LMFDB - Embedded newform 6032.2.a.bd.1.9

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 10 x^{10} + 98 x^{9} + 10 x^{8} - 585 x^{7} + 151 x^{6} + 1524 x^{5} - 445 x^{4} + \cdots + 68 $ x^12 - 6x^11 - 10x^10 + 98x^9 + 10x^8 - 585x^7 + 151x^6 + 1524x^5 - 445x^4 - 1567x^3 + 273x^2 + 424*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.19118$ 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)

$f(q)$	$=$	$q+1.19118 q^{3} +3.27457 q^{5} -4.85960 q^{7} -1.58109 q^{9} +O(q^{10})$	$q+1.19118 q^{3} +3.27457 q^{5} -4.85960 q^{7} -1.58109 q^{9} -3.40142 q^{11} +1.00000 q^{13} +3.90060 q^{15} +7.52221 q^{17} -0.394844 q^{19} -5.78867 q^{21} -1.27580 q^{23} +5.72281 q^{25} -5.45690 q^{27} +1.00000 q^{29} +6.58133 q^{31} -4.05171 q^{33} -15.9131 q^{35} -6.44280 q^{37} +1.19118 q^{39} -6.08242 q^{41} -12.4896 q^{43} -5.17739 q^{45} -3.24110 q^{47} +16.6158 q^{49} +8.96031 q^{51} -8.77374 q^{53} -11.1382 q^{55} -0.470330 q^{57} +8.60325 q^{59} -6.06448 q^{61} +7.68347 q^{63} +3.27457 q^{65} -9.34559 q^{67} -1.51970 q^{69} +7.89658 q^{71} -3.23039 q^{73} +6.81690 q^{75} +16.5296 q^{77} +3.97730 q^{79} -1.75689 q^{81} +15.9196 q^{83} +24.6320 q^{85} +1.19118 q^{87} -10.0776 q^{89} -4.85960 q^{91} +7.83956 q^{93} -1.29294 q^{95} -7.15828 q^{97} +5.37795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * 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
$3$	1.19118	0.687728	0.343864	−	0.939019i	$-0.388264\pi$
$3$	1.19118	0.687728	0.343864	+	0.939019i	$0.388264\pi$
$4$	0	0
$5$	3.27457	1.46443	0.732216	−	0.681072i	$-0.238486\pi$
$5$	3.27457	1.46443	0.732216	+	0.681072i	$0.238486\pi$
$6$	0	0
$7$	−4.85960	−1.83676	−0.918379	−	0.395702i	$-0.870501\pi$
$7$	−4.85960	−1.83676	−0.918379	+	0.395702i	$0.870501\pi$
$8$	0	0
$9$	−1.58109	−0.527030
$10$	0	0
$11$	−3.40142	−1.02557	−0.512784	−	0.858518i	$-0.671386\pi$
$11$	−3.40142	−1.02557	−0.512784	+	0.858518i	$0.671386\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.90060	1.00713
$16$	0	0
$17$	7.52221	1.82440	0.912202	−	0.409741i	$-0.134381\pi$
$17$	7.52221	1.82440	0.912202	+	0.409741i	$0.134381\pi$
$18$	0	0
$19$	−0.394844	−0.0905833	−0.0452917	−	0.998974i	$-0.514422\pi$
$19$	−0.394844	−0.0905833	−0.0452917	+	0.998974i	$0.514422\pi$
$20$	0	0
$21$	−5.78867	−1.26319
$22$	0	0
$23$	−1.27580	−0.266022	−0.133011	−	0.991115i	$-0.542465\pi$
$23$	−1.27580	−0.266022	−0.133011	+	0.991115i	$0.542465\pi$
$24$	0	0
$25$	5.72281	1.14456
$26$	0	0
$27$	−5.45690	−1.05018
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.58133	1.18204	0.591021	−	0.806656i	$-0.298725\pi$
$31$	6.58133	1.18204	0.591021	+	0.806656i	$0.298725\pi$
$32$	0	0
$33$	−4.05171	−0.705312
$34$	0	0
$35$	−15.9131	−2.68981
$36$	0	0
$37$	−6.44280	−1.05919	−0.529595	−	0.848251i	$-0.677656\pi$
$37$	−6.44280	−1.05919	−0.529595	+	0.848251i	$0.677656\pi$
$38$	0	0
$39$	1.19118	0.190741
$40$	0	0
$41$	−6.08242	−0.949914	−0.474957	−	0.880009i	$-0.657536\pi$
$41$	−6.08242	−0.949914	−0.474957	+	0.880009i	$0.657536\pi$
$42$	0	0
$43$	−12.4896	−1.90465	−0.952326	−	0.305083i	$-0.901316\pi$
$43$	−12.4896	−1.90465	−0.952326	+	0.305083i	$0.901316\pi$
$44$	0	0
$45$	−5.17739	−0.771800
$46$	0	0
$47$	−3.24110	−0.472763	−0.236381	−	0.971660i	$-0.575961\pi$
