LMFDB - Embedded newform 1045.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 $ x^7 - x^6 - 10x^5 + 8x^4 + 27x^3 - 16x^2 - 18*x + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.719047$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.719047 q^{2} +0.163114 q^{3} -1.48297 q^{4} -1.00000 q^{5} +0.117287 q^{6} +1.05678 q^{7} -2.50442 q^{8} -2.97339 q^{9} +O(q^{10})$	\(q+0.719047 q^{2} +0.163114 q^{3} -1.48297 q^{4} -1.00000 q^{5} +0.117287 q^{6} +1.05678 q^{7} -2.50442 q^{8} -2.97339 q^{9} -0.719047 q^{10} +1.00000 q^{11} -0.241893 q^{12} +3.88893 q^{13} +0.759873 q^{14} -0.163114 q^{15} +1.16515 q^{16} +5.65135 q^{17} -2.13801 q^{18} -1.00000 q^{19} +1.48297 q^{20} +0.172375 q^{21} +0.719047 q^{22} +2.78982 q^{23} -0.408506 q^{24} +1.00000 q^{25} +2.79632 q^{26} -0.974345 q^{27} -1.56717 q^{28} +0.890165 q^{29} -0.117287 q^{30} +6.13204 q^{31} +5.84664 q^{32} +0.163114 q^{33} +4.06359 q^{34} -1.05678 q^{35} +4.40946 q^{36} +3.51111 q^{37} -0.719047 q^{38} +0.634339 q^{39} +2.50442 q^{40} -2.28025 q^{41} +0.123946 q^{42} +3.91188 q^{43} -1.48297 q^{44} +2.97339 q^{45} +2.00601 q^{46} +7.75344 q^{47} +0.190052 q^{48} -5.88322 q^{49} +0.719047 q^{50} +0.921815 q^{51} -5.76717 q^{52} +7.61251 q^{53} -0.700600 q^{54} -1.00000 q^{55} -2.64662 q^{56} -0.163114 q^{57} +0.640071 q^{58} -2.17640 q^{59} +0.241893 q^{60} +0.475939 q^{61} +4.40923 q^{62} -3.14222 q^{63} +1.87372 q^{64} -3.88893 q^{65} +0.117287 q^{66} -0.383348 q^{67} -8.38079 q^{68} +0.455059 q^{69} -0.759873 q^{70} -2.51986 q^{71} +7.44663 q^{72} -10.7887 q^{73} +2.52465 q^{74} +0.163114 q^{75} +1.48297 q^{76} +1.05678 q^{77} +0.456120 q^{78} -4.27374 q^{79} -1.16515 q^{80} +8.76125 q^{81} -1.63960 q^{82} -1.39483 q^{83} -0.255628 q^{84} -5.65135 q^{85} +2.81283 q^{86} +0.145198 q^{87} -2.50442 q^{88} +8.36845 q^{89} +2.13801 q^{90} +4.10973 q^{91} -4.13723 q^{92} +1.00022 q^{93} +5.57509 q^{94} +1.00000 q^{95} +0.953669 q^{96} -6.75542 q^{97} -4.23031 q^{98} -2.97339 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) $ 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9	\( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9 - q^10 + 7 * q^11 + 13 * q^12 + q^13 + 12 * q^14 - 3 * q^15 + 3 * q^16 + q^17 + 7 * q^18 - 7 * q^19 - 7 * q^20 + 5 * q^21 + q^22 - 8 * q^23 + 25 * q^24 + 7 * q^25 + 12 * q^27 + 4 * q^28 + 11 * q^29 - 8 * q^30 + 7 * q^31 + 12 * q^32 + 3 * q^33 - 14 * q^34 + q^35 + 7 * q^36 - 17 * q^37 - q^38 + 30 * q^39 - 3 * q^40 + 17 * q^41 + 33 * q^42 - 3 * q^43 + 7 * q^44 - 2 * q^45 + 18 * q^46 + 14 * q^47 - 12 * q^48 + 6 * q^49 + q^50 + 8 * q^51 - 17 * q^52 + 7 * q^53 - 27 * q^54 - 7 * q^55 + 36 * q^56 - 3 * q^57 - 15 * q^58 + 35 * q^59 - 13 * q^60 + 17 * q^61 + 46 * q^62 - 22 * q^63 + 5 * q^64 - q^65 + 8 * q^66 + 4 * q^67 - 35 * q^68 - 4 * q^69 - 12 * q^70 + 10 * q^71 + 12 * q^72 + 22 * q^73 - 11 * q^74 + 3 * q^75 - 7 * q^76 - q^77 - 41 * q^78 + 11 * q^79 - 3 * q^80 - 21 * q^81 - 14 * q^82 + 39 * q^83 + 21 * q^84 - q^85 - 24 * q^86 - 2 * q^87 + 3 * q^88 + 18 * q^89 - 7 * q^90 - 22 * q^91 - 51 * q^92 + 10 * q^93 + 14 * q^94 + 7 * q^95 - 11 * q^96 - 4 * q^97 - 26 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.719047	0.508443	0.254222	−	0.967146i	$-0.418181\pi$
$2$	0.719047	0.508443	0.254222	+	0.967146i	$0.418181\pi$
$3$	0.163114	0.0941739	0.0470870	−	0.998891i	$-0.485006\pi$
$3$	0.163114	0.0941739	0.0470870	+	0.998891i	$0.485006\pi$
$4$	−1.48297	−0.741486
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.117287	0.0478821
$7$	1.05678	0.399424	0.199712	−	0.979855i	$-0.435999\pi$
$7$	1.05678	0.399424	0.199712	+	0.979855i	$0.435999\pi$
$8$	−2.50442	−0.885446
$9$	−2.97339	−0.991131
$10$	−0.719047	−0.227383
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.241893	−0.0698286
$13$	3.88893	1.07859	0.539297	−	0.842115i	$-0.318690\pi$
$13$	3.88893	1.07859	0.539297	+	0.842115i	$0.318690\pi$
$14$	0.759873	0.203085
$15$	−0.163114	−0.0421159
$16$	1.16515	0.291286
$17$	5.65135	1.37065	0.685327	−	0.728236i	$-0.259659\pi$
$17$	5.65135	1.37065	0.685327	+	0.728236i	$0.259659\pi$
$18$	−2.13801	−0.503934
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.48297	0.331602
$21$	0.172375	0.0376154
$22$	0.719047	0.153301
$23$	2.78982	0.581718	0.290859	−	0.956766i	$-0.406059\pi$
$23$	2.78982	0.581718	0.290859	+	0.956766i	$0.406059\pi$
$24$	−0.408506	−0.0833860
$25$	1.00000	0.200000
$26$	2.79632	0.548404
$27$	−0.974345	−0.187513
$28$	−1.56717	−0.296167
$29$	0.890165	0.165299	0.0826497	−	0.996579i	$-0.473662\pi$
$29$	0.890165	0.165299	0.0826497	+	0.996579i	$0.473662\pi$
$30$	−0.117287	−0.0214135
$31$	6.13204	1.10135	0.550673	−	0.834721i	$-0.314371\pi$
$31$	6.13204	1.10135	0.550673	+	0.834721i	$0.314371\pi$
$32$	5.84664	1.03355
$33$	0.163114	0.0283945
$34$	4.06359	0.696900
$35$	−1.05678	−0.178628
$36$	4.40946	0.734909
$37$	3.51111	0.577223	0.288612	−	0.957446i	$-0.406806\pi$
$37$	3.51111	0.577223	0.288612	+	0.957446i	$0.406806\pi$
$38$	−0.719047	−0.116645
$39$	0.634339	0.101576
$40$	2.50442	0.395984
$41$	−2.28025	−0.356115	−0.178057	−	0.984020i	$-0.556981\pi$
$41$	−2.28025	−0.356115	−0.178057	+	0.984020i	$0.556981\pi$
$42$	0.123946	0.0191253
$43$	3.91188	0.596556	0.298278	−	0.954479i	$-0.403588\pi$
$43$	3.91188	0.596556	0.298278	+	0.954479i	$0.403588\pi$
$44$	−1.48297	−0.223566
$45$	2.97339	0.443247
$46$	2.00601	0.295771
$47$	7.75344	1.13096	0.565478	−	0.824764i	$-0.308692\pi$
$47$	7.75344	1.13096	0.565478	+	0.824764i	$0.308692\pi$
$48$	0.190052	0.0274316
$49$	−5.88322	−0.840460
$50$	0.719047	0.101689
$51$	0.921815	0.129080
$52$	−5.76717	−0.799763
$53$	7.61251	1.04566	0.522830	−	0.852437i	$-0.324876\pi$
$53$	7.61251	1.04566	0.522830	+	0.852437i	$0.324876\pi$
$54$	−0.700600	−0.0953396
$55$	−1.00000	−0.134840
$56$	−2.64662	−0.353669
$57$	−0.163114	−0.0216050
$58$	0.640071	0.0840454
$59$	−2.17640	−0.283344	−0.141672	−	0.989914i	$-0.545248\pi$
