LMFDB - Embedded newform 3774.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3774,2,Mod(1,3774)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3774, 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("3774.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3774 = 2 \cdot 3 \cdot 17 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3774.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:	$30.1355417228$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1537472.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 6x^{2} + 13x + 4 $ x^5 - 2x^4 - 7x^3 + 6x^2 + 13x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.829366$ of defining polynomial
Character	$\chi$	$=$	3774.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.48279 q^{5} +1.00000 q^{6} +4.20706 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.48279 q^{5} +1.00000 q^{6} +4.20706 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.48279 q^{10} -0.164233 q^{11} -1.00000 q^{12} +1.48279 q^{13} -4.20706 q^{14} +2.48279 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.03111 q^{19} -2.48279 q^{20} -4.20706 q^{21} +0.164233 q^{22} -7.20706 q^{23} +1.00000 q^{24} +1.16423 q^{25} -1.48279 q^{26} -1.00000 q^{27} +4.20706 q^{28} -10.2166 q^{29} -2.48279 q^{30} -9.07394 q^{31} -1.00000 q^{32} +0.164233 q^{33} -1.00000 q^{34} -10.4452 q^{35} +1.00000 q^{36} +1.00000 q^{37} -4.03111 q^{38} -1.48279 q^{39} +2.48279 q^{40} +0.513902 q^{41} +4.20706 q^{42} +1.27573 q^{43} -0.164233 q^{44} -2.48279 q^{45} +7.20706 q^{46} +2.53662 q^{47} -1.00000 q^{48} +10.6993 q^{49} -1.16423 q^{50} -1.00000 q^{51} +1.48279 q^{52} +5.32805 q^{53} +1.00000 q^{54} +0.407756 q^{55} -4.20706 q^{56} -4.03111 q^{57} +10.2166 q^{58} +6.01831 q^{59} +2.48279 q^{60} +2.76801 q^{61} +9.07394 q^{62} +4.20706 q^{63} +1.00000 q^{64} -3.68145 q^{65} -0.164233 q^{66} +4.05274 q^{67} +1.00000 q^{68} +7.20706 q^{69} +10.4452 q^{70} +8.98678 q^{71} -1.00000 q^{72} -15.7061 q^{73} -1.00000 q^{74} -1.16423 q^{75} +4.03111 q^{76} -0.690938 q^{77} +1.48279 q^{78} -2.96557 q^{79} -2.48279 q^{80} +1.00000 q^{81} -0.513902 q^{82} -3.59115 q^{83} -4.20706 q^{84} -2.48279 q^{85} -1.27573 q^{86} +10.2166 q^{87} +0.164233 q^{88} +12.1578 q^{89} +2.48279 q^{90} +6.23817 q^{91} -7.20706 q^{92} +9.07394 q^{93} -2.53662 q^{94} -10.0084 q^{95} +1.00000 q^{96} -11.5967 q^{97} -10.6993 q^{98} -0.164233 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 4 q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 4 * q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 4 q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9} + 4 q^{10} - 2 q^{11} - 5 q^{12} - q^{13} - q^{14} + 4 q^{15} + 5 q^{16} + 5 q^{17} - 5 q^{18} + 4 q^{19} - 4 q^{20} - q^{21} + 2 q^{22} - 16 q^{23} + 5 q^{24} + 7 q^{25} + q^{26} - 5 q^{27} + q^{28} - 8 q^{29} - 4 q^{30} - 8 q^{31} - 5 q^{32} + 2 q^{33} - 5 q^{34} + 4 q^{35} + 5 q^{36} + 5 q^{37} - 4 q^{38} + q^{39} + 4 q^{40} - 22 q^{41} + q^{42} + 18 q^{43} - 2 q^{44} - 4 q^{45} + 16 q^{46} + 10 q^{47} - 5 q^{48} + 2 q^{49} - 7 q^{50} - 5 q^{51} - q^{52} - 6 q^{53} + 5 q^{54} + 16 q^{55} - q^{56} - 4 q^{57} + 8 q^{58} + 4 q^{59} + 4 q^{60} - 6 q^{61} + 8 q^{62} + q^{63} + 5 q^{64} - 28 q^{65} - 2 q^{66} + 11 q^{67} + 5 q^{68} + 16 q^{69} - 4 q^{70} - 5 q^{72} - 8 q^{73} - 5 q^{74} - 7 q^{75} + 4 q^{76} + 10 q^{77} - q^{78} + 2 q^{79} - 4 q^{80} + 5 q^{81} + 22 q^{82} + 11 q^{83} - q^{84} - 4 q^{85} - 18 q^{86} + 8 q^{87} + 2 q^{88} - 2 q^{89} + 4 q^{90} - 5 q^{91} - 16 q^{92} + 8 q^{93} - 10 q^{94} - 12 q^{95} + 5 q^{96} + 7 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 4 * q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9 + 4 * q^10 - 2 * q^11 - 5 * q^12 - q^13 - q^14 + 4 * q^15 + 5 * q^16 + 5 * q^17 - 5 * q^18 + 4 * q^19 - 4 * q^20 - q^21 + 2 * q^22 - 16 * q^23 + 5 * q^24 + 7 * q^25 + q^26 - 5 * q^27 + q^28 - 8 * q^29 - 4 * q^30 - 8 * q^31 - 5 * q^32 + 2 * q^33 - 5 * q^34 + 4 * q^35 + 5 * q^36 + 5 * q^37 - 4 * q^38 + q^39 + 4 * q^40 - 22 * q^41 + q^42 + 18 * q^43 - 2 * q^44 - 4 * q^45 + 16 * q^46 + 10 * q^47 - 5 * q^48 + 2 * q^49 - 7 * q^50 - 5 * q^51 - q^52 - 6 * q^53 + 5 * q^54 + 16 * q^55 - q^56 - 4 * q^57 + 8 * q^58 + 4 * q^59 + 4 * q^60 - 6 * q^61 + 8 * q^62 + q^63 + 5 * q^64 - 28 * q^65 - 2 * q^66 + 11 * q^67 + 5 * q^68 + 16 * q^69 - 4 * q^70 - 5 * q^72 - 8 * q^73 - 5 * q^74 - 7 * q^75 + 4 * q^76 + 10 * q^77 - q^78 + 2 * q^79 - 4 * q^80 + 5 * q^81 + 22 * q^82 + 11 * q^83 - q^84 - 4 * q^85 - 18 * q^86 + 8 * q^87 + 2 * q^88 - 2 * q^89 + 4 * q^90 - 5 * q^91 - 16 * q^92 + 8 * q^93 - 10 * q^94 - 12 * q^95 + 5 * q^96 + 7 * q^97 - 2 * q^98 - 2 * 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
$5$	−2.48279	−1.11034	−0.555168	−	0.831738i	$-0.687346\pi$
$5$	−2.48279	−1.11034	−0.555168	+	0.831738i	$0.687346\pi$
$6$	1.00000	0.408248
$7$	4.20706	1.59012	0.795059	−	0.606532i	$-0.207440\pi$
$7$	4.20706	1.59012	0.795059	+	0.606532i	$0.207440\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.48279	0.785126
$11$	−0.164233	−0.0495181	−0.0247591	−	0.999693i	$-0.507882\pi$
$11$	−0.164233	−0.0495181	−0.0247591	+	0.999693i	$0.507882\pi$
$12$	−1.00000	−0.288675
$13$	1.48279	0.411251	0.205626	−	0.978631i	$-0.434077\pi$
$13$	1.48279	0.411251	0.205626	+	0.978631i	$0.434077\pi$
$14$	−4.20706	−1.12438
$15$	2.48279	0.641053
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	4.03111	0.924801	0.462401	−	0.886671i	$-0.346988\pi$
$19$	4.03111	0.924801	0.462401	+	0.886671i	$0.346988\pi$
$20$	−2.48279	−0.555168
$21$	−4.20706	−0.918055
$22$	0.164233	0.0350146
$23$	−7.20706	−1.50278	−0.751388	−	0.659861i	$-0.770615\pi$
$23$	−7.20706	−1.50278	−0.751388	+	0.659861i	$0.770615\pi$
$24$	1.00000	0.204124
$25$	1.16423	0.232847
$26$	−1.48279	−0.290799
$27$	−1.00000	−0.192450
$28$	4.20706	0.795059
$29$	−10.2166	−1.89717	−0.948583	−	0.316529i	$-0.897483\pi$
$29$	−10.2166	−1.89717	−0.948583	+	0.316529i	$0.897483\pi$
$30$	−2.48279	−0.453293
$31$	−9.07394	−1.62973	−0.814864	−	0.579652i	$-0.803188\pi$
$31$	−9.07394	−1.62973	−0.814864	+	0.579652i	$0.803188\pi$
$32$	−1.00000	−0.176777
$33$	0.164233	0.0285893
