LMFDB - Embedded newform 3234.2.a.x.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3234,2,Mod(1,3234)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,-2,-2,2,0,2,0,-2,2,0,2,-2,-4,0,0,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$25.8236200137$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.46410 q^{10} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +3.46410 q^{15} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{18} +1.46410 q^{19} -3.46410 q^{20} -1.00000 q^{22} -6.92820 q^{23} +1.00000 q^{24} +7.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -3.46410 q^{30} +1.46410 q^{31} -1.00000 q^{32} -1.00000 q^{33} -3.46410 q^{34} +1.00000 q^{36} +8.92820 q^{37} -1.46410 q^{38} +2.00000 q^{39} +3.46410 q^{40} +3.46410 q^{41} +2.92820 q^{43} +1.00000 q^{44} -3.46410 q^{45} +6.92820 q^{46} -2.53590 q^{47} -1.00000 q^{48} -7.00000 q^{50} -3.46410 q^{51} -2.00000 q^{52} +12.9282 q^{53} +1.00000 q^{54} -3.46410 q^{55} -1.46410 q^{57} +6.00000 q^{58} +6.92820 q^{59} +3.46410 q^{60} -2.00000 q^{61} -1.46410 q^{62} +1.00000 q^{64} +6.92820 q^{65} +1.00000 q^{66} +1.07180 q^{67} +3.46410 q^{68} +6.92820 q^{69} -12.0000 q^{71} -1.00000 q^{72} +7.46410 q^{73} -8.92820 q^{74} -7.00000 q^{75} +1.46410 q^{76} -2.00000 q^{78} +2.92820 q^{79} -3.46410 q^{80} +1.00000 q^{81} -3.46410 q^{82} +16.3923 q^{83} -12.0000 q^{85} -2.92820 q^{86} +6.00000 q^{87} -1.00000 q^{88} -12.9282 q^{89} +3.46410 q^{90} -6.92820 q^{92} -1.46410 q^{93} +2.53590 q^{94} -5.07180 q^{95} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{18} - 4 q^{19} - 2 q^{22} + 2 q^{24} + 14 q^{25} + 4 q^{26} - 2 q^{27} - 12 q^{29} - 4 q^{31}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^18 - 4 * q^19 - 2 * q^22 + 2 * q^24 + 14 * q^25 + 4 * q^26 - 2 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^32 - 2 * q^33 + 2 * q^36 + 4 * q^37 + 4 * q^38 + 4 * q^39 - 8 * q^43 + 2 * q^44 - 12 * q^47 - 2 * q^48 - 14 * q^50 - 4 * q^52 + 12 * q^53 + 2 * q^54 + 4 * q^57 + 12 * q^58 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 2 * q^66 + 16 * q^67 - 24 * q^71 - 2 * q^72 + 8 * q^73 - 4 * q^74 - 14 * q^75 - 4 * q^76 - 4 * q^78 - 8 * q^79 + 2 * q^81 + 12 * q^83 - 24 * q^85 + 8 * q^86 + 12 * q^87 - 2 * q^88 - 12 * q^89 + 4 * q^93 + 12 * q^94 - 24 * q^95 + 2 * q^96 - 4 * q^97 + 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.46410	1.09545
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	−1.00000	−0.235702
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	−3.46410	−0.774597
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	1.00000	0.204124
$25$	7.00000	1.40000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−3.46410	−0.632456
$31$	1.46410	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$31$	1.46410	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−3.46410	−0.594089
$35$	0	0
$36$	1.00000	0.166667
$37$	8.92820	1.46779	0.733894	−	0.679264i	$-0.237701\pi$
$37$	8.92820	1.46779	0.733894	+	0.679264i	$0.237701\pi$
$38$	−1.46410	−0.237509
$39$	2.00000	0.320256
$40$	3.46410	0.547723
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	2.92820	0.446547	0.223273	−	0.974756i	$-0.428326\pi$
$43$	2.92820	0.446547	0.223273	+	0.974756i	$0.428326\pi$
$44$	1.00000	0.150756
$45$	−3.46410	−0.516398
$46$	6.92820	1.02151
$47$	−2.53590	−0.369899	−0.184949	−	0.982748i	$-0.559212\pi$
$47$	−2.53590	−0.369899	−0.184949	+	0.982748i	$0.559212\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−7.00000	−0.989949
$51$	−3.46410	−0.485071
$52$	−2.00000	−0.277350
$53$	12.9282	1.77583	0.887913	−	0.460012i	$-0.152155\pi$
$53$	12.9282	1.77583	0.887913	+	0.460012i	$0.152155\pi$
$54$	1.00000	0.136083
$55$	−3.46410	−0.467099
$56$	0	0
$57$	−1.46410	−0.193925
$58$	6.00000	0.787839
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	3.46410	0.447214
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−1.46410	−0.185941
$63$	0	0
$64$	1.00000	0.125000
$65$	6.92820	0.859338
$66$	1.00000	0.123091
$67$	1.07180	0.130941	0.0654704	−	0.997855i	$-0.479145\pi$
$67$	1.07180	0.130941	0.0654704	+	0.997855i	$0.479145\pi$
$68$	3.46410	0.420084
$69$	6.92820	0.834058
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	−8.92820	−1.03788
$75$	−7.00000	−0.808290
$76$	1.46410	0.167944
$77$	0	0
$78$	−2.00000	−0.226455
$79$	2.92820	0.329449	0.164724	−	0.986340i	$-0.447327\pi$
$79$	2.92820	0.329449	0.164724	+	0.986340i	$0.447327\pi$
$80$	−3.46410	−0.387298
$81$	1.00000	0.111111
$82$	−3.46410	−0.382546
$83$	16.3923	1.79929	0.899645	−	0.436623i	$-0.143826\pi$
$83$	16.3923	1.79929	0.899645	+	0.436623i	$0.143826\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	−2.92820	−0.315756
$87$	6.00000	0.643268
$88$	−1.00000	−0.106600
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	3.46410	0.365148
$91$	0	0
$92$	−6.92820	−0.722315
$93$	−1.46410	−0.151820
$94$	2.53590	0.261558
$95$	−5.07180	−0.520355
