LMFDB - Embedded newform 3850.2.a.bm.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{11} -2.73205 q^{12} +1.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +4.46410 q^{18} -2.73205 q^{19} +2.73205 q^{21} +1.00000 q^{22} -1.26795 q^{23} -2.73205 q^{24} +1.46410 q^{26} -4.00000 q^{27} -1.00000 q^{28} -4.73205 q^{29} +8.92820 q^{31} +1.00000 q^{32} -2.73205 q^{33} -3.46410 q^{34} +4.46410 q^{36} -3.26795 q^{37} -2.73205 q^{38} -4.00000 q^{39} -4.73205 q^{41} +2.73205 q^{42} +4.92820 q^{43} +1.00000 q^{44} -1.26795 q^{46} -2.73205 q^{48} +1.00000 q^{49} +9.46410 q^{51} +1.46410 q^{52} +4.73205 q^{53} -4.00000 q^{54} -1.00000 q^{56} +7.46410 q^{57} -4.73205 q^{58} -13.8564 q^{59} +2.00000 q^{61} +8.92820 q^{62} -4.46410 q^{63} +1.00000 q^{64} -2.73205 q^{66} +10.9282 q^{67} -3.46410 q^{68} +3.46410 q^{69} +9.46410 q^{71} +4.46410 q^{72} -11.4641 q^{73} -3.26795 q^{74} -2.73205 q^{76} -1.00000 q^{77} -4.00000 q^{78} +6.73205 q^{79} -2.46410 q^{81} -4.73205 q^{82} +4.39230 q^{83} +2.73205 q^{84} +4.92820 q^{86} +12.9282 q^{87} +1.00000 q^{88} -15.4641 q^{89} -1.46410 q^{91} -1.26795 q^{92} -24.3923 q^{93} -2.73205 q^{96} -5.80385 q^{97} +1.00000 q^{98} +4.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{18} - 2 q^{19} + 2 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} - 4 q^{26} - 8 q^{27} - 2 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{36} - 10 q^{37} - 2 q^{38} - 8 q^{39} - 6 q^{41} + 2 q^{42} - 4 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{48} + 2 q^{49} + 12 q^{51} - 4 q^{52} + 6 q^{53} - 8 q^{54} - 2 q^{56} + 8 q^{57} - 6 q^{58} + 4 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{66} + 8 q^{67} + 12 q^{71} + 2 q^{72} - 16 q^{73} - 10 q^{74} - 2 q^{76} - 2 q^{77} - 8 q^{78} + 10 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} + 2 q^{84} - 4 q^{86} + 12 q^{87} + 2 q^{88} - 24 q^{89} + 4 q^{91} - 6 q^{92} - 28 q^{93} - 2 q^{96} - 22 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^18 - 2 * q^19 + 2 * q^21 + 2 * q^22 - 6 * q^23 - 2 * q^24 - 4 * q^26 - 8 * q^27 - 2 * q^28 - 6 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^33 + 2 * q^36 - 10 * q^37 - 2 * q^38 - 8 * q^39 - 6 * q^41 + 2 * q^42 - 4 * q^43 + 2 * q^44 - 6 * q^46 - 2 * q^48 + 2 * q^49 + 12 * q^51 - 4 * q^52 + 6 * q^53 - 8 * q^54 - 2 * q^56 + 8 * q^57 - 6 * q^58 + 4 * q^61 + 4 * q^62 - 2 * q^63 + 2 * q^64 - 2 * q^66 + 8 * q^67 + 12 * q^71 + 2 * q^72 - 16 * q^73 - 10 * q^74 - 2 * q^76 - 2 * q^77 - 8 * q^78 + 10 * q^79 + 2 * q^81 - 6 * q^82 - 12 * q^83 + 2 * q^84 - 4 * q^86 + 12 * q^87 + 2 * q^88 - 24 * q^89 + 4 * q^91 - 6 * q^92 - 28 * q^93 - 2 * q^96 - 22 * q^97 + 2 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.73205	−1.11536
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	4.46410	1.48803
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.73205	−0.788675
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	4.46410	1.05220
$19$	−2.73205	−0.626775	−0.313388	−	0.949625i	$-0.601464\pi$
$19$	−2.73205	−0.626775	−0.313388	+	0.949625i	$0.601464\pi$
$20$	0	0
$21$	2.73205	0.596182
$22$	1.00000	0.213201
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	−2.73205	−0.557678
$25$	0	0
$26$	1.46410	0.287134
$27$	−4.00000	−0.769800
$28$	−1.00000	−0.188982
$29$	−4.73205	−0.878720	−0.439360	−	0.898311i	$-0.644795\pi$
$29$	−4.73205	−0.878720	−0.439360	+	0.898311i	$0.644795\pi$
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	1.00000	0.176777
$33$	−2.73205	−0.475589
$34$	−3.46410	−0.594089
$35$	0	0
$36$	4.46410	0.744017
$37$	−3.26795	−0.537248	−0.268624	−	0.963245i	$-0.586569\pi$
$37$	−3.26795	−0.537248	−0.268624	+	0.963245i	$0.586569\pi$
$38$	−2.73205	−0.443197
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−4.73205	−0.739022	−0.369511	−	0.929226i	$-0.620475\pi$
$41$	−4.73205	−0.739022	−0.369511	+	0.929226i	$0.620475\pi$
$42$	2.73205	0.421565
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−1.26795	−0.186949
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.73205	−0.394338
$49$	1.00000	0.142857
$50$	0	0
$51$	9.46410	1.32524
$52$	1.46410	0.203034
$53$	4.73205	0.649997	0.324999	−	0.945715i	$-0.394636\pi$
$53$	4.73205	0.649997	0.324999	+	0.945715i	$0.394636\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	7.46410	0.988644
$58$	−4.73205	−0.621349
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.92820	1.13388
$63$	−4.46410	−0.562424
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.73205	−0.336292
$67$	10.9282	1.33509	0.667546	−	0.744568i	$-0.267345\pi$
$67$	10.9282	1.33509	0.667546	+	0.744568i	$0.267345\pi$
$68$	−3.46410	−0.420084
$69$	3.46410	0.417029
$70$	0	0
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	4.46410	0.526099
$73$	−11.4641	−1.34177	−0.670886	−	0.741561i	$-0.734086\pi$
$73$	−11.4641	−1.34177	−0.670886	+	0.741561i	$0.734086\pi$
$74$	−3.26795	−0.379891
$75$	0	0
$76$	−2.73205	−0.313388
$77$	−1.00000	−0.113961
$78$	−4.00000	−0.452911
$79$	6.73205	0.757415	0.378707	−	0.925516i	$-0.376369\pi$
