LMFDB - Embedded newform 1248.2.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,2,Mod(1,1248)] 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("1248.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.96533017226$
Analytic rank:	$1$
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	$\chi$	$=$	1248.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.23607 q^{5} -3.23607 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} -3.23607 q^{15} -4.47214 q^{17} -0.763932 q^{19} +3.23607 q^{21} -6.47214 q^{23} +5.47214 q^{25} -1.00000 q^{27} -4.47214 q^{29} +5.70820 q^{31} +2.00000 q^{33} -10.4721 q^{35} -8.47214 q^{37} +1.00000 q^{39} -3.23607 q^{41} +2.47214 q^{43} +3.23607 q^{45} +10.9443 q^{47} +3.47214 q^{49} +4.47214 q^{51} -0.472136 q^{53} -6.47214 q^{55} +0.763932 q^{57} +0.472136 q^{59} -3.52786 q^{61} -3.23607 q^{63} -3.23607 q^{65} -11.2361 q^{67} +6.47214 q^{69} +4.47214 q^{71} -8.47214 q^{73} -5.47214 q^{75} +6.47214 q^{77} -8.94427 q^{79} +1.00000 q^{81} -16.4721 q^{83} -14.4721 q^{85} +4.47214 q^{87} +12.1803 q^{89} +3.23607 q^{91} -5.70820 q^{93} -2.47214 q^{95} +4.47214 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 4 q^{33} - 12 q^{35} - 8 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 - 6 * q^19 + 2 * q^21 - 4 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 4 * q^33 - 12 * q^35 - 8 * q^37 + 2 * q^39 - 2 * q^41 - 4 * q^43 + 2 * q^45 + 4 * q^47 - 2 * q^49 + 8 * q^53 - 4 * q^55 + 6 * q^57 - 8 * q^59 - 16 * q^61 - 2 * q^63 - 2 * q^65 - 18 * q^67 + 4 * q^69 - 8 * q^73 - 2 * q^75 + 4 * q^77 + 2 * q^81 - 24 * q^83 - 20 * q^85 + 2 * q^89 + 2 * q^91 + 2 * q^93 + 4 * q^95 - 4 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.23607	−0.835549
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−0.763932	−0.175258	−0.0876290	−	0.996153i	$-0.527929\pi$
$19$	−0.763932	−0.175258	−0.0876290	+	0.996153i	$0.527929\pi$
$20$	0	0
$21$	3.23607	0.706168
$22$	0	0
$23$	−6.47214	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$24$	0	0
$25$	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	$0.636298\pi$
$30$	0	0
$31$	5.70820	1.02522	0.512612	−	0.858620i	$-0.328678\pi$
$31$	5.70820	1.02522	0.512612	+	0.858620i	$0.328678\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−10.4721	−1.77011
$36$	0	0
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.23607	−0.505389	−0.252694	−	0.967546i	$-0.581317\pi$
$41$	−3.23607	−0.505389	−0.252694	+	0.967546i	$0.581317\pi$
$42$	0	0
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	0	0
$45$	3.23607	0.482405
$46$	0	0
$47$	10.9443	1.59639	0.798193	−	0.602402i	$-0.205789\pi$
$47$	10.9443	1.59639	0.798193	+	0.602402i	$0.205789\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	4.47214	0.626224
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	0	0
$57$	0.763932	0.101185
$58$	0	0
$59$	0.472136	0.0614669	0.0307334	−	0.999528i	$-0.490216\pi$
$59$	0.472136	0.0614669	0.0307334	+	0.999528i	$0.490216\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	−3.23607	−0.407706
$64$	0	0
$65$	−3.23607	−0.401385
$66$	0	0
$67$	−11.2361	−1.37270	−0.686352	−	0.727269i	$-0.740789\pi$
$67$	−11.2361	−1.37270	−0.686352	+	0.727269i	$0.740789\pi$
$68$	0	0
$69$	6.47214	0.779154
$70$	0	0
$71$	4.47214	0.530745	0.265372	−	0.964146i	$-0.414505\pi$
$71$	4.47214	0.530745	0.265372	+	0.964146i	$0.414505\pi$
$72$	0	0
$73$	−8.47214	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$73$	−8.47214	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$74$	0	0
$75$	−5.47214	−0.631868
$76$	0	0
$77$	6.47214	0.737568
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.4721	−1.80805	−0.904026	−	0.427478i	$-0.859402\pi$
$83$	−16.4721	−1.80805	−0.904026	+	0.427478i	$0.859402\pi$
$84$	0	0
$85$	−14.4721	−1.56972
$86$	0	0
$87$	4.47214	0.479463
$88$	0	0
$89$	12.1803	1.29111	0.645557	−	0.763712i	$-0.276625\pi$
$89$	12.1803	1.29111	0.645557	+	0.763712i	$0.276625\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	0	0
$93$	−5.70820	−0.591913
$94$	0	0
$95$	−2.47214	−0.253636
$96$	0	0
$97$	4.47214	0.454077	0.227038	−	0.973886i	$-0.427096\pi$
$97$	4.47214	0.454077	0.227038	+	0.973886i	$0.427096\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	14.9443	1.48701	0.743505	−	0.668730i	$-0.233162\pi$
