LMFDB - Embedded newform 1045.2.a.i.1.8

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			1.8
Root			$2.87791$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87791 q^{2} -3.37956 q^{3} +1.52654 q^{4} +1.00000 q^{5} -6.34651 q^{6} -0.818685 q^{7} -0.889109 q^{8} +8.42145 q^{9} +O(q^{10})$	\(q+1.87791 q^{2} -3.37956 q^{3} +1.52654 q^{4} +1.00000 q^{5} -6.34651 q^{6} -0.818685 q^{7} -0.889109 q^{8} +8.42145 q^{9} +1.87791 q^{10} +1.00000 q^{11} -5.15905 q^{12} +0.411092 q^{13} -1.53742 q^{14} -3.37956 q^{15} -4.72275 q^{16} -6.74644 q^{17} +15.8147 q^{18} -1.00000 q^{19} +1.52654 q^{20} +2.76680 q^{21} +1.87791 q^{22} -0.484830 q^{23} +3.00480 q^{24} +1.00000 q^{25} +0.771994 q^{26} -18.3221 q^{27} -1.24976 q^{28} +6.81675 q^{29} -6.34651 q^{30} -9.24353 q^{31} -7.09068 q^{32} -3.37956 q^{33} -12.6692 q^{34} -0.818685 q^{35} +12.8557 q^{36} -4.50941 q^{37} -1.87791 q^{38} -1.38931 q^{39} -0.889109 q^{40} -5.00288 q^{41} +5.19579 q^{42} -2.51246 q^{43} +1.52654 q^{44} +8.42145 q^{45} -0.910467 q^{46} -0.294134 q^{47} +15.9608 q^{48} -6.32976 q^{49} +1.87791 q^{50} +22.8000 q^{51} +0.627550 q^{52} -10.1669 q^{53} -34.4073 q^{54} +1.00000 q^{55} +0.727900 q^{56} +3.37956 q^{57} +12.8012 q^{58} -12.3329 q^{59} -5.15905 q^{60} +7.59567 q^{61} -17.3585 q^{62} -6.89451 q^{63} -3.87015 q^{64} +0.411092 q^{65} -6.34651 q^{66} -5.78082 q^{67} -10.2987 q^{68} +1.63852 q^{69} -1.53742 q^{70} -2.88001 q^{71} -7.48759 q^{72} +3.30583 q^{73} -8.46826 q^{74} -3.37956 q^{75} -1.52654 q^{76} -0.818685 q^{77} -2.60900 q^{78} +4.84475 q^{79} -4.72275 q^{80} +36.6565 q^{81} -9.39496 q^{82} +4.01132 q^{83} +4.22364 q^{84} -6.74644 q^{85} -4.71817 q^{86} -23.0376 q^{87} -0.889109 q^{88} +5.05047 q^{89} +15.8147 q^{90} -0.336555 q^{91} -0.740115 q^{92} +31.2391 q^{93} -0.552356 q^{94} -1.00000 q^{95} +23.9634 q^{96} +11.7319 q^{97} -11.8867 q^{98} +8.42145 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.87791	1.32788	0.663941	−	0.747785i	$-0.268883\pi$
$2$	1.87791	1.32788	0.663941	+	0.747785i	$0.268883\pi$
$3$	−3.37956	−1.95119	−0.975596	−	0.219573i	$-0.929534\pi$
$3$	−3.37956	−1.95119	−0.975596	+	0.219573i	$0.929534\pi$
$4$	1.52654	0.763272
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−6.34651	−2.59095
$7$	−0.818685	−0.309434	−0.154717	−	0.987959i	$-0.549447\pi$
$7$	−0.818685	−0.309434	−0.154717	+	0.987959i	$0.549447\pi$
$8$	−0.889109	−0.314347
$9$	8.42145	2.80715
$10$	1.87791	0.593847
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−5.15905	−1.48929
$13$	0.411092	0.114016	0.0570082	−	0.998374i	$-0.481844\pi$
$13$	0.411092	0.114016	0.0570082	+	0.998374i	$0.481844\pi$
$14$	−1.53742	−0.410892
$15$	−3.37956	−0.872600
$16$	−4.72275	−1.18069
$17$	−6.74644	−1.63625	−0.818126	−	0.575039i	$-0.804987\pi$
$17$	−6.74644	−1.63625	−0.818126	+	0.575039i	$0.804987\pi$
$18$	15.8147	3.72757
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.52654	0.341345
$21$	2.76680	0.603765
$22$	1.87791	0.400372
$23$	−0.484830	−0.101094	−0.0505471	−	0.998722i	$-0.516096\pi$
$23$	−0.484830	−0.101094	−0.0505471	+	0.998722i	$0.516096\pi$
$24$	3.00480	0.613352
$25$	1.00000	0.200000
$26$	0.771994	0.151400
$27$	−18.3221	−3.52610
$28$	−1.24976	−0.236182
$29$	6.81675	1.26584	0.632919	−	0.774218i	$-0.281857\pi$
$29$	6.81675	1.26584	0.632919	+	0.774218i	$0.281857\pi$
$30$	−6.34651	−1.15871
$31$	−9.24353	−1.66019	−0.830094	−	0.557624i	$-0.811713\pi$
$31$	−9.24353	−1.66019	−0.830094	+	0.557624i	$0.811713\pi$
$32$	−7.09068	−1.25347
$33$	−3.37956	−0.588307
$34$	−12.6692	−2.17275
$35$	−0.818685	−0.138383
$36$	12.8557	2.14262
$37$	−4.50941	−0.741342	−0.370671	−	0.928764i	$-0.620872\pi$
$37$	−4.50941	−0.741342	−0.370671	+	0.928764i	$0.620872\pi$
$38$	−1.87791	−0.304637
$39$	−1.38931	−0.222468
$40$	−0.889109	−0.140580
$41$	−5.00288	−0.781319	−0.390660	−	0.920535i	$-0.627753\pi$
$41$	−5.00288	−0.781319	−0.390660	+	0.920535i	$0.627753\pi$
$42$	5.19579	0.801728
$43$	−2.51246	−0.383147	−0.191573	−	0.981478i	$-0.561359\pi$
$43$	−2.51246	−0.383147	−0.191573	+	0.981478i	$0.561359\pi$
$44$	1.52654	0.230135
$45$	8.42145	1.25540
$46$	−0.910467	−0.134241
$47$	−0.294134	−0.0429038	−0.0214519	−	0.999770i	$-0.506829\pi$
$47$	−0.294134	−0.0429038	−0.0214519	+	0.999770i	$0.506829\pi$
$48$	15.9608	2.30375
$49$	−6.32976	−0.904251
$50$	1.87791	0.265576
$51$	22.8000	3.19264
$52$	0.627550	0.0870256
$53$	−10.1669	−1.39653	−0.698266	−	0.715838i	$-0.746045\pi$
$53$	−10.1669	−1.39653	−0.698266	+	0.715838i	$0.746045\pi$
$54$	−34.4073	−4.68224
$55$	1.00000	0.134840
$56$	0.727900	0.0972697
$57$	3.37956	0.447634
$58$	12.8012	1.68088
$59$	−12.3329	−1.60560	−0.802802	−	0.596245i	$-0.796659\pi$
$59$	−12.3329	−1.60560	−0.802802	+	0.596245i	$0.796659\pi$
$60$	−5.15905	−0.666031
$61$	7.59567	0.972526	0.486263	−	0.873813i	$-0.338360\pi$
$61$	7.59567	0.972526	0.486263	+	0.873813i	$0.338360\pi$
$62$	−17.3585	−2.20453
$63$	−6.89451	−0.868627
$64$	−3.87015	−0.483769
$65$	0.411092	0.0509897
$66$	−6.34651	−0.781202
$67$	−5.78082	−0.706240	−0.353120	−	0.935578i	$-0.614879\pi$
$67$	−5.78082	−0.706240	−0.353120	+	0.935578i	$0.614879\pi$
$68$	−10.2987	−1.24890
$69$	1.63852	0.197254
$70$	−1.53742	−0.183756
$71$	−2.88001	−0.341795	−0.170897	−	0.985289i	$-0.554667\pi$
$71$	−2.88001	−0.341795	−0.170897	+	0.985289i	$0.554667\pi$
$72$	−7.48759	−0.882420
$73$	3.30583	0.386918	0.193459	−	0.981108i	$-0.438029\pi$
$73$	3.30583	0.386918	0.193459	+	0.981108i	$0.438029\pi$
$74$	−8.46826	−0.984415
$75$	−3.37956	−0.390238
$76$	−1.52654	−0.175107
$77$	−0.818685	−0.0932978
