LMFDB - Embedded newform 8007.2.a.e.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.42820 q^{2} +1.00000 q^{3} +0.0397473 q^{4} -3.43580 q^{5} -1.42820 q^{6} +0.0177517 q^{7} +2.79963 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.42820 q^{2} +1.00000 q^{3} +0.0397473 q^{4} -3.43580 q^{5} -1.42820 q^{6} +0.0177517 q^{7} +2.79963 q^{8} +1.00000 q^{9} +4.90700 q^{10} -2.14032 q^{11} +0.0397473 q^{12} -1.57212 q^{13} -0.0253529 q^{14} -3.43580 q^{15} -4.07791 q^{16} -1.00000 q^{17} -1.42820 q^{18} -1.12471 q^{19} -0.136564 q^{20} +0.0177517 q^{21} +3.05680 q^{22} +2.11501 q^{23} +2.79963 q^{24} +6.80471 q^{25} +2.24530 q^{26} +1.00000 q^{27} +0.000705581 q^{28} +7.22554 q^{29} +4.90700 q^{30} -2.53455 q^{31} +0.224811 q^{32} -2.14032 q^{33} +1.42820 q^{34} -0.0609912 q^{35} +0.0397473 q^{36} +5.42717 q^{37} +1.60630 q^{38} -1.57212 q^{39} -9.61895 q^{40} +9.59589 q^{41} -0.0253529 q^{42} -11.5802 q^{43} -0.0850718 q^{44} -3.43580 q^{45} -3.02066 q^{46} -10.9992 q^{47} -4.07791 q^{48} -6.99968 q^{49} -9.71846 q^{50} -1.00000 q^{51} -0.0624875 q^{52} -4.78120 q^{53} -1.42820 q^{54} +7.35370 q^{55} +0.0496981 q^{56} -1.12471 q^{57} -10.3195 q^{58} +1.42869 q^{59} -0.136564 q^{60} +10.4637 q^{61} +3.61984 q^{62} +0.0177517 q^{63} +7.83475 q^{64} +5.40149 q^{65} +3.05680 q^{66} +13.3878 q^{67} -0.0397473 q^{68} +2.11501 q^{69} +0.0871075 q^{70} -5.47570 q^{71} +2.79963 q^{72} +10.9506 q^{73} -7.75107 q^{74} +6.80471 q^{75} -0.0447041 q^{76} -0.0379943 q^{77} +2.24530 q^{78} +3.38850 q^{79} +14.0109 q^{80} +1.00000 q^{81} -13.7048 q^{82} +8.56257 q^{83} +0.000705581 q^{84} +3.43580 q^{85} +16.5389 q^{86} +7.22554 q^{87} -5.99209 q^{88} -8.56091 q^{89} +4.90700 q^{90} -0.0279078 q^{91} +0.0840660 q^{92} -2.53455 q^{93} +15.7090 q^{94} +3.86427 q^{95} +0.224811 q^{96} +15.4548 q^{97} +9.99693 q^{98} -2.14032 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) $ 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.42820	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$2$	−1.42820	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.0397473	0.0198736
$5$	−3.43580	−1.53654	−0.768268	−	0.640129i	$-0.778881\pi$
$5$	−3.43580	−1.53654	−0.768268	+	0.640129i	$0.778881\pi$
$6$	−1.42820	−0.583059
$7$	0.0177517	0.00670951	0.00335475	−	0.999994i	$-0.498932\pi$
$7$	0.0177517	0.00670951	0.00335475	+	0.999994i	$0.498932\pi$
$8$	2.79963	0.989818
$9$	1.00000	0.333333
$10$	4.90700	1.55173
$11$	−2.14032	−0.645330	−0.322665	−	0.946513i	$-0.604579\pi$
$11$	−2.14032	−0.645330	−0.322665	+	0.946513i	$0.604579\pi$
$12$	0.0397473	0.0114741
$13$	−1.57212	−0.436028	−0.218014	−	0.975946i	$-0.569958\pi$
$13$	−1.57212	−0.436028	−0.218014	+	0.975946i	$0.569958\pi$
$14$	−0.0253529	−0.00677585
$15$	−3.43580	−0.887119
$16$	−4.07791	−1.01948
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.42820	−0.336629
$19$	−1.12471	−0.258026	−0.129013	−	0.991643i	$-0.541181\pi$
$19$	−1.12471	−0.258026	−0.129013	+	0.991643i	$0.541181\pi$
$20$	−0.136564	−0.0305366
$21$	0.0177517	0.00387374
$22$	3.05680	0.651711
$23$	2.11501	0.441011	0.220505	−	0.975386i	$-0.429229\pi$
$23$	2.11501	0.441011	0.220505	+	0.975386i	$0.429229\pi$
$24$	2.79963	0.571472
$25$	6.80471	1.36094
$26$	2.24530	0.440339
$27$	1.00000	0.192450
$28$	0.000705581 0	0.000133342 0
$29$	7.22554	1.34175	0.670874	−	0.741571i	$-0.265919\pi$
$29$	7.22554	1.34175	0.670874	+	0.741571i	$0.265919\pi$
$30$	4.90700	0.895891
$31$	−2.53455	−0.455219	−0.227610	−	0.973752i	$-0.573091\pi$
$31$	−2.53455	−0.455219	−0.227610	+	0.973752i	$0.573091\pi$
$32$	0.224811	0.0397414
$33$	−2.14032	−0.372582
$34$	1.42820	0.244934
$35$	−0.0609912	−0.0103094
$36$	0.0397473	0.00662455
$37$	5.42717	0.892222	0.446111	−	0.894978i	$-0.352809\pi$
$37$	5.42717	0.892222	0.446111	+	0.894978i	$0.352809\pi$
$38$	1.60630	0.260577
$39$	−1.57212	−0.251741
$40$	−9.61895	−1.52089
$41$	9.59589	1.49863	0.749313	−	0.662216i	$-0.230384\pi$
$41$	9.59589	1.49863	0.749313	+	0.662216i	$0.230384\pi$
$42$	−0.0253529	−0.00391204
$43$	−11.5802	−1.76597	−0.882986	−	0.469400i	$-0.844470\pi$
$43$	−11.5802	−1.76597	−0.882986	+	0.469400i	$0.844470\pi$
$44$	−0.0850718	−0.0128251
$45$	−3.43580	−0.512179
$46$	−3.02066	−0.445371
$47$	−10.9992	−1.60439	−0.802197	−	0.597059i	$-0.796336\pi$
$47$	−10.9992	−1.60439	−0.802197	+	0.597059i	$0.796336\pi$
$48$	−4.07791	−0.588596
$49$	−6.99968	−0.999955
$50$	−9.71846	−1.37440
$51$	−1.00000	−0.140028
$52$	−0.0624875	−0.00866546
$53$	−4.78120	−0.656749	−0.328374	−	0.944548i	$-0.606501\pi$
$53$	−4.78120	−0.656749	−0.328374	+	0.944548i	$0.606501\pi$
$54$	−1.42820	−0.194353
$55$	7.35370	0.991573
$56$	0.0496981	0.00664119
$57$	−1.12471	−0.148971
$58$	−10.3195	−1.35502
$59$	1.42869	0.185999	0.0929997	−	0.995666i	$-0.470354\pi$
$59$	1.42869	0.185999	0.0929997	+	0.995666i	$0.470354\pi$
$60$	−0.136564	−0.0176303
$61$	10.4637	1.33974	0.669872	−	0.742477i	$-0.266349\pi$
$61$	10.4637	1.33974	0.669872	+	0.742477i	$0.266349\pi$
$62$	3.61984	0.459720
$63$	0.0177517	0.00223650
$64$	7.83475	0.979344
$65$	5.40149	0.669972
$66$	3.05680	0.376266
$67$	13.3878	1.63558	0.817790	−	0.575517i	$-0.195199\pi$
$67$	13.3878	1.63558	0.817790	+	0.575517i	$0.195199\pi$
$68$	−0.0397473	−0.00482007
$69$	2.11501	0.254618
$70$	0.0871075	0.0104113
$71$	−5.47570	−0.649846	−0.324923	−	0.945740i	$-0.605338\pi$
$71$	−5.47570	−0.649846	−0.324923	+	0.945740i	$0.605338\pi$
$72$	2.79963	0.329939
$73$	10.9506	1.28167	0.640833	−	0.767680i	$-0.278589\pi$
$73$	10.9506	1.28167	0.640833	+	0.767680i	$0.278589\pi$
$74$	−7.75107	−0.901044
$75$	6.80471	0.785740
$76$	−0.0447041	−0.00512791
$77$	−0.0379943	−0.00432985