$47$	−3.24110	−0.472763	−0.236381	+	0.971660i	$0.575961\pi$
$48$	0	0
$49$	16.6158	2.37368
$50$	0	0
$51$	8.96031	1.25469
$52$	0	0
$53$	−8.77374	−1.20517	−0.602583	−	0.798056i	$-0.705862\pi$
$53$	−8.77374	−1.20517	−0.602583	+	0.798056i	$0.705862\pi$
$54$	0	0
$55$	−11.1382	−1.50187
$56$	0	0
$57$	−0.470330	−0.0622967
$58$	0	0
$59$	8.60325	1.12005	0.560024	−	0.828477i	$-0.310792\pi$
$59$	8.60325	1.12005	0.560024	+	0.828477i	$0.310792\pi$
$60$	0	0
$61$	−6.06448	−0.776477	−0.388238	−	0.921559i	$-0.626916\pi$
$61$	−6.06448	−0.776477	−0.388238	+	0.921559i	$0.626916\pi$
$62$	0	0
$63$	7.68347	0.968026
$64$	0	0
$65$	3.27457	0.406160
$66$	0	0
$67$	−9.34559	−1.14175	−0.570873	−	0.821038i	$-0.693395\pi$
$67$	−9.34559	−1.14175	−0.570873	+	0.821038i	$0.693395\pi$
$68$	0	0
$69$	−1.51970	−0.182951
$70$	0	0
$71$	7.89658	0.937151	0.468576	−	0.883423i	$-0.344767\pi$
$71$	7.89658	0.937151	0.468576	+	0.883423i	$0.344767\pi$
$72$	0	0
$73$	−3.23039	−0.378088	−0.189044	−	0.981969i	$-0.560539\pi$
$73$	−3.23039	−0.378088	−0.189044	+	0.981969i	$0.560539\pi$
$74$	0	0
$75$	6.81690	0.787148
$76$	0	0
$77$	16.5296	1.88372
$78$	0	0
$79$	3.97730	0.447482	0.223741	−	0.974649i	$-0.428173\pi$
$79$	3.97730	0.447482	0.223741	+	0.974649i	$0.428173\pi$
$80$	0	0
$81$	−1.75689	−0.195210
$82$	0	0
$83$	15.9196	1.74740	0.873702	−	0.486461i	$-0.161713\pi$
$83$	15.9196	1.74740	0.873702	+	0.486461i	$0.161713\pi$
$84$	0	0
$85$	24.6320	2.67172
$86$	0	0
$87$	1.19118	0.127708
$88$	0	0
$89$	−10.0776	−1.06823	−0.534114	−	0.845412i	$-0.679355\pi$
$89$	−10.0776	−1.06823	−0.534114	+	0.845412i	$0.679355\pi$
$90$	0	0
$91$	−4.85960	−0.509425
$92$	0	0
$93$	7.83956	0.812924
$94$	0	0
$95$	−1.29294	−0.132653
$96$	0	0
$97$	−7.15828	−0.726813	−0.363407	−	0.931631i	$-0.618386\pi$
$97$	−7.15828	−0.726813	−0.363407	+	0.931631i	$0.618386\pi$
$98$	0	0
$99$	5.37795	0.540505
$100$	0	0
$101$	−8.18065	−0.814005	−0.407002	−	0.913427i	$-0.633426\pi$
$101$	−8.18065	−0.814005	−0.407002	+	0.913427i	$0.633426\pi$
$102$	0	0
$103$	−4.74401	−0.467441	−0.233720	−	0.972304i	$-0.575090\pi$
$103$	−4.74401	−0.467441	−0.233720	+	0.972304i	$0.575090\pi$
$104$	0	0
$105$	−18.9554	−1.84986
$106$	0	0
$107$	−16.5149	−1.59656	−0.798279	−	0.602288i	$-0.794256\pi$
$107$	−16.5149	−1.59656	−0.798279	+	0.602288i	$0.794256\pi$
$108$	0	0
$109$	−16.5392	−1.58416	−0.792082	−	0.610415i	$-0.791003\pi$
$109$	−16.5392	−1.58416	−0.792082	+	0.610415i	$0.791003\pi$
$110$	0	0
$111$	−7.67454	−0.728435
$112$	0	0
$113$	−14.9633	−1.40763	−0.703814	−	0.710385i	$-0.748521\pi$
$113$	−14.9633	−1.40763	−0.703814	+	0.710385i	$0.748521\pi$
$114$	0	0
$115$	−4.17769	−0.389571
$116$	0	0
$117$	−1.58109	−0.146172
$118$	0	0
$119$	−36.5550	−3.35099
$120$	0	0
$121$	0.569675	0.0517886
$122$	0	0
$123$	−7.24526	−0.653283
$124$	0	0
$125$	2.36690	0.211702
$126$	0	0
$127$	−1.58798	−0.140911	−0.0704554	−	0.997515i	$-0.522445\pi$
$127$	−1.58798	−0.140911	−0.0704554	+	0.997515i	$0.522445\pi$
$128$	0	0
$129$	−14.8774	−1.30988
$130$	0	0
$131$	5.35682	0.468027	0.234014	−	0.972233i	$-0.424814\pi$
$131$	5.35682	0.468027	0.234014	+	0.972233i	$0.424814\pi$
$132$	0	0
$133$	1.91878	0.166380
$134$	0	0
$135$	−17.8690	−1.53792
$136$	0	0
