LMFDB - Embedded newform 5070.2.b.v.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.v.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.93900i q^{11} +1.00000 q^{12} -1.19806 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.74094 q^{17} +1.00000i q^{18} +0.911854i q^{19} +1.00000i q^{20} -1.19806i q^{21} +3.93900 q^{22} -1.24698 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.19806i q^{28} -3.47219 q^{29} -1.00000 q^{30} -2.15883i q^{31} +1.00000i q^{32} +3.93900i q^{33} -1.74094i q^{34} +1.19806 q^{35} -1.00000 q^{36} -4.80194i q^{37} -0.911854 q^{38} -1.00000 q^{40} +0.198062i q^{41} +1.19806 q^{42} -3.43296 q^{43} +3.93900i q^{44} -1.00000i q^{45} -1.24698i q^{46} -0.902165i q^{47} -1.00000 q^{48} +5.56465 q^{49} -1.00000i q^{50} +1.74094 q^{51} -6.91185 q^{53} -1.00000i q^{54} -3.93900 q^{55} +1.19806 q^{56} -0.911854i q^{57} -3.47219i q^{58} +10.2446i q^{59} -1.00000i q^{60} +9.92692 q^{61} +2.15883 q^{62} +1.19806i q^{63} -1.00000 q^{64} -3.93900 q^{66} -9.16852i q^{67} +1.74094 q^{68} +1.24698 q^{69} +1.19806i q^{70} +11.0707i q^{71} -1.00000i q^{72} -4.13706i q^{73} +4.80194 q^{74} +1.00000 q^{75} -0.911854i q^{76} +4.71917 q^{77} +2.73556 q^{79} -1.00000i q^{80} +1.00000 q^{81} -0.198062 q^{82} +15.5864i q^{83} +1.19806i q^{84} +1.74094i q^{85} -3.43296i q^{86} +3.47219 q^{87} -3.93900 q^{88} -8.57673i q^{89} +1.00000 q^{90} +1.24698 q^{92} +2.15883i q^{93} +0.902165 q^{94} +0.911854 q^{95} -1.00000i q^{96} +16.9976i q^{97} +5.56465i q^{98} -3.93900i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 18 q^{17} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 6 q^{27} - 8 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 16 q^{42} + 18 q^{43} - 6 q^{48} - 10 q^{49} - 18 q^{51} - 34 q^{53} - 4 q^{55} + 16 q^{56} + 2 q^{61} - 4 q^{62} - 6 q^{64} - 4 q^{66} - 18 q^{68} - 2 q^{69} + 20 q^{74} + 6 q^{75} + 6 q^{77} - 6 q^{79} + 6 q^{81} - 10 q^{82} + 8 q^{87} - 4 q^{88} + 6 q^{90} - 2 q^{92} + 42 q^{94} - 2 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 + 18 * q^17 + 4 * q^22 + 2 * q^23 - 6 * q^25 - 6 * q^27 - 8 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 16 * q^42 + 18 * q^43 - 6 * q^48 - 10 * q^49 - 18 * q^51 - 34 * q^53 - 4 * q^55 + 16 * q^56 + 2 * q^61 - 4 * q^62 - 6 * q^64 - 4 * q^66 - 18 * q^68 - 2 * q^69 + 20 * q^74 + 6 * q^75 + 6 * q^77 - 6 * q^79 + 6 * q^81 - 10 * q^82 + 8 * q^87 - 4 * q^88 + 6 * q^90 - 2 * q^92 + 42 * q^94 - 2 * 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
$11$	− 3.93900i	− 1.18765i	−0.804593	−	0.593827i	$-0.797616\pi$
$11$	− 3.93900i	− 1.18765i	0.804593	−	0.593827i	$-0.202384\pi$
$12$	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
$17$	−1.74094	−0.422240	−0.211120	−	0.977460i	$-0.567711\pi$
$17$	−1.74094	−0.422240	−0.211120	+	0.977460i	$0.567711\pi$
$18$	1.00000i	0.235702i
$19$	0.911854i	0.209194i	0.994515	+	0.104597i	$0.0333552\pi$
$19$	0.911854i	0.209194i	−0.994515	+	0.104597i	$0.966645\pi$
$20$	1.00000i	0.223607i
$21$	− 1.19806i	− 0.261439i
$22$	3.93900	0.839798
$23$	−1.24698	−0.260013	−0.130007	−	0.991513i	$-0.541500\pi$
$23$	−1.24698	−0.260013	−0.130007	+	0.991513i	$0.541500\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 1.19806i	− 0.226412i
$29$	−3.47219	−0.644769	−0.322385	−	0.946609i	$-0.604484\pi$
$29$	−3.47219	−0.644769	−0.322385	+	0.946609i	$0.604484\pi$
$30$	−1.00000	−0.182574
$31$	− 2.15883i	− 0.387738i	−0.981027	−	0.193869i	$-0.937896\pi$
$31$	− 2.15883i	− 0.387738i	0.981027	−	0.193869i	$-0.0621037\pi$
$32$	1.00000i	0.176777i
$33$	3.93900i	0.685692i
$34$	− 1.74094i	− 0.298569i
$35$	1.19806	0.202509
$36$	−1.00000	−0.166667
$37$	− 4.80194i	− 0.789434i	−0.918803	−	0.394717i	$-0.870843\pi$
$37$	− 4.80194i	− 0.789434i	0.918803	−	0.394717i	$-0.129157\pi$
$38$	−0.911854	−0.147922
$39$	0	0
$40$	−1.00000	−0.158114
$41$	0.198062i	0.0309321i	0.999880	+	0.0154661i	$0.00492320\pi$
$41$	0.198062i	0.0309321i	−0.999880	+	0.0154661i	$0.995077\pi$
$42$	1.19806	0.184865
$43$	−3.43296	−0.523522	−0.261761	−	0.965133i	$-0.584303\pi$
$43$	−3.43296	−0.523522	−0.261761	+	0.965133i	$0.584303\pi$
$44$	3.93900i	0.593827i
$45$	− 1.00000i	− 0.149071i
$46$	− 1.24698i	− 0.183857i
$47$	− 0.902165i	− 0.131594i	−0.997833	−	0.0657972i	$-0.979041\pi$
$47$	− 0.902165i	− 0.131594i	0.997833	−	0.0657972i	$-0.0209590\pi$
$48$	−1.00000	−0.144338
$49$	5.56465	0.794950
$50$	− 1.00000i	− 0.141421i
$51$	1.74094	0.243780
$52$	0	0
$53$	−6.91185	−0.949416	−0.474708	−	0.880143i	$-0.657446\pi$
$53$	−6.91185	−0.949416	−0.474708	+	0.880143i	$0.657446\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−3.93900	−0.531135
$56$	1.19806	0.160098
$57$	− 0.911854i	− 0.120778i
$58$	− 3.47219i	− 0.455921i
$59$	10.2446i	1.33373i	0.745178	+	0.666866i	$0.232365\pi$
$59$	10.2446i	1.33373i	−0.745178	+	0.666866i	$0.767635\pi$
$60$	− 1.00000i	− 0.129099i
$61$	9.92692	1.27101	0.635506	−	0.772096i	$-0.280792\pi$
$61$	9.92692	1.27101	0.635506	+	0.772096i	$0.280792\pi$
$62$	2.15883	0.274172
$63$	1.19806i	0.150942i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.93900	−0.484858
$67$	− 9.16852i	− 1.12011i	−0.828454	−	0.560057i	$-0.810779\pi$
$67$	− 9.16852i	− 1.12011i	0.828454	−	0.560057i	$-0.189221\pi$
$68$	1.74094	0.211120
$69$	1.24698	0.150119
$70$	1.19806i	0.143196i
$71$	11.0707i	1.31385i	0.753956	+	0.656924i	$0.228143\pi$
$71$	11.0707i	1.31385i	−0.753956	+	0.656924i	$0.771857\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.13706i	− 0.484207i	−0.970250	−	0.242103i	$-0.922163\pi$
$73$	− 4.13706i	− 0.484207i	0.970250	−	0.242103i	$-0.0778373\pi$
$74$	4.80194	0.558214
$75$	1.00000	0.115470
$76$	− 0.911854i	− 0.104597i
$77$	4.71917	0.537799
$78$	0	0
$79$	2.73556	0.307775	0.153887	−	0.988088i	$-0.450821\pi$