$101$	14.9443	1.48701	0.743505	+	0.668730i	$0.233162\pi$
$102$	0	0
$103$	10.4721	1.03185	0.515925	−	0.856634i	$-0.327448\pi$
$103$	10.4721	1.03185	0.515925	+	0.856634i	$0.327448\pi$
$104$	0	0
$105$	10.4721	1.02198
$106$	0	0
$107$	−5.52786	−0.534399	−0.267199	−	0.963641i	$-0.586098\pi$
$107$	−5.52786	−0.534399	−0.267199	+	0.963641i	$0.586098\pi$
$108$	0	0
$109$	−18.9443	−1.81453	−0.907266	−	0.420557i	$-0.861835\pi$
$109$	−18.9443	−1.81453	−0.907266	+	0.420557i	$0.861835\pi$
$110$	0	0
$111$	8.47214	0.804140
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−20.9443	−1.95306
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	14.4721	1.32666
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	3.23607	0.291786
$124$	0	0
$125$	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	$0.465016\pi$
$128$	0	0
$129$	−2.47214	−0.217659
$130$	0	0
$131$	−15.4164	−1.34694	−0.673469	−	0.739216i	$-0.735196\pi$
$131$	−15.4164	−1.34694	−0.673469	+	0.739216i	$0.735196\pi$
$132$	0	0
$133$	2.47214	0.214361
$134$	0	0
$135$	−3.23607	−0.278516
$136$	0	0
$137$	−11.2361	−0.959962	−0.479981	−	0.877279i	$-0.659356\pi$
$137$	−11.2361	−0.959962	−0.479981	+	0.877279i	$0.659356\pi$
$138$	0	0
$139$	−21.8885	−1.85656	−0.928281	−	0.371879i	$-0.878713\pi$
$139$	−21.8885	−1.85656	−0.928281	+	0.371879i	$0.878713\pi$
$140$	0	0
$141$	−10.9443	−0.921674
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−14.4721	−1.20185
$146$	0	0
$147$	−3.47214	−0.286377
$148$	0	0
$149$	21.7082	1.77841	0.889203	−	0.457513i	$-0.151260\pi$
$149$	21.7082	1.77841	0.889203	+	0.457513i	$0.151260\pi$
$150$	0	0
$151$	23.2361	1.89092	0.945462	−	0.325732i	$-0.105611\pi$
$151$	23.2361	1.89092	0.945462	+	0.325732i	$0.105611\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	18.4721	1.48372
$156$	0	0
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	0	0
$159$	0.472136	0.0374428
$160$	0	0
$161$	20.9443	1.65064
$162$	0	0
$163$	−7.23607	−0.566773	−0.283386	−	0.959006i	$-0.591458\pi$
$163$	−7.23607	−0.566773	−0.283386	+	0.959006i	$0.591458\pi$
$164$	0	0
$165$	6.47214	0.503855
$166$	0	0
$167$	−19.8885	−1.53902	−0.769511	−	0.638634i	$-0.779500\pi$
$167$	−19.8885	−1.53902	−0.769511	+	0.638634i	$0.779500\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.763932	−0.0584193
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−17.7082	−1.33861
$176$	0	0
$177$	−0.472136	−0.0354879
$178$	0	0
$179$	19.4164	1.45125	0.725625	−	0.688090i	$-0.241551\pi$
$179$	19.4164	1.45125	0.725625	+	0.688090i	$0.241551\pi$
$180$	0	0
$181$	−17.4164	−1.29455	−0.647276	−	0.762256i	$-0.724092\pi$
$181$	−17.4164	−1.29455	−0.647276	+	0.762256i	$0.724092\pi$
$182$	0	0
$183$	3.52786	0.260787
$184$	0	0
$185$	−27.4164	−2.01569
$186$	0	0
$187$	8.94427	0.654070
$188$	0	0
$189$	3.23607	0.235389
$190$	0	0
$191$	10.4721	0.757737	0.378869	−	0.925450i	$-0.376313\pi$
$191$	10.4721	0.757737	0.378869	+	0.925450i	$0.376313\pi$
$192$	0	0
$193$	−15.8885	−1.14368	−0.571841	−	0.820364i	$-0.693771\pi$
$193$	−15.8885	−1.14368	−0.571841	+	0.820364i	$0.693771\pi$
$194$	0	0
$195$	3.23607	0.231740
$196$	0	0
$197$	27.2361	1.94049	0.970245	−	0.242126i	$-0.0778448\pi$
$197$	27.2361	1.94049	0.970245	+	0.242126i	$0.0778448\pi$
$198$	0	0
$199$	18.4721	1.30945	0.654727	−	0.755865i	$-0.272783\pi$
$199$	18.4721	1.30945	0.654727	+	0.755865i	$0.272783\pi$
$200$	0	0
$201$	11.2361	0.792531
$202$	0	0
$203$	14.4721	1.01574
$204$	0	0
$205$	−10.4721	−0.731406
$206$	0	0
$207$	−6.47214	−0.449845
$208$	0	0
$209$	1.52786	0.105685
$210$	0	0
$211$	0.944272	0.0650064	0.0325032	−	0.999472i	$-0.489652\pi$
$211$	0.944272	0.0650064	0.0325032	+	0.999472i	$0.489652\pi$
$212$	0	0
$213$	−4.47214	−0.306426
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−18.4721	−1.25397
$218$	0	0
$219$	8.47214	0.572494
$220$	0	0
$221$	4.47214	0.300828
$222$	0	0
$223$	26.6525	1.78478	0.892391	−	0.451263i	$-0.149026\pi$
$223$	26.6525	1.78478	0.892391	+	0.451263i	$0.149026\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	0	0
$227$	6.94427	0.460908	0.230454	−	0.973083i	$-0.425979\pi$
$227$	6.94427	0.460908	0.230454	+	0.973083i	$0.425979\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−6.47214	−0.425835
$232$	0	0
$233$	9.41641	0.616889	0.308445	−	0.951242i	$-0.400192\pi$
