LMFDB - Embedded newform 3381.2.a.bc.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7997584.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 7x^{4} + 5x^{3} + 12x^{2} - 4x - 2 $ x^6 - x^5 - 7x^4 + 5x^3 + 12x^2 - 4x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.04340$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.04340 q^{2} -1.00000 q^{3} +2.17548 q^{4} +1.94349 q^{5} -2.04340 q^{6} +0.358585 q^{8} +1.00000 q^{9} +3.97133 q^{10} -5.66529 q^{11} -2.17548 q^{12} -6.17812 q^{13} -1.94349 q^{15} -3.61824 q^{16} +7.23463 q^{17} +2.04340 q^{18} +0.790220 q^{19} +4.22803 q^{20} -11.5765 q^{22} -1.00000 q^{23} -0.358585 q^{24} -1.22285 q^{25} -12.6244 q^{26} -1.00000 q^{27} -5.34085 q^{29} -3.97133 q^{30} +0.911316 q^{31} -8.11067 q^{32} +5.66529 q^{33} +14.7832 q^{34} +2.17548 q^{36} -0.973320 q^{37} +1.61473 q^{38} +6.17812 q^{39} +0.696907 q^{40} -1.51843 q^{41} -2.70342 q^{43} -12.3247 q^{44} +1.94349 q^{45} -2.04340 q^{46} -1.52839 q^{47} +3.61824 q^{48} -2.49877 q^{50} -7.23463 q^{51} -13.4404 q^{52} -8.95803 q^{53} -2.04340 q^{54} -11.0104 q^{55} -0.790220 q^{57} -10.9135 q^{58} -10.8931 q^{59} -4.22803 q^{60} -6.31775 q^{61} +1.86218 q^{62} -9.33688 q^{64} -12.0071 q^{65} +11.5765 q^{66} +7.31833 q^{67} +15.7388 q^{68} +1.00000 q^{69} -8.75692 q^{71} +0.358585 q^{72} -0.875251 q^{73} -1.98888 q^{74} +1.22285 q^{75} +1.71911 q^{76} +12.6244 q^{78} +5.96655 q^{79} -7.03200 q^{80} +1.00000 q^{81} -3.10277 q^{82} -0.273554 q^{83} +14.0604 q^{85} -5.52417 q^{86} +5.34085 q^{87} -2.03149 q^{88} +17.6270 q^{89} +3.97133 q^{90} -2.17548 q^{92} -0.911316 q^{93} -3.12312 q^{94} +1.53578 q^{95} +8.11067 q^{96} +3.09620 q^{97} -5.66529 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 6 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{8} + 6 q^{9} - 3 q^{10} - 14 q^{11} - 3 q^{12} - 3 q^{15} - 7 q^{16} + 15 q^{17} - q^{18} - q^{19} + 17 q^{20} - 6 q^{22} - 6 q^{23} + 3 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) $ 6 * q - q^2 - 6 * q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^8 + 6 * q^9 - 3 * q^10 - 14 * q^11 - 3 * q^12 - 3 * q^15 - 7 * q^16 + 15 * q^17 - q^18 - q^19 + 17 * q^20 - 6 * q^22 - 6 * q^23 + 3 * q^24 + 9 * q^25 - 15 * q^26 - 6 * q^27 - 6 * q^29 + 3 * q^30 - 11 * q^31 + 3 * q^32 + 14 * q^33 + 15 * q^34 + 3 * q^36 - 5 * q^37 + 14 * q^38 - 17 * q^40 + 18 * q^41 - 37 * q^43 - 10 * q^44 + 3 * q^45 + q^46 + 3 * q^47 + 7 * q^48 - 30 * q^50 - 15 * q^51 - 7 * q^52 - 15 * q^53 + q^54 - 2 * q^55 + q^57 + 4 * q^58 - 2 * q^59 - 17 * q^60 - 12 * q^61 + 36 * q^62 - 23 * q^64 - 17 * q^65 + 6 * q^66 - 10 * q^67 - q^68 + 6 * q^69 - 21 * q^71 - 3 * q^72 - 8 * q^73 - 16 * q^74 - 9 * q^75 + 18 * q^76 + 15 * q^78 - 17 * q^79 + 3 * q^80 + 6 * q^81 - 48 * q^82 + 12 * q^83 - 13 * q^85 + 22 * q^86 + 6 * q^87 - 2 * q^88 + 18 * q^89 - 3 * q^90 - 3 * q^92 + 11 * q^93 + 3 * q^94 - 16 * q^95 - 3 * q^96 - 2 * q^97 - 14 * q^99

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 $

$2$	2.04340	1.44490	0.722451	−	0.691422i	$-0.243015\pi$
$2$	2.04340	1.44490	0.722451	+	0.691422i	$0.243015\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.17548	1.08774
$5$	1.94349	0.869155	0.434577	−	0.900634i	$-0.356898\pi$
$5$	1.94349	0.869155	0.434577	+	0.900634i	$0.356898\pi$
$6$	−2.04340	−0.834215
$7$	0	0
$7$	0	0
$8$	0.358585	0.126779
$9$	1.00000	0.333333
$10$	3.97133	1.25584
$11$	−5.66529	−1.70815	−0.854074	−	0.520151i	$-0.825876\pi$
$11$	−5.66529	−1.70815	−0.854074	+	0.520151i	$0.825876\pi$
$12$	−2.17548	−0.628008
$13$	−6.17812	−1.71350	−0.856751	−	0.515730i	$-0.827521\pi$
$13$	−6.17812	−1.71350	−0.856751	+	0.515730i	$0.827521\pi$
$14$	0	0
$15$	−1.94349	−0.501807
$16$	−3.61824	−0.904559
$17$	7.23463	1.75466	0.877328	−	0.479892i	$-0.159324\pi$
$17$	7.23463	1.75466	0.877328	+	0.479892i	$0.159324\pi$
$18$	2.04340	0.481634
$19$	0.790220	0.181289	0.0906444	−	0.995883i	$-0.471107\pi$
$19$	0.790220	0.181289	0.0906444	+	0.995883i	$0.471107\pi$
$20$	4.22803	0.945417
$21$	0	0
$22$	−11.5765	−2.46811
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−0.358585	−0.0731960
$25$	−1.22285	−0.244570
$26$	−12.6244	−2.47584
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.34085	−0.991770	−0.495885	−	0.868388i	$-0.665156\pi$
$29$	−5.34085	−0.991770	−0.495885	+	0.868388i	$0.665156\pi$
$30$	−3.97133	−0.725062
$31$	0.911316	0.163677	0.0818386	−	0.996646i	$-0.473921\pi$
$31$	0.911316	0.163677	0.0818386	+	0.996646i	$0.473921\pi$
$32$	−8.11067	−1.43378
$33$	5.66529	0.986200
$34$	14.7832	2.53531
$35$	0	0
$36$	2.17548	0.362581
$37$	−0.973320	−0.160013	−0.0800064	−	0.996794i	$-0.525494\pi$
$37$	−0.973320	−0.160013	−0.0800064	+	0.996794i	$0.525494\pi$
$38$	1.61473	0.261945
$39$	6.17812	0.989291
$40$	0.696907	0.110191
$41$	−1.51843	−0.237140	−0.118570	−	0.992946i	$-0.537831\pi$
$41$	−1.51843	−0.237140	−0.118570	+	0.992946i	$0.537831\pi$
$42$	0	0
$43$	−2.70342	−0.412268	−0.206134	−	0.978524i	$-0.566088\pi$
$43$	−2.70342	−0.412268	−0.206134	+	0.978524i	$0.566088\pi$
$44$	−12.3247	−1.85803
$45$	1.94349	0.289718
$46$	−2.04340	−0.301283
$47$	−1.52839	−0.222939	−0.111470	−	0.993768i	$-0.535556\pi$
$47$	−1.52839	−0.222939	−0.111470	+	0.993768i	$0.535556\pi$
$48$	3.61824	0.522247
$49$	0	0
$50$	−2.49877	−0.353379
$51$	−7.23463	−1.01305
$52$	−13.4404	−1.86385
$53$	−8.95803	−1.23048	−0.615240	−	0.788340i	$-0.710941\pi$
$53$	−8.95803	−1.23048	−0.615240	+	0.788340i	$0.710941\pi$
$54$	−2.04340	−0.278072
$55$	−11.0104	−1.48465
$56$	0	0
$57$	−0.790220	−0.104667
$58$	−10.9135	−1.43301
