LMFDB - Embedded newform 6032.2.a.bb.1.5

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} - 10x^{8} + 21x^{7} + 28x^{6} - 67x^{5} - 20x^{4} + 76x^{3} - 8x^{2} - 26x + 8 $ x^10 - 2x^9 - 10x^8 + 21x^7 + 28x^6 - 67x^5 - 20x^4 + 76x^3 - 8x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.640377$ 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)

$f(q)$	$=$	$q+0.381396 q^{3} +0.331991 q^{5} -2.55994 q^{7} -2.85454 q^{9} +O(q^{10})$	$q+0.381396 q^{3} +0.331991 q^{5} -2.55994 q^{7} -2.85454 q^{9} -3.59922 q^{11} -1.00000 q^{13} +0.126620 q^{15} -2.36892 q^{17} +6.73773 q^{19} -0.976351 q^{21} +4.79707 q^{23} -4.88978 q^{25} -2.23290 q^{27} +1.00000 q^{29} -3.61871 q^{31} -1.37273 q^{33} -0.849877 q^{35} -5.90562 q^{37} -0.381396 q^{39} +3.31185 q^{41} +3.25774 q^{43} -0.947681 q^{45} +8.01911 q^{47} -0.446726 q^{49} -0.903496 q^{51} +0.428155 q^{53} -1.19491 q^{55} +2.56975 q^{57} +3.20338 q^{59} -9.83212 q^{61} +7.30743 q^{63} -0.331991 q^{65} -5.75184 q^{67} +1.82958 q^{69} +3.41146 q^{71} +9.63211 q^{73} -1.86495 q^{75} +9.21378 q^{77} -2.02095 q^{79} +7.71199 q^{81} -16.6345 q^{83} -0.786459 q^{85} +0.381396 q^{87} -7.10813 q^{89} +2.55994 q^{91} -1.38016 q^{93} +2.23687 q^{95} +10.5095 q^{97} +10.2741 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.381396	0.220199	0.110100	−	0.993921i	$-0.464883\pi$
$3$	0.381396	0.220199	0.110100	+	0.993921i	$0.464883\pi$
$4$	0	0
$5$	0.331991	0.148471	0.0742355	−	0.997241i	$-0.476348\pi$
$5$	0.331991	0.148471	0.0742355	+	0.997241i	$0.476348\pi$
$6$	0	0
$7$	−2.55994	−0.967565	−0.483782	−	0.875188i	$-0.660737\pi$
$7$	−2.55994	−0.967565	−0.483782	+	0.875188i	$0.660737\pi$
$8$	0	0
$9$	−2.85454	−0.951512
$10$	0	0
$11$	−3.59922	−1.08521	−0.542603	−	0.839989i	$-0.682561\pi$
$11$	−3.59922	−1.08521	−0.542603	+	0.839989i	$0.682561\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.126620	0.0326932
$16$	0	0
$17$	−2.36892	−0.574546	−0.287273	−	0.957849i	$-0.592749\pi$
$17$	−2.36892	−0.574546	−0.287273	+	0.957849i	$0.592749\pi$
$18$	0	0
$19$	6.73773	1.54574	0.772871	−	0.634564i	$-0.218820\pi$
$19$	6.73773	1.54574	0.772871	+	0.634564i	$0.218820\pi$
$20$	0	0
$21$	−0.976351	−0.213057
$22$	0	0
$23$	4.79707	1.00026	0.500129	−	0.865951i	$-0.333286\pi$
$23$	4.79707	1.00026	0.500129	+	0.865951i	$0.333286\pi$
$24$	0	0
$25$	−4.88978	−0.977956
$26$	0	0
$27$	−2.23290	−0.429722
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.61871	−0.649939	−0.324969	−	0.945725i	$-0.605354\pi$
$31$	−3.61871	−0.649939	−0.324969	+	0.945725i	$0.605354\pi$
$32$	0	0
$33$	−1.37273	−0.238962
$34$	0	0
$35$	−0.849877	−0.143655
$36$	0	0
$37$	−5.90562	−0.970877	−0.485439	−	0.874271i	$-0.661340\pi$
$37$	−5.90562	−0.970877	−0.485439	+	0.874271i	$0.661340\pi$
$38$	0	0
$39$	−0.381396	−0.0610723
$40$	0	0
$41$	3.31185	0.517224	0.258612	−	0.965981i	$-0.416735\pi$
$41$	3.31185	0.517224	0.258612	+	0.965981i	$0.416735\pi$
$42$	0	0
$43$	3.25774	0.496800	0.248400	−	0.968658i	$-0.420095\pi$
$43$	3.25774	0.496800	0.248400	+	0.968658i	$0.420095\pi$
$44$	0	0
$45$	−0.947681	−0.141272
$46$	0	0
$47$	8.01911	1.16971	0.584854	−	0.811139i	$-0.301152\pi$
$47$	8.01911	1.16971	0.584854	+	0.811139i	$0.301152\pi$
$48$	0	0
$49$	−0.446726	−0.0638181
$50$	0	0
$51$	−0.903496	−0.126515
$52$	0	0
$53$	0.428155	0.0588116	0.0294058	−	0.999568i	$-0.490638\pi$
$53$	0.428155	0.0588116	0.0294058	+	0.999568i	$0.490638\pi$
$54$	0	0
$55$	−1.19491	−0.161122
$56$	0	0
$57$	2.56975	0.340371
$58$	0	0
$59$	3.20338	0.417044	0.208522	−	0.978018i	$-0.433135\pi$
$59$	3.20338	0.417044	0.208522	+	0.978018i	$0.433135\pi$
$60$	0	0
$61$	−9.83212	−1.25887	−0.629437	−	0.777052i	$-0.716714\pi$
$61$	−9.83212	−1.25887	−0.629437	+	0.777052i	$0.716714\pi$
$62$	0	0
$63$	7.30743	0.920650
$64$	0	0
$65$	−0.331991	−0.0411785
$66$	0	0
$67$	−5.75184	−0.702699	−0.351349	−	0.936244i	$-0.614277\pi$
$67$	−5.75184	−0.702699	−0.351349	+	0.936244i	$0.614277\pi$
$68$	0	0
$69$	1.82958	0.220256
$70$	0	0
$71$	3.41146	0.404866	0.202433	−	0.979296i	$-0.435115\pi$
$71$	3.41146	0.404866	0.202433	+	0.979296i	$0.435115\pi$
$72$	0	0
$73$	9.63211	1.12735	0.563677	−	0.825996i	$-0.309386\pi$
$73$	9.63211	1.12735	0.563677	+	0.825996i	$0.309386\pi$
$74$	0	0
$75$	−1.86495	−0.215345
$76$	0	0
$77$	9.21378	1.05001
$78$	0	0
$79$	−2.02095	−0.227375	−0.113687	−	0.993517i	$-0.536266\pi$
$79$	−2.02095	−0.227375	−0.113687	+	0.993517i	$0.536266\pi$
$80$	0	0
$81$	7.71199	0.856888
$82$	0	0