$233$	9.41641	0.616889	0.308445	+	0.951242i	$0.400192\pi$
$234$	0	0
$235$	35.4164	2.31031
$236$	0	0
$237$	8.94427	0.580993
$238$	0	0
$239$	−3.52786	−0.228199	−0.114099	−	0.993469i	$-0.536398\pi$
$239$	−3.52786	−0.228199	−0.114099	+	0.993469i	$0.536398\pi$
$240$	0	0
$241$	22.3607	1.44038	0.720189	−	0.693778i	$-0.244055\pi$
$241$	22.3607	1.44038	0.720189	+	0.693778i	$0.244055\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	11.2361	0.717846
$246$	0	0
$247$	0.763932	0.0486078
$248$	0	0
$249$	16.4721	1.04388
$250$	0	0
$251$	11.0557	0.697831	0.348916	−	0.937154i	$-0.386550\pi$
$251$	11.0557	0.697831	0.348916	+	0.937154i	$0.386550\pi$
$252$	0	0
$253$	12.9443	0.813799
$254$	0	0
$255$	14.4721	0.906280
$256$	0	0
$257$	−3.52786	−0.220062	−0.110031	−	0.993928i	$-0.535095\pi$
$257$	−3.52786	−0.220062	−0.110031	+	0.993928i	$0.535095\pi$
$258$	0	0
$259$	27.4164	1.70357
$260$	0	0
$261$	−4.47214	−0.276818
$262$	0	0
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	−1.52786	−0.0938559
$266$	0	0
$267$	−12.1803	−0.745425
$268$	0	0
$269$	11.5279	0.702866	0.351433	−	0.936213i	$-0.385694\pi$
$269$	11.5279	0.702866	0.351433	+	0.936213i	$0.385694\pi$
$270$	0	0
$271$	−21.1246	−1.28323	−0.641614	−	0.767027i	$-0.721735\pi$
$271$	−21.1246	−1.28323	−0.641614	+	0.767027i	$0.721735\pi$
$272$	0	0
$273$	−3.23607	−0.195856
$274$	0	0
$275$	−10.9443	−0.659964
$276$	0	0
$277$	2.94427	0.176904	0.0884521	−	0.996080i	$-0.471808\pi$
$277$	2.94427	0.176904	0.0884521	+	0.996080i	$0.471808\pi$
$278$	0	0
$279$	5.70820	0.341741
$280$	0	0
$281$	9.70820	0.579143	0.289571	−	0.957156i	$-0.406487\pi$
$281$	9.70820	0.579143	0.289571	+	0.957156i	$0.406487\pi$
$282$	0	0
$283$	13.5279	0.804148	0.402074	−	0.915607i	$-0.368289\pi$
$283$	13.5279	0.804148	0.402074	+	0.915607i	$0.368289\pi$
$284$	0	0
$285$	2.47214	0.146437
$286$	0	0
$287$	10.4721	0.618151
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−4.47214	−0.262161
$292$	0	0
$293$	−12.1803	−0.711583	−0.355792	−	0.934565i	$-0.615789\pi$
$293$	−12.1803	−0.711583	−0.355792	+	0.934565i	$0.615789\pi$
$294$	0	0
$295$	1.52786	0.0889557
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	6.47214	0.374293
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−14.9443	−0.858526
$304$	0	0
$305$	−11.4164	−0.653702
$306$	0	0
$307$	11.2361	0.641276	0.320638	−	0.947202i	$-0.396103\pi$
$307$	11.2361	0.641276	0.320638	+	0.947202i	$0.396103\pi$
$308$	0	0
$309$	−10.4721	−0.595739
$310$	0	0
$311$	27.4164	1.55464	0.777321	−	0.629104i	$-0.216578\pi$
$311$	27.4164	1.55464	0.777321	+	0.629104i	$0.216578\pi$
$312$	0	0
$313$	−3.88854	−0.219793	−0.109897	−	0.993943i	$-0.535052\pi$
$313$	−3.88854	−0.219793	−0.109897	+	0.993943i	$0.535052\pi$
$314$	0	0
$315$	−10.4721	−0.590038
$316$	0	0
$317$	−4.76393	−0.267569	−0.133785	−	0.991010i	$-0.542713\pi$
$317$	−4.76393	−0.267569	−0.133785	+	0.991010i	$0.542713\pi$
$318$	0	0
$319$	8.94427	0.500783
$320$	0	0
$321$	5.52786	0.308535
$322$	0	0
$323$	3.41641	0.190094
$324$	0	0
$325$	−5.47214	−0.303539
$326$	0	0
$327$	18.9443	1.04762
$328$	0	0
$329$	−35.4164	−1.95257
$330$	0	0
$331$	−8.18034	−0.449632	−0.224816	−	0.974401i	$-0.572178\pi$
$331$	−8.18034	−0.449632	−0.224816	+	0.974401i	$0.572178\pi$
$332$	0	0
$333$	−8.47214	−0.464270
$334$	0	0
$335$	−36.3607	−1.98660
$336$	0	0
$337$	10.9443	0.596172	0.298086	−	0.954539i	$-0.403652\pi$
$337$	10.9443	0.596172	0.298086	+	0.954539i	$0.403652\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	−11.4164	−0.618233
$342$	0	0
$343$	11.4164	0.616428
$344$	0	0
$345$	20.9443	1.12760
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	30.3607	1.62517	0.812585	−	0.582843i	$-0.198060\pi$
$349$	30.3607	1.62517	0.812585	+	0.582843i	$0.198060\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	27.5967	1.46883	0.734413	−	0.678702i	$-0.237457\pi$
$353$	27.5967	1.46883	0.734413	+	0.678702i	$0.237457\pi$
$354$	0	0
$355$	14.4721	0.768101
$356$	0	0
$357$	−14.4721	−0.765947
$358$	0	0
$359$	−16.8328	−0.888402	−0.444201	−	0.895927i	$-0.646512\pi$
$359$	−16.8328	−0.888402	−0.444201	+	0.895927i	$0.646512\pi$
$360$	0	0
$361$	−18.4164	−0.969285
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−27.4164	−1.43504
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−3.23607	−0.168463