$59$	−10.8931	−1.41816	−0.709081	−	0.705127i	$-0.750890\pi$
$59$	−10.8931	−1.41816	−0.709081	+	0.705127i	$0.750890\pi$
$60$	−4.22803	−0.545837
$61$	−6.31775	−0.808905	−0.404452	−	0.914559i	$-0.632538\pi$
$61$	−6.31775	−0.808905	−0.404452	+	0.914559i	$0.632538\pi$
$62$	1.86218	0.236497
$63$	0	0
$64$	−9.33688	−1.16711
$65$	−12.0071	−1.48930
$66$	11.5765	1.42496
$67$	7.31833	0.894077	0.447038	−	0.894515i	$-0.352479\pi$
$67$	7.31833	0.894077	0.447038	+	0.894515i	$0.352479\pi$
$68$	15.7388	1.90861
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.75692	−1.03926	−0.519628	−	0.854393i	$-0.673929\pi$
$71$	−8.75692	−1.03926	−0.519628	+	0.854393i	$0.673929\pi$
$72$	0.358585	0.0422597
$73$	−0.875251	−0.102440	−0.0512202	−	0.998687i	$-0.516311\pi$
$73$	−0.875251	−0.102440	−0.0512202	+	0.998687i	$0.516311\pi$
$74$	−1.98888	−0.231203
$75$	1.22285	0.141202
$76$	1.71911	0.197196
$77$	0	0
$78$	12.6244	1.42943
$79$	5.96655	0.671289	0.335644	−	0.941989i	$-0.391046\pi$
$79$	5.96655	0.671289	0.335644	+	0.941989i	$0.391046\pi$
$80$	−7.03200	−0.786202
$81$	1.00000	0.111111
$82$	−3.10277	−0.342644
$83$	−0.273554	−0.0300265	−0.0150132	−	0.999887i	$-0.504779\pi$
$83$	−0.273554	−0.0300265	−0.0150132	+	0.999887i	$0.504779\pi$
$84$	0	0
$85$	14.0604	1.52507
$86$	−5.52417	−0.595686
$87$	5.34085	0.572599
$88$	−2.03149	−0.216558
$89$	17.6270	1.86846	0.934229	−	0.356672i	$-0.116089\pi$
$89$	17.6270	1.86846	0.934229	+	0.356672i	$0.116089\pi$
$90$	3.97133	0.418615
$91$	0	0
$92$	−2.17548	−0.226810
$93$	−0.911316	−0.0944991
$94$	−3.12312	−0.322125
$95$	1.53578	0.157568
$96$	8.11067	0.827792
$97$	3.09620	0.314372	0.157186	−	0.987569i	$-0.449758\pi$
$97$	3.09620	0.314372	0.157186	+	0.987569i	$0.449758\pi$
$98$	0	0
$99$	−5.66529	−0.569383
$100$	−2.66029	−0.266029
$101$	13.4276	1.33610	0.668050	−	0.744116i	$-0.267129\pi$
$101$	13.4276	1.33610	0.668050	+	0.744116i	$0.267129\pi$
$102$	−14.7832	−1.46376
$103$	6.33653	0.624357	0.312178	−	0.950023i	$-0.398941\pi$
$103$	6.33653	0.624357	0.312178	+	0.950023i	$0.398941\pi$
$104$	−2.21538	−0.217236
$105$	0	0
$106$	−18.3049	−1.77792
$107$	15.9609	1.54299	0.771497	−	0.636232i	$-0.219508\pi$
$107$	15.9609	1.54299	0.771497	+	0.636232i	$0.219508\pi$
$108$	−2.17548	−0.209336
$109$	−0.0470715	−0.00450863	−0.00225431	−	0.999997i	$-0.500718\pi$
$109$	−0.0470715	−0.00450863	−0.00225431	+	0.999997i	$0.500718\pi$
$110$	−22.4987	−2.14517
$111$	0.973320	0.0923835
$112$	0	0
$113$	−6.27141	−0.589965	−0.294982	−	0.955503i	$-0.595314\pi$
$113$	−6.27141	−0.589965	−0.294982	+	0.955503i	$0.595314\pi$
$114$	−1.61473	−0.151234
$115$	−1.94349	−0.181231
$116$	−11.6189	−1.07879
$117$	−6.17812	−0.571167
$118$	−22.2590	−2.04911
$119$	0	0
$120$	−0.696907	−0.0636186
$121$	21.0955	1.91777
$122$	−12.9097	−1.16879
$123$	1.51843	0.136913
$124$	1.98255	0.178039
$125$	−12.0940	−1.08172
$126$	0	0
$127$	−12.4325	−1.10320	−0.551601	−	0.834108i	$-0.685983\pi$
$127$	−12.4325	−1.10320	−0.551601	+	0.834108i	$0.685983\pi$
$128$	−2.85764	−0.252582
$129$	2.70342	0.238023
$130$	−24.5353	−2.15189
$131$	−3.31098	−0.289281	−0.144641	−	0.989484i	$-0.546203\pi$
$131$	−3.31098	−0.289281	−0.144641	+	0.989484i	$0.546203\pi$
$132$	12.3247	1.07273
$133$	0	0
$134$	14.9543	1.29185
$135$	−1.94349	−0.167269
$136$	2.59423	0.222454
$137$	−13.0442	−1.11444	−0.557220	−	0.830365i	$-0.688132\pi$
$137$	−13.0442	−1.11444	−0.557220	+	0.830365i	$0.688132\pi$
$138$	2.04340	0.173946
$139$	−11.3674	−0.964169	−0.482084	−	0.876125i	$-0.660120\pi$
$139$	−11.3674	−0.964169	−0.482084	+	0.876125i	$0.660120\pi$
$140$	0	0
$141$	1.52839	0.128714
$142$	−17.8939	−1.50162
$143$	35.0008	2.92692
$144$	−3.61824	−0.301520
$145$	−10.3799	−0.862002
$146$	−1.78849	−0.148016
$147$	0	0
$148$	−2.11744	−0.174053
$149$	16.2060	1.32765	0.663824	−	0.747889i	$-0.268932\pi$
$149$	16.2060	1.32765	0.663824	+	0.747889i	$0.268932\pi$
$150$	2.49877	0.204024
$151$	−12.9337	−1.05253	−0.526266	−	0.850320i	$-0.676408\pi$
$151$	−12.9337	−1.05253	−0.526266	+	0.850320i	$0.676408\pi$
$152$	0.283361	0.0229836
$153$	7.23463	0.584885
$154$	0	0
$155$	1.77113	0.142261
$156$	13.4404	1.07609
$157$	−9.73442	−0.776892	−0.388446	−	0.921472i	$-0.626988\pi$
$157$	−9.73442	−0.776892	−0.388446	+	0.921472i	$0.626988\pi$
$158$	12.1920	0.969947
$159$	8.95803	0.710418
$160$	−15.7630	−1.24618
$161$	0	0
$162$	2.04340	0.160545
$163$	−24.0922	−1.88705	−0.943525	−	0.331300i	$-0.892513\pi$
$163$	−24.0922	−1.88705	−0.943525	+	0.331300i	$0.892513\pi$
$164$	−3.30333	−0.257947
$165$	11.0104	0.857161
$166$	−0.558981	−0.0433853
$167$	−12.2582	−0.948566	−0.474283	−	0.880373i	$-0.657293\pi$
$167$	−12.2582	−0.948566	−0.474283	+	0.880373i	$0.657293\pi$
$168$	0	0
$169$	25.1692	1.93609
$170$	28.7311	2.20357
$171$	0.790220	0.0604296
$172$	−5.88125	−0.448441
$173$	−5.88796	−0.447653	−0.223826	−	0.974629i	$-0.571855\pi$
$173$	−5.88796	−0.447653	−0.223826	+	0.974629i	$0.571855\pi$
$174$	10.9135	0.827349
$175$	0	0
$176$	20.4984	1.54512
$177$	10.8931	0.818777
$178$	36.0190	2.69974
$179$	−16.9732	−1.26864	−0.634319	−	0.773071i	$-0.718719\pi$
$179$	−16.9732	−1.26864	−0.634319	+	0.773071i	$0.718719\pi$
$180$	4.22803	0.315139
$181$	19.1241	1.42149	0.710743	−	0.703452i	$-0.248359\pi$
$181$	19.1241	1.42149	0.710743	+	0.703452i	$0.248359\pi$
$182$	0	0
$183$	6.31775	0.467021
$184$	−0.358585	−0.0264353
$185$	−1.89164	−0.139076
$186$	−1.86218	−0.136542
$187$	−40.9863	−2.99721
$188$	−3.32500	−0.242500
$189$	0	0
$190$	3.13822	0.227670
$191$	−20.6871	−1.49686	−0.748432	−	0.663212i	$-0.769193\pi$
$191$	−20.6871	−1.49686	−0.748432	+	0.663212i	$0.769193\pi$
