LMFDB - Embedded newform 5054.2.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5054,2,Mod(1,5054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5054, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5054 = 2 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5054.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:	$40.3563931816$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.19520000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 $ x^8 - 2x^7 - 12x^6 + 16x^5 + 50x^4 - 24x^3 - 72x^2 - 32*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.368905$ of defining polynomial
Character	$\chi$	$=$	5054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.30521 q^{3} +1.00000 q^{4} +3.15124 q^{5} +2.30521 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.31400 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.30521 q^{3} +1.00000 q^{4} +3.15124 q^{5} +2.30521 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.31400 q^{9} -3.15124 q^{10} -2.86391 q^{11} -2.30521 q^{12} -6.74578 q^{13} +1.00000 q^{14} -7.26428 q^{15} +1.00000 q^{16} -3.61968 q^{17} -2.31400 q^{18} +3.15124 q^{20} +2.30521 q^{21} +2.86391 q^{22} +9.19510 q^{23} +2.30521 q^{24} +4.93032 q^{25} +6.74578 q^{26} +1.58137 q^{27} -1.00000 q^{28} +3.62627 q^{29} +7.26428 q^{30} -0.789075 q^{31} -1.00000 q^{32} +6.60192 q^{33} +3.61968 q^{34} -3.15124 q^{35} +2.31400 q^{36} +8.66278 q^{37} +15.5504 q^{39} -3.15124 q^{40} -1.81346 q^{41} -2.30521 q^{42} +5.87561 q^{43} -2.86391 q^{44} +7.29198 q^{45} -9.19510 q^{46} +5.19351 q^{47} -2.30521 q^{48} +1.00000 q^{49} -4.93032 q^{50} +8.34413 q^{51} -6.74578 q^{52} -0.209579 q^{53} -1.58137 q^{54} -9.02487 q^{55} +1.00000 q^{56} -3.62627 q^{58} -13.1224 q^{59} -7.26428 q^{60} +1.83856 q^{61} +0.789075 q^{62} -2.31400 q^{63} +1.00000 q^{64} -21.2576 q^{65} -6.60192 q^{66} +15.0535 q^{67} -3.61968 q^{68} -21.1966 q^{69} +3.15124 q^{70} +6.92468 q^{71} -2.31400 q^{72} -6.67063 q^{73} -8.66278 q^{74} -11.3654 q^{75} +2.86391 q^{77} -15.5504 q^{78} +1.08776 q^{79} +3.15124 q^{80} -10.5874 q^{81} +1.81346 q^{82} -8.02323 q^{83} +2.30521 q^{84} -11.4065 q^{85} -5.87561 q^{86} -8.35933 q^{87} +2.86391 q^{88} +5.21896 q^{89} -7.29198 q^{90} +6.74578 q^{91} +9.19510 q^{92} +1.81899 q^{93} -5.19351 q^{94} +2.30521 q^{96} -15.7121 q^{97} -1.00000 q^{98} -6.62709 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 - 6 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 8 * q^14 - 8 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 + 6 * q^20 + 4 * q^21 - 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 - 22 * q^27 - 8 * q^28 - 16 * q^29 + 8 * q^30 - 22 * q^31 - 8 * q^32 + 8 * q^33 - 2 * q^34 - 6 * q^35 + 8 * q^36 + 12 * q^37 + 8 * q^39 - 6 * q^40 - 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 - 20 * q^46 + 6 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 + 22 * q^54 + 18 * q^55 + 8 * q^56 + 16 * q^58 - 14 * q^59 - 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 - 32 * q^65 - 8 * q^66 + 20 * q^67 + 2 * q^68 - 40 * q^69 + 6 * q^70 - 8 * q^72 - 18 * q^73 - 12 * q^74 - 16 * q^75 - 4 * q^77 - 8 * q^78 - 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 + 4 * q^84 - 16 * q^85 + 28 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^90 + 6 * q^91 + 20 * q^92 + 16 * q^93 - 6 * q^94 + 4 * q^96 - 26 * q^97 - 8 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.30521	−1.33091	−0.665457	−	0.746436i	$-0.731763\pi$
$3$	−2.30521	−1.33091	−0.665457	+	0.746436i	$0.731763\pi$
$4$	1.00000	0.500000
$5$	3.15124	1.40928	0.704639	−	0.709566i	$-0.251109\pi$
$5$	3.15124	1.40928	0.704639	+	0.709566i	$0.251109\pi$
$6$	2.30521	0.941099
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	2.31400	0.771334
$10$	−3.15124	−0.996510
$11$	−2.86391	−0.863501	−0.431751	−	0.901993i	$-0.642104\pi$
$11$	−2.86391	−0.863501	−0.431751	+	0.901993i	$0.642104\pi$
$12$	−2.30521	−0.665457
$13$	−6.74578	−1.87094	−0.935471	−	0.353403i	$-0.885024\pi$
$13$	−6.74578	−1.87094	−0.935471	+	0.353403i	$0.885024\pi$
$14$	1.00000	0.267261
$15$	−7.26428	−1.87563
$16$	1.00000	0.250000
$17$	−3.61968	−0.877901	−0.438951	−	0.898511i	$-0.644650\pi$
$17$	−3.61968	−0.877901	−0.438951	+	0.898511i	$0.644650\pi$
$18$	−2.31400	−0.545415
$19$	0	0
$19$	0	0
$20$	3.15124	0.704639
$21$	2.30521	0.503038
$22$	2.86391	0.610587
$23$	9.19510	1.91731	0.958655	−	0.284571i	$-0.0918511\pi$
$23$	9.19510	1.91731	0.958655	+	0.284571i	$0.0918511\pi$
$24$	2.30521	0.470549
$25$	4.93032	0.986065
$26$	6.74578	1.32296
$27$	1.58137	0.304335
$28$	−1.00000	−0.188982
$29$	3.62627	0.673382	0.336691	−	0.941615i	$-0.390692\pi$
$29$	3.62627	0.673382	0.336691	+	0.941615i	$0.390692\pi$
$30$	7.26428	1.32627
$31$	−0.789075	−0.141722	−0.0708611	−	0.997486i	$-0.522575\pi$
$31$	−0.789075	−0.141722	−0.0708611	+	0.997486i	$0.522575\pi$
$32$	−1.00000	−0.176777
$33$	6.60192	1.14925
$34$	3.61968	0.620770
$35$	−3.15124	−0.532657
$36$	2.31400	0.385667
$37$	8.66278	1.42415	0.712076	−	0.702102i	$-0.247755\pi$
$37$	8.66278	1.42415	0.712076	+	0.702102i	$0.247755\pi$
$38$	0	0
$39$	15.5504	2.49006
$40$	−3.15124	−0.498255
$41$	−1.81346	−0.283215	−0.141608	−	0.989923i	$-0.545227\pi$
$41$	−1.81346	−0.283215	−0.141608	+	0.989923i	$0.545227\pi$
$42$	−2.30521	−0.355702
$43$	5.87561	0.896021	0.448011	−	0.894028i	$-0.352133\pi$
$43$	5.87561	0.896021	0.448011	+	0.894028i	$0.352133\pi$
$44$	−2.86391	−0.431751
$45$	7.29198	1.08702
$46$	−9.19510	−1.35574
$47$	5.19351	0.757551	0.378775	−	0.925489i	$-0.376345\pi$
$47$	5.19351	0.757551	0.378775	+	0.925489i	$0.376345\pi$
$48$	−2.30521	−0.332729
$49$	1.00000	0.142857
$50$	−4.93032	−0.697253
$51$	8.34413	1.16841
$52$	−6.74578	−0.935471
$53$	−0.209579	−0.0287878	−0.0143939	−	0.999896i	$-0.504582\pi$