$79$	2.73556	0.307775	0.153887	+	0.988088i	$0.450821\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−0.198062	−0.0218723
$83$	15.5864i	1.71083i	0.517942	+	0.855416i	$0.326698\pi$
$83$	15.5864i	1.71083i	−0.517942	+	0.855416i	$0.673302\pi$
$84$	1.19806i	0.130719i
$85$	1.74094i	0.188831i
$86$	− 3.43296i	− 0.370186i
$87$	3.47219	0.372258
$88$	−3.93900	−0.419899
$89$	− 8.57673i	− 0.909131i	−0.890713	−	0.454566i	$-0.849794\pi$
$89$	− 8.57673i	− 0.909131i	0.890713	−	0.454566i	$-0.150206\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	1.24698	0.130007
$93$	2.15883i	0.223861i
$94$	0.902165	0.0930512
$95$	0.911854	0.0935542
$96$	− 1.00000i	− 0.102062i
$97$	16.9976i	1.72585i	0.505336	+	0.862923i	$0.331369\pi$
$97$	16.9976i	1.72585i	−0.505336	+	0.862923i	$0.668631\pi$
$98$	5.56465i	0.562114i
$99$	− 3.93900i	− 0.395885i
$100$	1.00000	0.100000
$101$	−6.67025	−0.663715	−0.331857	−	0.943330i	$-0.607675\pi$
$101$	−6.67025	−0.663715	−0.331857	+	0.943330i	$0.607675\pi$
$102$	1.74094i	0.172379i
$103$	−10.0315	−0.988429	−0.494215	−	0.869340i	$-0.664544\pi$
$103$	−10.0315	−0.988429	−0.494215	+	0.869340i	$0.664544\pi$
$104$	0	0
$105$	−1.19806	−0.116919
$106$	− 6.91185i	− 0.671339i
$107$	−10.1032	−0.976714	−0.488357	−	0.872644i	$-0.662404\pi$
$107$	−10.1032	−0.976714	−0.488357	+	0.872644i	$0.662404\pi$
$108$	1.00000	0.0962250
$109$	10.5211i	1.00774i	0.863780	+	0.503870i	$0.168091\pi$
$109$	10.5211i	1.00774i	−0.863780	+	0.503870i	$0.831909\pi$
$110$	− 3.93900i	− 0.375569i
$111$	4.80194i	0.455780i
$112$	1.19806i	0.113206i
$113$	−11.2131	−1.05484	−0.527421	−	0.849604i	$-0.676841\pi$
$113$	−11.2131	−1.05484	−0.527421	+	0.849604i	$0.676841\pi$
$114$	0.911854	0.0854030
$115$	1.24698i	0.116281i
$116$	3.47219	0.322385
$117$	0	0
$118$	−10.2446	−0.943091
$119$	− 2.08575i	− 0.191201i
$120$	1.00000	0.0912871
$121$	−4.51573	−0.410521
$122$	9.92692i	0.898741i
$123$	− 0.198062i	− 0.0178587i
$124$	2.15883i	0.193869i
$125$	1.00000i	0.0894427i
$126$	−1.19806	−0.106732
$127$	9.32736	0.827669	0.413834	−	0.910352i	$-0.364189\pi$
$127$	9.32736	0.827669	0.413834	+	0.910352i	$0.364189\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	3.43296	0.302255
$130$	0	0
$131$	4.62133	0.403768	0.201884	−	0.979409i	$-0.435294\pi$
$131$	4.62133	0.403768	0.201884	+	0.979409i	$0.435294\pi$
$132$	− 3.93900i	− 0.342846i
$133$	−1.09246	−0.0947281
$134$	9.16852	0.792040
$135$	1.00000i	0.0860663i
$136$	1.74094i	0.149284i
$137$	− 0.0760644i	− 0.00649862i	−0.999995	−	0.00324931i	$-0.998966\pi$
$137$	− 0.0760644i	− 0.00649862i	0.999995	−	0.00324931i	$-0.00103429\pi$
$138$	1.24698i	0.106150i
$139$	18.7017	1.58626	0.793129	−	0.609053i	$-0.208451\pi$
$139$	18.7017	1.58626	0.793129	+	0.609053i	$0.208451\pi$
$140$	−1.19806	−0.101255
$141$	0.902165i	0.0759760i
$142$	−11.0707	−0.929031
$143$	0	0
$144$	1.00000	0.0833333
$145$	3.47219i	0.288350i
$146$	4.13706	0.342386
$147$	−5.56465	−0.458964
$148$	4.80194i	0.394717i
$149$	13.6799i	1.12070i	0.828254	+	0.560352i	$0.189334\pi$
$149$	13.6799i	1.12070i	−0.828254	+	0.560352i	$0.810666\pi$
$150$	1.00000i	0.0816497i
$151$	1.15883i	0.0943045i	0.998888	+	0.0471523i	$0.0150146\pi$
$151$	1.15883i	0.0943045i	−0.998888	+	0.0471523i	$0.984985\pi$
$152$	0.911854	0.0739611
$153$	−1.74094	−0.140747
$154$	4.71917i	0.380281i
$155$	−2.15883	−0.173402
$156$	0	0
$157$	−12.2664	−0.978962	−0.489481	−	0.872014i	$-0.662814\pi$
$157$	−12.2664	−0.978962	−0.489481	+	0.872014i	$0.662814\pi$
$158$	2.73556i	0.217630i
$159$	6.91185	0.548146
$160$	1.00000	0.0790569
$161$	− 1.49396i	− 0.117740i
$162$	1.00000i	0.0785674i
$163$	2.48858i	0.194921i	0.995239	+	0.0974604i	$0.0310719\pi$
$163$	2.48858i	0.194921i	−0.995239	+	0.0974604i	$0.968928\pi$
$164$	− 0.198062i	− 0.0154661i
$165$	3.93900	0.306651
$166$	−15.5864	−1.20974
$167$	− 7.27950i	− 0.563305i	−0.959516	−	0.281652i	$-0.909117\pi$
$167$	− 7.27950i	− 0.563305i	0.959516	−	0.281652i	$-0.0908825\pi$
$168$	−1.19806	−0.0924325
$169$	0	0
$170$	−1.74094	−0.133524
$171$	0.911854i	0.0697312i
$172$	3.43296	0.261761
$173$	−15.8485	−1.20494	−0.602468	−	0.798143i	$-0.705816\pi$
$173$	−15.8485	−1.20494	−0.602468	+	0.798143i	$0.705816\pi$
$174$	3.47219i	0.263226i
$175$	− 1.19806i	− 0.0905650i
$176$	− 3.93900i	− 0.296913i
$177$	− 10.2446i	− 0.770030i
$178$	8.57673	0.642853
$179$	−18.4819	−1.38140	−0.690700	−	0.723141i	$-0.742698\pi$
$179$	−18.4819	−1.38140	−0.690700	+	0.723141i	$0.742698\pi$
$180$	1.00000i	0.0745356i
$181$	−10.7041	−0.795630	−0.397815	−	0.917466i	$-0.630231\pi$
$181$	−10.7041	−0.795630	−0.397815	+	0.917466i	$0.630231\pi$
$182$	0	0
$183$	−9.92692	−0.733819
$184$	1.24698i	0.0919286i
$185$	−4.80194	−0.353045
$186$	−2.15883	−0.158293
$187$	6.85756i	0.501474i
$188$	0.902165i	0.0657972i
$189$	− 1.19806i	− 0.0871462i
$190$	0.911854i	0.0661528i
$191$	15.3623	1.11158	0.555788	−	0.831324i	$-0.312417\pi$
$191$	15.3623	1.11158	0.555788	+	0.831324i	$0.312417\pi$
$192$	1.00000	0.0721688
$193$	18.3207i	1.31875i	0.751815	+	0.659375i	$0.229179\pi$
$193$	18.3207i	1.31875i	−0.751815	+	0.659375i	$0.770821\pi$
$194$	−16.9976	−1.22036
$195$	0	0
$196$	−5.56465	−0.397475
$197$	− 2.31336i	− 0.164820i	−0.996599	−	0.0824099i	$-0.973738\pi$
$197$	− 2.31336i	− 0.164820i	0.996599	−	0.0824099i	$-0.0262617\pi$
$198$	3.93900	0.279933
$199$	−6.36765	−0.451391	−0.225695	−	0.974198i	$-0.572465\pi$
$199$	−6.36765	−0.451391	−0.225695	+	0.974198i	$0.572465\pi$
$200$	1.00000i	0.0707107i
$201$	9.16852i	0.646698i
$202$	− 6.67025i	− 0.469317i
$203$	− 4.15990i	− 0.291968i
$204$	−1.74094	−0.121890
$205$	0.198062	0.0138333
$206$	− 10.0315i	− 0.698925i
$207$	−1.24698	−0.0866711
$208$	0	0
$209$	3.59179	0.248450
$210$	− 1.19806i	− 0.0826742i
$211$	−11.6775	−0.803915	−0.401958	−	0.915658i	$-0.631670\pi$
