LMFDB - Embedded newform 3850.2.a.bz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.117688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 1 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.216294$ of defining polynomial
Character	$\chi$	$=$	3850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.98457 q^{3} +1.00000 q^{4} +2.98457 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.90764 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.98457 q^{3} +1.00000 q^{4} +2.98457 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.90764 q^{9} +1.00000 q^{11} -2.98457 q^{12} -0.645545 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.19632 q^{17} -5.90764 q^{18} +4.11939 q^{19} -2.98457 q^{21} -1.00000 q^{22} +3.27753 q^{23} +2.98457 q^{24} +0.645545 q^{26} -8.67804 q^{27} +1.00000 q^{28} +9.24666 q^{29} -0.907639 q^{31} -1.00000 q^{32} -2.98457 q^{33} -2.19632 q^{34} +5.90764 q^{36} -5.47385 q^{37} -4.11939 q^{38} +1.92667 q^{39} +2.36989 q^{41} +2.98457 q^{42} +11.6590 q^{43} +1.00000 q^{44} -3.27753 q^{46} +1.42023 q^{47} -2.98457 q^{48} +1.00000 q^{49} -6.55506 q^{51} -0.645545 q^{52} -7.44298 q^{53} +8.67804 q^{54} -1.00000 q^{56} -12.2946 q^{57} -9.24666 q^{58} +8.48928 q^{59} +4.06578 q^{61} +0.907639 q^{62} +5.90764 q^{63} +1.00000 q^{64} +2.98457 q^{66} +3.77282 q^{67} +2.19632 q^{68} -9.78200 q^{69} -8.72051 q^{71} -5.90764 q^{72} -7.77282 q^{73} +5.47385 q^{74} +4.11939 q^{76} +1.00000 q^{77} -1.92667 q^{78} +5.75925 q^{79} +8.17729 q^{81} -2.36989 q^{82} +3.70704 q^{83} -2.98457 q^{84} -11.6590 q^{86} -27.5973 q^{87} -1.00000 q^{88} -12.4933 q^{89} -0.645545 q^{91} +3.27753 q^{92} +2.70891 q^{93} -1.42023 q^{94} +2.98457 q^{96} +9.71262 q^{97} -1.00000 q^{98} +5.90764 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + 5 q^{7} - 5 q^{8} + 13 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 + 5 * q^7 - 5 * q^8 + 13 * q^9	$ 5 q - 5 q^{2} + 5 q^{4} + 5 q^{7} - 5 q^{8} + 13 q^{9} + 5 q^{11} - 10 q^{13} - 5 q^{14} + 5 q^{16} - 2 q^{17} - 13 q^{18} + 6 q^{19} - 5 q^{22} + 8 q^{23} + 10 q^{26} + 5 q^{28} + 8 q^{29} + 12 q^{31} - 5 q^{32} + 2 q^{34} + 13 q^{36} - 6 q^{37} - 6 q^{38} - 4 q^{39} + 20 q^{41} + 12 q^{43} + 5 q^{44} - 8 q^{46} - 10 q^{47} + 5 q^{49} - 16 q^{51} - 10 q^{52} + 14 q^{53} - 5 q^{56} + 12 q^{57} - 8 q^{58} + 36 q^{59} + 10 q^{61} - 12 q^{62} + 13 q^{63} + 5 q^{64} + 2 q^{67} - 2 q^{68} + 24 q^{69} + 16 q^{71} - 13 q^{72} - 22 q^{73} + 6 q^{74} + 6 q^{76} + 5 q^{77} + 4 q^{78} - 10 q^{79} + 25 q^{81} - 20 q^{82} + 12 q^{83} - 12 q^{86} - 32 q^{87} - 5 q^{88} + 14 q^{89} - 10 q^{91} + 8 q^{92} + 10 q^{94} - 2 q^{97} - 5 q^{98} + 13 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + 5 * q^7 - 5 * q^8 + 13 * q^9 + 5 * q^11 - 10 * q^13 - 5 * q^14 + 5 * q^16 - 2 * q^17 - 13 * q^18 + 6 * q^19 - 5 * q^22 + 8 * q^23 + 10 * q^26 + 5 * q^28 + 8 * q^29 + 12 * q^31 - 5 * q^32 + 2 * q^34 + 13 * q^36 - 6 * q^37 - 6 * q^38 - 4 * q^39 + 20 * q^41 + 12 * q^43 + 5 * q^44 - 8 * q^46 - 10 * q^47 + 5 * q^49 - 16 * q^51 - 10 * q^52 + 14 * q^53 - 5 * q^56 + 12 * q^57 - 8 * q^58 + 36 * q^59 + 10 * q^61 - 12 * q^62 + 13 * q^63 + 5 * q^64 + 2 * q^67 - 2 * q^68 + 24 * q^69 + 16 * q^71 - 13 * q^72 - 22 * q^73 + 6 * q^74 + 6 * q^76 + 5 * q^77 + 4 * q^78 - 10 * q^79 + 25 * q^81 - 20 * q^82 + 12 * q^83 - 12 * q^86 - 32 * q^87 - 5 * q^88 + 14 * q^89 - 10 * q^91 + 8 * q^92 + 10 * q^94 - 2 * q^97 - 5 * q^98 + 13 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.98457	−1.72314	−0.861570	−	0.507638i	$-0.830519\pi$
$3$	−2.98457	−1.72314	−0.861570	+	0.507638i	$0.830519\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.98457	1.21844
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	5.90764	1.96921
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.98457	−0.861570
$13$	−0.645545	−0.179042	−0.0895209	−	0.995985i	$-0.528534\pi$
$13$	−0.645545	−0.179042	−0.0895209	+	0.995985i	$0.528534\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.19632	0.532685	0.266343	−	0.963878i	$-0.414185\pi$
$17$	2.19632	0.532685	0.266343	+	0.963878i	$0.414185\pi$
$18$	−5.90764	−1.39244
$19$	4.11939	0.945053	0.472526	−	0.881317i	$-0.343342\pi$
$19$	4.11939	0.945053	0.472526	+	0.881317i	$0.343342\pi$
$20$	0	0
$21$	−2.98457	−0.651286
$22$	−1.00000	−0.213201
$23$	3.27753	0.683412	0.341706	−	0.939807i	$-0.388995\pi$
$23$	3.27753	0.683412	0.341706	+	0.939807i	$0.388995\pi$
$24$	2.98457	0.609222
$25$	0	0
$26$	0.645545	0.126602
$27$	−8.67804	−1.67009
$28$	1.00000	0.188982
$29$	9.24666	1.71706	0.858531	−	0.512762i	$-0.171378\pi$
$29$	9.24666	1.71706	0.858531	+	0.512762i	$0.171378\pi$
$30$	0	0
$31$	−0.907639	−0.163017	−0.0815084	−	0.996673i	$-0.525974\pi$
$31$	−0.907639	−0.163017	−0.0815084	+	0.996673i	$0.525974\pi$
$32$	−1.00000	−0.176777
$33$	−2.98457	−0.519546
$34$	−2.19632	−0.376665
$35$	0	0
$36$	5.90764	0.984607
$37$	−5.47385	−0.899895	−0.449947	−	0.893055i	$-0.648557\pi$
$37$	−5.47385	−0.899895	−0.449947	+	0.893055i	$0.648557\pi$
$38$	−4.11939	−0.668253
$39$	1.92667	0.308514
$40$	0	0
$41$	2.36989	0.370114	0.185057	−	0.982728i	$-0.440753\pi$
$41$	2.36989	0.370114	0.185057	+	0.982728i	$0.440753\pi$
$42$	2.98457	0.460529
$43$	11.6590	1.77798	0.888991	−	0.457924i	$-0.151407\pi$
$43$	11.6590	1.77798	0.888991	+	0.457924i	$0.151407\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−3.27753	−0.483245
$47$	1.42023	0.207162	0.103581	−	0.994621i	$-0.466970\pi$
$47$	1.42023	0.207162	0.103581	+	0.994621i	$0.466970\pi$
$48$	−2.98457	−0.430785