$79$	6.73205	0.757415	0.378707	+	0.925516i	$0.376369\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	−4.73205	−0.522568
$83$	4.39230	0.482118	0.241059	−	0.970510i	$-0.422505\pi$
$83$	4.39230	0.482118	0.241059	+	0.970510i	$0.422505\pi$
$84$	2.73205	0.298091
$85$	0	0
$86$	4.92820	0.531422
$87$	12.9282	1.38605
$88$	1.00000	0.106600
$89$	−15.4641	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$89$	−15.4641	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$90$	0	0
$91$	−1.46410	−0.153480
$92$	−1.26795	−0.132193
$93$	−24.3923	−2.52936
$94$	0	0
$95$	0	0
$96$	−2.73205	−0.278839
$97$	−5.80385	−0.589291	−0.294646	−	0.955607i	$-0.595202\pi$
$97$	−5.80385	−0.589291	−0.294646	+	0.955607i	$0.595202\pi$
$98$	1.00000	0.101015
$99$	4.46410	0.448659
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	9.46410	0.937086
$103$	−10.5359	−1.03813	−0.519066	−	0.854734i	$-0.673720\pi$
$103$	−10.5359	−1.03813	−0.519066	+	0.854734i	$0.673720\pi$
$104$	1.46410	0.143567
$105$	0	0
$106$	4.73205	0.459617
$107$	−0.928203	−0.0897328	−0.0448664	−	0.998993i	$-0.514286\pi$
$107$	−0.928203	−0.0897328	−0.0448664	+	0.998993i	$0.514286\pi$
$108$	−4.00000	−0.384900
$109$	−1.80385	−0.172777	−0.0863886	−	0.996262i	$-0.527533\pi$
$109$	−1.80385	−0.172777	−0.0863886	+	0.996262i	$0.527533\pi$
$110$	0	0
$111$	8.92820	0.847428
$112$	−1.00000	−0.0944911
$113$	−12.9282	−1.21618	−0.608092	−	0.793867i	$-0.708065\pi$
$113$	−12.9282	−1.21618	−0.608092	+	0.793867i	$0.708065\pi$
$114$	7.46410	0.699077
$115$	0	0
$116$	−4.73205	−0.439360
$117$	6.53590	0.604244
$118$	−13.8564	−1.27559
$119$	3.46410	0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	12.9282	1.16570
$124$	8.92820	0.801776
$125$	0	0
$126$	−4.46410	−0.397694
$127$	−9.85641	−0.874615	−0.437307	−	0.899312i	$-0.644068\pi$
$127$	−9.85641	−0.874615	−0.437307	+	0.899312i	$0.644068\pi$
$128$	1.00000	0.0883883
$129$	−13.4641	−1.18545
$130$	0	0
$131$	−17.6603	−1.54298	−0.771492	−	0.636239i	$-0.780489\pi$
$131$	−17.6603	−1.54298	−0.771492	+	0.636239i	$0.780489\pi$
$132$	−2.73205	−0.237795
$133$	2.73205	0.236899
$134$	10.9282	0.944053
$135$	0	0
$136$	−3.46410	−0.297044
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	3.46410	0.294884
$139$	4.19615	0.355913	0.177957	−	0.984038i	$-0.443051\pi$
$139$	4.19615	0.355913	0.177957	+	0.984038i	$0.443051\pi$
$140$	0	0
$141$	0	0
$142$	9.46410	0.794210
$143$	1.46410	0.122434
$144$	4.46410	0.372008
$145$	0	0
$146$	−11.4641	−0.948776
$147$	−2.73205	−0.225336
$148$	−3.26795	−0.268624
$149$	−7.26795	−0.595414	−0.297707	−	0.954657i	$-0.596222\pi$
$149$	−7.26795	−0.595414	−0.297707	+	0.954657i	$0.596222\pi$
$150$	0	0
$151$	−7.80385	−0.635068	−0.317534	−	0.948247i	$-0.602855\pi$
$151$	−7.80385	−0.635068	−0.317534	+	0.948247i	$0.602855\pi$
$152$	−2.73205	−0.221599
$153$	−15.4641	−1.25020
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−6.39230	−0.510161	−0.255081	−	0.966920i	$-0.582102\pi$
$157$	−6.39230	−0.510161	−0.255081	+	0.966920i	$0.582102\pi$
$158$	6.73205	0.535573
$159$	−12.9282	−1.02527
$160$	0	0
$161$	1.26795	0.0999284
$162$	−2.46410	−0.193598
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−4.73205	−0.369511
$165$	0	0
$166$	4.39230	0.340909
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	2.73205	0.210782
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−12.1962	−0.932663
$172$	4.92820	0.375772
$173$	−0.928203	−0.0705700	−0.0352850	−	0.999377i	$-0.511234\pi$
$173$	−0.928203	−0.0705700	−0.0352850	+	0.999377i	$0.511234\pi$
$174$	12.9282	0.980085
$175$	0	0
$176$	1.00000	0.0753778
$177$	37.8564	2.84546
$178$	−15.4641	−1.15908
$179$	−19.8564	−1.48414	−0.742069	−	0.670324i	$-0.766155\pi$
$179$	−19.8564	−1.48414	−0.742069	+	0.670324i	$0.766155\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	−1.46410	−0.108526
$183$	−5.46410	−0.403918
$184$	−1.26795	−0.0934745
$185$	0	0
$186$	−24.3923	−1.78853
$187$	−3.46410	−0.253320
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	−2.73205	−0.197169
$193$	25.4641	1.83295	0.916473	−	0.400096i	$-0.131023\pi$
$193$	25.4641	1.83295	0.916473	+	0.400096i	$0.131023\pi$
$194$	−5.80385	−0.416692
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.3923	1.59539	0.797693	−	0.603064i	$-0.206054\pi$
$197$	22.3923	1.59539	0.797693	+	0.603064i	$0.206054\pi$
$198$	4.46410	0.317250
$199$	−24.7846	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$200$	0	0
$201$	−29.8564	−2.10591
$202$	−6.00000	−0.422159
$203$	4.73205	0.332125
$204$	9.46410	0.662620
$205$	0	0
$206$	−10.5359	−0.734071
$207$	−5.66025	−0.393415
$208$	1.46410	0.101517
$209$	−2.73205	−0.188980
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	4.73205	0.324999
$213$	−25.8564	−1.77165
$214$	−0.928203	−0.0634507
$215$	0	0
$216$	−4.00000	−0.272166
$217$	−8.92820	−0.606086
$218$	−1.80385	−0.122172
$219$	31.3205	2.11644
$220$	0	0
$221$	−5.07180	−0.341166
$222$	8.92820	0.599222
$223$	8.39230	0.561990	0.280995	−	0.959709i	$-0.409336\pi$
$223$	8.39230	0.561990	0.280995	+	0.959709i	$0.409336\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−12.9282	−0.859971