$83$	−16.6345	−1.82587	−0.912937	−	0.408101i	$-0.866191\pi$
$83$	−16.6345	−1.82587	−0.912937	+	0.408101i	$0.866191\pi$
$84$	0	0
$85$	−0.786459	−0.0853035
$86$	0	0
$87$	0.381396	0.0408900
$88$	0	0
$89$	−7.10813	−0.753460	−0.376730	−	0.926323i	$-0.622952\pi$
$89$	−7.10813	−0.753460	−0.376730	+	0.926323i	$0.622952\pi$
$90$	0	0
$91$	2.55994	0.268354
$92$	0	0
$93$	−1.38016	−0.143116
$94$	0	0
$95$	2.23687	0.229498
$96$	0	0
$97$	10.5095	1.06708	0.533539	−	0.845775i	$-0.320862\pi$
$97$	10.5095	1.06708	0.533539	+	0.845775i	$0.320862\pi$
$98$	0	0
$99$	10.2741	1.03259
$100$	0	0
$101$	2.34296	0.233134	0.116567	−	0.993183i	$-0.462811\pi$
$101$	2.34296	0.233134	0.116567	+	0.993183i	$0.462811\pi$
$102$	0	0
$103$	15.4709	1.52439	0.762197	−	0.647345i	$-0.224121\pi$
$103$	15.4709	1.52439	0.762197	+	0.647345i	$0.224121\pi$
$104$	0	0
$105$	−0.324140	−0.0316328
$106$	0	0
$107$	7.47128	0.722276	0.361138	−	0.932512i	$-0.382388\pi$
$107$	7.47128	0.722276	0.361138	+	0.932512i	$0.382388\pi$
$108$	0	0
$109$	0.260534	0.0249546	0.0124773	−	0.999922i	$-0.496028\pi$
$109$	0.260534	0.0249546	0.0124773	+	0.999922i	$0.496028\pi$
$110$	0	0
$111$	−2.25238	−0.213787
$112$	0	0
$113$	−14.0072	−1.31769	−0.658845	−	0.752278i	$-0.728955\pi$
$113$	−14.0072	−1.31769	−0.658845	+	0.752278i	$0.728955\pi$
$114$	0	0
$115$	1.59258	0.148509
$116$	0	0
$117$	2.85454	0.263902
$118$	0	0
$119$	6.06427	0.555911
$120$	0	0
$121$	1.95441	0.177674
$122$	0	0
$123$	1.26313	0.113892
$124$	0	0
$125$	−3.28332	−0.293669
$126$	0	0
$127$	−1.39250	−0.123564	−0.0617822	−	0.998090i	$-0.519678\pi$
$127$	−1.39250	−0.123564	−0.0617822	+	0.998090i	$0.519678\pi$
$128$	0	0
$129$	1.24249	0.109395
$130$	0	0
$131$	21.4140	1.87095	0.935475	−	0.353393i	$-0.114972\pi$
$131$	21.4140	1.87095	0.935475	+	0.353393i	$0.114972\pi$
$132$	0	0
$133$	−17.2482	−1.49560
$134$	0	0
$135$	−0.741303	−0.0638012
$136$	0	0
$137$	20.4828	1.74996	0.874980	−	0.484159i	$-0.160874\pi$
$137$	20.4828	1.74996	0.874980	+	0.484159i	$0.160874\pi$
$138$	0	0
$139$	16.6097	1.40882	0.704410	−	0.709794i	$-0.251212\pi$
$139$	16.6097	1.40882	0.704410	+	0.709794i	$0.251212\pi$
$140$	0	0
$141$	3.05846	0.257569
$142$	0	0
$143$	3.59922	0.300982
$144$	0	0
$145$	0.331991	0.0275704
$146$	0	0
$147$	−0.170380	−0.0140527
$148$	0	0
$149$	9.09259	0.744894	0.372447	−	0.928054i	$-0.378519\pi$
$149$	9.09259	0.744894	0.372447	+	0.928054i	$0.378519\pi$
$150$	0	0
$151$	7.47430	0.608250	0.304125	−	0.952632i	$-0.401636\pi$
$151$	7.47430	0.608250	0.304125	+	0.952632i	$0.401636\pi$
$152$	0	0
$153$	6.76216	0.546688
$154$	0	0
$155$	−1.20138	−0.0964971
$156$	0	0
$157$	3.47750	0.277535	0.138767	−	0.990325i	$-0.455686\pi$
$157$	3.47750	0.277535	0.138767	+	0.990325i	$0.455686\pi$
$158$	0	0
$159$	0.163297	0.0129503
$160$	0	0
$161$	−12.2802	−0.967814
$162$	0	0
$163$	9.71347	0.760818	0.380409	−	0.924818i	$-0.375783\pi$
$163$	9.71347	0.760818	0.380409	+	0.924818i	$0.375783\pi$
$164$	0	0
$165$	−0.455735	−0.0354789
$166$	0	0
$167$	25.0581	1.93906	0.969529	−	0.244978i	$-0.0787808\pi$
$167$	25.0581	1.93906	0.969529	+	0.244978i	$0.0787808\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−19.2331	−1.47079
$172$	0	0
$173$	−8.74565	−0.664920	−0.332460	−	0.943117i	$-0.607879\pi$
$173$	−8.74565	−0.664920	−0.332460	+	0.943117i	$0.607879\pi$
$174$	0	0
$175$	12.5175	0.946236
$176$	0	0
$177$	1.22176	0.0918328
$178$	0	0
$179$	21.0653	1.57450	0.787248	−	0.616636i	$-0.211505\pi$
$179$	21.0653	1.57450	0.787248	+	0.616636i	$0.211505\pi$
$180$	0	0
$181$	24.2096	1.79949	0.899743	−	0.436421i	$-0.143754\pi$
$181$	24.2096	1.79949	0.899743	+	0.436421i	$0.143754\pi$
$182$	0	0
$183$	−3.74994	−0.277203
$184$	0	0
$185$	−1.96061	−0.144147
$186$	0	0
$187$	8.52625	0.623502
$188$	0	0
$189$	5.71608	0.415784
$190$	0	0
$191$	11.1179	0.804464	0.402232	−	0.915538i	$-0.368235\pi$
$191$	11.1179	0.804464	0.402232	+	0.915538i	$0.368235\pi$
$192$	0	0
$193$	1.38088	0.0993979	0.0496990	−	0.998764i	$-0.484174\pi$
$193$	1.38088	0.0993979	0.0496990	+	0.998764i	$0.484174\pi$
$194$	0	0
$195$	−0.126620	−0.00906747
$196$	0	0
$197$	10.7156	0.763457	0.381728	−	0.924275i	$-0.375329\pi$
$197$	10.7156	0.763457	0.381728	+	0.924275i	$0.375329\pi$
$198$	0	0
$199$	5.29837	0.375592	0.187796	−	0.982208i	$-0.439866\pi$
$199$	5.29837	0.375592	0.187796	+	0.982208i	$0.439866\pi$
$200$	0	0
$201$	−2.19373	−0.154734
$202$	0	0
$203$	−2.55994	−0.179672
$204$	0	0
$205$	1.09951	0.0767928
$206$	0	0
$207$	−13.6934	−0.951757
$208$	0	0
$209$	−24.2506	−1.67745
$210$	0	0