$78$	2.24530	0.254230
$79$	3.38850	0.381236	0.190618	−	0.981664i	$-0.438951\pi$
$79$	3.38850	0.381236	0.190618	+	0.981664i	$0.438951\pi$
$80$	14.0109	1.56647
$81$	1.00000	0.111111
$82$	−13.7048	−1.51345
$83$	8.56257	0.939864	0.469932	−	0.882703i	$-0.344278\pi$
$83$	8.56257	0.939864	0.469932	+	0.882703i	$0.344278\pi$
$84$	0.000705581 0	7.69852e−5 0
$85$	3.43580	0.372665
$86$	16.5389	1.78343
$87$	7.22554	0.774659
$88$	−5.99209	−0.638759
$89$	−8.56091	−0.907455	−0.453727	−	0.891141i	$-0.649906\pi$
$89$	−8.56091	−0.907455	−0.453727	+	0.891141i	$0.649906\pi$
$90$	4.90700	0.517243
$91$	−0.0279078	−0.00292553
$92$	0.0840660	0.00876449
$93$	−2.53455	−0.262821
$94$	15.7090	1.62026
$95$	3.86427	0.396465
$96$	0.224811	0.0229447
$97$	15.4548	1.56920	0.784600	−	0.620002i	$-0.212868\pi$
$97$	15.4548	1.56920	0.784600	+	0.620002i	$0.212868\pi$
$98$	9.99693	1.00984
$99$	−2.14032	−0.215110
$100$	0.270469	0.0270469
$101$	−10.8054	−1.07518	−0.537588	−	0.843207i	$-0.680665\pi$
$101$	−10.8054	−1.07518	−0.537588	+	0.843207i	$0.680665\pi$
$102$	1.42820	0.141413
$103$	13.8359	1.36329	0.681645	−	0.731683i	$-0.261265\pi$
$103$	13.8359	1.36329	0.681645	+	0.731683i	$0.261265\pi$
$104$	−4.40135	−0.431588
$105$	−0.0609912	−0.00595213
$106$	6.82850	0.663242
$107$	−16.7030	−1.61474	−0.807368	−	0.590049i	$-0.799109\pi$
$107$	−16.7030	−1.61474	−0.807368	+	0.590049i	$0.799109\pi$
$108$	0.0397473	0.00382468
$109$	5.21370	0.499382	0.249691	−	0.968326i	$-0.419671\pi$
$109$	5.21370	0.499382	0.249691	+	0.968326i	$0.419671\pi$
$110$	−10.5025	−1.00138
$111$	5.42717	0.515124
$112$	−0.0723899	−0.00684020
$113$	−6.98038	−0.656659	−0.328329	−	0.944563i	$-0.606486\pi$
$113$	−6.98038	−0.656659	−0.328329	+	0.944563i	$0.606486\pi$
$114$	1.60630	0.150444
$115$	−7.26676	−0.677629
$116$	0.287195	0.0266654
$117$	−1.57212	−0.145343
$118$	−2.04045	−0.187839
$119$	−0.0177517	−0.00162729
$120$	−9.61895	−0.878086
$121$	−6.41904	−0.583549
$122$	−14.9443	−1.35299
$123$	9.59589	0.865233
$124$	−0.100742	−0.00904686
$125$	−6.20061	−0.554600
$126$	−0.0253529	−0.00225862
$127$	13.9141	1.23468	0.617339	−	0.786697i	$-0.288211\pi$
$127$	13.9141	1.23468	0.617339	+	0.786697i	$0.288211\pi$
$128$	−11.6392	−1.02877
$129$	−11.5802	−1.01958
$130$	−7.71439	−0.676597
$131$	12.3653	1.08036	0.540181	−	0.841549i	$-0.318356\pi$
$131$	12.3653	1.08036	0.540181	+	0.841549i	$0.318356\pi$
$132$	−0.0850718	−0.00740455
$133$	−0.0199655	−0.00173122
$134$	−19.1204	−1.65175
$135$	−3.43580	−0.295706
$136$	−2.79963	−0.240066
$137$	−18.1813	−1.55334	−0.776668	−	0.629911i	$-0.783091\pi$
$137$	−18.1813	−1.55334	−0.776668	+	0.629911i	$0.783091\pi$
$138$	−3.02066	−0.257135
$139$	2.53697	0.215183	0.107592	−	0.994195i	$-0.465686\pi$
$139$	2.53697	0.215183	0.107592	+	0.994195i	$0.465686\pi$
$140$	−0.00242423	−0.000204885 0
$141$	−10.9992	−0.926298
$142$	7.82038	0.656272
$143$	3.36484	0.281382
$144$	−4.07791	−0.339826
$145$	−24.8255	−2.06164
$146$	−15.6396	−1.29434
$147$	−6.99968	−0.577324
$148$	0.215715	0.0177317
$149$	4.95984	0.406326	0.203163	−	0.979145i	$-0.434878\pi$
$149$	4.95984	0.406326	0.203163	+	0.979145i	$0.434878\pi$
$150$	−9.71846	−0.793509
$151$	8.66297	0.704982	0.352491	−	0.935815i	$-0.385335\pi$
$151$	8.66297	0.704982	0.352491	+	0.935815i	$0.385335\pi$
$152$	−3.14876	−0.255398
$153$	−1.00000	−0.0808452
$154$	0.0542633	0.00437266
$155$	8.70821	0.699460
$156$	−0.0624875	−0.00500301
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	−4.83945	−0.385006
$159$	−4.78120	−0.379174
$160$	−0.772407	−0.0610641
$161$	0.0375451	0.00295896
$162$	−1.42820	−0.112210
$163$	3.88060	0.303952	0.151976	−	0.988384i	$-0.451436\pi$
$163$	3.88060	0.303952	0.151976	+	0.988384i	$0.451436\pi$
$164$	0.381411	0.0297832
$165$	7.35370	0.572485
$166$	−12.2290	−0.949157
$167$	1.37018	0.106028	0.0530138	−	0.998594i	$-0.483117\pi$
$167$	1.37018	0.106028	0.0530138	+	0.998594i	$0.483117\pi$
$168$	0.0496981	0.00383429
$169$	−10.5284	−0.809880
$170$	−4.90700	−0.376349
$171$	−1.12471	−0.0860085
$172$	−0.460283	−0.0350963
$173$	−23.8813	−1.81566	−0.907831	−	0.419336i	$-0.862263\pi$
$173$	−23.8813	−1.81566	−0.907831	+	0.419336i	$0.862263\pi$
$174$	−10.3195	−0.782319
$175$	0.120795	0.00913125
$176$	8.72804	0.657900
$177$	1.42869	0.107387
$178$	12.2267	0.916428
$179$	21.4147	1.60061	0.800305	−	0.599593i	$-0.204671\pi$
$179$	21.4147	1.60061	0.800305	+	0.599593i	$0.204671\pi$
$180$	−0.136564	−0.0101789
$181$	9.64664	0.717029	0.358515	−	0.933524i	$-0.383283\pi$
$181$	9.64664	0.717029	0.358515	+	0.933524i	$0.383283\pi$
$182$	0.0398578	0.00295446
$183$	10.4637	0.773501
$184$	5.92125	0.436520
$185$	−18.6467	−1.37093
$186$	3.61984	0.265420
$187$	2.14032	0.156516
$188$	−0.437187	−0.0318852
$189$	0.0177517	0.00129125
$190$	−5.51894	−0.400386
$191$	−11.3109	−0.818424	−0.409212	−	0.912439i	$-0.634196\pi$
$191$	−11.3109	−0.818424	−0.409212	+	0.912439i	$0.634196\pi$
$192$	7.83475	0.565425
$193$	−17.7254	−1.27590	−0.637949	−	0.770078i	$-0.720217\pi$
$193$	−17.7254	−1.27590	−0.637949	+	0.770078i	$0.720217\pi$
$194$	−22.0725	−1.58472
$195$	5.40149	0.386809
$196$	−0.278218	−0.0198727
$197$	−13.1545	−0.937218	−0.468609	−	0.883406i	$-0.655245\pi$
$197$	−13.1545	−0.937218	−0.468609	+	0.883406i	$0.655245\pi$
$198$	3.05680	0.217237
$199$	19.7186	1.39781	0.698907	−	0.715213i	$-0.253670\pi$
$199$	19.7186	1.39781	0.698907	+	0.715213i	$0.253670\pi$
$200$	19.0506	1.34708
$201$	13.3878	0.944303
$202$	15.4322	1.08581
$203$	0.128265	0.00900247
$204$	−0.0397473	−0.00278287
$205$	−32.9696	−2.30269
$206$	−19.7604	−1.37677
$207$	2.11501	0.147004
$208$	6.41098	0.444521
$209$	2.40723	0.166512