$53$	−0.209579	−0.0287878	−0.0143939	+	0.999896i	$0.504582\pi$
$54$	−1.58137	−0.215198
$55$	−9.02487	−1.21691
$56$	1.00000	0.133631
$57$	0	0
$58$	−3.62627	−0.476153
$59$	−13.1224	−1.70839	−0.854194	−	0.519955i	$-0.825949\pi$
$59$	−13.1224	−1.70839	−0.854194	+	0.519955i	$0.825949\pi$
$60$	−7.26428	−0.937814
$61$	1.83856	0.235403	0.117702	−	0.993049i	$-0.462447\pi$
$61$	1.83856	0.235403	0.117702	+	0.993049i	$0.462447\pi$
$62$	0.789075	0.100213
$63$	−2.31400	−0.291537
$64$	1.00000	0.125000
$65$	−21.2576	−2.63668
$66$	−6.60192	−0.812640
$67$	15.0535	1.83908	0.919541	−	0.392994i	$-0.128561\pi$
$67$	15.0535	1.83908	0.919541	+	0.392994i	$0.128561\pi$
$68$	−3.61968	−0.438951
$69$	−21.1966	−2.55178
$70$	3.15124	0.376645
$71$	6.92468	0.821808	0.410904	−	0.911679i	$-0.365213\pi$
$71$	6.92468	0.821808	0.410904	+	0.911679i	$0.365213\pi$
$72$	−2.31400	−0.272708
$73$	−6.67063	−0.780738	−0.390369	−	0.920658i	$-0.627653\pi$
$73$	−6.67063	−0.780738	−0.390369	+	0.920658i	$0.627653\pi$
$74$	−8.66278	−1.00703
$75$	−11.3654	−1.31237
$76$	0	0
$77$	2.86391	0.326373
$78$	−15.5504	−1.76074
$79$	1.08776	0.122383	0.0611913	−	0.998126i	$-0.480510\pi$
$79$	1.08776	0.122383	0.0611913	+	0.998126i	$0.480510\pi$
$80$	3.15124	0.352320
$81$	−10.5874	−1.17638
$82$	1.81346	0.200263
$83$	−8.02323	−0.880664	−0.440332	−	0.897835i	$-0.645139\pi$
$83$	−8.02323	−0.880664	−0.440332	+	0.897835i	$0.645139\pi$
$84$	2.30521	0.251519
$85$	−11.4065	−1.23721
$86$	−5.87561	−0.633583
$87$	−8.35933	−0.896214
$88$	2.86391	0.305294
$89$	5.21896	0.553209	0.276604	−	0.960984i	$-0.410791\pi$
$89$	5.21896	0.553209	0.276604	+	0.960984i	$0.410791\pi$
$90$	−7.29198	−0.768642
$91$	6.74578	0.707150
$92$	9.19510	0.958655
$93$	1.81899	0.188620
$94$	−5.19351	−0.535669
$95$	0	0
$96$	2.30521	0.235275
$97$	−15.7121	−1.59533	−0.797663	−	0.603104i	$-0.793930\pi$
$97$	−15.7121	−1.59533	−0.797663	+	0.603104i	$0.793930\pi$
$98$	−1.00000	−0.101015
$99$	−6.62709	−0.666047
$100$	4.93032	0.493032
$101$	−10.8243	−1.07706	−0.538530	−	0.842607i	$-0.681020\pi$
$101$	−10.8243	−1.07706	−0.538530	+	0.842607i	$0.681020\pi$
$102$	−8.34413	−0.826192
$103$	17.0685	1.68181	0.840907	−	0.541180i	$-0.182022\pi$
$103$	17.0685	1.68181	0.840907	+	0.541180i	$0.182022\pi$
$104$	6.74578	0.661478
$105$	7.26428	0.708921
$106$	0.209579	0.0203561
$107$	2.41312	0.233285	0.116642	−	0.993174i	$-0.462787\pi$
$107$	2.41312	0.233285	0.116642	+	0.993174i	$0.462787\pi$
$108$	1.58137	0.152168
$109$	−12.9751	−1.24279	−0.621394	−	0.783498i	$-0.713433\pi$
$109$	−12.9751	−1.24279	−0.621394	+	0.783498i	$0.713433\pi$
$110$	9.02487	0.860488
$111$	−19.9695	−1.89542
$112$	−1.00000	−0.0944911
$113$	−15.6494	−1.47217	−0.736086	−	0.676888i	$-0.763328\pi$
$113$	−15.6494	−1.47217	−0.736086	+	0.676888i	$0.763328\pi$
$114$	0	0
$115$	28.9760	2.70202
$116$	3.62627	0.336691
$117$	−15.6097	−1.44312
$118$	13.1224	1.20801
$119$	3.61968	0.331816
$120$	7.26428	0.663135
$121$	−2.79803	−0.254366
$122$	−1.83856	−0.166455
$123$	4.18041	0.376935
$124$	−0.789075	−0.0708611
$125$	−0.219565	−0.0196385
$126$	2.31400	0.206148
$127$	−1.49804	−0.132929	−0.0664647	−	0.997789i	$-0.521172\pi$
$127$	−1.49804	−0.132929	−0.0664647	+	0.997789i	$0.521172\pi$
$128$	−1.00000	−0.0883883
$129$	−13.5445	−1.19253
$130$	21.2576	1.86441
$131$	−1.44493	−0.126244	−0.0631220	−	0.998006i	$-0.520106\pi$
$131$	−1.44493	−0.126244	−0.0631220	+	0.998006i	$0.520106\pi$
$132$	6.60192	0.574623
$133$	0	0
$134$	−15.0535	−1.30043
$135$	4.98329	0.428893
$136$	3.61968	0.310385
$137$	4.14601	0.354218	0.177109	−	0.984191i	$-0.443326\pi$
$137$	4.14601	0.354218	0.177109	+	0.984191i	$0.443326\pi$
$138$	21.1966	1.80438
$139$	16.3757	1.38896	0.694482	−	0.719510i	$-0.255633\pi$
$139$	16.3757	1.38896	0.694482	+	0.719510i	$0.255633\pi$
$140$	−3.15124	−0.266329
$141$	−11.9721	−1.00824
$142$	−6.92468	−0.581106
$143$	19.3193	1.61556
$144$	2.31400	0.192833
$145$	11.4273	0.948983
$146$	6.67063	0.552065
$147$	−2.30521	−0.190131
$148$	8.66278	0.712076
$149$	−15.7184	−1.28771	−0.643853	−	0.765150i	$-0.722665\pi$
$149$	−15.7184	−1.28771	−0.643853	+	0.765150i	$0.722665\pi$
$150$	11.3654	0.927984
$151$	−22.3656	−1.82008	−0.910042	−	0.414516i	$-0.863951\pi$
$151$	−22.3656	−1.82008	−0.910042	+	0.414516i	$0.863951\pi$
$152$	0	0
$153$	−8.37594	−0.677155
$154$	−2.86391	−0.230780
$155$	−2.48657	−0.199726
$156$	15.5504	1.24503
$157$	−13.5544	−1.08176	−0.540879	−	0.841100i	$-0.681908\pi$
$157$	−13.5544	−1.08176	−0.540879	+	0.841100i	$0.681908\pi$
$158$	−1.08776	−0.0865376
$159$	0.483123	0.0383141
$160$	−3.15124	−0.249128
$161$	−9.19510	−0.724675
$162$	10.5874	0.831825
$163$	2.05840	0.161227	0.0806133	−	0.996745i	$-0.474312\pi$
$163$	2.05840	0.161227	0.0806133	+	0.996745i	$0.474312\pi$
$164$	−1.81346	−0.141608
$165$	20.8042	1.61961
$166$	8.02323	0.622723
$167$	−8.87786	−0.686990	−0.343495	−	0.939155i	$-0.611611\pi$
$167$	−8.87786	−0.686990	−0.343495	+	0.939155i	$0.611611\pi$
$168$	−2.30521	−0.177851
$169$	32.5055	2.50043
$170$	11.4065	0.874838
$171$	0	0
$172$	5.87561	0.448011
$173$	19.2820	1.46598	0.732990	−	0.680239i	$-0.238124\pi$
$173$	19.2820	1.46598	0.732990	+	0.680239i	$0.238124\pi$
$174$	8.35933	0.633719
$175$	−4.93032	−0.372697
$176$	−2.86391	−0.215875
$177$	30.2498	2.27372
$178$	−5.21896	−0.391178
$179$	−10.3005	−0.769894	−0.384947	−	0.922939i	$-0.625780\pi$
$179$	−10.3005	−0.769894	−0.384947	+	0.922939i	$0.625780\pi$
$180$	7.29198	0.543512
$181$	−1.64346	−0.122157	−0.0610786	−	0.998133i	$-0.519454\pi$
$181$	−1.64346	−0.122157	−0.0610786	+	0.998133i	$0.519454\pi$
$182$	−6.74578	−0.500030
$183$	−4.23827	−0.313302
$184$	−9.19510	−0.677871
$185$	27.2985	2.00703
$186$	−1.81899	−0.133375
$187$	10.3664	0.758069
$188$	5.19351	0.378775