$211$	7.47659	0.514710	0.257355	−	0.966317i	$-0.417149\pi$
$211$	7.47659	0.514710	0.257355	+	0.966317i	$0.417149\pi$
$212$	0	0
$213$	1.30112	0.0891511
$214$	0	0
$215$	1.08154	0.0737604
$216$	0	0
$217$	9.26365	0.628858
$218$	0	0
$219$	3.67365	0.248242
$220$	0	0
$221$	2.36892	0.159350
$222$	0	0
$223$	−19.9024	−1.33277	−0.666383	−	0.745610i	$-0.732158\pi$
$223$	−19.9024	−1.33277	−0.666383	+	0.745610i	$0.732158\pi$
$224$	0	0
$225$	13.9581	0.930537
$226$	0	0
$227$	−4.97361	−0.330110	−0.165055	−	0.986284i	$-0.552780\pi$
$227$	−4.97361	−0.330110	−0.165055	+	0.986284i	$0.552780\pi$
$228$	0	0
$229$	7.53016	0.497607	0.248804	−	0.968554i	$-0.419963\pi$
$229$	7.53016	0.497607	0.248804	+	0.968554i	$0.419963\pi$
$230$	0	0
$231$	3.51410	0.231211
$232$	0	0
$233$	−12.2235	−0.800787	−0.400393	−	0.916343i	$-0.631126\pi$
$233$	−12.2235	−0.800787	−0.400393	+	0.916343i	$0.631126\pi$
$234$	0	0
$235$	2.66228	0.173668
$236$	0	0
$237$	−0.770783	−0.0500677
$238$	0	0
$239$	−19.3831	−1.25379	−0.626893	−	0.779105i	$-0.715674\pi$
$239$	−19.3831	−1.25379	−0.626893	+	0.779105i	$0.715674\pi$
$240$	0	0
$241$	19.7686	1.27341	0.636704	−	0.771108i	$-0.280297\pi$
$241$	19.7686	1.27341	0.636704	+	0.771108i	$0.280297\pi$
$242$	0	0
$243$	9.64002	0.618408
$244$	0	0
$245$	−0.148309	−0.00947514
$246$	0	0
$247$	−6.73773	−0.428711
$248$	0	0
$249$	−6.34434	−0.402056
$250$	0	0
$251$	−1.18019	−0.0744928	−0.0372464	−	0.999306i	$-0.511859\pi$
$251$	−1.18019	−0.0744928	−0.0372464	+	0.999306i	$0.511859\pi$
$252$	0	0
$253$	−17.2657	−1.08549
$254$	0	0
$255$	−0.299953	−0.0187838
$256$	0	0
$257$	11.0761	0.690910	0.345455	−	0.938435i	$-0.387725\pi$
$257$	11.0761	0.690910	0.345455	+	0.938435i	$0.387725\pi$
$258$	0	0
$259$	15.1180	0.939387
$260$	0	0
$261$	−2.85454	−0.176691
$262$	0	0
$263$	−2.59808	−0.160205	−0.0801023	−	0.996787i	$-0.525525\pi$
$263$	−2.59808	−0.160205	−0.0801023	+	0.996787i	$0.525525\pi$
$264$	0	0
$265$	0.142144	0.00873182
$266$	0	0
$267$	−2.71102	−0.165911
$268$	0	0
$269$	−19.3129	−1.17753	−0.588763	−	0.808306i	$-0.700385\pi$
$269$	−19.3129	−1.17753	−0.588763	+	0.808306i	$0.700385\pi$
$270$	0	0
$271$	−18.4071	−1.11815	−0.559074	−	0.829118i	$-0.688843\pi$
$271$	−18.4071	−1.11815	−0.559074	+	0.829118i	$0.688843\pi$
$272$	0	0
$273$	0.976351	0.0590914
$274$	0	0
$275$	17.5994	1.06128
$276$	0	0
$277$	−24.4486	−1.46897	−0.734487	−	0.678623i	$-0.762577\pi$
$277$	−24.4486	−1.46897	−0.734487	+	0.678623i	$0.762577\pi$
$278$	0	0
$279$	10.3297	0.618425
$280$	0	0
$281$	−7.13262	−0.425496	−0.212748	−	0.977107i	$-0.568241\pi$
$281$	−7.13262	−0.425496	−0.212748	+	0.977107i	$0.568241\pi$
$282$	0	0
$283$	−1.82816	−0.108673	−0.0543365	−	0.998523i	$-0.517304\pi$
$283$	−1.82816	−0.108673	−0.0543365	+	0.998523i	$0.517304\pi$
$284$	0	0
$285$	0.853133	0.0505353
$286$	0	0
$287$	−8.47812	−0.500448
$288$	0	0
$289$	−11.3882	−0.669897
$290$	0	0
$291$	4.00829	0.234970
$292$	0	0
$293$	−28.9738	−1.69267	−0.846335	−	0.532651i	$-0.821196\pi$
$293$	−28.9738	−1.69267	−0.846335	+	0.532651i	$0.821196\pi$
$294$	0	0
$295$	1.06349	0.0619189
$296$	0	0
$297$	8.03671	0.466337
$298$	0	0
$299$	−4.79707	−0.277422
$300$	0	0
$301$	−8.33959	−0.480686
$302$	0	0
$303$	0.893598	0.0513359
$304$	0	0
$305$	−3.26418	−0.186906
$306$	0	0
$307$	32.4174	1.85016	0.925080	−	0.379772i	$-0.123997\pi$
$307$	32.4174	1.85016	0.925080	+	0.379772i	$0.123997\pi$
$308$	0	0
$309$	5.90055	0.335670
$310$	0	0
$311$	−13.4785	−0.764297	−0.382149	−	0.924101i	$-0.624816\pi$
$311$	−13.4785	−0.764297	−0.382149	+	0.924101i	$0.624816\pi$
$312$	0	0
$313$	19.4781	1.10097	0.550483	−	0.834846i	$-0.314443\pi$
$313$	19.4781	1.10097	0.550483	+	0.834846i	$0.314443\pi$
$314$	0	0
$315$	2.42600	0.136690
$316$	0	0
$317$	−12.3228	−0.692119	−0.346060	−	0.938212i	$-0.612481\pi$
$317$	−12.3228	−0.692119	−0.346060	+	0.938212i	$0.612481\pi$
$318$	0	0
$319$	−3.59922	−0.201518
$320$	0	0
$321$	2.84952	0.159045
$322$	0	0
$323$	−15.9611	−0.888100
$324$	0	0
$325$	4.88978	0.271236
$326$	0	0
$327$	0.0993667	0.00549499
$328$	0	0
$329$	−20.5284	−1.13177
$330$	0	0
$331$	−23.9364	−1.31566	−0.657831	−	0.753165i	$-0.728526\pi$
$331$	−23.9364	−1.31566	−0.657831	+	0.753165i	$0.728526\pi$
$332$	0	0
$333$	16.8578	0.923802
$334$	0	0
$335$	−1.90956	−0.104330
$336$	0	0
$337$	12.6641	0.689858	0.344929	−	0.938629i	$-0.387903\pi$
$337$	12.6641	0.689858	0.344929	+	0.938629i	$0.387903\pi$
$338$	0	0
$339$	−5.34231	−0.290155
$340$	0	0
$341$	13.0245	0.705318
$342$	0	0
$343$	19.0631	1.02931
