LMFDB - Embedded newform 3850.2.a.ca.1.2

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			1.2
Root			$1.44272$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.63209 q^{3} +1.00000 q^{4} -1.63209 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.336288 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.63209 q^{3} +1.00000 q^{4} -1.63209 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.336288 q^{9} +1.00000 q^{11} -1.63209 q^{12} -2.35465 q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.69877 q^{17} -0.336288 q^{18} -5.40297 q^{19} +1.63209 q^{21} +1.00000 q^{22} -4.65045 q^{23} -1.63209 q^{24} -2.35465 q^{26} +5.44512 q^{27} -1.00000 q^{28} +1.38627 q^{29} +5.33629 q^{31} +1.00000 q^{32} -1.63209 q^{33} +5.69877 q^{34} -0.336288 q^{36} -1.04832 q^{37} -5.40297 q^{38} +3.84299 q^{39} +9.98673 q^{41} +1.63209 q^{42} -0.265838 q^{43} +1.00000 q^{44} -4.65045 q^{46} -9.07177 q^{47} -1.63209 q^{48} +1.00000 q^{49} -9.30089 q^{51} -2.35465 q^{52} -8.31250 q^{53} +5.44512 q^{54} -1.00000 q^{56} +8.81812 q^{57} +1.38627 q^{58} +6.58377 q^{59} +8.71713 q^{61} +5.33629 q^{62} +0.336288 q^{63} +1.00000 q^{64} -1.63209 q^{66} -2.43459 q^{67} +5.69877 q^{68} +7.58994 q^{69} +5.66205 q^{71} -0.336288 q^{72} +6.43459 q^{73} -1.04832 q^{74} -5.40297 q^{76} -1.00000 q^{77} +3.84299 q^{78} +11.7943 q^{79} -7.87805 q^{81} +9.98673 q^{82} +2.28254 q^{83} +1.63209 q^{84} -0.265838 q^{86} -2.26252 q^{87} +1.00000 q^{88} +3.22746 q^{89} +2.35465 q^{91} -4.65045 q^{92} -8.70929 q^{93} -9.07177 q^{94} -1.63209 q^{96} +15.8543 q^{97} +1.00000 q^{98} -0.336288 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.63209	−0.942287	−0.471143	−	0.882057i	$-0.656159\pi$
$3$	−1.63209	−0.942287	−0.471143	+	0.882057i	$0.656159\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.63209	−0.666297
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−0.336288	−0.112096
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.63209	−0.471143
$13$	−2.35465	−0.653061	−0.326531	−	0.945187i	$-0.605880\pi$
$13$	−2.35465	−0.653061	−0.326531	+	0.945187i	$0.605880\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.69877	1.38215	0.691077	−	0.722781i	$-0.257136\pi$
$17$	5.69877	1.38215	0.691077	+	0.722781i	$0.257136\pi$
$18$	−0.336288	−0.0792638
$19$	−5.40297	−1.23953	−0.619763	−	0.784789i	$-0.712771\pi$
$19$	−5.40297	−1.23953	−0.619763	+	0.784789i	$0.712771\pi$
$20$	0	0
$21$	1.63209	0.356151
$22$	1.00000	0.213201
$23$	−4.65045	−0.969685	−0.484843	−	0.874601i	$-0.661123\pi$
$23$	−4.65045	−0.969685	−0.484843	+	0.874601i	$0.661123\pi$
$24$	−1.63209	−0.333149
$25$	0	0
$26$	−2.35465	−0.461784
$27$	5.44512	1.04791
$28$	−1.00000	−0.188982
$29$	1.38627	0.257424	0.128712	−	0.991682i	$-0.458916\pi$
$29$	1.38627	0.257424	0.128712	+	0.991682i	$0.458916\pi$
$30$	0	0
$31$	5.33629	0.958426	0.479213	−	0.877699i	$-0.340922\pi$
$31$	5.33629	0.958426	0.479213	+	0.877699i	$0.340922\pi$
$32$	1.00000	0.176777
$33$	−1.63209	−0.284110
$34$	5.69877	0.977331
$35$	0	0
$36$	−0.336288	−0.0560479
$37$	−1.04832	−0.172343	−0.0861715	−	0.996280i	$-0.527463\pi$
$37$	−1.04832	−0.172343	−0.0861715	+	0.996280i	$0.527463\pi$
$38$	−5.40297	−0.876477
$39$	3.84299	0.615371
$40$	0	0
$41$	9.98673	1.55967	0.779833	−	0.625988i	$-0.215304\pi$
$41$	9.98673	1.55967	0.779833	+	0.625988i	$0.215304\pi$
$42$	1.63209	0.251837
$43$	−0.265838	−0.0405398	−0.0202699	−	0.999795i	$-0.506453\pi$
$43$	−0.265838	−0.0405398	−0.0202699	+	0.999795i	$0.506453\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.65045	−0.685671
$47$	−9.07177	−1.32325	−0.661627	−	0.749833i	$-0.730134\pi$
$47$	−9.07177	−1.32325	−0.661627	+	0.749833i	$0.730134\pi$
$48$	−1.63209	−0.235572
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.30089	−1.30239
$52$	−2.35465	−0.326531
$53$	−8.31250	−1.14181	−0.570905	−	0.821016i	$-0.693407\pi$
$53$	−8.31250	−1.14181	−0.570905	+	0.821016i	$0.693407\pi$
$54$	5.44512	0.740986
$55$	0	0
$56$	−1.00000	−0.133631
$57$	8.81812	1.16799
$58$	1.38627	0.182026
$59$	6.58377	0.857133	0.428567	−	0.903510i	$-0.359019\pi$
$59$	6.58377	0.857133	0.428567	+	0.903510i	$0.359019\pi$
$60$	0	0
$61$	8.71713	1.11611	0.558057	−	0.829803i	$-0.311547\pi$
$61$	8.71713	1.11611	0.558057	+	0.829803i	$0.311547\pi$
$62$	5.33629	0.677709
$63$	0.336288	0.0423683
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.63209	−0.200896
$67$	−2.43459	−0.297433	−0.148716	−	0.988880i	$-0.547514\pi$
$67$	−2.43459	−0.297433	−0.148716	+	0.988880i	$0.547514\pi$
$68$	5.69877	0.691077
$69$	7.58994	0.913721
$70$	0	0
$71$	5.66205	0.671962	0.335981	−	0.941869i	$-0.390932\pi$
$71$	5.66205	0.671962	0.335981	+	0.941869i	$0.390932\pi$
$72$	−0.336288	−0.0396319
$73$	6.43459	0.753112	0.376556	−	0.926394i	$-0.377108\pi$
$73$	6.43459	0.753112	0.376556	+	0.926394i	$0.377108\pi$
$74$	−1.04832	−0.121865
$75$	0	0
$76$	−5.40297	−0.619763
$77$	−1.00000	−0.113961
$78$	3.84299	0.435133
$79$	11.7943	1.32697	0.663483	−	0.748191i	$-0.269077\pi$
$79$	11.7943	1.32697	0.663483	+	0.748191i	$0.269077\pi$
$80$	0	0
$81$	−7.87805	−0.875339
$82$	9.98673	1.10285
$83$	2.28254	0.250541	0.125270	−	0.992123i	$-0.460020\pi$
