LMFDB - Embedded newform 8042.2.a.b.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.65492 q^{3} +1.00000 q^{4} +1.59716 q^{5} +2.65492 q^{6} +4.41180 q^{7} -1.00000 q^{8} +4.04858 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.65492 q^{3} +1.00000 q^{4} +1.59716 q^{5} +2.65492 q^{6} +4.41180 q^{7} -1.00000 q^{8} +4.04858 q^{9} -1.59716 q^{10} -5.64809 q^{11} -2.65492 q^{12} -5.21822 q^{13} -4.41180 q^{14} -4.24033 q^{15} +1.00000 q^{16} -3.45233 q^{17} -4.04858 q^{18} +3.74760 q^{19} +1.59716 q^{20} -11.7130 q^{21} +5.64809 q^{22} +5.40222 q^{23} +2.65492 q^{24} -2.44908 q^{25} +5.21822 q^{26} -2.78390 q^{27} +4.41180 q^{28} -2.32064 q^{29} +4.24033 q^{30} -6.81064 q^{31} -1.00000 q^{32} +14.9952 q^{33} +3.45233 q^{34} +7.04635 q^{35} +4.04858 q^{36} -7.41585 q^{37} -3.74760 q^{38} +13.8539 q^{39} -1.59716 q^{40} +10.7109 q^{41} +11.7130 q^{42} +10.4463 q^{43} -5.64809 q^{44} +6.46624 q^{45} -5.40222 q^{46} +7.13510 q^{47} -2.65492 q^{48} +12.4640 q^{49} +2.44908 q^{50} +9.16564 q^{51} -5.21822 q^{52} +9.25234 q^{53} +2.78390 q^{54} -9.02091 q^{55} -4.41180 q^{56} -9.94955 q^{57} +2.32064 q^{58} -1.99541 q^{59} -4.24033 q^{60} +8.37229 q^{61} +6.81064 q^{62} +17.8615 q^{63} +1.00000 q^{64} -8.33433 q^{65} -14.9952 q^{66} -15.1368 q^{67} -3.45233 q^{68} -14.3424 q^{69} -7.04635 q^{70} -9.08554 q^{71} -4.04858 q^{72} +4.67947 q^{73} +7.41585 q^{74} +6.50210 q^{75} +3.74760 q^{76} -24.9183 q^{77} -13.8539 q^{78} +2.22060 q^{79} +1.59716 q^{80} -4.75472 q^{81} -10.7109 q^{82} +15.1406 q^{83} -11.7130 q^{84} -5.51392 q^{85} -10.4463 q^{86} +6.16110 q^{87} +5.64809 q^{88} +4.63541 q^{89} -6.46624 q^{90} -23.0217 q^{91} +5.40222 q^{92} +18.0817 q^{93} -7.13510 q^{94} +5.98551 q^{95} +2.65492 q^{96} +2.06952 q^{97} -12.4640 q^{98} -22.8668 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) $ 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.65492	−1.53282	−0.766409	−	0.642353i	$-0.777958\pi$
$3$	−2.65492	−1.53282	−0.766409	+	0.642353i	$0.777958\pi$
$4$	1.00000	0.500000
$5$	1.59716	0.714272	0.357136	−	0.934052i	$-0.383753\pi$
$5$	1.59716	0.714272	0.357136	+	0.934052i	$0.383753\pi$
$6$	2.65492	1.08387
$7$	4.41180	1.66750	0.833752	−	0.552140i	$-0.186188\pi$
$7$	4.41180	1.66750	0.833752	+	0.552140i	$0.186188\pi$
$8$	−1.00000	−0.353553
$9$	4.04858	1.34953
$10$	−1.59716	−0.505066
$11$	−5.64809	−1.70296	−0.851482	−	0.524384i	$-0.824296\pi$
$11$	−5.64809	−1.70296	−0.851482	+	0.524384i	$0.824296\pi$
$12$	−2.65492	−0.766409
$13$	−5.21822	−1.44727	−0.723637	−	0.690181i	$-0.757531\pi$
$13$	−5.21822	−1.44727	−0.723637	+	0.690181i	$0.757531\pi$
$14$	−4.41180	−1.17910
$15$	−4.24033	−1.09485
$16$	1.00000	0.250000
$17$	−3.45233	−0.837312	−0.418656	−	0.908145i	$-0.637499\pi$
$17$	−3.45233	−0.837312	−0.418656	+	0.908145i	$0.637499\pi$
$18$	−4.04858	−0.954260
$19$	3.74760	0.859757	0.429879	−	0.902887i	$-0.358556\pi$
$19$	3.74760	0.859757	0.429879	+	0.902887i	$0.358556\pi$
$20$	1.59716	0.357136
$21$	−11.7130	−2.55598
$22$	5.64809	1.20418
$23$	5.40222	1.12644	0.563220	−	0.826307i	$-0.309562\pi$
$23$	5.40222	1.12644	0.563220	+	0.826307i	$0.309562\pi$
$24$	2.65492	0.541933
$25$	−2.44908	−0.489816
$26$	5.21822	1.02338
$27$	−2.78390	−0.535762
$28$	4.41180	0.833752
$29$	−2.32064	−0.430932	−0.215466	−	0.976511i	$-0.569127\pi$
$29$	−2.32064	−0.430932	−0.215466	+	0.976511i	$0.569127\pi$
$30$	4.24033	0.774174
$31$	−6.81064	−1.22323	−0.611613	−	0.791157i	$-0.709479\pi$
$31$	−6.81064	−1.22323	−0.611613	+	0.791157i	$0.709479\pi$
$32$	−1.00000	−0.176777
$33$	14.9952	2.61033
$34$	3.45233	0.592069
$35$	7.04635	1.19105
$36$	4.04858	0.674764
$37$	−7.41585	−1.21916	−0.609579	−	0.792725i	$-0.708661\pi$
$37$	−7.41585	−1.21916	−0.609579	+	0.792725i	$0.708661\pi$
$38$	−3.74760	−0.607940
$39$	13.8539	2.21841
$40$	−1.59716	−0.252533
$41$	10.7109	1.67277	0.836383	−	0.548145i	$-0.184666\pi$
$41$	10.7109	1.67277	0.836383	+	0.548145i	$0.184666\pi$
$42$	11.7130	1.80735
$43$	10.4463	1.59304	0.796521	−	0.604611i	$-0.206672\pi$
$43$	10.4463	1.59304	0.796521	+	0.604611i	$0.206672\pi$
$44$	−5.64809	−0.851482
$45$	6.46624	0.963930
$46$	−5.40222	−0.796513
$47$	7.13510	1.04076	0.520381	−	0.853934i	$-0.325790\pi$
$47$	7.13510	1.04076	0.520381	+	0.853934i	$0.325790\pi$
$48$	−2.65492	−0.383204
$49$	12.4640	1.78057
$50$	2.44908	0.346352
$51$	9.16564	1.28345
$52$	−5.21822	−0.723637
$53$	9.25234	1.27091	0.635453	−	0.772140i	$-0.280813\pi$
$53$	9.25234	1.27091	0.635453	+	0.772140i	$0.280813\pi$
$54$	2.78390	0.378841
$55$	−9.02091	−1.21638
$56$	−4.41180	−0.589551
$57$	−9.94955	−1.31785
$58$	2.32064	0.304715
$59$	−1.99541	−0.259780	−0.129890	−	0.991528i	$-0.541462\pi$
$59$	−1.99541	−0.259780	−0.129890	+	0.991528i	$0.541462\pi$
$60$	−4.24033	−0.547424
$61$	8.37229	1.07196	0.535981	−	0.844230i	$-0.319942\pi$
$61$	8.37229	1.07196	0.535981	+	0.844230i	$0.319942\pi$
$62$	6.81064	0.864952
$63$	17.8615	2.25034
$64$	1.00000	0.125000
$65$	−8.33433	−1.03375
$66$	−14.9952	−1.84578
$67$	−15.1368	−1.84926	−0.924628	−	0.380871i	$-0.875624\pi$
$67$	−15.1368	−1.84926	−0.924628	+	0.380871i	$0.875624\pi$
$68$	−3.45233	−0.418656
$69$	−14.3424	−1.72663
$70$	−7.04635	−0.842200
$71$	−9.08554	−1.07825	−0.539127	−	0.842224i	$-0.681246\pi$
$71$	−9.08554	−1.07825	−0.539127	+	0.842224i	$0.681246\pi$
$72$	−4.04858	−0.477130
$73$	4.67947	0.547690	0.273845	−	0.961774i	$-0.411704\pi$
$73$	4.67947	0.547690	0.273845	+	0.961774i	$0.411704\pi$
$74$	7.41585	0.862075
$75$	6.50210	0.750798
$76$	3.74760	0.429879
