LMFDB - Embedded newform 4009.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70369 q^{2} +1.93926 q^{3} +5.30992 q^{4} +1.68888 q^{5} -5.24315 q^{6} +4.83108 q^{7} -8.94900 q^{8} +0.760725 q^{9} +O(q^{10})$	\(q-2.70369 q^{2} +1.93926 q^{3} +5.30992 q^{4} +1.68888 q^{5} -5.24315 q^{6} +4.83108 q^{7} -8.94900 q^{8} +0.760725 q^{9} -4.56619 q^{10} -3.10973 q^{11} +10.2973 q^{12} -3.06975 q^{13} -13.0617 q^{14} +3.27517 q^{15} +13.5754 q^{16} -6.37810 q^{17} -2.05676 q^{18} +1.00000 q^{19} +8.96780 q^{20} +9.36872 q^{21} +8.40773 q^{22} -5.89111 q^{23} -17.3544 q^{24} -2.14770 q^{25} +8.29965 q^{26} -4.34253 q^{27} +25.6527 q^{28} -10.1889 q^{29} -8.85503 q^{30} -7.88562 q^{31} -18.8058 q^{32} -6.03057 q^{33} +17.2444 q^{34} +8.15910 q^{35} +4.03939 q^{36} +6.79489 q^{37} -2.70369 q^{38} -5.95304 q^{39} -15.1138 q^{40} +6.98418 q^{41} -25.3301 q^{42} -6.77347 q^{43} -16.5124 q^{44} +1.28477 q^{45} +15.9277 q^{46} +12.4136 q^{47} +26.3263 q^{48} +16.3393 q^{49} +5.80670 q^{50} -12.3688 q^{51} -16.3001 q^{52} -9.18206 q^{53} +11.7409 q^{54} -5.25194 q^{55} -43.2333 q^{56} +1.93926 q^{57} +27.5475 q^{58} -3.55429 q^{59} +17.3909 q^{60} +5.04153 q^{61} +21.3202 q^{62} +3.67512 q^{63} +23.6940 q^{64} -5.18443 q^{65} +16.3048 q^{66} -3.69827 q^{67} -33.8672 q^{68} -11.4244 q^{69} -22.0596 q^{70} +0.0274348 q^{71} -6.80773 q^{72} +1.57094 q^{73} -18.3712 q^{74} -4.16494 q^{75} +5.30992 q^{76} -15.0233 q^{77} +16.0952 q^{78} -16.4961 q^{79} +22.9273 q^{80} -10.7035 q^{81} -18.8830 q^{82} +7.74228 q^{83} +49.7472 q^{84} -10.7718 q^{85} +18.3134 q^{86} -19.7588 q^{87} +27.8290 q^{88} +2.26865 q^{89} -3.47362 q^{90} -14.8302 q^{91} -31.2813 q^{92} -15.2922 q^{93} -33.5626 q^{94} +1.68888 q^{95} -36.4693 q^{96} +0.814711 q^{97} -44.1765 q^{98} -2.36565 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−2.70369	−1.91180	−0.955898	−	0.293700i	$-0.905113\pi$
$2$	−2.70369	−1.91180	−0.955898	+	0.293700i	$0.905113\pi$
$3$	1.93926	1.11963	0.559816	−	0.828617i	$-0.310872\pi$
$3$	1.93926	1.11963	0.559816	+	0.828617i	$0.310872\pi$
$4$	5.30992	2.65496
$5$	1.68888	0.755288	0.377644	−	0.925951i	$-0.376734\pi$
$5$	1.68888	0.755288	0.377644	+	0.925951i	$0.376734\pi$
$6$	−5.24315	−2.14051
$7$	4.83108	1.82598	0.912989	−	0.407985i	$-0.133769\pi$
$7$	4.83108	1.82598	0.912989	+	0.407985i	$0.133769\pi$
$8$	−8.94900	−3.16395
$9$	0.760725	0.253575
$10$	−4.56619	−1.44396
$11$	−3.10973	−0.937618	−0.468809	−	0.883300i	$-0.655317\pi$
$11$	−3.10973	−0.937618	−0.468809	+	0.883300i	$0.655317\pi$
$12$	10.2973	2.97258
$13$	−3.06975	−0.851396	−0.425698	−	0.904865i	$-0.639971\pi$
$13$	−3.06975	−0.851396	−0.425698	+	0.904865i	$0.639971\pi$
$14$	−13.0617	−3.49089
$15$	3.27517	0.845645
$16$	13.5754	3.39386
$17$	−6.37810	−1.54692	−0.773459	−	0.633847i	$-0.781475\pi$
$17$	−6.37810	−1.54692	−0.773459	+	0.633847i	$0.781475\pi$
$18$	−2.05676	−0.484784
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	8.96780	2.00526
$21$	9.36872	2.04442
$22$	8.40773	1.79253
$23$	−5.89111	−1.22838	−0.614190	−	0.789158i	$-0.710517\pi$
$23$	−5.89111	−1.22838	−0.614190	+	0.789158i	$0.710517\pi$
$24$	−17.3544	−3.54246
$25$	−2.14770	−0.429539
$26$	8.29965	1.62769
$27$	−4.34253	−0.835721
$28$	25.6527	4.84790
$29$	−10.1889	−1.89202	−0.946011	−	0.324134i	$-0.894927\pi$
$29$	−10.1889	−1.89202	−0.946011	+	0.324134i	$0.894927\pi$
$30$	−8.85503	−1.61670
$31$	−7.88562	−1.41630	−0.708149	−	0.706063i	$-0.750470\pi$
$31$	−7.88562	−1.41630	−0.708149	+	0.706063i	$0.750470\pi$
$32$	−18.8058	−3.32442
$33$	−6.03057	−1.04979
$34$	17.2444	2.95739
$35$	8.15910	1.37914
$36$	4.03939	0.673232
$37$	6.79489	1.11707	0.558536	−	0.829480i	$-0.311363\pi$
$37$	6.79489	1.11707	0.558536	+	0.829480i	$0.311363\pi$
$38$	−2.70369	−0.438596
$39$	−5.95304	−0.953250
$40$	−15.1138	−2.38969
$41$	6.98418	1.09075	0.545373	−	0.838194i	$-0.316388\pi$
$41$	6.98418	1.09075	0.545373	+	0.838194i	$0.316388\pi$
$42$	−25.3301	−3.90852
$43$	−6.77347	−1.03294	−0.516472	−	0.856304i	$-0.672755\pi$
$43$	−6.77347	−1.03294	−0.516472	+	0.856304i	$0.672755\pi$
$44$	−16.5124	−2.48934
$45$	1.28477	0.191522
$46$	15.9277	2.34841
$47$	12.4136	1.81071	0.905357	−	0.424651i	$-0.139603\pi$
$47$	12.4136	1.81071	0.905357	+	0.424651i	$0.139603\pi$
$48$	26.3263	3.79988
$49$	16.3393	2.33419
$50$	5.80670	0.821192
$51$	−12.3688	−1.73198
$52$	−16.3001	−2.26042
$53$	−9.18206	−1.26125	−0.630627	−	0.776086i	$-0.717202\pi$
$53$	−9.18206	−1.26125	−0.630627	+	0.776086i	$0.717202\pi$
$54$	11.7409	1.59773
$55$	−5.25194	−0.708172
$56$	−43.2333	−5.77730
$57$	1.93926	0.256861
$58$	27.5475	3.61716
$59$	−3.55429	−0.462729	−0.231365	−	0.972867i	$-0.574319\pi$
$59$	−3.55429	−0.462729	−0.231365	+	0.972867i	$0.574319\pi$
$60$	17.3909	2.24515
$61$	5.04153	0.645501	0.322751	−	0.946484i	$-0.395392\pi$
$61$	5.04153	0.645501	0.322751	+	0.946484i	$0.395392\pi$
$62$	21.3202	2.70767
$63$	3.67512	0.463022
$64$	23.6940	2.96175
$65$	−5.18443	−0.643049
$66$	16.3048	2.00698
$67$	−3.69827	−0.451816	−0.225908	−	0.974149i	$-0.572535\pi$
$67$	−3.69827	−0.451816	−0.225908	+	0.974149i	$0.572535\pi$
$68$	−33.8672	−4.10701
$69$	−11.4244	−1.37533
$70$	−22.0596	−2.63663
$71$	0.0274348	0.00325591	0.00162795	−	0.999999i	$-0.499482\pi$
$71$	0.0274348	0.00325591	0.00162795	+	0.999999i	$0.499482\pi$
$72$	−6.80773	−0.802298
$73$	1.57094	0.183864	0.0919321	−	0.995765i	$-0.470696\pi$
$73$	1.57094	0.183864	0.0919321	+	0.995765i	$0.470696\pi$
$74$	−18.3712	−2.13561
$75$	−4.16494	−0.480926
