LMFDB - Embedded newform 4002.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
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} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 $ x^8 - 24x^6 - 3x^5 + 194x^4 + 39x^3 - 607x^2 - 104x + 600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.45069$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.40973 q^{5} -1.00000 q^{6} -0.551242 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.40973 q^{5} -1.00000 q^{6} -0.551242 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.40973 q^{10} +1.47581 q^{11} +1.00000 q^{12} -2.62354 q^{13} +0.551242 q^{14} -4.40973 q^{15} +1.00000 q^{16} -1.48725 q^{17} -1.00000 q^{18} +2.46932 q^{19} -4.40973 q^{20} -0.551242 q^{21} -1.47581 q^{22} -1.00000 q^{23} -1.00000 q^{24} +14.4457 q^{25} +2.62354 q^{26} +1.00000 q^{27} -0.551242 q^{28} +1.00000 q^{29} +4.40973 q^{30} +5.87228 q^{31} -1.00000 q^{32} +1.47581 q^{33} +1.48725 q^{34} +2.43083 q^{35} +1.00000 q^{36} +4.22032 q^{37} -2.46932 q^{38} -2.62354 q^{39} +4.40973 q^{40} -0.176142 q^{41} +0.551242 q^{42} +11.2284 q^{43} +1.47581 q^{44} -4.40973 q^{45} +1.00000 q^{46} -3.46051 q^{47} +1.00000 q^{48} -6.69613 q^{49} -14.4457 q^{50} -1.48725 q^{51} -2.62354 q^{52} -3.90957 q^{53} -1.00000 q^{54} -6.50794 q^{55} +0.551242 q^{56} +2.46932 q^{57} -1.00000 q^{58} +9.14724 q^{59} -4.40973 q^{60} -11.4953 q^{61} -5.87228 q^{62} -0.551242 q^{63} +1.00000 q^{64} +11.5691 q^{65} -1.47581 q^{66} -13.9867 q^{67} -1.48725 q^{68} -1.00000 q^{69} -2.43083 q^{70} -15.0454 q^{71} -1.00000 q^{72} +6.61444 q^{73} -4.22032 q^{74} +14.4457 q^{75} +2.46932 q^{76} -0.813530 q^{77} +2.62354 q^{78} -1.53217 q^{79} -4.40973 q^{80} +1.00000 q^{81} +0.176142 q^{82} -5.66460 q^{83} -0.551242 q^{84} +6.55839 q^{85} -11.2284 q^{86} +1.00000 q^{87} -1.47581 q^{88} -6.00851 q^{89} +4.40973 q^{90} +1.44620 q^{91} -1.00000 q^{92} +5.87228 q^{93} +3.46051 q^{94} -10.8890 q^{95} -1.00000 q^{96} -10.1549 q^{97} +6.69613 q^{98} +1.47581 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−4.40973	−1.97209	−0.986046	−	0.166476i	$-0.946761\pi$
$5$	−4.40973	−1.97209	−0.986046	+	0.166476i	$0.946761\pi$
$6$	−1.00000	−0.408248
$7$	−0.551242	−0.208350	−0.104175	−	0.994559i	$-0.533220\pi$
$7$	−0.551242	−0.208350	−0.104175	+	0.994559i	$0.533220\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	4.40973	1.39448
$11$	1.47581	0.444975	0.222487	−	0.974936i	$-0.428582\pi$
$11$	1.47581	0.444975	0.222487	+	0.974936i	$0.428582\pi$
$12$	1.00000	0.288675
$13$	−2.62354	−0.727638	−0.363819	−	0.931470i	$-0.618527\pi$
$13$	−2.62354	−0.727638	−0.363819	+	0.931470i	$0.618527\pi$
$14$	0.551242	0.147326
$15$	−4.40973	−1.13859
$16$	1.00000	0.250000
$17$	−1.48725	−0.360712	−0.180356	−	0.983601i	$-0.557725\pi$
$17$	−1.48725	−0.360712	−0.180356	+	0.983601i	$0.557725\pi$
$18$	−1.00000	−0.235702
$19$	2.46932	0.566502	0.283251	−	0.959046i	$-0.408587\pi$
$19$	2.46932	0.566502	0.283251	+	0.959046i	$0.408587\pi$
$20$	−4.40973	−0.986046
$21$	−0.551242	−0.120291
$22$	−1.47581	−0.314645
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	14.4457	2.88914
$26$	2.62354	0.514518
$27$	1.00000	0.192450
$28$	−0.551242	−0.104175
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	4.40973	0.805103
$31$	5.87228	1.05469	0.527346	−	0.849651i	$-0.323187\pi$
$31$	5.87228	1.05469	0.527346	+	0.849651i	$0.323187\pi$
$32$	−1.00000	−0.176777
$33$	1.47581	0.256906
$34$	1.48725	0.255062
$35$	2.43083	0.410885
$36$	1.00000	0.166667
$37$	4.22032	0.693816	0.346908	−	0.937899i	$-0.387232\pi$
$37$	4.22032	0.693816	0.346908	+	0.937899i	$0.387232\pi$
$38$	−2.46932	−0.400577
$39$	−2.62354	−0.420102
$40$	4.40973	0.697239
$41$	−0.176142	−0.0275088	−0.0137544	−	0.999905i	$-0.504378\pi$
$41$	−0.176142	−0.0275088	−0.0137544	+	0.999905i	$0.504378\pi$
$42$	0.551242	0.0850584
$43$	11.2284	1.71231	0.856154	−	0.516720i	$-0.172847\pi$
$43$	11.2284	1.71231	0.856154	+	0.516720i	$0.172847\pi$
$44$	1.47581	0.222487
$45$	−4.40973	−0.657364
$46$	1.00000	0.147442
$47$	−3.46051	−0.504767	−0.252384	−	0.967627i	$-0.581215\pi$
$47$	−3.46051	−0.504767	−0.252384	+	0.967627i	$0.581215\pi$
$48$	1.00000	0.144338
$49$	−6.69613	−0.956590
$50$	−14.4457	−2.04293
$51$	−1.48725	−0.208257
$52$	−2.62354	−0.363819
$53$	−3.90957	−0.537020	−0.268510	−	0.963277i	$-0.586531\pi$
$53$	−3.90957	−0.537020	−0.268510	+	0.963277i	$0.586531\pi$
$54$	−1.00000	−0.136083
$55$	−6.50794	−0.877530
$56$	0.551242	0.0736628
$57$	2.46932	0.327070
$58$	−1.00000	−0.131306
$59$	9.14724	1.19087	0.595434	−	0.803404i	$-0.296980\pi$
$59$	9.14724	1.19087	0.595434	+	0.803404i	$0.296980\pi$
$60$	−4.40973	−0.569294
$61$	−11.4953	−1.47182	−0.735910	−	0.677079i	$-0.763245\pi$
$61$	−11.4953	−1.47182	−0.735910	+	0.677079i	$0.763245\pi$
$62$	−5.87228	−0.745780
$63$	−0.551242	−0.0694499
$64$	1.00000	0.125000
$65$	11.5691	1.43497
$66$	−1.47581	−0.181660
$67$	−13.9867	−1.70874	−0.854372	−	0.519662i	$-0.826058\pi$
$67$	−13.9867	−1.70874	−0.854372	+	0.519662i	$0.826058\pi$
$68$	−1.48725	−0.180356
$69$	−1.00000	−0.120386
$70$	−2.43083	−0.290539
$71$	−15.0454	−1.78556	−0.892780	−	0.450494i	$-0.851248\pi$
$71$	−15.0454	−1.78556	−0.892780	+	0.450494i	$0.851248\pi$
$72$	−1.00000	−0.117851
$73$	6.61444	0.774162	0.387081	−	0.922046i	$-0.373483\pi$
$73$	6.61444	0.774162	0.387081	+	0.922046i	$0.373483\pi$
$74$	−4.22032	−0.490602
$75$	14.4457	1.66805
$76$	2.46932	0.283251
$77$	−0.813530	−0.0927103
$78$	2.62354	0.297057
$79$	−1.53217	−0.172382	−0.0861912	−	0.996279i	$-0.527470\pi$