$77$	−24.9183	−2.83970
$78$	−13.8539	−1.56865
$79$	2.22060	0.249837	0.124919	−	0.992167i	$-0.460133\pi$
$79$	2.22060	0.249837	0.124919	+	0.992167i	$0.460133\pi$
$80$	1.59716	0.178568
$81$	−4.75472	−0.528302
$82$	−10.7109	−1.18282
$83$	15.1406	1.66190	0.830951	−	0.556346i	$-0.187797\pi$
$83$	15.1406	1.66190	0.830951	+	0.556346i	$0.187797\pi$
$84$	−11.7130	−1.27799
$85$	−5.51392	−0.598068
$86$	−10.4463	−1.12645
$87$	6.16110	0.660539
$88$	5.64809	0.602089
$89$	4.63541	0.491353	0.245676	−	0.969352i	$-0.420990\pi$
$89$	4.63541	0.491353	0.245676	+	0.969352i	$0.420990\pi$
$90$	−6.46624	−0.681601
$91$	−23.0217	−2.41333
$92$	5.40222	0.563220
$93$	18.0817	1.87498
$94$	−7.13510	−0.735929
$95$	5.98551	0.614100
$96$	2.65492	0.270966
$97$	2.06952	0.210128	0.105064	−	0.994465i	$-0.466495\pi$
$97$	2.06952	0.210128	0.105064	+	0.994465i	$0.466495\pi$
$98$	−12.4640	−1.25905
$99$	−22.8668	−2.29820
$100$	−2.44908	−0.244908
$101$	−2.51403	−0.250155	−0.125078	−	0.992147i	$-0.539918\pi$
$101$	−2.51403	−0.250155	−0.125078	+	0.992147i	$0.539918\pi$
$102$	−9.16564	−0.907533
$103$	−14.2792	−1.40697	−0.703487	−	0.710708i	$-0.748375\pi$
$103$	−14.2792	−1.40697	−0.703487	+	0.710708i	$0.748375\pi$
$104$	5.21822	0.511688
$105$	−18.7075	−1.82566
$106$	−9.25234	−0.898666
$107$	−11.3686	−1.09905	−0.549524	−	0.835478i	$-0.685191\pi$
$107$	−11.3686	−1.09905	−0.549524	+	0.835478i	$0.685191\pi$
$108$	−2.78390	−0.267881
$109$	−14.9559	−1.43251	−0.716257	−	0.697836i	$-0.754146\pi$
$109$	−14.9559	−1.43251	−0.716257	+	0.697836i	$0.754146\pi$
$110$	9.02091	0.860110
$111$	19.6885	1.86875
$112$	4.41180	0.416876
$113$	−1.02919	−0.0968184	−0.0484092	−	0.998828i	$-0.515415\pi$
$113$	−1.02919	−0.0968184	−0.0484092	+	0.998828i	$0.515415\pi$
$114$	9.94955	0.931861
$115$	8.62820	0.804584
$116$	−2.32064	−0.215466
$117$	−21.1264	−1.95314
$118$	1.99541	0.183692
$119$	−15.2310	−1.39622
$120$	4.24033	0.387087
$121$	20.9010	1.90009
$122$	−8.37229	−0.757992
$123$	−28.4366	−2.56405
$124$	−6.81064	−0.611613
$125$	−11.8974	−1.06413
$126$	−17.8615	−1.59123
$127$	5.19911	0.461347	0.230673	−	0.973031i	$-0.425907\pi$
$127$	5.19911	0.461347	0.230673	+	0.973031i	$0.425907\pi$
$128$	−1.00000	−0.0883883
$129$	−27.7340	−2.44184
$130$	8.33433	0.730969
$131$	−15.1315	−1.32204	−0.661021	−	0.750367i	$-0.729877\pi$
$131$	−15.1315	−1.32204	−0.661021	+	0.750367i	$0.729877\pi$
$132$	14.9952	1.30517
$133$	16.5336	1.43365
$134$	15.1368	1.30762
$135$	−4.44634	−0.382680
$136$	3.45233	0.296034
$137$	12.9800	1.10896	0.554480	−	0.832197i	$-0.312917\pi$
$137$	12.9800	1.10896	0.554480	+	0.832197i	$0.312917\pi$
$138$	14.3424	1.22091
$139$	−1.39477	−0.118303	−0.0591516	−	0.998249i	$-0.518840\pi$
$139$	−1.39477	−0.118303	−0.0591516	+	0.998249i	$0.518840\pi$
$140$	7.04635	0.595525
$141$	−18.9431	−1.59530
$142$	9.08554	0.762441
$143$	29.4730	2.46465
$144$	4.04858	0.337382
$145$	−3.70643	−0.307802
$146$	−4.67947	−0.387276
$147$	−33.0908	−2.72928
$148$	−7.41585	−0.609579
$149$	14.8516	1.21669	0.608347	−	0.793671i	$-0.291833\pi$
$149$	14.8516	1.21669	0.608347	+	0.793671i	$0.291833\pi$
$150$	−6.50210	−0.530895
$151$	15.5105	1.26222	0.631112	−	0.775692i	$-0.282599\pi$
$151$	15.5105	1.26222	0.631112	+	0.775692i	$0.282599\pi$
$152$	−3.74760	−0.303970
$153$	−13.9770	−1.12998
$154$	24.9183	2.00797
$155$	−10.8777	−0.873716
$156$	13.8539	1.10920
$157$	−24.8243	−1.98120	−0.990598	−	0.136805i	$-0.956317\pi$
$157$	−24.8243	−1.98120	−0.990598	+	0.136805i	$0.956317\pi$
$158$	−2.22060	−0.176662
$159$	−24.5642	−1.94807
$160$	−1.59716	−0.126267
$161$	23.8335	1.87834
$162$	4.75472	0.373566
$163$	19.1996	1.50383	0.751914	−	0.659262i	$-0.229131\pi$
$163$	19.1996	1.50383	0.751914	+	0.659262i	$0.229131\pi$
$164$	10.7109	0.836383
$165$	23.9498	1.86449
$166$	−15.1406	−1.17514
$167$	−13.8252	−1.06982	−0.534912	−	0.844908i	$-0.679655\pi$
$167$	−13.8252	−1.06982	−0.534912	+	0.844908i	$0.679655\pi$
$168$	11.7130	0.903675
$169$	14.2298	1.09460
$170$	5.51392	0.422898
$171$	15.1725	1.16027
$172$	10.4463	0.796521
$173$	0.635650	0.0483276	0.0241638	−	0.999708i	$-0.492308\pi$
$173$	0.635650	0.0483276	0.0241638	+	0.999708i	$0.492308\pi$
$174$	−6.16110	−0.467072
$175$	−10.8048	−0.816770
$176$	−5.64809	−0.425741
$177$	5.29765	0.398196
$178$	−4.63541	−0.347439
$179$	5.01193	0.374609	0.187304	−	0.982302i	$-0.440025\pi$
$179$	5.01193	0.374609	0.187304	+	0.982302i	$0.440025\pi$
$180$	6.46624	0.481965
$181$	−8.81525	−0.655232	−0.327616	−	0.944811i	$-0.606245\pi$
$181$	−8.81525	−0.655232	−0.327616	+	0.944811i	$0.606245\pi$
$182$	23.0217	1.70648
$183$	−22.2277	−1.64312
$184$	−5.40222	−0.398257
$185$	−11.8443	−0.870810
$186$	−18.0817	−1.32581
$187$	19.4991	1.42591
$188$	7.13510	0.520381
$189$	−12.2820	−0.893386
$190$	−5.98551	−0.434234
$191$	7.66503	0.554622	0.277311	−	0.960780i	$-0.410557\pi$
$191$	7.66503	0.554622	0.277311	+	0.960780i	$0.410557\pi$
$192$	−2.65492	−0.191602
$193$	−26.6394	−1.91754	−0.958772	−	0.284177i	$-0.908280\pi$
$193$	−26.6394	−1.91754	−0.958772	+	0.284177i	$0.908280\pi$
$194$	−2.06952	−0.148583
$195$	22.1270	1.58454
$196$	12.4640	0.890284
$197$	9.11959	0.649744	0.324872	−	0.945758i	$-0.394679\pi$
$197$	9.11959	0.649744	0.324872	+	0.945758i	$0.394679\pi$
$198$	22.8668	1.62507
$199$	−4.92524	−0.349141	−0.174571	−	0.984645i	$-0.555854\pi$
$199$	−4.92524	−0.349141	−0.174571	+	0.984645i	$0.555854\pi$
$200$	2.44908	0.173176
$201$	40.1870	2.83457
$202$	2.51403	0.176886
$203$	−10.2382	−0.718580
$204$	9.16564	0.641723
$205$	17.1071	1.19481
