LMFDB - Embedded newform 3850.2.a.bu.1.3

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:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.31955$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.93923 q^{3} +1.00000 q^{4} +2.93923 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.63910 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.93923 q^{3} +1.00000 q^{4} +2.93923 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.63910 q^{9} +1.00000 q^{11} +2.93923 q^{12} +5.87847 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.63910 q^{17} +5.63910 q^{18} -5.57834 q^{19} +2.93923 q^{21} +1.00000 q^{22} -5.57834 q^{23} +2.93923 q^{24} +5.87847 q^{26} +7.75694 q^{27} +1.00000 q^{28} -9.45681 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.93923 q^{33} +4.63910 q^{34} +5.63910 q^{36} -2.30013 q^{37} -5.57834 q^{38} +17.2782 q^{39} -3.06077 q^{41} +2.93923 q^{42} -10.5176 q^{43} +1.00000 q^{44} -5.57834 q^{46} -8.51757 q^{47} +2.93923 q^{48} +1.00000 q^{49} +13.6354 q^{51} +5.87847 q^{52} +9.45681 q^{53} +7.75694 q^{54} +1.00000 q^{56} -16.3960 q^{57} -9.45681 q^{58} +7.23937 q^{59} +14.5176 q^{61} +6.00000 q^{62} +5.63910 q^{63} +1.00000 q^{64} +2.93923 q^{66} -8.00000 q^{67} +4.63910 q^{68} -16.3960 q^{69} -7.15667 q^{71} +5.63910 q^{72} -12.3960 q^{73} -2.30013 q^{74} -5.57834 q^{76} +1.00000 q^{77} +17.2782 q^{78} -10.8565 q^{79} +5.88216 q^{81} -3.06077 q^{82} +13.8785 q^{83} +2.93923 q^{84} -10.5176 q^{86} -27.7958 q^{87} +1.00000 q^{88} +3.87847 q^{89} +5.87847 q^{91} -5.57834 q^{92} +17.6354 q^{93} -8.51757 q^{94} +2.93923 q^{96} -7.57834 q^{97} +1.00000 q^{98} +5.63910 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9} + 3 q^{11} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} - 2 q^{19} + 3 q^{22} - 2 q^{23} - 12 q^{27} + 3 q^{28} + 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} + 11 q^{36} - 4 q^{37} - 2 q^{38} + 40 q^{39} - 18 q^{41} - 8 q^{43} + 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 4 q^{53} - 12 q^{54} + 3 q^{56} - 8 q^{57} + 4 q^{58} + 10 q^{59} + 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} - 24 q^{67} + 8 q^{68} - 8 q^{69} + 8 q^{71} + 11 q^{72} + 4 q^{73} - 4 q^{74} - 2 q^{76} + 3 q^{77} + 40 q^{78} - 6 q^{79} + 47 q^{81} - 18 q^{82} + 24 q^{83} - 8 q^{86} - 48 q^{87} + 3 q^{88} - 6 q^{89} - 2 q^{92} - 2 q^{94} - 8 q^{97} + 3 q^{98} + 11 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9 + 3 * q^11 + 3 * q^14 + 3 * q^16 + 8 * q^17 + 11 * q^18 - 2 * q^19 + 3 * q^22 - 2 * q^23 - 12 * q^27 + 3 * q^28 + 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 + 11 * q^36 - 4 * q^37 - 2 * q^38 + 40 * q^39 - 18 * q^41 - 8 * q^43 + 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 4 * q^53 - 12 * q^54 + 3 * q^56 - 8 * q^57 + 4 * q^58 + 10 * q^59 + 20 * q^61 + 18 * q^62 + 11 * q^63 + 3 * q^64 - 24 * q^67 + 8 * q^68 - 8 * q^69 + 8 * q^71 + 11 * q^72 + 4 * q^73 - 4 * q^74 - 2 * q^76 + 3 * q^77 + 40 * q^78 - 6 * q^79 + 47 * q^81 - 18 * q^82 + 24 * q^83 - 8 * q^86 - 48 * q^87 + 3 * q^88 - 6 * q^89 - 2 * q^92 - 2 * q^94 - 8 * q^97 + 3 * q^98 + 11 * 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
$3$	2.93923	1.69697	0.848484	−	0.529221i	$-0.177516\pi$
$3$	2.93923	1.69697	0.848484	+	0.529221i	$0.177516\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.93923	1.19994
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	5.63910	1.87970
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.93923	0.848484
$13$	5.87847	1.63039	0.815197	−	0.579184i	$-0.196629\pi$
$13$	5.87847	1.63039	0.815197	+	0.579184i	$0.196629\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.63910	1.12515	0.562574	−	0.826747i	$-0.309811\pi$
$17$	4.63910	1.12515	0.562574	+	0.826747i	$0.309811\pi$
$18$	5.63910	1.32915
$19$	−5.57834	−1.27976	−0.639879	−	0.768476i	$-0.721016\pi$
$19$	−5.57834	−1.27976	−0.639879	+	0.768476i	$0.721016\pi$
$20$	0	0
$21$	2.93923	0.641394
$22$	1.00000	0.213201
$23$	−5.57834	−1.16316	−0.581582	−	0.813488i	$-0.697566\pi$
$23$	−5.57834	−1.16316	−0.581582	+	0.813488i	$0.697566\pi$
$24$	2.93923	0.599969
$25$	0	0
$26$	5.87847	1.15286
$27$	7.75694	1.49282
$28$	1.00000	0.188982
$29$	−9.45681	−1.75608	−0.878042	−	0.478583i	$-0.841151\pi$
$29$	−9.45681	−1.75608	−0.878042	+	0.478583i	$0.841151\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	2.93923	0.511655
$34$	4.63910	0.795599
$35$	0	0
$36$	5.63910	0.939850
$37$	−2.30013	−0.378140	−0.189070	−	0.981964i	$-0.560547\pi$
$37$	−2.30013	−0.378140	−0.189070	+	0.981964i	$0.560547\pi$
$38$	−5.57834	−0.904926
$39$	17.2782	2.76673
$40$	0	0
$41$	−3.06077	−0.478011	−0.239006	−	0.971018i	$-0.576821\pi$
$41$	−3.06077	−0.478011	−0.239006	+	0.971018i	$0.576821\pi$
$42$	2.93923	0.453534
$43$	−10.5176	−1.60391	−0.801957	−	0.597381i	$-0.796208\pi$
$43$	−10.5176	−1.60391	−0.801957	+	0.597381i	$0.796208\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−5.57834	−0.822481
$47$	−8.51757	−1.24242	−0.621208	−	0.783646i	$-0.713358\pi$
$47$	−8.51757	−1.24242	−0.621208	+	0.783646i	$0.713358\pi$
$48$	2.93923	0.424242
$49$	1.00000	0.142857
$50$	0	0
$51$	13.6354	1.90934
$52$	5.87847	0.815197
$53$	9.45681	1.29899	0.649496	−	0.760365i	$-0.274980\pi$
$53$	9.45681	1.29899	0.649496	+	0.760365i	$0.274980\pi$
$54$	7.75694	1.05559
$55$	0	0
$56$	1.00000	0.133631
$57$	−16.3960	−2.17171
$58$	−9.45681	−1.24174
$59$	7.23937	0.942485	0.471243	−	0.882004i	$-0.343806\pi$