$192$	9.33688	0.673832
$193$	2.22901	0.160447	0.0802237	−	0.996777i	$-0.474437\pi$
$193$	2.22901	0.160447	0.0802237	+	0.996777i	$0.474437\pi$
$194$	6.32678	0.454237
$195$	12.0071	0.859847
$196$	0	0
$197$	19.3710	1.38013	0.690064	−	0.723748i	$-0.257582\pi$
$197$	19.3710	1.38013	0.690064	+	0.723748i	$0.257582\pi$
$198$	−11.5765	−0.822703
$199$	24.6840	1.74980	0.874902	−	0.484300i	$-0.160926\pi$
$199$	24.6840	1.74980	0.874902	+	0.484300i	$0.160926\pi$
$200$	−0.438496	−0.0310063
$201$	−7.31833	−0.516195
$202$	27.4381	1.93054
$203$	0	0
$204$	−15.7388	−1.10194
$205$	−2.95106	−0.206111
$206$	12.9481	0.902135
$207$	−1.00000	−0.0695048
$208$	22.3539	1.54996
$209$	−4.47682	−0.309668
$210$	0	0
$211$	9.90378	0.681804	0.340902	−	0.940099i	$-0.389267\pi$
$211$	9.90378	0.681804	0.340902	+	0.940099i	$0.389267\pi$
$212$	−19.4881	−1.33845
$213$	8.75692	0.600014
$214$	32.6144	2.22948
$215$	−5.25407	−0.358324
$216$	−0.358585	−0.0243987
$217$	0	0
$218$	−0.0961859	−0.00651453
$219$	0.875251	0.0591440
$220$	−23.9530	−1.61491
$221$	−44.6964	−3.00661
$222$	1.98888	0.133485
$223$	2.21461	0.148301	0.0741507	−	0.997247i	$-0.476375\pi$
$223$	2.21461	0.148301	0.0741507	+	0.997247i	$0.476375\pi$
$224$	0	0
$225$	−1.22285	−0.0815232
$226$	−12.8150	−0.852441
$227$	−5.58033	−0.370380	−0.185190	−	0.982703i	$-0.559290\pi$
$227$	−5.58033	−0.370380	−0.185190	+	0.982703i	$0.559290\pi$
$228$	−1.71911	−0.113851
$229$	6.97859	0.461158	0.230579	−	0.973054i	$-0.425938\pi$
$229$	6.97859	0.461158	0.230579	+	0.973054i	$0.425938\pi$
$230$	−3.97133	−0.261862
$231$	0	0
$232$	−1.91515	−0.125736
$233$	24.4042	1.59878	0.799388	−	0.600816i	$-0.205157\pi$
$233$	24.4042	1.59878	0.799388	+	0.600816i	$0.205157\pi$
$234$	−12.6244	−0.825281
$235$	−2.97042	−0.193769
$236$	−23.6978	−1.54260
$237$	−5.96655	−0.387569
$238$	0	0
$239$	0.424159	0.0274365	0.0137183	−	0.999906i	$-0.495633\pi$
$239$	0.424159	0.0274365	0.0137183	+	0.999906i	$0.495633\pi$
$240$	7.03200	0.453914
$241$	−16.3125	−1.05078	−0.525389	−	0.850862i	$-0.676080\pi$
$241$	−16.3125	−1.05078	−0.525389	+	0.850862i	$0.676080\pi$
$242$	43.1065	2.77099
$243$	−1.00000	−0.0641500
$244$	−13.7442	−0.879880
$245$	0	0
$246$	3.10277	0.197825
$247$	−4.88207	−0.310639
$248$	0.326785	0.0207508
$249$	0.273554	0.0173358
$250$	−24.7130	−1.56299
$251$	−15.9242	−1.00513	−0.502564	−	0.864540i	$-0.667610\pi$
$251$	−15.9242	−1.00513	−0.502564	+	0.864540i	$0.667610\pi$
$252$	0	0
$253$	5.66529	0.356174
$254$	−25.4045	−1.59402
$255$	−14.0604	−0.880498
$256$	12.8345	0.802154
$257$	−30.1569	−1.88114	−0.940569	−	0.339602i	$-0.889708\pi$
$257$	−30.1569	−1.88114	−0.940569	+	0.339602i	$0.889708\pi$
$258$	5.52417	0.343920
$259$	0	0
$260$	−26.1213	−1.61997
$261$	−5.34085	−0.330590
$262$	−6.76565	−0.417983
$263$	15.1656	0.935153	0.467576	−	0.883953i	$-0.345127\pi$
$263$	15.1656	0.935153	0.467576	+	0.883953i	$0.345127\pi$
$264$	2.03149	0.125030
$265$	−17.4098	−1.06948
$266$	0	0
$267$	−17.6270	−1.07876
$268$	15.9209	0.972525
$269$	0.470833	0.0287072	0.0143536	−	0.999897i	$-0.495431\pi$
$269$	0.470833	0.0287072	0.0143536	+	0.999897i	$0.495431\pi$
$270$	−3.97133	−0.241687
$271$	−7.28736	−0.442676	−0.221338	−	0.975197i	$-0.571042\pi$
$271$	−7.28736	−0.442676	−0.221338	+	0.975197i	$0.571042\pi$
$272$	−26.1766	−1.58719
$273$	0	0
$274$	−26.6545	−1.61026
$275$	6.92779	0.417761
$276$	2.17548	0.130949
$277$	2.96021	0.177862	0.0889310	−	0.996038i	$-0.471655\pi$
$277$	2.96021	0.177862	0.0889310	+	0.996038i	$0.471655\pi$
$278$	−23.2281	−1.39313
$279$	0.911316	0.0545591
$280$	0	0
$281$	7.16141	0.427214	0.213607	−	0.976920i	$-0.431479\pi$
$281$	7.16141	0.427214	0.213607	+	0.976920i	$0.431479\pi$
$282$	3.12312	0.185979
$283$	−20.0106	−1.18951	−0.594753	−	0.803908i	$-0.702750\pi$
$283$	−20.0106	−1.18951	−0.594753	+	0.803908i	$0.702750\pi$
$284$	−19.0505	−1.13044
$285$	−1.53578	−0.0909720
$286$	71.5207	4.22911
$287$	0	0
$288$	−8.11067	−0.477926
$289$	35.3399	2.07882
$290$	−21.2103	−1.24551
$291$	−3.09620	−0.181503
$292$	−1.90409	−0.111429
$293$	6.27910	0.366829	0.183414	−	0.983036i	$-0.441285\pi$
$293$	6.27910	0.366829	0.183414	+	0.983036i	$0.441285\pi$
$294$	0	0
$295$	−21.1707	−1.23260
$296$	−0.349019	−0.0202863
$297$	5.66529	0.328733
$298$	33.1154	1.91832
$299$	6.17812	0.357290
$300$	2.66029	0.153592
$301$	0	0
$302$	−26.4288	−1.52081
$303$	−13.4276	−0.771398
$304$	−2.85920	−0.163986
$305$	−12.2785	−0.703064
$306$	14.7832	0.845102
$307$	25.1923	1.43780	0.718900	−	0.695114i	$-0.244646\pi$
$307$	25.1923	1.43780	0.718900	+	0.695114i	$0.244646\pi$
$308$	0	0
$309$	−6.33653	−0.360473
$310$	3.61913	0.205553
$311$	11.9221	0.676038	0.338019	−	0.941139i	$-0.390243\pi$
$311$	11.9221	0.676038	0.338019	+	0.941139i	$0.390243\pi$
$312$	2.21538	0.125421
$313$	10.0423	0.567625	0.283812	−	0.958880i	$-0.408401\pi$
$313$	10.0423	0.567625	0.283812	+	0.958880i	$0.408401\pi$
$314$	−19.8913	−1.12253
$315$	0	0
$316$	12.9801	0.730189
$317$	−8.86210	−0.497745	−0.248872	−	0.968536i	$-0.580060\pi$
$317$	−8.86210	−0.497745	−0.248872	+	0.968536i	$0.580060\pi$
$318$	18.3049	1.02649
$319$	30.2574	1.69409
$320$	−18.1461	−1.01440
$321$	−15.9609	−0.890849
$322$	0	0
$323$	5.71695	0.318099
$324$	2.17548	0.120860
$325$	7.55490	0.419071
$326$	−49.2301	−2.72660
$327$	0.0470715	0.00260306
$328$	−0.544489	−0.0300644
$329$	0	0
$330$	22.4987	1.23851
$331$	−22.8992	−1.25865	−0.629327	−	0.777141i	$-0.716669\pi$
$331$	−22.8992	−1.25865	−0.629327	+	0.777141i	$0.716669\pi$
$332$	−0.595113	−0.0326611
$333$	−0.973320	−0.0533376
$334$	−25.0483	−1.37058