$206$	14.2792	0.994881
$207$	21.8713	1.52016
$208$	−5.21822	−0.361818
$209$	−21.1668	−1.46414
$210$	18.7075	1.29094
$211$	−14.0343	−0.966164	−0.483082	−	0.875575i	$-0.660483\pi$
$211$	−14.0343	−0.966164	−0.483082	+	0.875575i	$0.660483\pi$
$212$	9.25234	0.635453
$213$	24.1213	1.65277
$214$	11.3686	0.777144
$215$	16.6844	1.13786
$216$	2.78390	0.189421
$217$	−30.0472	−2.03973
$218$	14.9559	1.01294
$219$	−12.4236	−0.839509
$220$	−9.02091	−0.608190
$221$	18.0150	1.21182
$222$	−19.6885	−1.32140
$223$	−24.7103	−1.65472	−0.827361	−	0.561670i	$-0.810159\pi$
$223$	−24.7103	−1.65472	−0.827361	+	0.561670i	$0.810159\pi$
$224$	−4.41180	−0.294776
$225$	−9.91531	−0.661020
$226$	1.02919	0.0684610
$227$	−16.5010	−1.09521	−0.547604	−	0.836737i	$-0.684460\pi$
$227$	−16.5010	−1.09521	−0.547604	+	0.836737i	$0.684460\pi$
$228$	−9.94955	−0.658925
$229$	−8.11067	−0.535968	−0.267984	−	0.963423i	$-0.586357\pi$
$229$	−8.11067	−0.535968	−0.267984	+	0.963423i	$0.586357\pi$
$230$	−8.62820	−0.568927
$231$	66.1559	4.35274
$232$	2.32064	0.152357
$233$	22.4184	1.46868	0.734338	−	0.678784i	$-0.237493\pi$
$233$	22.4184	1.46868	0.734338	+	0.678784i	$0.237493\pi$
$234$	21.1264	1.38108
$235$	11.3959	0.743386
$236$	−1.99541	−0.129890
$237$	−5.89551	−0.382955
$238$	15.2310	0.987277
$239$	10.9653	0.709283	0.354642	−	0.935002i	$-0.384603\pi$
$239$	10.9653	0.709283	0.354642	+	0.935002i	$0.384603\pi$
$240$	−4.24033	−0.273712
$241$	21.0532	1.35615	0.678077	−	0.734991i	$-0.262814\pi$
$241$	21.0532	1.35615	0.678077	+	0.734991i	$0.262814\pi$
$242$	−20.9010	−1.34356
$243$	20.9751	1.34555
$244$	8.37229	0.535981
$245$	19.9070	1.27181
$246$	28.4366	1.81305
$247$	−19.5558	−1.24430
$248$	6.81064	0.432476
$249$	−40.1972	−2.54739
$250$	11.8974	0.752456
$251$	−24.8039	−1.56561	−0.782803	−	0.622269i	$-0.786211\pi$
$251$	−24.8039	−1.56561	−0.782803	+	0.622269i	$0.786211\pi$
$252$	17.8615	1.12517
$253$	−30.5122	−1.91829
$254$	−5.19911	−0.326221
$255$	14.6390	0.916729
$256$	1.00000	0.0625000
$257$	−26.9282	−1.67973	−0.839867	−	0.542793i	$-0.817367\pi$
$257$	−26.9282	−1.67973	−0.839867	+	0.542793i	$0.817367\pi$
$258$	27.7340	1.72664
$259$	−32.7172	−2.03295
$260$	−8.33433	−0.516873
$261$	−9.39530	−0.581554
$262$	15.1315	0.934825
$263$	19.4851	1.20151	0.600753	−	0.799435i	$-0.294868\pi$
$263$	19.4851	1.20151	0.600753	+	0.799435i	$0.294868\pi$
$264$	−14.9952	−0.922892
$265$	14.7775	0.907772
$266$	−16.5336	−1.01374
$267$	−12.3066	−0.753154
$268$	−15.1368	−0.924628
$269$	27.1622	1.65611	0.828055	−	0.560647i	$-0.189448\pi$
$269$	27.1622	1.65611	0.828055	+	0.560647i	$0.189448\pi$
$270$	4.44634	0.270596
$271$	−25.1438	−1.52738	−0.763688	−	0.645585i	$-0.776614\pi$
$271$	−25.1438	−1.52738	−0.763688	+	0.645585i	$0.776614\pi$
$272$	−3.45233	−0.209328
$273$	61.1208	3.69920
$274$	−12.9800	−0.784153
$275$	13.8326	0.834139
$276$	−14.3424	−0.863313
$277$	−30.5509	−1.83563	−0.917813	−	0.397012i	$-0.870047\pi$
$277$	−30.5509	−1.83563	−0.917813	+	0.397012i	$0.870047\pi$
$278$	1.39477	0.0836529
$279$	−27.5734	−1.65078
$280$	−7.04635	−0.421100
$281$	3.25214	0.194007	0.0970033	−	0.995284i	$-0.469074\pi$
$281$	3.25214	0.194007	0.0970033	+	0.995284i	$0.469074\pi$
$282$	18.9431	1.12805
$283$	−16.1387	−0.959345	−0.479672	−	0.877448i	$-0.659244\pi$
$283$	−16.1387	−0.959345	−0.479672	+	0.877448i	$0.659244\pi$
$284$	−9.08554	−0.539127
$285$	−15.8910	−0.941303
$286$	−29.4730	−1.74277
$287$	47.2545	2.78934
$288$	−4.04858	−0.238565
$289$	−5.08145	−0.298909
$290$	3.70643	0.217649
$291$	−5.49441	−0.322088
$292$	4.67947	0.273845
$293$	26.5947	1.55368	0.776840	−	0.629698i	$-0.216821\pi$
$293$	26.5947	1.55368	0.776840	+	0.629698i	$0.216821\pi$
$294$	33.0908	1.92990
$295$	−3.18699	−0.185554
$296$	7.41585	0.431037
$297$	15.7237	0.912384
$298$	−14.8516	−0.860332
$299$	−28.1899	−1.63027
$300$	6.50210	0.375399
$301$	46.0868	2.65640
$302$	−15.5105	−0.892527
$303$	6.67453	0.383442
$304$	3.74760	0.214939
$305$	13.3719	0.765672
$306$	13.9770	0.799014
$307$	2.37170	0.135360	0.0676800	−	0.997707i	$-0.478440\pi$
$307$	2.37170	0.135360	0.0676800	+	0.997707i	$0.478440\pi$
$308$	−24.9183	−1.41985
$309$	37.9102	2.15663
$310$	10.8777	0.617810
$311$	−24.3606	−1.38136	−0.690682	−	0.723159i	$-0.742689\pi$
$311$	−24.3606	−1.38136	−0.690682	+	0.723159i	$0.742689\pi$
$312$	−13.8539	−0.784325
$313$	5.25944	0.297281	0.148641	−	0.988891i	$-0.452510\pi$
$313$	5.25944	0.297281	0.148641	+	0.988891i	$0.452510\pi$
$314$	24.8243	1.40092
$315$	28.5277	1.60736
$316$	2.22060	0.124919
$317$	4.79293	0.269198	0.134599	−	0.990900i	$-0.457025\pi$
$317$	4.79293	0.269198	0.134599	+	0.990900i	$0.457025\pi$
$318$	24.5642	1.37749
$319$	13.1072	0.733861
$320$	1.59716	0.0892840
$321$	30.1828	1.68464
$322$	−23.8335	−1.32819
$323$	−12.9379	−0.719885
$324$	−4.75472	−0.264151
$325$	12.7798	0.708898
$326$	−19.1996	−1.06337
$327$	39.7066	2.19578
$328$	−10.7109	−0.591412
$329$	31.4786	1.73547
$330$	−23.9498	−1.31839
$331$	−6.37409	−0.350352	−0.175176	−	0.984537i	$-0.556049\pi$
$331$	−6.37409	−0.350352	−0.175176	+	0.984537i	$0.556049\pi$
$332$	15.1406	0.830951
$333$	−30.0237	−1.64529
$334$	13.8252	0.756479
$335$	−24.1759	−1.32087
$336$	−11.7130	−0.638994
$337$	5.66676	0.308688	0.154344	−	0.988017i	$-0.450674\pi$
$337$	5.66676	0.308688	0.154344	+	0.988017i	$0.450674\pi$
$338$	−14.2298	−0.773999
$339$	2.73242	0.148405
$340$	−5.51392	−0.299034
$341$	38.4671	2.08311
$342$	−15.1725	−0.820432
$343$	24.1059	1.30160
$344$	−10.4463	−0.563225
$345$	−22.9072	−1.23328
$346$	−0.635650	−0.0341727