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	7.46410	0.494322
$229$	−13.4641	−0.889733	−0.444866	−	0.895597i	$-0.646749\pi$
$229$	−13.4641	−0.889733	−0.444866	+	0.895597i	$0.646749\pi$
$230$	0	0
$231$	2.73205	0.179756
$232$	−4.73205	−0.310674
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	6.53590	0.427265
$235$	0	0
$236$	−13.8564	−0.901975
$237$	−18.3923	−1.19471
$238$	3.46410	0.224544
$239$	−3.80385	−0.246050	−0.123025	−	0.992404i	$-0.539260\pi$
$239$	−3.80385	−0.246050	−0.123025	+	0.992404i	$0.539260\pi$
$240$	0	0
$241$	−18.1962	−1.17212	−0.586059	−	0.810269i	$-0.699321\pi$
$241$	−18.1962	−1.17212	−0.586059	+	0.810269i	$0.699321\pi$
$242$	1.00000	0.0642824
$243$	18.7321	1.20166
$244$	2.00000	0.128037
$245$	0	0
$246$	12.9282	0.824272
$247$	−4.00000	−0.254514
$248$	8.92820	0.566941
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−4.46410	−0.281212
$253$	−1.26795	−0.0797153
$254$	−9.85641	−0.618446
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.9808	−1.43350	−0.716750	−	0.697330i	$-0.754371\pi$
$257$	−22.9808	−1.43350	−0.716750	+	0.697330i	$0.754371\pi$
$258$	−13.4641	−0.838238
$259$	3.26795	0.203060
$260$	0	0
$261$	−21.1244	−1.30756
$262$	−17.6603	−1.09105
$263$	−18.9282	−1.16716	−0.583582	−	0.812055i	$-0.698349\pi$
$263$	−18.9282	−1.16716	−0.583582	+	0.812055i	$0.698349\pi$
$264$	−2.73205	−0.168146
$265$	0	0
$266$	2.73205	0.167513
$267$	42.2487	2.58558
$268$	10.9282	0.667546
$269$	4.39230	0.267804	0.133902	−	0.990995i	$-0.457249\pi$
$269$	4.39230	0.267804	0.133902	+	0.990995i	$0.457249\pi$
$270$	0	0
$271$	24.3923	1.48173	0.740863	−	0.671656i	$-0.234417\pi$
$271$	24.3923	1.48173	0.740863	+	0.671656i	$0.234417\pi$
$272$	−3.46410	−0.210042
$273$	4.00000	0.242091
$274$	−7.85641	−0.474623
$275$	0	0
$276$	3.46410	0.208514
$277$	−27.8564	−1.67373	−0.836865	−	0.547410i	$-0.815614\pi$
$277$	−27.8564	−1.67373	−0.836865	+	0.547410i	$0.815614\pi$
$278$	4.19615	0.251668
$279$	39.8564	2.38614
$280$	0	0
$281$	−10.3923	−0.619953	−0.309976	−	0.950744i	$-0.600321\pi$
$281$	−10.3923	−0.619953	−0.309976	+	0.950744i	$0.600321\pi$
$282$	0	0
$283$	10.9282	0.649614	0.324807	−	0.945780i	$-0.394701\pi$
$283$	10.9282	0.649614	0.324807	+	0.945780i	$0.394701\pi$
$284$	9.46410	0.561591
$285$	0	0
$286$	1.46410	0.0865741
$287$	4.73205	0.279324
$288$	4.46410	0.263050
$289$	−5.00000	−0.294118
$290$	0	0
$291$	15.8564	0.929519
$292$	−11.4641	−0.670886
$293$	2.53590	0.148149	0.0740744	−	0.997253i	$-0.476400\pi$
$293$	2.53590	0.148149	0.0740744	+	0.997253i	$0.476400\pi$
$294$	−2.73205	−0.159336
$295$	0	0
$296$	−3.26795	−0.189946
$297$	−4.00000	−0.232104
$298$	−7.26795	−0.421021
$299$	−1.85641	−0.107359
$300$	0	0
$301$	−4.92820	−0.284057
$302$	−7.80385	−0.449061
$303$	16.3923	0.941713
$304$	−2.73205	−0.156694
$305$	0	0
$306$	−15.4641	−0.884024
$307$	−31.3205	−1.78756	−0.893778	−	0.448510i	$-0.851955\pi$
$307$	−31.3205	−1.78756	−0.893778	+	0.448510i	$0.851955\pi$
$308$	−1.00000	−0.0569803
$309$	28.7846	1.63750
$310$	0	0
$311$	7.85641	0.445496	0.222748	−	0.974876i	$-0.428497\pi$
$311$	7.85641	0.445496	0.222748	+	0.974876i	$0.428497\pi$
$312$	−4.00000	−0.226455
$313$	−27.2679	−1.54128	−0.770638	−	0.637273i	$-0.780062\pi$
$313$	−27.2679	−1.54128	−0.770638	+	0.637273i	$0.780062\pi$
$314$	−6.39230	−0.360739
$315$	0	0
$316$	6.73205	0.378707
$317$	32.4449	1.82229	0.911143	−	0.412091i	$-0.135202\pi$
$317$	32.4449	1.82229	0.911143	+	0.412091i	$0.135202\pi$
$318$	−12.9282	−0.724978
$319$	−4.73205	−0.264944
$320$	0	0
$321$	2.53590	0.141540
$322$	1.26795	0.0706600
$323$	9.46410	0.526597
$324$	−2.46410	−0.136895
$325$	0	0
$326$	−8.00000	−0.443079
$327$	4.92820	0.272530
$328$	−4.73205	−0.261284
$329$	0	0
$330$	0	0
$331$	−4.92820	−0.270879	−0.135439	−	0.990786i	$-0.543245\pi$
$331$	−4.92820	−0.270879	−0.135439	+	0.990786i	$0.543245\pi$
$332$	4.39230	0.241059
$333$	−14.5885	−0.799443
$334$	13.8564	0.758189
$335$	0	0
$336$	2.73205	0.149046
$337$	30.7846	1.67694	0.838472	−	0.544944i	$-0.183449\pi$
$337$	30.7846	1.67694	0.838472	+	0.544944i	$0.183449\pi$
$338$	−10.8564	−0.590511
$339$	35.3205	1.91835
$340$	0	0
$341$	8.92820	0.483489
$342$	−12.1962	−0.659492
$343$	−1.00000	−0.0539949
$344$	4.92820	0.265711
$345$	0	0
$346$	−0.928203	−0.0499005
$347$	−7.85641	−0.421754	−0.210877	−	0.977513i	$-0.567632\pi$
$347$	−7.85641	−0.421754	−0.210877	+	0.977513i	$0.567632\pi$
$348$	12.9282	0.693024
$349$	30.3923	1.62686	0.813431	−	0.581661i	$-0.197597\pi$
$349$	30.3923	1.62686	0.813431	+	0.581661i	$0.197597\pi$
$350$	0	0
$351$	−5.85641	−0.312592
$352$	1.00000	0.0533002
$353$	32.4449	1.72687	0.863433	−	0.504464i	$-0.168310\pi$
$353$	32.4449	1.72687	0.863433	+	0.504464i	$0.168310\pi$
$354$	37.8564	2.01205
$355$	0	0
$356$	−15.4641	−0.819596
$357$	−9.46410	−0.500893
$358$	−19.8564	−1.04944
$359$	−1.26795	−0.0669198	−0.0334599	−	0.999440i	$-0.510653\pi$
$359$	−1.26795	−0.0669198	−0.0334599	+	0.999440i	$0.510653\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	−11.8564	−0.623159
$363$	−2.73205	−0.143395
$364$	−1.46410	−0.0767398
$365$	0	0
$366$	−5.46410	−0.285613