$83$	2.28254	0.250541	0.125270	+	0.992123i	$0.460020\pi$
$84$	1.63209	0.178075
$85$	0	0
$86$	−0.265838	−0.0286660
$87$	−2.26252	−0.242567
$88$	1.00000	0.106600
$89$	3.22746	0.342110	0.171055	−	0.985261i	$-0.445282\pi$
$89$	3.22746	0.342110	0.171055	+	0.985261i	$0.445282\pi$
$90$	0	0
$91$	2.35465	0.246834
$92$	−4.65045	−0.484843
$93$	−8.70929	−0.903112
$94$	−9.07177	−0.935682
$95$	0	0
$96$	−1.63209	−0.166574
$97$	15.8543	1.60976	0.804878	−	0.593440i	$-0.202231\pi$
$97$	15.8543	1.60976	0.804878	+	0.593440i	$0.202231\pi$
$98$	1.00000	0.101015
$99$	−0.336288	−0.0337982
$100$	0	0
$101$	6.52052	0.648816	0.324408	−	0.945917i	$-0.394835\pi$
$101$	6.52052	0.648816	0.324408	+	0.945917i	$0.394835\pi$
$102$	−9.30089	−0.920926
$103$	10.9751	1.08141	0.540706	−	0.841212i	$-0.318157\pi$
$103$	10.9751	1.08141	0.540706	+	0.841212i	$0.318157\pi$
$104$	−2.35465	−0.230892
$105$	0	0
$106$	−8.31250	−0.807381
$107$	−0.410060	−0.0396420	−0.0198210	−	0.999804i	$-0.506310\pi$
$107$	−0.410060	−0.0396420	−0.0198210	+	0.999804i	$0.506310\pi$
$108$	5.44512	0.523957
$109$	5.87791	0.563001	0.281501	−	0.959561i	$-0.409168\pi$
$109$	5.87791	0.563001	0.281501	+	0.959561i	$0.409168\pi$
$110$	0	0
$111$	1.71095	0.162397
$112$	−1.00000	−0.0944911
$113$	−17.6355	−1.65901	−0.829505	−	0.558499i	$-0.811377\pi$
$113$	−17.6355	−1.65901	−0.829505	+	0.558499i	$0.811377\pi$
$114$	8.81812	0.825893
$115$	0	0
$116$	1.38627	0.128712
$117$	0.791839	0.0732055
$118$	6.58377	0.606085
$119$	−5.69877	−0.522405
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.71713	0.789212
$123$	−16.2992	−1.46965
$124$	5.33629	0.479213
$125$	0	0
$126$	0.336288	0.0299589
$127$	−7.04724	−0.625342	−0.312671	−	0.949862i	$-0.601224\pi$
$127$	−7.04724	−0.625342	−0.312671	+	0.949862i	$0.601224\pi$
$128$	1.00000	0.0883883
$129$	0.433870	0.0382002
$130$	0	0
$131$	−8.17551	−0.714298	−0.357149	−	0.934048i	$-0.616251\pi$
$131$	−8.17551	−0.714298	−0.357149	+	0.934048i	$0.616251\pi$
$132$	−1.63209	−0.142055
$133$	5.40297	0.468497
$134$	−2.43459	−0.210317
$135$	0	0
$136$	5.69877	0.488665
$137$	−0.0734349	−0.00627397	−0.00313698	−	0.999995i	$-0.500999\pi$
$137$	−0.0734349	−0.00627397	−0.00313698	+	0.999995i	$0.500999\pi$
$138$	7.58994	0.646099
$139$	−12.0313	−1.02048	−0.510240	−	0.860032i	$-0.670444\pi$
$139$	−12.0313	−1.02048	−0.510240	+	0.860032i	$0.670444\pi$
$140$	0	0
$141$	14.8059	1.24688
$142$	5.66205	0.475149
$143$	−2.35465	−0.196905
$144$	−0.336288	−0.0280240
$145$	0	0
$146$	6.43459	0.532531
$147$	−1.63209	−0.134612
$148$	−1.04832	−0.0861715
$149$	13.3863	1.09665	0.548323	−	0.836267i	$-0.315266\pi$
$149$	13.3863	1.09665	0.548323	+	0.836267i	$0.315266\pi$
$150$	0	0
$151$	12.7206	1.03518	0.517592	−	0.855627i	$-0.326828\pi$
$151$	12.7206	1.03518	0.517592	+	0.855627i	$0.326828\pi$
$152$	−5.40297	−0.438239
$153$	−1.91643	−0.154934
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	3.84299	0.307686
$157$	−8.25468	−0.658795	−0.329398	−	0.944191i	$-0.606846\pi$
$157$	−8.25468	−0.658795	−0.329398	+	0.944191i	$0.606846\pi$
$158$	11.7943	0.938307
$159$	13.5667	1.07591
$160$	0	0
$161$	4.65045	0.366507
$162$	−7.87805	−0.618958
$163$	11.7090	0.917116	0.458558	−	0.888664i	$-0.348366\pi$
$163$	11.7090	0.917116	0.458558	+	0.888664i	$0.348366\pi$
$164$	9.98673	0.779833
$165$	0	0
$166$	2.28254	0.177159
$167$	8.56507	0.662785	0.331393	−	0.943493i	$-0.392482\pi$
$167$	8.56507	0.662785	0.331393	+	0.943493i	$0.392482\pi$
$168$	1.63209	0.125918
$169$	−7.45564	−0.573511
$170$	0	0
$171$	1.81695	0.138946
$172$	−0.265838	−0.0202699
$173$	12.5572	0.954709	0.477354	−	0.878711i	$-0.341596\pi$
$173$	12.5572	0.954709	0.477354	+	0.878711i	$0.341596\pi$
$174$	−2.26252	−0.171521
$175$	0	0
$176$	1.00000	0.0753778
$177$	−10.7453	−0.807665
$178$	3.22746	0.241908
$179$	25.0928	1.87553	0.937763	−	0.347276i	$-0.112893\pi$
$179$	25.0928	1.87553	0.937763	+	0.347276i	$0.112893\pi$
$180$	0	0
$181$	19.2327	1.42955	0.714777	−	0.699352i	$-0.246528\pi$
$181$	19.2327	1.42955	0.714777	+	0.699352i	$0.246528\pi$
$182$	2.35465	0.174538
$183$	−14.2271	−1.05170
$184$	−4.65045	−0.342836
$185$	0	0
$186$	−8.70929	−0.638596
$187$	5.69877	0.416735
$188$	−9.07177	−0.661627
$189$	−5.44512	−0.396074
$190$	0	0
$191$	−6.21145	−0.449445	−0.224722	−	0.974423i	$-0.572148\pi$
$191$	−6.21145	−0.449445	−0.224722	+	0.974423i	$0.572148\pi$
$192$	−1.63209	−0.117786
$193$	26.2727	1.89115	0.945575	−	0.325403i	$-0.105500\pi$
$193$	26.2727	1.89115	0.945575	+	0.325403i	$0.105500\pi$
$194$	15.8543	1.13827
$195$	0	0
$196$	1.00000	0.0714286
$197$	1.34847	0.0960747	0.0480373	−	0.998846i	$-0.484703\pi$
$197$	1.34847	0.0960747	0.0480373	+	0.998846i	$0.484703\pi$
$198$	−0.336288	−0.0238989
$199$	5.80627	0.411596	0.205798	−	0.978595i	$-0.434021\pi$
$199$	5.80627	0.411596	0.205798	+	0.978595i	$0.434021\pi$
$200$	0	0
$201$	3.97347	0.280267
$202$	6.52052	0.458782
$203$	−1.38627	−0.0972971
$204$	−9.30089	−0.651193
$205$	0	0
$206$	10.9751	0.764674
$207$	1.56389	0.108698
$208$	−2.35465	−0.163265
$209$	−5.40297	−0.373731
$210$	0	0
$211$	−17.5785	−1.21015	−0.605076	−	0.796168i	$-0.706857\pi$
$211$	−17.5785	−1.21015	−0.605076	+	0.796168i	$0.706857\pi$
$212$	−8.31250	−0.570905
