LMFDB - Embedded newform 705.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [705,2,Mod(1,705)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(705, 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("705.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 705 = 3 \cdot 5 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	705.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:	$5.62945334250$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.14656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 2 $ x^4 - 2x^3 - 4x^2 + 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.385537$ of defining polynomial
Character	$\chi$	$=$	705.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.38554 q^{2} -1.00000 q^{3} -0.0802864 q^{4} -1.00000 q^{5} -1.38554 q^{6} +4.18757 q^{7} -2.88231 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.38554 q^{2} -1.00000 q^{3} -0.0802864 q^{4} -1.00000 q^{5} -1.38554 q^{6} +4.18757 q^{7} -2.88231 q^{8} +1.00000 q^{9} -1.38554 q^{10} +0.919714 q^{11} +0.0802864 q^{12} -0.614463 q^{13} +5.80203 q^{14} +1.00000 q^{15} -3.83298 q^{16} +6.98959 q^{17} +1.38554 q^{18} +1.12962 q^{19} +0.0802864 q^{20} -4.18757 q^{21} +1.27430 q^{22} +8.34814 q^{23} +2.88231 q^{24} +1.00000 q^{25} -0.851361 q^{26} -1.00000 q^{27} -0.336204 q^{28} +5.00000 q^{29} +1.38554 q^{30} -3.59544 q^{31} +0.453890 q^{32} -0.919714 q^{33} +9.68434 q^{34} -4.18757 q^{35} -0.0802864 q^{36} -3.75270 q^{37} +1.56513 q^{38} +0.614463 q^{39} +2.88231 q^{40} +6.08029 q^{41} -5.80203 q^{42} -8.21456 q^{43} -0.0738405 q^{44} -1.00000 q^{45} +11.5667 q^{46} +1.00000 q^{47} +3.83298 q^{48} +10.5357 q^{49} +1.38554 q^{50} -6.98959 q^{51} +0.0493330 q^{52} -1.49282 q^{53} -1.38554 q^{54} -0.919714 q^{55} -12.0699 q^{56} -1.12962 q^{57} +6.92769 q^{58} -2.57310 q^{59} -0.0802864 q^{60} -5.86974 q^{61} -4.98162 q^{62} +4.18757 q^{63} +8.29484 q^{64} +0.614463 q^{65} -1.27430 q^{66} -0.824368 q^{67} -0.561169 q^{68} -8.34814 q^{69} -5.80203 q^{70} +12.1232 q^{71} -2.88231 q^{72} +11.7732 q^{73} -5.19950 q^{74} -1.00000 q^{75} -0.0906930 q^{76} +3.85136 q^{77} +0.851361 q^{78} -4.06592 q^{79} +3.83298 q^{80} +1.00000 q^{81} +8.42446 q^{82} -14.4738 q^{83} +0.336204 q^{84} -6.98959 q^{85} -11.3816 q^{86} -5.00000 q^{87} -2.65090 q^{88} -16.7049 q^{89} -1.38554 q^{90} -2.57310 q^{91} -0.670242 q^{92} +3.59544 q^{93} +1.38554 q^{94} -1.12962 q^{95} -0.453890 q^{96} -1.93809 q^{97} +14.5976 q^{98} +0.919714 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 6 q^{8} + 4 q^{9} - 2 q^{10} + 8 q^{11} - 4 q^{12} - 6 q^{13} + 10 q^{14} + 4 q^{15} + 4 q^{16} - 2 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{20} + 12 q^{22} + 8 q^{23} - 6 q^{24} + 4 q^{25} + 8 q^{26} - 4 q^{27} + 4 q^{28} + 20 q^{29} + 2 q^{30} - 4 q^{31} + 14 q^{32} - 8 q^{33} + 8 q^{34} + 4 q^{36} + 8 q^{38} + 6 q^{39} - 6 q^{40} + 20 q^{41} - 10 q^{42} - 8 q^{43} + 32 q^{44} - 4 q^{45} - 2 q^{46} + 4 q^{47} - 4 q^{48} + 2 q^{50} + 2 q^{51} + 2 q^{52} + 10 q^{53} - 2 q^{54} - 8 q^{55} - 14 q^{56} - 2 q^{57} + 10 q^{58} + 10 q^{59} + 4 q^{60} - 20 q^{61} - 12 q^{62} + 4 q^{64} + 6 q^{65} - 12 q^{66} - 2 q^{68} - 8 q^{69} - 10 q^{70} + 18 q^{71} + 6 q^{72} - 4 q^{73} + 16 q^{74} - 4 q^{75} - 26 q^{76} + 4 q^{77} - 8 q^{78} + 20 q^{79} - 4 q^{80} + 4 q^{81} + 2 q^{82} - 28 q^{83} - 4 q^{84} + 2 q^{85} - 40 q^{86} - 20 q^{87} + 40 q^{88} - 2 q^{90} + 10 q^{91} - 36 q^{92} + 4 q^{93} + 2 q^{94} - 2 q^{95} - 14 q^{96} - 20 q^{97} + 4 q^{98} + 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 6 * q^8 + 4 * q^9 - 2 * q^10 + 8 * q^11 - 4 * q^12 - 6 * q^13 + 10 * q^14 + 4 * q^15 + 4 * q^16 - 2 * q^17 + 2 * q^18 + 2 * q^19 - 4 * q^20 + 12 * q^22 + 8 * q^23 - 6 * q^24 + 4 * q^25 + 8 * q^26 - 4 * q^27 + 4 * q^28 + 20 * q^29 + 2 * q^30 - 4 * q^31 + 14 * q^32 - 8 * q^33 + 8 * q^34 + 4 * q^36 + 8 * q^38 + 6 * q^39 - 6 * q^40 + 20 * q^41 - 10 * q^42 - 8 * q^43 + 32 * q^44 - 4 * q^45 - 2 * q^46 + 4 * q^47 - 4 * q^48 + 2 * q^50 + 2 * q^51 + 2 * q^52 + 10 * q^53 - 2 * q^54 - 8 * q^55 - 14 * q^56 - 2 * q^57 + 10 * q^58 + 10 * q^59 + 4 * q^60 - 20 * q^61 - 12 * q^62 + 4 * q^64 + 6 * q^65 - 12 * q^66 - 2 * q^68 - 8 * q^69 - 10 * q^70 + 18 * q^71 + 6 * q^72 - 4 * q^73 + 16 * q^74 - 4 * q^75 - 26 * q^76 + 4 * q^77 - 8 * q^78 + 20 * q^79 - 4 * q^80 + 4 * q^81 + 2 * q^82 - 28 * q^83 - 4 * q^84 + 2 * q^85 - 40 * q^86 - 20 * q^87 + 40 * q^88 - 2 * q^90 + 10 * q^91 - 36 * q^92 + 4 * q^93 + 2 * q^94 - 2 * q^95 - 14 * q^96 - 20 * q^97 + 4 * q^98 + 8 * 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 $

$2$	1.38554	0.979723	0.489861	−	0.871800i	$-0.337047\pi$
$2$	1.38554	0.979723	0.489861	+	0.871800i	$0.337047\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.0802864	−0.0401432
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.38554	−0.565643
$7$	4.18757	1.58275	0.791375	−	0.611330i	$-0.209365\pi$
$7$	4.18757	1.58275	0.791375	+	0.611330i	$0.209365\pi$
$8$	−2.88231	−1.01905
$9$	1.00000	0.333333
$10$	−1.38554	−0.438145
$11$	0.919714	0.277304	0.138652	−	0.990341i	$-0.455723\pi$
$11$	0.919714	0.277304	0.138652	+	0.990341i	$0.455723\pi$
$12$	0.0802864	0.0231767
$13$	−0.614463	−0.170421	−0.0852106	−	0.996363i	$-0.527156\pi$
$13$	−0.614463	−0.170421	−0.0852106	+	0.996363i	$0.527156\pi$
$14$	5.80203	1.55066
$15$	1.00000	0.258199
$16$	−3.83298	−0.958245
$17$	6.98959	1.69523	0.847613	−	0.530615i	$-0.178039\pi$
$17$	6.98959	1.69523	0.847613	+	0.530615i	$0.178039\pi$
$18$	1.38554	0.326574
$19$	1.12962	0.259152	0.129576	−	0.991569i	$-0.458638\pi$
$19$	1.12962	0.259152	0.129576	+	0.991569i	$0.458638\pi$
$20$	0.0802864	0.0179526
$21$	−4.18757	−0.913802
$22$	1.27430	0.271681
$23$	8.34814	1.74071	0.870354	−	0.492427i	$-0.163890\pi$
$23$	8.34814	1.74071	0.870354	+	0.492427i	$0.163890\pi$
$24$	2.88231	0.588350
$25$	1.00000	0.200000
$26$	−0.851361	−0.166966
$27$	−1.00000	−0.192450
$28$	−0.336204	−0.0635366
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	1.38554	0.252963
$31$	−3.59544	−0.645761	−0.322880	−	0.946440i	$-0.604651\pi$
$31$	−3.59544	−0.645761	−0.322880	+	0.946440i	$0.604651\pi$
$32$	0.453890	0.0802372
$33$	−0.919714	−0.160102