$370$	0	0
$371$	1.52786	0.0793227
$372$	0	0
$373$	−34.9443	−1.80935	−0.904673	−	0.426107i	$-0.859885\pi$
$373$	−34.9443	−1.80935	−0.904673	+	0.426107i	$0.859885\pi$
$374$	0	0
$375$	−1.52786	−0.0788986
$376$	0	0
$377$	4.47214	0.230327
$378$	0	0
$379$	−23.5967	−1.21208	−0.606042	−	0.795433i	$-0.707244\pi$
$379$	−23.5967	−1.21208	−0.606042	+	0.795433i	$0.707244\pi$
$380$	0	0
$381$	−2.47214	−0.126651
$382$	0	0
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	20.9443	1.06742
$386$	0	0
$387$	2.47214	0.125666
$388$	0	0
$389$	−11.8885	−0.602773	−0.301387	−	0.953502i	$-0.597449\pi$
$389$	−11.8885	−0.602773	−0.301387	+	0.953502i	$0.597449\pi$
$390$	0	0
$391$	28.9443	1.46377
$392$	0	0
$393$	15.4164	0.777655
$394$	0	0
$395$	−28.9443	−1.45634
$396$	0	0
$397$	−16.4721	−0.826713	−0.413356	−	0.910569i	$-0.635644\pi$
$397$	−16.4721	−0.826713	−0.413356	+	0.910569i	$0.635644\pi$
$398$	0	0
$399$	−2.47214	−0.123762
$400$	0	0
$401$	4.18034	0.208756	0.104378	−	0.994538i	$-0.466715\pi$
$401$	4.18034	0.208756	0.104378	+	0.994538i	$0.466715\pi$
$402$	0	0
$403$	−5.70820	−0.284346
$404$	0	0
$405$	3.23607	0.160802
$406$	0	0
$407$	16.9443	0.839896
$408$	0	0
$409$	−10.3607	−0.512303	−0.256151	−	0.966637i	$-0.582455\pi$
$409$	−10.3607	−0.512303	−0.256151	+	0.966637i	$0.582455\pi$
$410$	0	0
$411$	11.2361	0.554234
$412$	0	0
$413$	−1.52786	−0.0751813
$414$	0	0
$415$	−53.3050	−2.61664
$416$	0	0
$417$	21.8885	1.07189
$418$	0	0
$419$	−29.5279	−1.44253	−0.721265	−	0.692659i	$-0.756439\pi$
$419$	−29.5279	−1.44253	−0.721265	+	0.692659i	$0.756439\pi$
$420$	0	0
$421$	−9.05573	−0.441349	−0.220675	−	0.975347i	$-0.570826\pi$
$421$	−9.05573	−0.441349	−0.220675	+	0.975347i	$0.570826\pi$
$422$	0	0
$423$	10.9443	0.532129
$424$	0	0
$425$	−24.4721	−1.18707
$426$	0	0
$427$	11.4164	0.552479
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	0.472136	0.0226894	0.0113447	−	0.999936i	$-0.496389\pi$
$433$	0.472136	0.0226894	0.0113447	+	0.999936i	$0.496389\pi$
$434$	0	0
$435$	14.4721	0.693886
$436$	0	0
$437$	4.94427	0.236517
$438$	0	0
$439$	−13.5279	−0.645650	−0.322825	−	0.946459i	$-0.604632\pi$
$439$	−13.5279	−0.645650	−0.322825	+	0.946459i	$0.604632\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	21.5279	1.02282	0.511410	−	0.859337i	$-0.329123\pi$
$443$	21.5279	1.02282	0.511410	+	0.859337i	$0.329123\pi$
$444$	0	0
$445$	39.4164	1.86852
$446$	0	0
$447$	−21.7082	−1.02676
$448$	0	0
$449$	−36.5410	−1.72448	−0.862239	−	0.506502i	$-0.830938\pi$
$449$	−36.5410	−1.72448	−0.862239	+	0.506502i	$0.830938\pi$
$450$	0	0
$451$	6.47214	0.304761
$452$	0	0
$453$	−23.2361	−1.09173
$454$	0	0
$455$	10.4721	0.490941
$456$	0	0
$457$	−34.3607	−1.60732	−0.803662	−	0.595085i	$-0.797118\pi$
$457$	−34.3607	−1.60732	−0.803662	+	0.595085i	$0.797118\pi$
$458$	0	0
$459$	4.47214	0.208741
$460$	0	0
$461$	−22.6525	−1.05503	−0.527515	−	0.849545i	$-0.676876\pi$
$461$	−22.6525	−1.05503	−0.527515	+	0.849545i	$0.676876\pi$
$462$	0	0
$463$	17.7082	0.822970	0.411485	−	0.911417i	$-0.365010\pi$
$463$	17.7082	0.822970	0.411485	+	0.911417i	$0.365010\pi$
$464$	0	0
$465$	−18.4721	−0.856625
$466$	0	0
$467$	−1.88854	−0.0873914	−0.0436957	−	0.999045i	$-0.513913\pi$
$467$	−1.88854	−0.0873914	−0.0436957	+	0.999045i	$0.513913\pi$
$468$	0	0
$469$	36.3607	1.67898
$470$	0	0
$471$	−13.4164	−0.618195
$472$	0	0
$473$	−4.94427	−0.227338
$474$	0	0
$475$	−4.18034	−0.191807
$476$	0	0
$477$	−0.472136	−0.0216176
$478$	0	0
$479$	23.3050	1.06483	0.532415	−	0.846483i	$-0.321285\pi$
$479$	23.3050	1.06483	0.532415	+	0.846483i	$0.321285\pi$
$480$	0	0
$481$	8.47214	0.386296
$482$	0	0
$483$	−20.9443	−0.952997
$484$	0	0
$485$	14.4721	0.657146
$486$	0	0
$487$	−26.2918	−1.19140	−0.595698	−	0.803209i	$-0.703124\pi$
$487$	−26.2918	−1.19140	−0.595698	+	0.803209i	$0.703124\pi$
$488$	0	0
$489$	7.23607	0.327226
$490$	0	0
$491$	11.0557	0.498938	0.249469	−	0.968383i	$-0.419744\pi$
$491$	11.0557	0.498938	0.249469	+	0.968383i	$0.419744\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	−6.47214	−0.290901
$496$	0	0
$497$	−14.4721	−0.649164
$498$	0	0
$499$	−40.1803	−1.79872	−0.899360	−	0.437210i	$-0.855967\pi$
$499$	−40.1803	−1.79872	−0.899360	+	0.437210i	$0.855967\pi$
$500$	0	0
$501$	19.8885	0.888555
$502$	0	0
