LMFDB - Embedded newform 5070.2.b.u.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			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.u.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.554958i q^{11} +1.00000 q^{12} -1.19806 q^{14} +1.00000i q^{15} +1.00000 q^{16} +6.45473 q^{17} +1.00000i q^{18} +5.40581i q^{19} +1.00000i q^{20} -1.19806i q^{21} -0.554958 q^{22} -7.96077 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.19806i q^{28} -6.89977 q^{29} -1.00000 q^{30} -3.04892i q^{31} +1.00000i q^{32} -0.554958i q^{33} +6.45473i q^{34} +1.19806 q^{35} -1.00000 q^{36} -7.96615i q^{37} -5.40581 q^{38} -1.00000 q^{40} -3.08815i q^{41} +1.19806 q^{42} -1.15883 q^{43} -0.554958i q^{44} -1.00000i q^{45} -7.96077i q^{46} +12.2620i q^{47} -1.00000 q^{48} +5.56465 q^{49} -1.00000i q^{50} -6.45473 q^{51} +8.47219 q^{53} -1.00000i q^{54} +0.554958 q^{55} +1.19806 q^{56} -5.40581i q^{57} -6.89977i q^{58} +1.45712i q^{59} -1.00000i q^{60} +2.76271 q^{61} +3.04892 q^{62} +1.19806i q^{63} -1.00000 q^{64} +0.554958 q^{66} +3.34481i q^{67} -6.45473 q^{68} +7.96077 q^{69} +1.19806i q^{70} -2.35690i q^{71} -1.00000i q^{72} +13.3448i q^{73} +7.96615 q^{74} +1.00000 q^{75} -5.40581i q^{76} -0.664874 q^{77} -0.990311 q^{79} -1.00000i q^{80} +1.00000 q^{81} +3.08815 q^{82} -5.88471i q^{83} +1.19806i q^{84} -6.45473i q^{85} -1.15883i q^{86} +6.89977 q^{87} +0.554958 q^{88} +13.7845i q^{89} +1.00000 q^{90} +7.96077 q^{92} +3.04892i q^{93} -12.2620 q^{94} +5.40581 q^{95} -1.00000i q^{96} -7.07606i q^{97} +5.56465i q^{98} +0.554958i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 16 q^{14} + 6 q^{16} - 6 q^{17} - 4 q^{22} - 22 q^{23} - 6 q^{25} - 6 q^{27} + 4 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} - 6 q^{38} - 6 q^{40} + 16 q^{42} + 10 q^{43} - 6 q^{48} - 10 q^{49} + 6 q^{51} + 38 q^{53} + 4 q^{55} + 16 q^{56} - 18 q^{61} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 22 q^{69} + 16 q^{74} + 6 q^{75} - 6 q^{77} - 50 q^{79} + 6 q^{81} + 26 q^{82} - 4 q^{87} + 4 q^{88} + 6 q^{90} + 22 q^{92} - 14 q^{94} + 6 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 - 6 * q^17 - 4 * q^22 - 22 * q^23 - 6 * q^25 - 6 * q^27 + 4 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 - 6 * q^38 - 6 * q^40 + 16 * q^42 + 10 * q^43 - 6 * q^48 - 10 * q^49 + 6 * q^51 + 38 * q^53 + 4 * q^55 + 16 * q^56 - 18 * q^61 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 22 * q^69 + 16 * q^74 + 6 * q^75 - 6 * q^77 - 50 * q^79 + 6 * q^81 + 26 * q^82 - 4 * q^87 + 4 * q^88 + 6 * q^90 + 22 * q^92 - 14 * q^94 + 6 * 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
$7$	1.19806i	0.452825i	0.974032	+	0.226412i	$0.0726997\pi$
$7$	1.19806i	0.452825i	−0.974032	+	0.226412i	$0.927300\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0.554958i	0.167326i	0.996494	+	0.0836631i	$0.0266620\pi$
$11$	0.554958i	0.167326i	−0.996494	+	0.0836631i	$0.973338\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−1.19806	−0.320196
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	6.45473	1.56550	0.782751	−	0.622335i	$-0.213816\pi$
$17$	6.45473	1.56550	0.782751	+	0.622335i	$0.213816\pi$
$18$	1.00000i	0.235702i
$19$	5.40581i	1.24018i	0.784531	+	0.620089i	$0.212904\pi$
$19$	5.40581i	1.24018i	−0.784531	+	0.620089i	$0.787096\pi$
$20$	1.00000i	0.223607i
$21$	− 1.19806i	− 0.261439i
$22$	−0.554958	−0.118317
$23$	−7.96077	−1.65994	−0.829968	−	0.557811i	$-0.811641\pi$
$23$	−7.96077	−1.65994	−0.829968	+	0.557811i	$0.811641\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 1.19806i	− 0.226412i
$29$	−6.89977	−1.28126	−0.640628	−	0.767852i	$-0.721326\pi$
$29$	−6.89977	−1.28126	−0.640628	+	0.767852i	$0.721326\pi$
$30$	−1.00000	−0.182574
$31$	− 3.04892i	− 0.547602i	−0.961786	−	0.273801i	$-0.911719\pi$
$31$	− 3.04892i	− 0.547602i	0.961786	−	0.273801i	$-0.0882809\pi$
$32$	1.00000i	0.176777i
$33$	− 0.554958i	− 0.0966058i
$34$	6.45473i	1.10698i
$35$	1.19806	0.202509
$36$	−1.00000	−0.166667
$37$	− 7.96615i	− 1.30963i	−0.755791	−	0.654813i	$-0.772747\pi$
$37$	− 7.96615i	− 1.30963i	0.755791	−	0.654813i	$-0.227253\pi$
$38$	−5.40581	−0.876939
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 3.08815i	− 0.482287i	−0.970489	−	0.241144i	$-0.922477\pi$
$41$	− 3.08815i	− 0.482287i	0.970489	−	0.241144i	$-0.0775225\pi$
$42$	1.19806	0.184865
$43$	−1.15883	−0.176720	−0.0883602	−	0.996089i	$-0.528163\pi$
$43$	−1.15883	−0.176720	−0.0883602	+	0.996089i	$0.528163\pi$
$44$	− 0.554958i	− 0.0836631i
$45$	− 1.00000i	− 0.149071i
$46$	− 7.96077i	− 1.17375i
$47$	12.2620i	1.78860i	0.447465	+	0.894302i	$0.352327\pi$
$47$	12.2620i	1.78860i	−0.447465	+	0.894302i	$0.647673\pi$
$48$	−1.00000	−0.144338
$49$	5.56465	0.794950
$50$	− 1.00000i	− 0.141421i
$51$	−6.45473	−0.903843
$52$	0	0
$53$	8.47219	1.16374	0.581872	−	0.813280i	$-0.302320\pi$
$53$	8.47219	1.16374	0.581872	+	0.813280i	$0.302320\pi$
$54$	− 1.00000i	− 0.136083i
$55$	0.554958	0.0748305
$56$	1.19806	0.160098
$57$	− 5.40581i	− 0.716017i
$58$	− 6.89977i	− 0.905985i
$59$	1.45712i	0.189701i	0.995492	+	0.0948507i	$0.0302374\pi$
$59$	1.45712i	0.189701i	−0.995492	+	0.0948507i	$0.969763\pi$
$60$	− 1.00000i	− 0.129099i
$61$	2.76271	0.353729	0.176864	−	0.984235i	$-0.443405\pi$
$61$	2.76271	0.353729	0.176864	+	0.984235i	$0.443405\pi$
$62$	3.04892	0.387213
$63$	1.19806i	0.150942i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0.554958	0.0683106
$67$	3.34481i	0.408634i	0.978905	+	0.204317i	$0.0654973\pi$
$67$	3.34481i	0.408634i	−0.978905	+	0.204317i	$0.934503\pi$
$68$	−6.45473	−0.782751
$69$	7.96077	0.958364
$70$	1.19806i	0.143196i
$71$	− 2.35690i	− 0.279712i	−0.990172	−	0.139856i	$-0.955336\pi$
$71$	− 2.35690i	− 0.279712i	0.990172	−	0.139856i	$-0.0446640\pi$
$72$	− 1.00000i	− 0.117851i
