LMFDB - Embedded newform 3234.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.23607 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.23607 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.23607 q^{10} -1.00000 q^{11} +1.00000 q^{12} +3.23607 q^{15} +1.00000 q^{16} +0.763932 q^{17} +1.00000 q^{18} -5.70820 q^{19} +3.23607 q^{20} -1.00000 q^{22} +6.47214 q^{23} +1.00000 q^{24} +5.47214 q^{25} +1.00000 q^{27} -4.47214 q^{29} +3.23607 q^{30} +7.23607 q^{31} +1.00000 q^{32} -1.00000 q^{33} +0.763932 q^{34} +1.00000 q^{36} +6.94427 q^{37} -5.70820 q^{38} +3.23607 q^{40} +0.763932 q^{41} -2.47214 q^{43} -1.00000 q^{44} +3.23607 q^{45} +6.47214 q^{46} +9.70820 q^{47} +1.00000 q^{48} +5.47214 q^{50} +0.763932 q^{51} -6.00000 q^{53} +1.00000 q^{54} -3.23607 q^{55} -5.70820 q^{57} -4.47214 q^{58} +10.4721 q^{59} +3.23607 q^{60} +7.23607 q^{62} +1.00000 q^{64} -1.00000 q^{66} -11.4164 q^{67} +0.763932 q^{68} +6.47214 q^{69} +6.47214 q^{71} +1.00000 q^{72} +2.29180 q^{73} +6.94427 q^{74} +5.47214 q^{75} -5.70820 q^{76} -15.4164 q^{79} +3.23607 q^{80} +1.00000 q^{81} +0.763932 q^{82} +3.23607 q^{83} +2.47214 q^{85} -2.47214 q^{86} -4.47214 q^{87} -1.00000 q^{88} -2.47214 q^{89} +3.23607 q^{90} +6.47214 q^{92} +7.23607 q^{93} +9.70820 q^{94} -18.4721 q^{95} +1.00000 q^{96} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} + 2 q^{19} + 2 q^{20} - 2 q^{22} + 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} + 2 q^{30} + 10 q^{31} + 2 q^{32} - 2 q^{33} + 6 q^{34} + 2 q^{36} - 4 q^{37} + 2 q^{38} + 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} + 2 q^{45} + 4 q^{46} + 6 q^{47} + 2 q^{48} + 2 q^{50} + 6 q^{51} - 12 q^{53} + 2 q^{54} - 2 q^{55} + 2 q^{57} + 12 q^{59} + 2 q^{60} + 10 q^{62} + 2 q^{64} - 2 q^{66} + 4 q^{67} + 6 q^{68} + 4 q^{69} + 4 q^{71} + 2 q^{72} + 18 q^{73} - 4 q^{74} + 2 q^{75} + 2 q^{76} - 4 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} + 2 q^{83} - 4 q^{85} + 4 q^{86} - 2 q^{88} + 4 q^{89} + 2 q^{90} + 4 q^{92} + 10 q^{93} + 6 q^{94} - 28 q^{95} + 2 q^{96} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + 2 * q^15 + 2 * q^16 + 6 * q^17 + 2 * q^18 + 2 * q^19 + 2 * q^20 - 2 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 + 2 * q^30 + 10 * q^31 + 2 * q^32 - 2 * q^33 + 6 * q^34 + 2 * q^36 - 4 * q^37 + 2 * q^38 + 2 * q^40 + 6 * q^41 + 4 * q^43 - 2 * q^44 + 2 * q^45 + 4 * q^46 + 6 * q^47 + 2 * q^48 + 2 * q^50 + 6 * q^51 - 12 * q^53 + 2 * q^54 - 2 * q^55 + 2 * q^57 + 12 * q^59 + 2 * q^60 + 10 * q^62 + 2 * q^64 - 2 * q^66 + 4 * q^67 + 6 * q^68 + 4 * q^69 + 4 * q^71 + 2 * q^72 + 18 * q^73 - 4 * q^74 + 2 * q^75 + 2 * q^76 - 4 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 + 2 * q^83 - 4 * q^85 + 4 * q^86 - 2 * q^88 + 4 * q^89 + 2 * q^90 + 4 * q^92 + 10 * q^93 + 6 * q^94 - 28 * q^95 + 2 * q^96 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.23607	1.02333
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	3.23607	0.835549
$16$	1.00000	0.250000
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	1.00000	0.235702
$19$	−5.70820	−1.30955	−0.654776	−	0.755823i	$-0.727237\pi$
$19$	−5.70820	−1.30955	−0.654776	+	0.755823i	$0.727237\pi$
$20$	3.23607	0.723607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	6.47214	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$24$	1.00000	0.204124
$25$	5.47214	1.09443
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	3.23607	0.590822
$31$	7.23607	1.29964	0.649818	−	0.760090i	$-0.274845\pi$
$31$	7.23607	1.29964	0.649818	+	0.760090i	$0.274845\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	0.763932	0.131013
$35$	0	0
$36$	1.00000	0.166667
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	−5.70820	−0.925993
$39$	0	0
$40$	3.23607	0.511667
$41$	0.763932	0.119306	0.0596531	−	0.998219i	$-0.481001\pi$
$41$	0.763932	0.119306	0.0596531	+	0.998219i	$0.481001\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	−1.00000	−0.150756
$45$	3.23607	0.482405
$46$	6.47214	0.954264
$47$	9.70820	1.41609	0.708044	−	0.706169i	$-0.249578\pi$
$47$	9.70820	1.41609	0.708044	+	0.706169i	$0.249578\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	5.47214	0.773877
$51$	0.763932	0.106972
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	−3.23607	−0.436351
$56$	0	0
$57$	−5.70820	−0.756070
$58$	−4.47214	−0.587220
$59$	10.4721	1.36336	0.681678	−	0.731652i	$-0.261251\pi$
$59$	10.4721	1.36336	0.681678	+	0.731652i	$0.261251\pi$
$60$	3.23607	0.417775
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	7.23607	0.918982
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	−11.4164	−1.39474	−0.697368	−	0.716713i	$-0.745646\pi$
$67$	−11.4164	−1.39474	−0.697368	+	0.716713i	$0.745646\pi$
$68$	0.763932	0.0926404
$69$	6.47214	0.779154
$70$	0	0
$71$	6.47214	0.768101	0.384051	−	0.923312i	$-0.374529\pi$
$71$	6.47214	0.768101	0.384051	+	0.923312i	$0.374529\pi$
$72$	1.00000	0.117851
$73$	2.29180	0.268234	0.134117	−	0.990965i	$-0.457180\pi$
$73$	2.29180	0.268234	0.134117	+	0.990965i	$0.457180\pi$
$74$	6.94427	0.807255
$75$	5.47214	0.631868
$76$	−5.70820	−0.654776
$77$	0	0
$78$	0	0