$78$	−2.60900	−0.295411
$79$	4.84475	0.545077	0.272539	−	0.962145i	$-0.412137\pi$
$79$	4.84475	0.545077	0.272539	+	0.962145i	$0.412137\pi$
$80$	−4.72275	−0.528020
$81$	36.6565	4.07294
$82$	−9.39496	−1.03750
$83$	4.01132	0.440300	0.220150	−	0.975466i	$-0.429345\pi$
$83$	4.01132	0.440300	0.220150	+	0.975466i	$0.429345\pi$
$84$	4.22364	0.460836
$85$	−6.74644	−0.731754
$86$	−4.71817	−0.508774
$87$	−23.0376	−2.46989
$88$	−0.889109	−0.0947793
$89$	5.05047	0.535349	0.267675	−	0.963509i	$-0.413745\pi$
$89$	5.05047	0.535349	0.267675	+	0.963509i	$0.413745\pi$
$90$	15.8147	1.66702
$91$	−0.336555	−0.0352805
$92$	−0.740115	−0.0771623
$93$	31.2391	3.23934
$94$	−0.552356	−0.0569712
$95$	−1.00000	−0.102598
$96$	23.9634	2.44576
$97$	11.7319	1.19119	0.595597	−	0.803283i	$-0.296916\pi$
$97$	11.7319	1.19119	0.595597	+	0.803283i	$0.296916\pi$
$98$	−11.8867	−1.20074
$99$	8.42145	0.846388
$100$	1.52654	0.152654
$101$	8.47182	0.842977	0.421489	−	0.906834i	$-0.361508\pi$
$101$	8.47182	0.842977	0.421489	+	0.906834i	$0.361508\pi$
$102$	42.8164	4.23945
$103$	1.21381	0.119600	0.0597999	−	0.998210i	$-0.480954\pi$
$103$	1.21381	0.119600	0.0597999	+	0.998210i	$0.480954\pi$
$104$	−0.365506	−0.0358408
$105$	2.76680	0.270012
$106$	−19.0925	−1.85443
$107$	−7.50408	−0.725447	−0.362723	−	0.931897i	$-0.618153\pi$
$107$	−7.50408	−0.725447	−0.362723	+	0.931897i	$0.618153\pi$
$108$	−27.9695	−2.69137
$109$	6.97137	0.667736	0.333868	−	0.942620i	$-0.391646\pi$
$109$	6.97137	0.667736	0.333868	+	0.942620i	$0.391646\pi$
$110$	1.87791	0.179052
$111$	15.2398	1.44650
$112$	3.86644	0.365345
$113$	14.1527	1.33137	0.665685	−	0.746232i	$-0.268139\pi$
$113$	14.1527	1.33137	0.665685	+	0.746232i	$0.268139\pi$
$114$	6.34651	0.594406
$115$	−0.484830	−0.0452107
$116$	10.4061	0.966179
$117$	3.46199	0.320061
$118$	−23.1600	−2.13205
$119$	5.52320	0.506311
$120$	3.00480	0.274299
$121$	1.00000	0.0909091
$122$	14.2640	1.29140
$123$	16.9076	1.52450
$124$	−14.1107	−1.26717
$125$	1.00000	0.0894427
$126$	−12.9473	−1.15343
$127$	5.86848	0.520743	0.260372	−	0.965509i	$-0.416155\pi$
$127$	5.86848	0.520743	0.260372	+	0.965509i	$0.416155\pi$
$128$	6.91357	0.611079
$129$	8.49102	0.747593
$130$	0.771994	0.0677084
$131$	4.36451	0.381329	0.190664	−	0.981655i	$-0.438936\pi$
$131$	4.36451	0.381329	0.190664	+	0.981655i	$0.438936\pi$
$132$	−5.15905	−0.449038
$133$	0.818685	0.0709890
$134$	−10.8559	−0.937803
$135$	−18.3221	−1.57692
$136$	5.99832	0.514351
$137$	20.8079	1.77774	0.888871	−	0.458158i	$-0.151490\pi$
$137$	20.8079	1.77774	0.888871	+	0.458158i	$0.151490\pi$
$138$	3.07698	0.261930
$139$	−12.0810	−1.02470	−0.512348	−	0.858778i	$-0.671224\pi$
$139$	−12.0810	−1.02470	−0.512348	+	0.858778i	$0.671224\pi$
$140$	−1.24976	−0.105624
$141$	0.994043	0.0837135
$142$	−5.40841	−0.453863
$143$	0.411092	0.0343773
$144$	−39.7724	−3.31437
$145$	6.81675	0.566100
$146$	6.20804	0.513781
$147$	21.3918	1.76437
$148$	−6.88380	−0.565845
$149$	−17.9408	−1.46977	−0.734883	−	0.678194i	$-0.762763\pi$
$149$	−17.9408	−1.46977	−0.734883	+	0.678194i	$0.762763\pi$
$150$	−6.34651	−0.518191
$151$	−5.06925	−0.412530	−0.206265	−	0.978496i	$-0.566131\pi$
$151$	−5.06925	−0.412530	−0.206265	+	0.978496i	$0.566131\pi$
$152$	0.889109	0.0721162
$153$	−56.8148	−4.59320
$154$	−1.53742	−0.123888
$155$	−9.24353	−0.742458
$156$	−2.12085	−0.169804
$157$	−20.2477	−1.61594	−0.807971	−	0.589222i	$-0.799434\pi$
$157$	−20.2477	−1.61594	−0.807971	+	0.589222i	$0.799434\pi$
$158$	9.09800	0.723798
$159$	34.3597	2.72490
$160$	−7.09068	−0.560568
$161$	0.396923	0.0312819
$162$	68.8376	5.40839
$163$	6.34640	0.497088	0.248544	−	0.968621i	$-0.420048\pi$
$163$	6.34640	0.497088	0.248544	+	0.968621i	$0.420048\pi$
$164$	−7.63712	−0.596359
$165$	−3.37956	−0.263099
$166$	7.53290	0.584666
$167$	−17.8322	−1.37990	−0.689948	−	0.723859i	$-0.742367\pi$
$167$	−17.8322	−1.37990	−0.689948	+	0.723859i	$0.742367\pi$
$168$	−2.45998	−0.189792
$169$	−12.8310	−0.987000
$170$	−12.6692	−0.971683
$171$	−8.42145	−0.644004
$172$	−3.83538	−0.292445
$173$	7.28726	0.554040	0.277020	−	0.960864i	$-0.410653\pi$
$173$	7.28726	0.554040	0.277020	+	0.960864i	$0.410653\pi$
$174$	−43.2626	−3.27973
$175$	−0.818685	−0.0618867
$176$	−4.72275	−0.355991
$177$	41.6798	3.13284
$178$	9.48433	0.710881
$179$	13.4365	1.00429	0.502147	−	0.864782i	$-0.332544\pi$
$179$	13.4365	1.00429	0.502147	+	0.864782i	$0.332544\pi$
$180$	12.8557	0.958208
$181$	−17.7602	−1.32010	−0.660051	−	0.751221i	$-0.729465\pi$
$181$	−17.7602	−1.32010	−0.660051	+	0.751221i	$0.729465\pi$
$182$	−0.632020	−0.0468484
$183$	−25.6700	−1.89758
$184$	0.431067	0.0317787
$185$	−4.50941	−0.331538
$186$	58.6642	4.30147
$187$	−6.74644	−0.493348
$188$	−0.449008	−0.0327472
$189$	15.0001	1.09109
$190$	−1.87791	−0.136238
$191$	−23.4895	−1.69964	−0.849821	−	0.527071i	$-0.823290\pi$
$191$	−23.4895	−1.69964	−0.849821	+	0.527071i	$0.823290\pi$
$192$	13.0794	0.943927
$193$	−24.7392	−1.78077	−0.890385	−	0.455208i	$-0.849565\pi$
$193$	−24.7392	−1.78077	−0.890385	+	0.455208i	$0.849565\pi$
$194$	22.0314	1.58177
$195$	−1.38931	−0.0994907
$196$	−9.66265	−0.690189
$197$	−14.3150	−1.01990	−0.509949	−	0.860205i	$-0.670336\pi$
$197$	−14.3150	−1.01990	−0.509949	+	0.860205i	$0.670336\pi$
$198$	15.8147	1.12390
$199$	25.7462	1.82510	0.912549	−	0.408966i	$-0.134111\pi$
$199$	25.7462	1.82510	0.912549	+	0.408966i	$0.134111\pi$
$200$	−0.889109	−0.0628695
$201$	19.5367	1.37801
$202$	15.9093	1.11937
$203$	−5.58077	−0.391693
$204$	34.8052	2.43685
$205$	−5.00288	−0.349416
$206$	2.27942	0.158815
$207$	−4.08298	−0.283786