$213$	−9.24097	−0.633181
$214$	−0.410060	−0.0280311
$215$	0	0
$216$	5.44512	0.370493
$217$	−5.33629	−0.362251
$218$	5.87791	0.398102
$219$	−10.5018	−0.709647
$220$	0	0
$221$	−13.4186	−0.902632
$222$	1.71095	0.114832
$223$	23.1527	1.55042	0.775211	−	0.631702i	$-0.217643\pi$
$223$	23.1527	1.55042	0.775211	+	0.631702i	$0.217643\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−17.6355	−1.17310
$227$	−22.5336	−1.49561	−0.747803	−	0.663920i	$-0.768891\pi$
$227$	−22.5336	−1.49561	−0.747803	+	0.663920i	$0.768891\pi$
$228$	8.81812	0.583994
$229$	−2.64502	−0.174788	−0.0873938	−	0.996174i	$-0.527854\pi$
$229$	−2.64502	−0.174788	−0.0873938	+	0.996174i	$0.527854\pi$
$230$	0	0
$231$	1.63209	0.107384
$232$	1.38627	0.0910131
$233$	−11.7463	−0.769529	−0.384764	−	0.923015i	$-0.625717\pi$
$233$	−11.7463	−0.769529	−0.384764	+	0.923015i	$0.625717\pi$
$234$	0.791839	0.0517641
$235$	0	0
$236$	6.58377	0.428567
$237$	−19.2494	−1.25038
$238$	−5.69877	−0.369396
$239$	−2.79467	−0.180772	−0.0903861	−	0.995907i	$-0.528810\pi$
$239$	−2.79467	−0.180772	−0.0903861	+	0.995907i	$0.528810\pi$
$240$	0	0
$241$	16.7327	1.07785	0.538925	−	0.842353i	$-0.318830\pi$
$241$	16.7327	1.07785	0.538925	+	0.842353i	$0.318830\pi$
$242$	1.00000	0.0642824
$243$	−3.47768	−0.223093
$244$	8.71713	0.558057
$245$	0	0
$246$	−16.2992	−1.03920
$247$	12.7221	0.809487
$248$	5.33629	0.338855
$249$	−3.72530	−0.236081
$250$	0	0
$251$	−22.6063	−1.42690	−0.713449	−	0.700708i	$-0.752868\pi$
$251$	−22.6063	−1.42690	−0.713449	+	0.700708i	$0.752868\pi$
$252$	0.336288	0.0211841
$253$	−4.65045	−0.292371
$254$	−7.04724	−0.442183
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.3459	1.26914	0.634571	−	0.772865i	$-0.281177\pi$
$257$	20.3459	1.26914	0.634571	+	0.772865i	$0.281177\pi$
$258$	0.433870	0.0270116
$259$	1.04832	0.0651395
$260$	0	0
$261$	−0.466186	−0.0288562
$262$	−8.17551	−0.505085
$263$	23.2268	1.43222	0.716112	−	0.697985i	$-0.245920\pi$
$263$	23.2268	1.43222	0.716112	+	0.697985i	$0.245920\pi$
$264$	−1.63209	−0.100448
$265$	0	0
$266$	5.40297	0.331277
$267$	−5.26750	−0.322366
$268$	−2.43459	−0.148716
$269$	32.2813	1.96822	0.984112	−	0.177551i	$-0.0568175\pi$
$269$	32.2813	1.96822	0.984112	+	0.177551i	$0.0568175\pi$
$270$	0	0
$271$	1.15667	0.0702628	0.0351314	−	0.999383i	$-0.488815\pi$
$271$	1.15667	0.0702628	0.0351314	+	0.999383i	$0.488815\pi$
$272$	5.69877	0.345539
$273$	−3.84299	−0.232588
$274$	−0.0734349	−0.00443637
$275$	0	0
$276$	7.58994	0.456861
$277$	−14.2776	−0.857857	−0.428928	−	0.903338i	$-0.641109\pi$
$277$	−14.2776	−0.857857	−0.428928	+	0.903338i	$0.641109\pi$
$278$	−12.0313	−0.721589
$279$	−1.79453	−0.107436
$280$	0	0
$281$	−9.97347	−0.594967	−0.297484	−	0.954727i	$-0.596147\pi$
$281$	−9.97347	−0.594967	−0.297484	+	0.954727i	$0.596147\pi$
$282$	14.8059	0.881681
$283$	1.41335	0.0840150	0.0420075	−	0.999117i	$-0.486625\pi$
$283$	1.41335	0.0840150	0.0420075	+	0.999117i	$0.486625\pi$
$284$	5.66205	0.335981
$285$	0	0
$286$	−2.35465	−0.139233
$287$	−9.98673	−0.589498
$288$	−0.336288	−0.0198159
$289$	15.4760	0.910351
$290$	0	0
$291$	−25.8755	−1.51685
$292$	6.43459	0.376556
$293$	30.4615	1.77958	0.889789	−	0.456372i	$-0.150851\pi$
$293$	30.4615	1.77958	0.889789	+	0.456372i	$0.150851\pi$
$294$	−1.63209	−0.0951853
$295$	0	0
$296$	−1.04832	−0.0609325
$297$	5.44512	0.315958
$298$	13.3863	0.775446
$299$	10.9502	0.633264
$300$	0	0
$301$	0.265838	0.0153226
$302$	12.7206	0.731986
$303$	−10.6421	−0.611371
$304$	−5.40297	−0.309881
$305$	0	0
$306$	−1.91643	−0.109555
$307$	−10.5601	−0.602698	−0.301349	−	0.953514i	$-0.597437\pi$
$307$	−10.5601	−0.602698	−0.301349	+	0.953514i	$0.597437\pi$
$308$	−1.00000	−0.0569803
$309$	−17.9124	−1.01900
$310$	0	0
$311$	14.4795	0.821054	0.410527	−	0.911848i	$-0.365345\pi$
$311$	14.4795	0.821054	0.410527	+	0.911848i	$0.365345\pi$
$312$	3.84299	0.217567
$313$	4.16828	0.235605	0.117802	−	0.993037i	$-0.462415\pi$
$313$	4.16828	0.235605	0.117802	+	0.993037i	$0.462415\pi$
$314$	−8.25468	−0.465839
$315$	0	0
$316$	11.7943	0.663483
$317$	−9.19254	−0.516305	−0.258152	−	0.966104i	$-0.583114\pi$
$317$	−9.19254	−0.516305	−0.258152	+	0.966104i	$0.583114\pi$
$318$	13.5667	0.760785
$319$	1.38627	0.0776162
$320$	0	0
$321$	0.669254	0.0373541
$322$	4.65045	0.259159
$323$	−30.7903	−1.71322
$324$	−7.87805	−0.437669
$325$	0	0
$326$	11.7090	0.648499
$327$	−9.59326	−0.530509
$328$	9.98673	0.551425
$329$	9.07177	0.500143
$330$	0	0
$331$	−5.98493	−0.328962	−0.164481	−	0.986380i	$-0.552595\pi$
$331$	−5.98493	−0.328962	−0.164481	+	0.986380i	$0.552595\pi$
$332$	2.28254	0.125270
$333$	0.352538	0.0193189
$334$	8.56507	0.468660
$335$	0	0
$336$	1.63209	0.0890377
$337$	32.2878	1.75883	0.879413	−	0.476059i	$-0.157935\pi$
$337$	32.2878	1.75883	0.879413	+	0.476059i	$0.157935\pi$
$338$	−7.45564	−0.405533
$339$	28.7827	1.56326
$340$	0	0
$341$	5.33629	0.288976
$342$	1.81695	0.0982495
$343$	−1.00000	−0.0539949
$344$	−0.265838	−0.0143330
$345$	0	0
$346$	12.5572	0.675081
$347$	8.13934	0.436943	0.218471	−	0.975843i	$-0.429893\pi$
$347$	8.13934	0.436943	0.218471	+	0.975843i	$0.429893\pi$
$348$	−2.26252	−0.121284
$349$	−27.3788	−1.46556	−0.732778	−	0.680468i	$-0.761777\pi$