$335$	14.2231	0.777091
$336$	0	0
$337$	12.2517	0.667392	0.333696	−	0.942681i	$-0.391704\pi$
$337$	12.2517	0.667392	0.333696	+	0.942681i	$0.391704\pi$
$338$	51.4307	2.79746
$339$	6.27141	0.340616
$340$	30.5882	1.65888
$341$	−5.16287	−0.279585
$342$	1.61473	0.0873149
$343$	0	0
$344$	−0.969407	−0.0522669
$345$	1.94349	0.104634
$346$	−12.0314	−0.646815
$347$	−6.45833	−0.346701	−0.173351	−	0.984860i	$-0.555459\pi$
$347$	−6.45833	−0.346701	−0.173351	+	0.984860i	$0.555459\pi$
$348$	11.6189	0.622840
$349$	20.5047	1.09759	0.548795	−	0.835957i	$-0.315087\pi$
$349$	20.5047	1.09759	0.548795	+	0.835957i	$0.315087\pi$
$350$	0	0
$351$	6.17812	0.329764
$352$	45.9493	2.44911
$353$	14.5113	0.772357	0.386178	−	0.922424i	$-0.373795\pi$
$353$	14.5113	0.772357	0.386178	+	0.922424i	$0.373795\pi$
$354$	22.2590	1.18305
$355$	−17.0190	−0.903274
$356$	38.3473	2.03240
$357$	0	0
$358$	−34.6831	−1.83306
$359$	4.67780	0.246885	0.123442	−	0.992352i	$-0.460607\pi$
$359$	4.67780	0.246885	0.123442	+	0.992352i	$0.460607\pi$
$360$	0.696907	0.0367302
$361$	−18.3756	−0.967134
$362$	39.0782	2.05391
$363$	−21.0955	−1.10723
$364$	0	0
$365$	−1.70104	−0.0890366
$366$	12.9097	0.674800
$367$	−28.0660	−1.46503	−0.732515	−	0.680750i	$-0.761654\pi$
$367$	−28.0660	−1.46503	−0.732515	+	0.680750i	$0.761654\pi$
$368$	3.61824	0.188614
$369$	−1.51843	−0.0790465
$370$	−3.86537	−0.200951
$371$	0	0
$372$	−1.98255	−0.102791
$373$	14.6303	0.757527	0.378763	−	0.925494i	$-0.376349\pi$
$373$	14.6303	0.757527	0.378763	+	0.925494i	$0.376349\pi$
$374$	−83.7514	−4.33068
$375$	12.0940	0.624534
$376$	−0.548060	−0.0282640
$377$	32.9964	1.69940
$378$	0	0
$379$	−2.44369	−0.125524	−0.0627620	−	0.998029i	$-0.519991\pi$
$379$	−2.44369	−0.125524	−0.0627620	+	0.998029i	$0.519991\pi$
$380$	3.34107	0.171393
$381$	12.4325	0.636934
$382$	−42.2720	−2.16282
$383$	30.7792	1.57274	0.786371	−	0.617755i	$-0.211958\pi$
$383$	30.7792	1.57274	0.786371	+	0.617755i	$0.211958\pi$
$384$	2.85764	0.145828
$385$	0	0
$386$	4.55475	0.231831
$387$	−2.70342	−0.137423
$388$	6.73575	0.341956
$389$	12.1855	0.617830	0.308915	−	0.951090i	$-0.400034\pi$
$389$	12.1855	0.617830	0.308915	+	0.951090i	$0.400034\pi$
$390$	24.5353	1.24239
$391$	−7.23463	−0.365871
$392$	0	0
$393$	3.31098	0.167017
$394$	39.5827	1.99415
$395$	11.5959	0.583454
$396$	−12.3247	−0.619342
$397$	21.3044	1.06924	0.534618	−	0.845094i	$-0.320455\pi$
$397$	21.3044	1.06924	0.534618	+	0.845094i	$0.320455\pi$
$398$	50.4394	2.52830
$399$	0	0
$400$	4.42455	0.221228
$401$	7.40280	0.369678	0.184839	−	0.982769i	$-0.440824\pi$
$401$	7.40280	0.369678	0.184839	+	0.982769i	$0.440824\pi$
$402$	−14.9543	−0.745852
$403$	−5.63022	−0.280461
$404$	29.2116	1.45333
$405$	1.94349	0.0965728
$406$	0	0
$407$	5.51414	0.273326
$408$	−2.59423	−0.128434
$409$	26.4674	1.30873	0.654364	−	0.756180i	$-0.272936\pi$
$409$	26.4674	1.30873	0.654364	+	0.756180i	$0.272936\pi$
$410$	−6.03020	−0.297810
$411$	13.0442	0.643423
$412$	13.7850	0.679140
$413$	0	0
$414$	−2.04340	−0.100428
$415$	−0.531650	−0.0260977
$416$	50.1087	2.45678
$417$	11.3674	0.556663
$418$	−9.14794	−0.447440
$419$	−30.7294	−1.50123	−0.750614	−	0.660741i	$-0.770242\pi$
$419$	−30.7294	−1.50123	−0.750614	+	0.660741i	$0.770242\pi$
$420$	0	0
$421$	2.65571	0.129432	0.0647158	−	0.997904i	$-0.479386\pi$
$421$	2.65571	0.129432	0.0647158	+	0.997904i	$0.479386\pi$
$422$	20.2374	0.985141
$423$	−1.52839	−0.0743130
$424$	−3.21222	−0.155999
$425$	−8.84685	−0.429135
$426$	17.8939	0.866962
$427$	0	0
$428$	34.7226	1.67838
$429$	−35.0008	−1.68986
$430$	−10.7362	−0.517744
$431$	21.2541	1.02377	0.511887	−	0.859053i	$-0.328947\pi$
$431$	21.2541	1.02377	0.511887	+	0.859053i	$0.328947\pi$
$432$	3.61824	0.174082
$433$	−16.8730	−0.810864	−0.405432	−	0.914125i	$-0.632879\pi$
$433$	−16.8730	−0.810864	−0.405432	+	0.914125i	$0.632879\pi$
$434$	0	0
$435$	10.3799	0.497677
$436$	−0.102403	−0.00490423
$437$	−0.790220	−0.0378013
$438$	1.78849	0.0854573
$439$	−11.0905	−0.529320	−0.264660	−	0.964342i	$-0.585260\pi$
$439$	−11.0905	−0.529320	−0.264660	+	0.964342i	$0.585260\pi$
$440$	−3.94818	−0.188222
$441$	0	0
$442$	−91.3327	−4.34425
$443$	−12.7742	−0.606920	−0.303460	−	0.952844i	$-0.598142\pi$
$443$	−12.7742	−0.606920	−0.303460	+	0.952844i	$0.598142\pi$
$444$	2.11744	0.100489
$445$	34.2579	1.62398
$446$	4.52534	0.214281
$447$	−16.2060	−0.766518
$448$	0	0
$449$	38.7094	1.82681	0.913405	−	0.407052i	$-0.133443\pi$
$449$	38.7094	1.82681	0.913405	+	0.407052i	$0.133443\pi$
$450$	−2.49877	−0.117793
$451$	8.60237	0.405070
$452$	−13.6434	−0.641729
$453$	12.9337	0.607680
$454$	−11.4029	−0.535162
$455$	0	0
$456$	−0.283361	−0.0132696
$457$	−36.2907	−1.69761	−0.848803	−	0.528709i	$-0.822676\pi$
$457$	−36.2907	−1.69761	−0.848803	+	0.528709i	$0.822676\pi$
$458$	14.2601	0.666329
$459$	−7.23463	−0.337684
$460$	−4.22803	−0.197133
$461$	−3.70588	−0.172600	−0.0862999	−	0.996269i	$-0.527504\pi$
$461$	−3.70588	−0.172600	−0.0862999	+	0.996269i	$0.527504\pi$
$462$	0	0
$463$	−29.9926	−1.39387	−0.696937	−	0.717132i	$-0.745454\pi$
$463$	−29.9926	−1.39387	−0.696937	+	0.717132i	$0.745454\pi$
$464$	19.3244	0.897115
$465$	−1.77113	−0.0821343
$466$	49.8676	2.31007
$467$	28.0845	1.29959	0.649797	−	0.760108i	$-0.274854\pi$
$467$	28.0845	1.29959	0.649797	+	0.760108i	$0.274854\pi$
$468$	−13.4404	−0.621283
$469$	0	0
$470$	−6.06975	−0.279977
$471$	9.73442	0.448539
$472$	−3.90611	−0.179793
$473$	15.3156	0.704214
$474$	−12.1920	−0.559999
$475$	−0.966319	−0.0443377
$476$	0	0
$477$	−8.95803	−0.410160
$478$	0.866726	0.0396431
$479$	−12.8184	−0.585688	−0.292844	−	0.956160i	$-0.594602\pi$