$208$	−1.94149	−0.134618
$209$	−1.00000	−0.0691714
$210$	5.19579	0.358544
$211$	24.4544	1.68351	0.841755	−	0.539859i	$-0.181523\pi$
$211$	24.4544	1.68351	0.841755	+	0.539859i	$0.181523\pi$
$212$	−15.5202	−1.06593
$213$	9.73319	0.666907
$214$	−14.0920	−0.963308
$215$	−2.51246	−0.171348
$216$	16.2904	1.10842
$217$	7.56754	0.513718
$218$	13.0916	0.886675
$219$	−11.1722	−0.754951
$220$	1.52654	0.102920
$221$	−2.77341	−0.186560
$222$	28.6190	1.92078
$223$	−5.00757	−0.335332	−0.167666	−	0.985844i	$-0.553623\pi$
$223$	−5.00757	−0.335332	−0.167666	+	0.985844i	$0.553623\pi$
$224$	5.80503	0.387865
$225$	8.42145	0.561430
$226$	26.5774	1.76790
$227$	−12.2448	−0.812719	−0.406359	−	0.913713i	$-0.633202\pi$
$227$	−12.2448	−0.812719	−0.406359	+	0.913713i	$0.633202\pi$
$228$	5.15905	0.341666
$229$	5.64307	0.372904	0.186452	−	0.982464i	$-0.440301\pi$
$229$	5.64307	0.372904	0.186452	+	0.982464i	$0.440301\pi$
$230$	−0.910467	−0.0600344
$231$	2.76680	0.182042
$232$	−6.06083	−0.397913
$233$	26.6134	1.74350	0.871752	−	0.489948i	$-0.162984\pi$
$233$	26.6134	1.74350	0.871752	+	0.489948i	$0.162984\pi$
$234$	6.50131	0.425004
$235$	−0.294134	−0.0191872
$236$	−18.8267	−1.22551
$237$	−16.3731	−1.06355
$238$	10.3721	0.672322
$239$	7.23077	0.467719	0.233860	−	0.972270i	$-0.424864\pi$
$239$	7.23077	0.467719	0.233860	+	0.972270i	$0.424864\pi$
$240$	15.9608	1.03027
$241$	−10.7298	−0.691168	−0.345584	−	0.938388i	$-0.612319\pi$
$241$	−10.7298	−0.691168	−0.345584	+	0.938388i	$0.612319\pi$
$242$	1.87791	0.120717
$243$	−68.9165	−4.42100
$244$	11.5951	0.742301
$245$	−6.32976	−0.404393
$246$	31.7509	2.02436
$247$	−0.411092	−0.0261572
$248$	8.21850	0.521876
$249$	−13.5565	−0.859109
$250$	1.87791	0.118769
$251$	−24.0099	−1.51549	−0.757746	−	0.652550i	$-0.773699\pi$
$251$	−24.0099	−1.51549	−0.757746	+	0.652550i	$0.773699\pi$
$252$	−10.5248	−0.662998
$253$	−0.484830	−0.0304810
$254$	11.0205	0.691485
$255$	22.8000	1.42779
$256$	20.7234	1.29521
$257$	2.41874	0.150877	0.0754384	−	0.997150i	$-0.475964\pi$
$257$	2.41874	0.150877	0.0754384	+	0.997150i	$0.475964\pi$
$258$	15.9454	0.992715
$259$	3.69178	0.229396
$260$	0.627550	0.0389190
$261$	57.4069	3.55340
$262$	8.19615	0.506360
$263$	13.1054	0.808111	0.404055	−	0.914735i	$-0.367600\pi$
$263$	13.1054	0.808111	0.404055	+	0.914735i	$0.367600\pi$
$264$	3.00480	0.184933
$265$	−10.1669	−0.624548
$266$	1.53742	0.0942650
$267$	−17.0684	−1.04457
$268$	−8.82467	−0.539053
$269$	14.6974	0.896116	0.448058	−	0.894004i	$-0.352116\pi$
$269$	14.6974	0.896116	0.448058	+	0.894004i	$0.352116\pi$
$270$	−34.4073	−2.09396
$271$	15.1222	0.918609	0.459305	−	0.888279i	$-0.348099\pi$
$271$	15.1222	0.918609	0.459305	+	0.888279i	$0.348099\pi$
$272$	31.8618	1.93190
$273$	1.13741	0.0688391
$274$	39.0754	2.36063
$275$	1.00000	0.0603023
$276$	2.50126	0.150558
$277$	10.1124	0.607596	0.303798	−	0.952736i	$-0.401745\pi$
$277$	10.1124	0.607596	0.303798	+	0.952736i	$0.401745\pi$
$278$	−22.6870	−1.36068
$279$	−77.8440	−4.66040
$280$	0.727900	0.0435003
$281$	−23.3247	−1.39143	−0.695716	−	0.718317i	$-0.744913\pi$
$281$	−23.3247	−1.39143	−0.695716	+	0.718317i	$0.744913\pi$
$282$	1.86672	0.111162
$283$	4.19859	0.249580	0.124790	−	0.992183i	$-0.460174\pi$
$283$	4.19859	0.249580	0.124790	+	0.992183i	$0.460174\pi$
$284$	−4.39647	−0.260882
$285$	3.37956	0.200188
$286$	0.771994	0.0456490
$287$	4.09578	0.241766
$288$	−59.7138	−3.51867
$289$	28.5144	1.67732
$290$	12.8012	0.751714
$291$	−39.6487	−2.32425
$292$	5.04649	0.295323
$293$	−13.9932	−0.817489	−0.408744	−	0.912649i	$-0.634033\pi$
$293$	−13.9932	−0.817489	−0.408744	+	0.912649i	$0.634033\pi$
$294$	40.1719	2.34287
$295$	−12.3329	−0.718048
$296$	4.00935	0.233039
$297$	−18.3221	−1.06316
$298$	−33.6911	−1.95168
$299$	−0.199310	−0.0115264
$300$	−5.15905	−0.297858
$301$	2.05691	0.118559
$302$	−9.51960	−0.547791
$303$	−28.6310	−1.64481
$304$	4.72275	0.270868
$305$	7.59567	0.434927
$306$	−106.693	−6.09923
$307$	−25.4812	−1.45429	−0.727145	−	0.686484i	$-0.759153\pi$
$307$	−25.4812	−1.45429	−0.727145	+	0.686484i	$0.759153\pi$
$308$	−1.24976	−0.0712115
$309$	−4.10213	−0.233362
$310$	−17.3585	−0.985897
$311$	12.9273	0.733042	0.366521	−	0.930410i	$-0.380549\pi$
$311$	12.9273	0.733042	0.366521	+	0.930410i	$0.380549\pi$
$312$	1.23525	0.0699323
$313$	26.6036	1.50373	0.751863	−	0.659319i	$-0.229155\pi$
$313$	26.6036	1.50373	0.751863	+	0.659319i	$0.229155\pi$
$314$	−38.0234	−2.14578
$315$	−6.89451	−0.388462
$316$	7.39572	0.416042
$317$	10.5630	0.593275	0.296638	−	0.954990i	$-0.404135\pi$
$317$	10.5630	0.593275	0.296638	+	0.954990i	$0.404135\pi$
$318$	64.5244	3.61835
$319$	6.81675	0.381665
$320$	−3.87015	−0.216348
$321$	25.3605	1.41549
$322$	0.745386	0.0415387
$323$	6.74644	0.375382
$324$	55.9577	3.10876
$325$	0.411092	0.0228033
$326$	11.9180	0.660075
$327$	−23.5602	−1.30288
$328$	4.44811	0.245606
$329$	0.240803	0.0132759
$330$	−6.34651	−0.349364
$331$	2.91922	0.160455	0.0802274	−	0.996777i	$-0.474435\pi$
$331$	2.91922	0.160455	0.0802274	+	0.996777i	$0.474435\pi$
$332$	6.12345	0.336068
$333$	−37.9757	−2.08106
$334$	−33.4872	−1.83234
$335$	−5.78082	−0.315840
$336$	−13.0669	−0.712858
$337$	−12.8468	−0.699812	−0.349906	−	0.936785i	$-0.613786\pi$
$337$	−12.8468	−0.699812	−0.349906	+	0.936785i	$0.613786\pi$
$338$	−24.0955	−1.31062
$339$	−47.8298	−2.59776
$340$	−10.2987	−0.558527
$341$	−9.24353	−0.500565
$342$	−15.8147	−0.855162
$343$	10.9129	0.589239
$344$	2.23385	0.120441
$345$	1.63852	0.0882147
$346$	13.6848	0.735700
$347$	−2.11651	−0.113620	−0.0568102	−	0.998385i	$-0.518093\pi$