$79$	−15.4164	−1.73448	−0.867241	−	0.497889i	$-0.834109\pi$
$79$	−15.4164	−1.73448	−0.867241	+	0.497889i	$0.834109\pi$
$80$	3.23607	0.361803
$81$	1.00000	0.111111
$82$	0.763932	0.0843622
$83$	3.23607	0.355205	0.177602	−	0.984102i	$-0.443166\pi$
$83$	3.23607	0.355205	0.177602	+	0.984102i	$0.443166\pi$
$84$	0	0
$85$	2.47214	0.268141
$86$	−2.47214	−0.266577
$87$	−4.47214	−0.479463
$88$	−1.00000	−0.106600
$89$	−2.47214	−0.262046	−0.131023	−	0.991379i	$-0.541826\pi$
$89$	−2.47214	−0.262046	−0.131023	+	0.991379i	$0.541826\pi$
$90$	3.23607	0.341112
$91$	0	0
$92$	6.47214	0.674767
$93$	7.23607	0.750345
$94$	9.70820	1.00132
$95$	−18.4721	−1.89520
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	5.47214	0.547214
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0.763932	0.0756405
$103$	−15.2361	−1.50125	−0.750627	−	0.660726i	$-0.770249\pi$
$103$	−15.2361	−1.50125	−0.750627	+	0.660726i	$0.770249\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−16.9443	−1.63806	−0.819032	−	0.573747i	$-0.805489\pi$
$107$	−16.9443	−1.63806	−0.819032	+	0.573747i	$0.805489\pi$
$108$	1.00000	0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−3.23607	−0.308547
$111$	6.94427	0.659121
$112$	0	0
$113$	−13.4164	−1.26211	−0.631055	−	0.775738i	$-0.717378\pi$
$113$	−13.4164	−1.26211	−0.631055	+	0.775738i	$0.717378\pi$
$114$	−5.70820	−0.534622
$115$	20.9443	1.95306
$116$	−4.47214	−0.415227
$117$	0	0
$118$	10.4721	0.964038
$119$	0	0
$120$	3.23607	0.295411
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.763932	0.0688814
$124$	7.23607	0.649818
$125$	1.52786	0.136656
$126$	0	0
$127$	2.47214	0.219367	0.109683	−	0.993967i	$-0.465016\pi$
$127$	2.47214	0.219367	0.109683	+	0.993967i	$0.465016\pi$
$128$	1.00000	0.0883883
$129$	−2.47214	−0.217659
$130$	0	0
$131$	−11.2361	−0.981700	−0.490850	−	0.871244i	$-0.663314\pi$
$131$	−11.2361	−0.981700	−0.490850	+	0.871244i	$0.663314\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−11.4164	−0.986227
$135$	3.23607	0.278516
$136$	0.763932	0.0655066
$137$	−7.52786	−0.643149	−0.321574	−	0.946884i	$-0.604212\pi$
$137$	−7.52786	−0.643149	−0.321574	+	0.946884i	$0.604212\pi$
$138$	6.47214	0.550945
$139$	−12.1803	−1.03312	−0.516561	−	0.856250i	$-0.672788\pi$
$139$	−12.1803	−1.03312	−0.516561	+	0.856250i	$0.672788\pi$
$140$	0	0
$141$	9.70820	0.817578
$142$	6.47214	0.543130
$143$	0	0
$144$	1.00000	0.0833333
$145$	−14.4721	−1.20185
$146$	2.29180	0.189670
$147$	0	0
$148$	6.94427	0.570816
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	5.47214	0.446798
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−5.70820	−0.462996
$153$	0.763932	0.0617602
$154$	0	0
$155$	23.4164	1.88085
$156$	0	0
$157$	19.2361	1.53521	0.767603	−	0.640926i	$-0.221449\pi$
$157$	19.2361	1.53521	0.767603	+	0.640926i	$0.221449\pi$
$158$	−15.4164	−1.22646
$159$	−6.00000	−0.475831
$160$	3.23607	0.255834
$161$	0	0
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0.763932	0.0596531
$165$	−3.23607	−0.251928
$166$	3.23607	0.251168
$167$	−9.52786	−0.737288	−0.368644	−	0.929571i	$-0.620178\pi$
$167$	−9.52786	−0.737288	−0.368644	+	0.929571i	$0.620178\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	2.47214	0.189604
$171$	−5.70820	−0.436517
$172$	−2.47214	−0.188499
$173$	7.41641	0.563859	0.281930	−	0.959435i	$-0.409026\pi$
$173$	7.41641	0.563859	0.281930	+	0.959435i	$0.409026\pi$
$174$	−4.47214	−0.339032
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	10.4721	0.787134
$178$	−2.47214	−0.185294
$179$	−14.4721	−1.08170	−0.540849	−	0.841120i	$-0.681897\pi$
$179$	−14.4721	−1.08170	−0.540849	+	0.841120i	$0.681897\pi$
$180$	3.23607	0.241202
$181$	1.70820	0.126970	0.0634849	−	0.997983i	$-0.479779\pi$
$181$	1.70820	0.126970	0.0634849	+	0.997983i	$0.479779\pi$
$182$	0	0
$183$	0	0
$184$	6.47214	0.477132
$185$	22.4721	1.65218
$186$	7.23607	0.530574
$187$	−0.763932	−0.0558642
$188$	9.70820	0.708044
$189$	0	0
$190$	−18.4721	−1.34011
$191$	−1.52786	−0.110552	−0.0552762	−	0.998471i	$-0.517604\pi$
$191$	−1.52786	−0.110552	−0.0552762	+	0.998471i	$0.517604\pi$
$192$	1.00000	0.0721688
$193$	22.9443	1.65156	0.825782	−	0.563989i	$-0.190734\pi$
$193$	22.9443	1.65156	0.825782	+	0.563989i	$0.190734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.3607	1.59313	0.796566	−	0.604551i	$-0.206648\pi$
$197$	22.3607	1.59313	0.796566	+	0.604551i	$0.206648\pi$
$198$	−1.00000	−0.0710669
$199$	−17.1246	−1.21393	−0.606966	−	0.794728i	$-0.707614\pi$
$199$	−17.1246	−1.21393	−0.606966	+	0.794728i	$0.707614\pi$
$200$	5.47214	0.386938
$201$	−11.4164	−0.805251
$202$	12.0000	0.844317
$203$	0	0
$204$	0.763932	0.0534859
$205$	2.47214	0.172661
$206$	−15.2361	−1.06155
$207$	6.47214	0.449845
$208$	0	0
$209$	5.70820	0.394845
$210$	0	0
$211$	−23.4164	−1.61205	−0.806026	−	0.591880i	$-0.798386\pi$
$211$	−23.4164	−1.61205	−0.806026	+	0.591880i	$0.798386\pi$
$212$	−6.00000	−0.412082
$213$	6.47214	0.443463
$214$	−16.9443	−1.15829
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	6.00000	0.406371
$219$	2.29180	0.154865
$220$	−3.23607	−0.218176
$221$	0	0
$222$	6.94427	0.466069
$223$	2.29180	0.153470	0.0767350	−	0.997052i	$-0.475550\pi$