$367$	−29.4641	−1.53801	−0.769007	−	0.639241i	$-0.779249\pi$
$367$	−29.4641	−1.53801	−0.769007	+	0.639241i	$0.779249\pi$
$368$	−1.26795	−0.0660964
$369$	−21.1244	−1.09969
$370$	0	0
$371$	−4.73205	−0.245676
$372$	−24.3923	−1.26468
$373$	−30.3923	−1.57365	−0.786827	−	0.617174i	$-0.788278\pi$
$373$	−30.3923	−1.57365	−0.786827	+	0.617174i	$0.788278\pi$
$374$	−3.46410	−0.179124
$375$	0	0
$376$	0	0
$377$	−6.92820	−0.356821
$378$	4.00000	0.205738
$379$	23.7128	1.21805	0.609023	−	0.793153i	$-0.291562\pi$
$379$	23.7128	1.21805	0.609023	+	0.793153i	$0.291562\pi$
$380$	0	0
$381$	26.9282	1.37957
$382$	6.92820	0.354478
$383$	−16.3923	−0.837608	−0.418804	−	0.908077i	$-0.637551\pi$
$383$	−16.3923	−0.837608	−0.418804	+	0.908077i	$0.637551\pi$
$384$	−2.73205	−0.139419
$385$	0	0
$386$	25.4641	1.29609
$387$	22.0000	1.11832
$388$	−5.80385	−0.294646
$389$	−24.2487	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$389$	−24.2487	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$390$	0	0
$391$	4.39230	0.222128
$392$	1.00000	0.0505076
$393$	48.2487	2.43383
$394$	22.3923	1.12811
$395$	0	0
$396$	4.46410	0.224330
$397$	38.3923	1.92685	0.963427	−	0.267970i	$-0.0863528\pi$
$397$	38.3923	1.92685	0.963427	+	0.267970i	$0.0863528\pi$
$398$	−24.7846	−1.24234
$399$	−7.46410	−0.373672
$400$	0	0
$401$	−25.1769	−1.25728	−0.628638	−	0.777698i	$-0.716387\pi$
$401$	−25.1769	−1.25728	−0.628638	+	0.777698i	$0.716387\pi$
$402$	−29.8564	−1.48910
$403$	13.0718	0.651153
$404$	−6.00000	−0.298511
$405$	0	0
$406$	4.73205	0.234848
$407$	−3.26795	−0.161986
$408$	9.46410	0.468543
$409$	22.1962	1.09753	0.548765	−	0.835977i	$-0.315098\pi$
$409$	22.1962	1.09753	0.548765	+	0.835977i	$0.315098\pi$
$410$	0	0
$411$	21.4641	1.05875
$412$	−10.5359	−0.519066
$413$	13.8564	0.681829
$414$	−5.66025	−0.278186
$415$	0	0
$416$	1.46410	0.0717835
$417$	−11.4641	−0.561399
$418$	−2.73205	−0.133629
$419$	32.7846	1.60163	0.800816	−	0.598910i	$-0.204399\pi$
$419$	32.7846	1.60163	0.800816	+	0.598910i	$0.204399\pi$
$420$	0	0
$421$	10.7846	0.525610	0.262805	−	0.964849i	$-0.415352\pi$
$421$	10.7846	0.525610	0.262805	+	0.964849i	$0.415352\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	4.73205	0.229809
$425$	0	0
$426$	−25.8564	−1.25275
$427$	−2.00000	−0.0967868
$428$	−0.928203	−0.0448664
$429$	−4.00000	−0.193122
$430$	0	0
$431$	18.3397	0.883394	0.441697	−	0.897164i	$-0.354377\pi$
$431$	18.3397	0.883394	0.441697	+	0.897164i	$0.354377\pi$
$432$	−4.00000	−0.192450
$433$	−0.732051	−0.0351801	−0.0175901	−	0.999845i	$-0.505599\pi$
$433$	−0.732051	−0.0351801	−0.0175901	+	0.999845i	$0.505599\pi$
$434$	−8.92820	−0.428567
$435$	0	0
$436$	−1.80385	−0.0863886
$437$	3.46410	0.165710
$438$	31.3205	1.49655
$439$	36.3923	1.73691	0.868455	−	0.495768i	$-0.165113\pi$
$439$	36.3923	1.73691	0.868455	+	0.495768i	$0.165113\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	−5.07180	−0.241241
$443$	−20.7846	−0.987507	−0.493753	−	0.869602i	$-0.664375\pi$
$443$	−20.7846	−0.987507	−0.493753	+	0.869602i	$0.664375\pi$
$444$	8.92820	0.423714
$445$	0	0
$446$	8.39230	0.397387
$447$	19.8564	0.939176
$448$	−1.00000	−0.0472456
$449$	−35.3205	−1.66688	−0.833439	−	0.552612i	$-0.813631\pi$
$449$	−35.3205	−1.66688	−0.833439	+	0.552612i	$0.813631\pi$
$450$	0	0
$451$	−4.73205	−0.222824
$452$	−12.9282	−0.608092
$453$	21.3205	1.00172
$454$	−13.8564	−0.650313
$455$	0	0
$456$	7.46410	0.349539
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	−13.4641	−0.629136
$459$	13.8564	0.646762
$460$	0	0
$461$	29.3205	1.36559	0.682796	−	0.730609i	$-0.260764\pi$
$461$	29.3205	1.36559	0.682796	+	0.730609i	$0.260764\pi$
$462$	2.73205	0.127107
$463$	−12.9808	−0.603267	−0.301634	−	0.953424i	$-0.597532\pi$
$463$	−12.9808	−0.603267	−0.301634	+	0.953424i	$0.597532\pi$
$464$	−4.73205	−0.219680
$465$	0	0
$466$	−7.85641	−0.363941
$467$	−17.6603	−0.817219	−0.408610	−	0.912709i	$-0.633986\pi$
$467$	−17.6603	−0.817219	−0.408610	+	0.912709i	$0.633986\pi$
$468$	6.53590	0.302122
$469$	−10.9282	−0.504618
$470$	0	0
$471$	17.4641	0.804703
$472$	−13.8564	−0.637793
$473$	4.92820	0.226599
$474$	−18.3923	−0.844787
$475$	0	0
$476$	3.46410	0.158777
$477$	21.1244	0.967218
$478$	−3.80385	−0.173984
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	−4.78461	−0.218159
$482$	−18.1962	−0.828812
$483$	−3.46410	−0.157622
$484$	1.00000	0.0454545
$485$	0	0
$486$	18.7321	0.849703
$487$	−4.87564	−0.220937	−0.110468	−	0.993880i	$-0.535235\pi$
$487$	−4.87564	−0.220937	−0.110468	+	0.993880i	$0.535235\pi$
$488$	2.00000	0.0905357
$489$	21.8564	0.988381
$490$	0	0
$491$	42.9282	1.93732	0.968661	−	0.248385i	$-0.0798999\pi$
$491$	42.9282	1.93732	0.968661	+	0.248385i	$0.0798999\pi$
$492$	12.9282	0.582848
$493$	16.3923	0.738272
$494$	−4.00000	−0.179969
$495$	0	0
$496$	8.92820	0.400888
$497$	−9.46410	−0.424523
$498$	−12.0000	−0.537733
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	0	0
$501$	−37.8564	−1.69130
$502$	0	0
$503$	18.9282	0.843967	0.421983	−	0.906604i	$-0.361334\pi$
$503$	18.9282	0.843967	0.421983	+	0.906604i	$0.361334\pi$
$504$	−4.46410	−0.198847
$505$	0	0