$210$	0.0871075	0.00601099
$211$	7.41727	0.510626	0.255313	−	0.966858i	$-0.417822\pi$
$211$	7.41727	0.510626	0.255313	+	0.966858i	$0.417822\pi$
$212$	−0.190040	−0.0130520
$213$	−5.47570	−0.375189
$214$	23.8551	1.63070
$215$	39.7874	2.71348
$216$	2.79963	0.190491
$217$	−0.0449926	−0.00305430
$218$	−7.44620	−0.504320
$219$	10.9506	0.739970
$220$	0.292290	0.0197062
$221$	1.57212	0.105752
$222$	−7.75107	−0.520218
$223$	−5.73344	−0.383939	−0.191970	−	0.981401i	$-0.561488\pi$
$223$	−5.73344	−0.383939	−0.191970	+	0.981401i	$0.561488\pi$
$224$	0.00399078	0.000266645 0
$225$	6.80471	0.453647
$226$	9.96935	0.663152
$227$	−17.6673	−1.17262	−0.586311	−	0.810086i	$-0.699420\pi$
$227$	−17.6673	−1.17262	−0.586311	+	0.810086i	$0.699420\pi$
$228$	−0.0447041	−0.00296060
$229$	−10.6572	−0.704245	−0.352122	−	0.935954i	$-0.614540\pi$
$229$	−10.6572	−0.704245	−0.352122	+	0.935954i	$0.614540\pi$
$230$	10.3784	0.684329
$231$	−0.0379943	−0.00249984
$232$	20.2288	1.32809
$233$	5.83796	0.382457	0.191229	−	0.981546i	$-0.438753\pi$
$233$	5.83796	0.382457	0.191229	+	0.981546i	$0.438753\pi$
$234$	2.24530	0.146780
$235$	37.7909	2.46521
$236$	0.0567865	0.00369649
$237$	3.38850	0.220107
$238$	0.0253529	0.00164339
$239$	−1.52807	−0.0988425	−0.0494212	−	0.998778i	$-0.515738\pi$
$239$	−1.52807	−0.0988425	−0.0494212	+	0.998778i	$0.515738\pi$
$240$	14.0109	0.904399
$241$	−14.1203	−0.909567	−0.454784	−	0.890602i	$-0.650283\pi$
$241$	−14.1203	−0.909567	−0.454784	+	0.890602i	$0.650283\pi$
$242$	9.16765	0.589319
$243$	1.00000	0.0641500
$244$	0.415905	0.0266256
$245$	24.0495	1.53647
$246$	−13.7048	−0.873788
$247$	1.76818	0.112506
$248$	−7.09580	−0.450584
$249$	8.56257	0.542631
$250$	8.85570	0.560083
$251$	−9.25544	−0.584198	−0.292099	−	0.956388i	$-0.594354\pi$
$251$	−9.25544	−0.584198	−0.292099	+	0.956388i	$0.594354\pi$
$252$	0.000705581 0	4.44474e−5 0
$253$	−4.52680	−0.284598
$254$	−19.8721	−1.24689
$255$	3.43580	0.215158
$256$	0.953560	0.0595975
$257$	14.2047	0.886063	0.443032	−	0.896506i	$-0.353903\pi$
$257$	14.2047	0.886063	0.443032	+	0.896506i	$0.353903\pi$
$258$	16.5389	1.02967
$259$	0.0963415	0.00598637
$260$	0.214695	0.0133148
$261$	7.22554	0.447250
$262$	−17.6601	−1.09104
$263$	−19.5083	−1.20293	−0.601467	−	0.798898i	$-0.705417\pi$
$263$	−19.5083	−1.20293	−0.601467	+	0.798898i	$0.705417\pi$
$264$	−5.99209	−0.368788
$265$	16.4272	1.00912
$266$	0.0285146	0.00174834
$267$	−8.56091	−0.523919
$268$	0.532129	0.0325049
$269$	−4.24273	−0.258684	−0.129342	−	0.991600i	$-0.541286\pi$
$269$	−4.24273	−0.258684	−0.129342	+	0.991600i	$0.541286\pi$
$270$	4.90700	0.298630
$271$	−3.73401	−0.226825	−0.113412	−	0.993548i	$-0.536178\pi$
$271$	−3.73401	−0.226825	−0.113412	+	0.993548i	$0.536178\pi$
$272$	4.07791	0.247260
$273$	−0.0279078	−0.00168906
$274$	25.9665	1.56869
$275$	−14.5642	−0.878257
$276$	0.0840660	0.00506018
$277$	9.99835	0.600743	0.300371	−	0.953822i	$-0.402889\pi$
$277$	9.99835	0.600743	0.300371	+	0.953822i	$0.402889\pi$
$278$	−3.62330	−0.217311
$279$	−2.53455	−0.151740
$280$	−0.170753	−0.0102044
$281$	17.1755	1.02461	0.512303	−	0.858805i	$-0.328793\pi$
$281$	17.1755	1.02461	0.512303	+	0.858805i	$0.328793\pi$
$282$	15.7090	0.935457
$283$	−17.4753	−1.03880	−0.519401	−	0.854531i	$-0.673845\pi$
$283$	−17.4753	−1.03880	−0.519401	+	0.854531i	$0.673845\pi$
$284$	−0.217644	−0.0129148
$285$	3.86427	0.228899
$286$	−4.80566	−0.284164
$287$	0.170343	0.0100550
$288$	0.224811	0.0132471
$289$	1.00000	0.0588235
$290$	35.4557	2.08203
$291$	15.4548	0.905978
$292$	0.435255	0.0254714
$293$	21.0171	1.22783	0.613916	−	0.789371i	$-0.289593\pi$
$293$	21.0171	1.22783	0.613916	+	0.789371i	$0.289593\pi$
$294$	9.99693	0.583033
$295$	−4.90869	−0.285795
$296$	15.1941	0.883137
$297$	−2.14032	−0.124194
$298$	−7.08362	−0.410343
$299$	−3.32506	−0.192293
$300$	0.270469	0.0156155
$301$	−0.205569	−0.0118488
$302$	−12.3724	−0.711953
$303$	−10.8054	−0.620754
$304$	4.58646	0.263052
$305$	−35.9513	−2.05856
$306$	1.42820	0.0816446
$307$	10.8947	0.621795	0.310898	−	0.950443i	$-0.399370\pi$
$307$	10.8947	0.621795	0.310898	+	0.950443i	$0.399370\pi$
$308$	−0.00151017	−8.60498e−5 0
$309$	13.8359	0.787096
$310$	−12.4370	−0.706376
$311$	16.3943	0.929634	0.464817	−	0.885407i	$-0.346120\pi$
$311$	16.3943	0.929634	0.464817	+	0.885407i	$0.346120\pi$
$312$	−4.40135	−0.249178
$313$	11.8995	0.672599	0.336300	−	0.941755i	$-0.390825\pi$
$313$	11.8995	0.672599	0.336300	+	0.941755i	$0.390825\pi$
$314$	−1.42820	−0.0805978
$315$	−0.0609912	−0.00343647
$316$	0.134684	0.00757655
$317$	−12.2961	−0.690620	−0.345310	−	0.938489i	$-0.612226\pi$
$317$	−12.2961	−0.690620	−0.345310	+	0.938489i	$0.612226\pi$
$318$	6.82850	0.382923
$319$	−15.4650	−0.865871
$320$	−26.9186	−1.50480
$321$	−16.7030	−0.932268
$322$	−0.0536217	−0.00298822
$323$	1.12471	0.0625804
$324$	0.0397473	0.00220818
$325$	−10.6978	−0.593409
$326$	−5.54226	−0.306957
$327$	5.21370	0.288318
$328$	26.8649	1.48337
$329$	−0.195254	−0.0107647
$330$	−10.5025	−0.578146
$331$	2.79942	0.153870	0.0769351	−	0.997036i	$-0.475487\pi$
$331$	2.79942	0.153870	0.0769351	+	0.997036i	$0.475487\pi$
$332$	0.340339	0.0186785
$333$	5.42717	0.297407
$334$	−1.95689	−0.107076
$335$	−45.9978	−2.51313
$336$	−0.0723899	−0.00394919
$337$	1.24402	0.0677659	0.0338830	−	0.999426i	$-0.489213\pi$
$337$	1.24402	0.0677659	0.0338830	+	0.999426i	$0.489213\pi$
$338$	15.0367	0.817888
$339$	−6.98038	−0.379122
$340$	0.136564	0.00740620
$341$	5.42475	0.293767
$342$	1.60630	0.0868590
$343$	−0.248518	−0.0134187
$344$	−32.4204	−1.74799
$345$	−7.26676	−0.391229
$346$	34.1072	1.83362
$347$	4.98828	0.267785	0.133892	−	0.990996i	$-0.457252\pi$