$189$	−1.58137	−0.115028
$190$	0	0
$191$	−11.3392	−0.820477	−0.410239	−	0.911978i	$-0.634555\pi$
$191$	−11.3392	−0.820477	−0.410239	+	0.911978i	$0.634555\pi$
$192$	−2.30521	−0.166364
$193$	−1.50748	−0.108511	−0.0542555	−	0.998527i	$-0.517279\pi$
$193$	−1.50748	−0.108511	−0.0542555	+	0.998527i	$0.517279\pi$
$194$	15.7121	1.12807
$195$	49.0032	3.50919
$196$	1.00000	0.0714286
$197$	10.3413	0.736788	0.368394	−	0.929670i	$-0.379908\pi$
$197$	10.3413	0.736788	0.368394	+	0.929670i	$0.379908\pi$
$198$	6.62709	0.470967
$199$	−23.4790	−1.66438	−0.832189	−	0.554492i	$-0.812913\pi$
$199$	−23.4790	−1.66438	−0.832189	+	0.554492i	$0.812913\pi$
$200$	−4.93032	−0.348627
$201$	−34.7016	−2.44766
$202$	10.8243	0.761596
$203$	−3.62627	−0.254514
$204$	8.34413	0.584206
$205$	−5.71466	−0.399129
$206$	−17.0685	−1.18922
$207$	21.2775	1.47889
$208$	−6.74578	−0.467736
$209$	0	0
$210$	−7.26428	−0.501283
$211$	−21.0788	−1.45113	−0.725563	−	0.688156i	$-0.758421\pi$
$211$	−21.0788	−1.45113	−0.725563	+	0.688156i	$0.758421\pi$
$212$	−0.209579	−0.0143939
$213$	−15.9628	−1.09376
$214$	−2.41312	−0.164957
$215$	18.5155	1.26274
$216$	−1.58137	−0.107599
$217$	0.789075	0.0535659
$218$	12.9751	0.878784
$219$	15.3772	1.03910
$220$	−9.02487	−0.608457
$221$	24.4176	1.64250
$222$	19.9695	1.34027
$223$	9.94858	0.666206	0.333103	−	0.942890i	$-0.391904\pi$
$223$	9.94858	0.666206	0.333103	+	0.942890i	$0.391904\pi$
$224$	1.00000	0.0668153
$225$	11.4088	0.760585
$226$	15.6494	1.04098
$227$	11.1691	0.741322	0.370661	−	0.928768i	$-0.379131\pi$
$227$	11.1691	0.741322	0.370661	+	0.928768i	$0.379131\pi$
$228$	0	0
$229$	−17.4746	−1.15475	−0.577377	−	0.816478i	$-0.695924\pi$
$229$	−17.4746	−1.15475	−0.577377	+	0.816478i	$0.695924\pi$
$230$	−28.9760	−1.91062
$231$	−6.60192	−0.434374
$232$	−3.62627	−0.238077
$233$	−23.9983	−1.57218	−0.786089	−	0.618113i	$-0.787897\pi$
$233$	−23.9983	−1.57218	−0.786089	+	0.618113i	$0.787897\pi$
$234$	15.6097	1.02044
$235$	16.3660	1.06760
$236$	−13.1224	−0.854194
$237$	−2.50752	−0.162881
$238$	−3.61968	−0.234629
$239$	0.616095	0.0398519	0.0199259	−	0.999801i	$-0.493657\pi$
$239$	0.616095	0.0398519	0.0199259	+	0.999801i	$0.493657\pi$
$240$	−7.26428	−0.468907
$241$	13.9002	0.895391	0.447695	−	0.894186i	$-0.352245\pi$
$241$	13.9002	0.895391	0.447695	+	0.894186i	$0.352245\pi$
$242$	2.79803	0.179864
$243$	19.6621	1.26132
$244$	1.83856	0.117702
$245$	3.15124	0.201325
$246$	−4.18041	−0.266533
$247$	0	0
$248$	0.789075	0.0501063
$249$	18.4952	1.17209
$250$	0.219565	0.0138865
$251$	−0.0263156	−0.00166103	−0.000830513	−	1.00000i	$-0.500264\pi$
$251$	−0.0263156	−0.00166103	−0.000830513		1.00000i	$0.500264\pi$
$252$	−2.31400	−0.145768
$253$	−26.3339	−1.65560
$254$	1.49804	0.0939953
$255$	26.2944	1.64662
$256$	1.00000	0.0625000
$257$	10.3255	0.644088	0.322044	−	0.946725i	$-0.395630\pi$
$257$	10.3255	0.644088	0.322044	+	0.946725i	$0.395630\pi$
$258$	13.5445	0.843245
$259$	−8.66278	−0.538279
$260$	−21.2576	−1.31834
$261$	8.39120	0.519402
$262$	1.44493	0.0892680
$263$	−13.3076	−0.820583	−0.410292	−	0.911954i	$-0.634573\pi$
$263$	−13.3076	−0.820583	−0.410292	+	0.911954i	$0.634573\pi$
$264$	−6.60192	−0.406320
$265$	−0.660433	−0.0405700
$266$	0	0
$267$	−12.0308	−0.736274
$268$	15.0535	0.919541
$269$	3.35514	0.204567	0.102283	−	0.994755i	$-0.467385\pi$
$269$	3.35514	0.204567	0.102283	+	0.994755i	$0.467385\pi$
$270$	−4.98329	−0.303273
$271$	−9.21082	−0.559518	−0.279759	−	0.960070i	$-0.590254\pi$
$271$	−9.21082	−0.559518	−0.279759	+	0.960070i	$0.590254\pi$
$272$	−3.61968	−0.219475
$273$	−15.5504	−0.941156
$274$	−4.14601	−0.250470
$275$	−14.1200	−0.851468
$276$	−21.1966	−1.27589
$277$	21.7643	1.30769	0.653845	−	0.756628i	$-0.273155\pi$
$277$	21.7643	1.30769	0.653845	+	0.756628i	$0.273155\pi$
$278$	−16.3757	−0.982146
$279$	−1.82592	−0.109315
$280$	3.15124	0.188323
$281$	−17.6966	−1.05569	−0.527845	−	0.849341i	$-0.677000\pi$
$281$	−17.6966	−1.05569	−0.527845	+	0.849341i	$0.677000\pi$
$282$	11.9721	0.712930
$283$	−5.55850	−0.330418	−0.165209	−	0.986259i	$-0.552830\pi$
$283$	−5.55850	−0.330418	−0.165209	+	0.986259i	$0.552830\pi$
$284$	6.92468	0.410904
$285$	0	0
$286$	−19.3193	−1.14237
$287$	1.81346	0.107045
$288$	−2.31400	−0.136354
$289$	−3.89791	−0.229289
$290$	−11.4273	−0.671032
$291$	36.2198	2.12324
$292$	−6.67063	−0.390369
$293$	−3.47855	−0.203219	−0.101610	−	0.994824i	$-0.532399\pi$
$293$	−3.47855	−0.203219	−0.101610	+	0.994824i	$0.532399\pi$
$294$	2.30521	0.134443
$295$	−41.3518	−2.40759
$296$	−8.66278	−0.503514
$297$	−4.52891	−0.262794
$298$	15.7184	0.910545
$299$	−62.0281	−3.58718
$300$	−11.3654	−0.656184
$301$	−5.87561	−0.338664
$302$	22.3656	1.28699
$303$	24.9523	1.43347
$304$	0	0
$305$	5.79374	0.331749
$306$	8.37594	0.478821
$307$	−14.9887	−0.855452	−0.427726	−	0.903908i	$-0.640685\pi$
$307$	−14.9887	−0.855452	−0.427726	+	0.903908i	$0.640685\pi$
$308$	2.86391	0.163186
$309$	−39.3466	−2.23835
$310$	2.48657	0.141228
$311$	0.273351	0.0155003	0.00775015	−	0.999970i	$-0.497533\pi$
$311$	0.273351	0.0155003	0.00775015	+	0.999970i	$0.497533\pi$
$312$	−15.5504	−0.880371
$313$	−31.1586	−1.76119	−0.880593	−	0.473873i	$-0.842856\pi$
$313$	−31.1586	−1.76119	−0.880593	+	0.473873i	$0.842856\pi$
$314$	13.5544	0.764918
$315$	−7.29198	−0.410856
$316$	1.08776	0.0611913
$317$	−8.32630	−0.467652	−0.233826	−	0.972279i	$-0.575125\pi$
$317$	−8.32630	−0.467652	−0.233826	+	0.972279i	$0.575125\pi$
$318$	−0.483123	−0.0270922
$319$	−10.3853	−0.581466
$320$	3.15124	0.176160
$321$	−5.56275	−0.310482
$322$	9.19510	0.512423
$323$	0	0
$324$	−10.5874	−0.588189
$325$	−33.2589	−1.84487
$326$	−2.05840	−0.114004
$327$	29.9103	1.65405
$328$	1.81346	0.100132
$329$	−5.19351	−0.286327
$330$	−20.8042	−1.14524