$34$	−1.00000	−0.171499
$35$	−10.4452	−1.76557
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−4.03111	−0.653933
$39$	−1.48279	−0.237436
$40$	2.48279	0.392563
$41$	0.513902	0.0802580	0.0401290	−	0.999195i	$-0.487223\pi$
$41$	0.513902	0.0802580	0.0401290	+	0.999195i	$0.487223\pi$
$42$	4.20706	0.649163
$43$	1.27573	0.194547	0.0972734	−	0.995258i	$-0.468988\pi$
$43$	1.27573	0.194547	0.0972734	+	0.995258i	$0.468988\pi$
$44$	−0.164233	−0.0247591
$45$	−2.48279	−0.370112
$46$	7.20706	1.06262
$47$	2.53662	0.370004	0.185002	−	0.982738i	$-0.440771\pi$
$47$	2.53662	0.370004	0.185002	+	0.982738i	$0.440771\pi$
$48$	−1.00000	−0.144338
$49$	10.6993	1.52848
$50$	−1.16423	−0.164647
$51$	−1.00000	−0.140028
$52$	1.48279	0.205626
$53$	5.32805	0.731863	0.365932	−	0.930642i	$-0.380750\pi$
$53$	5.32805	0.731863	0.365932	+	0.930642i	$0.380750\pi$
$54$	1.00000	0.136083
$55$	0.407756	0.0549818
$56$	−4.20706	−0.562192
$57$	−4.03111	−0.533934
$58$	10.2166	1.34150
$59$	6.01831	0.783517	0.391759	−	0.920068i	$-0.371867\pi$
$59$	6.01831	0.783517	0.391759	+	0.920068i	$0.371867\pi$
$60$	2.48279	0.320526
$61$	2.76801	0.354407	0.177204	−	0.984174i	$-0.443295\pi$
$61$	2.76801	0.354407	0.177204	+	0.984174i	$0.443295\pi$
$62$	9.07394	1.15239
$63$	4.20706	0.530039
$64$	1.00000	0.125000
$65$	−3.68145	−0.456627
$66$	−0.164233	−0.0202157
$67$	4.05274	0.495121	0.247560	−	0.968872i	$-0.420371\pi$
$67$	4.05274	0.495121	0.247560	+	0.968872i	$0.420371\pi$
$68$	1.00000	0.121268
$69$	7.20706	0.867628
$70$	10.4452	1.24844
$71$	8.98678	1.06653	0.533267	−	0.845947i	$-0.320964\pi$
$71$	8.98678	1.06653	0.533267	+	0.845947i	$0.320964\pi$
$72$	−1.00000	−0.117851
$73$	−15.7061	−1.83826	−0.919131	−	0.393952i	$-0.871108\pi$
$73$	−15.7061	−1.83826	−0.919131	+	0.393952i	$0.871108\pi$
$74$	−1.00000	−0.116248
$75$	−1.16423	−0.134434
$76$	4.03111	0.462401
$77$	−0.690938	−0.0787397
$78$	1.48279	0.167893
$79$	−2.96557	−0.333653	−0.166827	−	0.985986i	$-0.553352\pi$
$79$	−2.96557	−0.333653	−0.166827	+	0.985986i	$0.553352\pi$
$80$	−2.48279	−0.277584
$81$	1.00000	0.111111
$82$	−0.513902	−0.0567510
$83$	−3.59115	−0.394180	−0.197090	−	0.980385i	$-0.563149\pi$
$83$	−3.59115	−0.394180	−0.197090	+	0.980385i	$0.563149\pi$
$84$	−4.20706	−0.459028
$85$	−2.48279	−0.269296
$86$	−1.27573	−0.137565
$87$	10.2166	1.09533
$88$	0.164233	0.0175073
$89$	12.1578	1.28872	0.644362	−	0.764721i	$-0.277123\pi$
$89$	12.1578	1.28872	0.644362	+	0.764721i	$0.277123\pi$
$90$	2.48279	0.261709
$91$	6.23817	0.653938
$92$	−7.20706	−0.751388
$93$	9.07394	0.940924
$94$	−2.53662	−0.261632
$95$	−10.0084	−1.02684
$96$	1.00000	0.102062
$97$	−11.5967	−1.17746	−0.588731	−	0.808329i	$-0.700372\pi$
$97$	−11.5967	−1.17746	−0.588731	+	0.808329i	$0.700372\pi$
$98$	−10.6993	−1.08080
$99$	−0.164233	−0.0165060
$100$	1.16423	0.116423
$101$	1.99560	0.198569	0.0992846	−	0.995059i	$-0.468345\pi$
$101$	1.99560	0.198569	0.0992846	+	0.995059i	$0.468345\pi$
$102$	1.00000	0.0990148
$103$	−13.8614	−1.36580	−0.682901	−	0.730510i	$-0.739282\pi$
$103$	−13.8614	−1.36580	−0.682901	+	0.730510i	$0.739282\pi$
$104$	−1.48279	−0.145399
$105$	10.4452	1.01935
$106$	−5.32805	−0.517506
$107$	−17.2777	−1.67030	−0.835149	−	0.550025i	$-0.814618\pi$
$107$	−17.2777	−1.67030	−0.835149	+	0.550025i	$0.814618\pi$
$108$	−1.00000	−0.0962250
$109$	5.99782	0.574487	0.287243	−	0.957858i	$-0.407261\pi$
$109$	5.99782	0.574487	0.287243	+	0.957858i	$0.407261\pi$
$110$	−0.407756	−0.0388780
$111$	−1.00000	−0.0949158
$112$	4.20706	0.397530
$113$	−4.81235	−0.452707	−0.226354	−	0.974045i	$-0.572681\pi$
$113$	−4.81235	−0.452707	−0.226354	+	0.974045i	$0.572681\pi$
$114$	4.03111	0.377548
$115$	17.8936	1.66859
$116$	−10.2166	−0.948583
$117$	1.48279	0.137084
$118$	−6.01831	−0.554030
$119$	4.20706	0.385660
$120$	−2.48279	−0.226646
$121$	−10.9730	−0.997548
$122$	−2.76801	−0.250604
$123$	−0.513902	−0.0463370
$124$	−9.07394	−0.814864
$125$	9.52339	0.851798
$126$	−4.20706	−0.374795
$127$	11.5461	1.02455	0.512277	−	0.858820i	$-0.328802\pi$
$127$	11.5461	1.02455	0.512277	+	0.858820i	$0.328802\pi$
$128$	−1.00000	−0.0883883
$129$	−1.27573	−0.112322
$130$	3.68145	0.322884
$131$	−12.4612	−1.08874	−0.544369	−	0.838846i	$-0.683231\pi$
$131$	−12.4612	−1.08874	−0.544369	+	0.838846i	$0.683231\pi$
$132$	0.164233	0.0142946
$133$	16.9591	1.47054
$134$	−4.05274	−0.350103
$135$	2.48279	0.213684
$136$	−1.00000	−0.0857493
$137$	−1.90335	−0.162614	−0.0813069	−	0.996689i	$-0.525909\pi$
$137$	−1.90335	−0.162614	−0.0813069	+	0.996689i	$0.525909\pi$
$138$	−7.20706	−0.613506
$139$	−2.38723	−0.202482	−0.101241	−	0.994862i	$-0.532281\pi$
$139$	−2.38723	−0.202482	−0.101241	+	0.994862i	$0.532281\pi$
$140$	−10.4452	−0.882783
$141$	−2.53662	−0.213622
$142$	−8.98678	−0.754154
$143$	−0.243523	−0.0203644
$144$	1.00000	0.0833333
$145$	25.3655	2.10649
$146$	15.7061	1.29985
$147$	−10.6993	−0.882466
$148$	1.00000	0.0821995
$149$	−17.6671	−1.44735	−0.723674	−	0.690142i	$-0.757548\pi$
$149$	−17.6671	−1.44735	−0.723674	+	0.690142i	$0.757548\pi$
$150$	1.16423	0.0950592
$151$	−12.4615	−1.01410	−0.507051	−	0.861916i	$-0.669265\pi$
$151$	−12.4615	−1.01410	−0.507051	+	0.861916i	$0.669265\pi$
$152$	−4.03111	−0.326967
$153$	1.00000	0.0808452
$154$	0.690938	0.0556773
$155$	22.5287	1.80955
$156$	−1.48279	−0.118718
$157$	13.4569	1.07398	0.536989	−	0.843589i	$-0.319562\pi$
$157$	13.4569	1.07398	0.536989	+	0.843589i	$0.319562\pi$
$158$	2.96557	0.235928
$159$	−5.32805	−0.422542
$160$	2.48279	0.196282
$161$	−30.3205	−2.38959
$162$	−1.00000	−0.0785674
$163$	−7.52230	−0.589192	−0.294596	−	0.955622i	$-0.595185\pi$
$163$	−7.52230	−0.589192	−0.294596	+	0.955622i	$0.595185\pi$
$164$	0.513902	0.0401290
$165$	−0.407756	−0.0317437
$166$	3.59115	0.278727
$167$	0.656932	0.0508349	0.0254175	−	0.999677i	$-0.491908\pi$
$167$	0.656932	0.0508349	0.0254175	+	0.999677i	$0.491908\pi$
$168$	4.20706	0.324582
$169$	−10.8013	−0.830872
$170$	2.48279	0.190421
$171$	4.03111	0.308267
$172$	1.27573	0.0972734
$173$	13.4881	1.02548	0.512739	−	0.858545i	$-0.328631\pi$
$173$	13.4881	1.02548	0.512739	+	0.858545i	$0.328631\pi$
$174$	−10.2166	−0.774515
$175$	4.89800	0.370254
$176$	−0.164233	−0.0123795
