LMFDB - Embedded newform 6045.2.a.bg.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 $ x^16 - 2x^15 - 27x^14 + 51x^13 + 294x^12 - 517x^11 - 1657x^10 + 2678x^9 + 5124x^8 - 7583x^7 - 8358x^6 + 11579x^5 + 6027x^4 - 8572x^3 - 780x^2 + 2178*x - 428
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.17778$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.17778 q^{2} -1.00000 q^{3} -0.612829 q^{4} -1.00000 q^{5} +1.17778 q^{6} -0.648351 q^{7} +3.07734 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.17778 q^{2} -1.00000 q^{3} -0.612829 q^{4} -1.00000 q^{5} +1.17778 q^{6} -0.648351 q^{7} +3.07734 q^{8} +1.00000 q^{9} +1.17778 q^{10} -3.99280 q^{11} +0.612829 q^{12} -1.00000 q^{13} +0.763616 q^{14} +1.00000 q^{15} -2.39878 q^{16} +3.70784 q^{17} -1.17778 q^{18} -2.56688 q^{19} +0.612829 q^{20} +0.648351 q^{21} +4.70265 q^{22} +5.59151 q^{23} -3.07734 q^{24} +1.00000 q^{25} +1.17778 q^{26} -1.00000 q^{27} +0.397328 q^{28} +0.446384 q^{29} -1.17778 q^{30} +1.00000 q^{31} -3.32944 q^{32} +3.99280 q^{33} -4.36702 q^{34} +0.648351 q^{35} -0.612829 q^{36} +0.136526 q^{37} +3.02322 q^{38} +1.00000 q^{39} -3.07734 q^{40} -11.4338 q^{41} -0.763616 q^{42} +0.971303 q^{43} +2.44690 q^{44} -1.00000 q^{45} -6.58558 q^{46} +2.09660 q^{47} +2.39878 q^{48} -6.57964 q^{49} -1.17778 q^{50} -3.70784 q^{51} +0.612829 q^{52} +8.06225 q^{53} +1.17778 q^{54} +3.99280 q^{55} -1.99520 q^{56} +2.56688 q^{57} -0.525744 q^{58} -3.19881 q^{59} -0.612829 q^{60} +5.68243 q^{61} -1.17778 q^{62} -0.648351 q^{63} +8.71893 q^{64} +1.00000 q^{65} -4.70265 q^{66} -2.48072 q^{67} -2.27227 q^{68} -5.59151 q^{69} -0.763616 q^{70} -14.3036 q^{71} +3.07734 q^{72} -15.0383 q^{73} -0.160798 q^{74} -1.00000 q^{75} +1.57306 q^{76} +2.58874 q^{77} -1.17778 q^{78} +13.5755 q^{79} +2.39878 q^{80} +1.00000 q^{81} +13.4665 q^{82} +4.65900 q^{83} -0.397328 q^{84} -3.70784 q^{85} -1.14398 q^{86} -0.446384 q^{87} -12.2872 q^{88} -6.58859 q^{89} +1.17778 q^{90} +0.648351 q^{91} -3.42664 q^{92} -1.00000 q^{93} -2.46934 q^{94} +2.56688 q^{95} +3.32944 q^{96} -4.82047 q^{97} +7.74939 q^{98} -3.99280 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * 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 $

$2$	−1.17778	−0.832818	−0.416409	−	0.909177i	$-0.636712\pi$
$2$	−1.17778	−0.832818	−0.416409	+	0.909177i	$0.636712\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.612829	−0.306414
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.17778	0.480828
$7$	−0.648351	−0.245054	−0.122527	−	0.992465i	$-0.539100\pi$
$7$	−0.648351	−0.245054	−0.122527	+	0.992465i	$0.539100\pi$
$8$	3.07734	1.08801
$9$	1.00000	0.333333
$10$	1.17778	0.372447
$11$	−3.99280	−1.20388	−0.601938	−	0.798543i	$-0.705605\pi$
$11$	−3.99280	−1.20388	−0.601938	+	0.798543i	$0.705605\pi$
$12$	0.612829	0.176908
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.763616	0.204085
$15$	1.00000	0.258199
$16$	−2.39878	−0.599696
$17$	3.70784	0.899282	0.449641	−	0.893209i	$-0.351552\pi$
$17$	3.70784	0.899282	0.449641	+	0.893209i	$0.351552\pi$
$18$	−1.17778	−0.277606
$19$	−2.56688	−0.588882	−0.294441	−	0.955670i	$-0.595133\pi$
$19$	−2.56688	−0.588882	−0.294441	+	0.955670i	$0.595133\pi$
$20$	0.612829	0.137033
$21$	0.648351	0.141482
$22$	4.70265	1.00261
$23$	5.59151	1.16591	0.582955	−	0.812504i	$-0.301896\pi$
$23$	5.59151	1.16591	0.582955	+	0.812504i	$0.301896\pi$
$24$	−3.07734	−0.628160
$25$	1.00000	0.200000
$26$	1.17778	0.230982
$27$	−1.00000	−0.192450
$28$	0.397328	0.0750879
$29$	0.446384	0.0828915	0.0414458	−	0.999141i	$-0.486804\pi$
$29$	0.446384	0.0828915	0.0414458	+	0.999141i	$0.486804\pi$
$30$	−1.17778	−0.215033
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.32944	−0.588568
$33$	3.99280	0.695058
$34$	−4.36702	−0.748938
$35$	0.648351	0.109591
$36$	−0.612829	−0.102138
$37$	0.136526	0.0224447	0.0112224	−	0.999937i	$-0.496428\pi$
$37$	0.136526	0.0224447	0.0112224	+	0.999937i	$0.496428\pi$
$38$	3.02322	0.490432
$39$	1.00000	0.160128
$40$	−3.07734	−0.486571
$41$	−11.4338	−1.78565	−0.892827	−	0.450400i	$-0.851281\pi$
$41$	−11.4338	−1.78565	−0.892827	+	0.450400i	$0.851281\pi$
$42$	−0.763616	−0.117829
$43$	0.971303	0.148122	0.0740612	−	0.997254i	$-0.476404\pi$
$43$	0.971303	0.148122	0.0740612	+	0.997254i	$0.476404\pi$
$44$	2.44690	0.368885
$45$	−1.00000	−0.149071
$46$	−6.58558	−0.970991
$47$	2.09660	0.305820	0.152910	−	0.988240i	$-0.451136\pi$
$47$	2.09660	0.305820	0.152910	+	0.988240i	$0.451136\pi$
$48$	2.39878	0.346235
$49$	−6.57964	−0.939949
$50$	−1.17778	−0.166564
$51$	−3.70784	−0.519201
$52$	0.612829	0.0849840
$53$	8.06225	1.10744	0.553718	−	0.832705i	$-0.313209\pi$
$53$	8.06225	1.10744	0.553718	+	0.832705i	$0.313209\pi$
$54$	1.17778	0.160276
$55$	3.99280	0.538390
$56$	−1.99520	−0.266620
$57$	2.56688	0.339991
$58$	−0.525744	−0.0690335
$59$	−3.19881	−0.416449	−0.208224	−	0.978081i	$-0.566768\pi$
$59$	−3.19881	−0.416449	−0.208224	+	0.978081i	$0.566768\pi$
$60$	−0.612829	−0.0791158
$61$	5.68243	0.727560	0.363780	−	0.931485i	$-0.381486\pi$
$61$	5.68243	0.727560	0.363780	+	0.931485i	$0.381486\pi$
$62$	−1.17778	−0.149579
$63$	−0.648351	−0.0816845
$64$	8.71893	1.08987
$65$	1.00000	0.124035
$66$	−4.70265	−0.578857
$67$	−2.48072	−0.303068	−0.151534	−	0.988452i	$-0.548421\pi$
$67$	−2.48072	−0.303068	−0.151534	+	0.988452i	$0.548421\pi$
$68$	−2.27227	−0.275553
$69$	−5.59151	−0.673138
$70$	−0.763616	−0.0912696
$71$	−14.3036	−1.69752	−0.848760	−	0.528778i	$-0.822650\pi$
$71$	−14.3036	−1.69752	−0.848760	+	0.528778i	$0.822650\pi$
$72$	3.07734	0.362668
$73$	−15.0383	−1.76010	−0.880048	−	0.474885i	$-0.842490\pi$