$73$	13.3448i	1.56189i	0.624598	+	0.780946i	$0.285263\pi$
$73$	13.3448i	1.56189i	−0.624598	+	0.780946i	$0.714737\pi$
$74$	7.96615	0.926046
$75$	1.00000	0.115470
$76$	− 5.40581i	− 0.620089i
$77$	−0.664874	−0.0757695
$78$	0	0
$79$	−0.990311	−0.111419	−0.0557094	−	0.998447i	$-0.517742\pi$
$79$	−0.990311	−0.111419	−0.0557094	+	0.998447i	$0.517742\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	3.08815	0.341029
$83$	− 5.88471i	− 0.645931i	−0.946411	−	0.322965i	$-0.895320\pi$
$83$	− 5.88471i	− 0.645931i	0.946411	−	0.322965i	$-0.104680\pi$
$84$	1.19806i	0.130719i
$85$	− 6.45473i	− 0.700114i
$86$	− 1.15883i	− 0.124960i
$87$	6.89977	0.739733
$88$	0.554958	0.0591587
$89$	13.7845i	1.46115i	0.682831	+	0.730576i	$0.260748\pi$
$89$	13.7845i	1.46115i	−0.682831	+	0.730576i	$0.739252\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	7.96077	0.829968
$93$	3.04892i	0.316158i
$94$	−12.2620	−1.26473
$95$	5.40581	0.554625
$96$	− 1.00000i	− 0.102062i
$97$	− 7.07606i	− 0.718465i	−0.933248	−	0.359233i	$-0.883038\pi$
$97$	− 7.07606i	− 0.718465i	0.933248	−	0.359233i	$-0.116962\pi$
$98$	5.56465i	0.562114i
$99$	0.554958i	0.0557754i
$100$	1.00000	0.100000
$101$	−4.09783	−0.407750	−0.203875	−	0.978997i	$-0.565354\pi$
$101$	−4.09783	−0.407750	−0.203875	+	0.978997i	$0.565354\pi$
$102$	− 6.45473i	− 0.639114i
$103$	4.16421	0.410312	0.205156	−	0.978729i	$-0.434230\pi$
$103$	4.16421	0.410312	0.205156	+	0.978729i	$0.434230\pi$
$104$	0	0
$105$	−1.19806	−0.116919
$106$	8.47219i	0.822892i
$107$	−4.98254	−0.481680	−0.240840	−	0.970565i	$-0.577423\pi$
$107$	−4.98254	−0.481680	−0.240840	+	0.970565i	$0.577423\pi$
$108$	1.00000	0.0962250
$109$	− 1.31336i	− 0.125797i	−0.998020	−	0.0628983i	$-0.979966\pi$
$109$	− 1.31336i	− 0.125797i	0.998020	−	0.0628983i	$-0.0200344\pi$
$110$	0.554958i	0.0529132i
$111$	7.96615i	0.756113i
$112$	1.19806i	0.113206i
$113$	3.55496	0.334422	0.167211	−	0.985921i	$-0.446524\pi$
$113$	3.55496	0.334422	0.167211	+	0.985921i	$0.446524\pi$
$114$	5.40581	0.506301
$115$	7.96077i	0.742346i
$116$	6.89977	0.640628
$117$	0	0
$118$	−1.45712	−0.134139
$119$	7.73317i	0.708898i
$120$	1.00000	0.0912871
$121$	10.6920	0.972002
$122$	2.76271i	0.250124i
$123$	3.08815i	0.278449i
$124$	3.04892i	0.273801i
$125$	1.00000i	0.0894427i
$126$	−1.19806	−0.106732
$127$	−21.8756	−1.94115	−0.970573	−	0.240806i	$-0.922588\pi$
$127$	−21.8756	−1.94115	−0.970573	+	0.240806i	$0.922588\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	1.15883	0.102030
$130$	0	0
$131$	−21.6528	−1.89181	−0.945907	−	0.324439i	$-0.894825\pi$
$131$	−21.6528	−1.89181	−0.945907	+	0.324439i	$0.894825\pi$
$132$	0.554958i	0.0483029i
$133$	−6.47650	−0.561584
$134$	−3.34481	−0.288948
$135$	1.00000i	0.0860663i
$136$	− 6.45473i	− 0.553489i
$137$	7.08815i	0.605581i	0.953057	+	0.302791i	$0.0979183\pi$
$137$	7.08815i	0.605581i	−0.953057	+	0.302791i	$0.902082\pi$
$138$	7.96077i	0.677666i
$139$	0.340502	0.0288810	0.0144405	−	0.999896i	$-0.495403\pi$
$139$	0.340502	0.0288810	0.0144405	+	0.999896i	$0.495403\pi$
$140$	−1.19806	−0.101255
$141$	− 12.2620i	− 1.03265i
$142$	2.35690	0.197786
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.89977i	0.572995i
$146$	−13.3448	−1.10442
$147$	−5.56465	−0.458964
$148$	7.96615i	0.654813i
$149$	− 7.40581i	− 0.606708i	−0.952878	−	0.303354i	$-0.901894\pi$
$149$	− 7.40581i	− 0.606708i	0.952878	−	0.303354i	$-0.0981064\pi$
$150$	1.00000i	0.0816497i
$151$	10.4209i	0.848039i	0.905653	+	0.424020i	$0.139381\pi$
$151$	10.4209i	0.848039i	−0.905653	+	0.424020i	$0.860619\pi$
$152$	5.40581	0.438469
$153$	6.45473	0.521834
$154$	− 0.664874i	− 0.0535771i
$155$	−3.04892	−0.244895
$156$	0	0
$157$	−19.0151	−1.51757	−0.758784	−	0.651343i	$-0.774206\pi$
$157$	−19.0151	−1.51757	−0.758784	+	0.651343i	$0.774206\pi$
$158$	− 0.990311i	− 0.0787849i
$159$	−8.47219	−0.671888
$160$	1.00000	0.0790569
$161$	− 9.53750i	− 0.751660i
$162$	1.00000i	0.0785674i
$163$	5.65279i	0.442761i	0.975188	+	0.221380i	$0.0710563\pi$
$163$	5.65279i	0.442761i	−0.975188	+	0.221380i	$0.928944\pi$
$164$	3.08815i	0.241144i
$165$	−0.554958	−0.0432034
$166$	5.88471	0.456742
$167$	2.15883i	0.167056i	0.996505	+	0.0835278i	$0.0266187\pi$
$167$	2.15883i	0.167056i	−0.996505	+	0.0835278i	$0.973381\pi$
$168$	−1.19806	−0.0924325
$169$	0	0
$170$	6.45473	0.495055
$171$	5.40581i	0.413393i
$172$	1.15883	0.0883602
$173$	−21.6722	−1.64770	−0.823852	−	0.566805i	$-0.808179\pi$
$173$	−21.6722	−1.64770	−0.823852	+	0.566805i	$0.808179\pi$
$174$	6.89977i	0.523070i
$175$	− 1.19806i	− 0.0905650i
$176$	0.554958i	0.0418315i
$177$	− 1.45712i	− 0.109524i
$178$	−13.7845	−1.03319
$179$	0.956459	0.0714891	0.0357446	−	0.999361i	$-0.488620\pi$
$179$	0.956459	0.0714891	0.0357446	+	0.999361i	$0.488620\pi$
$180$	1.00000i	0.0745356i
$181$	−0.405813	−0.0301639	−0.0150819	−	0.999886i	$-0.504801\pi$
$181$	−0.405813	−0.0301639	−0.0150819	+	0.999886i	$0.504801\pi$
$182$	0	0
$183$	−2.76271	−0.204225
$184$	7.96077i	0.586876i
$185$	−7.96615	−0.585683
$186$	−3.04892	−0.223557
$187$	3.58211i	0.261949i
$188$	− 12.2620i	− 0.894302i
$189$	− 1.19806i	− 0.0871462i
$190$	5.40581i	0.392179i
$191$	−19.0097	−1.37549	−0.687746	−	0.725951i	$-0.741400\pi$
$191$	−19.0097	−1.37549	−0.687746	+	0.725951i	$0.741400\pi$
$192$	1.00000	0.0721688
$193$	0.599564i	0.0431575i	0.999767	+	0.0215788i	$0.00686927\pi$
$193$	0.599564i	0.0431575i	−0.999767	+	0.0215788i	$0.993131\pi$
$194$	7.07606	0.508032
$195$	0	0
$196$	−5.56465	−0.397475
$197$	− 3.78448i	− 0.269633i	−0.990871	−	0.134816i	$-0.956956\pi$
$197$	− 3.78448i	− 0.269633i	0.990871	−	0.134816i	$-0.0430445\pi$
$198$	−0.554958	−0.0394392
$199$	10.2349	0.725533	0.362766	−	0.931880i	$-0.381832\pi$
$199$	10.2349	0.725533	0.362766	+	0.931880i	$0.381832\pi$
$200$	1.00000i	0.0707107i
$201$	− 3.34481i	− 0.235925i
$202$	− 4.09783i	− 0.288323i