$59$	7.23937	0.942485	0.471243	+	0.882004i	$0.343806\pi$
$60$	0	0
$61$	14.5176	1.85878	0.929392	−	0.369093i	$-0.120332\pi$
$61$	14.5176	1.85878	0.929392	+	0.369093i	$0.120332\pi$
$62$	6.00000	0.762001
$63$	5.63910	0.710460
$64$	1.00000	0.125000
$65$	0	0
$66$	2.93923	0.361795
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	4.63910	0.562574
$69$	−16.3960	−1.97385
$70$	0	0
$71$	−7.15667	−0.849341	−0.424670	−	0.905348i	$-0.639610\pi$
$71$	−7.15667	−0.849341	−0.424670	+	0.905348i	$0.639610\pi$
$72$	5.63910	0.664574
$73$	−12.3960	−1.45085	−0.725423	−	0.688303i	$-0.758356\pi$
$73$	−12.3960	−1.45085	−0.725423	+	0.688303i	$0.758356\pi$
$74$	−2.30013	−0.267385
$75$	0	0
$76$	−5.57834	−0.639879
$77$	1.00000	0.113961
$78$	17.2782	1.95637
$79$	−10.8565	−1.22146	−0.610728	−	0.791840i	$-0.709123\pi$
$79$	−10.8565	−1.22146	−0.610728	+	0.791840i	$0.709123\pi$
$80$	0	0
$81$	5.88216	0.653574
$82$	−3.06077	−0.338005
$83$	13.8785	1.52336	0.761680	−	0.647953i	$-0.224375\pi$
$83$	13.8785	1.52336	0.761680	+	0.647953i	$0.224375\pi$
$84$	2.93923	0.320697
$85$	0	0
$86$	−10.5176	−1.13414
$87$	−27.7958	−2.98002
$88$	1.00000	0.106600
$89$	3.87847	0.411117	0.205558	−	0.978645i	$-0.434099\pi$
$89$	3.87847	0.411117	0.205558	+	0.978645i	$0.434099\pi$
$90$	0	0
$91$	5.87847	0.616231
$92$	−5.57834	−0.581582
$93$	17.6354	1.82871
$94$	−8.51757	−0.878520
$95$	0	0
$96$	2.93923	0.299984
$97$	−7.57834	−0.769463	−0.384732	−	0.923028i	$-0.625706\pi$
$97$	−7.57834	−0.769463	−0.384732	+	0.923028i	$0.625706\pi$
$98$	1.00000	0.101015
$99$	5.63910	0.566751
$100$	0	0
$101$	11.7958	1.17372	0.586862	−	0.809687i	$-0.300363\pi$
$101$	11.7958	1.17372	0.586862	+	0.809687i	$0.300363\pi$
$102$	13.6354	1.35011
$103$	−17.1178	−1.68667	−0.843335	−	0.537388i	$-0.819411\pi$
$103$	−17.1178	−1.68667	−0.843335	+	0.537388i	$0.819411\pi$
$104$	5.87847	0.576431
$105$	0	0
$106$	9.45681	0.918526
$107$	−2.51757	−0.243383	−0.121691	−	0.992568i	$-0.538832\pi$
$107$	−2.51757	−0.243383	−0.121691	+	0.992568i	$0.538832\pi$
$108$	7.75694	0.746412
$109$	10.3001	0.986574	0.493287	−	0.869867i	$-0.335795\pi$
$109$	10.3001	0.986574	0.493287	+	0.869867i	$0.335795\pi$
$110$	0	0
$111$	−6.76063	−0.641691
$112$	1.00000	0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−16.3960	−1.53563
$115$	0	0
$116$	−9.45681	−0.878042
$117$	33.1493	3.06465
$118$	7.23937	0.666438
$119$	4.63910	0.425266
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.5176	1.31436
$123$	−8.99631	−0.811170
$124$	6.00000	0.538816
$125$	0	0
$126$	5.63910	0.502371
$127$	−10.4787	−0.929837	−0.464919	−	0.885353i	$-0.653916\pi$
$127$	−10.4787	−0.929837	−0.464919	+	0.885353i	$0.653916\pi$
$128$	1.00000	0.0883883
$129$	−30.9136	−2.72179
$130$	0	0
$131$	−1.57834	−0.137900	−0.0689499	−	0.997620i	$-0.521965\pi$
$131$	−1.57834	−0.137900	−0.0689499	+	0.997620i	$0.521965\pi$
$132$	2.93923	0.255828
$133$	−5.57834	−0.483703
$134$	−8.00000	−0.691095
$135$	0	0
$136$	4.63910	0.397800
$137$	3.27820	0.280076	0.140038	−	0.990146i	$-0.455278\pi$
$137$	3.27820	0.280076	0.140038	+	0.990146i	$0.455278\pi$
$138$	−16.3960	−1.39572
$139$	−16.0571	−1.36194	−0.680972	−	0.732310i	$-0.738442\pi$
$139$	−16.0571	−1.36194	−0.680972	+	0.732310i	$0.738442\pi$
$140$	0	0
$141$	−25.0351	−2.10834
$142$	−7.15667	−0.600575
$143$	5.87847	0.491582
$144$	5.63910	0.469925
$145$	0	0
$146$	−12.3960	−1.02590
$147$	2.93923	0.242424
$148$	−2.30013	−0.189070
$149$	10.7350	0.879446	0.439723	−	0.898133i	$-0.355077\pi$
$149$	10.7350	0.879446	0.439723	+	0.898133i	$0.355077\pi$
$150$	0	0
$151$	−2.17860	−0.177292	−0.0886461	−	0.996063i	$-0.528254\pi$
$151$	−2.17860	−0.177292	−0.0886461	+	0.996063i	$0.528254\pi$
$152$	−5.57834	−0.452463
$153$	26.1604	2.11494
$154$	1.00000	0.0805823
$155$	0	0
$156$	17.2782	1.38336
$157$	17.1567	1.36925	0.684626	−	0.728895i	$-0.259966\pi$
$157$	17.1567	1.36925	0.684626	+	0.728895i	$0.259966\pi$
$158$	−10.8565	−0.863700
$159$	27.7958	2.20435
$160$	0	0
$161$	−5.57834	−0.439635
$162$	5.88216	0.462146
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−3.06077	−0.239006
$165$	0	0
$166$	13.8785	1.07718
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	2.93923	0.226767
$169$	21.5564	1.65819
$170$	0	0
$171$	−31.4568	−2.40556
$172$	−10.5176	−0.801957
$173$	5.48243	0.416821	0.208411	−	0.978041i	$-0.433171\pi$
$173$	5.48243	0.416821	0.208411	+	0.978041i	$0.433171\pi$
$174$	−27.7958	−2.10719
$175$	0	0
$176$	1.00000	0.0753778
$177$	21.2782	1.59937
$178$	3.87847	0.290704
$179$	1.23937	0.0926347	0.0463174	−	0.998927i	$-0.485251\pi$
$179$	1.23937	0.0926347	0.0463174	+	0.998927i	$0.485251\pi$
$180$	0	0
$181$	8.55641	0.635993	0.317996	−	0.948092i	$-0.396990\pi$
$181$	8.55641	0.635993	0.317996	+	0.948092i	$0.396990\pi$
$182$	5.87847	0.435741
$183$	42.6706	3.15430
$184$	−5.57834	−0.411240
$185$	0	0
$186$	17.6354	1.29309
$187$	4.63910	0.339245
$188$	−8.51757	−0.621208
$189$	7.75694	0.564234
$190$	0	0
$191$	−1.27820	−0.0924875	−0.0462438	−	0.998930i	$-0.514725\pi$
$191$	−1.27820	−0.0924875	−0.0462438	+	0.998930i	$0.514725\pi$
$192$	2.93923	0.212121
$193$	10.6391	0.765819	0.382910	−	0.923786i	$-0.374922\pi$
$193$	10.6391	0.765819	0.382910	+	0.923786i	$0.374922\pi$
$194$	−7.57834	−0.544093
$195$	0	0
$196$	1.00000	0.0714286
$197$	−7.87847	−0.561318	−0.280659	−	0.959808i	$-0.590553\pi$
$197$	−7.87847	−0.561318	−0.280659	+	0.959808i	$0.590553\pi$
$198$	5.63910	0.400754