$137$	2.79055	0.238413	0.119206	−	0.992869i	$-0.461965\pi$
$137$	2.79055	0.238413	0.119206	+	0.992869i	$0.461965\pi$
$138$	0	0
$139$	5.92025	0.502149	0.251075	−	0.967968i	$-0.419216\pi$
$139$	5.92025	0.502149	0.251075	+	0.967968i	$0.419216\pi$
$140$	0	0
$141$	−3.86073	−0.325132
$142$	0	0
$143$	−3.40142	−0.284441
$144$	0	0
$145$	3.27457	0.271938
$146$	0	0
$147$	19.7924	1.63245
$148$	0	0
$149$	−13.3428	−1.09309	−0.546544	−	0.837430i	$-0.684057\pi$
$149$	−13.3428	−1.09309	−0.546544	+	0.837430i	$0.684057\pi$
$150$	0	0
$151$	7.37411	0.600096	0.300048	−	0.953924i	$-0.402997\pi$
$151$	7.37411	0.600096	0.300048	+	0.953924i	$0.402997\pi$
$152$	0	0
$153$	−11.8933	−0.961515
$154$	0	0
$155$	21.5510	1.73102
$156$	0	0
$157$	3.57695	0.285472	0.142736	−	0.989761i	$-0.454410\pi$
$157$	3.57695	0.285472	0.142736	+	0.989761i	$0.454410\pi$
$158$	0	0
$159$	−10.4511	−0.828826
$160$	0	0
$161$	6.19987	0.488618
$162$	0	0
$163$	−10.8385	−0.848934	−0.424467	−	0.905443i	$-0.639538\pi$
$163$	−10.8385	−0.848934	−0.424467	+	0.905443i	$0.639538\pi$
$164$	0	0
$165$	−13.2676	−1.03288
$166$	0	0
$167$	−19.9041	−1.54023	−0.770115	−	0.637906i	$-0.779801\pi$
$167$	−19.9041	−1.54023	−0.770115	+	0.637906i	$0.779801\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.624283	0.0477401
$172$	0	0
$173$	25.5194	1.94021	0.970103	−	0.242695i	$-0.0780313\pi$
$173$	25.5194	1.94021	0.970103	+	0.242695i	$0.0780313\pi$
$174$	0	0
$175$	−27.8106	−2.10228
$176$	0	0
$177$	10.2480	0.770288
$178$	0	0
$179$	−25.1453	−1.87945	−0.939723	−	0.341937i	$-0.888917\pi$
$179$	−25.1453	−1.87945	−0.939723	+	0.341937i	$0.888917\pi$
$180$	0	0
$181$	25.9697	1.93031	0.965155	−	0.261678i	$-0.0842759\pi$
$181$	25.9697	1.93031	0.965155	+	0.261678i	$0.0842759\pi$
$182$	0	0
$183$	−7.22389	−0.534005
$184$	0	0
$185$	−21.0974	−1.55111
$186$	0	0
$187$	−25.5862	−1.87105
$188$	0	0
$189$	26.5184	1.92893
$190$	0	0
$191$	8.17133	0.591257	0.295628	−	0.955303i	$-0.404471\pi$
$191$	8.17133	0.591257	0.295628	+	0.955303i	$0.404471\pi$
$192$	0	0
$193$	−15.8889	−1.14371	−0.571854	−	0.820355i	$-0.693776\pi$
$193$	−15.8889	−1.14371	−0.571854	+	0.820355i	$0.693776\pi$
$194$	0	0
$195$	3.90060	0.279328
$196$	0	0
$197$	−10.3566	−0.737874	−0.368937	−	0.929454i	$-0.620278\pi$
$197$	−10.3566	−0.737874	−0.368937	+	0.929454i	$0.620278\pi$
$198$	0	0
$199$	12.4875	0.885214	0.442607	−	0.896716i	$-0.354054\pi$
$199$	12.4875	0.885214	0.442607	+	0.896716i	$0.354054\pi$
$200$	0	0
$201$	−11.1323	−0.785211
$202$	0	0
$203$	−4.85960	−0.341077
$204$	0	0
$205$	−19.9173	−1.39109
$206$	0	0
$207$	2.01715	0.140202
$208$	0	0
$209$	1.34303	0.0928993
$210$	0	0
$211$	3.77098	0.259605	0.129802	−	0.991540i	$-0.458566\pi$
$211$	3.77098	0.259605	0.129802	+	0.991540i	$0.458566\pi$
$212$	0	0
$213$	9.40625	0.644505
$214$	0	0
$215$	−40.8982	−2.78923
$216$	0	0
$217$	−31.9827	−2.17113
$218$	0	0
$219$	−3.84798	−0.260022
$220$	0	0
$221$	7.52221	0.505999
$222$	0	0
$223$	3.41692	0.228814	0.114407	−	0.993434i	$-0.463503\pi$
$223$	3.41692	0.228814	0.114407	+	0.993434i	$0.463503\pi$
$224$	0	0
$225$	−9.04828	−0.603219
$226$	0	0
$227$	−18.5199	−1.22921	−0.614604	−	0.788836i	$-0.710684\pi$
$227$	−18.5199	−1.22921	−0.614604	+	0.788836i	$0.710684\pi$