$203$	− 8.26636i	− 0.580185i
$204$	6.45473	0.451922
$205$	−3.08815	−0.215685
$206$	4.16421i	0.290134i
$207$	−7.96077	−0.553312
$208$	0	0
$209$	−3.00000	−0.207514
$210$	− 1.19806i	− 0.0826742i
$211$	−8.71379	−0.599882	−0.299941	−	0.953958i	$-0.596967\pi$
$211$	−8.71379	−0.599882	−0.299941	+	0.953958i	$0.596967\pi$
$212$	−8.47219	−0.581872
$213$	2.35690i	0.161492i
$214$	− 4.98254i	− 0.340600i
$215$	1.15883i	0.0790318i
$216$	1.00000i	0.0680414i
$217$	3.65279	0.247968
$218$	1.31336	0.0889516
$219$	− 13.3448i	− 0.901759i
$220$	−0.554958	−0.0374153
$221$	0	0
$222$	−7.96615	−0.534653
$223$	15.0694i	1.00912i	0.863377	+	0.504559i	$0.168345\pi$
$223$	15.0694i	1.00912i	−0.863377	+	0.504559i	$0.831655\pi$
$224$	−1.19806	−0.0800489
$225$	−1.00000	−0.0666667
$226$	3.55496i	0.236472i
$227$	20.4577i	1.35783i	0.734219	+	0.678913i	$0.237549\pi$
$227$	20.4577i	1.35783i	−0.734219	+	0.678913i	$0.762451\pi$
$228$	5.40581i	0.358009i
$229$	18.7657i	1.24007i	0.784573	+	0.620036i	$0.212882\pi$
$229$	18.7657i	1.24007i	−0.784573	+	0.620036i	$0.787118\pi$
$230$	−7.96077	−0.524918
$231$	0.664874	0.0437455
$232$	6.89977i	0.452992i
$233$	−23.6353	−1.54840	−0.774201	−	0.632940i	$-0.781848\pi$
$233$	−23.6353	−1.54840	−0.774201	+	0.632940i	$0.781848\pi$
$234$	0	0
$235$	12.2620	0.799888
$236$	− 1.45712i	− 0.0948507i
$237$	0.990311	0.0643276
$238$	−7.73317	−0.501267
$239$	− 28.7506i	− 1.85972i	−0.367909	−	0.929862i	$-0.619926\pi$
$239$	− 28.7506i	− 1.85972i	0.367909	−	0.929862i	$-0.380074\pi$
$240$	1.00000i	0.0645497i
$241$	− 3.89440i	− 0.250860i	−0.992102	−	0.125430i	$-0.959969\pi$
$241$	− 3.89440i	− 0.250860i	0.992102	−	0.125430i	$-0.0400311\pi$
$242$	10.6920i	0.687309i
$243$	−1.00000	−0.0641500
$244$	−2.76271	−0.176864
$245$	− 5.56465i	− 0.355512i
$246$	−3.08815	−0.196893
$247$	0	0
$248$	−3.04892	−0.193606
$249$	5.88471i	0.372928i
$250$	−1.00000	−0.0632456
$251$	24.3327	1.53587	0.767934	−	0.640529i	$-0.221285\pi$
$251$	24.3327	1.53587	0.767934	+	0.640529i	$0.221285\pi$
$252$	− 1.19806i	− 0.0754708i
$253$	− 4.41789i	− 0.277751i
$254$	− 21.8756i	− 1.37260i
$255$	6.45473i	0.404211i
$256$	1.00000	0.0625000
$257$	−30.4426	−1.89896	−0.949480	−	0.313827i	$-0.898389\pi$
$257$	−30.4426	−1.89896	−0.949480	+	0.313827i	$0.898389\pi$
$258$	1.15883i	0.0721458i
$259$	9.54394	0.593032
$260$	0	0
$261$	−6.89977	−0.427085
$262$	− 21.6528i	− 1.33771i
$263$	−9.33944	−0.575894	−0.287947	−	0.957646i	$-0.592973\pi$
$263$	−9.33944	−0.575894	−0.287947	+	0.957646i	$0.592973\pi$
$264$	−0.554958	−0.0341553
$265$	− 8.47219i	− 0.520442i
$266$	− 6.47650i	− 0.397100i
$267$	− 13.7845i	− 0.843596i
$268$	− 3.34481i	− 0.204317i
$269$	26.7821	1.63293	0.816466	−	0.577393i	$-0.195930\pi$
$269$	26.7821	1.63293	0.816466	+	0.577393i	$0.195930\pi$
$270$	−1.00000	−0.0608581
$271$	− 2.58748i	− 0.157178i	−0.996907	−	0.0785892i	$-0.974958\pi$
$271$	− 2.58748i	− 0.157178i	0.996907	−	0.0785892i	$-0.0250415\pi$
$272$	6.45473	0.391376
$273$	0	0
$274$	−7.08815	−0.428211
$275$	− 0.554958i	− 0.0334652i
$276$	−7.96077	−0.479182
$277$	−11.0532	−0.664124	−0.332062	−	0.943258i	$-0.607744\pi$
$277$	−11.0532	−0.664124	−0.332062	+	0.943258i	$0.607744\pi$
$278$	0.340502i	0.0204220i
$279$	− 3.04892i	− 0.182534i
$280$	− 1.19806i	− 0.0715979i
$281$	10.5090i	0.626916i	0.949602	+	0.313458i	$0.101487\pi$
$281$	10.5090i	0.626916i	−0.949602	+	0.313458i	$0.898513\pi$
$282$	12.2620	0.730194
$283$	1.07069	0.0636458	0.0318229	−	0.999494i	$-0.489869\pi$
$283$	1.07069	0.0636458	0.0318229	+	0.999494i	$0.489869\pi$
$284$	2.35690i	0.139856i
$285$	−5.40581	−0.320213
$286$	0	0
$287$	3.69979	0.218392
$288$	1.00000i	0.0589256i
$289$	24.6635	1.45080
$290$	−6.89977	−0.405169
$291$	7.07606i	0.414806i
$292$	− 13.3448i	− 0.780946i
$293$	27.4142i	1.60155i	0.598963	+	0.800777i	$0.295580\pi$
$293$	27.4142i	1.60155i	−0.598963	+	0.800777i	$0.704420\pi$
$294$	− 5.56465i	− 0.324537i
$295$	1.45712	0.0848370
$296$	−7.96615	−0.463023
$297$	− 0.554958i	− 0.0322019i
$298$	7.40581	0.429007
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 1.38835i	− 0.0800234i
$302$	−10.4209	−0.599654
$303$	4.09783	0.235414
$304$	5.40581i	0.310045i
$305$	− 2.76271i	− 0.158192i
$306$	6.45473i	0.368992i
$307$	24.4969i	1.39811i	0.715066	+	0.699057i	$0.246397\pi$
$307$	24.4969i	1.39811i	−0.715066	+	0.699057i	$0.753603\pi$
$308$	0.664874	0.0378847
$309$	−4.16421	−0.236894
$310$	− 3.04892i	− 0.173167i
$311$	−16.4403	−0.932241	−0.466121	−	0.884721i	$-0.654349\pi$
$311$	−16.4403	−0.932241	−0.466121	+	0.884721i	$0.654349\pi$
$312$	0	0
$313$	−26.0320	−1.47142	−0.735709	−	0.677298i	$-0.763151\pi$
$313$	−26.0320	−1.47142	−0.735709	+	0.677298i	$0.763151\pi$
$314$	− 19.0151i	− 1.07308i
$315$	1.19806	0.0675032
$316$	0.990311	0.0557094
$317$	− 19.3545i	− 1.08706i	−0.839391	−	0.543529i	$-0.817088\pi$
$317$	− 19.3545i	− 1.08706i	0.839391	−	0.543529i	$-0.182912\pi$
$318$	− 8.47219i	− 0.475097i
$319$	− 3.82908i	− 0.214388i
$320$	1.00000i	0.0559017i
$321$	4.98254	0.278098
$322$	9.53750	0.531504
$323$	34.8931i	1.94150i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−5.65279	−0.313079
$327$	1.31336i	0.0726287i
$328$	−3.08815	−0.170514
$329$	−14.6907	−0.809924
$330$	− 0.554958i	− 0.0305494i
$331$	0.173899i	0.00955836i	0.999989	+	0.00477918i	$0.00152127\pi$
$331$	0.173899i	0.00955836i	−0.999989	+	0.00477918i	$0.998479\pi$
$332$	5.88471i	0.322965i
$333$	− 7.96615i	− 0.436542i
$334$	−2.15883	−0.118126
$335$	3.34481	0.182747
$336$	− 1.19806i	− 0.0653597i
$337$	14.3793	0.783288	0.391644	−	0.920117i	$-0.371906\pi$
$337$	14.3793	0.783288	0.391644	+	0.920117i	$0.371906\pi$
$338$	0	0
$339$	−3.55496	−0.193079
$340$	6.45473i	0.350057i
$341$	1.69202	0.0916281
$342$	−5.40581	−0.292313
$343$	15.0532i	0.812798i
$344$	1.15883i	0.0624801i
$345$	− 7.96077i	− 0.428594i
$346$	− 21.6722i	− 1.16510i