$59$	−2.17640	−0.283344	−0.141672	+	0.989914i	$0.545248\pi$
$60$	0.241893	0.0312283
$61$	0.475939	0.0609378	0.0304689	−	0.999536i	$-0.490300\pi$
$61$	0.475939	0.0609378	0.0304689	+	0.999536i	$0.490300\pi$
$62$	4.40923	0.559972
$63$	−3.14222	−0.395882
$64$	1.87372	0.234215
$65$	−3.88893	−0.482362
$66$	0.117287	0.0144370
$67$	−0.383348	−0.0468334	−0.0234167	−	0.999726i	$-0.507454\pi$
$67$	−0.383348	−0.0468334	−0.0234167	+	0.999726i	$0.507454\pi$
$68$	−8.38079	−1.01632
$69$	0.455059	0.0547827
$70$	−0.759873	−0.0908222
$71$	−2.51986	−0.299052	−0.149526	−	0.988758i	$-0.547775\pi$
$71$	−2.51986	−0.299052	−0.149526	+	0.988758i	$0.547775\pi$
$72$	7.44663	0.877594
$73$	−10.7887	−1.26272	−0.631358	−	0.775491i	$-0.717502\pi$
$73$	−10.7887	−1.26272	−0.631358	+	0.775491i	$0.717502\pi$
$74$	2.52465	0.293485
$75$	0.163114	0.0188348
$76$	1.48297	0.170108
$77$	1.05678	0.120431
$78$	0.456120	0.0516454
$79$	−4.27374	−0.480834	−0.240417	−	0.970670i	$-0.577284\pi$
$79$	−4.27374	−0.480834	−0.240417	+	0.970670i	$0.577284\pi$
$80$	−1.16515	−0.130267
$81$	8.76125	0.973472
$82$	−1.63960	−0.181064
$83$	−1.39483	−0.153102	−0.0765512	−	0.997066i	$-0.524391\pi$
$83$	−1.39483	−0.153102	−0.0765512	+	0.997066i	$0.524391\pi$
$84$	−0.255628	−0.0278913
$85$	−5.65135	−0.612975
$86$	2.81283	0.303315
$87$	0.145198	0.0155669
$88$	−2.50442	−0.266972
$89$	8.36845	0.887054	0.443527	−	0.896261i	$-0.353727\pi$
$89$	8.36845	0.887054	0.443527	+	0.896261i	$0.353727\pi$
$90$	2.13801	0.225366
$91$	4.10973	0.430817
$92$	−4.13723	−0.431336
$93$	1.00022	0.103718
$94$	5.57509	0.575026
$95$	1.00000	0.102598
$96$	0.953669	0.0973334
$97$	−6.75542	−0.685909	−0.342954	−	0.939352i	$-0.611428\pi$
$97$	−6.75542	−0.685909	−0.342954	+	0.939352i	$0.611428\pi$
$98$	−4.23031	−0.427326
$99$	−2.97339	−0.298837
$100$	−1.48297	−0.148297
$101$	9.99525	0.994565	0.497283	−	0.867589i	$-0.334331\pi$
$101$	9.99525	0.994565	0.497283	+	0.867589i	$0.334331\pi$
$102$	0.662828	0.0656298
$103$	8.95835	0.882692	0.441346	−	0.897337i	$-0.354501\pi$
$103$	8.95835	0.882692	0.441346	+	0.897337i	$0.354501\pi$
$104$	−9.73952	−0.955038
$105$	−0.172375	−0.0168221
$106$	5.47376	0.531658
$107$	−1.97814	−0.191234	−0.0956171	−	0.995418i	$-0.530482\pi$
$107$	−1.97814	−0.191234	−0.0956171	+	0.995418i	$0.530482\pi$
$108$	1.44492	0.139038
$109$	−0.846527	−0.0810826	−0.0405413	−	0.999178i	$-0.512908\pi$
$109$	−0.846527	−0.0810826	−0.0405413	+	0.999178i	$0.512908\pi$
$110$	−0.719047	−0.0685585
$111$	0.572712	0.0543594
$112$	1.23130	0.116347
$113$	1.91728	0.180362	0.0901812	−	0.995925i	$-0.471255\pi$
$113$	1.91728	0.180362	0.0901812	+	0.995925i	$0.471255\pi$
$114$	−0.117287	−0.0109849
$115$	−2.78982	−0.260152
$116$	−1.32009	−0.122567
$117$	−11.5633	−1.06903
$118$	−1.56494	−0.144064
$119$	5.97222	0.547473
$120$	0.408506	0.0372913
$121$	1.00000	0.0909091
$122$	0.342223	0.0309834
$123$	−0.371940	−0.0335367
$124$	−9.09364	−0.816633
$125$	−1.00000	−0.0894427
$126$	−2.25940	−0.201284
$127$	11.4902	1.01959	0.509795	−	0.860296i	$-0.329721\pi$
$127$	11.4902	1.01959	0.509795	+	0.860296i	$0.329721\pi$
$128$	−10.3460	−0.914464
$129$	0.638083	0.0561801
$130$	−2.79632	−0.245254
$131$	−1.00153	−0.0875044	−0.0437522	−	0.999042i	$-0.513931\pi$
$131$	−1.00153	−0.0875044	−0.0437522	+	0.999042i	$0.513931\pi$
$132$	−0.241893	−0.0210541
$133$	−1.05678	−0.0916343
$134$	−0.275645	−0.0238121
$135$	0.974345	0.0838582
$136$	−14.1534	−1.21364
$137$	13.7420	1.17406	0.587030	−	0.809565i	$-0.300297\pi$
$137$	13.7420	1.17406	0.587030	+	0.809565i	$0.300297\pi$
$138$	0.327209	0.0278539
$139$	−13.4541	−1.14116	−0.570582	−	0.821241i	$-0.693282\pi$
$139$	−13.4541	−1.14116	−0.570582	+	0.821241i	$0.693282\pi$
$140$	1.56717	0.132450
$141$	1.26469	0.106507
$142$	−1.81190	−0.152051
$143$	3.88893	0.325209
$144$	−3.46444	−0.288703
$145$	−0.890165	−0.0739242
$146$	−7.75755	−0.642020
$147$	−0.959636	−0.0791494
$148$	−5.20688	−0.428003
$149$	15.6201	1.27965	0.639823	−	0.768522i	$-0.279008\pi$
$149$	15.6201	1.27965	0.639823	+	0.768522i	$0.279008\pi$
$150$	0.117287	0.00957642
$151$	−13.2415	−1.07758	−0.538790	−	0.842440i	$-0.681118\pi$
$151$	−13.2415	−1.07758	−0.538790	+	0.842440i	$0.681118\pi$
$152$	2.50442	0.203135
$153$	−16.8037	−1.35850
$154$	0.759873	0.0612323
$155$	−6.13204	−0.492537
$156$	−0.940706	−0.0753168
$157$	−7.93760	−0.633490	−0.316745	−	0.948511i	$-0.602590\pi$
$157$	−7.93760	−0.633490	−0.316745	+	0.948511i	$0.602590\pi$
$158$	−3.07302	−0.244477
$159$	1.24171	0.0984739
$160$	−5.84664	−0.462217
$161$	2.94822	0.232353
$162$	6.29975	0.494955
$163$	4.86881	0.381355	0.190677	−	0.981653i	$-0.438932\pi$
$163$	4.86881	0.381355	0.190677	+	0.981653i	$0.438932\pi$
$164$	3.38154	0.264054
$165$	−0.163114	−0.0126984
$166$	−1.00295	−0.0778439
$167$	−6.78983	−0.525413	−0.262706	−	0.964876i	$-0.584615\pi$
$167$	−6.78983	−0.525413	−0.262706	+	0.964876i	$0.584615\pi$
$168$	−0.431700	−0.0333064
$169$	2.12377	0.163367
$170$	−4.06359	−0.311663
$171$	2.97339	0.227381
$172$	−5.80121	−0.442338
$173$	−14.6609	−1.11465	−0.557325	−	0.830295i	$-0.688172\pi$
$173$	−14.6609	−1.11465	−0.557325	+	0.830295i	$0.688172\pi$
$174$	0.104405	0.00791489
$175$	1.05678	0.0798849
$176$	1.16515	0.0878261
$177$	−0.355002	−0.0266836
$178$	6.01731	0.451017
$179$	−9.14657	−0.683647	−0.341823	−	0.939764i	$-0.611044\pi$
$179$	−9.14657	−0.683647	−0.341823	+	0.939764i	$0.611044\pi$
$180$	−4.40946	−0.328662
$181$	3.93642	0.292592	0.146296	−	0.989241i	$-0.453265\pi$
$181$	3.93642	0.292592	0.146296	+	0.989241i	$0.453265\pi$
$182$	2.95509	0.219046
$183$	0.0776324	0.00573875
$184$	−6.98689	−0.515080
$185$	−3.51111	−0.258142
$186$	0.719207	0.0527348
$187$	5.65135	0.413268
$188$	−11.4981	−0.838587
$189$	−1.02967	−0.0748972
$190$	0.719047	0.0521652
$191$	−22.0025	−1.59204	−0.796022	−	0.605267i	$-0.793066\pi$
$191$	−22.0025	−1.59204	−0.796022	+	0.605267i	$0.793066\pi$
$192$	0.305630	0.0220569
