LMFDB - Embedded newform 6032.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 $ x^10 - 2x^9 - 21x^8 + 40x^7 + 138x^6 - 243x^5 - 318x^4 + 448x^3 + 312x^2 - 240*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.22036$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22036 q^{3} -0.143149 q^{5} -1.44357 q^{7} +1.92998 q^{9} +O(q^{10})$	$q-2.22036 q^{3} -0.143149 q^{5} -1.44357 q^{7} +1.92998 q^{9} -4.98317 q^{11} +1.00000 q^{13} +0.317842 q^{15} -2.87775 q^{17} -1.75809 q^{19} +3.20523 q^{21} -2.19059 q^{23} -4.97951 q^{25} +2.37582 q^{27} -1.00000 q^{29} -10.4242 q^{31} +11.0644 q^{33} +0.206645 q^{35} -3.76655 q^{37} -2.22036 q^{39} +9.68357 q^{41} -8.39441 q^{43} -0.276276 q^{45} -0.154531 q^{47} -4.91612 q^{49} +6.38962 q^{51} +1.92431 q^{53} +0.713336 q^{55} +3.90358 q^{57} +4.22176 q^{59} -0.298213 q^{61} -2.78606 q^{63} -0.143149 q^{65} -0.522953 q^{67} +4.86390 q^{69} -5.65532 q^{71} -5.40765 q^{73} +11.0563 q^{75} +7.19353 q^{77} -1.00078 q^{79} -11.0651 q^{81} -0.385276 q^{83} +0.411947 q^{85} +2.22036 q^{87} -14.5128 q^{89} -1.44357 q^{91} +23.1454 q^{93} +0.251669 q^{95} -5.94582 q^{97} -9.61743 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.22036	−1.28192	−0.640962	−	0.767573i	$-0.721464\pi$
$3$	−2.22036	−1.28192	−0.640962	+	0.767573i	$0.721464\pi$
$4$	0	0
$5$	−0.143149	−0.0640183	−0.0320091	−	0.999488i	$-0.510191\pi$
$5$	−0.143149	−0.0640183	−0.0320091	+	0.999488i	$0.510191\pi$
$6$	0	0
$7$	−1.44357	−0.545617	−0.272808	−	0.962068i	$-0.587952\pi$
$7$	−1.44357	−0.545617	−0.272808	+	0.962068i	$0.587952\pi$
$8$	0	0
$9$	1.92998	0.643328
$10$	0	0
$11$	−4.98317	−1.50248	−0.751240	−	0.660029i	$-0.770544\pi$
$11$	−4.98317	−1.50248	−0.751240	+	0.660029i	$0.770544\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.317842	0.0820666
$16$	0	0
$17$	−2.87775	−0.697956	−0.348978	−	0.937131i	$-0.613471\pi$
$17$	−2.87775	−0.697956	−0.348978	+	0.937131i	$0.613471\pi$
$18$	0	0
$19$	−1.75809	−0.403333	−0.201666	−	0.979454i	$-0.564636\pi$
$19$	−1.75809	−0.403333	−0.201666	+	0.979454i	$0.564636\pi$
$20$	0	0
$21$	3.20523	0.699439
$22$	0	0
$23$	−2.19059	−0.456771	−0.228385	−	0.973571i	$-0.573345\pi$
$23$	−2.19059	−0.456771	−0.228385	+	0.973571i	$0.573345\pi$
$24$	0	0
$25$	−4.97951	−0.995902
$26$	0	0
$27$	2.37582	0.457226
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.4242	−1.87224	−0.936118	−	0.351686i	$-0.885608\pi$
$31$	−10.4242	−1.87224	−0.936118	+	0.351686i	$0.885608\pi$
$32$	0	0
$33$	11.0644	1.92607
$34$	0	0
$35$	0.206645	0.0349294
$36$	0	0
$37$	−3.76655	−0.619217	−0.309609	−	0.950864i	$-0.600198\pi$
$37$	−3.76655	−0.619217	−0.309609	+	0.950864i	$0.600198\pi$
$38$	0	0
$39$	−2.22036	−0.355542
$40$	0	0
$41$	9.68357	1.51232	0.756160	−	0.654387i	$-0.227073\pi$
$41$	9.68357	1.51232	0.756160	+	0.654387i	$0.227073\pi$
$42$	0	0
$43$	−8.39441	−1.28013	−0.640067	−	0.768319i	$-0.721094\pi$
$43$	−8.39441	−1.28013	−0.640067	+	0.768319i	$0.721094\pi$
$44$	0	0
$45$	−0.276276	−0.0411848
$46$	0	0
$47$	−0.154531	−0.0225407	−0.0112703	−	0.999936i	$-0.503588\pi$
$47$	−0.154531	−0.0225407	−0.0112703	+	0.999936i	$0.503588\pi$
$48$	0	0
$49$	−4.91612	−0.702303
$50$	0	0
$51$	6.38962	0.894726
$52$	0	0
$53$	1.92431	0.264324	0.132162	−	0.991228i	$-0.457808\pi$
$53$	1.92431	0.264324	0.132162	+	0.991228i	$0.457808\pi$
$54$	0	0
$55$	0.713336	0.0961862
$56$	0	0
$57$	3.90358	0.517042
$58$	0	0
$59$	4.22176	0.549626	0.274813	−	0.961498i	$-0.411384\pi$
$59$	4.22176	0.549626	0.274813	+	0.961498i	$0.411384\pi$
$60$	0	0
$61$	−0.298213	−0.0381822	−0.0190911	−	0.999818i	$-0.506077\pi$
$61$	−0.298213	−0.0381822	−0.0190911	+	0.999818i	$0.506077\pi$
$62$	0	0
$63$	−2.78606	−0.351011
$64$	0	0
$65$	−0.143149	−0.0177555
$66$	0	0
$67$	−0.522953	−0.0638888	−0.0319444	−	0.999490i	$-0.510170\pi$
$67$	−0.522953	−0.0638888	−0.0319444	+	0.999490i	$0.510170\pi$
$68$	0	0
$69$	4.86390	0.585545
$70$	0	0
$71$	−5.65532	−0.671163	−0.335581	−	0.942011i	$-0.608933\pi$
$71$	−5.65532	−0.671163	−0.335581	+	0.942011i	$0.608933\pi$
$72$	0	0
$73$	−5.40765	−0.632917	−0.316459	−	0.948606i	$-0.602494\pi$
$73$	−5.40765	−0.632917	−0.316459	+	0.948606i	$0.602494\pi$
$74$	0	0
$75$	11.0563	1.27667
$76$	0	0
$77$	7.19353	0.819778
$78$	0	0
$79$	−1.00078	−0.112596	−0.0562981	−	0.998414i	$-0.517930\pi$
$79$	−1.00078	−0.112596	−0.0562981	+	0.998414i	$0.517930\pi$
$80$	0	0
$81$	−11.0651	−1.22946
$82$	0	0
$83$	−0.385276	−0.0422896	−0.0211448	−	0.999776i	$-0.506731\pi$