$76$	5.30992	0.609090
$77$	−15.0233	−1.71207
$78$	16.0952	1.82242
$79$	−16.4961	−1.85596	−0.927979	−	0.372632i	$-0.878456\pi$
$79$	−16.4961	−1.85596	−0.927979	+	0.372632i	$0.878456\pi$
$80$	22.9273	2.56334
$81$	−10.7035	−1.18927
$82$	−18.8830	−2.08528
$83$	7.74228	0.849826	0.424913	−	0.905234i	$-0.360305\pi$
$83$	7.74228	0.849826	0.424913	+	0.905234i	$0.360305\pi$
$84$	49.7472	5.42786
$85$	−10.7718	−1.16837
$86$	18.3134	1.97478
$87$	−19.7588	−2.11837
$88$	27.8290	2.96658
$89$	2.26865	0.240476	0.120238	−	0.992745i	$-0.461634\pi$
$89$	2.26865	0.240476	0.120238	+	0.992745i	$0.461634\pi$
$90$	−3.47362	−0.366151
$91$	−14.8302	−1.55463
$92$	−31.2813	−3.26130
$93$	−15.2922	−1.58573
$94$	−33.5626	−3.46172
$95$	1.68888	0.173275
$96$	−36.4693	−3.72213
$97$	0.814711	0.0827214	0.0413607	−	0.999144i	$-0.486831\pi$
$97$	0.814711	0.0827214	0.0413607	+	0.999144i	$0.486831\pi$
$98$	−44.1765	−4.46250
$99$	−2.36565	−0.237756
$100$	−11.4041	−1.14041
$101$	−4.21383	−0.419291	−0.209646	−	0.977777i	$-0.567231\pi$
$101$	−4.21383	−0.419291	−0.209646	+	0.977777i	$0.567231\pi$
$102$	33.4413	3.31119
$103$	17.4058	1.71504	0.857522	−	0.514448i	$-0.172003\pi$
$103$	17.4058	1.71504	0.857522	+	0.514448i	$0.172003\pi$
$104$	27.4712	2.69377
$105$	15.8226	1.54413
$106$	24.8254	2.41126
$107$	−17.2405	−1.66670	−0.833351	−	0.552745i	$-0.813581\pi$
$107$	−17.2405	−1.66670	−0.833351	+	0.552745i	$0.813581\pi$
$108$	−23.0585	−2.21881
$109$	−3.40367	−0.326012	−0.163006	−	0.986625i	$-0.552119\pi$
$109$	−3.40367	−0.326012	−0.163006	+	0.986625i	$0.552119\pi$
$110$	14.1996	1.35388
$111$	13.1770	1.25071
$112$	65.5841	6.19711
$113$	5.36054	0.504277	0.252138	−	0.967691i	$-0.418866\pi$
$113$	5.36054	0.504277	0.252138	+	0.967691i	$0.418866\pi$
$114$	−5.24315	−0.491066
$115$	−9.94935	−0.927782
$116$	−54.1020	−5.02325
$117$	−2.33524	−0.215893
$118$	9.60969	0.884644
$119$	−30.8131	−2.82464
$120$	−29.3095	−2.67558
$121$	−1.32960	−0.120872
$122$	−13.6307	−1.23407
$123$	13.5441	1.22123
$124$	−41.8720	−3.76022
$125$	−12.0716	−1.07971
$126$	−9.93639	−0.885204
$127$	−9.28280	−0.823715	−0.411858	−	0.911248i	$-0.635120\pi$
$127$	−9.28280	−0.823715	−0.411858	+	0.911248i	$0.635120\pi$
$128$	−26.4497	−2.33784
$129$	−13.1355	−1.15652
$130$	14.0171	1.22938
$131$	9.21365	0.805000	0.402500	−	0.915420i	$-0.368141\pi$
$131$	9.21365	0.805000	0.402500	+	0.915420i	$0.368141\pi$
$132$	−32.0219	−2.78714
$133$	4.83108	0.418908
$134$	9.99898	0.863780
$135$	−7.33400	−0.631210
$136$	57.0777	4.89437
$137$	−0.919109	−0.0785248	−0.0392624	−	0.999229i	$-0.512501\pi$
$137$	−0.919109	−0.0785248	−0.0392624	+	0.999229i	$0.512501\pi$
$138$	30.8880	2.62936
$139$	5.62081	0.476751	0.238376	−	0.971173i	$-0.423385\pi$
$139$	5.62081	0.476751	0.238376	+	0.971173i	$0.423385\pi$
$140$	43.3242	3.66156
$141$	24.0732	2.02733
$142$	−0.0741750	−0.00622463
$143$	9.54609	0.798284
$144$	10.3272	0.860599
$145$	−17.2077	−1.42902
$146$	−4.24732	−0.351511
$147$	31.6862	2.61344
$148$	36.0803	2.96579
$149$	−0.930135	−0.0761996	−0.0380998	−	0.999274i	$-0.512130\pi$
$149$	−0.930135	−0.0761996	−0.0380998	+	0.999274i	$0.512130\pi$
$150$	11.2607	0.919432
$151$	−1.42515	−0.115977	−0.0579887	−	0.998317i	$-0.518469\pi$
$151$	−1.42515	−0.115977	−0.0579887	+	0.998317i	$0.518469\pi$
$152$	−8.94900	−0.725860
$153$	−4.85198	−0.392260
$154$	40.6184	3.27313
$155$	−13.3178	−1.06971
$156$	−31.6102	−2.53084
$157$	−16.4956	−1.31649	−0.658246	−	0.752803i	$-0.728701\pi$
$157$	−16.4956	−1.31649	−0.658246	+	0.752803i	$0.728701\pi$
$158$	44.6004	3.54821
$159$	−17.8064	−1.41214
$160$	−31.7606	−2.51090
$161$	−28.4604	−2.24300
$162$	28.9388	2.27365
$163$	8.10033	0.634467	0.317233	−	0.948347i	$-0.397246\pi$
$163$	8.10033	0.634467	0.317233	+	0.948347i	$0.397246\pi$
$164$	37.0855	2.89589
$165$	−10.1849	−0.792892
$166$	−20.9327	−1.62469
$167$	−9.27610	−0.717806	−0.358903	−	0.933375i	$-0.616849\pi$
$167$	−9.27610	−0.717806	−0.358903	+	0.933375i	$0.616849\pi$
$168$	−83.8407	−6.46845
$169$	−3.57663	−0.275125
$170$	29.1237	2.23368
$171$	0.760725	0.0581741
$172$	−35.9666	−2.74243
$173$	5.72964	0.435616	0.217808	−	0.975992i	$-0.430109\pi$
$173$	5.72964	0.435616	0.217808	+	0.975992i	$0.430109\pi$
$174$	53.4217	4.04989
$175$	−10.3757	−0.784329
$176$	−42.2159	−3.18215
$177$	−6.89269	−0.518086
$178$	−6.13371	−0.459741
$179$	−1.35140	−0.101008	−0.0505041	−	0.998724i	$-0.516083\pi$
$179$	−1.35140	−0.101008	−0.0505041	+	0.998724i	$0.516083\pi$
$180$	6.82203	0.508484
$181$	12.1641	0.904150	0.452075	−	0.891980i	$-0.350684\pi$
$181$	12.1641	0.904150	0.452075	+	0.891980i	$0.350684\pi$
$182$	40.0963	2.97213
$183$	9.77683	0.722724
$184$	52.7195	3.88653
$185$	11.4757	0.843712
$186$	41.3455	3.03160
$187$	19.8342	1.45042
$188$	65.9154	4.80738
$189$	−20.9791	−1.52601
$190$	−4.56619	−0.331266
$191$	21.4709	1.55358	0.776791	−	0.629759i	$-0.216846\pi$
$191$	21.4709	1.55358	0.776791	+	0.629759i	$0.216846\pi$
$192$	45.9488	3.31607
$193$	−1.28063	−0.0921816	−0.0460908	−	0.998937i	$-0.514676\pi$
$193$	−1.28063	−0.0921816	−0.0460908	+	0.998937i	$0.514676\pi$
$194$	−2.20272	−0.158146
$195$	−10.0540	−0.719978
$196$	86.7607	6.19719
$197$	21.2115	1.51126	0.755628	−	0.655000i	$-0.227332\pi$
$197$	21.2115	1.51126	0.755628	+	0.655000i	$0.227332\pi$
$198$	6.39597	0.454542
$199$	−15.6379	−1.10854	−0.554271	−	0.832337i	$-0.687003\pi$
$199$	−15.6379	−1.10854	−0.554271	+	0.832337i	$0.687003\pi$
$200$	19.2197	1.35904
$201$	−7.17191	−0.505868
$202$	11.3929	0.801599
$203$	−49.2232	−3.45479