$228$	0	0
$229$	−4.76621	−0.314960	−0.157480	−	0.987522i	$-0.550337\pi$
$229$	−4.76621	−0.314960	−0.157480	+	0.987522i	$0.550337\pi$
$230$	0	0
$231$	19.6897	1.29549
$232$	0	0
$233$	25.7138	1.68457	0.842285	−	0.539032i	$-0.181210\pi$
$233$	25.7138	1.68457	0.842285	+	0.539032i	$0.181210\pi$
$234$	0	0
$235$	−10.6132	−0.692329
$236$	0	0
$237$	4.73769	0.307746
$238$	0	0
$239$	−17.1233	−1.10761	−0.553806	−	0.832646i	$-0.686825\pi$
$239$	−17.1233	−1.10761	−0.553806	+	0.832646i	$0.686825\pi$
$240$	0	0
$241$	0.235635	0.0151786	0.00758929	−	0.999971i	$-0.497584\pi$
$241$	0.235635	0.0151786	0.00758929	+	0.999971i	$0.497584\pi$
$242$	0	0
$243$	14.2779	0.915930
$244$	0	0
$245$	54.4095	3.47609
$246$	0	0
$247$	−0.394844	−0.0251233
$248$	0	0
$249$	18.9631	1.20174
$250$	0	0
$251$	−6.10711	−0.385478	−0.192739	−	0.981250i	$-0.561737\pi$
$251$	−6.10711	−0.385478	−0.192739	+	0.981250i	$0.561737\pi$
$252$	0	0
$253$	4.33952	0.272823
$254$	0	0
$255$	29.3412	1.83741
$256$	0	0
$257$	10.8973	0.679753	0.339876	−	0.940470i	$-0.389615\pi$
$257$	10.8973	0.679753	0.339876	+	0.940470i	$0.389615\pi$
$258$	0	0
$259$	31.3095	1.94548
$260$	0	0
$261$	−1.58109	−0.0978670
$262$	0	0
$263$	−2.11438	−0.130378	−0.0651892	−	0.997873i	$-0.520765\pi$
$263$	−2.11438	−0.130378	−0.0651892	+	0.997873i	$0.520765\pi$
$264$	0	0
$265$	−28.7302	−1.76488
$266$	0	0
$267$	−12.0043	−0.734651
$268$	0	0
$269$	−6.09492	−0.371614	−0.185807	−	0.982586i	$-0.559490\pi$
$269$	−6.09492	−0.371614	−0.185807	+	0.982586i	$0.559490\pi$
$270$	0	0
$271$	12.7090	0.772018	0.386009	−	0.922495i	$-0.373853\pi$
$271$	12.7090	0.772018	0.386009	+	0.922495i	$0.373853\pi$
$272$	0	0
$273$	−5.78867	−0.350346
$274$	0	0
$275$	−19.4657	−1.17383
$276$	0	0
$277$	5.64710	0.339302	0.169651	−	0.985504i	$-0.445736\pi$
$277$	5.64710	0.339302	0.169651	+	0.985504i	$0.445736\pi$
$278$	0	0
$279$	−10.4057	−0.622972
$280$	0	0
$281$	2.56740	0.153158	0.0765792	−	0.997064i	$-0.475600\pi$
$281$	2.56740	0.153158	0.0765792	+	0.997064i	$0.475600\pi$
$282$	0	0
$283$	−13.5915	−0.807934	−0.403967	−	0.914774i	$-0.632369\pi$
$283$	−13.5915	−0.807934	−0.403967	+	0.914774i	$0.632369\pi$
$284$	0	0
$285$	−1.54013	−0.0912293
$286$	0	0
$287$	29.5582	1.74476
$288$	0	0
$289$	39.5836	2.32845
$290$	0	0
$291$	−8.52680	−0.499850
$292$	0	0
$293$	−0.896338	−0.0523646	−0.0261823	−	0.999657i	$-0.508335\pi$
$293$	−0.896338	−0.0523646	−0.0261823	+	0.999657i	$0.508335\pi$
$294$	0	0
$295$	28.1719	1.64023
$296$	0	0
$297$	18.5612	1.07703
$298$	0	0
$299$	−1.27580	−0.0737812
$300$	0	0
$301$	60.6947	3.49838
$302$	0	0
$303$	−9.74463	−0.559814
$304$	0	0
$305$	−19.8586	−1.13710
$306$	0	0
$307$	−12.9468	−0.738913	−0.369457	−	0.929248i	$-0.620456\pi$
$307$	−12.9468	−0.738913	−0.369457	+	0.929248i	$0.620456\pi$
$308$	0	0
$309$	−5.65097	−0.321472
$310$	0	0
$311$	−0.433374	−0.0245744	−0.0122872	−	0.999925i	$-0.503911\pi$
$311$	−0.433374	−0.0245744	−0.0122872	+	0.999925i	$0.503911\pi$
$312$	0	0
$313$	12.6109	0.712811	0.356405	−	0.934331i	$-0.384002\pi$
$313$	12.6109	0.712811	0.356405	+	0.934331i	$0.384002\pi$
$314$	0	0
$315$	25.1601	1.41761
$316$	0	0
$317$	−27.3175	−1.53431	−0.767153	−	0.641464i	$-0.778327\pi$
$317$	−27.3175	−1.53431	−0.767153	+	0.641464i	$0.778327\pi$