$211$	−11.6775	−0.803915	−0.401958	+	0.915658i	$0.631670\pi$
$212$	6.91185	0.474708
$213$	− 11.0707i	− 0.758551i
$214$	− 10.1032i	− 0.690641i
$215$	3.43296i	0.234126i
$216$	1.00000i	0.0680414i
$217$	2.58642	0.175577
$218$	−10.5211	−0.712579
$219$	4.13706i	0.279557i
$220$	3.93900	0.265567
$221$	0	0
$222$	−4.80194	−0.322285
$223$	11.4004i	0.763430i	0.924280	+	0.381715i	$0.124666\pi$
$223$	11.4004i	0.763430i	−0.924280	+	0.381715i	$0.875334\pi$
$224$	−1.19806	−0.0800489
$225$	−1.00000	−0.0666667
$226$	− 11.2131i	− 0.745886i
$227$	8.86725i	0.588540i	0.955722	+	0.294270i	$0.0950765\pi$
$227$	8.86725i	0.588540i	−0.955722	+	0.294270i	$0.904924\pi$
$228$	0.911854i	0.0603890i
$229$	− 12.6136i	− 0.833528i	−0.909015	−	0.416764i	$-0.863164\pi$
$229$	− 12.6136i	− 0.833528i	0.909015	−	0.416764i	$-0.136836\pi$
$230$	−1.24698	−0.0822234
$231$	−4.71917	−0.310498
$232$	3.47219i	0.227960i
$233$	23.4155	1.53400	0.767000	−	0.641647i	$-0.221748\pi$
$233$	23.4155	1.53400	0.767000	+	0.641647i	$0.221748\pi$
$234$	0	0
$235$	−0.902165	−0.0588508
$236$	− 10.2446i	− 0.666866i
$237$	−2.73556	−0.177694
$238$	2.08575	0.135199
$239$	8.87263i	0.573922i	0.957942	+	0.286961i	$0.0926451\pi$
$239$	8.87263i	0.573922i	−0.957942	+	0.286961i	$0.907355\pi$
$240$	1.00000i	0.0645497i
$241$	− 9.72886i	− 0.626691i	−0.949639	−	0.313345i	$-0.898550\pi$
$241$	− 9.72886i	− 0.626691i	0.949639	−	0.313345i	$-0.101450\pi$
$242$	− 4.51573i	− 0.290282i
$243$	−1.00000	−0.0641500
$244$	−9.92692	−0.635506
$245$	− 5.56465i	− 0.355512i
$246$	0.198062	0.0126280
$247$	0	0
$248$	−2.15883	−0.137086
$249$	− 15.5864i	− 0.987749i
$250$	−1.00000	−0.0632456
$251$	−29.8823	−1.88615	−0.943077	−	0.332573i	$-0.892083\pi$
$251$	−29.8823	−1.88615	−0.943077	+	0.332573i	$0.892083\pi$
$252$	− 1.19806i	− 0.0754708i
$253$	4.91185i	0.308806i
$254$	9.32736i	0.585250i
$255$	− 1.74094i	− 0.109022i
$256$	1.00000	0.0625000
$257$	8.52111	0.531532	0.265766	−	0.964038i	$-0.414375\pi$
$257$	8.52111	0.531532	0.265766	+	0.964038i	$0.414375\pi$
$258$	3.43296i	0.213727i
$259$	5.75302	0.357475
$260$	0	0
$261$	−3.47219	−0.214923
$262$	4.62133i	0.285507i
$263$	−22.6450	−1.39635	−0.698176	−	0.715926i	$-0.746005\pi$
$263$	−22.6450	−1.39635	−0.698176	+	0.715926i	$0.746005\pi$
$264$	3.93900	0.242429
$265$	6.91185i	0.424592i
$266$	− 1.09246i	− 0.0669829i
$267$	8.57673i	0.524887i
$268$	9.16852i	0.560057i
$269$	−9.02608	−0.550330	−0.275165	−	0.961397i	$-0.588732\pi$
$269$	−9.02608	−0.550330	−0.275165	+	0.961397i	$0.588732\pi$
$270$	−1.00000	−0.0608581
$271$	8.59611i	0.522176i	0.965315	+	0.261088i	$0.0840813\pi$
$271$	8.59611i	0.522176i	−0.965315	+	0.261088i	$0.915919\pi$
$272$	−1.74094	−0.105560
$273$	0	0
$274$	0.0760644	0.00459522
$275$	3.93900i	0.237531i
$276$	−1.24698	−0.0750594
$277$	−6.31527	−0.379448	−0.189724	−	0.981837i	$-0.560759\pi$
$277$	−6.31527	−0.379448	−0.189724	+	0.981837i	$0.560759\pi$
$278$	18.7017i	1.12165i
$279$	− 2.15883i	− 0.129246i
$280$	− 1.19806i	− 0.0715979i
$281$	− 21.7409i	− 1.29696i	−0.761234	−	0.648478i	$-0.775406\pi$
$281$	− 21.7409i	− 1.29696i	0.761234	−	0.648478i	$-0.224594\pi$
$282$	−0.902165	−0.0537232
$283$	−12.9487	−0.769720	−0.384860	−	0.922975i	$-0.625750\pi$
$283$	−12.9487	−0.769720	−0.384860	+	0.922975i	$0.625750\pi$
$284$	− 11.0707i	− 0.656924i
$285$	−0.911854	−0.0540136
$286$	0	0
$287$	−0.237291	−0.0140068
$288$	1.00000i	0.0589256i
$289$	−13.9691	−0.821714
$290$	−3.47219	−0.203894
$291$	− 16.9976i	− 0.996417i
$292$	4.13706i	0.242103i
$293$	− 27.3056i	− 1.59521i	−0.603181	−	0.797605i	$-0.706100\pi$
$293$	− 27.3056i	− 1.59521i	0.603181	−	0.797605i	$-0.293900\pi$
$294$	− 5.56465i	− 0.324537i
$295$	10.2446	0.596463
$296$	−4.80194	−0.279107
$297$	3.93900i	0.228564i
$298$	−13.6799	−0.792458
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 4.11290i	− 0.237064i
$302$	−1.15883	−0.0666834
$303$	6.67025	0.383196
$304$	0.911854i	0.0522984i
$305$	− 9.92692i	− 0.568414i
$306$	− 1.74094i	− 0.0995228i
$307$	24.9124i	1.42183i	0.703279	+	0.710914i	$0.251719\pi$
$307$	24.9124i	1.42183i	−0.703279	+	0.710914i	$0.748281\pi$
$308$	−4.71917	−0.268900
$309$	10.0315	0.570670
$310$	− 2.15883i	− 0.122614i
$311$	−21.3787	−1.21227	−0.606136	−	0.795361i	$-0.707281\pi$
$311$	−21.3787	−1.21227	−0.606136	+	0.795361i	$0.707281\pi$
$312$	0	0
$313$	−23.8713	−1.34929	−0.674643	−	0.738144i	$-0.735702\pi$
$313$	−23.8713	−1.34929	−0.674643	+	0.738144i	$0.735702\pi$
$314$	− 12.2664i	− 0.692231i
$315$	1.19806	0.0675032
$316$	−2.73556	−0.153887
$317$	− 12.6407i	− 0.709973i	−0.934871	−	0.354987i	$-0.884485\pi$
$317$	− 12.6407i	− 0.709973i	0.934871	−	0.354987i	$-0.115515\pi$
$318$	6.91185i	0.387598i
$319$	13.6770i	0.765763i
$320$	1.00000i	0.0559017i
$321$	10.1032	0.563906
$322$	1.49396	0.0832551
$323$	− 1.58748i	− 0.0883299i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−2.48858	−0.137830
$327$	− 10.5211i	− 0.581819i
$328$	0.198062	0.0109362
$329$	1.08085	0.0595892
$330$	3.93900i	0.216835i
$331$	20.5109i	1.12738i	0.825985	+	0.563692i	$0.190619\pi$
$331$	20.5109i	1.12738i	−0.825985	+	0.563692i	$0.809381\pi$
$332$	− 15.5864i	− 0.855416i
$333$	− 4.80194i	− 0.263145i
$334$	7.27950	0.398317
$335$	−9.16852	−0.500930
$336$	− 1.19806i	− 0.0653597i
$337$	−6.82371	−0.371711	−0.185856	−	0.982577i	$-0.559506\pi$
$337$	−6.82371	−0.371711	−0.185856	+	0.982577i	$0.559506\pi$
$338$	0	0
$339$	11.2131	0.609014
$340$	− 1.74094i	− 0.0944157i
$341$	−8.50365	−0.460498
$342$	−0.911854	−0.0493074
$343$	15.0532i	0.812798i
$344$	3.43296i	0.185093i
$345$	− 1.24698i	− 0.0671351i
$346$	− 15.8485i	− 0.852019i
$347$	−29.4741	−1.58225	−0.791127	−	0.611653i	$-0.790505\pi$
$347$	−29.4741	−1.58225	−0.791127	+	0.611653i	$0.790505\pi$
$348$	−3.47219	−0.186129
