LMFDB - Embedded newform 5070.2.a.bu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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, 0]))

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

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

chi := DirichletCharacter("5070.1");

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.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.15883 q^{11} -1.00000 q^{12} -1.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.35690 q^{17} +1.00000 q^{18} +0.198062 q^{19} +1.00000 q^{20} +1.69202 q^{21} +2.15883 q^{22} -3.74094 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +1.29590 q^{29} -1.00000 q^{30} +1.44504 q^{31} +1.00000 q^{32} -2.15883 q^{33} +2.35690 q^{34} -1.69202 q^{35} +1.00000 q^{36} +0.801938 q^{37} +0.198062 q^{38} +1.00000 q^{40} -1.89977 q^{41} +1.69202 q^{42} +12.5429 q^{43} +2.15883 q^{44} +1.00000 q^{45} -3.74094 q^{46} +8.87800 q^{47} -1.00000 q^{48} -4.13706 q^{49} +1.00000 q^{50} -2.35690 q^{51} -1.00969 q^{53} -1.00000 q^{54} +2.15883 q^{55} -1.69202 q^{56} -0.198062 q^{57} +1.29590 q^{58} +3.73125 q^{59} -1.00000 q^{60} -6.32304 q^{61} +1.44504 q^{62} -1.69202 q^{63} +1.00000 q^{64} -2.15883 q^{66} -7.56465 q^{67} +2.35690 q^{68} +3.74094 q^{69} -1.69202 q^{70} -4.18060 q^{71} +1.00000 q^{72} +11.9366 q^{73} +0.801938 q^{74} -1.00000 q^{75} +0.198062 q^{76} -3.65279 q^{77} +9.40581 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.89977 q^{82} +8.43296 q^{83} +1.69202 q^{84} +2.35690 q^{85} +12.5429 q^{86} -1.29590 q^{87} +2.15883 q^{88} -2.41119 q^{89} +1.00000 q^{90} -3.74094 q^{92} -1.44504 q^{93} +8.87800 q^{94} +0.198062 q^{95} -1.00000 q^{96} -0.0881460 q^{97} -4.13706 q^{98} +2.15883 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 2 q^{11} - 3 q^{12} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 5 q^{19} + 3 q^{20} - 2 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 3 q^{30} + 4 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 3 q^{36} - 2 q^{37} + 5 q^{38} + 3 q^{40} + 17 q^{41} + 19 q^{43} - 2 q^{44} + 3 q^{45} + 3 q^{46} + 7 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} - 2 q^{55} - 5 q^{57} - 10 q^{58} + 19 q^{59} - 3 q^{60} + q^{61} + 4 q^{62} + 3 q^{64} + 2 q^{66} - q^{67} + 3 q^{68} - 3 q^{69} - q^{71} + 3 q^{72} - 15 q^{73} - 2 q^{74} - 3 q^{75} + 5 q^{76} + 7 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} + 6 q^{83} + 3 q^{85} + 19 q^{86} + 10 q^{87} - 2 q^{88} + 9 q^{89} + 3 q^{90} + 3 q^{92} - 4 q^{93} + 7 q^{94} + 5 q^{95} - 3 q^{96} - 4 q^{97} - 7 q^{98} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 2 * q^11 - 3 * q^12 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 5 * q^19 + 3 * q^20 - 2 * q^22 + 3 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 10 * q^29 - 3 * q^30 + 4 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 3 * q^36 - 2 * q^37 + 5 * q^38 + 3 * q^40 + 17 * q^41 + 19 * q^43 - 2 * q^44 + 3 * q^45 + 3 * q^46 + 7 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 - 2 * q^55 - 5 * q^57 - 10 * q^58 + 19 * q^59 - 3 * q^60 + q^61 + 4 * q^62 + 3 * q^64 + 2 * q^66 - q^67 + 3 * q^68 - 3 * q^69 - q^71 + 3 * q^72 - 15 * q^73 - 2 * q^74 - 3 * q^75 + 5 * q^76 + 7 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 17 * q^82 + 6 * q^83 + 3 * q^85 + 19 * q^86 + 10 * q^87 - 2 * q^88 + 9 * q^89 + 3 * q^90 + 3 * q^92 - 4 * q^93 + 7 * q^94 + 5 * q^95 - 3 * q^96 - 4 * q^97 - 7 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−1.69202	−0.639524	−0.319762	−	0.947498i	$-0.603603\pi$
$7$	−1.69202	−0.639524	−0.319762	+	0.947498i	$0.603603\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	2.15883	0.650913	0.325456	−	0.945557i	$-0.394482\pi$
$11$	2.15883	0.650913	0.325456	+	0.945557i	$0.394482\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.69202	−0.452212
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	2.35690	0.571631	0.285816	−	0.958285i	$-0.407736\pi$
$17$	2.35690	0.571631	0.285816	+	0.958285i	$0.407736\pi$
$18$	1.00000	0.235702
$19$	0.198062	0.0454386	0.0227193	−	0.999742i	$-0.492768\pi$
$19$	0.198062	0.0454386	0.0227193	+	0.999742i	$0.492768\pi$
$20$	1.00000	0.223607
$21$	1.69202	0.369229
$22$	2.15883	0.460265
$23$	−3.74094	−0.780040	−0.390020	−	0.920806i	$-0.627532\pi$
$23$	−3.74094	−0.780040	−0.390020	+	0.920806i	$0.627532\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.69202	−0.319762
$29$	1.29590	0.240642	0.120321	−	0.992735i	$-0.461608\pi$
$29$	1.29590	0.240642	0.120321	+	0.992735i	$0.461608\pi$
$30$	−1.00000	−0.182574
$31$	1.44504	0.259537	0.129769	−	0.991544i	$-0.458577\pi$
$31$	1.44504	0.259537	0.129769	+	0.991544i	$0.458577\pi$
$32$	1.00000	0.176777
$33$	−2.15883	−0.375805
$34$	2.35690	0.404204
$35$	−1.69202	−0.286004
$36$	1.00000	0.166667
$37$	0.801938	0.131838	0.0659189	−	0.997825i	$-0.479002\pi$
$37$	0.801938	0.131838	0.0659189	+	0.997825i	$0.479002\pi$
$38$	0.198062	0.0321299
$39$	0	0
$40$	1.00000	0.158114
$41$	−1.89977	−0.296695	−0.148347	−	0.988935i	$-0.547395\pi$
$41$	−1.89977	−0.296695	−0.148347	+	0.988935i	$0.547395\pi$
$42$	1.69202	0.261085
$43$	12.5429	1.91277	0.956385	−	0.292108i	$-0.0943566\pi$
$43$	12.5429	1.91277	0.956385	+	0.292108i	$0.0943566\pi$
$44$	2.15883	0.325456
$45$	1.00000	0.149071
$46$	−3.74094	−0.551571
$47$	8.87800	1.29499	0.647495	−	0.762070i	$-0.275817\pi$
$47$	8.87800	1.29499	0.647495	+	0.762070i	$0.275817\pi$
$48$	−1.00000	−0.144338
$49$	−4.13706	−0.591009
$50$	1.00000	0.141421
$51$	−2.35690	−0.330031
$52$	0	0
$53$	−1.00969	−0.138691	−0.0693457	−	0.997593i	$-0.522091\pi$
$53$	−1.00969	−0.138691	−0.0693457	+	0.997593i	$0.522091\pi$
$54$	−1.00000	−0.136083
$55$	2.15883	0.291097
$56$	−1.69202	−0.226106
$57$	−0.198062	−0.0262340
$58$	1.29590	0.170160