$347$	4.98828	0.267785	0.133892	+	0.990996i	$0.457252\pi$
$348$	0.287195	0.0153953
$349$	−16.7777	−0.898090	−0.449045	−	0.893509i	$-0.648236\pi$
$349$	−16.7777	−0.898090	−0.449045	+	0.893509i	$0.648236\pi$
$350$	−0.172519	−0.00922154
$351$	−1.57212	−0.0839136
$352$	−0.481168	−0.0256463
$353$	−6.47133	−0.344434	−0.172217	−	0.985059i	$-0.555093\pi$
$353$	−6.47133	−0.344434	−0.172217	+	0.985059i	$0.555093\pi$
$354$	−2.04045	−0.108449
$355$	18.8134	0.998512
$356$	−0.340273	−0.0180344
$357$	−0.0177517	−0.000939519 0
$358$	−30.5844	−1.61644
$359$	−20.7359	−1.09440	−0.547199	−	0.837002i	$-0.684306\pi$
$359$	−20.7359	−1.09440	−0.547199	+	0.837002i	$0.684306\pi$
$360$	−9.61895	−0.506963
$361$	−17.7350	−0.933423
$362$	−13.7773	−0.724119
$363$	−6.41904	−0.336912
$364$	−0.00110926	−5.81410e−5 0
$365$	−37.6239	−1.96933
$366$	−14.9443	−0.781150
$367$	−22.6049	−1.17997	−0.589984	−	0.807415i	$-0.700866\pi$
$367$	−22.6049	−1.17997	−0.589984	+	0.807415i	$0.700866\pi$
$368$	−8.62484	−0.449601
$369$	9.59589	0.499542
$370$	26.6311	1.38449
$371$	−0.0848744	−0.00440646
$372$	−0.100742	−0.00522321
$373$	−33.3360	−1.72607	−0.863037	−	0.505141i	$-0.831440\pi$
$373$	−33.3360	−1.72607	−0.863037	+	0.505141i	$0.831440\pi$
$374$	−3.05680	−0.158063
$375$	−6.20061	−0.320198
$376$	−30.7936	−1.58806
$377$	−11.3594	−0.585040
$378$	−0.0253529	−0.00130401
$379$	−6.55025	−0.336464	−0.168232	−	0.985747i	$-0.553806\pi$
$379$	−6.55025	−0.336464	−0.168232	+	0.985747i	$0.553806\pi$
$380$	0.153594	0.00787921
$381$	13.9141	0.712842
$382$	16.1541	0.826517
$383$	−18.1074	−0.925243	−0.462622	−	0.886556i	$-0.653091\pi$
$383$	−18.1074	−0.925243	−0.462622	+	0.886556i	$0.653091\pi$
$384$	−11.6392	−0.593960
$385$	0.130541	0.00665297
$386$	25.3153	1.28851
$387$	−11.5802	−0.588657
$388$	0.614287	0.0311857
$389$	16.1574	0.819215	0.409607	−	0.912262i	$-0.365666\pi$
$389$	16.1574	0.819215	0.409607	+	0.912262i	$0.365666\pi$
$390$	−7.71439	−0.390634
$391$	−2.11501	−0.106961
$392$	−19.5965	−0.989773
$393$	12.3653	0.623747
$394$	18.7872	0.946485
$395$	−11.6422	−0.585783
$396$	−0.0850718	−0.00427502
$397$	−31.6739	−1.58967	−0.794833	−	0.606828i	$-0.792441\pi$
$397$	−31.6739	−1.58967	−0.794833	+	0.606828i	$0.792441\pi$
$398$	−28.1620	−1.41164
$399$	−0.0199655	−0.000999523 0
$400$	−27.7490	−1.38745
$401$	18.7676	0.937208	0.468604	−	0.883408i	$-0.344757\pi$
$401$	18.7676	0.937208	0.468604	+	0.883408i	$0.344757\pi$
$402$	−19.1204	−0.953640
$403$	3.98462	0.198488
$404$	−0.429485	−0.0213677
$405$	−3.43580	−0.170726
$406$	−0.183188	−0.00909149
$407$	−11.6159	−0.575778
$408$	−2.79963	−0.138602
$409$	−33.6880	−1.66577	−0.832883	−	0.553450i	$-0.813311\pi$
$409$	−33.6880	−1.66577	−0.832883	+	0.553450i	$0.813311\pi$
$410$	47.0870	2.32546
$411$	−18.1813	−0.896818
$412$	0.549939	0.0270935
$413$	0.0253616	0.00124796
$414$	−3.02066	−0.148457
$415$	−29.4193	−1.44413
$416$	−0.353431	−0.0173284
$417$	2.53697	0.124236
$418$	−3.43800	−0.168158
$419$	−26.0118	−1.27076	−0.635379	−	0.772200i	$-0.719156\pi$
$419$	−26.0118	−1.27076	−0.635379	+	0.772200i	$0.719156\pi$
$420$	−0.00242423	−0.000118291 0
$421$	34.0339	1.65871	0.829355	−	0.558722i	$-0.188708\pi$
$421$	34.0339	1.65871	0.829355	+	0.558722i	$0.188708\pi$
$422$	−10.5933	−0.515675
$423$	−10.9992	−0.534798
$424$	−13.3856	−0.650061
$425$	−6.80471	−0.330077
$426$	7.82038	0.378899
$427$	0.185749	0.00898902
$428$	−0.663897	−0.0320907
$429$	3.36484	0.162456
$430$	−56.8243	−2.74031
$431$	28.8515	1.38973	0.694864	−	0.719141i	$-0.255465\pi$
$431$	28.8515	1.38973	0.694864	+	0.719141i	$0.255465\pi$
$432$	−4.07791	−0.196199
$433$	6.65932	0.320026	0.160013	−	0.987115i	$-0.448846\pi$
$433$	6.65932	0.320026	0.160013	+	0.987115i	$0.448846\pi$
$434$	0.0642583	0.00308450
$435$	−24.8255	−1.19029
$436$	0.207230	0.00992454
$437$	−2.37877	−0.113792
$438$	−15.6396	−0.747287
$439$	−11.9295	−0.569365	−0.284682	−	0.958622i	$-0.591888\pi$
$439$	−11.9295	−0.569365	−0.284682	+	0.958622i	$0.591888\pi$
$440$	20.5876	0.981477
$441$	−6.99968	−0.333318
$442$	−2.24530	−0.106798
$443$	22.8934	1.08770	0.543850	−	0.839183i	$-0.316966\pi$
$443$	22.8934	1.08770	0.543850	+	0.839183i	$0.316966\pi$
$444$	0.215715	0.0102374
$445$	29.4136	1.39434
$446$	8.18848	0.387736
$447$	4.95984	0.234592
$448$	0.139080	0.00657092
$449$	31.9959	1.50998	0.754991	−	0.655735i	$-0.227641\pi$
$449$	31.9959	1.50998	0.754991	+	0.655735i	$0.227641\pi$
$450$	−9.71846	−0.458133
$451$	−20.5383	−0.967109
$452$	−0.277451	−0.0130502
$453$	8.66297	0.407022
$454$	25.2324	1.18422
$455$	0.0958856	0.00449519
$456$	−3.14876	−0.147454
$457$	22.5887	1.05666	0.528329	−	0.849040i	$-0.322819\pi$
$457$	22.5887	1.05666	0.528329	+	0.849040i	$0.322819\pi$
$458$	15.2205	0.711208
$459$	−1.00000	−0.0466760
$460$	−0.288834	−0.0134669
$461$	−15.1682	−0.706452	−0.353226	−	0.935538i	$-0.614915\pi$
$461$	−15.1682	−0.706452	−0.353226	+	0.935538i	$0.614915\pi$
$462$	0.0542633	0.00252456
$463$	−0.911350	−0.0423540	−0.0211770	−	0.999776i	$-0.506741\pi$
$463$	−0.911350	−0.0423540	−0.0211770	+	0.999776i	$0.506741\pi$
$464$	−29.4651	−1.36788
$465$	8.70821	0.403834
$466$	−8.33776	−0.386239
$467$	1.81262	0.0838781	0.0419391	−	0.999120i	$-0.486646\pi$
$467$	1.81262	0.0838781	0.0419391	+	0.999120i	$0.486646\pi$
$468$	−0.0624875	−0.00288849
$469$	0.237656	0.0109739
$470$	−53.9729	−2.48959
$471$	1.00000	0.0460776
$472$	3.99980	0.184106
$473$	24.7854	1.13963
$474$	−4.83945	−0.222283
$475$	−7.65331	−0.351158
$476$	−0.000705581 0	−3.23403e−5 0
$477$	−4.78120	−0.218916
$478$	2.18238	0.0998198
$479$	−35.9876	−1.64432	−0.822159	−	0.569259i	$-0.807230\pi$
$479$	−35.9876	−1.64432	−0.822159	+	0.569259i	$0.807230\pi$