$347$	−14.5274	−0.779874	−0.389937	−	0.920841i	$-0.627503\pi$
$347$	−14.5274	−0.779874	−0.389937	+	0.920841i	$0.627503\pi$
$348$	6.16110	0.330270
$349$	20.8620	1.11672	0.558360	−	0.829599i	$-0.311431\pi$
$349$	20.8620	1.11672	0.558360	+	0.829599i	$0.311431\pi$
$350$	10.8048	0.577543
$351$	14.5270	0.775395
$352$	5.64809	0.301044
$353$	−5.52688	−0.294166	−0.147083	−	0.989124i	$-0.546989\pi$
$353$	−5.52688	−0.294166	−0.147083	+	0.989124i	$0.546989\pi$
$354$	−5.29765	−0.281567
$355$	−14.5111	−0.770167
$356$	4.63541	0.245676
$357$	40.4370	2.14015
$358$	−5.01193	−0.264889
$359$	10.0070	0.528150	0.264075	−	0.964502i	$-0.414933\pi$
$359$	10.0070	0.528150	0.264075	+	0.964502i	$0.414933\pi$
$360$	−6.46624	−0.340801
$361$	−4.95553	−0.260817
$362$	8.81525	0.463319
$363$	−55.4903	−2.91249
$364$	−23.0217	−1.20667
$365$	7.47386	0.391200
$366$	22.2277	1.16186
$367$	−4.04260	−0.211022	−0.105511	−	0.994418i	$-0.533648\pi$
$367$	−4.04260	−0.211022	−0.105511	+	0.994418i	$0.533648\pi$
$368$	5.40222	0.281610
$369$	43.3641	2.25745
$370$	11.8443	0.615755
$371$	40.8194	2.11924
$372$	18.0817	0.937491
$373$	−34.0949	−1.76537	−0.882685	−	0.469966i	$-0.844266\pi$
$373$	−34.0949	−1.76537	−0.882685	+	0.469966i	$0.844266\pi$
$374$	−19.4991	−1.00827
$375$	31.5865	1.63112
$376$	−7.13510	−0.367965
$377$	12.1096	0.623676
$378$	12.2820	0.631719
$379$	−2.06608	−0.106127	−0.0530636	−	0.998591i	$-0.516899\pi$
$379$	−2.06608	−0.106127	−0.0530636	+	0.998591i	$0.516899\pi$
$380$	5.98551	0.307050
$381$	−13.8032	−0.707160
$382$	−7.66503	−0.392177
$383$	−10.3223	−0.527443	−0.263721	−	0.964599i	$-0.584950\pi$
$383$	−10.3223	−0.527443	−0.263721	+	0.964599i	$0.584950\pi$
$384$	2.65492	0.135483
$385$	−39.7984	−2.02832
$386$	26.6394	1.35591
$387$	42.2926	2.14985
$388$	2.06952	0.105064
$389$	−22.3296	−1.13216	−0.566078	−	0.824352i	$-0.691540\pi$
$389$	−22.3296	−1.13216	−0.566078	+	0.824352i	$0.691540\pi$
$390$	−22.1270	−1.12044
$391$	−18.6502	−0.943182
$392$	−12.4640	−0.629526
$393$	40.1728	2.02645
$394$	−9.11959	−0.459438
$395$	3.54665	0.178452
$396$	−22.8668	−1.14910
$397$	2.48301	0.124619	0.0623094	−	0.998057i	$-0.480153\pi$
$397$	2.48301	0.124619	0.0623094	+	0.998057i	$0.480153\pi$
$398$	4.92524	0.246880
$399$	−43.8954	−2.19752
$400$	−2.44908	−0.122454
$401$	2.11445	0.105591	0.0527953	−	0.998605i	$-0.483187\pi$
$401$	2.11445	0.105591	0.0527953	+	0.998605i	$0.483187\pi$
$402$	−40.1870	−2.00434
$403$	35.5394	1.77034
$404$	−2.51403	−0.125078
$405$	−7.59405	−0.377351
$406$	10.2382	0.508113
$407$	41.8854	2.07618
$408$	−9.16564	−0.453767
$409$	−2.70606	−0.133806	−0.0669031	−	0.997759i	$-0.521312\pi$
$409$	−2.70606	−0.133806	−0.0669031	+	0.997759i	$0.521312\pi$
$410$	−17.1071	−0.844858
$411$	−34.4609	−1.69983
$412$	−14.2792	−0.703487
$413$	−8.80335	−0.433185
$414$	−21.8713	−1.07492
$415$	24.1820	1.18705
$416$	5.21822	0.255844
$417$	3.70301	0.181337
$418$	21.1668	1.03530
$419$	1.39259	0.0680323	0.0340161	−	0.999421i	$-0.489170\pi$
$419$	1.39259	0.0680323	0.0340161	+	0.999421i	$0.489170\pi$
$420$	−18.7075	−0.912831
$421$	−37.8883	−1.84656	−0.923281	−	0.384125i	$-0.874503\pi$
$421$	−37.8883	−1.84656	−0.923281	+	0.384125i	$0.874503\pi$
$422$	14.0343	0.683181
$423$	28.8871	1.40454
$424$	−9.25234	−0.449333
$425$	8.45502	0.410129
$426$	−24.1213	−1.16868
$427$	36.9369	1.78750
$428$	−11.3686	−0.549524
$429$	−78.2483	−3.77786
$430$	−16.6844	−0.804591
$431$	4.36135	0.210079	0.105040	−	0.994468i	$-0.466503\pi$
$431$	4.36135	0.210079	0.105040	+	0.994468i	$0.466503\pi$
$432$	−2.78390	−0.133941
$433$	−8.11762	−0.390108	−0.195054	−	0.980792i	$-0.562488\pi$
$433$	−8.11762	−0.390108	−0.195054	+	0.980792i	$0.562488\pi$
$434$	30.0472	1.44231
$435$	9.84026	0.471804
$436$	−14.9559	−0.716257
$437$	20.2453	0.968465
$438$	12.4236	0.593623
$439$	32.7649	1.56378	0.781891	−	0.623415i	$-0.214255\pi$
$439$	32.7649	1.56378	0.781891	+	0.623415i	$0.214255\pi$
$440$	9.02091	0.430055
$441$	50.4614	2.40293
$442$	−18.0150	−0.856886
$443$	−29.4643	−1.39989	−0.699947	−	0.714195i	$-0.746793\pi$
$443$	−29.4643	−1.39989	−0.699947	+	0.714195i	$0.746793\pi$
$444$	19.6885	0.934373
$445$	7.40349	0.350959
$446$	24.7103	1.17007
$447$	−39.4299	−1.86497
$448$	4.41180	0.208438
$449$	2.60085	0.122742	0.0613708	−	0.998115i	$-0.480453\pi$
$449$	2.60085	0.122742	0.0613708	+	0.998115i	$0.480453\pi$
$450$	9.91531	0.467412
$451$	−60.4964	−2.84866
$452$	−1.02919	−0.0484092
$453$	−41.1790	−1.93476
$454$	16.5010	0.774430
$455$	−36.7694	−1.72378
$456$	9.94955	0.465931
$457$	−10.1756	−0.475993	−0.237997	−	0.971266i	$-0.576491\pi$
$457$	−10.1756	−0.475993	−0.237997	+	0.971266i	$0.576491\pi$
$458$	8.11067	0.378987
$459$	9.61094	0.448600
$460$	8.62820	0.402292
$461$	14.6226	0.681043	0.340522	−	0.940237i	$-0.389396\pi$
$461$	14.6226	0.681043	0.340522	+	0.940237i	$0.389396\pi$
$462$	−66.1559	−3.07785
$463$	−15.4574	−0.718367	−0.359184	−	0.933267i	$-0.616945\pi$
$463$	−15.4574	−0.718367	−0.359184	+	0.933267i	$0.616945\pi$
$464$	−2.32064	−0.107733
$465$	28.8793	1.33925
$466$	−22.4184	−1.03851
$467$	−17.0460	−0.788793	−0.394396	−	0.918940i	$-0.629046\pi$
$467$	−17.0460	−0.788793	−0.394396	+	0.918940i	$0.629046\pi$
$468$	−21.1264	−0.976568
$469$	−66.7806	−3.08364
$470$	−11.3959	−0.525653
$471$	65.9065	3.03681
$472$	1.99541	0.0918462
$473$	−59.0015	−2.71289
$474$	5.89551	0.270790
$475$	−9.17816	−0.421123
$476$	−15.2310	−0.698110
$477$	37.4589	1.71512
$478$	−10.9653	−0.501539
$479$	−2.65708	−0.121405	−0.0607025	−	0.998156i	$-0.519334\pi$
