LMFDB - Embedded newform 3328.2.a.be.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-1,0,-3,0,-1,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.146365 q^{3} -1.00000 q^{5} +0.146365 q^{7} -2.97858 q^{9} +2.68585 q^{11} -1.00000 q^{13} -0.146365 q^{15} +1.00000 q^{17} -4.00000 q^{19} +0.0214229 q^{21} +6.68585 q^{23} -4.00000 q^{25} -0.875057 q^{27} +4.39312 q^{29} +1.31415 q^{31} +0.393115 q^{33} -0.146365 q^{35} +3.97858 q^{37} -0.146365 q^{39} +6.39312 q^{41} -6.83221 q^{43} +2.97858 q^{45} -7.12494 q^{47} -6.97858 q^{49} +0.146365 q^{51} -8.97858 q^{53} -2.68585 q^{55} -0.585462 q^{57} -12.3503 q^{59} -8.35027 q^{61} -0.435961 q^{63} +1.00000 q^{65} -8.29273 q^{67} +0.978577 q^{69} -5.51806 q^{71} +6.97858 q^{73} -0.585462 q^{75} +0.393115 q^{77} +15.0361 q^{79} +8.80765 q^{81} -4.29273 q^{83} -1.00000 q^{85} +0.643000 q^{87} +5.37169 q^{89} -0.146365 q^{91} +0.192347 q^{93} +4.00000 q^{95} -10.3503 q^{97} -8.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{5} - q^{7} + 6 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 3 q^{17} - 12 q^{19} + 15 q^{21} + 8 q^{23} - 12 q^{25} - 19 q^{27} + 4 q^{29} + 16 q^{31} - 8 q^{33} + q^{35} - 3 q^{37} + q^{39}+ \cdots - 24 q^{99}+O(q^{100}) $ 3 * q - q^3 - 3 * q^5 - q^7 + 6 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 3 * q^17 - 12 * q^19 + 15 * q^21 + 8 * q^23 - 12 * q^25 - 19 * q^27 + 4 * q^29 + 16 * q^31 - 8 * q^33 + q^35 - 3 * q^37 + q^39 + 10 * q^41 - 7 * q^43 - 6 * q^45 - 5 * q^47 - 6 * q^49 - q^51 - 12 * q^53 + 4 * q^55 + 4 * q^57 + 2 * q^59 + 14 * q^61 - 22 * q^63 + 3 * q^65 - 22 * q^67 - 12 * q^69 + 9 * q^71 + 6 * q^73 + 4 * q^75 - 8 * q^77 - 6 * q^79 + 23 * q^81 - 10 * q^83 - 3 * q^85 - 40 * q^87 - 8 * q^89 + q^91 + 4 * q^93 + 12 * q^95 + 8 * q^97 - 24 * 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
$3$	0.146365	0.0845042	0.0422521	−	0.999107i	$-0.486547\pi$
$3$	0.146365	0.0845042	0.0422521	+	0.999107i	$0.486547\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0.146365	0.0553210	0.0276605	−	0.999617i	$-0.491194\pi$
$7$	0.146365	0.0553210	0.0276605	+	0.999617i	$0.491194\pi$
$8$	0	0
$9$	−2.97858	−0.992859
$10$	0	0
$11$	2.68585	0.809813	0.404907	−	0.914358i	$-0.367304\pi$
$11$	2.68585	0.809813	0.404907	+	0.914358i	$0.367304\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.146365	−0.0377914
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0.0214229	0.00467485
$22$	0	0
$23$	6.68585	1.39410	0.697048	−	0.717025i	$-0.254497\pi$
$23$	6.68585	1.39410	0.697048	+	0.717025i	$0.254497\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−0.875057	−0.168405
$28$	0	0
$29$	4.39312	0.815781	0.407891	−	0.913031i	$-0.366265\pi$
$29$	4.39312	0.815781	0.407891	+	0.913031i	$0.366265\pi$
$30$	0	0
$31$	1.31415	0.236029	0.118014	−	0.993012i	$-0.462347\pi$
$31$	1.31415	0.236029	0.118014	+	0.993012i	$0.462347\pi$
$32$	0	0
$33$	0.393115	0.0684326
$34$	0	0
$35$	−0.146365	−0.0247403
$36$	0	0
$37$	3.97858	0.654074	0.327037	−	0.945012i	$-0.393950\pi$
$37$	3.97858	0.654074	0.327037	+	0.945012i	$0.393950\pi$
$38$	0	0
$39$	−0.146365	−0.0234372
$40$	0	0
$41$	6.39312	0.998437	0.499218	−	0.866476i	$-0.333621\pi$
$41$	6.39312	0.998437	0.499218	+	0.866476i	$0.333621\pi$
$42$	0	0
$43$	−6.83221	−1.04190	−0.520951	−	0.853586i	$-0.674423\pi$
$43$	−6.83221	−1.04190	−0.520951	+	0.853586i	$0.674423\pi$
$44$	0	0
$45$	2.97858	0.444020
$46$	0	0
$47$	−7.12494	−1.03928	−0.519640	−	0.854385i	$-0.673934\pi$
$47$	−7.12494	−1.03928	−0.519640	+	0.854385i	$0.673934\pi$
$48$	0	0
$49$	−6.97858	−0.996940
$50$	0	0
$51$	0.146365	0.0204953
$52$	0	0
$53$	−8.97858	−1.23330	−0.616651	−	0.787236i	$-0.711511\pi$
$53$	−8.97858	−1.23330	−0.616651	+	0.787236i	$0.711511\pi$
$54$	0	0
$55$	−2.68585	−0.362159
$56$	0	0
$57$	−0.585462	−0.0775463
$58$	0	0
$59$	−12.3503	−1.60787	−0.803934	−	0.594718i	$-0.797264\pi$
$59$	−12.3503	−1.60787	−0.803934	+	0.594718i	$0.797264\pi$
$60$	0	0
$61$	−8.35027	−1.06914	−0.534571	−	0.845123i	$-0.679527\pi$
$61$	−8.35027	−1.06914	−0.534571	+	0.845123i	$0.679527\pi$
$62$	0	0
$63$	−0.435961	−0.0549259
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−8.29273	−1.01312	−0.506559	−	0.862205i	$-0.669083\pi$
$67$	−8.29273	−1.01312	−0.506559	+	0.862205i	$0.669083\pi$
$68$	0	0
$69$	0.978577	0.117807
$70$	0	0
$71$	−5.51806	−0.654873	−0.327436	−	0.944873i	$-0.606185\pi$
$71$	−5.51806	−0.654873	−0.327436	+	0.944873i	$0.606185\pi$
$72$	0	0
$73$	6.97858	0.816781	0.408390	−	0.912807i	$-0.366090\pi$
$73$	6.97858	0.816781	0.408390	+	0.912807i	$0.366090\pi$
$74$	0	0
$75$	−0.585462	−0.0676033
$76$	0	0
$77$	0.393115	0.0447996
$78$	0	0
$79$	15.0361	1.69170	0.845848	−	0.533425i	$-0.179095\pi$
$79$	15.0361	1.69170	0.845848	+	0.533425i	$0.179095\pi$