$479$	−12.8184	−0.585688	−0.292844	+	0.956160i	$0.594602\pi$
$480$	15.7630	0.719480
$481$	6.01329	0.274182
$482$	−33.3329	−1.51827
$483$	0	0
$484$	45.8929	2.08604
$485$	6.01744	0.273238
$486$	−2.04340	−0.0926905
$487$	9.33744	0.423120	0.211560	−	0.977365i	$-0.432146\pi$
$487$	9.33744	0.423120	0.211560	+	0.977365i	$0.432146\pi$
$488$	−2.26545	−0.102552
$489$	24.0922	1.08949
$490$	0	0
$491$	−1.98455	−0.0895616	−0.0447808	−	0.998997i	$-0.514259\pi$
$491$	−1.98455	−0.0895616	−0.0447808	+	0.998997i	$0.514259\pi$
$492$	3.30333	0.148926
$493$	−38.6391	−1.74022
$494$	−9.97603	−0.448843
$495$	−11.0104	−0.494882
$496$	−3.29736	−0.148056
$497$	0	0
$498$	0.558981	0.0250485
$499$	−40.8742	−1.82978	−0.914890	−	0.403702i	$-0.867723\pi$
$499$	−40.8742	−1.82978	−0.914890	+	0.403702i	$0.867723\pi$
$500$	−26.3104	−1.17664
$501$	12.2582	0.547655
$502$	−32.5396	−1.45231
$503$	−2.08575	−0.0929989	−0.0464994	−	0.998918i	$-0.514807\pi$
$503$	−2.08575	−0.0929989	−0.0464994	+	0.998918i	$0.514807\pi$
$504$	0	0
$505$	26.0965	1.16128
$506$	11.5765	0.514636
$507$	−25.1692	−1.11780
$508$	−27.0466	−1.20000
$509$	−17.4199	−0.772123	−0.386062	−	0.922473i	$-0.626165\pi$
$509$	−17.4199	−0.772123	−0.386062	+	0.922473i	$0.626165\pi$
$510$	−28.7311	−1.27223
$511$	0	0
$512$	31.9412	1.41162
$513$	−0.790220	−0.0348890
$514$	−61.6227	−2.71806
$515$	12.3150	0.542663
$516$	5.88125	0.258907
$517$	8.65879	0.380813
$518$	0	0
$519$	5.88796	0.258453
$520$	−4.30558	−0.188812
$521$	−13.2487	−0.580438	−0.290219	−	0.956960i	$-0.593728\pi$
$521$	−13.2487	−0.580438	−0.290219	+	0.956960i	$0.593728\pi$
$522$	−10.9135	−0.477670
$523$	6.49239	0.283892	0.141946	−	0.989874i	$-0.454664\pi$
$523$	6.49239	0.283892	0.141946	+	0.989874i	$0.454664\pi$
$524$	−7.20298	−0.314664
$525$	0	0
$526$	30.9894	1.35120
$527$	6.59303	0.287197
$528$	−20.4984	−0.892076
$529$	1.00000	0.0434783
$530$	−35.5753	−1.54529
$531$	−10.8931	−0.472721
$532$	0	0
$533$	9.38107	0.406339
$534$	−36.0190	−1.55870
$535$	31.0198	1.34110
$536$	2.62425	0.113350
$537$	16.9732	0.732449
$538$	0.962101	0.0414791
$539$	0	0
$540$	−4.22803	−0.181946
$541$	17.0384	0.732536	0.366268	−	0.930509i	$-0.380635\pi$
$541$	17.0384	0.732536	0.366268	+	0.930509i	$0.380635\pi$
$542$	−14.8910	−0.639623
$543$	−19.1241	−0.820695
$544$	−58.6777	−2.51579
$545$	−0.0914829	−0.00391870
$546$	0	0
$547$	−29.4820	−1.26056	−0.630280	−	0.776368i	$-0.717060\pi$
$547$	−29.4820	−1.26056	−0.630280	+	0.776368i	$0.717060\pi$
$548$	−28.3774	−1.21222
$549$	−6.31775	−0.269635
$550$	14.1562	0.603624
$551$	−4.22044	−0.179797
$552$	0.358585	0.0152624
$553$	0	0
$554$	6.04890	0.256993
$555$	1.89164	0.0802956
$556$	−24.7296	−1.04877
$557$	−10.2316	−0.433526	−0.216763	−	0.976224i	$-0.569550\pi$
$557$	−10.2316	−0.433526	−0.216763	+	0.976224i	$0.569550\pi$
$558$	1.86218	0.0788325
$559$	16.7020	0.706421
$560$	0	0
$561$	40.9863	1.73044
$562$	14.6336	0.617282
$563$	23.2599	0.980288	0.490144	−	0.871641i	$-0.336944\pi$
$563$	23.2599	0.980288	0.490144	+	0.871641i	$0.336944\pi$
$564$	3.32500	0.140008
$565$	−12.1884	−0.512771
$566$	−40.8897	−1.71872
$567$	0	0
$568$	−3.14010	−0.131756
$569$	19.6402	0.823362	0.411681	−	0.911328i	$-0.364942\pi$
$569$	19.6402	0.823362	0.411681	+	0.911328i	$0.364942\pi$
$570$	−3.13822	−0.131446
$571$	−41.5761	−1.73991	−0.869953	−	0.493134i	$-0.835851\pi$
$571$	−41.5761	−1.73991	−0.869953	+	0.493134i	$0.835851\pi$
$572$	76.1438	3.18373
$573$	20.6871	0.864215
$574$	0	0
$575$	1.22285	0.0509963
$576$	−9.33688	−0.389037
$577$	−1.41768	−0.0590187	−0.0295093	−	0.999565i	$-0.509394\pi$
$577$	−1.41768	−0.0590187	−0.0295093	+	0.999565i	$0.509394\pi$
$578$	72.2135	3.00369
$579$	−2.22901	−0.0926344
$580$	−22.5813	−0.937636
$581$	0	0
$582$	−6.32678	−0.262254
$583$	50.7499	2.10184
$584$	−0.313852	−0.0129873
$585$	−12.0071	−0.496433
$586$	12.8307	0.530032
$587$	28.5127	1.17685	0.588423	−	0.808553i	$-0.299749\pi$
$587$	28.5127	1.17685	0.588423	+	0.808553i	$0.299749\pi$
$588$	0	0
$589$	0.720140	0.0296728
$590$	−43.2601	−1.78099
$591$	−19.3710	−0.796817
$592$	3.52170	0.144741
$593$	−8.25387	−0.338946	−0.169473	−	0.985535i	$-0.554206\pi$
$593$	−8.25387	−0.338946	−0.169473	+	0.985535i	$0.554206\pi$
$594$	11.5765	0.474988
$595$	0	0
$596$	35.2559	1.44414
$597$	−24.6840	−1.01025
$598$	12.6244	0.516249
$599$	25.2430	1.03140	0.515700	−	0.856769i	$-0.327532\pi$
$599$	25.2430	1.03140	0.515700	+	0.856769i	$0.327532\pi$
$600$	0.438496	0.0179015
$601$	19.9481	0.813702	0.406851	−	0.913495i	$-0.366627\pi$
$601$	19.9481	0.813702	0.406851	+	0.913495i	$0.366627\pi$
$602$	0	0
$603$	7.31833	0.298026
$604$	−28.1371	−1.14488
$605$	40.9989	1.66684
$606$	−27.4381	−1.11459
$607$	28.0616	1.13899	0.569493	−	0.821996i	$-0.307139\pi$
$607$	28.0616	1.13899	0.569493	+	0.821996i	$0.307139\pi$
$608$	−6.40921	−0.259928
$609$	0	0
$610$	−25.0898	−1.01586
$611$	9.44260	0.382007
$612$	15.7388	0.636204
$613$	−34.9865	−1.41309	−0.706545	−	0.707668i	$-0.749747\pi$
$613$	−34.9865	−1.41309	−0.706545	+	0.707668i	$0.749747\pi$
$614$	51.4779	2.07748
$615$	2.95106	0.118998
$616$	0	0
$617$	−30.9952	−1.24782	−0.623911	−	0.781496i	$-0.714457\pi$
$617$	−30.9952	−1.24782	−0.623911	+	0.781496i	$0.714457\pi$
$618$	−12.9481	−0.520848
$619$	31.2143	1.25461	0.627305	−	0.778774i	$-0.284158\pi$
$619$	31.2143	1.25461	0.627305	+	0.778774i	$0.284158\pi$
$620$	3.85307	0.154743
$621$	1.00000	0.0401286
$622$	24.3615	0.976809
$623$	0	0
$624$	−22.3539	−0.894872
$625$	−17.3904	−0.695616
$626$	20.5205	0.820162
$627$	4.47682	0.178787
$628$	−21.1771	−0.845058