$177$	−6.01831	−0.452364
$178$	−12.1578	−0.911265
$179$	13.0073	0.972212	0.486106	−	0.873900i	$-0.338417\pi$
$179$	13.0073	0.972212	0.486106	+	0.873900i	$0.338417\pi$
$180$	−2.48279	−0.185056
$181$	17.6385	1.31106	0.655531	−	0.755168i	$-0.272445\pi$
$181$	17.6385	1.31106	0.655531	+	0.755168i	$0.272445\pi$
$182$	−6.23817	−0.462404
$183$	−2.76801	−0.204617
$184$	7.20706	0.531311
$185$	−2.48279	−0.182538
$186$	−9.07394	−0.665334
$187$	−0.164233	−0.0120099
$188$	2.53662	0.185002
$189$	−4.20706	−0.306018
$190$	10.0084	0.726086
$191$	−5.29531	−0.383155	−0.191578	−	0.981477i	$-0.561360\pi$
$191$	−5.29531	−0.383155	−0.191578	+	0.981477i	$0.561360\pi$
$192$	−1.00000	−0.0721688
$193$	4.96466	0.357364	0.178682	−	0.983907i	$-0.442817\pi$
$193$	4.96466	0.357364	0.178682	+	0.983907i	$0.442817\pi$
$194$	11.5967	0.832592
$195$	3.68145	0.263634
$196$	10.6993	0.764238
$197$	7.40353	0.527480	0.263740	−	0.964594i	$-0.415044\pi$
$197$	7.40353	0.527480	0.263740	+	0.964594i	$0.415044\pi$
$198$	0.164233	0.0116715
$199$	−11.8186	−0.837796	−0.418898	−	0.908033i	$-0.637583\pi$
$199$	−11.8186	−0.837796	−0.418898	+	0.908033i	$0.637583\pi$
$200$	−1.16423	−0.0823237
$201$	−4.05274	−0.285858
$202$	−1.99560	−0.140410
$203$	−42.9816	−3.01672
$204$	−1.00000	−0.0700140
$205$	−1.27591	−0.0891133
$206$	13.8614	0.965769
$207$	−7.20706	−0.500925
$208$	1.48279	0.102813
$209$	−0.662042	−0.0457944
$210$	−10.4452	−0.720789
$211$	−17.3797	−1.19647	−0.598234	−	0.801322i	$-0.704131\pi$
$211$	−17.3797	−1.19647	−0.598234	+	0.801322i	$0.704131\pi$
$212$	5.32805	0.365932
$213$	−8.98678	−0.615764
$214$	17.2777	1.18108
$215$	−3.16736	−0.216012
$216$	1.00000	0.0680414
$217$	−38.1746	−2.59146
$218$	−5.99782	−0.406223
$219$	15.7061	1.06132
$220$	0.407756	0.0274909
$221$	1.48279	0.0997431
$222$	1.00000	0.0671156
$223$	−20.1845	−1.35165	−0.675827	−	0.737060i	$-0.736213\pi$
$223$	−20.1845	−1.35165	−0.675827	+	0.737060i	$0.736213\pi$
$224$	−4.20706	−0.281096
$225$	1.16423	0.0776155
$226$	4.81235	0.320112
$227$	−15.9604	−1.05933	−0.529665	−	0.848207i	$-0.677682\pi$
$227$	−15.9604	−1.05933	−0.529665	+	0.848207i	$0.677682\pi$
$228$	−4.03111	−0.266967
$229$	19.0596	1.25950	0.629748	−	0.776800i	$-0.283158\pi$
$229$	19.0596	1.25950	0.629748	+	0.776800i	$0.283158\pi$
$230$	−17.8936	−1.17987
$231$	0.690938	0.0454604
$232$	10.2166	0.670749
$233$	27.2423	1.78471	0.892353	−	0.451339i	$-0.149053\pi$
$233$	27.2423	1.78471	0.892353	+	0.451339i	$0.149053\pi$
$234$	−1.48279	−0.0969328
$235$	−6.29788	−0.410828
$236$	6.01831	0.391759
$237$	2.96557	0.192635
$238$	−4.20706	−0.272703
$239$	−15.8275	−1.02380	−0.511899	−	0.859046i	$-0.671058\pi$
$239$	−15.8275	−1.02380	−0.511899	+	0.859046i	$0.671058\pi$
$240$	2.48279	0.160263
$241$	−27.7723	−1.78897	−0.894487	−	0.447095i	$-0.852459\pi$
$241$	−27.7723	−1.78897	−0.894487	+	0.447095i	$0.852459\pi$
$242$	10.9730	0.705373
$243$	−1.00000	−0.0641500
$244$	2.76801	0.177204
$245$	−26.5642	−1.69712
$246$	0.513902	0.0327652
$247$	5.97729	0.380326
$248$	9.07394	0.576196
$249$	3.59115	0.227580
$250$	−9.52339	−0.602312
$251$	2.43243	0.153533	0.0767667	−	0.997049i	$-0.475540\pi$
$251$	2.43243	0.153533	0.0767667	+	0.997049i	$0.475540\pi$
$252$	4.20706	0.265020
$253$	1.18364	0.0744146
$254$	−11.5461	−0.724469
$255$	2.48279	0.155478
$256$	1.00000	0.0625000
$257$	5.41302	0.337655	0.168828	−	0.985646i	$-0.446002\pi$
$257$	5.41302	0.337655	0.168828	+	0.985646i	$0.446002\pi$
$258$	1.27573	0.0794234
$259$	4.20706	0.261414
$260$	−3.68145	−0.228314
$261$	−10.2166	−0.632389
$262$	12.4612	0.769853
$263$	−7.32533	−0.451699	−0.225850	−	0.974162i	$-0.572516\pi$
$263$	−7.32533	−0.451699	−0.225850	+	0.974162i	$0.572516\pi$
$264$	−0.164233	−0.0101078
$265$	−13.2284	−0.812615
$266$	−16.9591	−1.03983
$267$	−12.1578	−0.744045
$268$	4.05274	0.247560
$269$	−6.08452	−0.370980	−0.185490	−	0.982646i	$-0.559387\pi$
$269$	−6.08452	−0.370980	−0.185490	+	0.982646i	$0.559387\pi$
$270$	−2.48279	−0.151098
$271$	−10.0127	−0.608226	−0.304113	−	0.952636i	$-0.598360\pi$
$271$	−10.0127	−0.608226	−0.304113	+	0.952636i	$0.598360\pi$
$272$	1.00000	0.0606339
$273$	−6.23817	−0.377551
$274$	1.90335	0.114985
$275$	−0.191205	−0.0115301
$276$	7.20706	0.433814
$277$	5.21694	0.313455	0.156728	−	0.987642i	$-0.449906\pi$
$277$	5.21694	0.313455	0.156728	+	0.987642i	$0.449906\pi$
$278$	2.38723	0.143176
$279$	−9.07394	−0.543243
$280$	10.4452	0.624222
$281$	−24.1683	−1.44176	−0.720879	−	0.693061i	$-0.756262\pi$
$281$	−24.1683	−1.44176	−0.720879	+	0.693061i	$0.756262\pi$
$282$	2.53662	0.151053
$283$	−30.5708	−1.81724	−0.908622	−	0.417619i	$-0.862865\pi$
$283$	−30.5708	−1.81724	−0.908622	+	0.417619i	$0.862865\pi$
$284$	8.98678	0.533267
$285$	10.0084	0.592846
$286$	0.243523	0.0143998
$287$	2.16201	0.127620
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−25.3655	−1.48951
$291$	11.5967	0.679808
$292$	−15.7061	−0.919131
$293$	5.89234	0.344234	0.172117	−	0.985076i	$-0.444939\pi$
$293$	5.89234	0.344234	0.172117	+	0.985076i	$0.444939\pi$
$294$	10.6993	0.623998
$295$	−14.9422	−0.869968
$296$	−1.00000	−0.0581238
$297$	0.164233	0.00952976
$298$	17.6671	1.02343
$299$	−10.6865	−0.618018
$300$	−1.16423	−0.0672170
$301$	5.36707	0.309353
$302$	12.4615	0.717079
$303$	−1.99560	−0.114644
$304$	4.03111	0.231200
$305$	−6.87238	−0.393511
$306$	−1.00000	−0.0571662
$307$	−8.50237	−0.485256	−0.242628	−	0.970119i	$-0.578009\pi$
$307$	−8.50237	−0.485256	−0.242628	+	0.970119i	$0.578009\pi$
$308$	−0.690938	−0.0393698
$309$	13.8614	0.788547
$310$	−22.5287	−1.27954
$311$	18.3439	1.04019	0.520095	−	0.854109i	$-0.325897\pi$
$311$	18.3439	1.04019	0.520095	+	0.854109i	$0.325897\pi$
$312$	1.48279	0.0839463
$313$	−22.1040	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$313$	−22.1040	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$314$	−13.4569	−0.759417
$315$	−10.4452	−0.588522
$316$	−2.96557	−0.166827
$317$	−32.5735	−1.82951	−0.914756	−	0.404007i	$-0.867617\pi$
$317$	−32.5735	−1.82951	−0.914756	+	0.404007i	$0.867617\pi$
$318$	5.32805	0.298782
$319$	1.67789	0.0939441
$320$	−2.48279	−0.138792
$321$	17.2777	0.964346
$322$	30.3205	1.68970
$323$	4.03111	0.224297