$83$	−0.385276	−0.0422896	−0.0211448	+	0.999776i	$0.506731\pi$
$84$	0	0
$85$	0.411947	0.0446819
$86$	0	0
$87$	2.22036	0.238047
$88$	0	0
$89$	−14.5128	−1.53835	−0.769176	−	0.639036i	$-0.779333\pi$
$89$	−14.5128	−1.53835	−0.769176	+	0.639036i	$0.779333\pi$
$90$	0	0
$91$	−1.44357	−0.151327
$92$	0	0
$93$	23.1454	2.40006
$94$	0	0
$95$	0.251669	0.0258207
$96$	0	0
$97$	−5.94582	−0.603707	−0.301853	−	0.953354i	$-0.597605\pi$
$97$	−5.94582	−0.603707	−0.301853	+	0.953354i	$0.597605\pi$
$98$	0	0
$99$	−9.61743	−0.966588
$100$	0	0
$101$	2.72004	0.270654	0.135327	−	0.990801i	$-0.456791\pi$
$101$	2.72004	0.270654	0.135327	+	0.990801i	$0.456791\pi$
$102$	0	0
$103$	−4.72249	−0.465321	−0.232661	−	0.972558i	$-0.574743\pi$
$103$	−4.72249	−0.465321	−0.232661	+	0.972558i	$0.574743\pi$
$104$	0	0
$105$	−0.458826	−0.0447769
$106$	0	0
$107$	−3.75246	−0.362764	−0.181382	−	0.983413i	$-0.558057\pi$
$107$	−3.75246	−0.362764	−0.181382	+	0.983413i	$0.558057\pi$
$108$	0	0
$109$	2.35204	0.225284	0.112642	−	0.993636i	$-0.464069\pi$
$109$	2.35204	0.225284	0.112642	+	0.993636i	$0.464069\pi$
$110$	0	0
$111$	8.36309	0.793789
$112$	0	0
$113$	−3.33799	−0.314012	−0.157006	−	0.987598i	$-0.550184\pi$
$113$	−3.33799	−0.314012	−0.157006	+	0.987598i	$0.550184\pi$
$114$	0	0
$115$	0.313582	0.0292417
$116$	0	0
$117$	1.92998	0.178427
$118$	0	0
$119$	4.15422	0.380816
$120$	0	0
$121$	13.8319	1.25745
$122$	0	0
$123$	−21.5010	−1.93868
$124$	0	0
$125$	1.42856	0.127774
$126$	0	0
$127$	6.58549	0.584368	0.292184	−	0.956362i	$-0.405618\pi$
$127$	6.58549	0.584368	0.292184	+	0.956362i	$0.405618\pi$
$128$	0	0
$129$	18.6386	1.64104
$130$	0	0
$131$	−11.7793	−1.02916	−0.514580	−	0.857443i	$-0.672052\pi$
$131$	−11.7793	−1.02916	−0.514580	+	0.857443i	$0.672052\pi$
$132$	0	0
$133$	2.53791	0.220065
$134$	0	0
$135$	−0.340096	−0.0292708
$136$	0	0
$137$	−5.75803	−0.491942	−0.245971	−	0.969277i	$-0.579107\pi$
$137$	−5.75803	−0.491942	−0.245971	+	0.969277i	$0.579107\pi$
$138$	0	0
$139$	−4.58588	−0.388969	−0.194485	−	0.980906i	$-0.562303\pi$
$139$	−4.58588	−0.388969	−0.194485	+	0.980906i	$0.562303\pi$
$140$	0	0
$141$	0.343114	0.0288954
$142$	0	0
$143$	−4.98317	−0.416713
$144$	0	0
$145$	0.143149	0.0118879
$146$	0	0
$147$	10.9155	0.900298
$148$	0	0
$149$	19.0788	1.56300	0.781499	−	0.623906i	$-0.214455\pi$
$149$	19.0788	1.56300	0.781499	+	0.623906i	$0.214455\pi$
$150$	0	0
$151$	3.14135	0.255640	0.127820	−	0.991797i	$-0.459202\pi$
$151$	3.14135	0.255640	0.127820	+	0.991797i	$0.459202\pi$
$152$	0	0
$153$	−5.55401	−0.449015
$154$	0	0
$155$	1.49221	0.119857
$156$	0	0
$157$	−7.53667	−0.601492	−0.300746	−	0.953704i	$-0.597236\pi$
$157$	−7.53667	−0.601492	−0.300746	+	0.953704i	$0.597236\pi$
$158$	0	0
$159$	−4.27265	−0.338843
$160$	0	0
$161$	3.16227	0.249222
$162$	0	0
$163$	13.0127	1.01923	0.509616	−	0.860402i	$-0.329787\pi$
$163$	13.0127	1.01923	0.509616	+	0.860402i	$0.329787\pi$
$164$	0	0
$165$	−1.58386	−0.123303
$166$	0	0
$167$	−4.35506	−0.337004	−0.168502	−	0.985701i	$-0.553893\pi$
$167$	−4.35506	−0.337004	−0.168502	+	0.985701i	$0.553893\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.39308	−0.259475
$172$	0	0
$173$	7.16628	0.544842	0.272421	−	0.962178i	$-0.412176\pi$
$173$	7.16628	0.544842	0.272421	+	0.962178i	$0.412176\pi$
$174$	0	0
$175$	7.18825	0.543380
$176$	0	0
$177$	−9.37381	−0.704579
$178$	0	0
$179$	−7.75789	−0.579852	−0.289926	−	0.957049i	$-0.593631\pi$
$179$	−7.75789	−0.579852	−0.289926	+	0.957049i	$0.593631\pi$
$180$	0	0
$181$	24.8281	1.84546	0.922729	−	0.385450i	$-0.125954\pi$
$181$	24.8281	1.84546	0.922729	+	0.385450i	$0.125954\pi$
$182$	0	0
$183$	0.662139	0.0489467
$184$	0	0
$185$	0.539179	0.0396412
$186$	0	0
$187$	14.3403	1.04867
$188$	0	0
$189$	−3.42965	−0.249470
$190$	0	0
$191$	−0.530841	−0.0384103	−0.0192051	−	0.999816i	$-0.506114\pi$
$191$	−0.530841	−0.0384103	−0.0192051	+	0.999816i	$0.506114\pi$
$192$	0	0
$193$	−11.2786	−0.811852	−0.405926	−	0.913906i	$-0.633051\pi$
$193$	−11.2786	−0.811852	−0.405926	+	0.913906i	$0.633051\pi$
$194$	0	0
$195$	0.317842	0.0227612
$196$	0	0
$197$	−23.6678	−1.68626	−0.843132	−	0.537706i	$-0.819291\pi$
$197$	−23.6678	−1.68626	−0.843132	+	0.537706i	$0.819291\pi$
$198$	0	0
$199$	19.2639	1.36558	0.682792	−	0.730613i	$-0.260766\pi$
$199$	19.2639	1.36558	0.682792	+	0.730613i	$0.260766\pi$
$200$	0	0
$201$	1.16114	0.0819006
$202$	0	0
$203$	1.44357	0.101318
$204$	0	0
$205$	−1.38620	−0.0968161
$206$	0	0
$207$	−4.22781	−0.293853
$208$	0	0
$209$	8.76084	0.606000
$210$	0	0