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	7.00000	0.700000
$101$	−7.85641	−0.781742	−0.390871	−	0.920446i	$-0.627826\pi$
$101$	−7.85641	−0.781742	−0.390871	+	0.920446i	$0.627826\pi$
$102$	3.46410	0.342997
$103$	6.53590	0.644001	0.322001	−	0.946739i	$-0.395645\pi$
$103$	6.53590	0.644001	0.322001	+	0.946739i	$0.395645\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−12.9282	−1.25570
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	−1.00000	−0.0962250
$109$	−11.8564	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$109$	−11.8564	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$110$	3.46410	0.330289
$111$	−8.92820	−0.847428
$112$	0	0
$113$	−19.8564	−1.86793	−0.933967	−	0.357360i	$-0.883677\pi$
$113$	−19.8564	−1.86793	−0.933967	+	0.357360i	$0.883677\pi$
$114$	1.46410	0.137126
$115$	24.0000	2.23801
$116$	−6.00000	−0.557086
$117$	−2.00000	−0.184900
$118$	−6.92820	−0.637793
$119$	0	0
$120$	−3.46410	−0.316228
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	−3.46410	−0.312348
$124$	1.46410	0.131480
$125$	−6.92820	−0.619677
$126$	0	0
$127$	2.92820	0.259836	0.129918	−	0.991525i	$-0.458529\pi$
$127$	2.92820	0.259836	0.129918	+	0.991525i	$0.458529\pi$
$128$	−1.00000	−0.0883883
$129$	−2.92820	−0.257814
$130$	−6.92820	−0.607644
$131$	−16.3923	−1.43220	−0.716101	−	0.697997i	$-0.754075\pi$
$131$	−16.3923	−1.43220	−0.716101	+	0.697997i	$0.754075\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−1.07180	−0.0925891
$135$	3.46410	0.298142
$136$	−3.46410	−0.297044
$137$	−19.8564	−1.69645	−0.848224	−	0.529638i	$-0.822328\pi$
$137$	−19.8564	−1.69645	−0.848224	+	0.529638i	$0.822328\pi$
$138$	−6.92820	−0.589768
$139$	6.53590	0.554368	0.277184	−	0.960817i	$-0.410599\pi$
$139$	6.53590	0.554368	0.277184	+	0.960817i	$0.410599\pi$
$140$	0	0
$141$	2.53590	0.213561
$142$	12.0000	1.00702
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	20.7846	1.72607
$146$	−7.46410	−0.617733
$147$	0	0
$148$	8.92820	0.733894
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	7.00000	0.571548
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−1.46410	−0.118754
$153$	3.46410	0.280056
$154$	0	0
$155$	−5.07180	−0.407377
$156$	2.00000	0.160128
$157$	−18.3923	−1.46787	−0.733933	−	0.679222i	$-0.762317\pi$
$157$	−18.3923	−1.46787	−0.733933	+	0.679222i	$0.762317\pi$
$158$	−2.92820	−0.232955
$159$	−12.9282	−1.02527
$160$	3.46410	0.273861
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	3.46410	0.270501
$165$	3.46410	0.269680
$166$	−16.3923	−1.27229
$167$	−5.07180	−0.392467	−0.196234	−	0.980557i	$-0.562871\pi$
$167$	−5.07180	−0.392467	−0.196234	+	0.980557i	$0.562871\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	1.46410	0.111963
$172$	2.92820	0.223273
$173$	−12.9282	−0.982913	−0.491457	−	0.870902i	$-0.663535\pi$
$173$	−12.9282	−0.982913	−0.491457	+	0.870902i	$0.663535\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	1.00000	0.0753778
$177$	−6.92820	−0.520756
$178$	12.9282	0.969010
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	−3.46410	−0.258199
$181$	14.3923	1.06977	0.534886	−	0.844924i	$-0.320355\pi$
$181$	14.3923	1.06977	0.534886	+	0.844924i	$0.320355\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	6.92820	0.510754
$185$	−30.9282	−2.27389
$186$	1.46410	0.107353
$187$	3.46410	0.253320
$188$	−2.53590	−0.184949
$189$	0	0
$190$	5.07180	0.367947
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	−1.00000	−0.0721688
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	2.00000	0.143592
$195$	−6.92820	−0.496139
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−1.00000	−0.0710669
$199$	20.3923	1.44557	0.722786	−	0.691072i	$-0.242861\pi$
$199$	20.3923	1.44557	0.722786	+	0.691072i	$0.242861\pi$
$200$	−7.00000	−0.494975
$201$	−1.07180	−0.0755987
$202$	7.85641	0.552775
$203$	0	0
$204$	−3.46410	−0.242536
$205$	−12.0000	−0.838116
$206$	−6.53590	−0.455378
$207$	−6.92820	−0.481543
$208$	−2.00000	−0.138675
$209$	1.46410	0.101274
$210$	0	0
$211$	−24.7846	−1.70624	−0.853121	−	0.521712i	$-0.825293\pi$
$211$	−24.7846	−1.70624	−0.853121	+	0.521712i	$0.825293\pi$
$212$	12.9282	0.887913
$213$	12.0000	0.822226
$214$	−6.92820	−0.473602
$215$	−10.1436	−0.691787
$216$	1.00000	0.0680414
$217$	0	0
$218$	11.8564	0.803017
$219$	−7.46410	−0.504377
$220$	−3.46410	−0.233550
$221$	−6.92820	−0.466041
$222$	8.92820	0.599222
$223$	−12.3923	−0.829850	−0.414925	−	0.909856i	$-0.636192\pi$
$223$	−12.3923	−0.829850	−0.414925	+	0.909856i	$0.636192\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	19.8564	1.32083
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	−1.46410	−0.0969625
$229$	−13.3205	−0.880244	−0.440122	−	0.897938i	$-0.645065\pi$
$229$	−13.3205	−0.880244	−0.440122	+	0.897938i	$0.645065\pi$
$230$	−24.0000	−1.58251
$231$	0	0
$232$	6.00000	0.393919
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	2.00000	0.130744
$235$	8.78461	0.573045
$236$	6.92820	0.450988