$503$	39.7771	1.77357	0.886786	−	0.462180i	$-0.152932\pi$
$503$	39.7771	1.77357	0.886786	+	0.462180i	$0.152932\pi$
$504$	0	0
$505$	48.3607	2.15202
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	16.1803	0.717181	0.358590	−	0.933495i	$-0.383257\pi$
$509$	16.1803	0.717181	0.358590	+	0.933495i	$0.383257\pi$
$510$	0	0
$511$	27.4164	1.21283
$512$	0	0
$513$	0.763932	0.0337284
$514$	0	0
$515$	33.8885	1.49331
$516$	0	0
$517$	−21.8885	−0.962657
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	18.9443	0.829964	0.414982	−	0.909830i	$-0.363788\pi$
$521$	18.9443	0.829964	0.414982	+	0.909830i	$0.363788\pi$
$522$	0	0
$523$	−23.0557	−1.00816	−0.504078	−	0.863658i	$-0.668168\pi$
$523$	−23.0557	−1.00816	−0.504078	+	0.863658i	$0.668168\pi$
$524$	0	0
$525$	17.7082	0.772849
$526$	0	0
$527$	−25.5279	−1.11201
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	0.472136	0.0204890
$532$	0	0
$533$	3.23607	0.140170
$534$	0	0
$535$	−17.8885	−0.773389
$536$	0	0
$537$	−19.4164	−0.837880
$538$	0	0
$539$	−6.94427	−0.299111
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	17.4164	0.747410
$544$	0	0
$545$	−61.3050	−2.62602
$546$	0	0
$547$	−0.944272	−0.0403742	−0.0201871	−	0.999796i	$-0.506426\pi$
$547$	−0.944272	−0.0403742	−0.0201871	+	0.999796i	$0.506426\pi$
$548$	0	0
$549$	−3.52786	−0.150566
$550$	0	0
$551$	3.41641	0.145544
$552$	0	0
$553$	28.9443	1.23084
$554$	0	0
$555$	27.4164	1.16376
$556$	0	0
$557$	13.7082	0.580835	0.290418	−	0.956900i	$-0.406206\pi$
$557$	13.7082	0.580835	0.290418	+	0.956900i	$0.406206\pi$
$558$	0	0
$559$	−2.47214	−0.104560
$560$	0	0
$561$	−8.94427	−0.377627
$562$	0	0
$563$	−1.52786	−0.0643918	−0.0321959	−	0.999482i	$-0.510250\pi$
$563$	−1.52786	−0.0643918	−0.0321959	+	0.999482i	$0.510250\pi$
$564$	0	0
$565$	−6.47214	−0.272285
$566$	0	0
$567$	−3.23607	−0.135902
$568$	0	0
$569$	30.3607	1.27279	0.636393	−	0.771365i	$-0.280426\pi$
$569$	30.3607	1.27279	0.636393	+	0.771365i	$0.280426\pi$
$570$	0	0
$571$	34.4721	1.44261	0.721307	−	0.692615i	$-0.243542\pi$
$571$	34.4721	1.44261	0.721307	+	0.692615i	$0.243542\pi$
$572$	0	0
$573$	−10.4721	−0.437480
$574$	0	0
$575$	−35.4164	−1.47697
$576$	0	0
$577$	−18.9443	−0.788660	−0.394330	−	0.918969i	$-0.629023\pi$
$577$	−18.9443	−0.788660	−0.394330	+	0.918969i	$0.629023\pi$
$578$	0	0
$579$	15.8885	0.660305
$580$	0	0
$581$	53.3050	2.21146
$582$	0	0
$583$	0.944272	0.0391077
$584$	0	0
$585$	−3.23607	−0.133795
$586$	0	0
$587$	−6.58359	−0.271734	−0.135867	−	0.990727i	$-0.543382\pi$
$587$	−6.58359	−0.271734	−0.135867	+	0.990727i	$0.543382\pi$
$588$	0	0
$589$	−4.36068	−0.179679
$590$	0	0
$591$	−27.2361	−1.12034
$592$	0	0
$593$	15.2361	0.625670	0.312835	−	0.949807i	$-0.398721\pi$
$593$	15.2361	0.625670	0.312835	+	0.949807i	$0.398721\pi$
$594$	0	0
$595$	46.8328	1.91996
$596$	0	0
$597$	−18.4721	−0.756014
$598$	0	0
$599$	24.9443	1.01920	0.509598	−	0.860413i	$-0.329794\pi$
$599$	24.9443	1.01920	0.509598	+	0.860413i	$0.329794\pi$
$600$	0	0
$601$	47.8885	1.95341	0.976707	−	0.214576i	$-0.0688371\pi$
$601$	47.8885	1.95341	0.976707	+	0.214576i	$0.0688371\pi$
$602$	0	0
$603$	−11.2361	−0.457568
$604$	0	0
$605$	−22.6525	−0.920954
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−14.4721	−0.586441
$610$	0	0
$611$	−10.9443	−0.442758
$612$	0	0
$613$	30.3607	1.22626	0.613128	−	0.789983i	$-0.289911\pi$
$613$	30.3607	1.22626	0.613128	+	0.789983i	$0.289911\pi$
$614$	0	0
$615$	10.4721	0.422277
$616$	0	0
$617$	3.59675	0.144800	0.0723998	−	0.997376i	$-0.476934\pi$
$617$	3.59675	0.144800	0.0723998	+	0.997376i	$0.476934\pi$
$618$	0	0
$619$	−38.6525	−1.55357	−0.776787	−	0.629763i	$-0.783152\pi$
$619$	−38.6525	−1.55357	−0.776787	+	0.629763i	$0.783152\pi$
$620$	0	0
$621$	6.47214	0.259718
$622$	0	0
$623$	−39.4164	−1.57919
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	−1.52786	−0.0610170
$628$	0	0
$629$	37.8885	1.51072
$630$	0	0
$631$	−18.6525	−0.742543	−0.371272	−	0.928524i	$-0.621078\pi$
$631$	−18.6525	−0.742543	−0.371272	+	0.928524i	$0.621078\pi$
$632$	0	0
$633$	−0.944272	−0.0375314
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	−3.47214	−0.137571
$638$	0	0
$639$	4.47214	0.176915
$640$	0	0
$641$	21.4164	0.845897	0.422949	−	0.906154i	$-0.360995\pi$
$641$	21.4164	0.845897	0.422949	+	0.906154i	$0.360995\pi$
$642$	0	0