$199$	0.517571	0.0366897	0.0183448	−	0.999832i	$-0.494160\pi$
$199$	0.517571	0.0366897	0.0183448	+	0.999832i	$0.494160\pi$
$200$	0	0
$201$	−23.5139	−1.65854
$202$	11.7958	0.829948
$203$	−9.45681	−0.663738
$204$	13.6354	0.954670
$205$	0	0
$206$	−17.1178	−1.19266
$207$	−31.4568	−2.18640
$208$	5.87847	0.407599
$209$	−5.57834	−0.385862
$210$	0	0
$211$	2.72180	0.187376	0.0936881	−	0.995602i	$-0.470134\pi$
$211$	2.72180	0.187376	0.0936881	+	0.995602i	$0.470134\pi$
$212$	9.45681	0.649496
$213$	−21.0351	−1.44130
$214$	−2.51757	−0.172098
$215$	0	0
$216$	7.75694	0.527793
$217$	6.00000	0.407307
$218$	10.3001	0.697613
$219$	−36.4349	−2.46204
$220$	0	0
$221$	27.2708	1.83443
$222$	−6.76063	−0.453744
$223$	22.3960	1.49975	0.749875	−	0.661580i	$-0.230114\pi$
$223$	22.3960	1.49975	0.749875	+	0.661580i	$0.230114\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	−13.0351	−0.865173	−0.432586	−	0.901592i	$-0.642399\pi$
$227$	−13.0351	−0.865173	−0.432586	+	0.901592i	$0.642399\pi$
$228$	−16.3960	−1.08585
$229$	−6.63910	−0.438724	−0.219362	−	0.975644i	$-0.570398\pi$
$229$	−6.63910	−0.438724	−0.219362	+	0.975644i	$0.570398\pi$
$230$	0	0
$231$	2.93923	0.193387
$232$	−9.45681	−0.620870
$233$	−19.0351	−1.24703	−0.623517	−	0.781810i	$-0.714297\pi$
$233$	−19.0351	−1.24703	−0.623517	+	0.781810i	$0.714297\pi$
$234$	33.1493	2.16704
$235$	0	0
$236$	7.23937	0.471243
$237$	−31.9099	−2.07277
$238$	4.63910	0.300708
$239$	22.6135	1.46274	0.731372	−	0.681979i	$-0.238880\pi$
$239$	22.6135	1.46274	0.731372	+	0.681979i	$0.238880\pi$
$240$	0	0
$241$	−17.9744	−1.15783	−0.578916	−	0.815387i	$-0.696524\pi$
$241$	−17.9744	−1.15783	−0.578916	+	0.815387i	$0.696524\pi$
$242$	1.00000	0.0642824
$243$	−5.98176	−0.383730
$244$	14.5176	0.929392
$245$	0	0
$246$	−8.99631	−0.573584
$247$	−32.7921	−2.08651
$248$	6.00000	0.381000
$249$	40.7921	2.58509
$250$	0	0
$251$	8.76063	0.552966	0.276483	−	0.961019i	$-0.410831\pi$
$251$	8.76063	0.552966	0.276483	+	0.961019i	$0.410831\pi$
$252$	5.63910	0.355230
$253$	−5.57834	−0.350707
$254$	−10.4787	−0.657494
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.09960	0.318104	0.159052	−	0.987270i	$-0.449156\pi$
$257$	5.09960	0.318104	0.159052	+	0.987270i	$0.449156\pi$
$258$	−30.9136	−1.92460
$259$	−2.30013	−0.142923
$260$	0	0
$261$	−53.3279	−3.30091
$262$	−1.57834	−0.0975100
$263$	11.7569	0.724964	0.362482	−	0.931991i	$-0.381929\pi$
$263$	11.7569	0.724964	0.362482	+	0.931991i	$0.381929\pi$
$264$	2.93923	0.180897
$265$	0	0
$266$	−5.57834	−0.342030
$267$	11.3997	0.697652
$268$	−8.00000	−0.488678
$269$	−11.6742	−0.711791	−0.355896	−	0.934526i	$-0.615824\pi$
$269$	−11.6742	−0.711791	−0.355896	+	0.934526i	$0.615824\pi$
$270$	0	0
$271$	17.6354	1.07127	0.535637	−	0.844448i	$-0.320071\pi$
$271$	17.6354	1.07127	0.535637	+	0.844448i	$0.320071\pi$
$272$	4.63910	0.281287
$273$	17.2782	1.04572
$274$	3.27820	0.198043
$275$	0	0
$276$	−16.3960	−0.986926
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−16.0571	−0.963039
$279$	33.8346	2.02563
$280$	0	0
$281$	1.15667	0.0690013	0.0345007	−	0.999405i	$-0.489016\pi$
$281$	1.15667	0.0690013	0.0345007	+	0.999405i	$0.489016\pi$
$282$	−25.0351	−1.49082
$283$	−17.2782	−1.02708	−0.513541	−	0.858065i	$-0.671667\pi$
$283$	−17.2782	−1.02708	−0.513541	+	0.858065i	$0.671667\pi$
$284$	−7.15667	−0.424670
$285$	0	0
$286$	5.87847	0.347601
$287$	−3.06077	−0.180671
$288$	5.63910	0.332287
$289$	4.52126	0.265957
$290$	0	0
$291$	−22.2745	−1.30575
$292$	−12.3960	−0.725423
$293$	−1.87847	−0.109741	−0.0548707	−	0.998493i	$-0.517475\pi$
$293$	−1.87847	−0.109741	−0.0548707	+	0.998493i	$0.517475\pi$
$294$	2.93923	0.171420
$295$	0	0
$296$	−2.30013	−0.133693
$297$	7.75694	0.450103
$298$	10.7350	0.621862
$299$	−32.7921	−1.89642
$300$	0	0
$301$	−10.5176	−0.606223
$302$	−2.17860	−0.125365
$303$	34.6706	1.99177
$304$	−5.57834	−0.319940
$305$	0	0
$306$	26.1604	1.49549
$307$	−8.60027	−0.490843	−0.245422	−	0.969416i	$-0.578926\pi$
$307$	−8.60027	−0.490843	−0.245422	+	0.969416i	$0.578926\pi$
$308$	1.00000	0.0569803
$309$	−50.3133	−2.86223
$310$	0	0
$311$	13.7569	0.780084	0.390042	−	0.920797i	$-0.372460\pi$
$311$	13.7569	0.780084	0.390042	+	0.920797i	$0.372460\pi$
$312$	17.2782	0.978186
$313$	−2.90040	−0.163940	−0.0819701	−	0.996635i	$-0.526121\pi$
$313$	−2.90040	−0.163940	−0.0819701	+	0.996635i	$0.526121\pi$
$314$	17.1567	0.968207
$315$	0	0
$316$	−10.8565	−0.610728
$317$	−19.3353	−1.08598	−0.542989	−	0.839740i	$-0.682707\pi$
$317$	−19.3353	−1.08598	−0.542989	+	0.839740i	$0.682707\pi$
$318$	27.7958	1.55871
$319$	−9.45681	−0.529479
$320$	0	0
$321$	−7.39973	−0.413013
$322$	−5.57834	−0.310869
$323$	−25.8785	−1.43992
$324$	5.88216	0.326787
$325$	0	0
$326$	8.00000	0.443079
$327$	30.2745	1.67418
$328$	−3.06077	−0.169002
$329$	−8.51757	−0.469589
$330$	0	0
$331$	10.5176	0.578098	0.289049	−	0.957314i	$-0.406661\pi$
$331$	10.5176	0.578098	0.289049	+	0.957314i	$0.406661\pi$
$332$	13.8785	0.761680
$333$	−12.9707	−0.710789
$334$	−8.00000	−0.437741
$335$	0	0
$336$	2.93923	0.160348
$337$	−7.27820	−0.396469	−0.198234	−	0.980155i	$-0.563521\pi$
$337$	−7.27820	−0.396469	−0.198234	+	0.980155i	$0.563521\pi$
$338$	21.5564	1.17251
$339$	41.1493	2.23492
$340$	0	0
$341$	6.00000	0.324918
$342$	−31.4568	−1.70099
$343$	1.00000	0.0539949
$344$	−10.5176	−0.567069
$345$	0	0
$346$	5.48243	0.294737
$347$	14.2745	0.766296	0.383148	−	0.923687i	$-0.374840\pi$
$347$	14.2745	0.766296	0.383148	+	0.923687i	$0.374840\pi$