$80$	0	0
$81$	8.80765	0.978628
$82$	0	0
$83$	−4.29273	−0.471188	−0.235594	−	0.971852i	$-0.575704\pi$
$83$	−4.29273	−0.471188	−0.235594	+	0.971852i	$0.575704\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	0.643000	0.0689369
$88$	0	0
$89$	5.37169	0.569398	0.284699	−	0.958617i	$-0.408106\pi$
$89$	5.37169	0.569398	0.284699	+	0.958617i	$0.408106\pi$
$90$	0	0
$91$	−0.146365	−0.0153433
$92$	0	0
$93$	0.192347	0.0199454
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−10.3503	−1.05091	−0.525455	−	0.850821i	$-0.676105\pi$
$97$	−10.3503	−1.05091	−0.525455	+	0.850821i	$0.676105\pi$
$98$	0	0
$99$	−8.00000	−0.804030
$100$	0	0
$101$	17.9572	1.78680	0.893402	−	0.449258i	$-0.148312\pi$
$101$	17.9572	1.78680	0.893402	+	0.449258i	$0.148312\pi$
$102$	0	0
$103$	−10.2499	−1.00995	−0.504976	−	0.863134i	$-0.668499\pi$
$103$	−10.2499	−1.00995	−0.504976	+	0.863134i	$0.668499\pi$
$104$	0	0
$105$	−0.0214229	−0.00209066
$106$	0	0
$107$	−14.6858	−1.41973	−0.709867	−	0.704336i	$-0.751245\pi$
$107$	−14.6858	−1.41973	−0.709867	+	0.704336i	$0.751245\pi$
$108$	0	0
$109$	−14.7648	−1.41421	−0.707106	−	0.707108i	$-0.750000\pi$
$109$	−14.7648	−1.41421	−0.707106	+	0.707108i	$0.750000\pi$
$110$	0	0
$111$	0.582326	0.0552720
$112$	0	0
$113$	3.95715	0.372258	0.186129	−	0.982525i	$-0.440406\pi$
$113$	3.95715	0.372258	0.186129	+	0.982525i	$0.440406\pi$
$114$	0	0
$115$	−6.68585	−0.623458
$116$	0	0
$117$	2.97858	0.275370
$118$	0	0
$119$	0.146365	0.0134173
$120$	0	0
$121$	−3.78623	−0.344203
$122$	0	0
$123$	0.935731	0.0843721
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−6.10038	−0.541322	−0.270661	−	0.962675i	$-0.587242\pi$
$127$	−6.10038	−0.541322	−0.270661	+	0.962675i	$0.587242\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−14.8322	−1.29590	−0.647948	−	0.761684i	$-0.724373\pi$
$131$	−14.8322	−1.29590	−0.647948	+	0.761684i	$0.724373\pi$
$132$	0	0
$133$	−0.585462	−0.0507660
$134$	0	0
$135$	0.875057	0.0753129
$136$	0	0
$137$	−9.76481	−0.834264	−0.417132	−	0.908846i	$-0.636965\pi$
$137$	−9.76481	−0.834264	−0.417132	+	0.908846i	$0.636965\pi$
$138$	0	0
$139$	−19.4752	−1.65187	−0.825933	−	0.563768i	$-0.809351\pi$
$139$	−19.4752	−1.65187	−0.825933	+	0.563768i	$0.809351\pi$
$140$	0	0
$141$	−1.04285	−0.0878235
$142$	0	0
$143$	−2.68585	−0.224602
$144$	0	0
$145$	−4.39312	−0.364828
$146$	0	0
$147$	−1.02142	−0.0842455
$148$	0	0
$149$	−11.9572	−0.979568	−0.489784	−	0.871844i	$-0.662924\pi$
$149$	−11.9572	−0.979568	−0.489784	+	0.871844i	$0.662924\pi$
$150$	0	0
$151$	−12.7894	−1.04078	−0.520392	−	0.853928i	$-0.674214\pi$
$151$	−12.7894	−1.04078	−0.520392	+	0.853928i	$0.674214\pi$
$152$	0	0
$153$	−2.97858	−0.240804
$154$	0	0
$155$	−1.31415	−0.105555
$156$	0	0
$157$	21.7220	1.73360	0.866801	−	0.498655i	$-0.166172\pi$
$157$	21.7220	1.73360	0.866801	+	0.498655i	$0.166172\pi$
$158$	0	0
$159$	−1.31415	−0.104219
$160$	0	0
$161$	0.978577	0.0771227
$162$	0	0
$163$	20.6430	1.61688	0.808442	−	0.588575i	$-0.200311\pi$
$163$	20.6430	1.61688	0.808442	+	0.588575i	$0.200311\pi$
$164$	0	0
$165$	−0.393115	−0.0306040
$166$	0	0
$167$	18.0147	1.39402	0.697009	−	0.717062i	$-0.254514\pi$
$167$	18.0147	1.39402	0.697009	+	0.717062i	$0.254514\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	11.9143	0.911110
$172$	0	0
$173$	−6.58546	−0.500683	−0.250342	−	0.968158i	$-0.580543\pi$
$173$	−6.58546	−0.500683	−0.250342	+	0.968158i	$0.580543\pi$
$174$	0	0
$175$	−0.585462	−0.0442568
$176$	0	0
$177$	−1.80765	−0.135872
$178$	0	0
$179$	−7.56090	−0.565128	−0.282564	−	0.959248i	$-0.591185\pi$
$179$	−7.56090	−0.565128	−0.282564	+	0.959248i	$0.591185\pi$
$180$	0	0
$181$	−0.628308	−0.0467017	−0.0233509	−	0.999727i	$-0.507433\pi$
$181$	−0.628308	−0.0467017	−0.0233509	+	0.999727i	$0.507433\pi$
$182$	0	0
$183$	−1.22219	−0.0903470
$184$	0	0
$185$	−3.97858	−0.292511
$186$	0	0
$187$	2.68585	0.196409
$188$	0	0
$189$	−0.128078	−0.00931632
$190$	0	0
$191$	−10.1004	−0.730838	−0.365419	−	0.930843i	$-0.619074\pi$
$191$	−10.1004	−0.730838	−0.365419	+	0.930843i	$0.619074\pi$
$192$	0	0
$193$	−3.37169	−0.242700	−0.121350	−	0.992610i	$-0.538722\pi$
$193$	−3.37169	−0.242700	−0.121350	+	0.992610i	$0.538722\pi$
$194$	0	0
$195$	0.146365	0.0104815
$196$	0	0
$197$	−14.7220	−1.04890	−0.524448	−	0.851442i	$-0.675728\pi$
$197$	−14.7220	−1.04890	−0.524448	+	0.851442i	$0.675728\pi$
$198$	0	0
$199$	13.6644	0.968645	0.484323	−	0.874889i	$-0.339066\pi$
$199$	13.6644	0.968645	0.484323	+	0.874889i	$0.339066\pi$
$200$	0	0
$201$	−1.21377	−0.0856127
$202$	0	0
$203$	0.643000	0.0451298
$204$	0	0
$205$	−6.39312	−0.446515
$206$	0	0
$207$	−19.9143	−1.38414
$208$	0	0
$209$	−10.7434	−0.743135
$210$	0	0
$211$	−1.46052	−0.100546	−0.0502731	−	0.998736i	$-0.516009\pi$
