LMFDB - Embedded newform 5070.2.b.y.1351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.y.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +0.307979i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +0.307979i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.335126i q^{11} -1.00000 q^{12} -0.307979 q^{14} -1.00000i q^{15} +1.00000 q^{16} +6.85086 q^{17} +1.00000i q^{18} -5.80194i q^{19} +1.00000i q^{20} +0.307979i q^{21} -0.335126 q^{22} -2.35690 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -0.307979i q^{28} -5.91185 q^{29} +1.00000 q^{30} -0.0609989i q^{31} +1.00000i q^{32} +0.335126i q^{33} +6.85086i q^{34} +0.307979 q^{35} -1.00000 q^{36} +7.07606i q^{37} +5.80194 q^{38} -1.00000 q^{40} -3.40581i q^{41} -0.307979 q^{42} +0.0489173 q^{43} -0.335126i q^{44} -1.00000i q^{45} -2.35690i q^{46} +7.00000i q^{47} +1.00000 q^{48} +6.90515 q^{49} -1.00000i q^{50} +6.85086 q^{51} +12.1739 q^{53} +1.00000i q^{54} +0.335126 q^{55} +0.307979 q^{56} -5.80194i q^{57} -5.91185i q^{58} -13.1347i q^{59} +1.00000i q^{60} +2.14675 q^{61} +0.0609989 q^{62} +0.307979i q^{63} -1.00000 q^{64} -0.335126 q^{66} -11.9608i q^{67} -6.85086 q^{68} -2.35690 q^{69} +0.307979i q^{70} -9.56465i q^{71} -1.00000i q^{72} +2.31336i q^{73} -7.07606 q^{74} -1.00000 q^{75} +5.80194i q^{76} -0.103211 q^{77} -0.0760644 q^{79} -1.00000i q^{80} +1.00000 q^{81} +3.40581 q^{82} -3.84117i q^{83} -0.307979i q^{84} -6.85086i q^{85} +0.0489173i q^{86} -5.91185 q^{87} +0.335126 q^{88} -4.41119i q^{89} +1.00000 q^{90} +2.35690 q^{92} -0.0609989i q^{93} -7.00000 q^{94} -5.80194 q^{95} +1.00000i q^{96} -6.13169i q^{97} +6.90515i q^{98} +0.335126i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} - 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} - 28 q^{29} + 6 q^{30} + 12 q^{35} - 6 q^{36} + 26 q^{38} - 6 q^{40} - 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} + 12 q^{56} - 42 q^{61} + 20 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} - 12 q^{74} - 6 q^{75} + 42 q^{77} + 30 q^{79} + 6 q^{81} - 6 q^{82} - 28 q^{87} + 6 q^{90} + 6 q^{92} - 42 q^{94} - 26 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 - 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 - 28 * q^29 + 6 * q^30 + 12 * q^35 - 6 * q^36 + 26 * q^38 - 6 * q^40 - 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 + 12 * q^56 - 42 * q^61 + 20 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 - 12 * q^74 - 6 * q^75 + 42 * q^77 + 30 * q^79 + 6 * q^81 - 6 * q^82 - 28 * q^87 + 6 * q^90 + 6 * q^92 - 42 * q^94 - 26 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	1.00000i	0.408248i
$7$	0.307979i	0.116405i	0.998305	+	0.0582025i	$0.0185369\pi$
$7$	0.307979i	0.116405i	−0.998305	+	0.0582025i	$0.981463\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0.335126i	0.101044i	0.998723	+	0.0505221i	$0.0160885\pi$
$11$	0.335126i	0.101044i	−0.998723	+	0.0505221i	$0.983911\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−0.307979	−0.0823107
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	6.85086	1.66158	0.830788	−	0.556589i	$-0.187890\pi$
$17$	6.85086	1.66158	0.830788	+	0.556589i	$0.187890\pi$
$18$	1.00000i	0.235702i
$19$	− 5.80194i	− 1.33106i	−0.746373	−	0.665528i	$-0.768206\pi$
$19$	− 5.80194i	− 1.33106i	0.746373	−	0.665528i	$-0.231794\pi$
$20$	1.00000i	0.223607i
$21$	0.307979i	0.0672064i
$22$	−0.335126	−0.0714490
$23$	−2.35690	−0.491447	−0.245723	−	0.969340i	$-0.579025\pi$
$23$	−2.35690	−0.491447	−0.245723	+	0.969340i	$0.579025\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 0.307979i	− 0.0582025i
$29$	−5.91185	−1.09780	−0.548902	−	0.835887i	$-0.684954\pi$
$29$	−5.91185	−1.09780	−0.548902	+	0.835887i	$0.684954\pi$
$30$	1.00000	0.182574
$31$	− 0.0609989i	− 0.0109557i	−0.999985	−	0.00547787i	$-0.998256\pi$
$31$	− 0.0609989i	− 0.0109557i	0.999985	−	0.00547787i	$-0.00174367\pi$
$32$	1.00000i	0.176777i
$33$	0.335126i	0.0583379i
$34$	6.85086i	1.17491i
$35$	0.307979	0.0520579
$36$	−1.00000	−0.166667
$37$	7.07606i	1.16330i	0.813440	+	0.581649i	$0.197592\pi$
$37$	7.07606i	1.16330i	−0.813440	+	0.581649i	$0.802408\pi$
$38$	5.80194	0.941199
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 3.40581i	− 0.531899i	−0.963987	−	0.265949i	$-0.914315\pi$
$41$	− 3.40581i	− 0.531899i	0.963987	−	0.265949i	$-0.0856854\pi$
$42$	−0.307979	−0.0475221
$43$	0.0489173	0.00745982	0.00372991	−	0.999993i	$-0.498813\pi$
$43$	0.0489173	0.00745982	0.00372991	+	0.999993i	$0.498813\pi$
$44$	− 0.335126i	− 0.0505221i
$45$	− 1.00000i	− 0.149071i
$46$	− 2.35690i	− 0.347505i
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	1.00000	0.144338
$49$	6.90515	0.986450
$50$	− 1.00000i	− 0.141421i
$51$	6.85086	0.959312
$52$	0	0
$53$	12.1739	1.67221	0.836107	−	0.548567i	$-0.184826\pi$
$53$	12.1739	1.67221	0.836107	+	0.548567i	$0.184826\pi$
$54$	1.00000i	0.136083i
$55$	0.335126	0.0451883
$56$	0.307979	0.0411554
$57$	− 5.80194i	− 0.768485i
$58$	− 5.91185i	− 0.776264i
$59$	− 13.1347i	− 1.70999i	−0.518638	−	0.854994i	$-0.673561\pi$
$59$	− 13.1347i	− 1.70999i	0.518638	−	0.854994i	$-0.326439\pi$
$60$	1.00000i	0.129099i
$61$	2.14675	0.274863	0.137432	−	0.990511i	$-0.456115\pi$
$61$	2.14675	0.274863	0.137432	+	0.990511i	$0.456115\pi$
$62$	0.0609989	0.00774687
$63$	0.307979i	0.0388016i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−0.335126	−0.0412511
$67$	− 11.9608i	− 1.46124i	−0.682784	−	0.730620i	$-0.739231\pi$
$67$	− 11.9608i	− 1.46124i	0.682784	−	0.730620i	$-0.260769\pi$
$68$	−6.85086	−0.830788
$69$	−2.35690	−0.283737
$70$	0.307979i	0.0368105i
$71$	− 9.56465i	− 1.13511i	−0.823334	−	0.567557i	$-0.807889\pi$
$71$	− 9.56465i	− 1.13511i	0.823334	−	0.567557i	$-0.192111\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.31336i	0.270758i	0.990794	+	0.135379i	$0.0432252\pi$
$73$	2.31336i	0.270758i	−0.990794	+	0.135379i	$0.956775\pi$