$479$	−2.65708	−0.121405	−0.0607025	+	0.998156i	$0.519334\pi$
$480$	4.24033	0.193544
$481$	38.6975	1.76445
$482$	−21.0532	−0.958945
$483$	−63.2759	−2.87915
$484$	20.9010	0.950043
$485$	3.30536	0.150088
$486$	−20.9751	−0.951450
$487$	−1.60071	−0.0725351	−0.0362675	−	0.999342i	$-0.511547\pi$
$487$	−1.60071	−0.0725351	−0.0362675	+	0.999342i	$0.511547\pi$
$488$	−8.37229	−0.378996
$489$	−50.9733	−2.30509
$490$	−19.9070	−0.899305
$491$	−4.81994	−0.217521	−0.108760	−	0.994068i	$-0.534688\pi$
$491$	−4.81994	−0.217521	−0.108760	+	0.994068i	$0.534688\pi$
$492$	−28.4366	−1.28202
$493$	8.01160	0.360824
$494$	19.5558	0.879856
$495$	−36.5219	−1.64154
$496$	−6.81064	−0.305807
$497$	−40.0836	−1.79799
$498$	40.1972	1.80128
$499$	29.0971	1.30257	0.651283	−	0.758835i	$-0.274231\pi$
$499$	29.0971	1.30257	0.651283	+	0.758835i	$0.274231\pi$
$500$	−11.8974	−0.532067
$501$	36.7047	1.63984
$502$	24.8039	1.10705
$503$	−30.4059	−1.35573	−0.677866	−	0.735186i	$-0.737095\pi$
$503$	−30.4059	−1.35573	−0.677866	+	0.735186i	$0.737095\pi$
$504$	−17.8615	−0.795616
$505$	−4.01530	−0.178679
$506$	30.5122	1.35643
$507$	−37.7789	−1.67782
$508$	5.19911	0.230673
$509$	41.7186	1.84914	0.924572	−	0.381008i	$-0.124423\pi$
$509$	41.7186	1.84914	0.924572	+	0.381008i	$0.124423\pi$
$510$	−14.6390	−0.648225
$511$	20.6449	0.913276
$512$	−1.00000	−0.0441942
$513$	−10.4329	−0.460626
$514$	26.9282	1.18775
$515$	−22.8062	−1.00496
$516$	−27.7340	−1.22092
$517$	−40.2997	−1.77238
$518$	32.7172	1.43751
$519$	−1.68760	−0.0740773
$520$	8.33433	0.365485
$521$	−32.1831	−1.40997	−0.704984	−	0.709223i	$-0.749046\pi$
$521$	−32.1831	−1.40997	−0.704984	+	0.709223i	$0.749046\pi$
$522$	9.39530	0.411221
$523$	−11.9799	−0.523844	−0.261922	−	0.965089i	$-0.584356\pi$
$523$	−11.9799	−0.523844	−0.261922	+	0.965089i	$0.584356\pi$
$524$	−15.1315	−0.661021
$525$	28.6860	1.25196
$526$	−19.4851	−0.849593
$527$	23.5125	1.02422
$528$	14.9952	0.652583
$529$	6.18394	0.268867
$530$	−14.7775	−0.641892
$531$	−8.07859	−0.350581
$532$	16.5336	0.716824
$533$	−55.8920	−2.42095
$534$	12.3066	0.532560
$535$	−18.1575	−0.785018
$536$	15.1368	0.653811
$537$	−13.3062	−0.574207
$538$	−27.1622	−1.17105
$539$	−70.3977	−3.03224
$540$	−4.44634	−0.191340
$541$	43.8317	1.88447	0.942236	−	0.334949i	$-0.108719\pi$
$541$	43.8317	1.88447	0.942236	+	0.334949i	$0.108719\pi$
$542$	25.1438	1.08002
$543$	23.4038	1.00435
$544$	3.45233	0.148017
$545$	−23.8869	−1.02320
$546$	−61.1208	−2.61573
$547$	−35.7701	−1.52942	−0.764710	−	0.644375i	$-0.777118\pi$
$547$	−35.7701	−1.52942	−0.764710	+	0.644375i	$0.777118\pi$
$548$	12.9800	0.554480
$549$	33.8959	1.44664
$550$	−13.8326	−0.589825
$551$	−8.69681	−0.370497
$552$	14.3424	0.610455
$553$	9.79684	0.416604
$554$	30.5509	1.29798
$555$	31.4456	1.33479
$556$	−1.39477	−0.0591516
$557$	−34.4451	−1.45949	−0.729743	−	0.683722i	$-0.760360\pi$
$557$	−34.4451	−1.45949	−0.729743	+	0.683722i	$0.760360\pi$
$558$	27.5734	1.16728
$559$	−54.5109	−2.30557
$560$	7.04635	0.297763
$561$	−51.7684	−2.18566
$562$	−3.25214	−0.137183
$563$	−12.2478	−0.516182	−0.258091	−	0.966121i	$-0.583093\pi$
$563$	−12.2478	−0.516182	−0.258091	+	0.966121i	$0.583093\pi$
$564$	−18.9431	−0.797648
$565$	−1.64379	−0.0691547
$566$	16.1387	0.678359
$567$	−20.9769	−0.880946
$568$	9.08554	0.381221
$569$	−16.4858	−0.691121	−0.345560	−	0.938397i	$-0.612311\pi$
$569$	−16.4858	−0.691121	−0.345560	+	0.938397i	$0.612311\pi$
$570$	15.8910	0.665602
$571$	−12.0676	−0.505011	−0.252506	−	0.967595i	$-0.581255\pi$
$571$	−12.0676	−0.505011	−0.252506	+	0.967595i	$0.581255\pi$
$572$	29.4730	1.23233
$573$	−20.3500	−0.850134
$574$	−47.2545	−1.97236
$575$	−13.2305	−0.551748
$576$	4.04858	0.168691
$577$	7.15568	0.297895	0.148947	−	0.988845i	$-0.452412\pi$
$577$	7.15568	0.297895	0.148947	+	0.988845i	$0.452412\pi$
$578$	5.08145	0.211360
$579$	70.7253	2.93924
$580$	−3.70643	−0.153901
$581$	66.7975	2.77123
$582$	5.49441	0.227750
$583$	−52.2581	−2.16431
$584$	−4.67947	−0.193638
$585$	−33.7422	−1.39507
$586$	−26.5947	−1.09862
$587$	−38.6713	−1.59614	−0.798068	−	0.602567i	$-0.794145\pi$
$587$	−38.6713	−1.59614	−0.798068	+	0.602567i	$0.794145\pi$
$588$	−33.0908	−1.36464
$589$	−25.5235	−1.05168
$590$	3.18699	0.131206
$591$	−24.2118	−0.995938
$592$	−7.41585	−0.304789
$593$	−27.1699	−1.11574	−0.557868	−	0.829930i	$-0.688380\pi$
$593$	−27.1699	−1.11574	−0.557868	+	0.829930i	$0.688380\pi$
$594$	−15.7237	−0.645153
$595$	−24.3263	−0.997281
$596$	14.8516	0.608347
$597$	13.0761	0.535169
$598$	28.1899	1.15277
$599$	−33.4886	−1.36831	−0.684153	−	0.729338i	$-0.739828\pi$
$599$	−33.4886	−1.36831	−0.684153	+	0.729338i	$0.739828\pi$
$600$	−6.50210	−0.265447
$601$	17.9912	0.733877	0.366939	−	0.930245i	$-0.380406\pi$
$601$	17.9912	0.733877	0.366939	+	0.930245i	$0.380406\pi$
$602$	−46.0868	−1.87836
$603$	−61.2827	−2.49562
$604$	15.5105	0.631112
$605$	33.3822	1.35718
$606$	−6.67453	−0.271134
$607$	17.4895	0.709876	0.354938	−	0.934890i	$-0.384502\pi$
$607$	17.4895	0.709876	0.354938	+	0.934890i	$0.384502\pi$
$608$	−3.74760	−0.151985
$609$	27.1815	1.10145
$610$	−13.3719	−0.541412
$611$	−37.2325	−1.50627
$612$	−13.9770	−0.564988
$613$	40.1360	1.62108	0.810540	−	0.585684i	$-0.199174\pi$
$613$	40.1360	1.62108	0.810540	+	0.585684i	$0.199174\pi$
$614$	−2.37170	−0.0957140
$615$	−45.4179	−1.83143
$616$	24.9183	1.00398
$617$	24.2294	0.975437	0.487719	−	0.873001i	$-0.337829\pi$
$617$	24.2294	0.975437	0.487719	+	0.873001i	$0.337829\pi$
$618$	−37.9102	−1.52497
$619$	19.0272	0.764766	0.382383	−	0.924004i	$-0.375104\pi$