$79$	−1.53217	−0.172382	−0.0861912	+	0.996279i	$0.527470\pi$
$80$	−4.40973	−0.493023
$81$	1.00000	0.111111
$82$	0.176142	0.0194517
$83$	−5.66460	−0.621771	−0.310885	−	0.950447i	$-0.600626\pi$
$83$	−5.66460	−0.621771	−0.310885	+	0.950447i	$0.600626\pi$
$84$	−0.551242	−0.0601454
$85$	6.55839	0.711358
$86$	−11.2284	−1.21079
$87$	1.00000	0.107211
$88$	−1.47581	−0.157322
$89$	−6.00851	−0.636901	−0.318451	−	0.947939i	$-0.603162\pi$
$89$	−6.00851	−0.636901	−0.318451	+	0.947939i	$0.603162\pi$
$90$	4.40973	0.464826
$91$	1.44620	0.151603
$92$	−1.00000	−0.104257
$93$	5.87228	0.608927
$94$	3.46051	0.356924
$95$	−10.8890	−1.11719
$96$	−1.00000	−0.102062
$97$	−10.1549	−1.03108	−0.515539	−	0.856866i	$-0.672408\pi$
$97$	−10.1549	−1.03108	−0.515539	+	0.856866i	$0.672408\pi$
$98$	6.69613	0.676412
$99$	1.47581	0.148325
$100$	14.4457	1.44457
$101$	12.8475	1.27837	0.639185	−	0.769053i	$-0.279272\pi$
$101$	12.8475	1.27837	0.639185	+	0.769053i	$0.279272\pi$
$102$	1.48725	0.147260
$103$	18.7277	1.84530	0.922649	−	0.385640i	$-0.126019\pi$
$103$	18.7277	1.84530	0.922649	+	0.385640i	$0.126019\pi$
$104$	2.62354	0.257259
$105$	2.43083	0.237224
$106$	3.90957	0.379731
$107$	6.71940	0.649589	0.324794	−	0.945785i	$-0.394705\pi$
$107$	6.71940	0.649589	0.324794	+	0.945785i	$0.394705\pi$
$108$	1.00000	0.0962250
$109$	−12.5528	−1.20234	−0.601170	−	0.799121i	$-0.705298\pi$
$109$	−12.5528	−1.20234	−0.601170	+	0.799121i	$0.705298\pi$
$110$	6.50794	0.620508
$111$	4.22032	0.400575
$112$	−0.551242	−0.0520874
$113$	−10.8654	−1.02213	−0.511065	−	0.859542i	$-0.670749\pi$
$113$	−10.8654	−1.02213	−0.511065	+	0.859542i	$0.670749\pi$
$114$	−2.46932	−0.231273
$115$	4.40973	0.411209
$116$	1.00000	0.0928477
$117$	−2.62354	−0.242546
$118$	−9.14724	−0.842071
$119$	0.819837	0.0751543
$120$	4.40973	0.402551
$121$	−8.82197	−0.801998
$122$	11.4953	1.04073
$123$	−0.176142	−0.0158822
$124$	5.87228	0.527346
$125$	−41.6531	−3.72556
$126$	0.551242	0.0491085
$127$	−14.9438	−1.32605	−0.663023	−	0.748599i	$-0.730727\pi$
$127$	−14.9438	−1.32605	−0.663023	+	0.748599i	$0.730727\pi$
$128$	−1.00000	−0.0883883
$129$	11.2284	0.988602
$130$	−11.5691	−1.01468
$131$	13.7058	1.19748	0.598740	−	0.800943i	$-0.295668\pi$
$131$	13.7058	1.19748	0.598740	+	0.800943i	$0.295668\pi$
$132$	1.47581	0.128453
$133$	−1.36119	−0.118030
$134$	13.9867	1.20826
$135$	−4.40973	−0.379529
$136$	1.48725	0.127531
$137$	9.28159	0.792980	0.396490	−	0.918039i	$-0.370228\pi$
$137$	9.28159	0.792980	0.396490	+	0.918039i	$0.370228\pi$
$138$	1.00000	0.0851257
$139$	4.17985	0.354530	0.177265	−	0.984163i	$-0.443275\pi$
$139$	4.17985	0.354530	0.177265	+	0.984163i	$0.443275\pi$
$140$	2.43083	0.205442
$141$	−3.46051	−0.291428
$142$	15.0454	1.26258
$143$	−3.87185	−0.323781
$144$	1.00000	0.0833333
$145$	−4.40973	−0.366208
$146$	−6.61444	−0.547415
$147$	−6.69613	−0.552288
$148$	4.22032	0.346908
$149$	−10.7775	−0.882929	−0.441464	−	0.897279i	$-0.645541\pi$
$149$	−10.7775	−0.882929	−0.441464	+	0.897279i	$0.645541\pi$
$150$	−14.4457	−1.17949
$151$	23.8315	1.93938	0.969688	−	0.244346i	$-0.0785732\pi$
$151$	23.8315	1.93938	0.969688	+	0.244346i	$0.0785732\pi$
$152$	−2.46932	−0.200289
$153$	−1.48725	−0.120237
$154$	0.813530	0.0655561
$155$	−25.8951	−2.07995
$156$	−2.62354	−0.210051
$157$	8.35619	0.666896	0.333448	−	0.942768i	$-0.391788\pi$
$157$	8.35619	0.666896	0.333448	+	0.942768i	$0.391788\pi$
$158$	1.53217	0.121893
$159$	−3.90957	−0.310049
$160$	4.40973	0.348620
$161$	0.551242	0.0434439
$162$	−1.00000	−0.0785674
$163$	−22.1395	−1.73410	−0.867048	−	0.498225i	$-0.833985\pi$
$163$	−22.1395	−1.73410	−0.867048	+	0.498225i	$0.833985\pi$
$164$	−0.176142	−0.0137544
$165$	−6.50794	−0.506642
$166$	5.66460	0.439658
$167$	−18.7616	−1.45182	−0.725909	−	0.687790i	$-0.758581\pi$
$167$	−18.7616	−1.45182	−0.725909	+	0.687790i	$0.758581\pi$
$168$	0.551242	0.0425292
$169$	−6.11705	−0.470542
$170$	−6.55839	−0.503006
$171$	2.46932	0.188834
$172$	11.2284	0.856154
$173$	−11.8748	−0.902821	−0.451411	−	0.892316i	$-0.649079\pi$
$173$	−11.8748	−0.902821	−0.451411	+	0.892316i	$0.649079\pi$
$174$	−1.00000	−0.0758098
$175$	−7.96308	−0.601952
$176$	1.47581	0.111244
$177$	9.14724	0.687548
$178$	6.00851	0.450357
$179$	−1.33732	−0.0999559	−0.0499780	−	0.998750i	$-0.515915\pi$
$179$	−1.33732	−0.0999559	−0.0499780	+	0.998750i	$0.515915\pi$
$180$	−4.40973	−0.328682
$181$	−24.0003	−1.78393	−0.891964	−	0.452107i	$-0.850672\pi$
$181$	−24.0003	−1.78393	−0.891964	+	0.452107i	$0.850672\pi$
$182$	−1.44620	−0.107200
$183$	−11.4953	−0.849756
$184$	1.00000	0.0737210
$185$	−18.6105	−1.36827
$186$	−5.87228	−0.430576
$187$	−2.19491	−0.160508
$188$	−3.46051	−0.252384
$189$	−0.551242	−0.0400969
$190$	10.8890	0.789975
$191$	−10.3196	−0.746697	−0.373349	−	0.927691i	$-0.621790\pi$
$191$	−10.3196	−0.746697	−0.373349	+	0.927691i	$0.621790\pi$
$192$	1.00000	0.0721688
$193$	12.6810	0.912797	0.456398	−	0.889776i	$-0.349139\pi$
$193$	12.6810	0.912797	0.456398	+	0.889776i	$0.349139\pi$
$194$	10.1549	0.729082
$195$	11.5691	0.828480
$196$	−6.69613	−0.478295
$197$	−11.2181	−0.799258	−0.399629	−	0.916677i	$-0.630861\pi$
$197$	−11.2181	−0.799258	−0.399629	+	0.916677i	$0.630861\pi$
$198$	−1.47581	−0.104882
$199$	14.4889	1.02709	0.513545	−	0.858063i	$-0.328332\pi$
$199$	14.4889	1.02709	0.513545	+	0.858063i	$0.328332\pi$
$200$	−14.4457	−1.02147
$201$	−13.9867	−0.986544
$202$	−12.8475	−0.903944
$203$	−0.551242	−0.0386896
$204$	−1.48725	−0.104129
$205$	0.776740	0.0542499
$206$	−18.7277	−1.30482
$207$	−1.00000	−0.0695048
