LMFDB - Embedded newform 8042.2.a.a.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.55257 q^{3} +1.00000 q^{4} +2.63171 q^{5} -1.55257 q^{6} -0.882040 q^{7} +1.00000 q^{8} -0.589531 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.55257 q^{3} +1.00000 q^{4} +2.63171 q^{5} -1.55257 q^{6} -0.882040 q^{7} +1.00000 q^{8} -0.589531 q^{9} +2.63171 q^{10} +5.91348 q^{11} -1.55257 q^{12} -0.542391 q^{13} -0.882040 q^{14} -4.08591 q^{15} +1.00000 q^{16} -4.94551 q^{17} -0.589531 q^{18} -6.50514 q^{19} +2.63171 q^{20} +1.36943 q^{21} +5.91348 q^{22} -5.08202 q^{23} -1.55257 q^{24} +1.92589 q^{25} -0.542391 q^{26} +5.57299 q^{27} -0.882040 q^{28} +9.49183 q^{29} -4.08591 q^{30} -1.26198 q^{31} +1.00000 q^{32} -9.18108 q^{33} -4.94551 q^{34} -2.32127 q^{35} -0.589531 q^{36} -9.40266 q^{37} -6.50514 q^{38} +0.842099 q^{39} +2.63171 q^{40} -1.75849 q^{41} +1.36943 q^{42} -7.78177 q^{43} +5.91348 q^{44} -1.55147 q^{45} -5.08202 q^{46} -1.60840 q^{47} -1.55257 q^{48} -6.22201 q^{49} +1.92589 q^{50} +7.67824 q^{51} -0.542391 q^{52} -6.48916 q^{53} +5.57299 q^{54} +15.5626 q^{55} -0.882040 q^{56} +10.0997 q^{57} +9.49183 q^{58} +3.01774 q^{59} -4.08591 q^{60} +6.91687 q^{61} -1.26198 q^{62} +0.519990 q^{63} +1.00000 q^{64} -1.42741 q^{65} -9.18108 q^{66} -6.48951 q^{67} -4.94551 q^{68} +7.89019 q^{69} -2.32127 q^{70} -3.38169 q^{71} -0.589531 q^{72} -11.2083 q^{73} -9.40266 q^{74} -2.99008 q^{75} -6.50514 q^{76} -5.21593 q^{77} +0.842099 q^{78} +10.2201 q^{79} +2.63171 q^{80} -6.88386 q^{81} -1.75849 q^{82} -7.97223 q^{83} +1.36943 q^{84} -13.0151 q^{85} -7.78177 q^{86} -14.7367 q^{87} +5.91348 q^{88} -7.41763 q^{89} -1.55147 q^{90} +0.478410 q^{91} -5.08202 q^{92} +1.95931 q^{93} -1.60840 q^{94} -17.1196 q^{95} -1.55257 q^{96} +1.29495 q^{97} -6.22201 q^{98} -3.48618 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.55257	−0.896376	−0.448188	−	0.893939i	$-0.647930\pi$
$3$	−1.55257	−0.896376	−0.448188	+	0.893939i	$0.647930\pi$
$4$	1.00000	0.500000
$5$	2.63171	1.17694	0.588468	−	0.808521i	$-0.299731\pi$
$5$	2.63171	1.17694	0.588468	+	0.808521i	$0.299731\pi$
$6$	−1.55257	−0.633833
$7$	−0.882040	−0.333380	−0.166690	−	0.986009i	$-0.553308\pi$
$7$	−0.882040	−0.333380	−0.166690	+	0.986009i	$0.553308\pi$
$8$	1.00000	0.353553
$9$	−0.589531	−0.196510
$10$	2.63171	0.832219
$11$	5.91348	1.78298	0.891491	−	0.453039i	$-0.149660\pi$
$11$	5.91348	1.78298	0.891491	+	0.453039i	$0.149660\pi$
$12$	−1.55257	−0.448188
$13$	−0.542391	−0.150432	−0.0752160	−	0.997167i	$-0.523965\pi$
$13$	−0.542391	−0.150432	−0.0752160	+	0.997167i	$0.523965\pi$
$14$	−0.882040	−0.235735
$15$	−4.08591	−1.05498
$16$	1.00000	0.250000
$17$	−4.94551	−1.19946	−0.599731	−	0.800202i	$-0.704726\pi$
$17$	−4.94551	−1.19946	−0.599731	+	0.800202i	$0.704726\pi$
$18$	−0.589531	−0.138954
$19$	−6.50514	−1.49238	−0.746191	−	0.665732i	$-0.768119\pi$
$19$	−6.50514	−1.49238	−0.746191	+	0.665732i	$0.768119\pi$
$20$	2.63171	0.588468
$21$	1.36943	0.298834
$22$	5.91348	1.26076
$23$	−5.08202	−1.05968	−0.529838	−	0.848099i	$-0.677747\pi$
$23$	−5.08202	−1.05968	−0.529838	+	0.848099i	$0.677747\pi$
$24$	−1.55257	−0.316917
$25$	1.92589	0.385178
$26$	−0.542391	−0.106372
$27$	5.57299	1.07252
$28$	−0.882040	−0.166690
$29$	9.49183	1.76259	0.881295	−	0.472567i	$-0.156673\pi$
$29$	9.49183	1.76259	0.881295	+	0.472567i	$0.156673\pi$
$30$	−4.08591	−0.745981
$31$	−1.26198	−0.226658	−0.113329	−	0.993558i	$-0.536151\pi$
$31$	−1.26198	−0.226658	−0.113329	+	0.993558i	$0.536151\pi$
$32$	1.00000	0.176777
$33$	−9.18108	−1.59822
$34$	−4.94551	−0.848147
$35$	−2.32127	−0.392367
$36$	−0.589531	−0.0982552
$37$	−9.40266	−1.54579	−0.772894	−	0.634535i	$-0.781191\pi$
$37$	−9.40266	−1.54579	−0.772894	+	0.634535i	$0.781191\pi$
$38$	−6.50514	−1.05527
$39$	0.842099	0.134844
$40$	2.63171	0.416110
$41$	−1.75849	−0.274630	−0.137315	−	0.990527i	$-0.543847\pi$
$41$	−1.75849	−0.274630	−0.137315	+	0.990527i	$0.543847\pi$
$42$	1.36943	0.211307
$43$	−7.78177	−1.18671	−0.593354	−	0.804942i	$-0.702197\pi$
$43$	−7.78177	−1.18671	−0.593354	+	0.804942i	$0.702197\pi$
$44$	5.91348	0.891491
$45$	−1.55147	−0.231280
$46$	−5.08202	−0.749303
$47$	−1.60840	−0.234609	−0.117305	−	0.993096i	$-0.537425\pi$
$47$	−1.60840	−0.234609	−0.117305	+	0.993096i	$0.537425\pi$
$48$	−1.55257	−0.224094
$49$	−6.22201	−0.888858
$50$	1.92589	0.272362
$51$	7.67824	1.07517
$52$	−0.542391	−0.0752160
$53$	−6.48916	−0.891354	−0.445677	−	0.895194i	$-0.647037\pi$
$53$	−6.48916	−0.891354	−0.445677	+	0.895194i	$0.647037\pi$
$54$	5.57299	0.758388
$55$	15.5626	2.09846
$56$	−0.882040	−0.117868
$57$	10.0997	1.33773
$58$	9.49183	1.24634
$59$	3.01774	0.392876	0.196438	−	0.980516i	$-0.437063\pi$
$59$	3.01774	0.392876	0.196438	+	0.980516i	$0.437063\pi$
$60$	−4.08591	−0.527488
$61$	6.91687	0.885615	0.442807	−	0.896617i	$-0.353983\pi$
$61$	6.91687	0.885615	0.442807	+	0.896617i	$0.353983\pi$
$62$	−1.26198	−0.160272
$63$	0.519990	0.0655126
$64$	1.00000	0.125000
$65$	−1.42741	−0.177049
$66$	−9.18108	−1.13011
$67$	−6.48951	−0.792820	−0.396410	−	0.918074i	$-0.629744\pi$
$67$	−6.48951	−0.792820	−0.396410	+	0.918074i	$0.629744\pi$
$68$	−4.94551	−0.599731
$69$	7.89019	0.949867
$70$	−2.32127	−0.277445
$71$	−3.38169	−0.401332	−0.200666	−	0.979660i	$-0.564311\pi$
$71$	−3.38169	−0.401332	−0.200666	+	0.979660i	$0.564311\pi$
$72$	−0.589531	−0.0694769
$73$	−11.2083	−1.31183	−0.655916	−	0.754834i	$-0.727717\pi$
$73$	−11.2083	−1.31183	−0.655916	+	0.754834i	$0.727717\pi$
$74$	−9.40266	−1.09304
$75$	−2.99008	−0.345264
$76$	−6.50514	−0.746191
$77$	−5.21593	−0.594410