$204$	−65.6774	−4.59834
$205$	11.7954	0.823827
$206$	−47.0598	−3.27881
$207$	−4.48151	−0.311487
$208$	−41.6733	−2.88952
$209$	−3.10973	−0.215104
$210$	−42.7794	−2.95206
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−48.7561	−3.34858
$213$	0.0532031	0.00364542
$214$	46.6129	3.18639
$215$	−11.4396	−0.780171
$216$	38.8613	2.64418
$217$	−38.0960	−2.58613
$218$	9.20246	0.623269
$219$	3.04645	0.205860
$220$	−27.8874	−1.88017
$221$	19.5792	1.31704
$222$	−35.6266	−2.39110
$223$	16.5049	1.10525	0.552626	−	0.833429i	$-0.313626\pi$
$223$	16.5049	1.10525	0.552626	+	0.833429i	$0.313626\pi$
$224$	−90.8522	−6.07032
$225$	−1.63381	−0.108920
$226$	−14.4932	−0.964074
$227$	13.6090	0.903259	0.451629	−	0.892206i	$-0.350843\pi$
$227$	13.6090	0.903259	0.451629	+	0.892206i	$0.350843\pi$
$228$	10.2973	0.681957
$229$	6.67579	0.441149	0.220574	−	0.975370i	$-0.429207\pi$
$229$	6.67579	0.441149	0.220574	+	0.975370i	$0.429207\pi$
$230$	26.8999	1.77373
$231$	−29.1342	−1.91689
$232$	91.1801	5.98626
$233$	−28.3415	−1.85672	−0.928358	−	0.371687i	$-0.878779\pi$
$233$	−28.3415	−1.85672	−0.928358	+	0.371687i	$0.878779\pi$
$234$	6.31375	0.412743
$235$	20.9651	1.36761
$236$	−18.8730	−1.22853
$237$	−31.9903	−2.07799
$238$	83.3091	5.40013
$239$	8.59965	0.556265	0.278132	−	0.960543i	$-0.410285\pi$
$239$	8.59965	0.556265	0.278132	+	0.960543i	$0.410285\pi$
$240$	44.4619	2.87000
$241$	−20.7898	−1.33919	−0.669594	−	0.742727i	$-0.733532\pi$
$241$	−20.7898	−1.33919	−0.669594	+	0.742727i	$0.733532\pi$
$242$	3.59481	0.231083
$243$	−7.72920	−0.495829
$244$	26.7701	1.71378
$245$	27.5951	1.76299
$246$	−36.6191	−2.33475
$247$	−3.06975	−0.195324
$248$	70.5684	4.48110
$249$	15.0143	0.951492
$250$	32.6378	2.06419
$251$	30.1707	1.90436	0.952180	−	0.305538i	$-0.0988363\pi$
$251$	30.1707	1.90436	0.952180	+	0.305538i	$0.0988363\pi$
$252$	19.5146	1.22931
$253$	18.3197	1.15175
$254$	25.0978	1.57478
$255$	−20.8894	−1.30814
$256$	24.1236	1.50773
$257$	−8.84419	−0.551685	−0.275843	−	0.961203i	$-0.588957\pi$
$257$	−8.84419	−0.551685	−0.275843	+	0.961203i	$0.588957\pi$
$258$	35.5143	2.21103
$259$	32.8267	2.03975
$260$	−27.5289	−1.70727
$261$	−7.75092	−0.479770
$262$	−24.9108	−1.53900
$263$	11.5914	0.714757	0.357378	−	0.933960i	$-0.383671\pi$
$263$	11.5914	0.714757	0.357378	+	0.933960i	$0.383671\pi$
$264$	53.9675	3.32147
$265$	−15.5074	−0.952610
$266$	−13.0617	−0.800866
$267$	4.39949	0.269245
$268$	−19.6376	−1.19955
$269$	26.6346	1.62394	0.811972	−	0.583697i	$-0.198394\pi$
$269$	26.6346	1.62394	0.811972	+	0.583697i	$0.198394\pi$
$270$	19.8288	1.20675
$271$	−4.54285	−0.275958	−0.137979	−	0.990435i	$-0.544061\pi$
$271$	−4.54285	−0.275958	−0.137979	+	0.990435i	$0.544061\pi$
$272$	−86.5856	−5.25002
$273$	−28.7596	−1.74061
$274$	2.48498	0.150123
$275$	6.67875	0.402744
$276$	−60.6626	−3.65146
$277$	−17.1463	−1.03022	−0.515112	−	0.857123i	$-0.672250\pi$
$277$	−17.1463	−1.03022	−0.515112	+	0.857123i	$0.672250\pi$
$278$	−15.1969	−0.911451
$279$	−5.99878	−0.359138
$280$	−73.0158	−4.36353
$281$	27.1929	1.62219	0.811097	−	0.584912i	$-0.198871\pi$
$281$	27.1929	1.62219	0.811097	+	0.584912i	$0.198871\pi$
$282$	−65.0865	−3.87585
$283$	8.18221	0.486382	0.243191	−	0.969978i	$-0.421806\pi$
$283$	8.18221	0.486382	0.243191	+	0.969978i	$0.421806\pi$
$284$	0.145677	0.00864431
$285$	3.27517	0.194004
$286$	−25.8096	−1.52616
$287$	33.7411	1.99168
$288$	−14.3060	−0.842990
$289$	23.6802	1.39295
$290$	46.5243	2.73200
$291$	1.57994	0.0926175
$292$	8.34156	0.488153
$293$	−3.24484	−0.189566	−0.0947828	−	0.995498i	$-0.530216\pi$
$293$	−3.24484	−0.189566	−0.0947828	+	0.995498i	$0.530216\pi$
$294$	−85.6696	−4.99635
$295$	−6.00276	−0.349494
$296$	−60.8074	−3.53436
$297$	13.5041	0.783587
$298$	2.51479	0.145678
$299$	18.0842	1.04584
$300$	−22.1155	−1.27684
$301$	−32.7232	−1.88613
$302$	3.85317	0.221725
$303$	−8.17170	−0.469452
$304$	13.5754	0.778605
$305$	8.51452	0.487540
$306$	13.1182	0.749920
$307$	−13.3760	−0.763408	−0.381704	−	0.924285i	$-0.624663\pi$
$307$	−13.3760	−0.763408	−0.381704	+	0.924285i	$0.624663\pi$
$308$	−79.7728	−4.54548
$309$	33.7543	1.92022
$310$	36.0072	2.04507
$311$	−14.1492	−0.802327	−0.401163	−	0.916007i	$-0.631394\pi$
$311$	−14.1492	−0.802327	−0.401163	+	0.916007i	$0.631394\pi$
$312$	53.2738	3.01603
$313$	−14.3166	−0.809219	−0.404610	−	0.914489i	$-0.632593\pi$
$313$	−14.3166	−0.809219	−0.404610	+	0.914489i	$0.632593\pi$
$314$	44.5989	2.51686
$315$	6.20683	0.349715
$316$	−87.5932	−4.92750
$317$	1.13777	0.0639033	0.0319517	−	0.999489i	$-0.489828\pi$
$317$	1.13777	0.0639033	0.0319517	+	0.999489i	$0.489828\pi$
$318$	48.1429	2.69972
$319$	31.6846	1.77399
$320$	40.0163	2.23698
$321$	−33.4338	−1.86609
$322$	76.9481	4.28815
$323$	−6.37810	−0.354887
$324$	−56.8346	−3.15748
$325$	6.59290	0.365708
$326$	−21.9008	−1.21297
$327$	−6.60060	−0.365014
$328$	−62.5014	−3.45106
$329$	59.9713	3.30632
$330$	27.5367	1.51585
$331$	27.0806	1.48849	0.744243	−	0.667909i	$-0.232810\pi$
$331$	27.0806	1.48849	0.744243	+	0.667909i	$0.232810\pi$
$332$	41.1109	2.25626
$333$	5.16904	0.283262
$334$	25.0797	1.37230
$335$	−6.24593	−0.341251
$336$	127.185	6.93849
$337$	−17.7083	−0.964634	−0.482317	−	0.875997i	$-0.660205\pi$
$337$	−17.7083	−0.964634	−0.482317	+	0.875997i	$0.660205\pi$
$338$	9.67008	0.525983
$339$	10.3955	0.564604
$340$	−57.1976	−3.10197
$341$	24.5221	1.32795
$342$	−2.05676	−0.111217
$343$	45.1191	2.43620
$344$	60.6158	3.26819
$345$	−19.2944	−1.03877
$346$	−15.4912	−0.832810