$59$	3.73125	0.485767	0.242884	−	0.970055i	$-0.421907\pi$
$59$	3.73125	0.485767	0.242884	+	0.970055i	$0.421907\pi$
$60$	−1.00000	−0.129099
$61$	−6.32304	−0.809583	−0.404791	−	0.914409i	$-0.632656\pi$
$61$	−6.32304	−0.809583	−0.404791	+	0.914409i	$0.632656\pi$
$62$	1.44504	0.183521
$63$	−1.69202	−0.213175
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.15883	−0.265734
$67$	−7.56465	−0.924169	−0.462084	−	0.886836i	$-0.652898\pi$
$67$	−7.56465	−0.924169	−0.462084	+	0.886836i	$0.652898\pi$
$68$	2.35690	0.285816
$69$	3.74094	0.450356
$70$	−1.69202	−0.202235
$71$	−4.18060	−0.496146	−0.248073	−	0.968741i	$-0.579797\pi$
$71$	−4.18060	−0.496146	−0.248073	+	0.968741i	$0.579797\pi$
$72$	1.00000	0.117851
$73$	11.9366	1.39707	0.698537	−	0.715574i	$-0.253835\pi$
$73$	11.9366	1.39707	0.698537	+	0.715574i	$0.253835\pi$
$74$	0.801938	0.0932234
$75$	−1.00000	−0.115470
$76$	0.198062	0.0227193
$77$	−3.65279	−0.416274
$78$	0	0
$79$	9.40581	1.05824	0.529118	−	0.848548i	$-0.322523\pi$
$79$	9.40581	1.05824	0.529118	+	0.848548i	$0.322523\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−1.89977	−0.209795
$83$	8.43296	0.925638	0.462819	−	0.886453i	$-0.346838\pi$
$83$	8.43296	0.925638	0.462819	+	0.886453i	$0.346838\pi$
$84$	1.69202	0.184615
$85$	2.35690	0.255641
$86$	12.5429	1.35253
$87$	−1.29590	−0.138935
$88$	2.15883	0.230132
$89$	−2.41119	−0.255586	−0.127793	−	0.991801i	$-0.540789\pi$
$89$	−2.41119	−0.255586	−0.127793	+	0.991801i	$0.540789\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−3.74094	−0.390020
$93$	−1.44504	−0.149844
$94$	8.87800	0.915696
$95$	0.198062	0.0203208
$96$	−1.00000	−0.102062
$97$	−0.0881460	−0.00894987	−0.00447494	−	0.999990i	$-0.501424\pi$
$97$	−0.0881460	−0.00894987	−0.00447494	+	0.999990i	$0.501424\pi$
$98$	−4.13706	−0.417907
$99$	2.15883	0.216971
$100$	1.00000	0.100000
$101$	−11.1099	−1.10548	−0.552739	−	0.833354i	$-0.686417\pi$
$101$	−11.1099	−1.10548	−0.552739	+	0.833354i	$0.686417\pi$
$102$	−2.35690	−0.233367
$103$	9.54825	0.940817	0.470409	−	0.882449i	$-0.344107\pi$
$103$	9.54825	0.940817	0.470409	+	0.882449i	$0.344107\pi$
$104$	0	0
$105$	1.69202	0.165124
$106$	−1.00969	−0.0980696
$107$	18.1468	1.75431	0.877156	−	0.480205i	$-0.159438\pi$
$107$	18.1468	1.75431	0.877156	+	0.480205i	$0.159438\pi$
$108$	−1.00000	−0.0962250
$109$	−2.24698	−0.215222	−0.107611	−	0.994193i	$-0.534320\pi$
$109$	−2.24698	−0.215222	−0.107611	+	0.994193i	$0.534320\pi$
$110$	2.15883	0.205837
$111$	−0.801938	−0.0761166
$112$	−1.69202	−0.159881
$113$	−11.8726	−1.11688	−0.558441	−	0.829544i	$-0.688600\pi$
$113$	−11.8726	−1.11688	−0.558441	+	0.829544i	$0.688600\pi$
$114$	−0.198062	−0.0185502
$115$	−3.74094	−0.348844
$116$	1.29590	0.120321
$117$	0	0
$118$	3.73125	0.343489
$119$	−3.98792	−0.365572
$120$	−1.00000	−0.0912871
$121$	−6.33944	−0.576312
$122$	−6.32304	−0.572462
$123$	1.89977	0.171297
$124$	1.44504	0.129769
$125$	1.00000	0.0894427
$126$	−1.69202	−0.150737
$127$	13.0097	1.15442	0.577212	−	0.816595i	$-0.304141\pi$
$127$	13.0097	1.15442	0.577212	+	0.816595i	$0.304141\pi$
$128$	1.00000	0.0883883
$129$	−12.5429	−1.10434
$130$	0	0
$131$	−7.82908	−0.684030	−0.342015	−	0.939694i	$-0.611109\pi$
$131$	−7.82908	−0.684030	−0.342015	+	0.939694i	$0.611109\pi$
$132$	−2.15883	−0.187902
$133$	−0.335126	−0.0290591
$134$	−7.56465	−0.653486
$135$	−1.00000	−0.0860663
$136$	2.35690	0.202102
$137$	5.36227	0.458130	0.229065	−	0.973411i	$-0.426433\pi$
$137$	5.36227	0.458130	0.229065	+	0.973411i	$0.426433\pi$
$138$	3.74094	0.318450
$139$	3.12200	0.264804	0.132402	−	0.991196i	$-0.457731\pi$
$139$	3.12200	0.264804	0.132402	+	0.991196i	$0.457731\pi$
$140$	−1.69202	−0.143002
$141$	−8.87800	−0.747663
$142$	−4.18060	−0.350828
$143$	0	0
$144$	1.00000	0.0833333
$145$	1.29590	0.107618
$146$	11.9366	0.987881
$147$	4.13706	0.341219
$148$	0.801938	0.0659189
$149$	4.24160	0.347486	0.173743	−	0.984791i	$-0.444414\pi$
$149$	4.24160	0.347486	0.173743	+	0.984791i	$0.444414\pi$
$150$	−1.00000	−0.0816497
$151$	22.9148	1.86478	0.932392	−	0.361450i	$-0.117718\pi$
$151$	22.9148	1.86478	0.932392	+	0.361450i	$0.117718\pi$
$152$	0.198062	0.0160650
$153$	2.35690	0.190544
$154$	−3.65279	−0.294350
$155$	1.44504	0.116069
$156$	0	0
$157$	16.0707	1.28258	0.641290	−	0.767298i	$-0.278399\pi$
$157$	16.0707	1.28258	0.641290	+	0.767298i	$0.278399\pi$
$158$	9.40581	0.748286
$159$	1.00969	0.0800735
$160$	1.00000	0.0790569
$161$	6.32975	0.498854
$162$	1.00000	0.0785674
$163$	−0.488582	−0.0382687	−0.0191344	−	0.999817i	$-0.506091\pi$
$163$	−0.488582	−0.0382687	−0.0191344	+	0.999817i	$0.506091\pi$
$164$	−1.89977	−0.148347
$165$	−2.15883	−0.168065
$166$	8.43296	0.654525
$167$	14.0610	1.08807	0.544036	−	0.839062i	$-0.316895\pi$
$167$	14.0610	1.08807	0.544036	+	0.839062i	$0.316895\pi$
$168$	1.69202	0.130542
$169$	0	0
$170$	2.35690	0.180766
$171$	0.198062	0.0151462
$172$	12.5429	0.956385
$173$	21.7942	1.65698	0.828490	−	0.560004i	$-0.189200\pi$
$173$	21.7942	1.65698	0.828490	+	0.560004i	$0.189200\pi$
$174$	−1.29590	−0.0982417
$175$	−1.69202	−0.127905
$176$	2.15883	0.162728
$177$	−3.73125	−0.280458
$178$	−2.41119	−0.180726
$179$	−14.1293	−1.05607	−0.528036	−	0.849222i	$-0.677072\pi$
$179$	−14.1293	−1.05607	−0.528036	+	0.849222i	$0.677072\pi$
$180$	1.00000	0.0745356
$181$	−5.63773	−0.419049	−0.209524	−	0.977803i	$-0.567192\pi$
$181$	−5.63773	−0.419049	−0.209524	+	0.977803i	$0.567192\pi$
$182$	0	0
$183$	6.32304	0.467413
$184$	−3.74094	−0.275786
$185$	0.801938	0.0589596
$186$	−1.44504	−0.105956
$187$	5.08815	0.372082
$188$	8.87800	0.647495
$189$	1.69202	0.123076
$190$	0.198062	0.0143689
$191$	−2.59419	−0.187709	−0.0938544	−	0.995586i	$-0.529919\pi$
$191$	−2.59419	−0.187709	−0.0938544	+	0.995586i	$0.529919\pi$
$192$	−1.00000	−0.0721688
$193$	−3.98361	−0.286746	−0.143373	−	0.989669i	$-0.545795\pi$