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.55506	−0.917891
$52$	−0.645545	−0.0895209
$53$	−7.44298	−1.02237	−0.511186	−	0.859470i	$-0.670794\pi$
$53$	−7.44298	−1.02237	−0.511186	+	0.859470i	$0.670794\pi$
$54$	8.67804	1.18093
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−12.2946	−1.62846
$58$	−9.24666	−1.21415
$59$	8.48928	1.10521	0.552605	−	0.833443i	$-0.313634\pi$
$59$	8.48928	1.10521	0.552605	+	0.833443i	$0.313634\pi$
$60$	0	0
$61$	4.06578	0.520569	0.260285	−	0.965532i	$-0.416184\pi$
$61$	4.06578	0.520569	0.260285	+	0.965532i	$0.416184\pi$
$62$	0.907639	0.115270
$63$	5.90764	0.744293
$64$	1.00000	0.125000
$65$	0	0
$66$	2.98457	0.367375
$67$	3.77282	0.460923	0.230461	−	0.973081i	$-0.425976\pi$
$67$	3.77282	0.460923	0.230461	+	0.973081i	$0.425976\pi$
$68$	2.19632	0.266343
$69$	−9.78200	−1.17761
$70$	0	0
$71$	−8.72051	−1.03493	−0.517467	−	0.855703i	$-0.673125\pi$
$71$	−8.72051	−1.03493	−0.517467	+	0.855703i	$0.673125\pi$
$72$	−5.90764	−0.696222
$73$	−7.77282	−0.909739	−0.454870	−	0.890558i	$-0.650314\pi$
$73$	−7.77282	−0.909739	−0.454870	+	0.890558i	$0.650314\pi$
$74$	5.47385	0.636322
$75$	0	0
$76$	4.11939	0.472526
$77$	1.00000	0.113961
$78$	−1.92667	−0.218153
$79$	5.75925	0.647967	0.323983	−	0.946063i	$-0.394978\pi$
$79$	5.75925	0.647967	0.323983	+	0.946063i	$0.394978\pi$
$80$	0	0
$81$	8.17729	0.908587
$82$	−2.36989	−0.261710
$83$	3.70704	0.406900	0.203450	−	0.979085i	$-0.434784\pi$
$83$	3.70704	0.406900	0.203450	+	0.979085i	$0.434784\pi$
$84$	−2.98457	−0.325643
$85$	0	0
$86$	−11.6590	−1.25722
$87$	−27.5973	−2.95874
$88$	−1.00000	−0.106600
$89$	−12.4933	−1.32429	−0.662145	−	0.749376i	$-0.730354\pi$
$89$	−12.4933	−1.32429	−0.662145	+	0.749376i	$0.730354\pi$
$90$	0	0
$91$	−0.645545	−0.0676715
$92$	3.27753	0.341706
$93$	2.70891	0.280901
$94$	−1.42023	−0.146486
$95$	0	0
$96$	2.98457	0.304611
$97$	9.71262	0.986168	0.493084	−	0.869982i	$-0.335870\pi$
$97$	9.71262	0.986168	0.493084	+	0.869982i	$0.335870\pi$
$98$	−1.00000	−0.101015
$99$	5.90764	0.593740
$100$	0	0
$101$	−13.2951	−1.32292	−0.661458	−	0.749983i	$-0.730062\pi$
$101$	−13.2951	−1.32292	−0.661458	+	0.749983i	$0.730062\pi$
$102$	6.55506	0.649047
$103$	−16.3679	−1.61278	−0.806390	−	0.591385i	$-0.798582\pi$
$103$	−16.3679	−1.61278	−0.806390	+	0.591385i	$0.798582\pi$
$104$	0.645545	0.0633009
$105$	0	0
$106$	7.44298	0.722926
$107$	17.7820	1.71905	0.859525	−	0.511093i	$-0.170759\pi$
$107$	17.7820	1.71905	0.859525	+	0.511093i	$0.170759\pi$
$108$	−8.67804	−0.835045
$109$	−11.2158	−1.07428	−0.537139	−	0.843494i	$-0.680495\pi$
$109$	−11.2158	−1.07428	−0.537139	+	0.843494i	$0.680495\pi$
$110$	0	0
$111$	16.3371	1.55064
$112$	1.00000	0.0944911
$113$	−11.9807	−1.12705	−0.563526	−	0.826098i	$-0.690556\pi$
$113$	−11.9807	−1.12705	−0.563526	+	0.826098i	$0.690556\pi$
$114$	12.2946	1.15149
$115$	0	0
$116$	9.24666	0.858531
$117$	−3.81365	−0.352572
$118$	−8.48928	−0.781501
$119$	2.19632	0.201336
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.06578	−0.368098
$123$	−7.07309	−0.637759
$124$	−0.907639	−0.0815084
$125$	0	0
$126$	−5.90764	−0.526294
$127$	15.4294	1.36914	0.684570	−	0.728947i	$-0.259990\pi$
$127$	15.4294	1.36914	0.684570	+	0.728947i	$0.259990\pi$
$128$	−1.00000	−0.0883883
$129$	−34.7971	−3.06371
$130$	0	0
$131$	−14.3739	−1.25586	−0.627928	−	0.778271i	$-0.716097\pi$
$131$	−14.3739	−1.25586	−0.627928	+	0.778271i	$0.716097\pi$
$132$	−2.98457	−0.259773
$133$	4.11939	0.357196
$134$	−3.77282	−0.325922
$135$	0	0
$136$	−2.19632	−0.188333
$137$	13.0484	1.11480	0.557399	−	0.830245i	$-0.311799\pi$
$137$	13.0484	1.11480	0.557399	+	0.830245i	$0.311799\pi$
$138$	9.78200	0.832699
$139$	−12.2509	−1.03911	−0.519556	−	0.854437i	$-0.673903\pi$
$139$	−12.2509	−1.03911	−0.519556	+	0.854437i	$0.673903\pi$
$140$	0	0
$141$	−4.23878	−0.356970
$142$	8.72051	0.731809
$143$	−0.645545	−0.0539832
$144$	5.90764	0.492303
$145$	0	0
$146$	7.77282	0.643283
$147$	−2.98457	−0.246163
$148$	−5.47385	−0.449947
$149$	21.2467	1.74059	0.870297	−	0.492527i	$-0.163927\pi$
$149$	21.2467	1.74059	0.870297	+	0.492527i	$0.163927\pi$
$150$	0	0
$151$	−16.9304	−1.37778	−0.688888	−	0.724868i	$-0.741901\pi$
$151$	−16.9304	−1.37778	−0.688888	+	0.724868i	$0.741901\pi$
$152$	−4.11939	−0.334127
$153$	12.9751	1.04897
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	1.92667	0.154257
$157$	−22.9541	−1.83194	−0.915970	−	0.401246i	$-0.868577\pi$
$157$	−22.9541	−1.83194	−0.915970	+	0.401246i	$0.868577\pi$
$158$	−5.75925	−0.458182
$159$	22.2141	1.76169
$160$	0	0
$161$	3.27753	0.258305
$162$	−8.17729	−0.642468
$163$	4.93235	0.386332	0.193166	−	0.981166i	$-0.438124\pi$
$163$	4.93235	0.386332	0.193166	+	0.981166i	$0.438124\pi$
$164$	2.36989	0.185057
$165$	0	0
$166$	−3.70704	−0.287722
$167$	3.41408	0.264189	0.132095	−	0.991237i	$-0.457830\pi$
$167$	3.41408	0.264189	0.132095	+	0.991237i	$0.457830\pi$
$168$	2.98457	0.230264
$169$	−12.5833	−0.967944
$170$	0	0
$171$	24.3359	1.86101
$172$	11.6590	0.888991
$173$	0.770945	0.0586139	0.0293069	−	0.999570i	$-0.490670\pi$
$173$	0.770945	0.0586139	0.0293069	+	0.999570i	$0.490670\pi$
$174$	27.5973	2.09214
$175$	0	0
$176$	1.00000	0.0753778
$177$	−25.3368	−1.90443
$178$	12.4933	0.936414
$179$	21.2307	1.58686	0.793428	−	0.608664i	$-0.208294\pi$
$179$	21.2307	1.58686	0.793428	+	0.608664i	$0.208294\pi$
$180$	0	0
$181$	0.177170	0.0131689	0.00658447	−	0.999978i	$-0.497904\pi$
$181$	0.177170	0.0131689	0.00658447	+	0.999978i	$0.497904\pi$
$182$	0.645545	0.0478510
$183$	−12.1346	−0.897014
$184$	−3.27753	−0.241623
$185$	0	0
$186$	−2.70891	−0.198627
$187$	2.19632	0.160611
