259539\pi$
$811$	39.0493	1.37121	0.685603	+	0.727975i	$0.259539\pi$
$812$	−96.0616	−3.37110
$813$	−56.8337	−1.99324
$814$	−9.15875	−0.321014
$815$	−2.05394	−0.0719462
$816$	46.3167	1.62141
$817$	4.31844	0.151083
$818$	89.3101	3.12265
$819$	−31.9014	−1.11472
$820$	19.0390	0.664870
$821$	−6.20214	−0.216456	−0.108228	−	0.994126i	$-0.534518\pi$
$821$	−6.20214	−0.216456	−0.108228	+	0.994126i	$0.534518\pi$
$822$	94.5721	3.29858
$823$	−30.5672	−1.06551	−0.532753	−	0.846271i	$-0.678843\pi$
$823$	−30.5672	−1.06551	−0.532753	+	0.846271i	$0.678843\pi$
$824$	22.1242	0.770733
$825$	−2.53274	−0.0881788
$826$	4.78593	0.166524
$827$	−34.2182	−1.18988	−0.594942	−	0.803769i	$-0.702825\pi$
$827$	−34.2182	−1.18988	−0.594942	+	0.803769i	$0.702825\pi$
$828$	−76.3437	−2.65313
$829$	14.5065	0.503832	0.251916	−	0.967749i	$-0.418939\pi$
$829$	14.5065	0.503832	0.251916	+	0.967749i	$0.418939\pi$
$830$	6.88226	0.238887
$831$	28.9501	1.00427
$832$	−11.5438	−0.400208
$833$	−2.80188	−0.0970795
$834$	−92.9223	−3.21764
$835$	11.0855	0.383630
$836$	−4.65426	−0.160971
$837$	−4.67110	−0.161457
$838$	13.2945	0.459250
$839$	−10.5381	−0.363816	−0.181908	−	0.983316i	$-0.558227\pi$
$839$	−10.5381	−0.363816	−0.181908	+	0.983316i	$0.558227\pi$
$840$	49.8996	1.72170
$841$	22.4486	0.774089
$842$	24.9966	0.861441
$843$	−28.2794	−0.973993
$844$	−120.573	−4.15031
$845$	−2.45940	−0.0846060
$846$	46.4602	1.59733
$847$	2.87748	0.0988715
$848$	20.0971	0.690138
$849$	75.6382	2.59590
$850$	5.64705	0.193692
$851$	−17.0547	−0.584629
$852$	172.646	5.91477
$853$	46.8866	1.60537	0.802684	−	0.596404i	$-0.203405\pi$
$853$	46.8866	1.60537	0.802684	+	0.596404i	$0.203405\pi$
$854$	−111.329	−3.80961
$855$	−3.41479	−0.116783
$856$	57.5587	1.96732
$857$	−17.1896	−0.587186	−0.293593	−	0.955930i	$-0.594851\pi$
$857$	−17.1896	−0.587186	−0.293593	+	0.955930i	$0.594851\pi$
$858$	−21.2116	−0.724152
$859$	12.6366	0.431155	0.215577	−	0.976487i	$-0.430837\pi$
$859$	12.6366	0.431155	0.215577	+	0.976487i	$0.430837\pi$
$860$	−20.0992	−0.685376
$861$	−29.8124	−1.01600
$862$	−0.507349	−0.0172804
$863$	7.20209	0.245162	0.122581	−	0.992459i	$-0.460883\pi$
$863$	7.20209	0.245162	0.122581	+	0.992459i	$0.460883\pi$
$864$	8.25218	0.280745
$865$	−23.7772	−0.808451
$866$	49.8169	1.69285
$867$	30.9190	1.05006
$868$	59.5475	2.02117
$869$	−4.11472	−0.139582
$870$	−46.8627	−1.58880
$871$	2.60431	0.0882436
$872$	49.1813	1.66549
$873$	51.9304	1.75758
$874$	−12.3911	−0.419134
$875$	2.87748	0.0972767
$876$	65.0393	2.19747
$877$	2.57936	0.0870988	0.0435494	−	0.999051i	$-0.486133\pi$
$877$	2.57936	0.0870988	0.0435494	+	0.999051i	$0.486133\pi$
$878$	−35.9284	−1.21253
$879$	50.8117	1.71384
$880$	8.35361	0.281600
$881$	9.80259	0.330258	0.165129	−	0.986272i	$-0.447196\pi$
$881$	9.80259	0.330258	0.165129	+	0.986272i	$0.447196\pi$
$882$	−11.2744	−0.379628
$883$	−10.5272	−0.354268	−0.177134	−	0.984187i	$-0.556683\pi$
$883$	−10.5272	−0.354268	−0.177134	+	0.984187i	$0.556683\pi$
$884$	33.0792	1.11257
$885$	1.63303	0.0548939
$886$	−75.9380	−2.55119
$887$	33.6576	1.13011	0.565056	−	0.825053i	$-0.308855\pi$