$193$	15.6976	1.12994	0.564968	−	0.825113i	$-0.308888\pi$
$193$	15.6976	1.12994	0.564968	+	0.825113i	$0.308888\pi$
$194$	−4.85746	−0.348746
$195$	−0.634339	−0.0454260
$196$	8.72465	0.623189
$197$	−9.97970	−0.711024	−0.355512	−	0.934672i	$-0.615694\pi$
$197$	−9.97970	−0.711024	−0.355512	+	0.934672i	$0.615694\pi$
$198$	−2.13801	−0.151942
$199$	−4.99525	−0.354104	−0.177052	−	0.984201i	$-0.556656\pi$
$199$	−4.99525	−0.354104	−0.177052	+	0.984201i	$0.556656\pi$
$200$	−2.50442	−0.177089
$201$	−0.0625295	−0.00441049
$202$	7.18706	0.505680
$203$	0.940706	0.0660247
$204$	−1.36702	−0.0957109
$205$	2.28025	0.159259
$206$	6.44147	0.448799
$207$	−8.29524	−0.576559
$208$	4.53117	0.314180
$209$	−1.00000	−0.0691714
$210$	−0.123946	−0.00855309
$211$	15.2151	1.04745	0.523727	−	0.851886i	$-0.324541\pi$
$211$	15.2151	1.04745	0.523727	+	0.851886i	$0.324541\pi$
$212$	−11.2891	−0.775341
$213$	−0.411024	−0.0281629
$214$	−1.42238	−0.0972317
$215$	−3.91188	−0.266788
$216$	2.44017	0.166032
$217$	6.48020	0.439905
$218$	−0.608693	−0.0412259
$219$	−1.75978	−0.118915
$220$	1.48297	0.0999819
$221$	21.9777	1.47838
$222$	0.411807	0.0276387
$223$	−19.1508	−1.28244	−0.641218	−	0.767359i	$-0.721570\pi$
$223$	−19.1508	−1.28244	−0.641218	+	0.767359i	$0.721570\pi$
$224$	6.17859	0.412825
$225$	−2.97339	−0.198226
$226$	1.37861	0.0917040
$227$	15.0978	1.00207	0.501037	−	0.865426i	$-0.332952\pi$
$227$	15.0978	1.00207	0.501037	+	0.865426i	$0.332952\pi$
$228$	0.241893	0.0160198
$229$	22.5618	1.49092	0.745461	−	0.666549i	$-0.232229\pi$
$229$	22.5618	1.49092	0.745461	+	0.666549i	$0.232229\pi$
$230$	−2.00601	−0.132273
$231$	0.172375	0.0113415
$232$	−2.22935	−0.146364
$233$	−0.917101	−0.0600813	−0.0300406	−	0.999549i	$-0.509564\pi$
$233$	−0.917101	−0.0600813	−0.0300406	+	0.999549i	$0.509564\pi$
$234$	−8.31457	−0.543541
$235$	−7.75344	−0.505779
$236$	3.22755	0.210095
$237$	−0.697108	−0.0452820
$238$	4.29431	0.278359
$239$	28.4833	1.84243	0.921215	−	0.389054i	$-0.127198\pi$
$239$	28.4833	1.84243	0.921215	+	0.389054i	$0.127198\pi$
$240$	−0.190052	−0.0122678
$241$	2.76910	0.178373	0.0891867	−	0.996015i	$-0.471573\pi$
$241$	2.76910	0.178373	0.0891867	+	0.996015i	$0.471573\pi$
$242$	0.719047	0.0462221
$243$	4.35212	0.279188
$244$	−0.705804	−0.0451845
$245$	5.88322	0.375865
$246$	−0.267443	−0.0170515
$247$	−3.88893	−0.247447
$248$	−15.3572	−0.975183
$249$	−0.227516	−0.0144183
$250$	−0.719047	−0.0454765
$251$	27.9920	1.76684	0.883419	−	0.468585i	$-0.155236\pi$
$251$	27.9920	1.76684	0.883419	+	0.468585i	$0.155236\pi$
$252$	4.65982	0.293541
$253$	2.78982	0.175395
$254$	8.26199	0.518403
$255$	−0.921815	−0.0577263
$256$	−11.1867	−0.699168
$257$	−16.4205	−1.02428	−0.512140	−	0.858902i	$-0.671147\pi$
$257$	−16.4205	−1.02428	−0.512140	+	0.858902i	$0.671147\pi$
$258$	0.458812	0.0285644
$259$	3.71046	0.230557
$260$	5.76717	0.357665
$261$	−2.64681	−0.163833
$262$	−0.720150	−0.0444910
$263$	−5.81131	−0.358340	−0.179170	−	0.983818i	$-0.557341\pi$
$263$	−5.81131	−0.358340	−0.179170	+	0.983818i	$0.557341\pi$
$264$	−0.408506	−0.0251418
$265$	−7.61251	−0.467633
$266$	−0.759873	−0.0465908
$267$	1.36501	0.0835374
$268$	0.568494	0.0347263
$269$	−16.2245	−0.989223	−0.494611	−	0.869114i	$-0.664690\pi$
$269$	−16.2245	−0.989223	−0.494611	+	0.869114i	$0.664690\pi$
$270$	0.700600	0.0426371
$271$	3.74012	0.227196	0.113598	−	0.993527i	$-0.463762\pi$
$271$	3.74012	0.227196	0.113598	+	0.993527i	$0.463762\pi$
$272$	6.58464	0.399253
$273$	0.670355	0.0405718
$274$	9.88116	0.596943
$275$	1.00000	0.0603023
$276$	−0.674840	−0.0406206
$277$	16.9533	1.01863	0.509313	−	0.860581i	$-0.329899\pi$
$277$	16.9533	1.01863	0.509313	+	0.860581i	$0.329899\pi$
$278$	−9.67414	−0.580217
$279$	−18.2330	−1.09158
$280$	2.64662	0.158166
$281$	−18.1398	−1.08213	−0.541064	−	0.840981i	$-0.681978\pi$
$281$	−18.1398	−1.08213	−0.541064	+	0.840981i	$0.681978\pi$
$282$	0.909375	0.0541525
$283$	−18.6616	−1.10932	−0.554658	−	0.832079i	$-0.687151\pi$
$283$	−18.6616	−1.10932	−0.554658	+	0.832079i	$0.687151\pi$
$284$	3.73687	0.221743
$285$	0.163114	0.00966204
$286$	2.79632	0.165350
$287$	−2.40971	−0.142241
$288$	−17.3844	−1.02438
$289$	14.9378	0.878692
$290$	−0.640071	−0.0375862
$291$	−1.10190	−0.0645947
$292$	15.9993	0.936286
$293$	−10.5405	−0.615780	−0.307890	−	0.951422i	$-0.599623\pi$
$293$	−10.5405	−0.615780	−0.307890	+	0.951422i	$0.599623\pi$
$294$	−0.690024	−0.0402430
$295$	2.17640	0.126715
$296$	−8.79330	−0.511100
$297$	−0.974345	−0.0565372
$298$	11.2316	0.650628
$299$	10.8494	0.627438
$300$	−0.241893	−0.0139657
$301$	4.13399	0.238279
$302$	−9.52128	−0.547888
$303$	1.63037	0.0936621
$304$	−1.16515	−0.0668257
$305$	−0.475939	−0.0272522
$306$	−12.0826	−0.690719
$307$	−7.13003	−0.406932	−0.203466	−	0.979082i	$-0.565221\pi$
$307$	−7.13003	−0.406932	−0.203466	+	0.979082i	$0.565221\pi$
$308$	−1.56717	−0.0892978
$309$	1.46123	0.0831266
$310$	−4.40923	−0.250427
$311$	19.8839	1.12751	0.563757	−	0.825941i	$-0.309356\pi$
$311$	19.8839	1.12751	0.563757	+	0.825941i	$0.309356\pi$
$312$	−1.58865	−0.0899397
$313$	−4.06293	−0.229651	−0.114825	−	0.993386i	$-0.536631\pi$
$313$	−4.06293	−0.229651	−0.114825	+	0.993386i	$0.536631\pi$
$314$	−5.70751	−0.322094
$315$	3.14222	0.177044
$316$	6.33784	0.356531
$317$	1.04954	0.0589479	0.0294739	−	0.999566i	$-0.490617\pi$
$317$	1.04954	0.0589479	0.0294739	+	0.999566i	$0.490617\pi$
$318$	0.892847	0.0500684
$319$	0.890165	0.0498397
$320$	−1.87372	−0.104744
$321$	−0.322663	−0.0180093
$322$	2.11991	0.118138
$323$	−5.65135	−0.314450
$324$	−12.9927	−0.721816
$325$	3.88893	0.215719
$326$	3.50091	0.193897
$327$	−0.138080	−0.00763587
$328$	5.71069	0.315320
$329$	8.19366	0.451731
$330$	−0.117287	−0.00645642
$331$	18.3670	1.00954	0.504772	−	0.863253i	$-0.331577\pi$
$331$	18.3670	1.00954	0.504772	+	0.863253i	$0.331577\pi$
$332$	2.06849	0.113523
$333$	−10.4399	−0.572104
$334$	−4.88221	−0.267142
$335$	0.383348	0.0209445
$336$	0.200842	0.0109568