$349$	34.5394i	1.84885i	0.381361	+	0.924426i	$0.375456\pi$
$349$	34.5394i	1.84885i	−0.381361	+	0.924426i	$0.624544\pi$
$350$	1.19806	0.0640391
$351$	0	0
$352$	3.93900	0.209949
$353$	31.2295i	1.66218i	0.556138	+	0.831090i	$0.312283\pi$
$353$	31.2295i	1.66218i	−0.556138	+	0.831090i	$0.687717\pi$
$354$	10.2446	0.544494
$355$	11.0707	0.587571
$356$	8.57673i	0.454566i
$357$	2.08575i	0.110390i
$358$	− 18.4819i	− 0.976798i
$359$	− 18.2524i	− 0.963323i	−0.876357	−	0.481661i	$-0.840034\pi$
$359$	− 18.2524i	− 0.963323i	0.876357	−	0.481661i	$-0.159966\pi$
$360$	−1.00000	−0.0527046
$361$	18.1685	0.956238
$362$	− 10.7041i	− 0.562595i
$363$	4.51573	0.237014
$364$	0	0
$365$	−4.13706	−0.216544
$366$	− 9.92692i	− 0.518888i
$367$	−19.3773	−1.01149	−0.505744	−	0.862683i	$-0.668782\pi$
$367$	−19.3773	−1.01149	−0.505744	+	0.862683i	$0.668782\pi$
$368$	−1.24698	−0.0650033
$369$	0.198062i	0.0103107i
$370$	− 4.80194i	− 0.249641i
$371$	− 8.28083i	− 0.429919i
$372$	− 2.15883i	− 0.111930i
$373$	−13.6280	−0.705633	−0.352817	−	0.935693i	$-0.614776\pi$
$373$	−13.6280	−0.705633	−0.352817	+	0.935693i	$0.614776\pi$
$374$	−6.85756	−0.354596
$375$	− 1.00000i	− 0.0516398i
$376$	−0.902165	−0.0465256
$377$	0	0
$378$	1.19806	0.0616217
$379$	− 1.32006i	− 0.0678069i	−0.999425	−	0.0339035i	$-0.989206\pi$
$379$	− 1.32006i	− 0.0678069i	0.999425	−	0.0339035i	$-0.0107939\pi$
$380$	−0.911854	−0.0467771
$381$	−9.32736	−0.477855
$382$	15.3623i	0.786002i
$383$	25.5308i	1.30456i	0.757977	+	0.652281i	$0.226188\pi$
$383$	25.5308i	1.30456i	−0.757977	+	0.652281i	$0.773812\pi$
$384$	1.00000i	0.0510310i
$385$	− 4.71917i	− 0.240511i
$386$	−18.3207	−0.932497
$387$	−3.43296	−0.174507
$388$	− 16.9976i	− 0.862923i
$389$	6.05967	0.307238	0.153619	−	0.988130i	$-0.450907\pi$
$389$	6.05967	0.307238	0.153619	+	0.988130i	$0.450907\pi$
$390$	0	0
$391$	2.17092	0.109788
$392$	− 5.56465i	− 0.281057i
$393$	−4.62133	−0.233115
$394$	2.31336	0.116545
$395$	− 2.73556i	− 0.137641i
$396$	3.93900i	0.197942i
$397$	− 5.99330i	− 0.300795i	−0.988626	−	0.150397i	$-0.951945\pi$
$397$	− 5.99330i	− 0.300795i	0.988626	−	0.150397i	$-0.0480553\pi$
$398$	− 6.36765i	− 0.319181i
$399$	1.09246	0.0546913
$400$	−1.00000	−0.0500000
$401$	− 17.3773i	− 0.867783i	−0.900965	−	0.433891i	$-0.857140\pi$
$401$	− 17.3773i	− 0.867783i	0.900965	−	0.433891i	$-0.142860\pi$
$402$	−9.16852	−0.457284
$403$	0	0
$404$	6.67025	0.331857
$405$	− 1.00000i	− 0.0496904i
$406$	4.15990	0.206452
$407$	−18.9148	−0.937574
$408$	− 1.74094i	− 0.0861893i
$409$	29.3672i	1.45211i	0.687635	+	0.726057i	$0.258649\pi$
$409$	29.3672i	1.45211i	−0.687635	+	0.726057i	$0.741351\pi$
$410$	0.198062i	0.00978160i
$411$	0.0760644i	0.00375198i
$412$	10.0315	0.494215
$413$	−12.2737	−0.603947
$414$	− 1.24698i	− 0.0612857i
$415$	15.5864	0.765107
$416$	0	0
$417$	−18.7017	−0.915827
$418$	3.59179i	0.175680i
$419$	−26.5472	−1.29692	−0.648458	−	0.761251i	$-0.724586\pi$
$419$	−26.5472	−1.29692	−0.648458	+	0.761251i	$0.724586\pi$
$420$	1.19806	0.0584595
$421$	− 31.0834i	− 1.51491i	−0.652887	−	0.757455i	$-0.726442\pi$
$421$	− 31.0834i	− 1.51491i	0.652887	−	0.757455i	$-0.273558\pi$
$422$	− 11.6775i	− 0.568454i
$423$	− 0.902165i	− 0.0438648i
$424$	6.91185i	0.335669i
$425$	1.74094	0.0844479
$426$	11.0707	0.536377
$427$	11.8931i	0.575546i
$428$	10.1032	0.488357
$429$	0	0
$430$	−3.43296	−0.165552
$431$	36.1226i	1.73996i	0.493084	+	0.869982i	$0.335870\pi$
$431$	36.1226i	1.73996i	−0.493084	+	0.869982i	$0.664130\pi$
$432$	−1.00000	−0.0481125
$433$	35.2868	1.69578	0.847888	−	0.530176i	$-0.177874\pi$
$433$	35.2868	1.69578	0.847888	+	0.530176i	$0.177874\pi$
$434$	2.58642i	0.124152i
$435$	− 3.47219i	− 0.166479i
$436$	− 10.5211i	− 0.503870i
$437$	− 1.13706i	− 0.0543931i
$438$	−4.13706	−0.197677
$439$	32.0170	1.52809	0.764044	−	0.645165i	$-0.223211\pi$
$439$	32.0170	1.52809	0.764044	+	0.645165i	$0.223211\pi$
$440$	3.93900i	0.187785i
$441$	5.56465	0.264983
$442$	0	0
$443$	−23.5948	−1.12102	−0.560511	−	0.828147i	$-0.689395\pi$
$443$	−23.5948	−1.12102	−0.560511	+	0.828147i	$0.689395\pi$
$444$	− 4.80194i	− 0.227890i
$445$	−8.57673	−0.406576
$446$	−11.4004	−0.539826
$447$	− 13.6799i	− 0.647039i
$448$	− 1.19806i	− 0.0566031i
$449$	19.8374i	0.936187i	0.883679	+	0.468093i	$0.155059\pi$
$449$	19.8374i	0.936187i	−0.883679	+	0.468093i	$0.844941\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	0.780167	0.0367367
$452$	11.2131	0.527421
$453$	− 1.15883i	− 0.0544468i
$454$	−8.86725	−0.416161
$455$	0	0
$456$	−0.911854	−0.0427015
$457$	− 26.5284i	− 1.24095i	−0.784228	−	0.620473i	$-0.786941\pi$
$457$	− 26.5284i	− 1.24095i	0.784228	−	0.620473i	$-0.213059\pi$
$458$	12.6136	0.589393
$459$	1.74094	0.0812601
$460$	− 1.24698i	− 0.0581407i
$461$	− 12.2717i	− 0.571552i	−0.958297	−	0.285776i	$-0.907749\pi$
$461$	− 12.2717i	− 0.571552i	0.958297	−	0.285776i	$-0.0922513\pi$
$462$	− 4.71917i	− 0.219556i
$463$	12.6455i	0.587686i	0.955854	+	0.293843i	$0.0949343\pi$
$463$	12.6455i	0.587686i	−0.955854	+	0.293843i	$0.905066\pi$
$464$	−3.47219	−0.161192
$465$	2.15883	0.100114
$466$	23.4155i	1.08470i
$467$	−11.3870	−0.526929	−0.263464	−	0.964669i	$-0.584865\pi$
$467$	−11.3870	−0.526929	−0.263464	+	0.964669i	$0.584865\pi$
$468$	0	0
$469$	10.9845	0.507215
$470$	− 0.902165i	− 0.0416138i
$471$	12.2664	0.565204
$472$	10.2446	0.471545
$473$	13.5224i	0.621762i
$474$	− 2.73556i	− 0.125649i
$475$	− 0.911854i	− 0.0418387i
$476$	2.08575i	0.0956003i
$477$	−6.91185	−0.316472
$478$	−8.87263	−0.405824
$479$	− 18.8431i	− 0.860963i	−0.902600	−	0.430481i	$-0.858344\pi$
$479$	− 18.8431i	− 0.860963i	0.902600	−	0.430481i	$-0.141656\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	9.72886	0.443137
$483$	1.49396i	0.0679775i
$484$	4.51573	0.205260
$485$	16.9976	0.771822