$348$	−27.7958	−1.49001
$349$	0.396041	0.0211996	0.0105998	−	0.999944i	$-0.496626\pi$
$349$	0.396041	0.0211996	0.0105998	+	0.999944i	$0.496626\pi$
$350$	0	0
$351$	45.5989	2.43389
$352$	1.00000	0.0533002
$353$	5.09960	0.271424	0.135712	−	0.990748i	$-0.456668\pi$
$353$	5.09960	0.271424	0.135712	+	0.990748i	$0.456668\pi$
$354$	21.2782	1.13092
$355$	0	0
$356$	3.87847	0.205558
$357$	13.6354	0.721662
$358$	1.23937	0.0655026
$359$	−18.3704	−0.969554	−0.484777	−	0.874638i	$-0.661099\pi$
$359$	−18.3704	−0.969554	−0.484777	+	0.874638i	$0.661099\pi$
$360$	0	0
$361$	12.1178	0.637781
$362$	8.55641	0.449715
$363$	2.93923	0.154270
$364$	5.87847	0.308116
$365$	0	0
$366$	42.6706	2.23043
$367$	−17.1178	−0.893544	−0.446772	−	0.894648i	$-0.647426\pi$
$367$	−17.1178	−0.893544	−0.446772	+	0.894648i	$0.647426\pi$
$368$	−5.57834	−0.290791
$369$	−17.2600	−0.898518
$370$	0	0
$371$	9.45681	0.490973
$372$	17.6354	0.914353
$373$	6.35721	0.329164	0.164582	−	0.986363i	$-0.447373\pi$
$373$	6.35721	0.329164	0.164582	+	0.986363i	$0.447373\pi$
$374$	4.63910	0.239882
$375$	0	0
$376$	−8.51757	−0.439260
$377$	−55.5915	−2.86311
$378$	7.75694	0.398974
$379$	−5.03514	−0.258638	−0.129319	−	0.991603i	$-0.541279\pi$
$379$	−5.03514	−0.258638	−0.129319	+	0.991603i	$0.541279\pi$
$380$	0	0
$381$	−30.7995	−1.57790
$382$	−1.27820	−0.0653986
$383$	7.43118	0.379716	0.189858	−	0.981812i	$-0.439197\pi$
$383$	7.43118	0.379716	0.189858	+	0.981812i	$0.439197\pi$
$384$	2.93923	0.149992
$385$	0	0
$386$	10.6391	0.541516
$387$	−59.3097	−3.01488
$388$	−7.57834	−0.384732
$389$	−5.15667	−0.261454	−0.130727	−	0.991418i	$-0.541731\pi$
$389$	−5.15667	−0.261454	−0.130727	+	0.991418i	$0.541731\pi$
$390$	0	0
$391$	−25.8785	−1.30873
$392$	1.00000	0.0505076
$393$	−4.63910	−0.234012
$394$	−7.87847	−0.396912
$395$	0	0
$396$	5.63910	0.283376
$397$	19.6354	0.985473	0.492736	−	0.870179i	$-0.335997\pi$
$397$	19.6354	0.985473	0.492736	+	0.870179i	$0.335997\pi$
$398$	0.517571	0.0259435
$399$	−16.3960	−0.820829
$400$	0	0
$401$	−5.11784	−0.255573	−0.127786	−	0.991802i	$-0.540787\pi$
$401$	−5.11784	−0.255573	−0.127786	+	0.991802i	$0.540787\pi$
$402$	−23.5139	−1.17277
$403$	35.2708	1.75696
$404$	11.7958	0.586862
$405$	0	0
$406$	−9.45681	−0.469333
$407$	−2.30013	−0.114013
$408$	13.6354	0.675053
$409$	2.21744	0.109645	0.0548226	−	0.998496i	$-0.482541\pi$
$409$	2.21744	0.109645	0.0548226	+	0.998496i	$0.482541\pi$
$410$	0	0
$411$	9.63541	0.475280
$412$	−17.1178	−0.843335
$413$	7.23937	0.356226
$414$	−31.4568	−1.54602
$415$	0	0
$416$	5.87847	0.288216
$417$	−47.1955	−2.31117
$418$	−5.57834	−0.272845
$419$	−4.51757	−0.220698	−0.110349	−	0.993893i	$-0.535197\pi$
$419$	−4.51757	−0.220698	−0.110349	+	0.993893i	$0.535197\pi$
$420$	0	0
$421$	−12.3133	−0.600116	−0.300058	−	0.953921i	$-0.597006\pi$
$421$	−12.3133	−0.600116	−0.300058	+	0.953921i	$0.597006\pi$
$422$	2.72180	0.132495
$423$	−48.0315	−2.33537
$424$	9.45681	0.459263
$425$	0	0
$426$	−21.0351	−1.01916
$427$	14.5176	0.702555
$428$	−2.51757	−0.121691
$429$	17.2782	0.834200
$430$	0	0
$431$	−2.17860	−0.104940	−0.0524698	−	0.998623i	$-0.516709\pi$
$431$	−2.17860	−0.104940	−0.0524698	+	0.998623i	$0.516709\pi$
$432$	7.75694	0.373206
$433$	−5.02193	−0.241339	−0.120669	−	0.992693i	$-0.538504\pi$
$433$	−5.02193	−0.241339	−0.120669	+	0.992693i	$0.538504\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	10.3001	0.493287
$437$	31.1178	1.48857
$438$	−36.4349	−1.74093
$439$	7.15667	0.341569	0.170785	−	0.985308i	$-0.445370\pi$
$439$	7.15667	0.341569	0.170785	+	0.985308i	$0.445370\pi$
$440$	0	0
$441$	5.63910	0.268529
$442$	27.2708	1.29714
$443$	19.5139	0.927132	0.463566	−	0.886062i	$-0.346570\pi$
$443$	19.5139	0.927132	0.463566	+	0.886062i	$0.346570\pi$
$444$	−6.76063	−0.320845
$445$	0	0
$446$	22.3960	1.06048
$447$	31.5527	1.49239
$448$	1.00000	0.0472456
$449$	1.36090	0.0642248	0.0321124	−	0.999484i	$-0.489777\pi$
$449$	1.36090	0.0642248	0.0321124	+	0.999484i	$0.489777\pi$
$450$	0	0
$451$	−3.06077	−0.144126
$452$	14.0000	0.658505
$453$	−6.40343	−0.300859
$454$	−13.0351	−0.611770
$455$	0	0
$456$	−16.3960	−0.767815
$457$	−4.47874	−0.209506	−0.104753	−	0.994498i	$-0.533405\pi$
$457$	−4.47874	−0.209506	−0.104753	+	0.994498i	$0.533405\pi$
$458$	−6.63910	−0.310225
$459$	35.9852	1.67965
$460$	0	0
$461$	−36.1530	−1.68381	−0.841906	−	0.539624i	$-0.818566\pi$
$461$	−36.1530	−1.68381	−0.841906	+	0.539624i	$0.818566\pi$
$462$	2.93923	0.136746
$463$	−21.0922	−0.980238	−0.490119	−	0.871655i	$-0.663047\pi$
$463$	−21.0922	−0.980238	−0.490119	+	0.871655i	$0.663047\pi$
$464$	−9.45681	−0.439021
$465$	0	0
$466$	−19.0351	−0.881786
$467$	−19.1311	−0.885279	−0.442640	−	0.896700i	$-0.645958\pi$
$467$	−19.1311	−0.885279	−0.442640	+	0.896700i	$0.645958\pi$
$468$	33.1493	1.53233
$469$	−8.00000	−0.369406
$470$	0	0
$471$	50.4275	2.32358
$472$	7.23937	0.333219
$473$	−10.5176	−0.483598
$474$	−31.9099	−1.46567
$475$	0	0
$476$	4.63910	0.212633
$477$	53.3279	2.44172
$478$	22.6135	1.03432
$479$	5.27820	0.241167	0.120584	−	0.992703i	$-0.461523\pi$
$479$	5.27820	0.241167	0.120584	+	0.992703i	$0.461523\pi$
$480$	0	0
$481$	−13.5213	−0.616517
$482$	−17.9744	−0.818710
$483$	−16.3960	−0.746046
$484$	1.00000	0.0454545
$485$	0	0
$486$	−5.98176	−0.271338
$487$	19.8140	0.897859	0.448929	−	0.893567i	$-0.351806\pi$
$487$	19.8140	0.897859	0.448929	+	0.893567i	$0.351806\pi$
$488$	14.5176	0.657180
$489$	23.5139	1.06333
$490$	0	0