$349$	−27.3788	−1.46556	−0.732778	+	0.680468i	$0.761777\pi$
$350$	0	0
$351$	−12.8213	−0.684352
$352$	1.00000	0.0533002
$353$	−36.3194	−1.93308	−0.966542	−	0.256507i	$-0.917428\pi$
$353$	−36.3194	−1.93308	−0.966542	+	0.256507i	$0.917428\pi$
$354$	−10.7453	−0.571106
$355$	0	0
$356$	3.22746	0.171055
$357$	9.30089	0.492256
$358$	25.0928	1.32620
$359$	−24.6940	−1.30330	−0.651651	−	0.758519i	$-0.725923\pi$
$359$	−24.6940	−1.30330	−0.651651	+	0.758519i	$0.725923\pi$
$360$	0	0
$361$	10.1921	0.536425
$362$	19.2327	1.01085
$363$	−1.63209	−0.0856624
$364$	2.35465	0.123417
$365$	0	0
$366$	−14.2271	−0.743664
$367$	−16.4067	−0.856425	−0.428212	−	0.903678i	$-0.640856\pi$
$367$	−16.4067	−0.856425	−0.428212	+	0.903678i	$0.640856\pi$
$368$	−4.65045	−0.242421
$369$	−3.35842	−0.174832
$370$	0	0
$371$	8.31250	0.431563
$372$	−8.70929	−0.451556
$373$	1.13082	0.0585514	0.0292757	−	0.999571i	$-0.490680\pi$
$373$	1.13082	0.0585514	0.0292757	+	0.999571i	$0.490680\pi$
$374$	5.69877	0.294676
$375$	0	0
$376$	−9.07177	−0.467841
$377$	−3.26418	−0.168114
$378$	−5.44512	−0.280067
$379$	5.57848	0.286547	0.143274	−	0.989683i	$-0.454237\pi$
$379$	5.57848	0.286547	0.143274	+	0.989683i	$0.454237\pi$
$380$	0	0
$381$	11.5017	0.589251
$382$	−6.21145	−0.317806
$383$	−28.3203	−1.44710	−0.723549	−	0.690273i	$-0.757491\pi$
$383$	−28.3203	−1.44710	−0.723549	+	0.690273i	$0.757491\pi$
$384$	−1.63209	−0.0832872
$385$	0	0
$386$	26.2727	1.33725
$387$	0.0893979	0.00454435
$388$	15.8543	0.804878
$389$	13.0934	0.663863	0.331931	−	0.943304i	$-0.392300\pi$
$389$	13.0934	0.663863	0.331931	+	0.943304i	$0.392300\pi$
$390$	0	0
$391$	−26.5018	−1.34025
$392$	1.00000	0.0505076
$393$	13.3432	0.673073
$394$	1.34847	0.0679351
$395$	0	0
$396$	−0.336288	−0.0168991
$397$	−6.33144	−0.317766	−0.158883	−	0.987297i	$-0.550789\pi$
$397$	−6.33144	−0.317766	−0.158883	+	0.987297i	$0.550789\pi$
$398$	5.80627	0.291042
$399$	−8.81812	−0.441458
$400$	0	0
$401$	−32.8774	−1.64182	−0.820909	−	0.571059i	$-0.806533\pi$
$401$	−32.8774	−1.64182	−0.820909	+	0.571059i	$0.806533\pi$
$402$	3.97347	0.198179
$403$	−12.5651	−0.625911
$404$	6.52052	0.324408
$405$	0	0
$406$	−1.38627	−0.0687994
$407$	−1.04832	−0.0519634
$408$	−9.30089	−0.460463
$409$	−30.9736	−1.53155	−0.765773	−	0.643111i	$-0.777644\pi$
$409$	−30.9736	−1.53155	−0.765773	+	0.643111i	$0.777644\pi$
$410$	0	0
$411$	0.119852	0.00591188
$412$	10.9751	0.540706
$413$	−6.58377	−0.323966
$414$	1.56389	0.0768609
$415$	0	0
$416$	−2.35465	−0.115446
$417$	19.6361	0.961585
$418$	−5.40297	−0.264268
$419$	−7.91806	−0.386822	−0.193411	−	0.981118i	$-0.561955\pi$
$419$	−7.91806	−0.386822	−0.193411	+	0.981118i	$0.561955\pi$
$420$	0	0
$421$	−2.92911	−0.142756	−0.0713781	−	0.997449i	$-0.522740\pi$
$421$	−2.92911	−0.142756	−0.0713781	+	0.997449i	$0.522740\pi$
$422$	−17.5785	−0.855707
$423$	3.05073	0.148331
$424$	−8.31250	−0.403691
$425$	0	0
$426$	−9.24097	−0.447726
$427$	−8.71713	−0.421851
$428$	−0.410060	−0.0198210
$429$	3.84299	0.185541
$430$	0	0
$431$	5.24959	0.252864	0.126432	−	0.991975i	$-0.459647\pi$
$431$	5.24959	0.252864	0.126432	+	0.991975i	$0.459647\pi$
$432$	5.44512	0.261978
$433$	−21.7801	−1.04669	−0.523344	−	0.852122i	$-0.675316\pi$
$433$	−21.7801	−1.04669	−0.523344	+	0.852122i	$0.675316\pi$
$434$	−5.33629	−0.256150
$435$	0	0
$436$	5.87791	0.281501
$437$	25.1262	1.20195
$438$	−10.5018	−0.501797
$439$	30.5509	1.45811	0.729057	−	0.684453i	$-0.239959\pi$
$439$	30.5509	1.45811	0.729057	+	0.684453i	$0.239959\pi$
$440$	0	0
$441$	−0.336288	−0.0160137
$442$	−13.4186	−0.638257
$443$	20.6359	0.980441	0.490221	−	0.871598i	$-0.336916\pi$
$443$	20.6359	0.980441	0.490221	+	0.871598i	$0.336916\pi$
$444$	1.71095	0.0811983
$445$	0	0
$446$	23.1527	1.09631
$447$	−21.8476	−1.03335
$448$	−1.00000	−0.0472456
$449$	36.7803	1.73577	0.867885	−	0.496766i	$-0.165479\pi$
$449$	36.7803	1.73577	0.867885	+	0.496766i	$0.165479\pi$
$450$	0	0
$451$	9.98673	0.470257
$452$	−17.6355	−0.829505
$453$	−20.7611	−0.975441
$454$	−22.5336	−1.05755
$455$	0	0
$456$	8.81812	0.412946
$457$	−34.3402	−1.60636	−0.803182	−	0.595733i	$-0.796862\pi$
$457$	−34.3402	−1.60636	−0.803182	+	0.595733i	$0.796862\pi$
$458$	−2.64502	−0.123594
$459$	31.0305	1.44838
$460$	0	0
$461$	16.2854	0.758488	0.379244	−	0.925297i	$-0.376184\pi$
$461$	16.2854	0.758488	0.379244	+	0.925297i	$0.376184\pi$
$462$	1.63209	0.0759316
$463$	−42.2329	−1.96273	−0.981365	−	0.192154i	$-0.938453\pi$
$463$	−42.2329	−1.96273	−0.981365	+	0.192154i	$0.938453\pi$
$464$	1.38627	0.0643560
$465$	0	0
$466$	−11.7463	−0.544139
$467$	19.5266	0.903585	0.451793	−	0.892123i	$-0.350785\pi$
$467$	19.5266	0.903585	0.451793	+	0.892123i	$0.350785\pi$
$468$	0.791839	0.0366028
$469$	2.43459	0.112419
$470$	0	0
$471$	13.4724	0.620774
$472$	6.58377	0.303042
$473$	−0.265838	−0.0122232
$474$	−19.2494	−0.884154
$475$	0	0
$476$	−5.69877	−0.261203
$477$	2.79539	0.127992
$478$	−2.79467	−0.127825
$479$	−3.23765	−0.147932	−0.0739659	−	0.997261i	$-0.523566\pi$
$479$	−3.23765	−0.147932	−0.0739659	+	0.997261i	$0.523566\pi$
$480$	0	0
$481$	2.46843	0.112551
$482$	16.7327	0.762156
$483$	−7.58994	−0.345354
$484$	1.00000	0.0454545
$485$	0	0
$486$	−3.47768	−0.157751