$486$	− 1.00000i	− 0.0453609i
$487$	27.6383i	1.25241i	0.779658	+	0.626206i	$0.215393\pi$
$487$	27.6383i	1.25241i	−0.779658	+	0.626206i	$0.784607\pi$
$488$	− 9.92692i	− 0.449371i
$489$	− 2.48858i	− 0.112538i
$490$	5.56465	0.251385
$491$	13.9866	0.631206	0.315603	−	0.948891i	$-0.397793\pi$
$491$	13.9866	0.631206	0.315603	+	0.948891i	$0.397793\pi$
$492$	0.198062i	0.00892934i
$493$	6.04487	0.272247
$494$	0	0
$495$	−3.93900	−0.177045
$496$	− 2.15883i	− 0.0969345i
$497$	−13.2634	−0.594944
$498$	15.5864	0.698444
$499$	− 8.87071i	− 0.397107i	−0.980090	−	0.198554i	$-0.936376\pi$
$499$	− 8.87071i	− 0.397107i	0.980090	−	0.198554i	$-0.0636244\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	7.27950i	0.325224i
$502$	− 29.8823i	− 1.33371i
$503$	1.99761	0.0890689	0.0445344	−	0.999008i	$-0.485820\pi$
$503$	1.99761	0.0890689	0.0445344	+	0.999008i	$0.485820\pi$
$504$	1.19806	0.0533659
$505$	6.67025i	0.296822i
$506$	−4.91185	−0.218359
$507$	0	0
$508$	−9.32736	−0.413834
$509$	− 21.7023i	− 0.961938i	−0.876738	−	0.480969i	$-0.840285\pi$
$509$	− 21.7023i	− 0.961938i	0.876738	−	0.480969i	$-0.159715\pi$
$510$	1.74094	0.0770901
$511$	4.95646	0.219261
$512$	1.00000i	0.0441942i
$513$	− 0.911854i	− 0.0402593i
$514$	8.52111i	0.375850i
$515$	10.0315i	0.442039i
$516$	−3.43296	−0.151128
$517$	−3.55363	−0.156288
$518$	5.75302i	0.252773i
$519$	15.8485	0.695670
$520$	0	0
$521$	11.8364	0.518561	0.259281	−	0.965802i	$-0.416515\pi$
$521$	11.8364	0.518561	0.259281	+	0.965802i	$0.416515\pi$
$522$	− 3.47219i	− 0.151974i
$523$	24.6112	1.07617	0.538086	−	0.842890i	$-0.319148\pi$
$523$	24.6112	1.07617	0.538086	+	0.842890i	$0.319148\pi$
$524$	−4.62133	−0.201884
$525$	1.19806i	0.0522877i
$526$	− 22.6450i	− 0.987370i
$527$	3.75840i	0.163718i
$528$	3.93900i	0.171423i
$529$	−21.4450	−0.932393
$530$	−6.91185	−0.300232
$531$	10.2446i	0.444577i
$532$	1.09246	0.0473641
$533$	0	0
$534$	−8.57673	−0.371151
$535$	10.1032i	0.436800i
$536$	−9.16852	−0.396020
$537$	18.4819	0.797552
$538$	− 9.02608i	− 0.389142i
$539$	− 21.9191i	− 0.944125i
$540$	− 1.00000i	− 0.0430331i
$541$	− 29.1282i	− 1.25232i	−0.779694	−	0.626160i	$-0.784626\pi$
$541$	− 29.1282i	− 1.25232i	0.779694	−	0.626160i	$-0.215374\pi$
$542$	−8.59611	−0.369234
$543$	10.7041	0.459357
$544$	− 1.74094i	− 0.0746421i
$545$	10.5211	0.450675
$546$	0	0
$547$	−10.1752	−0.435061	−0.217531	−	0.976053i	$-0.569800\pi$
$547$	−10.1752	−0.435061	−0.217531	+	0.976053i	$0.569800\pi$
$548$	0.0760644i	0.00324931i
$549$	9.92692	0.423671
$550$	−3.93900	−0.167960
$551$	− 3.16613i	− 0.134882i
$552$	− 1.24698i	− 0.0530750i
$553$	3.27737i	0.139368i
$554$	− 6.31527i	− 0.268310i
$555$	4.80194	0.203831
$556$	−18.7017	−0.793129
$557$	− 13.0398i	− 0.552515i	−0.961084	−	0.276257i	$-0.910906\pi$
$557$	− 13.0398i	− 0.552515i	0.961084	−	0.276257i	$-0.0890942\pi$
$558$	2.15883	0.0913907
$559$	0	0
$560$	1.19806	0.0506274
$561$	− 6.85756i	− 0.289526i
$562$	21.7409	0.917086
$563$	32.9221	1.38750	0.693751	−	0.720215i	$-0.255957\pi$
$563$	32.9221	1.38750	0.693751	+	0.720215i	$0.255957\pi$
$564$	− 0.902165i	− 0.0379880i
$565$	11.2131i	0.471740i
$566$	− 12.9487i	− 0.544274i
$567$	1.19806i	0.0503139i
$568$	11.0707	0.464516
$569$	−7.53319	−0.315808	−0.157904	−	0.987454i	$-0.550474\pi$
$569$	−7.53319	−0.315808	−0.157904	+	0.987454i	$0.550474\pi$
$570$	− 0.911854i	− 0.0381934i
$571$	−36.2422	−1.51669	−0.758344	−	0.651854i	$-0.773991\pi$
$571$	−36.2422	−1.51669	−0.758344	+	0.651854i	$0.773991\pi$
$572$	0	0
$573$	−15.3623	−0.641768
$574$	− 0.237291i	− 0.00990433i
$575$	1.24698	0.0520026
$576$	−1.00000	−0.0416667
$577$	− 14.2198i	− 0.591979i	−0.955191	−	0.295990i	$-0.904351\pi$
$577$	− 14.2198i	− 0.591979i	0.955191	−	0.295990i	$-0.0956494\pi$
$578$	− 13.9691i	− 0.581039i
$579$	− 18.3207i	− 0.761380i
$580$	− 3.47219i	− 0.144175i
$581$	−18.6735	−0.774707
$582$	16.9976	0.704573
$583$	27.2258i	1.12758i
$584$	−4.13706	−0.171193
$585$	0	0
$586$	27.3056	1.12798
$587$	22.3207i	0.921272i	0.887589	+	0.460636i	$0.152379\pi$
$587$	22.3207i	0.921272i	−0.887589	+	0.460636i	$0.847621\pi$
$588$	5.56465	0.229482
$589$	1.96854	0.0811123
$590$	10.2446i	0.421763i
$591$	2.31336i	0.0951587i
$592$	− 4.80194i	− 0.197358i
$593$	1.64742i	0.0676513i	0.999428	+	0.0338256i	$0.0107691\pi$
$593$	1.64742i	0.0676513i	−0.999428	+	0.0338256i	$0.989231\pi$
$594$	−3.93900	−0.161619
$595$	−2.08575	−0.0855075
$596$	− 13.6799i	− 0.560352i
$597$	6.36765	0.260611
$598$	0	0
$599$	9.41119	0.384531	0.192265	−	0.981343i	$-0.438417\pi$
$599$	9.41119	0.384531	0.192265	+	0.981343i	$0.438417\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	1.62565	0.0663115	0.0331557	−	0.999450i	$-0.489444\pi$
$601$	1.62565	0.0663115	0.0331557	+	0.999450i	$0.489444\pi$
$602$	4.11290	0.167629
$603$	− 9.16852i	− 0.373371i
$604$	− 1.15883i	− 0.0471523i
$605$	4.51573i	0.183591i
$606$	6.67025i	0.270960i
$607$	14.9282	0.605919	0.302959	−	0.953003i	$-0.402025\pi$
$607$	14.9282	0.605919	0.302959	+	0.953003i	$0.402025\pi$
$608$	−0.911854	−0.0369806
$609$	4.15990i	0.168568i
$610$	9.92692	0.401929
$611$	0	0
$612$	1.74094	0.0703733
$613$	28.8437i	1.16499i	0.812836	+	0.582493i	$0.197922\pi$
$613$	28.8437i	1.16499i	−0.812836	+	0.582493i	$0.802078\pi$
$614$	−24.9124	−1.00538
$615$	−0.198062	−0.00798664
$616$	− 4.71917i	− 0.190141i
$617$	− 39.7985i	− 1.60223i	−0.598513	−	0.801113i	$-0.704241\pi$
$617$	− 39.7985i	− 1.60223i	0.598513	−	0.801113i	$-0.295759\pi$
$618$	10.0315i	0.403524i
$619$	38.0465i	1.52922i	0.644494	+	0.764609i	$0.277068\pi$
$619$	38.0465i	1.52922i	−0.644494	+	0.764609i	$0.722932\pi$
$620$	2.15883	0.0867008
$621$	1.24698	0.0500396
$622$	− 21.3787i	− 0.857206i
$623$	10.2755	0.411677
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 23.8713i	− 0.954089i