$34$	9.68434	1.66085
$35$	−4.18757	−0.707828
$36$	−0.0802864	−0.0133811
$37$	−3.75270	−0.616939	−0.308470	−	0.951234i	$-0.599817\pi$
$37$	−3.75270	−0.616939	−0.308470	+	0.951234i	$0.599817\pi$
$38$	1.56513	0.253898
$39$	0.614463	0.0983928
$40$	2.88231	0.455734
$41$	6.08029	0.949581	0.474791	−	0.880099i	$-0.342524\pi$
$41$	6.08029	0.949581	0.474791	+	0.880099i	$0.342524\pi$
$42$	−5.80203	−0.895272
$43$	−8.21456	−1.25271	−0.626354	−	0.779539i	$-0.715454\pi$
$43$	−8.21456	−1.25271	−0.626354	+	0.779539i	$0.715454\pi$
$44$	−0.0738405	−0.0111319
$45$	−1.00000	−0.149071
$46$	11.5667	1.70541
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	3.83298	0.553243
$49$	10.5357	1.50510
$50$	1.38554	0.195945
$51$	−6.98959	−0.978739
$52$	0.0493330	0.00684125
$53$	−1.49282	−0.205054	−0.102527	−	0.994730i	$-0.532693\pi$
$53$	−1.49282	−0.205054	−0.102527	+	0.994730i	$0.532693\pi$
$54$	−1.38554	−0.188548
$55$	−0.919714	−0.124014
$56$	−12.0699	−1.61291
$57$	−1.12962	−0.149622
$58$	6.92769	0.909650
$59$	−2.57310	−0.334989	−0.167495	−	0.985873i	$-0.553568\pi$
$59$	−2.57310	−0.334989	−0.167495	+	0.985873i	$0.553568\pi$
$60$	−0.0802864	−0.0103649
$61$	−5.86974	−0.751543	−0.375772	−	0.926712i	$-0.622622\pi$
$61$	−5.86974	−0.751543	−0.375772	+	0.926712i	$0.622622\pi$
$62$	−4.98162	−0.632666
$63$	4.18757	0.527584
$64$	8.29484	1.03686
$65$	0.614463	0.0762147
$66$	−1.27430	−0.156855
$67$	−0.824368	−0.100713	−0.0503563	−	0.998731i	$-0.516036\pi$
$67$	−0.824368	−0.100713	−0.0503563	+	0.998731i	$0.516036\pi$
$68$	−0.561169	−0.0680517
$69$	−8.34814	−1.00500
$70$	−5.80203	−0.693475
$71$	12.1232	1.43876	0.719378	−	0.694619i	$-0.244427\pi$
$71$	12.1232	1.43876	0.719378	+	0.694619i	$0.244427\pi$
$72$	−2.88231	−0.339684
$73$	11.7732	1.37795	0.688977	−	0.724783i	$-0.258060\pi$
$73$	11.7732	1.37795	0.688977	+	0.724783i	$0.258060\pi$
$74$	−5.19950	−0.604429
$75$	−1.00000	−0.115470
$76$	−0.0906930	−0.0104032
$77$	3.85136	0.438903
$78$	0.851361	0.0963976
$79$	−4.06592	−0.457452	−0.228726	−	0.973491i	$-0.573456\pi$
$79$	−4.06592	−0.457452	−0.228726	+	0.973491i	$0.573456\pi$
$80$	3.83298	0.428540
$81$	1.00000	0.111111
$82$	8.42446	0.930326
$83$	−14.4738	−1.58871	−0.794353	−	0.607457i	$-0.792190\pi$
$83$	−14.4738	−1.58871	−0.794353	+	0.607457i	$0.792190\pi$
$84$	0.336204	0.0366829
$85$	−6.98959	−0.758128
$86$	−11.3816	−1.22731
$87$	−5.00000	−0.536056
$88$	−2.65090	−0.282587
$89$	−16.7049	−1.77071	−0.885357	−	0.464911i	$-0.846086\pi$
$89$	−16.7049	−1.77071	−0.885357	+	0.464911i	$0.846086\pi$
$90$	−1.38554	−0.146048
$91$	−2.57310	−0.269734
$92$	−0.670242	−0.0698775
$93$	3.59544	0.372830
$94$	1.38554	0.142907
$95$	−1.12962	−0.115896
$96$	−0.453890	−0.0463250
$97$	−1.93809	−0.196784	−0.0983918	−	0.995148i	$-0.531370\pi$
$97$	−1.93809	−0.196784	−0.0983918	+	0.995148i	$0.531370\pi$
$98$	14.5976	1.47458
$99$	0.919714	0.0924347
$100$	−0.0802864	−0.00802864
$101$	−4.94026	−0.491574	−0.245787	−	0.969324i	$-0.579046\pi$
$101$	−4.94026	−0.491574	−0.245787	+	0.969324i	$0.579046\pi$
$102$	−9.68434	−0.958893
$103$	−8.87191	−0.874175	−0.437088	−	0.899419i	$-0.643990\pi$
$103$	−8.87191	−0.874175	−0.437088	+	0.899419i	$0.643990\pi$
$104$	1.77107	0.173668
$105$	4.18757	0.408665
$106$	−2.06835	−0.200896
$107$	8.21456	0.794131	0.397066	−	0.917790i	$-0.370029\pi$
$107$	8.21456	0.794131	0.397066	+	0.917790i	$0.370029\pi$
$108$	0.0802864	0.00772556
$109$	−10.1692	−0.974031	−0.487016	−	0.873393i	$-0.661915\pi$
$109$	−10.1692	−0.974031	−0.487016	+	0.873393i	$0.661915\pi$
$110$	−1.27430	−0.121500
$111$	3.75270	0.356190
$112$	−16.0509	−1.51666
$113$	−6.01193	−0.565555	−0.282778	−	0.959186i	$-0.591256\pi$
$113$	−6.01193	−0.565555	−0.282778	+	0.959186i	$0.591256\pi$
$114$	−1.56513	−0.146588
$115$	−8.34814	−0.778468
$116$	−0.401432	−0.0372720
$117$	−0.614463	−0.0568071
$118$	−3.56513	−0.328197
$119$	29.2694	2.68312
$120$	−2.88231	−0.263118
$121$	−10.1541	−0.923102
$122$	−8.13274	−0.736304
$123$	−6.08029	−0.548241
$124$	0.288665	0.0259229
$125$	−1.00000	−0.0894427
$126$	5.80203	0.516886
$127$	−8.69628	−0.771670	−0.385835	−	0.922568i	$-0.626087\pi$
$127$	−8.69628	−0.771670	−0.385835	+	0.922568i	$0.626087\pi$
$128$	10.5850	0.935594
$129$	8.21456	0.723251
$130$	0.851361	0.0746693
$131$	12.1073	1.05782	0.528909	−	0.848679i	$-0.322601\pi$
$131$	12.1073	1.05782	0.528909	+	0.848679i	$0.322601\pi$
$132$	0.0738405	0.00642699
$133$	4.73035	0.410174
$134$	−1.14219	−0.0986705
$135$	1.00000	0.0860663
$136$	−20.1462	−1.72752
$137$	−17.9792	−1.53607	−0.768033	−	0.640411i	$-0.778764\pi$
$137$	−17.9792	−1.53607	−0.768033	+	0.640411i	$0.778764\pi$
$138$	−11.5667	−0.984619
$139$	9.65671	0.819071	0.409536	−	0.912294i	$-0.365691\pi$
$139$	9.65671	0.819071	0.409536	+	0.912294i	$0.365691\pi$
$140$	0.336204	0.0284145
$141$	−1.00000	−0.0842152
$142$	16.7971	1.40958
$143$	−0.565130	−0.0472585
$144$	−3.83298	−0.319415
$145$	−5.00000	−0.415227
$146$	16.3123	1.35001
$147$	−10.5357	−0.868970
$148$	0.301290	0.0247659
$149$	20.5529	1.68376	0.841881	−	0.539664i	$-0.181449\pi$
$149$	20.5529	1.68376	0.841881	+	0.539664i	$0.181449\pi$
$150$	−1.38554	−0.113129
$151$	9.64146	0.784611	0.392305	−	0.919835i	$-0.371678\pi$
$151$	9.64146	0.784611	0.392305	+	0.919835i	$0.371678\pi$
$152$	−3.25592	−0.264090
$153$	6.98959	0.565075
$154$	5.33620	0.430004
$155$	3.59544	0.288793
$156$	−0.0493330	−0.00394980
$157$	−12.6724	−1.01137	−0.505684	−	0.862719i	$-0.668760\pi$
$157$	−12.6724	−1.01137	−0.505684	+	0.862719i	$0.668760\pi$
$158$	−5.63348	−0.448176
$159$	1.49282	0.118388
$160$	−0.453890	−0.0358832
$161$	34.9584	2.75511
$162$	1.38554	0.108858
$163$	6.16057	0.482533	0.241267	−	0.970459i	$-0.422437\pi$
$163$	6.16057	0.482533	0.241267	+	0.970459i	$0.422437\pi$
$164$	−0.488164	−0.0381192
$165$	0.919714	0.0715996
$166$	−20.0540	−1.55649
$167$	−3.84159	−0.297272	−0.148636	−	0.988892i	$-0.547488\pi$
$167$	−3.84159	−0.297272	−0.148636	+	0.988892i	$0.547488\pi$
$168$	12.0699	0.931211
$169$	−12.6224	−0.970957
$170$	−9.68434	−0.742755
$171$	1.12962	0.0863841
$172$	0.659517	0.0502877
$173$	−14.2964	−1.08693	−0.543466	−	0.839431i	$-0.682889\pi$
$173$	−14.2964	−1.08693	−0.543466	+	0.839431i	$0.682889\pi$