$331$	5.22288	0.287075	0.143538	−	0.989645i	$-0.454152\pi$
$331$	5.22288	0.287075	0.143538	+	0.989645i	$0.454152\pi$
$332$	−8.02323	−0.440332
$333$	20.0457	1.09850
$334$	8.87786	0.485775
$335$	47.4373	2.59178
$336$	2.30521	0.125760
$337$	10.0330	0.546532	0.273266	−	0.961938i	$-0.411896\pi$
$337$	10.0330	0.546532	0.273266	+	0.961938i	$0.411896\pi$
$338$	−32.5055	−1.76807
$339$	36.0752	1.95934
$340$	−11.4065	−0.618604
$341$	2.25984	0.122377
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−5.87561	−0.316791
$345$	−66.7957	−3.59616
$346$	−19.2820	−1.03660
$347$	10.0313	0.538511	0.269255	−	0.963069i	$-0.413222\pi$
$347$	10.0313	0.538511	0.269255	+	0.963069i	$0.413222\pi$
$348$	−8.35933	−0.448107
$349$	−12.8029	−0.685325	−0.342662	−	0.939459i	$-0.611329\pi$
$349$	−12.8029	−0.685325	−0.342662	+	0.939459i	$0.611329\pi$
$350$	4.93032	0.263537
$351$	−10.6676	−0.569394
$352$	2.86391	0.152647
$353$	−17.6487	−0.939347	−0.469674	−	0.882840i	$-0.655628\pi$
$353$	−17.6487	−0.939347	−0.469674	+	0.882840i	$0.655628\pi$
$354$	−30.2498	−1.60776
$355$	21.8213	1.15816
$356$	5.21896	0.276604
$357$	−8.34413	−0.441618
$358$	10.3005	0.544397
$359$	−17.4994	−0.923582	−0.461791	−	0.886989i	$-0.652793\pi$
$359$	−17.4994	−0.923582	−0.461791	+	0.886989i	$0.652793\pi$
$360$	−7.29198	−0.384321
$361$	0	0
$362$	1.64346	0.0863782
$363$	6.45004	0.338539
$364$	6.74578	0.353575
$365$	−21.0208	−1.10028
$366$	4.23827	0.221538
$367$	−20.7154	−1.08133	−0.540667	−	0.841237i	$-0.681828\pi$
$367$	−20.7154	−1.08133	−0.540667	+	0.841237i	$0.681828\pi$
$368$	9.19510	0.479328
$369$	−4.19635	−0.218453
$370$	−27.2985	−1.41918
$371$	0.209579	0.0108808
$372$	1.81899	0.0943100
$373$	0.0845478	0.00437772	0.00218886	−	0.999998i	$-0.499303\pi$
$373$	0.0845478	0.00437772	0.00218886	+	0.999998i	$0.499303\pi$
$374$	−10.3664	−0.536036
$375$	0.506144	0.0261371
$376$	−5.19351	−0.267835
$377$	−24.4620	−1.25986
$378$	1.58137	0.0813370
$379$	14.3398	0.736584	0.368292	−	0.929710i	$-0.379943\pi$
$379$	14.3398	0.736584	0.368292	+	0.929710i	$0.379943\pi$
$380$	0	0
$381$	3.45330	0.176918
$382$	11.3392	0.580165
$383$	−38.2334	−1.95363	−0.976817	−	0.214075i	$-0.931326\pi$
$383$	−38.2334	−1.95363	−0.976817	+	0.214075i	$0.931326\pi$
$384$	2.30521	0.117637
$385$	9.02487	0.459950
$386$	1.50748	0.0767288
$387$	13.5962	0.691132
$388$	−15.7121	−0.797663
$389$	−13.5355	−0.686276	−0.343138	−	0.939285i	$-0.611490\pi$
$389$	−13.5355	−0.686276	−0.343138	+	0.939285i	$0.611490\pi$
$390$	−49.0032	−2.48137
$391$	−33.2833	−1.68321
$392$	−1.00000	−0.0505076
$393$	3.33087	0.168020
$394$	−10.3413	−0.520988
$395$	3.42780	0.172471
$396$	−6.62709	−0.333024
$397$	19.3503	0.971165	0.485583	−	0.874191i	$-0.338607\pi$
$397$	19.3503	0.971165	0.485583	+	0.874191i	$0.338607\pi$
$398$	23.4790	1.17689
$399$	0	0
$400$	4.93032	0.246516
$401$	19.8444	0.990984	0.495492	−	0.868612i	$-0.334988\pi$
$401$	19.8444	0.990984	0.495492	+	0.868612i	$0.334988\pi$
$402$	34.7016	1.73076
$403$	5.32293	0.265154
$404$	−10.8243	−0.538530
$405$	−33.3635	−1.65784
$406$	3.62627	0.179969
$407$	−24.8094	−1.22976
$408$	−8.34413	−0.413096
$409$	−18.5652	−0.917988	−0.458994	−	0.888439i	$-0.651790\pi$
$409$	−18.5652	−0.917988	−0.458994	+	0.888439i	$0.651790\pi$
$410$	5.71466	0.282227
$411$	−9.55743	−0.471433
$412$	17.0685	0.840907
$413$	13.1224	0.645710
$414$	−21.2775	−1.04573
$415$	−25.2831	−1.24110
$416$	6.74578	0.330739
$417$	−37.7493	−1.84859
$418$	0	0
$419$	5.88427	0.287465	0.143733	−	0.989617i	$-0.454089\pi$
$419$	5.88427	0.287465	0.143733	+	0.989617i	$0.454089\pi$
$420$	7.26428	0.354461
$421$	6.40040	0.311936	0.155968	−	0.987762i	$-0.450150\pi$
$421$	6.40040	0.311936	0.155968	+	0.987762i	$0.450150\pi$
$422$	21.0788	1.02610
$423$	12.0178	0.584324
$424$	0.209579	0.0101780
$425$	−17.8462	−0.865668
$426$	15.9628	0.773403
$427$	−1.83856	−0.0889741
$428$	2.41312	0.116642
$429$	−44.5351	−2.15017
$430$	−18.5155	−0.892894
$431$	24.7558	1.19245	0.596223	−	0.802819i	$-0.296667\pi$
$431$	24.7558	1.19245	0.596223	+	0.802819i	$0.296667\pi$
$432$	1.58137	0.0760838
$433$	15.8959	0.763910	0.381955	−	0.924181i	$-0.375251\pi$
$433$	15.8959	0.763910	0.381955	+	0.924181i	$0.375251\pi$
$434$	−0.789075	−0.0378768
$435$	−26.3423	−1.26301
$436$	−12.9751	−0.621394
$437$	0	0
$438$	−15.3772	−0.734752
$439$	−15.6112	−0.745081	−0.372541	−	0.928016i	$-0.621513\pi$
$439$	−15.6112	−0.745081	−0.372541	+	0.928016i	$0.621513\pi$
$440$	9.02487	0.430244
$441$	2.31400	0.110191
$442$	−24.4176	−1.16143
$443$	6.84424	0.325180	0.162590	−	0.986694i	$-0.448015\pi$
$443$	6.84424	0.325180	0.162590	+	0.986694i	$0.448015\pi$
$444$	−19.9695	−0.947712
$445$	16.4462	0.779625
$446$	−9.94858	−0.471079
$447$	36.2343	1.71383
$448$	−1.00000	−0.0472456
$449$	42.0306	1.98355	0.991774	−	0.127999i	$-0.0408554\pi$
$449$	42.0306	1.98355	0.991774	+	0.127999i	$0.0408554\pi$
$450$	−11.4088	−0.537815
$451$	5.19359	0.244557
$452$	−15.6494	−0.736086
$453$	51.5574	2.42238
$454$	−11.1691	−0.524194
$455$	21.2576	0.996571
$456$	0	0
$457$	4.26325	0.199426	0.0997132	−	0.995016i	$-0.468207\pi$
$457$	4.26325	0.199426	0.0997132	+	0.995016i	$0.468207\pi$
$458$	17.4746	0.816535
$459$	−5.72406	−0.267176
$460$	28.9760	1.35101
$461$	6.47322	0.301488	0.150744	−	0.988573i	$-0.451833\pi$
$461$	6.47322	0.301488	0.150744	+	0.988573i	$0.451833\pi$
$462$	6.60192	0.307149
$463$	16.1932	0.752561	0.376281	−	0.926506i	$-0.377203\pi$
$463$	16.1932	0.752561	0.376281	+	0.926506i	$0.377203\pi$
$464$	3.62627	0.168346
$465$	5.73206	0.265818
$466$	23.9983	1.11170
$467$	24.9856	1.15619	0.578097	−	0.815968i	$-0.303796\pi$
$467$	24.9856	1.15619	0.578097	+	0.815968i	$0.303796\pi$
$468$	−15.6097	−0.721560
$469$	−15.0535	−0.695108
$470$	−16.3660	−0.754907
$471$	31.2457	1.43973