$627$	−3.59179	−0.143442
$628$	12.2664	0.489481
$629$	8.35988i	0.333330i
$630$	1.19806i	0.0477319i
$631$	− 23.0901i	− 0.919201i	−0.888126	−	0.459600i	$-0.847993\pi$
$631$	− 23.0901i	− 0.919201i	0.888126	−	0.459600i	$-0.152007\pi$
$632$	− 2.73556i	− 0.108815i
$633$	11.6775	0.464141
$634$	12.6407	0.502027
$635$	− 9.32736i	− 0.370145i
$636$	−6.91185	−0.274073
$637$	0	0
$638$	−13.6770	−0.541476
$639$	11.0707i	0.437950i
$640$	−1.00000	−0.0395285
$641$	−23.3948	−0.924039	−0.462019	−	0.886870i	$-0.652875\pi$
$641$	−23.3948	−0.924039	−0.462019	+	0.886870i	$0.652875\pi$
$642$	10.1032i	0.398742i
$643$	− 6.00836i	− 0.236947i	−0.992957	−	0.118473i	$-0.962200\pi$
$643$	− 6.00836i	− 0.236947i	0.992957	−	0.118473i	$-0.0378000\pi$
$644$	1.49396i	0.0588702i
$645$	− 3.43296i	− 0.135173i
$646$	1.58748	0.0624586
$647$	4.38511	0.172396	0.0861982	−	0.996278i	$-0.472528\pi$
$647$	4.38511	0.172396	0.0861982	+	0.996278i	$0.472528\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	40.3534	1.58401
$650$	0	0
$651$	−2.58642	−0.101370
$652$	− 2.48858i	− 0.0974604i
$653$	−20.6160	−0.806765	−0.403382	−	0.915032i	$-0.632166\pi$
$653$	−20.6160	−0.806765	−0.403382	+	0.915032i	$0.632166\pi$
$654$	10.5211	0.411408
$655$	− 4.62133i	− 0.180570i
$656$	0.198062i	0.00773303i
$657$	− 4.13706i	− 0.161402i
$658$	1.08085i	0.0421359i
$659$	18.4910	0.720306	0.360153	−	0.932893i	$-0.382724\pi$
$659$	18.4910	0.720306	0.360153	+	0.932893i	$0.382724\pi$
$660$	−3.93900	−0.153325
$661$	− 17.6420i	− 0.686196i	−0.939300	−	0.343098i	$-0.888524\pi$
$661$	− 17.6420i	− 0.686196i	0.939300	−	0.343098i	$-0.111476\pi$
$662$	−20.5109	−0.797180
$663$	0	0
$664$	15.5864	0.604870
$665$	1.09246i	0.0423637i
$666$	4.80194	0.186071
$667$	4.32975	0.167649
$668$	7.27950i	0.281652i
$669$	− 11.4004i	− 0.440766i
$670$	− 9.16852i	− 0.354211i
$671$	− 39.1021i	− 1.50952i
$672$	1.19806	0.0462163
$673$	−17.1473	−0.660981	−0.330491	−	0.943809i	$-0.607214\pi$
$673$	−17.1473	−0.660981	−0.330491	+	0.943809i	$0.607214\pi$
$674$	− 6.82371i	− 0.262839i
$675$	1.00000	0.0384900
$676$	0	0
$677$	31.7982	1.22210	0.611052	−	0.791590i	$-0.290746\pi$
$677$	31.7982	1.22210	0.611052	+	0.791590i	$0.290746\pi$
$678$	11.2131i	0.430638i
$679$	−20.3642	−0.781506
$680$	1.74094	0.0667620
$681$	− 8.86725i	− 0.339794i
$682$	− 8.50365i	− 0.325622i
$683$	− 6.32783i	− 0.242128i	−0.992645	−	0.121064i	$-0.961369\pi$
$683$	− 6.32783i	− 0.242128i	0.992645	−	0.121064i	$-0.0386306\pi$
$684$	− 0.911854i	− 0.0348656i
$685$	−0.0760644	−0.00290627
$686$	−15.0532	−0.574735
$687$	12.6136i	0.481237i
$688$	−3.43296	−0.130880
$689$	0	0
$690$	1.24698	0.0474717
$691$	− 15.1825i	− 0.577570i	−0.957394	−	0.288785i	$-0.906749\pi$
$691$	− 15.1825i	− 0.577570i	0.957394	−	0.288785i	$-0.0932513\pi$
$692$	15.8485	0.602468
$693$	4.71917	0.179266
$694$	− 29.4741i	− 1.11882i
$695$	− 18.7017i	− 0.709396i
$696$	− 3.47219i	− 0.131613i
$697$	− 0.344814i	− 0.0130608i
$698$	−34.5394	−1.30734
$699$	−23.4155	−0.885656
$700$	1.19806i	0.0452825i
$701$	23.6528	0.893354	0.446677	−	0.894695i	$-0.352607\pi$
$701$	23.6528	0.893354	0.446677	+	0.894695i	$0.352607\pi$
$702$	0	0
$703$	4.37867	0.165145
$704$	3.93900i	0.148457i
$705$	0.902165	0.0339775
$706$	−31.2295	−1.17534
$707$	− 7.99138i	− 0.300547i
$708$	10.2446i	0.385015i
$709$	40.9353i	1.53736i	0.639636	+	0.768678i	$0.279085\pi$
$709$	40.9353i	1.53736i	−0.639636	+	0.768678i	$0.720915\pi$
$710$	11.0707i	0.415476i
$711$	2.73556	0.102592
$712$	−8.57673	−0.321426
$713$	2.69202i	0.100817i
$714$	−2.08575	−0.0780573
$715$	0	0
$716$	18.4819	0.690700
$717$	− 8.87263i	− 0.331354i
$718$	18.2524	0.681172
$719$	−4.55363	−0.169822	−0.0849109	−	0.996389i	$-0.527061\pi$
$719$	−4.55363	−0.169822	−0.0849109	+	0.996389i	$0.527061\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 12.0183i	− 0.447585i
$722$	18.1685i	0.676162i
$723$	9.72886i	0.361820i
$724$	10.7041	0.397815
$725$	3.47219	0.128954
$726$	4.51573i	0.167594i
$727$	−18.1612	−0.673563	−0.336781	−	0.941583i	$-0.609338\pi$
$727$	−18.1612	−0.673563	−0.336781	+	0.941583i	$0.609338\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 4.13706i	− 0.153120i
$731$	5.97657	0.221052
$732$	9.92692	0.366910
$733$	9.89115i	0.365338i	0.983174	+	0.182669i	$0.0584737\pi$
$733$	9.89115i	0.365338i	−0.983174	+	0.182669i	$0.941526\pi$
$734$	− 19.3773i	− 0.715231i
$735$	5.56465i	0.205255i
$736$	− 1.24698i	− 0.0459643i
$737$	−36.1148	−1.33031
$738$	−0.198062	−0.00729077
$739$	23.9608i	0.881411i	0.897652	+	0.440706i	$0.145272\pi$
$739$	23.9608i	0.881411i	−0.897652	+	0.440706i	$0.854728\pi$
$740$	4.80194	0.176523
$741$	0	0
$742$	8.28083	0.303999
$743$	46.6708i	1.71219i	0.516821	+	0.856094i	$0.327115\pi$
$743$	46.6708i	1.71219i	−0.516821	+	0.856094i	$0.672885\pi$
$744$	2.15883	0.0791467
$745$	13.6799	0.501194
$746$	− 13.6280i	− 0.498958i
$747$	15.5864i	0.570277i
$748$	− 6.85756i	− 0.250737i
$749$	− 12.1043i	− 0.442281i
$750$	1.00000	0.0365148
$751$	30.9168	1.12817	0.564084	−	0.825717i	$-0.309229\pi$
$751$	30.9168	1.12817	0.564084	+	0.825717i	$0.309229\pi$
$752$	− 0.902165i	− 0.0328986i
$753$	29.8823	1.08897
$754$	0	0
$755$	1.15883	0.0421743
$756$	1.19806i	0.0435731i
$757$	24.4566	0.888892	0.444446	−	0.895806i	$-0.353401\pi$
$757$	24.4566	0.888892	0.444446	+	0.895806i	$0.353401\pi$
$758$	1.32006	0.0479467
$759$	− 4.91185i	− 0.178289i
$760$	− 0.911854i	− 0.0330764i
$761$	20.5536i	0.745069i	0.928018	+	0.372534i	$0.121511\pi$
$761$	20.5536i	0.745069i	−0.928018	+	0.372534i	$0.878489\pi$
$762$	− 9.32736i	− 0.337894i
$763$	−12.6049	−0.456329
$764$	−15.3623	−0.555788
$765$	1.74094i	0.0629438i
$766$	−25.5308	−0.922465
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 21.0084i	− 0.757581i	−0.925482	−	0.378790i	$-0.876340\pi$