$188$	1.42023	0.103581
$189$	−8.67804	−0.631235
$190$	0	0
$191$	22.8483	1.65325	0.826623	−	0.562756i	$-0.190259\pi$
$191$	22.8483	1.65325	0.826623	+	0.562756i	$0.190259\pi$
$192$	−2.98457	−0.215393
$193$	12.3333	0.887771	0.443886	−	0.896083i	$-0.353600\pi$
$193$	12.3333	0.887771	0.443886	+	0.896083i	$0.353600\pi$
$194$	−9.71262	−0.697326
$195$	0	0
$196$	1.00000	0.0714286
$197$	−17.6257	−1.25578	−0.627891	−	0.778301i	$-0.716082\pi$
$197$	−17.6257	−1.25578	−0.627891	+	0.778301i	$0.716082\pi$
$198$	−5.90764	−0.419838
$199$	−2.59752	−0.184133	−0.0920666	−	0.995753i	$-0.529347\pi$
$199$	−2.59752	−0.184133	−0.0920666	+	0.995753i	$0.529347\pi$
$200$	0	0
$201$	−11.2602	−0.794235
$202$	13.2951	0.935442
$203$	9.24666	0.648988
$204$	−6.55506	−0.458946
$205$	0	0
$206$	16.3679	1.14041
$207$	19.3625	1.34578
$208$	−0.645545	−0.0447605
$209$	4.11939	0.284944
$210$	0	0
$211$	−14.2545	−0.981323	−0.490662	−	0.871350i	$-0.663245\pi$
$211$	−14.2545	−0.981323	−0.490662	+	0.871350i	$0.663245\pi$
$212$	−7.44298	−0.511186
$213$	26.0269	1.78334
$214$	−17.7820	−1.21555
$215$	0	0
$216$	8.67804	0.590466
$217$	−0.907639	−0.0616146
$218$	11.2158	0.759630
$219$	23.1985	1.56761
$220$	0	0
$221$	−1.41782	−0.0953730
$222$	−16.3371	−1.09647
$223$	0.241191	0.0161514	0.00807568	−	0.999967i	$-0.497429\pi$
$223$	0.241191	0.0161514	0.00807568	+	0.999967i	$0.497429\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	11.9807	0.796946
$227$	0.732226	0.0485995	0.0242998	−	0.999705i	$-0.492264\pi$
$227$	0.732226	0.0485995	0.0242998	+	0.999705i	$0.492264\pi$
$228$	−12.2946	−0.814229
$229$	4.99572	0.330126	0.165063	−	0.986283i	$-0.447217\pi$
$229$	4.99572	0.330126	0.165063	+	0.986283i	$0.447217\pi$
$230$	0	0
$231$	−2.98457	−0.196370
$232$	−9.24666	−0.607073
$233$	22.8744	1.49855	0.749275	−	0.662259i	$-0.230402\pi$
$233$	22.8744	1.49855	0.749275	+	0.662259i	$0.230402\pi$
$234$	3.81365	0.249306
$235$	0	0
$236$	8.48928	0.552605
$237$	−17.1889	−1.11654
$238$	−2.19632	−0.142366
$239$	−7.40052	−0.478700	−0.239350	−	0.970933i	$-0.576934\pi$
$239$	−7.40052	−0.478700	−0.239350	+	0.970933i	$0.576934\pi$
$240$	0	0
$241$	9.60299	0.618583	0.309291	−	0.950967i	$-0.399908\pi$
$241$	9.60299	0.618583	0.309291	+	0.950967i	$0.399908\pi$
$242$	−1.00000	−0.0642824
$243$	1.62848	0.104467
$244$	4.06578	0.260285
$245$	0	0
$246$	7.07309	0.450964
$247$	−2.65925	−0.169204
$248$	0.907639	0.0576352
$249$	−11.0639	−0.701147
$250$	0	0
$251$	22.7893	1.43845	0.719224	−	0.694778i	$-0.244497\pi$
$251$	22.7893	1.43845	0.719224	+	0.694778i	$0.244497\pi$
$252$	5.90764	0.372146
$253$	3.27753	0.206056
$254$	−15.4294	−0.968128
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.1751	1.88227	0.941135	−	0.338031i	$-0.109761\pi$
$257$	30.1751	1.88227	0.941135	+	0.338031i	$0.109761\pi$
$258$	34.7971	2.16637
$259$	−5.47385	−0.340128
$260$	0	0
$261$	54.6259	3.38126
$262$	14.3739	0.888025
$263$	13.7758	0.849455	0.424728	−	0.905321i	$-0.360370\pi$
$263$	13.7758	0.849455	0.424728	+	0.905321i	$0.360370\pi$
$264$	2.98457	0.183687
$265$	0	0
$266$	−4.11939	−0.252576
$267$	37.2872	2.28194
$268$	3.77282	0.230461
$269$	31.5935	1.92629	0.963144	−	0.268986i	$-0.0866887\pi$
$269$	31.5935	1.92629	0.963144	+	0.268986i	$0.0866887\pi$
$270$	0	0
$271$	−7.56793	−0.459719	−0.229860	−	0.973224i	$-0.573827\pi$
$271$	−7.56793	−0.459719	−0.229860	+	0.973224i	$0.573827\pi$
$272$	2.19632	0.133171
$273$	1.92667	0.116607
$274$	−13.0484	−0.788281
$275$	0	0
$276$	−9.78200	−0.588807
$277$	13.6995	0.823122	0.411561	−	0.911382i	$-0.364984\pi$
$277$	13.6995	0.823122	0.411561	+	0.911382i	$0.364984\pi$
$278$	12.2509	0.734763
$279$	−5.36201	−0.321015
$280$	0	0
$281$	5.26022	0.313799	0.156899	−	0.987615i	$-0.449850\pi$
$281$	5.26022	0.313799	0.156899	+	0.987615i	$0.449850\pi$
$282$	4.23878	0.252416
$283$	7.25267	0.431127	0.215563	−	0.976490i	$-0.430841\pi$
$283$	7.25267	0.431127	0.215563	+	0.976490i	$0.430841\pi$
$284$	−8.72051	−0.517467
$285$	0	0
$286$	0.645545	0.0381719
$287$	2.36989	0.139890
$288$	−5.90764	−0.348111
$289$	−12.1762	−0.716247
$290$	0	0
$291$	−28.9880	−1.69931
$292$	−7.77282	−0.454870
$293$	−5.67073	−0.331288	−0.165644	−	0.986186i	$-0.552970\pi$
$293$	−5.67073	−0.331288	−0.165644	+	0.986186i	$0.552970\pi$
$294$	2.98457	0.174063
$295$	0	0
$296$	5.47385	0.318161
$297$	−8.67804	−0.503551
$298$	−21.2467	−1.23079
$299$	−2.11579	−0.122359
$300$	0	0
$301$	11.6590	0.672014
$302$	16.9304	0.974234
$303$	39.6802	2.27957
$304$	4.11939	0.236263
$305$	0	0
$306$	−12.9751	−0.741734
$307$	3.99245	0.227861	0.113931	−	0.993489i	$-0.463656\pi$
$307$	3.99245	0.227861	0.113931	+	0.993489i	$0.463656\pi$
$308$	1.00000	0.0569803
$309$	48.8512	2.77905
$310$	0	0
$311$	−32.9958	−1.87102	−0.935511	−	0.353298i	$-0.885060\pi$
$311$	−32.9958	−1.87102	−0.935511	+	0.353298i	$0.885060\pi$
$312$	−1.92667	−0.109076
$313$	17.5660	0.992887	0.496444	−	0.868069i	$-0.334639\pi$
$313$	17.5660	0.992887	0.496444	+	0.868069i	$0.334639\pi$
$314$	22.9541	1.29538
$315$	0	0
$316$	5.75925	0.323983
$317$	8.64914	0.485784	0.242892	−	0.970053i	$-0.421904\pi$
$317$	8.64914	0.485784	0.242892	+	0.970053i	$0.421904\pi$
$318$	−22.2141	−1.24570
$319$	9.24666	0.517714
$320$	0	0
$321$	−53.0716	−2.96217
$322$	−3.27753	−0.182649
$323$	9.04749	0.503416
$324$	8.17729	0.454294
$325$	0	0
$326$	−4.93235	−0.273178
$327$	33.4743	1.85113
$328$	−2.36989	−0.130855
$329$	1.42023	0.0783000
$330$	0	0
$331$	23.2968	1.28051	0.640253	−	0.768164i	$-0.278829\pi$
$331$	23.2968	1.28051	0.640253	+	0.768164i	$0.278829\pi$
$332$	3.70704	0.203450
$333$	−32.3375	−1.77208