$73$	−15.0383	−1.76010	−0.880048	+	0.474885i	$0.842490\pi$
$74$	−0.160798	−0.0186924
$75$	−1.00000	−0.115470
$76$	1.57306	0.180442
$77$	2.58874	0.295014
$78$	−1.17778	−0.133358
$79$	13.5755	1.52736	0.763681	−	0.645594i	$-0.223390\pi$
$79$	13.5755	1.52736	0.763681	+	0.645594i	$0.223390\pi$
$80$	2.39878	0.268192
$81$	1.00000	0.111111
$82$	13.4665	1.48712
$83$	4.65900	0.511392	0.255696	−	0.966757i	$-0.417695\pi$
$83$	4.65900	0.511392	0.255696	+	0.966757i	$0.417695\pi$
$84$	−0.397328	−0.0433520
$85$	−3.70784	−0.402171
$86$	−1.14398	−0.123359
$87$	−0.446384	−0.0478574
$88$	−12.2872	−1.30982
$89$	−6.58859	−0.698389	−0.349194	−	0.937050i	$-0.613545\pi$
$89$	−6.58859	−0.698389	−0.349194	+	0.937050i	$0.613545\pi$
$90$	1.17778	0.124149
$91$	0.648351	0.0679656
$92$	−3.42664	−0.357252
$93$	−1.00000	−0.103695
$94$	−2.46934	−0.254692
$95$	2.56688	0.263356
$96$	3.32944	0.339810
$97$	−4.82047	−0.489445	−0.244722	−	0.969593i	$-0.578697\pi$
$97$	−4.82047	−0.489445	−0.244722	+	0.969593i	$0.578697\pi$
$98$	7.74939	0.782806
$99$	−3.99280	−0.401292
$100$	−0.612829	−0.0612829
$101$	11.8006	1.17421	0.587104	−	0.809512i	$-0.300268\pi$
$101$	11.8006	1.17421	0.587104	+	0.809512i	$0.300268\pi$
$102$	4.36702	0.432400
$103$	−17.1088	−1.68578	−0.842891	−	0.538085i	$-0.819148\pi$
$103$	−17.1088	−1.68578	−0.842891	+	0.538085i	$0.819148\pi$
$104$	−3.07734	−0.301758
$105$	−0.648351	−0.0632725
$106$	−9.49558	−0.922292
$107$	−15.6703	−1.51490	−0.757450	−	0.652893i	$-0.773555\pi$
$107$	−15.6703	−1.51490	−0.757450	+	0.652893i	$0.773555\pi$
$108$	0.612829	0.0589695
$109$	13.0698	1.25186	0.625929	−	0.779880i	$-0.284720\pi$
$109$	13.0698	1.25186	0.625929	+	0.779880i	$0.284720\pi$
$110$	−4.70265	−0.448381
$111$	−0.136526	−0.0129585
$112$	1.55525	0.146958
$113$	−11.3238	−1.06525	−0.532627	−	0.846350i	$-0.678795\pi$
$113$	−11.3238	−1.06525	−0.532627	+	0.846350i	$0.678795\pi$
$114$	−3.02322	−0.283151
$115$	−5.59151	−0.521411
$116$	−0.273557	−0.0253991
$117$	−1.00000	−0.0924500
$118$	3.76750	0.346826
$119$	−2.40398	−0.220372
$120$	3.07734	0.280922
$121$	4.94249	0.449317
$122$	−6.69266	−0.605925
$123$	11.4338	1.03095
$124$	−0.612829	−0.0550336
$125$	−1.00000	−0.0894427
$126$	0.763616	0.0680283
$127$	20.7868	1.84453	0.922263	−	0.386562i	$-0.126337\pi$
$127$	20.7868	1.84453	0.922263	+	0.386562i	$0.126337\pi$
$128$	−3.61011	−0.319092
$129$	−0.971303	−0.0855185
$130$	−1.17778	−0.103298
$131$	20.6954	1.80816	0.904082	−	0.427359i	$-0.140556\pi$
$131$	20.6954	1.80816	0.904082	+	0.427359i	$0.140556\pi$
$132$	−2.44690	−0.212976
$133$	1.66424	0.144308
$134$	2.92175	0.252400
$135$	1.00000	0.0860663
$136$	11.4103	0.978424
$137$	2.70713	0.231286	0.115643	−	0.993291i	$-0.463107\pi$
$137$	2.70713	0.231286	0.115643	+	0.993291i	$0.463107\pi$
$138$	6.58558	0.560602
$139$	11.4654	0.972487	0.486243	−	0.873823i	$-0.338367\pi$
$139$	11.4654	0.972487	0.486243	+	0.873823i	$0.338367\pi$
$140$	−0.397328	−0.0335803
$141$	−2.09660	−0.176565
$142$	16.8465	1.41373
$143$	3.99280	0.333895
$144$	−2.39878	−0.199899
$145$	−0.446384	−0.0370702
$146$	17.7118	1.46584
$147$	6.57964	0.542680
$148$	−0.0836670	−0.00687738
$149$	−18.7985	−1.54004	−0.770018	−	0.638023i	$-0.779753\pi$
$149$	−18.7985	−1.54004	−0.770018	+	0.638023i	$0.779753\pi$
$150$	1.17778	0.0961655
$151$	7.50794	0.610987	0.305494	−	0.952194i	$-0.401179\pi$
$151$	7.50794	0.610987	0.305494	+	0.952194i	$0.401179\pi$
$152$	−7.89917	−0.640707
$153$	3.70784	0.299761
$154$	−3.04897	−0.245693
$155$	−1.00000	−0.0803219
$156$	−0.612829	−0.0490656
$157$	−16.1990	−1.29282	−0.646411	−	0.762989i	$-0.723731\pi$
$157$	−16.1990	−1.29282	−0.646411	+	0.762989i	$0.723731\pi$
$158$	−15.9890	−1.27201
$159$	−8.06225	−0.639378
$160$	3.32944	0.263215
$161$	−3.62526	−0.285710
$162$	−1.17778	−0.0925353
$163$	−18.3692	−1.43878	−0.719392	−	0.694604i	$-0.755579\pi$
$163$	−18.3692	−1.43878	−0.719392	+	0.694604i	$0.755579\pi$
$164$	7.00694	0.547150
$165$	−3.99280	−0.310839
$166$	−5.48729	−0.425896
$167$	1.45419	0.112529	0.0562645	−	0.998416i	$-0.482081\pi$
$167$	1.45419	0.112529	0.0562645	+	0.998416i	$0.482081\pi$
$168$	1.99520	0.153933
$169$	1.00000	0.0769231
$170$	4.36702	0.334935
$171$	−2.56688	−0.196294
$172$	−0.595243	−0.0453868
$173$	−11.5414	−0.877473	−0.438736	−	0.898616i	$-0.644574\pi$
$173$	−11.5414	−0.877473	−0.438736	+	0.898616i	$0.644574\pi$
$174$	0.525744	0.0398565
$175$	−0.648351	−0.0490107
$176$	9.57787	0.721959
$177$	3.19881	0.240437
$178$	7.75992	0.581631
$179$	−14.6589	−1.09566	−0.547829	−	0.836591i	$-0.684545\pi$
$179$	−14.6589	−1.09566	−0.547829	+	0.836591i	$0.684545\pi$
$180$	0.612829	0.0456776
$181$	−12.9016	−0.958970	−0.479485	−	0.877550i	$-0.659177\pi$
$181$	−12.9016	−0.958970	−0.479485	+	0.877550i	$0.659177\pi$
$182$	−0.763616	−0.0566030
$183$	−5.68243	−0.420057
$184$	17.2070	1.26852
$185$	−0.136526	−0.0100376
$186$	1.17778	0.0863592
$187$	−14.8047	−1.08262
$188$	−1.28485	−0.0937077
$189$	0.648351	0.0471606
$190$	−3.02322	−0.219328
$191$	−24.9303	−1.80389	−0.901946	−	0.431848i	$-0.857862\pi$
$191$	−24.9303	−1.80389	−0.901946	+	0.431848i	$0.857862\pi$
$192$	−8.71893	−0.629234
$193$	10.4457	0.751900	0.375950	−	0.926640i	$-0.377316\pi$
$193$	10.4457	0.751900	0.375950	+	0.926640i	$0.377316\pi$
$194$	5.67747	0.407618
$195$	−1.00000	−0.0716115
$196$	4.03219	0.288014
$197$	−16.3547	−1.16522	−0.582612	−	0.812750i	$-0.697969\pi$
$197$	−16.3547	−1.16522	−0.582612	+	0.812750i	$0.697969\pi$
$198$	4.70265	0.334203
$199$	7.91588	0.561142	0.280571	−	0.959833i	$-0.409476\pi$
$199$	7.91588	0.561142	0.280571	+	0.959833i	$0.409476\pi$