$643$	−29.1246	−1.14856	−0.574281	−	0.818658i	$-0.694718\pi$
$643$	−29.1246	−1.14856	−0.574281	+	0.818658i	$0.694718\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−0.944272	−0.0370659
$650$	0	0
$651$	18.4721	0.723980
$652$	0	0
$653$	10.3607	0.405445	0.202722	−	0.979236i	$-0.435021\pi$
$653$	10.3607	0.405445	0.202722	+	0.979236i	$0.435021\pi$
$654$	0	0
$655$	−49.8885	−1.94931
$656$	0	0
$657$	−8.47214	−0.330530
$658$	0	0
$659$	−34.8328	−1.35689	−0.678447	−	0.734649i	$-0.737347\pi$
$659$	−34.8328	−1.35689	−0.678447	+	0.734649i	$0.737347\pi$
$660$	0	0
$661$	−36.2492	−1.40993	−0.704966	−	0.709241i	$-0.749038\pi$
$661$	−36.2492	−1.40993	−0.704966	+	0.709241i	$0.749038\pi$
$662$	0	0
$663$	−4.47214	−0.173683
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	28.9443	1.12073
$668$	0	0
$669$	−26.6525	−1.03044
$670$	0	0
$671$	7.05573	0.272383
$672$	0	0
$673$	18.5836	0.716345	0.358172	−	0.933655i	$-0.383400\pi$
$673$	18.5836	0.716345	0.358172	+	0.933655i	$0.383400\pi$
$674$	0	0
$675$	−5.47214	−0.210623
$676$	0	0
$677$	−34.3607	−1.32059	−0.660294	−	0.751007i	$-0.729568\pi$
$677$	−34.3607	−1.32059	−0.660294	+	0.751007i	$0.729568\pi$
$678$	0	0
$679$	−14.4721	−0.555390
$680$	0	0
$681$	−6.94427	−0.266105
$682$	0	0
$683$	−10.0000	−0.382639	−0.191320	−	0.981528i	$-0.561277\pi$
$683$	−10.0000	−0.382639	−0.191320	+	0.981528i	$0.561277\pi$
$684$	0	0
$685$	−36.3607	−1.38927
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	0.472136	0.0179869
$690$	0	0
$691$	27.5967	1.04983	0.524915	−	0.851155i	$-0.324097\pi$
$691$	27.5967	1.04983	0.524915	+	0.851155i	$0.324097\pi$
$692$	0	0
$693$	6.47214	0.245856
$694$	0	0
$695$	−70.8328	−2.68684
$696$	0	0
$697$	14.4721	0.548171
$698$	0	0
$699$	−9.41641	−0.356161
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	6.47214	0.244101
$704$	0	0
$705$	−35.4164	−1.33386
$706$	0	0
$707$	−48.3607	−1.81879
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−8.94427	−0.335436
$712$	0	0
$713$	−36.9443	−1.38357
$714$	0	0
$715$	6.47214	0.242044
$716$	0	0
$717$	3.52786	0.131750
$718$	0	0
$719$	−6.47214	−0.241370	−0.120685	−	0.992691i	$-0.538509\pi$
$719$	−6.47214	−0.241370	−0.120685	+	0.992691i	$0.538509\pi$
$720$	0	0
$721$	−33.8885	−1.26208
$722$	0	0
$723$	−22.3607	−0.831603
$724$	0	0
$725$	−24.4721	−0.908872
$726$	0	0
$727$	−10.4721	−0.388390	−0.194195	−	0.980963i	$-0.562209\pi$
$727$	−10.4721	−0.388390	−0.194195	+	0.980963i	$0.562209\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.0557	−0.408911
$732$	0	0
$733$	−7.88854	−0.291370	−0.145685	−	0.989331i	$-0.546539\pi$
$733$	−7.88854	−0.291370	−0.145685	+	0.989331i	$0.546539\pi$
$734$	0	0
$735$	−11.2361	−0.414449
$736$	0	0
$737$	22.4721	0.827772
$738$	0	0
$739$	−23.8197	−0.876220	−0.438110	−	0.898921i	$-0.644352\pi$
$739$	−23.8197	−0.876220	−0.438110	+	0.898921i	$0.644352\pi$
$740$	0	0
$741$	−0.763932	−0.0280637
$742$	0	0
$743$	7.52786	0.276171	0.138085	−	0.990420i	$-0.455905\pi$
$743$	7.52786	0.276171	0.138085	+	0.990420i	$0.455905\pi$
$744$	0	0
$745$	70.2492	2.57373
$746$	0	0
$747$	−16.4721	−0.602684
$748$	0	0
$749$	17.8885	0.653633
$750$	0	0
$751$	12.3607	0.451048	0.225524	−	0.974238i	$-0.427591\pi$
$751$	12.3607	0.451048	0.225524	+	0.974238i	$0.427591\pi$
$752$	0	0
$753$	−11.0557	−0.402893
$754$	0	0
$755$	75.1935	2.73657
$756$	0	0
$757$	10.5836	0.384667	0.192334	−	0.981330i	$-0.438394\pi$
$757$	10.5836	0.384667	0.192334	+	0.981330i	$0.438394\pi$
$758$	0	0
$759$	−12.9443	−0.469847
$760$	0	0
$761$	−32.1803	−1.16654	−0.583268	−	0.812280i	$-0.698226\pi$
$761$	−32.1803	−1.16654	−0.583268	+	0.812280i	$0.698226\pi$
$762$	0	0
$763$	61.3050	2.21939
$764$	0	0
$765$	−14.4721	−0.523241
$766$	0	0
$767$	−0.472136	−0.0170478
$768$	0	0
$769$	0.832816	0.0300321	0.0150161	−	0.999887i	$-0.495220\pi$
$769$	0.832816	0.0300321	0.0150161	+	0.999887i	$0.495220\pi$
$770$	0	0
$771$	3.52786	0.127053
$772$	0	0
$773$	21.7082	0.780790	0.390395	−	0.920647i	$-0.372338\pi$
$773$	21.7082	0.780790	0.390395	+	0.920647i	$0.372338\pi$
$774$	0	0
$775$	31.2361	1.12203
$776$	0	0
$777$	−27.4164	−0.983558
$778$	0	0
$779$	2.47214	0.0885735
$780$	0	0
$781$	−8.94427	−0.320051
$782$	0	0
$783$	4.47214	0.159821
$784$	0	0
$785$	43.4164	1.54960
$786$	0	0
$787$	−17.7082	−0.631229	−0.315615	−	0.948887i	$-0.602211\pi$