$347$	−26.3400	−1.41401	−0.707003	−	0.707210i	$-0.749953\pi$
$347$	−26.3400	−1.41401	−0.707003	+	0.707210i	$0.749953\pi$
$348$	−6.89977	−0.369867
$349$	− 8.44371i	− 0.451982i	−0.974129	−	0.225991i	$-0.927438\pi$
$349$	− 8.44371i	− 0.451982i	0.974129	−	0.225991i	$-0.0725619\pi$
$350$	1.19806	0.0640391
$351$	0	0
$352$	−0.554958	−0.0295794
$353$	10.9420i	0.582383i	0.956665	+	0.291192i	$0.0940517\pi$
$353$	10.9420i	0.582383i	−0.956665	+	0.291192i	$0.905948\pi$
$354$	1.45712	0.0774452
$355$	−2.35690	−0.125091
$356$	− 13.7845i	− 0.730576i
$357$	− 7.73317i	− 0.409283i
$358$	0.956459i	0.0505505i
$359$	8.07606i	0.426238i	0.977026	+	0.213119i	$0.0683623\pi$
$359$	8.07606i	0.426238i	−0.977026	+	0.213119i	$0.931638\pi$
$360$	−1.00000	−0.0527046
$361$	−10.2228	−0.538043
$362$	− 0.405813i	− 0.0213291i
$363$	−10.6920	−0.561186
$364$	0	0
$365$	13.3448	0.698500
$366$	− 2.76271i	− 0.144409i
$367$	17.1575	0.895615	0.447807	−	0.894130i	$-0.352205\pi$
$367$	17.1575	0.895615	0.447807	+	0.894130i	$0.352205\pi$
$368$	−7.96077	−0.414984
$369$	− 3.08815i	− 0.160762i
$370$	− 7.96615i	− 0.414140i
$371$	10.1502i	0.526973i
$372$	− 3.04892i	− 0.158079i
$373$	2.72455	0.141072	0.0705358	−	0.997509i	$-0.477529\pi$
$373$	2.72455	0.141072	0.0705358	+	0.997509i	$0.477529\pi$
$374$	−3.58211	−0.185226
$375$	− 1.00000i	− 0.0516398i
$376$	12.2620	0.632367
$377$	0	0
$378$	1.19806	0.0616217
$379$	26.8998i	1.38175i	0.722975	+	0.690874i	$0.242774\pi$
$379$	26.8998i	1.38175i	−0.722975	+	0.690874i	$0.757226\pi$
$380$	−5.40581	−0.277312
$381$	21.8756	1.12072
$382$	− 19.0097i	− 0.972620i
$383$	− 29.9511i	− 1.53043i	−0.643776	−	0.765214i	$-0.722633\pi$
$383$	− 29.9511i	− 1.53043i	0.643776	−	0.765214i	$-0.277367\pi$
$384$	1.00000i	0.0510310i
$385$	0.664874i	0.0338851i
$386$	−0.599564	−0.0305170
$387$	−1.15883	−0.0589068
$388$	7.07606i	0.359233i
$389$	−10.0140	−0.507730	−0.253865	−	0.967240i	$-0.581702\pi$
$389$	−10.0140	−0.507730	−0.253865	+	0.967240i	$0.581702\pi$
$390$	0	0
$391$	−51.3846	−2.59863
$392$	− 5.56465i	− 0.281057i
$393$	21.6528	1.09224
$394$	3.78448	0.190659
$395$	0.990311i	0.0498280i
$396$	− 0.554958i	− 0.0278877i
$397$	17.0489i	0.855661i	0.903859	+	0.427830i	$0.140722\pi$
$397$	17.0489i	0.855661i	−0.903859	+	0.427830i	$0.859278\pi$
$398$	10.2349i	0.513029i
$399$	6.47650	0.324231
$400$	−1.00000	−0.0500000
$401$	0.975246i	0.0487015i	0.999703	+	0.0243507i	$0.00775184\pi$
$401$	0.975246i	0.0487015i	−0.999703	+	0.0243507i	$0.992248\pi$
$402$	3.34481	0.166824
$403$	0	0
$404$	4.09783	0.203875
$405$	− 1.00000i	− 0.0496904i
$406$	8.26636	0.410252
$407$	4.42088	0.219135
$408$	6.45473i	0.319557i
$409$	37.7439i	1.86632i	0.359465	+	0.933158i	$0.382959\pi$
$409$	37.7439i	1.86632i	−0.359465	+	0.933158i	$0.617041\pi$
$410$	− 3.08815i	− 0.152513i
$411$	− 7.08815i	− 0.349632i
$412$	−4.16421	−0.205156
$413$	−1.74572	−0.0859015
$414$	− 7.96077i	− 0.391251i
$415$	−5.88471	−0.288869
$416$	0	0
$417$	−0.340502	−0.0166745
$418$	− 3.00000i	− 0.146735i
$419$	−21.5700	−1.05376	−0.526882	−	0.849938i	$-0.676639\pi$
$419$	−21.5700	−1.05376	−0.526882	+	0.849938i	$0.676639\pi$
$420$	1.19806	0.0584595
$421$	− 33.2403i	− 1.62003i	−0.586408	−	0.810016i	$-0.699458\pi$
$421$	− 33.2403i	− 1.62003i	0.586408	−	0.810016i	$-0.300542\pi$
$422$	− 8.71379i	− 0.424181i
$423$	12.2620i	0.596201i
$424$	− 8.47219i	− 0.411446i
$425$	−6.45473	−0.313100
$426$	−2.35690	−0.114192
$427$	3.30990i	0.160177i
$428$	4.98254	0.240840
$429$	0	0
$430$	−1.15883	−0.0558839
$431$	− 20.9041i	− 1.00691i	−0.864020	−	0.503457i	$-0.832061\pi$
$431$	− 20.9041i	− 1.00691i	0.864020	−	0.503457i	$-0.167939\pi$
$432$	−1.00000	−0.0481125
$433$	−36.5864	−1.75823	−0.879115	−	0.476609i	$-0.841866\pi$
$433$	−36.5864	−1.75823	−0.879115	+	0.476609i	$0.841866\pi$
$434$	3.65279i	0.175340i
$435$	− 6.89977i	− 0.330819i
$436$	1.31336i	0.0628983i
$437$	− 43.0344i	− 2.05862i
$438$	13.3448	0.637640
$439$	−27.5036	−1.31268	−0.656339	−	0.754466i	$-0.727896\pi$
$439$	−27.5036	−1.31268	−0.656339	+	0.754466i	$0.727896\pi$
$440$	− 0.554958i	− 0.0264566i
$441$	5.56465	0.264983
$442$	0	0
$443$	−4.34614	−0.206491	−0.103246	−	0.994656i	$-0.532923\pi$
$443$	−4.34614	−0.206491	−0.103246	+	0.994656i	$0.532923\pi$
$444$	− 7.96615i	− 0.378057i
$445$	13.7845	0.653447
$446$	−15.0694	−0.713555
$447$	7.40581i	0.350283i
$448$	− 1.19806i	− 0.0566031i
$449$	0.423272i	0.0199754i	0.999950	+	0.00998771i	$0.00317924\pi$
$449$	0.423272i	0.0199754i	−0.999950	+	0.00998771i	$0.996821\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	1.71379	0.0806993
$452$	−3.55496	−0.167211
$453$	− 10.4209i	− 0.489616i
$454$	−20.4577	−0.960128
$455$	0	0
$456$	−5.40581	−0.253150
$457$	18.6396i	0.871926i	0.899965	+	0.435963i	$0.143592\pi$
$457$	18.6396i	0.871926i	−0.899965	+	0.435963i	$0.856408\pi$
$458$	−18.7657	−0.876863
$459$	−6.45473	−0.301281
$460$	− 7.96077i	− 0.371173i
$461$	− 8.22223i	− 0.382947i	−0.981498	−	0.191474i	$-0.938673\pi$
$461$	− 8.22223i	− 0.382947i	0.981498	−	0.191474i	$-0.0613266\pi$
$462$	0.664874i	0.0309328i
$463$	34.1118i	1.58531i	0.609670	+	0.792656i	$0.291302\pi$
$463$	34.1118i	1.58531i	−0.609670	+	0.792656i	$0.708698\pi$
$464$	−6.89977	−0.320314
$465$	3.04892	0.141390
$466$	− 23.6353i	− 1.09489i
$467$	−13.9245	−0.644350	−0.322175	−	0.946680i	$-0.604414\pi$
$467$	−13.9245	−0.644350	−0.322175	+	0.946680i	$0.604414\pi$
$468$	0	0
$469$	−4.00730	−0.185040
$470$	12.2620i	0.565606i
$471$	19.0151	0.876168
$472$	1.45712	0.0670695
$473$	− 0.643104i	− 0.0295700i
$474$	0.990311i	0.0454865i
$475$	− 5.40581i	− 0.248036i
$476$	− 7.73317i	− 0.354449i
$477$	8.47219	0.387915
$478$	28.7506	1.31502
$479$	1.79225i	0.0818899i	0.999161	+	0.0409450i	$0.0130368\pi$
$479$	1.79225i	0.0818899i	−0.999161	+	0.0409450i	$0.986963\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	3.89440	0.177385