$78$	0.842099	0.0953489
$79$	10.2201	1.14985	0.574927	−	0.818205i	$-0.305030\pi$
$79$	10.2201	1.14985	0.574927	+	0.818205i	$0.305030\pi$
$80$	2.63171	0.294234
$81$	−6.88386	−0.764873
$82$	−1.75849	−0.194193
$83$	−7.97223	−0.875066	−0.437533	−	0.899202i	$-0.644148\pi$
$83$	−7.97223	−0.875066	−0.437533	+	0.899202i	$0.644148\pi$
$84$	1.36943	0.149417
$85$	−13.0151	−1.41169
$86$	−7.78177	−0.839129
$87$	−14.7367	−1.57994
$88$	5.91348	0.630379
$89$	−7.41763	−0.786267	−0.393134	−	0.919481i	$-0.628609\pi$
$89$	−7.41763	−0.786267	−0.393134	+	0.919481i	$0.628609\pi$
$90$	−1.55147	−0.163540
$91$	0.478410	0.0501510
$92$	−5.08202	−0.529838
$93$	1.95931	0.203171
$94$	−1.60840	−0.165894
$95$	−17.1196	−1.75644
$96$	−1.55257	−0.158458
$97$	1.29495	0.131482	0.0657410	−	0.997837i	$-0.479059\pi$
$97$	1.29495	0.131482	0.0657410	+	0.997837i	$0.479059\pi$
$98$	−6.22201	−0.628517
$99$	−3.48618	−0.350375
$100$	1.92589	0.192589
$101$	8.20757	0.816684	0.408342	−	0.912829i	$-0.366107\pi$
$101$	8.20757	0.816684	0.408342	+	0.912829i	$0.366107\pi$
$102$	7.67824	0.760259
$103$	0.559554	0.0551345	0.0275673	−	0.999620i	$-0.491224\pi$
$103$	0.559554	0.0551345	0.0275673	+	0.999620i	$0.491224\pi$
$104$	−0.542391	−0.0531858
$105$	3.60393	0.351708
$106$	−6.48916	−0.630283
$107$	0.00863711	0.000834981 0	0.000417490	−	1.00000i	$-0.499867\pi$
$107$	0.00863711	0.000834981 0	0.000417490		1.00000i	$0.499867\pi$
$108$	5.57299	0.536261
$109$	−7.33424	−0.702493	−0.351247	−	0.936283i	$-0.614242\pi$
$109$	−7.33424	−0.702493	−0.351247	+	0.936283i	$0.614242\pi$
$110$	15.5626	1.48383
$111$	14.5983	1.38561
$112$	−0.882040	−0.0833450
$113$	−13.1405	−1.23615	−0.618076	−	0.786118i	$-0.712088\pi$
$113$	−13.1405	−1.23615	−0.618076	+	0.786118i	$0.712088\pi$
$114$	10.0997	0.945921
$115$	−13.3744	−1.24717
$116$	9.49183	0.881295
$117$	0.319756	0.0295615
$118$	3.01774	0.277805
$119$	4.36214	0.399876
$120$	−4.08591	−0.372991
$121$	23.9693	2.17902
$122$	6.91687	0.626224
$123$	2.73018	0.246172
$124$	−1.26198	−0.113329
$125$	−8.09016	−0.723606
$126$	0.519990	0.0463244
$127$	−5.95706	−0.528604	−0.264302	−	0.964440i	$-0.585141\pi$
$127$	−5.95706	−0.528604	−0.264302	+	0.964440i	$0.585141\pi$
$128$	1.00000	0.0883883
$129$	12.0817	1.06374
$130$	−1.42741	−0.125193
$131$	−2.48326	−0.216963	−0.108482	−	0.994098i	$-0.534599\pi$
$131$	−2.48326	−0.216963	−0.108482	+	0.994098i	$0.534599\pi$
$132$	−9.18108	−0.799111
$133$	5.73779	0.497530
$134$	−6.48951	−0.560608
$135$	14.6665	1.26229
$136$	−4.94551	−0.424074
$137$	−6.15230	−0.525627	−0.262813	−	0.964847i	$-0.584650\pi$
$137$	−6.15230	−0.525627	−0.262813	+	0.964847i	$0.584650\pi$
$138$	7.89019	0.671657
$139$	−0.266066	−0.0225674	−0.0112837	−	0.999936i	$-0.503592\pi$
$139$	−0.266066	−0.0225674	−0.0112837	+	0.999936i	$0.503592\pi$
$140$	−2.32127	−0.196183
$141$	2.49715	0.210298
$142$	−3.38169	−0.283785
$143$	−3.20742	−0.268218
$144$	−0.589531	−0.0491276
$145$	24.9797	2.07445
$146$	−11.2083	−0.927605
$147$	9.66009	0.796751
$148$	−9.40266	−0.772894
$149$	21.3333	1.74769	0.873846	−	0.486203i	$-0.161618\pi$
$149$	21.3333	1.74769	0.873846	+	0.486203i	$0.161618\pi$
$150$	−2.99008	−0.244139
$151$	15.0180	1.22215	0.611075	−	0.791573i	$-0.290737\pi$
$151$	15.0180	1.22215	0.611075	+	0.791573i	$0.290737\pi$
$152$	−6.50514	−0.527636
$153$	2.91553	0.235707
$154$	−5.21593	−0.420311
$155$	−3.32116	−0.266762
$156$	0.842099	0.0674218
$157$	10.0160	0.799361	0.399681	−	0.916655i	$-0.369121\pi$
$157$	10.0160	0.799361	0.399681	+	0.916655i	$0.369121\pi$
$158$	10.2201	0.813070
$159$	10.0749	0.798988
$160$	2.63171	0.208055
$161$	4.48255	0.353274
$162$	−6.88386	−0.540847
$163$	10.0724	0.788929	0.394464	−	0.918911i	$-0.370930\pi$
$163$	10.0724	0.788929	0.394464	+	0.918911i	$0.370930\pi$
$164$	−1.75849	−0.137315
$165$	−24.1619	−1.88100
$166$	−7.97223	−0.618765
$167$	19.4175	1.50257	0.751284	−	0.659979i	$-0.229435\pi$
$167$	19.4175	1.50257	0.751284	+	0.659979i	$0.229435\pi$
$168$	1.36943	0.105654
$169$	−12.7058	−0.977370
$170$	−13.0151	−0.998215
$171$	3.83498	0.293268
$172$	−7.78177	−0.593354
$173$	−0.00547070	−0.000415929 0	−0.000207965	−	1.00000i	$-0.500066\pi$
$173$	−0.00547070	−0.000415929 0	−0.000207965		1.00000i	$0.500066\pi$
$174$	−14.7367	−1.11719
$175$	−1.69871	−0.128411
$176$	5.91348	0.445745
$177$	−4.68524	−0.352165
$178$	−7.41763	−0.555975
$179$	−21.0505	−1.57339	−0.786694	−	0.617343i	$-0.788209\pi$
$179$	−21.0505	−1.57339	−0.786694	+	0.617343i	$0.788209\pi$
$180$	−1.55147	−0.115640
$181$	15.3500	1.14096	0.570478	−	0.821313i	$-0.306758\pi$
$181$	15.3500	1.14096	0.570478	+	0.821313i	$0.306758\pi$
$182$	0.478410	0.0354621
$183$	−10.7389	−0.793844
$184$	−5.08202	−0.374652
$185$	−24.7451	−1.81929
$186$	1.95931	0.143664
$187$	−29.2452	−2.13862
$188$	−1.60840	−0.117305
$189$	−4.91560	−0.357558
$190$	−17.1196	−1.24199
$191$	−0.740802	−0.0536026	−0.0268013	−	0.999641i	$-0.508532\pi$
$191$	−0.740802	−0.0536026	−0.0268013	+	0.999641i	$0.508532\pi$
$192$	−1.55257	−0.112047
$193$	−6.69960	−0.482247	−0.241124	−	0.970494i	$-0.577516\pi$
$193$	−6.69960	−0.482247	−0.241124	+	0.970494i	$0.577516\pi$
$194$	1.29495	0.0929719
$195$	2.21616	0.158702
$196$	−6.22201	−0.444429
$197$	−14.5701	−1.03807	−0.519037	−	0.854752i	$-0.673709\pi$
$197$	−14.5701	−1.03807	−0.519037	+	0.854752i	$0.673709\pi$
$198$	−3.48618	−0.247752
$199$	18.2544	1.29402	0.647010	−	0.762481i	$-0.276019\pi$
$199$	18.2544	1.29402	0.647010	+	0.762481i	$0.276019\pi$
$200$	1.92589	0.136181
$201$	10.0754	0.710664
$202$	8.20757	0.577483
$203$	−8.37218	−0.587612
$204$	7.67824	0.537584