$347$	−2.11651	−0.113620	−0.0568102	+	0.998385i	$0.518093\pi$
$348$	−35.1680	−1.88520
$349$	−30.2265	−1.61799	−0.808994	−	0.587817i	$-0.799987\pi$
$349$	−30.2265	−1.61799	−0.808994	+	0.587817i	$0.799987\pi$
$350$	−1.53742	−0.0821783
$351$	−7.53209	−0.402033
$352$	−7.09068	−0.377935
$353$	26.6979	1.42099	0.710493	−	0.703704i	$-0.248472\pi$
$353$	26.6979	1.42099	0.710493	+	0.703704i	$0.248472\pi$
$354$	78.2708	4.16005
$355$	−2.88001	−0.152855
$356$	7.70977	0.408617
$357$	−18.6660	−0.987911
$358$	25.2326	1.33358
$359$	23.1290	1.22070	0.610351	−	0.792131i	$-0.291029\pi$
$359$	23.1290	1.22070	0.610351	+	0.792131i	$0.291029\pi$
$360$	−7.48759	−0.394630
$361$	1.00000	0.0526316
$362$	−33.3520	−1.75294
$363$	−3.37956	−0.177381
$364$	−0.513766	−0.0269286
$365$	3.30583	0.173035
$366$	−48.2060	−2.51977
$367$	−22.1243	−1.15488	−0.577441	−	0.816433i	$-0.695948\pi$
$367$	−22.1243	−1.15488	−0.577441	+	0.816433i	$0.695948\pi$
$368$	2.28973	0.119361
$369$	−42.1315	−2.19328
$370$	−8.46826	−0.440244
$371$	8.32349	0.432134
$372$	47.6878	2.47250
$373$	−11.5431	−0.597681	−0.298841	−	0.954303i	$-0.596600\pi$
$373$	−11.5431	−0.597681	−0.298841	+	0.954303i	$0.596600\pi$
$374$	−12.6692	−0.655109
$375$	−3.37956	−0.174520
$376$	0.261517	0.0134867
$377$	2.80231	0.144326
$378$	28.1687	1.44884
$379$	12.0741	0.620205	0.310103	−	0.950703i	$-0.399637\pi$
$379$	12.0741	0.620205	0.310103	+	0.950703i	$0.399637\pi$
$380$	−1.52654	−0.0783100
$381$	−19.8329	−1.01607
$382$	−44.1112	−2.25693
$383$	6.08982	0.311175	0.155588	−	0.987822i	$-0.450273\pi$
$383$	6.08982	0.311175	0.155588	+	0.987822i	$0.450273\pi$
$384$	−23.3648	−1.19233
$385$	−0.818685	−0.0417240
$386$	−46.4581	−2.36465
$387$	−21.1586	−1.07555
$388$	17.9093	0.909205
$389$	−9.41019	−0.477115	−0.238558	−	0.971128i	$-0.576675\pi$
$389$	−9.41019	−0.477115	−0.238558	+	0.971128i	$0.576675\pi$
$390$	−2.60900	−0.132112
$391$	3.27088	0.165415
$392$	5.62784	0.284249
$393$	−14.7501	−0.744046
$394$	−26.8822	−1.35431
$395$	4.84475	0.243766
$396$	12.8557	0.646024
$397$	1.02994	0.0516911	0.0258455	−	0.999666i	$-0.491772\pi$
$397$	1.02994	0.0516911	0.0258455	+	0.999666i	$0.491772\pi$
$398$	48.3490	2.42352
$399$	−2.76680	−0.138513
$400$	−4.72275	−0.236138
$401$	−26.1564	−1.30619	−0.653094	−	0.757277i	$-0.726529\pi$
$401$	−26.1564	−1.30619	−0.653094	+	0.757277i	$0.726529\pi$
$402$	36.6881	1.82983
$403$	−3.79994	−0.189289
$404$	12.9326	0.643421
$405$	36.6565	1.82148
$406$	−10.4802	−0.520122
$407$	−4.50941	−0.223523
$408$	−20.2717	−1.00360
$409$	−3.32785	−0.164551	−0.0822757	−	0.996610i	$-0.526219\pi$
$409$	−3.32785	−0.164551	−0.0822757	+	0.996610i	$0.526219\pi$
$410$	−9.39496	−0.463984
$411$	−70.3217	−3.46872
$412$	1.85293	0.0912872
$413$	10.0967	0.496828
$414$	−7.66746	−0.376835
$415$	4.01132	0.196908
$416$	−2.91493	−0.142916
$417$	40.8285	1.99938
$418$	−1.87791	−0.0918515
$419$	−5.88353	−0.287429	−0.143715	−	0.989619i	$-0.545905\pi$
$419$	−5.88353	−0.287429	−0.143715	+	0.989619i	$0.545905\pi$
$420$	4.22364	0.206092
$421$	21.2826	1.03725	0.518626	−	0.855001i	$-0.326444\pi$
$421$	21.2826	1.03725	0.518626	+	0.855001i	$0.326444\pi$
$422$	45.9232	2.23550
$423$	−2.47703	−0.120437
$424$	9.03949	0.438996
$425$	−6.74644	−0.327250
$426$	18.2781	0.885574
$427$	−6.21846	−0.300932
$428$	−11.4553	−0.553713
$429$	−1.38931	−0.0670766
$430$	−4.71817	−0.227531
$431$	−37.4139	−1.80216	−0.901082	−	0.433648i	$-0.857226\pi$
$431$	−37.4139	−1.80216	−0.901082	+	0.433648i	$0.857226\pi$
$432$	86.5309	4.16322
$433$	21.2168	1.01962	0.509808	−	0.860288i	$-0.329717\pi$
$433$	21.2168	1.01962	0.509808	+	0.860288i	$0.329717\pi$
$434$	14.2111	0.682157
$435$	−23.0376	−1.10457
$436$	10.6421	0.509664
$437$	0.484830	0.0231926
$438$	−20.9805	−1.00249
$439$	21.8320	1.04199	0.520993	−	0.853561i	$-0.325562\pi$
$439$	21.8320	1.04199	0.520993	+	0.853561i	$0.325562\pi$
$440$	−0.889109	−0.0423866
$441$	−53.3057	−2.53837
$442$	−5.20821	−0.247729
$443$	−21.7515	−1.03344	−0.516722	−	0.856153i	$-0.672848\pi$
$443$	−21.7515	−1.03344	−0.516722	+	0.856153i	$0.672848\pi$
$444$	23.2643	1.10407
$445$	5.05047	0.239415
$446$	−9.40376	−0.445281
$447$	60.6320	2.86779
$448$	3.16844	0.149695
$449$	16.7586	0.790887	0.395444	−	0.918490i	$-0.370591\pi$
$449$	16.7586	0.790887	0.395444	+	0.918490i	$0.370591\pi$
$450$	15.8147	0.745513
$451$	−5.00288	−0.235577
$452$	21.6047	1.01620
$453$	17.1319	0.804925
$454$	−22.9947	−1.07919
$455$	−0.336555	−0.0157779
$456$	−3.00480	−0.140713
$457$	35.3477	1.65349	0.826747	−	0.562573i	$-0.190189\pi$
$457$	35.3477	1.65349	0.826747	+	0.562573i	$0.190189\pi$
$458$	10.5972	0.495173
$459$	123.609	5.76958
$460$	−0.740115	−0.0345080
$461$	−27.4976	−1.28069	−0.640344	−	0.768088i	$-0.721208\pi$
$461$	−27.4976	−1.28069	−0.640344	+	0.768088i	$0.721208\pi$
$462$	5.19579	0.241730
$463$	23.6126	1.09737	0.548686	−	0.836029i	$-0.315128\pi$
$463$	23.6126	1.09737	0.548686	+	0.836029i	$0.315128\pi$
$464$	−32.1938	−1.49456
$465$	31.2391	1.44868
$466$	49.9776	2.31517
$467$	−4.19084	−0.193929	−0.0969645	−	0.995288i	$-0.530913\pi$
$467$	−4.19084	−0.193929	−0.0969645	+	0.995288i	$0.530913\pi$
$468$	5.28488	0.244294
$469$	4.73267	0.218534
$470$	−0.552356	−0.0254783
$471$	68.4284	3.15301
$472$	10.9653	0.504718
$473$	−2.51246	−0.115523
$474$	−30.7473	−1.41227
$475$	−1.00000	−0.0458831
$476$	8.43141	0.386453
$477$	−85.6201	−3.92028
$478$	13.5787	0.621076
$479$	39.1899	1.79063	0.895316	−	0.445433i	$-0.146950\pi$
$479$	39.1899	1.79063	0.895316	+	0.445433i	$0.146950\pi$
$480$	23.9634	1.09378
$481$	−1.85378	−0.0845252