$619$	19.0272	0.764766	0.382383	+	0.924004i	$0.375104\pi$
$620$	−10.8777	−0.436858
$621$	−15.0392	−0.603504
$622$	24.3606	0.976772
$623$	20.4505	0.819332
$624$	13.8539	0.554601
$625$	−6.75661	−0.270264
$626$	−5.25944	−0.210210
$627$	56.1960	2.24425
$628$	−24.8243	−0.990598
$629$	25.6019	1.02082
$630$	−28.5277	−1.13657
$631$	−48.9771	−1.94975	−0.974873	−	0.222760i	$-0.928493\pi$
$631$	−48.9771	−1.94975	−0.974873	+	0.222760i	$0.928493\pi$
$632$	−2.22060	−0.0883308
$633$	37.2600	1.48095
$634$	−4.79293	−0.190351
$635$	8.30382	0.329527
$636$	−24.5642	−0.974033
$637$	−65.0397	−2.57697
$638$	−13.1072	−0.518918
$639$	−36.7836	−1.45514
$640$	−1.59716	−0.0631333
$641$	15.1873	0.599861	0.299931	−	0.953961i	$-0.403037\pi$
$641$	15.1873	0.599861	0.299931	+	0.953961i	$0.403037\pi$
$642$	−30.1828	−1.19122
$643$	−32.3979	−1.27765	−0.638825	−	0.769352i	$-0.720579\pi$
$643$	−32.3979	−1.27765	−0.638825	+	0.769352i	$0.720579\pi$
$644$	23.8335	0.939171
$645$	−44.2956	−1.74414
$646$	12.9379	0.509036
$647$	2.70746	0.106441	0.0532206	−	0.998583i	$-0.483051\pi$
$647$	2.70746	0.106441	0.0532206	+	0.998583i	$0.483051\pi$
$648$	4.75472	0.186783
$649$	11.2703	0.442397
$650$	−12.7798	−0.501266
$651$	79.7727	3.12654
$652$	19.1996	0.751914
$653$	−16.4620	−0.644209	−0.322105	−	0.946704i	$-0.604390\pi$
$653$	−16.4620	−0.644209	−0.322105	+	0.946704i	$0.604390\pi$
$654$	−39.7066	−1.55265
$655$	−24.1674	−0.944297
$656$	10.7109	0.418192
$657$	18.9452	0.739124
$658$	−31.4786	−1.22716
$659$	4.84318	0.188664	0.0943318	−	0.995541i	$-0.469929\pi$
$659$	4.84318	0.188664	0.0943318	+	0.995541i	$0.469929\pi$
$660$	23.9498	0.932243
$661$	2.96466	0.115312	0.0576559	−	0.998337i	$-0.481637\pi$
$661$	2.96466	0.115312	0.0576559	+	0.998337i	$0.481637\pi$
$662$	6.37409	0.247736
$663$	−47.8283	−1.85750
$664$	−15.1406	−0.587571
$665$	26.4069	1.02401
$666$	30.0237	1.16339
$667$	−12.5366	−0.485419
$668$	−13.8252	−0.534912
$669$	65.6037	2.53639
$670$	24.1759	0.933997
$671$	−47.2875	−1.82551
$672$	11.7130	0.451837
$673$	34.7781	1.34060	0.670298	−	0.742092i	$-0.266166\pi$
$673$	34.7781	1.34060	0.670298	+	0.742092i	$0.266166\pi$
$674$	−5.66676	−0.218275
$675$	6.81800	0.262425
$676$	14.2298	0.547300
$677$	45.3966	1.74473	0.872367	−	0.488851i	$-0.162584\pi$
$677$	45.3966	1.74473	0.872367	+	0.488851i	$0.162584\pi$
$678$	−2.73242	−0.104938
$679$	9.13031	0.350389
$680$	5.51392	0.211449
$681$	43.8088	1.67875
$682$	−38.4671	−1.47298
$683$	25.5880	0.979098	0.489549	−	0.871976i	$-0.337161\pi$
$683$	25.5880	0.979098	0.489549	+	0.871976i	$0.337161\pi$
$684$	15.1725	0.580133
$685$	20.7312	0.792098
$686$	−24.1059	−0.920369
$687$	21.5332	0.821541
$688$	10.4463	0.398260
$689$	−48.2807	−1.83935
$690$	22.9072	0.872061
$691$	−43.6362	−1.66000	−0.830000	−	0.557764i	$-0.811660\pi$
$691$	−43.6362	−1.66000	−0.830000	+	0.557764i	$0.811660\pi$
$692$	0.635650	0.0241638
$693$	−100.884	−3.83225
$694$	14.5274	0.551454
$695$	−2.22768	−0.0845006
$696$	−6.16110	−0.233536
$697$	−36.9776	−1.40063
$698$	−20.8620	−0.789640
$699$	−59.5189	−2.25121
$700$	−10.8048	−0.408385
$701$	3.24302	0.122487	0.0612436	−	0.998123i	$-0.480493\pi$
$701$	3.24302	0.122487	0.0612436	+	0.998123i	$0.480493\pi$
$702$	−14.5270	−0.548287
$703$	−27.7916	−1.04818
$704$	−5.64809	−0.212871
$705$	−30.2552	−1.13948
$706$	5.52688	0.208007
$707$	−11.0914	−0.417134
$708$	5.29765	0.199098
$709$	21.6593	0.813433	0.406716	−	0.913554i	$-0.366674\pi$
$709$	21.6593	0.813433	0.406716	+	0.913554i	$0.366674\pi$
$710$	14.5111	0.544590
$711$	8.99029	0.337162
$712$	−4.63541	−0.173719
$713$	−36.7925	−1.37789
$714$	−40.4370	−1.51331
$715$	47.0731	1.76043
$716$	5.01193	0.187304
$717$	−29.1118	−1.08720
$718$	−10.0070	−0.373458
$719$	−13.4587	−0.501924	−0.250962	−	0.967997i	$-0.580747\pi$
$719$	−13.4587	−0.501924	−0.250962	+	0.967997i	$0.580747\pi$
$720$	6.46624	0.240982
$721$	−62.9971	−2.34613
$722$	4.95553	0.184426
$723$	−55.8944	−2.07873
$724$	−8.81525	−0.327616
$725$	5.68343	0.211077
$726$	55.4903	2.05944
$727$	−31.4176	−1.16521	−0.582607	−	0.812754i	$-0.697967\pi$
$727$	−31.4176	−1.16521	−0.582607	+	0.812754i	$0.697967\pi$
$728$	23.0217	0.853242
$729$	−41.4230	−1.53418
$730$	−7.47386	−0.276620
$731$	−36.0639	−1.33387
$732$	−22.2277	−0.821561
$733$	−40.6155	−1.50017	−0.750085	−	0.661342i	$-0.769987\pi$
$733$	−40.6155	−1.50017	−0.750085	+	0.661342i	$0.769987\pi$
$734$	4.04260	0.149215
$735$	−52.8513	−1.94945
$736$	−5.40222	−0.199128
$737$	85.4941	3.14922
$738$	−43.3641	−1.59626
$739$	−6.55016	−0.240951	−0.120476	−	0.992716i	$-0.538442\pi$
$739$	−6.55016	−0.240951	−0.120476	+	0.992716i	$0.538442\pi$
$740$	−11.8443	−0.435405
$741$	51.9189	1.90729
$742$	−40.8194	−1.49853
$743$	47.0728	1.72693	0.863467	−	0.504405i	$-0.168288\pi$
$743$	47.0728	1.72693	0.863467	+	0.504405i	$0.168288\pi$
$744$	−18.0817	−0.662906
$745$	23.7204	0.869050
$746$	34.0949	1.24830
$747$	61.2982	2.24278
$748$	19.4991	0.712956
$749$	−50.1561	−1.83266
$750$	−31.5865	−1.15338
$751$	46.7427	1.70566	0.852832	−	0.522185i	$-0.174883\pi$
$751$	46.7427	1.70566	0.852832	+	0.522185i	$0.174883\pi$
$752$	7.13510	0.260190
$753$	65.8522	2.39979
$754$	−12.1096	−0.441005
$755$	24.7727	0.901570
$756$	−12.2820	−0.446693
$757$	17.9081	0.650881	0.325441	−	0.945562i	$-0.394487\pi$
$757$	17.9081	0.650881	0.325441	+	0.945562i	$0.394487\pi$
$758$	2.06608	0.0750433
$759$	81.0074	2.94038
$760$	−5.98551	−0.217117
$761$	−9.63454	−0.349252	−0.174626	−	0.984635i	$-0.555872\pi$