$318$	0	0
$319$	−3.40142	−0.190443
$320$	0	0
$321$	−19.6722	−1.09800
$322$	0	0
$323$	−2.97010	−0.165261
$324$	0	0
$325$	5.72281	0.317445
$326$	0	0
$327$	−19.7011	−1.08947
$328$	0	0
$329$	15.7504	0.868350
$330$	0	0
$331$	31.4118	1.72655	0.863276	−	0.504733i	$-0.168409\pi$
$331$	31.4118	1.72655	0.863276	+	0.504733i	$0.168409\pi$
$332$	0	0
$333$	10.1866	0.558225
$334$	0	0
$335$	−30.6028	−1.67201
$336$	0	0
$337$	−11.1100	−0.605198	−0.302599	−	0.953118i	$-0.597854\pi$
$337$	−11.1100	−0.605198	−0.302599	+	0.953118i	$0.597854\pi$
$338$	0	0
$339$	−17.8240	−0.968065
$340$	0	0
$341$	−22.3859	−1.21226
$342$	0	0
$343$	−46.7288	−2.52312
$344$	0	0
$345$	−4.97638	−0.267919
$346$	0	0
$347$	−13.3548	−0.716925	−0.358463	−	0.933544i	$-0.616699\pi$
$347$	−13.3548	−0.716925	−0.358463	+	0.933544i	$0.616699\pi$
$348$	0	0
$349$	9.03752	0.483767	0.241884	−	0.970305i	$-0.422235\pi$
$349$	9.03752	0.483767	0.241884	+	0.970305i	$0.422235\pi$
$350$	0	0
$351$	−5.45690	−0.291268
$352$	0	0
$353$	11.3988	0.606697	0.303348	−	0.952880i	$-0.401895\pi$
$353$	11.3988	0.606697	0.303348	+	0.952880i	$0.401895\pi$
$354$	0	0
$355$	25.8579	1.37239
$356$	0	0
$357$	−43.5435	−2.30457
$358$	0	0
$359$	−11.5739	−0.610850	−0.305425	−	0.952216i	$-0.598798\pi$
$359$	−11.5739	−0.610850	−0.305425	+	0.952216i	$0.598798\pi$
$360$	0	0
$361$	−18.8441	−0.991795
$362$	0	0
$363$	0.678585	0.0356165
$364$	0	0
$365$	−10.5781	−0.553685
$366$	0	0
$367$	31.3314	1.63549	0.817743	−	0.575583i	$-0.195225\pi$
$367$	31.3314	1.63549	0.817743	+	0.575583i	$0.195225\pi$
$368$	0	0
$369$	9.61685	0.500633
$370$	0	0
$371$	42.6369	2.21360
$372$	0	0
$373$	24.1410	1.24998	0.624988	−	0.780635i	$-0.285104\pi$
$373$	24.1410	1.24998	0.624988	+	0.780635i	$0.285104\pi$
$374$	0	0
$375$	2.81941	0.145594
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	6.49722	0.333740	0.166870	−	0.985979i	$-0.446634\pi$
$379$	6.49722	0.333740	0.166870	+	0.985979i	$0.446634\pi$
$380$	0	0
$381$	−1.89158	−0.0969084
$382$	0	0
$383$	13.2110	0.675050	0.337525	−	0.941317i	$-0.390410\pi$
$383$	13.2110	0.675050	0.337525	+	0.941317i	$0.390410\pi$
$384$	0	0
$385$	54.1272	2.75858
$386$	0	0
$387$	19.7472	1.00381
$388$	0	0
$389$	−0.188728	−0.00956890	−0.00478445	−	0.999989i	$-0.501523\pi$
$389$	−0.188728	−0.00956890	−0.00478445	+	0.999989i	$0.501523\pi$
$390$	0	0
$391$	−9.59681	−0.485331
$392$	0	0
$393$	6.38093	0.321876
$394$	0	0
$395$	13.0240	0.655307
$396$	0	0
$397$	23.4062	1.17472	0.587361	−	0.809325i	$-0.300167\pi$
$397$	23.4062	1.17472	0.587361	+	0.809325i	$0.300167\pi$
$398$	0	0
$399$	2.28562	0.114424
$400$	0	0
$401$	−34.4448	−1.72009	−0.860045	−	0.510218i	$-0.829565\pi$
$401$	−34.4448	−1.72009	−0.860045	+	0.510218i	$0.829565\pi$
$402$	0	0
$403$	6.58133	0.327840
$404$	0	0
$405$	−5.75305	−0.285871
$406$	0	0
$407$	21.9147	1.08627
$408$	0	0
$409$	9.27188	0.458465	0.229232	−	0.973372i	$-0.426378\pi$
$409$	9.27188	0.458465	0.229232	+	0.973372i	$0.426378\pi$
$410$	0	0
$411$	3.32405	0.163963
$412$	0	0
$413$	−41.8084	−2.05726
$414$	0	0
$415$	52.1299	2.55896
$416$	0	0
$417$	7.05209	0.345342
$418$	0	0
$419$	−12.1487	−0.593504	−0.296752	−	0.954955i	$-0.595903\pi$
$419$	−12.1487	−0.593504	−0.296752	+	0.954955i	$0.595903\pi$