$491$	−6.47874	−0.292381	−0.146191	−	0.989256i	$-0.546701\pi$
$491$	−6.47874	−0.292381	−0.146191	+	0.989256i	$0.546701\pi$
$492$	−8.99631	−0.405585
$493$	−43.8711	−1.97585
$494$	−32.7921	−1.47539
$495$	0	0
$496$	6.00000	0.269408
$497$	−7.15667	−0.321021
$498$	40.7921	1.82794
$499$	−21.2394	−0.950805	−0.475402	−	0.879768i	$-0.657698\pi$
$499$	−21.2394	−0.950805	−0.475402	+	0.879768i	$0.657698\pi$
$500$	0	0
$501$	−23.5139	−1.05052
$502$	8.76063	0.391006
$503$	−42.2357	−1.88320	−0.941598	−	0.336739i	$-0.890676\pi$
$503$	−42.2357	−1.88320	−0.941598	+	0.336739i	$0.890676\pi$
$504$	5.63910	0.251186
$505$	0	0
$506$	−5.57834	−0.247987
$507$	63.3593	2.81389
$508$	−10.4787	−0.464919
$509$	38.8698	1.72287	0.861436	−	0.507867i	$-0.169566\pi$
$509$	38.8698	1.72287	0.861436	+	0.507867i	$0.169566\pi$
$510$	0	0
$511$	−12.3960	−0.548369
$512$	1.00000	0.0441942
$513$	−43.2708	−1.91045
$514$	5.09960	0.224934
$515$	0	0
$516$	−30.9136	−1.36090
$517$	−8.51757	−0.374602
$518$	−2.30013	−0.101062
$519$	16.1141	0.707332
$520$	0	0
$521$	25.9488	1.13684	0.568418	−	0.822740i	$-0.307556\pi$
$521$	25.9488	1.13684	0.568418	+	0.822740i	$0.307556\pi$
$522$	−53.3279	−2.33410
$523$	−42.7482	−1.86925	−0.934625	−	0.355636i	$-0.884264\pi$
$523$	−42.7482	−1.86925	−0.934625	+	0.355636i	$0.884264\pi$
$524$	−1.57834	−0.0689499
$525$	0	0
$526$	11.7569	0.512627
$527$	27.8346	1.21249
$528$	2.93923	0.127914
$529$	8.11784	0.352949
$530$	0	0
$531$	40.8235	1.77159
$532$	−5.57834	−0.241852
$533$	−17.9926	−0.779347
$534$	11.3997	0.493315
$535$	0	0
$536$	−8.00000	−0.345547
$537$	3.64279	0.157198
$538$	−11.6742	−0.503312
$539$	1.00000	0.0430730
$540$	0	0
$541$	−6.73501	−0.289561	−0.144780	−	0.989464i	$-0.546248\pi$
$541$	−6.73501	−0.289561	−0.144780	+	0.989464i	$0.546248\pi$
$542$	17.6354	0.757506
$543$	25.1493	1.07926
$544$	4.63910	0.198900
$545$	0	0
$546$	17.2782	0.739439
$547$	−23.7569	−1.01577	−0.507887	−	0.861424i	$-0.669573\pi$
$547$	−23.7569	−1.01577	−0.507887	+	0.861424i	$0.669573\pi$
$548$	3.27820	0.140038
$549$	81.8661	3.49396
$550$	0	0
$551$	52.7532	2.24736
$552$	−16.3960	−0.697862
$553$	−10.8565	−0.461667
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−16.0571	−0.680972
$557$	34.1918	1.44875	0.724377	−	0.689404i	$-0.242128\pi$
$557$	34.1918	1.44875	0.724377	+	0.689404i	$0.242128\pi$
$558$	33.8346	1.43233
$559$	−61.8272	−2.61501
$560$	0	0
$561$	13.6354	0.575687
$562$	1.15667	0.0487913
$563$	31.3485	1.32118	0.660591	−	0.750746i	$-0.270306\pi$
$563$	31.3485	1.32118	0.660591	+	0.750746i	$0.270306\pi$
$564$	−25.0351	−1.05417
$565$	0	0
$566$	−17.2782	−0.726257
$567$	5.88216	0.247028
$568$	−7.15667	−0.300287
$569$	−38.5490	−1.61606	−0.808030	−	0.589142i	$-0.799466\pi$
$569$	−38.5490	−1.61606	−0.808030	+	0.589142i	$0.799466\pi$
$570$	0	0
$571$	26.9136	1.12630	0.563150	−	0.826355i	$-0.309589\pi$
$571$	26.9136	1.12630	0.563150	+	0.826355i	$0.309589\pi$
$572$	5.87847	0.245791
$573$	−3.75694	−0.156948
$574$	−3.06077	−0.127754
$575$	0	0
$576$	5.63910	0.234963
$577$	15.3353	0.638416	0.319208	−	0.947685i	$-0.396583\pi$
$577$	15.3353	0.638416	0.319208	+	0.947685i	$0.396583\pi$
$578$	4.52126	0.188060
$579$	31.2708	1.29957
$580$	0	0
$581$	13.8785	0.575776
$582$	−22.2745	−0.923308
$583$	9.45681	0.391661
$584$	−12.3960	−0.512952
$585$	0	0
$586$	−1.87847	−0.0775989
$587$	21.2526	0.877188	0.438594	−	0.898685i	$-0.355477\pi$
$587$	21.2526	0.877188	0.438594	+	0.898685i	$0.355477\pi$
$588$	2.93923	0.121212
$589$	−33.4700	−1.37911
$590$	0	0
$591$	−23.1567	−0.952538
$592$	−2.30013	−0.0945349
$593$	−19.1955	−0.788265	−0.394133	−	0.919054i	$-0.628955\pi$
$593$	−19.1955	−0.788265	−0.394133	+	0.919054i	$0.628955\pi$
$594$	7.75694	0.318271
$595$	0	0
$596$	10.7350	0.439723
$597$	1.52126	0.0622612
$598$	−32.7921	−1.34097
$599$	3.39973	0.138909	0.0694547	−	0.997585i	$-0.477874\pi$
$599$	3.39973	0.138909	0.0694547	+	0.997585i	$0.477874\pi$
$600$	0	0
$601$	−7.90409	−0.322415	−0.161207	−	0.986921i	$-0.551539\pi$
$601$	−7.90409	−0.322415	−0.161207	+	0.986921i	$0.551539\pi$
$602$	−10.5176	−0.428664
$603$	−45.1128	−1.83714
$604$	−2.17860	−0.0886461
$605$	0	0
$606$	34.6706	1.40839
$607$	23.7181	0.962688	0.481344	−	0.876532i	$-0.340149\pi$
$607$	23.7181	0.962688	0.481344	+	0.876532i	$0.340149\pi$
$608$	−5.57834	−0.226231
$609$	−27.7958	−1.12634
$610$	0	0
$611$	−50.0703	−2.02563
$612$	26.1604	1.05747
$613$	−40.9136	−1.65249	−0.826243	−	0.563314i	$-0.809526\pi$
$613$	−40.9136	−1.65249	−0.826243	+	0.563314i	$0.809526\pi$
$614$	−8.60027	−0.347079
$615$	0	0
$616$	1.00000	0.0402911
$617$	30.3572	1.22214	0.611068	−	0.791578i	$-0.290740\pi$
$617$	30.3572	1.22214	0.611068	+	0.791578i	$0.290740\pi$
$618$	−50.3133	−2.02390
$619$	−34.9963	−1.40662	−0.703310	−	0.710883i	$-0.748295\pi$
$619$	−34.9963	−1.40662	−0.703310	+	0.710883i	$0.748295\pi$
$620$	0	0
$621$	−43.2708	−1.73640
$622$	13.7569	0.551603
$623$	3.87847	0.155388
$624$	17.2782	0.691682
$625$	0	0
$626$	−2.90040	−0.115923
$627$	−16.3960	−0.654795
$628$	17.1567	0.684626
$629$	−10.6706	−0.425463
$630$	0	0
$631$	38.2357	1.52214	0.761069	−	0.648671i	$-0.224675\pi$
$631$	38.2357	1.52214	0.761069	+	0.648671i	$0.224675\pi$
$632$	−10.8565	−0.431850
$633$	8.00000	0.317971
$634$	−19.3353	−0.767902
$635$	0	0
$636$	27.7958	1.10217
$637$	5.87847	0.232913
$638$	−9.45681	−0.374399
$639$	−40.3572	−1.59651
$640$	0	0
$641$	−25.5139	−1.00774	−0.503869	−	0.863780i	$-0.668091\pi$