$506$	−1.26795	−0.0563672
$507$	29.6603	1.31726
$508$	−9.85641	−0.437307
$509$	2.78461	0.123426	0.0617128	−	0.998094i	$-0.480344\pi$
$509$	2.78461	0.123426	0.0617128	+	0.998094i	$0.480344\pi$
$510$	0	0
$511$	11.4641	0.507142
$512$	1.00000	0.0441942
$513$	10.9282	0.482492
$514$	−22.9808	−1.01364
$515$	0	0
$516$	−13.4641	−0.592724
$517$	0	0
$518$	3.26795	0.143585
$519$	2.53590	0.111314
$520$	0	0
$521$	10.3923	0.455295	0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923	0.455295	0.227648	+	0.973744i	$0.426897\pi$
$522$	−21.1244	−0.924588
$523$	−19.3205	−0.844827	−0.422413	−	0.906403i	$-0.638817\pi$
$523$	−19.3205	−0.844827	−0.422413	+	0.906403i	$0.638817\pi$
$524$	−17.6603	−0.771492
$525$	0	0
$526$	−18.9282	−0.825309
$527$	−30.9282	−1.34725
$528$	−2.73205	−0.118897
$529$	−21.3923	−0.930100
$530$	0	0
$531$	−61.8564	−2.68434
$532$	2.73205	0.118449
$533$	−6.92820	−0.300094
$534$	42.2487	1.82828
$535$	0	0
$536$	10.9282	0.472026
$537$	54.2487	2.34100
$538$	4.39230	0.189366
$539$	1.00000	0.0430730
$540$	0	0
$541$	17.1244	0.736234	0.368117	−	0.929780i	$-0.380003\pi$
$541$	17.1244	0.736234	0.368117	+	0.929780i	$0.380003\pi$
$542$	24.3923	1.04774
$543$	32.3923	1.39009
$544$	−3.46410	−0.148522
$545$	0	0
$546$	4.00000	0.171184
$547$	−4.78461	−0.204575	−0.102288	−	0.994755i	$-0.532616\pi$
$547$	−4.78461	−0.204575	−0.102288	+	0.994755i	$0.532616\pi$
$548$	−7.85641	−0.335609
$549$	8.92820	0.381046
$550$	0	0
$551$	12.9282	0.550760
$552$	3.46410	0.147442
$553$	−6.73205	−0.286276
$554$	−27.8564	−1.18351
$555$	0	0
$556$	4.19615	0.177957
$557$	12.2487	0.518995	0.259497	−	0.965744i	$-0.416443\pi$
$557$	12.2487	0.518995	0.259497	+	0.965744i	$0.416443\pi$
$558$	39.8564	1.68726
$559$	7.21539	0.305178
$560$	0	0
$561$	9.46410	0.399575
$562$	−10.3923	−0.438373
$563$	20.7846	0.875967	0.437983	−	0.898983i	$-0.355693\pi$
$563$	20.7846	0.875967	0.437983	+	0.898983i	$0.355693\pi$
$564$	0	0
$565$	0	0
$566$	10.9282	0.459347
$567$	2.46410	0.103483
$568$	9.46410	0.397105
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−20.3923	−0.853391	−0.426696	−	0.904395i	$-0.640322\pi$
$571$	−20.3923	−0.853391	−0.426696	+	0.904395i	$0.640322\pi$
$572$	1.46410	0.0612172
$573$	−18.9282	−0.790737
$574$	4.73205	0.197512
$575$	0	0
$576$	4.46410	0.186004
$577$	−8.33975	−0.347188	−0.173594	−	0.984817i	$-0.555538\pi$
$577$	−8.33975	−0.347188	−0.173594	+	0.984817i	$0.555538\pi$
$578$	−5.00000	−0.207973
$579$	−69.5692	−2.89120
$580$	0	0
$581$	−4.39230	−0.182224
$582$	15.8564	0.657269
$583$	4.73205	0.195982
$584$	−11.4641	−0.474388
$585$	0	0
$586$	2.53590	0.104757
$587$	28.9808	1.19616	0.598082	−	0.801435i	$-0.295930\pi$
$587$	28.9808	1.19616	0.598082	+	0.801435i	$0.295930\pi$
$588$	−2.73205	−0.112668
$589$	−24.3923	−1.00507
$590$	0	0
$591$	−61.1769	−2.51648
$592$	−3.26795	−0.134312
$593$	−32.5359	−1.33609	−0.668045	−	0.744121i	$-0.732868\pi$
$593$	−32.5359	−1.33609	−0.668045	+	0.744121i	$0.732868\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−7.26795	−0.297707
$597$	67.7128	2.77130
$598$	−1.85641	−0.0759141
$599$	−32.1051	−1.31178	−0.655890	−	0.754857i	$-0.727706\pi$
$599$	−32.1051	−1.31178	−0.655890	+	0.754857i	$0.727706\pi$
$600$	0	0
$601$	28.4449	1.16029	0.580145	−	0.814513i	$-0.302996\pi$
$601$	28.4449	1.16029	0.580145	+	0.814513i	$0.302996\pi$
$602$	−4.92820	−0.200859
$603$	48.7846	1.98666
$604$	−7.80385	−0.317534
$605$	0	0
$606$	16.3923	0.665892
$607$	−34.7846	−1.41186	−0.705932	−	0.708280i	$-0.749472\pi$
$607$	−34.7846	−1.41186	−0.705932	+	0.708280i	$0.749472\pi$
$608$	−2.73205	−0.110799
$609$	−12.9282	−0.523877
$610$	0	0
$611$	0	0
$612$	−15.4641	−0.625099
$613$	−27.1769	−1.09767	−0.548833	−	0.835932i	$-0.684928\pi$
$613$	−27.1769	−1.09767	−0.548833	+	0.835932i	$0.684928\pi$
$614$	−31.3205	−1.26399
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−8.53590	−0.343642	−0.171821	−	0.985128i	$-0.554965\pi$
$617$	−8.53590	−0.343642	−0.171821	+	0.985128i	$0.554965\pi$
$618$	28.7846	1.15789
$619$	−10.9282	−0.439242	−0.219621	−	0.975585i	$-0.570482\pi$
$619$	−10.9282	−0.439242	−0.219621	+	0.975585i	$0.570482\pi$
$620$	0	0
$621$	5.07180	0.203524
$622$	7.85641	0.315013
$623$	15.4641	0.619556
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−27.2679	−1.08985
$627$	7.46410	0.298088
$628$	−6.39230	−0.255081
$629$	11.3205	0.451378
$630$	0	0
$631$	−12.7846	−0.508947	−0.254474	−	0.967080i	$-0.581902\pi$
$631$	−12.7846	−0.508947	−0.254474	+	0.967080i	$0.581902\pi$
$632$	6.73205	0.267787
$633$	−21.8564	−0.868714
$634$	32.4449	1.28855
$635$	0	0
$636$	−12.9282	−0.512637
$637$	1.46410	0.0580098
$638$	−4.73205	−0.187344
$639$	42.2487	1.67133
$640$	0	0
$641$	19.8564	0.784281	0.392140	−	0.919905i	$-0.371735\pi$
$641$	19.8564	0.784281	0.392140	+	0.919905i	$0.371735\pi$
$642$	2.53590	0.100084
$643$	14.7321	0.580975	0.290488	−	0.956879i	$-0.406182\pi$
$643$	14.7321	0.580975	0.290488	+	0.956879i	$0.406182\pi$
$644$	1.26795	0.0499642
$645$	0	0
$646$	9.46410	0.372360
$647$	−37.1769	−1.46158	−0.730788	−	0.682605i	$-0.760847\pi$
$647$	−37.1769	−1.46158	−0.730788	+	0.682605i	$0.760847\pi$