$420$	0	0
$421$	−20.8180	−1.01461	−0.507304	−	0.861767i	$-0.669358\pi$
$421$	−20.8180	−1.01461	−0.507304	+	0.861767i	$0.669358\pi$
$422$	0	0
$423$	5.12446	0.249160
$424$	0	0
$425$	43.0482	2.08814
$426$	0	0
$427$	29.4710	1.42620
$428$	0	0
$429$	−4.05171	−0.195618
$430$	0	0
$431$	−19.6869	−0.948286	−0.474143	−	0.880448i	$-0.657242\pi$
$431$	−19.6869	−0.948286	−0.474143	+	0.880448i	$0.657242\pi$
$432$	0	0
$433$	−17.8115	−0.855967	−0.427984	−	0.903787i	$-0.640776\pi$
$433$	−17.8115	−0.855967	−0.427984	+	0.903787i	$0.640776\pi$
$434$	0	0
$435$	3.90060	0.187020
$436$	0	0
$437$	0.503740	0.0240972
$438$	0	0
$439$	10.4334	0.497960	0.248980	−	0.968509i	$-0.419905\pi$
$439$	10.4334	0.497960	0.248980	+	0.968509i	$0.419905\pi$
$440$	0	0
$441$	−26.2710	−1.25100
$442$	0	0
$443$	26.2646	1.24787	0.623934	−	0.781477i	$-0.285533\pi$
$443$	26.2646	1.24787	0.623934	+	0.781477i	$0.285533\pi$
$444$	0	0
$445$	−33.0000	−1.56435
$446$	0	0
$447$	−15.8937	−0.751747
$448$	0	0
$449$	−6.77354	−0.319663	−0.159832	−	0.987144i	$-0.551095\pi$
$449$	−6.77354	−0.319663	−0.159832	+	0.987144i	$0.551095\pi$
$450$	0	0
$451$	20.6889	0.974201
$452$	0	0
$453$	8.78389	0.412703
$454$	0	0
$455$	−15.9131	−0.746019
$456$	0	0
$457$	−9.28773	−0.434462	−0.217231	−	0.976120i	$-0.569702\pi$
$457$	−9.28773	−0.434462	−0.217231	+	0.976120i	$0.569702\pi$
$458$	0	0
$459$	−41.0480	−1.91595
$460$	0	0
$461$	−3.62679	−0.168916	−0.0844582	−	0.996427i	$-0.526916\pi$
$461$	−3.62679	−0.168916	−0.0844582	+	0.996427i	$0.526916\pi$
$462$	0	0
$463$	34.7749	1.61613	0.808064	−	0.589095i	$-0.200516\pi$
$463$	34.7749	1.61613	0.808064	+	0.589095i	$0.200516\pi$
$464$	0	0
$465$	25.6712	1.19047
$466$	0	0
$467$	−10.1980	−0.471908	−0.235954	−	0.971764i	$-0.575821\pi$
$467$	−10.1980	−0.471908	−0.235954	+	0.971764i	$0.575821\pi$
$468$	0	0
$469$	45.4159	2.09711
$470$	0	0
$471$	4.26079	0.196327
$472$	0	0
$473$	42.4825	1.95335
$474$	0	0
$475$	−2.25962	−0.103678
$476$	0	0
$477$	13.8721	0.635158
$478$	0	0
$479$	9.99381	0.456629	0.228314	−	0.973587i	$-0.426679\pi$
$479$	9.99381	0.456629	0.228314	+	0.973587i	$0.426679\pi$
$480$	0	0
$481$	−6.44280	−0.293766
$482$	0	0
$483$	7.38516	0.336036
$484$	0	0
$485$	−23.4403	−1.06437
$486$	0	0
$487$	18.4607	0.836534	0.418267	−	0.908324i	$-0.362638\pi$
$487$	18.4607	0.836534	0.418267	+	0.908324i	$0.362638\pi$
$488$	0	0
$489$	−12.9106	−0.583836
$490$	0	0
$491$	23.9918	1.08274	0.541368	−	0.840786i	$-0.317906\pi$
$491$	23.9918	1.08274	0.541368	+	0.840786i	$0.317906\pi$
$492$	0	0
$493$	7.52221	0.338783
$494$	0	0
$495$	17.6105	0.791533
$496$	0	0
$497$	−38.3742	−1.72132
$498$	0	0
$499$	37.6537	1.68561	0.842806	−	0.538217i	$-0.180902\pi$
$499$	37.6537	1.68561	0.842806	+	0.538217i	$0.180902\pi$
$500$	0	0
$501$	−23.7094	−1.05926
$502$	0	0
$503$	−4.96616	−0.221430	−0.110715	−	0.993852i	$-0.535314\pi$
$503$	−4.96616	−0.221430	−0.110715	+	0.993852i	$0.535314\pi$
$504$	0	0
$505$	−26.7881	−1.19206
$506$	0	0
$507$	1.19118	0.0529022
$508$	0	0
$509$	20.7945	0.921700	0.460850	−	0.887478i	$-0.347545\pi$
$509$	20.7945	0.921700	0.460850	+	0.887478i	$0.347545\pi$
$510$	0	0
$511$	15.6984	0.694457
$512$	0	0
$513$	2.15462	0.0951290
$514$	0	0
$515$	−15.5346	−0.684536