$193$	−3.98361	−0.286746	−0.143373	+	0.989669i	$0.545795\pi$
$194$	−0.0881460	−0.00632851
$195$	0	0
$196$	−4.13706	−0.295505
$197$	−6.65040	−0.473821	−0.236911	−	0.971531i	$-0.576135\pi$
$197$	−6.65040	−0.473821	−0.236911	+	0.971531i	$0.576135\pi$
$198$	2.15883	0.153422
$199$	−1.01639	−0.0720502	−0.0360251	−	0.999351i	$-0.511470\pi$
$199$	−1.01639	−0.0720502	−0.0360251	+	0.999351i	$0.511470\pi$
$200$	1.00000	0.0707107
$201$	7.56465	0.533569
$202$	−11.1099	−0.781691
$203$	−2.19269	−0.153896
$204$	−2.35690	−0.165016
$205$	−1.89977	−0.132686
$206$	9.54825	0.665258
$207$	−3.74094	−0.260013
$208$	0	0
$209$	0.427583	0.0295766
$210$	1.69202	0.116761
$211$	10.1414	0.698161	0.349081	−	0.937093i	$-0.386494\pi$
$211$	10.1414	0.698161	0.349081	+	0.937093i	$0.386494\pi$
$212$	−1.00969	−0.0693457
$213$	4.18060	0.286450
$214$	18.1468	1.24049
$215$	12.5429	0.855417
$216$	−1.00000	−0.0680414
$217$	−2.44504	−0.165980
$218$	−2.24698	−0.152185
$219$	−11.9366	−0.806601
$220$	2.15883	0.145549
$221$	0	0
$222$	−0.801938	−0.0538225
$223$	−16.8388	−1.12761	−0.563804	−	0.825909i	$-0.690663\pi$
$223$	−16.8388	−1.12761	−0.563804	+	0.825909i	$0.690663\pi$
$224$	−1.69202	−0.113053
$225$	1.00000	0.0666667
$226$	−11.8726	−0.789755
$227$	18.3056	1.21498	0.607492	−	0.794326i	$-0.292176\pi$
$227$	18.3056	1.21498	0.607492	+	0.794326i	$0.292176\pi$
$228$	−0.198062	−0.0131170
$229$	5.14244	0.339822	0.169911	−	0.985459i	$-0.445652\pi$
$229$	5.14244	0.339822	0.169911	+	0.985459i	$0.445652\pi$
$230$	−3.74094	−0.246670
$231$	3.65279	0.240336
$232$	1.29590	0.0850798
$233$	−10.9215	−0.715494	−0.357747	−	0.933819i	$-0.616455\pi$
$233$	−10.9215	−0.715494	−0.357747	+	0.933819i	$0.616455\pi$
$234$	0	0
$235$	8.87800	0.579137
$236$	3.73125	0.242884
$237$	−9.40581	−0.610973
$238$	−3.98792	−0.258498
$239$	19.9041	1.28749	0.643744	−	0.765241i	$-0.277380\pi$
$239$	19.9041	1.28749	0.643744	+	0.765241i	$0.277380\pi$
$240$	−1.00000	−0.0645497
$241$	18.2228	1.17383	0.586917	−	0.809647i	$-0.300341\pi$
$241$	18.2228	1.17383	0.586917	+	0.809647i	$0.300341\pi$
$242$	−6.33944	−0.407514
$243$	−1.00000	−0.0641500
$244$	−6.32304	−0.404791
$245$	−4.13706	−0.264307
$246$	1.89977	0.121125
$247$	0	0
$248$	1.44504	0.0917603
$249$	−8.43296	−0.534417
$250$	1.00000	0.0632456
$251$	−17.0030	−1.07322	−0.536609	−	0.843831i	$-0.680295\pi$
$251$	−17.0030	−1.07322	−0.536609	+	0.843831i	$0.680295\pi$
$252$	−1.69202	−0.106587
$253$	−8.07606	−0.507738
$254$	13.0097	0.816300
$255$	−2.35690	−0.147595
$256$	1.00000	0.0625000
$257$	−7.65519	−0.477517	−0.238759	−	0.971079i	$-0.576740\pi$
$257$	−7.65519	−0.477517	−0.238759	+	0.971079i	$0.576740\pi$
$258$	−12.5429	−0.780885
$259$	−1.35690	−0.0843134
$260$	0	0
$261$	1.29590	0.0802140
$262$	−7.82908	−0.483682
$263$	21.0411	1.29745	0.648726	−	0.761022i	$-0.275302\pi$
$263$	21.0411	1.29745	0.648726	+	0.761022i	$0.275302\pi$
$264$	−2.15883	−0.132867
$265$	−1.00969	−0.0620247
$266$	−0.335126	−0.0205479
$267$	2.41119	0.147562
$268$	−7.56465	−0.462084
$269$	24.3183	1.48271	0.741355	−	0.671113i	$-0.234183\pi$
$269$	24.3183	1.48271	0.741355	+	0.671113i	$0.234183\pi$
$270$	−1.00000	−0.0608581
$271$	10.9922	0.667730	0.333865	−	0.942621i	$-0.391647\pi$
$271$	10.9922	0.667730	0.333865	+	0.942621i	$0.391647\pi$
$272$	2.35690	0.142908
$273$	0	0
$274$	5.36227	0.323947
$275$	2.15883	0.130183
$276$	3.74094	0.225178
$277$	−11.4601	−0.688571	−0.344286	−	0.938865i	$-0.611879\pi$
$277$	−11.4601	−0.688571	−0.344286	+	0.938865i	$0.611879\pi$
$278$	3.12200	0.187245
$279$	1.44504	0.0865124
$280$	−1.69202	−0.101118
$281$	−9.88231	−0.589529	−0.294765	−	0.955570i	$-0.595241\pi$
$281$	−9.88231	−0.589529	−0.294765	+	0.955570i	$0.595241\pi$
$282$	−8.87800	−0.528677
$283$	−21.3207	−1.26738	−0.633691	−	0.773587i	$-0.718461\pi$
$283$	−21.3207	−1.26738	−0.633691	+	0.773587i	$0.718461\pi$
$284$	−4.18060	−0.248073
$285$	−0.198062	−0.0117322
$286$	0	0
$287$	3.21446	0.189743
$288$	1.00000	0.0589256
$289$	−11.4450	−0.673238
$290$	1.29590	0.0760977
$291$	0.0881460	0.00516721
$292$	11.9366	0.698537
$293$	−18.7090	−1.09299	−0.546496	−	0.837462i	$-0.684039\pi$
$293$	−18.7090	−1.09299	−0.546496	+	0.837462i	$0.684039\pi$
$294$	4.13706	0.241278
$295$	3.73125	0.217242
$296$	0.801938	0.0466117
$297$	−2.15883	−0.125268
$298$	4.24160	0.245709
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−21.2228	−1.22326
$302$	22.9148	1.31860
$303$	11.1099	0.638248
$304$	0.198062	0.0113596
$305$	−6.32304	−0.362056
$306$	2.35690	0.134735
$307$	17.2241	0.983034	0.491517	−	0.870868i	$-0.336443\pi$
$307$	17.2241	0.983034	0.491517	+	0.870868i	$0.336443\pi$
$308$	−3.65279	−0.208137
$309$	−9.54825	−0.543181
$310$	1.44504	0.0820729
$311$	−13.5200	−0.766651	−0.383326	−	0.923613i	$-0.625221\pi$
$311$	−13.5200	−0.766651	−0.383326	+	0.923613i	$0.625221\pi$
$312$	0	0
$313$	24.3303	1.37523	0.687616	−	0.726074i	$-0.258657\pi$
$313$	24.3303	1.37523	0.687616	+	0.726074i	$0.258657\pi$
$314$	16.0707	0.906921
$315$	−1.69202	−0.0953346
$316$	9.40581	0.529118
$317$	34.7821	1.95356	0.976778	−	0.214253i	$-0.0687316\pi$
$317$	34.7821	1.95356	0.976778	+	0.214253i	$0.0687316\pi$
$318$	1.00969	0.0566205
$319$	2.79763	0.156637
$320$	1.00000	0.0559017
$321$	−18.1468	−1.01285
$322$	6.32975	0.352743
$323$	0.466812	0.0259741
$324$	1.00000	0.0555556
$325$	0	0
$326$	−0.488582	−0.0270601
$327$	2.24698	0.124258
$328$	−1.89977	−0.104897
$329$	−15.0218	−0.828177
$330$	−2.15883	−0.118840
$331$	−9.66919	−0.531467	−0.265733	−	0.964047i	$-0.585614\pi$
$331$	−9.66919	−0.531467	−0.265733	+	0.964047i	$0.585614\pi$
$332$	8.43296	0.462819
$333$	0.801938	0.0439459
$334$	14.0610	0.769384
$335$	−7.56465	−0.413301
$336$	1.69202	0.0923073
$337$	−17.3612	−0.945725	−0.472863	−	0.881136i	$-0.656779\pi$
$337$	−17.3612	−0.945725	−0.472863	+	0.881136i	$0.656779\pi$
$338$	0	0
$339$	11.8726	0.644832