$223$	2.29180	0.153470	0.0767350	+	0.997052i	$0.475550\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	−13.4164	−0.892446
$227$	22.6525	1.50350	0.751749	−	0.659450i	$-0.229211\pi$
$227$	22.6525	1.50350	0.751749	+	0.659450i	$0.229211\pi$
$228$	−5.70820	−0.378035
$229$	−6.29180	−0.415774	−0.207887	−	0.978153i	$-0.566659\pi$
$229$	−6.29180	−0.415774	−0.207887	+	0.978153i	$0.566659\pi$
$230$	20.9443	1.38102
$231$	0	0
$232$	−4.47214	−0.293610
$233$	−26.9443	−1.76518	−0.882589	−	0.470145i	$-0.844201\pi$
$233$	−26.9443	−1.76518	−0.882589	+	0.470145i	$0.844201\pi$
$234$	0	0
$235$	31.4164	2.04938
$236$	10.4721	0.681678
$237$	−15.4164	−1.00140
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	3.23607	0.208887
$241$	4.18034	0.269279	0.134640	−	0.990895i	$-0.457012\pi$
$241$	4.18034	0.269279	0.134640	+	0.990895i	$0.457012\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0.763932	0.0487065
$247$	0	0
$248$	7.23607	0.459491
$249$	3.23607	0.205077
$250$	1.52786	0.0966306
$251$	−2.47214	−0.156040	−0.0780199	−	0.996952i	$-0.524860\pi$
$251$	−2.47214	−0.156040	−0.0780199	+	0.996952i	$0.524860\pi$
$252$	0	0
$253$	−6.47214	−0.406900
$254$	2.47214	0.155116
$255$	2.47214	0.154811
$256$	1.00000	0.0625000
$257$	−13.5279	−0.843845	−0.421922	−	0.906632i	$-0.638645\pi$
$257$	−13.5279	−0.843845	−0.421922	+	0.906632i	$0.638645\pi$
$258$	−2.47214	−0.153908
$259$	0	0
$260$	0	0
$261$	−4.47214	−0.276818
$262$	−11.2361	−0.694167
$263$	−18.4721	−1.13904	−0.569520	−	0.821977i	$-0.692871\pi$
$263$	−18.4721	−1.13904	−0.569520	+	0.821977i	$0.692871\pi$
$264$	−1.00000	−0.0615457
$265$	−19.4164	−1.19274
$266$	0	0
$267$	−2.47214	−0.151292
$268$	−11.4164	−0.697368
$269$	14.2918	0.871386	0.435693	−	0.900095i	$-0.356503\pi$
$269$	14.2918	0.871386	0.435693	+	0.900095i	$0.356503\pi$
$270$	3.23607	0.196941
$271$	−6.47214	−0.393154	−0.196577	−	0.980488i	$-0.562983\pi$
$271$	−6.47214	−0.393154	−0.196577	+	0.980488i	$0.562983\pi$
$272$	0.763932	0.0463202
$273$	0	0
$274$	−7.52786	−0.454775
$275$	−5.47214	−0.329982
$276$	6.47214	0.389577
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−12.1803	−0.730528
$279$	7.23607	0.433212
$280$	0	0
$281$	−15.8885	−0.947831	−0.473916	−	0.880570i	$-0.657160\pi$
$281$	−15.8885	−0.947831	−0.473916	+	0.880570i	$0.657160\pi$
$282$	9.70820	0.578115
$283$	−10.2918	−0.611784	−0.305892	−	0.952066i	$-0.598955\pi$
$283$	−10.2918	−0.611784	−0.305892	+	0.952066i	$0.598955\pi$
$284$	6.47214	0.384051
$285$	−18.4721	−1.09419
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−16.4164	−0.965671
$290$	−14.4721	−0.849833
$291$	0	0
$292$	2.29180	0.134117
$293$	31.4164	1.83537	0.917683	−	0.397313i	$-0.130057\pi$
$293$	31.4164	1.83537	0.917683	+	0.397313i	$0.130057\pi$
$294$	0	0
$295$	33.8885	1.97307
$296$	6.94427	0.403628
$297$	−1.00000	−0.0580259
$298$	−6.00000	−0.347571
$299$	0	0
$300$	5.47214	0.315934
$301$	0	0
$302$	0	0
$303$	12.0000	0.689382
$304$	−5.70820	−0.327388
$305$	0	0
$306$	0.763932	0.0436711
$307$	23.5967	1.34674	0.673369	−	0.739307i	$-0.264847\pi$
$307$	23.5967	1.34674	0.673369	+	0.739307i	$0.264847\pi$
$308$	0	0
$309$	−15.2361	−0.866750
$310$	23.4164	1.32996
$311$	−22.6525	−1.28450	−0.642252	−	0.766494i	$-0.722000\pi$
$311$	−22.6525	−1.28450	−0.642252	+	0.766494i	$0.722000\pi$
$312$	0	0
$313$	−14.4721	−0.818013	−0.409007	−	0.912531i	$-0.634125\pi$
$313$	−14.4721	−0.818013	−0.409007	+	0.912531i	$0.634125\pi$
$314$	19.2361	1.08555
$315$	0	0
$316$	−15.4164	−0.867241
$317$	3.88854	0.218402	0.109201	−	0.994020i	$-0.465171\pi$
$317$	3.88854	0.218402	0.109201	+	0.994020i	$0.465171\pi$
$318$	−6.00000	−0.336463
$319$	4.47214	0.250392
$320$	3.23607	0.180902
$321$	−16.9443	−0.945737
$322$	0	0
$323$	−4.36068	−0.242635
$324$	1.00000	0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	6.00000	0.331801
$328$	0.763932	0.0421811
$329$	0	0
$330$	−3.23607	−0.178140
$331$	19.4164	1.06722	0.533611	−	0.845730i	$-0.320835\pi$
$331$	19.4164	1.06722	0.533611	+	0.845730i	$0.320835\pi$
$332$	3.23607	0.177602
$333$	6.94427	0.380544
$334$	−9.52786	−0.521342
$335$	−36.9443	−2.01848
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−13.0000	−0.707107
$339$	−13.4164	−0.728679
$340$	2.47214	0.134070
$341$	−7.23607	−0.391855
$342$	−5.70820	−0.308664
$343$	0	0
$344$	−2.47214	−0.133289
$345$	20.9443	1.12760
$346$	7.41641	0.398709
$347$	26.8328	1.44046	0.720231	−	0.693735i	$-0.244036\pi$
$347$	26.8328	1.44046	0.720231	+	0.693735i	$0.244036\pi$
$348$	−4.47214	−0.239732
$349$	27.4164	1.46757	0.733783	−	0.679384i	$-0.237753\pi$
$349$	27.4164	1.46757	0.733783	+	0.679384i	$0.237753\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	20.3607	1.08369	0.541845	−	0.840479i	$-0.317726\pi$
$353$	20.3607	1.08369	0.541845	+	0.840479i	$0.317726\pi$
$354$	10.4721	0.556588
$355$	20.9443	1.11161
$356$	−2.47214	−0.131023
$357$	0	0
$358$	−14.4721	−0.764876
$359$	10.4721	0.552698	0.276349	−	0.961057i	$-0.410875\pi$
$359$	10.4721	0.552698	0.276349	+	0.961057i	$0.410875\pi$
$360$	3.23607	0.170556
$361$	13.5836	0.714926
$362$	1.70820	0.0897812
$363$	1.00000	0.0524864
$364$	0	0