$487$	0.532755	0.0241415	0.0120707	−	0.999927i	$-0.496158\pi$
$487$	0.532755	0.0241415	0.0120707	+	0.999927i	$0.496158\pi$
$488$	8.71713	0.394606
$489$	−19.1101	−0.864186
$490$	0	0
$491$	−23.2710	−1.05021	−0.525104	−	0.851038i	$-0.675973\pi$
$491$	−23.2710	−1.05021	−0.525104	+	0.851038i	$0.675973\pi$
$492$	−16.2992	−0.734826
$493$	7.90003	0.355800
$494$	12.7221	0.572393
$495$	0	0
$496$	5.33629	0.239606
$497$	−5.66205	−0.253978
$498$	−3.72530	−0.166935
$499$	−11.4900	−0.514362	−0.257181	−	0.966363i	$-0.582794\pi$
$499$	−11.4900	−0.514362	−0.257181	+	0.966363i	$0.582794\pi$
$500$	0	0
$501$	−13.9790	−0.624534
$502$	−22.6063	−1.00897
$503$	40.0668	1.78649	0.893245	−	0.449570i	$-0.148423\pi$
$503$	40.0668	1.78649	0.893245	+	0.449570i	$0.148423\pi$
$504$	0.336288	0.0149794
$505$	0	0
$506$	−4.65045	−0.206738
$507$	12.1683	0.540412
$508$	−7.04724	−0.312671
$509$	31.5801	1.39976	0.699882	−	0.714259i	$-0.253236\pi$
$509$	31.5801	1.39976	0.699882	+	0.714259i	$0.253236\pi$
$510$	0	0
$511$	−6.43459	−0.284650
$512$	1.00000	0.0441942
$513$	−29.4198	−1.29892
$514$	20.3459	0.897419
$515$	0	0
$516$	0.433870	0.0191001
$517$	−9.07177	−0.398976
$518$	1.04832	0.0460606
$519$	−20.4945	−0.899610
$520$	0	0
$521$	26.3651	1.15508	0.577539	−	0.816363i	$-0.304013\pi$
$521$	26.3651	1.15508	0.577539	+	0.816363i	$0.304013\pi$
$522$	−0.466186	−0.0204044
$523$	−8.44997	−0.369491	−0.184746	−	0.982786i	$-0.559146\pi$
$523$	−8.44997	−0.369491	−0.184746	+	0.982786i	$0.559146\pi$
$524$	−8.17551	−0.357149
$525$	0	0
$526$	23.2268	1.01274
$527$	30.4103	1.32469
$528$	−1.63209	−0.0710275
$529$	−1.37334	−0.0597105
$530$	0	0
$531$	−2.21404	−0.0960811
$532$	5.40297	0.234248
$533$	−23.5152	−1.01856
$534$	−5.26750	−0.227947
$535$	0	0
$536$	−2.43459	−0.105158
$537$	−40.9537	−1.76728
$538$	32.2813	1.39174
$539$	1.00000	0.0430730
$540$	0	0
$541$	−35.8459	−1.54114	−0.770568	−	0.637358i	$-0.780027\pi$
$541$	−35.8459	−1.54114	−0.770568	+	0.637358i	$0.780027\pi$
$542$	1.15667	0.0496833
$543$	−31.3895	−1.34705
$544$	5.69877	0.244333
$545$	0	0
$546$	−3.84299	−0.164465
$547$	−26.3586	−1.12701	−0.563505	−	0.826112i	$-0.690548\pi$
$547$	−26.3586	−1.12701	−0.563505	+	0.826112i	$0.690548\pi$
$548$	−0.0734349	−0.00313698
$549$	−2.93146	−0.125112
$550$	0	0
$551$	−7.48997	−0.319084
$552$	7.58994	0.323049
$553$	−11.7943	−0.501546
$554$	−14.2776	−0.606596
$555$	0	0
$556$	−12.0313	−0.510240
$557$	−13.3710	−0.566548	−0.283274	−	0.959039i	$-0.591421\pi$
$557$	−13.3710	−0.566548	−0.283274	+	0.959039i	$0.591421\pi$
$558$	−1.79453	−0.0759684
$559$	0.625953	0.0264750
$560$	0	0
$561$	−9.30089	−0.392684
$562$	−9.97347	−0.420705
$563$	−47.0144	−1.98142	−0.990710	−	0.135995i	$-0.956577\pi$
$563$	−47.0144	−1.98142	−0.990710	+	0.135995i	$0.956577\pi$
$564$	14.8059	0.623442
$565$	0	0
$566$	1.41335	0.0594076
$567$	7.87805	0.330847
$568$	5.66205	0.237574
$569$	11.7220	0.491411	0.245705	−	0.969345i	$-0.420980\pi$
$569$	11.7220	0.491411	0.245705	+	0.969345i	$0.420980\pi$
$570$	0	0
$571$	27.3710	1.14544	0.572721	−	0.819751i	$-0.305888\pi$
$571$	27.3710	1.14544	0.572721	+	0.819751i	$0.305888\pi$
$572$	−2.35465	−0.0984527
$573$	10.1376	0.423506
$574$	−9.98673	−0.416838
$575$	0	0
$576$	−0.336288	−0.0140120
$577$	29.8910	1.24438	0.622189	−	0.782867i	$-0.286244\pi$
$577$	29.8910	1.24438	0.622189	+	0.782867i	$0.286244\pi$
$578$	15.4760	0.643715
$579$	−42.8794	−1.78201
$580$	0	0
$581$	−2.28254	−0.0946955
$582$	−25.8755	−1.07258
$583$	−8.31250	−0.344269
$584$	6.43459	0.266265
$585$	0	0
$586$	30.4615	1.25835
$587$	−12.8806	−0.531639	−0.265820	−	0.964023i	$-0.585643\pi$
$587$	−12.8806	−0.531639	−0.265820	+	0.964023i	$0.585643\pi$
$588$	−1.63209	−0.0673062
$589$	−28.8318	−1.18799
$590$	0	0
$591$	−2.20083	−0.0905299
$592$	−1.04832	−0.0430858
$593$	11.4913	0.471891	0.235946	−	0.971766i	$-0.424181\pi$
$593$	11.4913	0.471891	0.235946	+	0.971766i	$0.424181\pi$
$594$	5.44512	0.223416
$595$	0	0
$596$	13.3863	0.548323
$597$	−9.47635	−0.387841
$598$	10.9502	0.447785
$599$	−13.6621	−0.558216	−0.279108	−	0.960260i	$-0.590039\pi$
$599$	−13.6621	−0.558216	−0.279108	+	0.960260i	$0.590039\pi$
$600$	0	0
$601$	−18.6226	−0.759631	−0.379816	−	0.925062i	$-0.624013\pi$
$601$	−18.6226	−0.759631	−0.379816	+	0.925062i	$0.624013\pi$
$602$	0.265838	0.0108347
$603$	0.818723	0.0333410
$604$	12.7206	0.517592
$605$	0	0
$606$	−10.6421	−0.432304
$607$	−13.0950	−0.531509	−0.265754	−	0.964041i	$-0.585621\pi$
$607$	−13.0950	−0.531509	−0.265754	+	0.964041i	$0.585621\pi$
$608$	−5.40297	−0.219119
$609$	2.26252	0.0916818
$610$	0	0
$611$	21.3608	0.864166
$612$	−1.91643	−0.0774669
$613$	35.0203	1.41446	0.707228	−	0.706986i	$-0.249946\pi$
$613$	35.0203	1.41446	0.707228	+	0.706986i	$0.249946\pi$
$614$	−10.5601	−0.426172
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−9.66205	−0.388980	−0.194490	−	0.980905i	$-0.562305\pi$
$617$	−9.66205	−0.388980	−0.194490	+	0.980905i	$0.562305\pi$
$618$	−17.9124	−0.720542
$619$	−11.4598	−0.460609	−0.230304	−	0.973119i	$-0.573972\pi$
$619$	−11.4598	−0.460609	−0.230304	+	0.973119i	$0.573972\pi$
$620$	0	0
$621$	−25.3222	−1.01615
$622$	14.4795	0.580573
$623$	−3.22746	−0.129305
$624$	3.84299	0.153843
$625$	0	0