$211$	−1.46052	−0.100546	−0.0502731	+	0.998736i	$0.516009\pi$
$212$	0	0
$213$	−0.807653	−0.0553395
$214$	0	0
$215$	6.83221	0.465953
$216$	0	0
$217$	0.192347	0.0130573
$218$	0	0
$219$	1.02142	0.0690214
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	7.12494	0.477121	0.238561	−	0.971128i	$-0.423324\pi$
$223$	7.12494	0.477121	0.238561	+	0.971128i	$0.423324\pi$
$224$	0	0
$225$	11.9143	0.794287
$226$	0	0
$227$	−24.9357	−1.65504	−0.827521	−	0.561434i	$-0.810250\pi$
$227$	−24.9357	−1.65504	−0.827521	+	0.561434i	$0.810250\pi$
$228$	0	0
$229$	2.95715	0.195414	0.0977071	−	0.995215i	$-0.468849\pi$
$229$	2.95715	0.195414	0.0977071	+	0.995215i	$0.468849\pi$
$230$	0	0
$231$	0.0575385	0.00378576
$232$	0	0
$233$	5.19235	0.340162	0.170081	−	0.985430i	$-0.445597\pi$
$233$	5.19235	0.340162	0.170081	+	0.985430i	$0.445597\pi$
$234$	0	0
$235$	7.12494	0.464780
$236$	0	0
$237$	2.20077	0.142955
$238$	0	0
$239$	−14.3963	−0.931216	−0.465608	−	0.884991i	$-0.654164\pi$
$239$	−14.3963	−0.931216	−0.465608	+	0.884991i	$0.654164\pi$
$240$	0	0
$241$	23.3288	1.50274	0.751372	−	0.659879i	$-0.229393\pi$
$241$	23.3288	1.50274	0.751372	+	0.659879i	$0.229393\pi$
$242$	0	0
$243$	3.91431	0.251103
$244$	0	0
$245$	6.97858	0.445845
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−0.628308	−0.0398174
$250$	0	0
$251$	−15.2713	−0.963916	−0.481958	−	0.876194i	$-0.660074\pi$
$251$	−15.2713	−0.963916	−0.481958	+	0.876194i	$0.660074\pi$
$252$	0	0
$253$	17.9572	1.12896
$254$	0	0
$255$	−0.146365	−0.00916576
$256$	0	0
$257$	20.9143	1.30460	0.652299	−	0.757961i	$-0.273805\pi$
$257$	20.9143	1.30460	0.652299	+	0.757961i	$0.273805\pi$
$258$	0	0
$259$	0.582326	0.0361840
$260$	0	0
$261$	−13.0852	−0.809956
$262$	0	0
$263$	−13.0214	−0.802935	−0.401468	−	0.915873i	$-0.631500\pi$
$263$	−13.0214	−0.802935	−0.401468	+	0.915873i	$0.631500\pi$
$264$	0	0
$265$	8.97858	0.551550
$266$	0	0
$267$	0.786230	0.0481165
$268$	0	0
$269$	−24.3503	−1.48466	−0.742331	−	0.670033i	$-0.766280\pi$
$269$	−24.3503	−1.48466	−0.742331	+	0.670033i	$0.766280\pi$
$270$	0	0
$271$	25.5756	1.55361	0.776803	−	0.629743i	$-0.216840\pi$
$271$	25.5756	1.55361	0.776803	+	0.629743i	$0.216840\pi$
$272$	0	0
$273$	−0.0214229	−0.00129657
$274$	0	0
$275$	−10.7434	−0.647850
$276$	0	0
$277$	4.43596	0.266531	0.133266	−	0.991080i	$-0.457454\pi$
$277$	4.43596	0.266531	0.133266	+	0.991080i	$0.457454\pi$
$278$	0	0
$279$	−3.91431	−0.234344
$280$	0	0
$281$	−11.4145	−0.680934	−0.340467	−	0.940256i	$-0.610585\pi$
$281$	−11.4145	−0.680934	−0.340467	+	0.940256i	$0.610585\pi$
$282$	0	0
$283$	−19.4721	−1.15749	−0.578747	−	0.815507i	$-0.696458\pi$
$283$	−19.4721	−1.15749	−0.578747	+	0.815507i	$0.696458\pi$
$284$	0	0
$285$	0.585462	0.0346798
$286$	0	0
$287$	0.935731	0.0552345
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−1.51492	−0.0888063
$292$	0	0
$293$	10.8077	0.631390	0.315695	−	0.948861i	$-0.397762\pi$
$293$	10.8077	0.631390	0.315695	+	0.948861i	$0.397762\pi$
$294$	0	0
$295$	12.3503	0.719060
$296$	0	0
$297$	−2.35027	−0.136376
$298$	0	0
$299$	−6.68585	−0.386652
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	2.62831	0.150992
$304$	0	0
$305$	8.35027	0.478135
$306$	0	0
$307$	−4.93573	−0.281697	−0.140849	−	0.990031i	$-0.544983\pi$
$307$	−4.93573	−0.281697	−0.140849	+	0.990031i	$0.544983\pi$
$308$	0	0
$309$	−1.50023	−0.0853451
$310$	0	0
$311$	−0.786230	−0.0445830	−0.0222915	−	0.999752i	$-0.507096\pi$
$311$	−0.786230	−0.0445830	−0.0222915	+	0.999752i	$0.507096\pi$
$312$	0	0
$313$	13.9357	0.787694	0.393847	−	0.919176i	$-0.371144\pi$
$313$	13.9357	0.787694	0.393847	+	0.919176i	$0.371144\pi$
$314$	0	0
$315$	0.435961	0.0245636
$316$	0	0
$317$	19.9572	1.12091	0.560453	−	0.828186i	$-0.310627\pi$
$317$	19.9572	1.12091	0.560453	+	0.828186i	$0.310627\pi$
$318$	0	0
$319$	11.7992	0.660630
$320$	0	0
$321$	−2.14950	−0.119973
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−2.16106	−0.119507
$328$	0	0
$329$	−1.04285	−0.0574939
$330$	0	0
$331$	11.7073	0.643490	0.321745	−	0.946826i	$-0.395731\pi$
$331$	11.7073	0.643490	0.321745	+	0.946826i	$0.395731\pi$
$332$	0	0
$333$	−11.8505	−0.649403
$334$	0	0
$335$	8.29273	0.453080
$336$	0	0
$337$	−0.807653	−0.0439957	−0.0219978	−	0.999758i	$-0.507003\pi$
$337$	−0.807653	−0.0439957	−0.0219978	+	0.999758i	$0.507003\pi$
$338$	0	0
$339$	0.579191	0.0314573
$340$	0	0
$341$	3.52962	0.191139
$342$	0	0
$343$	−2.04598	−0.110473
$344$	0	0
$345$	−0.978577	−0.0526848
$346$	0	0
$347$	2.59702	0.139415	0.0697076	−	0.997567i	$-0.477793\pi$
$347$	2.59702	0.139415	0.0697076	+	0.997567i	$0.477793\pi$
$348$	0	0
$349$	16.1709	0.865610	0.432805	−	0.901488i	$-0.357524\pi$
$349$	16.1709	0.865610	0.432805	+	0.901488i	$0.357524\pi$
$350$	0	0