$208$	−2.62354	−0.181910
$209$	3.64426	0.252079
$210$	−2.43083	−0.167743
$211$	21.6426	1.48994	0.744968	−	0.667100i	$-0.232465\pi$
$211$	21.6426	1.48994	0.744968	+	0.667100i	$0.232465\pi$
$212$	−3.90957	−0.268510
$213$	−15.0454	−1.03089
$214$	−6.71940	−0.459328
$215$	−49.5140	−3.37683
$216$	−1.00000	−0.0680414
$217$	−3.23704	−0.219745
$218$	12.5528	0.850182
$219$	6.61444	0.446963
$220$	−6.50794	−0.438765
$221$	3.90187	0.262468
$222$	−4.22032	−0.283249
$223$	1.04791	0.0701736	0.0350868	−	0.999384i	$-0.488829\pi$
$223$	1.04791	0.0701736	0.0350868	+	0.999384i	$0.488829\pi$
$224$	0.551242	0.0368314
$225$	14.4457	0.963048
$226$	10.8654	0.722755
$227$	−10.1337	−0.672595	−0.336297	−	0.941756i	$-0.609175\pi$
$227$	−10.1337	−0.672595	−0.336297	+	0.941756i	$0.609175\pi$
$228$	2.46932	0.163535
$229$	−2.07214	−0.136931	−0.0684654	−	0.997653i	$-0.521810\pi$
$229$	−2.07214	−0.136931	−0.0684654	+	0.997653i	$0.521810\pi$
$230$	−4.40973	−0.290769
$231$	−0.813530	−0.0535263
$232$	−1.00000	−0.0656532
$233$	−25.9885	−1.70256	−0.851281	−	0.524710i	$-0.824174\pi$
$233$	−25.9885	−1.70256	−0.851281	+	0.524710i	$0.824174\pi$
$234$	2.62354	0.171506
$235$	15.2599	0.995447
$236$	9.14724	0.595434
$237$	−1.53217	−0.0995250
$238$	−0.819837	−0.0531421
$239$	−3.04112	−0.196714	−0.0983569	−	0.995151i	$-0.531359\pi$
$239$	−3.04112	−0.196714	−0.0983569	+	0.995151i	$0.531359\pi$
$240$	−4.40973	−0.284647
$241$	4.90079	0.315688	0.157844	−	0.987464i	$-0.449546\pi$
$241$	4.90079	0.315688	0.157844	+	0.987464i	$0.449546\pi$
$242$	8.82197	0.567098
$243$	1.00000	0.0641500
$244$	−11.4953	−0.735910
$245$	29.5281	1.88648
$246$	0.176142	0.0112304
$247$	−6.47836	−0.412208
$248$	−5.87228	−0.372890
$249$	−5.66460	−0.358979
$250$	41.6531	2.63437
$251$	15.5405	0.980907	0.490453	−	0.871467i	$-0.336831\pi$
$251$	15.5405	0.980907	0.490453	+	0.871467i	$0.336831\pi$
$252$	−0.551242	−0.0347250
$253$	−1.47581	−0.0927836
$254$	14.9438	0.937655
$255$	6.55839	0.410702
$256$	1.00000	0.0625000
$257$	−12.4808	−0.778529	−0.389265	−	0.921126i	$-0.627271\pi$
$257$	−12.4808	−0.778529	−0.389265	+	0.921126i	$0.627271\pi$
$258$	−11.2284	−0.699047
$259$	−2.32642	−0.144556
$260$	11.5691	0.717485
$261$	1.00000	0.0618984
$262$	−13.7058	−0.846746
$263$	−24.6452	−1.51969	−0.759844	−	0.650105i	$-0.774725\pi$
$263$	−24.6452	−1.51969	−0.759844	+	0.650105i	$0.774725\pi$
$264$	−1.47581	−0.0908300
$265$	17.2401	1.05905
$266$	1.36119	0.0834601
$267$	−6.00851	−0.367715
$268$	−13.9867	−0.854372
$269$	9.53343	0.581263	0.290632	−	0.956835i	$-0.406135\pi$
$269$	9.53343	0.581263	0.290632	+	0.956835i	$0.406135\pi$
$270$	4.40973	0.268368
$271$	14.8510	0.902134	0.451067	−	0.892490i	$-0.351044\pi$
$271$	14.8510	0.902134	0.451067	+	0.892490i	$0.351044\pi$
$272$	−1.48725	−0.0901781
$273$	1.44620	0.0875282
$274$	−9.28159	−0.560721
$275$	21.3192	1.28560
$276$	−1.00000	−0.0601929
$277$	−23.9020	−1.43613	−0.718067	−	0.695974i	$-0.754973\pi$
$277$	−23.9020	−1.43613	−0.718067	+	0.695974i	$0.754973\pi$
$278$	−4.17985	−0.250691
$279$	5.87228	0.351564
$280$	−2.43083	−0.145270
$281$	−7.50907	−0.447954	−0.223977	−	0.974594i	$-0.571904\pi$
$281$	−7.50907	−0.447954	−0.223977	+	0.974594i	$0.571904\pi$
$282$	3.46051	0.206070
$283$	−12.0219	−0.714626	−0.357313	−	0.933985i	$-0.616307\pi$
$283$	−12.0219	−0.714626	−0.357313	+	0.933985i	$0.616307\pi$
$284$	−15.0454	−0.892780
$285$	−10.8890	−0.645012
$286$	3.87185	0.228947
$287$	0.0970970	0.00573146
$288$	−1.00000	−0.0589256
$289$	−14.7881	−0.869887
$290$	4.40973	0.258948
$291$	−10.1549	−0.595293
$292$	6.61444	0.387081
$293$	−13.9017	−0.812148	−0.406074	−	0.913840i	$-0.633102\pi$
$293$	−13.9017	−0.812148	−0.406074	+	0.913840i	$0.633102\pi$
$294$	6.69613	0.390526
$295$	−40.3368	−2.34850
$296$	−4.22032	−0.245301
$297$	1.47581	0.0856354
$298$	10.7775	0.624325
$299$	2.62354	0.151723
$300$	14.4457	0.834024
$301$	−6.18954	−0.356759
$302$	−23.8315	−1.37135
$303$	12.8475	0.738068
$304$	2.46932	0.141625
$305$	50.6911	2.90256
$306$	1.48725	0.0850207
$307$	8.48620	0.484333	0.242167	−	0.970235i	$-0.422142\pi$
$307$	8.48620	0.484333	0.242167	+	0.970235i	$0.422142\pi$
$308$	−0.813530	−0.0463552
$309$	18.7277	1.06538
$310$	25.8951	1.47075
$311$	12.0620	0.683975	0.341987	−	0.939705i	$-0.388900\pi$
$311$	12.0620	0.683975	0.341987	+	0.939705i	$0.388900\pi$
$312$	2.62354	0.148529
$313$	−12.3576	−0.698495	−0.349248	−	0.937030i	$-0.613563\pi$
$313$	−12.3576	−0.698495	−0.349248	+	0.937030i	$0.613563\pi$
$314$	−8.35619	−0.471567
$315$	2.43083	0.136962
$316$	−1.53217	−0.0861912
$317$	5.92579	0.332825	0.166413	−	0.986056i	$-0.446782\pi$
$317$	5.92579	0.332825	0.166413	+	0.986056i	$0.446782\pi$
$318$	3.90957	0.219238
$319$	1.47581	0.0826297
$320$	−4.40973	−0.246511
$321$	6.71940	0.375040
$322$	−0.551242	−0.0307195
$323$	−3.67251	−0.204344
$324$	1.00000	0.0555556
$325$	−37.8989	−2.10225
$326$	22.1395	1.22619
$327$	−12.5528	−0.694171
$328$	0.176142	0.00972584
$329$	1.90758	0.105168
$330$	6.50794	0.358250
$331$	2.59893	0.142850	0.0714251	−	0.997446i	$-0.477245\pi$
$331$	2.59893	0.142850	0.0714251	+	0.997446i	$0.477245\pi$
$332$	−5.66460	−0.310885
$333$	4.22032	0.231272
$334$	18.7616	1.02659
$335$	61.6775	3.36980
$336$	−0.551242	−0.0300727
$337$	12.8005	0.697287	0.348644	−	0.937255i	$-0.386642\pi$
$337$	12.8005	0.697287	0.348644	+	0.937255i	$0.386642\pi$
$338$	6.11705	0.332724
$339$	−10.8654	−0.590127
$340$	6.55839	0.355679
$341$	8.66638	0.469311
$342$	−2.46932	−0.133526
$343$	7.54988	0.407655
$344$	−11.2284	−0.605393
$345$	4.40973	0.237412