$641$	−25.5139	−1.00774	−0.503869	+	0.863780i	$0.668091\pi$
$642$	−7.39973	−0.292044
$643$	37.8528	1.49277	0.746385	−	0.665514i	$-0.231788\pi$
$643$	37.8528	1.49277	0.746385	+	0.665514i	$0.231788\pi$
$644$	−5.57834	−0.219817
$645$	0	0
$646$	−25.8785	−1.01817
$647$	24.7094	0.971426	0.485713	−	0.874118i	$-0.338560\pi$
$647$	24.7094	0.971426	0.485713	+	0.874118i	$0.338560\pi$
$648$	5.88216	0.231073
$649$	7.23937	0.284170
$650$	0	0
$651$	17.6354	0.691186
$652$	8.00000	0.313304
$653$	−16.7789	−0.656608	−0.328304	−	0.944572i	$-0.606477\pi$
$653$	−16.7789	−0.656608	−0.328304	+	0.944572i	$0.606477\pi$
$654$	30.2745	1.18383
$655$	0	0
$656$	−3.06077	−0.119503
$657$	−69.9025	−2.72716
$658$	−8.51757	−0.332050
$659$	10.1215	0.394279	0.197139	−	0.980375i	$-0.436835\pi$
$659$	10.1215	0.394279	0.197139	+	0.980375i	$0.436835\pi$
$660$	0	0
$661$	23.1128	0.898984	0.449492	−	0.893284i	$-0.351605\pi$
$661$	23.1128	0.898984	0.449492	+	0.893284i	$0.351605\pi$
$662$	10.5176	0.408777
$663$	80.1553	3.11298
$664$	13.8785	0.538589
$665$	0	0
$666$	−12.9707	−0.502604
$667$	52.7532	2.04261
$668$	−8.00000	−0.309529
$669$	65.8272	2.54503
$670$	0	0
$671$	14.5176	0.560445
$672$	2.93923	0.113383
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−7.27820	−0.280346
$675$	0	0
$676$	21.5564	0.829093
$677$	−8.83092	−0.339400	−0.169700	−	0.985496i	$-0.554280\pi$
$677$	−8.83092	−0.339400	−0.169700	+	0.985496i	$0.554280\pi$
$678$	41.1493	1.58033
$679$	−7.57834	−0.290830
$680$	0	0
$681$	−38.3133	−1.46817
$682$	6.00000	0.229752
$683$	−11.3997	−0.436199	−0.218099	−	0.975927i	$-0.569986\pi$
$683$	−11.3997	−0.436199	−0.218099	+	0.975927i	$0.569986\pi$
$684$	−31.4568	−1.20278
$685$	0	0
$686$	1.00000	0.0381802
$687$	−19.5139	−0.744501
$688$	−10.5176	−0.400979
$689$	55.5915	2.11787
$690$	0	0
$691$	−0.0826952	−0.00314587	−0.00157294	−	0.999999i	$-0.500501\pi$
$691$	−0.0826952	−0.00314587	−0.00157294	+	0.999999i	$0.500501\pi$
$692$	5.48243	0.208411
$693$	5.63910	0.214212
$694$	14.2745	0.541853
$695$	0	0
$696$	−27.7958	−1.05360
$697$	−14.1992	−0.537833
$698$	0.396041	0.0149904
$699$	−55.9488	−2.11618
$700$	0	0
$701$	38.0571	1.43740	0.718698	−	0.695322i	$-0.244738\pi$
$701$	38.0571	1.43740	0.718698	+	0.695322i	$0.244738\pi$
$702$	45.5989	1.72102
$703$	12.8309	0.483927
$704$	1.00000	0.0376889
$705$	0	0
$706$	5.09960	0.191926
$707$	11.7958	0.443626
$708$	21.2782	0.799684
$709$	36.4275	1.36806	0.684032	−	0.729452i	$-0.260225\pi$
$709$	36.4275	1.36806	0.684032	+	0.729452i	$0.260225\pi$
$710$	0	0
$711$	−61.2211	−2.29597
$712$	3.87847	0.145352
$713$	−33.4700	−1.25346
$714$	13.6354	0.510292
$715$	0	0
$716$	1.23937	0.0463174
$717$	66.4663	2.48223
$718$	−18.3704	−0.685578
$719$	37.3097	1.39142	0.695708	−	0.718325i	$-0.255091\pi$
$719$	37.3097	1.39142	0.695708	+	0.718325i	$0.255091\pi$
$720$	0	0
$721$	−17.1178	−0.637502
$722$	12.1178	0.450979
$723$	−52.8309	−1.96480
$724$	8.55641	0.317996
$725$	0	0
$726$	2.93923	0.109085
$727$	−3.23937	−0.120142	−0.0600708	−	0.998194i	$-0.519133\pi$
$727$	−3.23937	−0.120142	−0.0600708	+	0.998194i	$0.519133\pi$
$728$	5.87847	0.217871
$729$	−35.2283	−1.30475
$730$	0	0
$731$	−48.7921	−1.80464
$732$	42.6706	1.57715
$733$	26.3522	0.973340	0.486670	−	0.873586i	$-0.338211\pi$
$733$	26.3522	0.973340	0.486670	+	0.873586i	$0.338211\pi$
$734$	−17.1178	−0.631831
$735$	0	0
$736$	−5.57834	−0.205620
$737$	−8.00000	−0.294684
$738$	−17.2600	−0.635348
$739$	−11.9223	−0.438570	−0.219285	−	0.975661i	$-0.570372\pi$
$739$	−11.9223	−0.438570	−0.219285	+	0.975661i	$0.570372\pi$
$740$	0	0
$741$	−96.3836	−3.54074
$742$	9.45681	0.347170
$743$	26.0703	0.956426	0.478213	−	0.878244i	$-0.341285\pi$
$743$	26.0703	0.956426	0.478213	+	0.878244i	$0.341285\pi$
$744$	17.6354	0.646545
$745$	0	0
$746$	6.35721	0.232754
$747$	78.2621	2.86346
$748$	4.63910	0.169622
$749$	−2.51757	−0.0919901
$750$	0	0
$751$	36.6003	1.33556	0.667781	−	0.744357i	$-0.267244\pi$
$751$	36.6003	1.33556	0.667781	+	0.744357i	$0.267244\pi$
$752$	−8.51757	−0.310604
$753$	25.7496	0.938366
$754$	−55.5915	−2.02452
$755$	0	0
$756$	7.75694	0.282117
$757$	43.0922	1.56621	0.783107	−	0.621887i	$-0.213634\pi$
$757$	43.0922	1.56621	0.783107	+	0.621887i	$0.213634\pi$
$758$	−5.03514	−0.182885
$759$	−16.3960	−0.595139
$760$	0	0
$761$	−44.0959	−1.59848	−0.799238	−	0.601015i	$-0.794763\pi$
$761$	−44.0959	−1.59848	−0.799238	+	0.601015i	$0.794763\pi$
$762$	−30.7995	−1.11575
$763$	10.3001	0.372890
$764$	−1.27820	−0.0462438
$765$	0	0
$766$	7.43118	0.268500
$767$	42.5564	1.53662
$768$	2.93923	0.106061
$769$	−8.33897	−0.300711	−0.150355	−	0.988632i	$-0.548042\pi$
$769$	−8.33897	−0.300711	−0.150355	+	0.988632i	$0.548042\pi$
$770$	0	0
$771$	14.9889	0.539813
$772$	10.6391	0.382910
$773$	19.2782	0.693389	0.346694	−	0.937978i	$-0.387304\pi$
$773$	19.2782	0.693389	0.346694	+	0.937978i	$0.387304\pi$
$774$	−59.3097	−2.13184
$775$	0	0
$776$	−7.57834	−0.272046
$777$	−6.76063	−0.242536
$778$	−5.15667	−0.184876
$779$	17.0740	0.611739
$780$	0	0
$781$	−7.15667	−0.256086
$782$	−25.8785	−0.925412
$783$	−73.3559	−2.62153
$784$	1.00000	0.0357143
$785$	0	0
$786$	−4.63910	−0.165471
$787$	−43.5915	−1.55387	−0.776935	−	0.629580i	$-0.783227\pi$
$787$	−43.5915	−1.55387	−0.776935	+	0.629580i	$0.783227\pi$
$788$	−7.87847	−0.280659
$789$	34.5564	1.23024
$790$	0	0
$791$	14.0000	0.497783
$792$	5.63910	0.200377
$793$	85.3411	3.03055
$794$	19.6354	0.696835
$795$	0	0
$796$	0.517571	0.0183448