$337$	−30.8428	−1.68012	−0.840058	−	0.542496i	$-0.817479\pi$
$337$	−30.8428	−1.68012	−0.840058	+	0.542496i	$0.817479\pi$
$338$	1.52709	0.0830628
$339$	0.312735	0.0169854
$340$	8.38079	0.454512
$341$	6.13204	0.332068
$342$	2.13801	0.115610
$343$	−13.6147	−0.735125
$344$	−9.79700	−0.528219
$345$	−0.455059	−0.0244996
$346$	−10.5419	−0.566736
$347$	12.7838	0.686269	0.343134	−	0.939286i	$-0.388511\pi$
$347$	12.7838	0.686269	0.343134	+	0.939286i	$0.388511\pi$
$348$	−0.215325	−0.0115426
$349$	−24.5164	−1.31233	−0.656166	−	0.754616i	$-0.727823\pi$
$349$	−24.5164	−1.31233	−0.656166	+	0.754616i	$0.727823\pi$
$350$	0.759873	0.0406169
$351$	−3.78916	−0.202250
$352$	5.84664	0.311627
$353$	−18.2615	−0.971963	−0.485981	−	0.873969i	$-0.661538\pi$
$353$	−18.2615	−0.971963	−0.485981	+	0.873969i	$0.661538\pi$
$354$	−0.255263	−0.0135671
$355$	2.51986	0.133740
$356$	−12.4102	−0.657738
$357$	0.974153	0.0515577
$358$	−6.57681	−0.347595
$359$	8.81005	0.464977	0.232488	−	0.972599i	$-0.425313\pi$
$359$	8.81005	0.464977	0.232488	+	0.972599i	$0.425313\pi$
$360$	−7.44663	−0.392472
$361$	1.00000	0.0526316
$362$	2.83047	0.148766
$363$	0.163114	0.00856127
$364$	−6.09462	−0.319445
$365$	10.7887	0.564704
$366$	0.0558213	0.00291783
$367$	−14.5509	−0.759552	−0.379776	−	0.925079i	$-0.623999\pi$
$367$	−14.5509	−0.759552	−0.379776	+	0.925079i	$0.623999\pi$
$368$	3.25055	0.169447
$369$	6.78007	0.352956
$370$	−2.52465	−0.131251
$371$	8.04474	0.417662
$372$	−1.48330	−0.0769055
$373$	−6.38135	−0.330414	−0.165207	−	0.986259i	$-0.552829\pi$
$373$	−6.38135	−0.330414	−0.165207	+	0.986259i	$0.552829\pi$
$374$	4.06359	0.210123
$375$	−0.163114	−0.00842317
$376$	−19.4179	−1.00140
$377$	3.46179	0.178291
$378$	−0.740378	−0.0380809
$379$	−11.7023	−0.601104	−0.300552	−	0.953765i	$-0.597171\pi$
$379$	−11.7023	−0.601104	−0.300552	+	0.953765i	$0.597171\pi$
$380$	−1.48297	−0.0760748
$381$	1.87421	0.0960188
$382$	−15.8208	−0.809464
$383$	29.3472	1.49957	0.749786	−	0.661680i	$-0.230156\pi$
$383$	29.3472	1.49957	0.749786	+	0.661680i	$0.230156\pi$
$384$	−1.68758	−0.0861187
$385$	−1.05678	−0.0538584
$386$	11.2873	0.574509
$387$	−11.6316	−0.591266
$388$	10.0181	0.508591
$389$	−13.9988	−0.709768	−0.354884	−	0.934910i	$-0.615480\pi$
$389$	−13.9988	−0.709768	−0.354884	+	0.934910i	$0.615480\pi$
$390$	−0.456120	−0.0230965
$391$	15.7663	0.797334
$392$	14.7341	0.744182
$393$	−0.163364	−0.00824064
$394$	−7.17588	−0.361515
$395$	4.27374	0.215035
$396$	4.40946	0.221584
$397$	−29.3475	−1.47291	−0.736455	−	0.676487i	$-0.763502\pi$
$397$	−29.3475	−1.47291	−0.736455	+	0.676487i	$0.763502\pi$
$398$	−3.59182	−0.180042
$399$	−0.172375	−0.00862956
$400$	1.16515	0.0582573
$401$	−1.30870	−0.0653533	−0.0326766	−	0.999466i	$-0.510403\pi$
$401$	−1.30870	−0.0653533	−0.0326766	+	0.999466i	$0.510403\pi$
$402$	−0.0449616	−0.00224248
$403$	23.8471	1.18791
$404$	−14.8227	−0.737456
$405$	−8.76125	−0.435350
$406$	0.676412	0.0335698
$407$	3.51111	0.174039
$408$	−2.30861	−0.114293
$409$	−0.0668127	−0.00330367	−0.00165184	−	0.999999i	$-0.500526\pi$
$409$	−0.0668127	−0.00330367	−0.00165184	+	0.999999i	$0.500526\pi$
$410$	1.63960	0.0809743
$411$	2.24152	0.110566
$412$	−13.2850	−0.654503
$413$	−2.29998	−0.113174
$414$	−5.96467	−0.293148
$415$	1.39483	0.0684695
$416$	22.7372	1.11478
$417$	−2.19456	−0.107468
$418$	−0.719047	−0.0351697
$419$	15.1746	0.741326	0.370663	−	0.928767i	$-0.379131\pi$
$419$	15.1746	0.741326	0.370663	+	0.928767i	$0.379131\pi$
$420$	0.255628	0.0124733
$421$	−3.96365	−0.193177	−0.0965883	−	0.995324i	$-0.530793\pi$
$421$	−3.96365	−0.193177	−0.0965883	+	0.995324i	$0.530793\pi$
$422$	10.9404	0.532570
$423$	−23.0540	−1.12093
$424$	−19.0649	−0.925875
$425$	5.65135	0.274131
$426$	−0.295546	−0.0143192
$427$	0.502962	0.0243400
$428$	2.93353	0.141797
$429$	0.634339	0.0306262
$430$	−2.81283	−0.135647
$431$	36.3363	1.75026	0.875129	−	0.483890i	$-0.160777\pi$
$431$	36.3363	1.75026	0.875129	+	0.483890i	$0.160777\pi$
$432$	−1.13525	−0.0546199
$433$	14.8147	0.711950	0.355975	−	0.934495i	$-0.384149\pi$
$433$	14.8147	0.711950	0.355975	+	0.934495i	$0.384149\pi$
$434$	4.65957	0.223667
$435$	−0.145198	−0.00696173
$436$	1.25538	0.0601216
$437$	−2.78982	−0.133455
$438$	−1.26537	−0.0604615
$439$	38.3087	1.82837	0.914187	−	0.405292i	$-0.132830\pi$
$439$	38.3087	1.82837	0.914187	+	0.405292i	$0.132830\pi$
$440$	2.50442	0.119394
$441$	17.4931	0.833006
$442$	15.8030	0.751672
$443$	1.84952	0.0878734	0.0439367	−	0.999034i	$-0.486010\pi$
$443$	1.84952	0.0878734	0.0439367	+	0.999034i	$0.486010\pi$
$444$	−0.849315	−0.0403067
$445$	−8.36845	−0.396703
$446$	−13.7704	−0.652045
$447$	2.54785	0.120509
$448$	1.98010	0.0935510
$449$	32.8639	1.55094	0.775472	−	0.631382i	$-0.217512\pi$
$449$	32.8639	1.55094	0.775472	+	0.631382i	$0.217512\pi$
$450$	−2.13801	−0.100787
$451$	−2.28025	−0.107373
$452$	−2.84327	−0.133736
$453$	−2.15988	−0.101480
$454$	10.8560	0.509498
$455$	−4.10973	−0.192667
$456$	0.408506	0.0191301
$457$	−10.8486	−0.507478	−0.253739	−	0.967273i	$-0.581660\pi$
$457$	−10.8486	−0.507478	−0.253739	+	0.967273i	$0.581660\pi$
$458$	16.2230	0.758049
$459$	−5.50636	−0.257015
$460$	4.13723	0.192899
$461$	2.26284	0.105391	0.0526954	−	0.998611i	$-0.483219\pi$
$461$	2.26284	0.105391	0.0526954	+	0.998611i	$0.483219\pi$
$462$	0.123946	0.00576649
$463$	−32.1970	−1.49632	−0.748160	−	0.663518i	$-0.769063\pi$
$463$	−32.1970	−1.49632	−0.748160	+	0.663518i	$0.769063\pi$
$464$	1.03717	0.0481495
$465$	−1.00022	−0.0463842
$466$	−0.659439	−0.0305479
$467$	37.3807	1.72977	0.864887	−	0.501967i	$-0.167390\pi$
$467$	37.3807	1.72977	0.864887	+	0.501967i	$0.167390\pi$
$468$	17.1481	0.792670
$469$	−0.405114	−0.0187064
$470$	−5.57509	−0.257160
$471$	−1.29473	−0.0596582
$472$	5.45063	0.250886
$473$	3.91188	0.179869
$474$	−0.501254	−0.0230233
$475$	−1.00000	−0.0458831
$476$	−8.85663	−0.405943
$477$	−22.6350	−1.03639
$478$	20.4808	0.936771
$479$	22.4365	1.02515	0.512574	−	0.858643i	$-0.328692\pi$