$787$	−17.7082	−0.631229	−0.315615	+	0.948887i	$0.602211\pi$
$788$	0	0
$789$	8.94427	0.318425
$790$	0	0
$791$	6.47214	0.230123
$792$	0	0
$793$	3.52786	0.125278
$794$	0	0
$795$	1.52786	0.0541878
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−48.9443	−1.73152
$800$	0	0
$801$	12.1803	0.430371
$802$	0	0
$803$	16.9443	0.597950
$804$	0	0
$805$	67.7771	2.38883
$806$	0	0
$807$	−11.5279	−0.405800
$808$	0	0
$809$	−13.0557	−0.459015	−0.229507	−	0.973307i	$-0.573712\pi$
$809$	−13.0557	−0.459015	−0.229507	+	0.973307i	$0.573712\pi$
$810$	0	0
$811$	−3.81966	−0.134126	−0.0670632	−	0.997749i	$-0.521363\pi$
$811$	−3.81966	−0.134126	−0.0670632	+	0.997749i	$0.521363\pi$
$812$	0	0
$813$	21.1246	0.740872
$814$	0	0
$815$	−23.4164	−0.820241
$816$	0	0
$817$	−1.88854	−0.0660718
$818$	0	0
$819$	3.23607	0.113077
$820$	0	0
$821$	−9.12461	−0.318451	−0.159226	−	0.987242i	$-0.550900\pi$
$821$	−9.12461	−0.318451	−0.159226	+	0.987242i	$0.550900\pi$
$822$	0	0
$823$	−17.3050	−0.603213	−0.301606	−	0.953433i	$-0.597523\pi$
$823$	−17.3050	−0.603213	−0.301606	+	0.953433i	$0.597523\pi$
$824$	0	0
$825$	10.9443	0.381031
$826$	0	0
$827$	−37.7771	−1.31364	−0.656819	−	0.754048i	$-0.728098\pi$
$827$	−37.7771	−1.31364	−0.656819	+	0.754048i	$0.728098\pi$
$828$	0	0
$829$	39.5279	1.37286	0.686430	−	0.727196i	$-0.259177\pi$
$829$	39.5279	1.37286	0.686430	+	0.727196i	$0.259177\pi$
$830$	0	0
$831$	−2.94427	−0.102136
$832$	0	0
$833$	−15.5279	−0.538009
$834$	0	0
$835$	−64.3607	−2.22729
$836$	0	0
$837$	−5.70820	−0.197304
$838$	0	0
$839$	−14.3607	−0.495786	−0.247893	−	0.968787i	$-0.579738\pi$
$839$	−14.3607	−0.495786	−0.247893	+	0.968787i	$0.579738\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	−9.70820	−0.334368
$844$	0	0
$845$	3.23607	0.111324
$846$	0	0
$847$	22.6525	0.778348
$848$	0	0
$849$	−13.5279	−0.464275
$850$	0	0
$851$	54.8328	1.87964
$852$	0	0
$853$	−11.5279	−0.394707	−0.197353	−	0.980332i	$-0.563235\pi$
$853$	−11.5279	−0.394707	−0.197353	+	0.980332i	$0.563235\pi$
$854$	0	0
$855$	−2.47214	−0.0845453
$856$	0	0
$857$	−37.4164	−1.27812	−0.639060	−	0.769157i	$-0.720676\pi$
$857$	−37.4164	−1.27812	−0.639060	+	0.769157i	$0.720676\pi$
$858$	0	0
$859$	32.9443	1.12404	0.562022	−	0.827122i	$-0.310024\pi$
$859$	32.9443	1.12404	0.562022	+	0.827122i	$0.310024\pi$
$860$	0	0
$861$	−10.4721	−0.356889
$862$	0	0
$863$	−47.8885	−1.63014	−0.815072	−	0.579359i	$-0.803303\pi$
$863$	−47.8885	−1.63014	−0.815072	+	0.579359i	$0.803303\pi$
$864$	0	0
$865$	6.47214	0.220059
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	17.8885	0.606827
$870$	0	0
$871$	11.2361	0.380720
$872$	0	0
$873$	4.47214	0.151359
$874$	0	0
$875$	−4.94427	−0.167147
$876$	0	0
$877$	27.3050	0.922023	0.461011	−	0.887394i	$-0.347487\pi$
$877$	27.3050	0.922023	0.461011	+	0.887394i	$0.347487\pi$
$878$	0	0
$879$	12.1803	0.410833
$880$	0	0
$881$	5.63932	0.189994	0.0949968	−	0.995478i	$-0.469716\pi$
$881$	5.63932	0.189994	0.0949968	+	0.995478i	$0.469716\pi$
$882$	0	0
$883$	3.63932	0.122473	0.0612364	−	0.998123i	$-0.480496\pi$
$883$	3.63932	0.122473	0.0612364	+	0.998123i	$0.480496\pi$
$884$	0	0
$885$	−1.52786	−0.0513586
$886$	0	0
$887$	17.3050	0.581043	0.290522	−	0.956868i	$-0.406171\pi$
$887$	17.3050	0.581043	0.290522	+	0.956868i	$0.406171\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−8.36068	−0.279779
$894$	0	0
$895$	62.8328	2.10027
$896$	0	0
$897$	−6.47214	−0.216098
$898$	0	0
$899$	−25.5279	−0.851402
$900$	0	0
$901$	2.11146	0.0703428
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−56.3607	−1.87349
$906$	0	0
$907$	−20.3607	−0.676065	−0.338033	−	0.941134i	$-0.609761\pi$
$907$	−20.3607	−0.676065	−0.338033	+	0.941134i	$0.609761\pi$
$908$	0	0
$909$	14.9443	0.495670
$910$	0	0
$911$	45.5279	1.50841	0.754203	−	0.656642i	$-0.228024\pi$
$911$	45.5279	1.50841	0.754203	+	0.656642i	$0.228024\pi$
$912$	0	0
$913$	32.9443	1.09030
$914$	0	0
$915$	11.4164	0.377415
$916$	0	0
$917$	49.8885	1.64746
$918$	0	0
$919$	13.8885	0.458141	0.229070	−	0.973410i	$-0.426431\pi$
$919$	13.8885	0.458141	0.229070	+	0.973410i	$0.426431\pi$
$920$	0	0
$921$	−11.2361	−0.370241
$922$	0	0
$923$	−4.47214	−0.147202
$924$	0	0
$925$	−46.3607	−1.52433
$926$	0	0
$927$	10.4721	0.343950
$928$	0	0