$174$	−6.92769	−0.525187
$175$	4.18757	0.316550
$176$	−3.52525	−0.265725
$177$	2.57310	0.193406
$178$	−23.1452	−1.73481
$179$	15.8633	1.18568	0.592839	−	0.805321i	$-0.298007\pi$
$179$	15.8633	1.18568	0.592839	+	0.805321i	$0.298007\pi$
$180$	0.0802864	0.00598419
$181$	−0.639928	−0.0475655	−0.0237827	−	0.999717i	$-0.507571\pi$
$181$	−0.639928	−0.0475655	−0.0237827	+	0.999717i	$0.507571\pi$
$182$	−3.56513	−0.264265
$183$	5.86974	0.433904
$184$	−24.0620	−1.77387
$185$	3.75270	0.275904
$186$	4.98162	0.365270
$187$	6.42842	0.470093
$188$	−0.0802864	−0.00585548
$189$	−4.18757	−0.304601
$190$	−1.56513	−0.113546
$191$	−7.31539	−0.529323	−0.264662	−	0.964341i	$-0.585260\pi$
$191$	−7.31539	−0.529323	−0.264662	+	0.964341i	$0.585260\pi$
$192$	−8.29484	−0.598629
$193$	9.10083	0.655092	0.327546	−	0.944835i	$-0.393778\pi$
$193$	9.10083	0.655092	0.327546	+	0.944835i	$0.393778\pi$
$194$	−2.68530	−0.192793
$195$	−0.614463	−0.0440026
$196$	−0.845873	−0.0604195
$197$	18.3950	1.31059	0.655296	−	0.755372i	$-0.272544\pi$
$197$	18.3950	1.31059	0.655296	+	0.755372i	$0.272544\pi$
$198$	1.27430	0.0905604
$199$	17.9798	1.27456	0.637278	−	0.770634i	$-0.280060\pi$
$199$	17.9798	1.27456	0.637278	+	0.770634i	$0.280060\pi$
$200$	−2.88231	−0.203810
$201$	0.824368	0.0581465
$202$	−6.84492	−0.481607
$203$	20.9378	1.46955
$204$	0.561169	0.0392897
$205$	−6.08029	−0.424666
$206$	−12.2924	−0.856449
$207$	8.34814	0.580236
$208$	2.35522	0.163305
$209$	1.03893	0.0718640
$210$	5.80203	0.400378
$211$	13.2375	0.911310	0.455655	−	0.890157i	$-0.349405\pi$
$211$	13.2375	0.911310	0.455655	+	0.890157i	$0.349405\pi$
$212$	0.119853	0.00823152
$213$	−12.1232	−0.830666
$214$	11.3816	0.778029
$215$	8.21456	0.560228
$216$	2.88231	0.196117
$217$	−15.0562	−1.02208
$218$	−14.0898	−0.954281
$219$	−11.7732	−0.795562
$220$	0.0738405	0.00497832
$221$	−4.29484	−0.288902
$222$	5.19950	0.348968
$223$	7.05974	0.472755	0.236378	−	0.971661i	$-0.424040\pi$
$223$	7.05974	0.472755	0.236378	+	0.971661i	$0.424040\pi$
$224$	1.90069	0.126995
$225$	1.00000	0.0666667
$226$	−8.32976	−0.554087
$227$	−24.3720	−1.61763	−0.808813	−	0.588065i	$-0.799890\pi$
$227$	−24.3720	−1.61763	−0.808813	+	0.588065i	$0.799890\pi$
$228$	0.0906930	0.00600629
$229$	19.8640	1.31265	0.656325	−	0.754479i	$-0.272110\pi$
$229$	19.8640	1.31265	0.656325	+	0.754479i	$0.272110\pi$
$230$	−11.5667	−0.762683
$231$	−3.85136	−0.253401
$232$	−14.4116	−0.946166
$233$	−0.923726	−0.0605153	−0.0302576	−	0.999542i	$-0.509633\pi$
$233$	−0.923726	−0.0605153	−0.0302576	+	0.999542i	$0.509633\pi$
$234$	−0.851361	−0.0556552
$235$	−1.00000	−0.0652328
$236$	0.206585	0.0134475
$237$	4.06592	0.264110
$238$	40.5538	2.62871
$239$	12.1073	0.783155	0.391577	−	0.920145i	$-0.371930\pi$
$239$	12.1073	0.783155	0.391577	+	0.920145i	$0.371930\pi$
$240$	−3.83298	−0.247418
$241$	−11.8870	−0.765707	−0.382853	−	0.923809i	$-0.625059\pi$
$241$	−11.8870	−0.765707	−0.382853	+	0.923809i	$0.625059\pi$
$242$	−14.0689	−0.904385
$243$	−1.00000	−0.0641500
$244$	0.471260	0.0301693
$245$	−10.5357	−0.673101
$246$	−8.42446	−0.537124
$247$	−0.694109	−0.0441651
$248$	10.3632	0.658064
$249$	14.4738	0.917240
$250$	−1.38554	−0.0876291
$251$	22.0858	1.39404	0.697022	−	0.717049i	$-0.254508\pi$
$251$	22.0858	1.39404	0.697022	+	0.717049i	$0.254508\pi$
$252$	−0.336204	−0.0211789
$253$	7.67790	0.482705
$254$	−12.0490	−0.756022
$255$	6.98959	0.437705
$256$	−1.92373	−0.120233
$257$	−5.60004	−0.349321	−0.174661	−	0.984629i	$-0.555883\pi$
$257$	−5.60004	−0.349321	−0.174661	+	0.984629i	$0.555883\pi$
$258$	11.3816	0.708586
$259$	−15.7147	−0.976461
$260$	−0.0493330	−0.00305950
$261$	5.00000	0.309492
$262$	16.7751	1.03637
$263$	−13.7892	−0.850278	−0.425139	−	0.905128i	$-0.639775\pi$
$263$	−13.7892	−0.850278	−0.425139	+	0.905128i	$0.639775\pi$
$264$	2.65090	0.163152
$265$	1.49282	0.0917030
$266$	6.55408	0.401857
$267$	16.7049	1.02232
$268$	0.0661855	0.00404292
$269$	−4.57374	−0.278866	−0.139433	−	0.990232i	$-0.544528\pi$
$269$	−4.57374	−0.278866	−0.139433	+	0.990232i	$0.544528\pi$
$270$	1.38554	0.0843211
$271$	−29.8057	−1.81057	−0.905284	−	0.424806i	$-0.860342\pi$
$271$	−29.8057	−1.81057	−0.905284	+	0.424806i	$0.860342\pi$
$272$	−26.7910	−1.62444
$273$	2.57310	0.155731
$274$	−24.9108	−1.50492
$275$	0.919714	0.0554608
$276$	0.670242	0.0403438
$277$	19.4187	1.16675	0.583377	−	0.812201i	$-0.301731\pi$
$277$	19.4187	1.16675	0.583377	+	0.812201i	$0.301731\pi$
$278$	13.3797	0.802463
$279$	−3.59544	−0.215254
$280$	12.0699	0.721313
$281$	17.0239	1.01556	0.507779	−	0.861487i	$-0.330467\pi$
$281$	17.0239	1.01556	0.507779	+	0.861487i	$0.330467\pi$
$282$	−1.38554	−0.0825075
$283$	−11.0303	−0.655684	−0.327842	−	0.944733i	$-0.606321\pi$
$283$	−11.0303	−0.655684	−0.327842	+	0.944733i	$0.606321\pi$
$284$	−0.973325	−0.0577562
$285$	1.12962	0.0669129
$286$	−0.783008	−0.0463003
$287$	25.4616	1.50295
$288$	0.453890	0.0267457
$289$	31.8544	1.87379
$290$	−6.92769	−0.406808
$291$	1.93809	0.113613
$292$	−0.945231	−0.0553154
$293$	−33.5084	−1.95758	−0.978792	−	0.204856i	$-0.934327\pi$
$293$	−33.5084	−1.95758	−0.978792	+	0.204856i	$0.934327\pi$
$294$	−14.5976	−0.851350
$295$	2.57310	0.149812
$296$	10.8164	0.628693
$297$	−0.919714	−0.0533672
$298$	28.4769	1.64962
$299$	−5.12962	−0.296654
$300$	0.0802864	0.00463533
$301$	−34.3990	−1.98273
$302$	13.3586	0.768701
$303$	4.94026	0.283811
$304$	−4.32981	−0.248332
$305$	5.86974	0.336100
$306$	9.68434	0.553617
$307$	9.98494	0.569871	0.284935	−	0.958547i	$-0.408028\pi$
$307$	9.98494	0.569871	0.284935	+	0.958547i	$0.408028\pi$
$308$	−0.309212	−0.0176190
$309$	8.87191	0.504705
$310$	4.98162	0.282937
$311$	12.9816	0.736120	0.368060	−	0.929802i	$-0.380022\pi$
$311$	12.9816	0.736120	0.368060	+	0.929802i	$0.380022\pi$
$312$	−1.77107	−0.100267
$313$	−21.9378	−1.24000	−0.619998	−	0.784603i	$-0.712867\pi$
$313$	−21.9378	−1.24000	−0.619998	+	0.784603i	$0.712867\pi$
$314$	−17.5581	−0.990861
$315$	−4.18757	−0.235943
$316$	0.326438	0.0183636
$317$	11.0856	0.622628	0.311314	−	0.950307i	$-0.399231\pi$
$317$	11.0856	0.622628	0.311314	+	0.950307i	$0.399231\pi$
$318$	2.06835	0.115987
$319$	4.59857	0.257470
$320$	−8.29484	−0.463696
$321$	−8.21456	−0.458492
$322$	48.4361	2.69924