$479$	22.4365	1.02515	0.512574	+	0.858643i	$0.328692\pi$
$480$	−0.953669	−0.0435288
$481$	13.6545	0.622590
$482$	1.99111	0.0906927
$483$	0.480897	0.0218816
$484$	−1.48297	−0.0674078
$485$	6.75542	0.306748
$486$	3.12938	0.141951
$487$	−7.74046	−0.350754	−0.175377	−	0.984501i	$-0.556114\pi$
$487$	−7.74046	−0.350754	−0.175377	+	0.984501i	$0.556114\pi$
$488$	−1.19195	−0.0539571
$489$	0.794172	0.0359137
$490$	4.23031	0.191106
$491$	2.94721	0.133006	0.0665028	−	0.997786i	$-0.478816\pi$
$491$	2.94721	0.133006	0.0665028	+	0.997786i	$0.478816\pi$
$492$	0.551576	0.0248670
$493$	5.03063	0.226568
$494$	−2.79632	−0.125813
$495$	2.97339	0.133644
$496$	7.14472	0.320807
$497$	−2.66293	−0.119449
$498$	−0.163595	−0.00733087
$499$	8.36252	0.374358	0.187179	−	0.982326i	$-0.440066\pi$
$499$	8.36252	0.374358	0.187179	+	0.982326i	$0.440066\pi$
$500$	1.48297	0.0663205
$501$	−1.10752	−0.0494802
$502$	20.1275	0.898336
$503$	3.93785	0.175580	0.0877901	−	0.996139i	$-0.472020\pi$
$503$	3.93785	0.175580	0.0877901	+	0.996139i	$0.472020\pi$
$504$	7.86943	0.350532
$505$	−9.99525	−0.444783
$506$	2.00601	0.0891782
$507$	0.346417	0.0153849
$508$	−17.0396	−0.756011
$509$	−26.3613	−1.16844	−0.584221	−	0.811595i	$-0.698600\pi$
$509$	−26.3613	−1.16844	−0.584221	+	0.811595i	$0.698600\pi$
$510$	−0.662828	−0.0293505
$511$	−11.4012	−0.504360
$512$	12.6482	0.558977
$513$	0.974345	0.0430184
$514$	−11.8071	−0.520788
$515$	−8.95835	−0.394752
$516$	−0.946259	−0.0416567
$517$	7.75344	0.340996
$518$	2.66800	0.117225
$519$	−2.39140	−0.104971
$520$	9.73952	0.427106
$521$	5.27680	0.231181	0.115590	−	0.993297i	$-0.463124\pi$
$521$	5.27680	0.231181	0.115590	+	0.993297i	$0.463124\pi$
$522$	−1.90318	−0.0833000
$523$	27.9611	1.22265	0.611327	−	0.791378i	$-0.290636\pi$
$523$	27.9611	1.22265	0.611327	+	0.791378i	$0.290636\pi$
$524$	1.48525	0.0648833
$525$	0.172375	0.00752308
$526$	−4.17860	−0.182196
$527$	34.6543	1.50956
$528$	0.190052	0.00827093
$529$	−15.2169	−0.661604
$530$	−5.47376	−0.237765
$531$	6.47131	0.280831
$532$	1.56717	0.0679455
$533$	−8.86771	−0.384103
$534$	0.981509	0.0424740
$535$	1.97814	0.0855226
$536$	0.960065	0.0414685
$537$	−1.49193	−0.0643817
$538$	−11.6662	−0.502963
$539$	−5.88322	−0.253408
$540$	−1.44492	−0.0621797
$541$	−20.8098	−0.894683	−0.447342	−	0.894363i	$-0.647629\pi$
$541$	−20.8098	−0.894683	−0.447342	+	0.894363i	$0.647629\pi$
$542$	2.68932	0.115516
$543$	0.642086	0.0275545
$544$	33.0414	1.41664
$545$	0.846527	0.0362612
$546$	0.482017	0.0206284
$547$	15.6205	0.667885	0.333942	−	0.942593i	$-0.391621\pi$
$547$	15.6205	0.667885	0.333942	+	0.942593i	$0.391621\pi$
$548$	−20.3790	−0.870549
$549$	−1.41515	−0.0603973
$550$	0.719047	0.0306603
$551$	−0.890165	−0.0379223
$552$	−1.13966	−0.0485072
$553$	−4.51640	−0.192057
$554$	12.1902	0.517914
$555$	−0.572712	−0.0243103
$556$	19.9521	0.846156
$557$	−19.8995	−0.843171	−0.421585	−	0.906789i	$-0.638526\pi$
$557$	−19.8995	−0.843171	−0.421585	+	0.906789i	$0.638526\pi$
$558$	−13.1104	−0.555006
$559$	15.2130	0.643443
$560$	−1.23130	−0.0520319
$561$	0.921815	0.0389190
$562$	−13.0434	−0.550201
$563$	−19.0668	−0.803570	−0.401785	−	0.915734i	$-0.631610\pi$
$563$	−19.0668	−0.803570	−0.401785	+	0.915734i	$0.631610\pi$
$564$	−1.87551	−0.0789730
$565$	−1.91728	−0.0806605
$566$	−13.4186	−0.564024
$567$	9.25870	0.388829
$568$	6.31078	0.264794
$569$	−4.26584	−0.178833	−0.0894166	−	0.995994i	$-0.528500\pi$
$569$	−4.26584	−0.178833	−0.0894166	+	0.995994i	$0.528500\pi$
$570$	0.117287	0.00491260
$571$	−4.97199	−0.208071	−0.104036	−	0.994574i	$-0.533176\pi$
$571$	−4.97199	−0.208071	−0.104036	+	0.994574i	$0.533176\pi$
$572$	−5.76717	−0.241137
$573$	−3.58892	−0.149929
$574$	−1.73270	−0.0723214
$575$	2.78982	0.116344
$576$	−5.57130	−0.232137
$577$	−19.1116	−0.795627	−0.397814	−	0.917466i	$-0.630231\pi$
$577$	−19.1116	−0.795627	−0.397814	+	0.917466i	$0.630231\pi$
$578$	10.7410	0.446765
$579$	2.56050	0.106411
$580$	1.32009	0.0548137
$581$	−1.47402	−0.0611528
$582$	−0.792321	−0.0328427
$583$	7.61251	0.315278
$584$	27.0193	1.11807
$585$	11.5633	0.478084
$586$	−7.57909	−0.313089
$587$	−2.43947	−0.100688	−0.0503438	−	0.998732i	$-0.516032\pi$
$587$	−2.43947	−0.100688	−0.0503438	+	0.998732i	$0.516032\pi$
$588$	1.42311	0.0586882
$589$	−6.13204	−0.252666
$590$	1.56494	0.0644275
$591$	−1.62783	−0.0669600
$592$	4.09096	0.168137
$593$	42.8982	1.76162	0.880809	−	0.473472i	$-0.157000\pi$
$593$	42.8982	1.76162	0.880809	+	0.473472i	$0.157000\pi$
$594$	−0.700600	−0.0287460
$595$	−5.97222	−0.244837
$596$	−23.1641	−0.948839
$597$	−0.814796	−0.0333474
$598$	7.80125	0.319017
$599$	11.9181	0.486962	0.243481	−	0.969906i	$-0.421711\pi$
$599$	11.9181	0.486962	0.243481	+	0.969906i	$0.421711\pi$
$600$	−0.408506	−0.0166772
$601$	−35.3304	−1.44116	−0.720579	−	0.693373i	$-0.756124\pi$
$601$	−35.3304	−1.44116	−0.720579	+	0.693373i	$0.756124\pi$
$602$	2.97253	0.121151
$603$	1.13984	0.0464181
$604$	19.6368	0.799010
$605$	−1.00000	−0.0406558
$606$	1.17231	0.0476219
$607$	2.12778	0.0863638	0.0431819	−	0.999067i	$-0.486250\pi$
$607$	2.12778	0.0863638	0.0431819	+	0.999067i	$0.486250\pi$
$608$	−5.84664	−0.237112
$609$	0.153442	0.00621780
$610$	−0.342223	−0.0138562
$611$	30.1526	1.21984
$612$	24.9194	1.00731
$613$	−25.7794	−1.04122	−0.520611	−	0.853794i	$-0.674296\pi$
$613$	−25.7794	−1.04122	−0.520611	+	0.853794i	$0.674296\pi$
$614$	−5.12683	−0.206902
$615$	0.371940	0.0149981
$616$	−2.64662	−0.106635
$617$	−25.0524	−1.00857	−0.504286	−	0.863537i	$-0.668244\pi$
$617$	−25.0524	−1.00857	−0.504286	+	0.863537i	$0.668244\pi$
$618$	1.05070	0.0422652
$619$	34.6510	1.39274	0.696372	−	0.717681i	$-0.254797\pi$
$619$	34.6510	1.39274	0.696372	+	0.717681i	$0.254797\pi$
$620$	9.09364	0.365209
$621$	−2.71825	−0.109080
$622$	14.2975	0.573276
$623$	8.84360	0.354311
$624$	0.739097	0.0295876
$625$	1.00000	0.0400000
$626$	−2.92144	−0.116764
$627$	−0.163114	−0.00651415