$334$	−3.41408	−0.186810
$335$	0	0
$336$	−2.98457	−0.162821
$337$	−22.9635	−1.25090	−0.625450	−	0.780265i	$-0.715084\pi$
$337$	−22.9635	−1.25090	−0.625450	+	0.780265i	$0.715084\pi$
$338$	12.5833	0.684440
$339$	35.7573	1.94207
$340$	0	0
$341$	−0.907639	−0.0491514
$342$	−24.3359	−1.31593
$343$	1.00000	0.0539949
$344$	−11.6590	−0.628612
$345$	0	0
$346$	−0.770945	−0.0414463
$347$	23.9098	1.28355	0.641773	−	0.766895i	$-0.278199\pi$
$347$	23.9098	1.28355	0.641773	+	0.766895i	$0.278199\pi$
$348$	−27.5973	−1.47937
$349$	2.29599	0.122902	0.0614508	−	0.998110i	$-0.480427\pi$
$349$	2.29599	0.122902	0.0614508	+	0.998110i	$0.480427\pi$
$350$	0	0
$351$	5.60207	0.299016
$352$	−1.00000	−0.0533002
$353$	−29.4353	−1.56668	−0.783342	−	0.621591i	$-0.786486\pi$
$353$	−29.4353	−1.56668	−0.783342	+	0.621591i	$0.786486\pi$
$354$	25.3368	1.34664
$355$	0	0
$356$	−12.4933	−0.662145
$357$	−6.55506	−0.346930
$358$	−21.2307	−1.12208
$359$	20.1906	1.06562	0.532810	−	0.846235i	$-0.321136\pi$
$359$	20.1906	1.06562	0.532810	+	0.846235i	$0.321136\pi$
$360$	0	0
$361$	−2.03063	−0.106875
$362$	−0.177170	−0.00931185
$363$	−2.98457	−0.156649
$364$	−0.645545	−0.0338357
$365$	0	0
$366$	12.1346	0.634285
$367$	−7.47429	−0.390155	−0.195077	−	0.980788i	$-0.562496\pi$
$367$	−7.47429	−0.390155	−0.195077	+	0.980788i	$0.562496\pi$
$368$	3.27753	0.170853
$369$	14.0004	0.728834
$370$	0	0
$371$	−7.44298	−0.386420
$372$	2.70891	0.140450
$373$	1.54563	0.0800298	0.0400149	−	0.999199i	$-0.487259\pi$
$373$	1.54563	0.0800298	0.0400149	+	0.999199i	$0.487259\pi$
$374$	−2.19632	−0.113569
$375$	0	0
$376$	−1.42023	−0.0732429
$377$	−5.96913	−0.307426
$378$	8.67804	0.446350
$379$	2.25454	0.115808	0.0579041	−	0.998322i	$-0.481558\pi$
$379$	2.25454	0.115808	0.0579041	+	0.998322i	$0.481558\pi$
$380$	0	0
$381$	−46.0501	−2.35922
$382$	−22.8483	−1.16902
$383$	8.73737	0.446459	0.223229	−	0.974766i	$-0.428340\pi$
$383$	8.73737	0.446459	0.223229	+	0.974766i	$0.428340\pi$
$384$	2.98457	0.152306
$385$	0	0
$386$	−12.3333	−0.627749
$387$	68.8772	3.50123
$388$	9.71262	0.493084
$389$	−17.3523	−0.879799	−0.439899	−	0.898047i	$-0.644986\pi$
$389$	−17.3523	−0.879799	−0.439899	+	0.898047i	$0.644986\pi$
$390$	0	0
$391$	7.19849	0.364043
$392$	−1.00000	−0.0505076
$393$	42.9000	2.16402
$394$	17.6257	0.887972
$395$	0	0
$396$	5.90764	0.296870
$397$	29.3505	1.47306	0.736530	−	0.676405i	$-0.236463\pi$
$397$	29.3505	1.47306	0.736530	+	0.676405i	$0.236463\pi$
$398$	2.59752	0.130202
$399$	−12.2946	−0.615500
$400$	0	0
$401$	4.45981	0.222712	0.111356	−	0.993781i	$-0.464481\pi$
$401$	4.45981	0.222712	0.111356	+	0.993781i	$0.464481\pi$
$402$	11.2602	0.561609
$403$	0.585922	0.0291868
$404$	−13.2951	−0.661458
$405$	0	0
$406$	−9.24666	−0.458904
$407$	−5.47385	−0.271328
$408$	6.55506	0.324524
$409$	−16.7783	−0.829633	−0.414816	−	0.909905i	$-0.636154\pi$
$409$	−16.7783	−0.829633	−0.414816	+	0.909905i	$0.636154\pi$
$410$	0	0
$411$	−38.9438	−1.92095
$412$	−16.3679	−0.806390
$413$	8.48928	0.417730
$414$	−19.3625	−0.951613
$415$	0	0
$416$	0.645545	0.0316504
$417$	36.5638	1.79054
$418$	−4.11939	−0.201486
$419$	27.6878	1.35264	0.676318	−	0.736610i	$-0.263575\pi$
$419$	27.6878	1.35264	0.676318	+	0.736610i	$0.263575\pi$
$420$	0	0
$421$	13.9262	0.678724	0.339362	−	0.940656i	$-0.389789\pi$
$421$	13.9262	0.678724	0.339362	+	0.940656i	$0.389789\pi$
$422$	14.2545	0.693900
$423$	8.39022	0.407947
$424$	7.44298	0.361463
$425$	0	0
$426$	−26.0269	−1.26101
$427$	4.06578	0.196757
$428$	17.7820	0.859525
$429$	1.92667	0.0930206
$430$	0	0
$431$	−21.5861	−1.03977	−0.519884	−	0.854237i	$-0.674025\pi$
$431$	−21.5861	−1.03977	−0.519884	+	0.854237i	$0.674025\pi$
$432$	−8.67804	−0.417523
$433$	−38.0435	−1.82826	−0.914128	−	0.405427i	$-0.867123\pi$
$433$	−38.0435	−1.82826	−0.914128	+	0.405427i	$0.867123\pi$
$434$	0.907639	0.0435681
$435$	0	0
$436$	−11.2158	−0.537139
$437$	13.5014	0.645860
$438$	−23.1985	−1.10847
$439$	−35.2169	−1.68081	−0.840404	−	0.541960i	$-0.817683\pi$
$439$	−35.2169	−1.68081	−0.840404	+	0.541960i	$0.817683\pi$
$440$	0	0
$441$	5.90764	0.281316
$442$	1.41782	0.0674389
$443$	−16.9479	−0.805220	−0.402610	−	0.915372i	$-0.631897\pi$
$443$	−16.9479	−0.805220	−0.402610	+	0.915372i	$0.631897\pi$
$444$	16.3371	0.775322
$445$	0	0
$446$	−0.241191	−0.0114207
$447$	−63.4121	−2.99929
$448$	1.00000	0.0472456
$449$	−22.0050	−1.03848	−0.519240	−	0.854628i	$-0.673785\pi$
$449$	−22.0050	−1.03848	−0.519240	+	0.854628i	$0.673785\pi$
$450$	0	0
$451$	2.36989	0.111594
$452$	−11.9807	−0.563526
$453$	50.5299	2.37410
$454$	−0.732226	−0.0343651
$455$	0	0
$456$	12.2946	0.575747
$457$	34.2014	1.59987	0.799936	−	0.600085i	$-0.204866\pi$
$457$	34.2014	1.59987	0.799936	+	0.600085i	$0.204866\pi$
$458$	−4.99572	−0.233435
$459$	−19.0597	−0.889632
$460$	0	0
$461$	17.0564	0.794394	0.397197	−	0.917733i	$-0.369983\pi$
$461$	17.0564	0.794394	0.397197	+	0.917733i	$0.369983\pi$
$462$	2.98457	0.138855
$463$	−19.2236	−0.893399	−0.446699	−	0.894684i	$-0.647401\pi$
$463$	−19.2236	−0.893399	−0.446699	+	0.894684i	$0.647401\pi$
$464$	9.24666	0.429265
$465$	0	0
$466$	−22.8744	−1.05963
$467$	24.7429	1.14497	0.572483	−	0.819916i	$-0.305980\pi$
$467$	24.7429	1.14497	0.572483	+	0.819916i	$0.305980\pi$
$468$	−3.81365	−0.176286
$469$	3.77282	0.174212
$470$	0	0
$471$	68.5082	3.15669
$472$	−8.48928	−0.390751
$473$	11.6590	0.536082
$474$	17.1889	0.789511
$475$	0	0
$476$	2.19632	0.100668
$477$	−43.9704	−2.01327
$478$	7.40052	0.338492
$479$	21.2294	0.969994	0.484997	−	0.874516i	$-0.338821\pi$
$479$	21.2294	0.969994	0.484997	+	0.874516i	$0.338821\pi$
$480$	0	0