$74$	−7.07606	−0.822576
$75$	−1.00000	−0.115470
$76$	5.80194i	0.665528i
$77$	−0.103211	−0.0117620
$78$	0	0
$79$	−0.0760644	−0.00855792	−0.00427896	−	0.999991i	$-0.501362\pi$
$79$	−0.0760644	−0.00855792	−0.00427896	+	0.999991i	$0.501362\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	3.40581	0.376109
$83$	− 3.84117i	− 0.421623i	−0.977527	−	0.210811i	$-0.932389\pi$
$83$	− 3.84117i	− 0.421623i	0.977527	−	0.210811i	$-0.0676106\pi$
$84$	− 0.307979i	− 0.0336032i
$85$	− 6.85086i	− 0.743080i
$86$	0.0489173i	0.00527489i
$87$	−5.91185	−0.633817
$88$	0.335126	0.0357245
$89$	− 4.41119i	− 0.467585i	−0.972286	−	0.233793i	$-0.924886\pi$
$89$	− 4.41119i	− 0.467585i	0.972286	−	0.233793i	$-0.0751137\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	2.35690	0.245723
$93$	− 0.0609989i	− 0.00632529i
$94$	−7.00000	−0.721995
$95$	−5.80194	−0.595266
$96$	1.00000i	0.102062i
$97$	− 6.13169i	− 0.622578i	−0.950315	−	0.311289i	$-0.899239\pi$
$97$	− 6.13169i	− 0.622578i	0.950315	−	0.311289i	$-0.100761\pi$
$98$	6.90515i	0.697525i
$99$	0.335126i	0.0336814i
$100$	1.00000	0.100000
$101$	−1.32975	−0.132315	−0.0661575	−	0.997809i	$-0.521074\pi$
$101$	−1.32975	−0.132315	−0.0661575	+	0.997809i	$0.521074\pi$
$102$	6.85086i	0.678336i
$103$	−8.92154	−0.879066	−0.439533	−	0.898227i	$-0.644856\pi$
$103$	−8.92154	−0.879066	−0.439533	+	0.898227i	$0.644856\pi$
$104$	0	0
$105$	0.307979	0.0300556
$106$	12.1739i	1.18243i
$107$	14.4644	1.39833	0.699164	−	0.714961i	$-0.253556\pi$
$107$	14.4644	1.39833	0.699164	+	0.714961i	$0.253556\pi$
$108$	−1.00000	−0.0962250
$109$	− 7.13706i	− 0.683607i	−0.939772	−	0.341803i	$-0.888962\pi$
$109$	− 7.13706i	− 0.683607i	0.939772	−	0.341803i	$-0.111038\pi$
$110$	0.335126i	0.0319530i
$111$	7.07606i	0.671630i
$112$	0.307979i	0.0291012i
$113$	−16.3424	−1.53737	−0.768683	−	0.639630i	$-0.779088\pi$
$113$	−16.3424	−1.53737	−0.768683	+	0.639630i	$0.779088\pi$
$114$	5.80194	0.543401
$115$	2.35690i	0.219782i
$116$	5.91185	0.548902
$117$	0	0
$118$	13.1347	1.20914
$119$	2.10992i	0.193416i
$120$	−1.00000	−0.0912871
$121$	10.8877	0.989790
$122$	2.14675i	0.194358i
$123$	− 3.40581i	− 0.307092i
$124$	0.0609989i	0.00547787i
$125$	1.00000i	0.0894427i
$126$	−0.307979	−0.0274369
$127$	0.911854	0.0809140	0.0404570	−	0.999181i	$-0.487119\pi$
$127$	0.911854	0.0809140	0.0404570	+	0.999181i	$0.487119\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0.0489173	0.00430693
$130$	0	0
$131$	13.0368	1.13903	0.569517	−	0.821980i	$-0.307130\pi$
$131$	13.0368	1.13903	0.569517	+	0.821980i	$0.307130\pi$
$132$	− 0.335126i	− 0.0291689i
$133$	1.78687	0.154941
$134$	11.9608	1.03325
$135$	− 1.00000i	− 0.0860663i
$136$	− 6.85086i	− 0.587456i
$137$	12.8877i	1.10107i	0.834812	+	0.550535i	$0.185576\pi$
$137$	12.8877i	1.10107i	−0.834812	+	0.550535i	$0.814424\pi$
$138$	− 2.35690i	− 0.200632i
$139$	−2.01208	−0.170663	−0.0853313	−	0.996353i	$-0.527195\pi$
$139$	−2.01208	−0.170663	−0.0853313	+	0.996353i	$0.527195\pi$
$140$	−0.307979	−0.0260289
$141$	7.00000i	0.589506i
$142$	9.56465	0.802647
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.91185i	0.490953i
$146$	−2.31336	−0.191455
$147$	6.90515	0.569527
$148$	− 7.07606i	− 0.581649i
$149$	− 3.44935i	− 0.282582i	−0.989968	−	0.141291i	$-0.954875\pi$
$149$	− 3.44935i	− 0.282582i	0.989968	−	0.141291i	$-0.0451253\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	2.84117i	0.231211i	0.993295	+	0.115605i	$0.0368808\pi$
$151$	2.84117i	0.231211i	−0.993295	+	0.115605i	$0.963119\pi$
$152$	−5.80194	−0.470599
$153$	6.85086	0.553859
$154$	− 0.103211i	− 0.00831702i
$155$	−0.0609989	−0.00489955
$156$	0	0
$157$	16.2228	1.29472	0.647361	−	0.762184i	$-0.275873\pi$
$157$	16.2228	1.29472	0.647361	+	0.762184i	$0.275873\pi$
$158$	− 0.0760644i	− 0.00605136i
$159$	12.1739	0.965453
$160$	1.00000	0.0790569
$161$	− 0.725873i	− 0.0572068i
$162$	1.00000i	0.0785674i
$163$	− 0.0924579i	− 0.00724186i	−0.999993	−	0.00362093i	$-0.998847\pi$
$163$	− 0.0924579i	− 0.00724186i	0.999993	−	0.00362093i	$-0.00115258\pi$
$164$	3.40581i	0.265949i
$165$	0.335126	0.0260895
$166$	3.84117	0.298132
$167$	− 9.64071i	− 0.746021i	−0.927827	−	0.373010i	$-0.878326\pi$
$167$	− 9.64071i	− 0.746021i	0.927827	−	0.373010i	$-0.121674\pi$
$168$	0.307979	0.0237611
$169$	0	0
$170$	6.85086	0.525437
$171$	− 5.80194i	− 0.443685i
$172$	−0.0489173	−0.00372991
$173$	−0.719169	−0.0546774	−0.0273387	−	0.999626i	$-0.508703\pi$
$173$	−0.719169	−0.0546774	−0.0273387	+	0.999626i	$0.508703\pi$
$174$	− 5.91185i	− 0.448176i
$175$	− 0.307979i	− 0.0232810i
$176$	0.335126i	0.0252610i
$177$	− 13.1347i	− 0.987262i
$178$	4.41119	0.330633
$179$	2.01208	0.150390	0.0751950	−	0.997169i	$-0.476042\pi$
$179$	2.01208	0.150390	0.0751950	+	0.997169i	$0.476042\pi$
$180$	1.00000i	0.0745356i
$181$	2.57135	0.191127	0.0955635	−	0.995423i	$-0.469535\pi$
$181$	2.57135	0.191127	0.0955635	+	0.995423i	$0.469535\pi$
$182$	0	0
$183$	2.14675	0.158692
$184$	2.35690i	0.173753i
$185$	7.07606	0.520243
$186$	0.0609989	0.00447266
$187$	2.29590i	0.167893i
$188$	− 7.00000i	− 0.510527i
$189$	0.307979i	0.0224021i
$190$	− 5.80194i	− 0.420917i
$191$	3.80194	0.275099	0.137549	−	0.990495i	$-0.456077\pi$
$191$	3.80194	0.275099	0.137549	+	0.990495i	$0.456077\pi$
$192$	−1.00000	−0.0721688
$193$	− 8.86294i	− 0.637968i	−0.947760	−	0.318984i	$-0.896658\pi$
$193$	− 8.86294i	− 0.637968i	0.947760	−	0.318984i	$-0.103342\pi$
$194$	6.13169	0.440229
$195$	0	0
$196$	−6.90515	−0.493225
$197$	22.6504i	1.61377i	0.590706	+	0.806887i	$0.298849\pi$
$197$	22.6504i	1.61377i	−0.590706	+	0.806887i	$0.701151\pi$
$198$	−0.335126	−0.0238163
$199$	5.46681	0.387532	0.193766	−	0.981048i	$-0.437930\pi$
$199$	5.46681	0.387532	0.193766	+	0.981048i	$0.437930\pi$
$200$	1.00000i	0.0707107i
$201$	− 11.9608i	− 0.843648i
$202$	− 1.32975i	− 0.0935608i
$203$	− 1.82072i	− 0.127790i
$204$	−6.85086	−0.479656