$629$	−7.04161	−0.280767
$630$	0	0
$631$	19.1609	0.762782	0.381391	−	0.924414i	$-0.375445\pi$
$631$	19.1609	0.762782	0.381391	+	0.924414i	$0.375445\pi$
$632$	2.13952	0.0851054
$633$	−9.90378	−0.393640
$634$	−18.1088	−0.719193
$635$	−24.1624	−0.958854
$636$	19.4881	0.772752
$637$	0	0
$638$	61.8281	2.44780
$639$	−8.75692	−0.346418
$640$	−5.55380	−0.219533
$641$	16.1599	0.638278	0.319139	−	0.947708i	$-0.396606\pi$
$641$	16.1599	0.638278	0.319139	+	0.947708i	$0.396606\pi$
$642$	−32.6144	−1.28719
$643$	−44.7116	−1.76325	−0.881627	−	0.471947i	$-0.843551\pi$
$643$	−44.7116	−1.76325	−0.881627	+	0.471947i	$0.843551\pi$
$644$	0	0
$645$	5.25407	0.206879
$646$	11.6820	0.459623
$647$	7.95077	0.312577	0.156288	−	0.987711i	$-0.450047\pi$
$647$	7.95077	0.312577	0.156288	+	0.987711i	$0.450047\pi$
$648$	0.358585	0.0140866
$649$	61.7126	2.42243
$650$	15.4377	0.605516
$651$	0	0
$652$	−52.4123	−2.05262
$653$	−42.7024	−1.67107	−0.835537	−	0.549434i	$-0.814843\pi$
$653$	−42.7024	−1.67107	−0.835537	+	0.549434i	$0.814843\pi$
$654$	0.0961859	0.00376117
$655$	−6.43485	−0.251430
$656$	5.49405	0.214507
$657$	−0.875251	−0.0341468
$658$	0	0
$659$	−31.2099	−1.21577	−0.607883	−	0.794027i	$-0.707981\pi$
$659$	−31.2099	−1.21577	−0.607883	+	0.794027i	$0.707981\pi$
$660$	23.9530	0.932370
$661$	5.89998	0.229483	0.114741	−	0.993395i	$-0.463396\pi$
$661$	5.89998	0.229483	0.114741	+	0.993395i	$0.463396\pi$
$662$	−46.7922	−1.81863
$663$	44.6964	1.73586
$664$	−0.0980926	−0.00380673
$665$	0	0
$666$	−1.98888	−0.0770677
$667$	5.34085	0.206798
$668$	−26.6675	−1.03180
$669$	−2.21461	−0.0856218
$670$	29.0635	1.12282
$671$	35.7919	1.38173
$672$	0	0
$673$	−6.61159	−0.254858	−0.127429	−	0.991848i	$-0.540673\pi$
$673$	−6.61159	−0.254858	−0.127429	+	0.991848i	$0.540673\pi$
$674$	25.0351	0.964316
$675$	1.22285	0.0470675
$676$	54.7551	2.10597
$677$	3.71673	0.142845	0.0714227	−	0.997446i	$-0.477246\pi$
$677$	3.71673	0.142845	0.0714227	+	0.997446i	$0.477246\pi$
$678$	12.8150	0.492157
$679$	0	0
$680$	5.04187	0.193347
$681$	5.58033	0.213839
$682$	−10.5498	−0.403973
$683$	29.5669	1.13135	0.565674	−	0.824629i	$-0.308616\pi$
$683$	29.5669	1.13135	0.565674	+	0.824629i	$0.308616\pi$
$684$	1.71911	0.0657318
$685$	−25.3513	−0.968622
$686$	0	0
$687$	−6.97859	−0.266250
$688$	9.78161	0.372920
$689$	55.3438	2.10843
$690$	3.97133	0.151186
$691$	0.603892	0.0229731	0.0114866	−	0.999934i	$-0.496344\pi$
$691$	0.603892	0.0229731	0.0114866	+	0.999934i	$0.496344\pi$
$692$	−12.8092	−0.486931
$693$	0	0
$694$	−13.1969	−0.500949
$695$	−22.0924	−0.838012
$696$	1.91515	0.0725936
$697$	−10.9853	−0.416098
$698$	41.8992	1.58591
$699$	−24.4042	−0.923053
$700$	0	0
$701$	8.15835	0.308136	0.154068	−	0.988060i	$-0.450762\pi$
$701$	8.15835	0.308136	0.154068	+	0.988060i	$0.450762\pi$
$702$	12.6244	0.476476
$703$	−0.769137	−0.0290085
$704$	52.8961	1.99360
$705$	2.97042	0.111872
$706$	29.6523	1.11598
$707$	0	0
$708$	23.6978	0.890618
$709$	−24.2270	−0.909863	−0.454932	−	0.890526i	$-0.650336\pi$
$709$	−24.2270	−0.909863	−0.454932	+	0.890526i	$0.650336\pi$
$710$	−34.7766	−1.30514
$711$	5.96655	0.223763
$712$	6.32079	0.236882
$713$	−0.911316	−0.0341290
$714$	0	0
$715$	68.0238	2.54394
$716$	−36.9250	−1.37995
$717$	−0.424159	−0.0158405
$718$	9.55861	0.356724
$719$	2.58504	0.0964058	0.0482029	−	0.998838i	$-0.484651\pi$
$719$	2.58504	0.0964058	0.0482029	+	0.998838i	$0.484651\pi$
$720$	−7.03200	−0.262067
$721$	0	0
$722$	−37.5486	−1.39741
$723$	16.3125	0.606667
$724$	41.6042	1.54621
$725$	6.53105	0.242557
$726$	−43.1065	−1.59983
$727$	−1.39574	−0.0517650	−0.0258825	−	0.999665i	$-0.508240\pi$
$727$	−1.39574	−0.0517650	−0.0258825	+	0.999665i	$0.508240\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−3.47591	−0.128649
$731$	−19.5582	−0.723388
$732$	13.7442	0.507999
$733$	20.4900	0.756815	0.378408	−	0.925639i	$-0.376472\pi$
$733$	20.4900	0.756815	0.378408	+	0.925639i	$0.376472\pi$
$734$	−57.3500	−2.11683
$735$	0	0
$736$	8.11067	0.298963
$737$	−41.4605	−1.52722
$738$	−3.10277	−0.114215
$739$	−47.4816	−1.74664	−0.873320	−	0.487148i	$-0.838037\pi$
$739$	−47.4816	−1.74664	−0.873320	+	0.487148i	$0.838037\pi$
$740$	−4.11523	−0.151279
$741$	4.88207	0.179347
$742$	0	0
$743$	−41.4450	−1.52047	−0.760235	−	0.649648i	$-0.774916\pi$
$743$	−41.4450	−1.52047	−0.760235	+	0.649648i	$0.774916\pi$
$744$	−0.326785	−0.0119805
$745$	31.4962	1.15393
$746$	29.8955	1.09455
$747$	−0.273554	−0.0100088
$748$	−89.1650	−3.26020
$749$	0	0
$750$	24.7130	0.902390
$751$	11.5886	0.422872	0.211436	−	0.977392i	$-0.432186\pi$
$751$	11.5886	0.422872	0.211436	+	0.977392i	$0.432186\pi$
$752$	5.53009	0.201662
$753$	15.9242	0.580311
$754$	67.4248	2.45547
$755$	−25.1366	−0.914814
$756$	0	0
$757$	−11.3101	−0.411073	−0.205536	−	0.978649i	$-0.565894\pi$
$757$	−11.3101	−0.411073	−0.205536	+	0.978649i	$0.565894\pi$
$758$	−4.99344	−0.181370
$759$	−5.66529	−0.205637
$760$	0.550710	0.0199763
$761$	−21.8257	−0.791183	−0.395591	−	0.918427i	$-0.629460\pi$
$761$	−21.8257	−0.791183	−0.395591	+	0.918427i	$0.629460\pi$
$762$	25.4045	0.920308
$763$	0	0
$764$	−45.0044	−1.62820
$765$	14.0604	0.508356
$766$	62.8941	2.27246
$767$	67.2990	2.43002
$768$	−12.8345	−0.463124
$769$	−24.1550	−0.871052	−0.435526	−	0.900176i	$-0.643438\pi$
$769$	−24.1550	−0.871052	−0.435526	+	0.900176i	$0.643438\pi$
$770$	0	0
$771$	30.1569	1.08608
$772$	4.84917	0.174525
$773$	14.7390	0.530124	0.265062	−	0.964231i	$-0.414608\pi$
$773$	14.7390	0.530124	0.265062	+	0.964231i	$0.414608\pi$
$774$	−5.52417	−0.198562
$775$	−1.11440	−0.0400305
$776$	1.11025	0.0398558