$205$	−4.62783	−0.323222
$206$	0.559554	0.0389860
$207$	2.99601	0.208237
$208$	−0.542391	−0.0376080
$209$	−38.4680	−2.66089
$210$	3.60393	0.248695
$211$	2.29284	0.157846	0.0789229	−	0.996881i	$-0.474852\pi$
$211$	2.29284	0.157846	0.0789229	+	0.996881i	$0.474852\pi$
$212$	−6.48916	−0.445677
$213$	5.25030	0.359745
$214$	0.00863711	0.000590420 0
$215$	−20.4793	−1.39668
$216$	5.57299	0.379194
$217$	1.11312	0.0755633
$218$	−7.33424	−0.496738
$219$	17.4016	1.17589
$220$	15.5626	1.04923
$221$	2.68240	0.180438
$222$	14.5983	0.979772
$223$	2.95621	0.197962	0.0989812	−	0.995089i	$-0.468442\pi$
$223$	2.95621	0.197962	0.0989812	+	0.995089i	$0.468442\pi$
$224$	−0.882040	−0.0589338
$225$	−1.13537	−0.0756915
$226$	−13.1405	−0.874092
$227$	17.2714	1.14634	0.573171	−	0.819436i	$-0.305713\pi$
$227$	17.2714	1.14634	0.573171	+	0.819436i	$0.305713\pi$
$228$	10.0997	0.668867
$229$	−12.4178	−0.820593	−0.410297	−	0.911952i	$-0.634575\pi$
$229$	−12.4178	−0.820593	−0.410297	+	0.911952i	$0.634575\pi$
$230$	−13.3744	−0.881882
$231$	8.09808	0.532815
$232$	9.49183	0.623169
$233$	−9.05913	−0.593483	−0.296742	−	0.954958i	$-0.595900\pi$
$233$	−9.05913	−0.593483	−0.296742	+	0.954958i	$0.595900\pi$
$234$	0.319756	0.0209031
$235$	−4.23284	−0.276120
$236$	3.01774	0.196438
$237$	−15.8674	−1.03070
$238$	4.36214	0.282755
$239$	11.5170	0.744973	0.372487	−	0.928038i	$-0.378505\pi$
$239$	11.5170	0.744973	0.372487	+	0.928038i	$0.378505\pi$
$240$	−4.08591	−0.263744
$241$	−2.75272	−0.177319	−0.0886593	−	0.996062i	$-0.528258\pi$
$241$	−2.75272	−0.177319	−0.0886593	+	0.996062i	$0.528258\pi$
$242$	23.9693	1.54080
$243$	−6.03132	−0.386909
$244$	6.91687	0.442807
$245$	−16.3745	−1.04613
$246$	2.73018	0.174070
$247$	3.52833	0.224502
$248$	−1.26198	−0.0801358
$249$	12.3774	0.784388
$250$	−8.09016	−0.511667
$251$	1.12731	0.0711555	0.0355777	−	0.999367i	$-0.488673\pi$
$251$	1.12731	0.0711555	0.0355777	+	0.999367i	$0.488673\pi$
$252$	0.519990	0.0327563
$253$	−30.0524	−1.88938
$254$	−5.95706	−0.373779
$255$	20.2069	1.26540
$256$	1.00000	0.0625000
$257$	0.288177	0.0179760	0.00898799	−	0.999960i	$-0.497139\pi$
$257$	0.288177	0.0179760	0.00898799	+	0.999960i	$0.497139\pi$
$258$	12.0817	0.752175
$259$	8.29352	0.515334
$260$	−1.42741	−0.0885245
$261$	−5.59573	−0.346367
$262$	−2.48326	−0.153416
$263$	−28.6909	−1.76916	−0.884579	−	0.466391i	$-0.845554\pi$
$263$	−28.6909	−1.76916	−0.884579	+	0.466391i	$0.845554\pi$
$264$	−9.18108	−0.565057
$265$	−17.0776	−1.04907
$266$	5.73779	0.351807
$267$	11.5164	0.704791
$268$	−6.48951	−0.396410
$269$	7.36450	0.449021	0.224511	−	0.974472i	$-0.427922\pi$
$269$	7.36450	0.449021	0.224511	+	0.974472i	$0.427922\pi$
$270$	14.6665	0.892574
$271$	−2.69132	−0.163486	−0.0817430	−	0.996653i	$-0.526049\pi$
$271$	−2.69132	−0.163486	−0.0817430	+	0.996653i	$0.526049\pi$
$272$	−4.94551	−0.299865
$273$	−0.742765	−0.0449542
$274$	−6.15230	−0.371674
$275$	11.3887	0.686766
$276$	7.89019	0.474934
$277$	14.8917	0.894755	0.447377	−	0.894345i	$-0.352358\pi$
$277$	14.8917	0.894755	0.447377	+	0.894345i	$0.352358\pi$
$278$	−0.266066	−0.0159576
$279$	0.743976	0.0445407
$280$	−2.32127	−0.138723
$281$	−11.3827	−0.679037	−0.339518	−	0.940599i	$-0.610264\pi$
$281$	−11.3827	−0.679037	−0.339518	+	0.940599i	$0.610264\pi$
$282$	2.49715	0.148703
$283$	−21.5999	−1.28398	−0.641991	−	0.766712i	$-0.721891\pi$
$283$	−21.5999	−1.28398	−0.641991	+	0.766712i	$0.721891\pi$
$284$	−3.38169	−0.200666
$285$	26.5794	1.57443
$286$	−3.20742	−0.189659
$287$	1.55106	0.0915561
$288$	−0.589531	−0.0347385
$289$	7.45804	0.438708
$290$	24.9797	1.46686
$291$	−2.01050	−0.117857
$292$	−11.2083	−0.655916
$293$	−5.88354	−0.343720	−0.171860	−	0.985121i	$-0.554978\pi$
$293$	−5.88354	−0.343720	−0.171860	+	0.985121i	$0.554978\pi$
$294$	9.66009	0.563388
$295$	7.94181	0.462390
$296$	−9.40266	−0.546518
$297$	32.9558	1.91229
$298$	21.3333	1.23580
$299$	2.75644	0.159409
$300$	−2.99008	−0.172632
$301$	6.86383	0.395625
$302$	15.0180	0.864190
$303$	−12.7428	−0.732056
$304$	−6.50514	−0.373095
$305$	18.2032	1.04231
$306$	2.91553	0.166670
$307$	27.9241	1.59371	0.796857	−	0.604168i	$-0.206495\pi$
$307$	27.9241	1.59371	0.796857	+	0.604168i	$0.206495\pi$
$308$	−5.21593	−0.297205
$309$	−0.868746	−0.0494212
$310$	−3.32116	−0.188629
$311$	−0.221177	−0.0125418	−0.00627089	−	0.999980i	$-0.501996\pi$
$311$	−0.221177	−0.0125418	−0.00627089	+	0.999980i	$0.501996\pi$
$312$	0.842099	0.0476744
$313$	15.9954	0.904115	0.452058	−	0.891989i	$-0.350690\pi$
$313$	15.9954	0.904115	0.452058	+	0.891989i	$0.350690\pi$
$314$	10.0160	0.565234
$315$	1.36846	0.0771041
$316$	10.2201	0.574927
$317$	−10.4611	−0.587555	−0.293778	−	0.955874i	$-0.594912\pi$
$317$	−10.4611	−0.587555	−0.293778	+	0.955874i	$0.594912\pi$
$318$	10.0749	0.564970
$319$	56.1298	3.14266
$320$	2.63171	0.147117
$321$	−0.0134097	−0.000748456 0
$322$	4.48255	0.249803
$323$	32.1712	1.79005
$324$	−6.88386	−0.382437
$325$	−1.04459	−0.0579432
$326$	10.0724	0.557857
$327$	11.3869	0.629698
$328$	−1.75849	−0.0970963
$329$	1.41867	0.0782140
$330$	−24.1619	−1.33007
$331$	−3.16315	−0.173862	−0.0869311	−	0.996214i	$-0.527706\pi$
$331$	−3.16315	−0.173862	−0.0869311	+	0.996214i	$0.527706\pi$
$332$	−7.97223	−0.437533
$333$	5.54316	0.303763
$334$	19.4175	1.06248
$335$	−17.0785	−0.933098
$336$	1.36943	0.0747084
$337$	−29.8037	−1.62351	−0.811755	−	0.583999i	$-0.801487\pi$
$337$	−29.8037	−1.62351	−0.811755	+	0.583999i	$0.801487\pi$
$338$	−12.7058	−0.691105
$339$	20.4015	1.10806
$340$	−13.0151	−0.705845
$341$	−7.46269	−0.404127
$342$	3.83498	0.207372