$626$	4.16828	0.166598
$627$	8.81812	0.352162
$628$	−8.25468	−0.329398
$629$	−5.97414	−0.238205
$630$	0	0
$631$	31.2718	1.24491	0.622455	−	0.782656i	$-0.286136\pi$
$631$	31.2718	1.24491	0.622455	+	0.782656i	$0.286136\pi$
$632$	11.7943	0.469153
$633$	28.6896	1.14031
$634$	−9.19254	−0.365083
$635$	0	0
$636$	13.5667	0.537956
$637$	−2.35465	−0.0932945
$638$	1.38627	0.0548830
$639$	−1.90408	−0.0753242
$640$	0	0
$641$	32.2406	1.27343	0.636714	−	0.771100i	$-0.280293\pi$
$641$	32.2406	1.27343	0.636714	+	0.771100i	$0.280293\pi$
$642$	0.669254	0.0264133
$643$	47.5158	1.87384	0.936920	−	0.349545i	$-0.113664\pi$
$643$	47.5158	1.87384	0.936920	+	0.349545i	$0.113664\pi$
$644$	4.65045	0.183253
$645$	0	0
$646$	−30.7903	−1.21143
$647$	−13.7498	−0.540562	−0.270281	−	0.962782i	$-0.587117\pi$
$647$	−13.7498	−0.540562	−0.270281	+	0.962782i	$0.587117\pi$
$648$	−7.87805	−0.309479
$649$	6.58377	0.258435
$650$	0	0
$651$	8.70929	0.341344
$652$	11.7090	0.458558
$653$	31.3695	1.22758	0.613791	−	0.789468i	$-0.289644\pi$
$653$	31.3695	1.22758	0.613791	+	0.789468i	$0.289644\pi$
$654$	−9.59326	−0.375126
$655$	0	0
$656$	9.98673	0.389916
$657$	−2.16387	−0.0844208
$658$	9.07177	0.353655
$659$	−43.9851	−1.71342	−0.856709	−	0.515800i	$-0.827495\pi$
$659$	−43.9851	−1.71342	−0.856709	+	0.515800i	$0.827495\pi$
$660$	0	0
$661$	−23.6692	−0.920626	−0.460313	−	0.887757i	$-0.652263\pi$
$661$	−23.6692	−0.920626	−0.460313	+	0.887757i	$0.652263\pi$
$662$	−5.98493	−0.232611
$663$	21.9003	0.850538
$664$	2.28254	0.0885796
$665$	0	0
$666$	0.352538	0.0136606
$667$	−6.44678	−0.249620
$668$	8.56507	0.331393
$669$	−37.7873	−1.46094
$670$	0	0
$671$	8.71713	0.336521
$672$	1.63209	0.0629592
$673$	8.80882	0.339555	0.169778	−	0.985482i	$-0.445695\pi$
$673$	8.80882	0.339555	0.169778	+	0.985482i	$0.445695\pi$
$674$	32.2878	1.24368
$675$	0	0
$676$	−7.45564	−0.286755
$677$	24.8062	0.953381	0.476690	−	0.879071i	$-0.341836\pi$
$677$	24.8062	0.953381	0.476690	+	0.879071i	$0.341836\pi$
$678$	28.7827	1.10539
$679$	−15.8543	−0.608431
$680$	0	0
$681$	36.7768	1.40929
$682$	5.33629	0.204337
$683$	6.06065	0.231904	0.115952	−	0.993255i	$-0.463008\pi$
$683$	6.06065	0.231904	0.115952	+	0.993255i	$0.463008\pi$
$684$	1.81695	0.0694729
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	4.31690	0.164700
$688$	−0.265838	−0.0101350
$689$	19.5730	0.745672
$690$	0	0
$691$	−47.1292	−1.79288	−0.896439	−	0.443167i	$-0.853855\pi$
$691$	−47.1292	−1.79288	−0.896439	+	0.443167i	$0.853855\pi$
$692$	12.5572	0.477354
$693$	0.336288	0.0127745
$694$	8.13934	0.308965
$695$	0	0
$696$	−2.26252	−0.0857604
$697$	56.9121	2.15570
$698$	−27.3788	−1.03630
$699$	19.1711	0.725117
$700$	0	0
$701$	2.82269	0.106611	0.0533057	−	0.998578i	$-0.483024\pi$
$701$	2.82269	0.106611	0.0533057	+	0.998578i	$0.483024\pi$
$702$	−12.8213	−0.483910
$703$	5.66405	0.213624
$704$	1.00000	0.0376889
$705$	0	0
$706$	−36.3194	−1.36690
$707$	−6.52052	−0.245229
$708$	−10.7453	−0.403833
$709$	9.45746	0.355183	0.177591	−	0.984104i	$-0.443170\pi$
$709$	9.45746	0.355183	0.177591	+	0.984104i	$0.443170\pi$
$710$	0	0
$711$	−3.96629	−0.148747
$712$	3.22746	0.120954
$713$	−24.8161	−0.929371
$714$	9.30089	0.348077
$715$	0	0
$716$	25.0928	0.937763
$717$	4.56115	0.170339
$718$	−24.6940	−0.921573
$719$	−49.0065	−1.82764	−0.913818	−	0.406125i	$-0.866880\pi$
$719$	−49.0065	−1.82764	−0.913818	+	0.406125i	$0.866880\pi$
$720$	0	0
$721$	−10.9751	−0.408735
$722$	10.1921	0.379309
$723$	−27.3093	−1.01564
$724$	19.2327	0.714777
$725$	0	0
$726$	−1.63209	−0.0605725
$727$	−21.6212	−0.801885	−0.400943	−	0.916103i	$-0.631317\pi$
$727$	−21.6212	−0.801885	−0.400943	+	0.916103i	$0.631317\pi$
$728$	2.35465	0.0872690
$729$	29.3100	1.08556
$730$	0	0
$731$	−1.51495	−0.0560323
$732$	−14.2271	−0.525850
$733$	−5.66700	−0.209316	−0.104658	−	0.994508i	$-0.533375\pi$
$733$	−5.66700	−0.209316	−0.104658	+	0.994508i	$0.533375\pi$
$734$	−16.4067	−0.605584
$735$	0	0
$736$	−4.65045	−0.171418
$737$	−2.43459	−0.0896794
$738$	−3.35842	−0.123625
$739$	−15.4683	−0.569012	−0.284506	−	0.958674i	$-0.591829\pi$
$739$	−15.4683	−0.569012	−0.284506	+	0.958674i	$0.591829\pi$
$740$	0	0
$741$	−20.7636	−0.762768
$742$	8.31250	0.305161
$743$	8.56219	0.314116	0.157058	−	0.987589i	$-0.449799\pi$
$743$	8.56219	0.314116	0.157058	+	0.987589i	$0.449799\pi$
$744$	−8.70929	−0.319298
$745$	0	0
$746$	1.13082	0.0414021
$747$	−0.767589	−0.0280846
$748$	5.69877	0.208368
$749$	0.410060	0.0149833
$750$	0	0
$751$	−6.43503	−0.234818	−0.117409	−	0.993084i	$-0.537459\pi$
$751$	−6.43503	−0.234818	−0.117409	+	0.993084i	$0.537459\pi$
$752$	−9.07177	−0.330814
$753$	36.8955	1.34455
$754$	−3.26418	−0.118874
$755$	0	0
$756$	−5.44512	−0.198037
$757$	−0.239064	−0.00868891	−0.00434446	−	0.999991i	$-0.501383\pi$
$757$	−0.239064	−0.00868891	−0.00434446	+	0.999991i	$0.501383\pi$
$758$	5.57848	0.202619
$759$	7.58994	0.275497
$760$	0	0
$761$	36.0595	1.30716	0.653578	−	0.756859i	$-0.273267\pi$
$761$	36.0595	1.30716	0.653578	+	0.756859i	$0.273267\pi$
$762$	11.5017	0.416663
$763$	−5.87791	−0.212794
$764$	−6.21145	−0.224722
$765$	0	0
$766$	−28.3203	−1.02325
$767$	−15.5024	−0.559761
$768$	−1.63209	−0.0588929