$929$	8.54102	0.280222	0.140111	−	0.990136i	$-0.455254\pi$
$929$	8.54102	0.280222	0.140111	+	0.990136i	$0.455254\pi$
$930$	0	0
$931$	−2.65248	−0.0869314
$932$	0	0
$933$	−27.4164	−0.897573
$934$	0	0
$935$	28.9443	0.946579
$936$	0	0
$937$	31.3050	1.02269	0.511344	−	0.859376i	$-0.329148\pi$
$937$	31.3050	1.02269	0.511344	+	0.859376i	$0.329148\pi$
$938$	0	0
$939$	3.88854	0.126898
$940$	0	0
$941$	−31.2361	−1.01827	−0.509133	−	0.860688i	$-0.670034\pi$
$941$	−31.2361	−1.01827	−0.509133	+	0.860688i	$0.670034\pi$
$942$	0	0
$943$	20.9443	0.682039
$944$	0	0
$945$	10.4721	0.340659
$946$	0	0
$947$	−41.4164	−1.34585	−0.672926	−	0.739710i	$-0.734963\pi$
$947$	−41.4164	−1.34585	−0.672926	+	0.739710i	$0.734963\pi$
$948$	0	0
$949$	8.47214	0.275017
$950$	0	0
$951$	4.76393	0.154481
$952$	0	0
$953$	31.5279	1.02129	0.510644	−	0.859792i	$-0.329407\pi$
$953$	31.5279	1.02129	0.510644	+	0.859792i	$0.329407\pi$
$954$	0	0
$955$	33.8885	1.09661
$956$	0	0
$957$	−8.94427	−0.289127
$958$	0	0
$959$	36.3607	1.17415
$960$	0	0
$961$	1.58359	0.0510836
$962$	0	0
$963$	−5.52786	−0.178133
$964$	0	0
$965$	−51.4164	−1.65515
$966$	0	0
$967$	−15.2361	−0.489959	−0.244979	−	0.969528i	$-0.578781\pi$
$967$	−15.2361	−0.489959	−0.244979	+	0.969528i	$0.578781\pi$
$968$	0	0
$969$	−3.41641	−0.109751
$970$	0	0
$971$	50.8328	1.63130	0.815651	−	0.578544i	$-0.196379\pi$
$971$	50.8328	1.63130	0.815651	+	0.578544i	$0.196379\pi$
$972$	0	0
$973$	70.8328	2.27080
$974$	0	0
$975$	5.47214	0.175249
$976$	0	0
$977$	−13.7082	−0.438564	−0.219282	−	0.975661i	$-0.570372\pi$
$977$	−13.7082	−0.438564	−0.219282	+	0.975661i	$0.570372\pi$
$978$	0	0
$979$	−24.3607	−0.778571
$980$	0	0
$981$	−18.9443	−0.604844
$982$	0	0
$983$	22.5836	0.720305	0.360152	−	0.932893i	$-0.382725\pi$
$983$	22.5836	0.720305	0.360152	+	0.932893i	$0.382725\pi$
$984$	0	0
$985$	88.1378	2.80830
$986$	0	0
$987$	35.4164	1.12732
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	40.9443	1.30064	0.650319	−	0.759661i	$-0.274635\pi$
$991$	40.9443	1.30064	0.650319	+	0.759661i	$0.274635\pi$
$992$	0	0
$993$	8.18034	0.259595
$994$	0	0
$995$	59.7771	1.89506
$996$	0	0
$997$	−59.8885	−1.89669	−0.948345	−	0.317242i	$-0.897243\pi$
$997$	−59.8885	−1.89669	−0.948345	+	0.317242i	$0.897243\pi$
$998$	0	0
$999$	8.47214	0.268047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.a.l.1.2	✓	2
3.2	odd	2		3744.2.a.r.1.1		2
4.3	odd	2		1248.2.a.n.1.2	yes	2
8.3	odd	2		2496.2.a.be.1.1		2
8.5	even	2		2496.2.a.bh.1.1		2
12.11	even	2		3744.2.a.s.1.1		2
24.5	odd	2		7488.2.a.cs.1.2		2
24.11	even	2		7488.2.a.ct.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.l.1.2	✓	2	1.1	even	1	trivial
1248.2.a.n.1.2	yes	2	4.3	odd	2
2496.2.a.be.1.1		2	8.3	odd	2
2496.2.a.bh.1.1		2	8.5	even	2
3744.2.a.r.1.1		2	3.2	odd	2
3744.2.a.s.1.1		2	12.11	even	2
7488.2.a.cs.1.2		2	24.5	odd	2
7488.2.a.ct.1.2		2	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.23607 q^{5} -3.23607 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} -3.23607 q^{15} -4.47214 q^{17} -0.763932 q^{19} +3.23607 q^{21} -6.47214 q^{23} +5.47214 q^{25} -1.00000 q^{27} -4.47214 q^{29} +5.70820 q^{31} +2.00000 q^{33} -10.4721 q^{35} -8.47214 q^{37} +1.00000 q^{39} -3.23607 q^{41} +2.47214 q^{43} +3.23607 q^{45} +10.9443 q^{47} +3.47214 q^{49} +4.47214 q^{51} -0.472136 q^{53} -6.47214 q^{55} +0.763932 q^{57} +0.472136 q^{59} -3.52786 q^{61} -3.23607 q^{63} -3.23607 q^{65} -11.2361 q^{67} +6.47214 q^{69} +4.47214 q^{71} -8.47214 q^{73} -5.47214 q^{75} +6.47214 q^{77} -8.94427 q^{79} +1.00000 q^{81} -16.4721 q^{83} -14.4721 q^{85} +4.47214 q^{87} +12.1803 q^{89} +3.23607 q^{91} -5.70820 q^{93} -2.47214 q^{95} +4.47214 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 4 q^{33} - 12 q^{35} - 8 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 - 6 * q^19 + 2 * q^21 - 4 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 4 * q^33 - 12 * q^35 - 8 * q^37 + 2 * q^39 - 2 * q^41 - 4 * q^43 + 2 * q^45 + 4 * q^47 - 2 * q^49 + 8 * q^53 - 4 * q^55 + 6 * q^57 - 8 * q^59 - 16 * q^61 - 2 * q^63 - 2 * q^65 - 18 * q^67 + 4 * q^69 - 8 * q^73 - 2 * q^75 + 4 * q^77 + 2 * q^81 - 24 * q^83 - 20 * q^85 + 2 * q^89 + 2 * q^91 + 2 * q^93 + 4 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

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