$483$	9.53750i	0.433971i
$484$	−10.6920	−0.486001
$485$	−7.07606	−0.321308
$486$	− 1.00000i	− 0.0453609i
$487$	0.126310i	0.00572364i	0.999996	+	0.00286182i	$0.000910947\pi$
$487$	0.126310i	0.00572364i	−0.999996	+	0.00286182i	$0.999089\pi$
$488$	− 2.76271i	− 0.125062i
$489$	− 5.65279i	− 0.255628i
$490$	5.56465	0.251385
$491$	17.6823	0.797993	0.398996	−	0.916953i	$-0.369359\pi$
$491$	17.6823	0.797993	0.398996	+	0.916953i	$0.369359\pi$
$492$	− 3.08815i	− 0.139224i
$493$	−44.5362	−2.00581
$494$	0	0
$495$	0.554958	0.0249435
$496$	− 3.04892i	− 0.136900i
$497$	2.82371	0.126661
$498$	−5.88471	−0.263700
$499$	21.1535i	0.946959i	0.880805	+	0.473479i	$0.157002\pi$
$499$	21.1535i	0.946959i	−0.880805	+	0.473479i	$0.842998\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 2.15883i	− 0.0964496i
$502$	24.3327i	1.08602i
$503$	−29.7187	−1.32509	−0.662546	−	0.749022i	$-0.730524\pi$
$503$	−29.7187	−1.32509	−0.662546	+	0.749022i	$0.730524\pi$
$504$	1.19806	0.0533659
$505$	4.09783i	0.182351i
$506$	4.41789	0.196399
$507$	0	0
$508$	21.8756	0.970573
$509$	− 0.786872i	− 0.0348775i	−0.999848	−	0.0174387i	$-0.994449\pi$
$509$	− 0.786872i	− 0.0348775i	0.999848	−	0.0174387i	$-0.00555121\pi$
$510$	−6.45473	−0.285820
$511$	−15.9879	−0.707264
$512$	1.00000i	0.0441942i
$513$	− 5.40581i	− 0.238672i
$514$	− 30.4426i	− 1.34277i
$515$	− 4.16421i	− 0.183497i
$516$	−1.15883	−0.0510148
$517$	−6.80492	−0.299280
$518$	9.54394i	0.419337i
$519$	21.6722	0.951303
$520$	0	0
$521$	24.1782	1.05927	0.529633	−	0.848227i	$-0.322330\pi$
$521$	24.1782	1.05927	0.529633	+	0.848227i	$0.322330\pi$
$522$	− 6.89977i	− 0.301995i
$523$	23.0750	1.00900	0.504500	−	0.863412i	$-0.331677\pi$
$523$	23.0750	1.00900	0.504500	+	0.863412i	$0.331677\pi$
$524$	21.6528	0.945907
$525$	1.19806i	0.0522877i
$526$	− 9.33944i	− 0.407219i
$527$	− 19.6799i	− 0.857272i
$528$	− 0.554958i	− 0.0241515i
$529$	40.3739	1.75539
$530$	8.47219	0.368008
$531$	1.45712i	0.0632338i
$532$	6.47650	0.280792
$533$	0	0
$534$	13.7845	0.596513
$535$	4.98254i	0.215414i
$536$	3.34481	0.144474
$537$	−0.956459	−0.0412743
$538$	26.7821i	1.15466i
$539$	3.08815i	0.133016i
$540$	− 1.00000i	− 0.0430331i
$541$	− 24.6122i	− 1.05816i	−0.848571	−	0.529081i	$-0.822537\pi$
$541$	− 24.6122i	− 1.05816i	0.848571	−	0.529081i	$-0.177463\pi$
$542$	2.58748	0.111142
$543$	0.405813	0.0174151
$544$	6.45473i	0.276744i
$545$	−1.31336	−0.0562580
$546$	0	0
$547$	−0.492894	−0.0210746	−0.0105373	−	0.999944i	$-0.503354\pi$
$547$	−0.492894	−0.0210746	−0.0105373	+	0.999944i	$0.503354\pi$
$548$	− 7.08815i	− 0.302791i
$549$	2.76271	0.117910
$550$	0.554958	0.0236635
$551$	− 37.2989i	− 1.58899i
$552$	− 7.96077i	− 0.338833i
$553$	− 1.18645i	− 0.0504532i
$554$	− 11.0532i	− 0.469607i
$555$	7.96615	0.338144
$556$	−0.340502	−0.0144405
$557$	− 12.6028i	− 0.533998i	−0.963697	−	0.266999i	$-0.913968\pi$
$557$	− 12.6028i	− 0.533998i	0.963697	−	0.266999i	$-0.0860321\pi$
$558$	3.04892	0.129071
$559$	0	0
$560$	1.19806	0.0506274
$561$	− 3.58211i	− 0.151237i
$562$	−10.5090	−0.443296
$563$	−24.5743	−1.03568	−0.517842	−	0.855476i	$-0.673265\pi$
$563$	−24.5743	−1.03568	−0.517842	+	0.855476i	$0.673265\pi$
$564$	12.2620i	0.516325i
$565$	− 3.55496i	− 0.149558i
$566$	1.07069i	0.0450044i
$567$	1.19806i	0.0503139i
$568$	−2.35690	−0.0988932
$569$	−18.9414	−0.794065	−0.397032	−	0.917805i	$-0.629960\pi$
$569$	−18.9414	−0.794065	−0.397032	+	0.917805i	$0.629960\pi$
$570$	− 5.40581i	− 0.226425i
$571$	17.2978	0.723891	0.361946	−	0.932199i	$-0.382113\pi$
$571$	17.2978	0.723891	0.361946	+	0.932199i	$0.382113\pi$
$572$	0	0
$573$	19.0097	0.794141
$574$	3.69979i	0.154426i
$575$	7.96077	0.331987
$576$	−1.00000	−0.0416667
$577$	42.5628i	1.77191i	0.463767	+	0.885957i	$0.346497\pi$
$577$	42.5628i	1.77191i	−0.463767	+	0.885957i	$0.653503\pi$
$578$	24.6635i	1.02587i
$579$	− 0.599564i	− 0.0249170i
$580$	− 6.89977i	− 0.286497i
$581$	7.05025	0.292493
$582$	−7.07606	−0.293312
$583$	4.70171i	0.194725i
$584$	13.3448	0.552212
$585$	0	0
$586$	−27.4142	−1.13247
$587$	− 43.2737i	− 1.78609i	−0.449963	−	0.893047i	$-0.648563\pi$
$587$	− 43.2737i	− 1.78609i	0.449963	−	0.893047i	$-0.351437\pi$
$588$	5.56465	0.229482
$589$	16.4819	0.679124
$590$	1.45712i	0.0599888i
$591$	3.78448i	0.155673i
$592$	− 7.96615i	− 0.327407i
$593$	19.8672i	0.815850i	0.913015	+	0.407925i	$0.133748\pi$
$593$	19.8672i	0.815850i	−0.913015	+	0.407925i	$0.866252\pi$
$594$	0.554958	0.0227702
$595$	7.73317	0.317029
$596$	7.40581i	0.303354i
$597$	−10.2349	−0.418886
$598$	0	0
$599$	22.3448	0.912984	0.456492	−	0.889727i	$-0.349106\pi$
$599$	22.3448	0.912984	0.456492	+	0.889727i	$0.349106\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	38.3696	1.56513	0.782564	−	0.622571i	$-0.213912\pi$
$601$	38.3696	1.56513	0.782564	+	0.622571i	$0.213912\pi$
$602$	1.38835	0.0565851
$603$	3.34481i	0.136211i
$604$	− 10.4209i	− 0.424020i
$605$	− 10.6920i	− 0.434692i
$606$	4.09783i	0.166463i
$607$	34.9778	1.41970	0.709852	−	0.704351i	$-0.248762\pi$
$607$	34.9778	1.41970	0.709852	+	0.704351i	$0.248762\pi$
$608$	−5.40581	−0.219235
$609$	8.26636i	0.334970i
$610$	2.76271	0.111859
$611$	0	0
$612$	−6.45473	−0.260917
$613$	− 28.1648i	− 1.13757i	−0.822488	−	0.568783i	$-0.807414\pi$
$613$	− 28.1648i	− 1.13757i	0.822488	−	0.568783i	$-0.192586\pi$
$614$	−24.4969	−0.988616
$615$	3.08815	0.124526
$616$	0.664874i	0.0267886i
$617$	24.0640i	0.968779i	0.874852	+	0.484390i	$0.160958\pi$
$617$	24.0640i	0.968779i	−0.874852	+	0.484390i	$0.839042\pi$
$618$	− 4.16421i	− 0.167509i
$619$	− 14.7651i	− 0.593460i	−0.954961	−	0.296730i	$-0.904104\pi$
$619$	− 14.7651i	− 0.593460i	0.954961	−	0.296730i	$-0.0958961\pi$
$620$	3.04892	0.122447
$621$	7.96077	0.319455
$622$	− 16.4403i	− 0.659194i
$623$	−16.5147	−0.661646