$347$	8.60374	0.461873	0.230937	−	0.972969i	$-0.425821\pi$
$347$	8.60374	0.461873	0.230937	+	0.972969i	$0.425821\pi$
$348$	−104.918	−5.62419
$349$	−24.3516	−1.30351	−0.651755	−	0.758430i	$-0.725967\pi$
$349$	−24.3516	−1.30351	−0.651755	+	0.758430i	$0.725967\pi$
$350$	28.0526	1.49948
$351$	13.3305	0.711529
$352$	58.4808	3.11704
$353$	−1.22951	−0.0654402	−0.0327201	−	0.999465i	$-0.510417\pi$
$353$	−1.22951	−0.0654402	−0.0327201	+	0.999465i	$0.510417\pi$
$354$	18.6357	0.990475
$355$	0.0463339	0.00245915
$356$	12.0463	0.638455
$357$	−59.7546	−3.16255
$358$	3.65376	0.193107
$359$	−0.967710	−0.0510738	−0.0255369	−	0.999674i	$-0.508130\pi$
$359$	−0.967710	−0.0510738	−0.0255369	+	0.999674i	$0.508130\pi$
$360$	−11.4974	−0.605967
$361$	1.00000	0.0526316
$362$	−32.8879	−1.72855
$363$	−2.57843	−0.135333
$364$	−78.7473	−4.12748
$365$	2.65312	0.138871
$366$	−26.4335	−1.38170
$367$	15.4235	0.805099	0.402550	−	0.915398i	$-0.368124\pi$
$367$	15.4235	0.805099	0.402550	+	0.915398i	$0.368124\pi$
$368$	−79.9744	−4.16896
$369$	5.31304	0.276586
$370$	−31.0268	−1.61300
$371$	−44.3593	−2.30302
$372$	−81.2007	−4.21006
$373$	20.8775	1.08100	0.540498	−	0.841345i	$-0.318236\pi$
$373$	20.8775	1.08100	0.540498	+	0.841345i	$0.318236\pi$
$374$	−53.6254	−2.77290
$375$	−23.4099	−1.20888
$376$	−111.090	−5.72901
$377$	31.2772	1.61086
$378$	56.7210	2.91741
$379$	21.2132	1.08965	0.544824	−	0.838550i	$-0.316596\pi$
$379$	21.2132	1.08965	0.544824	+	0.838550i	$0.316596\pi$
$380$	8.96780	0.460039
$381$	−18.0018	−0.922258
$382$	−58.0507	−2.97013
$383$	−36.6112	−1.87075	−0.935373	−	0.353663i	$-0.884936\pi$
$383$	−36.6112	−1.87075	−0.935373	+	0.353663i	$0.884936\pi$
$384$	−51.2928	−2.61752
$385$	−25.3726	−1.29311
$386$	3.46242	0.176232
$387$	−5.15275	−0.261929
$388$	4.32605	0.219622
$389$	−9.77670	−0.495698	−0.247849	−	0.968799i	$-0.579724\pi$
$389$	−9.77670	−0.495698	−0.247849	+	0.968799i	$0.579724\pi$
$390$	27.1827	1.37645
$391$	37.5741	1.90020
$392$	−146.221	−7.38527
$393$	17.8676	0.901304
$394$	−57.3493	−2.88921
$395$	−27.8599	−1.40178
$396$	−12.5614	−0.631235
$397$	11.3260	0.568438	0.284219	−	0.958759i	$-0.408266\pi$
$397$	11.3260	0.568438	0.284219	+	0.958759i	$0.408266\pi$
$398$	42.2800	2.11930
$399$	9.36872	0.469022
$400$	−29.1560	−1.45780
$401$	−22.0850	−1.10287	−0.551436	−	0.834217i	$-0.685920\pi$
$401$	−22.0850	−1.10287	−0.551436	+	0.834217i	$0.685920\pi$
$402$	19.3906	0.967115
$403$	24.2069	1.20583
$404$	−22.3751	−1.11320
$405$	−18.0768	−0.898245
$406$	133.084	6.60485
$407$	−21.1302	−1.04739
$408$	110.688	5.47989
$409$	−10.8790	−0.537931	−0.268966	−	0.963150i	$-0.586682\pi$
$409$	−10.8790	−0.537931	−0.268966	+	0.963150i	$0.586682\pi$
$410$	−31.8911	−1.57499
$411$	−1.78239	−0.0879188
$412$	92.4234	4.55338
$413$	−17.1711	−0.844933
$414$	12.1166	0.595499
$415$	13.0758	0.641863
$416$	57.7290	2.83040
$417$	10.9002	0.533786
$418$	8.40773	0.411236
$419$	−25.5771	−1.24952	−0.624762	−	0.780815i	$-0.714804\pi$
$419$	−25.5771	−1.24952	−0.624762	+	0.780815i	$0.714804\pi$
$420$	84.0168	4.09960
$421$	−7.61403	−0.371085	−0.185543	−	0.982636i	$-0.559404\pi$
$421$	−7.61403	−0.371085	−0.185543	+	0.982636i	$0.559404\pi$
$422$	−2.70369	−0.131613
$423$	9.44336	0.459152
$424$	82.1703	3.99054
$425$	13.6982	0.664462
$426$	−0.143845	−0.00696929
$427$	24.3560	1.17867
$428$	−91.5457	−4.42503
$429$	18.5123	0.893784
$430$	30.9290	1.49153
$431$	1.00617	0.0484657	0.0242329	−	0.999706i	$-0.492286\pi$
$431$	1.00617	0.0484657	0.0242329	+	0.999706i	$0.492286\pi$
$432$	−58.9518	−2.83632
$433$	−21.3721	−1.02708	−0.513540	−	0.858066i	$-0.671666\pi$
$433$	−21.3721	−1.02708	−0.513540	+	0.858066i	$0.671666\pi$
$434$	103.000	4.94415
$435$	−33.3702	−1.59998
$436$	−18.0732	−0.865551
$437$	−5.89111	−0.281810
$438$	−8.23666	−0.393563
$439$	−14.3007	−0.682536	−0.341268	−	0.939966i	$-0.610856\pi$
$439$	−14.3007	−0.682536	−0.341268	+	0.939966i	$0.610856\pi$
$440$	46.9997	2.24062
$441$	12.4297	0.591893
$442$	−52.9360	−2.51791
$443$	−0.899710	−0.0427465	−0.0213732	−	0.999772i	$-0.506804\pi$
$443$	−0.899710	−0.0427465	−0.0213732	+	0.999772i	$0.506804\pi$
$444$	69.9691	3.32059
$445$	3.83146	0.181629
$446$	−44.6242	−2.11302
$447$	−1.80377	−0.0853155
$448$	114.468	5.40809
$449$	6.11687	0.288673	0.144336	−	0.989529i	$-0.453895\pi$
$449$	6.11687	0.288673	0.144336	+	0.989529i	$0.453895\pi$
$450$	4.41730	0.208234
$451$	−21.7189	−1.02270
$452$	28.4640	1.33884
$453$	−2.76374	−0.129852
$454$	−36.7944	−1.72685
$455$	−25.0464	−1.17419
$456$	−17.3544	−0.812696
$457$	23.3192	1.09083	0.545413	−	0.838168i	$-0.316373\pi$
$457$	23.3192	1.09083	0.545413	+	0.838168i	$0.316373\pi$
$458$	−18.0493	−0.843387
$459$	27.6971	1.29279
$460$	−52.8303	−2.46323
$461$	6.24650	0.290928	0.145464	−	0.989364i	$-0.453532\pi$
$461$	6.24650	0.290928	0.145464	+	0.989364i	$0.453532\pi$
$462$	78.7696	3.66470
$463$	12.8265	0.596098	0.298049	−	0.954551i	$-0.403664\pi$
$463$	12.8265	0.596098	0.298049	+	0.954551i	$0.403664\pi$
$464$	−138.318	−6.42126
$465$	−25.8267	−1.19769
$466$	76.6267	3.54966
$467$	4.93835	0.228520	0.114260	−	0.993451i	$-0.463550\pi$
$467$	4.93835	0.228520	0.114260	+	0.993451i	$0.463550\pi$
$468$	−12.3999	−0.573187
$469$	−17.8667	−0.825006
$470$	−56.6830	−2.61459
$471$	−31.9892	−1.47399
$472$	31.8073	1.46405
$473$	21.0637	0.968508
$474$	86.4916	3.97269
$475$	−2.14770	−0.0985431
$476$	−163.615	−7.49930
$477$	−6.98503	−0.319822
$478$	−23.2508	−1.06346
$479$	−2.97377	−0.135875	−0.0679374	−	0.997690i	$-0.521642\pi$