$365$	7.41641	0.388193
$366$	0	0
$367$	−21.7082	−1.13316	−0.566580	−	0.824007i	$-0.691734\pi$
$367$	−21.7082	−1.13316	−0.566580	+	0.824007i	$0.691734\pi$
$368$	6.47214	0.337383
$369$	0.763932	0.0397687
$370$	22.4721	1.16827
$371$	0	0
$372$	7.23607	0.375173
$373$	25.4164	1.31601	0.658006	−	0.753013i	$-0.271400\pi$
$373$	25.4164	1.31601	0.658006	+	0.753013i	$0.271400\pi$
$374$	−0.763932	−0.0395020
$375$	1.52786	0.0788986
$376$	9.70820	0.500662
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−18.4721	−0.947601
$381$	2.47214	0.126651
$382$	−1.52786	−0.0781723
$383$	−30.6525	−1.56627	−0.783134	−	0.621853i	$-0.786380\pi$
$383$	−30.6525	−1.56627	−0.783134	+	0.621853i	$0.786380\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	22.9443	1.16783
$387$	−2.47214	−0.125666
$388$	0	0
$389$	10.9443	0.554897	0.277448	−	0.960741i	$-0.410511\pi$
$389$	10.9443	0.554897	0.277448	+	0.960741i	$0.410511\pi$
$390$	0	0
$391$	4.94427	0.250043
$392$	0	0
$393$	−11.2361	−0.566785
$394$	22.3607	1.12651
$395$	−49.8885	−2.51017
$396$	−1.00000	−0.0502519
$397$	24.1803	1.21358	0.606788	−	0.794864i	$-0.292458\pi$
$397$	24.1803	1.21358	0.606788	+	0.794864i	$0.292458\pi$
$398$	−17.1246	−0.858379
$399$	0	0
$400$	5.47214	0.273607
$401$	0.472136	0.0235773	0.0117887	−	0.999931i	$-0.496247\pi$
$401$	0.472136	0.0235773	0.0117887	+	0.999931i	$0.496247\pi$
$402$	−11.4164	−0.569399
$403$	0	0
$404$	12.0000	0.597022
$405$	3.23607	0.160802
$406$	0	0
$407$	−6.94427	−0.344215
$408$	0.763932	0.0378203
$409$	−7.23607	−0.357801	−0.178900	−	0.983867i	$-0.557254\pi$
$409$	−7.23607	−0.357801	−0.178900	+	0.983867i	$0.557254\pi$
$410$	2.47214	0.122090
$411$	−7.52786	−0.371322
$412$	−15.2361	−0.750627
$413$	0	0
$414$	6.47214	0.318088
$415$	10.4721	0.514057
$416$	0	0
$417$	−12.1803	−0.596474
$418$	5.70820	0.279197
$419$	−32.9443	−1.60943	−0.804717	−	0.593659i	$-0.797683\pi$
$419$	−32.9443	−1.60943	−0.804717	+	0.593659i	$0.797683\pi$
$420$	0	0
$421$	−22.9443	−1.11824	−0.559118	−	0.829088i	$-0.688860\pi$
$421$	−22.9443	−1.11824	−0.559118	+	0.829088i	$0.688860\pi$
$422$	−23.4164	−1.13989
$423$	9.70820	0.472029
$424$	−6.00000	−0.291386
$425$	4.18034	0.202776
$426$	6.47214	0.313576
$427$	0	0
$428$	−16.9443	−0.819032
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−13.5279	−0.651614	−0.325807	−	0.945436i	$-0.605636\pi$
$431$	−13.5279	−0.651614	−0.325807	+	0.945436i	$0.605636\pi$
$432$	1.00000	0.0481125
$433$	25.8885	1.24412	0.622062	−	0.782968i	$-0.286295\pi$
$433$	25.8885	1.24412	0.622062	+	0.782968i	$0.286295\pi$
$434$	0	0
$435$	−14.4721	−0.693886
$436$	6.00000	0.287348
$437$	−36.9443	−1.76728
$438$	2.29180	0.109506
$439$	−4.58359	−0.218763	−0.109381	−	0.994000i	$-0.534887\pi$
$439$	−4.58359	−0.218763	−0.109381	+	0.994000i	$0.534887\pi$
$440$	−3.23607	−0.154273
$441$	0	0
$442$	0	0
$443$	−19.4164	−0.922501	−0.461251	−	0.887270i	$-0.652599\pi$
$443$	−19.4164	−0.922501	−0.461251	+	0.887270i	$0.652599\pi$
$444$	6.94427	0.329561
$445$	−8.00000	−0.379236
$446$	2.29180	0.108520
$447$	−6.00000	−0.283790
$448$	0	0
$449$	28.4721	1.34368	0.671842	−	0.740695i	$-0.265504\pi$
$449$	28.4721	1.34368	0.671842	+	0.740695i	$0.265504\pi$
$450$	5.47214	0.257959
$451$	−0.763932	−0.0359722
$452$	−13.4164	−0.631055
$453$	0	0
$454$	22.6525	1.06313
$455$	0	0
$456$	−5.70820	−0.267311
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−6.29180	−0.293996
$459$	0.763932	0.0356573
$460$	20.9443	0.976532
$461$	−26.8328	−1.24973	−0.624864	−	0.780733i	$-0.714846\pi$
$461$	−26.8328	−1.24973	−0.624864	+	0.780733i	$0.714846\pi$
$462$	0	0
$463$	33.8885	1.57493	0.787467	−	0.616357i	$-0.211392\pi$
$463$	33.8885	1.57493	0.787467	+	0.616357i	$0.211392\pi$
$464$	−4.47214	−0.207614
$465$	23.4164	1.08591
$466$	−26.9443	−1.24817
$467$	−5.88854	−0.272489	−0.136245	−	0.990675i	$-0.543503\pi$
$467$	−5.88854	−0.272489	−0.136245	+	0.990675i	$0.543503\pi$
$468$	0	0
$469$	0	0
$470$	31.4164	1.44913
$471$	19.2361	0.886351
$472$	10.4721	0.482019
$473$	2.47214	0.113669
$474$	−15.4164	−0.708099
$475$	−31.2361	−1.43321
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	35.7771	1.63470	0.817348	−	0.576144i	$-0.195443\pi$
$479$	35.7771	1.63470	0.817348	+	0.576144i	$0.195443\pi$
$480$	3.23607	0.147706
$481$	0	0
$482$	4.18034	0.190409
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	19.0557	0.863497	0.431749	−	0.901994i	$-0.357897\pi$
$487$	19.0557	0.863497	0.431749	+	0.901994i	$0.357897\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	5.88854	0.265746	0.132873	−	0.991133i	$-0.457580\pi$
$491$	5.88854	0.265746	0.132873	+	0.991133i	$0.457580\pi$
$492$	0.763932	0.0344407
$493$	−3.41641	−0.153867
$494$	0	0
$495$	−3.23607	−0.145450
$496$	7.23607	0.324909
$497$	0	0
$498$	3.23607	0.145012
$499$	−19.4164	−0.869198	−0.434599	−	0.900624i	$-0.643110\pi$
$499$	−19.4164	−0.869198	−0.434599	+	0.900624i	$0.643110\pi$
$500$	1.52786	0.0683282
$501$	−9.52786	−0.425674
$502$	−2.47214	−0.110337
$503$	−21.3050	−0.949941	−0.474970	−	0.880002i	$-0.657541\pi$
$503$	−21.3050	−0.949941	−0.474970	+	0.880002i	$0.657541\pi$
$504$	0	0
$505$	38.8328	1.72804
$506$	−6.47214	−0.287722