$205$	−3.40581	−0.237872
$206$	− 8.92154i	− 0.621593i
$207$	−2.35690	−0.163816
$208$	0	0
$209$	1.94438	0.134495
$210$	0.307979i	0.0212525i
$211$	11.0422	0.760177	0.380089	−	0.924950i	$-0.375893\pi$
$211$	11.0422	0.760177	0.380089	+	0.924950i	$0.375893\pi$
$212$	−12.1739	−0.836107
$213$	− 9.56465i	− 0.655359i
$214$	14.4644i	0.988767i
$215$	− 0.0489173i	− 0.00333613i
$216$	− 1.00000i	− 0.0680414i
$217$	0.0187864	0.00127530
$218$	7.13706	0.483383
$219$	2.31336i	0.156322i
$220$	−0.335126	−0.0225942
$221$	0	0
$222$	−7.07606	−0.474914
$223$	− 18.3884i	− 1.23138i	−0.787990	−	0.615688i	$-0.788878\pi$
$223$	− 18.3884i	− 1.23138i	0.787990	−	0.615688i	$-0.211122\pi$
$224$	−0.307979	−0.0205777
$225$	−1.00000	−0.0666667
$226$	− 16.3424i	− 1.08708i
$227$	− 12.0073i	− 0.796952i	−0.917179	−	0.398476i	$-0.869539\pi$
$227$	− 12.0073i	− 0.796952i	0.917179	−	0.398476i	$-0.130461\pi$
$228$	5.80194i	0.384243i
$229$	25.4601i	1.68245i	0.540684	+	0.841226i	$0.318165\pi$
$229$	25.4601i	1.68245i	−0.540684	+	0.841226i	$0.681835\pi$
$230$	−2.35690	−0.155409
$231$	−0.103211	−0.00679082
$232$	5.91185i	0.388132i
$233$	2.10992	0.138225	0.0691126	−	0.997609i	$-0.477983\pi$
$233$	2.10992	0.138225	0.0691126	+	0.997609i	$0.477983\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	13.1347i	0.854994i
$237$	−0.0760644	−0.00494091
$238$	−2.10992	−0.136766
$239$	− 3.95838i	− 0.256046i	−0.991771	−	0.128023i	$-0.959137\pi$
$239$	− 3.95838i	− 0.256046i	0.991771	−	0.128023i	$-0.0408632\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 23.1564i	− 1.49164i	−0.666149	−	0.745819i	$-0.732058\pi$
$241$	− 23.1564i	− 1.49164i	0.666149	−	0.745819i	$-0.267942\pi$
$242$	10.8877i	0.699887i
$243$	1.00000	0.0641500
$244$	−2.14675	−0.137432
$245$	− 6.90515i	− 0.441154i
$246$	3.40581	0.217147
$247$	0	0
$248$	−0.0609989	−0.00387344
$249$	− 3.84117i	− 0.243424i
$250$	−1.00000	−0.0632456
$251$	5.54048	0.349712	0.174856	−	0.984594i	$-0.444054\pi$
$251$	5.54048	0.349712	0.174856	+	0.984594i	$0.444054\pi$
$252$	− 0.307979i	− 0.0194008i
$253$	− 0.789856i	− 0.0496578i
$254$	0.911854i	0.0572148i
$255$	− 6.85086i	− 0.429017i
$256$	1.00000	0.0625000
$257$	−6.60819	−0.412207	−0.206104	−	0.978530i	$-0.566078\pi$
$257$	−6.60819	−0.412207	−0.206104	+	0.978530i	$0.566078\pi$
$258$	0.0489173i	0.00304546i
$259$	−2.17928	−0.135414
$260$	0	0
$261$	−5.91185	−0.365935
$262$	13.0368i	0.805418i
$263$	−0.660563	−0.0407320	−0.0203660	−	0.999793i	$-0.506483\pi$
$263$	−0.660563	−0.0407320	−0.0203660	+	0.999793i	$0.506483\pi$
$264$	0.335126	0.0206256
$265$	− 12.1739i	− 0.747837i
$266$	1.78687i	0.109560i
$267$	− 4.41119i	− 0.269960i
$268$	11.9608i	0.730620i
$269$	−26.7928	−1.63359	−0.816794	−	0.576929i	$-0.804251\pi$
$269$	−26.7928	−1.63359	−0.816794	+	0.576929i	$0.804251\pi$
$270$	1.00000	0.0608581
$271$	14.0151i	0.851355i	0.904875	+	0.425677i	$0.139964\pi$
$271$	14.0151i	0.851355i	−0.904875	+	0.425677i	$0.860036\pi$
$272$	6.85086	0.415394
$273$	0	0
$274$	−12.8877	−0.778574
$275$	− 0.335126i	− 0.0202088i
$276$	2.35690	0.141868
$277$	23.4252	1.40748	0.703742	−	0.710456i	$-0.251511\pi$
$277$	23.4252	1.40748	0.703742	+	0.710456i	$0.251511\pi$
$278$	− 2.01208i	− 0.120677i
$279$	− 0.0609989i	− 0.00365191i
$280$	− 0.307979i	− 0.0184052i
$281$	8.40044i	0.501128i	0.968100	+	0.250564i	$0.0806161\pi$
$281$	8.40044i	0.501128i	−0.968100	+	0.250564i	$0.919384\pi$
$282$	−7.00000	−0.416844
$283$	31.6142	1.87927	0.939633	−	0.342183i	$-0.111166\pi$
$283$	31.6142	1.87927	0.939633	+	0.342183i	$0.111166\pi$
$284$	9.56465i	0.567557i
$285$	−5.80194	−0.343677
$286$	0	0
$287$	1.04892	0.0619156
$288$	1.00000i	0.0589256i
$289$	29.9342	1.76084
$290$	−5.91185	−0.347156
$291$	− 6.13169i	− 0.359446i
$292$	− 2.31336i	− 0.135379i
$293$	− 17.5254i	− 1.02385i	−0.859031	−	0.511923i	$-0.828933\pi$
$293$	− 17.5254i	− 1.02385i	0.859031	−	0.511923i	$-0.171067\pi$
$294$	6.90515i	0.402716i
$295$	−13.1347	−0.764730
$296$	7.07606	0.411288
$297$	0.335126i	0.0194460i
$298$	3.44935	0.199816
$299$	0	0
$300$	1.00000	0.0577350
$301$	0.0150655i	0 0.000868360i
$302$	−2.84117	−0.163491
$303$	−1.32975	−0.0763921
$304$	− 5.80194i	− 0.332764i
$305$	− 2.14675i	− 0.122923i
$306$	6.85086i	0.391637i
$307$	20.6732i	1.17988i	0.807446	+	0.589942i	$0.200849\pi$
$307$	20.6732i	1.17988i	−0.807446	+	0.589942i	$0.799151\pi$
$308$	0.103211	0.00588102
$309$	−8.92154	−0.507529
$310$	− 0.0609989i	− 0.00346451i
$311$	22.0248	1.24891	0.624455	−	0.781061i	$-0.285321\pi$
$311$	22.0248	1.24891	0.624455	+	0.781061i	$0.285321\pi$
$312$	0	0
$313$	21.7192	1.22764	0.613820	−	0.789446i	$-0.289632\pi$
$313$	21.7192	1.22764	0.613820	+	0.789446i	$0.289632\pi$
$314$	16.2228i	0.915506i
$315$	0.307979	0.0173526
$316$	0.0760644	0.00427896
$317$	16.4644i	0.924734i	0.886689	+	0.462367i	$0.153000\pi$
$317$	16.4644i	0.924734i	−0.886689	+	0.462367i	$0.847000\pi$
$318$	12.1739i	0.682678i
$319$	− 1.98121i	− 0.110927i
$320$	1.00000i	0.0559017i
$321$	14.4644	0.807325
$322$	0.725873	0.0404513
$323$	− 39.7482i	− 2.21165i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	0.0924579	0.00512077
$327$	− 7.13706i	− 0.394681i
$328$	−3.40581	−0.188055
$329$	−2.15585	−0.118856
$330$	0.335126i	0.0184481i
$331$	− 26.4373i	− 1.45312i	−0.687101	−	0.726562i	$-0.741117\pi$
$331$	− 26.4373i	− 1.45312i	0.687101	−	0.726562i	$-0.258883\pi$
$332$	3.84117i	0.210811i
$333$	7.07606i	0.387766i
$334$	9.64071	0.527516
$335$	−11.9608	−0.653487
$336$	0.307979i	0.0168016i
$337$	−27.4698	−1.49638	−0.748188	−	0.663487i	$-0.769076\pi$
$337$	−27.4698	−1.49638	−0.748188	+	0.663487i	$0.769076\pi$
$338$	0	0
$339$	−16.3424	−0.887598
$340$	6.85086i	0.371540i
$341$	0.0204423	0.00110701
$342$	5.80194	0.313733
$343$	4.28249i	0.231233i
$344$	− 0.0489173i	− 0.00263745i
$345$	2.35690i	0.126891i
$346$	− 0.719169i	− 0.0386627i
$347$	−33.1323	−1.77863	−0.889317	−	0.457291i	$-0.848820\pi$