$324$	1.00000	0.0555556
$325$	1.72631	0.0957584
$326$	7.52230	0.416622
$327$	−5.99782	−0.331680
$328$	−0.513902	−0.0283755
$329$	10.6717	0.588349
$330$	0.407756	0.0224462
$331$	21.8476	1.20085	0.600426	−	0.799680i	$-0.294998\pi$
$331$	21.8476	1.20085	0.600426	+	0.799680i	$0.294998\pi$
$332$	−3.59115	−0.197090
$333$	1.00000	0.0547997
$334$	−0.656932	−0.0359457
$335$	−10.0621	−0.549750
$336$	−4.20706	−0.229514
$337$	6.80371	0.370621	0.185311	−	0.982680i	$-0.440671\pi$
$337$	6.80371	0.370621	0.185311	+	0.982680i	$0.440671\pi$
$338$	10.8013	0.587516
$339$	4.81235	0.261371
$340$	−2.48279	−0.134648
$341$	1.49024	0.0807010
$342$	−4.03111	−0.217978
$343$	15.5633	0.840341
$344$	−1.27573	−0.0687827
$345$	−17.8936	−0.963359
$346$	−13.4881	−0.725122
$347$	15.4287	0.828256	0.414128	−	0.910219i	$-0.364087\pi$
$347$	15.4287	0.828256	0.414128	+	0.910219i	$0.364087\pi$
$348$	10.2166	0.547665
$349$	−19.0596	−1.02024	−0.510118	−	0.860104i	$-0.670398\pi$
$349$	−19.0596	−1.02024	−0.510118	+	0.860104i	$0.670398\pi$
$350$	−4.89800	−0.261809
$351$	−1.48279	−0.0791453
$352$	0.164233	0.00875365
$353$	−29.9397	−1.59353	−0.796765	−	0.604289i	$-0.793457\pi$
$353$	−29.9397	−1.59353	−0.796765	+	0.604289i	$0.793457\pi$
$354$	6.01831	0.319870
$355$	−22.3123	−1.18421
$356$	12.1578	0.644362
$357$	−4.20706	−0.222661
$358$	−13.0073	−0.687458
$359$	−9.56886	−0.505025	−0.252512	−	0.967594i	$-0.581257\pi$
$359$	−9.56886	−0.505025	−0.252512	+	0.967594i	$0.581257\pi$
$360$	2.48279	0.130854
$361$	−2.75012	−0.144743
$362$	−17.6385	−0.927061
$363$	10.9730	0.575935
$364$	6.23817	0.326969
$365$	38.9949	2.04109
$366$	2.76801	0.144686
$367$	20.7254	1.08186	0.540928	−	0.841069i	$-0.318073\pi$
$367$	20.7254	1.08186	0.540928	+	0.841069i	$0.318073\pi$
$368$	−7.20706	−0.375694
$369$	0.513902	0.0267527
$370$	2.48279	0.129074
$371$	22.4154	1.16375
$372$	9.07394	0.470462
$373$	22.4331	1.16154	0.580771	−	0.814067i	$-0.302751\pi$
$373$	22.4331	1.16154	0.580771	+	0.814067i	$0.302751\pi$
$374$	0.164233	0.00849229
$375$	−9.52339	−0.491786
$376$	−2.53662	−0.130816
$377$	−15.1490	−0.780212
$378$	4.20706	0.216388
$379$	27.2160	1.39799	0.698996	−	0.715126i	$-0.253631\pi$
$379$	27.2160	1.39799	0.698996	+	0.715126i	$0.253631\pi$
$380$	−10.0084	−0.513420
$381$	−11.5461	−0.591527
$382$	5.29531	0.270932
$383$	37.6320	1.92290	0.961451	−	0.274975i	$-0.0886695\pi$
$383$	37.6320	1.92290	0.961451	+	0.274975i	$0.0886695\pi$
$384$	1.00000	0.0510310
$385$	1.71545	0.0874275
$386$	−4.96466	−0.252695
$387$	1.27573	0.0648490
$388$	−11.5967	−0.588731
$389$	−8.22897	−0.417225	−0.208613	−	0.977998i	$-0.566895\pi$
$389$	−8.22897	−0.417225	−0.208613	+	0.977998i	$0.566895\pi$
$390$	−3.68145	−0.186417
$391$	−7.20706	−0.364477
$392$	−10.6993	−0.540398
$393$	12.4612	0.628583
$394$	−7.40353	−0.372985
$395$	7.36289	0.370467
$396$	−0.164233	−0.00825302
$397$	−13.5507	−0.680090	−0.340045	−	0.940409i	$-0.610442\pi$
$397$	−13.5507	−0.680090	−0.340045	+	0.940409i	$0.610442\pi$
$398$	11.8186	0.592411
$399$	−16.9591	−0.849019
$400$	1.16423	0.0582116
$401$	−22.9519	−1.14616	−0.573081	−	0.819499i	$-0.694252\pi$
$401$	−22.9519	−1.14616	−0.573081	+	0.819499i	$0.694252\pi$
$402$	4.05274	0.202132
$403$	−13.4547	−0.670227
$404$	1.99560	0.0992846
$405$	−2.48279	−0.123371
$406$	42.9816	2.13314
$407$	−0.164233	−0.00814073
$408$	1.00000	0.0495074
$409$	−23.9091	−1.18223	−0.591115	−	0.806587i	$-0.701312\pi$
$409$	−23.9091	−1.18223	−0.591115	+	0.806587i	$0.701312\pi$
$410$	1.27591	0.0630126
$411$	1.90335	0.0938851
$412$	−13.8614	−0.682901
$413$	25.3194	1.24589
$414$	7.20706	0.354208
$415$	8.91607	0.437673
$416$	−1.48279	−0.0726996
$417$	2.38723	0.116903
$418$	0.662042	0.0323815
$419$	10.3483	0.505549	0.252775	−	0.967525i	$-0.418657\pi$
$419$	10.3483	0.505549	0.252775	+	0.967525i	$0.418657\pi$
$420$	10.4452	0.509675
$421$	−34.4796	−1.68043	−0.840216	−	0.542252i	$-0.817572\pi$
$421$	−34.4796	−1.68043	−0.840216	+	0.542252i	$0.817572\pi$
$422$	17.3797	0.846030
$423$	2.53662	0.123335
$424$	−5.32805	−0.258753
$425$	1.16423	0.0564736
$426$	8.98678	0.435411
$427$	11.6452	0.563549
$428$	−17.2777	−0.835149
$429$	0.243523	0.0117574
$430$	3.16736	0.152744
$431$	−36.7758	−1.77143	−0.885715	−	0.464230i	$-0.846331\pi$
$431$	−36.7758	−1.77143	−0.885715	+	0.464230i	$0.846331\pi$
$432$	−1.00000	−0.0481125
$433$	−20.7293	−0.996188	−0.498094	−	0.867123i	$-0.665967\pi$
$433$	−20.7293	−0.996188	−0.498094	+	0.867123i	$0.665967\pi$
$434$	38.1746	1.83244
$435$	−25.3655	−1.21618
$436$	5.99782	0.287243
$437$	−29.0525	−1.38977
$438$	−15.7061	−0.750467
$439$	15.8379	0.755902	0.377951	−	0.925826i	$-0.376629\pi$
$439$	15.8379	0.755902	0.377951	+	0.925826i	$0.376629\pi$
$440$	−0.407756	−0.0194390
$441$	10.6993	0.509492
$442$	−1.48279	−0.0705290
$443$	−10.9850	−0.521912	−0.260956	−	0.965351i	$-0.584038\pi$
$443$	−10.9850	−0.521912	−0.260956	+	0.965351i	$0.584038\pi$
$444$	−1.00000	−0.0474579
$445$	−30.1852	−1.43092
$446$	20.1845	0.955764
$447$	17.6671	0.835627
$448$	4.20706	0.198765
$449$	−12.1304	−0.572469	−0.286235	−	0.958160i	$-0.592404\pi$
$449$	−12.1304	−0.572469	−0.286235	+	0.958160i	$0.592404\pi$
$450$	−1.16423	−0.0548825
$451$	−0.0843996	−0.00397422
$452$	−4.81235	−0.226354
$453$	12.4615	0.585492
$454$	15.9604	0.749059
$455$	−15.4881	−0.726091
$456$	4.03111	0.188774
$457$	−14.0108	−0.655396	−0.327698	−	0.944783i	$-0.606273\pi$
$457$	−14.0108	−0.655396	−0.327698	+	0.944783i	$0.606273\pi$
$458$	−19.0596	−0.890598
$459$	−1.00000	−0.0466760
$460$	17.8936	0.834293
$461$	−18.8347	−0.877219	−0.438609	−	0.898678i	$-0.644529\pi$
$461$	−18.8347	−0.877219	−0.438609	+	0.898678i	$0.644529\pi$
$462$	−0.690938	−0.0321453
$463$	32.6686	1.51824	0.759118	−	0.650953i	$-0.225630\pi$
$463$	32.6686	1.51824	0.759118	+	0.650953i	$0.225630\pi$
$464$	−10.2166	−0.474291
$465$	−22.5287	−1.04474
$466$	−27.2423	−1.26198
$467$	−24.2086	−1.12024	−0.560120	−	0.828411i	$-0.689245\pi$
$467$	−24.2086	−1.12024	−0.560120	+	0.828411i	$0.689245\pi$
$468$	1.48279	0.0685419
$469$	17.0501	0.787300
$470$	6.29788	0.290499
$471$	−13.4569	−0.620062
$472$	−6.01831	−0.277015
$473$	−0.209517	−0.00963359
$474$	−2.96557	−0.136213