$761$	−9.63454	−0.349252	−0.174626	+	0.984635i	$0.555872\pi$
$762$	13.8032	0.500038
$763$	−65.9824	−2.38872
$764$	7.66503	0.277311
$765$	−22.3236	−0.807110
$766$	10.3223	0.372958
$767$	10.4125	0.375973
$768$	−2.65492	−0.0958011
$769$	16.2707	0.586736	0.293368	−	0.956000i	$-0.405224\pi$
$769$	16.2707	0.586736	0.293368	+	0.956000i	$0.405224\pi$
$770$	39.7984	1.43424
$771$	71.4921	2.57472
$772$	−26.6394	−0.958772
$773$	−11.7713	−0.423385	−0.211692	−	0.977336i	$-0.567897\pi$
$773$	−11.7713	−0.423385	−0.211692	+	0.977336i	$0.567897\pi$
$774$	−42.2926	−1.52018
$775$	16.6798	0.599156
$776$	−2.06952	−0.0742915
$777$	86.8615	3.11614
$778$	22.3296	0.800555
$779$	40.1402	1.43817
$780$	22.1270	0.792272
$781$	51.3160	1.83623
$782$	18.6502	0.666930
$783$	6.46043	0.230877
$784$	12.4640	0.445142
$785$	−39.6484	−1.41511
$786$	−40.1728	−1.43292
$787$	−5.26219	−0.187577	−0.0937884	−	0.995592i	$-0.529898\pi$
$787$	−5.26219	−0.187577	−0.0937884	+	0.995592i	$0.529898\pi$
$788$	9.11959	0.324872
$789$	−51.7314	−1.84169
$790$	−3.54665	−0.126184
$791$	−4.54060	−0.161445
$792$	22.8668	0.812536
$793$	−43.6884	−1.55142
$794$	−2.48301	−0.0881187
$795$	−39.2329	−1.39145
$796$	−4.92524	−0.174571
$797$	20.2249	0.716403	0.358202	−	0.933644i	$-0.383390\pi$
$797$	20.2249	0.716403	0.358202	+	0.933644i	$0.383390\pi$
$798$	43.8954	1.55388
$799$	−24.6327	−0.871442
$800$	2.44908	0.0865881
$801$	18.7669	0.663094
$802$	−2.11445	−0.0746638
$803$	−26.4301	−0.932697
$804$	40.1870	1.41729
$805$	38.0659	1.34165
$806$	−35.5394	−1.25182
$807$	−72.1134	−2.53851
$808$	2.51403	0.0884432
$809$	−38.3176	−1.34718	−0.673588	−	0.739107i	$-0.735248\pi$
$809$	−38.3176	−1.34718	−0.673588	+	0.739107i	$0.735248\pi$
$810$	7.59405	0.266828
$811$	−25.6008	−0.898965	−0.449483	−	0.893289i	$-0.648392\pi$
$811$	−25.6008	−0.898965	−0.449483	+	0.893289i	$0.648392\pi$
$812$	−10.2382	−0.359290
$813$	66.7547	2.34119
$814$	−41.8854	−1.46808
$815$	30.6648	1.07414
$816$	9.16564	0.320861
$817$	39.1484	1.36963
$818$	2.70606	0.0946153
$819$	−93.2054	−3.25686
$820$	17.1071	0.597405
$821$	−23.5548	−0.822069	−0.411035	−	0.911620i	$-0.634833\pi$
$821$	−23.5548	−0.822069	−0.411035	+	0.911620i	$0.634833\pi$
$822$	34.4609	1.20196
$823$	4.98682	0.173830	0.0869149	−	0.996216i	$-0.472299\pi$
$823$	4.98682	0.173830	0.0869149	+	0.996216i	$0.472299\pi$
$824$	14.2792	0.497441
$825$	−36.7245	−1.27858
$826$	8.80335	0.306308
$827$	9.05855	0.314997	0.157498	−	0.987519i	$-0.449657\pi$
$827$	9.05855	0.314997	0.157498	+	0.987519i	$0.449657\pi$
$828$	21.8713	0.760081
$829$	31.8706	1.10691	0.553456	−	0.832879i	$-0.313309\pi$
$829$	31.8706	1.10691	0.553456	+	0.832879i	$0.313309\pi$
$830$	−24.1820	−0.839371
$831$	81.1101	2.81368
$832$	−5.21822	−0.180909
$833$	−43.0297	−1.49089
$834$	−3.70301	−0.128225
$835$	−22.0810	−0.764145
$836$	−21.1668	−0.732068
$837$	18.9601	0.655359
$838$	−1.39259	−0.0481061
$839$	−24.6478	−0.850937	−0.425468	−	0.904973i	$-0.639891\pi$
$839$	−24.6478	−0.850937	−0.425468	+	0.904973i	$0.639891\pi$
$840$	18.7075	0.645469
$841$	−23.6146	−0.814298
$842$	37.8883	1.30572
$843$	−8.63417	−0.297377
$844$	−14.0343	−0.483082
$845$	22.7273	0.781842
$846$	−28.8871	−0.993157
$847$	92.2108	3.16840
$848$	9.25234	0.317727
$849$	42.8468	1.47050
$850$	−8.45502	−0.290005
$851$	−40.0620	−1.37331
$852$	24.1213	0.826384
$853$	22.5119	0.770791	0.385396	−	0.922751i	$-0.374065\pi$
$853$	22.5119	0.770791	0.385396	+	0.922751i	$0.374065\pi$
$854$	−36.9369	−1.26395
$855$	24.2328	0.828746
$856$	11.3686	0.388572
$857$	45.8007	1.56452	0.782262	−	0.622950i	$-0.214066\pi$
$857$	45.8007	1.56452	0.782262	+	0.622950i	$0.214066\pi$
$858$	78.2483	2.67135
$859$	−24.4448	−0.834045	−0.417022	−	0.908896i	$-0.636926\pi$
$859$	−24.4448	−0.834045	−0.417022	+	0.908896i	$0.636926\pi$
$860$	16.6844	0.568932
$861$	−125.457	−4.27555
$862$	−4.36135	−0.148548
$863$	−13.3482	−0.454380	−0.227190	−	0.973851i	$-0.572954\pi$
$863$	−13.3482	−0.454380	−0.227190	+	0.973851i	$0.572954\pi$
$864$	2.78390	0.0947103
$865$	1.01523	0.0345190
$866$	8.11762	0.275848
$867$	13.4908	0.458172
$868$	−30.0472	−1.01987
$869$	−12.5422	−0.425464
$870$	−9.84026	−0.333616
$871$	78.9872	2.67638
$872$	14.9559	0.506470
$873$	8.37863	0.283574
$874$	−20.2453	−0.684808
$875$	−52.4888	−1.77445
$876$	−12.4236	−0.419755
$877$	−3.06327	−0.103439	−0.0517197	−	0.998662i	$-0.516470\pi$
$877$	−3.06327	−0.103439	−0.0517197	+	0.998662i	$0.516470\pi$
$878$	−32.7649	−1.10576
$879$	−70.6068	−2.38151
$880$	−9.02091	−0.304095
$881$	41.1072	1.38494	0.692468	−	0.721448i	$-0.256523\pi$
$881$	41.1072	1.38494	0.692468	+	0.721448i	$0.256523\pi$
$882$	−50.4614	−1.69912
$883$	−41.3248	−1.39069	−0.695346	−	0.718676i	$-0.744749\pi$
$883$	−41.3248	−1.39069	−0.695346	+	0.718676i	$0.744749\pi$
$884$	18.0150	0.605910
$885$	8.46119	0.284420
$886$	29.4643	0.989874
$887$	45.6088	1.53140	0.765698	−	0.643201i	$-0.222394\pi$
$887$	45.6088	1.53140	0.765698	+	0.643201i	$0.222394\pi$
$888$	−19.6885	−0.660701
$889$	22.9374	0.769297
$890$	−7.40349	−0.248166
$891$	26.8551	0.899680
$892$	−24.7103	−0.827361
$893$	26.7395	0.894802
$894$	39.4299	1.31873
$895$	8.00485	0.267573
$896$	−4.41180	−0.147388
$897$	74.8420	2.49890
$898$	−2.60085	−0.0867914
$899$	15.8050	0.527127
$900$	−9.91531	−0.330510
$901$	−31.9421	−1.06414
$902$	60.4964	2.01431
$903$	−122.357	−4.07178
$904$	1.02919	0.0342305
$905$	−14.0794	−0.468014
$906$	41.1790	1.36808
$907$	14.0812	0.467557	0.233779	−	0.972290i	$-0.424891\pi$
$907$	14.0812	0.467557	0.233779	+	0.972290i	$0.424891\pi$