$887$	33.6576	1.13011	0.565056	+	0.825053i	$0.308855\pi$
$888$	61.5703	2.06616
$889$	0.413417	0.0138655
$890$	1.82475	0.0611657
$891$	−7.58358	−0.254060
$892$	31.3960	1.05122
$893$	5.27433	0.176499
$894$	138.043	4.61684
$895$	−10.3606	−0.346315
$896$	18.8134	0.628511
$897$	−39.4987	−1.31882
$898$	−18.8526	−0.629118
$899$	−31.8924	−1.06367
$900$	15.8933	0.529777
$901$	−5.26661	−0.175456
$902$	−10.5522	−0.351350
$903$	31.4725	1.04734
$904$	−115.242	−3.83288
$905$	14.9465	0.496837
$906$	−123.083	−4.08917
$907$	9.49097	0.315143	0.157571	−	0.987508i	$-0.449634\pi$
$907$	9.49097	0.315143	0.157571	+	0.987508i	$0.449634\pi$
$908$	−41.0656	−1.36281
$909$	−67.2276	−2.22980
$910$	24.0988	0.798867
$911$	−38.4695	−1.27455	−0.637276	−	0.770636i	$-0.719939\pi$
$911$	−38.4695	−1.27455	−0.637276	+	0.770636i	$0.719939\pi$
$912$	21.1576	0.700597
$913$	−2.66797	−0.0882970
$914$	−31.6184	−1.04584
$915$	−37.9872	−1.25582
$916$	−47.7175	−1.57663
$917$	−65.4370	−2.16092
$918$	−5.93253	−0.195803
$919$	−12.5675	−0.414564	−0.207282	−	0.978281i	$-0.566462\pi$
$919$	−12.5675	−0.414564	−0.207282	+	0.978281i	$0.566462\pi$
$920$	32.8891	1.08432
$921$	69.4839	2.28957
$922$	21.3167	0.702028
$923$	47.5497	1.56512
$924$	−33.9199	−1.11588
$925$	3.55048	0.116739
$926$	−17.7025	−0.581742
$927$	−11.0341	−0.362409
$928$	56.3426	1.84954
$929$	−5.22161	−0.171315	−0.0856577	−	0.996325i	$-0.527299\pi$
$929$	−5.22161	−0.171315	−0.0856577	+	0.996325i	$0.527299\pi$
$930$	29.0497	0.952577
$931$	−1.27991	−0.0419473
$932$	−114.786	−3.75996
$933$	66.8369	2.18814
$934$	−69.3551	−2.26937
$935$	−2.18913	−0.0715922
$936$	75.9084	2.48114
$937$	−28.2092	−0.921554	−0.460777	−	0.887516i	$-0.652429\pi$
$937$	−28.2092	−0.921554	−0.460777	+	0.887516i	$0.652429\pi$
$938$	5.95419	0.194411
$939$	28.3793	0.926124
$940$	−24.5481	−0.800671
$941$	−14.8196	−0.483105	−0.241552	−	0.970388i	$-0.577657\pi$
$941$	−14.8196	−0.483105	−0.241552	+	0.970388i	$0.577657\pi$
$942$	−41.9223	−1.36590
$943$	−19.6495	−0.639876
$944$	−5.38615	−0.175304
$945$	−3.02295	−0.0983366
$946$	11.1398	0.362186
$947$	−41.1264	−1.33643	−0.668214	−	0.743970i	$-0.732941\pi$
$947$	−41.1264	−1.33643	−0.668214	+	0.743970i	$0.732941\pi$
$948$	48.5045	1.57535
$949$	17.9129	0.581479
$950$	2.57959	0.0836928
$951$	67.9492	2.20340
$952$	43.1298	1.39785
$953$	−18.9364	−0.613410	−0.306705	−	0.951805i	$-0.599226\pi$
$953$	−18.9364	−0.613410	−0.306705	+	0.951805i	$0.599226\pi$
$954$	−21.1921	−0.686118
$955$	9.79290	0.316891
$956$	110.544	3.57525
$957$	18.1668	0.587248
$958$	40.5848	1.31124
$959$	41.6519	1.34501
$960$	−9.00546	−0.290650
$961$	−11.2303	−0.362267
$962$	29.7351	0.958697
$963$	−28.7066	−0.925057
$964$	−4.81057	−0.154938
$965$	−4.90663	−0.157950
$966$	−90.3052	−2.90552
$967$	46.5459	1.49681	0.748407	−	0.663239i	$-0.230819\pi$
$967$	46.5459	1.49681	0.748407	+	0.663239i	$0.230819\pi$
$968$	−6.84689	−0.220067
$969$	−5.54451	−0.178115
$970$	−39.2291	−1.25957
$971$	3.49873	0.112280	0.0561399	−	0.998423i	$-0.482121\pi$
$971$	3.49873	0.112280	0.0561399	+	0.998423i	$0.482121\pi$
$972$	104.064	3.33786
$973$	−40.9253	−1.31200
$974$	92.6469	2.96860