$200$	3.07734	0.217601
$201$	2.48072	0.174976
$202$	−13.8986	−0.977901
$203$	−0.289414	−0.0203129
$204$	2.27227	0.159091
$205$	11.4338	0.798569
$206$	20.1505	1.40395
$207$	5.59151	0.388637
$208$	2.39878	0.166326
$209$	10.2490	0.708941
$210$	0.763616	0.0526945
$211$	13.6826	0.941948	0.470974	−	0.882147i	$-0.343902\pi$
$211$	13.6826	0.941948	0.470974	+	0.882147i	$0.343902\pi$
$212$	−4.94078	−0.339334
$213$	14.3036	0.980064
$214$	18.4561	1.26164
$215$	−0.971303	−0.0662423
$216$	−3.07734	−0.209387
$217$	−0.648351	−0.0440129
$218$	−15.3933	−1.04257
$219$	15.0383	1.01619
$220$	−2.44690	−0.164970
$221$	−3.70784	−0.249416
$222$	0.160798	0.0107920
$223$	6.79804	0.455230	0.227615	−	0.973751i	$-0.426907\pi$
$223$	6.79804	0.455230	0.227615	+	0.973751i	$0.426907\pi$
$224$	2.15865	0.144231
$225$	1.00000	0.0666667
$226$	13.3370	0.887163
$227$	10.7012	0.710263	0.355131	−	0.934816i	$-0.384436\pi$
$227$	10.7012	0.710263	0.355131	+	0.934816i	$0.384436\pi$
$228$	−1.57306	−0.104178
$229$	−0.850695	−0.0562155	−0.0281077	−	0.999605i	$-0.508948\pi$
$229$	−0.850695	−0.0562155	−0.0281077	+	0.999605i	$0.508948\pi$
$230$	6.58558	0.434240
$231$	−2.58874	−0.170326
$232$	1.37368	0.0901864
$233$	−5.53231	−0.362433	−0.181217	−	0.983443i	$-0.558004\pi$
$233$	−5.53231	−0.362433	−0.181217	+	0.983443i	$0.558004\pi$
$234$	1.17778	0.0769940
$235$	−2.09660	−0.136767
$236$	1.96032	0.127606
$237$	−13.5755	−0.881823
$238$	2.83136	0.183530
$239$	−15.1932	−0.982767	−0.491383	−	0.870943i	$-0.663509\pi$
$239$	−15.1932	−0.982767	−0.491383	+	0.870943i	$0.663509\pi$
$240$	−2.39878	−0.154841
$241$	−11.0779	−0.713589	−0.356794	−	0.934183i	$-0.616130\pi$
$241$	−11.0779	−0.713589	−0.356794	+	0.934183i	$0.616130\pi$
$242$	−5.82117	−0.374199
$243$	−1.00000	−0.0641500
$244$	−3.48235	−0.222935
$245$	6.57964	0.420358
$246$	−13.4665	−0.858592
$247$	2.56688	0.163327
$248$	3.07734	0.195412
$249$	−4.65900	−0.295252
$250$	1.17778	0.0744895
$251$	8.18659	0.516733	0.258367	−	0.966047i	$-0.416816\pi$
$251$	8.18659	0.516733	0.258367	+	0.966047i	$0.416816\pi$
$252$	0.397328	0.0250293
$253$	−22.3258	−1.40361
$254$	−24.4823	−1.53616
$255$	3.70784	0.232194
$256$	−13.1859	−0.824120
$257$	−8.11609	−0.506268	−0.253134	−	0.967431i	$-0.581461\pi$
$257$	−8.11609	−0.506268	−0.253134	+	0.967431i	$0.581461\pi$
$258$	1.14398	0.0712213
$259$	−0.0885167	−0.00550016
$260$	−0.612829	−0.0380060
$261$	0.446384	0.0276305
$262$	−24.3747	−1.50587
$263$	4.69962	0.289791	0.144896	−	0.989447i	$-0.453715\pi$
$263$	4.69962	0.289791	0.144896	+	0.989447i	$0.453715\pi$
$264$	12.2872	0.756227
$265$	−8.06225	−0.495260
$266$	−1.96011	−0.120182
$267$	6.58859	0.403215
$268$	1.52026	0.0928644
$269$	22.6749	1.38251	0.691257	−	0.722609i	$-0.257057\pi$
$269$	22.6749	1.38251	0.691257	+	0.722609i	$0.257057\pi$
$270$	−1.17778	−0.0716776
$271$	23.2353	1.41145	0.705723	−	0.708488i	$-0.250622\pi$
$271$	23.2353	1.41145	0.705723	+	0.708488i	$0.250622\pi$
$272$	−8.89430	−0.539296
$273$	−0.648351	−0.0392400
$274$	−3.18841	−0.192619
$275$	−3.99280	−0.240775
$276$	3.42664	0.206259
$277$	25.7234	1.54557	0.772783	−	0.634670i	$-0.218864\pi$
$277$	25.7234	1.54557	0.772783	+	0.634670i	$0.218864\pi$
$278$	−13.5038	−0.809904
$279$	1.00000	0.0598684
$280$	1.99520	0.119236
$281$	−27.7608	−1.65607	−0.828036	−	0.560675i	$-0.810542\pi$
$281$	−27.7608	−1.65607	−0.828036	+	0.560675i	$0.810542\pi$
$282$	2.46934	0.147047
$283$	−16.1395	−0.959396	−0.479698	−	0.877434i	$-0.659254\pi$
$283$	−16.1395	−0.959396	−0.479698	+	0.877434i	$0.659254\pi$
$284$	8.76563	0.520144
$285$	−2.56688	−0.152049
$286$	−4.70265	−0.278074
$287$	7.41309	0.437581
$288$	−3.32944	−0.196189
$289$	−3.25196	−0.191292
$290$	0.525744	0.0308727
$291$	4.82047	0.282581
$292$	9.21588	0.539319
$293$	22.3062	1.30314	0.651570	−	0.758589i	$-0.274111\pi$
$293$	22.3062	1.30314	0.651570	+	0.758589i	$0.274111\pi$
$294$	−7.74939	−0.451953
$295$	3.19881	0.186242
$296$	0.420137	0.0244200
$297$	3.99280	0.231686
$298$	22.1406	1.28257
$299$	−5.59151	−0.323365
$300$	0.612829	0.0353817
$301$	−0.629745	−0.0362979
$302$	−8.84272	−0.508841
$303$	−11.8006	−0.677929
$304$	6.15739	0.353150
$305$	−5.68243	−0.325375
$306$	−4.36702	−0.249646
$307$	33.6271	1.91920	0.959599	−	0.281371i	$-0.0907891\pi$
$307$	33.6271	1.91920	0.959599	+	0.281371i	$0.0907891\pi$
$308$	−1.58645	−0.0903965
$309$	17.1088	0.973286
$310$	1.17778	0.0668935
$311$	22.6224	1.28280	0.641399	−	0.767207i	$-0.278354\pi$
$311$	22.6224	1.28280	0.641399	+	0.767207i	$0.278354\pi$
$312$	3.07734	0.174220
$313$	30.6107	1.73022	0.865111	−	0.501581i	$-0.167248\pi$
$313$	30.6107	1.73022	0.865111	+	0.501581i	$0.167248\pi$
$314$	19.0789	1.07669
$315$	0.648351	0.0365304
$316$	−8.31945	−0.468006
$317$	21.4198	1.20305	0.601527	−	0.798853i	$-0.294559\pi$
$317$	21.4198	1.20305	0.601527	+	0.798853i	$0.294559\pi$
$318$	9.49558	0.532485
$319$	−1.78233	−0.0997911
$320$	−8.71893	−0.487403
$321$	15.6703	0.874628
$322$	4.26976	0.237945
$323$	−9.51756	−0.529571
$324$	−0.612829	−0.0340460
$325$	−1.00000	−0.0554700
$326$	21.6349	1.19824
$327$	−13.0698	−0.722760
$328$	−35.1856	−1.94280
$329$	−1.35933	−0.0749423
$330$	4.70265	0.258873
$331$	10.2001	0.560648	0.280324	−	0.959905i	$-0.409558\pi$
$331$	10.2001	0.560648	0.280324	+	0.959905i	$0.409558\pi$
$332$	−2.85517	−0.156698
$333$	0.136526	0.00748157
$334$	−1.71272	−0.0937161
$335$	2.48072	0.135536
$336$	−1.55525	−0.0848460
$337$	−1.82468	−0.0993965	−0.0496982	−	0.998764i	$-0.515826\pi$
$337$	−1.82468	−0.0993965	−0.0496982	+	0.998764i	$0.515826\pi$
$338$	−1.17778	−0.0640629
$339$	11.3238	0.615025