$346$	11.8748	0.638391
$347$	27.4861	1.47553	0.737767	−	0.675056i	$-0.235880\pi$
$347$	27.4861	1.47553	0.737767	+	0.675056i	$0.235880\pi$
$348$	1.00000	0.0536056
$349$	20.0071	1.07096	0.535479	−	0.844549i	$-0.320131\pi$
$349$	20.0071	1.07096	0.535479	+	0.844549i	$0.320131\pi$
$350$	7.96308	0.425645
$351$	−2.62354	−0.140034
$352$	−1.47581	−0.0786611
$353$	3.93851	0.209626	0.104813	−	0.994492i	$-0.466576\pi$
$353$	3.93851	0.209626	0.104813	+	0.994492i	$0.466576\pi$
$354$	−9.14724	−0.486170
$355$	66.3461	3.52129
$356$	−6.00851	−0.318451
$357$	0.819837	0.0433904
$358$	1.33732	0.0706795
$359$	−24.1124	−1.27260	−0.636302	−	0.771440i	$-0.719537\pi$
$359$	−24.1124	−1.27260	−0.636302	+	0.771440i	$0.719537\pi$
$360$	4.40973	0.232413
$361$	−12.9024	−0.679076
$362$	24.0003	1.26143
$363$	−8.82197	−0.463034
$364$	1.44620	0.0758016
$365$	−29.1679	−1.52672
$366$	11.4953	0.600868
$367$	15.9489	0.832528	0.416264	−	0.909244i	$-0.363339\pi$
$367$	15.9489	0.832528	0.416264	+	0.909244i	$0.363339\pi$
$368$	−1.00000	−0.0521286
$369$	−0.176142	−0.00916961
$370$	18.6105	0.967512
$371$	2.15512	0.111888
$372$	5.87228	0.304463
$373$	−23.4595	−1.21469	−0.607343	−	0.794440i	$-0.707764\pi$
$373$	−23.4595	−1.21469	−0.607343	+	0.794440i	$0.707764\pi$
$374$	2.19491	0.113496
$375$	−41.6531	−2.15095
$376$	3.46051	0.178462
$377$	−2.62354	−0.135119
$378$	0.551242	0.0283528
$379$	5.15148	0.264614	0.132307	−	0.991209i	$-0.457762\pi$
$379$	5.15148	0.264614	0.132307	+	0.991209i	$0.457762\pi$
$380$	−10.8890	−0.558596
$381$	−14.9438	−0.765592
$382$	10.3196	0.527995
$383$	11.1050	0.567438	0.283719	−	0.958908i	$-0.408432\pi$
$383$	11.1050	0.567438	0.283719	+	0.958908i	$0.408432\pi$
$384$	−1.00000	−0.0510310
$385$	3.58745	0.182833
$386$	−12.6810	−0.645445
$387$	11.2284	0.570770
$388$	−10.1549	−0.515539
$389$	−5.16503	−0.261878	−0.130939	−	0.991390i	$-0.541799\pi$
$389$	−5.16503	−0.261878	−0.130939	+	0.991390i	$0.541799\pi$
$390$	−11.5691	−0.585824
$391$	1.48725	0.0752137
$392$	6.69613	0.338206
$393$	13.7058	0.691366
$394$	11.2181	0.565161
$395$	6.75645	0.339954
$396$	1.47581	0.0741624
$397$	12.7545	0.640131	0.320065	−	0.947395i	$-0.396295\pi$
$397$	12.7545	0.640131	0.320065	+	0.947395i	$0.396295\pi$
$398$	−14.4889	−0.726262
$399$	−1.36119	−0.0681449
$400$	14.4457	0.722286
$401$	−2.73134	−0.136397	−0.0681984	−	0.997672i	$-0.521725\pi$
$401$	−2.73134	−0.136397	−0.0681984	+	0.997672i	$0.521725\pi$
$402$	13.9867	0.697592
$403$	−15.4061	−0.767434
$404$	12.8475	0.639185
$405$	−4.40973	−0.219121
$406$	0.551242	0.0273577
$407$	6.22840	0.308731
$408$	1.48725	0.0736301
$409$	−34.2894	−1.69550	−0.847752	−	0.530393i	$-0.822044\pi$
$409$	−34.2894	−1.69550	−0.847752	+	0.530393i	$0.822044\pi$
$410$	−0.776740	−0.0383605
$411$	9.28159	0.457827
$412$	18.7277	0.922649
$413$	−5.04234	−0.248117
$414$	1.00000	0.0491473
$415$	24.9794	1.22619
$416$	2.62354	0.128630
$417$	4.17985	0.204688
$418$	−3.64426	−0.178247
$419$	−8.93790	−0.436645	−0.218323	−	0.975877i	$-0.570058\pi$
$419$	−8.93790	−0.436645	−0.218323	+	0.975877i	$0.570058\pi$
$420$	2.43083	0.118612
$421$	−35.9832	−1.75371	−0.876857	−	0.480751i	$-0.840364\pi$
$421$	−35.9832	−1.75371	−0.876857	+	0.480751i	$0.840364\pi$
$422$	−21.6426	−1.05354
$423$	−3.46051	−0.168256
$424$	3.90957	0.189865
$425$	−21.4845	−1.04215
$426$	15.0454	0.728951
$427$	6.33668	0.306653
$428$	6.71940	0.324794
$429$	−3.87185	−0.186935
$430$	49.5140	2.38778
$431$	−9.00552	−0.433780	−0.216890	−	0.976196i	$-0.569591\pi$
$431$	−9.00552	−0.433780	−0.216890	+	0.976196i	$0.569591\pi$
$432$	1.00000	0.0481125
$433$	−16.8779	−0.811100	−0.405550	−	0.914073i	$-0.632920\pi$
$433$	−16.8779	−0.811100	−0.405550	+	0.914073i	$0.632920\pi$
$434$	3.23704	0.155383
$435$	−4.40973	−0.211430
$436$	−12.5528	−0.601170
$437$	−2.46932	−0.118124
$438$	−6.61444	−0.316050
$439$	10.2677	0.490048	0.245024	−	0.969517i	$-0.421204\pi$
$439$	10.2677	0.490048	0.245024	+	0.969517i	$0.421204\pi$
$440$	6.50794	0.310254
$441$	−6.69613	−0.318863
$442$	−3.90187	−0.185593
$443$	−10.8393	−0.514989	−0.257494	−	0.966280i	$-0.582897\pi$
$443$	−10.8393	−0.514989	−0.257494	+	0.966280i	$0.582897\pi$
$444$	4.22032	0.200287
$445$	26.4959	1.25603
$446$	−1.04791	−0.0496202
$447$	−10.7775	−0.509759
$448$	−0.551242	−0.0260437
$449$	13.0955	0.618017	0.309008	−	0.951059i	$-0.400003\pi$
$449$	13.0955	0.618017	0.309008	+	0.951059i	$0.400003\pi$
$450$	−14.4457	−0.680978
$451$	−0.259953	−0.0122407
$452$	−10.8654	−0.511065
$453$	23.8315	1.11970
$454$	10.1337	0.475596
$455$	−6.37737	−0.298976
$456$	−2.46932	−0.115637
$457$	−35.4656	−1.65901	−0.829506	−	0.558498i	$-0.811378\pi$
$457$	−35.4656	−1.65901	−0.829506	+	0.558498i	$0.811378\pi$
$458$	2.07214	0.0968247
$459$	−1.48725	−0.0694191
$460$	4.40973	0.205605
$461$	1.92528	0.0896693	0.0448347	−	0.998994i	$-0.485724\pi$
$461$	1.92528	0.0896693	0.0448347	+	0.998994i	$0.485724\pi$
$462$	0.813530	0.0378488
$463$	−11.7358	−0.545410	−0.272705	−	0.962098i	$-0.587918\pi$
$463$	−11.7358	−0.545410	−0.272705	+	0.962098i	$0.587918\pi$
$464$	1.00000	0.0464238
$465$	−25.8951	−1.20086
$466$	25.9885	1.20389
$467$	22.6109	1.04631	0.523153	−	0.852239i	$-0.324756\pi$
$467$	22.6109	1.04631	0.523153	+	0.852239i	$0.324756\pi$
$468$	−2.62354	−0.121273
$469$	7.71004	0.356016
$470$	−15.2599	−0.703888
$471$	8.35619	0.385033
$472$	−9.14724	−0.421036
$473$	16.5710	0.761934
$474$	1.53217	0.0703748
$475$	35.6711	1.63670
$476$	0.819837	0.0375772
$477$	−3.90957	−0.179007
$478$	3.04112	0.139098
$479$	14.5128	0.663107	0.331553	−	0.943437i	$-0.392427\pi$