$507$	−13.0000	−0.577350
$508$	2.47214	0.109683
$509$	11.2361	0.498030	0.249015	−	0.968500i	$-0.419893\pi$
$509$	11.2361	0.498030	0.249015	+	0.968500i	$0.419893\pi$
$510$	2.47214	0.109468
$511$	0	0
$512$	1.00000	0.0441942
$513$	−5.70820	−0.252023
$514$	−13.5279	−0.596689
$515$	−49.3050	−2.17264
$516$	−2.47214	−0.108830
$517$	−9.70820	−0.426966
$518$	0	0
$519$	7.41641	0.325544
$520$	0	0
$521$	−41.3050	−1.80960	−0.904801	−	0.425834i	$-0.859981\pi$
$521$	−41.3050	−1.80960	−0.904801	+	0.425834i	$0.859981\pi$
$522$	−4.47214	−0.195740
$523$	37.7082	1.64886	0.824432	−	0.565961i	$-0.191495\pi$
$523$	37.7082	1.64886	0.824432	+	0.565961i	$0.191495\pi$
$524$	−11.2361	−0.490850
$525$	0	0
$526$	−18.4721	−0.805423
$527$	5.52786	0.240798
$528$	−1.00000	−0.0435194
$529$	18.8885	0.821241
$530$	−19.4164	−0.843395
$531$	10.4721	0.454452
$532$	0	0
$533$	0	0
$534$	−2.47214	−0.106980
$535$	−54.8328	−2.37063
$536$	−11.4164	−0.493114
$537$	−14.4721	−0.624519
$538$	14.2918	0.616163
$539$	0	0
$540$	3.23607	0.139258
$541$	−5.41641	−0.232870	−0.116435	−	0.993198i	$-0.537147\pi$
$541$	−5.41641	−0.232870	−0.116435	+	0.993198i	$0.537147\pi$
$542$	−6.47214	−0.278002
$543$	1.70820	0.0733060
$544$	0.763932	0.0327533
$545$	19.4164	0.831708
$546$	0	0
$547$	29.5279	1.26252	0.631260	−	0.775571i	$-0.282538\pi$
$547$	29.5279	1.26252	0.631260	+	0.775571i	$0.282538\pi$
$548$	−7.52786	−0.321574
$549$	0	0
$550$	−5.47214	−0.233333
$551$	25.5279	1.08752
$552$	6.47214	0.275472
$553$	0	0
$554$	26.0000	1.10463
$555$	22.4721	0.953889
$556$	−12.1803	−0.516561
$557$	−37.4164	−1.58538	−0.792692	−	0.609622i	$-0.791321\pi$
$557$	−37.4164	−1.58538	−0.792692	+	0.609622i	$0.791321\pi$
$558$	7.23607	0.306327
$559$	0	0
$560$	0	0
$561$	−0.763932	−0.0322532
$562$	−15.8885	−0.670218
$563$	−11.2361	−0.473544	−0.236772	−	0.971565i	$-0.576089\pi$
$563$	−11.2361	−0.473544	−0.236772	+	0.971565i	$0.576089\pi$
$564$	9.70820	0.408789
$565$	−43.4164	−1.82654
$566$	−10.2918	−0.432596
$567$	0	0
$568$	6.47214	0.271565
$569$	−17.0557	−0.715013	−0.357507	−	0.933911i	$-0.616373\pi$
$569$	−17.0557	−0.715013	−0.357507	+	0.933911i	$0.616373\pi$
$570$	−18.4721	−0.773713
$571$	−26.4721	−1.10782	−0.553912	−	0.832575i	$-0.686866\pi$
$571$	−26.4721	−1.10782	−0.553912	+	0.832575i	$0.686866\pi$
$572$	0	0
$573$	−1.52786	−0.0638274
$574$	0	0
$575$	35.4164	1.47697
$576$	1.00000	0.0416667
$577$	12.5836	0.523862	0.261931	−	0.965087i	$-0.415641\pi$
$577$	12.5836	0.523862	0.261931	+	0.965087i	$0.415641\pi$
$578$	−16.4164	−0.682833
$579$	22.9443	0.953531
$580$	−14.4721	−0.600923
$581$	0	0
$582$	0	0
$583$	6.00000	0.248495
$584$	2.29180	0.0948352
$585$	0	0
$586$	31.4164	1.29780
$587$	−26.4721	−1.09262	−0.546311	−	0.837582i	$-0.683968\pi$
$587$	−26.4721	−1.09262	−0.546311	+	0.837582i	$0.683968\pi$
$588$	0	0
$589$	−41.3050	−1.70194
$590$	33.8885	1.39517
$591$	22.3607	0.919795
$592$	6.94427	0.285408
$593$	−23.2361	−0.954191	−0.477095	−	0.878851i	$-0.658310\pi$
$593$	−23.2361	−0.954191	−0.477095	+	0.878851i	$0.658310\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−17.1246	−0.700864
$598$	0	0
$599$	−37.3050	−1.52424	−0.762120	−	0.647436i	$-0.775841\pi$
$599$	−37.3050	−1.52424	−0.762120	+	0.647436i	$0.775841\pi$
$600$	5.47214	0.223399
$601$	13.7082	0.559169	0.279585	−	0.960121i	$-0.409803\pi$
$601$	13.7082	0.559169	0.279585	+	0.960121i	$0.409803\pi$
$602$	0	0
$603$	−11.4164	−0.464912
$604$	0	0
$605$	3.23607	0.131565
$606$	12.0000	0.487467
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−5.70820	−0.231498
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0.763932	0.0308801
$613$	3.52786	0.142489	0.0712445	−	0.997459i	$-0.477303\pi$
$613$	3.52786	0.142489	0.0712445	+	0.997459i	$0.477303\pi$
$614$	23.5967	0.952287
$615$	2.47214	0.0996861
$616$	0	0
$617$	−11.8885	−0.478615	−0.239307	−	0.970944i	$-0.576920\pi$
$617$	−11.8885	−0.478615	−0.239307	+	0.970944i	$0.576920\pi$
$618$	−15.2361	−0.612885
$619$	34.8328	1.40005	0.700025	−	0.714119i	$-0.253172\pi$
$619$	34.8328	1.40005	0.700025	+	0.714119i	$0.253172\pi$
$620$	23.4164	0.940426
$621$	6.47214	0.259718
$622$	−22.6525	−0.908282
$623$	0	0
$624$	0	0
$625$	−22.4164	−0.896656
$626$	−14.4721	−0.578423
$627$	5.70820	0.227964
$628$	19.2361	0.767603
$629$	5.30495	0.211522
$630$	0	0
$631$	−25.8885	−1.03061	−0.515303	−	0.857008i	$-0.672321\pi$
$631$	−25.8885	−1.03061	−0.515303	+	0.857008i	$0.672321\pi$
$632$	−15.4164	−0.613232
$633$	−23.4164	−0.930719
$634$	3.88854	0.154434
$635$	8.00000	0.317470
$636$	−6.00000	−0.237915
$637$	0	0
$638$	4.47214	0.177054
$639$	6.47214	0.256034
$640$	3.23607	0.127917
$641$	−0.111456	−0.00440225	−0.00220113	−	0.999998i	$-0.500701\pi$
$641$	−0.111456	−0.00440225	−0.00220113	+	0.999998i	$0.500701\pi$
$642$	−16.9443	−0.668737
$643$	−16.5836	−0.653993	−0.326997	−	0.945026i	$-0.606037\pi$
$643$	−16.5836	−0.653993	−0.326997	+	0.945026i	$0.606037\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	−4.36068	−0.171569
$647$	31.0132	1.21925	0.609626	−	0.792689i	$-0.291319\pi$
$647$	31.0132	1.21925	0.609626	+	0.792689i	$0.291319\pi$
$648$	1.00000	0.0392837
$649$	−10.4721	−0.411067