$237$	−2.92820	−0.190207
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	3.46410	0.223607
$241$	−30.3923	−1.95774	−0.978870	−	0.204482i	$-0.934449\pi$
$241$	−30.3923	−1.95774	−0.978870	+	0.204482i	$0.934449\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	3.46410	0.220863
$247$	−2.92820	−0.186317
$248$	−1.46410	−0.0929705
$249$	−16.3923	−1.03882
$250$	6.92820	0.438178
$251$	−17.0718	−1.07756	−0.538781	−	0.842446i	$-0.681115\pi$
$251$	−17.0718	−1.07756	−0.538781	+	0.842446i	$0.681115\pi$
$252$	0	0
$253$	−6.92820	−0.435572
$254$	−2.92820	−0.183732
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	0.928203	0.0578997	0.0289499	−	0.999581i	$-0.490784\pi$
$257$	0.928203	0.0578997	0.0289499	+	0.999581i	$0.490784\pi$
$258$	2.92820	0.182302
$259$	0	0
$260$	6.92820	0.429669
$261$	−6.00000	−0.371391
$262$	16.3923	1.01272
$263$	−18.9282	−1.16716	−0.583582	−	0.812055i	$-0.698349\pi$
$263$	−18.9282	−1.16716	−0.583582	+	0.812055i	$0.698349\pi$
$264$	1.00000	0.0615457
$265$	−44.7846	−2.75110
$266$	0	0
$267$	12.9282	0.791193
$268$	1.07180	0.0654704
$269$	24.2487	1.47847	0.739235	−	0.673448i	$-0.235187\pi$
$269$	24.2487	1.47847	0.739235	+	0.673448i	$0.235187\pi$
$270$	−3.46410	−0.210819
$271$	−16.7846	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$271$	−16.7846	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	19.8564	1.19957
$275$	7.00000	0.422116
$276$	6.92820	0.417029
$277$	15.8564	0.952719	0.476360	−	0.879251i	$-0.341956\pi$
$277$	15.8564	0.952719	0.476360	+	0.879251i	$0.341956\pi$
$278$	−6.53590	−0.391997
$279$	1.46410	0.0876535
$280$	0	0
$281$	19.8564	1.18453	0.592267	−	0.805742i	$-0.298233\pi$
$281$	19.8564	1.18453	0.592267	+	0.805742i	$0.298233\pi$
$282$	−2.53590	−0.151011
$283$	6.53590	0.388519	0.194259	−	0.980950i	$-0.437770\pi$
$283$	6.53590	0.388519	0.194259	+	0.980950i	$0.437770\pi$
$284$	−12.0000	−0.712069
$285$	5.07180	0.300427
$286$	2.00000	0.118262
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−5.00000	−0.294118
$290$	−20.7846	−1.22051
$291$	2.00000	0.117242
$292$	7.46410	0.436804
$293$	11.0718	0.646821	0.323411	−	0.946259i	$-0.395170\pi$
$293$	11.0718	0.646821	0.323411	+	0.946259i	$0.395170\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	−8.92820	−0.518941
$297$	−1.00000	−0.0580259
$298$	6.00000	0.347571
$299$	13.8564	0.801337
$300$	−7.00000	−0.404145
$301$	0	0
$302$	16.0000	0.920697
$303$	7.85641	0.451339
$304$	1.46410	0.0839720
$305$	6.92820	0.396708
$306$	−3.46410	−0.198030
$307$	−17.4641	−0.996729	−0.498364	−	0.866968i	$-0.666066\pi$
$307$	−17.4641	−0.996729	−0.498364	+	0.866968i	$0.666066\pi$
$308$	0	0
$309$	−6.53590	−0.371814
$310$	5.07180	0.288059
$311$	−7.60770	−0.431393	−0.215696	−	0.976460i	$-0.569202\pi$
$311$	−7.60770	−0.431393	−0.215696	+	0.976460i	$0.569202\pi$
$312$	−2.00000	−0.113228
$313$	3.07180	0.173628	0.0868141	−	0.996225i	$-0.472331\pi$
$313$	3.07180	0.173628	0.0868141	+	0.996225i	$0.472331\pi$
$314$	18.3923	1.03794
$315$	0	0
$316$	2.92820	0.164724
$317$	−0.928203	−0.0521331	−0.0260665	−	0.999660i	$-0.508298\pi$
$317$	−0.928203	−0.0521331	−0.0260665	+	0.999660i	$0.508298\pi$
$318$	12.9282	0.724978
$319$	−6.00000	−0.335936
$320$	−3.46410	−0.193649
$321$	−6.92820	−0.386695
$322$	0	0
$323$	5.07180	0.282202
$324$	1.00000	0.0555556
$325$	−14.0000	−0.776580
$326$	4.00000	0.221540
$327$	11.8564	0.655661
$328$	−3.46410	−0.191273
$329$	0	0
$330$	−3.46410	−0.190693
$331$	−22.9282	−1.26025	−0.630124	−	0.776495i	$-0.716996\pi$
$331$	−22.9282	−1.26025	−0.630124	+	0.776495i	$0.716996\pi$
$332$	16.3923	0.899645
$333$	8.92820	0.489263
$334$	5.07180	0.277516
$335$	−3.71281	−0.202853
$336$	0	0
$337$	7.07180	0.385225	0.192613	−	0.981275i	$-0.438304\pi$
$337$	7.07180	0.385225	0.192613	+	0.981275i	$0.438304\pi$
$338$	9.00000	0.489535
$339$	19.8564	1.07845
$340$	−12.0000	−0.650791
$341$	1.46410	0.0792855
$342$	−1.46410	−0.0791695
$343$	0	0
$344$	−2.92820	−0.157878
$345$	−24.0000	−1.29212
$346$	12.9282	0.695025
$347$	20.7846	1.11578	0.557888	−	0.829916i	$-0.311612\pi$
$347$	20.7846	1.11578	0.557888	+	0.829916i	$0.311612\pi$
$348$	6.00000	0.321634
$349$	30.7846	1.64786	0.823931	−	0.566690i	$-0.191776\pi$
$349$	30.7846	1.64786	0.823931	+	0.566690i	$0.191776\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−1.00000	−0.0533002
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	6.92820	0.368230
$355$	41.5692	2.20627
$356$	−12.9282	−0.685193
$357$	0	0
$358$	−20.7846	−1.09850
$359$	−18.9282	−0.998992	−0.499496	−	0.866316i	$-0.666482\pi$
$359$	−18.9282	−0.998992	−0.499496	+	0.866316i	$0.666482\pi$
$360$	3.46410	0.182574
$361$	−16.8564	−0.887179
$362$	−14.3923	−0.756443
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−25.8564	−1.35339
$366$	−2.00000	−0.104542
$367$	15.3205	0.799724	0.399862	−	0.916575i	$-0.369058\pi$
$367$	15.3205	0.799724	0.399862	+	0.916575i	$0.369058\pi$
$368$	−6.92820	−0.361158
$369$	3.46410	0.180334
$370$	30.9282	1.60788