$769$	0.427531	0.0154172	0.00770859	−	0.999970i	$-0.497546\pi$
$769$	0.427531	0.0154172	0.00770859	+	0.999970i	$0.497546\pi$
$770$	0	0
$771$	−33.2063	−1.19590
$772$	26.2727	0.945575
$773$	−21.0265	−0.756272	−0.378136	−	0.925750i	$-0.623435\pi$
$773$	−21.0265	−0.756272	−0.378136	+	0.925750i	$0.623435\pi$
$774$	0.0893979	0.00321334
$775$	0	0
$776$	15.8543	0.569135
$777$	−1.71095	−0.0613801
$778$	13.0934	0.469422
$779$	−53.9580	−1.93325
$780$	0	0
$781$	5.66205	0.202604
$782$	−26.5018	−0.947703
$783$	7.54840	0.269758
$784$	1.00000	0.0357143
$785$	0	0
$786$	13.3432	0.475935
$787$	−38.4029	−1.36891	−0.684457	−	0.729053i	$-0.739961\pi$
$787$	−38.4029	−1.36891	−0.684457	+	0.729053i	$0.739961\pi$
$788$	1.34847	0.0480373
$789$	−37.9082	−1.34957
$790$	0	0
$791$	17.6355	0.627047
$792$	−0.336288	−0.0119495
$793$	−20.5258	−0.728891
$794$	−6.33144	−0.224694
$795$	0	0
$796$	5.80627	0.205798
$797$	39.7230	1.40706	0.703531	−	0.710665i	$-0.251606\pi$
$797$	39.7230	1.40706	0.703531	+	0.710665i	$0.251606\pi$
$798$	−8.81812	−0.312158
$799$	−51.6979	−1.82894
$800$	0	0
$801$	−1.08535	−0.0383491
$802$	−32.8774	−1.16094
$803$	6.43459	0.227072
$804$	3.97347	0.140133
$805$	0	0
$806$	−12.5651	−0.442586
$807$	−52.6859	−1.85463
$808$	6.52052	0.229391
$809$	−41.1697	−1.44745	−0.723726	−	0.690088i	$-0.757572\pi$
$809$	−41.1697	−1.44745	−0.723726	+	0.690088i	$0.757572\pi$
$810$	0	0
$811$	−52.9916	−1.86079	−0.930394	−	0.366561i	$-0.880535\pi$
$811$	−52.9916	−1.86079	−0.930394	+	0.366561i	$0.880535\pi$
$812$	−1.38627	−0.0486486
$813$	−1.88779	−0.0662077
$814$	−1.04832	−0.0367437
$815$	0	0
$816$	−9.30089	−0.325596
$817$	1.43631	0.0502502
$818$	−30.9736	−1.08297
$819$	−0.791839	−0.0276691
$820$	0	0
$821$	22.8340	0.796913	0.398457	−	0.917187i	$-0.369546\pi$
$821$	22.8340	0.796913	0.398457	+	0.917187i	$0.369546\pi$
$822$	0.119852	0.00418033
$823$	−36.4295	−1.26985	−0.634927	−	0.772573i	$-0.718970\pi$
$823$	−36.4295	−1.26985	−0.634927	+	0.772573i	$0.718970\pi$
$824$	10.9751	0.382337
$825$	0	0
$826$	−6.58377	−0.229079
$827$	−39.2003	−1.36313	−0.681563	−	0.731759i	$-0.738700\pi$
$827$	−39.2003	−1.36313	−0.681563	+	0.731759i	$0.738700\pi$
$828$	1.56389	0.0543489
$829$	−16.8967	−0.586848	−0.293424	−	0.955982i	$-0.594795\pi$
$829$	−16.8967	−0.586848	−0.293424	+	0.955982i	$0.594795\pi$
$830$	0	0
$831$	23.3023	0.808347
$832$	−2.35465	−0.0816327
$833$	5.69877	0.197451
$834$	19.6361	0.679943
$835$	0	0
$836$	−5.40297	−0.186866
$837$	29.0567	1.00435
$838$	−7.91806	−0.273525
$839$	29.9878	1.03529	0.517647	−	0.855594i	$-0.326808\pi$
$839$	29.9878	1.03529	0.517647	+	0.855594i	$0.326808\pi$
$840$	0	0
$841$	−27.0783	−0.933733
$842$	−2.92911	−0.100944
$843$	16.2776	0.560630
$844$	−17.5785	−0.605076
$845$	0	0
$846$	3.05073	0.104886
$847$	−1.00000	−0.0343604
$848$	−8.31250	−0.285452
$849$	−2.30671	−0.0791662
$850$	0	0
$851$	4.87517	0.167119
$852$	−9.24097	−0.316590
$853$	12.7572	0.436797	0.218399	−	0.975860i	$-0.429917\pi$
$853$	12.7572	0.436797	0.218399	+	0.975860i	$0.429917\pi$
$854$	−8.71713	−0.298294
$855$	0	0
$856$	−0.410060	−0.0140156
$857$	17.6696	0.603582	0.301791	−	0.953374i	$-0.402416\pi$
$857$	17.6696	0.603582	0.301791	+	0.953374i	$0.402416\pi$
$858$	3.84299	0.131198
$859$	37.0383	1.26373	0.631865	−	0.775078i	$-0.282290\pi$
$859$	37.0383	1.26373	0.631865	+	0.775078i	$0.282290\pi$
$860$	0	0
$861$	16.2992	0.555476
$862$	5.24959	0.178802
$863$	−26.0277	−0.885992	−0.442996	−	0.896524i	$-0.646084\pi$
$863$	−26.0277	−0.885992	−0.442996	+	0.896524i	$0.646084\pi$
$864$	5.44512	0.185247
$865$	0	0
$866$	−21.7801	−0.740120
$867$	−25.2581	−0.857812
$868$	−5.33629	−0.181125
$869$	11.7943	0.400095
$870$	0	0
$871$	5.73260	0.194242
$872$	5.87791	0.199051
$873$	−5.33159	−0.180447
$874$	25.1262	0.849907
$875$	0	0
$876$	−10.5018	−0.354824
$877$	−18.0204	−0.608505	−0.304252	−	0.952591i	$-0.598407\pi$
$877$	−18.0204	−0.608505	−0.304252	+	0.952591i	$0.598407\pi$
$878$	30.5509	1.03104
$879$	−49.7158	−1.67687
$880$	0	0
$881$	−14.3317	−0.482849	−0.241424	−	0.970420i	$-0.577615\pi$
$881$	−14.3317	−0.482849	−0.241424	+	0.970420i	$0.577615\pi$
$882$	−0.336288	−0.0113234
$883$	6.13264	0.206380	0.103190	−	0.994662i	$-0.467095\pi$
$883$	6.13264	0.206380	0.103190	+	0.994662i	$0.467095\pi$
$884$	−13.4186	−0.451316
$885$	0	0
$886$	20.6359	0.693277
$887$	0.215600	0.00723915	0.00361957	−	0.999993i	$-0.498848\pi$
$887$	0.215600	0.00723915	0.00361957	+	0.999993i	$0.498848\pi$
$888$	1.71095	0.0574158
$889$	7.04724	0.236357
$890$	0	0
$891$	−7.87805	−0.263925
$892$	23.1527	0.775211
$893$	49.0145	1.64021
$894$	−21.8476	−0.730692
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−17.8716	−0.596716
$898$	36.7803	1.22737
$899$	7.39754	0.246722
$900$	0	0
$901$	−47.3710	−1.57816
$902$	9.98673	0.332522
$903$	−0.433870	−0.0144383
$904$	−17.6355	−0.586549
$905$	0	0
$906$	−20.7611	−0.689741
$907$	15.2282	0.505643	0.252822	−	0.967513i	$-0.418641\pi$
$907$	15.2282	0.505643	0.252822	+	0.967513i	$0.418641\pi$
$908$	−22.5336	−0.747803
$909$	−2.19277	−0.0727296
$910$	0	0
$911$	14.1908	0.470164	0.235082	−	0.971976i	$-0.424464\pi$
$911$	14.1908	0.470164	0.235082	+	0.971976i	$0.424464\pi$
$912$	8.81812	0.291997
$913$	2.28254	0.0755409