$481$	3.53361	0.161119
$482$	−9.60299	−0.437404
$483$	−9.78200	−0.445096
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.62848	−0.0738693
$487$	−8.41476	−0.381309	−0.190655	−	0.981657i	$-0.561061\pi$
$487$	−8.41476	−0.381309	−0.190655	+	0.981657i	$0.561061\pi$
$488$	−4.06578	−0.184049
$489$	−14.7209	−0.665704
$490$	0	0
$491$	35.9615	1.62292	0.811459	−	0.584409i	$-0.198674\pi$
$491$	35.9615	1.62292	0.811459	+	0.584409i	$0.198674\pi$
$492$	−7.07309	−0.318880
$493$	20.3086	0.914653
$494$	2.65925	0.119645
$495$	0	0
$496$	−0.907639	−0.0407542
$497$	−8.72051	−0.391168
$498$	11.0639	0.495785
$499$	34.0906	1.52610	0.763052	−	0.646337i	$-0.223700\pi$
$499$	34.0906	1.52610	0.763052	+	0.646337i	$0.223700\pi$
$500$	0	0
$501$	−10.1895	−0.455235
$502$	−22.7893	−1.01714
$503$	29.4642	1.31374	0.656872	−	0.754002i	$-0.271879\pi$
$503$	29.4642	1.31374	0.656872	+	0.754002i	$0.271879\pi$
$504$	−5.90764	−0.263147
$505$	0	0
$506$	−3.27753	−0.145704
$507$	37.5556	1.66790
$508$	15.4294	0.684570
$509$	−18.4083	−0.815932	−0.407966	−	0.912997i	$-0.633762\pi$
$509$	−18.4083	−0.815932	−0.407966	+	0.912997i	$0.633762\pi$
$510$	0	0
$511$	−7.77282	−0.343849
$512$	−1.00000	−0.0441942
$513$	−35.7482	−1.57832
$514$	−30.1751	−1.33097
$515$	0	0
$516$	−34.7971	−1.53186
$517$	1.42023	0.0624618
$518$	5.47385	0.240507
$519$	−2.30094	−0.101000
$520$	0	0
$521$	−42.0313	−1.84142	−0.920712	−	0.390243i	$-0.872391\pi$
$521$	−42.0313	−1.84142	−0.920712	+	0.390243i	$0.872391\pi$
$522$	−54.6259	−2.39091
$523$	−17.5801	−0.768725	−0.384362	−	0.923182i	$-0.625579\pi$
$523$	−17.5801	−0.768725	−0.384362	+	0.923182i	$0.625579\pi$
$524$	−14.3739	−0.627928
$525$	0	0
$526$	−13.7758	−0.600655
$527$	−1.99346	−0.0868367
$528$	−2.98457	−0.129887
$529$	−12.2578	−0.532948
$530$	0	0
$531$	50.1516	2.17639
$532$	4.11939	0.178598
$533$	−1.52987	−0.0662660
$534$	−37.2872	−1.61357
$535$	0	0
$536$	−3.77282	−0.162961
$537$	−63.3644	−2.73438
$538$	−31.5935	−1.36209
$539$	1.00000	0.0430730
$540$	0	0
$541$	7.92579	0.340756	0.170378	−	0.985379i	$-0.445501\pi$
$541$	7.92579	0.340756	0.170378	+	0.985379i	$0.445501\pi$
$542$	7.56793	0.325071
$543$	−0.528776	−0.0226919
$544$	−2.19632	−0.0941663
$545$	0	0
$546$	−1.92667	−0.0824539
$547$	10.0381	0.429197	0.214598	−	0.976702i	$-0.431156\pi$
$547$	10.0381	0.429197	0.214598	+	0.976702i	$0.431156\pi$
$548$	13.0484	0.557399
$549$	24.0191	1.02511
$550$	0	0
$551$	38.0906	1.62271
$552$	9.78200	0.416350
$553$	5.75925	0.244908
$554$	−13.6995	−0.582035
$555$	0	0
$556$	−12.2509	−0.519556
$557$	−17.6529	−0.747976	−0.373988	−	0.927434i	$-0.622010\pi$
$557$	−17.6529	−0.747976	−0.373988	+	0.927434i	$0.622010\pi$
$558$	5.36201	0.226992
$559$	−7.52641	−0.318333
$560$	0	0
$561$	−6.55506	−0.276755
$562$	−5.26022	−0.221889
$563$	−12.2767	−0.517402	−0.258701	−	0.965957i	$-0.583294\pi$
$563$	−12.2767	−0.517402	−0.258701	+	0.965957i	$0.583294\pi$
$564$	−4.23878	−0.178485
$565$	0	0
$566$	−7.25267	−0.304853
$567$	8.17729	0.343414
$568$	8.72051	0.365904
$569$	−22.1924	−0.930353	−0.465177	−	0.885218i	$-0.654009\pi$
$569$	−22.1924	−0.930353	−0.465177	+	0.885218i	$0.654009\pi$
$570$	0	0
$571$	−3.65286	−0.152867	−0.0764337	−	0.997075i	$-0.524353\pi$
$571$	−3.65286	−0.152867	−0.0764337	+	0.997075i	$0.524353\pi$
$572$	−0.645545	−0.0269916
$573$	−68.1923	−2.84878
$574$	−2.36989	−0.0989172
$575$	0	0
$576$	5.90764	0.246152
$577$	−10.8116	−0.450091	−0.225046	−	0.974348i	$-0.572253\pi$
$577$	−10.8116	−0.450091	−0.225046	+	0.974348i	$0.572253\pi$
$578$	12.1762	0.506463
$579$	−36.8096	−1.52975
$580$	0	0
$581$	3.70704	0.153794
$582$	28.9880	1.20159
$583$	−7.44298	−0.308257
$584$	7.77282	0.321641
$585$	0	0
$586$	5.67073	0.234256
$587$	−3.66743	−0.151371	−0.0756856	−	0.997132i	$-0.524115\pi$
$587$	−3.66743	−0.151371	−0.0756856	+	0.997132i	$0.524115\pi$
$588$	−2.98457	−0.123081
$589$	−3.73892	−0.154060
$590$	0	0
$591$	52.6052	2.16389
$592$	−5.47385	−0.224974
$593$	24.1037	0.989821	0.494911	−	0.868944i	$-0.335201\pi$
$593$	24.1037	0.989821	0.494911	+	0.868944i	$0.335201\pi$
$594$	8.67804	0.356064
$595$	0	0
$596$	21.2467	0.870297
$597$	7.75247	0.317287
$598$	2.11579	0.0865211
$599$	0.720507	0.0294391	0.0147195	−	0.999892i	$-0.495314\pi$
$599$	0.720507	0.0294391	0.0147195	+	0.999892i	$0.495314\pi$
$600$	0	0
$601$	7.96958	0.325086	0.162543	−	0.986701i	$-0.448030\pi$
$601$	7.96958	0.325086	0.162543	+	0.986701i	$0.448030\pi$
$602$	−11.6590	−0.475186
$603$	22.2884	0.907655
$604$	−16.9304	−0.688888
$605$	0	0
$606$	−39.6802	−1.61190
$607$	−20.5758	−0.835147	−0.417574	−	0.908643i	$-0.637119\pi$
$607$	−20.5758	−0.835147	−0.417574	+	0.908643i	$0.637119\pi$
$608$	−4.11939	−0.167063
$609$	−27.5973	−1.11830
$610$	0	0
$611$	−0.916824	−0.0370907
$612$	12.9751	0.524485
$613$	6.32370	0.255412	0.127706	−	0.991812i	$-0.459239\pi$
$613$	6.32370	0.255412	0.127706	+	0.991812i	$0.459239\pi$
$614$	−3.99245	−0.161122
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−4.72051	−0.190040	−0.0950202	−	0.995475i	$-0.530292\pi$
$617$	−4.72051	−0.190040	−0.0950202	+	0.995475i	$0.530292\pi$
$618$	−48.8512	−1.96508
$619$	33.9574	1.36486	0.682432	−	0.730949i	$-0.260922\pi$
$619$	33.9574	1.36486	0.682432	+	0.730949i	$0.260922\pi$
$620$	0	0
$621$	−28.4425	−1.14136
$622$	32.9958	1.32301
$623$	−12.4933	−0.500534
$624$	1.92667	0.0771286
$625$	0	0
$626$	−17.5660	−0.702077
$627$	−12.2946	−0.490999
$628$	−22.9541	−0.915970
$629$	−12.0223	−0.479360
$630$	0	0
$631$	8.60846	0.342697	0.171349	−	0.985210i	$-0.445188\pi$
$631$	8.60846	0.342697	0.171349	+	0.985210i	$0.445188\pi$
$632$	−5.75925	−0.229091