$479$	14.5128	0.663107	0.331553	+	0.943437i	$0.392427\pi$
$480$	4.40973	0.201276
$481$	−11.0722	−0.504847
$482$	−4.90079	−0.223225
$483$	0.551242	0.0250824
$484$	−8.82197	−0.400999
$485$	44.7805	2.03338
$486$	−1.00000	−0.0453609
$487$	−35.8058	−1.62252	−0.811258	−	0.584688i	$-0.801217\pi$
$487$	−35.8058	−1.62252	−0.811258	+	0.584688i	$0.801217\pi$
$488$	11.4953	0.520367
$489$	−22.1395	−1.00118
$490$	−29.5281	−1.33395
$491$	26.8408	1.21131	0.605654	−	0.795728i	$-0.292912\pi$
$491$	26.8408	1.21131	0.605654	+	0.795728i	$0.292912\pi$
$492$	−0.176142	−0.00794111
$493$	−1.48725	−0.0669826
$494$	6.47836	0.291475
$495$	−6.50794	−0.292510
$496$	5.87228	0.263673
$497$	8.29364	0.372021
$498$	5.66460	0.253837
$499$	−17.7232	−0.793398	−0.396699	−	0.917949i	$-0.629844\pi$
$499$	−17.7232	−0.793398	−0.396699	+	0.917949i	$0.629844\pi$
$500$	−41.6531	−1.86278
$501$	−18.7616	−0.838208
$502$	−15.5405	−0.693606
$503$	−5.71020	−0.254605	−0.127303	−	0.991864i	$-0.540632\pi$
$503$	−5.71020	−0.254605	−0.127303	+	0.991864i	$0.540632\pi$
$504$	0.551242	0.0245543
$505$	−56.6538	−2.52106
$506$	1.47581	0.0656079
$507$	−6.11705	−0.271668
$508$	−14.9438	−0.663023
$509$	26.2838	1.16501	0.582504	−	0.812828i	$-0.302073\pi$
$509$	26.2838	1.16501	0.582504	+	0.812828i	$0.302073\pi$
$510$	−6.55839	−0.290410
$511$	−3.64616	−0.161296
$512$	−1.00000	−0.0441942
$513$	2.46932	0.109023
$514$	12.4808	0.550503
$515$	−82.5842	−3.63910
$516$	11.2284	0.494301
$517$	−5.10707	−0.224609
$518$	2.32642	0.102217
$519$	−11.8748	−0.521244
$520$	−11.5691	−0.507338
$521$	−5.74010	−0.251478	−0.125739	−	0.992063i	$-0.540130\pi$
$521$	−5.74010	−0.251478	−0.125739	+	0.992063i	$0.540130\pi$
$522$	−1.00000	−0.0437688
$523$	26.4650	1.15723	0.578617	−	0.815599i	$-0.303593\pi$
$523$	26.4650	1.15723	0.578617	+	0.815599i	$0.303593\pi$
$524$	13.7058	0.598740
$525$	−7.96308	−0.347537
$526$	24.6452	1.07458
$527$	−8.73357	−0.380440
$528$	1.47581	0.0642265
$529$	1.00000	0.0434783
$530$	−17.2401	−0.748864
$531$	9.14724	0.396956
$532$	−1.36119	−0.0590152
$533$	0.462116	0.0200165
$534$	6.00851	0.260014
$535$	−29.6307	−1.28105
$536$	13.9867	0.604132
$537$	−1.33732	−0.0577096
$538$	−9.53343	−0.411015
$539$	−9.88224	−0.425658
$540$	−4.40973	−0.189765
$541$	−23.2870	−1.00119	−0.500593	−	0.865683i	$-0.666885\pi$
$541$	−23.2870	−1.00119	−0.500593	+	0.865683i	$0.666885\pi$
$542$	−14.8510	−0.637905
$543$	−24.0003	−1.02995
$544$	1.48725	0.0637655
$545$	55.3544	2.37112
$546$	−1.44620	−0.0618918
$547$	9.56612	0.409018	0.204509	−	0.978865i	$-0.434440\pi$
$547$	9.56612	0.409018	0.204509	+	0.978865i	$0.434440\pi$
$548$	9.28159	0.396490
$549$	−11.4953	−0.490607
$550$	−21.3192	−0.909053
$551$	2.46932	0.105197
$552$	1.00000	0.0425628
$553$	0.844595	0.0359158
$554$	23.9020	1.01550
$555$	−18.6105	−0.789970
$556$	4.17985	0.177265
$557$	−17.3040	−0.733195	−0.366597	−	0.930380i	$-0.619477\pi$
$557$	−17.3040	−0.733195	−0.366597	+	0.930380i	$0.619477\pi$
$558$	−5.87228	−0.248593
$559$	−29.4580	−1.24594
$560$	2.43083	0.102721
$561$	−2.19491	−0.0926692
$562$	7.50907	0.316751
$563$	18.1919	0.766699	0.383349	−	0.923603i	$-0.374770\pi$
$563$	18.1919	0.766699	0.383349	+	0.923603i	$0.374770\pi$
$564$	−3.46051	−0.145714
$565$	47.9134	2.01573
$566$	12.0219	0.505317
$567$	−0.551242	−0.0231500
$568$	15.0454	0.631290
$569$	24.4337	1.02431	0.512157	−	0.858892i	$-0.328846\pi$
$569$	24.4337	1.02431	0.512157	+	0.858892i	$0.328846\pi$
$570$	10.8890	0.456092
$571$	−38.9830	−1.63139	−0.815693	−	0.578485i	$-0.803644\pi$
$571$	−38.9830	−1.63139	−0.815693	+	0.578485i	$0.803644\pi$
$572$	−3.87185	−0.161890
$573$	−10.3196	−0.431106
$574$	−0.0970970	−0.00405275
$575$	−14.4457	−0.602428
$576$	1.00000	0.0416667
$577$	−7.61844	−0.317160	−0.158580	−	0.987346i	$-0.550692\pi$
$577$	−7.61844	−0.317160	−0.158580	+	0.987346i	$0.550692\pi$
$578$	14.7881	0.615103
$579$	12.6810	0.527003
$580$	−4.40973	−0.183104
$581$	3.12256	0.129546
$582$	10.1549	0.420935
$583$	−5.76979	−0.238960
$584$	−6.61444	−0.273708
$585$	11.5691	0.478323
$586$	13.9017	0.574275
$587$	−12.6532	−0.522252	−0.261126	−	0.965305i	$-0.584094\pi$
$587$	−12.6532	−0.522252	−0.261126	+	0.965305i	$0.584094\pi$
$588$	−6.69613	−0.276144
$589$	14.5005	0.597485
$590$	40.3368	1.66064
$591$	−11.2181	−0.461452
$592$	4.22032	0.173454
$593$	−37.6916	−1.54781	−0.773904	−	0.633303i	$-0.781699\pi$
$593$	−37.6916	−1.54781	−0.773904	+	0.633303i	$0.781699\pi$
$594$	−1.47581	−0.0605534
$595$	−3.61526	−0.148211
$596$	−10.7775	−0.441464
$597$	14.4889	0.592990
$598$	−2.62354	−0.107284
$599$	27.0764	1.10631	0.553156	−	0.833078i	$-0.313423\pi$
$599$	27.0764	1.10631	0.553156	+	0.833078i	$0.313423\pi$
$600$	−14.4457	−0.589744
$601$	42.3032	1.72558	0.862792	−	0.505560i	$-0.168714\pi$
$601$	42.3032	1.72558	0.862792	+	0.505560i	$0.168714\pi$
$602$	6.18954	0.252267
$603$	−13.9867	−0.569581
$604$	23.8315	0.969688
$605$	38.9025	1.58161
$606$	−12.8475	−0.521893
$607$	−40.2988	−1.63568	−0.817839	−	0.575447i	$-0.804828\pi$
$607$	−40.2988	−1.63568	−0.817839	+	0.575447i	$0.804828\pi$
$608$	−2.46932	−0.100144
$609$	−0.551242	−0.0223374
$610$	−50.6911	−2.05242
$611$	9.07878	0.367288
$612$	−1.48725	−0.0601187
$613$	19.1251	0.772455	0.386228	−	0.922404i	$-0.373778\pi$
$613$	19.1251	0.772455	0.386228	+	0.922404i	$0.373778\pi$
$614$	−8.48620	−0.342475
$615$	0.776740	0.0313212
$616$	0.813530	0.0327781
$617$	−36.8285	−1.48266	−0.741329	−	0.671141i	$-0.765804\pi$
$617$	−36.8285	−1.48266	−0.741329	+	0.671141i	$0.765804\pi$
$618$	−18.7277	−0.753340