$340$	2.35690	0.127821
$341$	3.11960	0.168936
$342$	0.198062	0.0107100
$343$	18.8442	1.01749
$344$	12.5429	0.676267
$345$	3.74094	0.201405
$346$	21.7942	1.17166
$347$	8.86294	0.475787	0.237894	−	0.971291i	$-0.423543\pi$
$347$	8.86294	0.475787	0.237894	+	0.971291i	$0.423543\pi$
$348$	−1.29590	−0.0694674
$349$	2.78554	0.149107	0.0745534	−	0.997217i	$-0.476247\pi$
$349$	2.78554	0.149107	0.0745534	+	0.997217i	$0.476247\pi$
$350$	−1.69202	−0.0904424
$351$	0	0
$352$	2.15883	0.115066
$353$	21.0640	1.12112	0.560561	−	0.828113i	$-0.310585\pi$
$353$	21.0640	1.12112	0.560561	+	0.828113i	$0.310585\pi$
$354$	−3.73125	−0.198314
$355$	−4.18060	−0.221883
$356$	−2.41119	−0.127793
$357$	3.98792	0.211063
$358$	−14.1293	−0.746756
$359$	17.3274	0.914503	0.457251	−	0.889337i	$-0.348834\pi$
$359$	17.3274	0.914503	0.457251	+	0.889337i	$0.348834\pi$
$360$	1.00000	0.0527046
$361$	−18.9608	−0.997935
$362$	−5.63773	−0.296312
$363$	6.33944	0.332734
$364$	0	0
$365$	11.9366	0.624791
$366$	6.32304	0.330511
$367$	−30.8049	−1.60800	−0.804002	−	0.594627i	$-0.797300\pi$
$367$	−30.8049	−1.60800	−0.804002	+	0.594627i	$0.797300\pi$
$368$	−3.74094	−0.195010
$369$	−1.89977	−0.0988982
$370$	0.801938	0.0416908
$371$	1.70841	0.0886965
$372$	−1.44504	−0.0749219
$373$	−25.6668	−1.32898	−0.664488	−	0.747299i	$-0.731350\pi$
$373$	−25.6668	−1.32898	−0.664488	+	0.747299i	$0.731350\pi$
$374$	5.08815	0.263102
$375$	−1.00000	−0.0516398
$376$	8.87800	0.457848
$377$	0	0
$378$	1.69202	0.0870282
$379$	5.46144	0.280535	0.140268	−	0.990114i	$-0.455204\pi$
$379$	5.46144	0.280535	0.140268	+	0.990114i	$0.455204\pi$
$380$	0.198062	0.0101604
$381$	−13.0097	−0.666507
$382$	−2.59419	−0.132730
$383$	−25.9221	−1.32456	−0.662280	−	0.749257i	$-0.730411\pi$
$383$	−25.9221	−1.32456	−0.662280	+	0.749257i	$0.730411\pi$
$384$	−1.00000	−0.0510310
$385$	−3.65279	−0.186164
$386$	−3.98361	−0.202760
$387$	12.5429	0.637590
$388$	−0.0881460	−0.00447494
$389$	−32.0006	−1.62249	−0.811247	−	0.584703i	$-0.801211\pi$
$389$	−32.0006	−1.62249	−0.811247	+	0.584703i	$0.801211\pi$
$390$	0	0
$391$	−8.81700	−0.445895
$392$	−4.13706	−0.208953
$393$	7.82908	0.394925
$394$	−6.65040	−0.335042
$395$	9.40581	0.473258
$396$	2.15883	0.108485
$397$	0.457123	0.0229424	0.0114712	−	0.999934i	$-0.496349\pi$
$397$	0.457123	0.0229424	0.0114712	+	0.999934i	$0.496349\pi$
$398$	−1.01639	−0.0509472
$399$	0.335126	0.0167773
$400$	1.00000	0.0500000
$401$	23.2553	1.16132	0.580658	−	0.814147i	$-0.302795\pi$
$401$	23.2553	1.16132	0.580658	+	0.814147i	$0.302795\pi$
$402$	7.56465	0.377290
$403$	0	0
$404$	−11.1099	−0.552739
$405$	1.00000	0.0496904
$406$	−2.19269	−0.108821
$407$	1.73125	0.0858149
$408$	−2.35690	−0.116684
$409$	39.0629	1.93154	0.965768	−	0.259406i	$-0.0835266\pi$
$409$	39.0629	1.93154	0.965768	+	0.259406i	$0.0835266\pi$
$410$	−1.89977	−0.0938231
$411$	−5.36227	−0.264501
$412$	9.54825	0.470409
$413$	−6.31336	−0.310660
$414$	−3.74094	−0.183857
$415$	8.43296	0.413958
$416$	0	0
$417$	−3.12200	−0.152885
$418$	0.427583	0.0209138
$419$	−18.1269	−0.885557	−0.442779	−	0.896631i	$-0.646007\pi$
$419$	−18.1269	−0.885557	−0.442779	+	0.896631i	$0.646007\pi$
$420$	1.69202	0.0825622
$421$	−20.4916	−0.998698	−0.499349	−	0.866401i	$-0.666427\pi$
$421$	−20.4916	−0.998698	−0.499349	+	0.866401i	$0.666427\pi$
$422$	10.1414	0.493674
$423$	8.87800	0.431663
$424$	−1.00969	−0.0490348
$425$	2.35690	0.114326
$426$	4.18060	0.202551
$427$	10.6987	0.517748
$428$	18.1468	0.877156
$429$	0	0
$430$	12.5429	0.604871
$431$	−22.2989	−1.07410	−0.537050	−	0.843551i	$-0.680461\pi$
$431$	−22.2989	−1.07410	−0.537050	+	0.843551i	$0.680461\pi$
$432$	−1.00000	−0.0481125
$433$	7.47650	0.359298	0.179649	−	0.983731i	$-0.442504\pi$
$433$	7.47650	0.359298	0.179649	+	0.983731i	$0.442504\pi$
$434$	−2.44504	−0.117366
$435$	−1.29590	−0.0621335
$436$	−2.24698	−0.107611
$437$	−0.740939	−0.0354439
$438$	−11.9366	−0.570353
$439$	−19.3924	−0.925549	−0.462774	−	0.886476i	$-0.653146\pi$
$439$	−19.3924	−0.925549	−0.462774	+	0.886476i	$0.653146\pi$
$440$	2.15883	0.102918
$441$	−4.13706	−0.197003
$442$	0	0
$443$	10.0935	0.479558	0.239779	−	0.970828i	$-0.422925\pi$
$443$	10.0935	0.479558	0.239779	+	0.970828i	$0.422925\pi$
$444$	−0.801938	−0.0380583
$445$	−2.41119	−0.114301
$446$	−16.8388	−0.797339
$447$	−4.24160	−0.200621
$448$	−1.69202	−0.0799405
$449$	−2.78448	−0.131408	−0.0657039	−	0.997839i	$-0.520929\pi$
$449$	−2.78448	−0.131408	−0.0657039	+	0.997839i	$0.520929\pi$
$450$	1.00000	0.0471405
$451$	−4.10129	−0.193122
$452$	−11.8726	−0.558441
$453$	−22.9148	−1.07663
$454$	18.3056	0.859124
$455$	0	0
$456$	−0.198062	−0.00927512
$457$	−13.1056	−0.613054	−0.306527	−	0.951862i	$-0.599167\pi$
$457$	−13.1056	−0.613054	−0.306527	+	0.951862i	$0.599167\pi$
$458$	5.14244	0.240290
$459$	−2.35690	−0.110010
$460$	−3.74094	−0.174422
$461$	16.2174	0.755321	0.377661	−	0.925944i	$-0.376729\pi$
$461$	16.2174	0.755321	0.377661	+	0.925944i	$0.376729\pi$
$462$	3.65279	0.169943
$463$	15.5415	0.722277	0.361139	−	0.932512i	$-0.382388\pi$
$463$	15.5415	0.722277	0.361139	+	0.932512i	$0.382388\pi$
$464$	1.29590	0.0601605
$465$	−1.44504	−0.0670122
$466$	−10.9215	−0.505931
$467$	−10.1578	−0.470045	−0.235023	−	0.971990i	$-0.575516\pi$
$467$	−10.1578	−0.470045	−0.235023	+	0.971990i	$0.575516\pi$
$468$	0	0
$469$	12.7995	0.591028
$470$	8.87800	0.409512
$471$	−16.0707	−0.740498
$472$	3.73125	0.171745
$473$	27.0780	1.24505
$474$	−9.40581	−0.432023
$475$	0.198062	0.00908772
$476$	−3.98792	−0.182786
$477$	−1.00969	−0.0462305
$478$	19.9041	0.910392
$479$	31.4263	1.43590	0.717951	−	0.696094i	$-0.245080\pi$
$479$	31.4263	1.43590	0.717951	+	0.696094i	$0.245080\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	18.2228	0.830027
$483$	−6.32975	−0.288014
$484$	−6.33944	−0.288156
$485$	−0.0881460	−0.00400250