$479$	−2.97377	−0.135875	−0.0679374	+	0.997690i	$0.521642\pi$
$480$	−61.5920	−2.81128
$481$	−20.8586	−0.951071
$482$	56.2091	2.56025
$483$	−55.1921	−2.51133
$484$	−7.06006	−0.320912
$485$	1.37595	0.0624785
$486$	20.8973	0.947923
$487$	−14.6507	−0.663888	−0.331944	−	0.943299i	$-0.607704\pi$
$487$	−14.6507	−0.663888	−0.331944	+	0.943299i	$0.607704\pi$
$488$	−45.1166	−2.04233
$489$	15.7086	0.710369
$490$	−74.6086	−3.37047
$491$	−16.8368	−0.759835	−0.379918	−	0.925020i	$-0.624048\pi$
$491$	−16.8368	−0.759835	−0.379918	+	0.925020i	$0.624048\pi$
$492$	71.9183	3.24233
$493$	64.9856	2.92680
$494$	8.29965	0.373419
$495$	−3.99529	−0.179575
$496$	−107.051	−4.80672
$497$	0.132540	0.00594521
$498$	−40.5939	−1.81906
$499$	28.0698	1.25657	0.628287	−	0.777981i	$-0.283756\pi$
$499$	28.0698	1.25657	0.628287	+	0.777981i	$0.283756\pi$
$500$	−64.0992	−2.86660
$501$	−17.9888	−0.803678
$502$	−81.5722	−3.64075
$503$	26.3219	1.17364	0.586818	−	0.809719i	$-0.300381\pi$
$503$	26.3219	1.17364	0.586818	+	0.809719i	$0.300381\pi$
$504$	−32.8887	−1.46498
$505$	−7.11663	−0.316686
$506$	−49.5308	−2.20191
$507$	−6.93601	−0.308039
$508$	−49.2910	−2.18693
$509$	−14.2330	−0.630865	−0.315432	−	0.948948i	$-0.602150\pi$
$509$	−14.2330	−0.630865	−0.315432	+	0.948948i	$0.602150\pi$
$510$	56.4783	2.50090
$511$	7.58932	0.335732
$512$	−12.3233	−0.544620
$513$	−4.34253	−0.191728
$514$	23.9119	1.05471
$515$	29.3962	1.29535
$516$	−69.7486	−3.07051
$517$	−38.6030	−1.69776
$518$	−88.7530	−3.89958
$519$	11.1113	0.487730
$520$	46.3955	2.03458
$521$	−23.6952	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$521$	−23.6952	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$522$	20.9561	0.917221
$523$	−19.8050	−0.866010	−0.433005	−	0.901391i	$-0.642547\pi$
$523$	−19.8050	−0.866010	−0.433005	+	0.901391i	$0.642547\pi$
$524$	48.9238	2.13724
$525$	−20.1212	−0.878160
$526$	−31.3395	−1.36647
$527$	50.2953	2.19090
$528$	−81.8676	−3.56283
$529$	11.7051	0.508920
$530$	41.9271	1.82120
$531$	−2.70384	−0.117337
$532$	25.6527	1.11218
$533$	−21.4397	−0.928656
$534$	−11.8949	−0.514741
$535$	−29.1171	−1.25884
$536$	33.0959	1.42952
$537$	−2.62071	−0.113092
$538$	−72.0118	−3.10465
$539$	−50.8109	−2.18858
$540$	−38.9430	−1.67584
$541$	−0.823610	−0.0354098	−0.0177049	−	0.999843i	$-0.505636\pi$
$541$	−0.823610	−0.0354098	−0.0177049	+	0.999843i	$0.505636\pi$
$542$	12.2824	0.527576
$543$	23.5893	1.01231
$544$	119.945	5.14261
$545$	−5.74838	−0.246233
$546$	77.7570	3.32769
$547$	−9.33900	−0.399307	−0.199653	−	0.979867i	$-0.563982\pi$
$547$	−9.33900	−0.399307	−0.199653	+	0.979867i	$0.563982\pi$
$548$	−4.88040	−0.208480
$549$	3.83522	0.163683
$550$	−18.0573	−0.769964
$551$	−10.1889	−0.434060
$552$	102.237	4.35149
$553$	−79.6941	−3.38894
$554$	46.3584	1.96958
$555$	22.2544	0.944647
$556$	29.8461	1.26576
$557$	7.81568	0.331161	0.165580	−	0.986196i	$-0.447050\pi$
$557$	7.81568	0.331161	0.165580	+	0.986196i	$0.447050\pi$
$558$	16.2188	0.686598
$559$	20.7929	0.879445
$560$	110.763	4.68061
$561$	38.4636	1.62393
$562$	−73.5212	−3.10130
$563$	3.82521	0.161213	0.0806067	−	0.996746i	$-0.474314\pi$
$563$	3.82521	0.161213	0.0806067	+	0.996746i	$0.474314\pi$
$564$	127.827	5.38249
$565$	9.05328	0.380875
$566$	−22.1221	−0.929862
$567$	−51.7093	−2.17159
$568$	−0.245514	−0.0103015
$569$	−40.5066	−1.69813	−0.849063	−	0.528291i	$-0.822833\pi$
$569$	−40.5066	−1.69813	−0.849063	+	0.528291i	$0.822833\pi$
$570$	−8.85503	−0.370896
$571$	−11.6873	−0.489099	−0.244549	−	0.969637i	$-0.578640\pi$
$571$	−11.6873	−0.489099	−0.244549	+	0.969637i	$0.578640\pi$
$572$	50.6890	2.11941
$573$	41.6377	1.73944
$574$	−91.2255	−3.80768
$575$	12.6523	0.527638
$576$	18.0246	0.751026
$577$	6.50862	0.270958	0.135479	−	0.990780i	$-0.456743\pi$
$577$	6.50862	0.270958	0.135479	+	0.990780i	$0.456743\pi$
$578$	−64.0239	−2.66304
$579$	−2.48347	−0.103209
$580$	−91.3716	−3.79400
$581$	37.4036	1.55176
$582$	−4.27165	−0.177066
$583$	28.5537	1.18257
$584$	−14.0583	−0.581737
$585$	−3.94393	−0.163061
$586$	8.77303	0.362411
$587$	−2.69292	−0.111149	−0.0555743	−	0.998455i	$-0.517699\pi$
$587$	−2.69292	−0.111149	−0.0555743	+	0.998455i	$0.517699\pi$
$588$	168.251	6.93857
$589$	−7.88562	−0.324921
$590$	16.2296	0.668161
$591$	41.1346	1.69205
$592$	92.2436	3.79119
$593$	−42.8992	−1.76166	−0.880829	−	0.473434i	$-0.843014\pi$
$593$	−42.8992	−1.76166	−0.880829	+	0.473434i	$0.843014\pi$
$594$	−36.5109	−1.49806
$595$	−52.0396	−2.13341
$596$	−4.93895	−0.202307
$597$	−30.3259	−1.24116
$598$	−48.8941	−1.99943
$599$	−26.9412	−1.10079	−0.550394	−	0.834905i	$-0.685522\pi$
$599$	−26.9412	−1.10079	−0.550394	+	0.834905i	$0.685522\pi$
$600$	37.2721	1.52163
$601$	−27.3720	−1.11653	−0.558264	−	0.829663i	$-0.688532\pi$
$601$	−27.3720	−1.11653	−0.558264	+	0.829663i	$0.688532\pi$
$602$	88.4733	3.60590
$603$	−2.81337	−0.114569
$604$	−7.56746	−0.307915
$605$	−2.24552	−0.0912935
$606$	22.0937	0.897496
$607$	−6.84939	−0.278008	−0.139004	−	0.990292i	$-0.544390\pi$
$607$	−6.84939	−0.278008	−0.139004	+	0.990292i	$0.544390\pi$
$608$	−18.8058	−0.762675
$609$	−95.4565	−3.86809
$610$	−23.0206	−0.932076
$611$	−38.1068	−1.54163
$612$	−25.7637	−1.04143
$613$	39.6831	1.60279	0.801393	−	0.598138i	$-0.204093\pi$
$613$	39.6831	1.60279	0.801393	+	0.598138i	$0.204093\pi$
$614$	36.1645	1.45948
$615$	22.8744	0.922383
$616$	134.444	5.41690
$617$	−3.24438	−0.130614	−0.0653068	−	0.997865i	$-0.520803\pi$
$617$	−3.24438	−0.130614	−0.0653068	+	0.997865i	$0.520803\pi$
$618$	−91.2612	−3.67106