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 26.0320i	− 1.04045i
$627$	3.00000	0.119808
$628$	19.0151	0.758784
$629$	− 51.4193i	− 2.05022i
$630$	1.19806i	0.0477319i
$631$	− 22.9788i	− 0.914772i	−0.889268	−	0.457386i	$-0.848786\pi$
$631$	− 22.9788i	− 0.914772i	0.889268	−	0.457386i	$-0.151214\pi$
$632$	0.990311i	0.0393925i
$633$	8.71379	0.346342
$634$	19.3545	0.768666
$635$	21.8756i	0.868107i
$636$	8.47219	0.335944
$637$	0	0
$638$	3.82908	0.151595
$639$	− 2.35690i	− 0.0932374i
$640$	−1.00000	−0.0395285
$641$	5.34913	0.211278	0.105639	−	0.994405i	$-0.466311\pi$
$641$	5.34913	0.211278	0.105639	+	0.994405i	$0.466311\pi$
$642$	4.98254i	0.196645i
$643$	− 39.2984i	− 1.54978i	−0.632097	−	0.774889i	$-0.717806\pi$
$643$	− 39.2984i	− 1.54978i	0.632097	−	0.774889i	$-0.282194\pi$
$644$	9.53750i	0.375830i
$645$	− 1.15883i	− 0.0456290i
$646$	−34.8931	−1.37285
$647$	−0.328684	−0.0129219	−0.00646095	−	0.999979i	$-0.502057\pi$
$647$	−0.328684	−0.0129219	−0.00646095	+	0.999979i	$0.502057\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−0.808643	−0.0317420
$650$	0	0
$651$	−3.65279	−0.143164
$652$	− 5.65279i	− 0.221380i
$653$	41.6426	1.62960	0.814801	−	0.579741i	$-0.196846\pi$
$653$	41.6426	1.62960	0.814801	+	0.579741i	$0.196846\pi$
$654$	−1.31336	−0.0513563
$655$	21.6528i	0.846045i
$656$	− 3.08815i	− 0.120572i
$657$	13.3448i	0.520631i
$658$	− 14.6907i	− 0.572703i
$659$	−4.24698	−0.165439	−0.0827194	−	0.996573i	$-0.526361\pi$
$659$	−4.24698	−0.165439	−0.0827194	+	0.996573i	$0.526361\pi$
$660$	0.554958	0.0216017
$661$	29.7071i	1.15547i	0.816224	+	0.577736i	$0.196064\pi$
$661$	29.7071i	1.15547i	−0.816224	+	0.577736i	$0.803936\pi$
$662$	−0.173899	−0.00675878
$663$	0	0
$664$	−5.88471	−0.228371
$665$	6.47650i	0.251148i
$666$	7.96615	0.308682
$667$	54.9275	2.12680
$668$	− 2.15883i	− 0.0835278i
$669$	− 15.0694i	− 0.582615i
$670$	3.34481i	0.129221i
$671$	1.53319i	0.0591881i
$672$	1.19806	0.0462163
$673$	21.7004	0.836488	0.418244	−	0.908335i	$-0.362646\pi$
$673$	21.7004	0.836488	0.418244	+	0.908335i	$0.362646\pi$
$674$	14.3793i	0.553868i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−12.5168	−0.481059	−0.240530	−	0.970642i	$-0.577321\pi$
$677$	−12.5168	−0.481059	−0.240530	+	0.970642i	$0.577321\pi$
$678$	− 3.55496i	− 0.136527i
$679$	8.47757	0.325339
$680$	−6.45473	−0.247528
$681$	− 20.4577i	− 0.783941i
$682$	1.69202i	0.0647909i
$683$	− 3.82908i	− 0.146516i	−0.997313	−	0.0732579i	$-0.976660\pi$
$683$	− 3.82908i	− 0.146516i	0.997313	−	0.0732579i	$-0.0233396\pi$
$684$	− 5.40581i	− 0.206696i
$685$	7.08815	0.270824
$686$	−15.0532	−0.574735
$687$	− 18.7657i	− 0.715956i
$688$	−1.15883	−0.0441801
$689$	0	0
$690$	7.96077	0.303061
$691$	− 27.4239i	− 1.04325i	−0.853174	−	0.521626i	$-0.825325\pi$
$691$	− 27.4239i	− 1.04325i	0.853174	−	0.521626i	$-0.174675\pi$
$692$	21.6722	0.823852
$693$	−0.664874	−0.0252565
$694$	− 26.3400i	− 0.999854i
$695$	− 0.340502i	− 0.0129160i
$696$	− 6.89977i	− 0.261535i
$697$	− 19.9332i	− 0.755022i
$698$	8.44371	0.319599
$699$	23.6353	0.893970
$700$	1.19806i	0.0452825i
$701$	−31.5803	−1.19277	−0.596386	−	0.802698i	$-0.703397\pi$
$701$	−31.5803	−1.19277	−0.596386	+	0.802698i	$0.703397\pi$
$702$	0	0
$703$	43.0635	1.62417
$704$	− 0.554958i	− 0.0209158i
$705$	−12.2620	−0.461815
$706$	−10.9420	−0.411807
$707$	− 4.90946i	− 0.184639i
$708$	1.45712i	0.0547621i
$709$	28.9788i	1.08832i	0.838981	+	0.544161i	$0.183152\pi$
$709$	28.9788i	1.08832i	−0.838981	+	0.544161i	$0.816848\pi$
$710$	− 2.35690i	− 0.0884527i
$711$	−0.990311	−0.0371396
$712$	13.7845	0.516595
$713$	24.2717i	0.908984i
$714$	7.73317	0.289407
$715$	0	0
$716$	−0.956459	−0.0357446
$717$	28.7506i	1.07371i
$718$	−8.07606	−0.301396
$719$	−27.0670	−1.00943	−0.504714	−	0.863287i	$-0.668402\pi$
$719$	−27.0670	−1.00943	−0.504714	+	0.863287i	$0.668402\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	4.98898i	0.185799i
$722$	− 10.2228i	− 0.380454i
$723$	3.89440i	0.144834i
$724$	0.405813	0.0150819
$725$	6.89977	0.256251
$726$	− 10.6920i	− 0.396818i
$727$	26.6612	0.988807	0.494404	−	0.869232i	$-0.335386\pi$
$727$	26.6612	0.988807	0.494404	+	0.869232i	$0.335386\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	13.3448i	0.493914i
$731$	−7.47996	−0.276656
$732$	2.76271	0.102113
$733$	51.2984i	1.89475i	0.320126	+	0.947375i	$0.396275\pi$
$733$	51.2984i	1.89475i	−0.320126	+	0.947375i	$0.603725\pi$
$734$	17.1575i	0.633295i
$735$	5.56465i	0.205255i
$736$	− 7.96077i	− 0.293438i
$737$	−1.85623	−0.0683752
$738$	3.08815	0.113676
$739$	28.6456i	1.05375i	0.849944	+	0.526873i	$0.176636\pi$
$739$	28.6456i	1.05375i	−0.849944	+	0.526873i	$0.823364\pi$
$740$	7.96615	0.292841
$741$	0	0
$742$	−10.1502	−0.372626
$743$	43.6674i	1.60200i	0.598664	+	0.801000i	$0.295699\pi$
$743$	43.6674i	1.60200i	−0.598664	+	0.801000i	$0.704301\pi$
$744$	3.04892	0.111779
$745$	−7.40581	−0.271328
$746$	2.72455i	0.0997527i
$747$	− 5.88471i	− 0.215310i
$748$	− 3.58211i	− 0.130975i
$749$	− 5.96940i	− 0.218117i
$750$	1.00000	0.0365148
$751$	20.1535	0.735410	0.367705	−	0.929942i	$-0.380144\pi$
$751$	20.1535	0.735410	0.367705	+	0.929942i	$0.380144\pi$
$752$	12.2620i	0.447151i
$753$	−24.3327	−0.886734
$754$	0	0
$755$	10.4209	0.379255
$756$	1.19806i	0.0435731i
$757$	−41.8562	−1.52129	−0.760645	−	0.649168i	$-0.775117\pi$
$757$	−41.8562	−1.52129	−0.760645	+	0.649168i	$0.775117\pi$
$758$	−26.8998	−0.977044
$759$	4.41789i	0.160359i
$760$	− 5.40581i	− 0.196089i
$761$	− 32.8665i	− 1.19141i	−0.803203	−	0.595705i	$-0.796873\pi$
$761$	− 32.8665i	− 1.19141i	0.803203	−	0.595705i	$-0.203127\pi$
$762$	21.8756i	0.792470i
$763$	1.57348	0.0569639
$764$	19.0097	0.687746
$765$	− 6.45473i	− 0.233371i
$766$	29.9511	1.08218
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	37.8877i	1.36627i	0.730294	+	0.683133i	$0.239383\pi$