$344$	0	0
$345$	0.607406	0.0327016
$346$	0	0
$347$	−1.65094	−0.0886270	−0.0443135	−	0.999018i	$-0.514110\pi$
$347$	−1.65094	−0.0886270	−0.0443135	+	0.999018i	$0.514110\pi$
$348$	0	0
$349$	−10.4669	−0.560282	−0.280141	−	0.959959i	$-0.590381\pi$
$349$	−10.4669	−0.560282	−0.280141	+	0.959959i	$0.590381\pi$
$350$	0	0
$351$	2.23290	0.119183
$352$	0	0
$353$	−29.8975	−1.59129	−0.795643	−	0.605766i	$-0.792867\pi$
$353$	−29.8975	−1.59129	−0.795643	+	0.605766i	$0.792867\pi$
$354$	0	0
$355$	1.13257	0.0601108
$356$	0	0
$357$	2.31289	0.122411
$358$	0	0
$359$	9.28796	0.490200	0.245100	−	0.969498i	$-0.421179\pi$
$359$	9.28796	0.490200	0.245100	+	0.969498i	$0.421179\pi$
$360$	0	0
$361$	26.3970	1.38932
$362$	0	0
$363$	0.745405	0.0391236
$364$	0	0
$365$	3.19778	0.167379
$366$	0	0
$367$	−5.36697	−0.280154	−0.140077	−	0.990141i	$-0.544735\pi$
$367$	−5.36697	−0.280154	−0.140077	+	0.990141i	$0.544735\pi$
$368$	0	0
$369$	−9.45379	−0.492145
$370$	0	0
$371$	−1.09605	−0.0569040
$372$	0	0
$373$	−20.8450	−1.07931	−0.539656	−	0.841885i	$-0.681446\pi$
$373$	−20.8450	−1.07931	−0.539656	+	0.841885i	$0.681446\pi$
$374$	0	0
$375$	−1.25225	−0.0646658
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−21.4885	−1.10379	−0.551896	−	0.833913i	$-0.686095\pi$
$379$	−21.4885	−1.10379	−0.551896	+	0.833913i	$0.686095\pi$
$380$	0	0
$381$	−0.531095	−0.0272088
$382$	0	0
$383$	2.92547	0.149484	0.0747422	−	0.997203i	$-0.476187\pi$
$383$	2.92547	0.149484	0.0747422	+	0.997203i	$0.476187\pi$
$384$	0	0
$385$	3.05890	0.155896
$386$	0	0
$387$	−9.29933	−0.472711
$388$	0	0
$389$	18.8654	0.956514	0.478257	−	0.878220i	$-0.341269\pi$
$389$	18.8654	0.956514	0.478257	+	0.878220i	$0.341269\pi$
$390$	0	0
$391$	−11.3638	−0.574694
$392$	0	0
$393$	8.16722	0.411982
$394$	0	0
$395$	−0.670938	−0.0337585
$396$	0	0
$397$	35.9359	1.80357	0.901786	−	0.432183i	$-0.142257\pi$
$397$	35.9359	1.80357	0.901786	+	0.432183i	$0.142257\pi$
$398$	0	0
$399$	−6.57839	−0.329331
$400$	0	0
$401$	−4.46291	−0.222867	−0.111434	−	0.993772i	$-0.535544\pi$
$401$	−4.46291	−0.222867	−0.111434	+	0.993772i	$0.535544\pi$
$402$	0	0
$403$	3.61871	0.180261
$404$	0	0
$405$	2.56031	0.127223
$406$	0	0
$407$	21.2556	1.05360
$408$	0	0
$409$	−11.3643	−0.561927	−0.280963	−	0.959719i	$-0.590654\pi$
$409$	−11.3643	−0.561927	−0.280963	+	0.959719i	$0.590654\pi$
$410$	0	0
$411$	7.81205	0.385340
$412$	0	0
$413$	−8.20044	−0.403517
$414$	0	0
$415$	−5.52251	−0.271089
$416$	0	0
$417$	6.33489	0.310221
$418$	0	0
$419$	16.7730	0.819416	0.409708	−	0.912217i	$-0.365631\pi$
$419$	16.7730	0.819416	0.409708	+	0.912217i	$0.365631\pi$
$420$	0	0
$421$	−4.61533	−0.224937	−0.112469	−	0.993655i	$-0.535876\pi$
$421$	−4.61533	−0.224937	−0.112469	+	0.993655i	$0.535876\pi$
$422$	0	0
$423$	−22.8909	−1.11299
$424$	0	0
$425$	11.5835	0.561881
$426$	0	0
$427$	25.1696	1.21804
$428$	0	0
$429$	1.37273	0.0662761
$430$	0	0
$431$	−15.7827	−0.760227	−0.380114	−	0.924940i	$-0.624115\pi$
$431$	−15.7827	−0.760227	−0.380114	+	0.924940i	$0.624115\pi$
$432$	0	0
$433$	34.1104	1.63924	0.819620	−	0.572908i	$-0.194185\pi$
$433$	34.1104	1.63924	0.819620	+	0.572908i	$0.194185\pi$
$434$	0	0
$435$	0.126620	0.00607098
$436$	0	0
$437$	32.3213	1.54614
$438$	0	0
$439$	10.7531	0.513219	0.256609	−	0.966515i	$-0.417395\pi$
$439$	10.7531	0.513219	0.256609	+	0.966515i	$0.417395\pi$
$440$	0	0
$441$	1.27520	0.0607237
$442$	0	0
$443$	24.1389	1.14687	0.573436	−	0.819250i	$-0.305610\pi$
$443$	24.1389	1.14687	0.573436	+	0.819250i	$0.305610\pi$
$444$	0	0
$445$	−2.35984	−0.111867
$446$	0	0
$447$	3.46788	0.164025
$448$	0	0
$449$	2.05026	0.0967577	0.0483789	−	0.998829i	$-0.484595\pi$
$449$	2.05026	0.0967577	0.0483789	+	0.998829i	$0.484595\pi$
$450$	0	0
$451$	−11.9201	−0.561295
$452$	0	0
$453$	2.85067	0.133936
$454$	0	0
$455$	0.849877	0.0398428
$456$	0	0
$457$	24.7831	1.15930	0.579652	−	0.814864i	$-0.303188\pi$
$457$	24.7831	1.15930	0.579652	+	0.814864i	$0.303188\pi$
$458$	0	0
$459$	5.28955	0.246895
$460$	0	0
$461$	10.7068	0.498663	0.249332	−	0.968418i	$-0.419789\pi$
$461$	10.7068	0.498663	0.249332	+	0.968418i	$0.419789\pi$
$462$	0	0
$463$	12.7853	0.594182	0.297091	−	0.954849i	$-0.403983\pi$
$463$	12.7853	0.594182	0.297091	+	0.954849i	$0.403983\pi$
$464$	0	0
$465$	−0.458202	−0.0212486
$466$	0	0
$467$	32.1482	1.48764	0.743820	−	0.668380i	$-0.233012\pi$
$467$	32.1482	1.48764	0.743820	+	0.668380i	$0.233012\pi$
$468$	0	0
$469$	14.7243	0.679907
$470$	0	0
$471$	1.32631	0.0611130
$472$	0	0
$473$	−11.7253	−0.539131