$633$	42.5436	1.69096
$634$	−8.64914	−0.343501
$635$	0	0
$636$	22.2141	0.880845
$637$	−0.645545	−0.0255774
$638$	−9.24666	−0.366079
$639$	−51.5176	−2.03801
$640$	0	0
$641$	38.3857	1.51614	0.758071	−	0.652171i	$-0.226142\pi$
$641$	38.3857	1.51614	0.758071	+	0.652171i	$0.226142\pi$
$642$	53.0716	2.09457
$643$	9.28943	0.366339	0.183170	−	0.983081i	$-0.441364\pi$
$643$	9.28943	0.366339	0.183170	+	0.983081i	$0.441364\pi$
$644$	3.27753	0.129153
$645$	0	0
$646$	−9.04749	−0.355969
$647$	5.05472	0.198722	0.0993608	−	0.995051i	$-0.468320\pi$
$647$	5.05472	0.198722	0.0993608	+	0.995051i	$0.468320\pi$
$648$	−8.17729	−0.321234
$649$	8.48928	0.333233
$650$	0	0
$651$	2.70891	0.106171
$652$	4.93235	0.193166
$653$	19.2425	0.753017	0.376508	−	0.926413i	$-0.377125\pi$
$653$	19.2425	0.753017	0.376508	+	0.926413i	$0.377125\pi$
$654$	−33.4743	−1.30895
$655$	0	0
$656$	2.36989	0.0925286
$657$	−45.9190	−1.79147
$658$	−1.42023	−0.0553664
$659$	31.0853	1.21091	0.605456	−	0.795879i	$-0.292991\pi$
$659$	31.0853	1.21091	0.605456	+	0.795879i	$0.292991\pi$
$660$	0	0
$661$	10.6462	0.414088	0.207044	−	0.978332i	$-0.433616\pi$
$661$	10.6462	0.414088	0.207044	+	0.978332i	$0.433616\pi$
$662$	−23.2968	−0.905455
$663$	4.23158	0.164341
$664$	−3.70704	−0.143861
$665$	0	0
$666$	32.3375	1.25305
$667$	30.3062	1.17346
$668$	3.41408	0.132095
$669$	−0.719851	−0.0278311
$670$	0	0
$671$	4.06578	0.156958
$672$	2.98457	0.115132
$673$	3.47539	0.133966	0.0669831	−	0.997754i	$-0.478663\pi$
$673$	3.47539	0.133966	0.0669831	+	0.997754i	$0.478663\pi$
$674$	22.9635	0.884519
$675$	0	0
$676$	−12.5833	−0.483972
$677$	−35.8279	−1.37698	−0.688489	−	0.725247i	$-0.741726\pi$
$677$	−35.8279	−1.37698	−0.688489	+	0.725247i	$0.741726\pi$
$678$	−35.7573	−1.37325
$679$	9.71262	0.372736
$680$	0	0
$681$	−2.18538	−0.0837438
$682$	0.907639	0.0347553
$683$	−23.0980	−0.883822	−0.441911	−	0.897059i	$-0.645699\pi$
$683$	−23.0980	−0.883822	−0.441911	+	0.897059i	$0.645699\pi$
$684$	24.3359	0.930505
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−14.9101	−0.568854
$688$	11.6590	0.444496
$689$	4.80478	0.183047
$690$	0	0
$691$	28.1773	1.07192	0.535958	−	0.844244i	$-0.319950\pi$
$691$	28.1773	1.07192	0.535958	+	0.844244i	$0.319950\pi$
$692$	0.770945	0.0293069
$693$	5.90764	0.224413
$694$	−23.9098	−0.907604
$695$	0	0
$696$	27.5973	1.04607
$697$	5.20503	0.197154
$698$	−2.29599	−0.0869045
$699$	−68.2701	−2.58221
$700$	0	0
$701$	−28.4283	−1.07372	−0.536861	−	0.843671i	$-0.680390\pi$
$701$	−28.4283	−1.07372	−0.536861	+	0.843671i	$0.680390\pi$
$702$	−5.60207	−0.211436
$703$	−22.5489	−0.850448
$704$	1.00000	0.0376889
$705$	0	0
$706$	29.4353	1.10781
$707$	−13.2951	−0.500015
$708$	−25.3368	−0.952216
$709$	−25.8645	−0.971362	−0.485681	−	0.874136i	$-0.661428\pi$
$709$	−25.8645	−0.971362	−0.485681	+	0.874136i	$0.661428\pi$
$710$	0	0
$711$	34.0236	1.27598
$712$	12.4933	0.468207
$713$	−2.97481	−0.111408
$714$	6.55506	0.245317
$715$	0	0
$716$	21.2307	0.793428
$717$	22.0873	0.824867
$718$	−20.1906	−0.753507
$719$	11.6336	0.433860	0.216930	−	0.976187i	$-0.430396\pi$
$719$	11.6336	0.433860	0.216930	+	0.976187i	$0.430396\pi$
$720$	0	0
$721$	−16.3679	−0.609573
$722$	2.03063	0.0755722
$723$	−28.6608	−1.06591
$724$	0.177170	0.00658447
$725$	0	0
$726$	2.98457	0.110768
$727$	−0.707579	−0.0262427	−0.0131213	−	0.999914i	$-0.504177\pi$
$727$	−0.707579	−0.0262427	−0.0131213	+	0.999914i	$0.504177\pi$
$728$	0.645545	0.0239255
$729$	−29.3922	−1.08860
$730$	0	0
$731$	25.6069	0.947105
$732$	−12.1346	−0.448507
$733$	−14.1270	−0.521794	−0.260897	−	0.965367i	$-0.584018\pi$
$733$	−14.1270	−0.521794	−0.260897	+	0.965367i	$0.584018\pi$
$734$	7.47429	0.275881
$735$	0	0
$736$	−3.27753	−0.120811
$737$	3.77282	0.138973
$738$	−14.0004	−0.515364
$739$	7.31802	0.269198	0.134599	−	0.990900i	$-0.457025\pi$
$739$	7.31802	0.269198	0.134599	+	0.990900i	$0.457025\pi$
$740$	0	0
$741$	7.93671	0.291562
$742$	7.44298	0.273240
$743$	10.1775	0.373375	0.186688	−	0.982419i	$-0.440225\pi$
$743$	10.1775	0.373375	0.186688	+	0.982419i	$0.440225\pi$
$744$	−2.70891	−0.0993135
$745$	0	0
$746$	−1.54563	−0.0565896
$747$	21.8999	0.801274
$748$	2.19632	0.0803053
$749$	17.7820	0.649740
$750$	0	0
$751$	−42.2657	−1.54230	−0.771149	−	0.636655i	$-0.780318\pi$
$751$	−42.2657	−1.54230	−0.771149	+	0.636655i	$0.780318\pi$
$752$	1.42023	0.0517906
$753$	−68.0162	−2.47865
$754$	5.96913	0.217383
$755$	0	0
$756$	−8.67804	−0.315617
$757$	−28.4914	−1.03554	−0.517768	−	0.855521i	$-0.673237\pi$
$757$	−28.4914	−1.03554	−0.517768	+	0.855521i	$0.673237\pi$
$758$	−2.25454	−0.0818887
$759$	−9.78200	−0.355064
$760$	0	0
$761$	20.1357	0.729920	0.364960	−	0.931023i	$-0.381083\pi$
$761$	20.1357	0.729920	0.364960	+	0.931023i	$0.381083\pi$
$762$	46.0501	1.66822
$763$	−11.2158	−0.406039
$764$	22.8483	0.826623
$765$	0	0
$766$	−8.73737	−0.315694
$767$	−5.48021	−0.197879
$768$	−2.98457	−0.107696
$769$	42.1624	1.52041	0.760207	−	0.649681i	$-0.225097\pi$
$769$	42.1624	1.52041	0.760207	+	0.649681i	$0.225097\pi$
$770$	0	0
$771$	−90.0595	−3.24342
$772$	12.3333	0.443886
$773$	−15.7433	−0.566249	−0.283124	−	0.959083i	$-0.591371\pi$
$773$	−15.7433	−0.566249	−0.283124	+	0.959083i	$0.591371\pi$
$774$	−68.8772	−2.47574
$775$	0	0
$776$	−9.71262	−0.348663
$777$	16.3371	0.586089
$778$	17.3523	0.622112
$779$	9.76249	0.349778
$780$	0	0
$781$	−8.72051	−0.312044
$782$	−7.19849	−0.257418
$783$	−80.2429	−2.86765
$784$	1.00000	0.0357143
$785$	0	0
$786$	−42.9000	−1.53019
$787$	43.1295	1.53740	0.768700	−	0.639609i	$-0.220904\pi$
$787$	43.1295	1.53740	0.768700	+	0.639609i	$0.220904\pi$