$480$	−0.772407	−0.0352554
$481$	−8.53217	−0.389034
$482$	20.1665	0.918561
$483$	0.0375451	0.00170836
$484$	−0.255139	−0.0115972
$485$	−53.0997	−2.41113
$486$	−1.42820	−0.0647843
$487$	3.92136	0.177694	0.0888469	−	0.996045i	$-0.471682\pi$
$487$	3.92136	0.177694	0.0888469	+	0.996045i	$0.471682\pi$
$488$	29.2945	1.32610
$489$	3.88060	0.175487
$490$	−34.3474	−1.55166
$491$	−21.6489	−0.977000	−0.488500	−	0.872564i	$-0.662456\pi$
$491$	−21.6489	−0.977000	−0.488500	+	0.872564i	$0.662456\pi$
$492$	0.381411	0.0171953
$493$	−7.22554	−0.325422
$494$	−2.52530	−0.113619
$495$	7.35370	0.330524
$496$	10.3357	0.464086
$497$	−0.0972029	−0.00436015
$498$	−12.2290	−0.547996
$499$	−3.63515	−0.162732	−0.0813658	−	0.996684i	$-0.525928\pi$
$499$	−3.63515	−0.162732	−0.0813658	+	0.996684i	$0.525928\pi$
$500$	−0.246457	−0.0110219
$501$	1.37018	0.0612151
$502$	13.2186	0.589975
$503$	−15.9087	−0.709333	−0.354666	−	0.934993i	$-0.615406\pi$
$503$	−15.9087	−0.709333	−0.354666	+	0.934993i	$0.615406\pi$
$504$	0.0496981	0.00221373
$505$	37.1251	1.65205
$506$	6.46517	0.287412
$507$	−10.5284	−0.467584
$508$	0.553048	0.0245375
$509$	−19.1996	−0.851005	−0.425503	−	0.904957i	$-0.639903\pi$
$509$	−19.1996	−0.851005	−0.425503	+	0.904957i	$0.639903\pi$
$510$	−4.90700	−0.217285
$511$	0.194391	0.00859935
$512$	21.9165	0.968583
$513$	−1.12471	−0.0496570
$514$	−20.2871	−0.894824
$515$	−47.5373	−2.09474
$516$	−0.460283	−0.0202628
$517$	23.5417	1.03536
$518$	−0.137595	−0.00604556
$519$	−23.8813	−1.04827
$520$	15.1222	0.663151
$521$	−6.79497	−0.297693	−0.148847	−	0.988860i	$-0.547556\pi$
$521$	−6.79497	−0.297693	−0.148847	+	0.988860i	$0.547556\pi$
$522$	−10.3195	−0.451672
$523$	23.7900	1.04026	0.520131	−	0.854086i	$-0.325883\pi$
$523$	23.7900	1.04026	0.520131	+	0.854086i	$0.325883\pi$
$524$	0.491487	0.0214707
$525$	0.120795	0.00527193
$526$	27.8617	1.21483
$527$	2.53455	0.110407
$528$	8.72804	0.379839
$529$	−18.5267	−0.805510
$530$	−23.4613	−1.01910
$531$	1.42869	0.0619998
$532$	−0.000793572 0	−3.44057e−5 0
$533$	−15.0859	−0.653443
$534$	12.2267	0.529100
$535$	57.3880	2.48110
$536$	37.4808	1.61893
$537$	21.4147	0.924112
$538$	6.05946	0.261242
$539$	14.9816	0.645301
$540$	−0.136564	−0.00587676
$541$	18.4019	0.791162	0.395581	−	0.918431i	$-0.370543\pi$
$541$	18.4019	0.791162	0.395581	+	0.918431i	$0.370543\pi$
$542$	5.33290	0.229068
$543$	9.64664	0.413977
$544$	−0.224811	−0.00963871
$545$	−17.9132	−0.767318
$546$	0.0398578	0.00170576
$547$	−21.1291	−0.903413	−0.451707	−	0.892166i	$-0.649185\pi$
$547$	−21.1291	−0.903413	−0.451707	+	0.892166i	$0.649185\pi$
$548$	−0.722658	−0.0308704
$549$	10.4637	0.446581
$550$	20.8006	0.886941
$551$	−8.12662	−0.346205
$552$	5.92125	0.252025
$553$	0.0601517	0.00255791
$554$	−14.2796	−0.606683
$555$	−18.6467	−0.791507
$556$	0.100838	0.00427648
$557$	−34.6864	−1.46971	−0.734855	−	0.678224i	$-0.762750\pi$
$557$	−34.6864	−1.46971	−0.734855	+	0.678224i	$0.762750\pi$
$558$	3.61984	0.153240
$559$	18.2056	0.770013
$560$	0.248717	0.0105102
$561$	2.14032	0.0903643
$562$	−24.5300	−1.03474
$563$	13.5564	0.571332	0.285666	−	0.958329i	$-0.407785\pi$
$563$	13.5564	0.571332	0.285666	+	0.958329i	$0.407785\pi$
$564$	−0.437187	−0.0184089
$565$	23.9832	1.00898
$566$	24.9582	1.04907
$567$	0.0177517	0.000745501 0
$568$	−15.3299	−0.643229
$569$	−35.1773	−1.47471	−0.737354	−	0.675506i	$-0.763925\pi$
$569$	−35.1773	−1.47471	−0.737354	+	0.675506i	$0.763925\pi$
$570$	−5.51894	−0.231163
$571$	−33.0205	−1.38186	−0.690932	−	0.722920i	$-0.742799\pi$
$571$	−33.0205	−1.38186	−0.690932	+	0.722920i	$0.742799\pi$
$572$	0.133743	0.00559209
$573$	−11.3109	−0.472518
$574$	−0.243284	−0.0101545
$575$	14.3920	0.600190
$576$	7.83475	0.326448
$577$	−12.2284	−0.509074	−0.254537	−	0.967063i	$-0.581923\pi$
$577$	−12.2284	−0.509074	−0.254537	+	0.967063i	$0.581923\pi$
$578$	−1.42820	−0.0594052
$579$	−17.7254	−0.736640
$580$	−0.986745	−0.0409724
$581$	0.152000	0.00630602
$582$	−22.0725	−0.914937
$583$	10.2333	0.423820
$584$	30.6575	1.26862
$585$	5.40149	0.223324
$586$	−30.0166	−1.23997
$587$	−5.87170	−0.242351	−0.121175	−	0.992631i	$-0.538666\pi$
$587$	−5.87170	−0.242351	−0.121175	+	0.992631i	$0.538666\pi$
$588$	−0.278218	−0.0114735
$589$	2.85063	0.117458
$590$	7.01057	0.288621
$591$	−13.1545	−0.541103
$592$	−22.1315	−0.909601
$593$	−29.9397	−1.22948	−0.614739	−	0.788731i	$-0.710738\pi$
$593$	−29.9397	−1.22948	−0.614739	+	0.788731i	$0.710738\pi$
$594$	3.05680	0.125422
$595$	0.0609912	0.00250040
$596$	0.197140	0.00807517
$597$	19.7186	0.807028
$598$	4.74884	0.194194
$599$	−25.8178	−1.05489	−0.527444	−	0.849590i	$-0.676850\pi$
$599$	−25.8178	−1.05489	−0.527444	+	0.849590i	$0.676850\pi$
$600$	19.0506	0.777739
$601$	−44.8575	−1.82977	−0.914887	−	0.403709i	$-0.867721\pi$
$601$	−44.8575	−1.82977	−0.914887	+	0.403709i	$0.867721\pi$
$602$	0.293593	0.0119660
$603$	13.3878	0.545193
$604$	0.344329	0.0140106
$605$	22.0545	0.896643
$606$	15.4322	0.626891
$607$	30.1202	1.22254	0.611271	−	0.791421i	$-0.290659\pi$
$607$	30.1202	1.22254	0.611271	+	0.791421i	$0.290659\pi$
$608$	−0.252847	−0.0102543
$609$	0.128265	0.00519758
$610$	51.3455	2.07892
$611$	17.2920	0.699561
$612$	−0.0397473	−0.00160669
$613$	36.1798	1.46129	0.730645	−	0.682758i	$-0.239220\pi$
$613$	36.1798	1.46129	0.730645	+	0.682758i	$0.239220\pi$
$614$	−15.5598	−0.627944
$615$	−32.9696	−1.32946
$616$	−0.106370	−0.00428576
$617$	−31.0597	−1.25042	−0.625209	−	0.780457i	$-0.714986\pi$
$617$	−31.0597	−1.25042	−0.625209	+	0.780457i	$0.714986\pi$
$618$	−19.7604	−0.794878
$619$	−35.1436	−1.41254	−0.706269	−	0.707943i	$-0.749623\pi$