$474$	0	0
$475$	−32.9460	−1.51167
$476$	0	0
$477$	−1.22218	−0.0559600
$478$	0	0
$479$	−36.3371	−1.66028	−0.830142	−	0.557552i	$-0.811741\pi$
$479$	−36.3371	−1.66028	−0.830142	+	0.557552i	$0.811741\pi$
$480$	0	0
$481$	5.90562	0.269273
$482$	0	0
$483$	−4.68362	−0.213112
$484$	0	0
$485$	3.48906	0.158430
$486$	0	0
$487$	−22.7685	−1.03174	−0.515870	−	0.856667i	$-0.672531\pi$
$487$	−22.7685	−1.03174	−0.515870	+	0.856667i	$0.672531\pi$
$488$	0	0
$489$	3.70468	0.167532
$490$	0	0
$491$	−31.8270	−1.43633	−0.718167	−	0.695871i	$-0.755019\pi$
$491$	−31.8270	−1.43633	−0.718167	+	0.695871i	$0.755019\pi$
$492$	0	0
$493$	−2.36892	−0.106691
$494$	0	0
$495$	3.41092	0.153309
$496$	0	0
$497$	−8.73311	−0.391734
$498$	0	0
$499$	6.75902	0.302575	0.151288	−	0.988490i	$-0.451658\pi$
$499$	6.75902	0.302575	0.151288	+	0.988490i	$0.451658\pi$
$500$	0	0
$501$	9.55709	0.426979
$502$	0	0
$503$	−24.2778	−1.08249	−0.541246	−	0.840864i	$-0.682047\pi$
$503$	−24.2778	−1.08249	−0.541246	+	0.840864i	$0.682047\pi$
$504$	0	0
$505$	0.777844	0.0346136
$506$	0	0
$507$	0.381396	0.0169384
$508$	0	0
$509$	24.9060	1.10394	0.551969	−	0.833865i	$-0.313877\pi$
$509$	24.9060	1.10394	0.551969	+	0.833865i	$0.313877\pi$
$510$	0	0
$511$	−24.6576	−1.09079
$512$	0	0
$513$	−15.0447	−0.664239
$514$	0	0
$515$	5.13621	0.226328
$516$	0	0
$517$	−28.8626	−1.26937
$518$	0	0
$519$	−3.33556	−0.146415
$520$	0	0
$521$	3.00035	0.131448	0.0657240	−	0.997838i	$-0.479064\pi$
$521$	3.00035	0.131448	0.0657240	+	0.997838i	$0.479064\pi$
$522$	0	0
$523$	8.00082	0.349852	0.174926	−	0.984582i	$-0.444031\pi$
$523$	8.00082	0.349852	0.174926	+	0.984582i	$0.444031\pi$
$524$	0	0
$525$	4.77414	0.208361
$526$	0	0
$527$	8.57241	0.373420
$528$	0	0
$529$	0.0118449	0.000514995 0
$530$	0	0
$531$	−9.14415	−0.396822
$532$	0	0
$533$	−3.31185	−0.143452
$534$	0	0
$535$	2.48040	0.107237
$536$	0	0
$537$	8.03424	0.346703
$538$	0	0
$539$	1.60787	0.0692558
$540$	0	0
$541$	−32.2021	−1.38447	−0.692237	−	0.721670i	$-0.743375\pi$
$541$	−32.2021	−1.38447	−0.692237	+	0.721670i	$0.743375\pi$
$542$	0	0
$543$	9.23346	0.396246
$544$	0	0
$545$	0.0864950	0.00370504
$546$	0	0
$547$	−35.9624	−1.53764	−0.768821	−	0.639465i	$-0.779156\pi$
$547$	−35.9624	−1.53764	−0.768821	+	0.639465i	$0.779156\pi$
$548$	0	0
$549$	28.0661	1.19783
$550$	0	0
$551$	6.73773	0.287037
$552$	0	0
$553$	5.17350	0.220000
$554$	0	0
$555$	−0.747771	−0.0317411
$556$	0	0
$557$	−40.6114	−1.72076	−0.860380	−	0.509652i	$-0.829774\pi$
$557$	−40.6114	−1.72076	−0.860380	+	0.509652i	$0.829774\pi$
$558$	0	0
$559$	−3.25774	−0.137788
$560$	0	0
$561$	3.25188	0.137295
$562$	0	0
$563$	30.5509	1.28757	0.643783	−	0.765209i	$-0.277364\pi$
$563$	30.5509	1.28757	0.643783	+	0.765209i	$0.277364\pi$
$564$	0	0
$565$	−4.65028	−0.195639
$566$	0	0
$567$	−19.7422	−0.829095
$568$	0	0
$569$	36.5627	1.53279	0.766394	−	0.642371i	$-0.222049\pi$
$569$	36.5627	1.53279	0.766394	+	0.642371i	$0.222049\pi$
$570$	0	0
$571$	16.1583	0.676206	0.338103	−	0.941109i	$-0.390215\pi$
$571$	16.1583	0.676206	0.338103	+	0.941109i	$0.390215\pi$
$572$	0	0
$573$	4.24033	0.177142
$574$	0	0
$575$	−23.4566	−0.978208
$576$	0	0
$577$	−11.8724	−0.494254	−0.247127	−	0.968983i	$-0.579486\pi$
$577$	−11.8724	−0.494254	−0.247127	+	0.968983i	$0.579486\pi$
$578$	0	0
$579$	0.526663	0.0218874
$580$	0	0
$581$	42.5833	1.76665
$582$	0	0
$583$	−1.54103	−0.0638228
$584$	0	0
$585$	0.947681	0.0391818
$586$	0	0
$587$	22.8467	0.942985	0.471492	−	0.881870i	$-0.343716\pi$
$587$	22.8467	0.942985	0.471492	+	0.881870i	$0.343716\pi$
$588$	0	0
$589$	−24.3819	−1.00464
$590$	0	0
$591$	4.08690	0.168113
$592$	0	0
$593$	6.90002	0.283350	0.141675	−	0.989913i	$-0.454751\pi$
$593$	6.90002	0.283350	0.141675	+	0.989913i	$0.454751\pi$
$594$	0	0
$595$	2.01329	0.0825367
$596$	0	0
$597$	2.02078	0.0827050
$598$	0	0
$599$	26.5656	1.08544	0.542720	−	0.839914i	$-0.317395\pi$
$599$	26.5656	1.08544	0.542720	+	0.839914i	$0.317395\pi$
$600$	0	0
$601$	35.4834	1.44740	0.723700	−	0.690115i	$-0.242440\pi$
$601$	35.4834	1.44740	0.723700	+	0.690115i	$0.242440\pi$
$602$	0	0
$603$	16.4188	0.668627
$604$	0	0
$605$	0.648847	0.0263794
$606$	0	0
$607$	−26.2559	−1.06569	−0.532847	−	0.846212i	$-0.678878\pi$
$607$	−26.2559	−1.06569	−0.532847	+	0.846212i	$0.678878\pi$
$608$	0	0
$609$	−0.976351	−0.0395637
$610$	0	0
$611$	−8.01911	−0.324419
$612$	0	0
$613$	35.4765	1.43288	0.716441	−	0.697648i	$-0.245770\pi$
$613$	35.4765	1.43288	0.716441	+	0.697648i	$0.245770\pi$
$614$	0	0