$472$	13.1224	0.604006
$473$	−16.8272	−0.773715
$474$	2.50752	0.115174
$475$	0	0
$476$	3.61968	0.165908
$477$	−0.484965	−0.0222050
$478$	−0.616095	−0.0281795
$479$	35.2299	1.60970	0.804848	−	0.593481i	$-0.202247\pi$
$479$	35.2299	1.60970	0.804848	+	0.593481i	$0.202247\pi$
$480$	7.26428	0.331567
$481$	−58.4372	−2.66451
$482$	−13.9002	−0.633137
$483$	21.1966	0.964481
$484$	−2.79803	−0.127183
$485$	−49.5127	−2.24826
$486$	−19.6621	−0.891890
$487$	−12.3178	−0.558175	−0.279087	−	0.960266i	$-0.590032\pi$
$487$	−12.3178	−0.558175	−0.279087	+	0.960266i	$0.590032\pi$
$488$	−1.83856	−0.0832277
$489$	−4.74506	−0.214579
$490$	−3.15124	−0.142359
$491$	−11.7203	−0.528928	−0.264464	−	0.964396i	$-0.585195\pi$
$491$	−11.7203	−0.528928	−0.264464	+	0.964396i	$0.585195\pi$
$492$	4.18041	0.188468
$493$	−13.1260	−0.591163
$494$	0	0
$495$	−20.8836	−0.938646
$496$	−0.789075	−0.0354305
$497$	−6.92468	−0.310614
$498$	−18.4952	−0.828792
$499$	0.815844	0.0365222	0.0182611	−	0.999833i	$-0.494187\pi$
$499$	0.815844	0.0365222	0.0182611	+	0.999833i	$0.494187\pi$
$500$	−0.219565	−0.00981924
$501$	20.4654	0.914325
$502$	0.0263156	0.00117452
$503$	44.2443	1.97276	0.986379	−	0.164491i	$-0.0525983\pi$
$503$	44.2443	1.97276	0.986379	+	0.164491i	$0.0525983\pi$
$504$	2.31400	0.103074
$505$	−34.1100	−1.51788
$506$	26.3339	1.17069
$507$	−74.9321	−3.32785
$508$	−1.49804	−0.0664647
$509$	26.2671	1.16427	0.582134	−	0.813093i	$-0.302218\pi$
$509$	26.2671	1.16427	0.582134	+	0.813093i	$0.302218\pi$
$510$	−26.2944	−1.16433
$511$	6.67063	0.295091
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.3255	−0.455439
$515$	53.7871	2.37014
$516$	−13.5445	−0.596264
$517$	−14.8737	−0.654146
$518$	8.66278	0.380621
$519$	−44.4490	−1.95109
$520$	21.2576	0.932207
$521$	4.00494	0.175460	0.0877299	−	0.996144i	$-0.472039\pi$
$521$	4.00494	0.175460	0.0877299	+	0.996144i	$0.472039\pi$
$522$	−8.39120	−0.367273
$523$	−22.2696	−0.973780	−0.486890	−	0.873463i	$-0.661869\pi$
$523$	−22.2696	−0.973780	−0.486890	+	0.873463i	$0.661869\pi$
$524$	−1.44493	−0.0631220
$525$	11.3654	0.496029
$526$	13.3076	0.580240
$527$	2.85620	0.124418
$528$	6.60192	0.287312
$529$	61.5498	2.67608
$530$	0.660433	0.0286874
$531$	−30.3652	−1.31774
$532$	0	0
$533$	12.2332	0.529879
$534$	12.0308	0.520624
$535$	7.60432	0.328763
$536$	−15.0535	−0.650214
$537$	23.7448	1.02466
$538$	−3.35514	−0.144650
$539$	−2.86391	−0.123357
$540$	4.98329	0.214447
$541$	10.6022	0.455824	0.227912	−	0.973682i	$-0.426810\pi$
$541$	10.6022	0.455824	0.227912	+	0.973682i	$0.426810\pi$
$542$	9.21082	0.395639
$543$	3.78852	0.162581
$544$	3.61968	0.155193
$545$	−40.8876	−1.75143
$546$	15.5504	0.665498
$547$	−9.54622	−0.408167	−0.204083	−	0.978953i	$-0.565421\pi$
$547$	−9.54622	−0.408167	−0.204083	+	0.978953i	$0.565421\pi$
$548$	4.14601	0.177109
$549$	4.25443	0.181575
$550$	14.1200	0.602079
$551$	0	0
$552$	21.1966	0.902189
$553$	−1.08776	−0.0462563
$554$	−21.7643	−0.924677
$555$	−62.9288	−2.67118
$556$	16.3757	0.694482
$557$	36.9689	1.56642	0.783211	−	0.621756i	$-0.213580\pi$
$557$	36.9689	1.56642	0.783211	+	0.621756i	$0.213580\pi$
$558$	1.82592	0.0772974
$559$	−39.6355	−1.67640
$560$	−3.15124	−0.133164
$561$	−23.8968	−1.00892
$562$	17.6966	0.746485
$563$	−7.55515	−0.318412	−0.159206	−	0.987245i	$-0.550893\pi$
$563$	−7.55515	−0.318412	−0.159206	+	0.987245i	$0.550893\pi$
$564$	−11.9721	−0.504118
$565$	−49.3151	−2.07470
$566$	5.55850	0.233641
$567$	10.5874	0.444629
$568$	−6.92468	−0.290553
$569$	−40.7408	−1.70794	−0.853972	−	0.520318i	$-0.825813\pi$
$569$	−40.7408	−1.70794	−0.853972	+	0.520318i	$0.825813\pi$
$570$	0	0
$571$	20.8304	0.871723	0.435862	−	0.900014i	$-0.356444\pi$
$571$	20.8304	0.871723	0.435862	+	0.900014i	$0.356444\pi$
$572$	19.3193	0.807780
$573$	26.1393	1.09199
$574$	−1.81346	−0.0756924
$575$	45.3348	1.89059
$576$	2.31400	0.0964167
$577$	−13.5759	−0.565171	−0.282586	−	0.959242i	$-0.591192\pi$
$577$	−13.5759	−0.565171	−0.282586	+	0.959242i	$0.591192\pi$
$578$	3.89791	0.162132
$579$	3.47507	0.144419
$580$	11.4273	0.474491
$581$	8.02323	0.332860
$582$	−36.2198	−1.50136
$583$	0.600214	0.0248583
$584$	6.67063	0.276033
$585$	−49.1901	−2.03376
$586$	3.47855	0.143698
$587$	−5.49606	−0.226847	−0.113423	−	0.993547i	$-0.536182\pi$
$587$	−5.49606	−0.226847	−0.113423	+	0.993547i	$0.536182\pi$
$588$	−2.30521	−0.0950653
$589$	0	0
$590$	41.3518	1.70243
$591$	−23.8389	−0.980602
$592$	8.66278	0.356038
$593$	−2.73118	−0.112156	−0.0560781	−	0.998426i	$-0.517860\pi$
$593$	−2.73118	−0.112156	−0.0560781	+	0.998426i	$0.517860\pi$
$594$	4.52891	0.185823
$595$	11.4065	0.467620
$596$	−15.7184	−0.643853
$597$	54.1240	2.21515
$598$	62.0281	2.53652
$599$	−15.6211	−0.638261	−0.319130	−	0.947711i	$-0.603391\pi$
$599$	−15.6211	−0.638261	−0.319130	+	0.947711i	$0.603391\pi$
$600$	11.3654	0.463992
$601$	−37.0488	−1.51125	−0.755626	−	0.655004i	$-0.772667\pi$
$601$	−37.0488	−1.51125	−0.755626	+	0.655004i	$0.772667\pi$
$602$	5.87561	0.239472
$603$	34.8339	1.41855
$604$	−22.3656	−0.910042
$605$	−8.81726	−0.358472
$606$	−24.9523	−1.01362
$607$	−4.79845	−0.194763	−0.0973815	−	0.995247i	$-0.531047\pi$
$607$	−4.79845	−0.194763	−0.0973815	+	0.995247i	$0.531047\pi$
$608$	0	0
$609$	8.35933	0.338737
$610$	−5.79374	−0.234582
$611$	−35.0342	−1.41733
$612$	−8.37594	−0.338577
$613$	−34.1292	−1.37846	−0.689232	−	0.724541i	$-0.742052\pi$
$613$	−34.1292	−1.37846	−0.689232	+	0.724541i	$0.742052\pi$
$614$	14.9887	0.604896
$615$	13.1735	0.531207
$616$	−2.86391	−0.115390
$617$	−31.7928	−1.27993	−0.639964	−	0.768405i	$-0.721051\pi$
$617$	−31.7928	−1.27993	−0.639964	+	0.768405i	$0.721051\pi$
$618$	39.3466	1.58275
$619$	−32.3107	−1.29868	−0.649339	−	0.760499i	$-0.724954\pi$
$619$	−32.3107	−1.29868	−0.649339	+	0.760499i	$0.724954\pi$