$628$	11.7712	0.469723
$629$	19.8425	0.791173
$630$	2.25940	0.0900167
$631$	−9.38162	−0.373476	−0.186738	−	0.982410i	$-0.559792\pi$
$631$	−9.38162	−0.373476	−0.186738	+	0.982410i	$0.559792\pi$
$632$	10.7033	0.425753
$633$	2.48180	0.0986428
$634$	0.754667	0.0299717
$635$	−11.4902	−0.455974
$636$	−1.84142	−0.0730169
$637$	−22.8794	−0.906516
$638$	0.640071	0.0253406
$639$	7.49252	0.296400
$640$	10.3460	0.408961
$641$	−45.9392	−1.81449	−0.907245	−	0.420601i	$-0.861819\pi$
$641$	−45.9392	−1.81449	−0.907245	+	0.420601i	$0.861819\pi$
$642$	−0.232010	−0.00915670
$643$	−20.4152	−0.805097	−0.402548	−	0.915399i	$-0.631875\pi$
$643$	−20.4152	−0.805097	−0.402548	+	0.915399i	$0.631875\pi$
$644$	−4.37213	−0.172286
$645$	−0.638083	−0.0251245
$646$	−4.06359	−0.159880
$647$	17.2966	0.680000	0.340000	−	0.940425i	$-0.389573\pi$
$647$	17.2966	0.680000	0.340000	+	0.940425i	$0.389573\pi$
$648$	−21.9419	−0.861958
$649$	−2.17640	−0.0854314
$650$	2.79632	0.109681
$651$	1.05701	0.0414276
$652$	−7.22031	−0.282769
$653$	17.4074	0.681204	0.340602	−	0.940208i	$-0.389369\pi$
$653$	17.4074	0.681204	0.340602	+	0.940208i	$0.389369\pi$
$654$	−0.0992864	−0.00388241
$655$	1.00153	0.0391332
$656$	−2.65682	−0.103731
$657$	32.0789	1.25152
$658$	5.89163	0.229680
$659$	19.7517	0.769415	0.384708	−	0.923038i	$-0.374302\pi$
$659$	19.7517	0.769415	0.384708	+	0.923038i	$0.374302\pi$
$660$	0.241893	0.00941569
$661$	−8.59246	−0.334208	−0.167104	−	0.985939i	$-0.553442\pi$
$661$	−8.59246	−0.334208	−0.167104	+	0.985939i	$0.553442\pi$
$662$	13.2068	0.513295
$663$	3.58487	0.139225
$664$	3.49324	0.135564
$665$	1.05678	0.0409801
$666$	−7.50679	−0.290882
$667$	2.48340	0.0961577
$668$	10.0691	0.389586
$669$	−3.12377	−0.120772
$670$	0.275645	0.0106491
$671$	0.475939	0.0183734
$672$	1.00782	0.0388773
$673$	−13.8476	−0.533787	−0.266893	−	0.963726i	$-0.585997\pi$
$673$	−13.8476	−0.533787	−0.266893	+	0.963726i	$0.585997\pi$
$674$	−22.1775	−0.854244
$675$	−0.974345	−0.0375025
$676$	−3.14949	−0.121134
$677$	42.0501	1.61612	0.808058	−	0.589102i	$-0.200519\pi$
$677$	42.0501	1.61612	0.808058	+	0.589102i	$0.200519\pi$
$678$	0.224871	0.00863613
$679$	−7.13897	−0.273969
$680$	14.1534	0.542757
$681$	2.46266	0.0943693
$682$	4.40923	0.168838
$683$	−37.7020	−1.44263	−0.721313	−	0.692609i	$-0.756461\pi$
$683$	−37.7020	−1.44263	−0.721313	+	0.692609i	$0.756461\pi$
$684$	−4.40946	−0.168600
$685$	−13.7420	−0.525056
$686$	−9.78961	−0.373769
$687$	3.68014	0.140406
$688$	4.55791	0.173769
$689$	29.6045	1.12784
$690$	−0.327209	−0.0124566
$691$	−48.8445	−1.85813	−0.929066	−	0.369913i	$-0.879387\pi$
$691$	−48.8445	−1.85813	−0.929066	+	0.369913i	$0.879387\pi$
$692$	21.7417	0.826497
$693$	−3.14222	−0.119363
$694$	9.19213	0.348929
$695$	13.4541	0.510344
$696$	−0.363638	−0.0137837
$697$	−12.8865	−0.488110
$698$	−17.6284	−0.667246
$699$	−0.149592	−0.00565809
$700$	−1.56717	−0.0592335
$701$	−23.7536	−0.897160	−0.448580	−	0.893743i	$-0.648070\pi$
$701$	−23.7536	−0.897160	−0.448580	+	0.893743i	$0.648070\pi$
$702$	−2.72458	−0.102833
$703$	−3.51111	−0.132424
$704$	1.87372	0.0706184
$705$	−1.26469	−0.0476312
$706$	−13.1309	−0.494188
$707$	10.5628	0.397254
$708$	0.526458	0.0197855
$709$	27.0551	1.01607	0.508037	−	0.861335i	$-0.330371\pi$
$709$	27.0551	1.01607	0.508037	+	0.861335i	$0.330371\pi$
$710$	1.81190	0.0679992
$711$	12.7075	0.476569
$712$	−20.9581	−0.785439
$713$	17.1073	0.640673
$714$	0.700462	0.0262141
$715$	−3.88893	−0.145438
$716$	13.5641	0.506914
$717$	4.64602	0.173509
$718$	6.33485	0.236414
$719$	−31.5836	−1.17787	−0.588934	−	0.808181i	$-0.700452\pi$
$719$	−31.5836	−1.17787	−0.588934	+	0.808181i	$0.700452\pi$
$720$	3.46444	0.129112
$721$	9.46698	0.352569
$722$	0.719047	0.0267602
$723$	0.451679	0.0167981
$724$	−5.83760	−0.216953
$725$	0.890165	0.0330599
$726$	0.117287	0.00435292
$727$	−13.1743	−0.488607	−0.244303	−	0.969699i	$-0.578559\pi$
$727$	−13.1743	−0.488607	−0.244303	+	0.969699i	$0.578559\pi$
$728$	−10.2925	−0.381466
$729$	−25.5739	−0.947180
$730$	7.75755	0.287120
$731$	22.1074	0.817672
$732$	−0.115127	−0.00425520
$733$	13.5227	0.499472	0.249736	−	0.968314i	$-0.419656\pi$
$733$	13.5227	0.499472	0.249736	+	0.968314i	$0.419656\pi$
$734$	−10.4628	−0.386189
$735$	0.959636	0.0353967
$736$	16.3111	0.601234
$737$	−0.383348	−0.0141208
$738$	4.87519	0.179458
$739$	20.7838	0.764544	0.382272	−	0.924050i	$-0.375142\pi$
$739$	20.7838	0.764544	0.382272	+	0.924050i	$0.375142\pi$
$740$	5.20688	0.191409
$741$	−0.634339	−0.0233030
$742$	5.78454	0.212357
$743$	39.5592	1.45129	0.725644	−	0.688070i	$-0.241542\pi$
$743$	39.5592	1.45129	0.725644	+	0.688070i	$0.241542\pi$
$744$	−2.50498	−0.0918369
$745$	−15.6201	−0.572275
$746$	−4.58849	−0.167997
$747$	4.14738	0.151745
$748$	−8.38079	−0.306432
$749$	−2.09046	−0.0763836
$750$	−0.117287	−0.00428271
$751$	−45.2335	−1.65059	−0.825297	−	0.564698i	$-0.808993\pi$
$751$	−45.2335	−1.65059	−0.825297	+	0.564698i	$0.808993\pi$
$752$	9.03388	0.329432
$753$	4.56588	0.166390
$754$	2.48919	0.0906509
$755$	13.2415	0.481908
$756$	1.52696	0.0555352
$757$	9.13639	0.332068	0.166034	−	0.986120i	$-0.446904\pi$
$757$	9.13639	0.332068	0.166034	+	0.986120i	$0.446904\pi$
$758$	−8.41447	−0.305627
$759$	0.455059	0.0165176
$760$	−2.50442	−0.0908449
$761$	4.46096	0.161710	0.0808549	−	0.996726i	$-0.474235\pi$
$761$	4.46096	0.161710	0.0808549	+	0.996726i	$0.474235\pi$
$762$	1.34765	0.0488201
$763$	−0.894591	−0.0323864
$764$	32.6291	1.18048
$765$	16.8037	0.607539
$766$	21.1020	0.762447
$767$	−8.46389	−0.305613
$768$	−1.82471	−0.0658434
$769$	−49.3765	−1.78056	−0.890282	−	0.455410i	$-0.849493\pi$
$769$	−49.3765	−1.78056	−0.890282	+	0.455410i	$0.849493\pi$
$770$	−0.759873	−0.0273839
$771$	−2.67841	−0.0964605
$772$	−23.2791	−0.837832
$773$	28.1145	1.01121	0.505605	−	0.862765i	$-0.331269\pi$
$773$	28.1145	1.01121	0.505605	+	0.862765i	$0.331269\pi$
$774$	−8.36364	−0.300625
$775$	6.13204	0.220269