$211$	−7.72923	−0.532102	−0.266051	−	0.963959i	$-0.585719\pi$
$211$	−7.72923	−0.532102	−0.266051	+	0.963959i	$0.585719\pi$
$212$	0	0
$213$	12.5568	0.860379
$214$	0	0
$215$	1.20165	0.0819520
$216$	0	0
$217$	15.0480	1.02152
$218$	0	0
$219$	12.0069	0.811352
$220$	0	0
$221$	−2.87775	−0.193578
$222$	0	0
$223$	−12.5116	−0.837841	−0.418921	−	0.908023i	$-0.637591\pi$
$223$	−12.5116	−0.837841	−0.418921	+	0.908023i	$0.637591\pi$
$224$	0	0
$225$	−9.61038	−0.640692
$226$	0	0
$227$	10.6531	0.707069	0.353535	−	0.935421i	$-0.384980\pi$
$227$	10.6531	0.707069	0.353535	+	0.935421i	$0.384980\pi$
$228$	0	0
$229$	6.88031	0.454664	0.227332	−	0.973817i	$-0.427000\pi$
$229$	6.88031	0.454664	0.227332	+	0.973817i	$0.427000\pi$
$230$	0	0
$231$	−15.9722	−1.05089
$232$	0	0
$233$	−5.81280	−0.380809	−0.190405	−	0.981706i	$-0.560980\pi$
$233$	−5.81280	−0.380809	−0.190405	+	0.981706i	$0.560980\pi$
$234$	0	0
$235$	0.0221210	0.00144302
$236$	0	0
$237$	2.22208	0.144340
$238$	0	0
$239$	−10.7617	−0.696117	−0.348058	−	0.937473i	$-0.613159\pi$
$239$	−10.7617	−0.696117	−0.348058	+	0.937473i	$0.613159\pi$
$240$	0	0
$241$	18.7251	1.20619	0.603095	−	0.797670i	$-0.293934\pi$
$241$	18.7251	1.20619	0.603095	+	0.797670i	$0.293934\pi$
$242$	0	0
$243$	17.4411	1.11884
$244$	0	0
$245$	0.703738	0.0449602
$246$	0	0
$247$	−1.75809	−0.111864
$248$	0	0
$249$	0.855451	0.0542120
$250$	0	0
$251$	2.57962	0.162824	0.0814119	−	0.996681i	$-0.474057\pi$
$251$	2.57962	0.162824	0.0814119	+	0.996681i	$0.474057\pi$
$252$	0	0
$253$	10.9161	0.686289
$254$	0	0
$255$	−0.914670	−0.0572788
$256$	0	0
$257$	−1.43466	−0.0894919	−0.0447459	−	0.998998i	$-0.514248\pi$
$257$	−1.43466	−0.0894919	−0.0447459	+	0.998998i	$0.514248\pi$
$258$	0	0
$259$	5.43727	0.337855
$260$	0	0
$261$	−1.92998	−0.119463
$262$	0	0
$263$	1.79056	0.110411	0.0552054	−	0.998475i	$-0.482419\pi$
$263$	1.79056	0.110411	0.0552054	+	0.998475i	$0.482419\pi$
$264$	0	0
$265$	−0.275463	−0.0169216
$266$	0	0
$267$	32.2236	1.97205
$268$	0	0
$269$	−10.1277	−0.617498	−0.308749	−	0.951144i	$-0.599910\pi$
$269$	−10.1277	−0.617498	−0.308749	+	0.951144i	$0.599910\pi$
$270$	0	0
$271$	14.4219	0.876067	0.438034	−	0.898959i	$-0.355675\pi$
$271$	14.4219	0.876067	0.438034	+	0.898959i	$0.355675\pi$
$272$	0	0
$273$	3.20523	0.193989
$274$	0	0
$275$	24.8137	1.49632
$276$	0	0
$277$	5.57870	0.335192	0.167596	−	0.985856i	$-0.446400\pi$
$277$	5.57870	0.335192	0.167596	+	0.985856i	$0.446400\pi$
$278$	0	0
$279$	−20.1185	−1.20446
$280$	0	0
$281$	−22.8212	−1.36140	−0.680700	−	0.732562i	$-0.738324\pi$
$281$	−22.8212	−1.36140	−0.680700	+	0.732562i	$0.738324\pi$
$282$	0	0
$283$	20.3066	1.20710	0.603552	−	0.797323i	$-0.293751\pi$
$283$	20.3066	1.20710	0.603552	+	0.797323i	$0.293751\pi$
$284$	0	0
$285$	−0.558794	−0.0331001
$286$	0	0
$287$	−13.9789	−0.825146
$288$	0	0
$289$	−8.71857	−0.512857
$290$	0	0
$291$	13.2019	0.773906
$292$	0	0
$293$	−13.1844	−0.770242	−0.385121	−	0.922866i	$-0.625840\pi$
$293$	−13.1844	−0.770242	−0.385121	+	0.922866i	$0.625840\pi$
$294$	0	0
$295$	−0.604342	−0.0351861
$296$	0	0
$297$	−11.8391	−0.686973
$298$	0	0
$299$	−2.19059	−0.126685
$300$	0	0
$301$	12.1179	0.698463
$302$	0	0
$303$	−6.03946	−0.346958
$304$	0	0
$305$	0.0426889	0.00244436
$306$	0	0
$307$	28.8033	1.64389	0.821944	−	0.569568i	$-0.192889\pi$
$307$	28.8033	1.64389	0.821944	+	0.569568i	$0.192889\pi$
$308$	0	0
$309$	10.4856	0.596506
$310$	0	0
$311$	−23.7557	−1.34706	−0.673530	−	0.739160i	$-0.735223\pi$
$311$	−23.7557	−1.34706	−0.673530	+	0.739160i	$0.735223\pi$
$312$	0	0
$313$	−0.532605	−0.0301046	−0.0150523	−	0.999887i	$-0.504791\pi$
$313$	−0.532605	−0.0301046	−0.0150523	+	0.999887i	$0.504791\pi$
$314$	0	0
$315$	0.398822	0.0224711
$316$	0	0
$317$	23.2541	1.30608	0.653040	−	0.757323i	$-0.273493\pi$
$317$	23.2541	1.30608	0.653040	+	0.757323i	$0.273493\pi$
$318$	0	0
$319$	4.98317	0.279004
$320$	0	0
$321$	8.33180	0.465036
$322$	0	0
$323$	5.05933	0.281509
$324$	0	0
$325$	−4.97951	−0.276213
$326$	0	0
$327$	−5.22236	−0.288797
$328$	0	0
$329$	0.223076	0.0122986
$330$	0	0
$331$	19.6129	1.07802	0.539011	−	0.842299i	$-0.318798\pi$
$331$	19.6129	1.07802	0.539011	+	0.842299i	$0.318798\pi$
$332$	0	0
$333$	−7.26939	−0.398360
$334$	0	0
$335$	0.0748603	0.00409005
$336$	0	0
$337$	4.57765	0.249361	0.124680	−	0.992197i	$-0.460209\pi$
$337$	4.57765	0.249361	0.124680	+	0.992197i	$0.460209\pi$
$338$	0	0
$339$	7.41153	0.402539
$340$	0	0
$341$	51.9454	2.81300
$342$	0	0
$343$	17.2017	0.928805
$344$	0	0
$345$	−0.696264	−0.0374856
$346$	0	0