$340$	2.27227	0.123231
$341$	−3.99280	−0.216222
$342$	3.02322	0.163477
$343$	8.80437	0.475391
$344$	2.98903	0.161158
$345$	5.59151	0.301037
$346$	13.5932	0.730775
$347$	23.9164	1.28390	0.641950	−	0.766747i	$-0.278126\pi$
$347$	23.9164	1.28390	0.641950	+	0.766747i	$0.278126\pi$
$348$	0.273557	0.0146642
$349$	33.5649	1.79669	0.898343	−	0.439294i	$-0.144772\pi$
$349$	33.5649	1.79669	0.898343	+	0.439294i	$0.144772\pi$
$350$	0.763616	0.0408170
$351$	1.00000	0.0533761
$352$	13.2938	0.708562
$353$	−25.0080	−1.33104	−0.665522	−	0.746379i	$-0.731791\pi$
$353$	−25.0080	−1.33104	−0.665522	+	0.746379i	$0.731791\pi$
$354$	−3.76750	−0.200240
$355$	14.3036	0.759154
$356$	4.03768	0.213996
$357$	2.40398	0.127232
$358$	17.2650	0.912483
$359$	13.6876	0.722401	0.361201	−	0.932488i	$-0.382367\pi$
$359$	13.6876	0.722401	0.361201	+	0.932488i	$0.382367\pi$
$360$	−3.07734	−0.162190
$361$	−12.4111	−0.653218
$362$	15.1953	0.798647
$363$	−4.94249	−0.259413
$364$	−0.397328	−0.0208256
$365$	15.0383	0.787139
$366$	6.69266	0.349831
$367$	12.8963	0.673180	0.336590	−	0.941651i	$-0.390726\pi$
$367$	12.8963	0.673180	0.336590	+	0.941651i	$0.390726\pi$
$368$	−13.4128	−0.699191
$369$	−11.4338	−0.595218
$370$	0.160798	0.00835948
$371$	−5.22717	−0.271381
$372$	0.612829	0.0317737
$373$	1.40066	0.0725236	0.0362618	−	0.999342i	$-0.488455\pi$
$373$	1.40066	0.0725236	0.0362618	+	0.999342i	$0.488455\pi$
$374$	17.4367	0.901629
$375$	1.00000	0.0516398
$376$	6.45195	0.332734
$377$	−0.446384	−0.0229900
$378$	−0.763616	−0.0392762
$379$	−25.0311	−1.28576	−0.642882	−	0.765966i	$-0.722261\pi$
$379$	−25.0311	−1.28576	−0.642882	+	0.765966i	$0.722261\pi$
$380$	−1.57306	−0.0806961
$381$	−20.7868	−1.06494
$382$	29.3625	1.50231
$383$	17.6468	0.901711	0.450855	−	0.892597i	$-0.351119\pi$
$383$	17.6468	0.901711	0.450855	+	0.892597i	$0.351119\pi$
$384$	3.61011	0.184228
$385$	−2.58874	−0.131934
$386$	−12.3028	−0.626196
$387$	0.971303	0.0493741
$388$	2.95412	0.149973
$389$	28.3881	1.43933	0.719667	−	0.694319i	$-0.244294\pi$
$389$	28.3881	1.43933	0.719667	+	0.694319i	$0.244294\pi$
$390$	1.17778	0.0596393
$391$	20.7324	1.04848
$392$	−20.2478	−1.02267
$393$	−20.6954	−1.04394
$394$	19.2623	0.970420
$395$	−13.5755	−0.683057
$396$	2.44690	0.122962
$397$	−3.52259	−0.176794	−0.0883969	−	0.996085i	$-0.528174\pi$
$397$	−3.52259	−0.176794	−0.0883969	+	0.996085i	$0.528174\pi$
$398$	−9.32318	−0.467329
$399$	−1.66424	−0.0833161
$400$	−2.39878	−0.119939
$401$	−4.31997	−0.215729	−0.107865	−	0.994166i	$-0.534401\pi$
$401$	−4.31997	−0.215729	−0.107865	+	0.994166i	$0.534401\pi$
$402$	−2.92175	−0.145723
$403$	−1.00000	−0.0498135
$404$	−7.23177	−0.359794
$405$	−1.00000	−0.0496904
$406$	0.340866	0.0169169
$407$	−0.545121	−0.0270207
$408$	−11.4103	−0.564893
$409$	19.0528	0.942103	0.471051	−	0.882106i	$-0.343875\pi$
$409$	19.0528	0.942103	0.471051	+	0.882106i	$0.343875\pi$
$410$	−13.4665	−0.665062
$411$	−2.70713	−0.133533
$412$	10.4848	0.516548
$413$	2.07395	0.102052
$414$	−6.58558	−0.323664
$415$	−4.65900	−0.228701
$416$	3.32944	0.163239
$417$	−11.4654	−0.561465
$418$	−12.0711	−0.590419
$419$	29.1473	1.42394	0.711968	−	0.702211i	$-0.247804\pi$
$419$	29.1473	1.42394	0.711968	+	0.702211i	$0.247804\pi$
$420$	0.397328	0.0193876
$421$	28.8866	1.40784	0.703922	−	0.710277i	$-0.251430\pi$
$421$	28.8866	1.40784	0.703922	+	0.710277i	$0.251430\pi$
$422$	−16.1151	−0.784471
$423$	2.09660	0.101940
$424$	24.8103	1.20490
$425$	3.70784	0.179856
$426$	−16.8465	−0.816215
$427$	−3.68420	−0.178291
$428$	9.60318	0.464187
$429$	−3.99280	−0.192774
$430$	1.14398	0.0551678
$431$	−14.6285	−0.704629	−0.352314	−	0.935882i	$-0.614605\pi$
$431$	−14.6285	−0.704629	−0.352314	+	0.935882i	$0.614605\pi$
$432$	2.39878	0.115412
$433$	−21.3793	−1.02742	−0.513712	−	0.857963i	$-0.671730\pi$
$433$	−21.3793	−1.02742	−0.513712	+	0.857963i	$0.671730\pi$
$434$	0.763616	0.0366547
$435$	0.446384	0.0214025
$436$	−8.00953	−0.383587
$437$	−14.3527	−0.686584
$438$	−17.7118	−0.846303
$439$	32.0017	1.52736	0.763678	−	0.645598i	$-0.223392\pi$
$439$	32.0017	1.52736	0.763678	+	0.645598i	$0.223392\pi$
$440$	12.2872	0.585771
$441$	−6.57964	−0.313316
$442$	4.36702	0.207718
$443$	−22.3522	−1.06198	−0.530991	−	0.847377i	$-0.678180\pi$
$443$	−22.3522	−1.06198	−0.530991	+	0.847377i	$0.678180\pi$
$444$	0.0836670	0.00397066
$445$	6.58859	0.312329
$446$	−8.00661	−0.379124
$447$	18.7985	0.889140
$448$	−5.65292	−0.267075
$449$	−14.0831	−0.664620	−0.332310	−	0.943170i	$-0.607828\pi$
$449$	−14.0831	−0.664620	−0.332310	+	0.943170i	$0.607828\pi$
$450$	−1.17778	−0.0555212
$451$	45.6528	2.14971
$452$	6.93955	0.326409
$453$	−7.50794	−0.352754
$454$	−12.6037	−0.591519
$455$	−0.648351	−0.0303951
$456$	7.89917	0.369912
$457$	36.2553	1.69595	0.847976	−	0.530034i	$-0.177821\pi$
$457$	36.2553	1.69595	0.847976	+	0.530034i	$0.177821\pi$
$458$	1.00193	0.0468173
$459$	−3.70784	−0.173067
$460$	3.42664	0.159768
$461$	2.77994	0.129475	0.0647374	−	0.997902i	$-0.479379\pi$
$461$	2.77994	0.129475	0.0647374	+	0.997902i	$0.479379\pi$
$462$	3.04897	0.141851
$463$	−6.33850	−0.294575	−0.147287	−	0.989094i	$-0.547054\pi$
$463$	−6.33850	−0.294575	−0.147287	+	0.989094i	$0.547054\pi$
$464$	−1.07078	−0.0497097
$465$	1.00000	0.0463739
$466$	6.51585	0.301841
$467$	−14.5800	−0.674682	−0.337341	−	0.941382i	$-0.609528\pi$
$467$	−14.5800	−0.674682	−0.337341	+	0.941382i	$0.609528\pi$
$468$	0.612829	0.0283280
$469$	1.60838	0.0742679
$470$	2.46934	0.113902
$471$	16.1990	0.746411
$472$	−9.84382	−0.453099
$473$	−3.87822	−0.178321
$474$	15.9890	0.734398
$475$	−2.56688	−0.117776