$615$	0.419347	0.0169097
$616$	0	0
$617$	20.9826	0.844726	0.422363	−	0.906427i	$-0.361201\pi$
$617$	20.9826	0.844726	0.422363	+	0.906427i	$0.361201\pi$
$618$	0	0
$619$	−1.99852	−0.0803272	−0.0401636	−	0.999193i	$-0.512788\pi$
$619$	−1.99852	−0.0803272	−0.0401636	+	0.999193i	$0.512788\pi$
$620$	0	0
$621$	−10.7114	−0.429832
$622$	0	0
$623$	18.1964	0.729022
$624$	0	0
$625$	23.3589	0.934355
$626$	0	0
$627$	−9.24909	−0.369373
$628$	0	0
$629$	13.9899	0.557814
$630$	0	0
$631$	9.06953	0.361052	0.180526	−	0.983570i	$-0.442220\pi$
$631$	9.06953	0.361052	0.180526	+	0.983570i	$0.442220\pi$
$632$	0	0
$633$	2.85155	0.113339
$634$	0	0
$635$	−0.462298	−0.0183457
$636$	0	0
$637$	0.446726	0.0176999
$638$	0	0
$639$	−9.73813	−0.385235
$640$	0	0
$641$	−21.9630	−0.867488	−0.433744	−	0.901036i	$-0.642808\pi$
$641$	−21.9630	−0.867488	−0.433744	+	0.901036i	$0.642808\pi$
$642$	0	0
$643$	−30.1914	−1.19063	−0.595317	−	0.803491i	$-0.702973\pi$
$643$	−30.1914	−1.19063	−0.595317	+	0.803491i	$0.702973\pi$
$644$	0	0
$645$	0.412496	0.0162420
$646$	0	0
$647$	−35.4651	−1.39428	−0.697139	−	0.716936i	$-0.745544\pi$
$647$	−35.4651	−1.39428	−0.697139	+	0.716936i	$0.745544\pi$
$648$	0	0
$649$	−11.5297	−0.452579
$650$	0	0
$651$	3.53313	0.138474
$652$	0	0
$653$	−1.27154	−0.0497593	−0.0248796	−	0.999690i	$-0.507920\pi$
$653$	−1.27154	−0.0497593	−0.0248796	+	0.999690i	$0.507920\pi$
$654$	0	0
$655$	7.10926	0.277782
$656$	0	0
$657$	−27.4952	−1.07269
$658$	0	0
$659$	−0.523516	−0.0203933	−0.0101967	−	0.999948i	$-0.503246\pi$
$659$	−0.523516	−0.0203933	−0.0101967	+	0.999948i	$0.503246\pi$
$660$	0	0
$661$	6.29814	0.244969	0.122485	−	0.992470i	$-0.460914\pi$
$661$	6.29814	0.244969	0.122485	+	0.992470i	$0.460914\pi$
$662$	0	0
$663$	0.903496	0.0350889
$664$	0	0
$665$	−5.72624	−0.222054
$666$	0	0
$667$	4.79707	0.185743
$668$	0	0
$669$	−7.59072	−0.293474
$670$	0	0
$671$	35.3880	1.36614
$672$	0	0
$673$	16.1992	0.624432	0.312216	−	0.950011i	$-0.398929\pi$
$673$	16.1992	0.624432	0.312216	+	0.950011i	$0.398929\pi$
$674$	0	0
$675$	10.9184	0.420249
$676$	0	0
$677$	43.8852	1.68665	0.843324	−	0.537406i	$-0.180596\pi$
$677$	43.8852	1.68665	0.843324	+	0.537406i	$0.180596\pi$
$678$	0	0
$679$	−26.9037	−1.03247
$680$	0	0
$681$	−1.89692	−0.0726900
$682$	0	0
$683$	19.2615	0.737022	0.368511	−	0.929623i	$-0.379868\pi$
$683$	19.2615	0.737022	0.368511	+	0.929623i	$0.379868\pi$
$684$	0	0
$685$	6.80010	0.259818
$686$	0	0
$687$	2.87198	0.109573
$688$	0	0
$689$	−0.428155	−0.0163114
$690$	0	0
$691$	−14.4389	−0.549280	−0.274640	−	0.961547i	$-0.588559\pi$
$691$	−14.4389	−0.549280	−0.274640	+	0.961547i	$0.588559\pi$
$692$	0	0
$693$	−26.3011	−0.999095
$694$	0	0
$695$	5.51429	0.209169
$696$	0	0
$697$	−7.84549	−0.297169
$698$	0	0
$699$	−4.66199	−0.176333
$700$	0	0
$701$	−3.37972	−0.127650	−0.0638251	−	0.997961i	$-0.520330\pi$
$701$	−3.37972	−0.127650	−0.0638251	+	0.997961i	$0.520330\pi$
$702$	0	0
$703$	−39.7904	−1.50073
$704$	0	0
$705$	1.01538	0.0382415
$706$	0	0
$707$	−5.99784	−0.225572
$708$	0	0
$709$	−20.0337	−0.752383	−0.376191	−	0.926542i	$-0.622766\pi$
$709$	−20.0337	−0.752383	−0.376191	+	0.926542i	$0.622766\pi$
$710$	0	0
$711$	5.76887	0.216350
$712$	0	0
$713$	−17.3592	−0.650106
$714$	0	0
$715$	1.19491	0.0446871
$716$	0	0
$717$	−7.39263	−0.276083
$718$	0	0
$719$	−18.4541	−0.688221	−0.344111	−	0.938929i	$-0.611820\pi$
$719$	−18.4541	−0.688221	−0.344111	+	0.938929i	$0.611820\pi$
$720$	0	0
$721$	−39.6045	−1.47495
$722$	0	0
$723$	7.53968	0.280404
$724$	0	0
$725$	−4.88978	−0.181602
$726$	0	0
$727$	14.1964	0.526515	0.263257	−	0.964726i	$-0.415203\pi$
$727$	14.1964	0.526515	0.263257	+	0.964726i	$0.415203\pi$
$728$	0	0
$729$	−19.4593	−0.720715
$730$	0	0
$731$	−7.71730	−0.285435
$732$	0	0
$733$	40.2422	1.48638	0.743189	−	0.669082i	$-0.233312\pi$
$733$	40.2422	1.48638	0.743189	+	0.669082i	$0.233312\pi$
$734$	0	0
$735$	−0.0565647	−0.00208642
$736$	0	0
$737$	20.7022	0.762574
$738$	0	0
$739$	−9.76346	−0.359155	−0.179577	−	0.983744i	$-0.557473\pi$
$739$	−9.76346	−0.359155	−0.179577	+	0.983744i	$0.557473\pi$
$740$	0	0
$741$	−2.56975	−0.0944020
$742$	0	0
$743$	−24.8896	−0.913111	−0.456555	−	0.889695i	$-0.650917\pi$
$743$	−24.8896	−0.913111	−0.456555	+	0.889695i	$0.650917\pi$
$744$	0	0
$745$	3.01866	0.110595
$746$	0	0
$747$	47.4838	1.73734
$748$	0	0
$749$	−19.1260	−0.698849
$750$	0	0
$751$	−13.2293	−0.482742	−0.241371	−	0.970433i	$-0.577597\pi$
$751$	−13.2293	−0.482742	−0.241371	+	0.970433i	$0.577597\pi$
$752$	0	0