$347$	−2.32897	−0.125026	−0.0625128	−	0.998044i	$-0.519911\pi$
$347$	−2.32897	−0.125026	−0.0625128	+	0.998044i	$0.519911\pi$
$348$	0	0
$349$	−27.0861	−1.44989	−0.724944	−	0.688808i	$-0.758134\pi$
$349$	−27.0861	−1.44989	−0.724944	+	0.688808i	$0.758134\pi$
$350$	0	0
$351$	2.37582	0.126812
$352$	0	0
$353$	−18.4406	−0.981492	−0.490746	−	0.871303i	$-0.663276\pi$
$353$	−18.4406	−0.981492	−0.490746	+	0.871303i	$0.663276\pi$
$354$	0	0
$355$	0.809554	0.0429667
$356$	0	0
$357$	−9.22384	−0.488178
$358$	0	0
$359$	2.53886	0.133996	0.0669979	−	0.997753i	$-0.478658\pi$
$359$	2.53886	0.133996	0.0669979	+	0.997753i	$0.478658\pi$
$360$	0	0
$361$	−15.9091	−0.837323
$362$	0	0
$363$	−30.7118	−1.61195
$364$	0	0
$365$	0.774101	0.0405183
$366$	0	0
$367$	5.72207	0.298690	0.149345	−	0.988785i	$-0.452284\pi$
$367$	5.72207	0.298690	0.149345	+	0.988785i	$0.452284\pi$
$368$	0	0
$369$	18.6891	0.972918
$370$	0	0
$371$	−2.77787	−0.144220
$372$	0	0
$373$	7.80105	0.403923	0.201962	−	0.979393i	$-0.435268\pi$
$373$	7.80105	0.403923	0.201962	+	0.979393i	$0.435268\pi$
$374$	0	0
$375$	−3.17191	−0.163797
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	34.5290	1.77364	0.886818	−	0.462120i	$-0.152911\pi$
$379$	34.5290	1.77364	0.886818	+	0.462120i	$0.152911\pi$
$380$	0	0
$381$	−14.6221	−0.749115
$382$	0	0
$383$	5.27441	0.269510	0.134755	−	0.990879i	$-0.456975\pi$
$383$	5.27441	0.269510	0.134755	+	0.990879i	$0.456975\pi$
$384$	0	0
$385$	−1.02975	−0.0524808
$386$	0	0
$387$	−16.2011	−0.823547
$388$	0	0
$389$	−7.22981	−0.366566	−0.183283	−	0.983060i	$-0.558672\pi$
$389$	−7.22981	−0.366566	−0.183283	+	0.983060i	$0.558672\pi$
$390$	0	0
$391$	6.30398	0.318806
$392$	0	0
$393$	26.1542	1.31930
$394$	0	0
$395$	0.143260	0.00720821
$396$	0	0
$397$	29.2591	1.46847	0.734237	−	0.678893i	$-0.237540\pi$
$397$	29.2591	1.46847	0.734237	+	0.678893i	$0.237540\pi$
$398$	0	0
$399$	−5.63507	−0.282107
$400$	0	0
$401$	−4.20876	−0.210175	−0.105088	−	0.994463i	$-0.533512\pi$
$401$	−4.20876	−0.210175	−0.105088	+	0.994463i	$0.533512\pi$
$402$	0	0
$403$	−10.4242	−0.519265
$404$	0	0
$405$	1.58396	0.0787077
$406$	0	0
$407$	18.7694	0.930362
$408$	0	0
$409$	30.6676	1.51642	0.758208	−	0.652013i	$-0.226075\pi$
$409$	30.6676	1.51642	0.758208	+	0.652013i	$0.226075\pi$
$410$	0	0
$411$	12.7849	0.630632
$412$	0	0
$413$	−6.09439	−0.299885
$414$	0	0
$415$	0.0551520	0.00270731
$416$	0	0
$417$	10.1823	0.498629
$418$	0	0
$419$	0.630109	0.0307829	0.0153914	−	0.999882i	$-0.495101\pi$
$419$	0.630109	0.0307829	0.0153914	+	0.999882i	$0.495101\pi$
$420$	0	0
$421$	−26.1924	−1.27654	−0.638271	−	0.769812i	$-0.720350\pi$
$421$	−26.1924	−1.27654	−0.638271	+	0.769812i	$0.720350\pi$
$422$	0	0
$423$	−0.298243	−0.0145011
$424$	0	0
$425$	14.3298	0.695096
$426$	0	0
$427$	0.430490	0.0208329
$428$	0	0
$429$	11.0644	0.534195
$430$	0	0
$431$	−6.49939	−0.313065	−0.156532	−	0.987673i	$-0.550032\pi$
$431$	−6.49939	−0.313065	−0.156532	+	0.987673i	$0.550032\pi$
$432$	0	0
$433$	2.82879	0.135943	0.0679715	−	0.997687i	$-0.478347\pi$
$433$	2.82879	0.135943	0.0679715	+	0.997687i	$0.478347\pi$
$434$	0	0
$435$	−0.317842	−0.0152394
$436$	0	0
$437$	3.85126	0.184231
$438$	0	0
$439$	15.6483	0.746855	0.373427	−	0.927659i	$-0.378182\pi$
$439$	15.6483	0.746855	0.373427	+	0.927659i	$0.378182\pi$
$440$	0	0
$441$	−9.48803	−0.451811
$442$	0	0
$443$	−0.911226	−0.0432937	−0.0216468	−	0.999766i	$-0.506891\pi$
$443$	−0.911226	−0.0432937	−0.0216468	+	0.999766i	$0.506891\pi$
$444$	0	0
$445$	2.07750	0.0984827
$446$	0	0
$447$	−42.3618	−2.00364
$448$	0	0
$449$	21.4359	1.01162	0.505812	−	0.862644i	$-0.331193\pi$
$449$	21.4359	1.01162	0.505812	+	0.862644i	$0.331193\pi$
$450$	0	0
$451$	−48.2548	−2.27223
$452$	0	0
$453$	−6.97493	−0.327711
$454$	0	0
$455$	0.206645	0.00968768
$456$	0	0
$457$	35.8383	1.67645	0.838223	−	0.545328i	$-0.183595\pi$
$457$	35.8383	1.67645	0.838223	+	0.545328i	$0.183595\pi$
$458$	0	0
$459$	−6.83699	−0.319124
$460$	0	0
$461$	−20.5466	−0.956951	−0.478476	−	0.878101i	$-0.658810\pi$
$461$	−20.5466	−0.956951	−0.478476	+	0.878101i	$0.658810\pi$
$462$	0	0
$463$	34.6590	1.61074	0.805369	−	0.592773i	$-0.201967\pi$
$463$	34.6590	1.61074	0.805369	+	0.592773i	$0.201967\pi$
$464$	0	0
$465$	−3.31324	−0.153648
$466$	0	0
$467$	5.58263	0.258333	0.129167	−	0.991623i	$-0.458770\pi$
$467$	5.58263	0.258333	0.129167	+	0.991623i	$0.458770\pi$
$468$	0	0
$469$	0.754916	0.0348588
$470$	0	0
$471$	16.7341	0.771066
$472$	0	0
$473$	41.8307	1.92338
$474$	0	0
$475$	8.75441	0.401680