$371$	0	0
$372$	−1.46410	−0.0759101
$373$	−25.7128	−1.33136	−0.665679	−	0.746238i	$-0.731858\pi$
$373$	−25.7128	−1.33136	−0.665679	+	0.746238i	$0.731858\pi$
$374$	−3.46410	−0.179124
$375$	6.92820	0.357771
$376$	2.53590	0.130779
$377$	12.0000	0.618031
$378$	0	0
$379$	9.85641	0.506290	0.253145	−	0.967428i	$-0.418535\pi$
$379$	9.85641	0.506290	0.253145	+	0.967428i	$0.418535\pi$
$380$	−5.07180	−0.260178
$381$	−2.92820	−0.150016
$382$	20.7846	1.06343
$383$	−21.4641	−1.09676	−0.548382	−	0.836228i	$-0.684756\pi$
$383$	−21.4641	−1.09676	−0.548382	+	0.836228i	$0.684756\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−26.0000	−1.32337
$387$	2.92820	0.148849
$388$	−2.00000	−0.101535
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	6.92820	0.350823
$391$	−24.0000	−1.21373
$392$	0	0
$393$	16.3923	0.826882
$394$	−18.0000	−0.906827
$395$	−10.1436	−0.510380
$396$	1.00000	0.0502519
$397$	−18.3923	−0.923083	−0.461542	−	0.887119i	$-0.652704\pi$
$397$	−18.3923	−0.923083	−0.461542	+	0.887119i	$0.652704\pi$
$398$	−20.3923	−1.02217
$399$	0	0
$400$	7.00000	0.350000
$401$	31.8564	1.59083	0.795417	−	0.606063i	$-0.207252\pi$
$401$	31.8564	1.59083	0.795417	+	0.606063i	$0.207252\pi$
$402$	1.07180	0.0534564
$403$	−2.92820	−0.145864
$404$	−7.85641	−0.390871
$405$	−3.46410	−0.172133
$406$	0	0
$407$	8.92820	0.442555
$408$	3.46410	0.171499
$409$	−25.3205	−1.25202	−0.626009	−	0.779816i	$-0.715313\pi$
$409$	−25.3205	−1.25202	−0.626009	+	0.779816i	$0.715313\pi$
$410$	12.0000	0.592638
$411$	19.8564	0.979444
$412$	6.53590	0.322001
$413$	0	0
$414$	6.92820	0.340503
$415$	−56.7846	−2.78745
$416$	2.00000	0.0980581
$417$	−6.53590	−0.320064
$418$	−1.46410	−0.0716116
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	17.7128	0.863270	0.431635	−	0.902048i	$-0.357937\pi$
$421$	17.7128	0.863270	0.431635	+	0.902048i	$0.357937\pi$
$422$	24.7846	1.20650
$423$	−2.53590	−0.123300
$424$	−12.9282	−0.627849
$425$	24.2487	1.17624
$426$	−12.0000	−0.581402
$427$	0	0
$428$	6.92820	0.334887
$429$	2.00000	0.0965609
$430$	10.1436	0.489168
$431$	−5.07180	−0.244300	−0.122150	−	0.992512i	$-0.538979\pi$
$431$	−5.07180	−0.244300	−0.122150	+	0.992512i	$0.538979\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	−20.7846	−0.996546
$436$	−11.8564	−0.567819
$437$	−10.1436	−0.485234
$438$	7.46410	0.356649
$439$	−16.7846	−0.801086	−0.400543	−	0.916278i	$-0.631178\pi$
$439$	−16.7846	−0.801086	−0.400543	+	0.916278i	$0.631178\pi$
$440$	3.46410	0.165145
$441$	0	0
$442$	6.92820	0.329541
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	−8.92820	−0.423714
$445$	44.7846	2.12299
$446$	12.3923	0.586793
$447$	6.00000	0.283790
$448$	0	0
$449$	7.85641	0.370767	0.185383	−	0.982666i	$-0.440647\pi$
$449$	7.85641	0.370767	0.185383	+	0.982666i	$0.440647\pi$
$450$	−7.00000	−0.329983
$451$	3.46410	0.163118
$452$	−19.8564	−0.933967
$453$	16.0000	0.751746
$454$	16.3923	0.769329
$455$	0	0
$456$	1.46410	0.0685628
$457$	−11.8564	−0.554619	−0.277310	−	0.960781i	$-0.589443\pi$
$457$	−11.8564	−0.554619	−0.277310	+	0.960781i	$0.589443\pi$
$458$	13.3205	0.622426
$459$	−3.46410	−0.161690
$460$	24.0000	1.11901
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−6.00000	−0.278543
$465$	5.07180	0.235199
$466$	−19.8564	−0.919830
$467$	−39.7128	−1.83769	−0.918845	−	0.394619i	$-0.870877\pi$
$467$	−39.7128	−1.83769	−0.918845	+	0.394619i	$0.870877\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−8.78461	−0.405204
$471$	18.3923	0.847473
$472$	−6.92820	−0.318896
$473$	2.92820	0.134639
$474$	2.92820	0.134497
$475$	10.2487	0.470243
$476$	0	0
$477$	12.9282	0.591942
$478$	24.0000	1.09773
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	−3.46410	−0.158114
$481$	−17.8564	−0.814182
$482$	30.3923	1.38433
$483$	0	0
$484$	1.00000	0.0454545
$485$	6.92820	0.314594
$486$	1.00000	0.0453609
$487$	−0.784610	−0.0355541	−0.0177770	−	0.999842i	$-0.505659\pi$
$487$	−0.784610	−0.0355541	−0.0177770	+	0.999842i	$0.505659\pi$
$488$	2.00000	0.0905357
$489$	4.00000	0.180886
$490$	0	0
$491$	25.8564	1.16688	0.583442	−	0.812155i	$-0.301706\pi$
$491$	25.8564	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$492$	−3.46410	−0.156174
$493$	−20.7846	−0.936092
$494$	2.92820	0.131746
$495$	−3.46410	−0.155700
$496$	1.46410	0.0657401
$497$	0	0
$498$	16.3923	0.734557
$499$	−22.9282	−1.02641	−0.513204	−	0.858267i	$-0.671541\pi$
$499$	−22.9282	−1.02641	−0.513204	+	0.858267i	$0.671541\pi$
$500$	−6.92820	−0.309839
$501$	5.07180	0.226591
$502$	17.0718	0.761952
$503$	−5.07180	−0.226140	−0.113070	−	0.993587i	$-0.536068\pi$
$503$	−5.07180	−0.226140	−0.113070	+	0.993587i	$0.536068\pi$
$504$	0	0
$505$	27.2154	1.21107
$506$	6.92820	0.307996
$507$	9.00000	0.399704
$508$	2.92820	0.129918
$509$	6.67949	0.296063	0.148032	−	0.988983i	$-0.452706\pi$
$509$	6.67949	0.296063	0.148032	+	0.988983i	$0.452706\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−1.46410	−0.0646417