$486$	−1.00000	−0.0453609
$487$	−24.1323	−1.09354	−0.546769	−	0.837284i	$-0.684142\pi$
$487$	−24.1323	−1.09354	−0.546769	+	0.837284i	$0.684142\pi$
$488$	−6.32304	−0.286231
$489$	0.488582	0.0220945
$490$	−4.13706	−0.186893
$491$	−19.1099	−0.862418	−0.431209	−	0.902252i	$-0.641913\pi$
$491$	−19.1099	−0.862418	−0.431209	+	0.902252i	$0.641913\pi$
$492$	1.89977	0.0856484
$493$	3.05429	0.137558
$494$	0	0
$495$	2.15883	0.0970324
$496$	1.44504	0.0648843
$497$	7.07367	0.317298
$498$	−8.43296	−0.377890
$499$	−34.9724	−1.56558	−0.782789	−	0.622287i	$-0.786204\pi$
$499$	−34.9724	−1.56558	−0.782789	+	0.622287i	$0.786204\pi$
$500$	1.00000	0.0447214
$501$	−14.0610	−0.628199
$502$	−17.0030	−0.758880
$503$	29.5579	1.31792	0.658962	−	0.752176i	$-0.270996\pi$
$503$	29.5579	1.31792	0.658962	+	0.752176i	$0.270996\pi$
$504$	−1.69202	−0.0753686
$505$	−11.1099	−0.494385
$506$	−8.07606	−0.359025
$507$	0	0
$508$	13.0097	0.577212
$509$	24.4838	1.08523	0.542613	−	0.839983i	$-0.317435\pi$
$509$	24.4838	1.08523	0.542613	+	0.839983i	$0.317435\pi$
$510$	−2.35690	−0.104365
$511$	−20.1970	−0.893463
$512$	1.00000	0.0441942
$513$	−0.198062	−0.00874466
$514$	−7.65519	−0.337656
$515$	9.54825	0.420746
$516$	−12.5429	−0.552169
$517$	19.1661	0.842925
$518$	−1.35690	−0.0596186
$519$	−21.7942	−0.956658
$520$	0	0
$521$	−33.8713	−1.48393	−0.741964	−	0.670439i	$-0.766106\pi$
$521$	−33.8713	−1.48393	−0.741964	+	0.670439i	$0.766106\pi$
$522$	1.29590	0.0567199
$523$	−42.5870	−1.86220	−0.931100	−	0.364764i	$-0.881150\pi$
$523$	−42.5870	−1.86220	−0.931100	+	0.364764i	$0.881150\pi$
$524$	−7.82908	−0.342015
$525$	1.69202	0.0738459
$526$	21.0411	0.917438
$527$	3.40581	0.148360
$528$	−2.15883	−0.0939512
$529$	−9.00538	−0.391538
$530$	−1.00969	−0.0438581
$531$	3.73125	0.161922
$532$	−0.335126	−0.0145295
$533$	0	0
$534$	2.41119	0.104342
$535$	18.1468	0.784553
$536$	−7.56465	−0.326743
$537$	14.1293	0.609724
$538$	24.3183	1.04843
$539$	−8.93123	−0.384695
$540$	−1.00000	−0.0430331
$541$	−3.23298	−0.138997	−0.0694983	−	0.997582i	$-0.522140\pi$
$541$	−3.23298	−0.138997	−0.0694983	+	0.997582i	$0.522140\pi$
$542$	10.9922	0.472157
$543$	5.63773	0.241938
$544$	2.35690	0.101051
$545$	−2.24698	−0.0962500
$546$	0	0
$547$	−5.51573	−0.235836	−0.117918	−	0.993023i	$-0.537622\pi$
$547$	−5.51573	−0.235836	−0.117918	+	0.993023i	$0.537622\pi$
$548$	5.36227	0.229065
$549$	−6.32304	−0.269861
$550$	2.15883	0.0920530
$551$	0.256668	0.0109344
$552$	3.74094	0.159225
$553$	−15.9148	−0.676768
$554$	−11.4601	−0.486893
$555$	−0.801938	−0.0340404
$556$	3.12200	0.132402
$557$	30.3502	1.28598	0.642989	−	0.765875i	$-0.277694\pi$
$557$	30.3502	1.28598	0.642989	+	0.765875i	$0.277694\pi$
$558$	1.44504	0.0611735
$559$	0	0
$560$	−1.69202	−0.0715010
$561$	−5.08815	−0.214822
$562$	−9.88231	−0.416860
$563$	6.34375	0.267357	0.133679	−	0.991025i	$-0.457321\pi$
$563$	6.34375	0.267357	0.133679	+	0.991025i	$0.457321\pi$
$564$	−8.87800	−0.373831
$565$	−11.8726	−0.499485
$566$	−21.3207	−0.896174
$567$	−1.69202	−0.0710582
$568$	−4.18060	−0.175414
$569$	24.7275	1.03663	0.518316	−	0.855189i	$-0.326559\pi$
$569$	24.7275	1.03663	0.518316	+	0.855189i	$0.326559\pi$
$570$	−0.198062	−0.00829592
$571$	13.6262	0.570240	0.285120	−	0.958492i	$-0.407967\pi$
$571$	13.6262	0.570240	0.285120	+	0.958492i	$0.407967\pi$
$572$	0	0
$573$	2.59419	0.108374
$574$	3.21446	0.134169
$575$	−3.74094	−0.156008
$576$	1.00000	0.0416667
$577$	−16.2828	−0.677860	−0.338930	−	0.940812i	$-0.610065\pi$
$577$	−16.2828	−0.677860	−0.338930	+	0.940812i	$0.610065\pi$
$578$	−11.4450	−0.476051
$579$	3.98361	0.165553
$580$	1.29590	0.0538092
$581$	−14.2687	−0.591967
$582$	0.0881460	0.00365377
$583$	−2.17975	−0.0902760
$584$	11.9366	0.493940
$585$	0	0
$586$	−18.7090	−0.772862
$587$	−11.0344	−0.455440	−0.227720	−	0.973727i	$-0.573127\pi$
$587$	−11.0344	−0.455440	−0.227720	+	0.973727i	$0.573127\pi$
$588$	4.13706	0.170610
$589$	0.286208	0.0117930
$590$	3.73125	0.153613
$591$	6.65040	0.273561
$592$	0.801938	0.0329594
$593$	−25.4905	−1.04677	−0.523385	−	0.852096i	$-0.675331\pi$
$593$	−25.4905	−1.04677	−0.523385	+	0.852096i	$0.675331\pi$
$594$	−2.15883	−0.0885780
$595$	−3.98792	−0.163489
$596$	4.24160	0.173743
$597$	1.01639	0.0415982
$598$	0	0
$599$	−0.501729	−0.0205001	−0.0102500	−	0.999947i	$-0.503263\pi$
$599$	−0.501729	−0.0205001	−0.0102500	+	0.999947i	$0.503263\pi$
$600$	−1.00000	−0.0408248
$601$	−18.3744	−0.749506	−0.374753	−	0.927125i	$-0.622272\pi$
$601$	−18.3744	−0.749506	−0.374753	+	0.927125i	$0.622272\pi$
$602$	−21.2228	−0.864977
$603$	−7.56465	−0.308056
$604$	22.9148	0.932392
$605$	−6.33944	−0.257735
$606$	11.1099	0.451309
$607$	−17.1395	−0.695669	−0.347835	−	0.937556i	$-0.613083\pi$
$607$	−17.1395	−0.695669	−0.347835	+	0.937556i	$0.613083\pi$
$608$	0.198062	0.00803249
$609$	2.19269	0.0888521
$610$	−6.32304	−0.256013
$611$	0	0
$612$	2.35690	0.0952719
$613$	1.15751	0.0467512	0.0233756	−	0.999727i	$-0.492559\pi$
$613$	1.15751	0.0467512	0.0233756	+	0.999727i	$0.492559\pi$
$614$	17.2241	0.695110
$615$	1.89977	0.0766062
$616$	−3.65279	−0.147175
$617$	7.71618	0.310642	0.155321	−	0.987864i	$-0.450359\pi$
$617$	7.71618	0.310642	0.155321	+	0.987864i	$0.450359\pi$
$618$	−9.54825	−0.384087
$619$	16.1142	0.647686	0.323843	−	0.946111i	$-0.395025\pi$
$619$	16.1142	0.647686	0.323843	+	0.946111i	$0.395025\pi$
$620$	1.44504	0.0580343
$621$	3.74094	0.150119
$622$	−13.5200	−0.542104
$623$	4.07979	0.163453
$624$	0	0
$625$	1.00000	0.0400000
$626$	24.3303	0.972436
$627$	−0.427583	−0.0170760
$628$	16.0707	0.641290
$629$	1.89008	0.0753626
$630$	−1.69202	−0.0674117
$631$	−7.77586	−0.309552	−0.154776	−	0.987950i	$-0.549466\pi$
$631$	−7.77586	−0.309552	−0.154776	+	0.987950i	$0.549466\pi$
$632$	9.40581	0.374143
$633$	−10.1414	−0.403083
$634$	34.7821	1.38137
$635$	13.0097	0.516274