$343$	11.6623	0.629707
$344$	−7.78177	−0.419565
$345$	20.7647	1.11793
$346$	−0.00547070	−0.000294107 0
$347$	−5.26729	−0.282763	−0.141382	−	0.989955i	$-0.545154\pi$
$347$	−5.26729	−0.282763	−0.141382	+	0.989955i	$0.545154\pi$
$348$	−14.7367	−0.789971
$349$	−24.9944	−1.33792	−0.668960	−	0.743298i	$-0.733260\pi$
$349$	−24.9944	−1.33792	−0.668960	+	0.743298i	$0.733260\pi$
$350$	−1.69871	−0.0908000
$351$	−3.02274	−0.161342
$352$	5.91348	0.315190
$353$	−22.8772	−1.21763	−0.608814	−	0.793313i	$-0.708355\pi$
$353$	−22.8772	−1.21763	−0.608814	+	0.793313i	$0.708355\pi$
$354$	−4.68524	−0.249018
$355$	−8.89962	−0.472343
$356$	−7.41763	−0.393134
$357$	−6.77251	−0.358439
$358$	−21.0505	−1.11255
$359$	0.243368	0.0128445	0.00642223	−	0.999979i	$-0.497956\pi$
$359$	0.243368	0.0128445	0.00642223	+	0.999979i	$0.497956\pi$
$360$	−1.55147	−0.0817699
$361$	23.3168	1.22720
$362$	15.3500	0.806778
$363$	−37.2139	−1.95322
$364$	0.478410	0.0250755
$365$	−29.4970	−1.54394
$366$	−10.7389	−0.561332
$367$	−10.2883	−0.537046	−0.268523	−	0.963273i	$-0.586536\pi$
$367$	−10.2883	−0.537046	−0.268523	+	0.963273i	$0.586536\pi$
$368$	−5.08202	−0.264919
$369$	1.03668	0.0539676
$370$	−24.7451	−1.28643
$371$	5.72370	0.297160
$372$	1.95931	0.101585
$373$	−29.5349	−1.52926	−0.764629	−	0.644471i	$-0.777078\pi$
$373$	−29.5349	−1.52926	−0.764629	+	0.644471i	$0.777078\pi$
$374$	−29.2452	−1.51223
$375$	12.5605	0.648623
$376$	−1.60840	−0.0829469
$377$	−5.14828	−0.265150
$378$	−4.91560	−0.252831
$379$	−31.9775	−1.64257	−0.821286	−	0.570517i	$-0.806743\pi$
$379$	−31.9775	−1.64257	−0.821286	+	0.570517i	$0.806743\pi$
$380$	−17.1196	−0.878218
$381$	9.24874	0.473827
$382$	−0.740802	−0.0379027
$383$	18.6199	0.951432	0.475716	−	0.879599i	$-0.342189\pi$
$383$	18.6199	0.951432	0.475716	+	0.879599i	$0.342189\pi$
$384$	−1.55257	−0.0792292
$385$	−13.7268	−0.699583
$386$	−6.69960	−0.341000
$387$	4.58759	0.233201
$388$	1.29495	0.0657410
$389$	−23.6433	−1.19876	−0.599381	−	0.800464i	$-0.704587\pi$
$389$	−23.6433	−1.19876	−0.599381	+	0.800464i	$0.704587\pi$
$390$	2.21616	0.112220
$391$	25.1332	1.27104
$392$	−6.22201	−0.314259
$393$	3.85542	0.194480
$394$	−14.5701	−0.734029
$395$	26.8964	1.35330
$396$	−3.48618	−0.175187
$397$	4.67201	0.234481	0.117241	−	0.993104i	$-0.462595\pi$
$397$	4.67201	0.234481	0.117241	+	0.993104i	$0.462595\pi$
$398$	18.2544	0.915010
$399$	−8.90832	−0.445974
$400$	1.92589	0.0962945
$401$	14.7675	0.737454	0.368727	−	0.929538i	$-0.379794\pi$
$401$	14.7675	0.737454	0.368727	+	0.929538i	$0.379794\pi$
$402$	10.0754	0.502516
$403$	0.684486	0.0340967
$404$	8.20757	0.408342
$405$	−18.1163	−0.900207
$406$	−8.37218	−0.415504
$407$	−55.6024	−2.75611
$408$	7.67824	0.380129
$409$	−23.0521	−1.13985	−0.569927	−	0.821695i	$-0.693029\pi$
$409$	−23.0521	−1.13985	−0.569927	+	0.821695i	$0.693029\pi$
$410$	−4.62783	−0.228552
$411$	9.55187	0.471159
$412$	0.559554	0.0275673
$413$	−2.66177	−0.130977
$414$	2.99601	0.147246
$415$	−20.9806	−1.02990
$416$	−0.542391	−0.0265929
$417$	0.413086	0.0202289
$418$	−38.4680	−1.88153
$419$	−39.7993	−1.94432	−0.972161	−	0.234314i	$-0.924716\pi$
$419$	−39.7993	−1.94432	−0.972161	+	0.234314i	$0.924716\pi$
$420$	3.60393	0.175854
$421$	−9.88650	−0.481838	−0.240919	−	0.970545i	$-0.577449\pi$
$421$	−9.88650	−0.481838	−0.240919	+	0.970545i	$0.577449\pi$
$422$	2.29284	0.111614
$423$	0.948203	0.0461032
$424$	−6.48916	−0.315141
$425$	−9.52451	−0.462006
$426$	5.25030	0.254378
$427$	−6.10096	−0.295246
$428$	0.00863711	0.000417490 0
$429$	4.97973	0.240424
$430$	−20.4793	−0.987601
$431$	6.91554	0.333110	0.166555	−	0.986032i	$-0.446736\pi$
$431$	6.91554	0.333110	0.166555	+	0.986032i	$0.446736\pi$
$432$	5.57299	0.268131
$433$	2.07242	0.0995942	0.0497971	−	0.998759i	$-0.484143\pi$
$433$	2.07242	0.0995942	0.0497971	+	0.998759i	$0.484143\pi$
$434$	1.11312	0.0534313
$435$	−38.7828	−1.85949
$436$	−7.33424	−0.351247
$437$	33.0593	1.58144
$438$	17.4016	0.831482
$439$	13.0518	0.622930	0.311465	−	0.950258i	$-0.399180\pi$
$439$	13.0518	0.622930	0.311465	+	0.950258i	$0.399180\pi$
$440$	15.5626	0.741916
$441$	3.66807	0.174670
$442$	2.68240	0.127589
$443$	23.0493	1.09510	0.547552	−	0.836771i	$-0.315560\pi$
$443$	23.0493	1.09510	0.547552	+	0.836771i	$0.315560\pi$
$444$	14.5983	0.692803
$445$	−19.5210	−0.925386
$446$	2.95621	0.139981
$447$	−33.1214	−1.56659
$448$	−0.882040	−0.0416725
$449$	−1.94533	−0.0918059	−0.0459030	−	0.998946i	$-0.514617\pi$
$449$	−1.94533	−0.0918059	−0.0459030	+	0.998946i	$0.514617\pi$
$450$	−1.13537	−0.0535220
$451$	−10.3988	−0.489660
$452$	−13.1405	−0.618076
$453$	−23.3165	−1.09551
$454$	17.2714	0.810586
$455$	1.25904	0.0590245
$456$	10.0997	0.472960
$457$	36.3128	1.69864	0.849320	−	0.527878i	$-0.177012\pi$
$457$	36.3128	1.69864	0.849320	+	0.527878i	$0.177012\pi$
$458$	−12.4178	−0.580247
$459$	−27.5613	−1.28645
$460$	−13.3744	−0.623585
$461$	6.56578	0.305799	0.152899	−	0.988242i	$-0.451139\pi$
$461$	6.56578	0.305799	0.152899	+	0.988242i	$0.451139\pi$
$462$	8.09808	0.376757
$463$	−18.5160	−0.860509	−0.430255	−	0.902708i	$-0.641576\pi$
$463$	−18.5160	−0.860509	−0.430255	+	0.902708i	$0.641576\pi$
$464$	9.49183	0.440647
$465$	5.15633	0.239119
$466$	−9.05913	−0.419656
$467$	−29.8689	−1.38217	−0.691085	−	0.722774i	$-0.742867\pi$
$467$	−29.8689	−1.38217	−0.691085	+	0.722774i	$0.742867\pi$
$468$	0.319756	0.0147807
$469$	5.72401	0.264310
$470$	−4.23284	−0.195246
$471$	−15.5505	−0.716528
$472$	3.01774	0.138903
$473$	−46.0173	−2.11588
$474$	−15.8674	−0.728816
$475$	−12.5282	−0.574833
$476$	4.36214	0.199938
$477$	3.82556	0.175160