$619$	25.2137	1.01342	0.506712	−	0.862115i	$-0.330861\pi$
$619$	25.2137	1.01342	0.506712	+	0.862115i	$0.330861\pi$
$620$	−25.8951	−1.03997
$621$	−1.00000	−0.0401286
$622$	−12.0620	−0.483643
$623$	3.31214	0.132698
$624$	−2.62354	−0.105026
$625$	111.450	4.45801
$626$	12.3576	0.493911
$627$	3.64426	0.145538
$628$	8.35619	0.333448
$629$	−6.27669	−0.250268
$630$	−2.43083	−0.0968465
$631$	2.87385	0.114406	0.0572030	−	0.998363i	$-0.481782\pi$
$631$	2.87385	0.114406	0.0572030	+	0.998363i	$0.481782\pi$
$632$	1.53217	0.0609464
$633$	21.6426	0.860215
$634$	−5.92579	−0.235343
$635$	65.8980	2.61508
$636$	−3.90957	−0.155024
$637$	17.5676	0.696052
$638$	−1.47581	−0.0584280
$639$	−15.0454	−0.595186
$640$	4.40973	0.174310
$641$	−24.4191	−0.964495	−0.482247	−	0.876035i	$-0.660179\pi$
$641$	−24.4191	−0.964495	−0.482247	+	0.876035i	$0.660179\pi$
$642$	−6.71940	−0.265193
$643$	−17.8372	−0.703430	−0.351715	−	0.936107i	$-0.614401\pi$
$643$	−17.8372	−0.703430	−0.351715	+	0.936107i	$0.614401\pi$
$644$	0.551242	0.0217220
$645$	−49.5140	−1.94961
$646$	3.67251	0.144493
$647$	−42.0735	−1.65408	−0.827040	−	0.562143i	$-0.809977\pi$
$647$	−42.0735	−1.65408	−0.827040	+	0.562143i	$0.809977\pi$
$648$	−1.00000	−0.0392837
$649$	13.4996	0.529906
$650$	37.8989	1.48652
$651$	−3.23704	−0.126870
$652$	−22.1395	−0.867048
$653$	31.3197	1.22563	0.612817	−	0.790225i	$-0.290036\pi$
$653$	31.3197	1.22563	0.612817	+	0.790225i	$0.290036\pi$
$654$	12.5528	0.490853
$655$	−60.4388	−2.36154
$656$	−0.176142	−0.00687721
$657$	6.61444	0.258054
$658$	−1.90758	−0.0743651
$659$	−37.6295	−1.46584	−0.732918	−	0.680317i	$-0.761842\pi$
$659$	−37.6295	−1.46584	−0.732918	+	0.680317i	$0.761842\pi$
$660$	−6.50794	−0.253321
$661$	−41.2593	−1.60480	−0.802400	−	0.596787i	$-0.796444\pi$
$661$	−41.2593	−1.60480	−0.802400	+	0.596787i	$0.796444\pi$
$662$	−2.59893	−0.101010
$663$	3.90187	0.151536
$664$	5.66460	0.219829
$665$	6.00250	0.232767
$666$	−4.22032	−0.163534
$667$	−1.00000	−0.0387202
$668$	−18.7616	−0.725909
$669$	1.04791	0.0405147
$670$	−61.6775	−2.38281
$671$	−16.9649	−0.654922
$672$	0.551242	0.0212646
$673$	6.76825	0.260897	0.130448	−	0.991455i	$-0.458358\pi$
$673$	6.76825	0.260897	0.130448	+	0.991455i	$0.458358\pi$
$674$	−12.8005	−0.493056
$675$	14.4457	0.556016
$676$	−6.11705	−0.235271
$677$	−3.98166	−0.153028	−0.0765138	−	0.997069i	$-0.524379\pi$
$677$	−3.98166	−0.153028	−0.0765138	+	0.997069i	$0.524379\pi$
$678$	10.8654	0.417283
$679$	5.59782	0.214825
$680$	−6.55839	−0.251503
$681$	−10.1337	−0.388323
$682$	−8.66638	−0.331853
$683$	46.2909	1.77127	0.885637	−	0.464379i	$-0.153722\pi$
$683$	46.2909	1.77127	0.885637	+	0.464379i	$0.153722\pi$
$684$	2.46932	0.0944169
$685$	−40.9293	−1.56383
$686$	−7.54988	−0.288256
$687$	−2.07214	−0.0790570
$688$	11.2284	0.428077
$689$	10.2569	0.390757
$690$	−4.40973	−0.167876
$691$	−17.3548	−0.660206	−0.330103	−	0.943945i	$-0.607083\pi$
$691$	−17.3548	−0.660206	−0.330103	+	0.943945i	$0.607083\pi$
$692$	−11.8748	−0.451411
$693$	−0.813530	−0.0309034
$694$	−27.4861	−1.04336
$695$	−18.4320	−0.699166
$696$	−1.00000	−0.0379049
$697$	0.261969	0.00992277
$698$	−20.0071	−0.757281
$699$	−25.9885	−0.982975
$700$	−7.96308	−0.300976
$701$	33.6790	1.27204	0.636018	−	0.771674i	$-0.280580\pi$
$701$	33.6790	1.27204	0.636018	+	0.771674i	$0.280580\pi$
$702$	2.62354	0.0990191
$703$	10.4213	0.393048
$704$	1.47581	0.0556218
$705$	15.2599	0.574722
$706$	−3.93851	−0.148228
$707$	−7.08206	−0.266348
$708$	9.14724	0.343774
$709$	−2.08144	−0.0781701	−0.0390850	−	0.999236i	$-0.512444\pi$
$709$	−2.08144	−0.0781701	−0.0390850	+	0.999236i	$0.512444\pi$
$710$	−66.3461	−2.48992
$711$	−1.53217	−0.0574608
$712$	6.00851	0.225179
$713$	−5.87228	−0.219918
$714$	−0.819837	−0.0306816
$715$	17.0738	0.638525
$716$	−1.33732	−0.0499780
$717$	−3.04112	−0.113573
$718$	24.1124	0.899867
$719$	−33.1031	−1.23454	−0.617269	−	0.786752i	$-0.711761\pi$
$719$	−33.1031	−1.23454	−0.617269	+	0.786752i	$0.711761\pi$
$720$	−4.40973	−0.164341
$721$	−10.3235	−0.384467
$722$	12.9024	0.480179
$723$	4.90079	0.182262
$724$	−24.0003	−0.891964
$725$	14.4457	0.536500
$726$	8.82197	0.327414
$727$	47.0295	1.74423	0.872114	−	0.489302i	$-0.162748\pi$
$727$	47.0295	1.74423	0.872114	+	0.489302i	$0.162748\pi$
$728$	−1.44620	−0.0535999
$729$	1.00000	0.0370370
$730$	29.1679	1.07955
$731$	−16.6994	−0.617651
$732$	−11.4953	−0.424878
$733$	10.1938	0.376515	0.188257	−	0.982120i	$-0.439716\pi$
$733$	10.1938	0.376515	0.188257	+	0.982120i	$0.439716\pi$
$734$	−15.9489	−0.588686
$735$	29.5281	1.08916
$736$	1.00000	0.0368605
$737$	−20.6417	−0.760348
$738$	0.176142	0.00648389
$739$	−6.78095	−0.249441	−0.124721	−	0.992192i	$-0.539803\pi$
$739$	−6.78095	−0.249441	−0.124721	+	0.992192i	$0.539803\pi$
$740$	−18.6105	−0.684134
$741$	−6.47836	−0.237989
$742$	−2.15512	−0.0791168
$743$	9.46070	0.347079	0.173540	−	0.984827i	$-0.444479\pi$
$743$	9.46070	0.347079	0.173540	+	0.984827i	$0.444479\pi$
$744$	−5.87228	−0.215288
$745$	47.5259	1.74122
$746$	23.4595	0.858912
$747$	−5.66460	−0.207257
$748$	−2.19491	−0.0802539
$749$	−3.70401	−0.135342
$750$	41.6531	1.52095
$751$	48.8132	1.78122	0.890610	−	0.454767i	$-0.150278\pi$
$751$	48.8132	1.78122	0.890610	+	0.454767i	$0.150278\pi$
$752$	−3.46051	−0.126192
$753$	15.5405	0.566327
$754$	2.62354	0.0955436
$755$	−105.090	−3.82463
$756$	−0.551242	−0.0200485
$757$	−14.2333	−0.517318	−0.258659	−	0.965969i	$-0.583281\pi$
$757$	−14.2333	−0.517318	−0.258659	+	0.965969i	$0.583281\pi$
$758$	−5.15148	−0.187110
$759$	−1.47581	−0.0535686
$760$	10.8890	0.394987