$351$	0.875057	0.0467071
$352$	0	0
$353$	−5.32885	−0.283626	−0.141813	−	0.989893i	$-0.545293\pi$
$353$	−5.32885	−0.283626	−0.141813	+	0.989893i	$0.545293\pi$
$354$	0	0
$355$	5.51806	0.292868
$356$	0	0
$357$	0.0214229	0.00113382
$358$	0	0
$359$	−34.6002	−1.82613	−0.913063	−	0.407818i	$-0.866290\pi$
$359$	−34.6002	−1.82613	−0.913063	+	0.407818i	$0.866290\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−0.554173	−0.0290866
$364$	0	0
$365$	−6.97858	−0.365275
$366$	0	0
$367$	−14.2499	−0.743838	−0.371919	−	0.928265i	$-0.621300\pi$
$367$	−14.2499	−0.743838	−0.371919	+	0.928265i	$0.621300\pi$
$368$	0	0
$369$	−19.0424	−0.991307
$370$	0	0
$371$	−1.31415	−0.0682275
$372$	0	0
$373$	15.1281	0.783302	0.391651	−	0.920114i	$-0.371904\pi$
$373$	15.1281	0.783302	0.391651	+	0.920114i	$0.371904\pi$
$374$	0	0
$375$	1.31729	0.0680245
$376$	0	0
$377$	−4.39312	−0.226257
$378$	0	0
$379$	−5.22846	−0.268568	−0.134284	−	0.990943i	$-0.542873\pi$
$379$	−5.22846	−0.268568	−0.134284	+	0.990943i	$0.542873\pi$
$380$	0	0
$381$	−0.892886	−0.0457439
$382$	0	0
$383$	27.3257	1.39628	0.698139	−	0.715962i	$-0.254012\pi$
$383$	27.3257	1.39628	0.698139	+	0.715962i	$0.254012\pi$
$384$	0	0
$385$	−0.393115	−0.0200350
$386$	0	0
$387$	20.3503	1.03446
$388$	0	0
$389$	9.02142	0.457404	0.228702	−	0.973496i	$-0.426552\pi$
$389$	9.02142	0.457404	0.228702	+	0.973496i	$0.426552\pi$
$390$	0	0
$391$	6.68585	0.338118
$392$	0	0
$393$	−2.17092	−0.109509
$394$	0	0
$395$	−15.0361	−0.756549
$396$	0	0
$397$	−35.0852	−1.76088	−0.880439	−	0.474160i	$-0.842752\pi$
$397$	−35.0852	−1.76088	−0.880439	+	0.474160i	$0.842752\pi$
$398$	0	0
$399$	−0.0856914	−0.00428994
$400$	0	0
$401$	−1.06427	−0.0531470	−0.0265735	−	0.999647i	$-0.508460\pi$
$401$	−1.06427	−0.0531470	−0.0265735	+	0.999647i	$0.508460\pi$
$402$	0	0
$403$	−1.31415	−0.0654627
$404$	0	0
$405$	−8.80765	−0.437656
$406$	0	0
$407$	10.6858	0.529678
$408$	0	0
$409$	31.7220	1.56855	0.784275	−	0.620413i	$-0.213035\pi$
$409$	31.7220	1.56855	0.784275	+	0.620413i	$0.213035\pi$
$410$	0	0
$411$	−1.42923	−0.0704988
$412$	0	0
$413$	−1.80765	−0.0889488
$414$	0	0
$415$	4.29273	0.210722
$416$	0	0
$417$	−2.85050	−0.139590
$418$	0	0
$419$	−3.06740	−0.149852	−0.0749262	−	0.997189i	$-0.523872\pi$
$419$	−3.06740	−0.149852	−0.0749262	+	0.997189i	$0.523872\pi$
$420$	0	0
$421$	2.21377	0.107893	0.0539463	−	0.998544i	$-0.482820\pi$
$421$	2.21377	0.107893	0.0539463	+	0.998544i	$0.482820\pi$
$422$	0	0
$423$	21.2222	1.03186
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−1.22219	−0.0591460
$428$	0	0
$429$	−0.393115	−0.0189798
$430$	0	0
$431$	18.8898	0.909887	0.454944	−	0.890520i	$-0.349659\pi$
$431$	18.8898	0.909887	0.454944	+	0.890520i	$0.349659\pi$
$432$	0	0
$433$	−21.8929	−1.05210	−0.526052	−	0.850452i	$-0.676328\pi$
$433$	−21.8929	−1.05210	−0.526052	+	0.850452i	$0.676328\pi$
$434$	0	0
$435$	−0.643000	−0.0308295
$436$	0	0
$437$	−26.7434	−1.27931
$438$	0	0
$439$	−13.8077	−0.659003	−0.329502	−	0.944155i	$-0.606881\pi$
$439$	−13.8077	−0.659003	−0.329502	+	0.944155i	$0.606881\pi$
$440$	0	0
$441$	20.7862	0.989820
$442$	0	0
$443$	25.5756	1.21513	0.607567	−	0.794269i	$-0.292146\pi$
$443$	25.5756	1.21513	0.607567	+	0.794269i	$0.292146\pi$
$444$	0	0
$445$	−5.37169	−0.254643
$446$	0	0
$447$	−1.75011	−0.0827776
$448$	0	0
$449$	−24.3074	−1.14714	−0.573569	−	0.819157i	$-0.694442\pi$
$449$	−24.3074	−1.14714	−0.573569	+	0.819157i	$0.694442\pi$
$450$	0	0
$451$	17.1709	0.808547
$452$	0	0
$453$	−1.87192	−0.0879506
$454$	0	0
$455$	0.146365	0.00686172
$456$	0	0
$457$	−12.2008	−0.570728	−0.285364	−	0.958419i	$-0.592114\pi$
$457$	−12.2008	−0.570728	−0.285364	+	0.958419i	$0.592114\pi$
$458$	0	0
$459$	−0.875057	−0.0408442
$460$	0	0
$461$	−8.02142	−0.373595	−0.186797	−	0.982398i	$-0.559811\pi$
$461$	−8.02142	−0.373595	−0.186797	+	0.982398i	$0.559811\pi$
$462$	0	0
$463$	−36.0575	−1.67574	−0.837868	−	0.545873i	$-0.816198\pi$
$463$	−36.0575	−1.67574	−0.837868	+	0.545873i	$0.816198\pi$
$464$	0	0
$465$	−0.192347	−0.00891987
$466$	0	0
$467$	19.1856	0.887804	0.443902	−	0.896075i	$-0.353594\pi$
$467$	19.1856	0.887804	0.443902	+	0.896075i	$0.353594\pi$
$468$	0	0
$469$	−1.21377	−0.0560467
$470$	0	0
$471$	3.17935	0.146497
$472$	0	0
$473$	−18.3503	−0.843746
$474$	0	0
$475$	16.0000	0.734130
$476$	0	0
$477$	26.7434	1.22450
$478$	0	0
$479$	−11.9112	−0.544235	−0.272118	−	0.962264i	$-0.587724\pi$
$479$	−11.9112	−0.544235	−0.272118	+	0.962264i	$0.587724\pi$
$480$	0	0
$481$	−3.97858	−0.181408
$482$	0	0
$483$	0.143230	0.00651719
$484$	0	0
$485$	10.3503	0.469982
$486$	0	0
$487$	15.8568	0.718539	0.359269	−	0.933234i	$-0.383026\pi$
$487$	15.8568	0.718539	0.359269	+	0.933234i	$0.383026\pi$
$488$	0	0
$489$	3.02142	0.136633
$490$	0	0