$475$	4.69316	0.215337
$476$	4.20706	0.192830
$477$	5.32805	0.243954
$478$	15.8275	0.723934
$479$	20.2978	0.927430	0.463715	−	0.885984i	$-0.346516\pi$
$479$	20.2978	0.927430	0.463715	+	0.885984i	$0.346516\pi$
$480$	−2.48279	−0.113323
$481$	1.48279	0.0676093
$482$	27.7723	1.26499
$483$	30.3205	1.37963
$484$	−10.9730	−0.498774
$485$	28.7920	1.30738
$486$	1.00000	0.0453609
$487$	32.8057	1.48657	0.743283	−	0.668977i	$-0.233267\pi$
$487$	32.8057	1.48657	0.743283	+	0.668977i	$0.233267\pi$
$488$	−2.76801	−0.125302
$489$	7.52230	0.340170
$490$	26.5642	1.20005
$491$	32.9062	1.48504	0.742518	−	0.669826i	$-0.233631\pi$
$491$	32.9062	1.48504	0.742518	+	0.669826i	$0.233631\pi$
$492$	−0.513902	−0.0231685
$493$	−10.2166	−0.460130
$494$	−5.97729	−0.268931
$495$	0.407756	0.0183273
$496$	−9.07394	−0.407432
$497$	37.8079	1.69592
$498$	−3.59115	−0.160923
$499$	−7.87933	−0.352727	−0.176364	−	0.984325i	$-0.556433\pi$
$499$	−7.87933	−0.352727	−0.176364	+	0.984325i	$0.556433\pi$
$500$	9.52339	0.425899
$501$	−0.656932	−0.0293496
$502$	−2.43243	−0.108565
$503$	38.5115	1.71714	0.858571	−	0.512694i	$-0.171353\pi$
$503$	38.5115	1.71714	0.858571	+	0.512694i	$0.171353\pi$
$504$	−4.20706	−0.187397
$505$	−4.95464	−0.220479
$506$	−1.18364	−0.0526191
$507$	10.8013	0.479704
$508$	11.5461	0.512277
$509$	4.89486	0.216961	0.108481	−	0.994099i	$-0.465401\pi$
$509$	4.89486	0.216961	0.108481	+	0.994099i	$0.465401\pi$
$510$	−2.48279	−0.109940
$511$	−66.0765	−2.92305
$512$	−1.00000	−0.0441942
$513$	−4.03111	−0.177978
$514$	−5.41302	−0.238758
$515$	34.4149	1.51650
$516$	−1.27573	−0.0561609
$517$	−0.416596	−0.0183219
$518$	−4.20706	−0.184848
$519$	−13.4881	−0.592060
$520$	3.68145	0.161442
$521$	4.72485	0.206999	0.103500	−	0.994629i	$-0.466996\pi$
$521$	4.72485	0.206999	0.103500	+	0.994629i	$0.466996\pi$
$522$	10.2166	0.447166
$523$	2.55289	0.111630	0.0558150	−	0.998441i	$-0.482224\pi$
$523$	2.55289	0.111630	0.0558150	+	0.998441i	$0.482224\pi$
$524$	−12.4612	−0.544369
$525$	−4.89800	−0.213766
$526$	7.32533	0.319400
$527$	−9.07394	−0.395267
$528$	0.164233	0.00714732
$529$	28.9417	1.25833
$530$	13.2284	0.574605
$531$	6.01831	0.261172
$532$	16.9591	0.735272
$533$	0.762007	0.0330062
$534$	12.1578	0.526119
$535$	42.8968	1.85459
$536$	−4.05274	−0.175052
$537$	−13.0073	−0.561307
$538$	6.08452	0.262322
$539$	−1.75718	−0.0756873
$540$	2.48279	0.106842
$541$	−25.4566	−1.09447	−0.547233	−	0.836980i	$-0.684319\pi$
$541$	−25.4566	−1.09447	−0.547233	+	0.836980i	$0.684319\pi$
$542$	10.0127	0.430080
$543$	−17.6385	−0.756942
$544$	−1.00000	−0.0428746
$545$	−14.8913	−0.637873
$546$	6.23817	0.266969
$547$	41.8657	1.79005	0.895023	−	0.446019i	$-0.147159\pi$
$547$	41.8657	1.79005	0.895023	+	0.446019i	$0.147159\pi$
$548$	−1.90335	−0.0813069
$549$	2.76801	0.118136
$550$	0.191205	0.00815303
$551$	−41.1841	−1.75450
$552$	−7.20706	−0.306753
$553$	−12.4763	−0.530548
$554$	−5.21694	−0.221646
$555$	2.48279	0.105388
$556$	−2.38723	−0.101241
$557$	41.1092	1.74185	0.870927	−	0.491413i	$-0.163520\pi$
$557$	41.1092	1.74185	0.870927	+	0.491413i	$0.163520\pi$
$558$	9.07394	0.384130
$559$	1.89164	0.0800076
$560$	−10.4452	−0.441392
$561$	0.164233	0.00693392
$562$	24.1683	1.01948
$563$	−18.2024	−0.767140	−0.383570	−	0.923512i	$-0.625306\pi$
$563$	−18.2024	−0.767140	−0.383570	+	0.923512i	$0.625306\pi$
$564$	−2.53662	−0.106811
$565$	11.9480	0.502657
$566$	30.5708	1.28499
$567$	4.20706	0.176680
$568$	−8.98678	−0.377077
$569$	−17.5573	−0.736038	−0.368019	−	0.929818i	$-0.619964\pi$
$569$	−17.5573	−0.736038	−0.368019	+	0.929818i	$0.619964\pi$
$570$	−10.0084	−0.419206
$571$	17.7134	0.741283	0.370642	−	0.928776i	$-0.379138\pi$
$571$	17.7134	0.741283	0.370642	+	0.928776i	$0.379138\pi$
$572$	−0.243523	−0.0101822
$573$	5.29531	0.221215
$574$	−2.16201	−0.0902407
$575$	−8.39069	−0.349916
$576$	1.00000	0.0416667
$577$	35.2795	1.46870	0.734352	−	0.678769i	$-0.237486\pi$
$577$	35.2795	1.46870	0.734352	+	0.678769i	$0.237486\pi$
$578$	−1.00000	−0.0415945
$579$	−4.96466	−0.206324
$580$	25.3655	1.05325
$581$	−15.1082	−0.626793
$582$	−11.5967	−0.480697
$583$	−0.875041	−0.0362405
$584$	15.7061	0.649924
$585$	−3.68145	−0.152209
$586$	−5.89234	−0.243410
$587$	−43.6743	−1.80263	−0.901316	−	0.433161i	$-0.857398\pi$
$587$	−43.6743	−1.80263	−0.901316	+	0.433161i	$0.857398\pi$
$588$	−10.6993	−0.441233
$589$	−36.5781	−1.50717
$590$	14.9422	0.615160
$591$	−7.40353	−0.304541
$592$	1.00000	0.0410997
$593$	2.11135	0.0867028	0.0433514	−	0.999060i	$-0.486196\pi$
$593$	2.11135	0.0867028	0.0433514	+	0.999060i	$0.486196\pi$
$594$	−0.164233	−0.00673856
$595$	−10.4452	−0.428213
$596$	−17.6671	−0.723674
$597$	11.8186	0.483701
$598$	10.6865	0.437005
$599$	−43.5453	−1.77921	−0.889606	−	0.456729i	$-0.849021\pi$
$599$	−43.5453	−1.77921	−0.889606	+	0.456729i	$0.849021\pi$
$600$	1.16423	0.0475296
$601$	−28.8140	−1.17535	−0.587674	−	0.809098i	$-0.699956\pi$
$601$	−28.8140	−1.17535	−0.587674	+	0.809098i	$0.699956\pi$
$602$	−5.36707	−0.218745
$603$	4.05274	0.165040
$604$	−12.4615	−0.507051
$605$	27.2437	1.10761
$606$	1.99560	0.0810656
$607$	−25.9683	−1.05402	−0.527010	−	0.849859i	$-0.676687\pi$
$607$	−25.9683	−1.05402	−0.527010	+	0.849859i	$0.676687\pi$
$608$	−4.03111	−0.163483
$609$	42.9816	1.74170
$610$	6.87238	0.278254
$611$	3.76126	0.152164
$612$	1.00000	0.0404226
$613$	−38.3734	−1.54989	−0.774944	−	0.632030i	$-0.782222\pi$
$613$	−38.3734	−1.54989	−0.774944	+	0.632030i	$0.782222\pi$
$614$	8.50237	0.343128
$615$	1.27591	0.0514496
$616$	0.690938	0.0278387
$617$	−13.6863	−0.550989	−0.275495	−	0.961303i	$-0.588842\pi$
$617$	−13.6863	−0.550989	−0.275495	+	0.961303i	$0.588842\pi$
$618$	−13.8614	−0.557587
$619$	18.0734	0.726430	0.363215	−	0.931705i	$-0.381679\pi$
$619$	18.0734	0.726430	0.363215	+	0.931705i	$0.381679\pi$
$620$	22.5287	0.904773
$621$	7.20706	0.289209
$622$	−18.3439	−0.735525
$623$	51.1485	2.04922
$624$	−1.48279	−0.0593590
$625$	−29.4657	−1.17863
$626$	22.1040	0.883452
$627$	0.662042	0.0264394
$628$	13.4569	0.536989
$629$	1.00000	0.0398726
$630$	10.4452	0.416148
$631$	−29.7874	−1.18582	−0.592908	−	0.805270i	$-0.702020\pi$
$631$	−29.7874	−1.18582	−0.592908	+	0.805270i	$0.702020\pi$