$482$	−20.1496	−0.917789
$483$	−1.34143	−0.0610370
$484$	1.52654	0.0693883
$485$	11.7319	0.532718
$486$	−129.419	−5.87056
$487$	4.99748	0.226457	0.113229	−	0.993569i	$-0.463881\pi$
$487$	4.99748	0.226457	0.113229	+	0.993569i	$0.463881\pi$
$488$	−6.75338	−0.305711
$489$	−21.4481	−0.969914
$490$	−11.8867	−0.536987
$491$	−5.09162	−0.229782	−0.114891	−	0.993378i	$-0.536652\pi$
$491$	−5.09162	−0.229782	−0.114891	+	0.993378i	$0.536652\pi$
$492$	25.8101	1.16361
$493$	−45.9888	−2.07123
$494$	−0.771994	−0.0347337
$495$	8.42145	0.378516
$496$	43.6549	1.96016
$497$	2.35782	0.105763
$498$	−25.4579	−1.14080
$499$	17.7176	0.793148	0.396574	−	0.918003i	$-0.370199\pi$
$499$	17.7176	0.793148	0.396574	+	0.918003i	$0.370199\pi$
$500$	1.52654	0.0682691
$501$	60.2650	2.69244
$502$	−45.0884	−2.01239
$503$	−4.29901	−0.191683	−0.0958417	−	0.995397i	$-0.530554\pi$
$503$	−4.29901	−0.191683	−0.0958417	+	0.995397i	$0.530554\pi$
$504$	6.12997	0.273051
$505$	8.47182	0.376991
$506$	−0.910467	−0.0404752
$507$	43.3632	1.92583
$508$	8.95848	0.397468
$509$	−17.3611	−0.769516	−0.384758	−	0.923017i	$-0.625715\pi$
$509$	−17.3611	−0.769516	−0.384758	+	0.923017i	$0.625715\pi$
$510$	42.8164	1.89594
$511$	−2.70643	−0.119725
$512$	25.0895	1.10881
$513$	18.3221	0.808942
$514$	4.54217	0.200347
$515$	1.21381	0.0534867
$516$	12.9619	0.570616
$517$	−0.294134	−0.0129360
$518$	6.93283	0.304611
$519$	−24.6278	−1.08104
$520$	−0.365506	−0.0160285
$521$	−15.9277	−0.697806	−0.348903	−	0.937159i	$-0.613446\pi$
$521$	−15.9277	−0.697806	−0.348903	+	0.937159i	$0.613446\pi$
$522$	107.805	4.71850
$523$	29.3262	1.28235	0.641173	−	0.767396i	$-0.278448\pi$
$523$	29.3262	1.28235	0.641173	+	0.767396i	$0.278448\pi$
$524$	6.66261	0.291057
$525$	2.76680	0.120753
$526$	24.6107	1.07308
$527$	62.3609	2.71648
$528$	15.9608	0.694606
$529$	−22.7649	−0.989780
$530$	−19.0925	−0.829327
$531$	−103.861	−4.50717
$532$	1.24976	0.0541839
$533$	−2.05665	−0.0890833
$534$	−32.0529	−1.38707
$535$	−7.50408	−0.324430
$536$	5.13978	0.222005
$537$	−45.4096	−1.95957
$538$	27.6004	1.18994
$539$	−6.32976	−0.272642
$540$	−27.9695	−1.20362
$541$	−11.6752	−0.501957	−0.250978	−	0.967993i	$-0.580752\pi$
$541$	−11.6752	−0.501957	−0.250978	+	0.967993i	$0.580752\pi$
$542$	28.3982	1.21981
$543$	60.0216	2.57577
$544$	47.8368	2.05099
$545$	6.97137	0.298621
$546$	2.13595	0.0914103
$547$	−17.0424	−0.728679	−0.364339	−	0.931266i	$-0.618705\pi$
$547$	−17.0424	−0.728679	−0.364339	+	0.931266i	$0.618705\pi$
$548$	31.7642	1.35690
$549$	63.9666	2.73003
$550$	1.87791	0.0800743
$551$	−6.81675	−0.290403
$552$	−1.45682	−0.0620063
$553$	−3.96632	−0.168665
$554$	18.9902	0.806816
$555$	15.2398	0.646895
$556$	−18.4422	−0.782122
$557$	−31.2790	−1.32533	−0.662667	−	0.748914i	$-0.730576\pi$
$557$	−31.2790	−1.32533	−0.662667	+	0.748914i	$0.730576\pi$
$558$	−146.184	−6.18846
$559$	−1.03285	−0.0436850
$560$	3.86644	0.163387
$561$	22.8000	0.962617
$562$	−43.8016	−1.84766
$563$	−29.6265	−1.24861	−0.624305	−	0.781181i	$-0.714618\pi$
$563$	−29.6265	−1.24861	−0.624305	+	0.781181i	$0.714618\pi$
$564$	1.51745	0.0638962
$565$	14.1527	0.595407
$566$	7.88457	0.331413
$567$	−30.0101	−1.26031
$568$	2.56065	0.107442
$569$	−25.5456	−1.07093	−0.535464	−	0.844558i	$-0.679863\pi$
$569$	−25.5456	−1.07093	−0.535464	+	0.844558i	$0.679863\pi$
$570$	6.34651	0.265826
$571$	−25.8832	−1.08318	−0.541589	−	0.840643i	$-0.682177\pi$
$571$	−25.8832	−1.08318	−0.541589	+	0.840643i	$0.682177\pi$
$572$	0.627550	0.0262392
$573$	79.3844	3.31633
$574$	7.69151	0.321037
$575$	−0.484830	−0.0202188
$576$	−32.5923	−1.35801
$577$	−28.0864	−1.16925	−0.584626	−	0.811303i	$-0.698759\pi$
$577$	−28.0864	−1.16925	−0.584626	+	0.811303i	$0.698759\pi$
$578$	53.5475	2.22728
$579$	83.6079	3.47462
$580$	10.4061	0.432088
$581$	−3.28401	−0.136244
$582$	−74.4567	−3.08633
$583$	−10.1669	−0.421070
$584$	−2.93924	−0.121627
$585$	3.46199	0.143136
$586$	−26.2779	−1.08553
$587$	−23.3285	−0.962872	−0.481436	−	0.876481i	$-0.659885\pi$
$587$	−23.3285	−0.962872	−0.481436	+	0.876481i	$0.659885\pi$
$588$	32.6555	1.34669
$589$	9.24353	0.380873
$590$	−23.1600	−0.953484
$591$	48.3783	1.99002
$592$	21.2968	0.875293
$593$	−24.1321	−0.990985	−0.495492	−	0.868612i	$-0.665012\pi$
$593$	−24.1321	−0.990985	−0.495492	+	0.868612i	$0.665012\pi$
$594$	−34.4073	−1.41175
$595$	5.52320	0.226429
$596$	−27.3874	−1.12183
$597$	−87.0109	−3.56112
$598$	−0.374286	−0.0153057
$599$	0.0811542	0.00331587	0.00165793	−	0.999999i	$-0.499472\pi$
$599$	0.0811542	0.00331587	0.00165793	+	0.999999i	$0.499472\pi$
$600$	3.00480	0.122670
$601$	−3.29903	−0.134570	−0.0672851	−	0.997734i	$-0.521434\pi$
$601$	−3.29903	−0.134570	−0.0672851	+	0.997734i	$0.521434\pi$
$602$	3.86270	0.157432
$603$	−48.6829	−1.98252
$604$	−7.73844	−0.314872
$605$	1.00000	0.0406558
$606$	−53.7665	−2.18412
$607$	−29.4199	−1.19412	−0.597058	−	0.802198i	$-0.703664\pi$
$607$	−29.4199	−1.19412	−0.597058	+	0.802198i	$0.703664\pi$
$608$	7.09068	0.287565
$609$	18.8606	0.764268
$610$	14.2640	0.577532
$611$	−0.120916	−0.00489174
$612$	−86.7302	−3.50586
$613$	−35.4607	−1.43224	−0.716122	−	0.697975i	$-0.754084\pi$
$613$	−35.4607	−1.43224	−0.716122	+	0.697975i	$0.754084\pi$
$614$	−47.8514	−1.93113
$615$	16.9076	0.681779
$616$	0.727900	0.0293279
$617$	20.6501	0.831343	0.415671	−	0.909515i	$-0.363547\pi$
$617$	20.6501	0.831343	0.415671	+	0.909515i	$0.363547\pi$
$618$	−7.70344	−0.309878
$619$	36.3198	1.45982	0.729908	−	0.683546i	$-0.239563\pi$
$619$	36.3198	1.45982	0.729908	+	0.683546i	$0.239563\pi$
$620$	−14.1107	−0.566697
$621$	8.88313	0.356468