$620$	−2.48657	−0.0998630
$621$	14.5409	0.583505
$622$	−0.273351	−0.0109604
$623$	−5.21896	−0.209093
$624$	15.5504	0.622516
$625$	−25.3435	−1.01374
$626$	31.1586	1.24535
$627$	0	0
$628$	−13.5544	−0.540879
$629$	−31.3565	−1.25027
$630$	7.29198	0.290519
$631$	−12.4170	−0.494311	−0.247156	−	0.968976i	$-0.579496\pi$
$631$	−12.4170	−0.494311	−0.247156	+	0.968976i	$0.579496\pi$
$632$	−1.08776	−0.0432688
$633$	48.5912	1.93133
$634$	8.32630	0.330680
$635$	−4.72068	−0.187334
$636$	0.483123	0.0191571
$637$	−6.74578	−0.267277
$638$	10.3853	0.411159
$639$	16.0237	0.633888
$640$	−3.15124	−0.124564
$641$	4.78304	0.188919	0.0944593	−	0.995529i	$-0.469888\pi$
$641$	4.78304	0.188919	0.0944593	+	0.995529i	$0.469888\pi$
$642$	5.56275	0.219544
$643$	7.71296	0.304169	0.152085	−	0.988367i	$-0.451401\pi$
$643$	7.71296	0.304169	0.152085	+	0.988367i	$0.451401\pi$
$644$	−9.19510	−0.362338
$645$	−42.6820	−1.68060
$646$	0	0
$647$	−17.9076	−0.704021	−0.352011	−	0.935996i	$-0.614502\pi$
$647$	−17.9076	−0.704021	−0.352011	+	0.935996i	$0.614502\pi$
$648$	10.5874	0.415912
$649$	37.5813	1.47519
$650$	33.2589	1.30452
$651$	−1.81899	−0.0712917
$652$	2.05840	0.0806133
$653$	−12.9721	−0.507638	−0.253819	−	0.967252i	$-0.581687\pi$
$653$	−12.9721	−0.507638	−0.253819	+	0.967252i	$0.581687\pi$
$654$	−29.9103	−1.16959
$655$	−4.55332	−0.177913
$656$	−1.81346	−0.0708038
$657$	−15.4358	−0.602210
$658$	5.19351	0.202464
$659$	19.7138	0.767940	0.383970	−	0.923346i	$-0.374557\pi$
$659$	19.7138	0.767940	0.383970	+	0.923346i	$0.374557\pi$
$660$	20.8042	0.809804
$661$	−42.6877	−1.66036	−0.830180	−	0.557496i	$-0.811762\pi$
$661$	−42.6877	−1.66036	−0.830180	+	0.557496i	$0.811762\pi$
$662$	−5.22288	−0.202993
$663$	−56.2877	−2.18603
$664$	8.02323	0.311362
$665$	0	0
$666$	−20.0457	−0.776754
$667$	33.3439	1.29108
$668$	−8.87786	−0.343495
$669$	−22.9336	−0.886664
$670$	−47.4373	−1.83266
$671$	−5.26547	−0.203271
$672$	−2.30521	−0.0889255
$673$	−20.8336	−0.803076	−0.401538	−	0.915842i	$-0.631524\pi$
$673$	−20.8336	−0.803076	−0.401538	+	0.915842i	$0.631524\pi$
$674$	−10.0330	−0.386457
$675$	7.79668	0.300094
$676$	32.5055	1.25021
$677$	−38.1961	−1.46799	−0.733997	−	0.679153i	$-0.762347\pi$
$677$	−38.1961	−1.46799	−0.733997	+	0.679153i	$0.762347\pi$
$678$	−36.0752	−1.38546
$679$	15.7121	0.602976
$680$	11.4065	0.437419
$681$	−25.7472	−0.986636
$682$	−2.25984	−0.0865338
$683$	16.3374	0.625135	0.312567	−	0.949896i	$-0.398811\pi$
$683$	16.3374	0.625135	0.312567	+	0.949896i	$0.398811\pi$
$684$	0	0
$685$	13.0651	0.499191
$686$	1.00000	0.0381802
$687$	40.2827	1.53688
$688$	5.87561	0.224005
$689$	1.41377	0.0538604
$690$	66.7957	2.54287
$691$	−13.1849	−0.501578	−0.250789	−	0.968042i	$-0.580690\pi$
$691$	−13.1849	−0.501578	−0.250789	+	0.968042i	$0.580690\pi$
$692$	19.2820	0.732990
$693$	6.62709	0.251742
$694$	−10.0313	−0.380784
$695$	51.6036	1.95744
$696$	8.35933	0.316859
$697$	6.56415	0.248635
$698$	12.8029	0.484598
$699$	55.3211	2.09244
$700$	−4.93032	−0.186349
$701$	−16.8187	−0.635234	−0.317617	−	0.948219i	$-0.602883\pi$
$701$	−16.8187	−0.635234	−0.317617	+	0.948219i	$0.602883\pi$
$702$	10.6676	0.402622
$703$	0	0
$704$	−2.86391	−0.107938
$705$	−37.7271	−1.42088
$706$	17.6487	0.664219
$707$	10.8243	0.407090
$708$	30.2498	1.13686
$709$	−36.4238	−1.36792	−0.683961	−	0.729518i	$-0.739744\pi$
$709$	−36.4238	−1.36792	−0.683961	+	0.729518i	$0.739744\pi$
$710$	−21.8213	−0.818940
$711$	2.51708	0.0943979
$712$	−5.21896	−0.195589
$713$	−7.25562	−0.271725
$714$	8.34413	0.312271
$715$	60.8798	2.27677
$716$	−10.3005	−0.384947
$717$	−1.42023	−0.0530394
$718$	17.4994	0.653071
$719$	−12.9847	−0.484248	−0.242124	−	0.970245i	$-0.577844\pi$
$719$	−12.9847	−0.484248	−0.242124	+	0.970245i	$0.577844\pi$
$720$	7.29198	0.271756
$721$	−17.0685	−0.635666
$722$	0	0
$723$	−32.0429	−1.19169
$724$	−1.64346	−0.0610786
$725$	17.8787	0.663998
$726$	−6.45004	−0.239384
$727$	33.3184	1.23571	0.617856	−	0.786291i	$-0.288001\pi$
$727$	33.3184	1.23571	0.617856	+	0.786291i	$0.288001\pi$
$728$	−6.74578	−0.250015
$729$	−13.5631	−0.502336
$730$	21.0208	0.778014
$731$	−21.2678	−0.786619
$732$	−4.23827	−0.156651
$733$	52.2909	1.93141	0.965704	−	0.259645i	$-0.0836055\pi$
$733$	52.2909	1.93141	0.965704	+	0.259645i	$0.0836055\pi$
$734$	20.7154	0.764619
$735$	−7.26428	−0.267947
$736$	−9.19510	−0.338936
$737$	−43.1120	−1.58805
$738$	4.19635	0.154470
$739$	4.33272	0.159382	0.0796909	−	0.996820i	$-0.474607\pi$
$739$	4.33272	0.159382	0.0796909	+	0.996820i	$0.474607\pi$
$740$	27.2985	1.00351
$741$	0	0
$742$	−0.209579	−0.00769387
$743$	−33.8202	−1.24074	−0.620372	−	0.784307i	$-0.713018\pi$
$743$	−33.8202	−1.24074	−0.620372	+	0.784307i	$0.713018\pi$
$744$	−1.81899	−0.0666873
$745$	−49.5326	−1.81473
$746$	−0.0845478	−0.00309551
$747$	−18.5658	−0.679286
$748$	10.3664	0.379034
$749$	−2.41312	−0.0881734
$750$	−0.506144	−0.0184818
$751$	47.4282	1.73068	0.865340	−	0.501185i	$-0.167103\pi$
$751$	47.4282	1.73068	0.865340	+	0.501185i	$0.167103\pi$
$752$	5.19351	0.189388
$753$	0.0606630	0.00221068
$754$	24.4620	0.890855
$755$	−70.4793	−2.56501
$756$	−1.58137	−0.0575140
$757$	−12.7315	−0.462735	−0.231367	−	0.972866i	$-0.574320\pi$
$757$	−12.7315	−0.462735	−0.231367	+	0.972866i	$0.574320\pi$
$758$	−14.3398	−0.520844
$759$	60.7053	2.20346
$760$	0	0
$761$	−42.1557	−1.52814	−0.764072	−	0.645131i	$-0.776803\pi$
$761$	−42.1557	−1.52814	−0.764072	+	0.645131i	$0.776803\pi$
$762$	−3.45330	−0.125100
$763$	12.9751	0.469730
$764$	−11.3392	−0.410239
$765$	−26.3946	−0.954300
$766$	38.2334	1.38143
$767$	88.5206	3.19629
$768$	−2.30521	−0.0831822
$769$	27.6643	0.997601	0.498800	−	0.866717i	$-0.333774\pi$
$769$	27.6643	0.997601	0.498800	+	0.866717i	$0.333774\pi$
$770$	−9.02487	−0.325234
$771$	−23.8025	−0.857226