$619$	−33.6128	−1.35101	−0.675506	−	0.737355i	$-0.736075\pi$
$619$	−33.6128	−1.35101	−0.675506	+	0.737355i	$0.736075\pi$
$620$	−70.7167	−2.84005
$621$	25.5823	1.02658
$622$	38.2550	1.53388
$623$	10.9600	0.439104
$624$	−80.8152	−3.23520
$625$	−9.64891	−0.385956
$626$	38.7075	1.54706
$627$	−6.03057	−0.240838
$628$	−87.5903	−3.49523
$629$	−43.3385	−1.72802
$630$	−16.7813	−0.668584
$631$	21.0771	0.839065	0.419533	−	0.907740i	$-0.362194\pi$
$631$	21.0771	0.839065	0.419533	+	0.907740i	$0.362194\pi$
$632$	147.624	5.87216
$633$	1.93926	0.0770786
$634$	−3.07616	−0.122170
$635$	−15.6775	−0.622143
$636$	−94.5506	−3.74918
$637$	−50.1577	−1.98732
$638$	−85.6651	−3.39151
$639$	0.0208703	0.000825617 0
$640$	−44.6702	−1.76575
$641$	18.3880	0.726282	0.363141	−	0.931734i	$-0.381704\pi$
$641$	18.3880	0.726282	0.363141	+	0.931734i	$0.381704\pi$
$642$	90.3945	3.56759
$643$	25.2660	0.996393	0.498197	−	0.867064i	$-0.333996\pi$
$643$	25.2660	0.996393	0.498197	+	0.867064i	$0.333996\pi$
$644$	−151.123	−5.95507
$645$	−22.1843	−0.873504
$646$	17.2444	0.678472
$647$	−34.5925	−1.35997	−0.679987	−	0.733224i	$-0.738014\pi$
$647$	−34.5925	−1.35997	−0.679987	+	0.733224i	$0.738014\pi$
$648$	95.7854	3.76281
$649$	11.0529	0.433863
$650$	−17.8251	−0.699159
$651$	−73.8781	−2.89551
$652$	43.0122	1.68449
$653$	47.1087	1.84350	0.921752	−	0.387779i	$-0.126758\pi$
$653$	47.1087	1.84350	0.921752	+	0.387779i	$0.126758\pi$
$654$	17.8459	0.697832
$655$	15.5607	0.608007
$656$	94.8133	3.70184
$657$	1.19505	0.0466234
$658$	−162.144	−6.32101
$659$	−31.5202	−1.22785	−0.613927	−	0.789363i	$-0.710411\pi$
$659$	−31.5202	−1.22785	−0.613927	+	0.789363i	$0.710411\pi$
$660$	−54.0809	−2.10510
$661$	−39.7020	−1.54423	−0.772114	−	0.635484i	$-0.780801\pi$
$661$	−39.7020	−1.54423	−0.772114	+	0.635484i	$0.780801\pi$
$662$	−73.2176	−2.84568
$663$	37.9691	1.47460
$664$	−69.2857	−2.68881
$665$	8.15910	0.316396
$666$	−13.9755	−0.541538
$667$	60.0236	2.32412
$668$	−49.2554	−1.90575
$669$	32.0074	1.23748
$670$	16.8870	0.652403
$671$	−15.6778	−0.605234
$672$	−176.186	−6.79652
$673$	44.3464	1.70943	0.854714	−	0.519100i	$-0.173733\pi$
$673$	44.3464	1.70943	0.854714	+	0.519100i	$0.173733\pi$
$674$	47.8778	1.84418
$675$	9.32645	0.358975
$676$	−18.9916	−0.730447
$677$	−4.79223	−0.184180	−0.0920902	−	0.995751i	$-0.529355\pi$
$677$	−4.79223	−0.184180	−0.0920902	+	0.995751i	$0.529355\pi$
$678$	−28.1061	−1.07941
$679$	3.93594	0.151047
$680$	96.3971	3.69666
$681$	26.3913	1.01132
$682$	−66.3001	−2.53876
$683$	10.5939	0.405364	0.202682	−	0.979245i	$-0.435034\pi$
$683$	10.5939	0.405364	0.202682	+	0.979245i	$0.435034\pi$
$684$	4.03939	0.154450
$685$	−1.55226	−0.0593089
$686$	−121.988	−4.65752
$687$	12.9461	0.493924
$688$	−91.9529	−3.50567
$689$	28.1867	1.07383
$690$	52.1659	1.98592
$691$	−36.1210	−1.37411	−0.687054	−	0.726607i	$-0.741096\pi$
$691$	−36.1210	−1.37411	−0.687054	+	0.726607i	$0.741096\pi$
$692$	30.4240	1.15655
$693$	−11.4286	−0.434138
$694$	−23.2618	−0.883007
$695$	9.49286	0.360085
$696$	176.822	6.70241
$697$	−44.5458	−1.68729
$698$	65.8390	2.49204
$699$	−54.9616	−2.07884
$700$	−55.0942	−2.08236
$701$	13.5453	0.511598	0.255799	−	0.966730i	$-0.417661\pi$
$701$	13.5453	0.511598	0.255799	+	0.966730i	$0.417661\pi$
$702$	−36.0415	−1.36030
$703$	6.79489	0.256274
$704$	−73.6819	−2.77699
$705$	40.6567	1.53122
$706$	3.32421	0.125108
$707$	−20.3573	−0.765616
$708$	−36.5997	−1.37550
$709$	12.1210	0.455215	0.227608	−	0.973753i	$-0.426910\pi$
$709$	12.1210	0.455215	0.227608	+	0.973753i	$0.426910\pi$
$710$	−0.125272	−0.00470139
$711$	−12.5490	−0.470625
$712$	−20.3021	−0.760854
$713$	46.4550	1.73975
$714$	161.558	6.04615
$715$	16.1222	0.602935
$716$	−7.17582	−0.268173
$717$	16.6769	0.622812
$718$	2.61639	0.0976426
$719$	34.8151	1.29838	0.649192	−	0.760625i	$-0.275107\pi$
$719$	34.8151	1.29838	0.649192	+	0.760625i	$0.275107\pi$
$720$	17.4413	0.650000
$721$	84.0888	3.13163
$722$	−2.70369	−0.100621
$723$	−40.3168	−1.49940
$724$	64.5904	2.40048
$725$	21.8826	0.812698
$726$	6.97127	0.258728
$727$	23.6593	0.877474	0.438737	−	0.898616i	$-0.355426\pi$
$727$	23.6593	0.877474	0.438737	+	0.898616i	$0.355426\pi$
$728$	132.716	4.91877
$729$	17.1215	0.634129
$730$	−7.17320	−0.265492
$731$	43.2019	1.59788
$732$	51.9142	1.91880
$733$	5.18279	0.191431	0.0957153	−	0.995409i	$-0.469486\pi$
$733$	5.18279	0.191431	0.0957153	+	0.995409i	$0.469486\pi$
$734$	−41.7003	−1.53918
$735$	53.5141	1.97390
$736$	110.787	4.08366
$737$	11.5006	0.423631
$738$	−14.3648	−0.528775
$739$	−39.7187	−1.46108	−0.730538	−	0.682872i	$-0.760731\pi$
$739$	−39.7187	−1.46108	−0.730538	+	0.682872i	$0.760731\pi$
$740$	60.9352	2.24002
$741$	−5.95304	−0.218690
$742$	119.934	4.40290
$743$	−12.1228	−0.444741	−0.222370	−	0.974962i	$-0.571379\pi$
$743$	−12.1228	−0.444741	−0.222370	+	0.974962i	$0.571379\pi$
$744$	136.850	5.01718
$745$	−1.57088	−0.0575527
$746$	−56.4463	−2.06665
$747$	5.88975	0.215495
$748$	105.318	3.85080
$749$	−83.2902	−3.04336
$750$	63.2931	2.31114
$751$	24.8330	0.906171	0.453085	−	0.891467i	$-0.350323\pi$
$751$	24.8330	0.906171	0.453085	+	0.891467i	$0.350323\pi$
$752$	168.521	6.14532
$753$	58.5089	2.13218
$754$	−84.5639	−3.07964
$755$	−2.40691	−0.0875963
$756$	−111.398	−4.05149
$757$	13.3690	0.485906	0.242953	−	0.970038i	$-0.421884\pi$
$757$	13.3690	0.485906	0.242953	+	0.970038i	$0.421884\pi$
$758$	−57.3538	−2.08318
$759$	35.5267	1.28954
$760$	−15.1138	−0.548233
$761$	−19.9923	−0.724720	−0.362360	−	0.932038i	$-0.618029\pi$