$476$	0	0
$477$	3.71389	0.170047
$478$	0	0
$479$	−28.2843	−1.29234	−0.646171	−	0.763193i	$-0.723631\pi$
$479$	−28.2843	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$480$	0	0
$481$	−3.76655	−0.171740
$482$	0	0
$483$	−7.02136	−0.319483
$484$	0	0
$485$	0.851140	0.0386483
$486$	0	0
$487$	26.0050	1.17840	0.589200	−	0.807987i	$-0.299443\pi$
$487$	26.0050	1.17840	0.589200	+	0.807987i	$0.299443\pi$
$488$	0	0
$489$	−28.8928	−1.30658
$490$	0	0
$491$	7.84471	0.354027	0.177013	−	0.984208i	$-0.443356\pi$
$491$	7.84471	0.354027	0.177013	+	0.984208i	$0.443356\pi$
$492$	0	0
$493$	2.87775	0.129607
$494$	0	0
$495$	1.37673	0.0618793
$496$	0	0
$497$	8.16382	0.366197
$498$	0	0
$499$	15.2694	0.683552	0.341776	−	0.939781i	$-0.388972\pi$
$499$	15.2694	0.683552	0.341776	+	0.939781i	$0.388972\pi$
$500$	0	0
$501$	9.66978	0.432014
$502$	0	0
$503$	15.3409	0.684018	0.342009	−	0.939697i	$-0.388893\pi$
$503$	15.3409	0.684018	0.342009	+	0.939697i	$0.388893\pi$
$504$	0	0
$505$	−0.389372	−0.0173268
$506$	0	0
$507$	−2.22036	−0.0986095
$508$	0	0
$509$	22.1921	0.983645	0.491823	−	0.870695i	$-0.336331\pi$
$509$	22.1921	0.983645	0.491823	+	0.870695i	$0.336331\pi$
$510$	0	0
$511$	7.80630	0.345330
$512$	0	0
$513$	−4.17689	−0.184414
$514$	0	0
$515$	0.676021	0.0297891
$516$	0	0
$517$	0.770054	0.0338669
$518$	0	0
$519$	−15.9117	−0.698446
$520$	0	0
$521$	34.1380	1.49561	0.747807	−	0.663917i	$-0.231107\pi$
$521$	34.1380	1.49561	0.747807	+	0.663917i	$0.231107\pi$
$522$	0	0
$523$	23.6732	1.03516	0.517579	−	0.855635i	$-0.326833\pi$
$523$	23.6732	1.03516	0.517579	+	0.855635i	$0.326833\pi$
$524$	0	0
$525$	−15.9605	−0.696572
$526$	0	0
$527$	29.9981	1.30674
$528$	0	0
$529$	−18.2013	−0.791361
$530$	0	0
$531$	8.14793	0.353590
$532$	0	0
$533$	9.68357	0.419442
$534$	0	0
$535$	0.537162	0.0232235
$536$	0	0
$537$	17.2253	0.743326
$538$	0	0
$539$	24.4978	1.05520
$540$	0	0
$541$	−19.5389	−0.840042	−0.420021	−	0.907514i	$-0.637977\pi$
$541$	−19.5389	−0.840042	−0.420021	+	0.907514i	$0.637977\pi$
$542$	0	0
$543$	−55.1272	−2.36574
$544$	0	0
$545$	−0.336692	−0.0144223
$546$	0	0
$547$	−24.1641	−1.03318	−0.516591	−	0.856232i	$-0.672799\pi$
$547$	−24.1641	−1.03318	−0.516591	+	0.856232i	$0.672799\pi$
$548$	0	0
$549$	−0.575546	−0.0245637
$550$	0	0
$551$	1.75809	0.0748970
$552$	0	0
$553$	1.44469	0.0614343
$554$	0	0
$555$	−1.19717	−0.0508170
$556$	0	0
$557$	22.7780	0.965137	0.482568	−	0.875858i	$-0.339704\pi$
$557$	22.7780	0.965137	0.482568	+	0.875858i	$0.339704\pi$
$558$	0	0
$559$	−8.39441	−0.355046
$560$	0	0
$561$	−31.8406	−1.34431
$562$	0	0
$563$	1.60288	0.0675532	0.0337766	−	0.999429i	$-0.489247\pi$
$563$	1.60288	0.0675532	0.0337766	+	0.999429i	$0.489247\pi$
$564$	0	0
$565$	0.477831	0.0201025
$566$	0	0
$567$	15.9732	0.670812
$568$	0	0
$569$	−44.5585	−1.86799	−0.933994	−	0.357289i	$-0.883701\pi$
$569$	−44.5585	−1.86799	−0.933994	+	0.357289i	$0.883701\pi$
$570$	0	0
$571$	13.0843	0.547559	0.273780	−	0.961792i	$-0.411726\pi$
$571$	13.0843	0.547559	0.273780	+	0.961792i	$0.411726\pi$
$572$	0	0
$573$	1.17866	0.0492391
$574$	0	0
$575$	10.9081	0.454899
$576$	0	0
$577$	34.1108	1.42005	0.710026	−	0.704175i	$-0.248683\pi$
$577$	34.1108	1.42005	0.710026	+	0.704175i	$0.248683\pi$
$578$	0	0
$579$	25.0425	1.04073
$580$	0	0
$581$	0.556172	0.0230739
$582$	0	0
$583$	−9.58914	−0.397142
$584$	0	0
$585$	−0.276276	−0.0114226
$586$	0	0
$587$	19.9004	0.821376	0.410688	−	0.911776i	$-0.365289\pi$
$587$	19.9004	0.821376	0.410688	+	0.911776i	$0.365289\pi$
$588$	0	0
$589$	18.3266	0.755134
$590$	0	0
$591$	52.5511	2.16166
$592$	0	0
$593$	−33.6155	−1.38042	−0.690212	−	0.723607i	$-0.742483\pi$
$593$	−33.6155	−1.38042	−0.690212	+	0.723607i	$0.742483\pi$
$594$	0	0
$595$	−0.594673	−0.0243792
$596$	0	0
$597$	−42.7728	−1.75057
$598$	0	0
$599$	−22.0826	−0.902270	−0.451135	−	0.892456i	$-0.648981\pi$
$599$	−22.0826	−0.902270	−0.451135	+	0.892456i	$0.648981\pi$
$600$	0	0
$601$	−44.1656	−1.80155	−0.900776	−	0.434284i	$-0.857001\pi$
$601$	−44.1656	−1.80155	−0.900776	+	0.434284i	$0.857001\pi$
$602$	0	0
$603$	−1.00929	−0.0411015
$604$	0	0
$605$	−1.98003	−0.0804997
$606$	0	0
$607$	45.1920	1.83429	0.917143	−	0.398558i	$-0.130489\pi$
$607$	45.1920	1.83429	0.917143	+	0.398558i	$0.130489\pi$
$608$	0	0
$609$	−3.20523	−0.129883
$610$	0	0
$611$	−0.154531	−0.00625166
$612$	0	0
$613$	−28.1228	−1.13587	−0.567935	−	0.823073i	$-0.692258\pi$
$613$	−28.1228	−1.13587	−0.567935	+	0.823073i	$0.692258\pi$
$614$	0	0
$615$	3.07785	0.124111
$616$	0	0