$797$	51.1493	1.81180	0.905900	−	0.423491i	$-0.139195\pi$
$797$	51.1493	1.81180	0.905900	+	0.423491i	$0.139195\pi$
$798$	−16.3960	−0.580414
$799$	−39.5139	−1.39790
$800$	0	0
$801$	21.8711	0.772777
$802$	−5.11784	−0.180717
$803$	−12.3960	−0.437447
$804$	−23.5139	−0.829271
$805$	0	0
$806$	35.2708	1.24236
$807$	−34.3133	−1.20789
$808$	11.7958	0.414974
$809$	45.1567	1.58762	0.793812	−	0.608163i	$-0.208093\pi$
$809$	45.1567	1.58762	0.793812	+	0.608163i	$0.208093\pi$
$810$	0	0
$811$	39.2914	1.37971	0.689854	−	0.723948i	$-0.257675\pi$
$811$	39.2914	1.37971	0.689854	+	0.723948i	$0.257675\pi$
$812$	−9.45681	−0.331869
$813$	51.8346	1.81792
$814$	−2.30013	−0.0806196
$815$	0	0
$816$	13.6354	0.477335
$817$	58.6706	2.05262
$818$	2.21744	0.0775309
$819$	33.1493	1.15833
$820$	0	0
$821$	25.6486	0.895143	0.447572	−	0.894248i	$-0.352289\pi$
$821$	25.6486	0.895143	0.447572	+	0.894248i	$0.352289\pi$
$822$	9.63541	0.336073
$823$	51.6486	1.80036	0.900179	−	0.435520i	$-0.143436\pi$
$823$	51.6486	1.80036	0.900179	+	0.435520i	$0.143436\pi$
$824$	−17.1178	−0.596328
$825$	0	0
$826$	7.23937	0.251890
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−31.4568	−1.09320
$829$	5.36090	0.186192	0.0930958	−	0.995657i	$-0.470324\pi$
$829$	5.36090	0.186192	0.0930958	+	0.995657i	$0.470324\pi$
$830$	0	0
$831$	−64.6632	−2.24314
$832$	5.87847	0.203799
$833$	4.63910	0.160735
$834$	−47.1955	−1.63425
$835$	0	0
$836$	−5.57834	−0.192931
$837$	46.5416	1.60871
$838$	−4.51757	−0.156057
$839$	16.9649	0.585692	0.292846	−	0.956160i	$-0.405398\pi$
$839$	16.9649	0.585692	0.292846	+	0.956160i	$0.405398\pi$
$840$	0	0
$841$	60.4312	2.08383
$842$	−12.3133	−0.424346
$843$	3.39973	0.117093
$844$	2.72180	0.0936881
$845$	0	0
$846$	−48.0315	−1.65136
$847$	1.00000	0.0343604
$848$	9.45681	0.324748
$849$	−50.7847	−1.74293
$850$	0	0
$851$	12.8309	0.439838
$852$	−21.0351	−0.720652
$853$	−31.0666	−1.06370	−0.531850	−	0.846839i	$-0.678503\pi$
$853$	−31.0666	−1.06370	−0.531850	+	0.846839i	$0.678503\pi$
$854$	14.5176	0.496781
$855$	0	0
$856$	−2.51757	−0.0860488
$857$	−37.5966	−1.28427	−0.642137	−	0.766590i	$-0.721952\pi$
$857$	−37.5966	−1.28427	−0.642137	+	0.766590i	$0.721952\pi$
$858$	17.2782	0.589868
$859$	−14.0388	−0.478999	−0.239499	−	0.970897i	$-0.576983\pi$
$859$	−14.0388	−0.478999	−0.239499	+	0.970897i	$0.576983\pi$
$860$	0	0
$861$	−8.99631	−0.306593
$862$	−2.17860	−0.0742035
$863$	−38.8053	−1.32095	−0.660474	−	0.750849i	$-0.729645\pi$
$863$	−38.8053	−1.32095	−0.660474	+	0.750849i	$0.729645\pi$
$864$	7.75694	0.263896
$865$	0	0
$866$	−5.02193	−0.170652
$867$	13.2891	0.451320
$868$	6.00000	0.203653
$869$	−10.8565	−0.368283
$870$	0	0
$871$	−47.0278	−1.59347
$872$	10.3001	0.348807
$873$	−42.7350	−1.44636
$874$	31.1178	1.05258
$875$	0	0
$876$	−36.4349	−1.23102
$877$	−31.7131	−1.07087	−0.535437	−	0.844575i	$-0.679853\pi$
$877$	−31.7131	−1.07087	−0.535437	+	0.844575i	$0.679853\pi$
$878$	7.15667	0.241526
$879$	−5.52126	−0.186228
$880$	0	0
$881$	2.24306	0.0755706	0.0377853	−	0.999286i	$-0.487970\pi$
$881$	2.24306	0.0755706	0.0377853	+	0.999286i	$0.487970\pi$
$882$	5.63910	0.189878
$883$	−24.4349	−0.822299	−0.411150	−	0.911568i	$-0.634873\pi$
$883$	−24.4349	−0.822299	−0.411150	+	0.911568i	$0.634873\pi$
$884$	27.2708	0.917217
$885$	0	0
$886$	19.5139	0.655582
$887$	−35.5915	−1.19505	−0.597524	−	0.801851i	$-0.703849\pi$
$887$	−35.5915	−1.19505	−0.597524	+	0.801851i	$0.703849\pi$
$888$	−6.76063	−0.226872
$889$	−10.4787	−0.351446
$890$	0	0
$891$	5.88216	0.197060
$892$	22.3960	0.749875
$893$	47.5139	1.58999
$894$	31.5527	1.05528
$895$	0	0
$896$	1.00000	0.0334077
$897$	−96.3836	−3.21816
$898$	1.36090	0.0454138
$899$	−56.7408	−1.89241
$900$	0	0
$901$	43.8711	1.46156
$902$	−3.06077	−0.101912
$903$	−30.9136	−1.02874
$904$	14.0000	0.465633
$905$	0	0
$906$	−6.40343	−0.212740
$907$	−17.3923	−0.577503	−0.288752	−	0.957404i	$-0.593240\pi$
$907$	−17.3923	−0.577503	−0.288752	+	0.957404i	$0.593240\pi$
$908$	−13.0351	−0.432586
$909$	66.5176	2.20625
$910$	0	0
$911$	52.1141	1.72662	0.863309	−	0.504675i	$-0.168388\pi$
$911$	52.1141	1.72662	0.863309	+	0.504675i	$0.168388\pi$
$912$	−16.3960	−0.542927
$913$	13.8785	0.459310
$914$	−4.47874	−0.148143
$915$	0	0
$916$	−6.63910	−0.219362
$917$	−1.57834	−0.0521213
$918$	35.9852	1.18769
$919$	12.1347	0.400288	0.200144	−	0.979766i	$-0.435859\pi$
$919$	12.1347	0.400288	0.200144	+	0.979766i	$0.435859\pi$
$920$	0	0
$921$	−25.2782	−0.832945
$922$	−36.1530	−1.19064
$923$	−42.0703	−1.38476
$924$	2.93923	0.0966937
$925$	0	0
$926$	−21.0922	−0.693133
$927$	−96.5292	−3.17044
$928$	−9.45681	−0.310435
$929$	23.8008	0.780879	0.390439	−	0.920629i	$-0.372323\pi$
$929$	23.8008	0.780879	0.390439	+	0.920629i	$0.372323\pi$
$930$	0	0
$931$	−5.57834	−0.182823
$932$	−19.0351	−0.623517
$933$	40.4349	1.32378
$934$	−19.1311	−0.625987
$935$	0	0
$936$	33.1493	1.08352
$937$	−9.16170	−0.299300	−0.149650	−	0.988739i	$-0.547815\pi$
$937$	−9.16170	−0.299300	−0.149650	+	0.988739i	$0.547815\pi$
$938$	−8.00000	−0.261209
$939$	−8.52496	−0.278201
$940$	0	0
$941$	0.996308	0.0324787	0.0162393	−	0.999868i	$-0.494831\pi$
$941$	0.996308	0.0324787	0.0162393	+	0.999868i	$0.494831\pi$
$942$	50.4275	1.64302
$943$	17.0740	0.556005
$944$	7.23937	0.235621
$945$	0	0
$946$	−10.5176	−0.341956
$947$	18.8359	0.612086	0.306043	−	0.952018i	$-0.400995\pi$
$947$	18.8359	0.612086	0.306043	+	0.952018i	$0.400995\pi$
$948$	−31.9099	−1.03639
$949$	−72.8698	−2.36545
$950$	0	0