$761$	−19.9923	−0.724720	−0.362360	+	0.932038i	$0.618029\pi$
$762$	48.6711	1.76317
$763$	−16.4434	−0.595291
$764$	114.009	4.12470
$765$	−8.19440	−0.296269
$766$	98.9853	3.57648
$767$	10.9108	0.393966
$768$	46.7819	1.68810
$769$	−3.48179	−0.125557	−0.0627784	−	0.998027i	$-0.519996\pi$
$769$	−3.48179	−0.125557	−0.0627784	+	0.998027i	$0.519996\pi$
$770$	68.5995	2.47215
$771$	−17.1512	−0.617684
$772$	−6.80004	−0.244739
$773$	19.6712	0.707523	0.353761	−	0.935336i	$-0.384903\pi$
$773$	19.6712	0.707523	0.353761	+	0.935336i	$0.384903\pi$
$774$	13.9314	0.500755
$775$	16.9359	0.608356
$776$	−7.29085	−0.261726
$777$	63.6594	2.28377
$778$	26.4331	0.947674
$779$	6.98418	0.250234
$780$	−53.3857	−1.91152
$781$	−0.0853146	−0.00305280
$782$	−101.589	−3.63280
$783$	44.2454	1.58120
$784$	221.814	7.92193
$785$	−27.8590	−0.994331
$786$	−48.3085	−1.72311
$787$	14.6941	0.523787	0.261893	−	0.965097i	$-0.415653\pi$
$787$	14.6941	0.523787	0.261893	+	0.965097i	$0.415653\pi$
$788$	112.631	4.01233
$789$	22.4787	0.800264
$790$	75.3245	2.67992
$791$	25.8972	0.920798
$792$	21.1702	0.752250
$793$	−15.4762	−0.549577
$794$	−30.6221	−1.08674
$795$	−30.0728	−1.06657
$796$	−83.0361	−2.94314
$797$	−27.8038	−0.984860	−0.492430	−	0.870352i	$-0.663891\pi$
$797$	−27.8038	−0.984860	−0.492430	+	0.870352i	$0.663891\pi$
$798$	−25.3301	−0.896675
$799$	−79.1754	−2.80103
$800$	40.3891	1.42797
$801$	1.72582	0.0609787
$802$	59.7109	2.10846
$803$	−4.88518	−0.172394
$804$	−38.0823	−1.34306
$805$	−48.0661	−1.69411
$806$	−65.4478	−2.30530
$807$	51.6515	1.81822
$808$	37.7095	1.32662
$809$	−1.19402	−0.0419797	−0.0209898	−	0.999780i	$-0.506682\pi$
$809$	−1.19402	−0.0419797	−0.0209898	+	0.999780i	$0.506682\pi$
$810$	48.8741	1.71726
$811$	−26.3752	−0.926157	−0.463078	−	0.886317i	$-0.653255\pi$
$811$	−26.3752	−0.926157	−0.463078	+	0.886317i	$0.653255\pi$
$812$	−261.371	−9.17234
$813$	−8.80976	−0.308972
$814$	57.1296	2.00239
$815$	13.6805	0.479206
$816$	−167.912	−5.87809
$817$	−6.77347	−0.236974
$818$	29.4134	1.02841
$819$	−11.2817	−0.394215
$820$	62.6327	2.18723
$821$	−4.05064	−0.141368	−0.0706841	−	0.997499i	$-0.522518\pi$
$821$	−4.05064	−0.141368	−0.0706841	+	0.997499i	$0.522518\pi$
$822$	4.81903	0.168083
$823$	46.8677	1.63371	0.816853	−	0.576846i	$-0.195717\pi$
$823$	46.8677	1.63371	0.816853	+	0.576846i	$0.195717\pi$
$824$	−155.764	−5.42631
$825$	12.9518	0.450925
$826$	46.4252	1.61534
$827$	3.72316	0.129467	0.0647335	−	0.997903i	$-0.479380\pi$
$827$	3.72316	0.129467	0.0647335	+	0.997903i	$0.479380\pi$
$828$	−23.7965	−0.826985
$829$	−44.3216	−1.53935	−0.769677	−	0.638434i	$-0.779583\pi$
$829$	−44.3216	−1.53935	−0.769677	+	0.638434i	$0.779583\pi$
$830$	−35.3527	−1.22711
$831$	−33.2512	−1.15347
$832$	−72.7347	−2.52162
$833$	−104.214	−3.61080
$834$	−29.4708	−1.02049
$835$	−15.6662	−0.542151
$836$	−16.5124	−0.571094
$837$	34.2436	1.18363
$838$	69.1525	2.38883
$839$	−5.24491	−0.181075	−0.0905373	−	0.995893i	$-0.528858\pi$
$839$	−5.24491	−0.181075	−0.0905373	+	0.995893i	$0.528858\pi$
$840$	−141.596	−4.88554
$841$	74.8127	2.57975
$842$	20.5860	0.709439
$843$	52.7341	1.81626
$844$	5.30992	0.182775
$845$	−6.04048	−0.207799
$846$	−25.5319	−0.877805
$847$	−6.42339	−0.220710
$848$	−124.651	−4.28052
$849$	15.8674	0.544568
$850$	−37.0357	−1.27032
$851$	−40.0294	−1.37219
$852$	0.282505	0.00967845
$853$	−17.4012	−0.595807	−0.297903	−	0.954596i	$-0.596287\pi$
$853$	−17.4012	−0.595807	−0.297903	+	0.954596i	$0.596287\pi$
$854$	−65.8511	−2.25338
$855$	1.28477	0.0439382
$856$	154.285	5.27336
$857$	32.8660	1.12268	0.561340	−	0.827585i	$-0.310286\pi$
$857$	32.8660	1.12268	0.561340	+	0.827585i	$0.310286\pi$
$858$	−50.0516	−1.70873
$859$	−24.3426	−0.830558	−0.415279	−	0.909694i	$-0.636316\pi$
$859$	−24.3426	−0.830558	−0.415279	+	0.909694i	$0.636316\pi$
$860$	−60.7432	−2.07133
$861$	65.4328	2.22994
$862$	−2.72038	−0.0926565
$863$	7.13513	0.242883	0.121441	−	0.992599i	$-0.461248\pi$
$863$	7.13513	0.242883	0.121441	+	0.992599i	$0.461248\pi$
$864$	81.6647	2.77829
$865$	9.67665	0.329016
$866$	57.7836	1.96357
$867$	45.9220	1.55959
$868$	−202.287	−6.86607
$869$	51.2984	1.74018
$870$	90.2226	3.05883
$871$	11.3528	0.384674
$872$	30.4594	1.03149
$873$	0.619771	0.0209761
$874$	15.9277	0.538763
$875$	−58.3188	−1.97153
$876$	16.1764	0.546551
$877$	58.0896	1.96155	0.980773	−	0.195153i	$-0.0625203\pi$
$877$	58.0896	1.96155	0.980773	+	0.195153i	$0.0625203\pi$
$878$	38.6647	1.30487
$879$	−6.29258	−0.212244
$880$	−71.2975	−2.40344
$881$	32.0010	1.07814	0.539071	−	0.842261i	$-0.318776\pi$
$881$	32.0010	1.07814	0.539071	+	0.842261i	$0.318776\pi$
$882$	−33.6062	−1.13158
$883$	−26.0480	−0.876586	−0.438293	−	0.898832i	$-0.644417\pi$
$883$	−26.0480	−0.876586	−0.438293	+	0.898832i	$0.644417\pi$
$884$	103.964	3.49669
$885$	−11.6409	−0.391304
$886$	2.43253	0.0817226
$887$	52.1626	1.75145	0.875725	−	0.482811i	$-0.160384\pi$
$887$	52.1626	1.75145	0.875725	+	0.482811i	$0.160384\pi$
$888$	−117.921	−3.95718
$889$	−44.8460	−1.50409
$890$	−10.3591	−0.347237
$891$	33.2849	1.11509
$892$	87.6400	2.93440
$893$	12.4136	0.415406
$894$	4.87684	0.163106
$895$	−2.28234	−0.0762903
$896$	−127.780	−4.26885
$897$	35.0700	1.17095
$898$	−16.5381	−0.551884
$899$	80.3454	2.67967
$900$	−8.67539	−0.289180
$901$	58.5642	1.95105
$902$	58.7211	1.95520
$903$	−63.4588	−2.11178
$904$	−47.9715	−1.59551
$905$	20.5436	0.682894
$906$	7.47229	0.248250
$907$	51.0635	1.69553	0.847767	−	0.530368i	$-0.177946\pi$
$907$	51.0635	1.69553	0.847767	+	0.530368i	$0.177946\pi$