$323$	7.89558	0.439322
$324$	−0.0802864	−0.00446035
$325$	−0.614463	−0.0340843
$326$	8.53570	0.472749
$327$	10.1692	0.562357
$328$	−17.5253	−0.967673
$329$	4.18757	0.230868
$330$	1.27430	0.0701478
$331$	7.88716	0.433518	0.216759	−	0.976225i	$-0.430451\pi$
$331$	7.88716	0.433518	0.216759	+	0.976225i	$0.430451\pi$
$332$	1.16205	0.0637757
$333$	−3.75270	−0.205646
$334$	−5.32267	−0.291244
$335$	0.824368	0.0450401
$336$	16.0509	0.875646
$337$	−22.0819	−1.20288	−0.601438	−	0.798920i	$-0.705405\pi$
$337$	−22.0819	−1.20288	−0.601438	+	0.798920i	$0.705405\pi$
$338$	−17.4889	−0.951268
$339$	6.01193	0.326523
$340$	0.561169	0.0304337
$341$	−3.30678	−0.179072
$342$	1.56513	0.0846325
$343$	14.8060	0.799448
$344$	23.6769	1.27657
$345$	8.34814	0.449449
$346$	−19.8082	−1.06489
$347$	−24.5149	−1.31603	−0.658014	−	0.753006i	$-0.728603\pi$
$347$	−24.5149	−1.31603	−0.658014	+	0.753006i	$0.728603\pi$
$348$	0.401432	0.0215190
$349$	−28.2103	−1.51006	−0.755031	−	0.655689i	$-0.772378\pi$
$349$	−28.2103	−1.51006	−0.755031	+	0.655689i	$0.772378\pi$
$350$	5.80203	0.310131
$351$	0.614463	0.0327976
$352$	0.417449	0.0222501
$353$	−32.9115	−1.75170	−0.875852	−	0.482580i	$-0.839700\pi$
$353$	−32.9115	−1.75170	−0.875852	+	0.482580i	$0.839700\pi$
$354$	3.56513	0.189485
$355$	−12.1232	−0.643431
$356$	1.34117	0.0710821
$357$	−29.2694	−1.54910
$358$	21.9792	1.16164
$359$	−17.7819	−0.938490	−0.469245	−	0.883068i	$-0.655474\pi$
$359$	−17.7819	−0.938490	−0.469245	+	0.883068i	$0.655474\pi$
$360$	2.88231	0.151911
$361$	−17.7240	−0.932840
$362$	−0.886645	−0.0466010
$363$	10.1541	0.532953
$364$	0.206585	0.0108280
$365$	−11.7732	−0.616240
$366$	8.13274	0.425105
$367$	14.0079	0.731208	0.365604	−	0.930771i	$-0.380863\pi$
$367$	14.0079	0.731208	0.365604	+	0.930771i	$0.380863\pi$
$368$	−31.9983	−1.66802
$369$	6.08029	0.316527
$370$	5.19950	0.270309
$371$	−6.25127	−0.324550
$372$	−0.288665	−0.0149666
$373$	21.9577	1.13693	0.568463	−	0.822709i	$-0.307538\pi$
$373$	21.9577	1.13693	0.568463	+	0.822709i	$0.307538\pi$
$374$	8.90682	0.460561
$375$	1.00000	0.0516398
$376$	−2.88231	−0.148644
$377$	−3.07231	−0.158232
$378$	−5.80203	−0.298424
$379$	−5.10415	−0.262183	−0.131091	−	0.991370i	$-0.541848\pi$
$379$	−5.10415	−0.262183	−0.131091	+	0.991370i	$0.541848\pi$
$380$	0.0906930	0.00465245
$381$	8.69628	0.445524
$382$	−10.1357	−0.518590
$383$	6.43635	0.328882	0.164441	−	0.986387i	$-0.447418\pi$
$383$	6.43635	0.328882	0.164441	+	0.986387i	$0.447418\pi$
$384$	−10.5850	−0.540165
$385$	−3.85136	−0.196284
$386$	12.6095	0.641809
$387$	−8.21456	−0.417569
$388$	0.155602	0.00789952
$389$	−36.7518	−1.86339	−0.931696	−	0.363239i	$-0.881671\pi$
$389$	−36.7518	−1.86339	−0.931696	+	0.363239i	$0.881671\pi$
$390$	−0.851361	−0.0431103
$391$	58.3501	2.95089
$392$	−30.3672	−1.53378
$393$	−12.1073	−0.610731
$394$	25.4870	1.28402
$395$	4.06592	0.204579
$396$	−0.0738405	−0.00371062
$397$	−20.7423	−1.04103	−0.520514	−	0.853853i	$-0.674260\pi$
$397$	−20.7423	−1.04103	−0.520514	+	0.853853i	$0.674260\pi$
$398$	24.9117	1.24871
$399$	−4.73035	−0.236814
$400$	−3.83298	−0.191649
$401$	37.7899	1.88714	0.943568	−	0.331178i	$-0.107446\pi$
$401$	37.7899	1.88714	0.943568	+	0.331178i	$0.107446\pi$
$402$	1.14219	0.0569674
$403$	2.20927	0.110051
$404$	0.396635	0.0197334
$405$	−1.00000	−0.0496904
$406$	29.0101	1.43975
$407$	−3.45140	−0.171080
$408$	20.1462	0.997386
$409$	−1.49678	−0.0740109	−0.0370054	−	0.999315i	$-0.511782\pi$
$409$	−1.49678	−0.0740109	−0.0370054	+	0.999315i	$0.511782\pi$
$410$	−8.42446	−0.416055
$411$	17.9792	0.886848
$412$	0.712293	0.0350922
$413$	−10.7750	−0.530205
$414$	11.5667	0.568470
$415$	14.4738	0.710491
$416$	−0.278898	−0.0136741
$417$	−9.65671	−0.472891
$418$	1.43947	0.0704068
$419$	−2.85379	−0.139417	−0.0697085	−	0.997567i	$-0.522207\pi$
$419$	−2.85379	−0.139417	−0.0697085	+	0.997567i	$0.522207\pi$
$420$	−0.336204	−0.0164051
$421$	−27.0334	−1.31753	−0.658763	−	0.752351i	$-0.728920\pi$
$421$	−27.0334	−1.31753	−0.658763	+	0.752351i	$0.728920\pi$
$422$	18.3411	0.892831
$423$	1.00000	0.0486217
$424$	4.30277	0.208961
$425$	6.98959	0.339045
$426$	−16.7971	−0.813823
$427$	−24.5799	−1.18951
$428$	−0.659517	−0.0318790
$429$	0.565130	0.0272847
$430$	11.3816	0.548868
$431$	−6.47444	−0.311863	−0.155931	−	0.987768i	$-0.549838\pi$
$431$	−6.47444	−0.311863	−0.155931	+	0.987768i	$0.549838\pi$
$432$	3.83298	0.184414
$433$	−37.4885	−1.80158	−0.900792	−	0.434251i	$-0.857013\pi$
$433$	−37.4885	−1.80158	−0.900792	+	0.434251i	$0.857013\pi$
$434$	−20.8609	−1.00135
$435$	5.00000	0.239732
$436$	0.816447	0.0391007
$437$	9.43022	0.451109
$438$	−16.3123	−0.779430
$439$	31.0135	1.48019	0.740097	−	0.672500i	$-0.234780\pi$
$439$	31.0135	1.48019	0.740097	+	0.672500i	$0.234780\pi$
$440$	2.65090	0.126377
$441$	10.5357	0.501700
$442$	−5.95067	−0.283044
$443$	7.04971	0.334942	0.167471	−	0.985877i	$-0.446440\pi$
$443$	7.04971	0.334942	0.167471	+	0.985877i	$0.446440\pi$
$444$	−0.301290	−0.0142986
$445$	16.7049	0.791888
$446$	9.78153	0.463169
$447$	−20.5529	−0.972120
$448$	34.7352	1.64108
$449$	−5.58166	−0.263415	−0.131708	−	0.991289i	$-0.542046\pi$
$449$	−5.58166	−0.263415	−0.131708	+	0.991289i	$0.542046\pi$
$450$	1.38554	0.0653149
$451$	5.59212	0.263323
$452$	0.482676	0.0227032
$453$	−9.64146	−0.452995
$454$	−33.7683	−1.58483
$455$	2.57310	0.120629
$456$	3.25592	0.152472
$457$	−1.07141	−0.0501183	−0.0250591	−	0.999686i	$-0.507977\pi$
$457$	−1.07141	−0.0501183	−0.0250591	+	0.999686i	$0.507977\pi$
$458$	27.5223	1.28603
$459$	−6.98959	−0.326246
$460$	0.670242	0.0312502
$461$	5.04754	0.235087	0.117544	−	0.993068i	$-0.462498\pi$
$461$	5.04754	0.235087	0.117544	+	0.993068i	$0.462498\pi$
$462$	−5.33620	−0.248263
$463$	6.75454	0.313910	0.156955	−	0.987606i	$-0.449832\pi$
$463$	6.75454	0.313910	0.156955	+	0.987606i	$0.449832\pi$
$464$	−19.1649	−0.889708
$465$	−3.59544	−0.166735
$466$	−1.27986	−0.0592882
$467$	19.5324	0.903851	0.451925	−	0.892056i	$-0.350737\pi$
$467$	19.5324	0.903851	0.451925	+	0.892056i	$0.350737\pi$
$468$	0.0493330	0.00228042
$469$	−3.45210	−0.159403
$470$	−1.38554	−0.0639101
$471$	12.6724	0.583914
$472$	7.41649	0.341372
$473$	−7.55504	−0.347381
$474$	5.63348	0.258754