$478$	11.5170	0.526776
$479$	31.0213	1.41740	0.708699	−	0.705511i	$-0.249282\pi$
$479$	31.0213	1.41740	0.708699	+	0.705511i	$0.249282\pi$
$480$	−4.08591	−0.186495
$481$	5.09991	0.232536
$482$	−2.75272	−0.125383
$483$	−6.95946	−0.316666
$484$	23.9693	1.08951
$485$	3.40793	0.154746
$486$	−6.03132	−0.273586
$487$	19.7355	0.894299	0.447150	−	0.894459i	$-0.352439\pi$
$487$	19.7355	0.894299	0.447150	+	0.894459i	$0.352439\pi$
$488$	6.91687	0.313112
$489$	−15.6380	−0.707177
$490$	−16.3745	−0.739725
$491$	−20.3203	−0.917040	−0.458520	−	0.888684i	$-0.651620\pi$
$491$	−20.3203	−0.917040	−0.458520	+	0.888684i	$0.651620\pi$
$492$	2.73018	0.123086
$493$	−46.9419	−2.11416
$494$	3.52833	0.158747
$495$	−9.17462	−0.412368
$496$	−1.26198	−0.0566646
$497$	2.98278	0.133796
$498$	12.3774	0.554646
$499$	−7.99526	−0.357917	−0.178958	−	0.983857i	$-0.557273\pi$
$499$	−7.99526	−0.357917	−0.178958	+	0.983857i	$0.557273\pi$
$500$	−8.09016	−0.361803
$501$	−30.1469	−1.34687
$502$	1.12731	0.0503145
$503$	42.6353	1.90101	0.950507	−	0.310704i	$-0.100565\pi$
$503$	42.6353	1.90101	0.950507	+	0.310704i	$0.100565\pi$
$504$	0.519990	0.0231622
$505$	21.5999	0.961185
$506$	−30.0524	−1.33599
$507$	19.7266	0.876091
$508$	−5.95706	−0.264302
$509$	22.6908	1.00575	0.502876	−	0.864359i	$-0.332275\pi$
$509$	22.6908	1.00575	0.502876	+	0.864359i	$0.332275\pi$
$510$	20.2069	0.894776
$511$	9.88616	0.437338
$512$	1.00000	0.0441942
$513$	−36.2531	−1.60061
$514$	0.288177	0.0127109
$515$	1.47258	0.0648898
$516$	12.0817	0.531868
$517$	−9.51125	−0.418304
$518$	8.29352	0.364396
$519$	0.00849363	0.000372829 0
$520$	−1.42741	−0.0625963
$521$	−3.94159	−0.172684	−0.0863420	−	0.996266i	$-0.527518\pi$
$521$	−3.94159	−0.172684	−0.0863420	+	0.996266i	$0.527518\pi$
$522$	−5.59573	−0.244919
$523$	15.8239	0.691932	0.345966	−	0.938247i	$-0.387551\pi$
$523$	15.8239	0.691932	0.345966	+	0.938247i	$0.387551\pi$
$524$	−2.48326	−0.108482
$525$	2.63737	0.115104
$526$	−28.6909	−1.25098
$527$	6.24113	0.271868
$528$	−9.18108	−0.399555
$529$	2.82696	0.122911
$530$	−17.0776	−0.741802
$531$	−1.77905	−0.0772042
$532$	5.73779	0.248765
$533$	0.953788	0.0413132
$534$	11.5164	0.498362
$535$	0.0227303	0.000982719 0
$536$	−6.48951	−0.280304
$537$	32.6823	1.41035
$538$	7.36450	0.317506
$539$	−36.7937	−1.58482
$540$	14.6665	0.631145
$541$	6.35580	0.273257	0.136629	−	0.990622i	$-0.456373\pi$
$541$	6.35580	0.273257	0.136629	+	0.990622i	$0.456373\pi$
$542$	−2.69132	−0.115602
$543$	−23.8319	−1.02273
$544$	−4.94551	−0.212037
$545$	−19.3016	−0.826789
$546$	−0.742765	−0.0317874
$547$	−8.82179	−0.377193	−0.188596	−	0.982055i	$-0.560394\pi$
$547$	−8.82179	−0.377193	−0.188596	+	0.982055i	$0.560394\pi$
$548$	−6.15230	−0.262813
$549$	−4.07771	−0.174033
$550$	11.3887	0.485617
$551$	−61.7457	−2.63045
$552$	7.89019	0.335829
$553$	−9.01456	−0.383338
$554$	14.8917	0.632687
$555$	38.4184	1.63077
$556$	−0.266066	−0.0112837
$557$	−14.5363	−0.615922	−0.307961	−	0.951399i	$-0.599647\pi$
$557$	−14.5363	−0.615922	−0.307961	+	0.951399i	$0.599647\pi$
$558$	0.743976	0.0314950
$559$	4.22076	0.178519
$560$	−2.32127	−0.0980917
$561$	45.4051	1.91701
$562$	−11.3827	−0.480152
$563$	34.8655	1.46941	0.734704	−	0.678388i	$-0.237321\pi$
$563$	34.8655	1.46941	0.734704	+	0.678388i	$0.237321\pi$
$564$	2.49715	0.105149
$565$	−34.5819	−1.45487
$566$	−21.5999	−0.907912
$567$	6.07184	0.254993
$568$	−3.38169	−0.141892
$569$	26.8211	1.12440	0.562200	−	0.827001i	$-0.309955\pi$
$569$	26.8211	1.12440	0.562200	+	0.827001i	$0.309955\pi$
$570$	26.5794	1.11329
$571$	−12.0883	−0.505878	−0.252939	−	0.967482i	$-0.581397\pi$
$571$	−12.0883	−0.505878	−0.252939	+	0.967482i	$0.581397\pi$
$572$	−3.20742	−0.134109
$573$	1.15015	0.0480480
$574$	1.55106	0.0647399
$575$	−9.78742	−0.408164
$576$	−0.589531	−0.0245638
$577$	33.8792	1.41041	0.705205	−	0.709003i	$-0.250855\pi$
$577$	33.8792	1.41041	0.705205	+	0.709003i	$0.250855\pi$
$578$	7.45804	0.310214
$579$	10.4016	0.432275
$580$	24.9797	1.03723
$581$	7.03183	0.291729
$582$	−2.01050	−0.0833377
$583$	−38.3735	−1.58927
$584$	−11.2083	−0.463802
$585$	0.841505	0.0347920
$586$	−5.88354	−0.243047
$587$	−44.5269	−1.83782	−0.918911	−	0.394465i	$-0.870930\pi$
$587$	−44.5269	−1.83782	−0.918911	+	0.394465i	$0.870930\pi$
$588$	9.66009	0.398375
$589$	8.20935	0.338260
$590$	7.94181	0.326959
$591$	22.6210	0.930504
$592$	−9.40266	−0.386447
$593$	3.97893	0.163395	0.0816976	−	0.996657i	$-0.473966\pi$
$593$	3.97893	0.163395	0.0816976	+	0.996657i	$0.473966\pi$
$594$	32.9558	1.35219
$595$	11.4799	0.470629
$596$	21.3333	0.873846
$597$	−28.3412	−1.15993
$598$	2.75644	0.112719
$599$	12.5086	0.511090	0.255545	−	0.966797i	$-0.417745\pi$
$599$	12.5086	0.511090	0.255545	+	0.966797i	$0.417745\pi$
$600$	−2.99008	−0.122069
$601$	−32.0039	−1.30546	−0.652732	−	0.757589i	$-0.726377\pi$
$601$	−32.0039	−1.30546	−0.652732	+	0.757589i	$0.726377\pi$
$602$	6.86383	0.279749
$603$	3.82577	0.155797
$604$	15.0180	0.611075
$605$	63.0801	2.56457
$606$	−12.7428	−0.517642
$607$	27.9887	1.13603	0.568013	−	0.823020i	$-0.307712\pi$
$607$	27.9887	1.13603	0.568013	+	0.823020i	$0.307712\pi$
$608$	−6.50514	−0.263818
$609$	12.9984	0.526721
$610$	18.2032	0.737026
$611$	0.872382	0.0352928
$612$	2.91553	0.117853
$613$	−33.7759	−1.36420	−0.682098	−	0.731261i	$-0.738932\pi$
$613$	−33.7759	−1.36420	−0.682098	+	0.731261i	$0.738932\pi$
$614$	27.9241	1.12693
$615$	7.18503	0.289728
$616$	−5.21593	−0.210156
$617$	−18.2996	−0.736714	−0.368357	−	0.929684i	$-0.620080\pi$
$617$	−18.2996	−0.736714	−0.368357	+	0.929684i	$0.620080\pi$
$618$	−0.868746	−0.0349461