$619$	−35.1436	−1.41254	−0.706269	+	0.707943i	$0.749623\pi$
$620$	0.346128	0.0139008
$621$	2.11501	0.0848726
$622$	−23.4143	−0.938826
$623$	−0.151971	−0.00608857
$624$	6.41098	0.256644
$625$	−12.7195	−0.508780
$626$	−16.9948	−0.679250
$627$	2.40723	0.0961356
$628$	0.0397473	0.00158609
$629$	−5.42717	−0.216396
$630$	0.0871075	0.00347045
$631$	20.8299	0.829225	0.414612	−	0.909998i	$-0.363917\pi$
$631$	20.8299	0.829225	0.414612	+	0.909998i	$0.363917\pi$
$632$	9.48655	0.377355
$633$	7.41727	0.294810
$634$	17.5613	0.697449
$635$	−47.8061	−1.89713
$636$	−0.190040	−0.00753557
$637$	11.0044	0.436008
$638$	22.0870	0.874433
$639$	−5.47570	−0.216615
$640$	39.9899	1.58074
$641$	−46.0706	−1.81968	−0.909839	−	0.414960i	$-0.863795\pi$
$641$	−46.0706	−1.81968	−0.909839	+	0.414960i	$0.863795\pi$
$642$	23.8551	0.941486
$643$	13.5403	0.533979	0.266990	−	0.963699i	$-0.413971\pi$
$643$	13.5403	0.533979	0.266990	+	0.963699i	$0.413971\pi$
$644$	0.00149231	5.88054e−5 0
$645$	39.7874	1.56663
$646$	−1.60630	−0.0631992
$647$	−49.3679	−1.94085	−0.970427	−	0.241397i	$-0.922395\pi$
$647$	−49.3679	−1.94085	−0.970427	+	0.241397i	$0.922395\pi$
$648$	2.79963	0.109980
$649$	−3.05785	−0.120031
$650$	15.2786	0.599276
$651$	−0.0449926	−0.00176340
$652$	0.154243	0.00604063
$653$	8.33714	0.326258	0.163129	−	0.986605i	$-0.447841\pi$
$653$	8.33714	0.326258	0.163129	+	0.986605i	$0.447841\pi$
$654$	−7.44620	−0.291169
$655$	−42.4847	−1.66001
$656$	−39.1312	−1.52782
$657$	10.9506	0.427222
$658$	0.278861	0.0108711
$659$	14.8314	0.577749	0.288874	−	0.957367i	$-0.406719\pi$
$659$	14.8314	0.577749	0.288874	+	0.957367i	$0.406719\pi$
$660$	0.292290	0.0113774
$661$	−38.9353	−1.51441	−0.757203	−	0.653179i	$-0.773435\pi$
$661$	−38.9353	−1.51441	−0.757203	+	0.653179i	$0.773435\pi$
$662$	−3.99813	−0.155392
$663$	1.57212	0.0610561
$664$	23.9720	0.930294
$665$	0.0685973	0.00266009
$666$	−7.75107	−0.300348
$667$	15.2821	0.591726
$668$	0.0544609	0.00210715
$669$	−5.73344	−0.221667
$670$	65.6939	2.53798
$671$	−22.3957	−0.864577
$672$	0.00399078	0.000153948 0
$673$	−17.3054	−0.667076	−0.333538	−	0.942737i	$-0.608242\pi$
$673$	−17.3054	−0.667076	−0.333538	+	0.942737i	$0.608242\pi$
$674$	−1.77670	−0.0684360
$675$	6.80471	0.261913
$676$	−0.418477	−0.0160953
$677$	−16.4340	−0.631611	−0.315806	−	0.948824i	$-0.602275\pi$
$677$	−16.4340	−0.631611	−0.315806	+	0.948824i	$0.602275\pi$
$678$	9.96935	0.382871
$679$	0.274349	0.0105286
$680$	9.61895	0.368870
$681$	−17.6673	−0.677013
$682$	−7.74761	−0.296671
$683$	7.26276	0.277902	0.138951	−	0.990299i	$-0.455627\pi$
$683$	7.26276	0.277902	0.138951	+	0.990299i	$0.455627\pi$
$684$	−0.0447041	−0.00170930
$685$	62.4673	2.38675
$686$	0.354933	0.0135514
$687$	−10.6572	−0.406596
$688$	47.2233	1.80037
$689$	7.51663	0.286361
$690$	10.3784	0.395098
$691$	−12.3929	−0.471447	−0.235724	−	0.971820i	$-0.575746\pi$
$691$	−12.3929	−0.471447	−0.235724	+	0.971820i	$0.575746\pi$
$692$	−0.949217	−0.0360838
$693$	−0.0379943	−0.00144328
$694$	−7.12425	−0.270433
$695$	−8.71653	−0.330637
$696$	20.2288	0.766771
$697$	−9.59589	−0.363470
$698$	23.9619	0.906970
$699$	5.83796	0.220812
$700$	0.00480127	0.000181471 0
$701$	20.5757	0.777132	0.388566	−	0.921421i	$-0.372971\pi$
$701$	20.5757	0.777132	0.388566	+	0.921421i	$0.372971\pi$
$702$	2.24530	0.0847434
$703$	−6.10398	−0.230216
$704$	−16.7689	−0.632001
$705$	37.7909	1.42329
$706$	9.24233	0.347840
$707$	−0.191814	−0.00721391
$708$	0.0567865	0.00213417
$709$	−0.00689544	−0.000258964 0	−0.000129482	−	1.00000i	$-0.500041\pi$
$709$	−0.00689544	−0.000258964 0	−0.000129482		1.00000i	$0.500041\pi$
$710$	−26.8692	−1.00838
$711$	3.38850	0.127079
$712$	−23.9674	−0.898215
$713$	−5.36061	−0.200756
$714$	0.0253529	0.000948809 0
$715$	−11.5609	−0.432354
$716$	0.851176	0.0318099
$717$	−1.52807	−0.0570667
$718$	29.6149	1.10522
$719$	22.3294	0.832748	0.416374	−	0.909194i	$-0.363301\pi$
$719$	22.3294	0.832748	0.416374	+	0.909194i	$0.363301\pi$
$720$	14.0109	0.522155
$721$	0.245610	0.00914700
$722$	25.3291	0.942652
$723$	−14.1203	−0.525139
$724$	0.383428	0.0142500
$725$	49.1677	1.82604
$726$	9.16765	0.340243
$727$	16.6132	0.616149	0.308075	−	0.951362i	$-0.400315\pi$
$727$	16.6132	0.616149	0.308075	+	0.951362i	$0.400315\pi$
$728$	−0.0781315	−0.00289574
$729$	1.00000	0.0370370
$730$	53.7344	1.98880
$731$	11.5802	0.428311
$732$	0.415905	0.0153723
$733$	−34.4739	−1.27332	−0.636660	−	0.771144i	$-0.719685\pi$
$733$	−34.4739	−1.27332	−0.636660	+	0.771144i	$0.719685\pi$
$734$	32.2843	1.19164
$735$	24.0495	0.887079
$736$	0.475479	0.0175264
$737$	−28.6542	−1.05549
$738$	−13.7048	−0.504482
$739$	−11.3500	−0.417515	−0.208758	−	0.977967i	$-0.566942\pi$
$739$	−11.3500	−0.417515	−0.208758	+	0.977967i	$0.566942\pi$
$740$	−0.741154	−0.0272454
$741$	1.76818	0.0649556
$742$	0.121217	0.00445003
$743$	38.9822	1.43012	0.715058	−	0.699065i	$-0.246400\pi$
$743$	38.9822	1.43012	0.715058	+	0.699065i	$0.246400\pi$
$744$	−7.09580	−0.260145
$745$	−17.0410	−0.624334
$746$	47.6104	1.74314
$747$	8.56257	0.313288
$748$	0.0850718	0.00311053
$749$	−0.296506	−0.0108341
$750$	8.85570	0.323364
$751$	4.25670	0.155329	0.0776645	−	0.996980i	$-0.475254\pi$
$751$	4.25670	0.155329	0.0776645	+	0.996980i	$0.475254\pi$
$752$	44.8537	1.63565
$753$	−9.25544	−0.337287
$754$	16.2235	0.590825
$755$	−29.7642	−1.08323
$756$	0.000705581 0	2.56617e−5 0
$757$	9.05927	0.329265	0.164632	−	0.986355i	$-0.447356\pi$
$757$	9.05927	0.329265	0.164632	+	0.986355i	$0.447356\pi$
$758$	9.35505	0.339791
$759$	−4.52680	−0.164313
$760$	10.8185	0.392429
$761$	7.34113	0.266116	0.133058	−	0.991108i	$-0.457520\pi$
$761$	7.34113	0.266116	0.133058	+	0.991108i	$0.457520\pi$