$772$	−1.50748	−0.0542555
$773$	15.3328	0.551481	0.275741	−	0.961232i	$-0.411077\pi$
$773$	15.3328	0.551481	0.275741	+	0.961232i	$0.411077\pi$
$774$	−13.5962	−0.488704
$775$	−3.89040	−0.139747
$776$	15.7121	0.564033
$777$	19.9695	0.716403
$778$	13.5355	0.485271
$779$	0	0
$780$	49.0032	1.75460
$781$	−19.8316	−0.709632
$782$	33.2833	1.19021
$783$	5.73449	0.204934
$784$	1.00000	0.0357143
$785$	−42.7132	−1.52450
$786$	−3.33087	−0.118808
$787$	−9.14195	−0.325875	−0.162938	−	0.986636i	$-0.552097\pi$
$787$	−9.14195	−0.325875	−0.162938	+	0.986636i	$0.552097\pi$
$788$	10.3413	0.368394
$789$	30.6769	1.09213
$790$	−3.42780	−0.121956
$791$	15.6494	0.556429
$792$	6.62709	0.235483
$793$	−12.4025	−0.440426
$794$	−19.3503	−0.686718
$795$	1.52244	0.0539953
$796$	−23.4790	−0.832189
$797$	37.5107	1.32870	0.664349	−	0.747422i	$-0.268709\pi$
$797$	37.5107	1.32870	0.664349	+	0.747422i	$0.268709\pi$
$798$	0	0
$799$	−18.7988	−0.665055
$800$	−4.93032	−0.174313
$801$	12.0767	0.426709
$802$	−19.8444	−0.700732
$803$	19.1041	0.674168
$804$	−34.7016	−1.22383
$805$	−28.9760	−1.02127
$806$	−5.32293	−0.187492
$807$	−7.73431	−0.272261
$808$	10.8243	0.380798
$809$	46.1739	1.62339	0.811693	−	0.584084i	$-0.198546\pi$
$809$	46.1739	1.62339	0.811693	+	0.584084i	$0.198546\pi$
$810$	33.3635	1.17227
$811$	12.6144	0.442953	0.221476	−	0.975166i	$-0.428912\pi$
$811$	12.6144	0.442953	0.221476	+	0.975166i	$0.428912\pi$
$812$	−3.62627	−0.127257
$813$	21.2329	0.744670
$814$	24.8094	0.869569
$815$	6.48653	0.227213
$816$	8.34413	0.292103
$817$	0	0
$818$	18.5652	0.649115
$819$	15.6097	0.545448
$820$	−5.71466	−0.199564
$821$	−40.0794	−1.39878	−0.699390	−	0.714740i	$-0.746545\pi$
$821$	−40.0794	−1.39878	−0.699390	+	0.714740i	$0.746545\pi$
$822$	9.55743	0.333354
$823$	−24.2247	−0.844419	−0.422209	−	0.906498i	$-0.638745\pi$
$823$	−24.2247	−0.844419	−0.422209	+	0.906498i	$0.638745\pi$
$824$	−17.0685	−0.594611
$825$	32.5496	1.13323
$826$	−13.1224	−0.456586
$827$	−13.1161	−0.456092	−0.228046	−	0.973650i	$-0.573234\pi$
$827$	−13.1161	−0.456092	−0.228046	+	0.973650i	$0.573234\pi$
$828$	21.2775	0.739443
$829$	−34.2148	−1.18833	−0.594165	−	0.804344i	$-0.702517\pi$
$829$	−34.2148	−1.18833	−0.594165	+	0.804344i	$0.702517\pi$
$830$	25.2831	0.877590
$831$	−50.1713	−1.74042
$832$	−6.74578	−0.233868
$833$	−3.61968	−0.125414
$834$	37.7493	1.30715
$835$	−27.9763	−0.968160
$836$	0	0
$837$	−1.24782	−0.0431311
$838$	−5.88427	−0.203269
$839$	29.1251	1.00551	0.502756	−	0.864429i	$-0.332320\pi$
$839$	29.1251	1.00551	0.502756	+	0.864429i	$0.332320\pi$
$840$	−7.26428	−0.250641
$841$	−15.8501	−0.546557
$842$	−6.40040	−0.220572
$843$	40.7943	1.40503
$844$	−21.0788	−0.725563
$845$	102.433	3.52379
$846$	−12.0178	−0.413180
$847$	2.79803	0.0961413
$848$	−0.209579	−0.00719696
$849$	12.8135	0.439758
$850$	17.8462	0.612120
$851$	79.6551	2.73054
$852$	−15.9628	−0.546878
$853$	13.2308	0.453013	0.226507	−	0.974010i	$-0.427270\pi$
$853$	13.2308	0.453013	0.226507	+	0.974010i	$0.427270\pi$
$854$	1.83856	0.0629142
$855$	0	0
$856$	−2.41312	−0.0824787
$857$	53.8424	1.83922	0.919611	−	0.392830i	$-0.128504\pi$
$857$	53.8424	1.83922	0.919611	+	0.392830i	$0.128504\pi$
$858$	44.5351	1.52040
$859$	12.2700	0.418648	0.209324	−	0.977846i	$-0.432874\pi$
$859$	12.2700	0.418648	0.209324	+	0.977846i	$0.432874\pi$
$860$	18.5155	0.631372
$861$	−4.18041	−0.142468
$862$	−24.7558	−0.843187
$863$	−22.2017	−0.755754	−0.377877	−	0.925856i	$-0.623346\pi$
$863$	−22.2017	−0.755754	−0.377877	+	0.925856i	$0.623346\pi$
$864$	−1.58137	−0.0537994
$865$	60.7621	2.06597
$866$	−15.8959	−0.540166
$867$	8.98551	0.305164
$868$	0.789075	0.0267830
$869$	−3.11525	−0.105678
$870$	26.3423	0.893086
$871$	−101.548	−3.44082
$872$	12.9751	0.439392
$873$	−36.3579	−1.23053
$874$	0	0
$875$	0.219565	0.00742265
$876$	15.3772	0.519548
$877$	50.0986	1.69171	0.845855	−	0.533412i	$-0.179091\pi$
$877$	50.0986	1.69171	0.845855	+	0.533412i	$0.179091\pi$
$878$	15.6112	0.526852
$879$	8.01880	0.270467
$880$	−9.02487	−0.304228
$881$	17.6634	0.595096	0.297548	−	0.954707i	$-0.403831\pi$
$881$	17.6634	0.595096	0.297548	+	0.954707i	$0.403831\pi$
$882$	−2.31400	−0.0779165
$883$	43.7666	1.47286	0.736431	−	0.676512i	$-0.236509\pi$
$883$	43.7666	1.47286	0.736431	+	0.676512i	$0.236509\pi$
$884$	24.4176	0.821252
$885$	95.3246	3.20430
$886$	−6.84424	−0.229937
$887$	−25.1621	−0.844862	−0.422431	−	0.906395i	$-0.638823\pi$
$887$	−25.1621	−0.844862	−0.422431	+	0.906395i	$0.638823\pi$
$888$	19.9695	0.670134
$889$	1.49804	0.0502426
$890$	−16.4462	−0.551278
$891$	30.3214	1.01580
$892$	9.94858	0.333103
$893$	0	0
$894$	−36.2343	−1.21186
$895$	−32.4593	−1.08500
$896$	1.00000	0.0334077
$897$	142.988	4.77423
$898$	−42.0306	−1.40258
$899$	−2.86140	−0.0954331
$900$	11.4088	0.380293
$901$	0.758607	0.0252729
$902$	−5.19359	−0.172928
$903$	13.5445	0.450733
$904$	15.6494	0.520491
$905$	−5.17893	−0.172153
$906$	−51.5574	−1.71288
$907$	−20.8683	−0.692920	−0.346460	−	0.938065i	$-0.612616\pi$
$907$	−20.8683	−0.692920	−0.346460	+	0.938065i	$0.612616\pi$
$908$	11.1691	0.370661
$909$	−25.0475	−0.830772
$910$	−21.2576	−0.704682
$911$	−44.1205	−1.46178	−0.730889	−	0.682496i	$-0.760895\pi$
$911$	−44.1205	−1.46178	−0.730889	+	0.682496i	$0.760895\pi$
$912$	0	0
$913$	22.9778	0.760454
$914$	−4.26325	−0.141016
$915$	−13.3558	−0.441529
$916$	−17.4746	−0.577377
$917$	1.44493	0.0477158
$918$	5.72406	0.188922
$919$	−20.0363	−0.660938	−0.330469	−	0.943817i	$-0.607207\pi$
$919$	−20.0363	−0.660938	−0.330469	+	0.943817i	$0.607207\pi$
$920$	−28.9760	−0.955309
$921$	34.5522	1.13853
$922$	−6.47322	−0.213184
$923$	−46.7123	−1.53756
$924$	−6.60192	−0.217187
$925$	42.7103	1.40431
$926$	−16.1932	−0.532141
$927$	39.4966	1.29724
$928$	−3.62627	−0.119038