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	2.94427	0.115218	0.0576091	−	0.998339i	$-0.481652\pi$
$653$	2.94427	0.115218	0.0576091	+	0.998339i	$0.481652\pi$
$654$	6.00000	0.234619
$655$	−36.3607	−1.42073
$656$	0.763932	0.0298265
$657$	2.29180	0.0894115
$658$	0	0
$659$	−21.8885	−0.852657	−0.426328	−	0.904569i	$-0.640193\pi$
$659$	−21.8885	−0.852657	−0.426328	+	0.904569i	$0.640193\pi$
$660$	−3.23607	−0.125964
$661$	43.5967	1.69572	0.847858	−	0.530223i	$-0.177892\pi$
$661$	43.5967	1.69572	0.847858	+	0.530223i	$0.177892\pi$
$662$	19.4164	0.754640
$663$	0	0
$664$	3.23607	0.125584
$665$	0	0
$666$	6.94427	0.269085
$667$	−28.9443	−1.12073
$668$	−9.52786	−0.368644
$669$	2.29180	0.0886060
$670$	−36.9443	−1.42728
$671$	0	0
$672$	0	0
$673$	17.0557	0.657450	0.328725	−	0.944426i	$-0.393381\pi$
$673$	17.0557	0.657450	0.328725	+	0.944426i	$0.393381\pi$
$674$	−18.0000	−0.693334
$675$	5.47214	0.210623
$676$	−13.0000	−0.500000
$677$	37.5279	1.44231	0.721156	−	0.692772i	$-0.243611\pi$
$677$	37.5279	1.44231	0.721156	+	0.692772i	$0.243611\pi$
$678$	−13.4164	−0.515254
$679$	0	0
$680$	2.47214	0.0948021
$681$	22.6525	0.868045
$682$	−7.23607	−0.277083
$683$	40.3607	1.54436	0.772179	−	0.635405i	$-0.219167\pi$
$683$	40.3607	1.54436	0.772179	+	0.635405i	$0.219167\pi$
$684$	−5.70820	−0.218259
$685$	−24.3607	−0.930774
$686$	0	0
$687$	−6.29180	−0.240047
$688$	−2.47214	−0.0942493
$689$	0	0
$690$	20.9443	0.797335
$691$	26.4721	1.00705	0.503524	−	0.863981i	$-0.332037\pi$
$691$	26.4721	1.00705	0.503524	+	0.863981i	$0.332037\pi$
$692$	7.41641	0.281930
$693$	0	0
$694$	26.8328	1.01856
$695$	−39.4164	−1.49515
$696$	−4.47214	−0.169516
$697$	0.583592	0.0221051
$698$	27.4164	1.03773
$699$	−26.9443	−1.01913
$700$	0	0
$701$	−43.8885	−1.65765	−0.828824	−	0.559510i	$-0.810989\pi$
$701$	−43.8885	−1.65765	−0.828824	+	0.559510i	$0.810989\pi$
$702$	0	0
$703$	−39.6393	−1.49503
$704$	−1.00000	−0.0376889
$705$	31.4164	1.18321
$706$	20.3607	0.766284
$707$	0	0
$708$	10.4721	0.393567
$709$	22.9443	0.861690	0.430845	−	0.902426i	$-0.358216\pi$
$709$	22.9443	0.861690	0.430845	+	0.902426i	$0.358216\pi$
$710$	20.9443	0.786025
$711$	−15.4164	−0.578160
$712$	−2.47214	−0.0926472
$713$	46.8328	1.75390
$714$	0	0
$715$	0	0
$716$	−14.4721	−0.540849
$717$	0	0
$718$	10.4721	0.390817
$719$	−33.7082	−1.25710	−0.628552	−	0.777768i	$-0.716352\pi$
$719$	−33.7082	−1.25710	−0.628552	+	0.777768i	$0.716352\pi$
$720$	3.23607	0.120601
$721$	0	0
$722$	13.5836	0.505529
$723$	4.18034	0.155469
$724$	1.70820	0.0634849
$725$	−24.4721	−0.908872
$726$	1.00000	0.0371135
$727$	−40.7639	−1.51185	−0.755925	−	0.654658i	$-0.772813\pi$
$727$	−40.7639	−1.51185	−0.755925	+	0.654658i	$0.772813\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	7.41641	0.274494
$731$	−1.88854	−0.0698503
$732$	0	0
$733$	−30.8328	−1.13884	−0.569418	−	0.822048i	$-0.692831\pi$
$733$	−30.8328	−1.13884	−0.569418	+	0.822048i	$0.692831\pi$
$734$	−21.7082	−0.801264
$735$	0	0
$736$	6.47214	0.238566
$737$	11.4164	0.420529
$738$	0.763932	0.0281207
$739$	16.5836	0.610037	0.305019	−	0.952346i	$-0.401337\pi$
$739$	16.5836	0.610037	0.305019	+	0.952346i	$0.401337\pi$
$740$	22.4721	0.826092
$741$	0	0
$742$	0	0
$743$	9.88854	0.362775	0.181388	−	0.983412i	$-0.441941\pi$
$743$	9.88854	0.362775	0.181388	+	0.983412i	$0.441941\pi$
$744$	7.23607	0.265287
$745$	−19.4164	−0.711362
$746$	25.4164	0.930561
$747$	3.23607	0.118402
$748$	−0.763932	−0.0279321
$749$	0	0
$750$	1.52786	0.0557897
$751$	38.8328	1.41703	0.708515	−	0.705696i	$-0.249366\pi$
$751$	38.8328	1.41703	0.708515	+	0.705696i	$0.249366\pi$
$752$	9.70820	0.354022
$753$	−2.47214	−0.0900896
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.05573	0.183754	0.0918768	−	0.995770i	$-0.470713\pi$
$757$	5.05573	0.183754	0.0918768	+	0.995770i	$0.470713\pi$
$758$	−4.00000	−0.145287
$759$	−6.47214	−0.234924
$760$	−18.4721	−0.670055
$761$	17.1246	0.620767	0.310383	−	0.950611i	$-0.399543\pi$
$761$	17.1246	0.620767	0.310383	+	0.950611i	$0.399543\pi$
$762$	2.47214	0.0895560
$763$	0	0
$764$	−1.52786	−0.0552762
$765$	2.47214	0.0893803
$766$	−30.6525	−1.10752
$767$	0	0
$768$	1.00000	0.0360844
$769$	−46.0689	−1.66129	−0.830643	−	0.556805i	$-0.812027\pi$
$769$	−46.0689	−1.66129	−0.830643	+	0.556805i	$0.812027\pi$
$770$	0	0
$771$	−13.5279	−0.487194
$772$	22.9443	0.825782
$773$	−43.9574	−1.58104	−0.790519	−	0.612437i	$-0.790189\pi$
$773$	−43.9574	−1.58104	−0.790519	+	0.612437i	$0.790189\pi$
$774$	−2.47214	−0.0888591
$775$	39.5967	1.42236
$776$	0	0
$777$	0	0
$778$	10.9443	0.392371
$779$	−4.36068	−0.156238
$780$	0	0
$781$	−6.47214	−0.231591
$782$	4.94427	0.176807
$783$	−4.47214	−0.159821
$784$	0	0
$785$	62.2492	2.22177
$786$	−11.2361	−0.400777
$787$	−44.5410	−1.58772	−0.793858	−	0.608103i	$-0.791931\pi$
$787$	−44.5410	−1.58772	−0.793858	+	0.608103i	$0.791931\pi$
$788$	22.3607	0.796566
$789$	−18.4721	−0.657625
$790$	−49.8885	−1.77495
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	24.1803	0.858128
$795$	−19.4164	−0.688629
$796$	−17.1246	−0.606966
$797$	−53.1246	−1.88177	−0.940885	−	0.338726i	$-0.890004\pi$
$797$	−53.1246	−1.88177	−0.940885	+	0.338726i	$0.890004\pi$