$632$	2.96557	0.117964
$633$	17.3797	0.690781
$634$	32.5735	1.29366
$635$	−28.6666	−1.13760
$636$	−5.32805	−0.211271
$637$	15.8648	0.628588
$638$	−1.67789	−0.0664285
$639$	8.98678	0.355511
$640$	2.48279	0.0981408
$641$	−20.9135	−0.826035	−0.413017	−	0.910723i	$-0.635525\pi$
$641$	−20.9135	−0.826035	−0.413017	+	0.910723i	$0.635525\pi$
$642$	−17.2777	−0.681896
$643$	1.35460	0.0534202	0.0267101	−	0.999643i	$-0.491497\pi$
$643$	1.35460	0.0534202	0.0267101	+	0.999643i	$0.491497\pi$
$644$	−30.3205	−1.19480
$645$	3.16736	0.124715
$646$	−4.03111	−0.158602
$647$	13.3457	0.524676	0.262338	−	0.964976i	$-0.415507\pi$
$647$	13.3457	0.524676	0.262338	+	0.964976i	$0.415507\pi$
$648$	−1.00000	−0.0392837
$649$	−0.988405	−0.0387983
$650$	−1.72631	−0.0677114
$651$	38.1746	1.49618
$652$	−7.52230	−0.294596
$653$	−29.7211	−1.16308	−0.581539	−	0.813519i	$-0.697549\pi$
$653$	−29.7211	−1.16308	−0.581539	+	0.813519i	$0.697549\pi$
$654$	5.99782	0.234533
$655$	30.9384	1.20886
$656$	0.513902	0.0200645
$657$	−15.7061	−0.612754
$658$	−10.6717	−0.416026
$659$	−30.9741	−1.20658	−0.603290	−	0.797522i	$-0.706144\pi$
$659$	−30.9741	−1.20658	−0.603290	+	0.797522i	$0.706144\pi$
$660$	−0.407756	−0.0158719
$661$	37.4778	1.45772	0.728860	−	0.684663i	$-0.240051\pi$
$661$	37.4778	1.45772	0.728860	+	0.684663i	$0.240051\pi$
$662$	−21.8476	−0.849131
$663$	−1.48279	−0.0575867
$664$	3.59115	0.139364
$665$	−42.1059	−1.63280
$666$	−1.00000	−0.0387492
$667$	73.6313	2.85101
$668$	0.656932	0.0254175
$669$	20.1845	0.780378
$670$	10.0621	0.388732
$671$	−0.454598	−0.0175496
$672$	4.20706	0.162291
$673$	20.2412	0.780240	0.390120	−	0.920764i	$-0.372433\pi$
$673$	20.2412	0.780240	0.390120	+	0.920764i	$0.372433\pi$
$674$	−6.80371	−0.262069
$675$	−1.16423	−0.0448113
$676$	−10.8013	−0.415436
$677$	−14.8246	−0.569754	−0.284877	−	0.958564i	$-0.591953\pi$
$677$	−14.8246	−0.569754	−0.284877	+	0.958564i	$0.591953\pi$
$678$	−4.81235	−0.184817
$679$	−48.7878	−1.87230
$680$	2.48279	0.0952105
$681$	15.9604	0.611604
$682$	−1.49024	−0.0570642
$683$	0.00443724	0.000169786 0	8.48932e−5	−	1.00000i	$-0.499973\pi$
$683$	0.00443724	0.000169786 0	8.48932e−5		1.00000i	$0.499973\pi$
$684$	4.03111	0.154134
$685$	4.72560	0.180556
$686$	−15.5633	−0.594211
$687$	−19.0596	−0.727170
$688$	1.27573	0.0486367
$689$	7.90036	0.300980
$690$	17.8936	0.681197
$691$	−41.4543	−1.57700	−0.788498	−	0.615038i	$-0.789141\pi$
$691$	−41.4543	−1.57700	−0.788498	+	0.615038i	$0.789141\pi$
$692$	13.4881	0.512739
$693$	−0.690938	−0.0262466
$694$	−15.4287	−0.585665
$695$	5.92697	0.224823
$696$	−10.2166	−0.387257
$697$	0.513902	0.0194654
$698$	19.0596	0.721416
$699$	−27.2423	−1.03040
$700$	4.89800	0.185127
$701$	19.3505	0.730858	0.365429	−	0.930839i	$-0.380922\pi$
$701$	19.3505	0.730858	0.365429	+	0.930839i	$0.380922\pi$
$702$	1.48279	0.0559642
$703$	4.03111	0.152036
$704$	−0.164233	−0.00618976
$705$	6.29788	0.237192
$706$	29.9397	1.12680
$707$	8.39559	0.315749
$708$	−6.01831	−0.226182
$709$	−13.5212	−0.507800	−0.253900	−	0.967231i	$-0.581713\pi$
$709$	−13.5212	−0.507800	−0.253900	+	0.967231i	$0.581713\pi$
$710$	22.3123	0.837364
$711$	−2.96557	−0.111218
$712$	−12.1578	−0.455632
$713$	65.3964	2.44911
$714$	4.20706	0.157445
$715$	0.604615	0.0226113
$716$	13.0073	0.486106
$717$	15.8275	0.591090
$718$	9.56886	0.357107
$719$	−2.75189	−0.102628	−0.0513142	−	0.998683i	$-0.516341\pi$
$719$	−2.75189	−0.102628	−0.0513142	+	0.998683i	$0.516341\pi$
$720$	−2.48279	−0.0925280
$721$	−58.3157	−2.17179
$722$	2.75012	0.102349
$723$	27.7723	1.03286
$724$	17.6385	0.655531
$725$	−11.8944	−0.441749
$726$	−10.9730	−0.407247
$727$	11.5052	0.426703	0.213352	−	0.976976i	$-0.431562\pi$
$727$	11.5052	0.426703	0.213352	+	0.976976i	$0.431562\pi$
$728$	−6.23817	−0.231202
$729$	1.00000	0.0370370
$730$	−38.9949	−1.44327
$731$	1.27573	0.0471846
$732$	−2.76801	−0.102309
$733$	8.27674	0.305708	0.152854	−	0.988249i	$-0.451154\pi$
$733$	8.27674	0.305708	0.152854	+	0.988249i	$0.451154\pi$
$734$	−20.7254	−0.764988
$735$	26.5642	0.979835
$736$	7.20706	0.265656
$737$	−0.665593	−0.0245174
$738$	−0.513902	−0.0189170
$739$	14.6275	0.538080	0.269040	−	0.963129i	$-0.413294\pi$
$739$	14.6275	0.538080	0.269040	+	0.963129i	$0.413294\pi$
$740$	−2.48279	−0.0912691
$741$	−5.97729	−0.219581
$742$	−22.4154	−0.822895
$743$	38.4380	1.41015	0.705077	−	0.709131i	$-0.250912\pi$
$743$	38.4380	1.41015	0.705077	+	0.709131i	$0.250912\pi$
$744$	−9.07394	−0.332667
$745$	43.8637	1.60704
$746$	−22.4331	−0.821334
$747$	−3.59115	−0.131393
$748$	−0.164233	−0.00600495
$749$	−72.6882	−2.65597
$750$	9.52339	0.347745
$751$	−33.3469	−1.21684	−0.608422	−	0.793613i	$-0.708197\pi$
$751$	−33.3469	−1.21684	−0.608422	+	0.793613i	$0.708197\pi$
$752$	2.53662	0.0925009
$753$	−2.43243	−0.0886426
$754$	15.1490	0.551693
$755$	30.9393	1.12599
$756$	−4.20706	−0.153009
$757$	−8.09970	−0.294389	−0.147194	−	0.989108i	$-0.547024\pi$
$757$	−8.09970	−0.294389	−0.147194	+	0.989108i	$0.547024\pi$
$758$	−27.2160	−0.988529
$759$	−1.18364	−0.0429633
$760$	10.0084	0.363043
$761$	−52.3214	−1.89665	−0.948324	−	0.317303i	$-0.897223\pi$
$761$	−52.3214	−1.89665	−0.948324	+	0.317303i	$0.897223\pi$
$762$	11.5461	0.418273
$763$	25.2332	0.913502
$764$	−5.29531	−0.191578
$765$	−2.48279	−0.0897654
$766$	−37.6320	−1.35970
$767$	8.92388	0.322222
$768$	−1.00000	−0.0360844
$769$	16.2785	0.587016	0.293508	−	0.955957i	$-0.405177\pi$
$769$	16.2785	0.587016	0.293508	+	0.955957i	$0.405177\pi$
$770$	−1.71545	−0.0618206
$771$	−5.41302	−0.194945
$772$	4.96466	0.178682
$773$	−8.14210	−0.292851	−0.146425	−	0.989222i	$-0.546777\pi$
$773$	−8.14210	−0.292851	−0.146425	+	0.989222i	$0.546777\pi$
$774$	−1.27573	−0.0458551
$775$	−10.5642	−0.379477
$776$	11.5967	0.416296
$777$	−4.20706	−0.150927
$778$	8.22897	0.295023
$779$	2.07160	0.0742227
$780$	3.68145	0.131817
$781$	−1.47593	−0.0528128
$782$	7.20706	0.257724
$783$	10.2166	0.365110
$784$	10.6993	0.382119
$785$	−33.4106	−1.19248
$786$	−12.4612	−0.444475
$787$	15.5630	0.554761	0.277380	−	0.960760i	$-0.410534\pi$
$787$	15.5630	0.554761	0.277380	+	0.960760i	$0.410534\pi$
$788$	7.40353	0.263740
$789$	7.32533	0.260789