$347$	−33.1323	−1.77863	−0.889317	+	0.457291i	$0.848820\pi$
$348$	5.91185	0.316909
$349$	26.3739i	1.41176i	0.708331	+	0.705881i	$0.249449\pi$
$349$	26.3739i	1.41176i	−0.708331	+	0.705881i	$0.750551\pi$
$350$	0.307979	0.0164621
$351$	0	0
$352$	−0.335126	−0.0178623
$353$	− 21.0291i	− 1.11926i	−0.828741	−	0.559632i	$-0.810942\pi$
$353$	− 21.0291i	− 1.11926i	0.828741	−	0.559632i	$-0.189058\pi$
$354$	13.1347	0.698100
$355$	−9.56465	−0.507639
$356$	4.41119i	0.233793i
$357$	2.10992i	0.111669i
$358$	2.01208i	0.106342i
$359$	− 30.8877i	− 1.63019i	−0.579327	−	0.815095i	$-0.696685\pi$
$359$	− 30.8877i	− 1.63019i	0.579327	−	0.815095i	$-0.303315\pi$
$360$	−1.00000	−0.0527046
$361$	−14.6625	−0.771710
$362$	2.57135i	0.135147i
$363$	10.8877	0.571456
$364$	0	0
$365$	2.31336	0.121087
$366$	2.14675i	0.112213i
$367$	11.9584	0.624222	0.312111	−	0.950046i	$-0.398964\pi$
$367$	11.9584	0.624222	0.312111	+	0.950046i	$0.398964\pi$
$368$	−2.35690	−0.122862
$369$	− 3.40581i	− 0.177300i
$370$	7.07606i	0.367867i
$371$	3.74930i	0.194654i
$372$	0.0609989i	0.00316265i
$373$	25.0315	1.29608	0.648040	−	0.761606i	$-0.275589\pi$
$373$	25.0315	1.29608	0.648040	+	0.761606i	$0.275589\pi$
$374$	−2.29590	−0.118718
$375$	1.00000i	0.0516398i
$376$	7.00000	0.360997
$377$	0	0
$378$	−0.307979	−0.0158407
$379$	34.4989i	1.77209i	0.463602	+	0.886044i	$0.346557\pi$
$379$	34.4989i	1.77209i	−0.463602	+	0.886044i	$0.653443\pi$
$380$	5.80194	0.297633
$381$	0.911854	0.0467157
$382$	3.80194i	0.194524i
$383$	− 10.4644i	− 0.534707i	−0.963599	−	0.267353i	$-0.913851\pi$
$383$	− 10.4644i	− 0.534707i	0.963599	−	0.267353i	$-0.0861491\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	0.103211i	0.00526014i
$386$	8.86294	0.451112
$387$	0.0489173	0.00248661
$388$	6.13169i	0.311289i
$389$	−25.2218	−1.27879	−0.639397	−	0.768877i	$-0.720816\pi$
$389$	−25.2218	−1.27879	−0.639397	+	0.768877i	$0.720816\pi$
$390$	0	0
$391$	−16.1468	−0.816576
$392$	− 6.90515i	− 0.348763i
$393$	13.0368	0.657621
$394$	−22.6504	−1.14111
$395$	0.0760644i	0.00382722i
$396$	− 0.335126i	− 0.0168407i
$397$	21.3860i	1.07333i	0.843795	+	0.536665i	$0.180316\pi$
$397$	21.3860i	1.07333i	−0.843795	+	0.536665i	$0.819684\pi$
$398$	5.46681i	0.274027i
$399$	1.78687	0.0894555
$400$	−1.00000	−0.0500000
$401$	− 11.8062i	− 0.589576i	−0.955563	−	0.294788i	$-0.904751\pi$
$401$	− 11.8062i	− 0.589576i	0.955563	−	0.294788i	$-0.0952490\pi$
$402$	11.9608	0.596549
$403$	0	0
$404$	1.32975	0.0661575
$405$	− 1.00000i	− 0.0496904i
$406$	1.82072	0.0903610
$407$	−2.37137	−0.117544
$408$	− 6.85086i	− 0.339168i
$409$	5.65950i	0.279844i	0.990163	+	0.139922i	$0.0446852\pi$
$409$	5.65950i	0.279844i	−0.990163	+	0.139922i	$0.955315\pi$
$410$	− 3.40581i	− 0.168201i
$411$	12.8877i	0.635703i
$412$	8.92154	0.439533
$413$	4.04520	0.199051
$414$	− 2.35690i	− 0.115835i
$415$	−3.84117	−0.188555
$416$	0	0
$417$	−2.01208	−0.0985321
$418$	1.94438i	0.0951026i
$419$	17.0084	0.830913	0.415456	−	0.909613i	$-0.363622\pi$
$419$	17.0084	0.830913	0.415456	+	0.909613i	$0.363622\pi$
$420$	−0.307979	−0.0150278
$421$	10.2851i	0.501267i	0.968082	+	0.250634i	$0.0806389\pi$
$421$	10.2851i	0.501267i	−0.968082	+	0.250634i	$0.919361\pi$
$422$	11.0422i	0.537526i
$423$	7.00000i	0.340352i
$424$	− 12.1739i	− 0.591217i
$425$	−6.85086	−0.332315
$426$	9.56465	0.463409
$427$	0.661154i	0.0319955i
$428$	−14.4644	−0.699164
$429$	0	0
$430$	0.0489173	0.00235900
$431$	35.9657i	1.73241i	0.499693	+	0.866203i	$0.333446\pi$
$431$	35.9657i	1.73241i	−0.499693	+	0.866203i	$0.666554\pi$
$432$	1.00000	0.0481125
$433$	−7.63342	−0.366839	−0.183419	−	0.983035i	$-0.558717\pi$
$433$	−7.63342	−0.366839	−0.183419	+	0.983035i	$0.558717\pi$
$434$	0.0187864i	0 0.000901774i
$435$	5.91185i	0.283452i
$436$	7.13706i	0.341803i
$437$	13.6746i	0.654143i
$438$	−2.31336	−0.110536
$439$	34.4373	1.64360	0.821801	−	0.569775i	$-0.192970\pi$
$439$	34.4373	1.64360	0.821801	+	0.569775i	$0.192970\pi$
$440$	− 0.335126i	− 0.0159765i
$441$	6.90515	0.328817
$442$	0	0
$443$	33.6431	1.59843	0.799216	−	0.601044i	$-0.205248\pi$
$443$	33.6431	1.59843	0.799216	+	0.601044i	$0.205248\pi$
$444$	− 7.07606i	− 0.335815i
$445$	−4.41119	−0.209110
$446$	18.3884	0.870714
$447$	− 3.44935i	− 0.163149i
$448$	− 0.307979i	− 0.0145506i
$449$	− 27.9729i	− 1.32012i	−0.751213	−	0.660060i	$-0.770531\pi$
$449$	− 27.9729i	− 1.32012i	0.751213	−	0.660060i	$-0.229469\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	1.14138	0.0537453
$452$	16.3424	0.768683
$453$	2.84117i	0.133490i
$454$	12.0073	0.563530
$455$	0	0
$456$	−5.80194	−0.271701
$457$	6.18060i	0.289116i	0.989496	+	0.144558i	$0.0461761\pi$
$457$	6.18060i	0.289116i	−0.989496	+	0.144558i	$0.953824\pi$
$458$	−25.4601	−1.18967
$459$	6.85086	0.319771
$460$	− 2.35690i	− 0.109891i
$461$	− 38.1062i	− 1.77478i	−0.461017	−	0.887391i	$-0.652515\pi$
$461$	− 38.1062i	− 1.77478i	0.461017	−	0.887391i	$-0.347485\pi$
$462$	− 0.103211i	− 0.00480183i
$463$	− 2.06829i	− 0.0961218i	−0.998844	−	0.0480609i	$-0.984696\pi$
$463$	− 2.06829i	− 0.0961218i	0.998844	−	0.0480609i	$-0.0153042\pi$
$464$	−5.91185	−0.274451
$465$	−0.0609989	−0.00282876
$466$	2.10992i	0.0977400i
$467$	4.27114	0.197645	0.0988225	−	0.995105i	$-0.468492\pi$
$467$	4.27114	0.197645	0.0988225	+	0.995105i	$0.468492\pi$
$468$	0	0
$469$	3.68366	0.170096
$470$	7.00000i	0.322886i
$471$	16.2228	0.747508
$472$	−13.1347	−0.604572
$473$	0.0163935i	0 0.000753772i
$474$	− 0.0760644i	− 0.00349375i
$475$	5.80194i	0.266211i
$476$	− 2.10992i	− 0.0967079i
$477$	12.1739	0.557405
$478$	3.95838	0.181052
$479$	− 38.7560i	− 1.77081i	−0.464823	−	0.885404i	$-0.653882\pi$
$479$	− 38.7560i	− 1.77081i	0.464823	−	0.885404i	$-0.346118\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	23.1564	1.05475
$483$	− 0.725873i	− 0.0330284i
$484$	−10.8877	−0.494895
$485$	−6.13169	−0.278426
$486$	1.00000i	0.0453609i