$491$	3.91117	0.176509	0.0882544	−	0.996098i	$-0.471871\pi$
$491$	3.91117	0.176509	0.0882544	+	0.996098i	$0.471871\pi$
$492$	0	0
$493$	4.39312	0.197856
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	−0.807653	−0.0362282
$498$	0	0
$499$	23.1793	1.03765	0.518825	−	0.854880i	$-0.326370\pi$
$499$	23.1793	1.03765	0.518825	+	0.854880i	$0.326370\pi$
$500$	0	0
$501$	2.63673	0.117800
$502$	0	0
$503$	−18.5426	−0.826774	−0.413387	−	0.910555i	$-0.635654\pi$
$503$	−18.5426	−0.826774	−0.413387	+	0.910555i	$0.635654\pi$
$504$	0	0
$505$	−17.9572	−0.799083
$506$	0	0
$507$	0.146365	0.00650032
$508$	0	0
$509$	25.9143	1.14863	0.574316	−	0.818634i	$-0.305268\pi$
$509$	25.9143	1.14863	0.574316	+	0.818634i	$0.305268\pi$
$510$	0	0
$511$	1.02142	0.0451851
$512$	0	0
$513$	3.50023	0.154539
$514$	0	0
$515$	10.2499	0.451664
$516$	0	0
$517$	−19.1365	−0.841622
$518$	0	0
$519$	−0.963884	−0.0423098
$520$	0	0
$521$	12.7648	0.559236	0.279618	−	0.960111i	$-0.409792\pi$
$521$	12.7648	0.559236	0.279618	+	0.960111i	$0.409792\pi$
$522$	0	0
$523$	27.1856	1.18874	0.594372	−	0.804190i	$-0.297401\pi$
$523$	27.1856	1.18874	0.594372	+	0.804190i	$0.297401\pi$
$524$	0	0
$525$	−0.0856914	−0.00373988
$526$	0	0
$527$	1.31415	0.0572454
$528$	0	0
$529$	21.7005	0.943502
$530$	0	0
$531$	36.7862	1.59639
$532$	0	0
$533$	−6.39312	−0.276917
$534$	0	0
$535$	14.6858	0.634924
$536$	0	0
$537$	−1.10666	−0.0477557
$538$	0	0
$539$	−18.7434	−0.807335
$540$	0	0
$541$	6.76481	0.290842	0.145421	−	0.989370i	$-0.453546\pi$
$541$	6.76481	0.290842	0.145421	+	0.989370i	$0.453546\pi$
$542$	0	0
$543$	−0.0919626	−0.00394649
$544$	0	0
$545$	14.7648	0.632455
$546$	0	0
$547$	−15.3832	−0.657740	−0.328870	−	0.944375i	$-0.606668\pi$
$547$	−15.3832	−0.657740	−0.328870	+	0.944375i	$0.606668\pi$
$548$	0	0
$549$	24.8719	1.06151
$550$	0	0
$551$	−17.5725	−0.748612
$552$	0	0
$553$	2.20077	0.0935862
$554$	0	0
$555$	−0.582326	−0.0247184
$556$	0	0
$557$	−7.59388	−0.321763	−0.160882	−	0.986974i	$-0.551434\pi$
$557$	−7.59388	−0.321763	−0.160882	+	0.986974i	$0.551434\pi$
$558$	0	0
$559$	6.83221	0.288972
$560$	0	0
$561$	0.393115	0.0165973
$562$	0	0
$563$	−14.9754	−0.631140	−0.315570	−	0.948902i	$-0.602196\pi$
$563$	−14.9754	−0.631140	−0.315570	+	0.948902i	$0.602196\pi$
$564$	0	0
$565$	−3.95715	−0.166479
$566$	0	0
$567$	1.28914	0.0541386
$568$	0	0
$569$	4.25662	0.178447	0.0892233	−	0.996012i	$-0.471562\pi$
$569$	4.25662	0.178447	0.0892233	+	0.996012i	$0.471562\pi$
$570$	0	0
$571$	43.5903	1.82420	0.912098	−	0.409971i	$-0.134461\pi$
$571$	43.5903	1.82420	0.912098	+	0.409971i	$0.134461\pi$
$572$	0	0
$573$	−1.47835	−0.0617589
$574$	0	0
$575$	−26.7434	−1.11528
$576$	0	0
$577$	43.5934	1.81482	0.907409	−	0.420249i	$-0.138057\pi$
$577$	43.5934	1.81482	0.907409	+	0.420249i	$0.138057\pi$
$578$	0	0
$579$	−0.493499	−0.0205091
$580$	0	0
$581$	−0.628308	−0.0260666
$582$	0	0
$583$	−24.1151	−0.998744
$584$	0	0
$585$	−2.97858	−0.123149
$586$	0	0
$587$	1.31415	0.0542409	0.0271205	−	0.999632i	$-0.491366\pi$
$587$	1.31415	0.0542409	0.0271205	+	0.999632i	$0.491366\pi$
$588$	0	0
$589$	−5.25662	−0.216595
$590$	0	0
$591$	−2.15479	−0.0886361
$592$	0	0
$593$	−0.777809	−0.0319408	−0.0159704	−	0.999872i	$-0.505084\pi$
$593$	−0.777809	−0.0319408	−0.0159704	+	0.999872i	$0.505084\pi$
$594$	0	0
$595$	−0.146365	−0.00600040
$596$	0	0
$597$	2.00000	0.0818546
$598$	0	0
$599$	−3.36327	−0.137420	−0.0687098	−	0.997637i	$-0.521888\pi$
$599$	−3.36327	−0.137420	−0.0687098	+	0.997637i	$0.521888\pi$
$600$	0	0
$601$	−4.17092	−0.170136	−0.0850678	−	0.996375i	$-0.527111\pi$
$601$	−4.17092	−0.170136	−0.0850678	+	0.996375i	$0.527111\pi$
$602$	0	0
$603$	24.7005	1.00588
$604$	0	0
$605$	3.78623	0.153932
$606$	0	0
$607$	−34.8929	−1.41626	−0.708129	−	0.706083i	$-0.750461\pi$
$607$	−34.8929	−1.41626	−0.708129	+	0.706083i	$0.750461\pi$
$608$	0	0
$609$	0.0941131	0.00381365
$610$	0	0
$611$	7.12494	0.288244
$612$	0	0
$613$	17.9143	0.723552	0.361776	−	0.932265i	$-0.382171\pi$
$613$	17.9143	0.723552	0.361776	+	0.932265i	$0.382171\pi$
$614$	0	0
$615$	−0.935731	−0.0377323
$616$	0	0
$617$	−23.9143	−0.962754	−0.481377	−	0.876514i	$-0.659863\pi$
$617$	−23.9143	−0.962754	−0.481377	+	0.876514i	$0.659863\pi$
$618$	0	0
$619$	27.9656	1.12403	0.562016	−	0.827127i	$-0.310026\pi$
$619$	27.9656	1.12403	0.562016	+	0.827127i	$0.310026\pi$
$620$	0	0
$621$	−5.85050	−0.234772
$622$	0	0
$623$	0.786230	0.0314997
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−1.57246	−0.0627980
$628$	0	0
$629$	3.97858	0.158636
$630$	0	0
$631$	28.4966	1.13443	0.567217	−	0.823569i	$-0.308020\pi$
$631$	28.4966	1.13443	0.567217	+	0.823569i	$0.308020\pi$
$632$	0	0
$633$	−0.213770	−0.00849658
$634$	0	0
$635$	6.10038	0.242086
$636$	0	0
$637$	6.97858	0.276501
$638$	0	0