$475$	1.12962	0.0518305
$476$	−2.34993	−0.107709
$477$	−1.49282	−0.0683514
$478$	16.7751	0.767275
$479$	−10.9843	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$479$	−10.9843	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$480$	0.453890	0.0207171
$481$	2.30589	0.105140
$482$	−16.4698	−0.750180
$483$	−34.9584	−1.59066
$484$	0.815238	0.0370563
$485$	1.93809	0.0880043
$486$	−1.38554	−0.0628492
$487$	−25.9801	−1.17727	−0.588635	−	0.808399i	$-0.700334\pi$
$487$	−25.9801	−1.17727	−0.588635	+	0.808399i	$0.700334\pi$
$488$	16.9184	0.765862
$489$	−6.16057	−0.278591
$490$	−14.5976	−0.659453
$491$	29.1879	1.31723	0.658617	−	0.752479i	$-0.271142\pi$
$491$	29.1879	1.31723	0.658617	+	0.752479i	$0.271142\pi$
$492$	0.488164	0.0220081
$493$	34.9480	1.57398
$494$	−0.961714	−0.0432696
$495$	−0.919714	−0.0413381
$496$	13.7813	0.618797
$497$	50.7666	2.27719
$498$	20.0540	0.898641
$499$	−2.73795	−0.122568	−0.0612838	−	0.998120i	$-0.519519\pi$
$499$	−2.73795	−0.122568	−0.0612838	+	0.998120i	$0.519519\pi$
$500$	0.0802864	0.00359051
$501$	3.84159	0.171630
$502$	30.6007	1.36578
$503$	6.50871	0.290209	0.145105	−	0.989416i	$-0.453648\pi$
$503$	6.50871	0.290209	0.145105	+	0.989416i	$0.453648\pi$
$504$	−12.0699	−0.537635
$505$	4.94026	0.219839
$506$	10.6380	0.472917
$507$	12.6224	0.560582
$508$	0.698192	0.0309773
$509$	−10.4054	−0.461213	−0.230607	−	0.973047i	$-0.574071\pi$
$509$	−10.4054	−0.461213	−0.230607	+	0.973047i	$0.574071\pi$
$510$	9.68434	0.428830
$511$	49.3012	2.18096
$512$	−23.8355	−1.05339
$513$	−1.12962	−0.0498739
$514$	−7.75907	−0.342238
$515$	8.87191	0.390943
$516$	−0.659517	−0.0290336
$517$	0.919714	0.0404490
$518$	−21.7732	−0.956661
$519$	14.2964	0.627541
$520$	−1.77107	−0.0776668
$521$	−10.2017	−0.446943	−0.223472	−	0.974710i	$-0.571739\pi$
$521$	−10.2017	−0.446943	−0.223472	+	0.974710i	$0.571739\pi$
$522$	6.92769	0.303217
$523$	10.8838	0.475917	0.237959	−	0.971275i	$-0.423522\pi$
$523$	10.8838	0.475917	0.237959	+	0.971275i	$0.423522\pi$
$524$	−0.972049	−0.0424642
$525$	−4.18757	−0.182760
$526$	−19.1054	−0.833037
$527$	−25.1307	−1.09471
$528$	3.52525	0.153417
$529$	46.6914	2.03006
$530$	2.06835	0.0898435
$531$	−2.57310	−0.111663
$532$	−0.379783	−0.0164657
$533$	−3.73611	−0.161829
$534$	23.1452	1.00159
$535$	−8.21456	−0.355146
$536$	2.37609	0.102631
$537$	−15.8633	−0.684552
$538$	−6.33709	−0.273211
$539$	9.68983	0.417371
$540$	−0.0802864	−0.00345497
$541$	−4.50138	−0.193529	−0.0967647	−	0.995307i	$-0.530849\pi$
$541$	−4.50138	−0.193529	−0.0967647	+	0.995307i	$0.530849\pi$
$542$	−41.2969	−1.77386
$543$	0.639928	0.0274620
$544$	3.17251	0.136020
$545$	10.1692	0.435600
$546$	3.56513	0.152573
$547$	42.9758	1.83751	0.918756	−	0.394825i	$-0.129195\pi$
$547$	42.9758	1.83751	0.918756	+	0.394825i	$0.129195\pi$
$548$	1.44348	0.0616626
$549$	−5.86974	−0.250514
$550$	1.27430	0.0543362
$551$	5.64810	0.240617
$552$	24.0620	1.02414
$553$	−17.0263	−0.724032
$554$	26.9053	1.14310
$555$	−3.75270	−0.159293
$556$	−0.775302	−0.0328801
$557$	27.3479	1.15877	0.579383	−	0.815055i	$-0.303293\pi$
$557$	27.3479	1.15877	0.579383	+	0.815055i	$0.303293\pi$
$558$	−4.98162	−0.210889
$559$	5.04754	0.213488
$560$	16.0509	0.678273
$561$	−6.42842	−0.271408
$562$	23.5872	0.994966
$563$	−21.5293	−0.907350	−0.453675	−	0.891167i	$-0.649887\pi$
$563$	−21.5293	−0.907350	−0.453675	+	0.891167i	$0.649887\pi$
$564$	0.0802864	0.00338067
$565$	6.01193	0.252924
$566$	−15.2829	−0.642389
$567$	4.18757	0.175861
$568$	−34.9428	−1.46617
$569$	9.20811	0.386024	0.193012	−	0.981196i	$-0.438174\pi$
$569$	9.20811	0.386024	0.193012	+	0.981196i	$0.438174\pi$
$570$	1.56513	0.0655561
$571$	29.9531	1.25350	0.626749	−	0.779221i	$-0.284385\pi$
$571$	29.9531	1.25350	0.626749	+	0.779221i	$0.284385\pi$
$572$	0.0453722	0.00189711
$573$	7.31539	0.305605
$574$	35.2780	1.47247
$575$	8.34814	0.348141
$576$	8.29484	0.345619
$577$	41.6419	1.73357	0.866787	−	0.498678i	$-0.166181\pi$
$577$	41.6419	1.73357	0.866787	+	0.498678i	$0.166181\pi$
$578$	44.1355	1.83579
$579$	−9.10083	−0.378218
$580$	0.401432	0.0166685
$581$	−60.6100	−2.51453
$582$	2.68530	0.111309
$583$	−1.37296	−0.0568623
$584$	−33.9342	−1.40421
$585$	0.614463	0.0254049
$586$	−46.4272	−1.91789
$587$	−36.9519	−1.52517	−0.762585	−	0.646888i	$-0.776070\pi$
$587$	−36.9519	−1.52517	−0.762585	+	0.646888i	$0.776070\pi$
$588$	0.845873	0.0348832
$589$	−4.06148	−0.167350
$590$	3.56513	0.146774
$591$	−18.3950	−0.756671
$592$	14.3840	0.591179
$593$	−27.6951	−1.13730	−0.568651	−	0.822579i	$-0.692535\pi$
$593$	−27.6951	−1.13730	−0.568651	+	0.822579i	$0.692535\pi$
$594$	−1.27430	−0.0522851
$595$	−29.2694	−1.19993
$596$	−1.65012	−0.0675915
$597$	−17.9798	−0.735865
$598$	−7.10728	−0.290638
$599$	−6.30033	−0.257425	−0.128712	−	0.991682i	$-0.541084\pi$
$599$	−6.30033	−0.257425	−0.128712	+	0.991682i	$0.541084\pi$
$600$	2.88231	0.117670
$601$	−21.1106	−0.861119	−0.430560	−	0.902562i	$-0.641684\pi$
$601$	−21.1106	−0.861119	−0.430560	+	0.902562i	$0.641684\pi$
$602$	−47.6611	−1.94252
$603$	−0.824368	−0.0335709
$604$	−0.774077	−0.0314968
$605$	10.1541	0.412824
$606$	6.84492	0.278056
$607$	−40.9648	−1.66271	−0.831355	−	0.555741i	$-0.812435\pi$
$607$	−40.9648	−1.66271	−0.831355	+	0.555741i	$0.812435\pi$
$608$	0.512723	0.0207937
$609$	−20.9378	−0.848444
$610$	8.13274	0.329285
$611$	−0.614463	−0.0248585
$612$	−0.561169	−0.0226839
$613$	−48.1149	−1.94334	−0.971672	−	0.236334i	$-0.924054\pi$
$613$	−48.1149	−1.94334	−0.971672	+	0.236334i	$0.924054\pi$
$614$	13.8345	0.558315
$615$	6.08029	0.245181
$616$	−11.1008	−0.447265
$617$	−3.74809	−0.150893	−0.0754463	−	0.997150i	$-0.524038\pi$
$617$	−3.74809	−0.150893	−0.0754463	+	0.997150i	$0.524038\pi$
$618$	12.2924	0.494471
$619$	−15.6789	−0.630186	−0.315093	−	0.949061i	$-0.602036\pi$
$619$	−15.6789	−0.630186	−0.315093	+	0.949061i	$0.602036\pi$
$620$	−0.288665	−0.0115931
$621$	−8.34814	−0.334999
$622$	17.9865	0.721194
$623$	−69.9528	−2.80260
$624$	−2.35522	−0.0942844
$625$	1.00000	0.0400000
$626$	−30.3956	−1.21485
$627$	−1.03893	−0.0414907
$628$	1.01742	0.0405995
$629$	−26.2298	−1.04585
$630$	−5.80203	−0.231158
$631$	−30.2501	−1.20424	−0.602119	−	0.798407i	$-0.705677\pi$