$487$	33.4620i	1.51631i	0.652075	+	0.758155i	$0.273899\pi$
$487$	33.4620i	1.51631i	−0.652075	+	0.758155i	$0.726101\pi$
$488$	− 2.14675i	− 0.0971789i
$489$	− 0.0924579i	− 0.00418109i
$490$	6.90515	0.311943
$491$	−35.1487	−1.58624	−0.793119	−	0.609067i	$-0.791544\pi$
$491$	−35.1487	−1.58624	−0.793119	+	0.609067i	$0.791544\pi$
$492$	3.40581i	0.153546i
$493$	−40.5013	−1.82408
$494$	0	0
$495$	0.335126	0.0150628
$496$	− 0.0609989i	− 0.00273893i
$497$	2.94571	0.132133
$498$	3.84117	0.172127
$499$	9.61463i	0.430410i	0.976569	+	0.215205i	$0.0690420\pi$
$499$	9.61463i	0.430410i	−0.976569	+	0.215205i	$0.930958\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 9.64071i	− 0.430715i
$502$	5.54048i	0.247284i
$503$	18.5265	0.826055	0.413028	−	0.910719i	$-0.364471\pi$
$503$	18.5265	0.826055	0.413028	+	0.910719i	$0.364471\pi$
$504$	0.307979	0.0137185
$505$	1.32975i	0.0591730i
$506$	0.789856	0.0351134
$507$	0	0
$508$	−0.911854	−0.0404570
$509$	− 5.43296i	− 0.240812i	−0.992725	−	0.120406i	$-0.961580\pi$
$509$	− 5.43296i	− 0.240812i	0.992725	−	0.120406i	$-0.0384196\pi$
$510$	6.85086	0.303361
$511$	−0.712464	−0.0315175
$512$	1.00000i	0.0441942i
$513$	− 5.80194i	− 0.256162i
$514$	− 6.60819i	− 0.291475i
$515$	8.92154i	0.393130i
$516$	−0.0489173	−0.00215347
$517$	−2.34588	−0.103172
$518$	− 2.17928i	− 0.0957519i
$519$	−0.719169	−0.0315680
$520$	0	0
$521$	−38.8001	−1.69986	−0.849932	−	0.526892i	$-0.823357\pi$
$521$	−38.8001	−1.69986	−0.849932	+	0.526892i	$0.823357\pi$
$522$	− 5.91185i	− 0.258755i
$523$	32.0629	1.40201	0.701007	−	0.713155i	$-0.252734\pi$
$523$	32.0629	1.40201	0.701007	+	0.713155i	$0.252734\pi$
$524$	−13.0368	−0.569517
$525$	− 0.307979i	− 0.0134413i
$526$	− 0.660563i	− 0.0288019i
$527$	− 0.417895i	− 0.0182038i
$528$	0.335126i	0.0145845i
$529$	−17.4450	−0.758480
$530$	12.1739	0.528800
$531$	− 13.1347i	− 0.569996i
$532$	−1.78687	−0.0774707
$533$	0	0
$534$	4.41119	0.190891
$535$	− 14.4644i	− 0.625351i
$536$	−11.9608	−0.516627
$537$	2.01208	0.0868277
$538$	− 26.7928i	− 1.15512i
$539$	2.31409i	0.0996750i
$540$	1.00000i	0.0430331i
$541$	31.8334i	1.36862i	0.729189	+	0.684312i	$0.239897\pi$
$541$	31.8334i	1.36862i	−0.729189	+	0.684312i	$0.760103\pi$
$542$	−14.0151	−0.601999
$543$	2.57135	0.110347
$544$	6.85086i	0.293728i
$545$	−7.13706	−0.305718
$546$	0	0
$547$	−44.4010	−1.89845	−0.949225	−	0.314597i	$-0.898131\pi$
$547$	−44.4010	−1.89845	−0.949225	+	0.314597i	$0.898131\pi$
$548$	− 12.8877i	− 0.550535i
$549$	2.14675	0.0916211
$550$	0.335126	0.0142898
$551$	34.3002i	1.46124i
$552$	2.35690i	0.100316i
$553$	− 0.0234262i	0 0.000996184i
$554$	23.4252i	0.995241i
$555$	7.07606	0.300362
$556$	2.01208	0.0853313
$557$	− 7.73423i	− 0.327710i	−0.986484	−	0.163855i	$-0.947607\pi$
$557$	− 7.73423i	− 0.327710i	0.986484	−	0.163855i	$-0.0523929\pi$
$558$	0.0609989	0.00258229
$559$	0	0
$560$	0.307979	0.0130145
$561$	2.29590i	0.0969328i
$562$	−8.40044	−0.354351
$563$	−44.5803	−1.87884	−0.939418	−	0.342774i	$-0.888633\pi$
$563$	−44.5803	−1.87884	−0.939418	+	0.342774i	$0.888633\pi$
$564$	− 7.00000i	− 0.294753i
$565$	16.3424i	0.687531i
$566$	31.6142i	1.32884i
$567$	0.307979i	0.0129339i
$568$	−9.56465	−0.401324
$569$	−24.9065	−1.04413	−0.522067	−	0.852905i	$-0.674839\pi$
$569$	−24.9065	−1.04413	−0.522067	+	0.852905i	$0.674839\pi$
$570$	− 5.80194i	− 0.243016i
$571$	0.873690	0.0365628	0.0182814	−	0.999833i	$-0.494181\pi$
$571$	0.873690	0.0365628	0.0182814	+	0.999833i	$0.494181\pi$
$572$	0	0
$573$	3.80194	0.158828
$574$	1.04892i	0.0437810i
$575$	2.35690	0.0982894
$576$	−1.00000	−0.0416667
$577$	20.0871i	0.836236i	0.908393	+	0.418118i	$0.137310\pi$
$577$	20.0871i	0.836236i	−0.908393	+	0.418118i	$0.862690\pi$
$578$	29.9342i	1.24510i
$579$	− 8.86294i	− 0.368331i
$580$	− 5.91185i	− 0.245476i
$581$	1.18300	0.0490790
$582$	6.13169	0.254167
$583$	4.07979i	0.168967i
$584$	2.31336	0.0957273
$585$	0	0
$586$	17.5254	0.723968
$587$	16.0901i	0.664108i	0.943260	+	0.332054i	$0.107742\pi$
$587$	16.0901i	0.664108i	−0.943260	+	0.332054i	$0.892258\pi$
$588$	−6.90515	−0.284764
$589$	−0.353912	−0.0145827
$590$	− 13.1347i	− 0.540746i
$591$	22.6504i	0.931713i
$592$	7.07606i	0.290824i
$593$	4.85517i	0.199378i	0.995019	+	0.0996889i	$0.0317848\pi$
$593$	4.85517i	0.199378i	−0.995019	+	0.0996889i	$0.968215\pi$
$594$	−0.335126	−0.0137504
$595$	2.10992	0.0864981
$596$	3.44935i	0.141291i
$597$	5.46681	0.223742
$598$	0	0
$599$	−34.4319	−1.40685	−0.703425	−	0.710770i	$-0.748347\pi$
$599$	−34.4319	−1.40685	−0.703425	+	0.710770i	$0.748347\pi$
$600$	1.00000i	0.0408248i
$601$	−24.5244	−1.00037	−0.500185	−	0.865919i	$-0.666735\pi$
$601$	−24.5244	−1.00037	−0.500185	+	0.865919i	$0.666735\pi$
$602$	−0.0150655	−0.000614024 0
$603$	− 11.9608i	− 0.487080i
$604$	− 2.84117i	− 0.115605i
$605$	− 10.8877i	− 0.442648i
$606$	− 1.32975i	− 0.0540174i
$607$	17.6420	0.716068	0.358034	−	0.933708i	$-0.383447\pi$
$607$	17.6420	0.716068	0.358034	+	0.933708i	$0.383447\pi$
$608$	5.80194	0.235300
$609$	− 1.82072i	− 0.0737795i
$610$	2.14675	0.0869194
$611$	0	0
$612$	−6.85086	−0.276929
$613$	− 44.5018i	− 1.79741i	−0.438551	−	0.898706i	$-0.644508\pi$
$613$	− 44.5018i	− 1.79741i	0.438551	−	0.898706i	$-0.355492\pi$
$614$	−20.6732	−0.834304
$615$	−3.40581	−0.137336
$616$	0.103211i	0.00415851i
$617$	− 32.1618i	− 1.29479i	−0.762156	−	0.647393i	$-0.775859\pi$
$617$	− 32.1618i	− 1.29479i	0.762156	−	0.647393i	$-0.224141\pi$
$618$	− 8.92154i	− 0.358877i
$619$	24.8340i	0.998162i	0.866555	+	0.499081i	$0.166329\pi$
$619$	24.8340i	0.998162i	−0.866555	+	0.499081i	$0.833671\pi$
$620$	0.0609989	0.00244978
$621$	−2.35690	−0.0945790
$622$	22.0248i	0.883112i
$623$	1.35855	0.0544292
$624$	0	0
$625$	1.00000	0.0400000
$626$	21.7192i	0.868073i
$627$	1.94438	0.0776510
$628$	−16.2228	−0.647361
$629$	48.4771i	1.93291i
$630$	0.307979i	0.0122702i
$631$	10.5418i	0.419663i	0.977738	+	0.209831i	$0.0672915\pi$