$790$	−7.36289	−0.261960
$791$	−20.2458	−0.719858
$792$	0.164233	0.00583577
$793$	4.10437	0.145750
$794$	13.5507	0.480896
$795$	13.2284	0.469163
$796$	−11.8186	−0.418898
$797$	−33.6485	−1.19189	−0.595945	−	0.803026i	$-0.703222\pi$
$797$	−33.6485	−1.19189	−0.595945	+	0.803026i	$0.703222\pi$
$798$	16.9591	0.600347
$799$	2.53662	0.0897390
$800$	−1.16423	−0.0411619
$801$	12.1578	0.429574
$802$	22.9519	0.810458
$803$	2.57946	0.0910273
$804$	−4.05274	−0.142929
$805$	75.2794	2.65325
$806$	13.4547	0.473922
$807$	6.08452	0.214185
$808$	−1.99560	−0.0702048
$809$	−20.6256	−0.725157	−0.362578	−	0.931953i	$-0.618103\pi$
$809$	−20.6256	−0.725157	−0.362578	+	0.931953i	$0.618103\pi$
$810$	2.48279	0.0872363
$811$	18.9430	0.665180	0.332590	−	0.943072i	$-0.392077\pi$
$811$	18.9430	0.665180	0.332590	+	0.943072i	$0.392077\pi$
$812$	−42.9816	−1.50836
$813$	10.0127	0.351159
$814$	0.164233	0.00575636
$815$	18.6763	0.654201
$816$	−1.00000	−0.0350070
$817$	5.14261	0.179917
$818$	23.9091	0.835964
$819$	6.23817	0.217979
$820$	−1.27591	−0.0445567
$821$	5.87005	0.204866	0.102433	−	0.994740i	$-0.467337\pi$
$821$	5.87005	0.204866	0.102433	+	0.994740i	$0.467337\pi$
$822$	−1.90335	−0.0663868
$823$	29.2560	1.01980	0.509900	−	0.860233i	$-0.329682\pi$
$823$	29.2560	1.01980	0.509900	+	0.860233i	$0.329682\pi$
$824$	13.8614	0.482884
$825$	0.191205	0.00665692
$826$	−25.3194	−0.880974
$827$	40.2327	1.39903	0.699513	−	0.714620i	$-0.253400\pi$
$827$	40.2327	1.39903	0.699513	+	0.714620i	$0.253400\pi$
$828$	−7.20706	−0.250463
$829$	16.0608	0.557813	0.278906	−	0.960318i	$-0.410028\pi$
$829$	16.0608	0.557813	0.278906	+	0.960318i	$0.410028\pi$
$830$	−8.91607	−0.309481
$831$	−5.21694	−0.180973
$832$	1.48279	0.0514064
$833$	10.6993	0.370710
$834$	−2.38723	−0.0826629
$835$	−1.63102	−0.0564439
$836$	−0.662042	−0.0228972
$837$	9.07394	0.313641
$838$	−10.3483	−0.357477
$839$	42.6251	1.47158	0.735792	−	0.677208i	$-0.236810\pi$
$839$	42.6251	1.47158	0.735792	+	0.677208i	$0.236810\pi$
$840$	−10.4452	−0.360395
$841$	75.3779	2.59924
$842$	34.4796	1.18824
$843$	24.1683	0.832399
$844$	−17.3797	−0.598234
$845$	26.8174	0.922548
$846$	−2.53662	−0.0872107
$847$	−46.1642	−1.58622
$848$	5.32805	0.182966
$849$	30.5708	1.04919
$850$	−1.16423	−0.0399329
$851$	−7.20706	−0.247055
$852$	−8.98678	−0.307882
$853$	−11.9652	−0.409680	−0.204840	−	0.978795i	$-0.565667\pi$
$853$	−11.9652	−0.409680	−0.204840	+	0.978795i	$0.565667\pi$
$854$	−11.6452	−0.398490
$855$	−10.0084	−0.342280
$856$	17.2777	0.590539
$857$	−12.8002	−0.437247	−0.218624	−	0.975809i	$-0.570157\pi$
$857$	−12.8002	−0.437247	−0.218624	+	0.975809i	$0.570157\pi$
$858$	−0.243523	−0.00831372
$859$	−30.2636	−1.03258	−0.516291	−	0.856413i	$-0.672688\pi$
$859$	−30.2636	−1.03258	−0.516291	+	0.856413i	$0.672688\pi$
$860$	−3.16736	−0.108006
$861$	−2.16201	−0.0736813
$862$	36.7758	1.25259
$863$	−29.3807	−1.00013	−0.500066	−	0.865988i	$-0.666691\pi$
$863$	−29.3807	−1.00013	−0.500066	+	0.865988i	$0.666691\pi$
$864$	1.00000	0.0340207
$865$	−33.4880	−1.13863
$866$	20.7293	0.704411
$867$	−1.00000	−0.0339618
$868$	−38.1746	−1.29573
$869$	0.487045	0.0165219
$870$	25.3655	0.859972
$871$	6.00935	0.203619
$872$	−5.99782	−0.203112
$873$	−11.5967	−0.392487
$874$	29.0525	0.982715
$875$	40.0655	1.35446
$876$	15.7061	0.530661
$877$	7.12660	0.240648	0.120324	−	0.992735i	$-0.461607\pi$
$877$	7.12660	0.240648	0.120324	+	0.992735i	$0.461607\pi$
$878$	−15.8379	−0.534503
$879$	−5.89234	−0.198744
$880$	0.407756	0.0137454
$881$	1.95049	0.0657138	0.0328569	−	0.999460i	$-0.489539\pi$
$881$	1.95049	0.0657138	0.0328569	+	0.999460i	$0.489539\pi$
$882$	−10.6993	−0.360265
$883$	21.8023	0.733707	0.366853	−	0.930279i	$-0.380435\pi$
$883$	21.8023	0.733707	0.366853	+	0.930279i	$0.380435\pi$
$884$	1.48279	0.0498715
$885$	14.9422	0.502276
$886$	10.9850	0.369048
$887$	−25.8922	−0.869376	−0.434688	−	0.900581i	$-0.643141\pi$
$887$	−25.8922	−0.869376	−0.434688	+	0.900581i	$0.643141\pi$
$888$	1.00000	0.0335578
$889$	48.5753	1.62916
$890$	30.1852	1.01181
$891$	−0.164233	−0.00550201
$892$	−20.1845	−0.675827
$893$	10.2254	0.342180
$894$	−17.6671	−0.590877
$895$	−32.2944	−1.07948
$896$	−4.20706	−0.140548
$897$	10.6865	0.356813
$898$	12.1304	0.404797
$899$	92.7044	3.09186
$900$	1.16423	0.0388078
$901$	5.32805	0.177503
$902$	0.0843996	0.00281020
$903$	−5.36707	−0.178605
$904$	4.81235	0.160056
$905$	−43.7927	−1.45572
$906$	−12.4615	−0.414006
$907$	1.89342	0.0628700	0.0314350	−	0.999506i	$-0.489992\pi$
$907$	1.89342	0.0628700	0.0314350	+	0.999506i	$0.489992\pi$
$908$	−15.9604	−0.529665
$909$	1.99560	0.0661898
$910$	15.4881	0.513424
$911$	53.5423	1.77394	0.886968	−	0.461830i	$-0.152807\pi$
$911$	53.5423	1.77394	0.886968	+	0.461830i	$0.152807\pi$
$912$	−4.03111	−0.133484
$913$	0.589786	0.0195191
$914$	14.0108	0.463435
$915$	6.87238	0.227194
$916$	19.0596	0.629748
$917$	−52.4248	−1.73122
$918$	1.00000	0.0330049
$919$	−37.6502	−1.24196	−0.620982	−	0.783825i	$-0.713266\pi$
$919$	−37.6502	−1.24196	−0.620982	+	0.783825i	$0.713266\pi$
$920$	−17.8936	−0.589934
$921$	8.50237	0.280163
$922$	18.8347	0.620287
$923$	13.3255	0.438614
$924$	0.690938	0.0227302
$925$	1.16423	0.0382797
$926$	−32.6686	−1.07356
$927$	−13.8614	−0.455268
$928$	10.2166	0.335375
$929$	21.4870	0.704965	0.352483	−	0.935818i	$-0.385338\pi$
$929$	21.4870	0.704965	0.352483	+	0.935818i	$0.385338\pi$
$930$	22.5287	0.738744
$931$	43.1303	1.41354
$932$	27.2423	0.892353
$933$	−18.3439	−0.600553
$934$	24.2086	0.792129
$935$	0.407756	0.0133350
$936$	−1.48279	−0.0484664
$937$	−17.3686	−0.567408	−0.283704	−	0.958912i	$-0.591563\pi$
$937$	−17.3686	−0.567408	−0.283704	+	0.958912i	$0.591563\pi$
$938$	−17.0501	−0.556705
$939$	22.1040	0.721336
$940$	−6.29788	−0.205414
$941$	−22.4298	−0.731190	−0.365595	−	0.930774i	$-0.619135\pi$
$941$	−22.4298	−0.731190	−0.365595	+	0.930774i	$0.619135\pi$
$942$	13.4569	0.438450
$943$	−3.70372	−0.120610
$944$	6.01831	0.195879
$945$	10.4452	0.339783
$946$	0.209517	0.00681198
$947$	41.2671	1.34100	0.670500	−	0.741910i	$-0.266080\pi$
$947$	41.2671	1.34100	0.670500	+	0.741910i	$0.266080\pi$
$948$	2.96557	0.0963174
$949$	−23.2888	−0.755987
$950$	−4.69316	−0.152266