$777$	0	0
$778$	24.8999	0.892703
$779$	−1.19990	−0.0429908
$780$	26.1213	0.935292
$781$	49.6105	1.77520
$782$	−14.7832	−0.528648
$783$	5.34085	0.190866
$784$	0	0
$785$	−18.9188	−0.675239
$786$	6.76565	0.241323
$787$	23.8103	0.848746	0.424373	−	0.905487i	$-0.360495\pi$
$787$	23.8103	0.848746	0.424373	+	0.905487i	$0.360495\pi$
$788$	42.1414	1.50122
$789$	−15.1656	−0.539911
$790$	23.6951	0.843034
$791$	0	0
$792$	−2.03149	−0.0721859
$793$	39.0318	1.38606
$794$	43.5334	1.54494
$795$	17.4098	0.617464
$796$	53.6997	1.90334
$797$	19.8938	0.704674	0.352337	−	0.935873i	$-0.385387\pi$
$797$	19.8938	0.704674	0.352337	+	0.935873i	$0.385387\pi$
$798$	0	0
$799$	−11.0574	−0.391181
$800$	9.91812	0.350659
$801$	17.6270	0.622820
$802$	15.1269	0.534149
$803$	4.95855	0.174983
$804$	−15.9209	−0.561488
$805$	0	0
$806$	−11.5048	−0.405239
$807$	−0.470833	−0.0165741
$808$	4.81496	0.169390
$809$	−47.9127	−1.68452	−0.842260	−	0.539071i	$-0.818775\pi$
$809$	−47.9127	−1.68452	−0.842260	+	0.539071i	$0.818775\pi$
$810$	3.97133	0.139538
$811$	−48.3843	−1.69900	−0.849500	−	0.527588i	$-0.823097\pi$
$811$	−48.3843	−1.69900	−0.849500	+	0.527588i	$0.823097\pi$
$812$	0	0
$813$	7.28736	0.255579
$814$	11.2676	0.394929
$815$	−46.8230	−1.64014
$816$	26.1766	0.916364
$817$	−2.13629	−0.0747395
$818$	54.0835	1.89098
$819$	0	0
$820$	−6.41999	−0.224196
$821$	−8.16367	−0.284914	−0.142457	−	0.989801i	$-0.545500\pi$
$821$	−8.16367	−0.284914	−0.142457	+	0.989801i	$0.545500\pi$
$822$	26.6545	0.929683
$823$	−33.2648	−1.15954	−0.579769	−	0.814781i	$-0.696857\pi$
$823$	−33.2648	−1.15954	−0.579769	+	0.814781i	$0.696857\pi$
$824$	2.27219	0.0791554
$825$	−6.92779	−0.241195
$826$	0	0
$827$	31.2711	1.08740	0.543701	−	0.839279i	$-0.317023\pi$
$827$	31.2711	1.08740	0.543701	+	0.839279i	$0.317023\pi$
$828$	−2.17548	−0.0756033
$829$	−17.8076	−0.618484	−0.309242	−	0.950983i	$-0.600075\pi$
$829$	−17.8076	−0.618484	−0.309242	+	0.950983i	$0.600075\pi$
$830$	−1.08637	−0.0377086
$831$	−2.96021	−0.102689
$832$	57.6844	1.99985
$833$	0	0
$834$	23.2281	0.804324
$835$	−23.8236	−0.824451
$836$	−9.73926	−0.336839
$837$	−0.911316	−0.0314997
$838$	−62.7924	−2.16913
$839$	29.7947	1.02863	0.514313	−	0.857603i	$-0.328047\pi$
$839$	29.7947	1.02863	0.514313	+	0.857603i	$0.328047\pi$
$840$	0	0
$841$	−0.475352	−0.0163915
$842$	5.42669	0.187016
$843$	−7.16141	−0.246652
$844$	21.5455	0.741628
$845$	48.9160	1.68276
$846$	−3.12312	−0.107375
$847$	0	0
$848$	32.4123	1.11304
$849$	20.0106	0.686762
$850$	−18.0777	−0.620059
$851$	0.973320	0.0333650
$852$	19.0505	0.652661
$853$	−43.0902	−1.47538	−0.737690	−	0.675140i	$-0.764083\pi$
$853$	−43.0902	−1.47538	−0.737690	+	0.675140i	$0.764083\pi$
$854$	0	0
$855$	1.53578	0.0525227
$856$	5.72333	0.195620
$857$	48.0096	1.63998	0.819988	−	0.572381i	$-0.193980\pi$
$857$	48.0096	1.63998	0.819988	+	0.572381i	$0.193980\pi$
$858$	−71.5207	−2.44168
$859$	11.7497	0.400896	0.200448	−	0.979704i	$-0.435760\pi$
$859$	11.7497	0.400896	0.200448	+	0.979704i	$0.435760\pi$
$860$	−11.4301	−0.389765
$861$	0	0
$862$	43.4306	1.47925
$863$	−5.60759	−0.190885	−0.0954424	−	0.995435i	$-0.530427\pi$
$863$	−5.60759	−0.190885	−0.0954424	+	0.995435i	$0.530427\pi$
$864$	8.11067	0.275931
$865$	−11.4432	−0.389080
$866$	−34.4783	−1.17162
$867$	−35.3399	−1.20021
$868$	0	0
$869$	−33.8022	−1.14666
$870$	21.2103	0.719095
$871$	−45.2135	−1.53200
$872$	−0.0168791	−0.000571600 0
$873$	3.09620	0.104791
$874$	−1.61473	−0.0546192
$875$	0	0
$876$	1.90409	0.0643334
$877$	16.7638	0.566075	0.283037	−	0.959109i	$-0.408658\pi$
$877$	16.7638	0.566075	0.283037	+	0.959109i	$0.408658\pi$
$878$	−22.6623	−0.764816
$879$	−6.27910	−0.211789
$880$	39.8383	1.34295
$881$	−39.4239	−1.32823	−0.664113	−	0.747632i	$-0.731191\pi$
$881$	−39.4239	−1.32823	−0.664113	+	0.747632i	$0.731191\pi$
$882$	0	0
$883$	−17.4928	−0.588679	−0.294339	−	0.955701i	$-0.595100\pi$
$883$	−17.4928	−0.588679	−0.294339	+	0.955701i	$0.595100\pi$
$884$	−97.2364	−3.27041
$885$	21.1707	0.711644
$886$	−26.1028	−0.876940
$887$	2.75212	0.0924073	0.0462037	−	0.998932i	$-0.485288\pi$
$887$	2.75212	0.0924073	0.0462037	+	0.998932i	$0.485288\pi$
$888$	0.349019	0.0117123
$889$	0	0
$890$	70.0026	2.34649
$891$	−5.66529	−0.189794
$892$	4.81785	0.161314
$893$	−1.20777	−0.0404164
$894$	−33.1154	−1.10754
$895$	−32.9873	−1.10264
$896$	0	0
$897$	−6.17812	−0.206281
$898$	79.0988	2.63956
$899$	−4.86720	−0.162330
$900$	−2.66029	−0.0886763
$901$	−64.8081	−2.15907
$902$	17.5781	0.585286
$903$	0	0
$904$	−2.24884	−0.0747952
$905$	37.1675	1.23549
$906$	26.4288	0.878038
$907$	−29.1794	−0.968886	−0.484443	−	0.874823i	$-0.660978\pi$
$907$	−29.1794	−0.968886	−0.484443	+	0.874823i	$0.660978\pi$
$908$	−12.1399	−0.402878
$909$	13.4276	0.445367
$910$	0	0
$911$	−6.24020	−0.206747	−0.103374	−	0.994643i	$-0.532964\pi$
$911$	−6.24020	−0.206747	−0.103374	+	0.994643i	$0.532964\pi$
$912$	2.85920	0.0946776
$913$	1.54976	0.0512897
$914$	−74.1564	−2.45288
$915$	12.2785	0.405914
$916$	15.1818	0.501621
$917$	0	0
$918$	−14.7832	−0.487920
$919$	−36.4022	−1.20080	−0.600398	−	0.799701i	$-0.704991\pi$
$919$	−36.4022	−1.20080	−0.600398	+	0.799701i	$0.704991\pi$
$920$	−0.696907	−0.0229763
$921$	−25.1923	−0.830114
$922$	−7.57259	−0.249390
$923$	54.1013	1.78077
$924$	0	0
$925$	1.19022	0.0391343
$926$	−61.2869	−2.01401
$927$	6.33653	0.208119
$928$	43.3179	1.42198
$929$	−0.0513202	−0.00168376	−0.000841881	−	1.00000i	$-0.500268\pi$
$929$	−0.0513202	−0.00168376	−0.000841881		1.00000i	$0.500268\pi$
$930$	−3.61913	−0.118676
$931$	0	0
$932$	53.0911	1.73906
$933$	−11.9221	−0.390311