$631$	10.5418i	0.419663i	−0.977738	+	0.209831i	$0.932708\pi$
$632$	0.0760644i	0.00302568i
$633$	11.0422	0.438889
$634$	−16.4644	−0.653886
$635$	− 0.911854i	− 0.0361858i
$636$	−12.1739	−0.482726
$637$	0	0
$638$	1.98121	0.0784370
$639$	− 9.56465i	− 0.378372i
$640$	−1.00000	−0.0395285
$641$	−33.5663	−1.32579	−0.662895	−	0.748713i	$-0.730672\pi$
$641$	−33.5663	−1.32579	−0.662895	+	0.748713i	$0.730672\pi$
$642$	14.4644i	0.570865i
$643$	− 8.24160i	− 0.325017i	−0.986707	−	0.162509i	$-0.948041\pi$
$643$	− 8.24160i	− 0.325017i	0.986707	−	0.162509i	$-0.0519585\pi$
$644$	0.725873i	0.0286034i
$645$	− 0.0489173i	− 0.00192612i
$646$	39.7482	1.56387
$647$	3.66056	0.143912	0.0719558	−	0.997408i	$-0.477076\pi$
$647$	3.66056	0.143912	0.0719558	+	0.997408i	$0.477076\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	4.40176	0.172784
$650$	0	0
$651$	0.0187864	0.000736295 0
$652$	0.0924579i	0.00362093i
$653$	−44.6064	−1.74558	−0.872791	−	0.488093i	$-0.837693\pi$
$653$	−44.6064	−1.74558	−0.872791	+	0.488093i	$0.837693\pi$
$654$	7.13706	0.279081
$655$	− 13.0368i	− 0.509391i
$656$	− 3.40581i	− 0.132975i
$657$	2.31336i	0.0902526i
$658$	− 2.15585i	− 0.0840438i
$659$	−2.51035	−0.0977895	−0.0488947	−	0.998804i	$-0.515570\pi$
$659$	−2.51035	−0.0977895	−0.0488947	+	0.998804i	$0.515570\pi$
$660$	−0.335126	−0.0130447
$661$	40.3096i	1.56786i	0.620847	+	0.783932i	$0.286789\pi$
$661$	40.3096i	1.56786i	−0.620847	+	0.783932i	$0.713211\pi$
$662$	26.4373	1.02751
$663$	0	0
$664$	−3.84117	−0.149066
$665$	− 1.78687i	− 0.0692919i
$666$	−7.07606	−0.274192
$667$	13.9336	0.539512
$668$	9.64071i	0.373010i
$669$	− 18.3884i	− 0.710935i
$670$	− 11.9608i	− 0.462085i
$671$	0.719432i	0.0277733i
$672$	−0.307979	−0.0118805
$673$	−42.6340	−1.64342	−0.821710	−	0.569906i	$-0.806980\pi$
$673$	−42.6340	−1.64342	−0.821710	+	0.569906i	$0.806980\pi$
$674$	− 27.4698i	− 1.05810i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	18.5254	0.711990	0.355995	−	0.934488i	$-0.384142\pi$
$677$	18.5254	0.711990	0.355995	+	0.934488i	$0.384142\pi$
$678$	− 16.3424i	− 0.627627i
$679$	1.88843	0.0724712
$680$	−6.85086	−0.262718
$681$	− 12.0073i	− 0.460121i
$682$	0.0204423i	0 0.000782776i
$683$	48.1769i	1.84344i	0.387859	+	0.921719i	$0.373215\pi$
$683$	48.1769i	1.84344i	−0.387859	+	0.921719i	$0.626785\pi$
$684$	5.80194i	0.221843i
$685$	12.8877	0.492413
$686$	−4.28249	−0.163506
$687$	25.4601i	0.971364i
$688$	0.0489173	0.00186496
$689$	0	0
$690$	−2.35690	−0.0897255
$691$	− 37.0084i	− 1.40786i	−0.710267	−	0.703932i	$-0.751426\pi$
$691$	− 37.0084i	− 1.40786i	0.710267	−	0.703932i	$-0.248574\pi$
$692$	0.719169	0.0273387
$693$	−0.103211	−0.00392068
$694$	− 33.1323i	− 1.25768i
$695$	2.01208i	0.0763226i
$696$	5.91185i	0.224088i
$697$	− 23.3327i	− 0.883790i
$698$	−26.3739	−0.998266
$699$	2.10992	0.0798044
$700$	0.307979i	0.0116405i
$701$	−9.25188	−0.349439	−0.174719	−	0.984618i	$-0.555902\pi$
$701$	−9.25188	−0.349439	−0.174719	+	0.984618i	$0.555902\pi$
$702$	0	0
$703$	41.0549	1.54841
$704$	− 0.335126i	− 0.0126305i
$705$	7.00000	0.263635
$706$	21.0291	0.791439
$707$	− 0.409534i	− 0.0154021i
$708$	13.1347i	0.493631i
$709$	− 15.0573i	− 0.565488i	−0.959195	−	0.282744i	$-0.908755\pi$
$709$	− 15.0573i	− 0.565488i	0.959195	−	0.282744i	$-0.0912447\pi$
$710$	− 9.56465i	− 0.358955i
$711$	−0.0760644	−0.00285264
$712$	−4.41119	−0.165316
$713$	0.143768i	0.00538416i
$714$	−2.10992	−0.0789616
$715$	0	0
$716$	−2.01208	−0.0751950
$717$	− 3.95838i	− 0.147828i
$718$	30.8877	1.15272
$719$	−24.8364	−0.926241	−0.463120	−	0.886295i	$-0.653270\pi$
$719$	−24.8364	−0.926241	−0.463120	+	0.886295i	$0.653270\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 2.74764i	− 0.102328i
$722$	− 14.6625i	− 0.545681i
$723$	− 23.1564i	− 0.861197i
$724$	−2.57135	−0.0955635
$725$	5.91185	0.219561
$726$	10.8877i	0.404080i
$727$	24.7590	0.918260	0.459130	−	0.888369i	$-0.348161\pi$
$727$	24.7590	0.918260	0.459130	+	0.888369i	$0.348161\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.31336i	0.0856211i
$731$	0.335126	0.0123951
$732$	−2.14675	−0.0793462
$733$	15.4359i	0.570140i	0.958507	+	0.285070i	$0.0920168\pi$
$733$	15.4359i	0.570140i	−0.958507	+	0.285070i	$0.907983\pi$
$734$	11.9584i	0.441392i
$735$	− 6.90515i	− 0.254700i
$736$	− 2.35690i	− 0.0868763i
$737$	4.00836	0.147650
$738$	3.40581	0.125370
$739$	− 20.3327i	− 0.747952i	−0.927438	−	0.373976i	$-0.877994\pi$
$739$	− 20.3327i	− 0.747952i	0.927438	−	0.373976i	$-0.122006\pi$
$740$	−7.07606	−0.260121
$741$	0	0
$742$	−3.74930	−0.137641
$743$	− 17.8243i	− 0.653910i	−0.945040	−	0.326955i	$-0.893977\pi$
$743$	− 17.8243i	− 0.653910i	0.945040	−	0.326955i	$-0.106023\pi$
$744$	−0.0609989	−0.00223633
$745$	−3.44935	−0.126375
$746$	25.0315i	0.916467i
$747$	− 3.84117i	− 0.140541i
$748$	− 2.29590i	− 0.0839463i
$749$	4.45473i	0.162772i
$750$	−1.00000	−0.0365148
$751$	−48.0616	−1.75379	−0.876896	−	0.480680i	$-0.840390\pi$
$751$	−48.0616	−1.75379	−0.876896	+	0.480680i	$0.840390\pi$
$752$	7.00000i	0.255264i
$753$	5.54048	0.201906
$754$	0	0
$755$	2.84117	0.103401
$756$	− 0.307979i	− 0.0112011i
$757$	32.9748	1.19849	0.599244	−	0.800566i	$-0.295468\pi$
$757$	32.9748	1.19849	0.599244	+	0.800566i	$0.295468\pi$
$758$	−34.4989	−1.25306
$759$	− 0.789856i	− 0.0286700i
$760$	5.80194i	0.210458i
$761$	11.9946i	0.434805i	0.976082	+	0.217402i	$0.0697584\pi$
$761$	11.9946i	0.434805i	−0.976082	+	0.217402i	$0.930242\pi$
$762$	0.911854i	0.0330330i
$763$	2.19806	0.0795752
$764$	−3.80194	−0.137549
$765$	− 6.85086i	− 0.247693i
$766$	10.4644	0.378095
$767$	0	0
$768$	1.00000	0.0360844
$769$	18.5036i	0.667259i	0.942704	+	0.333629i	$0.108273\pi$
$769$	18.5036i	0.667259i	−0.942704	+	0.333629i	$0.891727\pi$
$770$	−0.103211	−0.00371948
$771$	−6.60819	−0.237988
$772$	8.86294i	0.318984i
$773$	23.4946i	0.845040i	0.906354	+	0.422520i	$0.138854\pi$