$769$	− 21.0084i	− 0.757581i	0.925482	−	0.378790i	$-0.123660\pi$
$770$	4.71917	0.170067
$771$	−8.52111	−0.306880
$772$	− 18.3207i	− 0.659375i
$773$	− 36.2626i	− 1.30428i	−0.758100	−	0.652138i	$-0.773872\pi$
$773$	− 36.2626i	− 1.30428i	0.758100	−	0.652138i	$-0.226128\pi$
$774$	− 3.43296i	− 0.123395i
$775$	2.15883i	0.0775476i
$776$	16.9976	0.610179
$777$	−5.75302	−0.206388
$778$	6.05967i	0.217250i
$779$	−0.180604	−0.00647081
$780$	0	0
$781$	43.6075	1.56040
$782$	2.17092i	0.0776318i
$783$	3.47219	0.124086
$784$	5.56465	0.198737
$785$	12.2664i	0.437805i
$786$	− 4.62133i	− 0.164838i
$787$	1.55927i	0.0555820i	0.999614	+	0.0277910i	$0.00884728\pi$
$787$	1.55927i	0.0555820i	−0.999614	+	0.0277910i	$0.991153\pi$
$788$	2.31336i	0.0824099i
$789$	22.6450	0.806184
$790$	2.73556	0.0973269
$791$	− 13.4340i	− 0.477659i
$792$	−3.93900	−0.139966
$793$	0	0
$794$	5.99330	0.212694
$795$	− 6.91185i	− 0.245138i
$796$	6.36765	0.225695
$797$	−5.16793	−0.183058	−0.0915288	−	0.995802i	$-0.529175\pi$
$797$	−5.16793	−0.183058	−0.0915288	+	0.995802i	$0.529175\pi$
$798$	1.09246i	0.0386726i
$799$	1.57061i	0.0555644i
$800$	− 1.00000i	− 0.0353553i
$801$	− 8.57673i	− 0.303044i
$802$	17.3773	0.613615
$803$	−16.2959	−0.575070
$804$	− 9.16852i	− 0.323349i
$805$	−1.49396	−0.0526551
$806$	0	0
$807$	9.02608	0.317733
$808$	6.67025i	0.234659i
$809$	−31.8605	−1.12016	−0.560079	−	0.828440i	$-0.689229\pi$
$809$	−31.8605	−1.12016	−0.560079	+	0.828440i	$0.689229\pi$
$810$	1.00000	0.0351364
$811$	− 15.3056i	− 0.537452i	−0.963217	−	0.268726i	$-0.913397\pi$
$811$	− 15.3056i	− 0.537452i	0.963217	−	0.268726i	$-0.0866026\pi$
$812$	4.15990i	0.145984i
$813$	− 8.59611i	− 0.301479i
$814$	− 18.9148i	− 0.662965i
$815$	2.48858	0.0871712
$816$	1.74094	0.0609450
$817$	− 3.13036i	− 0.109517i
$818$	−29.3672	−1.02680
$819$	0	0
$820$	−0.198062	−0.00691663
$821$	32.0248i	1.11767i	0.829278	+	0.558836i	$0.188752\pi$
$821$	32.0248i	1.11767i	−0.829278	+	0.558836i	$0.811248\pi$
$822$	−0.0760644	−0.00265305
$823$	36.8571	1.28476	0.642379	−	0.766387i	$-0.277948\pi$
$823$	36.8571	1.28476	0.642379	+	0.766387i	$0.277948\pi$
$824$	10.0315i	0.349462i
$825$	− 3.93900i	− 0.137138i
$826$	− 12.2737i	− 0.427055i
$827$	37.9269i	1.31885i	0.751771	+	0.659424i	$0.229200\pi$
$827$	37.9269i	1.31885i	−0.751771	+	0.659424i	$0.770800\pi$
$828$	1.24698	0.0433355
$829$	34.2349	1.18903	0.594514	−	0.804086i	$-0.297345\pi$
$829$	34.2349	1.18903	0.594514	+	0.804086i	$0.297345\pi$
$830$	15.5864i	0.541012i
$831$	6.31527	0.219074
$832$	0	0
$833$	−9.68771	−0.335659
$834$	− 18.7017i	− 0.647587i
$835$	−7.27950	−0.251918
$836$	−3.59179	−0.124225
$837$	2.15883i	0.0746202i
$838$	− 26.5472i	− 0.917057i
$839$	− 18.7318i	− 0.646695i	−0.946280	−	0.323347i	$-0.895192\pi$
$839$	− 18.7318i	− 0.646695i	0.946280	−	0.323347i	$-0.104808\pi$
$840$	1.19806i	0.0413371i
$841$	−16.9439	−0.584273
$842$	31.0834	1.07120
$843$	21.7409i	0.748798i
$844$	11.6775	0.401958
$845$	0	0
$846$	0.902165	0.0310171
$847$	− 5.41013i	− 0.185894i
$848$	−6.91185	−0.237354
$849$	12.9487	0.444398
$850$	1.74094i	0.0597137i
$851$	5.98792i	0.205263i
$852$	11.0707i	0.379276i
$853$	4.64071i	0.158895i	0.996839	+	0.0794475i	$0.0253156\pi$
$853$	4.64071i	0.158895i	−0.996839	+	0.0794475i	$0.974684\pi$
$854$	−11.8931	−0.406972
$855$	0.911854	0.0311847
$856$	10.1032i	0.345321i
$857$	−11.1347	−0.380353	−0.190177	−	0.981750i	$-0.560906\pi$
$857$	−11.1347	−0.380353	−0.190177	+	0.981750i	$0.560906\pi$
$858$	0	0
$859$	28.9661	0.988312	0.494156	−	0.869373i	$-0.335477\pi$
$859$	28.9661	0.988312	0.494156	+	0.869373i	$0.335477\pi$
$860$	− 3.43296i	− 0.117063i
$861$	0.237291	0.00808685
$862$	−36.1226	−1.23034
$863$	33.1371i	1.12800i	0.825775	+	0.563999i	$0.190738\pi$
$863$	33.1371i	1.12800i	−0.825775	+	0.563999i	$0.809262\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	15.8485i	0.538864i
$866$	35.2868i	1.19909i
$867$	13.9691	0.474417
$868$	−2.58642	−0.0877887
$869$	− 10.7754i	− 0.365530i
$870$	3.47219	0.117718
$871$	0	0
$872$	10.5211	0.356290
$873$	16.9976i	0.575282i
$874$	1.13706	0.0384617
$875$	−1.19806	−0.0405019
$876$	− 4.13706i	− 0.139778i
$877$	19.1371i	0.646213i	0.946363	+	0.323106i	$0.104727\pi$
$877$	19.1371i	0.646213i	−0.946363	+	0.323106i	$0.895273\pi$
$878$	32.0170i	1.08052i
$879$	27.3056i	0.920995i
$880$	−3.93900	−0.132784
$881$	40.3387	1.35905	0.679523	−	0.733655i	$-0.262187\pi$
$881$	40.3387	1.35905	0.679523	+	0.733655i	$0.262187\pi$
$882$	5.56465i	0.187371i
$883$	4.31336	0.145156	0.0725780	−	0.997363i	$-0.476877\pi$
$883$	4.31336	0.145156	0.0725780	+	0.997363i	$0.476877\pi$
$884$	0	0
$885$	−10.2446	−0.344368
$886$	− 23.5948i	− 0.792682i
$887$	−9.38404	−0.315085	−0.157543	−	0.987512i	$-0.550357\pi$
$887$	−9.38404	−0.315085	−0.157543	+	0.987512i	$0.550357\pi$
$888$	4.80194	0.161142
$889$	11.1748i	0.374789i
$890$	− 8.57673i	− 0.287493i
$891$	− 3.93900i	− 0.131962i
$892$	− 11.4004i	− 0.381715i
$893$	0.822643	0.0275287
$894$	13.6799	0.457526
$895$	18.4819i	0.617781i
$896$	1.19806	0.0400245
$897$	0	0
$898$	−19.8374	−0.661984
$899$	7.49588i	0.250002i
$900$	1.00000	0.0333333
$901$	12.0331	0.400881
$902$	0.780167i	0.0259767i
$903$	4.11290i	0.136869i
$904$	11.2131i	0.372943i
$905$	10.7041i	0.355816i
$906$	1.15883	0.0384997
$907$	−14.2131	−0.471939	−0.235970	−	0.971760i	$-0.575827\pi$
$907$	−14.2131	−0.471939	−0.235970	+	0.971760i	$0.575827\pi$
$908$	− 8.86725i	− 0.294270i
$909$	−6.67025	−0.221238
$910$	0	0
$911$	40.2707	1.33423	0.667113	−	0.744956i	$-0.267530\pi$
$911$	40.2707	1.33423	0.667113	+	0.744956i	$0.267530\pi$
$912$	− 0.911854i	− 0.0301945i
$913$	61.3949	2.03188
$914$	26.5284	0.877482
$915$	9.92692i	0.328174i
$916$	12.6136i	0.416764i
$917$	5.53665i	0.182836i
$918$	1.74094i	0.0574595i