$762$	−19.8721	−0.719890
$763$	0.0925520	0.00335061
$764$	−0.449575	−0.0162651
$765$	3.43580	0.124222
$766$	25.8609	0.934392
$767$	−2.24607	−0.0811010
$768$	0.953560	0.0344086
$769$	−30.9731	−1.11692	−0.558460	−	0.829532i	$-0.688607\pi$
$769$	−30.9731	−1.11692	−0.558460	+	0.829532i	$0.688607\pi$
$770$	−0.186438	−0.00671875
$771$	14.2047	0.511569
$772$	−0.704534	−0.0253567
$773$	39.3182	1.41418	0.707089	−	0.707124i	$-0.250008\pi$
$773$	39.3182	1.41418	0.707089	+	0.707124i	$0.250008\pi$
$774$	16.5389	0.594478
$775$	−17.2469	−0.619526
$776$	43.2678	1.55322
$777$	0.0963415	0.00345623
$778$	−23.0760	−0.827315
$779$	−10.7926	−0.386684
$780$	0.214695	0.00768730
$781$	11.7197	0.419365
$782$	3.02066	0.108018
$783$	7.22554	0.258220
$784$	28.5441	1.01943
$785$	−3.43580	−0.122629
$786$	−17.6601	−0.629915
$787$	20.9054	0.745196	0.372598	−	0.927993i	$-0.378467\pi$
$787$	20.9054	0.745196	0.372598	+	0.927993i	$0.378467\pi$
$788$	−0.522855	−0.0186259
$789$	−19.5083	−0.694514
$790$	16.6274	0.591575
$791$	−0.123913	−0.00440586
$792$	−5.99209	−0.212920
$793$	−16.4503	−0.584166
$794$	45.2365	1.60538
$795$	16.4272	0.582614
$796$	0.783760	0.0277796
$797$	−47.2798	−1.67474	−0.837369	−	0.546638i	$-0.815907\pi$
$797$	−47.2798	−1.67474	−0.837369	+	0.546638i	$0.815907\pi$
$798$	0.0285146	0.00100941
$799$	10.9992	0.389123
$800$	1.52978	0.0540858
$801$	−8.56091	−0.302485
$802$	−26.8038	−0.946475
$803$	−23.4377	−0.827098
$804$	0.532129	0.0187667
$805$	−0.128997	−0.00454655
$806$	−5.69083	−0.200451
$807$	−4.24273	−0.149351
$808$	−30.2511	−1.06423
$809$	−21.2575	−0.747375	−0.373687	−	0.927555i	$-0.621907\pi$
$809$	−21.2575	−0.747375	−0.373687	+	0.927555i	$0.621907\pi$
$810$	4.90700	0.172414
$811$	−32.9937	−1.15857	−0.579283	−	0.815126i	$-0.696667\pi$
$811$	−32.9937	−1.15857	−0.579283	+	0.815126i	$0.696667\pi$
$812$	0.00509820	0.000178912 0
$813$	−3.73401	−0.130957
$814$	16.5898	0.581471
$815$	−13.3330	−0.467033
$816$	4.07791	0.142756
$817$	13.0244	0.455666
$818$	48.1131	1.68224
$819$	−0.0279078	−0.000975178 0
$820$	−1.31045	−0.0457629
$821$	−9.92574	−0.346411	−0.173205	−	0.984886i	$-0.555412\pi$
$821$	−9.92574	−0.346411	−0.173205	+	0.984886i	$0.555412\pi$
$822$	25.9665	0.905686
$823$	24.3703	0.849495	0.424747	−	0.905312i	$-0.360363\pi$
$823$	24.3703	0.849495	0.424747	+	0.905312i	$0.360363\pi$
$824$	38.7353	1.34941
$825$	−14.5642	−0.507062
$826$	−0.0362214	−0.00126030
$827$	41.9660	1.45930	0.729650	−	0.683821i	$-0.239683\pi$
$827$	41.9660	1.45930	0.729650	+	0.683821i	$0.239683\pi$
$828$	0.0840660	0.00292150
$829$	39.9557	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$829$	39.9557	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$830$	42.0165	1.45841
$831$	9.99835	0.346839
$832$	−12.3172	−0.427022
$833$	6.99968	0.242525
$834$	−3.62330	−0.125465
$835$	−4.70766	−0.162915
$836$	0.0956809	0.00330919
$837$	−2.53455	−0.0876069
$838$	37.1499	1.28332
$839$	6.13011	0.211635	0.105817	−	0.994386i	$-0.466254\pi$
$839$	6.13011	0.211635	0.105817	+	0.994386i	$0.466254\pi$
$840$	−0.170753	−0.00589153
$841$	23.2084	0.800289
$842$	−48.6071	−1.67511
$843$	17.1755	0.591556
$844$	0.294816	0.0101480
$845$	36.1736	1.24441
$846$	15.7090	0.540086
$847$	−0.113949	−0.00391532
$848$	19.4973	0.669541
$849$	−17.4753	−0.599752
$850$	9.71846	0.333341
$851$	11.4785	0.393479
$852$	−0.217644	−0.00745637
$853$	2.89569	0.0991466	0.0495733	−	0.998770i	$-0.484214\pi$
$853$	2.89569	0.0991466	0.0495733	+	0.998770i	$0.484214\pi$
$854$	−0.265286	−0.00907790
$855$	3.86427	0.132155
$856$	−46.7620	−1.59829
$857$	15.3289	0.523625	0.261813	−	0.965119i	$-0.415680\pi$
$857$	15.3289	0.523625	0.261813	+	0.965119i	$0.415680\pi$
$858$	−4.80566	−0.164062
$859$	−7.10231	−0.242328	−0.121164	−	0.992633i	$-0.538663\pi$
$859$	−7.10231	−0.242328	−0.121164	+	0.992633i	$0.538663\pi$
$860$	1.58144	0.0539267
$861$	0.170343	0.00580528
$862$	−41.2056	−1.40347
$863$	15.8374	0.539112	0.269556	−	0.962985i	$-0.413123\pi$
$863$	15.8374	0.539112	0.269556	+	0.962985i	$0.413123\pi$
$864$	0.224811	0.00764824
$865$	82.0514	2.78983
$866$	−9.51082	−0.323191
$867$	1.00000	0.0339618
$868$	−0.00178833	−6.07000e−5 0
$869$	−7.25248	−0.246023
$870$	35.4557	1.20206
$871$	−21.0472	−0.713159
$872$	14.5964	0.494297
$873$	15.4548	0.523067
$874$	3.39735	0.114917
$875$	−0.110071	−0.00372109
$876$	0.435255	0.0147059
$877$	−33.2347	−1.12226	−0.561128	−	0.827729i	$-0.689632\pi$
$877$	−33.2347	−1.12226	−0.561128	+	0.827729i	$0.689632\pi$
$878$	17.0377	0.574995
$879$	21.0171	0.708890
$880$	−29.9878	−1.01089
$881$	−13.1617	−0.443430	−0.221715	−	0.975111i	$-0.571165\pi$
$881$	−13.1617	−0.443430	−0.221715	+	0.975111i	$0.571165\pi$
$882$	9.99693	0.336614
$883$	−7.78730	−0.262063	−0.131032	−	0.991378i	$-0.541829\pi$
$883$	−7.78730	−0.262063	−0.131032	+	0.991378i	$0.541829\pi$
$884$	0.0624875	0.00210168
$885$	−4.90869	−0.165004
$886$	−32.6963	−1.09845
$887$	23.2948	0.782162	0.391081	−	0.920356i	$-0.372101\pi$
$887$	23.2948	0.782162	0.391081	+	0.920356i	$0.372101\pi$
$888$	15.1941	0.509879
$889$	0.246999	0.00828408
$890$	−42.0084	−1.40812
$891$	−2.14032	−0.0717034
$892$	−0.227889	−0.00763027
$893$	12.3709	0.413975
$894$	−7.08362	−0.236912
$895$	−73.5766	−2.45939
$896$	−0.206615	−0.00690254
$897$	−3.32506	−0.111020
$898$	−45.6965	−1.52491
$899$	−18.3135	−0.610789
$900$	0.270469	0.00901562
$901$	4.78120	0.159285
$902$	29.3327	0.976672
$903$	−0.205569	−0.00684091
$904$	−19.5425	−0.649972
$905$	−33.1439	−1.10174
$906$	−12.3724	−0.411046
$907$	−37.0970	−1.23179	−0.615893	−	0.787830i	$-0.711204\pi$
$907$	−37.0970	−1.23179	−0.615893	+	0.787830i	$0.711204\pi$
$908$	−0.702228	−0.0233043