$753$	−0.450119	−0.0164033
$754$	0	0
$755$	2.48140	0.0903075
$756$	0	0
$757$	31.2585	1.13611	0.568055	−	0.822990i	$-0.307696\pi$
$757$	31.2585	1.13611	0.568055	+	0.822990i	$0.307696\pi$
$758$	0	0
$759$	−6.58508	−0.239023
$760$	0	0
$761$	−29.0039	−1.05139	−0.525695	−	0.850673i	$-0.676195\pi$
$761$	−29.0039	−1.05139	−0.525695	+	0.850673i	$0.676195\pi$
$762$	0	0
$763$	−0.666950	−0.0241452
$764$	0	0
$765$	2.24498	0.0811673
$766$	0	0
$767$	−3.20338	−0.115667
$768$	0	0
$769$	16.7270	0.603191	0.301596	−	0.953436i	$-0.402481\pi$
$769$	16.7270	0.603191	0.301596	+	0.953436i	$0.402481\pi$
$770$	0	0
$771$	4.22439	0.152138
$772$	0	0
$773$	−41.0836	−1.47767	−0.738837	−	0.673884i	$-0.764625\pi$
$773$	−41.0836	−1.47767	−0.738837	+	0.673884i	$0.764625\pi$
$774$	0	0
$775$	17.6947	0.635612
$776$	0	0
$777$	5.76595	0.206852
$778$	0	0
$779$	22.3143	0.799494
$780$	0	0
$781$	−12.2786	−0.439363
$782$	0	0
$783$	−2.23290	−0.0797973
$784$	0	0
$785$	1.15450	0.0412059
$786$	0	0
$787$	5.10953	0.182135	0.0910675	−	0.995845i	$-0.470972\pi$
$787$	5.10953	0.182135	0.0910675	+	0.995845i	$0.470972\pi$
$788$	0	0
$789$	−0.990899	−0.0352770
$790$	0	0
$791$	35.8576	1.27495
$792$	0	0
$793$	9.83212	0.349149
$794$	0	0
$795$	0.0542131	0.00192274
$796$	0	0
$797$	−18.6665	−0.661203	−0.330601	−	0.943770i	$-0.607252\pi$
$797$	−18.6665	−0.661203	−0.330601	+	0.943770i	$0.607252\pi$
$798$	0	0
$799$	−18.9966	−0.672051
$800$	0	0
$801$	20.2904	0.716926
$802$	0	0
$803$	−34.6681	−1.22341
$804$	0	0
$805$	−4.07691	−0.143692
$806$	0	0
$807$	−7.36586	−0.259290
$808$	0	0
$809$	−48.7538	−1.71409	−0.857046	−	0.515240i	$-0.827703\pi$
$809$	−48.7538	−1.71409	−0.857046	+	0.515240i	$0.827703\pi$
$810$	0	0
$811$	33.4569	1.17483	0.587416	−	0.809285i	$-0.300145\pi$
$811$	33.4569	1.17483	0.587416	+	0.809285i	$0.300145\pi$
$812$	0	0
$813$	−7.02038	−0.246216
$814$	0	0
$815$	3.22479	0.112959
$816$	0	0
$817$	21.9497	0.767924
$818$	0	0
$819$	−7.30743	−0.255342
$820$	0	0
$821$	25.3649	0.885243	0.442621	−	0.896709i	$-0.354049\pi$
$821$	25.3649	0.885243	0.442621	+	0.896709i	$0.354049\pi$
$822$	0	0
$823$	−8.16169	−0.284498	−0.142249	−	0.989831i	$-0.545433\pi$
$823$	−8.16169	−0.284498	−0.142249	+	0.989831i	$0.545433\pi$
$824$	0	0
$825$	6.71236	0.233694
$826$	0	0
$827$	33.5619	1.16706	0.583531	−	0.812091i	$-0.301671\pi$
$827$	33.5619	1.16706	0.583531	+	0.812091i	$0.301671\pi$
$828$	0	0
$829$	−19.6341	−0.681922	−0.340961	−	0.940077i	$-0.610752\pi$
$829$	−19.6341	−0.681922	−0.340961	+	0.940077i	$0.610752\pi$
$830$	0	0
$831$	−9.32460	−0.323467
$832$	0	0
$833$	1.05826	0.0366664
$834$	0	0
$835$	8.31909	0.287894
$836$	0	0
$837$	8.08021	0.279293
$838$	0	0
$839$	−45.7160	−1.57829	−0.789146	−	0.614206i	$-0.789476\pi$
$839$	−45.7160	−1.57829	−0.789146	+	0.614206i	$0.789476\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.72036	−0.0936940
$844$	0	0
$845$	0.331991	0.0114208
$846$	0	0
$847$	−5.00316	−0.171911
$848$	0	0
$849$	−0.697255	−0.0239297
$850$	0	0
$851$	−28.3296	−0.971127
$852$	0	0
$853$	37.3496	1.27883	0.639414	−	0.768863i	$-0.279177\pi$
$853$	37.3496	1.27883	0.639414	+	0.768863i	$0.279177\pi$
$854$	0	0
$855$	−6.38522	−0.218370
$856$	0	0
$857$	32.3656	1.10559	0.552794	−	0.833318i	$-0.313561\pi$
$857$	32.3656	1.10559	0.552794	+	0.833318i	$0.313561\pi$
$858$	0	0
$859$	48.2773	1.64720	0.823600	−	0.567171i	$-0.191962\pi$
$859$	48.2773	1.64720	0.823600	+	0.567171i	$0.191962\pi$
$860$	0	0
$861$	−3.23353	−0.110198
$862$	0	0
$863$	−27.6258	−0.940393	−0.470196	−	0.882562i	$-0.655817\pi$
$863$	−27.6258	−0.940393	−0.470196	+	0.882562i	$0.655817\pi$
$864$	0	0
$865$	−2.90348	−0.0987213
$866$	0	0
$867$	−4.34344	−0.147511
$868$	0	0
$869$	7.27385	0.246748
$870$	0	0
$871$	5.75184	0.194894
$872$	0	0
$873$	−29.9998	−1.01534
$874$	0	0
$875$	8.40509	0.284144
$876$	0	0
$877$	5.70482	0.192638	0.0963190	−	0.995351i	$-0.469293\pi$
$877$	5.70482	0.192638	0.0963190	+	0.995351i	$0.469293\pi$
$878$	0	0
$879$	−11.0505	−0.372725
$880$	0	0
$881$	23.0295	0.775883	0.387942	−	0.921684i	$-0.373186\pi$
$881$	23.0295	0.775883	0.387942	+	0.921684i	$0.373186\pi$
$882$	0	0
$883$	27.4016	0.922137	0.461069	−	0.887364i	$-0.347466\pi$
$883$	27.4016	0.922137	0.461069	+	0.887364i	$0.347466\pi$
$884$	0	0
$885$	0.405612	0.0136345
$886$	0	0
$887$	3.24278	0.108882	0.0544409	−	0.998517i	$-0.482662\pi$
$887$	3.24278	0.108882	0.0544409	+	0.998517i	$0.482662\pi$
$888$	0	0
$889$	3.56471	0.119557
$890$	0	0
$891$	−27.7572	−0.929900
$892$	0	0
$893$	54.0306	1.80807