$769$	37.8877i	1.36627i	−0.730294	+	0.683133i	$0.760617\pi$
$770$	−0.664874	−0.0239604
$771$	30.4426	1.09637
$772$	− 0.599564i	− 0.0215788i
$773$	12.4523i	0.447879i	0.974603	+	0.223940i	$0.0718919\pi$
$773$	12.4523i	0.447879i	−0.974603	+	0.223940i	$0.928108\pi$
$774$	− 1.15883i	− 0.0416534i
$775$	3.04892i	0.109520i
$776$	−7.07606	−0.254016
$777$	−9.54394	−0.342387
$778$	− 10.0140i	− 0.359019i
$779$	16.6939	0.598122
$780$	0	0
$781$	1.30798	0.0468032
$782$	− 51.3846i	− 1.83751i
$783$	6.89977	0.246578
$784$	5.56465	0.198737
$785$	19.0151i	0.678677i
$786$	21.6528i	0.772330i
$787$	− 6.20536i	− 0.221197i	−0.993865	−	0.110599i	$-0.964723\pi$
$787$	− 6.20536i	− 0.221197i	0.993865	−	0.110599i	$-0.0352768\pi$
$788$	3.78448i	0.134816i
$789$	9.33944	0.332493
$790$	−0.990311	−0.0352337
$791$	4.25906i	0.151435i
$792$	0.554958	0.0197196
$793$	0	0
$794$	−17.0489	−0.605043
$795$	8.47219i	0.300478i
$796$	−10.2349	−0.362766
$797$	−33.0627	−1.17114	−0.585570	−	0.810622i	$-0.699129\pi$
$797$	−33.0627	−1.17114	−0.585570	+	0.810622i	$0.699129\pi$
$798$	6.47650i	0.229266i
$799$	79.1482i	2.80006i
$800$	− 1.00000i	− 0.0353553i
$801$	13.7845i	0.487051i
$802$	−0.975246	−0.0344371
$803$	−7.40581	−0.261345
$804$	3.34481i	0.117963i
$805$	−9.53750	−0.336153
$806$	0	0
$807$	−26.7821	−0.942774
$808$	4.09783i	0.144161i
$809$	−52.5454	−1.84740	−0.923699	−	0.383120i	$-0.874850\pi$
$809$	−52.5454	−1.84740	−0.923699	+	0.383120i	$0.874850\pi$
$810$	1.00000	0.0351364
$811$	9.96508i	0.349921i	0.984575	+	0.174961i	$0.0559798\pi$
$811$	9.96508i	0.349921i	−0.984575	+	0.174961i	$0.944020\pi$
$812$	8.26636i	0.290092i
$813$	2.58748i	0.0907470i
$814$	4.42088i	0.154952i
$815$	5.65279	0.198009
$816$	−6.45473	−0.225961
$817$	− 6.26444i	− 0.219165i
$818$	−37.7439	−1.31969
$819$	0	0
$820$	3.08815	0.107843
$821$	− 28.6950i	− 1.00146i	−0.865603	−	0.500731i	$-0.833064\pi$
$821$	− 28.6950i	− 1.00146i	0.865603	−	0.500731i	$-0.166936\pi$
$822$	7.08815	0.247227
$823$	−37.4053	−1.30387	−0.651934	−	0.758276i	$-0.726042\pi$
$823$	−37.4053	−1.30387	−0.651934	+	0.758276i	$0.726042\pi$
$824$	− 4.16421i	− 0.145067i
$825$	0.554958i	0.0193212i
$826$	− 1.74572i	− 0.0607415i
$827$	− 36.3666i	− 1.26459i	−0.774728	−	0.632295i	$-0.782113\pi$
$827$	− 36.3666i	− 1.26459i	0.774728	−	0.632295i	$-0.217887\pi$
$828$	7.96077	0.276656
$829$	32.6370	1.13353	0.566765	−	0.823880i	$-0.308195\pi$
$829$	32.6370	1.13353	0.566765	+	0.823880i	$0.308195\pi$
$830$	− 5.88471i	− 0.204261i
$831$	11.0532	0.383432
$832$	0	0
$833$	35.9183	1.24450
$834$	− 0.340502i	− 0.0117906i
$835$	2.15883	0.0747095
$836$	3.00000	0.103757
$837$	3.04892i	0.105386i
$838$	− 21.5700i	− 0.745124i
$839$	− 11.6810i	− 0.403273i	−0.979460	−	0.201637i	$-0.935374\pi$
$839$	− 11.6810i	− 0.403273i	0.979460	−	0.201637i	$-0.0646260\pi$
$840$	1.19806i	0.0413371i
$841$	18.6069	0.641616
$842$	33.2403	1.14554
$843$	− 10.5090i	− 0.361950i
$844$	8.71379	0.299941
$845$	0	0
$846$	−12.2620	−0.421578
$847$	12.8097i	0.440147i
$848$	8.47219	0.290936
$849$	−1.07069	−0.0367459
$850$	− 6.45473i	− 0.221395i
$851$	63.4167i	2.17390i
$852$	− 2.35690i	− 0.0807459i
$853$	− 45.1756i	− 1.54678i	−0.633930	−	0.773391i	$-0.718559\pi$
$853$	− 45.1756i	− 1.54678i	0.633930	−	0.773391i	$-0.281441\pi$
$854$	−3.30990	−0.113262
$855$	5.40581	0.184875
$856$	4.98254i	0.170300i
$857$	16.1602	0.552021	0.276010	−	0.961155i	$-0.410988\pi$
$857$	16.1602	0.552021	0.276010	+	0.961155i	$0.410988\pi$
$858$	0	0
$859$	24.9071	0.849818	0.424909	−	0.905236i	$-0.360306\pi$
$859$	24.9071	0.849818	0.424909	+	0.905236i	$0.360306\pi$
$860$	− 1.15883i	− 0.0395159i
$861$	−3.69979	−0.126089
$862$	20.9041	0.711996
$863$	25.9030i	0.881749i	0.897569	+	0.440875i	$0.145332\pi$
$863$	25.9030i	0.881749i	−0.897569	+	0.440875i	$0.854668\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	21.6722i	0.736876i
$866$	− 36.5864i	− 1.24326i
$867$	−24.6635	−0.837618
$868$	−3.65279	−0.123984
$869$	− 0.549581i	− 0.0186433i
$870$	6.89977	0.233924
$871$	0	0
$872$	−1.31336	−0.0444758
$873$	− 7.07606i	− 0.239488i
$874$	43.0344	1.45566
$875$	−1.19806	−0.0405019
$876$	13.3448i	0.450879i
$877$	21.5875i	0.728957i	0.931212	+	0.364479i	$0.118753\pi$
$877$	21.5875i	0.728957i	−0.931212	+	0.364479i	$0.881247\pi$
$878$	− 27.5036i	− 0.928203i
$879$	− 27.4142i	− 0.924657i
$880$	0.554958	0.0187076
$881$	−0.468140	−0.0157720	−0.00788602	−	0.999969i	$-0.502510\pi$
$881$	−0.468140	−0.0157720	−0.00788602	+	0.999969i	$0.502510\pi$
$882$	5.56465i	0.187371i
$883$	43.6034	1.46737	0.733686	−	0.679489i	$-0.237798\pi$
$883$	43.6034	1.46737	0.733686	+	0.679489i	$0.237798\pi$
$884$	0	0
$885$	−1.45712	−0.0489807
$886$	− 4.34614i	− 0.146012i
$887$	24.1038	0.809326	0.404663	−	0.914466i	$-0.367389\pi$
$887$	24.1038	0.809326	0.404663	+	0.914466i	$0.367389\pi$
$888$	7.96615	0.267326
$889$	− 26.2083i	− 0.879000i
$890$	13.7845i	0.462057i
$891$	0.554958i	0.0185918i
$892$	− 15.0694i	− 0.504559i
$893$	−66.2863	−2.21819
$894$	−7.40581	−0.247687
$895$	− 0.956459i	− 0.0319709i
$896$	1.19806	0.0400245
$897$	0	0
$898$	−0.423272	−0.0141248
$899$	21.0368i	0.701618i
$900$	1.00000	0.0333333
$901$	54.6857	1.82184
$902$	1.71379i	0.0570630i
$903$	1.38835i	0.0462016i
$904$	− 3.55496i	− 0.118236i
$905$	0.405813i	0.0134897i
$906$	10.4209	0.346211
$907$	−55.0635	−1.82835	−0.914177	−	0.405315i	$-0.867162\pi$
$907$	−55.0635	−1.82835	−0.914177	+	0.405315i	$0.867162\pi$
$908$	− 20.4577i	− 0.678913i
$909$	−4.09783	−0.135917
$910$	0	0
$911$	7.17151	0.237603	0.118801	−	0.992918i	$-0.462095\pi$
$911$	7.17151	0.237603	0.118801	+	0.992918i	$0.462095\pi$
$912$	− 5.40581i	− 0.179004i
$913$	3.26577	0.108081
$914$	−18.6396	−0.616545
$915$	2.76271i	0.0913323i
$916$	− 18.7657i	− 0.620036i
$917$	− 25.9414i	− 0.856660i
$918$	− 6.45473i	− 0.213038i
$919$	−32.1081	−1.05915	−0.529574	−	0.848263i	$-0.677648\pi$