$908$	−16.5010	−0.547604
$909$	−10.1783	−0.337591
$910$	36.7694	1.21889
$911$	−19.1817	−0.635517	−0.317758	−	0.948172i	$-0.602930\pi$
$911$	−19.1817	−0.635517	−0.317758	+	0.948172i	$0.602930\pi$
$912$	−9.94955	−0.329463
$913$	−85.5158	−2.83016
$914$	10.1756	0.336578
$915$	−35.5013	−1.17364
$916$	−8.11067	−0.267984
$917$	−66.7570	−2.20451
$918$	−9.61094	−0.317208
$919$	19.2672	0.635566	0.317783	−	0.948163i	$-0.397062\pi$
$919$	19.2672	0.635566	0.317783	+	0.948163i	$0.397062\pi$
$920$	−8.62820	−0.284463
$921$	−6.29666	−0.207482
$922$	−14.6226	−0.481570
$923$	47.4103	1.56053
$924$	66.1559	2.17637
$925$	18.1620	0.597163
$926$	15.4574	0.507962
$927$	−57.8107	−1.89875
$928$	2.32064	0.0761787
$929$	20.8116	0.682805	0.341403	−	0.939917i	$-0.389098\pi$
$929$	20.8116	0.682805	0.341403	+	0.939917i	$0.389098\pi$
$930$	−28.8793	−0.946990
$931$	46.7099	1.53086
$932$	22.4184	0.734338
$933$	64.6754	2.11738
$934$	17.0460	0.557761
$935$	31.1431	1.01849
$936$	21.1264	0.690538
$937$	25.9270	0.846997	0.423499	−	0.905897i	$-0.360802\pi$
$937$	25.9270	0.846997	0.423499	+	0.905897i	$0.360802\pi$
$938$	66.7806	2.18046
$939$	−13.9634	−0.455678
$940$	11.3959	0.371693
$941$	−24.5770	−0.801186	−0.400593	−	0.916256i	$-0.631196\pi$
$941$	−24.5770	−0.801186	−0.400593	+	0.916256i	$0.631196\pi$
$942$	−65.9065	−2.14735
$943$	57.8628	1.88427
$944$	−1.99541	−0.0649451
$945$	−19.6164	−0.638120
$946$	59.0015	1.91830
$947$	−9.47395	−0.307862	−0.153931	−	0.988082i	$-0.549193\pi$
$947$	−9.47395	−0.307862	−0.153931	+	0.988082i	$0.549193\pi$
$948$	−5.89551	−0.191477
$949$	−24.4185	−0.792658
$950$	9.17816	0.297779
$951$	−12.7248	−0.412631
$952$	15.2310	0.493638
$953$	−16.1509	−0.523179	−0.261590	−	0.965179i	$-0.584247\pi$
$953$	−16.1509	−0.523179	−0.261590	+	0.965179i	$0.584247\pi$
$954$	−37.4589	−1.21278
$955$	12.2423	0.396151
$956$	10.9653	0.354642
$957$	−34.7985	−1.12487
$958$	2.65708	0.0858464
$959$	57.2653	1.84919
$960$	−4.24033	−0.136856
$961$	15.3848	0.496282
$962$	−38.6975	−1.24766
$963$	−46.0269	−1.48319
$964$	21.0532	0.678077
$965$	−42.5473	−1.36965
$966$	63.2759	2.03587
$967$	4.30463	0.138428	0.0692139	−	0.997602i	$-0.477951\pi$
$967$	4.30463	0.138428	0.0692139	+	0.997602i	$0.477951\pi$
$968$	−20.9010	−0.671782
$969$	34.3491	1.10345
$970$	−3.30536	−0.106129
$971$	−7.58236	−0.243329	−0.121665	−	0.992571i	$-0.538823\pi$
$971$	−7.58236	−0.243329	−0.121665	+	0.992571i	$0.538823\pi$
$972$	20.9751	0.672777
$973$	−6.15346	−0.197271
$974$	1.60071	0.0512900
$975$	−33.9294	−1.08661
$976$	8.37229	0.267991
$977$	−14.6162	−0.467614	−0.233807	−	0.972283i	$-0.575118\pi$
$977$	−14.6162	−0.467614	−0.233807	+	0.972283i	$0.575118\pi$
$978$	50.9733	1.62995
$979$	−26.1812	−0.836756
$980$	19.9070	0.635904
$981$	−60.5502	−1.93322
$982$	4.81994	0.153811
$983$	−12.9559	−0.413230	−0.206615	−	0.978422i	$-0.566245\pi$
$983$	−12.9559	−0.413230	−0.206615	+	0.978422i	$0.566245\pi$
$984$	28.4366	0.906527
$985$	14.5654	0.464094
$986$	−8.01160	−0.255141
$987$	−83.5731	−2.66016
$988$	−19.5558	−0.622152
$989$	56.4330	1.79446
$990$	36.5219	1.16074
$991$	−30.3342	−0.963597	−0.481799	−	0.876282i	$-0.660016\pi$
$991$	−30.3342	−0.963597	−0.481799	+	0.876282i	$0.660016\pi$
$992$	6.81064	0.216238
$993$	16.9227	0.537025
$994$	40.0836	1.27137
$995$	−7.86640	−0.249382
$996$	−40.1972	−1.27370
$997$	−18.6435	−0.590446	−0.295223	−	0.955428i	$-0.595394\pi$
$997$	−18.6435	−0.590446	−0.295223	+	0.955428i	$0.595394\pi$
$998$	−29.0971	−0.921053
$999$	20.6450	0.653179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.b.1.14	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.b.1.14	✓	82	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.65492 q^{3} +1.00000 q^{4} +1.59716 q^{5} +2.65492 q^{6} +4.41180 q^{7} -1.00000 q^{8} +4.04858 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.65492 q^{3} +1.00000 q^{4} +1.59716 q^{5} +2.65492 q^{6} +4.41180 q^{7} -1.00000 q^{8} +4.04858 q^{9} -1.59716 q^{10} -5.64809 q^{11} -2.65492 q^{12} -5.21822 q^{13} -4.41180 q^{14} -4.24033 q^{15} +1.00000 q^{16} -3.45233 q^{17} -4.04858 q^{18} +3.74760 q^{19} +1.59716 q^{20} -11.7130 q^{21} +5.64809 q^{22} +5.40222 q^{23} +2.65492 q^{24} -2.44908 q^{25} +5.21822 q^{26} -2.78390 q^{27} +4.41180 q^{28} -2.32064 q^{29} +4.24033 q^{30} -6.81064 q^{31} -1.00000 q^{32} +14.9952 q^{33} +3.45233 q^{34} +7.04635 q^{35} +4.04858 q^{36} -7.41585 q^{37} -3.74760 q^{38} +13.8539 q^{39} -1.59716 q^{40} +10.7109 q^{41} +11.7130 q^{42} +10.4463 q^{43} -5.64809 q^{44} +6.46624 q^{45} -5.40222 q^{46} +7.13510 q^{47} -2.65492 q^{48} +12.4640 q^{49} +2.44908 q^{50} +9.16564 q^{51} -5.21822 q^{52} +9.25234 q^{53} +2.78390 q^{54} -9.02091 q^{55} -4.41180 q^{56} -9.94955 q^{57} +2.32064 q^{58} -1.99541 q^{59} -4.24033 q^{60} +8.37229 q^{61} +6.81064 q^{62} +17.8615 q^{63} +1.00000 q^{64} -8.33433 q^{65} -14.9952 q^{66} -15.1368 q^{67} -3.45233 q^{68} -14.3424 q^{69} -7.04635 q^{70} -9.08554 q^{71} -4.04858 q^{72} +4.67947 q^{73} +7.41585 q^{74} +6.50210 q^{75} +3.74760 q^{76} -24.9183 q^{77} -13.8539 q^{78} +2.22060 q^{79} +1.59716 q^{80} -4.75472 q^{81} -10.7109 q^{82} +15.1406 q^{83} -11.7130 q^{84} -5.51392 q^{85} -10.4463 q^{86} +6.16110 q^{87} +5.64809 q^{88} +4.63541 q^{89} -6.46624 q^{90} -23.0217 q^{91} +5.40222 q^{92} +18.0817 q^{93} -7.13510 q^{94} +5.98551 q^{95} +2.65492 q^{96} +2.06952 q^{97} -12.4640 q^{98} -22.8668 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * q^99

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