$788$	−17.6257	−0.627891
$789$	−41.1149	−1.46373
$790$	0	0
$791$	−11.9807	−0.425986
$792$	−5.90764	−0.209919
$793$	−2.62464	−0.0932037
$794$	−29.3505	−1.04161
$795$	0	0
$796$	−2.59752	−0.0920666
$797$	14.2722	0.505546	0.252773	−	0.967526i	$-0.418657\pi$
$797$	14.2722	0.505546	0.252773	+	0.967526i	$0.418657\pi$
$798$	12.2946	0.435224
$799$	3.11928	0.110352
$800$	0	0
$801$	−73.8060	−2.60781
$802$	−4.45981	−0.157481
$803$	−7.77282	−0.274297
$804$	−11.2602	−0.397117
$805$	0	0
$806$	−0.585922	−0.0206382
$807$	−94.2928	−3.31926
$808$	13.2951	0.467721
$809$	−30.4594	−1.07090	−0.535448	−	0.844568i	$-0.679857\pi$
$809$	−30.4594	−1.07090	−0.535448	+	0.844568i	$0.679857\pi$
$810$	0	0
$811$	−31.3991	−1.10257	−0.551286	−	0.834316i	$-0.685863\pi$
$811$	−31.3991	−1.10257	−0.551286	+	0.834316i	$0.685863\pi$
$812$	9.24666	0.324494
$813$	22.5870	0.792161
$814$	5.47385	0.191858
$815$	0	0
$816$	−6.55506	−0.229473
$817$	48.0280	1.68029
$818$	16.7783	0.586639
$819$	−3.81365	−0.133260
$820$	0	0
$821$	53.8985	1.88107	0.940535	−	0.339697i	$-0.110325\pi$
$821$	53.8985	1.88107	0.940535	+	0.339697i	$0.110325\pi$
$822$	38.9438	1.35832
$823$	−9.86274	−0.343793	−0.171897	−	0.985115i	$-0.554990\pi$
$823$	−9.86274	−0.343793	−0.171897	+	0.985115i	$0.554990\pi$
$824$	16.3679	0.570204
$825$	0	0
$826$	−8.48928	−0.295380
$827$	−13.0361	−0.453309	−0.226654	−	0.973975i	$-0.572779\pi$
$827$	−13.0361	−0.453309	−0.226654	+	0.973975i	$0.572779\pi$
$828$	19.3625	0.672892
$829$	−14.7261	−0.511458	−0.255729	−	0.966749i	$-0.582315\pi$
$829$	−14.7261	−0.511458	−0.255729	+	0.966749i	$0.582315\pi$
$830$	0	0
$831$	−40.8870	−1.41836
$832$	−0.645545	−0.0223802
$833$	2.19632	0.0760979
$834$	−36.5638	−1.26610
$835$	0	0
$836$	4.11939	0.142472
$837$	7.87653	0.272253
$838$	−27.6878	−0.956458
$839$	7.46663	0.257777	0.128888	−	0.991659i	$-0.458859\pi$
$839$	7.46663	0.257777	0.128888	+	0.991659i	$0.458859\pi$
$840$	0	0
$841$	56.5007	1.94830
$842$	−13.9262	−0.479930
$843$	−15.6995	−0.540719
$844$	−14.2545	−0.490662
$845$	0	0
$846$	−8.39022	−0.288462
$847$	1.00000	0.0343604
$848$	−7.44298	−0.255593
$849$	−21.6461	−0.742892
$850$	0	0
$851$	−17.9407	−0.614999
$852$	26.0269	0.891668
$853$	−55.8463	−1.91214	−0.956070	−	0.293139i	$-0.905300\pi$
$853$	−55.8463	−1.91214	−0.956070	+	0.293139i	$0.905300\pi$
$854$	−4.06578	−0.139128
$855$	0	0
$856$	−17.7820	−0.607776
$857$	25.4303	0.868683	0.434341	−	0.900748i	$-0.356981\pi$
$857$	25.4303	0.868683	0.434341	+	0.900748i	$0.356981\pi$
$858$	−1.92667	−0.0657755
$859$	−11.7029	−0.399297	−0.199648	−	0.979868i	$-0.563980\pi$
$859$	−11.7029	−0.399297	−0.199648	+	0.979868i	$0.563980\pi$
$860$	0	0
$861$	−7.07309	−0.241050
$862$	21.5861	0.735226
$863$	41.6444	1.41759	0.708795	−	0.705414i	$-0.249239\pi$
$863$	41.6444	1.41759	0.708795	+	0.705414i	$0.249239\pi$
$864$	8.67804	0.295233
$865$	0	0
$866$	38.0435	1.29277
$867$	36.3407	1.23419
$868$	−0.907639	−0.0308073
$869$	5.75925	0.195369
$870$	0	0
$871$	−2.43552	−0.0825245
$872$	11.2158	0.379815
$873$	57.3787	1.94197
$874$	−13.5014	−0.456692
$875$	0	0
$876$	23.1985	0.783804
$877$	0.527932	0.0178270	0.00891350	−	0.999960i	$-0.497163\pi$
$877$	0.527932	0.0178270	0.00891350	+	0.999960i	$0.497163\pi$
$878$	35.2169	1.18851
$879$	16.9247	0.570855
$880$	0	0
$881$	19.2992	0.650206	0.325103	−	0.945679i	$-0.394601\pi$
$881$	19.2992	0.650206	0.325103	+	0.945679i	$0.394601\pi$
$882$	−5.90764	−0.198921
$883$	36.9934	1.24493	0.622464	−	0.782649i	$-0.286132\pi$
$883$	36.9934	1.24493	0.622464	+	0.782649i	$0.286132\pi$
$884$	−1.41782	−0.0476865
$885$	0	0
$886$	16.9479	0.569377
$887$	−59.3309	−1.99214	−0.996069	−	0.0885812i	$-0.971767\pi$
$887$	−59.3309	−1.99214	−0.996069	+	0.0885812i	$0.971767\pi$
$888$	−16.3371	−0.548236
$889$	15.4294	0.517486
$890$	0	0
$891$	8.17729	0.273949
$892$	0.241191	0.00807568
$893$	5.85049	0.195779
$894$	63.4121	2.12082
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	6.31472	0.210842
$898$	22.0050	0.734317
$899$	−8.39263	−0.279910
$900$	0	0
$901$	−16.3471	−0.544602
$902$	−2.36989	−0.0789087
$903$	−34.7971	−1.15798
$904$	11.9807	0.398473
$905$	0	0
$906$	−50.5299	−1.67874
$907$	−36.0766	−1.19790	−0.598952	−	0.800785i	$-0.704416\pi$
$907$	−36.0766	−1.19790	−0.598952	+	0.800785i	$0.704416\pi$
$908$	0.732226	0.0242998
$909$	−78.5429	−2.60510
$910$	0	0
$911$	15.8341	0.524608	0.262304	−	0.964985i	$-0.415518\pi$
$911$	15.8341	0.524608	0.262304	+	0.964985i	$0.415518\pi$
$912$	−12.2946	−0.407115
$913$	3.70704	0.122685
$914$	−34.2014	−1.13128
$915$	0	0
$916$	4.99572	0.165063
$917$	−14.3739	−0.474669
$918$	19.0597	0.629065
$919$	20.6798	0.682164	0.341082	−	0.940033i	$-0.389207\pi$
$919$	20.6798	0.682164	0.341082	+	0.940033i	$0.389207\pi$
$920$	0	0
$921$	−11.9157	−0.392637
$922$	−17.0564	−0.561721
$923$	5.62948	0.185296
$924$	−2.98457	−0.0981850
$925$	0	0
$926$	19.2236	0.631728
$927$	−96.6958	−3.17591
$928$	−9.24666	−0.303537
$929$	3.59880	0.118073	0.0590364	−	0.998256i	$-0.481197\pi$
$929$	3.59880	0.118073	0.0590364	+	0.998256i	$0.481197\pi$
$930$	0	0
$931$	4.11939	0.135008
$932$	22.8744	0.749275
$933$	98.4783	3.22403
$934$	−24.7429	−0.809614
$935$	0	0
$936$	3.81365	0.124653
$937$	−4.45479	−0.145532	−0.0727659	−	0.997349i	$-0.523183\pi$
$937$	−4.45479	−0.145532	−0.0727659	+	0.997349i	$0.523183\pi$
$938$	−3.77282	−0.123187
$939$	−52.4268	−1.71088
$940$	0	0
$941$	−11.3838	−0.371101	−0.185551	−	0.982635i	$-0.559407\pi$
$941$	−11.3838	−0.371101	−0.185551	+	0.982635i	$0.559407\pi$
$942$	−68.5082	−2.23212
$943$	7.76738	0.252941
$944$	8.48928	0.276302