$951$	−56.8309	−1.84287
$952$	4.63910	0.150354
$953$	−21.4386	−0.694463	−0.347232	−	0.937779i	$-0.612878\pi$
$953$	−21.4386	−0.694463	−0.347232	+	0.937779i	$0.612878\pi$
$954$	53.3279	1.72655
$955$	0	0
$956$	22.6135	0.731372
$957$	−27.7958	−0.898510
$958$	5.27820	0.170531
$959$	3.27820	0.105859
$960$	0	0
$961$	5.00000	0.161290
$962$	−13.5213	−0.435943
$963$	−14.1968	−0.457487
$964$	−17.9744	−0.578916
$965$	0	0
$966$	−16.3960	−0.527534
$967$	−31.5139	−1.01342	−0.506709	−	0.862117i	$-0.669138\pi$
$967$	−31.5139	−1.01342	−0.506709	+	0.862117i	$0.669138\pi$
$968$	1.00000	0.0321412
$969$	−76.0629	−2.44349
$970$	0	0
$971$	16.1968	0.519781	0.259891	−	0.965638i	$-0.416313\pi$
$971$	16.1968	0.519781	0.259891	+	0.965638i	$0.416313\pi$
$972$	−5.98176	−0.191865
$973$	−16.0571	−0.514766
$974$	19.8140	0.634882
$975$	0	0
$976$	14.5176	0.464696
$977$	−17.5139	−0.560319	−0.280159	−	0.959954i	$-0.590387\pi$
$977$	−17.5139	−0.560319	−0.280159	+	0.959954i	$0.590387\pi$
$978$	23.5139	0.751891
$979$	3.87847	0.123956
$980$	0	0
$981$	58.0835	1.85446
$982$	−6.47874	−0.206745
$983$	−41.1178	−1.31146	−0.655728	−	0.754997i	$-0.727638\pi$
$983$	−41.1178	−1.31146	−0.655728	+	0.754997i	$0.727638\pi$
$984$	−8.99631	−0.286792
$985$	0	0
$986$	−43.8711	−1.39714
$987$	−25.0351	−0.796877
$988$	−32.7921	−1.04326
$989$	58.6706	1.86562
$990$	0	0
$991$	−2.79947	−0.0889280	−0.0444640	−	0.999011i	$-0.514158\pi$
$991$	−2.79947	−0.0889280	−0.0444640	+	0.999011i	$0.514158\pi$
$992$	6.00000	0.190500
$993$	30.9136	0.981014
$994$	−7.15667	−0.226996
$995$	0	0
$996$	40.7921	1.29255
$997$	−11.7181	−0.371116	−0.185558	−	0.982633i	$-0.559409\pi$
$997$	−11.7181	−0.371116	−0.185558	+	0.982633i	$0.559409\pi$
$998$	−21.2394	−0.672320
$999$	−17.8420	−0.564496

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bu.1.3		3
5.2	odd	4		3850.2.c.z.1849.4		6
5.3	odd	4		3850.2.c.z.1849.3		6
5.4	even	2		770.2.a.l.1.1	✓	3
15.14	odd	2		6930.2.a.cl.1.1		3
20.19	odd	2		6160.2.a.bi.1.3		3
35.34	odd	2		5390.2.a.bz.1.3		3
55.54	odd	2		8470.2.a.cl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.l.1.1	✓	3	5.4	even	2
3850.2.a.bu.1.3		3	1.1	even	1	trivial
3850.2.c.z.1849.3		6	5.3	odd	4
3850.2.c.z.1849.4		6	5.2	odd	4
5390.2.a.bz.1.3		3	35.34	odd	2
6160.2.a.bi.1.3		3	20.19	odd	2
6930.2.a.cl.1.1		3	15.14	odd	2
8470.2.a.cl.1.1		3	55.54	odd	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.93923 q^{3} +1.00000 q^{4} +2.93923 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.63910 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.93923 q^{3} +1.00000 q^{4} +2.93923 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.63910 q^{9} +1.00000 q^{11} +2.93923 q^{12} +5.87847 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.63910 q^{17} +5.63910 q^{18} -5.57834 q^{19} +2.93923 q^{21} +1.00000 q^{22} -5.57834 q^{23} +2.93923 q^{24} +5.87847 q^{26} +7.75694 q^{27} +1.00000 q^{28} -9.45681 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.93923 q^{33} +4.63910 q^{34} +5.63910 q^{36} -2.30013 q^{37} -5.57834 q^{38} +17.2782 q^{39} -3.06077 q^{41} +2.93923 q^{42} -10.5176 q^{43} +1.00000 q^{44} -5.57834 q^{46} -8.51757 q^{47} +2.93923 q^{48} +1.00000 q^{49} +13.6354 q^{51} +5.87847 q^{52} +9.45681 q^{53} +7.75694 q^{54} +1.00000 q^{56} -16.3960 q^{57} -9.45681 q^{58} +7.23937 q^{59} +14.5176 q^{61} +6.00000 q^{62} +5.63910 q^{63} +1.00000 q^{64} +2.93923 q^{66} -8.00000 q^{67} +4.63910 q^{68} -16.3960 q^{69} -7.15667 q^{71} +5.63910 q^{72} -12.3960 q^{73} -2.30013 q^{74} -5.57834 q^{76} +1.00000 q^{77} +17.2782 q^{78} -10.8565 q^{79} +5.88216 q^{81} -3.06077 q^{82} +13.8785 q^{83} +2.93923 q^{84} -10.5176 q^{86} -27.7958 q^{87} +1.00000 q^{88} +3.87847 q^{89} +5.87847 q^{91} -5.57834 q^{92} +17.6354 q^{93} -8.51757 q^{94} +2.93923 q^{96} -7.57834 q^{97} +1.00000 q^{98} +5.63910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{7} + 3 q^{8} + 11 q^{9} + 3 q^{11} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} - 2 q^{19} + 3 q^{22} - 2 q^{23} - 12 q^{27} + 3 q^{28} + 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} + 11 q^{36} - 4 q^{37} - 2 q^{38} + 40 q^{39} - 18 q^{41} - 8 q^{43} + 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 4 q^{53} - 12 q^{54} + 3 q^{56} - 8 q^{57} + 4 q^{58} + 10 q^{59} + 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} - 24 q^{67} + 8 q^{68} - 8 q^{69} + 8 q^{71} + 11 q^{72} + 4 q^{73} - 4 q^{74} - 2 q^{76} + 3 q^{77} + 40 q^{78} - 6 q^{79} + 47 q^{81} - 18 q^{82} + 24 q^{83} - 8 q^{86} - 48 q^{87} + 3 q^{88} - 6 q^{89} - 2 q^{92} - 2 q^{94} - 8 q^{97} + 3 q^{98} + 11 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^7 + 3 * q^8 + 11 * q^9 + 3 * q^11 + 3 * q^14 + 3 * q^16 + 8 * q^17 + 11 * q^18 - 2 * q^19 + 3 * q^22 - 2 * q^23 - 12 * q^27 + 3 * q^28 + 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 + 11 * q^36 - 4 * q^37 - 2 * q^38 + 40 * q^39 - 18 * q^41 - 8 * q^43 + 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 4 * q^53 - 12 * q^54 + 3 * q^56 - 8 * q^57 + 4 * q^58 + 10 * q^59 + 20 * q^61 + 18 * q^62 + 11 * q^63 + 3 * q^64 - 24 * q^67 + 8 * q^68 - 8 * q^69 + 8 * q^71 + 11 * q^72 + 4 * q^73 - 4 * q^74 - 2 * q^76 + 3 * q^77 + 40 * q^78 - 6 * q^79 + 47 * q^81 - 18 * q^82 + 24 * q^83 - 8 * q^86 - 48 * q^87 + 3 * q^88 - 6 * q^89 - 2 * q^92 - 2 * q^94 - 8 * q^97 + 3 * q^98 + 11 * q^99

Introduction

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