$776$	16.9184	0.607335
$777$	0.605229	0.0217125
$778$	−10.0658	−0.360877
$779$	2.28025	0.0816983
$780$	0.940706	0.0336827
$781$	−2.51986	−0.0901675
$782$	11.3367	0.405399
$783$	−0.867327	−0.0309958
$784$	−6.85481	−0.244815
$785$	7.93760	0.283305
$786$	−0.117467	−0.00418990
$787$	−29.8108	−1.06264	−0.531320	−	0.847171i	$-0.678304\pi$
$787$	−29.8108	−1.06264	−0.531320	+	0.847171i	$0.678304\pi$
$788$	14.7996	0.527214
$789$	−0.947906	−0.0337463
$790$	3.07302	0.109333
$791$	2.02614	0.0720412
$792$	7.44663	0.264604
$793$	1.85089	0.0657272
$794$	−21.1023	−0.748891
$795$	−1.24171	−0.0440388
$796$	7.40782	0.262563
$797$	−50.0474	−1.77277	−0.886385	−	0.462948i	$-0.846792\pi$
$797$	−50.0474	−1.77277	−0.886385	+	0.462948i	$0.846792\pi$
$798$	−0.123946	−0.00438764
$799$	43.8174	1.55015
$800$	5.84664	0.206710
$801$	−24.8827	−0.879187
$802$	−0.941016	−0.0332284
$803$	−10.7887	−0.380723
$804$	0.0927294	0.00327031
$805$	−2.94822	−0.103911
$806$	17.1472	0.603983
$807$	−2.64644	−0.0931590
$808$	−25.0323	−0.880634
$809$	−27.8648	−0.979675	−0.489837	−	0.871814i	$-0.662944\pi$
$809$	−27.8648	−0.979675	−0.489837	+	0.871814i	$0.662944\pi$
$810$	−6.29975	−0.221351
$811$	49.6627	1.74389	0.871947	−	0.489600i	$-0.162857\pi$
$811$	49.6627	1.74389	0.871947	+	0.489600i	$0.162857\pi$
$812$	−1.39504	−0.0489563
$813$	0.610066	0.0213960
$814$	2.52465	0.0884891
$815$	−4.86881	−0.170547
$816$	1.07405	0.0375992
$817$	−3.91188	−0.136859
$818$	−0.0480415	−0.00167973
$819$	−12.2199	−0.426996
$820$	−3.38154	−0.118088
$821$	−21.1699	−0.738836	−0.369418	−	0.929263i	$-0.620443\pi$
$821$	−21.1699	−0.738836	−0.369418	+	0.929263i	$0.620443\pi$
$822$	1.61176	0.0562165
$823$	−14.6145	−0.509430	−0.254715	−	0.967016i	$-0.581982\pi$
$823$	−14.6145	−0.509430	−0.254715	+	0.967016i	$0.581982\pi$
$824$	−22.4355	−0.781577
$825$	0.163114	0.00567890
$826$	−1.65379	−0.0575428
$827$	23.5899	0.820302	0.410151	−	0.912018i	$-0.365476\pi$
$827$	23.5899	0.820302	0.410151	+	0.912018i	$0.365476\pi$
$828$	12.3016	0.427510
$829$	18.9224	0.657204	0.328602	−	0.944469i	$-0.393423\pi$
$829$	18.9224	0.657204	0.328602	+	0.944469i	$0.393423\pi$
$830$	1.00295	0.0348128
$831$	2.76533	0.0959281
$832$	7.28675	0.252623
$833$	−33.2481	−1.15198
$834$	−1.57799	−0.0546413
$835$	6.78983	0.234972
$836$	1.48297	0.0512896
$837$	−5.97472	−0.206516
$838$	10.9112	0.376922
$839$	−30.5617	−1.05511	−0.527553	−	0.849522i	$-0.676890\pi$
$839$	−30.5617	−1.05511	−0.527553	+	0.849522i	$0.676890\pi$
$840$	0.431700	0.0148951
$841$	−28.2076	−0.972676
$842$	−2.85005	−0.0982193
$843$	−2.95885	−0.101908
$844$	−22.5636	−0.776671
$845$	−2.12377	−0.0730600
$846$	−16.5769	−0.569927
$847$	1.05678	0.0363113
$848$	8.86968	0.304586
$849$	−3.04397	−0.104469
$850$	4.06359	0.139380
$851$	9.79538	0.335781
$852$	0.609537	0.0208824
$853$	35.0082	1.19866	0.599329	−	0.800503i	$-0.295434\pi$
$853$	35.0082	1.19866	0.599329	+	0.800503i	$0.295434\pi$
$854$	0.361653	0.0123755
$855$	−2.97339	−0.101688
$856$	4.95410	0.169328
$857$	−20.5089	−0.700569	−0.350285	−	0.936643i	$-0.613915\pi$
$857$	−20.5089	−0.700569	−0.350285	+	0.936643i	$0.613915\pi$
$858$	0.456120	0.0155717
$859$	−39.8526	−1.35975	−0.679877	−	0.733326i	$-0.737967\pi$
$859$	−39.8526	−1.35975	−0.679877	+	0.733326i	$0.737967\pi$
$860$	5.80121	0.197820
$861$	−0.393058	−0.0133954
$862$	26.1275	0.889906
$863$	26.4251	0.899520	0.449760	−	0.893149i	$-0.351509\pi$
$863$	26.4251	0.899520	0.449760	+	0.893149i	$0.351509\pi$
$864$	−5.69664	−0.193804
$865$	14.6609	0.498486
$866$	10.6525	0.361986
$867$	2.43656	0.0827499
$868$	−9.60995	−0.326183
$869$	−4.27374	−0.144977
$870$	−0.104405	−0.00353964
$871$	−1.49081	−0.0505143
$872$	2.12006	0.0717943
$873$	20.0865	0.679825
$874$	−2.00601	−0.0678545
$875$	−1.05678	−0.0357256
$876$	2.60970	0.0881737
$877$	−25.0363	−0.845415	−0.422707	−	0.906266i	$-0.638920\pi$
$877$	−25.0363	−0.845415	−0.422707	+	0.906266i	$0.638920\pi$
$878$	27.5458	0.929625
$879$	−1.71930	−0.0579904
$880$	−1.16515	−0.0392770
$881$	35.9933	1.21264	0.606322	−	0.795219i	$-0.292644\pi$
$881$	35.9933	1.21264	0.606322	+	0.795219i	$0.292644\pi$
$882$	12.5784	0.423536
$883$	−9.56372	−0.321845	−0.160922	−	0.986967i	$-0.551447\pi$
$883$	−9.56372	−0.321845	−0.160922	+	0.986967i	$0.551447\pi$
$884$	−32.5923	−1.09620
$885$	0.355002	0.0119333
$886$	1.32989	0.0446786
$887$	19.4058	0.651583	0.325792	−	0.945442i	$-0.394369\pi$
$887$	19.4058	0.651583	0.325792	+	0.945442i	$0.394369\pi$
$888$	−1.43431	−0.0481323
$889$	12.1426	0.407249
$890$	−6.01731	−0.201701
$891$	8.76125	0.293513
$892$	28.4001	0.950907
$893$	−7.75344	−0.259459
$894$	1.83203	0.0612722
$895$	9.14657	0.305736
$896$	−10.9334	−0.365259
$897$	1.76969	0.0590884
$898$	23.6307	0.788567
$899$	5.45853	0.182052
$900$	4.40946	0.146982
$901$	43.0210	1.43324
$902$	−1.63960	−0.0545929
$903$	0.674312	0.0224397
$904$	−4.80167	−0.159701
$905$	−3.93642	−0.130851
$906$	−1.55305	−0.0515968
$907$	−41.6012	−1.38134	−0.690672	−	0.723168i	$-0.742685\pi$
$907$	−41.6012	−1.38134	−0.690672	+	0.723168i	$0.742685\pi$
$908$	−22.3896	−0.743024
$909$	−29.7198	−0.985744
$910$	−2.95509	−0.0979604
$911$	−41.9843	−1.39100	−0.695501	−	0.718525i	$-0.744818\pi$
$911$	−41.9843	−1.39100	−0.695501	+	0.718525i	$0.744818\pi$
$912$	−0.190052	−0.00629324
$913$	−1.39483	−0.0461621
$914$	−7.80068	−0.258023
$915$	−0.0776324	−0.00256645
$916$	−33.4584	−1.10550
$917$	−1.05840	−0.0349514
$918$	−3.95933	−0.130678
$919$	−23.2518	−0.767007	−0.383504	−	0.923539i	$-0.625283\pi$
$919$	−23.2518	−0.767007	−0.383504	+	0.923539i	$0.625283\pi$
$920$	6.98689	0.230351
$921$	−1.16301	−0.0383224
$922$	1.62709	0.0535852
$923$	−9.79954	−0.322556
$924$	−0.255628	−0.00840953
$925$	3.51111	0.115445
$926$	−23.1512	−0.760794
$927$	−26.6367	−0.874864
$928$	5.20447	0.170845
$929$	−11.0442	−0.362348	−0.181174	−	0.983451i	$-0.557990\pi$
$929$	−11.0442	−0.362348	−0.181174	+	0.983451i	$0.557990\pi$
$930$	−0.719207	−0.0235837