$619$	5.98223	0.240446	0.120223	−	0.992747i	$-0.461639\pi$
$619$	5.98223	0.240446	0.120223	+	0.992747i	$0.461639\pi$
$620$	−3.32116	−0.133381
$621$	−28.3221	−1.13653
$622$	−0.221177	−0.00886838
$623$	6.54265	0.262126
$624$	0.842099	0.0337109
$625$	−30.9204	−1.23682
$626$	15.9954	0.639306
$627$	59.7242	2.38516
$628$	10.0160	0.399681
$629$	46.5009	1.85411
$630$	1.36846	0.0545209
$631$	27.0906	1.07846	0.539229	−	0.842159i	$-0.318716\pi$
$631$	27.0906	1.07846	0.539229	+	0.842159i	$0.318716\pi$
$632$	10.2201	0.406535
$633$	−3.55979	−0.141489
$634$	−10.4611	−0.415464
$635$	−15.6772	−0.622132
$636$	10.0749	0.399494
$637$	3.37476	0.133713
$638$	56.1298	2.22220
$639$	1.99361	0.0788660
$640$	2.63171	0.104027
$641$	−19.7405	−0.779701	−0.389851	−	0.920878i	$-0.627473\pi$
$641$	−19.7405	−0.779701	−0.389851	+	0.920878i	$0.627473\pi$
$642$	−0.0134097	−0.000529239 0
$643$	−7.04426	−0.277798	−0.138899	−	0.990307i	$-0.544356\pi$
$643$	−7.04426	−0.277798	−0.138899	+	0.990307i	$0.544356\pi$
$644$	4.48255	0.176637
$645$	31.7956	1.25195
$646$	32.1712	1.26576
$647$	−2.08754	−0.0820696	−0.0410348	−	0.999158i	$-0.513065\pi$
$647$	−2.08754	−0.0820696	−0.0410348	+	0.999158i	$0.513065\pi$
$648$	−6.88386	−0.270424
$649$	17.8453	0.700491
$650$	−1.04459	−0.0409720
$651$	−1.72819	−0.0677331
$652$	10.0724	0.394464
$653$	4.67434	0.182921	0.0914606	−	0.995809i	$-0.470846\pi$
$653$	4.67434	0.182921	0.0914606	+	0.995809i	$0.470846\pi$
$654$	11.3869	0.445264
$655$	−6.53520	−0.255352
$656$	−1.75849	−0.0686575
$657$	6.60764	0.257789
$658$	1.41867	0.0553057
$659$	−21.7121	−0.845784	−0.422892	−	0.906180i	$-0.638985\pi$
$659$	−21.7121	−0.845784	−0.422892	+	0.906180i	$0.638985\pi$
$660$	−24.1619	−0.940502
$661$	9.91003	0.385455	0.192728	−	0.981252i	$-0.438267\pi$
$661$	9.91003	0.385455	0.192728	+	0.981252i	$0.438267\pi$
$662$	−3.16315	−0.122939
$663$	−4.16460	−0.161740
$664$	−7.97223	−0.309383
$665$	15.1002	0.585561
$666$	5.54316	0.214793
$667$	−48.2377	−1.86777
$668$	19.4175	0.751284
$669$	−4.58972	−0.177449
$670$	−17.0785	−0.659800
$671$	40.9028	1.57903
$672$	1.36943	0.0528268
$673$	26.9480	1.03877	0.519384	−	0.854541i	$-0.326161\pi$
$673$	26.9480	1.03877	0.519384	+	0.854541i	$0.326161\pi$
$674$	−29.8037	−1.14799
$675$	10.7330	0.413112
$676$	−12.7058	−0.488685
$677$	−27.5716	−1.05966	−0.529831	−	0.848103i	$-0.677744\pi$
$677$	−27.5716	−1.05966	−0.529831	+	0.848103i	$0.677744\pi$
$678$	20.4015	0.783515
$679$	−1.14220	−0.0438335
$680$	−13.0151	−0.499108
$681$	−26.8150	−1.02755
$682$	−7.46269	−0.285761
$683$	−43.6122	−1.66878	−0.834388	−	0.551178i	$-0.814178\pi$
$683$	−43.6122	−1.66878	−0.834388	+	0.551178i	$0.814178\pi$
$684$	3.83498	0.146634
$685$	−16.1911	−0.618629
$686$	11.6623	0.445270
$687$	19.2795	0.735560
$688$	−7.78177	−0.296677
$689$	3.51966	0.134088
$690$	20.7647	0.790498
$691$	9.32740	0.354831	0.177416	−	0.984136i	$-0.443226\pi$
$691$	9.32740	0.354831	0.177416	+	0.984136i	$0.443226\pi$
$692$	−0.00547070	−0.000207965 0
$693$	3.07495	0.116808
$694$	−5.26729	−0.199944
$695$	−0.700208	−0.0265604
$696$	−14.7367	−0.558594
$697$	8.69662	0.329408
$698$	−24.9944	−0.946052
$699$	14.0649	0.531984
$700$	−1.69871	−0.0642053
$701$	−8.74867	−0.330433	−0.165216	−	0.986257i	$-0.552832\pi$
$701$	−8.74867	−0.330433	−0.165216	+	0.986257i	$0.552832\pi$
$702$	−3.02274	−0.114086
$703$	61.1656	2.30690
$704$	5.91348	0.222873
$705$	6.57178	0.247507
$706$	−22.8772	−0.860993
$707$	−7.23941	−0.272266
$708$	−4.68524	−0.176082
$709$	−41.2119	−1.54775	−0.773873	−	0.633341i	$-0.781683\pi$
$709$	−41.2119	−1.54775	−0.773873	+	0.633341i	$0.781683\pi$
$710$	−8.89962	−0.333997
$711$	−6.02509	−0.225958
$712$	−7.41763	−0.277987
$713$	6.41341	0.240184
$714$	−6.77251	−0.253455
$715$	−8.44099	−0.315675
$716$	−21.0505	−0.786694
$717$	−17.8809	−0.667776
$718$	0.243368	0.00908240
$719$	37.0291	1.38095	0.690475	−	0.723356i	$-0.257401\pi$
$719$	37.0291	1.38095	0.690475	+	0.723356i	$0.257401\pi$
$720$	−1.55147	−0.0578200
$721$	−0.493549	−0.0183807
$722$	23.3168	0.867762
$723$	4.27379	0.158944
$724$	15.3500	0.570478
$725$	18.2802	0.678911
$726$	−37.2139	−1.38114
$727$	47.1436	1.74846	0.874230	−	0.485512i	$-0.161367\pi$
$727$	47.1436	1.74846	0.874230	+	0.485512i	$0.161367\pi$
$728$	0.478410	0.0177311
$729$	30.0156	1.11169
$730$	−29.4970	−1.09173
$731$	38.4848	1.42341
$732$	−10.7389	−0.396922
$733$	27.9809	1.03350	0.516748	−	0.856137i	$-0.327142\pi$
$733$	27.9809	1.03350	0.516748	+	0.856137i	$0.327142\pi$
$734$	−10.2883	−0.379749
$735$	25.4225	0.937725
$736$	−5.08202	−0.187326
$737$	−38.3756	−1.41358
$738$	1.03668	0.0381609
$739$	20.7881	0.764703	0.382351	−	0.924017i	$-0.375114\pi$
$739$	20.7881	0.764703	0.382351	+	0.924017i	$0.375114\pi$
$740$	−24.7451	−0.909646
$741$	−5.47797	−0.201238
$742$	5.72370	0.210124
$743$	−48.8925	−1.79369	−0.896845	−	0.442344i	$-0.854147\pi$
$743$	−48.8925	−1.79369	−0.896845	+	0.442344i	$0.854147\pi$
$744$	1.95931	0.0718318
$745$	56.1430	2.05692
$746$	−29.5349	−1.08135
$747$	4.69988	0.171960
$748$	−29.2452	−1.06931
$749$	−0.00761827	−0.000278366 0
$750$	12.5605	0.458646
$751$	−37.6155	−1.37261	−0.686304	−	0.727315i	$-0.740768\pi$
$751$	−37.6155	−1.37261	−0.686304	+	0.727315i	$0.740768\pi$
$752$	−1.60840	−0.0586523
$753$	−1.75023	−0.0637820
$754$	−5.14828	−0.187489
$755$	39.5231	1.43839
$756$	−4.91560	−0.178779
$757$	20.9612	0.761847	0.380923	−	0.924607i	$-0.375606\pi$
$757$	20.9612	0.761847	0.380923	+	0.924607i	$0.375606\pi$
$758$	−31.9775	−1.16147
$759$	46.6585	1.69360
$760$	−17.1196	−0.620994