$636$	1.00969	0.0400368
$637$	0	0
$638$	2.79763	0.110759
$639$	−4.18060	−0.165382
$640$	1.00000	0.0395285
$641$	−35.1293	−1.38752	−0.693762	−	0.720204i	$-0.744048\pi$
$641$	−35.1293	−1.38752	−0.693762	+	0.720204i	$0.744048\pi$
$642$	−18.1468	−0.716195
$643$	−14.5459	−0.573633	−0.286816	−	0.957986i	$-0.592597\pi$
$643$	−14.5459	−0.573633	−0.286816	+	0.957986i	$0.592597\pi$
$644$	6.32975	0.249427
$645$	−12.5429	−0.493875
$646$	0.466812	0.0183665
$647$	36.2234	1.42409	0.712045	−	0.702134i	$-0.247769\pi$
$647$	36.2234	1.42409	0.712045	+	0.702134i	$0.247769\pi$
$648$	1.00000	0.0392837
$649$	8.05515	0.316192
$650$	0	0
$651$	2.44504	0.0958287
$652$	−0.488582	−0.0191344
$653$	19.4926	0.762806	0.381403	−	0.924409i	$-0.375441\pi$
$653$	19.4926	0.762806	0.381403	+	0.924409i	$0.375441\pi$
$654$	2.24698	0.0878639
$655$	−7.82908	−0.305908
$656$	−1.89977	−0.0741737
$657$	11.9366	0.465691
$658$	−15.0218	−0.585610
$659$	38.3400	1.49352	0.746758	−	0.665096i	$-0.231609\pi$
$659$	38.3400	1.49352	0.746758	+	0.665096i	$0.231609\pi$
$660$	−2.15883	−0.0840325
$661$	13.5985	0.528920	0.264460	−	0.964397i	$-0.414806\pi$
$661$	13.5985	0.528920	0.264460	+	0.964397i	$0.414806\pi$
$662$	−9.66919	−0.375804
$663$	0	0
$664$	8.43296	0.327262
$665$	−0.335126	−0.0129956
$666$	0.801938	0.0310745
$667$	−4.84787	−0.187710
$668$	14.0610	0.544036
$669$	16.8388	0.651025
$670$	−7.56465	−0.292248
$671$	−13.6504	−0.526968
$672$	1.69202	0.0652711
$673$	−49.7090	−1.91614	−0.958071	−	0.286532i	$-0.907498\pi$
$673$	−49.7090	−1.91614	−0.958071	+	0.286532i	$0.907498\pi$
$674$	−17.3612	−0.668729
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−50.2573	−1.93154	−0.965772	−	0.259394i	$-0.916477\pi$
$677$	−50.2573	−1.93154	−0.965772	+	0.259394i	$0.916477\pi$
$678$	11.8726	0.455965
$679$	0.149145	0.00572366
$680$	2.35690	0.0903828
$681$	−18.3056	−0.701472
$682$	3.11960	0.119456
$683$	−21.5415	−0.824264	−0.412132	−	0.911124i	$-0.635216\pi$
$683$	−21.5415	−0.824264	−0.412132	+	0.911124i	$0.635216\pi$
$684$	0.198062	0.00757310
$685$	5.36227	0.204882
$686$	18.8442	0.719473
$687$	−5.14244	−0.196196
$688$	12.5429	0.478193
$689$	0	0
$690$	3.74094	0.142415
$691$	−50.8165	−1.93315	−0.966576	−	0.256380i	$-0.917470\pi$
$691$	−50.8165	−1.93315	−0.966576	+	0.256380i	$0.917470\pi$
$692$	21.7942	0.828490
$693$	−3.65279	−0.138758
$694$	8.86294	0.336432
$695$	3.12200	0.118424
$696$	−1.29590	−0.0491208
$697$	−4.47757	−0.169600
$698$	2.78554	0.105434
$699$	10.9215	0.413091
$700$	−1.69202	−0.0639524
$701$	−13.2024	−0.498647	−0.249323	−	0.968420i	$-0.580208\pi$
$701$	−13.2024	−0.498647	−0.249323	+	0.968420i	$0.580208\pi$
$702$	0	0
$703$	0.158834	0.00599052
$704$	2.15883	0.0813641
$705$	−8.87800	−0.334365
$706$	21.0640	0.792753
$707$	18.7982	0.706980
$708$	−3.73125	−0.140229
$709$	0.430567	0.0161703	0.00808515	−	0.999967i	$-0.497426\pi$
$709$	0.430567	0.0161703	0.00808515	+	0.999967i	$0.497426\pi$
$710$	−4.18060	−0.156895
$711$	9.40581	0.352746
$712$	−2.41119	−0.0903632
$713$	−5.40581	−0.202449
$714$	3.98792	0.149244
$715$	0	0
$716$	−14.1293	−0.528036
$717$	−19.9041	−0.743332
$718$	17.3274	0.646651
$719$	8.24459	0.307471	0.153736	−	0.988112i	$-0.450870\pi$
$719$	8.24459	0.307471	0.153736	+	0.988112i	$0.450870\pi$
$720$	1.00000	0.0372678
$721$	−16.1558	−0.601675
$722$	−18.9608	−0.705647
$723$	−18.2228	−0.677714
$724$	−5.63773	−0.209524
$725$	1.29590	0.0481284
$726$	6.33944	0.235279
$727$	21.2664	0.788726	0.394363	−	0.918955i	$-0.370965\pi$
$727$	21.2664	0.788726	0.394363	+	0.918955i	$0.370965\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	11.9366	0.441794
$731$	29.5623	1.09340
$732$	6.32304	0.233706
$733$	38.9178	1.43746	0.718731	−	0.695288i	$-0.244723\pi$
$733$	38.9178	1.43746	0.718731	+	0.695288i	$0.244723\pi$
$734$	−30.8049	−1.13703
$735$	4.13706	0.152598
$736$	−3.74094	−0.137893
$737$	−16.3308	−0.601553
$738$	−1.89977	−0.0699316
$739$	4.48486	0.164978	0.0824891	−	0.996592i	$-0.473713\pi$
$739$	4.48486	0.164978	0.0824891	+	0.996592i	$0.473713\pi$
$740$	0.801938	0.0294798
$741$	0	0
$742$	1.70841	0.0627179
$743$	26.8558	0.985242	0.492621	−	0.870244i	$-0.336039\pi$
$743$	26.8558	0.985242	0.492621	+	0.870244i	$0.336039\pi$
$744$	−1.44504	−0.0529778
$745$	4.24160	0.155400
$746$	−25.6668	−0.939728
$747$	8.43296	0.308546
$748$	5.08815	0.186041
$749$	−30.7047	−1.12193
$750$	−1.00000	−0.0365148
$751$	−5.64012	−0.205811	−0.102905	−	0.994691i	$-0.532814\pi$
$751$	−5.64012	−0.205811	−0.102905	+	0.994691i	$0.532814\pi$
$752$	8.87800	0.323747
$753$	17.0030	0.619623
$754$	0	0
$755$	22.9148	0.833956
$756$	1.69202	0.0615382
$757$	−35.6644	−1.29624	−0.648122	−	0.761536i	$-0.724445\pi$
$757$	−35.6644	−1.29624	−0.648122	+	0.761536i	$0.724445\pi$
$758$	5.46144	0.198368
$759$	8.07606	0.293143
$760$	0.198062	0.00718447
$761$	2.68425	0.0973041	0.0486520	−	0.998816i	$-0.484507\pi$
$761$	2.68425	0.0973041	0.0486520	+	0.998816i	$0.484507\pi$
$762$	−13.0097	−0.471291
$763$	3.80194	0.137639
$764$	−2.59419	−0.0938544
$765$	2.35690	0.0852137
$766$	−25.9221	−0.936605
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	48.7644	1.75849	0.879244	−	0.476372i	$-0.158048\pi$
$769$	48.7644	1.75849	0.879244	+	0.476372i	$0.158048\pi$
$770$	−3.65279	−0.131638
$771$	7.65519	0.275695
$772$	−3.98361	−0.143373
$773$	−46.6152	−1.67663	−0.838316	−	0.545184i	$-0.816460\pi$
$773$	−46.6152	−1.67663	−0.838316	+	0.545184i	$0.816460\pi$
$774$	12.5429	0.450844
$775$	1.44504	0.0519074
$776$	−0.0881460	−0.00316426
$777$	1.35690	0.0486784
$778$	−32.0006	−1.14728
$779$	−0.376273	−0.0134814
$780$	0	0
$781$	−9.02523	−0.322948
$782$	−8.81700	−0.315295
$783$	−1.29590	−0.0463116
$784$	−4.13706	−0.147752
$785$	16.0707	0.573587
$786$	7.82908	0.279254
$787$	−27.4045	−0.976864	−0.488432	−	0.872602i	$-0.662431\pi$
$787$	−27.4045	−0.976864	−0.488432	+	0.872602i	$0.662431\pi$