$894$	0	0
$895$	6.99350	0.233767
$896$	0	0
$897$	−1.82958	−0.0610880
$898$	0	0
$899$	−3.61871	−0.120691
$900$	0	0
$901$	−1.01426	−0.0337900
$902$	0	0
$903$	−3.18069	−0.105847
$904$	0	0
$905$	8.03738	0.267171
$906$	0	0
$907$	−33.3468	−1.10726	−0.553630	−	0.832762i	$-0.686758\pi$
$907$	−33.3468	−1.10726	−0.553630	+	0.832762i	$0.686758\pi$
$908$	0	0
$909$	−6.68808	−0.221829
$910$	0	0
$911$	−36.1876	−1.19895	−0.599475	−	0.800394i	$-0.704624\pi$
$911$	−36.1876	−1.19895	−0.599475	+	0.800394i	$0.704624\pi$
$912$	0	0
$913$	59.8713	1.98145
$914$	0	0
$915$	−1.24495	−0.0411567
$916$	0	0
$917$	−54.8185	−1.81027
$918$	0	0
$919$	23.2944	0.768411	0.384205	−	0.923248i	$-0.374475\pi$
$919$	23.2944	0.768411	0.384205	+	0.923248i	$0.374475\pi$
$920$	0	0
$921$	12.3639	0.407404
$922$	0	0
$923$	−3.41146	−0.112290
$924$	0	0
$925$	28.8772	0.949476
$926$	0	0
$927$	−44.1623	−1.45048
$928$	0	0
$929$	−12.4521	−0.408540	−0.204270	−	0.978915i	$-0.565482\pi$
$929$	−12.4521	−0.408540	−0.204270	+	0.978915i	$0.565482\pi$
$930$	0	0
$931$	−3.00992	−0.0986462
$932$	0	0
$933$	−5.14066	−0.168298
$934$	0	0
$935$	2.83064	0.0925719
$936$	0	0
$937$	−40.6552	−1.32815	−0.664075	−	0.747666i	$-0.731174\pi$
$937$	−40.6552	−1.32815	−0.664075	+	0.747666i	$0.731174\pi$
$938$	0	0
$939$	7.42887	0.242432
$940$	0	0
$941$	15.6285	0.509474	0.254737	−	0.967010i	$-0.418011\pi$
$941$	15.6285	0.509474	0.254737	+	0.967010i	$0.418011\pi$
$942$	0	0
$943$	15.8872	0.517357
$944$	0	0
$945$	1.89769	0.0617318
$946$	0	0
$947$	0.850047	0.0276228	0.0138114	−	0.999905i	$-0.495604\pi$
$947$	0.850047	0.0276228	0.0138114	+	0.999905i	$0.495604\pi$
$948$	0	0
$949$	−9.63211	−0.312672
$950$	0	0
$951$	−4.69989	−0.152404
$952$	0	0
$953$	−5.25530	−0.170236	−0.0851180	−	0.996371i	$-0.527127\pi$
$953$	−5.25530	−0.170236	−0.0851180	+	0.996371i	$0.527127\pi$
$954$	0	0
$955$	3.69105	0.119440
$956$	0	0
$957$	−1.37273	−0.0443741
$958$	0	0
$959$	−52.4346	−1.69320
$960$	0	0
$961$	−17.9050	−0.577580
$962$	0	0
$963$	−21.3271	−0.687255
$964$	0	0
$965$	0.458440	0.0147577
$966$	0	0
$967$	−52.2185	−1.67923	−0.839616	−	0.543180i	$-0.817220\pi$
$967$	−52.2185	−1.67923	−0.839616	+	0.543180i	$0.817220\pi$
$968$	0	0
$969$	−6.08751	−0.195559
$970$	0	0
$971$	23.8550	0.765543	0.382771	−	0.923843i	$-0.374970\pi$
$971$	23.8550	0.765543	0.382771	+	0.923843i	$0.374970\pi$
$972$	0	0
$973$	−42.5199	−1.36312
$974$	0	0
$975$	1.86495	0.0597261
$976$	0	0
$977$	−45.7480	−1.46361	−0.731803	−	0.681516i	$-0.761321\pi$
$977$	−45.7480	−1.46361	−0.731803	+	0.681516i	$0.761321\pi$
$978$	0	0
$979$	25.5837	0.817660
$980$	0	0
$981$	−0.743704	−0.0237446
$982$	0	0
$983$	−0.281102	−0.00896575	−0.00448288	−	0.999990i	$-0.501427\pi$
$983$	−0.281102	−0.00896575	−0.00448288	+	0.999990i	$0.501427\pi$
$984$	0	0
$985$	3.55749	0.113351
$986$	0	0
$987$	−7.82947	−0.249215
$988$	0	0
$989$	15.6276	0.496928
$990$	0	0
$991$	−17.1894	−0.546039	−0.273019	−	0.962009i	$-0.588022\pi$
$991$	−17.1894	−0.546039	−0.273019	+	0.962009i	$0.588022\pi$
$992$	0	0
$993$	−9.12925	−0.289708
$994$	0	0
$995$	1.75901	0.0557645
$996$	0	0
$997$	−34.4611	−1.09139	−0.545697	−	0.837982i	$-0.683735\pi$
$997$	−34.4611	−1.09139	−0.545697	+	0.837982i	$0.683735\pi$
$998$	0	0
$999$	13.1866	0.417207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bb.1.5		10
4.3	odd	2		3016.2.a.f.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.f.1.6	✓	10	4.3	odd	2
6032.2.a.bb.1.5		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)

\(f(q)\)	\(=\)	\(q+0.381396 q^{3} +0.331991 q^{5} -2.55994 q^{7} -2.85454 q^{9} +O(q^{10})\)	\(q+0.381396 q^{3} +0.331991 q^{5} -2.55994 q^{7} -2.85454 q^{9} -3.59922 q^{11} -1.00000 q^{13} +0.126620 q^{15} -2.36892 q^{17} +6.73773 q^{19} -0.976351 q^{21} +4.79707 q^{23} -4.88978 q^{25} -2.23290 q^{27} +1.00000 q^{29} -3.61871 q^{31} -1.37273 q^{33} -0.849877 q^{35} -5.90562 q^{37} -0.381396 q^{39} +3.31185 q^{41} +3.25774 q^{43} -0.947681 q^{45} +8.01911 q^{47} -0.446726 q^{49} -0.903496 q^{51} +0.428155 q^{53} -1.19491 q^{55} +2.56975 q^{57} +3.20338 q^{59} -9.83212 q^{61} +7.30743 q^{63} -0.331991 q^{65} -5.75184 q^{67} +1.82958 q^{69} +3.41146 q^{71} +9.63211 q^{73} -1.86495 q^{75} +9.21378 q^{77} -2.02095 q^{79} +7.71199 q^{81} -16.6345 q^{83} -0.786459 q^{85} +0.381396 q^{87} -7.10813 q^{89} +2.55994 q^{91} -1.38016 q^{93} +2.23687 q^{95} +10.5095 q^{97} +10.2741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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