$919$	7.03577	0.232089	0.116044	−	0.993244i	$-0.462979\pi$
$919$	7.03577	0.232089	0.116044	+	0.993244i	$0.462979\pi$
$920$	1.24698	0.0411117
$921$	− 24.9124i	− 0.820893i
$922$	12.2717	0.404148
$923$	0	0
$924$	4.71917	0.155249
$925$	4.80194i	0.157887i
$926$	−12.6455	−0.415557
$927$	−10.0315	−0.329476
$928$	− 3.47219i	− 0.113980i
$929$	− 21.3575i	− 0.700716i	−0.936616	−	0.350358i	$-0.886060\pi$
$929$	− 21.3575i	− 0.700716i	0.936616	−	0.350358i	$-0.113940\pi$
$930$	2.15883i	0.0707909i
$931$	5.07415i	0.166298i
$932$	−23.4155	−0.767000
$933$	21.3787	0.699906
$934$	− 11.3870i	− 0.372595i
$935$	6.85756	0.224266
$936$	0	0
$937$	41.7396	1.36357	0.681787	−	0.731551i	$-0.261203\pi$
$937$	41.7396	1.36357	0.681787	+	0.731551i	$0.261203\pi$
$938$	10.9845i	0.358655i
$939$	23.8713	0.779010
$940$	0.902165	0.0294254
$941$	− 26.2078i	− 0.854348i	−0.904169	−	0.427174i	$-0.859509\pi$
$941$	− 26.2078i	− 0.854348i	0.904169	−	0.427174i	$-0.140491\pi$
$942$	12.2664i	0.399660i
$943$	− 0.246980i	− 0.00804276i
$944$	10.2446i	0.333433i
$945$	−1.19806	−0.0389730
$946$	−13.5224	−0.439652
$947$	3.15239i	0.102439i	0.998687	+	0.0512195i	$0.0163108\pi$
$947$	3.15239i	0.102439i	−0.998687	+	0.0512195i	$0.983689\pi$
$948$	2.73556	0.0888469
$949$	0	0
$950$	0.911854	0.0295845
$951$	12.6407i	0.409903i
$952$	−2.08575	−0.0675996
$953$	3.56060	0.115339	0.0576695	−	0.998336i	$-0.481633\pi$
$953$	3.56060	0.115339	0.0576695	+	0.998336i	$0.481633\pi$
$954$	− 6.91185i	− 0.223780i
$955$	− 15.3623i	− 0.497111i
$956$	− 8.87263i	− 0.286961i
$957$	− 13.6770i	− 0.442113i
$958$	18.8431	0.608792
$959$	0.0911299	0.00294274
$960$	− 1.00000i	− 0.0322749i
$961$	26.3394	0.849659
$962$	0	0
$963$	−10.1032	−0.325571
$964$	9.72886i	0.313345i
$965$	18.3207	0.589763
$966$	−1.49396	−0.0480673
$967$	− 7.26875i	− 0.233747i	−0.993147	−	0.116874i	$-0.962713\pi$
$967$	− 7.26875i	− 0.233747i	0.993147	−	0.116874i	$-0.0372872\pi$
$968$	4.51573i	0.145141i
$969$	1.58748i	0.0509973i
$970$	16.9976i	0.545760i
$971$	−9.53511	−0.305996	−0.152998	−	0.988226i	$-0.548893\pi$
$971$	−9.53511	−0.305996	−0.152998	+	0.988226i	$0.548893\pi$
$972$	1.00000	0.0320750
$973$	22.4058i	0.718297i
$974$	−27.6383	−0.885589
$975$	0	0
$976$	9.92692	0.317753
$977$	− 17.5700i	− 0.562115i	−0.959691	−	0.281057i	$-0.909315\pi$
$977$	− 17.5700i	− 0.562115i	0.959691	−	0.281057i	$-0.0906851\pi$
$978$	2.48858	0.0795761
$979$	−33.7837	−1.07973
$980$	5.56465i	0.177756i
$981$	10.5211i	0.335913i
$982$	13.9866i	0.446330i
$983$	43.4935i	1.38723i	0.720347	+	0.693613i	$0.243982\pi$
$983$	43.4935i	1.38723i	−0.720347	+	0.693613i	$0.756018\pi$
$984$	−0.198062	−0.00631399
$985$	−2.31336	−0.0737096
$986$	6.04487i	0.192508i
$987$	−1.08085	−0.0344038
$988$	0	0
$989$	4.28083	0.136123
$990$	− 3.93900i	− 0.125190i
$991$	38.8232	1.23326	0.616630	−	0.787253i	$-0.288497\pi$
$991$	38.8232	1.23326	0.616630	+	0.787253i	$0.288497\pi$
$992$	2.15883	0.0685430
$993$	− 20.5109i	− 0.650895i
$994$	− 13.2634i	− 0.420689i
$995$	6.36765i	0.201868i
$996$	15.5864i	0.493875i
$997$	−21.9807	−0.696137	−0.348069	−	0.937469i	$-0.613162\pi$
$997$	−21.9807	−0.696137	−0.348069	+	0.937469i	$0.613162\pi$
$998$	8.87071	0.280797
$999$	4.80194i	0.151927i

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.19806i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.93900i q^{11} +1.00000 q^{12} -1.19806 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.74094 q^{17} +1.00000i q^{18} +0.911854i q^{19} +1.00000i q^{20} -1.19806i q^{21} +3.93900 q^{22} -1.24698 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.19806i q^{28} -3.47219 q^{29} -1.00000 q^{30} -2.15883i q^{31} +1.00000i q^{32} +3.93900i q^{33} -1.74094i q^{34} +1.19806 q^{35} -1.00000 q^{36} -4.80194i q^{37} -0.911854 q^{38} -1.00000 q^{40} +0.198062i q^{41} +1.19806 q^{42} -3.43296 q^{43} +3.93900i q^{44} -1.00000i q^{45} -1.24698i q^{46} -0.902165i q^{47} -1.00000 q^{48} +5.56465 q^{49} -1.00000i q^{50} +1.74094 q^{51} -6.91185 q^{53} -1.00000i q^{54} -3.93900 q^{55} +1.19806 q^{56} -0.911854i q^{57} -3.47219i q^{58} +10.2446i q^{59} -1.00000i q^{60} +9.92692 q^{61} +2.15883 q^{62} +1.19806i q^{63} -1.00000 q^{64} -3.93900 q^{66} -9.16852i q^{67} +1.74094 q^{68} +1.24698 q^{69} +1.19806i q^{70} +11.0707i q^{71} -1.00000i q^{72} -4.13706i q^{73} +4.80194 q^{74} +1.00000 q^{75} -0.911854i q^{76} +4.71917 q^{77} +2.73556 q^{79} -1.00000i q^{80} +1.00000 q^{81} -0.198062 q^{82} +15.5864i q^{83} +1.19806i q^{84} +1.74094i q^{85} -3.43296i q^{86} +3.47219 q^{87} -3.93900 q^{88} -8.57673i q^{89} +1.00000 q^{90} +1.24698 q^{92} +2.15883i q^{93} +0.902165 q^{94} +0.911854 q^{95} -1.00000i q^{96} +16.9976i q^{97} +5.56465i q^{98} -3.93900i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} + 18 q^{17} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 6 q^{27} - 8 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 16 q^{42} + 18 q^{43} - 6 q^{48} - 10 q^{49} - 18 q^{51} - 34 q^{53} - 4 q^{55} + 16 q^{56} + 2 q^{61} - 4 q^{62} - 6 q^{64} - 4 q^{66} - 18 q^{68} - 2 q^{69} + 20 q^{74} + 6 q^{75} + 6 q^{77} - 6 q^{79} + 6 q^{81} - 10 q^{82} + 8 q^{87} - 4 q^{88} + 6 q^{90} - 2 q^{92} + 42 q^{94} - 2 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 + 18 * q^17 + 4 * q^22 + 2 * q^23 - 6 * q^25 - 6 * q^27 - 8 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 16 * q^42 + 18 * q^43 - 6 * q^48 - 10 * q^49 - 18 * q^51 - 34 * q^53 - 4 * q^55 + 16 * q^56 + 2 * q^61 - 4 * q^62 - 6 * q^64 - 4 * q^66 - 18 * q^68 - 2 * q^69 + 20 * q^74 + 6 * q^75 + 6 * q^77 - 6 * q^79 + 6 * q^81 - 10 * q^82 + 8 * q^87 - 4 * q^88 + 6 * q^90 - 2 * q^92 + 42 * q^94 - 2 * q^95

Introduction

L-functions

Modular forms

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