$648$	−2.46410	−0.0967991
$649$	−13.8564	−0.543912
$650$	0	0
$651$	24.3923	0.956010
$652$	−8.00000	−0.313304
$653$	9.80385	0.383654	0.191827	−	0.981429i	$-0.438559\pi$
$653$	9.80385	0.383654	0.191827	+	0.981429i	$0.438559\pi$
$654$	4.92820	0.192708
$655$	0	0
$656$	−4.73205	−0.184756
$657$	−51.1769	−1.99660
$658$	0	0
$659$	6.24871	0.243415	0.121708	−	0.992566i	$-0.461163\pi$
$659$	6.24871	0.243415	0.121708	+	0.992566i	$0.461163\pi$
$660$	0	0
$661$	3.85641	0.149997	0.0749984	−	0.997184i	$-0.476105\pi$
$661$	3.85641	0.149997	0.0749984	+	0.997184i	$0.476105\pi$
$662$	−4.92820	−0.191540
$663$	13.8564	0.538138
$664$	4.39230	0.170454
$665$	0	0
$666$	−14.5885	−0.565291
$667$	6.00000	0.232321
$668$	13.8564	0.536120
$669$	−22.9282	−0.886456
$670$	0	0
$671$	2.00000	0.0772091
$672$	2.73205	0.105391
$673$	20.1436	0.776478	0.388239	−	0.921559i	$-0.373083\pi$
$673$	20.1436	0.776478	0.388239	+	0.921559i	$0.373083\pi$
$674$	30.7846	1.18578
$675$	0	0
$676$	−10.8564	−0.417554
$677$	−19.8564	−0.763144	−0.381572	−	0.924339i	$-0.624617\pi$
$677$	−19.8564	−0.763144	−0.381572	+	0.924339i	$0.624617\pi$
$678$	35.3205	1.35648
$679$	5.80385	0.222731
$680$	0	0
$681$	37.8564	1.45066
$682$	8.92820	0.341879
$683$	−42.2487	−1.61660	−0.808301	−	0.588769i	$-0.799613\pi$
$683$	−42.2487	−1.61660	−0.808301	+	0.588769i	$0.799613\pi$
$684$	−12.1962	−0.466332
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	36.7846	1.40342
$688$	4.92820	0.187886
$689$	6.92820	0.263944
$690$	0	0
$691$	31.3205	1.19149	0.595744	−	0.803174i	$-0.296857\pi$
$691$	31.3205	1.19149	0.595744	+	0.803174i	$0.296857\pi$
$692$	−0.928203	−0.0352850
$693$	−4.46410	−0.169577
$694$	−7.85641	−0.298225
$695$	0	0
$696$	12.9282	0.490042
$697$	16.3923	0.620903
$698$	30.3923	1.15037
$699$	21.4641	0.811847
$700$	0	0
$701$	14.1962	0.536181	0.268091	−	0.963394i	$-0.413607\pi$
$701$	14.1962	0.536181	0.268091	+	0.963394i	$0.413607\pi$
$702$	−5.85641	−0.221036
$703$	8.92820	0.336734
$704$	1.00000	0.0376889
$705$	0	0
$706$	32.4449	1.22108
$707$	6.00000	0.225653
$708$	37.8564	1.42273
$709$	−2.39230	−0.0898449	−0.0449224	−	0.998990i	$-0.514304\pi$
$709$	−2.39230	−0.0898449	−0.0449224	+	0.998990i	$0.514304\pi$
$710$	0	0
$711$	30.0526	1.12706
$712$	−15.4641	−0.579542
$713$	−11.3205	−0.423956
$714$	−9.46410	−0.354185
$715$	0	0
$716$	−19.8564	−0.742069
$717$	10.3923	0.388108
$718$	−1.26795	−0.0473194
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	10.5359	0.392377
$722$	−11.5359	−0.429322
$723$	49.7128	1.84884
$724$	−11.8564	−0.440640
$725$	0	0
$726$	−2.73205	−0.101396
$727$	38.6410	1.43312	0.716558	−	0.697528i	$-0.245717\pi$
$727$	38.6410	1.43312	0.716558	+	0.697528i	$0.245717\pi$
$728$	−1.46410	−0.0542632
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−17.0718	−0.631423
$732$	−5.46410	−0.201959
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−29.4641	−1.08754
$735$	0	0
$736$	−1.26795	−0.0467372
$737$	10.9282	0.402546
$738$	−21.1244	−0.777598
$739$	18.1436	0.667423	0.333711	−	0.942675i	$-0.391699\pi$
$739$	18.1436	0.667423	0.333711	+	0.942675i	$0.391699\pi$
$740$	0	0
$741$	10.9282	0.401458
$742$	−4.73205	−0.173719
$743$	−3.71281	−0.136210	−0.0681049	−	0.997678i	$-0.521695\pi$
$743$	−3.71281	−0.136210	−0.0681049	+	0.997678i	$0.521695\pi$
$744$	−24.3923	−0.894265
$745$	0	0
$746$	−30.3923	−1.11274
$747$	19.6077	0.717408
$748$	−3.46410	−0.126660
$749$	0.928203	0.0339158
$750$	0	0
$751$	2.24871	0.0820566	0.0410283	−	0.999158i	$-0.486937\pi$
$751$	2.24871	0.0820566	0.0410283	+	0.999158i	$0.486937\pi$
$752$	0	0
$753$	0	0
$754$	−6.92820	−0.252310
$755$	0	0
$756$	4.00000	0.145479
$757$	−23.3731	−0.849509	−0.424754	−	0.905309i	$-0.639640\pi$
$757$	−23.3731	−0.849509	−0.424754	+	0.905309i	$0.639640\pi$
$758$	23.7128	0.861288
$759$	3.46410	0.125739
$760$	0	0
$761$	19.2679	0.698463	0.349231	−	0.937037i	$-0.386443\pi$
$761$	19.2679	0.698463	0.349231	+	0.937037i	$0.386443\pi$
$762$	26.9282	0.975506
$763$	1.80385	0.0653037
$764$	6.92820	0.250654
$765$	0	0
$766$	−16.3923	−0.592278
$767$	−20.2872	−0.732528
$768$	−2.73205	−0.0985844
$769$	−24.4449	−0.881504	−0.440752	−	0.897629i	$-0.645288\pi$
$769$	−24.4449	−0.881504	−0.440752	+	0.897629i	$0.645288\pi$
$770$	0	0
$771$	62.7846	2.26113
$772$	25.4641	0.916473
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	22.0000	0.790774
$775$	0	0
$776$	−5.80385	−0.208346
$777$	−8.92820	−0.320298
$778$	−24.2487	−0.869358
$779$	12.9282	0.463201
$780$	0	0
$781$	9.46410	0.338652
$782$	4.39230	0.157069
$783$	18.9282	0.676439
$784$	1.00000	0.0357143
$785$	0	0
$786$	48.2487	1.72097
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	22.3923	0.797693
$789$	51.7128	1.84102
$790$	0	0
$791$	12.9282	0.459674
$792$	4.46410	0.158625
$793$	2.92820	0.103984
$794$	38.3923	1.36249
$795$	0	0
$796$	−24.7846	−0.878467
$797$	−6.67949	−0.236600	−0.118300	−	0.992978i	$-0.537744\pi$
$797$	−6.67949	−0.236600	−0.118300	+	0.992978i	$0.537744\pi$
$798$	−7.46410	−0.264226
$799$	0	0
$800$	0	0
$801$	−69.0333	−2.43917
$802$	−25.1769	−0.889028
$803$	−11.4641	−0.404559