$617$	−4.00914	−0.161402	−0.0807010	−	0.996738i	$-0.525716\pi$
$617$	−4.00914	−0.161402	−0.0807010	+	0.996738i	$0.525716\pi$
$618$	0	0
$619$	12.8380	0.516003	0.258001	−	0.966145i	$-0.416936\pi$
$619$	12.8380	0.516003	0.258001	+	0.966145i	$0.416936\pi$
$620$	0	0
$621$	−5.20445	−0.208847
$622$	0	0
$623$	20.9502	0.839351
$624$	0	0
$625$	24.6930	0.987722
$626$	0	0
$627$	−19.4522	−0.776845
$628$	0	0
$629$	10.8392	0.432186
$630$	0	0
$631$	−36.8073	−1.46527	−0.732637	−	0.680619i	$-0.761711\pi$
$631$	−36.8073	−1.46527	−0.732637	+	0.680619i	$0.761711\pi$
$632$	0	0
$633$	17.1617	0.682115
$634$	0	0
$635$	−0.942709	−0.0374102
$636$	0	0
$637$	−4.91612	−0.194784
$638$	0	0
$639$	−10.9147	−0.431778
$640$	0	0
$641$	−19.4670	−0.768902	−0.384451	−	0.923145i	$-0.625609\pi$
$641$	−19.4670	−0.768902	−0.384451	+	0.923145i	$0.625609\pi$
$642$	0	0
$643$	42.4337	1.67342	0.836711	−	0.547645i	$-0.184476\pi$
$643$	42.4337	1.67342	0.836711	+	0.547645i	$0.184476\pi$
$644$	0	0
$645$	−2.66810	−0.105056
$646$	0	0
$647$	−27.7495	−1.09095	−0.545473	−	0.838128i	$-0.683650\pi$
$647$	−27.7495	−1.09095	−0.545473	+	0.838128i	$0.683650\pi$
$648$	0	0
$649$	−21.0377	−0.825803
$650$	0	0
$651$	−33.4119	−1.30951
$652$	0	0
$653$	45.3331	1.77402	0.887011	−	0.461749i	$-0.152778\pi$
$653$	45.3331	1.77402	0.887011	+	0.461749i	$0.152778\pi$
$654$	0	0
$655$	1.68619	0.0658850
$656$	0	0
$657$	−10.4367	−0.407174
$658$	0	0
$659$	−28.7137	−1.11853	−0.559263	−	0.828990i	$-0.688916\pi$
$659$	−28.7137	−1.11853	−0.559263	+	0.828990i	$0.688916\pi$
$660$	0	0
$661$	−21.4482	−0.834238	−0.417119	−	0.908852i	$-0.636960\pi$
$661$	−21.4482	−0.834238	−0.417119	+	0.908852i	$0.636960\pi$
$662$	0	0
$663$	6.38962	0.248152
$664$	0	0
$665$	−0.363300	−0.0140882
$666$	0	0
$667$	2.19059	0.0848202
$668$	0	0
$669$	27.7803	1.07405
$670$	0	0
$671$	1.48604	0.0573681
$672$	0	0
$673$	−17.3517	−0.668860	−0.334430	−	0.942421i	$-0.608544\pi$
$673$	−17.3517	−0.668860	−0.334430	+	0.942421i	$0.608544\pi$
$674$	0	0
$675$	−11.8304	−0.455352
$676$	0	0
$677$	−6.77037	−0.260206	−0.130103	−	0.991500i	$-0.541531\pi$
$677$	−6.77037	−0.260206	−0.130103	+	0.991500i	$0.541531\pi$
$678$	0	0
$679$	8.58319	0.329393
$680$	0	0
$681$	−23.6536	−0.906409
$682$	0	0
$683$	−9.75697	−0.373340	−0.186670	−	0.982423i	$-0.559770\pi$
$683$	−9.75697	−0.373340	−0.186670	+	0.982423i	$0.559770\pi$
$684$	0	0
$685$	0.824257	0.0314933
$686$	0	0
$687$	−15.2767	−0.582844
$688$	0	0
$689$	1.92431	0.0733103
$690$	0	0
$691$	−37.9866	−1.44508	−0.722538	−	0.691331i	$-0.757025\pi$
$691$	−37.9866	−1.44508	−0.722538	+	0.691331i	$0.757025\pi$
$692$	0	0
$693$	13.8834	0.527387
$694$	0	0
$695$	0.656465	0.0249011
$696$	0	0
$697$	−27.8669	−1.05553
$698$	0	0
$699$	12.9065	0.488169
$700$	0	0
$701$	13.3091	0.502677	0.251338	−	0.967899i	$-0.419129\pi$
$701$	13.3091	0.502677	0.251338	+	0.967899i	$0.419129\pi$
$702$	0	0
$703$	6.62192	0.249751
$704$	0	0
$705$	−0.0491165	−0.00184984
$706$	0	0
$707$	−3.92656	−0.147673
$708$	0	0
$709$	13.4338	0.504517	0.252259	−	0.967660i	$-0.418827\pi$
$709$	13.4338	0.504517	0.252259	+	0.967660i	$0.418827\pi$
$710$	0	0
$711$	−1.93148	−0.0724363
$712$	0	0
$713$	22.8351	0.855182
$714$	0	0
$715$	0.713336	0.0266773
$716$	0	0
$717$	23.8948	0.892368
$718$	0	0
$719$	29.1738	1.08800	0.544000	−	0.839085i	$-0.316909\pi$
$719$	29.1738	1.08800	0.544000	+	0.839085i	$0.316909\pi$
$720$	0	0
$721$	6.81723	0.253887
$722$	0	0
$723$	−41.5764	−1.54624
$724$	0	0
$725$	4.97951	0.184934
$726$	0	0
$727$	−18.3875	−0.681954	−0.340977	−	0.940072i	$-0.610758\pi$
$727$	−18.3875	−0.681954	−0.340977	+	0.940072i	$0.610758\pi$
$728$	0	0
$729$	−5.53003	−0.204816
$730$	0	0
$731$	24.1570	0.893478
$732$	0	0
$733$	−31.1361	−1.15004	−0.575019	−	0.818140i	$-0.695006\pi$
$733$	−31.1361	−1.15004	−0.575019	+	0.818140i	$0.695006\pi$
$734$	0	0
$735$	−1.56255	−0.0576355
$736$	0	0
$737$	2.60596	0.0959917
$738$	0	0
$739$	−22.8661	−0.841144	−0.420572	−	0.907259i	$-0.638171\pi$
$739$	−22.8661	−0.841144	−0.420572	+	0.907259i	$0.638171\pi$
$740$	0	0
$741$	3.90358	0.143402
$742$	0	0
$743$	−6.69152	−0.245488	−0.122744	−	0.992438i	$-0.539169\pi$
$743$	−6.69152	−0.245488	−0.122744	+	0.992438i	$0.539169\pi$
$744$	0	0
$745$	−2.73112	−0.100060
$746$	0	0
$747$	−0.743577	−0.0272061
$748$	0	0
$749$	5.41693	0.197930
$750$	0	0
$751$	−19.6880	−0.718427	−0.359213	−	0.933255i	$-0.616955\pi$
$751$	−19.6880	−0.718427	−0.359213	+	0.933255i	$0.616955\pi$
$752$	0	0
$753$	−5.72767	−0.208728
$754$	0	0
$755$	−0.449682	−0.0163656