$514$	−0.928203	−0.0409413
$515$	−22.6410	−0.997682
$516$	−2.92820	−0.128907
$517$	−2.53590	−0.111529
$518$	0	0
$519$	12.9282	0.567485
$520$	−6.92820	−0.303822
$521$	−23.0718	−1.01079	−0.505397	−	0.862887i	$-0.668654\pi$
$521$	−23.0718	−1.01079	−0.505397	+	0.862887i	$0.668654\pi$
$522$	6.00000	0.262613
$523$	−3.60770	−0.157753	−0.0788767	−	0.996884i	$-0.525133\pi$
$523$	−3.60770	−0.157753	−0.0788767	+	0.996884i	$0.525133\pi$
$524$	−16.3923	−0.716101
$525$	0	0
$526$	18.9282	0.825309
$527$	5.07180	0.220931
$528$	−1.00000	−0.0435194
$529$	25.0000	1.08696
$530$	44.7846	1.94532
$531$	6.92820	0.300658
$532$	0	0
$533$	−6.92820	−0.300094
$534$	−12.9282	−0.559458
$535$	−24.0000	−1.03761
$536$	−1.07180	−0.0462946
$537$	−20.7846	−0.896922
$538$	−24.2487	−1.04544
$539$	0	0
$540$	3.46410	0.149071
$541$	29.7128	1.27745	0.638727	−	0.769434i	$-0.279461\pi$
$541$	29.7128	1.27745	0.638727	+	0.769434i	$0.279461\pi$
$542$	16.7846	0.720961
$543$	−14.3923	−0.617633
$544$	−3.46410	−0.148522
$545$	41.0718	1.75932
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	−19.8564	−0.848224
$549$	−2.00000	−0.0853579
$550$	−7.00000	−0.298481
$551$	−8.78461	−0.374237
$552$	−6.92820	−0.294884
$553$	0	0
$554$	−15.8564	−0.673674
$555$	30.9282	1.31283
$556$	6.53590	0.277184
$557$	−33.7128	−1.42846	−0.714229	−	0.699912i	$-0.753222\pi$
$557$	−33.7128	−1.42846	−0.714229	+	0.699912i	$0.753222\pi$
$558$	−1.46410	−0.0619804
$559$	−5.85641	−0.247700
$560$	0	0
$561$	−3.46410	−0.146254
$562$	−19.8564	−0.837592
$563$	−30.2487	−1.27483	−0.637416	−	0.770520i	$-0.719997\pi$
$563$	−30.2487	−1.27483	−0.637416	+	0.770520i	$0.719997\pi$
$564$	2.53590	0.106781
$565$	68.7846	2.89379
$566$	−6.53590	−0.274724
$567$	0	0
$568$	12.0000	0.503509
$569$	−4.14359	−0.173708	−0.0868542	−	0.996221i	$-0.527681\pi$
$569$	−4.14359	−0.173708	−0.0868542	+	0.996221i	$0.527681\pi$
$570$	−5.07180	−0.212434
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	−2.00000	−0.0836242
$573$	20.7846	0.868290
$574$	0	0
$575$	−48.4974	−2.02248
$576$	1.00000	0.0416667
$577$	−10.7846	−0.448969	−0.224485	−	0.974478i	$-0.572070\pi$
$577$	−10.7846	−0.448969	−0.224485	+	0.974478i	$0.572070\pi$
$578$	5.00000	0.207973
$579$	−26.0000	−1.08052
$580$	20.7846	0.863034
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	12.9282	0.535431
$584$	−7.46410	−0.308867
$585$	6.92820	0.286446
$586$	−11.0718	−0.457372
$587$	−20.7846	−0.857873	−0.428936	−	0.903335i	$-0.641112\pi$
$587$	−20.7846	−0.857873	−0.428936	+	0.903335i	$0.641112\pi$
$588$	0	0
$589$	2.14359	0.0883252
$590$	24.0000	0.988064
$591$	−18.0000	−0.740421
$592$	8.92820	0.366947
$593$	27.4641	1.12782	0.563908	−	0.825838i	$-0.309297\pi$
$593$	27.4641	1.12782	0.563908	+	0.825838i	$0.309297\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−20.3923	−0.834601
$598$	−13.8564	−0.566631
$599$	−25.8564	−1.05646	−0.528232	−	0.849100i	$-0.677145\pi$
$599$	−25.8564	−1.05646	−0.528232	+	0.849100i	$0.677145\pi$
$600$	7.00000	0.285774
$601$	2.39230	0.0975842	0.0487921	−	0.998809i	$-0.484463\pi$
$601$	2.39230	0.0975842	0.0487921	+	0.998809i	$0.484463\pi$
$602$	0	0
$603$	1.07180	0.0436469
$604$	−16.0000	−0.651031
$605$	−3.46410	−0.140836
$606$	−7.85641	−0.319145
$607$	−35.7128	−1.44954	−0.724769	−	0.688992i	$-0.758054\pi$
$607$	−35.7128	−1.44954	−0.724769	+	0.688992i	$0.758054\pi$
$608$	−1.46410	−0.0593772
$609$	0	0
$610$	−6.92820	−0.280515
$611$	5.07180	0.205183
$612$	3.46410	0.140028
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	17.4641	0.704794
$615$	12.0000	0.483887
$616$	0	0
$617$	7.85641	0.316287	0.158144	−	0.987416i	$-0.449449\pi$
$617$	7.85641	0.316287	0.158144	+	0.987416i	$0.449449\pi$
$618$	6.53590	0.262912
$619$	−23.7128	−0.953098	−0.476549	−	0.879148i	$-0.658113\pi$
$619$	−23.7128	−0.953098	−0.476549	+	0.879148i	$0.658113\pi$
$620$	−5.07180	−0.203688
$621$	6.92820	0.278019
$622$	7.60770	0.305041
$623$	0	0
$624$	2.00000	0.0800641
$625$	−11.0000	−0.440000
$626$	−3.07180	−0.122774
$627$	−1.46410	−0.0584706
$628$	−18.3923	−0.733933
$629$	30.9282	1.23319
$630$	0	0
$631$	40.7846	1.62361	0.811805	−	0.583929i	$-0.198485\pi$
$631$	40.7846	1.62361	0.811805	+	0.583929i	$0.198485\pi$
$632$	−2.92820	−0.116478
$633$	24.7846	0.985100
$634$	0.928203	0.0368637
$635$	−10.1436	−0.402536
$636$	−12.9282	−0.512637
$637$	0	0
$638$	6.00000	0.237542
$639$	−12.0000	−0.474713
$640$	3.46410	0.136931
$641$	4.14359	0.163662	0.0818311	−	0.996646i	$-0.473923\pi$
$641$	4.14359	0.163662	0.0818311	+	0.996646i	$0.473923\pi$
$642$	6.92820	0.273434
$643$	−1.07180	−0.0422675	−0.0211338	−	0.999777i	$-0.506728\pi$
$643$	−1.07180	−0.0422675	−0.0211338	+	0.999777i	$0.506728\pi$
$644$	0	0
$645$	10.1436	0.399404
$646$	−5.07180	−0.199547
$647$	−35.3205	−1.38859	−0.694296	−	0.719689i	$-0.744284\pi$
$647$	−35.3205	−1.38859	−0.694296	+	0.719689i	$0.744284\pi$
$648$	−1.00000	−0.0392837
$649$	6.92820	0.271956
$650$	14.0000	0.549125
$651$	0	0