$804$	−29.8564	−1.05295
$805$	0	0
$806$	13.0718	0.460434
$807$	−12.0000	−0.422420
$808$	−6.00000	−0.211079
$809$	5.32051	0.187059	0.0935296	−	0.995617i	$-0.470185\pi$
$809$	5.32051	0.187059	0.0935296	+	0.995617i	$0.470185\pi$
$810$	0	0
$811$	−4.58846	−0.161123	−0.0805613	−	0.996750i	$-0.525671\pi$
$811$	−4.58846	−0.161123	−0.0805613	+	0.996750i	$0.525671\pi$
$812$	4.73205	0.166062
$813$	−66.6410	−2.33720
$814$	−3.26795	−0.114542
$815$	0	0
$816$	9.46410	0.331310
$817$	−13.4641	−0.471049
$818$	22.1962	0.776070
$819$	−6.53590	−0.228383
$820$	0	0
$821$	52.0526	1.81665	0.908323	−	0.418269i	$-0.137363\pi$
$821$	52.0526	1.81665	0.908323	+	0.418269i	$0.137363\pi$
$822$	21.4641	0.748647
$823$	24.1962	0.843425	0.421712	−	0.906730i	$-0.361429\pi$
$823$	24.1962	0.843425	0.421712	+	0.906730i	$0.361429\pi$
$824$	−10.5359	−0.367035
$825$	0	0
$826$	13.8564	0.482126
$827$	−53.5692	−1.86278	−0.931392	−	0.364017i	$-0.881405\pi$
$827$	−53.5692	−1.86278	−0.931392	+	0.364017i	$0.881405\pi$
$828$	−5.66025	−0.196707
$829$	40.1051	1.39291	0.696454	−	0.717601i	$-0.254760\pi$
$829$	40.1051	1.39291	0.696454	+	0.717601i	$0.254760\pi$
$830$	0	0
$831$	76.1051	2.64006
$832$	1.46410	0.0507586
$833$	−3.46410	−0.120024
$834$	−11.4641	−0.396969
$835$	0	0
$836$	−2.73205	−0.0944900
$837$	−35.7128	−1.23442
$838$	32.7846	1.13253
$839$	−38.7846	−1.33899	−0.669497	−	0.742815i	$-0.733490\pi$
$839$	−38.7846	−1.33899	−0.669497	+	0.742815i	$0.733490\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	10.7846	0.371662
$843$	28.3923	0.977883
$844$	8.00000	0.275371
$845$	0	0
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	4.73205	0.162499
$849$	−29.8564	−1.02467
$850$	0	0
$851$	4.14359	0.142041
$852$	−25.8564	−0.885826
$853$	−20.9282	−0.716568	−0.358284	−	0.933613i	$-0.616638\pi$
$853$	−20.9282	−0.716568	−0.358284	+	0.933613i	$0.616638\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−0.928203	−0.0317253
$857$	19.1769	0.655071	0.327535	−	0.944839i	$-0.393782\pi$
$857$	19.1769	0.655071	0.327535	+	0.944839i	$0.393782\pi$
$858$	−4.00000	−0.136558
$859$	40.7846	1.39155	0.695776	−	0.718258i	$-0.255060\pi$
$859$	40.7846	1.39155	0.695776	+	0.718258i	$0.255060\pi$
$860$	0	0
$861$	−12.9282	−0.440592
$862$	18.3397	0.624654
$863$	−24.5885	−0.837001	−0.418500	−	0.908217i	$-0.637444\pi$
$863$	−24.5885	−0.837001	−0.418500	+	0.908217i	$0.637444\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	−0.732051	−0.0248761
$867$	13.6603	0.463927
$868$	−8.92820	−0.303043
$869$	6.73205	0.228369
$870$	0	0
$871$	16.0000	0.542139
$872$	−1.80385	−0.0610860
$873$	−25.9090	−0.876886
$874$	3.46410	0.117175
$875$	0	0
$876$	31.3205	1.05822
$877$	23.1769	0.782629	0.391314	−	0.920257i	$-0.372021\pi$
$877$	23.1769	0.782629	0.391314	+	0.920257i	$0.372021\pi$
$878$	36.3923	1.22818
$879$	−6.92820	−0.233682
$880$	0	0
$881$	24.9282	0.839853	0.419926	−	0.907558i	$-0.362056\pi$
$881$	24.9282	0.839853	0.419926	+	0.907558i	$0.362056\pi$
$882$	4.46410	0.150314
$883$	41.1769	1.38571	0.692857	−	0.721075i	$-0.256352\pi$
$883$	41.1769	1.38571	0.692857	+	0.721075i	$0.256352\pi$
$884$	−5.07180	−0.170583
$885$	0	0
$886$	−20.7846	−0.698273
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	8.92820	0.299611
$889$	9.85641	0.330573
$890$	0	0
$891$	−2.46410	−0.0825505
$892$	8.39230	0.280995
$893$	0	0
$894$	19.8564	0.664098
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	5.07180	0.169342
$898$	−35.3205	−1.17866
$899$	−42.2487	−1.40907
$900$	0	0
$901$	−16.3923	−0.546107
$902$	−4.73205	−0.157560
$903$	13.4641	0.448057
$904$	−12.9282	−0.429986
$905$	0	0
$906$	21.3205	0.708326
$907$	15.3205	0.508709	0.254355	−	0.967111i	$-0.418137\pi$
$907$	15.3205	0.508709	0.254355	+	0.967111i	$0.418137\pi$
$908$	−13.8564	−0.459841
$909$	−26.7846	−0.888389
$910$	0	0
$911$	14.5359	0.481596	0.240798	−	0.970575i	$-0.422591\pi$
$911$	14.5359	0.481596	0.240798	+	0.970575i	$0.422591\pi$
$912$	7.46410	0.247161
$913$	4.39230	0.145364
$914$	34.0000	1.12462
$915$	0	0
$916$	−13.4641	−0.444866
$917$	17.6603	0.583193
$918$	13.8564	0.457330
$919$	24.9808	0.824039	0.412020	−	0.911175i	$-0.364824\pi$
$919$	24.9808	0.824039	0.412020	+	0.911175i	$0.364824\pi$
$920$	0	0
$921$	85.5692	2.81960
$922$	29.3205	0.965620
$923$	13.8564	0.456089
$924$	2.73205	0.0898779
$925$	0	0
$926$	−12.9808	−0.426574
$927$	−47.0333	−1.54478
$928$	−4.73205	−0.155337
$929$	26.1051	0.856481	0.428241	−	0.903665i	$-0.359134\pi$
$929$	26.1051	0.856481	0.428241	+	0.903665i	$0.359134\pi$
$930$	0	0
$931$	−2.73205	−0.0895393
$932$	−7.85641	−0.257345
$933$	−21.4641	−0.702703
$934$	−17.6603	−0.577861
$935$	0	0
$936$	6.53590	0.213633
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−10.9282	−0.356818
$939$	74.4974	2.43113
$940$	0	0
$941$	−35.5692	−1.15952	−0.579762	−	0.814786i	$-0.696854\pi$
$941$	−35.5692	−1.15952	−0.579762	+	0.814786i	$0.696854\pi$
$942$	17.4641	0.569011
$943$	6.00000	0.195387
$944$	−13.8564	−0.450988
$945$	0	0
$946$	4.92820	0.160230
$947$	25.1769	0.818140	0.409070	−	0.912503i	$-0.365853\pi$
$947$	25.1769	0.818140	0.409070	+	0.912503i	$0.365853\pi$