$622$	24.2764	0.973394
$623$	−4.13475	−0.165655
$624$	6.56138	0.262665
$625$	1.00000	0.0400000
$626$	49.9592	1.99677
$627$	3.37956	0.134967
$628$	−30.9090	−1.23340
$629$	30.4224	1.21302
$630$	−12.9473	−0.515832
$631$	38.4669	1.53134	0.765671	−	0.643232i	$-0.222407\pi$
$631$	38.4669	1.53134	0.765671	+	0.643232i	$0.222407\pi$
$632$	−4.30751	−0.171344
$633$	−82.6452	−3.28485
$634$	19.8363	0.787800
$635$	5.86848	0.232883
$636$	52.4516	2.07984
$637$	−2.60211	−0.103100
$638$	12.8012	0.506806
$639$	−24.2539	−0.959469
$640$	6.91357	0.273283
$641$	4.72684	0.186699	0.0933494	−	0.995633i	$-0.470243\pi$
$641$	4.72684	0.186699	0.0933494	+	0.995633i	$0.470243\pi$
$642$	47.6247	1.87960
$643$	32.5276	1.28276	0.641382	−	0.767222i	$-0.278361\pi$
$643$	32.5276	1.28276	0.641382	+	0.767222i	$0.278361\pi$
$644$	0.605920	0.0238766
$645$	8.49102	0.334334
$646$	12.6692	0.498463
$647$	−2.49180	−0.0979626	−0.0489813	−	0.998800i	$-0.515597\pi$
$647$	−2.49180	−0.0979626	−0.0489813	+	0.998800i	$0.515597\pi$
$648$	−32.5916	−1.28032
$649$	−12.3329	−0.484108
$650$	0.771994	0.0302801
$651$	−25.5750	−1.00236
$652$	9.68805	0.379413
$653$	−13.8678	−0.542687	−0.271344	−	0.962483i	$-0.587468\pi$
$653$	−13.8678	−0.542687	−0.271344	+	0.962483i	$0.587468\pi$
$654$	−44.2439	−1.73007
$655$	4.36451	0.170535
$656$	23.6274	0.922494
$657$	27.8398	1.08614
$658$	0.452205	0.0176288
$659$	3.37583	0.131504	0.0657519	−	0.997836i	$-0.479055\pi$
$659$	3.37583	0.131504	0.0657519	+	0.997836i	$0.479055\pi$
$660$	−5.15905	−0.200816
$661$	17.9273	0.697290	0.348645	−	0.937255i	$-0.386642\pi$
$661$	17.9273	0.697290	0.348645	+	0.937255i	$0.386642\pi$
$662$	5.48203	0.213065
$663$	9.37291	0.364014
$664$	−3.56650	−0.138407
$665$	0.818685	0.0317472
$666$	−71.3150	−2.76340
$667$	−3.30497	−0.127969
$668$	−27.2216	−1.05324
$669$	16.9234	0.654296
$670$	−10.8559	−0.419398
$671$	7.59567	0.293228
$672$	−19.6185	−0.756799
$673$	37.7631	1.45566	0.727831	−	0.685757i	$-0.240529\pi$
$673$	37.7631	1.45566	0.727831	+	0.685757i	$0.240529\pi$
$674$	−24.1252	−0.929268
$675$	−18.3221	−0.705219
$676$	−19.5871	−0.753349
$677$	15.8218	0.608081	0.304041	−	0.952659i	$-0.401664\pi$
$677$	15.8218	0.608081	0.304041	+	0.952659i	$0.401664\pi$
$678$	−89.8201	−3.44952
$679$	−9.60473	−0.368596
$680$	5.99832	0.230025
$681$	41.3822	1.58577
$682$	−17.3585	−0.664692
$683$	−36.3891	−1.39239	−0.696195	−	0.717853i	$-0.745125\pi$
$683$	−36.3891	−1.39239	−0.696195	+	0.717853i	$0.745125\pi$
$684$	−12.8557	−0.491550
$685$	20.8079	0.795030
$686$	20.4934	0.782441
$687$	−19.0711	−0.727608
$688$	11.8657	0.452377
$689$	−4.17954	−0.159228
$690$	3.07698	0.117139
$691$	26.3533	1.00253	0.501264	−	0.865295i	$-0.332869\pi$
$691$	26.3533	1.00253	0.501264	+	0.865295i	$0.332869\pi$
$692$	11.1243	0.422883
$693$	−6.89451	−0.261901
$694$	−3.97462	−0.150875
$695$	−12.0810	−0.458258
$696$	20.4830	0.776405
$697$	33.7516	1.27843
$698$	−56.7626	−2.14850
$699$	−89.9418	−3.40191
$700$	−1.24976	−0.0472364
$701$	43.0566	1.62622	0.813112	−	0.582108i	$-0.197772\pi$
$701$	43.0566	1.62622	0.813112	+	0.582108i	$0.197772\pi$
$702$	−14.1446	−0.533853
$703$	4.50941	0.170075
$704$	−3.87015	−0.145862
$705$	0.994043	0.0374378
$706$	50.1362	1.88690
$707$	−6.93575	−0.260846
$708$	63.6260	2.39121
$709$	−14.7306	−0.553218	−0.276609	−	0.960983i	$-0.589211\pi$
$709$	−14.7306	−0.553218	−0.276609	+	0.960983i	$0.589211\pi$
$710$	−5.40841	−0.202974
$711$	40.7998	1.53011
$712$	−4.49042	−0.168286
$713$	4.48154	0.167835
$714$	−35.0531	−1.31183
$715$	0.411092	0.0153740
$716$	20.5115	0.766549
$717$	−24.4368	−0.912610
$718$	43.4341	1.62095
$719$	12.8598	0.479591	0.239796	−	0.970823i	$-0.422920\pi$
$719$	12.8598	0.479591	0.239796	+	0.970823i	$0.422920\pi$
$720$	−39.7724	−1.48223
$721$	−0.993724	−0.0370082
$722$	1.87791	0.0698885
$723$	36.2621	1.34860
$724$	−27.1116	−1.00760
$725$	6.81675	0.253168
$726$	−6.34651	−0.235541
$727$	35.3905	1.31256	0.656280	−	0.754517i	$-0.272129\pi$
$727$	35.3905	1.31256	0.656280	+	0.754517i	$0.272129\pi$
$728$	0.299234	0.0110903
$729$	122.938	4.55327
$730$	6.20804	0.229770
$731$	16.9502	0.626924
$732$	−39.1864	−1.44837
$733$	−23.6397	−0.873153	−0.436576	−	0.899667i	$-0.643809\pi$
$733$	−23.6397	−0.873153	−0.436576	+	0.899667i	$0.643809\pi$
$734$	−41.5475	−1.53355
$735$	21.3918	0.789049
$736$	3.43778	0.126718
$737$	−5.78082	−0.212939
$738$	−79.1192	−2.91242
$739$	20.8503	0.766991	0.383496	−	0.923543i	$-0.374720\pi$
$739$	20.8503	0.766991	0.383496	+	0.923543i	$0.374720\pi$
$740$	−6.88380	−0.253054
$741$	1.38931	0.0510377
$742$	15.6308	0.573823
$743$	−20.5383	−0.753476	−0.376738	−	0.926320i	$-0.622954\pi$
$743$	−20.5383	−0.753476	−0.376738	+	0.926320i	$0.622954\pi$
$744$	−27.7750	−1.01828
$745$	−17.9408	−0.657299
$746$	−21.6770	−0.793651
$747$	33.7811	1.23599
$748$	−10.2987	−0.376559
$749$	6.14347	0.224478
$750$	−6.34651	−0.231742
$751$	−29.0687	−1.06073	−0.530366	−	0.847769i	$-0.677946\pi$
$751$	−29.0687	−1.06073	−0.530366	+	0.847769i	$0.677946\pi$
$752$	1.38912	0.0506560
$753$	81.1430	2.95701
$754$	5.26249	0.191649
$755$	−5.06925	−0.184489
$756$	22.8982	0.832801
$757$	20.0944	0.730345	0.365172	−	0.930940i	$-0.381010\pi$
$757$	20.0944	0.730345	0.365172	+	0.930940i	$0.381010\pi$
$758$	22.6741	0.823560
$759$	1.63852	0.0594743
$760$	0.889109	0.0322514
$761$	−43.0391	−1.56017	−0.780084	−	0.625675i	$-0.784824\pi$
$761$	−43.0391	−1.56017	−0.780084	+	0.625675i	$0.784824\pi$
$762$	−37.2444	−1.34922
$763$	−5.70735	−0.206620
$764$	−35.8578	−1.29729
$765$	−56.8148	−2.05414
$766$	11.4361	0.413204
$767$	−5.06995	−0.183065