$919$	−32.1081	−1.05915	−0.529574	+	0.848263i	$0.677648\pi$
$920$	7.96077	0.262459
$921$	− 24.4969i	− 0.807202i
$922$	8.22223	0.270785
$923$	0	0
$924$	−0.664874	−0.0218728
$925$	7.96615i	0.261925i
$926$	−34.1118	−1.12098
$927$	4.16421	0.136771
$928$	− 6.89977i	− 0.226496i
$929$	− 3.46011i	− 0.113522i	−0.998388	−	0.0567612i	$-0.981923\pi$
$929$	− 3.46011i	− 0.113522i	0.998388	−	0.0567612i	$-0.0180774\pi$
$930$	3.04892i	0.0999779i
$931$	30.0814i	0.985879i
$932$	23.6353	0.774201
$933$	16.4403	0.538230
$934$	− 13.9245i	− 0.455624i
$935$	3.58211	0.117147
$936$	0	0
$937$	13.3918	0.437491	0.218746	−	0.975782i	$-0.429803\pi$
$937$	13.3918	0.437491	0.218746	+	0.975782i	$0.429803\pi$
$938$	− 4.00730i	− 0.130843i
$939$	26.0320	0.849524
$940$	−12.2620	−0.399944
$941$	− 47.0060i	− 1.53235i	−0.642632	−	0.766175i	$-0.722157\pi$
$941$	− 47.0060i	− 1.53235i	0.642632	−	0.766175i	$-0.277843\pi$
$942$	19.0151i	0.619544i
$943$	24.5840i	0.800566i
$944$	1.45712i	0.0474253i
$945$	−1.19806	−0.0389730
$946$	0.643104	0.0209091
$947$	44.9687i	1.46129i	0.682760	+	0.730643i	$0.260779\pi$
$947$	44.9687i	1.46129i	−0.682760	+	0.730643i	$0.739221\pi$
$948$	−0.990311	−0.0321638
$949$	0	0
$950$	5.40581	0.175388
$951$	19.3545i	0.627613i
$952$	7.73317	0.250633
$953$	12.1691	0.394196	0.197098	−	0.980384i	$-0.436848\pi$
$953$	12.1691	0.394196	0.197098	+	0.980384i	$0.436848\pi$
$954$	8.47219i	0.274297i
$955$	19.0097i	0.615139i
$956$	28.7506i	0.929862i
$957$	3.82908i	0.123777i
$958$	−1.79225	−0.0579049
$959$	−8.49204	−0.274222
$960$	− 1.00000i	− 0.0322749i
$961$	21.7041	0.700132
$962$	0	0
$963$	−4.98254	−0.160560
$964$	3.89440i	0.125430i
$965$	0.599564	0.0193006
$966$	−9.53750	−0.306864
$967$	− 21.8799i	− 0.703611i	−0.936073	−	0.351805i	$-0.885568\pi$
$967$	− 21.8799i	− 0.703611i	0.936073	−	0.351805i	$-0.114432\pi$
$968$	− 10.6920i	− 0.343655i
$969$	− 34.8931i	− 1.12093i
$970$	− 7.07606i	− 0.227199i
$971$	−36.6450	−1.17599	−0.587997	−	0.808863i	$-0.700083\pi$
$971$	−36.6450	−1.17599	−0.587997	+	0.808863i	$0.700083\pi$
$972$	1.00000	0.0320750
$973$	0.407943i	0.0130781i
$974$	−0.126310	−0.00404722
$975$	0	0
$976$	2.76271	0.0884322
$977$	− 7.36957i	− 0.235773i	−0.993027	−	0.117887i	$-0.962388\pi$
$977$	− 7.36957i	− 0.235773i	0.993027	−	0.117887i	$-0.0376120\pi$
$978$	5.65279	0.180756
$979$	−7.64981	−0.244489
$980$	5.56465i	0.177756i
$981$	− 1.31336i	− 0.0419322i
$982$	17.6823i	0.564266i
$983$	45.5612i	1.45318i	0.687073	+	0.726588i	$0.258895\pi$
$983$	45.5612i	1.45318i	−0.687073	+	0.726588i	$0.741105\pi$
$984$	3.08815	0.0984465
$985$	−3.78448	−0.120584
$986$	− 44.5362i	− 1.41832i
$987$	14.6907	0.467610
$988$	0	0
$989$	9.22521	0.293345
$990$	0.554958i	0.0176377i
$991$	35.4693	1.12672	0.563360	−	0.826211i	$-0.309508\pi$
$991$	35.4693	1.12672	0.563360	+	0.826211i	$0.309508\pi$
$992$	3.04892	0.0968032
$993$	− 0.173899i	− 0.00551852i
$994$	2.82371i	0.0895626i
$995$	− 10.2349i	− 0.324468i
$996$	− 5.88471i	− 0.186464i
$997$	−3.74632	−0.118647	−0.0593235	−	0.998239i	$-0.518894\pi$
$997$	−3.74632	−0.118647	−0.0593235	+	0.998239i	$0.518894\pi$
$998$	−21.1535	−0.669601
$999$	7.96615i	0.252038i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bm.1.1	✓	3	13.8	odd	4
5070.2.a.br.1.3	yes	3	13.5	odd	4
5070.2.b.u.1351.3		6	13.12	even	2	inner
5070.2.b.u.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.554958i q^{11} +1.00000 q^{12} -1.19806 q^{14} +1.00000i q^{15} +1.00000 q^{16} +6.45473 q^{17} +1.00000i q^{18} +5.40581i q^{19} +1.00000i q^{20} -1.19806i q^{21} -0.554958 q^{22} -7.96077 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.19806i q^{28} -6.89977 q^{29} -1.00000 q^{30} -3.04892i q^{31} +1.00000i q^{32} -0.554958i q^{33} +6.45473i q^{34} +1.19806 q^{35} -1.00000 q^{36} -7.96615i q^{37} -5.40581 q^{38} -1.00000 q^{40} -3.08815i q^{41} +1.19806 q^{42} -1.15883 q^{43} -0.554958i q^{44} -1.00000i q^{45} -7.96077i q^{46} +12.2620i q^{47} -1.00000 q^{48} +5.56465 q^{49} -1.00000i q^{50} -6.45473 q^{51} +8.47219 q^{53} -1.00000i q^{54} +0.554958 q^{55} +1.19806 q^{56} -5.40581i q^{57} -6.89977i q^{58} +1.45712i q^{59} -1.00000i q^{60} +2.76271 q^{61} +3.04892 q^{62} +1.19806i q^{63} -1.00000 q^{64} +0.554958 q^{66} +3.34481i q^{67} -6.45473 q^{68} +7.96077 q^{69} +1.19806i q^{70} -2.35690i q^{71} -1.00000i q^{72} +13.3448i q^{73} +7.96615 q^{74} +1.00000 q^{75} -5.40581i q^{76} -0.664874 q^{77} -0.990311 q^{79} -1.00000i q^{80} +1.00000 q^{81} +3.08815 q^{82} -5.88471i q^{83} +1.19806i q^{84} -6.45473i q^{85} -1.15883i q^{86} +6.89977 q^{87} +0.554958 q^{88} +13.7845i q^{89} +1.00000 q^{90} +7.96077 q^{92} +3.04892i q^{93} -12.2620 q^{94} +5.40581 q^{95} -1.00000i q^{96} -7.07606i q^{97} +5.56465i q^{98} +0.554958i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} - 6 q^{17} - 4 q^{22} - 22 q^{23} - 6 q^{25} - 6 q^{27} + 4 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} - 6 q^{38} - 6 q^{40} + 16 q^{42} + 10 q^{43} - 6 q^{48} - 10 q^{49} + 6 q^{51} + 38 q^{53} + 4 q^{55} + 16 q^{56} - 18 q^{61} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 22 q^{69} + 16 q^{74} + 6 q^{75} - 6 q^{77} - 50 q^{79} + 6 q^{81} + 26 q^{82} - 4 q^{87} + 4 q^{88} + 6 q^{90} + 22 q^{92} - 14 q^{94} + 6 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 - 6 * q^17 - 4 * q^22 - 22 * q^23 - 6 * q^25 - 6 * q^27 + 4 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 - 6 * q^38 - 6 * q^40 + 16 * q^42 + 10 * q^43 - 6 * q^48 - 10 * q^49 + 6 * q^51 + 38 * q^53 + 4 * q^55 + 16 * q^56 - 18 * q^61 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 22 * q^69 + 16 * q^74 + 6 * q^75 - 6 * q^77 - 50 * q^79 + 6 * q^81 + 26 * q^82 - 4 * q^87 + 4 * q^88 + 6 * q^90 + 22 * q^92 - 14 * q^94 + 6 * q^95

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