$788$	−6.65040	−0.236911
$789$	−21.0411	−0.749085
$790$	9.40581	0.334644
$791$	20.0887	0.714273
$792$	2.15883	0.0767108
$793$	0	0
$794$	0.457123	0.0162227
$795$	1.00969	0.0358100
$796$	−1.01639	−0.0360251
$797$	−5.70278	−0.202003	−0.101001	−	0.994886i	$-0.532205\pi$
$797$	−5.70278	−0.202003	−0.101001	+	0.994886i	$0.532205\pi$
$798$	0.335126	0.0118633
$799$	20.9245	0.740257
$800$	1.00000	0.0353553
$801$	−2.41119	−0.0851952
$802$	23.2553	0.821175
$803$	25.7692	0.909374
$804$	7.56465	0.266785
$805$	6.32975	0.223094
$806$	0	0
$807$	−24.3183	−0.856043
$808$	−11.1099	−0.390845
$809$	−29.6165	−1.04126	−0.520631	−	0.853782i	$-0.674303\pi$
$809$	−29.6165	−1.04126	−0.520631	+	0.853782i	$0.674303\pi$
$810$	1.00000	0.0351364
$811$	−48.8901	−1.71676	−0.858382	−	0.513012i	$-0.828530\pi$
$811$	−48.8901	−1.71676	−0.858382	+	0.513012i	$0.828530\pi$
$812$	−2.19269	−0.0769482
$813$	−10.9922	−0.385514
$814$	1.73125	0.0606803
$815$	−0.488582	−0.0171143
$816$	−2.35690	−0.0825079
$817$	2.48427	0.0869136
$818$	39.0629	1.36580
$819$	0	0
$820$	−1.89977	−0.0663429
$821$	−19.7942	−0.690821	−0.345411	−	0.938452i	$-0.612260\pi$
$821$	−19.7942	−0.690821	−0.345411	+	0.938452i	$0.612260\pi$
$822$	−5.36227	−0.187031
$823$	49.6118	1.72936	0.864679	−	0.502325i	$-0.167522\pi$
$823$	49.6118	1.72936	0.864679	+	0.502325i	$0.167522\pi$
$824$	9.54825	0.332629
$825$	−2.15883	−0.0751609
$826$	−6.31336	−0.219670
$827$	26.0683	0.906483	0.453242	−	0.891388i	$-0.350267\pi$
$827$	26.0683	0.906483	0.453242	+	0.891388i	$0.350267\pi$
$828$	−3.74094	−0.130007
$829$	46.2215	1.60534	0.802669	−	0.596424i	$-0.203412\pi$
$829$	46.2215	1.60534	0.802669	+	0.596424i	$0.203412\pi$
$830$	8.43296	0.292712
$831$	11.4601	0.397547
$832$	0	0
$833$	−9.75063	−0.337839
$834$	−3.12200	−0.108106
$835$	14.0610	0.486601
$836$	0.427583	0.0147883
$837$	−1.44504	−0.0499480
$838$	−18.1269	−0.626183
$839$	33.2693	1.14859	0.574293	−	0.818650i	$-0.305277\pi$
$839$	33.2693	1.14859	0.574293	+	0.818650i	$0.305277\pi$
$840$	1.69202	0.0583803
$841$	−27.3207	−0.942091
$842$	−20.4916	−0.706186
$843$	9.88231	0.340365
$844$	10.1414	0.349081
$845$	0	0
$846$	8.87800	0.305232
$847$	10.7265	0.368566
$848$	−1.00969	−0.0346729
$849$	21.3207	0.731723
$850$	2.35690	0.0808409
$851$	−3.00000	−0.102839
$852$	4.18060	0.143225
$853$	−39.3002	−1.34561	−0.672807	−	0.739818i	$-0.734911\pi$
$853$	−39.3002	−1.34561	−0.672807	+	0.739818i	$0.734911\pi$
$854$	10.6987	0.366103
$855$	0.198062	0.00677359
$856$	18.1468	0.620243
$857$	−39.7566	−1.35806	−0.679030	−	0.734111i	$-0.737599\pi$
$857$	−39.7566	−1.35806	−0.679030	+	0.734111i	$0.737599\pi$
$858$	0	0
$859$	−3.43860	−0.117324	−0.0586618	−	0.998278i	$-0.518683\pi$
$859$	−3.43860	−0.117324	−0.0586618	+	0.998278i	$0.518683\pi$
$860$	12.5429	0.427709
$861$	−3.21446	−0.109548
$862$	−22.2989	−0.759503
$863$	21.2301	0.722681	0.361341	−	0.932434i	$-0.382319\pi$
$863$	21.2301	0.722681	0.361341	+	0.932434i	$0.382319\pi$
$864$	−1.00000	−0.0340207
$865$	21.7942	0.741024
$866$	7.47650	0.254062
$867$	11.4450	0.388694
$868$	−2.44504	−0.0829901
$869$	20.3056	0.688820
$870$	−1.29590	−0.0439350
$871$	0	0
$872$	−2.24698	−0.0760923
$873$	−0.0881460	−0.00298329
$874$	−0.740939	−0.0250626
$875$	−1.69202	−0.0572008
$876$	−11.9366	−0.403301
$877$	−5.74227	−0.193903	−0.0969513	−	0.995289i	$-0.530909\pi$
$877$	−5.74227	−0.193903	−0.0969513	+	0.995289i	$0.530909\pi$
$878$	−19.3924	−0.654462
$879$	18.7090	0.631039
$880$	2.15883	0.0727743
$881$	12.6069	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$881$	12.6069	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$882$	−4.13706	−0.139302
$883$	25.8086	0.868530	0.434265	−	0.900785i	$-0.357008\pi$
$883$	25.8086	0.868530	0.434265	+	0.900785i	$0.357008\pi$
$884$	0	0
$885$	−3.73125	−0.125425
$886$	10.0935	0.339099
$887$	−20.2392	−0.679566	−0.339783	−	0.940504i	$-0.610354\pi$
$887$	−20.2392	−0.679566	−0.339783	+	0.940504i	$0.610354\pi$
$888$	−0.801938	−0.0269113
$889$	−22.0127	−0.738281
$890$	−2.41119	−0.0808233
$891$	2.15883	0.0723236
$892$	−16.8388	−0.563804
$893$	1.75840	0.0588425
$894$	−4.24160	−0.141860
$895$	−14.1293	−0.472290
$896$	−1.69202	−0.0565265
$897$	0	0
$898$	−2.78448	−0.0929193
$899$	1.87263	0.0624556
$900$	1.00000	0.0333333
$901$	−2.37973	−0.0792803
$902$	−4.10129	−0.136558
$903$	21.2228	0.706251
$904$	−11.8726	−0.394878
$905$	−5.63773	−0.187404
$906$	−22.9148	−0.761294
$907$	−3.52350	−0.116996	−0.0584979	−	0.998288i	$-0.518631\pi$
$907$	−3.52350	−0.116996	−0.0584979	+	0.998288i	$0.518631\pi$
$908$	18.3056	0.607492
$909$	−11.1099	−0.368493
$910$	0	0
$911$	−34.7426	−1.15107	−0.575537	−	0.817776i	$-0.695207\pi$
$911$	−34.7426	−1.15107	−0.575537	+	0.817776i	$0.695207\pi$
$912$	−0.198062	−0.00655850
$913$	18.2054	0.602509
$914$	−13.1056	−0.433495
$915$	6.32304	0.209033
$916$	5.14244	0.169911
$917$	13.2470	0.437454
$918$	−2.35690	−0.0777892
$919$	−24.1430	−0.796405	−0.398203	−	0.917298i	$-0.630366\pi$
$919$	−24.1430	−0.796405	−0.398203	+	0.917298i	$0.630366\pi$
$920$	−3.74094	−0.123335
$921$	−17.2241	−0.567555
$922$	16.2174	0.534093
$923$	0	0
$924$	3.65279	0.120168
$925$	0.801938	0.0263676
$926$	15.5415	0.510727
$927$	9.54825	0.313606
$928$	1.29590	0.0425399
$929$	−56.9700	−1.86912	−0.934562	−	0.355800i	$-0.884209\pi$
$929$	−56.9700	−1.86912	−0.934562	+	0.355800i	$0.884209\pi$
$930$	−1.44504	−0.0473848
$931$	−0.819396	−0.0268546
$932$	−10.9215	−0.357747
$933$	13.5200	0.442626
$934$	−10.1578	−0.332372
$935$	5.08815	0.166400
$936$	0	0
$937$	−45.7966	−1.49611	−0.748054	−	0.663638i	$-0.769012\pi$
$937$	−45.7966	−1.49611	−0.748054	+	0.663638i	$0.769012\pi$
$938$	12.7995	0.417920
$939$	−24.3303	−0.793991
$940$	8.87800	0.289569
$941$	−7.10262	−0.231539	−0.115769	−	0.993276i	$-0.536933\pi$