$798$	0	0
$799$	7.41641	0.262374
$800$	5.47214	0.193469
$801$	−2.47214	−0.0873486
$802$	0.472136	0.0166717
$803$	−2.29180	−0.0808757
$804$	−11.4164	−0.402626
$805$	0	0
$806$	0	0
$807$	14.2918	0.503095
$808$	12.0000	0.422159
$809$	−8.83282	−0.310545	−0.155273	−	0.987872i	$-0.549626\pi$
$809$	−8.83282	−0.310545	−0.155273	+	0.987872i	$0.549626\pi$
$810$	3.23607	0.113704
$811$	31.5967	1.10951	0.554756	−	0.832013i	$-0.312812\pi$
$811$	31.5967	1.10951	0.554756	+	0.832013i	$0.312812\pi$
$812$	0	0
$813$	−6.47214	−0.226988
$814$	−6.94427	−0.243397
$815$	−38.8328	−1.36025
$816$	0.763932	0.0267430
$817$	14.1115	0.493697
$818$	−7.23607	−0.253003
$819$	0	0
$820$	2.47214	0.0863307
$821$	−41.7771	−1.45803	−0.729015	−	0.684498i	$-0.760022\pi$
$821$	−41.7771	−1.45803	−0.729015	+	0.684498i	$0.760022\pi$
$822$	−7.52786	−0.262564
$823$	22.8328	0.795902	0.397951	−	0.917407i	$-0.369721\pi$
$823$	22.8328	0.795902	0.397951	+	0.917407i	$0.369721\pi$
$824$	−15.2361	−0.530774
$825$	−5.47214	−0.190515
$826$	0	0
$827$	−52.7214	−1.83330	−0.916651	−	0.399689i	$-0.869118\pi$
$827$	−52.7214	−1.83330	−0.916651	+	0.399689i	$0.869118\pi$
$828$	6.47214	0.224922
$829$	37.4853	1.30192	0.650959	−	0.759113i	$-0.274367\pi$
$829$	37.4853	1.30192	0.650959	+	0.759113i	$0.274367\pi$
$830$	10.4721	0.363493
$831$	26.0000	0.901930
$832$	0	0
$833$	0	0
$834$	−12.1803	−0.421771
$835$	−30.8328	−1.06701
$836$	5.70820	0.197422
$837$	7.23607	0.250115
$838$	−32.9443	−1.13804
$839$	−20.7639	−0.716851	−0.358425	−	0.933558i	$-0.616686\pi$
$839$	−20.7639	−0.716851	−0.358425	+	0.933558i	$0.616686\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−22.9443	−0.790712
$843$	−15.8885	−0.547231
$844$	−23.4164	−0.806026
$845$	−42.0689	−1.44721
$846$	9.70820	0.333775
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−10.2918	−0.353214
$850$	4.18034	0.143384
$851$	44.9443	1.54067
$852$	6.47214	0.221732
$853$	−1.88854	−0.0646625	−0.0323313	−	0.999477i	$-0.510293\pi$
$853$	−1.88854	−0.0646625	−0.0323313	+	0.999477i	$0.510293\pi$
$854$	0	0
$855$	−18.4721	−0.631734
$856$	−16.9443	−0.579143
$857$	−6.87539	−0.234859	−0.117429	−	0.993081i	$-0.537465\pi$
$857$	−6.87539	−0.234859	−0.117429	+	0.993081i	$0.537465\pi$
$858$	0	0
$859$	16.5836	0.565825	0.282912	−	0.959146i	$-0.408699\pi$
$859$	16.5836	0.565825	0.282912	+	0.959146i	$0.408699\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−13.5279	−0.460761
$863$	0.360680	0.0122777	0.00613884	−	0.999981i	$-0.498046\pi$
$863$	0.360680	0.0122777	0.00613884	+	0.999981i	$0.498046\pi$
$864$	1.00000	0.0340207
$865$	24.0000	0.816024
$866$	25.8885	0.879729
$867$	−16.4164	−0.557530
$868$	0	0
$869$	15.4164	0.522966
$870$	−14.4721	−0.490651
$871$	0	0
$872$	6.00000	0.203186
$873$	0	0
$874$	−36.9443	−1.24966
$875$	0	0
$876$	2.29180	0.0774326
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	−4.58359	−0.154689
$879$	31.4164	1.05965
$880$	−3.23607	−0.109088
$881$	26.8328	0.904021	0.452010	−	0.892013i	$-0.350707\pi$
$881$	26.8328	0.904021	0.452010	+	0.892013i	$0.350707\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	33.8885	1.13915
$886$	−19.4164	−0.652307
$887$	19.0557	0.639829	0.319914	−	0.947446i	$-0.396346\pi$
$887$	19.0557	0.639829	0.319914	+	0.947446i	$0.396346\pi$
$888$	6.94427	0.233035
$889$	0	0
$890$	−8.00000	−0.268161
$891$	−1.00000	−0.0335013
$892$	2.29180	0.0767350
$893$	−55.4164	−1.85444
$894$	−6.00000	−0.200670
$895$	−46.8328	−1.56545
$896$	0	0
$897$	0	0
$898$	28.4721	0.950127
$899$	−32.3607	−1.07929
$900$	5.47214	0.182405
$901$	−4.58359	−0.152702
$902$	−0.763932	−0.0254362
$903$	0	0
$904$	−13.4164	−0.446223
$905$	5.52786	0.183752
$906$	0	0
$907$	29.3050	0.973055	0.486527	−	0.873665i	$-0.338263\pi$
$907$	29.3050	0.973055	0.486527	+	0.873665i	$0.338263\pi$
$908$	22.6525	0.751749
$909$	12.0000	0.398015
$910$	0	0
$911$	−4.58359	−0.151861	−0.0759306	−	0.997113i	$-0.524193\pi$
$911$	−4.58359	−0.151861	−0.0759306	+	0.997113i	$0.524193\pi$
$912$	−5.70820	−0.189018
$913$	−3.23607	−0.107098
$914$	18.0000	0.595387
$915$	0	0
$916$	−6.29180	−0.207887
$917$	0	0
$918$	0.763932	0.0252135
$919$	53.6656	1.77027	0.885133	−	0.465338i	$-0.154067\pi$
$919$	53.6656	1.77027	0.885133	+	0.465338i	$0.154067\pi$
$920$	20.9443	0.690512
$921$	23.5967	0.777539
$922$	−26.8328	−0.883692
$923$	0	0
$924$	0	0
$925$	38.0000	1.24943
$926$	33.8885	1.11365
$927$	−15.2361	−0.500418
$928$	−4.47214	−0.146805
$929$	36.3607	1.19296	0.596478	−	0.802630i	$-0.296566\pi$
$929$	36.3607	1.19296	0.596478	+	0.802630i	$0.296566\pi$
$930$	23.4164	0.767854
$931$	0	0
$932$	−26.9443	−0.882589
$933$	−22.6525	−0.741609
$934$	−5.88854	−0.192679
$935$	−2.47214	−0.0808475
$936$	0	0
$937$	−47.5967	−1.55492	−0.777459	−	0.628934i	$-0.783492\pi$
$937$	−47.5967	−1.55492	−0.777459	+	0.628934i	$0.783492\pi$
$938$	0	0
$939$	−14.4721	−0.472280
$940$	31.4164	1.02469
$941$	52.3607	1.70691	0.853455	−	0.521167i	$-0.174503\pi$
$941$	52.3607	1.70691	0.853455	+	0.521167i	$0.174503\pi$
$942$	19.2361	0.626745
$943$	4.94427	0.161008
$944$	10.4721	0.340839
$945$	0	0
$946$	2.47214	0.0803761
$947$	−2.11146	−0.0686131	−0.0343066	−	0.999411i	$-0.510922\pi$