$516$	0	0
$517$	11.0243	0.484850
$518$	0	0
$519$	30.3982	1.33433
$520$	0	0
$521$	−26.0480	−1.14118	−0.570592	−	0.821233i	$-0.693286\pi$
$521$	−26.0480	−1.14118	−0.570592	+	0.821233i	$0.693286\pi$
$522$	0	0
$523$	−37.9702	−1.66032	−0.830159	−	0.557526i	$-0.811751\pi$
$523$	−37.9702	−1.66032	−0.830159	+	0.557526i	$0.811751\pi$
$524$	0	0
$525$	−33.1274	−1.44580
$526$	0	0
$527$	49.5062	2.15652
$528$	0	0
$529$	−21.3723	−0.929232
$530$	0	0
$531$	−13.6025	−0.590299
$532$	0	0
$533$	−6.08242	−0.263459
$534$	0	0
$535$	−54.0793	−2.33805
$536$	0	0
$537$	−29.9525	−1.29255
$538$	0	0
$539$	−56.5172	−2.43437
$540$	0	0
$541$	5.39379	0.231897	0.115949	−	0.993255i	$-0.463009\pi$
$541$	5.39379	0.231897	0.115949	+	0.993255i	$0.463009\pi$
$542$	0	0
$543$	30.9346	1.32753
$544$	0	0
$545$	−54.1586	−2.31990
$546$	0	0
$547$	−28.1770	−1.20476	−0.602381	−	0.798209i	$-0.705781\pi$
$547$	−28.1770	−1.20476	−0.602381	+	0.798209i	$0.705781\pi$
$548$	0	0
$549$	9.58848	0.409227
$550$	0	0
$551$	−0.394844	−0.0168209
$552$	0	0
$553$	−19.3281	−0.821916
$554$	0	0
$555$	−25.1308	−1.06674
$556$	0	0
$557$	39.6463	1.67987	0.839934	−	0.542688i	$-0.182593\pi$
$557$	39.6463	1.67987	0.839934	+	0.542688i	$0.182593\pi$
$558$	0	0
$559$	−12.4896	−0.528255
$560$	0	0
$561$	−30.4778	−1.28677
$562$	0	0
$563$	−1.50678	−0.0635031	−0.0317515	−	0.999496i	$-0.510109\pi$
$563$	−1.50678	−0.0635031	−0.0317515	+	0.999496i	$0.510109\pi$
$564$	0	0
$565$	−48.9983	−2.06137
$566$	0	0
$567$	8.53777	0.358553
$568$	0	0
$569$	44.4693	1.86425	0.932125	−	0.362137i	$-0.117953\pi$
$569$	44.4693	1.86425	0.932125	+	0.362137i	$0.117953\pi$
$570$	0	0
$571$	3.20730	0.134221	0.0671106	−	0.997746i	$-0.478622\pi$
$571$	3.20730	0.134221	0.0671106	+	0.997746i	$0.478622\pi$
$572$	0	0
$573$	9.73353	0.406624
$574$	0	0
$575$	−7.30114	−0.304479
$576$	0	0
$577$	−4.23461	−0.176289	−0.0881446	−	0.996108i	$-0.528094\pi$
$577$	−4.23461	−0.176289	−0.0881446	+	0.996108i	$0.528094\pi$
$578$	0	0
$579$	−18.9265	−0.786560
$580$	0	0
$581$	−77.3630	−3.20956
$582$	0	0
$583$	29.8432	1.23598
$584$	0	0
$585$	−5.17739	−0.214059
$586$	0	0
$587$	36.2601	1.49662	0.748308	−	0.663352i	$-0.230867\pi$
$587$	36.2601	1.49662	0.748308	+	0.663352i	$0.230867\pi$
$588$	0	0
$589$	−2.59860	−0.107073
$590$	0	0
$591$	−12.3365	−0.507457
$592$	0	0
$593$	−11.9753	−0.491765	−0.245882	−	0.969300i	$-0.579078\pi$
$593$	−11.9753	−0.491765	−0.245882	+	0.969300i	$0.579078\pi$
$594$	0	0
$595$	−119.702	−4.90730
$596$	0	0
$597$	14.8748	0.608787
$598$	0	0
$599$	7.25995	0.296634	0.148317	−	0.988940i	$-0.452614\pi$
$599$	7.25995	0.296634	0.148317	+	0.988940i	$0.452614\pi$
$600$	0	0
$601$	38.1375	1.55566	0.777830	−	0.628475i	$-0.216321\pi$
$601$	38.1375	1.55566	0.777830	+	0.628475i	$0.216321\pi$
$602$	0	0
$603$	14.7762	0.601734
$604$	0	0
$605$	1.86544	0.0758409
$606$	0	0
$607$	34.5524	1.40244	0.701220	−	0.712945i	$-0.252639\pi$
$607$	34.5524	1.40244	0.701220	+	0.712945i	$0.252639\pi$
$608$	0	0
$609$	−5.78867	−0.234569
$610$	0	0
$611$	−3.24110	−0.131121
$612$	0	0
$613$	−10.2239	−0.412940	−0.206470	−	0.978453i	$-0.566198\pi$
$613$	−10.2239	−0.412940	−0.206470	+	0.978453i	$0.566198\pi$
$614$	0	0
$615$	−23.7251	−0.956689
$616$	0	0