$476$	1.47323	0.0675252
$477$	8.06225	0.369145
$478$	17.8943	0.818466
$479$	11.7767	0.538092	0.269046	−	0.963127i	$-0.413292\pi$
$479$	11.7767	0.538092	0.269046	+	0.963127i	$0.413292\pi$
$480$	−3.32944	−0.151968
$481$	−0.136526	−0.00622505
$482$	13.0473	0.594290
$483$	3.62526	0.164955
$484$	−3.02890	−0.137677
$485$	4.82047	0.218886
$486$	1.17778	0.0534253
$487$	18.4544	0.836248	0.418124	−	0.908390i	$-0.362688\pi$
$487$	18.4544	0.836248	0.418124	+	0.908390i	$0.362688\pi$
$488$	17.4868	0.791589
$489$	18.3692	0.830682
$490$	−7.74939	−0.350082
$491$	13.1097	0.591634	0.295817	−	0.955245i	$-0.404408\pi$
$491$	13.1097	0.591634	0.295817	+	0.955245i	$0.404408\pi$
$492$	−7.00694	−0.315897
$493$	1.65512	0.0745429
$494$	−3.02322	−0.136021
$495$	3.99280	0.179463
$496$	−2.39878	−0.107709
$497$	9.27372	0.415983
$498$	5.48729	0.245891
$499$	−8.70670	−0.389765	−0.194883	−	0.980827i	$-0.562433\pi$
$499$	−8.70670	−0.389765	−0.194883	+	0.980827i	$0.562433\pi$
$500$	0.612829	0.0274065
$501$	−1.45419	−0.0649686
$502$	−9.64202	−0.430345
$503$	3.96398	0.176745	0.0883727	−	0.996087i	$-0.471833\pi$
$503$	3.96398	0.176745	0.0883727	+	0.996087i	$0.471833\pi$
$504$	−1.99520	−0.0888732
$505$	−11.8006	−0.525122
$506$	26.2949	1.16895
$507$	−1.00000	−0.0444116
$508$	−12.7387	−0.565189
$509$	14.6281	0.648378	0.324189	−	0.945992i	$-0.394909\pi$
$509$	14.6281	0.648378	0.324189	+	0.945992i	$0.394909\pi$
$510$	−4.36702	−0.193375
$511$	9.75007	0.431318
$512$	22.7504	1.00543
$513$	2.56688	0.113330
$514$	9.55899	0.421629
$515$	17.1088	0.753904
$516$	0.595243	0.0262041
$517$	−8.37130	−0.368169
$518$	0.104253	0.00458063
$519$	11.5414	0.506609
$520$	3.07734	0.134950
$521$	−9.93295	−0.435171	−0.217585	−	0.976041i	$-0.569818\pi$
$521$	−9.93295	−0.435171	−0.217585	+	0.976041i	$0.569818\pi$
$522$	−0.525744	−0.0230112
$523$	−37.9250	−1.65834	−0.829171	−	0.558994i	$-0.811187\pi$
$523$	−37.9250	−1.65834	−0.829171	+	0.558994i	$0.811187\pi$
$524$	−12.6827	−0.554047
$525$	0.648351	0.0282963
$526$	−5.53513	−0.241343
$527$	3.70784	0.161516
$528$	−9.57787	−0.416823
$529$	8.26496	0.359346
$530$	9.49558	0.412461
$531$	−3.19881	−0.138816
$532$	−1.01989	−0.0442179
$533$	11.4338	0.495251
$534$	−7.75992	−0.335805
$535$	15.6703	0.677484
$536$	−7.63402	−0.329740
$537$	14.6589	0.632578
$538$	−26.7061	−1.15138
$539$	26.2712	1.13158
$540$	−0.612829	−0.0263719
$541$	37.0362	1.59231	0.796155	−	0.605092i	$-0.206864\pi$
$541$	37.0362	1.59231	0.796155	+	0.605092i	$0.206864\pi$
$542$	−27.3662	−1.17548
$543$	12.9016	0.553662
$544$	−12.3450	−0.529289
$545$	−13.0698	−0.559848
$546$	0.763616	0.0326797
$547$	−20.9579	−0.896093	−0.448047	−	0.894010i	$-0.647880\pi$
$547$	−20.9579	−0.896093	−0.448047	+	0.894010i	$0.647880\pi$
$548$	−1.65901	−0.0708693
$549$	5.68243	0.242520
$550$	4.70265	0.200522
$551$	−1.14581	−0.0488133
$552$	−17.2070	−0.732378
$553$	−8.80168	−0.374286
$554$	−30.2965	−1.28718
$555$	0.136526	0.00579520
$556$	−7.02636	−0.297984
$557$	−7.38300	−0.312828	−0.156414	−	0.987692i	$-0.549993\pi$
$557$	−7.38300	−0.312828	−0.156414	+	0.987692i	$0.549993\pi$
$558$	−1.17778	−0.0498595
$559$	−0.971303	−0.0410818
$560$	−1.55525	−0.0657214
$561$	14.8047	0.625053
$562$	32.6962	1.37921
$563$	19.0287	0.801966	0.400983	−	0.916086i	$-0.368669\pi$
$563$	19.0287	0.801966	0.400983	+	0.916086i	$0.368669\pi$
$564$	1.28485	0.0541022
$565$	11.3238	0.476396
$566$	19.0089	0.799002
$567$	−0.648351	−0.0272282
$568$	−44.0170	−1.84691
$569$	−3.93694	−0.165045	−0.0825225	−	0.996589i	$-0.526298\pi$
$569$	−3.93694	−0.165045	−0.0825225	+	0.996589i	$0.526298\pi$
$570$	3.02322	0.126629
$571$	36.7268	1.53697	0.768484	−	0.639869i	$-0.221011\pi$
$571$	36.7268	1.53697	0.768484	+	0.639869i	$0.221011\pi$
$572$	−2.44690	−0.102310
$573$	24.9303	1.04148
$574$	−8.73101	−0.364425
$575$	5.59151	0.233182
$576$	8.71893	0.363289
$577$	−10.3041	−0.428966	−0.214483	−	0.976728i	$-0.568807\pi$
$577$	−10.3041	−0.428966	−0.214483	+	0.976728i	$0.568807\pi$
$578$	3.83010	0.159311
$579$	−10.4457	−0.434110
$580$	0.273557	0.0113588
$581$	−3.02067	−0.125318
$582$	−5.67747	−0.235338
$583$	−32.1910	−1.33321
$584$	−46.2779	−1.91499
$585$	1.00000	0.0413449
$586$	−26.2718	−1.08528
$587$	31.1295	1.28485	0.642425	−	0.766348i	$-0.277928\pi$
$587$	31.1295	1.28485	0.642425	+	0.766348i	$0.277928\pi$
$588$	−4.03219	−0.166285
$589$	−2.56688	−0.105766
$590$	−3.76750	−0.155105
$591$	16.3547	0.672743
$592$	−0.327496	−0.0134600
$593$	−21.9867	−0.902886	−0.451443	−	0.892300i	$-0.649091\pi$
$593$	−21.9867	−0.902886	−0.451443	+	0.892300i	$0.649091\pi$
$594$	−4.70265	−0.192952
$595$	2.40398	0.0985535
$596$	11.5203	0.471889
$597$	−7.91588	−0.323975
$598$	6.58558	0.269304
$599$	32.7779	1.33927	0.669635	−	0.742690i	$-0.266451\pi$
$599$	32.7779	1.33927	0.669635	+	0.742690i	$0.266451\pi$
$600$	−3.07734	−0.125632
$601$	2.83988	0.115841	0.0579206	−	0.998321i	$-0.481553\pi$
$601$	2.83988	0.115841	0.0579206	+	0.998321i	$0.481553\pi$
$602$	0.741703	0.0302296
$603$	−2.48072	−0.101023
$604$	−4.60108	−0.187215
$605$	−4.94249	−0.200941
$606$	13.8986	0.564591
$607$	25.5640	1.03761	0.518806	−	0.854892i	$-0.326377\pi$
$607$	25.5640	1.03761	0.518806	+	0.854892i	$0.326377\pi$
$608$	8.54627	0.346597
$609$	0.289414	0.0117276
$610$	6.69266	0.270978
$611$	−2.09660	−0.0848192
$612$	−2.27227	−0.0918510
$613$	−29.7236	−1.20052	−0.600262	−	0.799804i	$-0.704937\pi$
$613$	−29.7236	−1.20052	−0.600262	+	0.799804i	$0.704937\pi$
$614$	−39.6054	−1.59834
$615$	−11.4338	−0.461054
$616$	7.96643	0.320977
$617$	29.0913	1.17117	0.585585	−	0.810611i	$-0.300865\pi$
$617$	29.0913	1.17117	0.585585	+	0.810611i	$0.300865\pi$