$639$	16.4360	0.650197
$640$	0	0
$641$	13.9143	0.549582	0.274791	−	0.961504i	$-0.411391\pi$
$641$	13.9143	0.549582	0.274791	+	0.961504i	$0.411391\pi$
$642$	0	0
$643$	19.0361	0.750711	0.375356	−	0.926881i	$-0.377521\pi$
$643$	19.0361	0.750711	0.375356	+	0.926881i	$0.377521\pi$
$644$	0	0
$645$	1.00000	0.0393750
$646$	0	0
$647$	17.6069	0.692198	0.346099	−	0.938198i	$-0.387506\pi$
$647$	17.6069	0.692198	0.346099	+	0.938198i	$0.387506\pi$
$648$	0	0
$649$	−33.1709	−1.30207
$650$	0	0
$651$	0.0281529	0.00110340
$652$	0	0
$653$	−33.8715	−1.32549	−0.662746	−	0.748844i	$-0.730609\pi$
$653$	−33.8715	−1.32549	−0.662746	+	0.748844i	$0.730609\pi$
$654$	0	0
$655$	14.8322	0.579542
$656$	0	0
$657$	−20.7862	−0.810948
$658$	0	0
$659$	−27.1856	−1.05900	−0.529501	−	0.848310i	$-0.677621\pi$
$659$	−27.1856	−1.05900	−0.529501	+	0.848310i	$0.677621\pi$
$660$	0	0
$661$	−48.2730	−1.87760	−0.938801	−	0.344460i	$-0.888062\pi$
$661$	−48.2730	−1.87760	−0.938801	+	0.344460i	$0.888062\pi$
$662$	0	0
$663$	−0.146365	−0.00568436
$664$	0	0
$665$	0.585462	0.0227032
$666$	0	0
$667$	29.3717	1.13728
$668$	0	0
$669$	1.04285	0.0403187
$670$	0	0
$671$	−22.4275	−0.865806
$672$	0	0
$673$	5.78623	0.223043	0.111521	−	0.993762i	$-0.464428\pi$
$673$	5.78623	0.223043	0.111521	+	0.993762i	$0.464428\pi$
$674$	0	0
$675$	3.50023	0.134724
$676$	0	0
$677$	37.3288	1.43466	0.717332	−	0.696731i	$-0.245363\pi$
$677$	37.3288	1.43466	0.717332	+	0.696731i	$0.245363\pi$
$678$	0	0
$679$	−1.51492	−0.0581374
$680$	0	0
$681$	−3.64973	−0.139858
$682$	0	0
$683$	17.6644	0.675910	0.337955	−	0.941162i	$-0.390265\pi$
$683$	17.6644	0.675910	0.337955	+	0.941162i	$0.390265\pi$
$684$	0	0
$685$	9.76481	0.373094
$686$	0	0
$687$	0.432825	0.0165133
$688$	0	0
$689$	8.97858	0.342057
$690$	0	0
$691$	44.4998	1.69285	0.846426	−	0.532507i	$-0.178750\pi$
$691$	44.4998	1.69285	0.846426	+	0.532507i	$0.178750\pi$
$692$	0	0
$693$	−1.17092	−0.0444797
$694$	0	0
$695$	19.4752	0.738737
$696$	0	0
$697$	6.39312	0.242157
$698$	0	0
$699$	0.759980	0.0287451
$700$	0	0
$701$	17.0643	0.644509	0.322254	−	0.946653i	$-0.395559\pi$
$701$	17.0643	0.644509	0.322254	+	0.946653i	$0.395559\pi$
$702$	0	0
$703$	−15.9143	−0.600220
$704$	0	0
$705$	1.04285	0.0392758
$706$	0	0
$707$	2.62831	0.0988477
$708$	0	0
$709$	−0.427539	−0.0160566	−0.00802829	−	0.999968i	$-0.502556\pi$
$709$	−0.427539	−0.0160566	−0.00802829	+	0.999968i	$0.502556\pi$
$710$	0	0
$711$	−44.7862	−1.67961
$712$	0	0
$713$	8.78623	0.329047
$714$	0	0
$715$	2.68585	0.100445
$716$	0	0
$717$	−2.10711	−0.0786916
$718$	0	0
$719$	37.3780	1.39396	0.696981	−	0.717089i	$-0.254526\pi$
$719$	37.3780	1.39396	0.696981	+	0.717089i	$0.254526\pi$
$720$	0	0
$721$	−1.50023	−0.0558715
$722$	0	0
$723$	3.41454	0.126988
$724$	0	0
$725$	−17.5725	−0.652625
$726$	0	0
$727$	−14.4851	−0.537222	−0.268611	−	0.963249i	$-0.586565\pi$
$727$	−14.4851	−0.537222	−0.268611	+	0.963249i	$0.586565\pi$
$728$	0	0
$729$	−25.8500	−0.957409
$730$	0	0
$731$	−6.83221	−0.252698
$732$	0	0
$733$	0.299461	0.0110608	0.00553042	−	0.999985i	$-0.498240\pi$
$733$	0.299461	0.0110608	0.00553042	+	0.999985i	$0.498240\pi$
$734$	0	0
$735$	1.02142	0.0376757
$736$	0	0
$737$	−22.2730	−0.820436
$738$	0	0
$739$	18.9786	0.698138	0.349069	−	0.937097i	$-0.386498\pi$
$739$	18.9786	0.698138	0.349069	+	0.937097i	$0.386498\pi$
$740$	0	0
$741$	0.585462	0.0215075
$742$	0	0
$743$	26.7465	0.981235	0.490617	−	0.871375i	$-0.336771\pi$
$743$	26.7465	0.981235	0.490617	+	0.871375i	$0.336771\pi$
$744$	0	0
$745$	11.9572	0.438076
$746$	0	0
$747$	12.7862	0.467824
$748$	0	0
$749$	−2.14950	−0.0785411
$750$	0	0
$751$	−14.1923	−0.517886	−0.258943	−	0.965893i	$-0.583374\pi$
$751$	−14.1923	−0.517886	−0.258943	+	0.965893i	$0.583374\pi$
$752$	0	0
$753$	−2.23519	−0.0814549
$754$	0	0
$755$	12.7894	0.465453
$756$	0	0
$757$	−45.5934	−1.65712	−0.828561	−	0.559899i	$-0.810840\pi$
$757$	−45.5934	−1.65712	−0.828561	+	0.559899i	$0.810840\pi$
$758$	0	0
$759$	2.62831	0.0954015
$760$	0	0
$761$	37.6363	1.36431	0.682157	−	0.731206i	$-0.261042\pi$
$761$	37.6363	1.36431	0.682157	+	0.731206i	$0.261042\pi$
$762$	0	0
$763$	−2.16106	−0.0782356
$764$	0	0
$765$	2.97858	0.107691
$766$	0	0
$767$	12.3503	0.445942
$768$	0	0
$769$	0.700539	0.0252621	0.0126310	−	0.999920i	$-0.495979\pi$
$769$	0.700539	0.0252621	0.0126310	+	0.999920i	$0.495979\pi$
$770$	0	0
$771$	3.06113	0.110244
$772$	0	0
$773$	−5.08569	−0.182920	−0.0914598	−	0.995809i	$-0.529153\pi$
$773$	−5.08569	−0.182920	−0.0914598	+	0.995809i	$0.529153\pi$
$774$	0	0
$775$	−5.25662	−0.188823
$776$	0	0
$777$	0.0852325	0.00305770
$778$	0	0
$779$	−25.5725	−0.916228
$780$	0	0
$781$	−14.8207	−0.530325
$782$	0	0
$783$	−3.84423	−0.137381
$784$	0	0
$785$	−21.7220	−0.775290
$786$	0	0