$931$	5.88322	0.192815
$932$	1.36003	0.0445494
$933$	3.24335	0.106182
$934$	26.8785	0.879492
$935$	−5.65135	−0.184819
$936$	28.9594	0.946568
$937$	−11.1871	−0.365468	−0.182734	−	0.983162i	$-0.558495\pi$
$937$	−11.1871	−0.365468	−0.182734	+	0.983162i	$0.558495\pi$
$938$	−0.291296	−0.00951115
$939$	−0.662722	−0.0216271
$940$	11.4981	0.375028
$941$	10.2267	0.333380	0.166690	−	0.986009i	$-0.446692\pi$
$941$	10.2267	0.333380	0.166690	+	0.986009i	$0.446692\pi$
$942$	−0.930975	−0.0303328
$943$	−6.36148	−0.207158
$944$	−2.53583	−0.0825342
$945$	1.02967	0.0334950
$946$	2.81283	0.0914529
$947$	−21.1154	−0.686157	−0.343079	−	0.939307i	$-0.611470\pi$
$947$	−21.1154	−0.686157	−0.343079	+	0.939307i	$0.611470\pi$
$948$	1.03379	0.0335760
$949$	−41.9563	−1.36196
$950$	−0.719047	−0.0233290
$951$	0.171194	0.00555136
$952$	−14.9570	−0.484758
$953$	−15.7726	−0.510925	−0.255463	−	0.966819i	$-0.582228\pi$
$953$	−15.7726	−0.510925	−0.255463	+	0.966819i	$0.582228\pi$
$954$	−16.2756	−0.526943
$955$	22.0025	0.711984
$956$	−42.2399	−1.36614
$957$	0.145198	0.00469360
$958$	16.1329	0.521230
$959$	14.5223	0.468948
$960$	−0.305630	−0.00986415
$961$	6.60189	0.212964
$962$	9.81820	0.316552
$963$	5.88180	0.189538
$964$	−4.10650	−0.132261
$965$	−15.6976	−0.505323
$966$	0.345787	0.0111255
$967$	−45.7966	−1.47272	−0.736359	−	0.676591i	$-0.763457\pi$
$967$	−45.7966	−1.47272	−0.736359	+	0.676591i	$0.763457\pi$
$968$	−2.50442	−0.0804951
$969$	−0.921815	−0.0296130
$970$	4.85746	0.155964
$971$	0.966326	0.0310109	0.0155054	−	0.999880i	$-0.495064\pi$
$971$	0.966326	0.0310109	0.0155054	+	0.999880i	$0.495064\pi$
$972$	−6.45406	−0.207014
$973$	−14.2180	−0.455808
$974$	−5.56575	−0.178338
$975$	0.634339	0.0203151
$976$	0.554538	0.0177503
$977$	−42.9904	−1.37539	−0.687693	−	0.726002i	$-0.741376\pi$
$977$	−42.9904	−1.37539	−0.687693	+	0.726002i	$0.741376\pi$
$978$	0.571047	0.0182601
$979$	8.36845	0.267457
$980$	−8.72465	−0.278699
$981$	2.51706	0.0803635
$982$	2.11918	0.0676258
$983$	1.22382	0.0390337	0.0195169	−	0.999810i	$-0.493787\pi$
$983$	1.22382	0.0390337	0.0195169	+	0.999810i	$0.493787\pi$
$984$	0.931495	0.0296950
$985$	9.97970	0.317980
$986$	3.61726	0.115197
$987$	1.33650	0.0425413
$988$	5.76717	0.183478
$989$	10.9135	0.347028
$990$	2.13801	0.0679504
$991$	−54.2437	−1.72311	−0.861554	−	0.507666i	$-0.830508\pi$
$991$	−54.2437	−1.72311	−0.861554	+	0.507666i	$0.830508\pi$
$992$	35.8518	1.13830
$993$	2.99592	0.0950727
$994$	−1.91477	−0.0607328
$995$	4.99525	0.158360
$996$	0.337400	0.0106909
$997$	−36.8826	−1.16808	−0.584041	−	0.811724i	$-0.698529\pi$
$997$	−36.8826	−1.16808	−0.584041	+	0.811724i	$0.698529\pi$
$998$	6.01305	0.190340
$999$	−3.42103	−0.108237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.h.1.4	✓	7
3.2	odd	2		9405.2.a.bd.1.4		7
5.4	even	2		5225.2.a.m.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.h.1.4	✓	7	1.1	even	1	trivial
5225.2.a.m.1.4		7	5.4	even	2
9405.2.a.bd.1.4		7	3.2	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.719047 q^{2} +0.163114 q^{3} -1.48297 q^{4} -1.00000 q^{5} +0.117287 q^{6} +1.05678 q^{7} -2.50442 q^{8} -2.97339 q^{9} +O(q^{10})\)	\(q+0.719047 q^{2} +0.163114 q^{3} -1.48297 q^{4} -1.00000 q^{5} +0.117287 q^{6} +1.05678 q^{7} -2.50442 q^{8} -2.97339 q^{9} -0.719047 q^{10} +1.00000 q^{11} -0.241893 q^{12} +3.88893 q^{13} +0.759873 q^{14} -0.163114 q^{15} +1.16515 q^{16} +5.65135 q^{17} -2.13801 q^{18} -1.00000 q^{19} +1.48297 q^{20} +0.172375 q^{21} +0.719047 q^{22} +2.78982 q^{23} -0.408506 q^{24} +1.00000 q^{25} +2.79632 q^{26} -0.974345 q^{27} -1.56717 q^{28} +0.890165 q^{29} -0.117287 q^{30} +6.13204 q^{31} +5.84664 q^{32} +0.163114 q^{33} +4.06359 q^{34} -1.05678 q^{35} +4.40946 q^{36} +3.51111 q^{37} -0.719047 q^{38} +0.634339 q^{39} +2.50442 q^{40} -2.28025 q^{41} +0.123946 q^{42} +3.91188 q^{43} -1.48297 q^{44} +2.97339 q^{45} +2.00601 q^{46} +7.75344 q^{47} +0.190052 q^{48} -5.88322 q^{49} +0.719047 q^{50} +0.921815 q^{51} -5.76717 q^{52} +7.61251 q^{53} -0.700600 q^{54} -1.00000 q^{55} -2.64662 q^{56} -0.163114 q^{57} +0.640071 q^{58} -2.17640 q^{59} +0.241893 q^{60} +0.475939 q^{61} +4.40923 q^{62} -3.14222 q^{63} +1.87372 q^{64} -3.88893 q^{65} +0.117287 q^{66} -0.383348 q^{67} -8.38079 q^{68} +0.455059 q^{69} -0.759873 q^{70} -2.51986 q^{71} +7.44663 q^{72} -10.7887 q^{73} +2.52465 q^{74} +0.163114 q^{75} +1.48297 q^{76} +1.05678 q^{77} +0.456120 q^{78} -4.27374 q^{79} -1.16515 q^{80} +8.76125 q^{81} -1.63960 q^{82} -1.39483 q^{83} -0.255628 q^{84} -5.65135 q^{85} +2.81283 q^{86} +0.145198 q^{87} -2.50442 q^{88} +8.36845 q^{89} +2.13801 q^{90} +4.10973 q^{91} -4.13723 q^{92} +1.00022 q^{93} +5.57509 q^{94} +1.00000 q^{95} +0.953669 q^{96} -6.75542 q^{97} -4.23031 q^{98} -2.97339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9	\( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9 - q^10 + 7 * q^11 + 13 * q^12 + q^13 + 12 * q^14 - 3 * q^15 + 3 * q^16 + q^17 + 7 * q^18 - 7 * q^19 - 7 * q^20 + 5 * q^21 + q^22 - 8 * q^23 + 25 * q^24 + 7 * q^25 + 12 * q^27 + 4 * q^28 + 11 * q^29 - 8 * q^30 + 7 * q^31 + 12 * q^32 + 3 * q^33 - 14 * q^34 + q^35 + 7 * q^36 - 17 * q^37 - q^38 + 30 * q^39 - 3 * q^40 + 17 * q^41 + 33 * q^42 - 3 * q^43 + 7 * q^44 - 2 * q^45 + 18 * q^46 + 14 * q^47 - 12 * q^48 + 6 * q^49 + q^50 + 8 * q^51 - 17 * q^52 + 7 * q^53 - 27 * q^54 - 7 * q^55 + 36 * q^56 - 3 * q^57 - 15 * q^58 + 35 * q^59 - 13 * q^60 + 17 * q^61 + 46 * q^62 - 22 * q^63 + 5 * q^64 - q^65 + 8 * q^66 + 4 * q^67 - 35 * q^68 - 4 * q^69 - 12 * q^70 + 10 * q^71 + 12 * q^72 + 22 * q^73 - 11 * q^74 + 3 * q^75 - 7 * q^76 - q^77 - 41 * q^78 + 11 * q^79 - 3 * q^80 - 21 * q^81 - 14 * q^82 + 39 * q^83 + 21 * q^84 - q^85 - 24 * q^86 - 2 * q^87 + 3 * q^88 + 18 * q^89 - 7 * q^90 - 22 * q^91 - 51 * q^92 + 10 * q^93 + 14 * q^94 + 7 * q^95 - 11 * q^96 - 4 * q^97 - 26 * q^98 + 2 * q^99

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