$914$	−34.3402	−1.13587
$915$	0	0
$916$	−2.64502	−0.0873938
$917$	8.17551	0.269979
$918$	31.0305	1.02416
$919$	28.4172	0.937396	0.468698	−	0.883358i	$-0.344723\pi$
$919$	28.4172	0.937396	0.468698	+	0.883358i	$0.344723\pi$
$920$	0	0
$921$	17.2350	0.567914
$922$	16.2854	0.536332
$923$	−13.3321	−0.438832
$924$	1.63209	0.0536918
$925$	0	0
$926$	−42.2329	−1.38786
$927$	−3.69080	−0.121222
$928$	1.38627	0.0455066
$929$	4.10750	0.134763	0.0673814	−	0.997727i	$-0.478536\pi$
$929$	4.10750	0.134763	0.0673814	+	0.997727i	$0.478536\pi$
$930$	0	0
$931$	−5.40297	−0.177075
$932$	−11.7463	−0.384764
$933$	−23.6317	−0.773669
$934$	19.5266	0.638931
$935$	0	0
$936$	0.791839	0.0258821
$937$	25.9029	0.846211	0.423106	−	0.906080i	$-0.360940\pi$
$937$	25.9029	0.846211	0.423106	+	0.906080i	$0.360940\pi$
$938$	2.43459	0.0794923
$939$	−6.80300	−0.222007
$940$	0	0
$941$	6.75120	0.220083	0.110041	−	0.993927i	$-0.464902\pi$
$941$	6.75120	0.220083	0.110041	+	0.993927i	$0.464902\pi$
$942$	13.4724	0.438953
$943$	−46.4428	−1.51239
$944$	6.58377	0.214283
$945$	0	0
$946$	−0.265838	−0.00864312
$947$	−32.7332	−1.06369	−0.531843	−	0.846843i	$-0.678501\pi$
$947$	−32.7332	−1.06369	−0.531843	+	0.846843i	$0.678501\pi$
$948$	−19.2494	−0.625191
$949$	−15.1512	−0.491828
$950$	0	0
$951$	15.0030	0.486507
$952$	−5.69877	−0.184698
$953$	−35.3432	−1.14488	−0.572439	−	0.819948i	$-0.694002\pi$
$953$	−35.3432	−1.14488	−0.572439	+	0.819948i	$0.694002\pi$
$954$	2.79539	0.0905041
$955$	0	0
$956$	−2.79467	−0.0903861
$957$	−2.26252	−0.0731367
$958$	−3.23765	−0.104604
$959$	0.0734349	0.00237134
$960$	0	0
$961$	−2.52403	−0.0814204
$962$	2.46843	0.0795853
$963$	0.137898	0.00444370
$964$	16.7327	0.538925
$965$	0	0
$966$	−7.58994	−0.244202
$967$	12.5883	0.404813	0.202407	−	0.979302i	$-0.435124\pi$
$967$	12.5883	0.404813	0.202407	+	0.979302i	$0.435124\pi$
$968$	1.00000	0.0321412
$969$	50.2524	1.61434
$970$	0	0
$971$	20.1466	0.646534	0.323267	−	0.946308i	$-0.395219\pi$
$971$	20.1466	0.646534	0.323267	+	0.946308i	$0.395219\pi$
$972$	−3.47768	−0.111547
$973$	12.0313	0.385705
$974$	0.532755	0.0170706
$975$	0	0
$976$	8.71713	0.279028
$977$	−20.2177	−0.646820	−0.323410	−	0.946259i	$-0.604829\pi$
$977$	−20.2177	−0.646820	−0.323410	+	0.946259i	$0.604829\pi$
$978$	−19.1101	−0.611072
$979$	3.22746	0.103150
$980$	0	0
$981$	−1.97667	−0.0631101
$982$	−23.2710	−0.742609
$983$	60.7312	1.93703	0.968513	−	0.248964i	$-0.0800902\pi$
$983$	60.7312	1.93703	0.968513	+	0.248964i	$0.0800902\pi$
$984$	−16.2992	−0.519601
$985$	0	0
$986$	7.90003	0.251588
$987$	−14.8059	−0.471278
$988$	12.7221	0.404743
$989$	1.23626	0.0393109
$990$	0	0
$991$	−3.08728	−0.0980707	−0.0490354	−	0.998797i	$-0.515615\pi$
$991$	−3.08728	−0.0980707	−0.0490354	+	0.998797i	$0.515615\pi$
$992$	5.33629	0.169427
$993$	9.76794	0.309976
$994$	−5.66205	−0.179589
$995$	0	0
$996$	−3.72530	−0.118041
$997$	45.3243	1.43543	0.717717	−	0.696334i	$-0.245187\pi$
$997$	45.3243	1.43543	0.717717	+	0.696334i	$0.245187\pi$
$998$	−11.4900	−0.363709
$999$	−5.70823	−0.180601

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.ca.1.2		5
5.2	odd	4		770.2.c.f.309.9	yes	10
5.3	odd	4		770.2.c.f.309.2	✓	10
5.4	even	2		3850.2.a.bz.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.c.f.309.2	✓	10	5.3	odd	4
770.2.c.f.309.9	yes	10	5.2	odd	4
3850.2.a.bz.1.4		5	5.4	even	2
3850.2.a.ca.1.2		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.63209 q^{3} +1.00000 q^{4} -1.63209 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.336288 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.63209 q^{3} +1.00000 q^{4} -1.63209 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.336288 q^{9} +1.00000 q^{11} -1.63209 q^{12} -2.35465 q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.69877 q^{17} -0.336288 q^{18} -5.40297 q^{19} +1.63209 q^{21} +1.00000 q^{22} -4.65045 q^{23} -1.63209 q^{24} -2.35465 q^{26} +5.44512 q^{27} -1.00000 q^{28} +1.38627 q^{29} +5.33629 q^{31} +1.00000 q^{32} -1.63209 q^{33} +5.69877 q^{34} -0.336288 q^{36} -1.04832 q^{37} -5.40297 q^{38} +3.84299 q^{39} +9.98673 q^{41} +1.63209 q^{42} -0.265838 q^{43} +1.00000 q^{44} -4.65045 q^{46} -9.07177 q^{47} -1.63209 q^{48} +1.00000 q^{49} -9.30089 q^{51} -2.35465 q^{52} -8.31250 q^{53} +5.44512 q^{54} -1.00000 q^{56} +8.81812 q^{57} +1.38627 q^{58} +6.58377 q^{59} +8.71713 q^{61} +5.33629 q^{62} +0.336288 q^{63} +1.00000 q^{64} -1.63209 q^{66} -2.43459 q^{67} +5.69877 q^{68} +7.58994 q^{69} +5.66205 q^{71} -0.336288 q^{72} +6.43459 q^{73} -1.04832 q^{74} -5.40297 q^{76} -1.00000 q^{77} +3.84299 q^{78} +11.7943 q^{79} -7.87805 q^{81} +9.98673 q^{82} +2.28254 q^{83} +1.63209 q^{84} -0.265838 q^{86} -2.26252 q^{87} +1.00000 q^{88} +3.22746 q^{89} +2.35465 q^{91} -4.65045 q^{92} -8.70929 q^{93} -9.07177 q^{94} -1.63209 q^{96} +15.8543 q^{97} +1.00000 q^{98} -0.336288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{7} + 5 q^{8} + 13 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

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Representations

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Database

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