$761$	−44.1859	−1.60174	−0.800869	−	0.598840i	$-0.795628\pi$
$761$	−44.1859	−1.60174	−0.800869	+	0.598840i	$0.795628\pi$
$762$	9.24874	0.335047
$763$	6.46910	0.234197
$764$	−0.740802	−0.0268013
$765$	7.67283	0.277412
$766$	18.6199	0.672764
$767$	−1.63679	−0.0591012
$768$	−1.55257	−0.0560235
$769$	−28.5036	−1.02787	−0.513934	−	0.857830i	$-0.671812\pi$
$769$	−28.5036	−1.02787	−0.513934	+	0.857830i	$0.671812\pi$
$770$	−13.7268	−0.494680
$771$	−0.447414	−0.0161132
$772$	−6.69960	−0.241124
$773$	32.6834	1.17554	0.587769	−	0.809028i	$-0.300006\pi$
$773$	32.6834	1.17554	0.587769	+	0.809028i	$0.300006\pi$
$774$	4.58759	0.164898
$775$	−2.43043	−0.0873038
$776$	1.29495	0.0464859
$777$	−12.8763	−0.461933
$778$	−23.6433	−0.847653
$779$	11.4392	0.409853
$780$	2.21616	0.0793512
$781$	−19.9975	−0.715568
$782$	25.1332	0.898761
$783$	52.8979	1.89042
$784$	−6.22201	−0.222214
$785$	26.3591	0.940797
$786$	3.85542	0.137518
$787$	−5.16397	−0.184076	−0.0920378	−	0.995756i	$-0.529338\pi$
$787$	−5.16397	−0.184076	−0.0920378	+	0.995756i	$0.529338\pi$
$788$	−14.5701	−0.519037
$789$	44.5446	1.58583
$790$	26.8964	0.956931
$791$	11.5904	0.412108
$792$	−3.48618	−0.123876
$793$	−3.75165	−0.133225
$794$	4.67201	0.165803
$795$	26.5141	0.940358
$796$	18.2544	0.647010
$797$	−16.2077	−0.574105	−0.287052	−	0.957915i	$-0.592675\pi$
$797$	−16.2077	−0.574105	−0.287052	+	0.957915i	$0.592675\pi$
$798$	−8.90832	−0.315351
$799$	7.95436	0.281405
$800$	1.92589	0.0680905
$801$	4.37292	0.154510
$802$	14.7675	0.521459
$803$	−66.2800	−2.33897
$804$	10.0754	0.355332
$805$	11.7968	0.415781
$806$	0.684486	0.0241100
$807$	−11.4339	−0.402492
$808$	8.20757	0.288741
$809$	26.0475	0.915783	0.457891	−	0.889008i	$-0.348605\pi$
$809$	26.0475	0.915783	0.457891	+	0.889008i	$0.348605\pi$
$810$	−18.1163	−0.636542
$811$	−42.0203	−1.47553	−0.737766	−	0.675056i	$-0.764119\pi$
$811$	−42.0203	−1.47553	−0.737766	+	0.675056i	$0.764119\pi$
$812$	−8.37218	−0.293806
$813$	4.17846	0.146545
$814$	−55.6024	−1.94886
$815$	26.5075	0.928518
$816$	7.67824	0.268792
$817$	50.6215	1.77102
$818$	−23.0521	−0.805999
$819$	−0.282038	−0.00985520
$820$	−4.62783	−0.161611
$821$	−53.1911	−1.85638	−0.928191	−	0.372104i	$-0.878636\pi$
$821$	−53.1911	−1.85638	−0.928191	+	0.372104i	$0.878636\pi$
$822$	9.55187	0.333160
$823$	21.7285	0.757407	0.378703	−	0.925518i	$-0.376370\pi$
$823$	21.7285	0.757407	0.378703	+	0.925518i	$0.376370\pi$
$824$	0.559554	0.0194930
$825$	−17.6818	−0.615600
$826$	−2.66177	−0.0926147
$827$	4.42385	0.153832	0.0769162	−	0.997038i	$-0.475493\pi$
$827$	4.42385	0.153832	0.0769162	+	0.997038i	$0.475493\pi$
$828$	2.99601	0.104119
$829$	30.9185	1.07384	0.536922	−	0.843632i	$-0.319587\pi$
$829$	30.9185	1.07384	0.536922	+	0.843632i	$0.319587\pi$
$830$	−20.9806	−0.728247
$831$	−23.1204	−0.802036
$832$	−0.542391	−0.0188040
$833$	30.7710	1.06615
$834$	0.413086	0.0143040
$835$	51.1011	1.76843
$836$	−38.4680	−1.33044
$837$	−7.03300	−0.243096
$838$	−39.7993	−1.37484
$839$	−10.8624	−0.375010	−0.187505	−	0.982264i	$-0.560040\pi$
$839$	−10.8624	−0.375010	−0.187505	+	0.982264i	$0.560040\pi$
$840$	3.60393	0.124348
$841$	61.0949	2.10672
$842$	−9.88650	−0.340711
$843$	17.6725	0.608672
$844$	2.29284	0.0789229
$845$	−33.4380	−1.15030
$846$	0.948203	0.0325999
$847$	−21.1419	−0.726443
$848$	−6.48916	−0.222839
$849$	33.5354	1.15093
$850$	−9.52451	−0.326688
$851$	47.7845	1.63803
$852$	5.25030	0.179872
$853$	37.7761	1.29343	0.646714	−	0.762733i	$-0.276143\pi$
$853$	37.7761	1.29343	0.646714	+	0.762733i	$0.276143\pi$
$854$	−6.10096	−0.208770
$855$	10.0926	0.345158
$856$	0.00863711	0.000295210 0
$857$	−45.5128	−1.55469	−0.777343	−	0.629077i	$-0.783433\pi$
$857$	−45.5128	−1.55469	−0.777343	+	0.629077i	$0.783433\pi$
$858$	4.97973	0.170005
$859$	27.2305	0.929094	0.464547	−	0.885549i	$-0.346217\pi$
$859$	27.2305	0.929094	0.464547	+	0.885549i	$0.346217\pi$
$860$	−20.4793	−0.698340
$861$	−2.40812	−0.0820687
$862$	6.91554	0.235544
$863$	1.49875	0.0510181	0.0255091	−	0.999675i	$-0.491879\pi$
$863$	1.49875	0.0510181	0.0255091	+	0.999675i	$0.491879\pi$
$864$	5.57299	0.189597
$865$	−0.0143973	−0.000489522 0
$866$	2.07242	0.0704237
$867$	−11.5791	−0.393248
$868$	1.11312	0.0377816
$869$	60.4365	2.05017
$870$	−38.7828	−1.31486
$871$	3.51985	0.119266
$872$	−7.33424	−0.248369
$873$	−0.763412	−0.0258376
$874$	33.0593	1.11825
$875$	7.13585	0.241236
$876$	17.4016	0.587947
$877$	−13.4100	−0.452823	−0.226412	−	0.974032i	$-0.572699\pi$
$877$	−13.4100	−0.452823	−0.226412	+	0.974032i	$0.572699\pi$
$878$	13.0518	0.440478
$879$	9.13460	0.308102
$880$	15.5626	0.524614
$881$	−46.6340	−1.57114	−0.785570	−	0.618773i	$-0.787630\pi$
$881$	−46.6340	−1.57114	−0.785570	+	0.618773i	$0.787630\pi$
$882$	3.66807	0.123510
$883$	37.6874	1.26828	0.634141	−	0.773218i	$-0.281354\pi$
$883$	37.6874	1.26828	0.634141	+	0.773218i	$0.281354\pi$
$884$	2.68240	0.0902188
$885$	−12.3302	−0.414475
$886$	23.0493	0.774356
$887$	−5.27908	−0.177254	−0.0886271	−	0.996065i	$-0.528248\pi$
$887$	−5.27908	−0.177254	−0.0886271	+	0.996065i	$0.528248\pi$
$888$	14.5983	0.489886
$889$	5.25436	0.176226
$890$	−19.5210	−0.654347
$891$	−40.7076	−1.36376
$892$	2.95621	0.0989812
$893$	10.4629	0.350127
$894$	−33.1214	−1.10775
$895$	−55.3988	−1.85178
$896$	−0.882040	−0.0294669
$897$	−4.27956	−0.142890
$898$	−1.94533	−0.0649166
$899$	−11.9785	−0.399505
$900$	−1.13537	−0.0378458
$901$	32.0922	1.06915
$902$	−10.3988	−0.346242
$903$	−10.6566	−0.354628
$904$	−13.1405	−0.437046
$905$	40.3967	1.34283
$906$	−23.3165	−0.774639
$907$	49.7010	1.65030	0.825148	−	0.564917i	$-0.191092\pi$