$652$	−4.00000	−0.156652
$653$	21.7128	0.849688	0.424844	−	0.905267i	$-0.360329\pi$
$653$	21.7128	0.849688	0.424844	+	0.905267i	$0.360329\pi$
$654$	−11.8564	−0.463622
$655$	56.7846	2.21876
$656$	3.46410	0.135250
$657$	7.46410	0.291202
$658$	0	0
$659$	−30.9282	−1.20479	−0.602396	−	0.798197i	$-0.705787\pi$
$659$	−30.9282	−1.20479	−0.602396	+	0.798197i	$0.705787\pi$
$660$	3.46410	0.134840
$661$	4.24871	0.165256	0.0826279	−	0.996580i	$-0.473669\pi$
$661$	4.24871	0.165256	0.0826279	+	0.996580i	$0.473669\pi$
$662$	22.9282	0.891130
$663$	6.92820	0.269069
$664$	−16.3923	−0.636145
$665$	0	0
$666$	−8.92820	−0.345961
$667$	41.5692	1.60957
$668$	−5.07180	−0.196234
$669$	12.3923	0.479114
$670$	3.71281	0.143438
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	7.07180	0.272598	0.136299	−	0.990668i	$-0.456479\pi$
$673$	7.07180	0.272598	0.136299	+	0.990668i	$0.456479\pi$
$674$	−7.07180	−0.272395
$675$	−7.00000	−0.269430
$676$	−9.00000	−0.346154
$677$	38.7846	1.49061	0.745307	−	0.666722i	$-0.232303\pi$
$677$	38.7846	1.49061	0.745307	+	0.666722i	$0.232303\pi$
$678$	−19.8564	−0.762581
$679$	0	0
$680$	12.0000	0.460179
$681$	16.3923	0.628154
$682$	−1.46410	−0.0560633
$683$	44.7846	1.71364	0.856818	−	0.515619i	$-0.172438\pi$
$683$	44.7846	1.71364	0.856818	+	0.515619i	$0.172438\pi$
$684$	1.46410	0.0559813
$685$	68.7846	2.62812
$686$	0	0
$687$	13.3205	0.508209
$688$	2.92820	0.111637
$689$	−25.8564	−0.985051
$690$	24.0000	0.913664
$691$	46.9282	1.78523	0.892616	−	0.450817i	$-0.148867\pi$
$691$	46.9282	1.78523	0.892616	+	0.450817i	$0.148867\pi$
$692$	−12.9282	−0.491457
$693$	0	0
$694$	−20.7846	−0.788973
$695$	−22.6410	−0.858823
$696$	−6.00000	−0.227429
$697$	12.0000	0.454532
$698$	−30.7846	−1.16521
$699$	−19.8564	−0.751038
$700$	0	0
$701$	−9.71281	−0.366848	−0.183424	−	0.983034i	$-0.558718\pi$
$701$	−9.71281	−0.366848	−0.183424	+	0.983034i	$0.558718\pi$
$702$	−2.00000	−0.0754851
$703$	13.0718	0.493012
$704$	1.00000	0.0376889
$705$	−8.78461	−0.330848
$706$	−0.928203	−0.0349334
$707$	0	0
$708$	−6.92820	−0.260378
$709$	5.21539	0.195868	0.0979340	−	0.995193i	$-0.468777\pi$
$709$	5.21539	0.195868	0.0979340	+	0.995193i	$0.468777\pi$
$710$	−41.5692	−1.56007
$711$	2.92820	0.109816
$712$	12.9282	0.484505
$713$	−10.1436	−0.379881
$714$	0	0
$715$	6.92820	0.259100
$716$	20.7846	0.776757
$717$	24.0000	0.896296
$718$	18.9282	0.706394
$719$	−25.1769	−0.938940	−0.469470	−	0.882948i	$-0.655555\pi$
$719$	−25.1769	−0.938940	−0.469470	+	0.882948i	$0.655555\pi$
$720$	−3.46410	−0.129099
$721$	0	0
$722$	16.8564	0.627330
$723$	30.3923	1.13030
$724$	14.3923	0.534886
$725$	−42.0000	−1.55984
$726$	1.00000	0.0371135
$727$	29.1769	1.08211	0.541056	−	0.840987i	$-0.318025\pi$
$727$	29.1769	1.08211	0.541056	+	0.840987i	$0.318025\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	25.8564	0.956989
$731$	10.1436	0.375174
$732$	2.00000	0.0739221
$733$	−39.8564	−1.47213	−0.736065	−	0.676911i	$-0.763318\pi$
$733$	−39.8564	−1.47213	−0.736065	+	0.676911i	$0.763318\pi$
$734$	−15.3205	−0.565490
$735$	0	0
$736$	6.92820	0.255377
$737$	1.07180	0.0394801
$738$	−3.46410	−0.127515
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	−30.9282	−1.13694
$741$	2.92820	0.107570
$742$	0	0
$743$	41.5692	1.52503	0.762513	−	0.646972i	$-0.223965\pi$
$743$	41.5692	1.52503	0.762513	+	0.646972i	$0.223965\pi$
$744$	1.46410	0.0536766
$745$	20.7846	0.761489
$746$	25.7128	0.941413
$747$	16.3923	0.599763
$748$	3.46410	0.126660
$749$	0	0
$750$	−6.92820	−0.252982
$751$	16.7846	0.612479	0.306240	−	0.951954i	$-0.400929\pi$
$751$	16.7846	0.612479	0.306240	+	0.951954i	$0.400929\pi$
$752$	−2.53590	−0.0924747
$753$	17.0718	0.622131
$754$	−12.0000	−0.437014
$755$	55.4256	2.01715
$756$	0	0
$757$	−18.7846	−0.682738	−0.341369	−	0.939929i	$-0.610891\pi$
$757$	−18.7846	−0.682738	−0.341369	+	0.939929i	$0.610891\pi$
$758$	−9.85641	−0.358001
$759$	6.92820	0.251478
$760$	5.07180	0.183973
$761$	46.3923	1.68172	0.840860	−	0.541253i	$-0.182050\pi$
$761$	46.3923	1.68172	0.840860	+	0.541253i	$0.182050\pi$
$762$	2.92820	0.106078
$763$	0	0
$764$	−20.7846	−0.751961
$765$	−12.0000	−0.433861
$766$	21.4641	0.775530
$767$	−13.8564	−0.500326
$768$	−1.00000	−0.0360844
$769$	−35.4641	−1.27887	−0.639434	−	0.768846i	$-0.720831\pi$
$769$	−35.4641	−1.27887	−0.639434	+	0.768846i	$0.720831\pi$
$770$	0	0
$771$	−0.928203	−0.0334284
$772$	26.0000	0.935760
$773$	39.4641	1.41943	0.709713	−	0.704491i	$-0.248825\pi$
$773$	39.4641	1.41943	0.709713	+	0.704491i	$0.248825\pi$
$774$	−2.92820	−0.105252
$775$	10.2487	0.368145
$776$	2.00000	0.0717958
$777$	0	0
$778$	30.0000	1.07555
$779$	5.07180	0.181716
$780$	−6.92820	−0.248069
$781$	−12.0000	−0.429394
$782$	24.0000	0.858238
$783$	6.00000	0.214423
$784$	0	0
$785$	63.7128	2.27401
$786$	−16.3923	−0.584694
$787$	29.1769	1.04004	0.520022	−	0.854153i	$-0.325924\pi$
$787$	29.1769	1.04004	0.520022	+	0.854153i	$0.325924\pi$
$788$	18.0000	0.641223
$789$	18.9282	0.673862