$761$	40.5303	1.46922	0.734612	−	0.678488i	$-0.237364\pi$
$761$	40.5303	1.46922	0.734612	+	0.678488i	$0.237364\pi$
$762$	14.9438	0.541356
$763$	6.91962	0.250507
$764$	−10.3196	−0.373349
$765$	6.55839	0.237119
$766$	−11.1050	−0.401239
$767$	−23.9981	−0.866522
$768$	1.00000	0.0360844
$769$	−15.5560	−0.560964	−0.280482	−	0.959859i	$-0.590494\pi$
$769$	−15.5560	−0.560964	−0.280482	+	0.959859i	$0.590494\pi$
$770$	−3.58745	−0.129283
$771$	−12.4808	−0.449484
$772$	12.6810	0.456398
$773$	28.0880	1.01026	0.505128	−	0.863044i	$-0.331445\pi$
$773$	28.0880	1.01026	0.505128	+	0.863044i	$0.331445\pi$
$774$	−11.2284	−0.403595
$775$	84.8292	3.04716
$776$	10.1549	0.364541
$777$	−2.32642	−0.0834597
$778$	5.16503	0.185175
$779$	−0.434953	−0.0155838
$780$	11.5691	0.414240
$781$	−22.2042	−0.794528
$782$	−1.48725	−0.0531841
$783$	1.00000	0.0357371
$784$	−6.69613	−0.239148
$785$	−36.8485	−1.31518
$786$	−13.7058	−0.488869
$787$	−35.7497	−1.27434	−0.637169	−	0.770724i	$-0.719895\pi$
$787$	−35.7497	−1.27434	−0.637169	+	0.770724i	$0.719895\pi$
$788$	−11.2181	−0.399629
$789$	−24.6452	−0.877392
$790$	−6.75645	−0.240384
$791$	5.98945	0.212960
$792$	−1.47581	−0.0524408
$793$	30.1583	1.07095
$794$	−12.7545	−0.452641
$795$	17.2401	0.611445
$796$	14.4889	0.513545
$797$	29.7454	1.05364	0.526818	−	0.849978i	$-0.323385\pi$
$797$	29.7454	1.05364	0.526818	+	0.849978i	$0.323385\pi$
$798$	1.36119	0.0481857
$799$	5.14666	0.182076
$800$	−14.4457	−0.510733
$801$	−6.00851	−0.212300
$802$	2.73134	0.0964471
$803$	9.76168	0.344482
$804$	−13.9867	−0.493272
$805$	−2.43083	−0.0856754
$806$	15.4061	0.542658
$807$	9.53343	0.335593
$808$	−12.8475	−0.451972
$809$	−40.6118	−1.42783	−0.713917	−	0.700230i	$-0.753081\pi$
$809$	−40.6118	−1.42783	−0.713917	+	0.700230i	$0.753081\pi$
$810$	4.40973	0.154942
$811$	−9.99785	−0.351072	−0.175536	−	0.984473i	$-0.556166\pi$
$811$	−9.99785	−0.351072	−0.175536	+	0.984473i	$0.556166\pi$
$812$	−0.551242	−0.0193448
$813$	14.8510	0.520847
$814$	−6.22840	−0.218305
$815$	97.6290	3.41979
$816$	−1.48725	−0.0520643
$817$	27.7264	0.970026
$818$	34.2894	1.19890
$819$	1.44620	0.0505344
$820$	0.776740	0.0271250
$821$	54.1700	1.89055	0.945273	−	0.326281i	$-0.105796\pi$
$821$	54.1700	1.89055	0.945273	+	0.326281i	$0.105796\pi$
$822$	−9.28159	−0.323733
$823$	14.2877	0.498037	0.249018	−	0.968499i	$-0.419892\pi$
$823$	14.2877	0.498037	0.249018	+	0.968499i	$0.419892\pi$
$824$	−18.7277	−0.652412
$825$	21.3192	0.742239
$826$	5.04234	0.175445
$827$	14.8112	0.515038	0.257519	−	0.966273i	$-0.417095\pi$
$827$	14.8112	0.515038	0.257519	+	0.966273i	$0.417095\pi$
$828$	−1.00000	−0.0347524
$829$	56.9386	1.97756	0.988780	−	0.149380i	$-0.0477279\pi$
$829$	56.9386	1.97756	0.988780	+	0.149380i	$0.0477279\pi$
$830$	−24.9794	−0.867046
$831$	−23.9020	−0.829152
$832$	−2.62354	−0.0909548
$833$	9.95886	0.345054
$834$	−4.17985	−0.144736
$835$	82.7337	2.86312
$836$	3.64426	0.126039
$837$	5.87228	0.202976
$838$	8.93790	0.308755
$839$	30.6948	1.05970	0.529852	−	0.848090i	$-0.322248\pi$
$839$	30.6948	1.05970	0.529852	+	0.848090i	$0.322248\pi$
$840$	−2.43083	−0.0838715
$841$	1.00000	0.0344828
$842$	35.9832	1.24006
$843$	−7.50907	−0.258626
$844$	21.6426	0.744968
$845$	26.9745	0.927952
$846$	3.46051	0.118975
$847$	4.86304	0.167096
$848$	−3.90957	−0.134255
$849$	−12.0219	−0.412590
$850$	21.4845	0.736911
$851$	−4.22032	−0.144671
$852$	−15.0454	−0.515447
$853$	−44.2827	−1.51621	−0.758106	−	0.652131i	$-0.773875\pi$
$853$	−44.2827	−1.51621	−0.758106	+	0.652131i	$0.773875\pi$
$854$	−6.33668	−0.216837
$855$	−10.8890	−0.372398
$856$	−6.71940	−0.229664
$857$	−22.5936	−0.771783	−0.385892	−	0.922544i	$-0.626106\pi$
$857$	−22.5936	−0.771783	−0.385892	+	0.922544i	$0.626106\pi$
$858$	3.87185	0.132183
$859$	35.7087	1.21837	0.609183	−	0.793030i	$-0.291498\pi$
$859$	35.7087	1.21837	0.609183	+	0.793030i	$0.291498\pi$
$860$	−49.5140	−1.68841
$861$	0.0970970	0.00330906
$862$	9.00552	0.306729
$863$	−40.2004	−1.36844	−0.684219	−	0.729277i	$-0.739857\pi$
$863$	−40.2004	−1.36844	−0.684219	+	0.729277i	$0.739857\pi$
$864$	−1.00000	−0.0340207
$865$	52.3645	1.78045
$866$	16.8779	0.573535
$867$	−14.7881	−0.502229
$868$	−3.23704	−0.109872
$869$	−2.26119	−0.0767057
$870$	4.40973	0.149504
$871$	36.6946	1.24335
$872$	12.5528	0.425091
$873$	−10.1549	−0.343692
$874$	2.46932	0.0835261
$875$	22.9609	0.776220
$876$	6.61444	0.223481
$877$	−25.3805	−0.857039	−0.428520	−	0.903532i	$-0.640965\pi$
$877$	−25.3805	−0.857039	−0.428520	+	0.903532i	$0.640965\pi$
$878$	−10.2677	−0.346517
$879$	−13.9017	−0.468894
$880$	−6.50794	−0.219383
$881$	17.9112	0.603443	0.301721	−	0.953396i	$-0.402439\pi$
$881$	17.9112	0.603443	0.301721	+	0.953396i	$0.402439\pi$
$882$	6.69613	0.225471
$883$	28.5778	0.961719	0.480860	−	0.876798i	$-0.340325\pi$
$883$	28.5778	0.961719	0.480860	+	0.876798i	$0.340325\pi$
$884$	3.90187	0.131234
$885$	−40.3368	−1.35591
$886$	10.8393	0.364152
$887$	28.1153	0.944018	0.472009	−	0.881594i	$-0.343529\pi$
$887$	28.1153	0.944018	0.472009	+	0.881594i	$0.343529\pi$
$888$	−4.22032	−0.141625
$889$	8.23763	0.276281
$890$	−26.4959	−0.888145
$891$	1.47581	0.0494416
$892$	1.04791	0.0350868
$893$	−8.54512	−0.285952
$894$	10.7775	0.360454
$895$	5.89722	0.197122
$896$	0.551242	0.0184157
$897$	2.62354	0.0875974
$898$	−13.0955	−0.437004
$899$	5.87228	0.195851
$900$	14.4457	0.481524
$901$	5.81452	0.193710
$902$	0.259953	0.00865550
$903$	−6.18954	−0.205975
$904$	10.8654	0.361377
$905$	105.835	3.51807
$906$	−23.8315	−0.791747
$907$	−37.7943	−1.25494	−0.627470	−	0.778641i	$-0.715909\pi$