$951$	32.5735	1.05627
$952$	−4.20706	−0.136352
$953$	17.0573	0.552541	0.276271	−	0.961080i	$-0.410901\pi$
$953$	17.0573	0.552541	0.276271	+	0.961080i	$0.410901\pi$
$954$	−5.32805	−0.172502
$955$	13.1471	0.425431
$956$	−15.8275	−0.511899
$957$	−1.67789	−0.0542386
$958$	−20.2978	−0.655792
$959$	−8.00749	−0.258575
$960$	2.48279	0.0801316
$961$	51.3364	1.65601
$962$	−1.48279	−0.0478070
$963$	−17.2777	−0.556766
$964$	−27.7723	−0.894487
$965$	−12.3262	−0.396794
$966$	−30.3205	−0.975546
$967$	−12.0710	−0.388178	−0.194089	−	0.980984i	$-0.562175\pi$
$967$	−12.0710	−0.388178	−0.194089	+	0.980984i	$0.562175\pi$
$968$	10.9730	0.352686
$969$	−4.03111	−0.129498
$970$	−28.7920	−0.924457
$971$	−25.3082	−0.812180	−0.406090	−	0.913833i	$-0.633108\pi$
$971$	−25.3082	−0.812180	−0.406090	+	0.913833i	$0.633108\pi$
$972$	−1.00000	−0.0320750
$973$	−10.0432	−0.321970
$974$	−32.8057	−1.05116
$975$	−1.72631	−0.0552862
$976$	2.76801	0.0886018
$977$	−31.4650	−1.00665	−0.503327	−	0.864096i	$-0.667891\pi$
$977$	−31.4650	−1.00665	−0.503327	+	0.864096i	$0.667891\pi$
$978$	−7.52230	−0.240537
$979$	−1.99671	−0.0638151
$980$	−26.5642	−0.848562
$981$	5.99782	0.191496
$982$	−32.9062	−1.05008
$983$	43.0283	1.37239	0.686194	−	0.727418i	$-0.259280\pi$
$983$	43.0283	1.37239	0.686194	+	0.727418i	$0.259280\pi$
$984$	0.513902	0.0163826
$985$	−18.3814	−0.585680
$986$	10.2166	0.325361
$987$	−10.6717	−0.339684
$988$	5.97729	0.190163
$989$	−9.19426	−0.292360
$990$	−0.407756	−0.0129593
$991$	60.2781	1.91480	0.957399	−	0.288769i	$-0.0932460\pi$
$991$	60.2781	1.91480	0.957399	+	0.288769i	$0.0932460\pi$
$992$	9.07394	0.288098
$993$	−21.8476	−0.693313
$994$	−37.8079	−1.19919
$995$	29.3430	0.930235
$996$	3.59115	0.113790
$997$	6.54466	0.207271	0.103636	−	0.994615i	$-0.466952\pi$
$997$	6.54466	0.207271	0.103636	+	0.994615i	$0.466952\pi$
$998$	7.87933	0.249416
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3774.2.a.y.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3774.2.a.y.1.2	✓	5	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:	\( N \)	\(=\)	\( 3774 = 2 \cdot 3 \cdot 17 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3774.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.48279 q^{5} +1.00000 q^{6} +4.20706 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.48279 q^{5} +1.00000 q^{6} +4.20706 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.48279 q^{10} -0.164233 q^{11} -1.00000 q^{12} +1.48279 q^{13} -4.20706 q^{14} +2.48279 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.03111 q^{19} -2.48279 q^{20} -4.20706 q^{21} +0.164233 q^{22} -7.20706 q^{23} +1.00000 q^{24} +1.16423 q^{25} -1.48279 q^{26} -1.00000 q^{27} +4.20706 q^{28} -10.2166 q^{29} -2.48279 q^{30} -9.07394 q^{31} -1.00000 q^{32} +0.164233 q^{33} -1.00000 q^{34} -10.4452 q^{35} +1.00000 q^{36} +1.00000 q^{37} -4.03111 q^{38} -1.48279 q^{39} +2.48279 q^{40} +0.513902 q^{41} +4.20706 q^{42} +1.27573 q^{43} -0.164233 q^{44} -2.48279 q^{45} +7.20706 q^{46} +2.53662 q^{47} -1.00000 q^{48} +10.6993 q^{49} -1.16423 q^{50} -1.00000 q^{51} +1.48279 q^{52} +5.32805 q^{53} +1.00000 q^{54} +0.407756 q^{55} -4.20706 q^{56} -4.03111 q^{57} +10.2166 q^{58} +6.01831 q^{59} +2.48279 q^{60} +2.76801 q^{61} +9.07394 q^{62} +4.20706 q^{63} +1.00000 q^{64} -3.68145 q^{65} -0.164233 q^{66} +4.05274 q^{67} +1.00000 q^{68} +7.20706 q^{69} +10.4452 q^{70} +8.98678 q^{71} -1.00000 q^{72} -15.7061 q^{73} -1.00000 q^{74} -1.16423 q^{75} +4.03111 q^{76} -0.690938 q^{77} +1.48279 q^{78} -2.96557 q^{79} -2.48279 q^{80} +1.00000 q^{81} -0.513902 q^{82} -3.59115 q^{83} -4.20706 q^{84} -2.48279 q^{85} -1.27573 q^{86} +10.2166 q^{87} +0.164233 q^{88} +12.1578 q^{89} +2.48279 q^{90} +6.23817 q^{91} -7.20706 q^{92} +9.07394 q^{93} -2.53662 q^{94} -10.0084 q^{95} +1.00000 q^{96} -11.5967 q^{97} -10.6993 q^{98} -0.164233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 4 q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 4 * q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 4 q^{5} + 5 q^{6} + q^{7} - 5 q^{8} + 5 q^{9} + 4 q^{10} - 2 q^{11} - 5 q^{12} - q^{13} - q^{14} + 4 q^{15} + 5 q^{16} + 5 q^{17} - 5 q^{18} + 4 q^{19} - 4 q^{20} - q^{21} + 2 q^{22} - 16 q^{23} + 5 q^{24} + 7 q^{25} + q^{26} - 5 q^{27} + q^{28} - 8 q^{29} - 4 q^{30} - 8 q^{31} - 5 q^{32} + 2 q^{33} - 5 q^{34} + 4 q^{35} + 5 q^{36} + 5 q^{37} - 4 q^{38} + q^{39} + 4 q^{40} - 22 q^{41} + q^{42} + 18 q^{43} - 2 q^{44} - 4 q^{45} + 16 q^{46} + 10 q^{47} - 5 q^{48} + 2 q^{49} - 7 q^{50} - 5 q^{51} - q^{52} - 6 q^{53} + 5 q^{54} + 16 q^{55} - q^{56} - 4 q^{57} + 8 q^{58} + 4 q^{59} + 4 q^{60} - 6 q^{61} + 8 q^{62} + q^{63} + 5 q^{64} - 28 q^{65} - 2 q^{66} + 11 q^{67} + 5 q^{68} + 16 q^{69} - 4 q^{70} - 5 q^{72} - 8 q^{73} - 5 q^{74} - 7 q^{75} + 4 q^{76} + 10 q^{77} - q^{78} + 2 q^{79} - 4 q^{80} + 5 q^{81} + 22 q^{82} + 11 q^{83} - q^{84} - 4 q^{85} - 18 q^{86} + 8 q^{87} + 2 q^{88} - 2 q^{89} + 4 q^{90} - 5 q^{91} - 16 q^{92} + 8 q^{93} - 10 q^{94} - 12 q^{95} + 5 q^{96} + 7 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 - 4 * q^5 + 5 * q^6 + q^7 - 5 * q^8 + 5 * q^9 + 4 * q^10 - 2 * q^11 - 5 * q^12 - q^13 - q^14 + 4 * q^15 + 5 * q^16 + 5 * q^17 - 5 * q^18 + 4 * q^19 - 4 * q^20 - q^21 + 2 * q^22 - 16 * q^23 + 5 * q^24 + 7 * q^25 + q^26 - 5 * q^27 + q^28 - 8 * q^29 - 4 * q^30 - 8 * q^31 - 5 * q^32 + 2 * q^33 - 5 * q^34 + 4 * q^35 + 5 * q^36 + 5 * q^37 - 4 * q^38 + q^39 + 4 * q^40 - 22 * q^41 + q^42 + 18 * q^43 - 2 * q^44 - 4 * q^45 + 16 * q^46 + 10 * q^47 - 5 * q^48 + 2 * q^49 - 7 * q^50 - 5 * q^51 - q^52 - 6 * q^53 + 5 * q^54 + 16 * q^55 - q^56 - 4 * q^57 + 8 * q^58 + 4 * q^59 + 4 * q^60 - 6 * q^61 + 8 * q^62 + q^63 + 5 * q^64 - 28 * q^65 - 2 * q^66 + 11 * q^67 + 5 * q^68 + 16 * q^69 - 4 * q^70 - 5 * q^72 - 8 * q^73 - 5 * q^74 - 7 * q^75 + 4 * q^76 + 10 * q^77 - q^78 + 2 * q^79 - 4 * q^80 + 5 * q^81 + 22 * q^82 + 11 * q^83 - q^84 - 4 * q^85 - 18 * q^86 + 8 * q^87 + 2 * q^88 - 2 * q^89 + 4 * q^90 - 5 * q^91 - 16 * q^92 + 8 * q^93 - 10 * q^94 - 12 * q^95 + 5 * q^96 + 7 * q^97 - 2 * q^98 - 2 * q^99

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