$941$	−7.10262	−0.231539	−0.115769	+	0.993276i	$0.536933\pi$
$942$	−16.0707	−0.523611
$943$	7.10693	0.231434
$944$	3.73125	0.121442
$945$	1.69202	0.0550415
$946$	27.0780	0.880381
$947$	−11.2524	−0.365652	−0.182826	−	0.983145i	$-0.558525\pi$
$947$	−11.2524	−0.365652	−0.182826	+	0.983145i	$0.558525\pi$
$948$	−9.40581	−0.305487
$949$	0	0
$950$	0.198062	0.00642599
$951$	−34.7821	−1.12789
$952$	−3.98792	−0.129249
$953$	−1.61702	−0.0523805	−0.0261902	−	0.999657i	$-0.508338\pi$
$953$	−1.61702	−0.0523805	−0.0261902	+	0.999657i	$0.508338\pi$
$954$	−1.00969	−0.0326899
$955$	−2.59419	−0.0839459
$956$	19.9041	0.643744
$957$	−2.79763	−0.0904344
$958$	31.4263	1.01534
$959$	−9.07308	−0.292985
$960$	−1.00000	−0.0322749
$961$	−28.9119	−0.932640
$962$	0	0
$963$	18.1468	0.584771
$964$	18.2228	0.586917
$965$	−3.98361	−0.128237
$966$	−6.32975	−0.203656
$967$	−24.1588	−0.776896	−0.388448	−	0.921471i	$-0.626989\pi$
$967$	−24.1588	−0.776896	−0.388448	+	0.921471i	$0.626989\pi$
$968$	−6.33944	−0.203757
$969$	−0.466812	−0.0149962
$970$	−0.0881460	−0.00283020
$971$	43.2669	1.38850	0.694251	−	0.719733i	$-0.255736\pi$
$971$	43.2669	1.38850	0.694251	+	0.719733i	$0.255736\pi$
$972$	−1.00000	−0.0320750
$973$	−5.28249	−0.169349
$974$	−24.1323	−0.773248
$975$	0	0
$976$	−6.32304	−0.202396
$977$	24.9782	0.799124	0.399562	−	0.916706i	$-0.369162\pi$
$977$	24.9782	0.799124	0.399562	+	0.916706i	$0.369162\pi$
$978$	0.488582	0.0156231
$979$	−5.20536	−0.166364
$980$	−4.13706	−0.132154
$981$	−2.24698	−0.0717405
$982$	−19.1099	−0.609822
$983$	−19.3134	−0.616000	−0.308000	−	0.951386i	$-0.599660\pi$
$983$	−19.3134	−0.616000	−0.308000	+	0.951386i	$0.599660\pi$
$984$	1.89977	0.0605625
$985$	−6.65040	−0.211899
$986$	3.05429	0.0972685
$987$	15.0218	0.478148
$988$	0	0
$989$	−46.9221	−1.49204
$990$	2.15883	0.0686122
$991$	−27.5090	−0.873853	−0.436926	−	0.899497i	$-0.643933\pi$
$991$	−27.5090	−0.873853	−0.436926	+	0.899497i	$0.643933\pi$
$992$	1.44504	0.0458801
$993$	9.66919	0.306842
$994$	7.07367	0.224363
$995$	−1.01639	−0.0322218
$996$	−8.43296	−0.267209
$997$	−23.1933	−0.734538	−0.367269	−	0.930115i	$-0.619707\pi$
$997$	−23.1933	−0.734538	−0.367269	+	0.930115i	$0.619707\pi$
$998$	−34.9724	−1.10703
$999$	−0.801938	−0.0253722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bu.1.1	yes	3
13.5	odd	4		5070.2.b.t.1351.1		6
13.8	odd	4		5070.2.b.t.1351.6		6
13.12	even	2		5070.2.a.bj.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bj.1.3	✓	3	13.12	even	2
5070.2.a.bu.1.1	yes	3	1.1	even	1	trivial
5070.2.b.t.1351.1		6	13.5	odd	4
5070.2.b.t.1351.6		6	13.8	odd	4

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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.15883 q^{11} -1.00000 q^{12} -1.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.35690 q^{17} +1.00000 q^{18} +0.198062 q^{19} +1.00000 q^{20} +1.69202 q^{21} +2.15883 q^{22} -3.74094 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +1.29590 q^{29} -1.00000 q^{30} +1.44504 q^{31} +1.00000 q^{32} -2.15883 q^{33} +2.35690 q^{34} -1.69202 q^{35} +1.00000 q^{36} +0.801938 q^{37} +0.198062 q^{38} +1.00000 q^{40} -1.89977 q^{41} +1.69202 q^{42} +12.5429 q^{43} +2.15883 q^{44} +1.00000 q^{45} -3.74094 q^{46} +8.87800 q^{47} -1.00000 q^{48} -4.13706 q^{49} +1.00000 q^{50} -2.35690 q^{51} -1.00969 q^{53} -1.00000 q^{54} +2.15883 q^{55} -1.69202 q^{56} -0.198062 q^{57} +1.29590 q^{58} +3.73125 q^{59} -1.00000 q^{60} -6.32304 q^{61} +1.44504 q^{62} -1.69202 q^{63} +1.00000 q^{64} -2.15883 q^{66} -7.56465 q^{67} +2.35690 q^{68} +3.74094 q^{69} -1.69202 q^{70} -4.18060 q^{71} +1.00000 q^{72} +11.9366 q^{73} +0.801938 q^{74} -1.00000 q^{75} +0.198062 q^{76} -3.65279 q^{77} +9.40581 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.89977 q^{82} +8.43296 q^{83} +1.69202 q^{84} +2.35690 q^{85} +12.5429 q^{86} -1.29590 q^{87} +2.15883 q^{88} -2.41119 q^{89} +1.00000 q^{90} -3.74094 q^{92} -1.44504 q^{93} +8.87800 q^{94} +0.198062 q^{95} -1.00000 q^{96} -0.0881460 q^{97} -4.13706 q^{98} +2.15883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 2 q^{11} - 3 q^{12} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 5 q^{19} + 3 q^{20} - 2 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 3 q^{30} + 4 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 3 q^{36} - 2 q^{37} + 5 q^{38} + 3 q^{40} + 17 q^{41} + 19 q^{43} - 2 q^{44} + 3 q^{45} + 3 q^{46} + 7 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} - 2 q^{55} - 5 q^{57} - 10 q^{58} + 19 q^{59} - 3 q^{60} + q^{61} + 4 q^{62} + 3 q^{64} + 2 q^{66} - q^{67} + 3 q^{68} - 3 q^{69} - q^{71} + 3 q^{72} - 15 q^{73} - 2 q^{74} - 3 q^{75} + 5 q^{76} + 7 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} + 6 q^{83} + 3 q^{85} + 19 q^{86} + 10 q^{87} - 2 q^{88} + 9 q^{89} + 3 q^{90} + 3 q^{92} - 4 q^{93} + 7 q^{94} + 5 q^{95} - 3 q^{96} - 4 q^{97} - 7 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 2 * q^11 - 3 * q^12 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 5 * q^19 + 3 * q^20 - 2 * q^22 + 3 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 10 * q^29 - 3 * q^30 + 4 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 3 * q^36 - 2 * q^37 + 5 * q^38 + 3 * q^40 + 17 * q^41 + 19 * q^43 - 2 * q^44 + 3 * q^45 + 3 * q^46 + 7 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 - 2 * q^55 - 5 * q^57 - 10 * q^58 + 19 * q^59 - 3 * q^60 + q^61 + 4 * q^62 + 3 * q^64 + 2 * q^66 - q^67 + 3 * q^68 - 3 * q^69 - q^71 + 3 * q^72 - 15 * q^73 - 2 * q^74 - 3 * q^75 + 5 * q^76 + 7 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 17 * q^82 + 6 * q^83 + 3 * q^85 + 19 * q^86 + 10 * q^87 - 2 * q^88 + 9 * q^89 + 3 * q^90 + 3 * q^92 - 4 * q^93 + 7 * q^94 + 5 * q^95 - 3 * q^96 - 4 * q^97 - 7 * q^98 - 2 * q^99

Introduction

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