$618$	−20.1505	−0.810570
$619$	26.0692	1.04781	0.523904	−	0.851777i	$-0.324475\pi$
$619$	26.0692	1.04781	0.523904	+	0.851777i	$0.324475\pi$
$620$	0.612829	0.0246118
$621$	−5.59151	−0.224379
$622$	−26.6443	−1.06834
$623$	4.27172	0.171143
$624$	−2.39878	−0.0960282
$625$	1.00000	0.0400000
$626$	−36.0528	−1.44096
$627$	−10.2490	−0.409307
$628$	9.92722	0.396139
$629$	0.506216	0.0201841
$630$	−0.763616	−0.0304232
$631$	42.3666	1.68659	0.843295	−	0.537452i	$-0.180613\pi$
$631$	42.3666	1.68659	0.843295	+	0.537452i	$0.180613\pi$
$632$	41.7765	1.66178
$633$	−13.6826	−0.543834
$634$	−25.2278	−1.00192
$635$	−20.7868	−0.824898
$636$	4.94078	0.195915
$637$	6.57964	0.260695
$638$	2.09919	0.0831078
$639$	−14.3036	−0.565840
$640$	3.61011	0.142702
$641$	19.5855	0.773582	0.386791	−	0.922167i	$-0.373583\pi$
$641$	19.5855	0.773582	0.386791	+	0.922167i	$0.373583\pi$
$642$	−18.4561	−0.728406
$643$	40.7636	1.60756	0.803780	−	0.594927i	$-0.202819\pi$
$643$	40.7636	1.60756	0.803780	+	0.594927i	$0.202819\pi$
$644$	2.22166	0.0875457
$645$	0.971303	0.0382450
$646$	11.2096	0.441036
$647$	21.4228	0.842218	0.421109	−	0.907010i	$-0.361641\pi$
$647$	21.4228	0.842218	0.421109	+	0.907010i	$0.361641\pi$
$648$	3.07734	0.120889
$649$	12.7722	0.501353
$650$	1.17778	0.0461964
$651$	0.648351	0.0254109
$652$	11.2571	0.440864
$653$	−39.4143	−1.54240	−0.771201	−	0.636591i	$-0.780344\pi$
$653$	−39.4143	−1.54240	−0.771201	+	0.636591i	$0.780344\pi$
$654$	15.3933	0.601927
$655$	−20.6954	−0.808635
$656$	27.4271	1.07085
$657$	−15.0383	−0.586699
$658$	1.60100	0.0624133
$659$	−38.9681	−1.51798	−0.758992	−	0.651100i	$-0.774308\pi$
$659$	−38.9681	−1.51798	−0.758992	+	0.651100i	$0.774308\pi$
$660$	2.44690	0.0952456
$661$	−22.1634	−0.862055	−0.431027	−	0.902339i	$-0.641849\pi$
$661$	−22.1634	−0.862055	−0.431027	+	0.902339i	$0.641849\pi$
$662$	−12.0135	−0.466917
$663$	3.70784	0.144000
$664$	14.3373	0.556397
$665$	−1.66424	−0.0645363
$666$	−0.160798	−0.00623079
$667$	2.49596	0.0966440
$668$	−0.891172	−0.0344805
$669$	−6.79804	−0.262827
$670$	−2.92175	−0.112877
$671$	−22.6888	−0.875892
$672$	−2.15865	−0.0832716
$673$	−47.1948	−1.81923	−0.909613	−	0.415458i	$-0.863621\pi$
$673$	−47.1948	−1.81923	−0.909613	+	0.415458i	$0.863621\pi$
$674$	2.14907	0.0827792
$675$	−1.00000	−0.0384900
$676$	−0.612829	−0.0235703
$677$	−40.7291	−1.56535	−0.782673	−	0.622433i	$-0.786144\pi$
$677$	−40.7291	−1.56535	−0.782673	+	0.622433i	$0.786144\pi$
$678$	−13.3370	−0.512204
$679$	3.12536	0.119940
$680$	−11.4103	−0.437564
$681$	−10.7012	−0.410070
$682$	4.70265	0.180074
$683$	−25.3552	−0.970189	−0.485095	−	0.874462i	$-0.661215\pi$
$683$	−25.3552	−0.970189	−0.485095	+	0.874462i	$0.661215\pi$
$684$	1.57306	0.0601473
$685$	−2.70713	−0.103434
$686$	−10.3696	−0.395914
$687$	0.850695	0.0324560
$688$	−2.32995	−0.0888284
$689$	−8.06225	−0.307147
$690$	−6.58558	−0.250709
$691$	9.43310	0.358852	0.179426	−	0.983771i	$-0.442576\pi$
$691$	9.43310	0.358852	0.179426	+	0.983771i	$0.442576\pi$
$692$	7.07287	0.268870
$693$	2.58874	0.0983380
$694$	−28.1683	−1.06925
$695$	−11.4654	−0.434909
$696$	−1.37368	−0.0520691
$697$	−42.3945	−1.60581
$698$	−39.5321	−1.49631
$699$	5.53231	0.209251
$700$	0.397328	0.0150176
$701$	25.0308	0.945400	0.472700	−	0.881223i	$-0.343279\pi$
$701$	25.0308	0.945400	0.472700	+	0.881223i	$0.343279\pi$
$702$	−1.17778	−0.0444525
$703$	−0.350445	−0.0132173
$704$	−34.8130	−1.31206
$705$	2.09660	0.0789624
$706$	29.4540	1.10852
$707$	−7.65095	−0.287744
$708$	−1.96032	−0.0736733
$709$	48.9254	1.83743	0.918716	−	0.394919i	$-0.129227\pi$
$709$	48.9254	1.83743	0.918716	+	0.394919i	$0.129227\pi$
$710$	−16.8465	−0.632237
$711$	13.5755	0.509121
$712$	−20.2753	−0.759851
$713$	5.59151	0.209404
$714$	−2.83136	−0.105961
$715$	−3.99280	−0.149322
$716$	8.98339	0.335725
$717$	15.1932	0.567401
$718$	−16.1210	−0.601629
$719$	−48.8407	−1.82145	−0.910725	−	0.413013i	$-0.864477\pi$
$719$	−48.8407	−1.82145	−0.910725	+	0.413013i	$0.864477\pi$
$720$	2.39878	0.0893974
$721$	11.0925	0.413107
$722$	14.6176	0.544011
$723$	11.0779	0.411991
$724$	7.90648	0.293842
$725$	0.446384	0.0165783
$726$	5.82117	0.216044
$727$	34.0269	1.26199	0.630994	−	0.775787i	$-0.282647\pi$
$727$	34.0269	1.26199	0.630994	+	0.775787i	$0.282647\pi$
$728$	1.99520	0.0739470
$729$	1.00000	0.0370370
$730$	−17.7118	−0.655543
$731$	3.60143	0.133204
$732$	3.48235	0.128711
$733$	24.6946	0.912115	0.456057	−	0.889950i	$-0.349261\pi$
$733$	24.6946	0.912115	0.456057	+	0.889950i	$0.349261\pi$
$734$	−15.1890	−0.560636
$735$	−6.57964	−0.242694
$736$	−18.6166	−0.686217
$737$	9.90502	0.364856
$738$	13.4665	0.495708
$739$	34.1451	1.25605	0.628024	−	0.778194i	$-0.283864\pi$
$739$	34.1451	1.25605	0.628024	+	0.778194i	$0.283864\pi$
$740$	0.0836670	0.00307566
$741$	−2.56688	−0.0942966
$742$	6.15646	0.226011
$743$	−27.1442	−0.995822	−0.497911	−	0.867228i	$-0.665899\pi$
$743$	−27.1442	−0.995822	−0.497911	+	0.867228i	$0.665899\pi$
$744$	−3.07734	−0.112821
$745$	18.7985	0.688725
$746$	−1.64968	−0.0603990
$747$	4.65900	0.170464
$748$	9.07272	0.331732
$749$	10.1598	0.371232
$750$	−1.17778	−0.0430065
$751$	−26.7332	−0.975507	−0.487754	−	0.872981i	$-0.662184\pi$
$751$	−26.7332	−0.975507	−0.487754	+	0.872981i	$0.662184\pi$
$752$	−5.02928	−0.183399
$753$	−8.18659	−0.298336
$754$	0.525744	0.0191465
$755$	−7.50794	−0.273242
$756$	−0.397328	−0.0144507
$757$	8.71369	0.316704	0.158352	−	0.987383i	$-0.449382\pi$
$757$	8.71369	0.316704	0.158352	+	0.987383i	$0.449382\pi$
$758$	29.4812	1.07081
$759$	22.3258	0.810375
$760$	7.89917	0.286533