$790$	10.1436	0.360893
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	4.00000	0.142044
$794$	18.3923	0.652718
$795$	44.7846	1.58835
$796$	20.3923	0.722786
$797$	39.4641	1.39789	0.698945	−	0.715175i	$-0.253653\pi$
$797$	39.4641	1.39789	0.698945	+	0.715175i	$0.253653\pi$
$798$	0	0
$799$	−8.78461	−0.310777
$800$	−7.00000	−0.247487
$801$	−12.9282	−0.456796
$802$	−31.8564	−1.12489
$803$	7.46410	0.263402
$804$	−1.07180	−0.0377994
$805$	0	0
$806$	2.92820	0.103142
$807$	−24.2487	−0.853595
$808$	7.85641	0.276387
$809$	24.9282	0.876429	0.438214	−	0.898870i	$-0.355611\pi$
$809$	24.9282	0.876429	0.438214	+	0.898870i	$0.355611\pi$
$810$	3.46410	0.121716
$811$	−45.1769	−1.58638	−0.793188	−	0.608977i	$-0.791580\pi$
$811$	−45.1769	−1.58638	−0.793188	+	0.608977i	$0.791580\pi$
$812$	0	0
$813$	16.7846	0.588662
$814$	−8.92820	−0.312933
$815$	13.8564	0.485369
$816$	−3.46410	−0.121268
$817$	4.28719	0.149990
$818$	25.3205	0.885311
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−19.8564	−0.692572
$823$	−0.784610	−0.0273498	−0.0136749	−	0.999906i	$-0.504353\pi$
$823$	−0.784610	−0.0273498	−0.0136749	+	0.999906i	$0.504353\pi$
$824$	−6.53590	−0.227689
$825$	−7.00000	−0.243709
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−6.92820	−0.240772
$829$	4.24871	0.147564	0.0737819	−	0.997274i	$-0.476493\pi$
$829$	4.24871	0.147564	0.0737819	+	0.997274i	$0.476493\pi$
$830$	56.7846	1.97102
$831$	−15.8564	−0.550053
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	6.53590	0.226320
$835$	17.5692	0.608008
$836$	1.46410	0.0506370
$837$	−1.46410	−0.0506068
$838$	−36.0000	−1.24360
$839$	26.5359	0.916121	0.458060	−	0.888921i	$-0.348544\pi$
$839$	26.5359	0.916121	0.458060	+	0.888921i	$0.348544\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−17.7128	−0.610424
$843$	−19.8564	−0.683891
$844$	−24.7846	−0.853121
$845$	31.1769	1.07252
$846$	2.53590	0.0871860
$847$	0	0
$848$	12.9282	0.443956
$849$	−6.53590	−0.224311
$850$	−24.2487	−0.831724
$851$	−61.8564	−2.12041
$852$	12.0000	0.411113
$853$	−5.71281	−0.195603	−0.0978015	−	0.995206i	$-0.531181\pi$
$853$	−5.71281	−0.195603	−0.0978015	+	0.995206i	$0.531181\pi$
$854$	0	0
$855$	−5.07180	−0.173452
$856$	−6.92820	−0.236801
$857$	−29.3205	−1.00157	−0.500785	−	0.865572i	$-0.666955\pi$
$857$	−29.3205	−1.00157	−0.500785	+	0.865572i	$0.666955\pi$
$858$	−2.00000	−0.0682789
$859$	−11.2154	−0.382664	−0.191332	−	0.981525i	$-0.561281\pi$
$859$	−11.2154	−0.382664	−0.191332	+	0.981525i	$0.561281\pi$
$860$	−10.1436	−0.345894
$861$	0	0
$862$	5.07180	0.172746
$863$	3.21539	0.109453	0.0547266	−	0.998501i	$-0.482571\pi$
$863$	3.21539	0.109453	0.0547266	+	0.998501i	$0.482571\pi$
$864$	1.00000	0.0340207
$865$	44.7846	1.52272
$866$	2.00000	0.0679628
$867$	5.00000	0.169809
$868$	0	0
$869$	2.92820	0.0993325
$870$	20.7846	0.704664
$871$	−2.14359	−0.0726329
$872$	11.8564	0.401509
$873$	−2.00000	−0.0676897
$874$	10.1436	0.343112
$875$	0	0
$876$	−7.46410	−0.252189
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	16.7846	0.566453
$879$	−11.0718	−0.373442
$880$	−3.46410	−0.116775
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	−14.1436	−0.475970	−0.237985	−	0.971269i	$-0.576487\pi$
$883$	−14.1436	−0.475970	−0.237985	+	0.971269i	$0.576487\pi$
$884$	−6.92820	−0.233021
$885$	24.0000	0.806751
$886$	−20.7846	−0.698273
$887$	−37.8564	−1.27109	−0.635547	−	0.772062i	$-0.719225\pi$
$887$	−37.8564	−1.27109	−0.635547	+	0.772062i	$0.719225\pi$
$888$	8.92820	0.299611
$889$	0	0
$890$	−44.7846	−1.50118
$891$	1.00000	0.0335013
$892$	−12.3923	−0.414925
$893$	−3.71281	−0.124245
$894$	−6.00000	−0.200670
$895$	−72.0000	−2.40669
$896$	0	0
$897$	−13.8564	−0.462652
$898$	−7.85641	−0.262172
$899$	−8.78461	−0.292983
$900$	7.00000	0.233333
$901$	44.7846	1.49199
$902$	−3.46410	−0.115342
$903$	0	0
$904$	19.8564	0.660414
$905$	−49.8564	−1.65728
$906$	−16.0000	−0.531564
$907$	1.07180	0.0355884	0.0177942	−	0.999842i	$-0.494336\pi$
$907$	1.07180	0.0355884	0.0177942	+	0.999842i	$0.494336\pi$
$908$	−16.3923	−0.543998
$909$	−7.85641	−0.260581
$910$	0	0
$911$	48.4974	1.60679	0.803396	−	0.595446i	$-0.203024\pi$
$911$	48.4974	1.60679	0.803396	+	0.595446i	$0.203024\pi$
$912$	−1.46410	−0.0484812
$913$	16.3923	0.542506
$914$	11.8564	0.392175
$915$	−6.92820	−0.229039
$916$	−13.3205	−0.440122
$917$	0	0
$918$	3.46410	0.114332
$919$	−5.85641	−0.193185	−0.0965925	−	0.995324i	$-0.530794\pi$
$919$	−5.85641	−0.193185	−0.0965925	+	0.995324i	$0.530794\pi$
$920$	−24.0000	−0.791257
$921$	17.4641	0.575462
$922$	−6.00000	−0.197599
$923$	24.0000	0.789970
$924$	0	0
$925$	62.4974	2.05490
$926$	−8.00000	−0.262896
$927$	6.53590	0.214667
$928$	6.00000	0.196960
$929$	28.6410	0.939681	0.469841	−	0.882751i	$-0.344311\pi$
$929$	28.6410	0.939681	0.469841	+	0.882751i	$0.344311\pi$
$930$	−5.07180	−0.166311
$931$	0	0
$932$	19.8564	0.650418
$933$	7.60770	0.249065
$934$	39.7128	1.29944
$935$	−12.0000	−0.392442