$934$	57.3878	1.87779
$935$	−79.6564	−2.60504
$936$	−2.21538	−0.0724121
$937$	−8.46139	−0.276421	−0.138211	−	0.990403i	$-0.544135\pi$
$937$	−8.46139	−0.276421	−0.138211	+	0.990403i	$0.544135\pi$
$938$	0	0
$939$	−10.0423	−0.327718
$940$	−6.46210	−0.210770
$941$	−6.94143	−0.226284	−0.113142	−	0.993579i	$-0.536092\pi$
$941$	−6.94143	−0.226284	−0.113142	+	0.993579i	$0.536092\pi$
$942$	19.8913	0.648094
$943$	1.51843	0.0494470
$944$	39.4139	1.28281
$945$	0	0
$946$	31.2960	1.01752
$947$	6.05341	0.196709	0.0983547	−	0.995151i	$-0.468642\pi$
$947$	6.05341	0.196709	0.0983547	+	0.995151i	$0.468642\pi$
$948$	−12.9801	−0.421575
$949$	5.40740	0.175532
$950$	−1.97458	−0.0640637
$951$	8.86210	0.287373
$952$	0	0
$953$	−52.7135	−1.70756	−0.853778	−	0.520637i	$-0.825695\pi$
$953$	−52.7135	−1.70756	−0.853778	+	0.520637i	$0.825695\pi$
$954$	−18.3049	−0.592641
$955$	−40.2051	−1.30101
$956$	0.922751	0.0298439
$957$	−30.2574	−0.978084
$958$	−26.1931	−0.846262
$959$	0	0
$960$	18.1461	0.585664
$961$	−30.1695	−0.973210
$962$	12.2876	0.396167
$963$	15.9609	0.514332
$964$	−35.4875	−1.14298
$965$	4.33205	0.139454
$966$	0	0
$967$	−16.9757	−0.545900	−0.272950	−	0.962028i	$-0.587999\pi$
$967$	−16.9757	−0.545900	−0.272950	+	0.962028i	$0.587999\pi$
$968$	7.56454	0.243133
$969$	−5.71695	−0.183655
$970$	12.2960	0.394802
$971$	−29.0124	−0.931051	−0.465525	−	0.885034i	$-0.654135\pi$
$971$	−29.0124	−0.931051	−0.465525	+	0.885034i	$0.654135\pi$
$972$	−2.17548	−0.0697787
$973$	0	0
$974$	19.0801	0.611367
$975$	−7.55490	−0.241951
$976$	22.8591	0.731702
$977$	−32.1866	−1.02974	−0.514871	−	0.857268i	$-0.672160\pi$
$977$	−32.1866	−1.02974	−0.514871	+	0.857268i	$0.672160\pi$
$978$	49.2301	1.57421
$979$	−99.8621	−3.19161
$980$	0	0
$981$	−0.0470715	−0.00150288
$982$	−4.05523	−0.129408
$983$	6.03861	0.192602	0.0963009	−	0.995352i	$-0.469299\pi$
$983$	6.03861	0.192602	0.0963009	+	0.995352i	$0.469299\pi$
$984$	0.544489	0.0173577
$985$	37.6474	1.19955
$986$	−78.9551	−2.51444
$987$	0	0
$988$	−10.6209	−0.337895
$989$	2.70342	0.0859637
$990$	−22.4987	−0.715056
$991$	20.8108	0.661075	0.330538	−	0.943793i	$-0.392770\pi$
$991$	20.8108	0.661075	0.330538	+	0.943793i	$0.392770\pi$
$992$	−7.39139	−0.234677
$993$	22.8992	0.726684
$994$	0	0
$995$	47.9732	1.52085
$996$	0.595113	0.0188569
$997$	−53.1594	−1.68358	−0.841788	−	0.539809i	$-0.818496\pi$
$997$	−53.1594	−1.68358	−0.841788	+	0.539809i	$0.818496\pi$
$998$	−83.5224	−2.64385
$999$	0.973320	0.0307945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bc.1.6		6
7.2	even	3		483.2.i.f.277.1	✓	12
7.4	even	3		483.2.i.f.415.1	yes	12
7.6	odd	2		3381.2.a.bd.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.f.277.1	✓	12	7.2	even	3
483.2.i.f.415.1	yes	12	7.4	even	3
3381.2.a.bc.1.6		6	1.1	even	1	trivial
3381.2.a.bd.1.6		6	7.6	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+2.04340 q^{2} -1.00000 q^{3} +2.17548 q^{4} +1.94349 q^{5} -2.04340 q^{6} +0.358585 q^{8} +1.00000 q^{9} +3.97133 q^{10} -5.66529 q^{11} -2.17548 q^{12} -6.17812 q^{13} -1.94349 q^{15} -3.61824 q^{16} +7.23463 q^{17} +2.04340 q^{18} +0.790220 q^{19} +4.22803 q^{20} -11.5765 q^{22} -1.00000 q^{23} -0.358585 q^{24} -1.22285 q^{25} -12.6244 q^{26} -1.00000 q^{27} -5.34085 q^{29} -3.97133 q^{30} +0.911316 q^{31} -8.11067 q^{32} +5.66529 q^{33} +14.7832 q^{34} +2.17548 q^{36} -0.973320 q^{37} +1.61473 q^{38} +6.17812 q^{39} +0.696907 q^{40} -1.51843 q^{41} -2.70342 q^{43} -12.3247 q^{44} +1.94349 q^{45} -2.04340 q^{46} -1.52839 q^{47} +3.61824 q^{48} -2.49877 q^{50} -7.23463 q^{51} -13.4404 q^{52} -8.95803 q^{53} -2.04340 q^{54} -11.0104 q^{55} -0.790220 q^{57} -10.9135 q^{58} -10.8931 q^{59} -4.22803 q^{60} -6.31775 q^{61} +1.86218 q^{62} -9.33688 q^{64} -12.0071 q^{65} +11.5765 q^{66} +7.31833 q^{67} +15.7388 q^{68} +1.00000 q^{69} -8.75692 q^{71} +0.358585 q^{72} -0.875251 q^{73} -1.98888 q^{74} +1.22285 q^{75} +1.71911 q^{76} +12.6244 q^{78} +5.96655 q^{79} -7.03200 q^{80} +1.00000 q^{81} -3.10277 q^{82} -0.273554 q^{83} +14.0604 q^{85} -5.52417 q^{86} +5.34085 q^{87} -2.03149 q^{88} +17.6270 q^{89} +3.97133 q^{90} -2.17548 q^{92} -0.911316 q^{93} -3.12312 q^{94} +1.53578 q^{95} +8.11067 q^{96} +3.09620 q^{97} -5.66529 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 6 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{8} + 6 q^{9} - 3 q^{10} - 14 q^{11} - 3 q^{12} - 3 q^{15} - 7 q^{16} + 15 q^{17} - q^{18} - q^{19} + 17 q^{20} - 6 q^{22} - 6 q^{23} + 3 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) \) 6 * q - q^2 - 6 * q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^8 + 6 * q^9 - 3 * q^10 - 14 * q^11 - 3 * q^12 - 3 * q^15 - 7 * q^16 + 15 * q^17 - q^18 - q^19 + 17 * q^20 - 6 * q^22 - 6 * q^23 + 3 * q^24 + 9 * q^25 - 15 * q^26 - 6 * q^27 - 6 * q^29 + 3 * q^30 - 11 * q^31 + 3 * q^32 + 14 * q^33 + 15 * q^34 + 3 * q^36 - 5 * q^37 + 14 * q^38 - 17 * q^40 + 18 * q^41 - 37 * q^43 - 10 * q^44 + 3 * q^45 + q^46 + 3 * q^47 + 7 * q^48 - 30 * q^50 - 15 * q^51 - 7 * q^52 - 15 * q^53 + q^54 - 2 * q^55 + q^57 + 4 * q^58 - 2 * q^59 - 17 * q^60 - 12 * q^61 + 36 * q^62 - 23 * q^64 - 17 * q^65 + 6 * q^66 - 10 * q^67 - q^68 + 6 * q^69 - 21 * q^71 - 3 * q^72 - 8 * q^73 - 16 * q^74 - 9 * q^75 + 18 * q^76 + 15 * q^78 - 17 * q^79 + 3 * q^80 + 6 * q^81 - 48 * q^82 + 12 * q^83 - 13 * q^85 + 22 * q^86 + 6 * q^87 - 2 * q^88 + 18 * q^89 - 3 * q^90 - 3 * q^92 + 11 * q^93 + 3 * q^94 - 16 * q^95 - 3 * q^96 - 2 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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