$907$	49.7010	1.65030	0.825148	+	0.564917i	$0.191092\pi$
$908$	17.2714	0.573171
$909$	−4.83862	−0.160487
$910$	1.25904	0.0417367
$911$	58.0575	1.92353	0.961766	−	0.273872i	$-0.0883044\pi$
$911$	58.0575	1.92353	0.961766	+	0.273872i	$0.0883044\pi$
$912$	10.0997	0.334434
$913$	−47.1436	−1.56023
$914$	36.3128	1.20112
$915$	−28.2617	−0.934303
$916$	−12.4178	−0.410297
$917$	2.19033	0.0723311
$918$	−27.5613	−0.909658
$919$	−23.3875	−0.771482	−0.385741	−	0.922607i	$-0.626054\pi$
$919$	−23.3875	−0.771482	−0.385741	+	0.922607i	$0.626054\pi$
$920$	−13.3744	−0.440941
$921$	−43.3541	−1.42857
$922$	6.56578	0.216232
$923$	1.83420	0.0603733
$924$	8.09808	0.266407
$925$	−18.1085	−0.595404
$926$	−18.5160	−0.608472
$927$	−0.329875	−0.0108345
$928$	9.49183	0.311585
$929$	36.0194	1.18176	0.590879	−	0.806760i	$-0.298781\pi$
$929$	36.0194	1.18176	0.590879	+	0.806760i	$0.298781\pi$
$930$	5.15633	0.169083
$931$	40.4750	1.32651
$932$	−9.05913	−0.296742
$933$	0.343392	0.0112422
$934$	−29.8689	−0.977342
$935$	−76.9648	−2.51702
$936$	0.319756	0.0104516
$937$	53.8068	1.75779	0.878897	−	0.477012i	$-0.158280\pi$
$937$	53.8068	1.75779	0.878897	+	0.477012i	$0.158280\pi$
$938$	5.72401	0.186895
$939$	−24.8340	−0.810427
$940$	−4.23284	−0.138060
$941$	−7.32756	−0.238872	−0.119436	−	0.992842i	$-0.538109\pi$
$941$	−7.32756	−0.238872	−0.119436	+	0.992842i	$0.538109\pi$
$942$	−15.5505	−0.506662
$943$	8.93668	0.291018
$944$	3.01774	0.0982190
$945$	−12.9364	−0.420822
$946$	−46.0173	−1.49615
$947$	−35.9251	−1.16741	−0.583705	−	0.811966i	$-0.698397\pi$
$947$	−35.9251	−1.16741	−0.583705	+	0.811966i	$0.698397\pi$
$948$	−15.8674	−0.515351
$949$	6.07927	0.197342
$950$	−12.5282	−0.406468
$951$	16.2416	0.526670
$952$	4.36214	0.141378
$953$	47.8518	1.55007	0.775036	−	0.631917i	$-0.217732\pi$
$953$	47.8518	1.55007	0.775036	+	0.631917i	$0.217732\pi$
$954$	3.82556	0.123857
$955$	−1.94958	−0.0630868
$956$	11.5170	0.372487
$957$	−87.1453	−2.81701
$958$	31.0213	1.00225
$959$	5.42658	0.175233
$960$	−4.08591	−0.131872
$961$	−29.4074	−0.948626
$962$	5.09991	0.164428
$963$	−0.00509184	−0.000164082 0
$964$	−2.75272	−0.0886593
$965$	−17.6314	−0.567574
$966$	−6.95946	−0.223917
$967$	35.0205	1.12618	0.563092	−	0.826394i	$-0.309612\pi$
$967$	35.0205	1.12618	0.563092	+	0.826394i	$0.309612\pi$
$968$	23.9693	0.770401
$969$	−49.9480	−1.60456
$970$	3.40793	0.109422
$971$	12.7728	0.409897	0.204949	−	0.978773i	$-0.434297\pi$
$971$	12.7728	0.409897	0.204949	+	0.978773i	$0.434297\pi$
$972$	−6.03132	−0.193455
$973$	0.234681	0.00752352
$974$	19.7355	0.632365
$975$	1.62179	0.0519388
$976$	6.91687	0.221404
$977$	−11.3687	−0.363716	−0.181858	−	0.983325i	$-0.558211\pi$
$977$	−11.3687	−0.363716	−0.181858	+	0.983325i	$0.558211\pi$
$978$	−15.6380	−0.500049
$979$	−43.8640	−1.40190
$980$	−16.3745	−0.523064
$981$	4.32377	0.138047
$982$	−20.3203	−0.648445
$983$	−43.0587	−1.37336	−0.686679	−	0.726961i	$-0.740932\pi$
$983$	−43.0587	−1.37336	−0.686679	+	0.726961i	$0.740932\pi$
$984$	2.73018	0.0870348
$985$	−38.3442	−1.22175
$986$	−46.9419	−1.49494
$987$	−2.20259	−0.0701092
$988$	3.52833	0.112251
$989$	39.5471	1.25752
$990$	−9.17462	−0.291588
$991$	56.8558	1.80608	0.903042	−	0.429551i	$-0.141328\pi$
$991$	56.8558	1.80608	0.903042	+	0.429551i	$0.141328\pi$
$992$	−1.26198	−0.0400679
$993$	4.91100	0.155846
$994$	2.98278	0.0946082
$995$	48.0403	1.52298
$996$	12.3774	0.392194
$997$	14.4490	0.457604	0.228802	−	0.973473i	$-0.426519\pi$
$997$	14.4490	0.457604	0.228802	+	0.973473i	$0.426519\pi$
$998$	−7.99526	−0.253085
$999$	−52.4009	−1.65789

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.19	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.19	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.55257 q^{3} +1.00000 q^{4} +2.63171 q^{5} -1.55257 q^{6} -0.882040 q^{7} +1.00000 q^{8} -0.589531 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.55257 q^{3} +1.00000 q^{4} +2.63171 q^{5} -1.55257 q^{6} -0.882040 q^{7} +1.00000 q^{8} -0.589531 q^{9} +2.63171 q^{10} +5.91348 q^{11} -1.55257 q^{12} -0.542391 q^{13} -0.882040 q^{14} -4.08591 q^{15} +1.00000 q^{16} -4.94551 q^{17} -0.589531 q^{18} -6.50514 q^{19} +2.63171 q^{20} +1.36943 q^{21} +5.91348 q^{22} -5.08202 q^{23} -1.55257 q^{24} +1.92589 q^{25} -0.542391 q^{26} +5.57299 q^{27} -0.882040 q^{28} +9.49183 q^{29} -4.08591 q^{30} -1.26198 q^{31} +1.00000 q^{32} -9.18108 q^{33} -4.94551 q^{34} -2.32127 q^{35} -0.589531 q^{36} -9.40266 q^{37} -6.50514 q^{38} +0.842099 q^{39} +2.63171 q^{40} -1.75849 q^{41} +1.36943 q^{42} -7.78177 q^{43} +5.91348 q^{44} -1.55147 q^{45} -5.08202 q^{46} -1.60840 q^{47} -1.55257 q^{48} -6.22201 q^{49} +1.92589 q^{50} +7.67824 q^{51} -0.542391 q^{52} -6.48916 q^{53} +5.57299 q^{54} +15.5626 q^{55} -0.882040 q^{56} +10.0997 q^{57} +9.49183 q^{58} +3.01774 q^{59} -4.08591 q^{60} +6.91687 q^{61} -1.26198 q^{62} +0.519990 q^{63} +1.00000 q^{64} -1.42741 q^{65} -9.18108 q^{66} -6.48951 q^{67} -4.94551 q^{68} +7.89019 q^{69} -2.32127 q^{70} -3.38169 q^{71} -0.589531 q^{72} -11.2083 q^{73} -9.40266 q^{74} -2.99008 q^{75} -6.50514 q^{76} -5.21593 q^{77} +0.842099 q^{78} +10.2201 q^{79} +2.63171 q^{80} -6.88386 q^{81} -1.75849 q^{82} -7.97223 q^{83} +1.36943 q^{84} -13.0151 q^{85} -7.78177 q^{86} -14.7367 q^{87} +5.91348 q^{88} -7.41763 q^{89} -1.55147 q^{90} +0.478410 q^{91} -5.08202 q^{92} +1.95931 q^{93} -1.60840 q^{94} -17.1196 q^{95} -1.55257 q^{96} +1.29495 q^{97} -6.22201 q^{98} -3.48618 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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