$975$	8.22288	0.263343
$976$	125.291	4.01047
$977$	38.3694	1.22755	0.613773	−	0.789482i	$-0.289651\pi$
$977$	38.3694	1.22755	0.613773	+	0.789482i	$0.289651\pi$
$978$	−13.4192	−0.429100
$979$	−0.707380	−0.0226080
$980$	5.95702	0.190290
$981$	−24.5285	−0.783134
$982$	−68.6998	−2.19230
$983$	−5.67706	−0.181070	−0.0905351	−	0.995893i	$-0.528858\pi$
$983$	−5.67706	−0.181070	−0.0905351	+	0.995893i	$0.528858\pi$
$984$	70.9378	2.26141
$985$	0.934222	0.0297668
$986$	−40.5050	−1.28994
$987$	38.4389	1.22352
$988$	15.1107	0.480734
$989$	20.7437	0.659611
$990$	−8.80874	−0.279960
$991$	−21.9133	−0.696100	−0.348050	−	0.937476i	$-0.613156\pi$
$991$	−21.9133	−0.696100	−0.348050	+	0.937476i	$0.613156\pi$
$992$	−34.9262	−1.10891
$993$	−8.43151	−0.267566
$994$	108.712	3.44814
$995$	4.45374	0.141193
$996$	31.4502	0.996537
$997$	−13.9667	−0.442330	−0.221165	−	0.975236i	$-0.570986\pi$
$997$	−13.9667	−0.442330	−0.221165	+	0.975236i	$0.570986\pi$
$998$	25.3642	0.802891
$999$	−3.72997	−0.118011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.i.1.2	✓	8
3.2	odd	2		9405.2.a.bf.1.7		8
5.4	even	2		5225.2.a.o.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.i.1.2	✓	8	1.1	even	1	trivial
5225.2.a.o.1.7		8	5.4	even	2
9405.2.a.bf.1.7		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57959 q^{2} -2.53274 q^{3} +4.65426 q^{4} +1.00000 q^{5} +6.53343 q^{6} +2.87748 q^{7} -6.84689 q^{8} +3.41479 q^{9} +O(q^{10})\)	\(q-2.57959 q^{2} -2.53274 q^{3} +4.65426 q^{4} +1.00000 q^{5} +6.53343 q^{6} +2.87748 q^{7} -6.84689 q^{8} +3.41479 q^{9} -2.57959 q^{10} +1.00000 q^{11} -11.7880 q^{12} -3.24663 q^{13} -7.42271 q^{14} -2.53274 q^{15} +8.35361 q^{16} -2.18913 q^{17} -8.80874 q^{18} -1.00000 q^{19} +4.65426 q^{20} -7.28793 q^{21} -2.57959 q^{22} -4.80351 q^{23} +17.3414 q^{24} +1.00000 q^{25} +8.37495 q^{26} -1.05055 q^{27} +13.3926 q^{28} -7.17277 q^{29} +6.53343 q^{30} +4.44632 q^{31} -7.85508 q^{32} -2.53274 q^{33} +5.64705 q^{34} +2.87748 q^{35} +15.8933 q^{36} +3.55048 q^{37} +2.57959 q^{38} +8.22288 q^{39} -6.84689 q^{40} +4.09066 q^{41} +18.7998 q^{42} -4.31844 q^{43} +4.65426 q^{44} +3.41479 q^{45} +12.3911 q^{46} -5.27433 q^{47} -21.1576 q^{48} +1.27991 q^{49} -2.57959 q^{50} +5.54451 q^{51} -15.1107 q^{52} +2.40580 q^{53} +2.70999 q^{54} +1.00000 q^{55} -19.7018 q^{56} +2.53274 q^{57} +18.5028 q^{58} -0.644769 q^{59} -11.7880 q^{60} +14.9985 q^{61} -11.4697 q^{62} +9.82600 q^{63} +3.55562 q^{64} -3.24663 q^{65} +6.53343 q^{66} -0.802158 q^{67} -10.1888 q^{68} +12.1661 q^{69} -7.42271 q^{70} -14.6459 q^{71} -23.3807 q^{72} -5.51740 q^{73} -9.15875 q^{74} -2.53274 q^{75} -4.65426 q^{76} +2.87748 q^{77} -21.2116 q^{78} -4.11472 q^{79} +8.35361 q^{80} -7.58358 q^{81} -10.5522 q^{82} -2.66797 q^{83} -33.9199 q^{84} -2.18913 q^{85} +11.1398 q^{86} +18.1668 q^{87} -6.84689 q^{88} -0.707380 q^{89} -8.80874 q^{90} -9.34212 q^{91} -22.3568 q^{92} -11.2614 q^{93} +13.6056 q^{94} -1.00000 q^{95} +19.8949 q^{96} +15.2075 q^{97} -3.30163 q^{98} +3.41479 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

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