$787$	−14.3074	−0.510005	−0.255002	−	0.966940i	$-0.582076\pi$
$787$	−14.3074	−0.510005	−0.255002	+	0.966940i	$0.582076\pi$
$788$	0	0
$789$	−1.90589	−0.0678514
$790$	0	0
$791$	0.579191	0.0205937
$792$	0	0
$793$	8.35027	0.296527
$794$	0	0
$795$	1.31415	0.0466082
$796$	0	0
$797$	2.54262	0.0900641	0.0450320	−	0.998986i	$-0.485661\pi$
$797$	2.54262	0.0900641	0.0450320	+	0.998986i	$0.485661\pi$
$798$	0	0
$799$	−7.12494	−0.252062
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	18.7434	0.661440
$804$	0	0
$805$	−0.978577	−0.0344903
$806$	0	0
$807$	−3.56404	−0.125460
$808$	0	0
$809$	33.1495	1.16547	0.582737	−	0.812661i	$-0.301982\pi$
$809$	33.1495	1.16547	0.582737	+	0.812661i	$0.301982\pi$
$810$	0	0
$811$	−1.28600	−0.0451576	−0.0225788	−	0.999745i	$-0.507188\pi$
$811$	−1.28600	−0.0451576	−0.0225788	+	0.999745i	$0.507188\pi$
$812$	0	0
$813$	3.74338	0.131286
$814$	0	0
$815$	−20.6430	−0.723093
$816$	0	0
$817$	27.3288	0.956115
$818$	0	0
$819$	0.435961	0.0152337
$820$	0	0
$821$	11.7005	0.408352	0.204176	−	0.978934i	$-0.434549\pi$
$821$	11.7005	0.408352	0.204176	+	0.978934i	$0.434549\pi$
$822$	0	0
$823$	38.9786	1.35871	0.679354	−	0.733811i	$-0.262260\pi$
$823$	38.9786	1.35871	0.679354	+	0.733811i	$0.262260\pi$
$824$	0	0
$825$	−1.57246	−0.0547461
$826$	0	0
$827$	−43.5296	−1.51367	−0.756837	−	0.653604i	$-0.773256\pi$
$827$	−43.5296	−1.51367	−0.756837	+	0.653604i	$0.773256\pi$
$828$	0	0
$829$	35.0852	1.21856	0.609280	−	0.792955i	$-0.291458\pi$
$829$	35.0852	1.21856	0.609280	+	0.792955i	$0.291458\pi$
$830$	0	0
$831$	0.649272	0.0225230
$832$	0	0
$833$	−6.97858	−0.241793
$834$	0	0
$835$	−18.0147	−0.623424
$836$	0	0
$837$	−1.14996	−0.0397484
$838$	0	0
$839$	10.4851	0.361985	0.180993	−	0.983484i	$-0.442069\pi$
$839$	10.4851	0.361985	0.180993	+	0.983484i	$0.442069\pi$
$840$	0	0
$841$	−9.70054	−0.334501
$842$	0	0
$843$	−1.67069	−0.0575418
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−0.554173	−0.0190416
$848$	0	0
$849$	−2.85004	−0.0978131
$850$	0	0
$851$	26.6002	0.911842
$852$	0	0
$853$	40.5296	1.38771	0.693854	−	0.720116i	$-0.255911\pi$
$853$	40.5296	1.38771	0.693854	+	0.720116i	$0.255911\pi$
$854$	0	0
$855$	−11.9143	−0.407461
$856$	0	0
$857$	−5.61531	−0.191815	−0.0959076	−	0.995390i	$-0.530575\pi$
$857$	−5.61531	−0.191815	−0.0959076	+	0.995390i	$0.530575\pi$
$858$	0	0
$859$	−27.4721	−0.937335	−0.468668	−	0.883375i	$-0.655266\pi$
$859$	−27.4721	−0.937335	−0.468668	+	0.883375i	$0.655266\pi$
$860$	0	0
$861$	0.136959	0.00466754
$862$	0	0
$863$	42.8898	1.45998	0.729992	−	0.683456i	$-0.239524\pi$
$863$	42.8898	1.45998	0.729992	+	0.683456i	$0.239524\pi$
$864$	0	0
$865$	6.58546	0.223912
$866$	0	0
$867$	−2.34185	−0.0795333
$868$	0	0
$869$	40.3847	1.36996
$870$	0	0
$871$	8.29273	0.280988
$872$	0	0
$873$	30.8291	1.04341
$874$	0	0
$875$	1.31729	0.0445325
$876$	0	0
$877$	−31.7005	−1.07045	−0.535226	−	0.844709i	$-0.679773\pi$
$877$	−31.7005	−1.07045	−0.535226	+	0.844709i	$0.679773\pi$
$878$	0	0
$879$	1.58187	0.0533551
$880$	0	0
$881$	−44.4011	−1.49591	−0.747955	−	0.663749i	$-0.768964\pi$
$881$	−44.4011	−1.49591	−0.747955	+	0.663749i	$0.768964\pi$
$882$	0	0
$883$	−1.28287	−0.0431719	−0.0215859	−	0.999767i	$-0.506872\pi$
$883$	−1.28287	−0.0431719	−0.0215859	+	0.999767i	$0.506872\pi$
$884$	0	0
$885$	1.80765	0.0607636
$886$	0	0
$887$	−41.5212	−1.39415	−0.697073	−	0.717001i	$-0.745514\pi$
$887$	−41.5212	−1.39415	−0.697073	+	0.717001i	$0.745514\pi$
$888$	0	0
$889$	−0.892886	−0.0299464
$890$	0	0
$891$	23.6560	0.792506
$892$	0	0
$893$	28.4998	0.953708
$894$	0	0
$895$	7.56090	0.252733
$896$	0	0
$897$	−0.978577	−0.0326737
$898$	0	0
$899$	5.77323	0.192548
$900$	0	0
$901$	−8.97858	−0.299120
$902$	0	0
$903$	−0.146365	−0.00487074
$904$	0	0
$905$	0.628308	0.0208857
$906$	0	0
$907$	41.9834	1.39404	0.697018	−	0.717054i	$-0.254510\pi$
$907$	41.9834	1.39404	0.697018	+	0.717054i	$0.254510\pi$
$908$	0	0
$909$	−53.4868	−1.77404
$910$	0	0
$911$	−37.5443	−1.24390	−0.621949	−	0.783058i	$-0.713659\pi$
$911$	−37.5443	−1.24390	−0.621949	+	0.783058i	$0.713659\pi$
$912$	0	0
$913$	−11.5296	−0.381575
$914$	0	0
$915$	1.22219	0.0404044
$916$	0	0
$917$	−2.17092	−0.0716902
$918$	0	0
$919$	−48.7005	−1.60648	−0.803241	−	0.595654i	$-0.796893\pi$
$919$	−48.7005	−1.60648	−0.803241	+	0.595654i	$0.796893\pi$
$920$	0	0
$921$	−0.722421	−0.0238046
$922$	0	0
$923$	5.51806	0.181629
$924$	0	0
$925$	−15.9143	−0.523259
$926$	0	0
$927$	30.5301	1.00274
$928$	0	0
$929$	−11.1281	−0.365100	−0.182550	−	0.983197i	$-0.558435\pi$
$929$	−11.1281	−0.365100	−0.182550	+	0.983197i	$0.558435\pi$
$930$	0	0
$931$	27.9143	0.914855
$932$	0	0
$933$	−0.115077	−0.00376745
$934$	0	0
$935$	−2.68585	−0.0878366
$936$	0	0
$937$	−17.6153	−0.575467	−0.287733	−	0.957711i	$-0.592902\pi$