$631$	−30.2501	−1.20424	−0.602119	+	0.798407i	$0.705677\pi$
$632$	11.7193	0.466167
$633$	−13.2375	−0.526145
$634$	15.3595	0.610003
$635$	8.69628	0.345101
$636$	−0.119853	−0.00475247
$637$	−6.47380	−0.256501
$638$	6.37149	0.252250
$639$	12.1232	0.479585
$640$	−10.5850	−0.418410
$641$	−23.5134	−0.928724	−0.464362	−	0.885645i	$-0.653716\pi$
$641$	−23.5134	−0.928724	−0.464362	+	0.885645i	$0.653716\pi$
$642$	−11.3816	−0.449195
$643$	−24.5259	−0.967209	−0.483604	−	0.875287i	$-0.660673\pi$
$643$	−24.5259	−0.967209	−0.483604	+	0.875287i	$0.660673\pi$
$644$	−2.80668	−0.110599
$645$	−8.21456	−0.323448
$646$	10.9396	0.430414
$647$	10.2318	0.402253	0.201126	−	0.979565i	$-0.435540\pi$
$647$	10.2318	0.402253	0.201126	+	0.979565i	$0.435540\pi$
$648$	−2.88231	−0.113228
$649$	−2.36652	−0.0928939
$650$	−0.851361	−0.0333931
$651$	15.0562	0.590097
$652$	−0.494610	−0.0193704
$653$	15.4266	0.603691	0.301845	−	0.953357i	$-0.402397\pi$
$653$	15.4266	0.603691	0.301845	+	0.953357i	$0.402397\pi$
$654$	14.0898	0.550954
$655$	−12.1073	−0.473071
$656$	−23.3056	−0.909932
$657$	11.7732	0.459318
$658$	5.80203	0.226187
$659$	−40.8238	−1.59027	−0.795135	−	0.606432i	$-0.792600\pi$
$659$	−40.8238	−1.59027	−0.795135	+	0.606432i	$0.792600\pi$
$660$	−0.0738405	−0.00287424
$661$	−28.2581	−1.09911	−0.549556	−	0.835457i	$-0.685203\pi$
$661$	−28.2581	−1.09911	−0.549556	+	0.835457i	$0.685203\pi$
$662$	10.9280	0.424727
$663$	4.29484	0.166798
$664$	41.7180	1.61897
$665$	−4.73035	−0.183435
$666$	−5.19950	−0.201476
$667$	41.7407	1.61621
$668$	0.308428	0.0119334
$669$	−7.05974	−0.272945
$670$	1.14219	0.0441268
$671$	−5.39848	−0.208406
$672$	−1.90069	−0.0733209
$673$	−13.6620	−0.526631	−0.263316	−	0.964710i	$-0.584816\pi$
$673$	−13.6620	−0.526631	−0.263316	+	0.964710i	$0.584816\pi$
$674$	−30.5952	−1.17848
$675$	−1.00000	−0.0384900
$676$	1.01341	0.0389773
$677$	50.5626	1.94328	0.971640	−	0.236466i	$-0.0759892\pi$
$677$	50.5626	1.94328	0.971640	+	0.236466i	$0.0759892\pi$
$678$	8.32976	0.319902
$679$	−8.11589	−0.311459
$680$	20.1462	0.772572
$681$	24.3720	0.933937
$682$	−4.58166	−0.175441
$683$	20.6092	0.788590	0.394295	−	0.918984i	$-0.370989\pi$
$683$	20.6092	0.788590	0.394295	+	0.918984i	$0.370989\pi$
$684$	−0.0906930	−0.00346773
$685$	17.9792	0.686949
$686$	20.5142	0.783238
$687$	−19.8640	−0.757858
$688$	31.4862	1.20040
$689$	0.917280	0.0349456
$690$	11.5667	0.440335
$691$	25.9719	0.988018	0.494009	−	0.869457i	$-0.335531\pi$
$691$	25.9719	0.988018	0.494009	+	0.869457i	$0.335531\pi$
$692$	1.14780	0.0436329
$693$	3.85136	0.146301
$694$	−33.9663	−1.28934
$695$	−9.65671	−0.366300
$696$	14.4116	0.546269
$697$	42.4987	1.60975
$698$	−39.0864	−1.47944
$699$	0.923726	0.0349385
$700$	−0.336204	−0.0127073
$701$	−37.7609	−1.42621	−0.713105	−	0.701057i	$-0.752712\pi$
$701$	−37.7609	−1.42621	−0.713105	+	0.701057i	$0.752712\pi$
$702$	0.851361	0.0321325
$703$	−4.23912	−0.159881
$704$	7.62888	0.287524
$705$	1.00000	0.0376622
$706$	−45.6001	−1.71618
$707$	−20.6877	−0.778040
$708$	−0.206585	−0.00776394
$709$	27.1125	1.01823	0.509116	−	0.860698i	$-0.329973\pi$
$709$	27.1125	1.01823	0.509116	+	0.860698i	$0.329973\pi$
$710$	−16.7971	−0.630384
$711$	−4.06592	−0.152484
$712$	48.1487	1.80445
$713$	−30.0153	−1.12408
$714$	−40.5538	−1.51769
$715$	0.565130	0.0211347
$716$	−1.27361	−0.0475969
$717$	−12.1073	−0.452155
$718$	−24.6374	−0.919460
$719$	5.44412	0.203032	0.101516	−	0.994834i	$-0.467631\pi$
$719$	5.44412	0.203032	0.101516	+	0.994834i	$0.467631\pi$
$720$	3.83298	0.142847
$721$	−37.1517	−1.38360
$722$	−24.5572	−0.913925
$723$	11.8870	0.442081
$724$	0.0513775	0.00190943
$725$	5.00000	0.185695
$726$	14.0689	0.522147
$727$	32.3623	1.20025	0.600126	−	0.799905i	$-0.295117\pi$
$727$	32.3623	1.20025	0.600126	+	0.799905i	$0.295117\pi$
$728$	7.41649	0.274873
$729$	1.00000	0.0370370
$730$	−16.3123	−0.603744
$731$	−57.4164	−2.12362
$732$	−0.471260	−0.0174183
$733$	−22.6016	−0.834810	−0.417405	−	0.908721i	$-0.637060\pi$
$733$	−22.6016	−0.834810	−0.417405	+	0.908721i	$0.637060\pi$
$734$	19.4085	0.716381
$735$	10.5357	0.388615
$736$	3.78914	0.139669
$737$	−0.758183	−0.0279280
$738$	8.42446	0.310109
$739$	−10.6215	−0.390717	−0.195359	−	0.980732i	$-0.562587\pi$
$739$	−10.6215	−0.390717	−0.195359	+	0.980732i	$0.562587\pi$
$740$	−0.301290	−0.0110756
$741$	0.694109	0.0254987
$742$	−8.66136	−0.317969
$743$	10.2836	0.377269	0.188634	−	0.982047i	$-0.439594\pi$
$743$	10.2836	0.377269	0.188634	+	0.982047i	$0.439594\pi$
$744$	−10.3632	−0.379933
$745$	−20.5529	−0.753001
$746$	30.4232	1.11387
$747$	−14.4738	−0.529569
$748$	−0.516115	−0.0188710
$749$	34.3990	1.25691
$750$	1.38554	0.0505927
$751$	15.6206	0.570005	0.285003	−	0.958527i	$-0.408006\pi$
$751$	15.6206	0.570005	0.285003	+	0.958527i	$0.408006\pi$
$752$	−3.83298	−0.139774
$753$	−22.0858	−0.804852
$754$	−4.25680	−0.155024
$755$	−9.64146	−0.350888
$756$	0.336204	0.0122276
$757$	30.0073	1.09064	0.545318	−	0.838230i	$-0.316409\pi$
$757$	30.0073	1.09064	0.545318	+	0.838230i	$0.316409\pi$
$758$	−7.07200	−0.256866
$759$	−7.67790	−0.278690
$760$	3.25592	0.118105
$761$	−5.88716	−0.213409	−0.106705	−	0.994291i	$-0.534030\pi$
$761$	−5.88716	−0.213409	−0.106705	+	0.994291i	$0.534030\pi$
$762$	12.0490	0.436490
$763$	−42.5841	−1.54165
$764$	0.587326	0.0212487
$765$	−6.98959	−0.252709
$766$	8.91780	0.322213
$767$	1.58108	0.0570893
$768$	1.92373	0.0694165
$769$	−28.8029	−1.03866	−0.519329	−	0.854574i	$-0.673818\pi$
$769$	−28.8029	−1.03866	−0.519329	+	0.854574i	$0.673818\pi$
$770$	−5.33620	−0.192303
$771$	5.60004	0.201681
$772$	−0.730673	−0.0262975
$773$	−24.5651	−0.883546	−0.441773	−	0.897127i	$-0.645650\pi$
$773$	−24.5651	−0.883546	−0.441773	+	0.897127i	$0.645650\pi$
$774$	−11.3816	−0.409102
$775$	−3.59544	−0.129152
$776$	5.58619	0.200533
$777$	15.7147	0.563760
$778$	−50.9210	−1.82561
$779$	6.86841	0.246086
$780$	0.0493330	0.00176640
$781$	11.1498	0.398973
$782$	80.8462	2.89106
$783$	−5.00000	−0.178685
$784$	−40.3832	−1.44226
$785$	12.6724	0.452298
$786$	−16.7751	−0.598348
$787$	−51.2953	−1.82848	−0.914239	−	0.405174i	$-0.867211\pi$
$787$	−51.2953	−1.82848	−0.914239	+	0.405174i	$0.867211\pi$
$788$	−1.47687	−0.0526113
$789$	13.7892	0.490908