$768$	−70.0359	−2.52720
$769$	−54.0387	−1.94869	−0.974343	−	0.225070i	$-0.927739\pi$
$769$	−54.0387	−1.94869	−0.974343	+	0.225070i	$0.927739\pi$
$770$	−1.53742	−0.0554046
$771$	−8.17428	−0.294390
$772$	−37.7655	−1.35921
$773$	20.2486	0.728290	0.364145	−	0.931342i	$-0.381361\pi$
$773$	20.2486	0.728290	0.364145	+	0.931342i	$0.381361\pi$
$774$	−39.7339	−1.42820
$775$	−9.24353	−0.332037
$776$	−10.4309	−0.374449
$777$	−12.4766	−0.447596
$778$	−17.6715	−0.633553
$779$	5.00288	0.179247
$780$	−2.12085	−0.0759385
$781$	−2.88001	−0.103055
$782$	6.14241	0.219652
$783$	−124.897	−4.46347
$784$	29.8939	1.06764
$785$	−20.2477	−0.722672
$786$	−27.6994	−0.988005
$787$	11.9516	0.426029	0.213014	−	0.977049i	$-0.431672\pi$
$787$	11.9516	0.426029	0.213014	+	0.977049i	$0.431672\pi$
$788$	−21.8524	−0.778460
$789$	−44.2904	−1.57678
$790$	9.09800	0.323692
$791$	−11.5866	−0.411971
$792$	−7.48759	−0.266060
$793$	3.12252	0.110884
$794$	1.93413	0.0686397
$795$	34.3597	1.21861
$796$	39.3027	1.39305
$797$	−36.1482	−1.28044	−0.640218	−	0.768194i	$-0.721156\pi$
$797$	−36.1482	−1.28044	−0.640218	+	0.768194i	$0.721156\pi$
$798$	−5.19579	−0.183929
$799$	1.98435	0.0702014
$800$	−7.09068	−0.250693
$801$	42.5323	1.50281
$802$	−49.1193	−1.73446
$803$	3.30583	0.116660
$804$	29.8235	1.05180
$805$	0.396923	0.0139897
$806$	−7.13595	−0.251353
$807$	−49.6708	−1.74849
$808$	−7.53237	−0.264988
$809$	−6.86396	−0.241324	−0.120662	−	0.992694i	$-0.538502\pi$
$809$	−6.86396	−0.241324	−0.120662	+	0.992694i	$0.538502\pi$
$810$	68.8376	2.41871
$811$	−10.8353	−0.380477	−0.190239	−	0.981738i	$-0.560926\pi$
$811$	−10.8353	−0.380477	−0.190239	+	0.981738i	$0.560926\pi$
$812$	−8.51928	−0.298968
$813$	−51.1065	−1.79238
$814$	−8.46826	−0.296812
$815$	6.34640	0.222305
$816$	−107.679	−3.76951
$817$	2.51246	0.0878999
$818$	−6.24940	−0.218505
$819$	−2.83428	−0.0990378
$820$	−7.63712	−0.266700
$821$	−20.3913	−0.711661	−0.355830	−	0.934551i	$-0.615802\pi$
$821$	−20.3913	−0.711661	−0.355830	+	0.934551i	$0.615802\pi$
$822$	−132.058	−4.60605
$823$	35.7264	1.24534	0.622671	−	0.782483i	$-0.286047\pi$
$823$	35.7264	1.24534	0.622671	+	0.782483i	$0.286047\pi$
$824$	−1.07921	−0.0375959
$825$	−3.37956	−0.117661
$826$	18.9608	0.659729
$827$	−10.5199	−0.365813	−0.182907	−	0.983130i	$-0.558551\pi$
$827$	−10.5199	−0.365813	−0.182907	+	0.983130i	$0.558551\pi$
$828$	−6.23284	−0.216606
$829$	−12.0740	−0.419347	−0.209674	−	0.977771i	$-0.567240\pi$
$829$	−12.0740	−0.419347	−0.209674	+	0.977771i	$0.567240\pi$
$830$	7.53290	0.261471
$831$	−34.1755	−1.18554
$832$	−1.59099	−0.0551577
$833$	42.7033	1.47958
$834$	76.6722	2.65494
$835$	−17.8322	−0.617108
$836$	−1.52654	−0.0527966
$837$	169.361	5.85398
$838$	−11.0487	−0.381672
$839$	35.4044	1.22230	0.611148	−	0.791516i	$-0.290708\pi$
$839$	35.4044	1.22230	0.611148	+	0.791516i	$0.290708\pi$
$840$	−2.45998	−0.0848775
$841$	17.4681	0.602347
$842$	39.9668	1.37735
$843$	78.8272	2.71495
$844$	37.3307	1.28498
$845$	−12.8310	−0.441400
$846$	−4.65164	−0.159927
$847$	−0.818685	−0.0281303
$848$	48.0158	1.64887
$849$	−14.1894	−0.486979
$850$	−12.6692	−0.434550
$851$	2.18630	0.0749453
$852$	14.8581	0.509031
$853$	−34.2446	−1.17251	−0.586256	−	0.810126i	$-0.699399\pi$
$853$	−34.2446	−1.17251	−0.586256	+	0.810126i	$0.699399\pi$
$854$	−11.6777	−0.399603
$855$	−8.42145	−0.288008
$856$	6.67194	0.228042
$857$	46.1746	1.57729	0.788647	−	0.614846i	$-0.210782\pi$
$857$	46.1746	1.57729	0.788647	+	0.614846i	$0.210782\pi$
$858$	−2.60900	−0.0890699
$859$	49.0717	1.67430	0.837152	−	0.546970i	$-0.184219\pi$
$859$	49.0717	1.67430	0.837152	+	0.546970i	$0.184219\pi$
$860$	−3.83538	−0.130785
$861$	−13.8420	−0.471733
$862$	−70.2599	−2.39306
$863$	−24.5186	−0.834622	−0.417311	−	0.908764i	$-0.637027\pi$
$863$	−24.5186	−0.834622	−0.417311	+	0.908764i	$0.637027\pi$
$864$	129.916	4.41985
$865$	7.28726	0.247774
$866$	39.8433	1.35393
$867$	−96.3663	−3.27277
$868$	11.5522	0.392106
$869$	4.84475	0.164347
$870$	−43.2626	−1.46674
$871$	−2.37645	−0.0805230
$872$	−6.19831	−0.209901
$873$	98.7996	3.34386
$874$	0.910467	0.0307970
$875$	−0.818685	−0.0276766
$876$	−17.0549	−0.576232
$877$	−16.3666	−0.552662	−0.276331	−	0.961063i	$-0.589119\pi$
$877$	−16.3666	−0.552662	−0.276331	+	0.961063i	$0.589119\pi$
$878$	40.9986	1.38364
$879$	47.2908	1.59508
$880$	−4.72275	−0.159204
$881$	−3.27376	−0.110296	−0.0551480	−	0.998478i	$-0.517563\pi$
$881$	−3.27376	−0.110296	−0.0551480	+	0.998478i	$0.517563\pi$
$882$	−100.103	−3.37065
$883$	34.4824	1.16042	0.580212	−	0.814466i	$-0.302970\pi$
$883$	34.4824	1.16042	0.580212	+	0.814466i	$0.302970\pi$
$884$	−4.23373	−0.142396
$885$	41.6798	1.40105
$886$	−40.8473	−1.37229
$887$	29.8217	1.00131	0.500657	−	0.865646i	$-0.333092\pi$
$887$	29.8217	1.00131	0.500657	+	0.865646i	$0.333092\pi$
$888$	−13.5499	−0.454704
$889$	−4.80443	−0.161135
$890$	9.48433	0.317916
$891$	36.6565	1.22804
$892$	−7.64427	−0.255949
$893$	0.294134	0.00984280
$894$	113.861	3.80809
$895$	13.4365	0.449134
$896$	−5.66003	−0.189088
$897$	0.673581	0.0224902
$898$	31.4711	1.05021
$899$	−63.0108	−2.10153
$900$	12.8557	0.428524
$901$	68.5904	2.28508
$902$	−9.39496	−0.312818
$903$	−6.95147	−0.231330
$904$	−12.5833	−0.418513
$905$	−17.7602	−0.590367
$906$	32.1721	1.06885
$907$	7.77868	0.258287	0.129143	−	0.991626i	$-0.458777\pi$
$907$	7.77868	0.258287	0.129143	+	0.991626i	$0.458777\pi$
$908$	−18.6923	−0.620325
$909$	71.3450	2.36636
$910$	−0.632020	−0.0209512
$911$	−42.5468	−1.40964	−0.704820	−	0.709387i	$-0.748972\pi$
$911$	−42.5468	−1.40964	−0.704820	+	0.709387i	$0.748972\pi$