$947$	−2.11146	−0.0686131	−0.0343066	+	0.999411i	$0.510922\pi$
$948$	−15.4164	−0.500702
$949$	0	0
$950$	−31.2361	−1.01343
$951$	3.88854	0.126095
$952$	0	0
$953$	−13.0557	−0.422917	−0.211458	−	0.977387i	$-0.567821\pi$
$953$	−13.0557	−0.422917	−0.211458	+	0.977387i	$0.567821\pi$
$954$	−6.00000	−0.194257
$955$	−4.94427	−0.159993
$956$	0	0
$957$	4.47214	0.144564
$958$	35.7771	1.15591
$959$	0	0
$960$	3.23607	0.104444
$961$	21.3607	0.689054
$962$	0	0
$963$	−16.9443	−0.546022
$964$	4.18034	0.134640
$965$	74.2492	2.39017
$966$	0	0
$967$	1.16718	0.0375341	0.0187671	−	0.999824i	$-0.494026\pi$
$967$	1.16718	0.0375341	0.0187671	+	0.999824i	$0.494026\pi$
$968$	1.00000	0.0321412
$969$	−4.36068	−0.140085
$970$	0	0
$971$	56.9443	1.82743	0.913714	−	0.406357i	$-0.133201\pi$
$971$	56.9443	1.82743	0.913714	+	0.406357i	$0.133201\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	19.0557	0.610585
$975$	0	0
$976$	0	0
$977$	23.8885	0.764262	0.382131	−	0.924108i	$-0.375190\pi$
$977$	23.8885	0.764262	0.382131	+	0.924108i	$0.375190\pi$
$978$	−12.0000	−0.383718
$979$	2.47214	0.0790098
$980$	0	0
$981$	6.00000	0.191565
$982$	5.88854	0.187911
$983$	−38.2918	−1.22132	−0.610659	−	0.791893i	$-0.709096\pi$
$983$	−38.2918	−1.22132	−0.610659	+	0.791893i	$0.709096\pi$
$984$	0.763932	0.0243533
$985$	72.3607	2.30560
$986$	−3.41641	−0.108801
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	−3.23607	−0.102849
$991$	22.8328	0.725308	0.362654	−	0.931924i	$-0.381871\pi$
$991$	22.8328	0.725308	0.362654	+	0.931924i	$0.381871\pi$
$992$	7.23607	0.229745
$993$	19.4164	0.616161
$994$	0	0
$995$	−55.4164	−1.75682
$996$	3.23607	0.102539
$997$	26.2492	0.831321	0.415661	−	0.909520i	$-0.363550\pi$
$997$	26.2492	0.831321	0.415661	+	0.909520i	$0.363550\pi$
$998$	−19.4164	−0.614616
$999$	6.94427	0.219707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.be.1.2	yes	2
3.2	odd	2		9702.2.a.ck.1.1		2
7.6	odd	2		3234.2.a.bb.1.1	✓	2
21.20	even	2		9702.2.a.cw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bb.1.1	✓	2	7.6	odd	2
3234.2.a.be.1.2	yes	2	1.1	even	1	trivial
9702.2.a.ck.1.1		2	3.2	odd	2
9702.2.a.cw.1.2		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.23607 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.23607 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.23607 q^{10} -1.00000 q^{11} +1.00000 q^{12} +3.23607 q^{15} +1.00000 q^{16} +0.763932 q^{17} +1.00000 q^{18} -5.70820 q^{19} +3.23607 q^{20} -1.00000 q^{22} +6.47214 q^{23} +1.00000 q^{24} +5.47214 q^{25} +1.00000 q^{27} -4.47214 q^{29} +3.23607 q^{30} +7.23607 q^{31} +1.00000 q^{32} -1.00000 q^{33} +0.763932 q^{34} +1.00000 q^{36} +6.94427 q^{37} -5.70820 q^{38} +3.23607 q^{40} +0.763932 q^{41} -2.47214 q^{43} -1.00000 q^{44} +3.23607 q^{45} +6.47214 q^{46} +9.70820 q^{47} +1.00000 q^{48} +5.47214 q^{50} +0.763932 q^{51} -6.00000 q^{53} +1.00000 q^{54} -3.23607 q^{55} -5.70820 q^{57} -4.47214 q^{58} +10.4721 q^{59} +3.23607 q^{60} +7.23607 q^{62} +1.00000 q^{64} -1.00000 q^{66} -11.4164 q^{67} +0.763932 q^{68} +6.47214 q^{69} +6.47214 q^{71} +1.00000 q^{72} +2.29180 q^{73} +6.94427 q^{74} +5.47214 q^{75} -5.70820 q^{76} -15.4164 q^{79} +3.23607 q^{80} +1.00000 q^{81} +0.763932 q^{82} +3.23607 q^{83} +2.47214 q^{85} -2.47214 q^{86} -4.47214 q^{87} -1.00000 q^{88} -2.47214 q^{89} +3.23607 q^{90} +6.47214 q^{92} +7.23607 q^{93} +9.70820 q^{94} -18.4721 q^{95} +1.00000 q^{96} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} + 2 q^{19} + 2 q^{20} - 2 q^{22} + 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} + 2 q^{30} + 10 q^{31} + 2 q^{32} - 2 q^{33} + 6 q^{34} + 2 q^{36} - 4 q^{37} + 2 q^{38} + 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} + 2 q^{45} + 4 q^{46} + 6 q^{47} + 2 q^{48} + 2 q^{50} + 6 q^{51} - 12 q^{53} + 2 q^{54} - 2 q^{55} + 2 q^{57} + 12 q^{59} + 2 q^{60} + 10 q^{62} + 2 q^{64} - 2 q^{66} + 4 q^{67} + 6 q^{68} + 4 q^{69} + 4 q^{71} + 2 q^{72} + 18 q^{73} - 4 q^{74} + 2 q^{75} + 2 q^{76} - 4 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} + 2 q^{83} - 4 q^{85} + 4 q^{86} - 2 q^{88} + 4 q^{89} + 2 q^{90} + 4 q^{92} + 10 q^{93} + 6 q^{94} - 28 q^{95} + 2 q^{96} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + 2 * q^15 + 2 * q^16 + 6 * q^17 + 2 * q^18 + 2 * q^19 + 2 * q^20 - 2 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 + 2 * q^30 + 10 * q^31 + 2 * q^32 - 2 * q^33 + 6 * q^34 + 2 * q^36 - 4 * q^37 + 2 * q^38 + 2 * q^40 + 6 * q^41 + 4 * q^43 - 2 * q^44 + 2 * q^45 + 4 * q^46 + 6 * q^47 + 2 * q^48 + 2 * q^50 + 6 * q^51 - 12 * q^53 + 2 * q^54 - 2 * q^55 + 2 * q^57 + 12 * q^59 + 2 * q^60 + 10 * q^62 + 2 * q^64 - 2 * q^66 + 4 * q^67 + 6 * q^68 + 4 * q^69 + 4 * q^71 + 2 * q^72 + 18 * q^73 - 4 * q^74 + 2 * q^75 + 2 * q^76 - 4 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 + 2 * q^83 - 4 * q^85 + 4 * q^86 - 2 * q^88 + 4 * q^89 + 2 * q^90 + 4 * q^92 + 10 * q^93 + 6 * q^94 - 28 * q^95 + 2 * q^96 - 2 * q^99

Introduction

L-functions

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Database

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