$790$	5.63348	0.200430
$791$	−25.1754	−0.895133
$792$	−2.65090	−0.0941958
$793$	3.60674	0.128079
$794$	−28.7393	−1.01992
$795$	−1.49282	−0.0529447
$796$	−1.44353	−0.0511647
$797$	32.4937	1.15098	0.575492	−	0.817807i	$-0.304811\pi$
$797$	32.4937	1.15098	0.575492	+	0.817807i	$0.304811\pi$
$798$	−6.55408	−0.232012
$799$	6.98959	0.247274
$800$	0.453890	0.0160474
$801$	−16.7049	−0.590238
$802$	52.3593	1.84887
$803$	10.8280	0.382112
$804$	−0.0661855	−0.00233418
$805$	−34.9584	−1.23212
$806$	3.06102	0.107820
$807$	4.57374	0.161003
$808$	14.2394	0.500940
$809$	49.2124	1.73022	0.865109	−	0.501584i	$-0.167249\pi$
$809$	49.2124	1.73022	0.865109	+	0.501584i	$0.167249\pi$
$810$	−1.38554	−0.0486828
$811$	6.70917	0.235591	0.117795	−	0.993038i	$-0.462417\pi$
$811$	6.70917	0.235591	0.117795	+	0.993038i	$0.462417\pi$
$812$	−1.68102	−0.0589923
$813$	29.8057	1.04533
$814$	−4.78205	−0.167611
$815$	−6.16057	−0.215795
$816$	26.7910	0.937872
$817$	−9.27932	−0.324642
$818$	−2.07384	−0.0725101
$819$	−2.57310	−0.0899115
$820$	0.488164	0.0170474
$821$	45.5348	1.58918	0.794589	−	0.607148i	$-0.207687\pi$
$821$	45.5348	1.58918	0.794589	+	0.607148i	$0.207687\pi$
$822$	24.9108	0.868865
$823$	−8.76979	−0.305696	−0.152848	−	0.988250i	$-0.548844\pi$
$823$	−8.76979	−0.305696	−0.152848	+	0.988250i	$0.548844\pi$
$824$	25.5716	0.890830
$825$	−0.919714	−0.0320203
$826$	−14.9292	−0.519454
$827$	49.4171	1.71840	0.859201	−	0.511639i	$-0.170961\pi$
$827$	49.4171	1.71840	0.859201	+	0.511639i	$0.170961\pi$
$828$	−0.670242	−0.0232925
$829$	−56.2875	−1.95495	−0.977474	−	0.211058i	$-0.932309\pi$
$829$	−56.2875	−1.95495	−0.977474	+	0.211058i	$0.932309\pi$
$830$	20.0540	0.696084
$831$	−19.4187	−0.673626
$832$	−5.09687	−0.176702
$833$	73.6403	2.55148
$834$	−13.3797	−0.463302
$835$	3.84159	0.132944
$836$	−0.0834116	−0.00288485
$837$	3.59544	0.124277
$838$	−3.95404	−0.136590
$839$	22.8330	0.788282	0.394141	−	0.919050i	$-0.371042\pi$
$839$	22.8330	0.788282	0.394141	+	0.919050i	$0.371042\pi$
$840$	−12.0699	−0.416450
$841$	−4.00000	−0.137931
$842$	−37.4557	−1.29081
$843$	−17.0239	−0.586333
$844$	−1.06279	−0.0365829
$845$	12.6224	0.434225
$846$	1.38554	0.0476358
$847$	−42.5211	−1.46104
$848$	5.72194	0.196492
$849$	11.0303	0.378559
$850$	9.68434	0.332170
$851$	−31.3280	−1.07391
$852$	0.973325	0.0333456
$853$	−12.4069	−0.424805	−0.212402	−	0.977182i	$-0.568129\pi$
$853$	−12.4069	−0.424805	−0.212402	+	0.977182i	$0.568129\pi$
$854$	−34.0564	−1.16539
$855$	−1.12962	−0.0386322
$856$	−23.6769	−0.809261
$857$	−28.5268	−0.974458	−0.487229	−	0.873274i	$-0.661992\pi$
$857$	−28.5268	−0.974458	−0.487229	+	0.873274i	$0.661992\pi$
$858$	0.783008	0.0267315
$859$	−27.2313	−0.929121	−0.464561	−	0.885541i	$-0.653788\pi$
$859$	−27.2313	−0.929121	−0.464561	+	0.885541i	$0.653788\pi$
$860$	−0.659517	−0.0224893
$861$	−25.4616	−0.867729
$862$	−8.97057	−0.305539
$863$	8.61981	0.293422	0.146711	−	0.989179i	$-0.453131\pi$
$863$	8.61981	0.293422	0.146711	+	0.989179i	$0.453131\pi$
$864$	−0.453890	−0.0154417
$865$	14.2964	0.486091
$866$	−51.9418	−1.76505
$867$	−31.8544	−1.08183
$868$	1.20880	0.0410295
$869$	−3.73948	−0.126853
$870$	6.92769	0.234871
$871$	0.506544	0.0171636
$872$	29.3108	0.992589
$873$	−1.93809	−0.0655945
$874$	13.0659	0.441961
$875$	−4.18757	−0.141566
$876$	0.945231	0.0319364
$877$	28.3424	0.957056	0.478528	−	0.878072i	$-0.341171\pi$
$877$	28.3424	0.957056	0.478528	+	0.878072i	$0.341171\pi$
$878$	42.9704	1.45018
$879$	33.5084	1.13021
$880$	3.52525	0.118836
$881$	12.8920	0.434344	0.217172	−	0.976133i	$-0.430317\pi$
$881$	12.8920	0.434344	0.217172	+	0.976133i	$0.430317\pi$
$882$	14.5976	0.491527
$883$	48.1627	1.62081	0.810403	−	0.585873i	$-0.199248\pi$
$883$	48.1627	1.62081	0.810403	+	0.585873i	$0.199248\pi$
$884$	0.344817	0.0115975
$885$	−2.57310	−0.0864939
$886$	9.76763	0.328150
$887$	−17.0254	−0.571659	−0.285829	−	0.958281i	$-0.592269\pi$
$887$	−17.0254	−0.571659	−0.285829	+	0.958281i	$0.592269\pi$
$888$	−10.8164	−0.362976
$889$	−36.4162	−1.22136
$890$	23.1452	0.775830
$891$	0.919714	0.0308116
$892$	−0.566801	−0.0189779
$893$	1.12962	0.0378013
$894$	−28.4769	−0.952408
$895$	−15.8633	−0.530251
$896$	44.3255	1.48081
$897$	5.12962	0.171273
$898$	−7.73360	−0.258074
$899$	−17.9772	−0.599574
$900$	−0.0802864	−0.00267621
$901$	−10.4342	−0.347613
$902$	7.74809	0.257983
$903$	34.3990	1.14473
$904$	17.3283	0.576330
$905$	0.639928	0.0212719
$906$	−13.3586	−0.443810
$907$	−19.3378	−0.642101	−0.321050	−	0.947062i	$-0.604036\pi$
$907$	−19.3378	−0.642101	−0.321050	+	0.947062i	$0.604036\pi$
$908$	1.95674	0.0649367
$909$	−4.94026	−0.163858
$910$	3.56513	0.118183
$911$	−46.8527	−1.55230	−0.776149	−	0.630549i	$-0.782830\pi$
$911$	−46.8527	−1.55230	−0.776149	+	0.630549i	$0.782830\pi$
$912$	4.32981	0.143374
$913$	−13.3117	−0.440555
$914$	−1.48447	−0.0491020
$915$	−5.86974	−0.194048
$916$	−1.59481	−0.0526939
$917$	50.7000	1.67426
$918$	−9.68434	−0.319631
$919$	26.0103	0.858002	0.429001	−	0.903304i	$-0.358866\pi$
$919$	26.0103	0.858002	0.429001	+	0.903304i	$0.358866\pi$
$920$	24.0620	0.793299
$921$	−9.98494	−0.329015
$922$	6.99355	0.230320
$923$	−7.44924	−0.245195
$924$	0.309212	0.0101723
$925$	−3.75270	−0.123388
$926$	9.35867	0.307545
$927$	−8.87191	−0.291392
$928$	2.26945	0.0744983
$929$	57.6243	1.89059	0.945297	−	0.326212i	$-0.105772\pi$
$929$	57.6243	1.89059	0.945297	+	0.326212i	$0.105772\pi$
$930$	−4.98162	−0.163354
$931$	11.9013	0.390050
$932$	0.0741626	0.00242928
$933$	−12.9816	−0.424999
$934$	27.0628	0.885523
$935$	−6.42842	−0.210232
$936$	1.77107	0.0578894
$937$	61.0181	1.99337	0.996687	−	0.0813275i	$-0.0259160\pi$
$937$	61.0181	1.99337	0.996687	+	0.0813275i	$0.0259160\pi$
$938$	−4.78301	−0.156171
$939$	21.9378	0.715912
$940$	0.0802864	0.00261865
$941$	43.9597	1.43304	0.716522	−	0.697565i	$-0.245733\pi$
$941$	43.9597	1.43304	0.716522	+	0.697565i	$0.245733\pi$
$942$	17.5581	0.572074
$943$	50.7591	1.65294
$944$	9.86265	0.321002
$945$	4.18757	0.136222
$946$	−10.4678	−0.340337
$947$	44.4156	1.44331	0.721657	−	0.692251i	$-0.243381\pi$
$947$	44.4156	1.44331	0.721657	+	0.692251i	$0.243381\pi$
$948$	−0.326438	−0.0106022
$949$	−7.23422	−0.234833
$950$	1.56513	0.0507795