$907$	−37.7943	−1.25494	−0.627470	+	0.778641i	$0.715909\pi$
$908$	−10.1337	−0.336297
$909$	12.8475	0.426124
$910$	6.37737	0.211408
$911$	−40.2558	−1.33374	−0.666868	−	0.745176i	$-0.732365\pi$
$911$	−40.2558	−1.33374	−0.666868	+	0.745176i	$0.732365\pi$
$912$	2.46932	0.0817675
$913$	−8.35989	−0.276672
$914$	35.4656	1.17310
$915$	50.6911	1.67580
$916$	−2.07214	−0.0684654
$917$	−7.55520	−0.249495
$918$	1.48725	0.0490867
$919$	−45.3190	−1.49494	−0.747469	−	0.664297i	$-0.768731\pi$
$919$	−45.3190	−1.49494	−0.747469	+	0.664297i	$0.768731\pi$
$920$	−4.40973	−0.145384
$921$	8.48620	0.279630
$922$	−1.92528	−0.0634058
$923$	39.4721	1.29924
$924$	−0.813530	−0.0267632
$925$	60.9655	2.00453
$926$	11.7358	0.385663
$927$	18.7277	0.615099
$928$	−1.00000	−0.0328266
$929$	6.17934	0.202738	0.101369	−	0.994849i	$-0.467678\pi$
$929$	6.17934	0.202738	0.101369	+	0.994849i	$0.467678\pi$
$930$	25.8951	0.849135
$931$	−16.5349	−0.541910
$932$	−25.9885	−0.851281
$933$	12.0620	0.394893
$934$	−22.6109	−0.739850
$935$	9.67896	0.316536
$936$	2.62354	0.0857530
$937$	−21.5453	−0.703853	−0.351926	−	0.936028i	$-0.614473\pi$
$937$	−21.5453	−0.703853	−0.351926	+	0.936028i	$0.614473\pi$
$938$	−7.71004	−0.251742
$939$	−12.3576	−0.403276
$940$	15.2599	0.497724
$941$	−29.7555	−0.970001	−0.485001	−	0.874514i	$-0.661181\pi$
$941$	−29.7555	−0.970001	−0.485001	+	0.874514i	$0.661181\pi$
$942$	−8.35619	−0.272259
$943$	0.176142	0.00573599
$944$	9.14724	0.297717
$945$	2.43083	0.0790748
$946$	−16.5710	−0.538768
$947$	29.5397	0.959910	0.479955	−	0.877293i	$-0.340653\pi$
$947$	29.5397	0.959910	0.479955	+	0.877293i	$0.340653\pi$
$948$	−1.53217	−0.0497625
$949$	−17.3532	−0.563310
$950$	−35.6711	−1.15732
$951$	5.92579	0.192157
$952$	−0.819837	−0.0265711
$953$	13.7123	0.444184	0.222092	−	0.975026i	$-0.428711\pi$
$953$	13.7123	0.444184	0.222092	+	0.975026i	$0.428711\pi$
$954$	3.90957	0.126577
$955$	45.5065	1.47256
$956$	−3.04112	−0.0983569
$957$	1.47581	0.0477063
$958$	−14.5128	−0.468887
$959$	−5.11640	−0.165217
$960$	−4.40973	−0.142323
$961$	3.48361	0.112375
$962$	11.0722	0.356981
$963$	6.71940	0.216530
$964$	4.90079	0.157844
$965$	−55.9197	−1.80012
$966$	−0.551242	−0.0177359
$967$	−23.5452	−0.757163	−0.378581	−	0.925568i	$-0.623588\pi$
$967$	−23.5452	−0.757163	−0.378581	+	0.925568i	$0.623588\pi$
$968$	8.82197	0.283549
$969$	−3.67251	−0.117978
$970$	−44.7805	−1.43782
$971$	55.3083	1.77493	0.887465	−	0.460876i	$-0.152465\pi$
$971$	55.3083	1.77493	0.887465	+	0.460876i	$0.152465\pi$
$972$	1.00000	0.0320750
$973$	−2.30411	−0.0738663
$974$	35.8058	1.14729
$975$	−37.8989	−1.21374
$976$	−11.4953	−0.367955
$977$	6.34352	0.202947	0.101474	−	0.994838i	$-0.467644\pi$
$977$	6.34352	0.202947	0.101474	+	0.994838i	$0.467644\pi$
$978$	22.1395	0.707942
$979$	−8.86745	−0.283405
$980$	29.5281	0.943242
$981$	−12.5528	−0.400780
$982$	−26.8408	−0.856524
$983$	39.7126	1.26664	0.633318	−	0.773892i	$-0.281693\pi$
$983$	39.7126	1.26664	0.633318	+	0.773892i	$0.281693\pi$
$984$	0.176142	0.00561521
$985$	49.4689	1.57621
$986$	1.48725	0.0473638
$987$	1.90758	0.0607189
$988$	−6.47836	−0.206104
$989$	−11.2284	−0.357041
$990$	6.50794	0.206836
$991$	−25.5234	−0.810776	−0.405388	−	0.914145i	$-0.632864\pi$
$991$	−25.5234	−0.810776	−0.405388	+	0.914145i	$0.632864\pi$
$992$	−5.87228	−0.186445
$993$	2.59893	0.0824746
$994$	−8.29364	−0.263058
$995$	−63.8920	−2.02551
$996$	−5.66460	−0.179490
$997$	27.0545	0.856824	0.428412	−	0.903583i	$-0.359073\pi$
$997$	27.0545	0.856824	0.428412	+	0.903583i	$0.359073\pi$
$998$	17.7232	0.561017
$999$	4.22032	0.133525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bi.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bi.1.1	✓	8	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.40973 q^{5} -1.00000 q^{6} -0.551242 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.40973 q^{5} -1.00000 q^{6} -0.551242 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.40973 q^{10} +1.47581 q^{11} +1.00000 q^{12} -2.62354 q^{13} +0.551242 q^{14} -4.40973 q^{15} +1.00000 q^{16} -1.48725 q^{17} -1.00000 q^{18} +2.46932 q^{19} -4.40973 q^{20} -0.551242 q^{21} -1.47581 q^{22} -1.00000 q^{23} -1.00000 q^{24} +14.4457 q^{25} +2.62354 q^{26} +1.00000 q^{27} -0.551242 q^{28} +1.00000 q^{29} +4.40973 q^{30} +5.87228 q^{31} -1.00000 q^{32} +1.47581 q^{33} +1.48725 q^{34} +2.43083 q^{35} +1.00000 q^{36} +4.22032 q^{37} -2.46932 q^{38} -2.62354 q^{39} +4.40973 q^{40} -0.176142 q^{41} +0.551242 q^{42} +11.2284 q^{43} +1.47581 q^{44} -4.40973 q^{45} +1.00000 q^{46} -3.46051 q^{47} +1.00000 q^{48} -6.69613 q^{49} -14.4457 q^{50} -1.48725 q^{51} -2.62354 q^{52} -3.90957 q^{53} -1.00000 q^{54} -6.50794 q^{55} +0.551242 q^{56} +2.46932 q^{57} -1.00000 q^{58} +9.14724 q^{59} -4.40973 q^{60} -11.4953 q^{61} -5.87228 q^{62} -0.551242 q^{63} +1.00000 q^{64} +11.5691 q^{65} -1.47581 q^{66} -13.9867 q^{67} -1.48725 q^{68} -1.00000 q^{69} -2.43083 q^{70} -15.0454 q^{71} -1.00000 q^{72} +6.61444 q^{73} -4.22032 q^{74} +14.4457 q^{75} +2.46932 q^{76} -0.813530 q^{77} +2.62354 q^{78} -1.53217 q^{79} -4.40973 q^{80} +1.00000 q^{81} +0.176142 q^{82} -5.66460 q^{83} -0.551242 q^{84} +6.55839 q^{85} -11.2284 q^{86} +1.00000 q^{87} -1.47581 q^{88} -6.00851 q^{89} +4.40973 q^{90} +1.44620 q^{91} -1.00000 q^{92} +5.87228 q^{93} +3.46051 q^{94} -10.8890 q^{95} -1.00000 q^{96} -10.1549 q^{97} +6.69613 q^{98} +1.47581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * q^99

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