$761$	−9.01951	−0.326957	−0.163479	−	0.986547i	$-0.552271\pi$
$761$	−9.01951	−0.326957	−0.163479	+	0.986547i	$0.552271\pi$
$762$	24.4823	0.886900
$763$	−8.47380	−0.306772
$764$	15.2780	0.552739
$765$	−3.70784	−0.134057
$766$	−20.7841	−0.750961
$767$	3.19881	0.115502
$768$	13.1859	0.475806
$769$	29.7747	1.07370	0.536852	−	0.843677i	$-0.319613\pi$
$769$	29.7747	1.07370	0.536852	+	0.843677i	$0.319613\pi$
$770$	3.04897	0.109877
$771$	8.11609	0.292294
$772$	−6.40144	−0.230393
$773$	−29.7040	−1.06838	−0.534189	−	0.845365i	$-0.679383\pi$
$773$	−29.7040	−1.06838	−0.534189	+	0.845365i	$0.679383\pi$
$774$	−1.14398	−0.0411197
$775$	1.00000	0.0359211
$776$	−14.8342	−0.532518
$777$	0.0885167	0.00317552
$778$	−33.4350	−1.19870
$779$	29.3491	1.05154
$780$	0.612829	0.0219428
$781$	57.1113	2.04360
$782$	−24.4182	−0.873195
$783$	−0.446384	−0.0159525
$784$	15.7831	0.563683
$785$	16.1990	0.578168
$786$	24.3747	0.869415
$787$	−11.6960	−0.416918	−0.208459	−	0.978031i	$-0.566845\pi$
$787$	−11.6960	−0.416918	−0.208459	+	0.978031i	$0.566845\pi$
$788$	10.0226	0.357042
$789$	−4.69962	−0.167311
$790$	15.9890	0.568862
$791$	7.34180	0.261044
$792$	−12.2872	−0.436608
$793$	−5.68243	−0.201789
$794$	4.14885	0.147237
$795$	8.06225	0.285939
$796$	−4.85108	−0.171942
$797$	21.0823	0.746773	0.373387	−	0.927676i	$-0.378197\pi$
$797$	21.0823	0.746773	0.373387	+	0.927676i	$0.378197\pi$
$798$	1.96011	0.0693871
$799$	7.77384	0.275019
$800$	−3.32944	−0.117714
$801$	−6.58859	−0.232796
$802$	5.08799	0.179663
$803$	60.0449	2.11894
$804$	−1.52026	−0.0536153
$805$	3.62526	0.127774
$806$	1.17778	0.0414856
$807$	−22.6749	−0.798195
$808$	36.3146	1.27754
$809$	−29.8063	−1.04793	−0.523967	−	0.851738i	$-0.675549\pi$
$809$	−29.8063	−1.04793	−0.523967	+	0.851738i	$0.675549\pi$
$810$	1.17778	0.0413831
$811$	14.5146	0.509676	0.254838	−	0.966984i	$-0.417978\pi$
$811$	14.5146	0.509676	0.254838	+	0.966984i	$0.417978\pi$
$812$	0.177361	0.00622415
$813$	−23.2353	−0.814899
$814$	0.642034	0.0225033
$815$	18.3692	0.643444
$816$	8.89430	0.311363
$817$	−2.49322	−0.0872266
$818$	−22.4401	−0.784600
$819$	0.648351	0.0226552
$820$	−7.00694	−0.244693
$821$	−32.3524	−1.12911	−0.564553	−	0.825397i	$-0.690951\pi$
$821$	−32.3524	−1.12911	−0.564553	+	0.825397i	$0.690951\pi$
$822$	3.18841	0.111209
$823$	3.77348	0.131535	0.0657676	−	0.997835i	$-0.479050\pi$
$823$	3.77348	0.131535	0.0657676	+	0.997835i	$0.479050\pi$
$824$	−52.6497	−1.83414
$825$	3.99280	0.139012
$826$	−2.44266	−0.0849910
$827$	−37.1596	−1.29217	−0.646083	−	0.763267i	$-0.723594\pi$
$827$	−37.1596	−1.29217	−0.646083	+	0.763267i	$0.723594\pi$
$828$	−3.42664	−0.119084
$829$	−20.0786	−0.697358	−0.348679	−	0.937242i	$-0.613370\pi$
$829$	−20.0786	−0.697358	−0.348679	+	0.937242i	$0.613370\pi$
$830$	5.48729	0.190467
$831$	−25.7234	−0.892334
$832$	−8.71893	−0.302274
$833$	−24.3962	−0.845279
$834$	13.5038	0.467598
$835$	−1.45419	−0.0503245
$836$	−6.28091	−0.217230
$837$	−1.00000	−0.0345651
$838$	−34.3291	−1.18588
$839$	54.1289	1.86874	0.934369	−	0.356308i	$-0.115964\pi$
$839$	54.1289	1.86874	0.934369	+	0.356308i	$0.115964\pi$
$840$	−1.99520	−0.0688409
$841$	−28.8007	−0.993129
$842$	−34.0221	−1.17248
$843$	27.7608	0.956134
$844$	−8.38508	−0.288626
$845$	−1.00000	−0.0344010
$846$	−2.46934	−0.0848975
$847$	−3.20446	−0.110107
$848$	−19.3396	−0.664124
$849$	16.1395	0.553907
$850$	−4.36702	−0.149788
$851$	0.763386	0.0261685
$852$	−8.76563	−0.300306
$853$	41.8462	1.43279	0.716394	−	0.697696i	$-0.245791\pi$
$853$	41.8462	1.43279	0.716394	+	0.697696i	$0.245791\pi$
$854$	4.33919	0.148484
$855$	2.56688	0.0877854
$856$	−48.2228	−1.64822
$857$	14.0310	0.479291	0.239646	−	0.970860i	$-0.422969\pi$
$857$	14.0310	0.479291	0.239646	+	0.970860i	$0.422969\pi$
$858$	4.70265	0.160546
$859$	−14.4186	−0.491955	−0.245977	−	0.969276i	$-0.579109\pi$
$859$	−14.4186	−0.491955	−0.245977	+	0.969276i	$0.579109\pi$
$860$	0.595243	0.0202976
$861$	−7.41309	−0.252637
$862$	17.2292	0.586828
$863$	−23.3042	−0.793283	−0.396642	−	0.917973i	$-0.629824\pi$
$863$	−23.3042	−0.793283	−0.396642	+	0.917973i	$0.629824\pi$
$864$	3.32944	0.113270
$865$	11.5414	0.392418
$866$	25.1802	0.855656
$867$	3.25196	0.110442
$868$	0.397328	0.0134862
$869$	−54.2043	−1.83875
$870$	−0.525744	−0.0178244
$871$	2.48072	0.0840559
$872$	40.2202	1.36203
$873$	−4.82047	−0.163148
$874$	16.9044	0.571799
$875$	0.648351	0.0219183
$876$	−9.21588	−0.311376
$877$	33.2073	1.12133	0.560665	−	0.828043i	$-0.310546\pi$
$877$	33.2073	1.12133	0.560665	+	0.828043i	$0.310546\pi$
$878$	−37.6910	−1.27201
$879$	−22.3062	−0.752368
$880$	−9.57787	−0.322870
$881$	10.3761	0.349580	0.174790	−	0.984606i	$-0.444075\pi$
$881$	10.3761	0.349580	0.174790	+	0.984606i	$0.444075\pi$
$882$	7.74939	0.260935
$883$	28.3463	0.953930	0.476965	−	0.878922i	$-0.341737\pi$
$883$	28.3463	0.953930	0.476965	+	0.878922i	$0.341737\pi$
$884$	2.27227	0.0764246
$885$	−3.19881	−0.107527
$886$	26.3260	0.884438
$887$	−29.5835	−0.993316	−0.496658	−	0.867946i	$-0.665440\pi$
$887$	−29.5835	−0.993316	−0.496658	+	0.867946i	$0.665440\pi$
$888$	−0.420137	−0.0140989
$889$	−13.4771	−0.452008
$890$	−7.75992	−0.260113
$891$	−3.99280	−0.133764
$892$	−4.16603	−0.139489
$893$	−5.38171	−0.180092
$894$	−22.1406	−0.740492
$895$	14.6589	0.489993
$896$	2.34062	0.0781946
$897$	5.59151	0.186695
$898$	16.5868	0.553508
$899$	0.446384	0.0148878
$900$	−0.612829	−0.0204276
$901$	29.8935	0.995897
$902$	−53.7690	−1.79031
$903$	0.629745	0.0209566
$904$	−34.8473	−1.15900
$905$	12.9016	0.428864
$906$	8.84272	0.293780
$907$	10.7814	0.357992	0.178996	−	0.983850i	$-0.442715\pi$