$936$	2.00000	0.0653720
$937$	−6.39230	−0.208827	−0.104414	−	0.994534i	$-0.533297\pi$
$937$	−6.39230	−0.208827	−0.104414	+	0.994534i	$0.533297\pi$
$938$	0	0
$939$	−3.07180	−0.100244
$940$	8.78461	0.286522
$941$	24.9282	0.812636	0.406318	−	0.913732i	$-0.366812\pi$
$941$	24.9282	0.812636	0.406318	+	0.913732i	$0.366812\pi$
$942$	−18.3923	−0.599254
$943$	−24.0000	−0.781548
$944$	6.92820	0.225494
$945$	0	0
$946$	−2.92820	−0.0952041
$947$	22.1436	0.719570	0.359785	−	0.933035i	$-0.382850\pi$
$947$	22.1436	0.719570	0.359785	+	0.933035i	$0.382850\pi$
$948$	−2.92820	−0.0951036
$949$	−14.9282	−0.484590
$950$	−10.2487	−0.332512
$951$	0.928203	0.0300991
$952$	0	0
$953$	14.7846	0.478920	0.239460	−	0.970906i	$-0.423030\pi$
$953$	14.7846	0.478920	0.239460	+	0.970906i	$0.423030\pi$
$954$	−12.9282	−0.418566
$955$	72.0000	2.32987
$956$	−24.0000	−0.776215
$957$	6.00000	0.193952
$958$	13.8564	0.447680
$959$	0	0
$960$	3.46410	0.111803
$961$	−28.8564	−0.930852
$962$	17.8564	0.575714
$963$	6.92820	0.223258
$964$	−30.3923	−0.978870
$965$	−90.0666	−2.89935
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−1.00000	−0.0321412
$969$	−5.07180	−0.162930
$970$	−6.92820	−0.222451
$971$	−39.7128	−1.27444	−0.637222	−	0.770680i	$-0.719917\pi$
$971$	−39.7128	−1.27444	−0.637222	+	0.770680i	$0.719917\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	0.784610	0.0251405
$975$	14.0000	0.448359
$976$	−2.00000	−0.0640184
$977$	35.5692	1.13796	0.568980	−	0.822351i	$-0.307338\pi$
$977$	35.5692	1.13796	0.568980	+	0.822351i	$0.307338\pi$
$978$	−4.00000	−0.127906
$979$	−12.9282	−0.413187
$980$	0	0
$981$	−11.8564	−0.378546
$982$	−25.8564	−0.825111
$983$	11.3205	0.361068	0.180534	−	0.983569i	$-0.442217\pi$
$983$	11.3205	0.361068	0.180534	+	0.983569i	$0.442217\pi$
$984$	3.46410	0.110432
$985$	−62.3538	−1.98676
$986$	20.7846	0.661917
$987$	0	0
$988$	−2.92820	−0.0931586
$989$	−20.2872	−0.645095
$990$	3.46410	0.110096
$991$	−29.8564	−0.948420	−0.474210	−	0.880412i	$-0.657266\pi$
$991$	−29.8564	−0.948420	−0.474210	+	0.880412i	$0.657266\pi$
$992$	−1.46410	−0.0464853
$993$	22.9282	0.727605
$994$	0	0
$995$	−70.6410	−2.23947
$996$	−16.3923	−0.519410
$997$	44.6410	1.41380	0.706898	−	0.707316i	$-0.250094\pi$
$997$	44.6410	1.41380	0.706898	+	0.707316i	$0.250094\pi$
$998$	22.9282	0.725780
$999$	−8.92820	−0.282476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.x.1.1		2
3.2	odd	2		9702.2.a.dd.1.2		2
7.6	odd	2		462.2.a.h.1.2	✓	2
21.20	even	2		1386.2.a.p.1.1		2
28.27	even	2		3696.2.a.bc.1.2		2
77.76	even	2		5082.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.h.1.2	✓	2	7.6	odd	2
1386.2.a.p.1.1		2	21.20	even	2
3234.2.a.x.1.1		2	1.1	even	1	trivial
3696.2.a.bc.1.2		2	28.27	even	2
5082.2.a.bu.1.2		2	77.76	even	2
9702.2.a.dd.1.2		2	3.2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.46410 q^{10} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +3.46410 q^{15} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{18} +1.46410 q^{19} -3.46410 q^{20} -1.00000 q^{22} -6.92820 q^{23} +1.00000 q^{24} +7.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -3.46410 q^{30} +1.46410 q^{31} -1.00000 q^{32} -1.00000 q^{33} -3.46410 q^{34} +1.00000 q^{36} +8.92820 q^{37} -1.46410 q^{38} +2.00000 q^{39} +3.46410 q^{40} +3.46410 q^{41} +2.92820 q^{43} +1.00000 q^{44} -3.46410 q^{45} +6.92820 q^{46} -2.53590 q^{47} -1.00000 q^{48} -7.00000 q^{50} -3.46410 q^{51} -2.00000 q^{52} +12.9282 q^{53} +1.00000 q^{54} -3.46410 q^{55} -1.46410 q^{57} +6.00000 q^{58} +6.92820 q^{59} +3.46410 q^{60} -2.00000 q^{61} -1.46410 q^{62} +1.00000 q^{64} +6.92820 q^{65} +1.00000 q^{66} +1.07180 q^{67} +3.46410 q^{68} +6.92820 q^{69} -12.0000 q^{71} -1.00000 q^{72} +7.46410 q^{73} -8.92820 q^{74} -7.00000 q^{75} +1.46410 q^{76} -2.00000 q^{78} +2.92820 q^{79} -3.46410 q^{80} +1.00000 q^{81} -3.46410 q^{82} +16.3923 q^{83} -12.0000 q^{85} -2.92820 q^{86} +6.00000 q^{87} -1.00000 q^{88} -12.9282 q^{89} +3.46410 q^{90} -6.92820 q^{92} -1.46410 q^{93} +2.53590 q^{94} -5.07180 q^{95} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{18} - 4 q^{19} - 2 q^{22} + 2 q^{24} + 14 q^{25} + 4 q^{26} - 2 q^{27} - 12 q^{29} - 4 q^{31}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^18 - 4 * q^19 - 2 * q^22 + 2 * q^24 + 14 * q^25 + 4 * q^26 - 2 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^32 - 2 * q^33 + 2 * q^36 + 4 * q^37 + 4 * q^38 + 4 * q^39 - 8 * q^43 + 2 * q^44 - 12 * q^47 - 2 * q^48 - 14 * q^50 - 4 * q^52 + 12 * q^53 + 2 * q^54 + 4 * q^57 + 12 * q^58 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 2 * q^66 + 16 * q^67 - 24 * q^71 - 2 * q^72 + 8 * q^73 - 4 * q^74 - 14 * q^75 - 4 * q^76 - 4 * q^78 - 8 * q^79 + 2 * q^81 + 12 * q^83 - 24 * q^85 + 8 * q^86 + 12 * q^87 - 2 * q^88 - 12 * q^89 + 4 * q^93 + 12 * q^94 - 24 * q^95 + 2 * q^96 - 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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