$929$	56.9868	1.86968	0.934839	−	0.355073i	$-0.115544\pi$
$929$	56.9868	1.86968	0.934839	+	0.355073i	$0.115544\pi$
$930$	−5.73206	−0.187962
$931$	0	0
$932$	−23.9983	−0.786089
$933$	−0.630132	−0.0206296
$934$	−24.9856	−0.817552
$935$	32.6671	1.06833
$936$	15.6097	0.510220
$937$	−13.0183	−0.425288	−0.212644	−	0.977130i	$-0.568207\pi$
$937$	−13.0183	−0.425288	−0.212644	+	0.977130i	$0.568207\pi$
$938$	15.0535	0.491515
$939$	71.8271	2.34399
$940$	16.3660	0.533800
$941$	44.6448	1.45538	0.727690	−	0.685906i	$-0.240594\pi$
$941$	44.6448	1.45538	0.727690	+	0.685906i	$0.240594\pi$
$942$	−31.2457	−1.01804
$943$	−16.6750	−0.543011
$944$	−13.1224	−0.427097
$945$	−4.98329	−0.162106
$946$	16.8272	0.547099
$947$	23.8473	0.774932	0.387466	−	0.921884i	$-0.373351\pi$
$947$	23.8473	0.774932	0.387466	+	0.921884i	$0.373351\pi$
$948$	−2.50752	−0.0814405
$949$	44.9986	1.46072
$950$	0	0
$951$	19.1939	0.622404
$952$	−3.61968	−0.117315
$953$	14.4371	0.467662	0.233831	−	0.972277i	$-0.424874\pi$
$953$	14.4371	0.467662	0.233831	+	0.972277i	$0.424874\pi$
$954$	0.484965	0.0157013
$955$	−35.7326	−1.15628
$956$	0.616095	0.0199259
$957$	23.9404	0.773882
$958$	−35.2299	−1.13823
$959$	−4.14601	−0.133882
$960$	−7.26428	−0.234454
$961$	−30.3774	−0.979915
$962$	58.4372	1.88409
$963$	5.58396	0.179940
$964$	13.9002	0.447695
$965$	−4.75044	−0.152922
$966$	−21.1966	−0.681991
$967$	−9.56791	−0.307683	−0.153842	−	0.988096i	$-0.549165\pi$
$967$	−9.56791	−0.307683	−0.153842	+	0.988096i	$0.549165\pi$
$968$	2.79803	0.0899320
$969$	0	0
$970$	49.5127	1.58976
$971$	−14.5828	−0.467985	−0.233992	−	0.972238i	$-0.575179\pi$
$971$	−14.5828	−0.467985	−0.233992	+	0.972238i	$0.575179\pi$
$972$	19.6621	0.630662
$973$	−16.3757	−0.524979
$974$	12.3178	0.394689
$975$	76.6688	2.45537
$976$	1.83856	0.0588508
$977$	−47.9851	−1.53518	−0.767590	−	0.640941i	$-0.778544\pi$
$977$	−47.9851	−1.53518	−0.767590	+	0.640941i	$0.778544\pi$
$978$	4.74506	0.151730
$979$	−14.9466	−0.477696
$980$	3.15124	0.100663
$981$	−30.0244	−0.958604
$982$	11.7203	0.374008
$983$	−36.8003	−1.17375	−0.586874	−	0.809679i	$-0.699641\pi$
$983$	−36.8003	−1.17375	−0.586874	+	0.809679i	$0.699641\pi$
$984$	−4.18041	−0.133267
$985$	32.5880	1.03834
$986$	13.1260	0.418015
$987$	11.9721	0.381077
$988$	0	0
$989$	54.0268	1.71795
$990$	20.8836	0.663723
$991$	−32.0191	−1.01712	−0.508560	−	0.861027i	$-0.669822\pi$
$991$	−32.0191	−1.01712	−0.508560	+	0.861027i	$0.669822\pi$
$992$	0.789075	0.0250532
$993$	−12.0398	−0.382073
$994$	6.92468	0.219637
$995$	−73.9879	−2.34557
$996$	18.4952	0.586044
$997$	−14.8172	−0.469264	−0.234632	−	0.972084i	$-0.575389\pi$
$997$	−14.8172	−0.469264	−0.234632	+	0.972084i	$0.575389\pi$
$998$	−0.815844	−0.0258251
$999$	13.6991	0.433420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bf.1.3	✓	8
19.18	odd	2		5054.2.a.bi.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.bf.1.3	✓	8	1.1	even	1	trivial
5054.2.a.bi.1.6	yes	8	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.30521 q^{3} +1.00000 q^{4} +3.15124 q^{5} +2.30521 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.31400 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.30521 q^{3} +1.00000 q^{4} +3.15124 q^{5} +2.30521 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.31400 q^{9} -3.15124 q^{10} -2.86391 q^{11} -2.30521 q^{12} -6.74578 q^{13} +1.00000 q^{14} -7.26428 q^{15} +1.00000 q^{16} -3.61968 q^{17} -2.31400 q^{18} +3.15124 q^{20} +2.30521 q^{21} +2.86391 q^{22} +9.19510 q^{23} +2.30521 q^{24} +4.93032 q^{25} +6.74578 q^{26} +1.58137 q^{27} -1.00000 q^{28} +3.62627 q^{29} +7.26428 q^{30} -0.789075 q^{31} -1.00000 q^{32} +6.60192 q^{33} +3.61968 q^{34} -3.15124 q^{35} +2.31400 q^{36} +8.66278 q^{37} +15.5504 q^{39} -3.15124 q^{40} -1.81346 q^{41} -2.30521 q^{42} +5.87561 q^{43} -2.86391 q^{44} +7.29198 q^{45} -9.19510 q^{46} +5.19351 q^{47} -2.30521 q^{48} +1.00000 q^{49} -4.93032 q^{50} +8.34413 q^{51} -6.74578 q^{52} -0.209579 q^{53} -1.58137 q^{54} -9.02487 q^{55} +1.00000 q^{56} -3.62627 q^{58} -13.1224 q^{59} -7.26428 q^{60} +1.83856 q^{61} +0.789075 q^{62} -2.31400 q^{63} +1.00000 q^{64} -21.2576 q^{65} -6.60192 q^{66} +15.0535 q^{67} -3.61968 q^{68} -21.1966 q^{69} +3.15124 q^{70} +6.92468 q^{71} -2.31400 q^{72} -6.67063 q^{73} -8.66278 q^{74} -11.3654 q^{75} +2.86391 q^{77} -15.5504 q^{78} +1.08776 q^{79} +3.15124 q^{80} -10.5874 q^{81} +1.81346 q^{82} -8.02323 q^{83} +2.30521 q^{84} -11.4065 q^{85} -5.87561 q^{86} -8.35933 q^{87} +2.86391 q^{88} +5.21896 q^{89} -7.29198 q^{90} +6.74578 q^{91} +9.19510 q^{92} +1.81899 q^{93} -5.19351 q^{94} +2.30521 q^{96} -15.7121 q^{97} -1.00000 q^{98} -6.62709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 - 6 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 8 * q^14 - 8 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 + 6 * q^20 + 4 * q^21 - 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 - 22 * q^27 - 8 * q^28 - 16 * q^29 + 8 * q^30 - 22 * q^31 - 8 * q^32 + 8 * q^33 - 2 * q^34 - 6 * q^35 + 8 * q^36 + 12 * q^37 + 8 * q^39 - 6 * q^40 - 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 - 20 * q^46 + 6 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 + 22 * q^54 + 18 * q^55 + 8 * q^56 + 16 * q^58 - 14 * q^59 - 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 - 32 * q^65 - 8 * q^66 + 20 * q^67 + 2 * q^68 - 40 * q^69 + 6 * q^70 - 8 * q^72 - 18 * q^73 - 12 * q^74 - 16 * q^75 - 4 * q^77 - 8 * q^78 - 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 + 4 * q^84 - 16 * q^85 + 28 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^90 + 6 * q^91 + 20 * q^92 + 16 * q^93 - 6 * q^94 + 4 * q^96 - 26 * q^97 - 8 * q^98 + 4 * q^99

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