$912$	−15.9608	−0.528516
$913$	4.01132	0.132755
$914$	66.3797	2.19565
$915$	−25.6700	−0.848626
$916$	8.61439	0.284627
$917$	−3.57315	−0.117996
$918$	232.127	7.66133
$919$	−22.8711	−0.754449	−0.377225	−	0.926122i	$-0.623122\pi$
$919$	−22.8711	−0.754449	−0.377225	+	0.926122i	$0.623122\pi$
$920$	0.431067	0.0142119
$921$	86.1154	2.83760
$922$	−51.6379	−1.70060
$923$	−1.18395	−0.0389702
$924$	4.22364	0.138947
$925$	−4.50941	−0.148268
$926$	44.3424	1.45718
$927$	10.2220	0.335735
$928$	−48.3354	−1.58669
$929$	−23.5368	−0.772219	−0.386109	−	0.922453i	$-0.626181\pi$
$929$	−23.5368	−0.772219	−0.386109	+	0.922453i	$0.626181\pi$
$930$	58.6642	1.92368
$931$	6.32976	0.207449
$932$	40.6265	1.33077
$933$	−43.6888	−1.43031
$934$	−7.87002	−0.257515
$935$	−6.74644	−0.220632
$936$	−3.07809	−0.100610
$937$	23.0465	0.752895	0.376447	−	0.926438i	$-0.377146\pi$
$937$	23.0465	0.752895	0.376447	+	0.926438i	$0.377146\pi$
$938$	8.88752	0.290188
$939$	−89.9087	−2.93406
$940$	−0.449008	−0.0146450
$941$	−36.0742	−1.17599	−0.587993	−	0.808866i	$-0.700082\pi$
$941$	−36.0742	−1.17599	−0.587993	+	0.808866i	$0.700082\pi$
$942$	128.502	4.18683
$943$	2.42555	0.0789868
$944$	58.2452	1.89572
$945$	15.0001	0.487952
$946$	−4.71817	−0.153401
$947$	−35.9989	−1.16981	−0.584903	−	0.811103i	$-0.698867\pi$
$947$	−35.9989	−1.16981	−0.584903	+	0.811103i	$0.698867\pi$
$948$	−24.9943	−0.811778
$949$	1.35900	0.0441150
$950$	−1.87791	−0.0609274
$951$	−35.6982	−1.15759
$952$	−4.91073	−0.159158
$953$	−28.3167	−0.917269	−0.458635	−	0.888625i	$-0.651661\pi$
$953$	−28.3167	−0.917269	−0.458635	+	0.888625i	$0.651661\pi$
$954$	−160.787	−5.20567
$955$	−23.4895	−0.760103
$956$	11.0381	0.356997
$957$	−23.0376	−0.744701
$958$	73.5950	2.37775
$959$	−17.0351	−0.550093
$960$	13.0794	0.422137
$961$	54.4429	1.75622
$962$	−3.48123	−0.112240
$963$	−63.1952	−2.03644
$964$	−16.3795	−0.527549
$965$	−24.7392	−0.796385
$966$	−2.51908	−0.0810500
$967$	2.75238	0.0885107	0.0442553	−	0.999020i	$-0.485908\pi$
$967$	2.75238	0.0885107	0.0442553	+	0.999020i	$0.485908\pi$
$968$	−0.889109	−0.0285770
$969$	−22.8000	−0.732442
$970$	22.0314	0.707387
$971$	33.2226	1.06616	0.533082	−	0.846063i	$-0.321034\pi$
$971$	33.2226	1.06616	0.533082	+	0.846063i	$0.321034\pi$
$972$	−105.204	−3.37442
$973$	9.89052	0.317076
$974$	9.38481	0.300709
$975$	−1.38931	−0.0444936
$976$	−35.8725	−1.14825
$977$	−17.0948	−0.546912	−0.273456	−	0.961884i	$-0.588167\pi$
$977$	−17.0948	−0.546912	−0.273456	+	0.961884i	$0.588167\pi$
$978$	−40.2775	−1.28793
$979$	5.05047	0.161414
$980$	−9.66265	−0.308662
$981$	58.7091	1.87444
$982$	−9.56161	−0.305123
$983$	19.3351	0.616694	0.308347	−	0.951274i	$-0.400224\pi$
$983$	19.3351	0.616694	0.308347	+	0.951274i	$0.400224\pi$
$984$	−15.0327	−0.479224
$985$	−14.3150	−0.456112
$986$	−86.3627	−2.75035
$987$	−0.813808	−0.0259038
$988$	−0.627550	−0.0199650
$989$	1.21812	0.0387339
$990$	15.8147	0.502625
$991$	7.92345	0.251697	0.125848	−	0.992050i	$-0.459835\pi$
$991$	7.92345	0.251697	0.125848	+	0.992050i	$0.459835\pi$
$992$	65.5430	2.08099
$993$	−9.86568	−0.313078
$994$	4.42778	0.140441
$995$	25.7462	0.816209
$996$	−20.6946	−0.655734
$997$	−15.7784	−0.499707	−0.249853	−	0.968284i	$-0.580382\pi$
$997$	−15.7784	−0.499707	−0.249853	+	0.968284i	$0.580382\pi$
$998$	33.2720	1.05321
$999$	82.6220	2.61404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.i.1.8	✓	8
3.2	odd	2		9405.2.a.bf.1.1		8
5.4	even	2		5225.2.a.o.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.i.1.8	✓	8	1.1	even	1	trivial
5225.2.a.o.1.1		8	5.4	even	2
9405.2.a.bf.1.1		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)

\(f(q)\)	\(=\)	\(q+1.87791 q^{2} -3.37956 q^{3} +1.52654 q^{4} +1.00000 q^{5} -6.34651 q^{6} -0.818685 q^{7} -0.889109 q^{8} +8.42145 q^{9} +O(q^{10})\)	\(q+1.87791 q^{2} -3.37956 q^{3} +1.52654 q^{4} +1.00000 q^{5} -6.34651 q^{6} -0.818685 q^{7} -0.889109 q^{8} +8.42145 q^{9} +1.87791 q^{10} +1.00000 q^{11} -5.15905 q^{12} +0.411092 q^{13} -1.53742 q^{14} -3.37956 q^{15} -4.72275 q^{16} -6.74644 q^{17} +15.8147 q^{18} -1.00000 q^{19} +1.52654 q^{20} +2.76680 q^{21} +1.87791 q^{22} -0.484830 q^{23} +3.00480 q^{24} +1.00000 q^{25} +0.771994 q^{26} -18.3221 q^{27} -1.24976 q^{28} +6.81675 q^{29} -6.34651 q^{30} -9.24353 q^{31} -7.09068 q^{32} -3.37956 q^{33} -12.6692 q^{34} -0.818685 q^{35} +12.8557 q^{36} -4.50941 q^{37} -1.87791 q^{38} -1.38931 q^{39} -0.889109 q^{40} -5.00288 q^{41} +5.19579 q^{42} -2.51246 q^{43} +1.52654 q^{44} +8.42145 q^{45} -0.910467 q^{46} -0.294134 q^{47} +15.9608 q^{48} -6.32976 q^{49} +1.87791 q^{50} +22.8000 q^{51} +0.627550 q^{52} -10.1669 q^{53} -34.4073 q^{54} +1.00000 q^{55} +0.727900 q^{56} +3.37956 q^{57} +12.8012 q^{58} -12.3329 q^{59} -5.15905 q^{60} +7.59567 q^{61} -17.3585 q^{62} -6.89451 q^{63} -3.87015 q^{64} +0.411092 q^{65} -6.34651 q^{66} -5.78082 q^{67} -10.2987 q^{68} +1.63852 q^{69} -1.53742 q^{70} -2.88001 q^{71} -7.48759 q^{72} +3.30583 q^{73} -8.46826 q^{74} -3.37956 q^{75} -1.52654 q^{76} -0.818685 q^{77} -2.60900 q^{78} +4.84475 q^{79} -4.72275 q^{80} +36.6565 q^{81} -9.39496 q^{82} +4.01132 q^{83} +4.22364 q^{84} -6.74644 q^{85} -4.71817 q^{86} -23.0376 q^{87} -0.889109 q^{88} +5.05047 q^{89} +15.8147 q^{90} -0.336555 q^{91} -0.740115 q^{92} +31.2391 q^{93} -0.552356 q^{94} -1.00000 q^{95} +23.9634 q^{96} +11.7319 q^{97} -11.8867 q^{98} +8.42145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 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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