$908$	72.2626	2.39812
$909$	−3.20556	−0.106322
$910$	67.7176	2.24482
$911$	−32.5104	−1.07712	−0.538559	−	0.842588i	$-0.681031\pi$
$911$	−32.5104	−1.07712	−0.538559	+	0.842588i	$0.681031\pi$
$912$	26.3263	0.871751
$913$	−24.0764	−0.796812
$914$	−63.0478	−2.08544
$915$	16.5118	0.545865
$916$	35.4480	1.17123
$917$	44.5119	1.46991
$918$	−74.8844	−2.47155
$919$	−52.3578	−1.72712	−0.863562	−	0.504243i	$-0.831772\pi$
$919$	−52.3578	−1.72712	−0.863562	+	0.504243i	$0.831772\pi$
$920$	89.0367	2.93545
$921$	−25.9395	−0.854736
$922$	−16.8886	−0.556196
$923$	−0.0842179	−0.00277207
$924$	−154.700	−5.08926
$925$	−14.5934	−0.479827
$926$	−34.6789	−1.13962
$927$	13.2410	0.434892
$928$	191.609	6.28988
$929$	48.3484	1.58626	0.793129	−	0.609053i	$-0.208450\pi$
$929$	48.3484	1.58626	0.793129	+	0.609053i	$0.208450\pi$
$930$	69.8274	2.28973
$931$	16.3393	0.535500
$932$	−150.491	−4.92951
$933$	−27.4389	−0.898310
$934$	−13.3518	−0.436883
$935$	33.4974	1.09548
$936$	20.8980	0.683074
$937$	25.3696	0.828788	0.414394	−	0.910098i	$-0.363994\pi$
$937$	25.3696	0.828788	0.414394	+	0.910098i	$0.363994\pi$
$938$	48.3059	1.57724
$939$	−27.7635	−0.906028
$940$	111.323	3.63096
$941$	−22.3776	−0.729489	−0.364745	−	0.931108i	$-0.618844\pi$
$941$	−22.3776	−0.729489	−0.364745	+	0.931108i	$0.618844\pi$
$942$	86.4888	2.81796
$943$	−41.1445	−1.33985
$944$	−48.2511	−1.57044
$945$	−35.4312	−1.15258
$946$	−56.9495	−1.85159
$947$	9.70100	0.315240	0.157620	−	0.987500i	$-0.449618\pi$
$947$	9.70100	0.315240	0.157620	+	0.987500i	$0.449618\pi$
$948$	−169.866	−5.51699
$949$	−4.82238	−0.156541
$950$	5.80670	0.188394
$951$	2.20642	0.0715482
$952$	275.747	8.93700
$953$	−56.5303	−1.83119	−0.915597	−	0.402096i	$-0.868282\pi$
$953$	−56.5303	−1.83119	−0.915597	+	0.402096i	$0.868282\pi$
$954$	18.8853	0.611435
$955$	36.2617	1.17340
$956$	45.6635	1.47686
$957$	61.4446	1.98622
$958$	8.04014	0.259765
$959$	−4.44029	−0.143384
$960$	77.6019	2.50459
$961$	31.1829	1.00590
$962$	56.3952	1.81825
$963$	−13.1153	−0.422634
$964$	−110.392	−3.55549
$965$	−2.16282	−0.0696237
$966$	149.222	4.80115
$967$	−13.2578	−0.426343	−0.213172	−	0.977015i	$-0.568379\pi$
$967$	−13.2578	−0.426343	−0.213172	+	0.977015i	$0.568379\pi$
$968$	11.8986	0.382434
$969$	−12.3688	−0.397343
$970$	−3.72013	−0.119446
$971$	−28.1519	−0.903438	−0.451719	−	0.892160i	$-0.649189\pi$
$971$	−28.1519	−0.903438	−0.451719	+	0.892160i	$0.649189\pi$
$972$	−41.0415	−1.31641
$973$	27.1546	0.870537
$974$	39.6110	1.26922
$975$	12.7853	0.409458
$976$	68.4410	2.19074
$977$	−35.8616	−1.14732	−0.573658	−	0.819095i	$-0.694476\pi$
$977$	−35.8616	−1.14732	−0.573658	+	0.819095i	$0.694476\pi$
$978$	−42.4713	−1.35808
$979$	−7.05487	−0.225475
$980$	146.528	4.68067
$981$	−2.58926	−0.0826686
$982$	45.5215	1.45265
$983$	−4.65839	−0.148580	−0.0742898	−	0.997237i	$-0.523669\pi$
$983$	−4.65839	−0.148580	−0.0742898	+	0.997237i	$0.523669\pi$
$984$	−121.206	−3.86392
$985$	35.8236	1.14143
$986$	−175.701	−5.59545
$987$	116.300	3.70186
$988$	−16.3001	−0.518577
$989$	39.9033	1.26885
$990$	10.8020	0.343310
$991$	−32.7571	−1.04056	−0.520281	−	0.853995i	$-0.674173\pi$
$991$	−32.7571	−1.04056	−0.520281	+	0.853995i	$0.674173\pi$
$992$	148.295	4.70837
$993$	52.5164	1.66656
$994$	−0.358346	−0.0113660
$995$	−26.4105	−0.837268
$996$	79.7247	2.52617
$997$	11.5364	0.365363	0.182681	−	0.983172i	$-0.441522\pi$
$997$	11.5364	0.365363	0.182681	+	0.983172i	$0.441522\pi$
$998$	−75.8918	−2.40231
$999$	−29.5070	−0.933561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.2	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.2	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70369 q^{2} +1.93926 q^{3} +5.30992 q^{4} +1.68888 q^{5} -5.24315 q^{6} +4.83108 q^{7} -8.94900 q^{8} +0.760725 q^{9} +O(q^{10})\)	\(q-2.70369 q^{2} +1.93926 q^{3} +5.30992 q^{4} +1.68888 q^{5} -5.24315 q^{6} +4.83108 q^{7} -8.94900 q^{8} +0.760725 q^{9} -4.56619 q^{10} -3.10973 q^{11} +10.2973 q^{12} -3.06975 q^{13} -13.0617 q^{14} +3.27517 q^{15} +13.5754 q^{16} -6.37810 q^{17} -2.05676 q^{18} +1.00000 q^{19} +8.96780 q^{20} +9.36872 q^{21} +8.40773 q^{22} -5.89111 q^{23} -17.3544 q^{24} -2.14770 q^{25} +8.29965 q^{26} -4.34253 q^{27} +25.6527 q^{28} -10.1889 q^{29} -8.85503 q^{30} -7.88562 q^{31} -18.8058 q^{32} -6.03057 q^{33} +17.2444 q^{34} +8.15910 q^{35} +4.03939 q^{36} +6.79489 q^{37} -2.70369 q^{38} -5.95304 q^{39} -15.1138 q^{40} +6.98418 q^{41} -25.3301 q^{42} -6.77347 q^{43} -16.5124 q^{44} +1.28477 q^{45} +15.9277 q^{46} +12.4136 q^{47} +26.3263 q^{48} +16.3393 q^{49} +5.80670 q^{50} -12.3688 q^{51} -16.3001 q^{52} -9.18206 q^{53} +11.7409 q^{54} -5.25194 q^{55} -43.2333 q^{56} +1.93926 q^{57} +27.5475 q^{58} -3.55429 q^{59} +17.3909 q^{60} +5.04153 q^{61} +21.3202 q^{62} +3.67512 q^{63} +23.6940 q^{64} -5.18443 q^{65} +16.3048 q^{66} -3.69827 q^{67} -33.8672 q^{68} -11.4244 q^{69} -22.0596 q^{70} +0.0274348 q^{71} -6.80773 q^{72} +1.57094 q^{73} -18.3712 q^{74} -4.16494 q^{75} +5.30992 q^{76} -15.0233 q^{77} +16.0952 q^{78} -16.4961 q^{79} +22.9273 q^{80} -10.7035 q^{81} -18.8830 q^{82} +7.74228 q^{83} +49.7472 q^{84} -10.7718 q^{85} +18.3134 q^{86} -19.7588 q^{87} +27.8290 q^{88} +2.26865 q^{89} -3.47362 q^{90} -14.8302 q^{91} -31.2813 q^{92} -15.2922 q^{93} -33.5626 q^{94} +1.68888 q^{95} -36.4693 q^{96} +0.814711 q^{97} -44.1765 q^{98} -2.36565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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