$937$	−17.6153	−0.575467	−0.287733	+	0.957711i	$0.592902\pi$
$938$	0	0
$939$	2.03971	0.0665634
$940$	0	0
$941$	35.6577	1.16241	0.581204	−	0.813758i	$-0.302582\pi$
$941$	35.6577	1.16241	0.581204	+	0.813758i	$0.302582\pi$
$942$	0	0
$943$	42.7434	1.39192
$944$	0	0
$945$	0.128078	0.00416638
$946$	0	0
$947$	−22.7715	−0.739976	−0.369988	−	0.929037i	$-0.620638\pi$
$947$	−22.7715	−0.739976	−0.369988	+	0.929037i	$0.620638\pi$
$948$	0	0
$949$	−6.97858	−0.226534
$950$	0	0
$951$	2.92104	0.0947212
$952$	0	0
$953$	1.74338	0.0564738	0.0282369	−	0.999601i	$-0.491011\pi$
$953$	1.74338	0.0564738	0.0282369	+	0.999601i	$0.491011\pi$
$954$	0	0
$955$	10.1004	0.326841
$956$	0	0
$957$	1.72700	0.0558260
$958$	0	0
$959$	−1.42923	−0.0461523
$960$	0	0
$961$	−29.2730	−0.944290
$962$	0	0
$963$	43.7429	1.40960
$964$	0	0
$965$	3.37169	0.108539
$966$	0	0
$967$	33.8402	1.08823	0.544113	−	0.839012i	$-0.316866\pi$
$967$	33.8402	1.08823	0.544113	+	0.839012i	$0.316866\pi$
$968$	0	0
$969$	−0.585462	−0.0188077
$970$	0	0
$971$	40.0031	1.28376	0.641881	−	0.766804i	$-0.278154\pi$
$971$	40.0031	1.28376	0.641881	+	0.766804i	$0.278154\pi$
$972$	0	0
$973$	−2.85050	−0.0913828
$974$	0	0
$975$	0.585462	0.0187498
$976$	0	0
$977$	38.7005	1.23814	0.619070	−	0.785336i	$-0.287510\pi$
$977$	38.7005	1.23814	0.619070	+	0.785336i	$0.287510\pi$
$978$	0	0
$979$	14.4275	0.461106
$980$	0	0
$981$	43.9781	1.40411
$982$	0	0
$983$	−23.8536	−0.760813	−0.380406	−	0.924819i	$-0.624216\pi$
$983$	−23.8536	−0.760813	−0.380406	+	0.924819i	$0.624216\pi$
$984$	0	0
$985$	14.7220	0.469081
$986$	0	0
$987$	−0.152637	−0.00485848
$988$	0	0
$989$	−45.6791	−1.45251
$990$	0	0
$991$	−3.01300	−0.0957111	−0.0478556	−	0.998854i	$-0.515239\pi$
$991$	−3.01300	−0.0957111	−0.0478556	+	0.998854i	$0.515239\pi$
$992$	0	0
$993$	1.71354	0.0543776
$994$	0	0
$995$	−13.6644	−0.433191
$996$	0	0
$997$	7.13650	0.226015	0.113008	−	0.993594i	$-0.463952\pi$
$997$	7.13650	0.226015	0.113008	+	0.993594i	$0.463952\pi$
$998$	0	0
$999$	−3.48148	−0.110149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.be.1.2		3
4.3	odd	2		3328.2.a.bg.1.2		3
8.3	odd	2		3328.2.a.bf.1.2		3
8.5	even	2		3328.2.a.bh.1.2		3
16.3	odd	4		416.2.b.c.209.3		6
16.5	even	4		104.2.b.c.53.5	✓	6
16.11	odd	4		416.2.b.c.209.4		6
16.13	even	4		104.2.b.c.53.6	yes	6
48.5	odd	4		936.2.g.c.469.2		6
48.11	even	4		3744.2.g.c.1873.5		6
48.29	odd	4		936.2.g.c.469.1		6
48.35	even	4		3744.2.g.c.1873.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.c.53.5	✓	6	16.5	even	4
104.2.b.c.53.6	yes	6	16.13	even	4
416.2.b.c.209.3		6	16.3	odd	4
416.2.b.c.209.4		6	16.11	odd	4
936.2.g.c.469.1		6	48.29	odd	4
936.2.g.c.469.2		6	48.5	odd	4
3328.2.a.be.1.2		3	1.1	even	1	trivial
3328.2.a.bf.1.2		3	8.3	odd	2
3328.2.a.bg.1.2		3	4.3	odd	2
3328.2.a.bh.1.2		3	8.5	even	2
3744.2.g.c.1873.2		6	48.35	even	4
3744.2.g.c.1873.5		6	48.11	even	4

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q+0.146365 q^{3} -1.00000 q^{5} +0.146365 q^{7} -2.97858 q^{9} +2.68585 q^{11} -1.00000 q^{13} -0.146365 q^{15} +1.00000 q^{17} -4.00000 q^{19} +0.0214229 q^{21} +6.68585 q^{23} -4.00000 q^{25} -0.875057 q^{27} +4.39312 q^{29} +1.31415 q^{31} +0.393115 q^{33} -0.146365 q^{35} +3.97858 q^{37} -0.146365 q^{39} +6.39312 q^{41} -6.83221 q^{43} +2.97858 q^{45} -7.12494 q^{47} -6.97858 q^{49} +0.146365 q^{51} -8.97858 q^{53} -2.68585 q^{55} -0.585462 q^{57} -12.3503 q^{59} -8.35027 q^{61} -0.435961 q^{63} +1.00000 q^{65} -8.29273 q^{67} +0.978577 q^{69} -5.51806 q^{71} +6.97858 q^{73} -0.585462 q^{75} +0.393115 q^{77} +15.0361 q^{79} +8.80765 q^{81} -4.29273 q^{83} -1.00000 q^{85} +0.643000 q^{87} +5.37169 q^{89} -0.146365 q^{91} +0.192347 q^{93} +4.00000 q^{95} -10.3503 q^{97} -8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} - q^{7} + 6 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 3 q^{17} - 12 q^{19} + 15 q^{21} + 8 q^{23} - 12 q^{25} - 19 q^{27} + 4 q^{29} + 16 q^{31} - 8 q^{33} + q^{35} - 3 q^{37} + q^{39}+ \cdots - 24 q^{99}+O(q^{100}) \) 3 * q - q^3 - 3 * q^5 - q^7 + 6 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 3 * q^17 - 12 * q^19 + 15 * q^21 + 8 * q^23 - 12 * q^25 - 19 * q^27 + 4 * q^29 + 16 * q^31 - 8 * q^33 + q^35 - 3 * q^37 + q^39 + 10 * q^41 - 7 * q^43 - 6 * q^45 - 5 * q^47 - 6 * q^49 - q^51 - 12 * q^53 + 4 * q^55 + 4 * q^57 + 2 * q^59 + 14 * q^61 - 22 * q^63 + 3 * q^65 - 22 * q^67 - 12 * q^69 + 9 * q^71 + 6 * q^73 + 4 * q^75 - 8 * q^77 - 6 * q^79 + 23 * q^81 - 10 * q^83 - 3 * q^85 - 40 * q^87 - 8 * q^89 + q^91 + 4 * q^93 + 12 * q^95 + 8 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

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Database

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