$907$	10.7814	0.357992	0.178996	+	0.983850i	$0.442715\pi$
$908$	−6.55799	−0.217635
$909$	11.8006	0.391403
$910$	0.763616	0.0253136
$911$	30.4435	1.00864	0.504319	−	0.863518i	$-0.331744\pi$
$911$	30.4435	1.00864	0.504319	+	0.863518i	$0.331744\pi$
$912$	−6.15739	−0.203891
$913$	−18.6025	−0.615652
$914$	−42.7009	−1.41242
$915$	5.68243	0.187855
$916$	0.521330	0.0172252
$917$	−13.4179	−0.443097
$918$	4.36702	0.144133
$919$	−56.1738	−1.85300	−0.926502	−	0.376290i	$-0.877200\pi$
$919$	−56.1738	−1.85300	−0.926502	+	0.376290i	$0.877200\pi$
$920$	−17.2070	−0.567298
$921$	−33.6271	−1.10805
$922$	−3.27417	−0.107829
$923$	14.3036	0.470807
$924$	1.58645	0.0521905
$925$	0.136526	0.00448894
$926$	7.46537	0.245327
$927$	−17.1088	−0.561927
$928$	−1.48621	−0.0487873
$929$	−30.1106	−0.987895	−0.493948	−	0.869492i	$-0.664447\pi$
$929$	−30.1106	−0.987895	−0.493948	+	0.869492i	$0.664447\pi$
$930$	−1.17778	−0.0386210
$931$	16.8891	0.553519
$932$	3.39036	0.111055
$933$	−22.6224	−0.740624
$934$	17.1721	0.561887
$935$	14.8047	0.484164
$936$	−3.07734	−0.100586
$937$	19.5596	0.638986	0.319493	−	0.947589i	$-0.396487\pi$
$937$	19.5596	0.638986	0.319493	+	0.947589i	$0.396487\pi$
$938$	−1.89432	−0.0618516
$939$	−30.6107	−0.998944
$940$	1.28485	0.0419073
$941$	−19.1542	−0.624409	−0.312205	−	0.950015i	$-0.601067\pi$
$941$	−19.1542	−0.624409	−0.312205	+	0.950015i	$0.601067\pi$
$942$	−19.0789	−0.621625
$943$	−63.9320	−2.08191
$944$	7.67324	0.249743
$945$	−0.648351	−0.0210908
$946$	4.56770	0.148509
$947$	45.6214	1.48250	0.741249	−	0.671230i	$-0.234234\pi$
$947$	45.6214	1.48250	0.741249	+	0.671230i	$0.234234\pi$
$948$	8.31945	0.270203
$949$	15.0383	0.488163
$950$	3.02322	0.0980863
$951$	−21.4198	−0.694583
$952$	−7.39787	−0.239766
$953$	22.7441	0.736755	0.368378	−	0.929676i	$-0.379913\pi$
$953$	22.7441	0.736755	0.368378	+	0.929676i	$0.379913\pi$
$954$	−9.49558	−0.307431
$955$	24.9303	0.806725
$956$	9.31083	0.301134
$957$	1.78233	0.0576144
$958$	−13.8704	−0.448132
$959$	−1.75517	−0.0566774
$960$	8.71893	0.281402
$961$	1.00000	0.0322581
$962$	0.160798	0.00518433
$963$	−15.6703	−0.504967
$964$	6.78884	0.218654
$965$	−10.4457	−0.336260
$966$	−4.26976	−0.137377
$967$	−0.256316	−0.00824257	−0.00412129	−	0.999992i	$-0.501312\pi$
$967$	−0.256316	−0.00824257	−0.00412129	+	0.999992i	$0.501312\pi$
$968$	15.2097	0.488859
$969$	9.51756	0.305748
$970$	−5.67747	−0.182292
$971$	17.4528	0.560085	0.280043	−	0.959988i	$-0.409651\pi$
$971$	17.4528	0.560085	0.280043	+	0.959988i	$0.409651\pi$
$972$	0.612829	0.0196565
$973$	−7.43363	−0.238311
$974$	−21.7353	−0.696443
$975$	1.00000	0.0320256
$976$	−13.6309	−0.436315
$977$	−13.9181	−0.445281	−0.222640	−	0.974901i	$-0.571468\pi$
$977$	−13.9181	−0.445281	−0.222640	+	0.974901i	$0.571468\pi$
$978$	−21.6349	−0.691807
$979$	26.3069	0.840773
$980$	−4.03219	−0.128804
$981$	13.0698	0.417286
$982$	−15.4404	−0.492723
$983$	−10.7726	−0.343593	−0.171797	−	0.985132i	$-0.554957\pi$
$983$	−10.7726	−0.343593	−0.171797	+	0.985132i	$0.554957\pi$
$984$	35.1856	1.12168
$985$	16.3547	0.521104
$986$	−1.94937	−0.0620806
$987$	1.35933	0.0432680
$988$	−1.57306	−0.0500456
$989$	5.43105	0.172697
$990$	−4.70265	−0.149460
$991$	−27.8895	−0.885939	−0.442969	−	0.896537i	$-0.646075\pi$
$991$	−27.8895	−0.885939	−0.442969	+	0.896537i	$0.646075\pi$
$992$	−3.32944	−0.105710
$993$	−10.2001	−0.323690
$994$	−10.9224	−0.346438
$995$	−7.91588	−0.250950
$996$	2.85517	0.0904695
$997$	0.222231	0.00703814	0.00351907	−	0.999994i	$-0.498880\pi$
$997$	0.222231	0.00703814	0.00351907	+	0.999994i	$0.498880\pi$
$998$	10.2546	0.324604
$999$	−0.136526	−0.00431949

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.6	✓	16	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.17778 q^{2} -1.00000 q^{3} -0.612829 q^{4} -1.00000 q^{5} +1.17778 q^{6} -0.648351 q^{7} +3.07734 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.17778 q^{2} -1.00000 q^{3} -0.612829 q^{4} -1.00000 q^{5} +1.17778 q^{6} -0.648351 q^{7} +3.07734 q^{8} +1.00000 q^{9} +1.17778 q^{10} -3.99280 q^{11} +0.612829 q^{12} -1.00000 q^{13} +0.763616 q^{14} +1.00000 q^{15} -2.39878 q^{16} +3.70784 q^{17} -1.17778 q^{18} -2.56688 q^{19} +0.612829 q^{20} +0.648351 q^{21} +4.70265 q^{22} +5.59151 q^{23} -3.07734 q^{24} +1.00000 q^{25} +1.17778 q^{26} -1.00000 q^{27} +0.397328 q^{28} +0.446384 q^{29} -1.17778 q^{30} +1.00000 q^{31} -3.32944 q^{32} +3.99280 q^{33} -4.36702 q^{34} +0.648351 q^{35} -0.612829 q^{36} +0.136526 q^{37} +3.02322 q^{38} +1.00000 q^{39} -3.07734 q^{40} -11.4338 q^{41} -0.763616 q^{42} +0.971303 q^{43} +2.44690 q^{44} -1.00000 q^{45} -6.58558 q^{46} +2.09660 q^{47} +2.39878 q^{48} -6.57964 q^{49} -1.17778 q^{50} -3.70784 q^{51} +0.612829 q^{52} +8.06225 q^{53} +1.17778 q^{54} +3.99280 q^{55} -1.99520 q^{56} +2.56688 q^{57} -0.525744 q^{58} -3.19881 q^{59} -0.612829 q^{60} +5.68243 q^{61} -1.17778 q^{62} -0.648351 q^{63} +8.71893 q^{64} +1.00000 q^{65} -4.70265 q^{66} -2.48072 q^{67} -2.27227 q^{68} -5.59151 q^{69} -0.763616 q^{70} -14.3036 q^{71} +3.07734 q^{72} -15.0383 q^{73} -0.160798 q^{74} -1.00000 q^{75} +1.57306 q^{76} +2.58874 q^{77} -1.17778 q^{78} +13.5755 q^{79} +2.39878 q^{80} +1.00000 q^{81} +13.4665 q^{82} +4.65900 q^{83} -0.397328 q^{84} -3.70784 q^{85} -1.14398 q^{86} -0.446384 q^{87} -12.2872 q^{88} -6.58859 q^{89} +1.17778 q^{90} +0.648351 q^{91} -3.42664 q^{92} -1.00000 q^{93} -2.46934 q^{94} +2.56688 q^{95} +3.32944 q^{96} -4.82047 q^{97} +7.74939 q^{98} -3.99280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

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