$773$	23.4946i	0.845040i	−0.906354	+	0.422520i	$0.861146\pi$
$774$	0.0489173i	0.00175830i
$775$	0.0609989i	0.00219115i
$776$	−6.13169	−0.220115
$777$	−2.17928	−0.0781811
$778$	− 25.2218i	− 0.904244i
$779$	−19.7603	−0.707987
$780$	0	0
$781$	3.20536	0.114697
$782$	− 16.1468i	− 0.577407i
$783$	−5.91185	−0.211272
$784$	6.90515	0.246612
$785$	− 16.2228i	− 0.579017i
$786$	13.0368i	0.465009i
$787$	7.23682i	0.257965i	0.991647	+	0.128982i	$0.0411711\pi$
$787$	7.23682i	0.257965i	−0.991647	+	0.128982i	$0.958829\pi$
$788$	− 22.6504i	− 0.806887i
$789$	−0.660563	−0.0235166
$790$	−0.0760644	−0.00270625
$791$	− 5.03311i	− 0.178957i
$792$	0.335126	0.0119082
$793$	0	0
$794$	−21.3860	−0.758959
$795$	− 12.1739i	− 0.431764i
$796$	−5.46681	−0.193766
$797$	46.7079	1.65448	0.827240	−	0.561849i	$-0.189910\pi$
$797$	46.7079	1.65448	0.827240	+	0.561849i	$0.189910\pi$
$798$	1.78687i	0.0632546i
$799$	47.9560i	1.69656i
$800$	− 1.00000i	− 0.0353553i
$801$	− 4.41119i	− 0.155862i
$802$	11.8062	0.416893
$803$	−0.775265	−0.0273585
$804$	11.9608i	0.421824i
$805$	−0.725873	−0.0255837
$806$	0	0
$807$	−26.7928	−0.943153
$808$	1.32975i	0.0467804i
$809$	−9.95838	−0.350118	−0.175059	−	0.984558i	$-0.556012\pi$
$809$	−9.95838	−0.350118	−0.175059	+	0.984558i	$0.556012\pi$
$810$	1.00000	0.0351364
$811$	36.2452i	1.27274i	0.771384	+	0.636370i	$0.219565\pi$
$811$	36.2452i	1.27274i	−0.771384	+	0.636370i	$0.780435\pi$
$812$	1.82072i	0.0638949i
$813$	14.0151i	0.491530i
$814$	− 2.37137i	− 0.0831165i
$815$	−0.0924579	−0.00323866
$816$	6.85086	0.239828
$817$	− 0.283815i	− 0.00992944i
$818$	−5.65950	−0.197880
$819$	0	0
$820$	3.40581	0.118936
$821$	1.37867i	0.0481158i	0.999711	+	0.0240579i	$0.00765860\pi$
$821$	1.37867i	0.0481158i	−0.999711	+	0.0240579i	$0.992341\pi$
$822$	−12.8877	−0.449510
$823$	23.8713	0.832101	0.416051	−	0.909341i	$-0.363414\pi$
$823$	23.8713	0.832101	0.416051	+	0.909341i	$0.363414\pi$
$824$	8.92154i	0.310797i
$825$	− 0.335126i	− 0.0116676i
$826$	4.04520i	0.140750i
$827$	− 21.9571i	− 0.763521i	−0.924261	−	0.381761i	$-0.875318\pi$
$827$	− 21.9571i	− 0.763521i	0.924261	−	0.381761i	$-0.124682\pi$
$828$	2.35690	0.0819078
$829$	27.4862	0.954635	0.477317	−	0.878731i	$-0.341609\pi$
$829$	27.4862	0.954635	0.477317	+	0.878731i	$0.341609\pi$
$830$	− 3.84117i	− 0.133329i
$831$	23.4252	0.812611
$832$	0	0
$833$	47.3062	1.63906
$834$	− 2.01208i	− 0.0696727i
$835$	−9.64071	−0.333631
$836$	−1.94438	−0.0672477
$837$	− 0.0609989i	− 0.00210843i
$838$	17.0084i	0.587544i
$839$	20.5749i	0.710325i	0.934805	+	0.355163i	$0.115575\pi$
$839$	20.5749i	0.710325i	−0.934805	+	0.355163i	$0.884425\pi$
$840$	− 0.307979i	− 0.0106263i
$841$	5.95002	0.205173
$842$	−10.2851	−0.354449
$843$	8.40044i	0.289326i
$844$	−11.0422	−0.380089
$845$	0	0
$846$	−7.00000	−0.240665
$847$	3.35317i	0.115216i
$848$	12.1739	0.418053
$849$	31.6142	1.08499
$850$	− 6.85086i	− 0.234982i
$851$	− 16.6775i	− 0.571699i
$852$	9.56465i	0.327679i
$853$	− 50.3812i	− 1.72502i	−0.506041	−	0.862509i	$-0.668892\pi$
$853$	− 50.3812i	− 1.72502i	0.506041	−	0.862509i	$-0.331108\pi$
$854$	−0.661154	−0.0226242
$855$	−5.80194	−0.198422
$856$	− 14.4644i	− 0.494384i
$857$	−21.6045	−0.737995	−0.368997	−	0.929430i	$-0.620299\pi$
$857$	−21.6045	−0.737995	−0.368997	+	0.929430i	$0.620299\pi$
$858$	0	0
$859$	−38.2959	−1.30664	−0.653320	−	0.757082i	$-0.726624\pi$
$859$	−38.2959	−1.30664	−0.653320	+	0.757082i	$0.726624\pi$
$860$	0.0489173i	0.00166807i
$861$	1.04892	0.0357470
$862$	−35.9657	−1.22500
$863$	− 8.61894i	− 0.293392i	−0.989182	−	0.146696i	$-0.953136\pi$
$863$	− 8.61894i	− 0.293392i	0.989182	−	0.146696i	$-0.0468639\pi$
$864$	1.00000i	0.0340207i
$865$	0.719169i	0.0244525i
$866$	− 7.63342i	− 0.259394i
$867$	29.9342	1.01662
$868$	−0.0187864	−0.000637651 0
$869$	− 0.0254911i	0 0.000864727i
$870$	−5.91185	−0.200431
$871$	0	0
$872$	−7.13706	−0.241691
$873$	− 6.13169i	− 0.207526i
$874$	−13.6746	−0.462549
$875$	−0.307979	−0.0104116
$876$	− 2.31336i	− 0.0781610i
$877$	33.1021i	1.11778i	0.829242	+	0.558890i	$0.188773\pi$
$877$	33.1021i	1.11778i	−0.829242	+	0.558890i	$0.811227\pi$
$878$	34.4373i	1.16220i
$879$	− 17.5254i	− 0.591118i
$880$	0.335126	0.0112971
$881$	−20.8267	−0.701669	−0.350835	−	0.936437i	$-0.614102\pi$
$881$	−20.8267	−0.701669	−0.350835	+	0.936437i	$0.614102\pi$
$882$	6.90515i	0.232508i
$883$	−0.785807	−0.0264445	−0.0132223	−	0.999913i	$-0.504209\pi$
$883$	−0.785807	−0.0264445	−0.0132223	+	0.999913i	$0.504209\pi$
$884$	0	0
$885$	−13.1347	−0.441517
$886$	33.6431i	1.13026i
$887$	−44.2392	−1.48541	−0.742704	−	0.669620i	$-0.766457\pi$
$887$	−44.2392	−1.48541	−0.742704	+	0.669620i	$0.766457\pi$
$888$	7.07606	0.237457
$889$	0.280831i	0.00941878i
$890$	− 4.41119i	− 0.147863i
$891$	0.335126i	0.0112271i
$892$	18.3884i	0.615688i
$893$	40.6136	1.35908
$894$	3.44935	0.115364
$895$	− 2.01208i	− 0.0672565i
$896$	0.307979	0.0102888
$897$	0	0
$898$	27.9729	0.933466
$899$	0.360617i	0.0120272i
$900$	1.00000	0.0333333
$901$	83.4016	2.77851
$902$	1.14138i	0.0380036i
$903$	0.0150655i	0 0.000501348i
$904$	16.3424i	0.543541i
$905$	− 2.57135i	− 0.0854746i
$906$	−2.84117	−0.0943914
$907$	−8.44371	−0.280369	−0.140184	−	0.990125i	$-0.544770\pi$
$907$	−8.44371	−0.280369	−0.140184	+	0.990125i	$0.544770\pi$
$908$	12.0073i	0.398476i
$909$	−1.32975	−0.0441050
$910$	0	0
$911$	−47.2801	−1.56646	−0.783230	−	0.621732i	$-0.786429\pi$
$911$	−47.2801	−1.56646	−0.783230	+	0.621732i	$0.786429\pi$
$912$	− 5.80194i	− 0.192121i
$913$	1.28727	0.0426025
$914$	−6.18060	−0.204436
$915$	− 2.14675i	− 0.0709694i
$916$	− 25.4601i	− 0.841226i
$917$	4.01507i	0.132589i
$918$	6.85086i	0.226112i
$919$	−11.1927	−0.369213	−0.184606	−	0.982813i	$-0.559101\pi$
$919$	−11.1927	−0.369213	−0.184606	+	0.982813i	$0.559101\pi$
$920$	2.35690	0.0777046
$921$	20.6732i	0.681206i
$922$	38.1062	1.25496
$923$	0	0
$924$	0.103211	0.00339541
$925$	− 7.07606i	− 0.232660i