$951$	−11.0856	−0.359475
$952$	−84.3636	−2.73424
$953$	−56.6835	−1.83616	−0.918079	−	0.396397i	$-0.870260\pi$
$953$	−56.6835	−1.83616	−0.918079	+	0.396397i	$0.870260\pi$
$954$	−2.06835	−0.0669654
$955$	7.31539	0.236720
$956$	−0.972049	−0.0314383
$957$	−4.59857	−0.148651
$958$	−15.2192	−0.491709
$959$	−75.2890	−2.43121
$960$	8.29484	0.267715
$961$	−18.0728	−0.582993
$962$	3.19490	0.103008
$963$	8.21456	0.264710
$964$	0.954361	0.0307379
$965$	−9.10083	−0.292966
$966$	−48.4361	−1.55841
$967$	9.38822	0.301905	0.150952	−	0.988541i	$-0.451766\pi$
$967$	9.38822	0.301905	0.150952	+	0.988541i	$0.451766\pi$
$968$	29.2674	0.940689
$969$	−7.89558	−0.253643
$970$	2.68530	0.0862198
$971$	−1.85360	−0.0594848	−0.0297424	−	0.999558i	$-0.509469\pi$
$971$	−1.85360	−0.0594848	−0.0297424	+	0.999558i	$0.509469\pi$
$972$	0.0802864	0.00257519
$973$	40.4381	1.29639
$974$	−35.9964	−1.15340
$975$	0.614463	0.0196786
$976$	22.4986	0.720163
$977$	39.6717	1.26921	0.634605	−	0.772837i	$-0.281163\pi$
$977$	39.6717	1.26921	0.634605	+	0.772837i	$0.281163\pi$
$978$	−8.53570	−0.272942
$979$	−15.3637	−0.491026
$980$	0.845873	0.0270204
$981$	−10.1692	−0.324677
$982$	40.4410	1.29052
$983$	3.32976	0.106203	0.0531014	−	0.998589i	$-0.483089\pi$
$983$	3.32976	0.106203	0.0531014	+	0.998589i	$0.483089\pi$
$984$	17.5253	0.558686
$985$	−18.3950	−0.586115
$986$	48.4217	1.54206
$987$	−4.18757	−0.133292
$988$	0.0557275	0.00177293
$989$	−68.5763	−2.18060
$990$	−1.27430	−0.0404998
$991$	23.6961	0.752731	0.376365	−	0.926471i	$-0.377174\pi$
$991$	23.6961	0.752731	0.376365	+	0.926471i	$0.377174\pi$
$992$	−1.63194	−0.0518140
$993$	−7.88716	−0.250292
$994$	70.3390	2.23102
$995$	−17.9798	−0.569999
$996$	−1.16205	−0.0368209
$997$	−12.5238	−0.396633	−0.198317	−	0.980138i	$-0.563547\pi$
$997$	−12.5238	−0.396633	−0.198317	+	0.980138i	$0.563547\pi$
$998$	−3.79354	−0.120082
$999$	3.75270	0.118730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	705.2.a.j.1.3	✓	4
3.2	odd	2		2115.2.a.p.1.2		4
5.4	even	2		3525.2.a.u.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.j.1.3	✓	4	1.1	even	1	trivial
2115.2.a.p.1.2		4	3.2	odd	2
3525.2.a.u.1.2		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 705 = 3 \cdot 5 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	705.a (trivial)

\(f(q)\)	\(=\)	\(q+1.38554 q^{2} -1.00000 q^{3} -0.0802864 q^{4} -1.00000 q^{5} -1.38554 q^{6} +4.18757 q^{7} -2.88231 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.38554 q^{2} -1.00000 q^{3} -0.0802864 q^{4} -1.00000 q^{5} -1.38554 q^{6} +4.18757 q^{7} -2.88231 q^{8} +1.00000 q^{9} -1.38554 q^{10} +0.919714 q^{11} +0.0802864 q^{12} -0.614463 q^{13} +5.80203 q^{14} +1.00000 q^{15} -3.83298 q^{16} +6.98959 q^{17} +1.38554 q^{18} +1.12962 q^{19} +0.0802864 q^{20} -4.18757 q^{21} +1.27430 q^{22} +8.34814 q^{23} +2.88231 q^{24} +1.00000 q^{25} -0.851361 q^{26} -1.00000 q^{27} -0.336204 q^{28} +5.00000 q^{29} +1.38554 q^{30} -3.59544 q^{31} +0.453890 q^{32} -0.919714 q^{33} +9.68434 q^{34} -4.18757 q^{35} -0.0802864 q^{36} -3.75270 q^{37} +1.56513 q^{38} +0.614463 q^{39} +2.88231 q^{40} +6.08029 q^{41} -5.80203 q^{42} -8.21456 q^{43} -0.0738405 q^{44} -1.00000 q^{45} +11.5667 q^{46} +1.00000 q^{47} +3.83298 q^{48} +10.5357 q^{49} +1.38554 q^{50} -6.98959 q^{51} +0.0493330 q^{52} -1.49282 q^{53} -1.38554 q^{54} -0.919714 q^{55} -12.0699 q^{56} -1.12962 q^{57} +6.92769 q^{58} -2.57310 q^{59} -0.0802864 q^{60} -5.86974 q^{61} -4.98162 q^{62} +4.18757 q^{63} +8.29484 q^{64} +0.614463 q^{65} -1.27430 q^{66} -0.824368 q^{67} -0.561169 q^{68} -8.34814 q^{69} -5.80203 q^{70} +12.1232 q^{71} -2.88231 q^{72} +11.7732 q^{73} -5.19950 q^{74} -1.00000 q^{75} -0.0906930 q^{76} +3.85136 q^{77} +0.851361 q^{78} -4.06592 q^{79} +3.83298 q^{80} +1.00000 q^{81} +8.42446 q^{82} -14.4738 q^{83} +0.336204 q^{84} -6.98959 q^{85} -11.3816 q^{86} -5.00000 q^{87} -2.65090 q^{88} -16.7049 q^{89} -1.38554 q^{90} -2.57310 q^{91} -0.670242 q^{92} +3.59544 q^{93} +1.38554 q^{94} -1.12962 q^{95} -0.453890 q^{96} -1.93809 q^{97} +14.5976 q^{98} +0.919714 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 6 q^{8} + 4 q^{9} - 2 q^{10} + 8 q^{11} - 4 q^{12} - 6 q^{13} + 10 q^{14} + 4 q^{15} + 4 q^{16} - 2 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{20} + 12 q^{22} + 8 q^{23} - 6 q^{24} + 4 q^{25} + 8 q^{26} - 4 q^{27} + 4 q^{28} + 20 q^{29} + 2 q^{30} - 4 q^{31} + 14 q^{32} - 8 q^{33} + 8 q^{34} + 4 q^{36} + 8 q^{38} + 6 q^{39} - 6 q^{40} + 20 q^{41} - 10 q^{42} - 8 q^{43} + 32 q^{44} - 4 q^{45} - 2 q^{46} + 4 q^{47} - 4 q^{48} + 2 q^{50} + 2 q^{51} + 2 q^{52} + 10 q^{53} - 2 q^{54} - 8 q^{55} - 14 q^{56} - 2 q^{57} + 10 q^{58} + 10 q^{59} + 4 q^{60} - 20 q^{61} - 12 q^{62} + 4 q^{64} + 6 q^{65} - 12 q^{66} - 2 q^{68} - 8 q^{69} - 10 q^{70} + 18 q^{71} + 6 q^{72} - 4 q^{73} + 16 q^{74} - 4 q^{75} - 26 q^{76} + 4 q^{77} - 8 q^{78} + 20 q^{79} - 4 q^{80} + 4 q^{81} + 2 q^{82} - 28 q^{83} - 4 q^{84} + 2 q^{85} - 40 q^{86} - 20 q^{87} + 40 q^{88} - 2 q^{90} + 10 q^{91} - 36 q^{92} + 4 q^{93} + 2 q^{94} - 2 q^{95} - 14 q^{96} - 20 q^{97} + 4 q^{98} + 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 6 * q^8 + 4 * q^9 - 2 * q^10 + 8 * q^11 - 4 * q^12 - 6 * q^13 + 10 * q^14 + 4 * q^15 + 4 * q^16 - 2 * q^17 + 2 * q^18 + 2 * q^19 - 4 * q^20 + 12 * q^22 + 8 * q^23 - 6 * q^24 + 4 * q^25 + 8 * q^26 - 4 * q^27 + 4 * q^28 + 20 * q^29 + 2 * q^30 - 4 * q^31 + 14 * q^32 - 8 * q^33 + 8 * q^34 + 4 * q^36 + 8 * q^38 + 6 * q^39 - 6 * q^40 + 20 * q^41 - 10 * q^42 - 8 * q^43 + 32 * q^44 - 4 * q^45 - 2 * q^46 + 4 * q^47 - 4 * q^48 + 2 * q^50 + 2 * q^51 + 2 * q^52 + 10 * q^53 - 2 * q^54 - 8 * q^55 - 14 * q^56 - 2 * q^57 + 10 * q^58 + 10 * q^59 + 4 * q^60 - 20 * q^61 - 12 * q^62 + 4 * q^64 + 6 * q^65 - 12 * q^66 - 2 * q^68 - 8 * q^69 - 10 * q^70 + 18 * q^71 + 6 * q^72 - 4 * q^73 + 16 * q^74 - 4 * q^75 - 26 * q^76 + 4 * q^77 - 8 * q^78 + 20 * q^79 - 4 * q^80 + 4 * q^81 + 2 * q^82 - 28 * q^83 - 4 * q^84 + 2 * q^85 - 40 * q^86 - 20 * q^87 + 40 * q^88 - 2 * q^90 + 10 * q^91 - 36 * q^92 + 4 * q^93 + 2 * q^94 - 2 * q^95 - 14 * q^96 - 20 * q^97 + 4 * q^98 + 8 * q^99

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