$756$	0	0
$757$	−9.68857	−0.352137	−0.176068	−	0.984378i	$-0.556338\pi$
$757$	−9.68857	−0.352137	−0.176068	+	0.984378i	$0.556338\pi$
$758$	0	0
$759$	−24.2376	−0.879770
$760$	0	0
$761$	10.6701	0.386789	0.193395	−	0.981121i	$-0.438050\pi$
$761$	10.6701	0.386789	0.193395	+	0.981121i	$0.438050\pi$
$762$	0	0
$763$	−3.39532	−0.122919
$764$	0	0
$765$	0.795052	0.0287452
$766$	0	0
$767$	4.22176	0.152439
$768$	0	0
$769$	−3.03078	−0.109293	−0.0546464	−	0.998506i	$-0.517403\pi$
$769$	−3.03078	−0.109293	−0.0546464	+	0.998506i	$0.517403\pi$
$770$	0	0
$771$	3.18547	0.114722
$772$	0	0
$773$	8.34440	0.300127	0.150064	−	0.988676i	$-0.452052\pi$
$773$	8.34440	0.300127	0.150064	+	0.988676i	$0.452052\pi$
$774$	0	0
$775$	51.9072	1.86456
$776$	0	0
$777$	−12.0727	−0.433105
$778$	0	0
$779$	−17.0245	−0.609968
$780$	0	0
$781$	28.1814	1.00841
$782$	0	0
$783$	−2.37582	−0.0849047
$784$	0	0
$785$	1.07887	0.0385065
$786$	0	0
$787$	−2.40470	−0.0857183	−0.0428591	−	0.999081i	$-0.513647\pi$
$787$	−2.40470	−0.0857183	−0.0428591	+	0.999081i	$0.513647\pi$
$788$	0	0
$789$	−3.97569	−0.141538
$790$	0	0
$791$	4.81861	0.171330
$792$	0	0
$793$	−0.298213	−0.0105899
$794$	0	0
$795$	0.611627	0.0216922
$796$	0	0
$797$	−43.8307	−1.55256	−0.776282	−	0.630386i	$-0.782897\pi$
$797$	−43.8307	−1.55256	−0.776282	+	0.630386i	$0.782897\pi$
$798$	0	0
$799$	0.444701	0.0157324
$800$	0	0
$801$	−28.0095	−0.989666
$802$	0	0
$803$	26.9472	0.950946
$804$	0	0
$805$	−0.452676	−0.0159547
$806$	0	0
$807$	22.4872	0.791585
$808$	0	0
$809$	−14.0772	−0.494929	−0.247464	−	0.968897i	$-0.579597\pi$
$809$	−14.0772	−0.494929	−0.247464	+	0.968897i	$0.579597\pi$
$810$	0	0
$811$	21.0679	0.739794	0.369897	−	0.929073i	$-0.379393\pi$
$811$	21.0679	0.739794	0.369897	+	0.929073i	$0.379393\pi$
$812$	0	0
$813$	−32.0217	−1.12305
$814$	0	0
$815$	−1.86276	−0.0652495
$816$	0	0
$817$	14.7581	0.516320
$818$	0	0
$819$	−2.78606	−0.0973528
$820$	0	0
$821$	28.8449	1.00669	0.503347	−	0.864084i	$-0.332102\pi$
$821$	28.8449	1.00669	0.503347	+	0.864084i	$0.332102\pi$
$822$	0	0
$823$	−7.88419	−0.274826	−0.137413	−	0.990514i	$-0.543879\pi$
$823$	−7.88419	−0.274826	−0.137413	+	0.990514i	$0.543879\pi$
$824$	0	0
$825$	−55.0953	−1.91817
$826$	0	0
$827$	−27.4360	−0.954042	−0.477021	−	0.878892i	$-0.658283\pi$
$827$	−27.4360	−0.954042	−0.477021	+	0.878892i	$0.658283\pi$
$828$	0	0
$829$	−38.8695	−1.34999	−0.674996	−	0.737821i	$-0.735855\pi$
$829$	−38.8695	−1.34999	−0.674996	+	0.737821i	$0.735855\pi$
$830$	0	0
$831$	−12.3867	−0.429690
$832$	0	0
$833$	14.1473	0.490176
$834$	0	0
$835$	0.623423	0.0215744
$836$	0	0
$837$	−24.7659	−0.856035
$838$	0	0
$839$	54.3186	1.87529	0.937644	−	0.347597i	$-0.113002\pi$
$839$	54.3186	1.87529	0.937644	+	0.347597i	$0.113002\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	50.6713	1.74521
$844$	0	0
$845$	−0.143149	−0.00492448
$846$	0	0
$847$	−19.9673	−0.686085
$848$	0	0
$849$	−45.0880	−1.54742
$850$	0	0
$851$	8.25099	0.282840
$852$	0	0
$853$	−33.3800	−1.14291	−0.571454	−	0.820634i	$-0.693621\pi$
$853$	−33.3800	−1.14291	−0.571454	+	0.820634i	$0.693621\pi$
$854$	0	0
$855$	0.485717	0.0166112
$856$	0	0
$857$	−3.84410	−0.131312	−0.0656560	−	0.997842i	$-0.520914\pi$
$857$	−3.84410	−0.131312	−0.0656560	+	0.997842i	$0.520914\pi$
$858$	0	0
$859$	0.00794910	0.000271220 0	0.000135610	−	1.00000i	$-0.499957\pi$
$859$	0.00794910	0.000271220 0	0.000135610		1.00000i	$0.499957\pi$
$860$	0	0
$861$	31.0381	1.05777
$862$	0	0
$863$	−20.0505	−0.682527	−0.341263	−	0.939968i	$-0.610855\pi$
$863$	−20.0505	−0.682527	−0.341263	+	0.939968i	$0.610855\pi$
$864$	0	0
$865$	−1.02585	−0.0348799
$866$	0	0
$867$	19.3583	0.657444
$868$	0	0
$869$	4.98703	0.169173
$870$	0	0
$871$	−0.522953	−0.0177196
$872$	0	0
$873$	−11.4754	−0.388382
$874$	0	0
$875$	−2.06222	−0.0697157
$876$	0	0
$877$	−16.7122	−0.564330	−0.282165	−	0.959366i	$-0.591053\pi$
$877$	−16.7122	−0.564330	−0.282165	+	0.959366i	$0.591053\pi$
$878$	0	0
$879$	29.2741	0.987391
$880$	0	0
$881$	−36.8756	−1.24237	−0.621185	−	0.783664i	$-0.713349\pi$
$881$	−36.8756	−1.24237	−0.621185	+	0.783664i	$0.713349\pi$
$882$	0	0
$883$	44.9694	1.51334	0.756671	−	0.653796i	$-0.226824\pi$
$883$	44.9694	1.51334	0.756671	+	0.653796i	$0.226824\pi$
$884$	0	0
$885$	1.34185	0.0451059
$886$	0	0
$887$	−9.28862	−0.311881	−0.155941	−	0.987766i	$-0.549841\pi$
$887$	−9.28862	−0.311881	−0.155941	+	0.987766i	$0.549841\pi$
$888$	0	0
$889$	−9.50660	−0.318841
$890$	0	0
$891$	55.1393	1.84724
$892$	0	0
$893$	0.271679	0.00909139