$909$	−10.8054	−0.358392
$910$	−0.136944	−0.00453963
$911$	−38.7110	−1.28255	−0.641276	−	0.767310i	$-0.721595\pi$
$911$	−38.7110	−1.28255	−0.641276	+	0.767310i	$0.721595\pi$
$912$	4.58646	0.151873
$913$	−18.3266	−0.606523
$914$	−32.2612	−1.06711
$915$	−35.9513	−1.18851
$916$	−0.423593	−0.0139959
$917$	0.219505	0.00724870
$918$	1.42820	0.0471375
$919$	1.12584	0.0371381	0.0185691	−	0.999828i	$-0.494089\pi$
$919$	1.12584	0.0371381	0.0185691	+	0.999828i	$0.494089\pi$
$920$	−20.3442	−0.670729
$921$	10.8947	0.358994
$922$	21.6631	0.713437
$923$	8.60847	0.283351
$924$	−0.00151017	−4.96809e−5 0
$925$	36.9303	1.21426
$926$	1.30159	0.0427728
$927$	13.8359	0.454430
$928$	1.62438	0.0533230
$929$	52.9710	1.73792	0.868961	−	0.494881i	$-0.164788\pi$
$929$	52.9710	1.73792	0.868961	+	0.494881i	$0.164788\pi$
$930$	−12.4370	−0.407827
$931$	7.87260	0.258014
$932$	0.232043	0.00760082
$933$	16.3943	0.536725
$934$	−2.58878	−0.0847075
$935$	−7.35370	−0.240492
$936$	−4.40135	−0.143863
$937$	−2.48383	−0.0811433	−0.0405716	−	0.999177i	$-0.512918\pi$
$937$	−2.48383	−0.0811433	−0.0405716	+	0.999177i	$0.512918\pi$
$938$	−0.339420	−0.0110824
$939$	11.8995	0.388325
$940$	1.50209	0.0489927
$941$	−15.5052	−0.505456	−0.252728	−	0.967537i	$-0.581328\pi$
$941$	−15.5052	−0.505456	−0.252728	+	0.967537i	$0.581328\pi$
$942$	−1.42820	−0.0465332
$943$	20.2954	0.660911
$944$	−5.82607	−0.189622
$945$	−0.0609912	−0.00198404
$946$	−35.3985	−1.15090
$947$	−52.1726	−1.69538	−0.847692	−	0.530489i	$-0.822008\pi$
$947$	−52.1726	−1.69538	−0.847692	+	0.530489i	$0.822008\pi$
$948$	0.134684	0.00437433
$949$	−17.2156	−0.558842
$950$	10.9304	0.354630
$951$	−12.2961	−0.398730
$952$	−0.0496981	−0.00161073
$953$	16.1006	0.521550	0.260775	−	0.965400i	$-0.416022\pi$
$953$	16.1006	0.521550	0.260775	+	0.965400i	$0.416022\pi$
$954$	6.82850	0.221081
$955$	38.8618	1.25754
$956$	−0.0607365	−0.00196436
$957$	−15.4650	−0.499911
$958$	51.3974	1.66058
$959$	−0.322749	−0.0104221
$960$	−26.9186	−0.868795
$961$	−24.5760	−0.792776
$962$	12.1856	0.392880
$963$	−16.7030	−0.538245
$964$	−0.561243	−0.0180764
$965$	60.9007	1.96046
$966$	−0.0536217	−0.00172525
$967$	−30.0760	−0.967179	−0.483589	−	0.875295i	$-0.660667\pi$
$967$	−30.0760	−0.967179	−0.483589	+	0.875295i	$0.660667\pi$
$968$	−17.9709	−0.577607
$969$	1.12471	0.0361308
$970$	75.8368	2.43497
$971$	−54.2862	−1.74213	−0.871064	−	0.491169i	$-0.836570\pi$
$971$	−54.2862	−1.74213	−0.871064	+	0.491169i	$0.836570\pi$
$972$	0.0397473	0.00127489
$973$	0.0450356	0.00144377
$974$	−5.60047	−0.179451
$975$	−10.6978	−0.342605
$976$	−42.6702	−1.36584
$977$	−52.6742	−1.68520	−0.842599	−	0.538541i	$-0.818976\pi$
$977$	−52.6742	−1.68520	−0.842599	+	0.538541i	$0.818976\pi$
$978$	−5.54226	−0.177222
$979$	18.3231	0.585608
$980$	0.955902	0.0305352
$981$	5.21370	0.166461
$982$	30.9189	0.986660
$983$	57.1798	1.82375	0.911876	−	0.410465i	$-0.134634\pi$
$983$	57.1798	1.82375	0.911876	+	0.410465i	$0.134634\pi$
$984$	26.8649	0.856423
$985$	45.1961	1.44007
$986$	10.3195	0.328640
$987$	−0.195254	−0.00621500
$988$	0.0702802	0.00223591
$989$	−24.4924	−0.778812
$990$	−10.5025	−0.333793
$991$	−37.7418	−1.19891	−0.599454	−	0.800409i	$-0.704616\pi$
$991$	−37.7418	−1.19891	−0.599454	+	0.800409i	$0.704616\pi$
$992$	−0.569796	−0.0180911
$993$	2.79942	0.0888370
$994$	0.138825	0.00440326
$995$	−67.7491	−2.14779
$996$	0.340339	0.0107840
$997$	23.7173	0.751135	0.375568	−	0.926795i	$-0.377448\pi$
$997$	23.7173	0.751135	0.375568	+	0.926795i	$0.377448\pi$
$998$	5.19171	0.164341
$999$	5.42717	0.171708

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.e.1.14	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.e.1.14	✓	46	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-1.42820 q^{2} +1.00000 q^{3} +0.0397473 q^{4} -3.43580 q^{5} -1.42820 q^{6} +0.0177517 q^{7} +2.79963 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.42820 q^{2} +1.00000 q^{3} +0.0397473 q^{4} -3.43580 q^{5} -1.42820 q^{6} +0.0177517 q^{7} +2.79963 q^{8} +1.00000 q^{9} +4.90700 q^{10} -2.14032 q^{11} +0.0397473 q^{12} -1.57212 q^{13} -0.0253529 q^{14} -3.43580 q^{15} -4.07791 q^{16} -1.00000 q^{17} -1.42820 q^{18} -1.12471 q^{19} -0.136564 q^{20} +0.0177517 q^{21} +3.05680 q^{22} +2.11501 q^{23} +2.79963 q^{24} +6.80471 q^{25} +2.24530 q^{26} +1.00000 q^{27} +0.000705581 q^{28} +7.22554 q^{29} +4.90700 q^{30} -2.53455 q^{31} +0.224811 q^{32} -2.14032 q^{33} +1.42820 q^{34} -0.0609912 q^{35} +0.0397473 q^{36} +5.42717 q^{37} +1.60630 q^{38} -1.57212 q^{39} -9.61895 q^{40} +9.59589 q^{41} -0.0253529 q^{42} -11.5802 q^{43} -0.0850718 q^{44} -3.43580 q^{45} -3.02066 q^{46} -10.9992 q^{47} -4.07791 q^{48} -6.99968 q^{49} -9.71846 q^{50} -1.00000 q^{51} -0.0624875 q^{52} -4.78120 q^{53} -1.42820 q^{54} +7.35370 q^{55} +0.0496981 q^{56} -1.12471 q^{57} -10.3195 q^{58} +1.42869 q^{59} -0.136564 q^{60} +10.4637 q^{61} +3.61984 q^{62} +0.0177517 q^{63} +7.83475 q^{64} +5.40149 q^{65} +3.05680 q^{66} +13.3878 q^{67} -0.0397473 q^{68} +2.11501 q^{69} +0.0871075 q^{70} -5.47570 q^{71} +2.79963 q^{72} +10.9506 q^{73} -7.75107 q^{74} +6.80471 q^{75} -0.0447041 q^{76} -0.0379943 q^{77} +2.24530 q^{78} +3.38850 q^{79} +14.0109 q^{80} +1.00000 q^{81} -13.7048 q^{82} +8.56257 q^{83} +0.000705581 q^{84} +3.43580 q^{85} +16.5389 q^{86} +7.22554 q^{87} -5.99209 q^{88} -8.56091 q^{89} +4.90700 q^{90} -0.0279078 q^{91} +0.0840660 q^{92} -2.53455 q^{93} +15.7090 q^{94} +3.86427 q^{95} +0.224811 q^{96} +15.4548 q^{97} +9.99693 q^{98} -2.14032 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

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