$945$	0	0
$946$	−11.6590	−0.379067
$947$	37.2828	1.21153	0.605763	−	0.795645i	$-0.292868\pi$
$947$	37.2828	1.21153	0.605763	+	0.795645i	$0.292868\pi$
$948$	−17.1889	−0.558269
$949$	5.01770	0.162881
$950$	0	0
$951$	−25.8139	−0.837075
$952$	−2.19632	−0.0711830
$953$	−20.9000	−0.677016	−0.338508	−	0.940963i	$-0.609922\pi$
$953$	−20.9000	−0.677016	−0.338508	+	0.940963i	$0.609922\pi$
$954$	43.9704	1.42360
$955$	0	0
$956$	−7.40052	−0.239350
$957$	−27.5973	−0.892093
$958$	−21.2294	−0.685889
$959$	13.0484	0.421354
$960$	0	0
$961$	−30.1762	−0.973426
$962$	−3.53361	−0.113928
$963$	105.050	3.38518
$964$	9.60299	0.309291
$965$	0	0
$966$	9.78200	0.314731
$967$	10.1228	0.325526	0.162763	−	0.986665i	$-0.447959\pi$
$967$	10.1228	0.325526	0.162763	+	0.986665i	$0.447959\pi$
$968$	−1.00000	−0.0321412
$969$	−27.0028	−0.867456
$970$	0	0
$971$	16.0301	0.514430	0.257215	−	0.966354i	$-0.417195\pi$
$971$	16.0301	0.514430	0.257215	+	0.966354i	$0.417195\pi$
$972$	1.62848	0.0522335
$973$	−12.2509	−0.392747
$974$	8.41476	0.269626
$975$	0	0
$976$	4.06578	0.130142
$977$	39.1714	1.25320	0.626602	−	0.779340i	$-0.284445\pi$
$977$	39.1714	1.25320	0.626602	+	0.779340i	$0.284445\pi$
$978$	14.7209	0.470723
$979$	−12.4933	−0.399288
$980$	0	0
$981$	−66.2589	−2.11548
$982$	−35.9615	−1.14758
$983$	−34.0134	−1.08486	−0.542429	−	0.840102i	$-0.682495\pi$
$983$	−34.0134	−1.08486	−0.542429	+	0.840102i	$0.682495\pi$
$984$	7.07309	0.225482
$985$	0	0
$986$	−20.3086	−0.646758
$987$	−4.23878	−0.134922
$988$	−2.65925	−0.0846020
$989$	38.2127	1.21509
$990$	0	0
$991$	−59.2099	−1.88086	−0.940432	−	0.339981i	$-0.889579\pi$
$991$	−59.2099	−1.88086	−0.940432	+	0.339981i	$0.889579\pi$
$992$	0.907639	0.0288176
$993$	−69.5308	−2.20649
$994$	8.72051	0.276598
$995$	0	0
$996$	−11.0639	−0.350573
$997$	−36.2726	−1.14877	−0.574383	−	0.818587i	$-0.694758\pi$
$997$	−36.2726	−1.14877	−0.574383	+	0.818587i	$0.694758\pi$
$998$	−34.0906	−1.07912
$999$	47.5023	1.50291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bz.1.1		5
5.2	odd	4		770.2.c.f.309.5	✓	10
5.3	odd	4		770.2.c.f.309.6	yes	10
5.4	even	2		3850.2.a.ca.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.c.f.309.5	✓	10	5.2	odd	4
770.2.c.f.309.6	yes	10	5.3	odd	4
3850.2.a.bz.1.1		5	1.1	even	1	trivial
3850.2.a.ca.1.5		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.98457 q^{3} +1.00000 q^{4} +2.98457 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.90764 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.98457 q^{3} +1.00000 q^{4} +2.98457 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.90764 q^{9} +1.00000 q^{11} -2.98457 q^{12} -0.645545 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.19632 q^{17} -5.90764 q^{18} +4.11939 q^{19} -2.98457 q^{21} -1.00000 q^{22} +3.27753 q^{23} +2.98457 q^{24} +0.645545 q^{26} -8.67804 q^{27} +1.00000 q^{28} +9.24666 q^{29} -0.907639 q^{31} -1.00000 q^{32} -2.98457 q^{33} -2.19632 q^{34} +5.90764 q^{36} -5.47385 q^{37} -4.11939 q^{38} +1.92667 q^{39} +2.36989 q^{41} +2.98457 q^{42} +11.6590 q^{43} +1.00000 q^{44} -3.27753 q^{46} +1.42023 q^{47} -2.98457 q^{48} +1.00000 q^{49} -6.55506 q^{51} -0.645545 q^{52} -7.44298 q^{53} +8.67804 q^{54} -1.00000 q^{56} -12.2946 q^{57} -9.24666 q^{58} +8.48928 q^{59} +4.06578 q^{61} +0.907639 q^{62} +5.90764 q^{63} +1.00000 q^{64} +2.98457 q^{66} +3.77282 q^{67} +2.19632 q^{68} -9.78200 q^{69} -8.72051 q^{71} -5.90764 q^{72} -7.77282 q^{73} +5.47385 q^{74} +4.11939 q^{76} +1.00000 q^{77} -1.92667 q^{78} +5.75925 q^{79} +8.17729 q^{81} -2.36989 q^{82} +3.70704 q^{83} -2.98457 q^{84} -11.6590 q^{86} -27.5973 q^{87} -1.00000 q^{88} -12.4933 q^{89} -0.645545 q^{91} +3.27753 q^{92} +2.70891 q^{93} -1.42023 q^{94} +2.98457 q^{96} +9.71262 q^{97} -1.00000 q^{98} +5.90764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + 5 q^{7} - 5 q^{8} + 13 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 + 5 * q^7 - 5 * q^8 + 13 * q^9	\( 5 q - 5 q^{2} + 5 q^{4} + 5 q^{7} - 5 q^{8} + 13 q^{9} + 5 q^{11} - 10 q^{13} - 5 q^{14} + 5 q^{16} - 2 q^{17} - 13 q^{18} + 6 q^{19} - 5 q^{22} + 8 q^{23} + 10 q^{26} + 5 q^{28} + 8 q^{29} + 12 q^{31} - 5 q^{32} + 2 q^{34} + 13 q^{36} - 6 q^{37} - 6 q^{38} - 4 q^{39} + 20 q^{41} + 12 q^{43} + 5 q^{44} - 8 q^{46} - 10 q^{47} + 5 q^{49} - 16 q^{51} - 10 q^{52} + 14 q^{53} - 5 q^{56} + 12 q^{57} - 8 q^{58} + 36 q^{59} + 10 q^{61} - 12 q^{62} + 13 q^{63} + 5 q^{64} + 2 q^{67} - 2 q^{68} + 24 q^{69} + 16 q^{71} - 13 q^{72} - 22 q^{73} + 6 q^{74} + 6 q^{76} + 5 q^{77} + 4 q^{78} - 10 q^{79} + 25 q^{81} - 20 q^{82} + 12 q^{83} - 12 q^{86} - 32 q^{87} - 5 q^{88} + 14 q^{89} - 10 q^{91} + 8 q^{92} + 10 q^{94} - 2 q^{97} - 5 q^{98} + 13 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + 5 * q^7 - 5 * q^8 + 13 * q^9 + 5 * q^11 - 10 * q^13 - 5 * q^14 + 5 * q^16 - 2 * q^17 - 13 * q^18 + 6 * q^19 - 5 * q^22 + 8 * q^23 + 10 * q^26 + 5 * q^28 + 8 * q^29 + 12 * q^31 - 5 * q^32 + 2 * q^34 + 13 * q^36 - 6 * q^37 - 6 * q^38 - 4 * q^39 + 20 * q^41 + 12 * q^43 + 5 * q^44 - 8 * q^46 - 10 * q^47 + 5 * q^49 - 16 * q^51 - 10 * q^52 + 14 * q^53 - 5 * q^56 + 12 * q^57 - 8 * q^58 + 36 * q^59 + 10 * q^61 - 12 * q^62 + 13 * q^63 + 5 * q^64 + 2 * q^67 - 2 * q^68 + 24 * q^69 + 16 * q^71 - 13 * q^72 - 22 * q^73 + 6 * q^74 + 6 * q^76 + 5 * q^77 + 4 * q^78 - 10 * q^79 + 25 * q^81 - 20 * q^82 + 12 * q^83 - 12 * q^86 - 32 * q^87 - 5 * q^88 + 14 * q^89 - 10 * q^91 + 8 * q^92 + 10 * q^94 - 2 * q^97 - 5 * q^98 + 13 * q^99

Introduction

L-functions

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