$894$	0	0
$895$	1.11054	0.0371211
$896$	0	0
$897$	4.86390	0.162401
$898$	0	0
$899$	10.4242	0.347665
$900$	0	0
$901$	−5.53767	−0.184487
$902$	0	0
$903$	−26.9060	−0.895376
$904$	0	0
$905$	−3.55412	−0.118143
$906$	0	0
$907$	7.69143	0.255390	0.127695	−	0.991813i	$-0.459242\pi$
$907$	7.69143	0.255390	0.127695	+	0.991813i	$0.459242\pi$
$908$	0	0
$909$	5.24964	0.174119
$910$	0	0
$911$	6.20928	0.205723	0.102861	−	0.994696i	$-0.467200\pi$
$911$	6.20928	0.205723	0.102861	+	0.994696i	$0.467200\pi$
$912$	0	0
$913$	1.91990	0.0635393
$914$	0	0
$915$	−0.0947847	−0.00313349
$916$	0	0
$917$	17.0041	0.561526
$918$	0	0
$919$	−22.3174	−0.736184	−0.368092	−	0.929789i	$-0.619989\pi$
$919$	−22.3174	−0.736184	−0.368092	+	0.929789i	$0.619989\pi$
$920$	0	0
$921$	−63.9535	−2.10734
$922$	0	0
$923$	−5.65532	−0.186147
$924$	0	0
$925$	18.7556	0.616680
$926$	0	0
$927$	−9.11434	−0.299354
$928$	0	0
$929$	−23.8221	−0.781577	−0.390788	−	0.920481i	$-0.627798\pi$
$929$	−23.8221	−0.781577	−0.390788	+	0.920481i	$0.627798\pi$
$930$	0	0
$931$	8.64296	0.283262
$932$	0	0
$933$	52.7461	1.72683
$934$	0	0
$935$	−2.05280	−0.0671338
$936$	0	0
$937$	−29.3747	−0.959630	−0.479815	−	0.877370i	$-0.659296\pi$
$937$	−29.3747	−0.959630	−0.479815	+	0.877370i	$0.659296\pi$
$938$	0	0
$939$	1.18257	0.0385918
$940$	0	0
$941$	−44.3441	−1.44558	−0.722788	−	0.691070i	$-0.757140\pi$
$941$	−44.3441	−1.44558	−0.722788	+	0.691070i	$0.757140\pi$
$942$	0	0
$943$	−21.2128	−0.690783
$944$	0	0
$945$	0.490951	0.0159706
$946$	0	0
$947$	−19.1001	−0.620671	−0.310335	−	0.950627i	$-0.600441\pi$
$947$	−19.1001	−0.620671	−0.310335	+	0.950627i	$0.600441\pi$
$948$	0	0
$949$	−5.40765	−0.175540
$950$	0	0
$951$	−51.6324	−1.67429
$952$	0	0
$953$	−19.5997	−0.634898	−0.317449	−	0.948275i	$-0.602826\pi$
$953$	−19.5997	−0.634898	−0.317449	+	0.948275i	$0.602826\pi$
$954$	0	0
$955$	0.0759895	0.00245896
$956$	0	0
$957$	−11.0644	−0.357661
$958$	0	0
$959$	8.31209	0.268412
$960$	0	0
$961$	77.6633	2.50527
$962$	0	0
$963$	−7.24219	−0.233376
$964$	0	0
$965$	1.61452	0.0519734
$966$	0	0
$967$	−21.7778	−0.700327	−0.350164	−	0.936689i	$-0.613874\pi$
$967$	−21.7778	−0.700327	−0.350164	+	0.936689i	$0.613874\pi$
$968$	0	0
$969$	−11.2335	−0.360872
$970$	0	0
$971$	−53.8569	−1.72835	−0.864175	−	0.503191i	$-0.832159\pi$
$971$	−53.8569	−1.72835	−0.864175	+	0.503191i	$0.832159\pi$
$972$	0	0
$973$	6.62002	0.212228
$974$	0	0
$975$	11.0563	0.354085
$976$	0	0
$977$	0.568046	0.0181734	0.00908670	−	0.999959i	$-0.497108\pi$
$977$	0.568046	0.0181734	0.00908670	+	0.999959i	$0.497108\pi$
$978$	0	0
$979$	72.3196	2.31135
$980$	0	0
$981$	4.53940	0.144932
$982$	0	0
$983$	−54.0551	−1.72409	−0.862045	−	0.506831i	$-0.830817\pi$
$983$	−54.0551	−1.72409	−0.862045	+	0.506831i	$0.830817\pi$
$984$	0	0
$985$	3.38803	0.107952
$986$	0	0
$987$	−0.495308	−0.0157658
$988$	0	0
$989$	18.3887	0.584728
$990$	0	0
$991$	−32.9337	−1.04617	−0.523087	−	0.852279i	$-0.675220\pi$
$991$	−32.9337	−1.04617	−0.523087	+	0.852279i	$0.675220\pi$
$992$	0	0
$993$	−43.5476	−1.38194
$994$	0	0
$995$	−2.75762	−0.0874223
$996$	0	0
$997$	−48.3674	−1.53181	−0.765905	−	0.642954i	$-0.777709\pi$
$997$	−48.3674	−1.53181	−0.765905	+	0.642954i	$0.777709\pi$
$998$	0	0
$999$	−8.94863	−0.283122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.ba.1.3		10
4.3	odd	2		3016.2.a.g.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.g.1.8	✓	10	4.3	odd	2
6032.2.a.ba.1.3		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22036 q^{3} -0.143149 q^{5} -1.44357 q^{7} +1.92998 q^{9} +O(q^{10})\)	\(q-2.22036 q^{3} -0.143149 q^{5} -1.44357 q^{7} +1.92998 q^{9} -4.98317 q^{11} +1.00000 q^{13} +0.317842 q^{15} -2.87775 q^{17} -1.75809 q^{19} +3.20523 q^{21} -2.19059 q^{23} -4.97951 q^{25} +2.37582 q^{27} -1.00000 q^{29} -10.4242 q^{31} +11.0644 q^{33} +0.206645 q^{35} -3.76655 q^{37} -2.22036 q^{39} +9.68357 q^{41} -8.39441 q^{43} -0.276276 q^{45} -0.154531 q^{47} -4.91612 q^{49} +6.38962 q^{51} +1.92431 q^{53} +0.713336 q^{55} +3.90358 q^{57} +4.22176 q^{59} -0.298213 q^{61} -2.78606 q^{63} -0.143149 q^{65} -0.522953 q^{67} +4.86390 q^{69} -5.65532 q^{71} -5.40765 q^{73} +11.0563 q^{75} +7.19353 q^{77} -1.00078 q^{79} -11.0651 q^{81} -0.385276 q^{83} +0.411947 q^{85} +2.22036 q^{87} -14.5128 q^{89} -1.44357 q^{91} +23.1454 q^{93} +0.251669 q^{95} -5.94582 q^{97} -9.61743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * q^97

Introduction

L-functions

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