$926$	2.06829	0.0679684
$927$	−8.92154	−0.293022
$928$	− 5.91185i	− 0.194066i
$929$	− 1.11828i	− 0.0366895i	−0.999832	−	0.0183447i	$-0.994160\pi$
$929$	− 1.11828i	− 0.0366895i	0.999832	−	0.0183447i	$-0.00583964\pi$
$930$	− 0.0609989i	− 0.00200023i
$931$	− 40.0632i	− 1.31302i
$932$	−2.10992	−0.0691126
$933$	22.0248	0.721058
$934$	4.27114i	0.139756i
$935$	2.29590	0.0750839
$936$	0	0
$937$	−29.9051	−0.976959	−0.488479	−	0.872575i	$-0.662448\pi$
$937$	−29.9051	−0.976959	−0.488479	+	0.872575i	$0.662448\pi$
$938$	3.68366i	0.120276i
$939$	21.7192	0.708778
$940$	−7.00000	−0.228315
$941$	32.7646i	1.06810i	0.845454	+	0.534048i	$0.179330\pi$
$941$	32.7646i	1.06810i	−0.845454	+	0.534048i	$0.820670\pi$
$942$	16.2228i	0.528568i
$943$	8.02715i	0.261400i
$944$	− 13.1347i	− 0.427497i
$945$	0.307979	0.0100185
$946$	−0.0163935	−0.000532997 0
$947$	− 6.81535i	− 0.221469i	−0.993850	−	0.110735i	$-0.964680\pi$
$947$	− 6.81535i	− 0.221469i	0.993850	−	0.110735i	$-0.0353203\pi$
$948$	0.0760644	0.00247046
$949$	0	0
$950$	−5.80194	−0.188240
$951$	16.4644i	0.533895i
$952$	2.10992	0.0683828
$953$	−23.6886	−0.767348	−0.383674	−	0.923469i	$-0.625341\pi$
$953$	−23.6886	−0.767348	−0.383674	+	0.923469i	$0.625341\pi$
$954$	12.1739i	0.394145i
$955$	− 3.80194i	− 0.123028i
$956$	3.95838i	0.128023i
$957$	− 1.98121i	− 0.0640435i
$958$	38.7560	1.25215
$959$	−3.96913	−0.128170
$960$	1.00000i	0.0322749i
$961$	30.9963	0.999880
$962$	0	0
$963$	14.4644	0.466109
$964$	23.1564i	0.745819i
$965$	−8.86294	−0.285308
$966$	0.725873	0.0233546
$967$	28.9075i	0.929604i	0.885415	+	0.464802i	$0.153874\pi$
$967$	28.9075i	0.929604i	−0.885415	+	0.464802i	$0.846126\pi$
$968$	− 10.8877i	− 0.349944i
$969$	− 39.7482i	− 1.27690i
$970$	− 6.13169i	− 0.196877i
$971$	−47.2717	−1.51702	−0.758511	−	0.651660i	$-0.774073\pi$
$971$	−47.2717	−1.51702	−0.758511	+	0.651660i	$0.774073\pi$
$972$	−1.00000	−0.0320750
$973$	− 0.619678i	− 0.0198660i
$974$	−33.4620	−1.07219
$975$	0	0
$976$	2.14675	0.0687159
$977$	33.9527i	1.08624i	0.839654	+	0.543122i	$0.182758\pi$
$977$	33.9527i	1.08624i	−0.839654	+	0.543122i	$0.817242\pi$
$978$	0.0924579	0.00295648
$979$	1.47830	0.0472468
$980$	6.90515i	0.220577i
$981$	− 7.13706i	− 0.227869i
$982$	− 35.1487i	− 1.12164i
$983$	− 22.9366i	− 0.731564i	−0.930701	−	0.365782i	$-0.880802\pi$
$983$	− 22.9366i	− 0.731564i	0.930701	−	0.365782i	$-0.119198\pi$
$984$	−3.40581	−0.108573
$985$	22.6504	0.721702
$986$	− 40.5013i	− 1.28982i
$987$	−2.15585	−0.0686215
$988$	0	0
$989$	−0.115293	−0.00366611
$990$	0.335126i	0.0106510i
$991$	−5.75781	−0.182903	−0.0914514	−	0.995810i	$-0.529151\pi$
$991$	−5.75781	−0.182903	−0.0914514	+	0.995810i	$0.529151\pi$
$992$	0.0609989	0.00193672
$993$	− 26.4373i	− 0.838961i
$994$	2.94571i	0.0934321i
$995$	− 5.46681i	− 0.173310i
$996$	3.84117i	0.121712i
$997$	3.86353	0.122359	0.0611796	−	0.998127i	$-0.480514\pi$
$997$	3.86353	0.122359	0.0611796	+	0.998127i	$0.480514\pi$
$998$	−9.61463	−0.304346
$999$	7.07606i	0.223877i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.y.1351.4		6
13.5	odd	4		5070.2.a.bv.1.3	yes	3
13.8	odd	4		5070.2.a.bq.1.1	✓	3
13.12	even	2	inner	5070.2.b.y.1351.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bq.1.1	✓	3	13.8	odd	4
5070.2.a.bv.1.3	yes	3	13.5	odd	4
5070.2.b.y.1351.3		6	13.12	even	2	inner
5070.2.b.y.1351.4		6	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +0.307979i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +0.307979i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.335126i q^{11} -1.00000 q^{12} -0.307979 q^{14} -1.00000i q^{15} +1.00000 q^{16} +6.85086 q^{17} +1.00000i q^{18} -5.80194i q^{19} +1.00000i q^{20} +0.307979i q^{21} -0.335126 q^{22} -2.35690 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -0.307979i q^{28} -5.91185 q^{29} +1.00000 q^{30} -0.0609989i q^{31} +1.00000i q^{32} +0.335126i q^{33} +6.85086i q^{34} +0.307979 q^{35} -1.00000 q^{36} +7.07606i q^{37} +5.80194 q^{38} -1.00000 q^{40} -3.40581i q^{41} -0.307979 q^{42} +0.0489173 q^{43} -0.335126i q^{44} -1.00000i q^{45} -2.35690i q^{46} +7.00000i q^{47} +1.00000 q^{48} +6.90515 q^{49} -1.00000i q^{50} +6.85086 q^{51} +12.1739 q^{53} +1.00000i q^{54} +0.335126 q^{55} +0.307979 q^{56} -5.80194i q^{57} -5.91185i q^{58} -13.1347i q^{59} +1.00000i q^{60} +2.14675 q^{61} +0.0609989 q^{62} +0.307979i q^{63} -1.00000 q^{64} -0.335126 q^{66} -11.9608i q^{67} -6.85086 q^{68} -2.35690 q^{69} +0.307979i q^{70} -9.56465i q^{71} -1.00000i q^{72} +2.31336i q^{73} -7.07606 q^{74} -1.00000 q^{75} +5.80194i q^{76} -0.103211 q^{77} -0.0760644 q^{79} -1.00000i q^{80} +1.00000 q^{81} +3.40581 q^{82} -3.84117i q^{83} -0.307979i q^{84} -6.85086i q^{85} +0.0489173i q^{86} -5.91185 q^{87} +0.335126 q^{88} -4.41119i q^{89} +1.00000 q^{90} +2.35690 q^{92} -0.0609989i q^{93} -7.00000 q^{94} -5.80194 q^{95} +1.00000i q^{96} -6.13169i q^{97} +6.90515i q^{98} +0.335126i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} - 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} - 28 q^{29} + 6 q^{30} + 12 q^{35} - 6 q^{36} + 26 q^{38} - 6 q^{40} - 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} + 12 q^{56} - 42 q^{61} + 20 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} - 12 q^{74} - 6 q^{75} + 42 q^{77} + 30 q^{79} + 6 q^{81} - 6 q^{82} - 28 q^{87} + 6 q^{90} + 6 q^{92} - 42 q^{94} - 26 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 - 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 - 28 * q^29 + 6 * q^30 + 12 * q^35 - 6 * q^36 + 26 * q^38 - 6 * q^40 - 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 + 12 * q^56 - 42 * q^61 + 20 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 - 12 * q^74 - 6 * q^75 + 42 * q^77 + 30 * q^79 + 6 * q^81 - 6 * q^82 - 28 * q^87 + 6 * q^90 + 6 * q^92 - 42 * q^94 - 26 * q^95

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