$948$	−18.3923	−0.597354
$949$	−16.7846	−0.544851
$950$	0	0
$951$	−88.6410	−2.87438
$952$	3.46410	0.112272
$953$	11.3205	0.366707	0.183354	−	0.983047i	$-0.441305\pi$
$953$	11.3205	0.366707	0.183354	+	0.983047i	$0.441305\pi$
$954$	21.1244	0.683926
$955$	0	0
$956$	−3.80385	−0.123025
$957$	12.9282	0.417909
$958$	13.8564	0.447680
$959$	7.85641	0.253697
$960$	0	0
$961$	48.7128	1.57138
$962$	−4.78461	−0.154262
$963$	−4.14359	−0.133525
$964$	−18.1962	−0.586059
$965$	0	0
$966$	−3.46410	−0.111456
$967$	26.1436	0.840721	0.420361	−	0.907357i	$-0.361904\pi$
$967$	26.1436	0.840721	0.420361	+	0.907357i	$0.361904\pi$
$968$	1.00000	0.0321412
$969$	−25.8564	−0.830627
$970$	0	0
$971$	49.8564	1.59997	0.799984	−	0.600021i	$-0.204841\pi$
$971$	49.8564	1.59997	0.799984	+	0.600021i	$0.204841\pi$
$972$	18.7321	0.600831
$973$	−4.19615	−0.134522
$974$	−4.87564	−0.156226
$975$	0	0
$976$	2.00000	0.0640184
$977$	0.928203	0.0296959	0.0148479	−	0.999890i	$-0.495274\pi$
$977$	0.928203	0.0296959	0.0148479	+	0.999890i	$0.495274\pi$
$978$	21.8564	0.698891
$979$	−15.4641	−0.494235
$980$	0	0
$981$	−8.05256	−0.257098
$982$	42.9282	1.36989
$983$	7.60770	0.242648	0.121324	−	0.992613i	$-0.461286\pi$
$983$	7.60770	0.242648	0.121324	+	0.992613i	$0.461286\pi$
$984$	12.9282	0.412136
$985$	0	0
$986$	16.3923	0.522037
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−6.24871	−0.198697
$990$	0	0
$991$	−22.9282	−0.728338	−0.364169	−	0.931333i	$-0.618647\pi$
$991$	−22.9282	−0.728338	−0.364169	+	0.931333i	$0.618647\pi$
$992$	8.92820	0.283471
$993$	13.4641	0.427270
$994$	−9.46410	−0.300183
$995$	0	0
$996$	−12.0000	−0.380235
$997$	23.8564	0.755540	0.377770	−	0.925899i	$-0.376691\pi$
$997$	23.8564	0.755540	0.377770	+	0.925899i	$0.376691\pi$
$998$	2.00000	0.0633089
$999$	13.0718	0.413573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.bm.1.1		2
5.2	odd	4		3850.2.c.s.1849.4		4
5.3	odd	4		3850.2.c.s.1849.1		4
5.4	even	2		770.2.a.h.1.2	✓	2
15.14	odd	2		6930.2.a.ca.1.1		2
20.19	odd	2		6160.2.a.v.1.1		2
35.34	odd	2		5390.2.a.bk.1.1		2
55.54	odd	2		8470.2.a.ce.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.h.1.2	✓	2	5.4	even	2
3850.2.a.bm.1.1		2	1.1	even	1	trivial
3850.2.c.s.1849.1		4	5.3	odd	4
3850.2.c.s.1849.4		4	5.2	odd	4
5390.2.a.bk.1.1		2	35.34	odd	2
6160.2.a.v.1.1		2	20.19	odd	2
6930.2.a.ca.1.1		2	15.14	odd	2
8470.2.a.ce.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{11} -2.73205 q^{12} +1.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +4.46410 q^{18} -2.73205 q^{19} +2.73205 q^{21} +1.00000 q^{22} -1.26795 q^{23} -2.73205 q^{24} +1.46410 q^{26} -4.00000 q^{27} -1.00000 q^{28} -4.73205 q^{29} +8.92820 q^{31} +1.00000 q^{32} -2.73205 q^{33} -3.46410 q^{34} +4.46410 q^{36} -3.26795 q^{37} -2.73205 q^{38} -4.00000 q^{39} -4.73205 q^{41} +2.73205 q^{42} +4.92820 q^{43} +1.00000 q^{44} -1.26795 q^{46} -2.73205 q^{48} +1.00000 q^{49} +9.46410 q^{51} +1.46410 q^{52} +4.73205 q^{53} -4.00000 q^{54} -1.00000 q^{56} +7.46410 q^{57} -4.73205 q^{58} -13.8564 q^{59} +2.00000 q^{61} +8.92820 q^{62} -4.46410 q^{63} +1.00000 q^{64} -2.73205 q^{66} +10.9282 q^{67} -3.46410 q^{68} +3.46410 q^{69} +9.46410 q^{71} +4.46410 q^{72} -11.4641 q^{73} -3.26795 q^{74} -2.73205 q^{76} -1.00000 q^{77} -4.00000 q^{78} +6.73205 q^{79} -2.46410 q^{81} -4.73205 q^{82} +4.39230 q^{83} +2.73205 q^{84} +4.92820 q^{86} +12.9282 q^{87} +1.00000 q^{88} -15.4641 q^{89} -1.46410 q^{91} -1.26795 q^{92} -24.3923 q^{93} -2.73205 q^{96} -5.80385 q^{97} +1.00000 q^{98} +4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{18} - 2 q^{19} + 2 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} - 4 q^{26} - 8 q^{27} - 2 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{36} - 10 q^{37} - 2 q^{38} - 8 q^{39} - 6 q^{41} + 2 q^{42} - 4 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{48} + 2 q^{49} + 12 q^{51} - 4 q^{52} + 6 q^{53} - 8 q^{54} - 2 q^{56} + 8 q^{57} - 6 q^{58} + 4 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{66} + 8 q^{67} + 12 q^{71} + 2 q^{72} - 16 q^{73} - 10 q^{74} - 2 q^{76} - 2 q^{77} - 8 q^{78} + 10 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} + 2 q^{84} - 4 q^{86} + 12 q^{87} + 2 q^{88} - 24 q^{89} + 4 q^{91} - 6 q^{92} - 28 q^{93} - 2 q^{96} - 22 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^18 - 2 * q^19 + 2 * q^21 + 2 * q^22 - 6 * q^23 - 2 * q^24 - 4 * q^26 - 8 * q^27 - 2 * q^28 - 6 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^33 + 2 * q^36 - 10 * q^37 - 2 * q^38 - 8 * q^39 - 6 * q^41 + 2 * q^42 - 4 * q^43 + 2 * q^44 - 6 * q^46 - 2 * q^48 + 2 * q^49 + 12 * q^51 - 4 * q^52 + 6 * q^53 - 8 * q^54 - 2 * q^56 + 8 * q^57 - 6 * q^58 + 4 * q^61 + 4 * q^62 - 2 * q^63 + 2 * q^64 - 2 * q^66 + 8 * q^67 + 12 * q^71 + 2 * q^72 - 16 * q^73 - 10 * q^74 - 2 * q^76 - 2 * q^77 - 8 * q^78 + 10 * q^79 + 2 * q^81 - 6 * q^82 - 12 * q^83 + 2 * q^84 - 4 * q^86 + 12 * q^87 + 2 * q^88 - 24 * q^89 + 4 * q^91 - 6 * q^92 - 28 * q^93 - 2 * q^96 - 22 * q^97 + 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more