$617$	3.62290	0.145852	0.0729262	−	0.997337i	$-0.476766\pi$
$617$	3.62290	0.145852	0.0729262	+	0.997337i	$0.476766\pi$
$618$	0	0
$619$	16.0996	0.647098	0.323549	−	0.946211i	$-0.395124\pi$
$619$	16.0996	0.647098	0.323549	+	0.946211i	$0.395124\pi$
$620$	0	0
$621$	6.96190	0.279371
$622$	0	0
$623$	48.9734	1.96208
$624$	0	0
$625$	−20.8635	−0.834539
$626$	0	0
$627$	1.59979	0.0638895
$628$	0	0
$629$	−48.4641	−1.93239
$630$	0	0
$631$	−12.7608	−0.507998	−0.253999	−	0.967205i	$-0.581746\pi$
$631$	−12.7608	−0.507998	−0.253999	+	0.967205i	$0.581746\pi$
$632$	0	0
$633$	4.49192	0.178538
$634$	0	0
$635$	−5.19997	−0.206354
$636$	0	0
$637$	16.6158	0.658340
$638$	0	0
$639$	−12.4852	−0.493907
$640$	0	0
$641$	7.27058	0.287171	0.143585	−	0.989638i	$-0.454137\pi$
$641$	7.27058	0.287171	0.143585	+	0.989638i	$0.454137\pi$
$642$	0	0
$643$	−13.3875	−0.527953	−0.263976	−	0.964529i	$-0.585034\pi$
$643$	−13.3875	−0.527953	−0.263976	+	0.964529i	$0.585034\pi$
$644$	0	0
$645$	−48.7171	−1.91823
$646$	0	0
$647$	−12.1880	−0.479160	−0.239580	−	0.970877i	$-0.577010\pi$
$647$	−12.1880	−0.479160	−0.239580	+	0.970877i	$0.577010\pi$
$648$	0	0
$649$	−29.2633	−1.14868
$650$	0	0
$651$	−38.0971	−1.49314
$652$	0	0
$653$	8.76773	0.343108	0.171554	−	0.985175i	$-0.445121\pi$
$653$	8.76773	0.343108	0.171554	+	0.985175i	$0.445121\pi$
$654$	0	0
$655$	17.5413	0.685394
$656$	0	0
$657$	5.10754	0.199264
$658$	0	0
$659$	−35.1732	−1.37015	−0.685076	−	0.728472i	$-0.740231\pi$
$659$	−35.1732	−1.37015	−0.685076	+	0.728472i	$0.740231\pi$
$660$	0	0
$661$	−36.0277	−1.40132	−0.700658	−	0.713497i	$-0.747110\pi$
$661$	−36.0277	−1.40132	−0.700658	+	0.713497i	$0.747110\pi$
$662$	0	0
$663$	8.96031	0.347989
$664$	0	0
$665$	6.28319	0.243652
$666$	0	0
$667$	−1.27580	−0.0493990
$668$	0	0
$669$	4.07017	0.157362
$670$	0	0
$671$	20.6279	0.796329
$672$	0	0
$673$	−7.79627	−0.300524	−0.150262	−	0.988646i	$-0.548012\pi$
$673$	−7.79627	−0.300524	−0.150262	+	0.988646i	$0.548012\pi$
$674$	0	0
$675$	−31.2288	−1.20200
$676$	0	0
$677$	40.0509	1.53928	0.769641	−	0.638476i	$-0.220435\pi$
$677$	40.0509	1.53928	0.769641	+	0.638476i	$0.220435\pi$
$678$	0	0
$679$	34.7864	1.33498
$680$	0	0
$681$	−22.0605	−0.845361
$682$	0	0
$683$	−36.5201	−1.39740	−0.698701	−	0.715414i	$-0.746238\pi$
$683$	−36.5201	−1.39740	−0.698701	+	0.715414i	$0.746238\pi$
$684$	0	0
$685$	9.13786	0.349140
$686$	0	0
$687$	−5.67741	−0.216607
$688$	0	0
$689$	−8.77374	−0.334253
$690$	0	0
$691$	9.98610	0.379889	0.189945	−	0.981795i	$-0.439169\pi$
$691$	9.98610	0.379889	0.189945	+	0.981795i	$0.439169\pi$
$692$	0	0
$693$	−26.1347	−0.992776
$694$	0	0
$695$	19.3863	0.735364
$696$	0	0
$697$	−45.7532	−1.73303
$698$	0	0
$699$	30.6298	1.15853
$700$	0	0
$701$	36.5598	1.38084	0.690422	−	0.723407i	$-0.257425\pi$
$701$	36.5598	1.38084	0.690422	+	0.723407i	$0.257425\pi$
$702$	0	0
$703$	2.54390	0.0959450
$704$	0	0
$705$	−12.6422	−0.476134
$706$	0	0
$707$	39.7547	1.49513
$708$	0	0
$709$	−19.4714	−0.731265	−0.365632	−	0.930759i	$-0.619147\pi$
$709$	−19.4714	−0.731265	−0.365632	+	0.930759i	$0.619147\pi$
$710$	0	0
$711$	−6.28848	−0.235836
$712$	0	0
$713$	−8.39644	−0.314449
$714$	0	0
$715$	−11.1382	−0.416545
$716$	0	0
$717$	−20.3969	−0.761736
$718$	0	0
$719$	26.9544	1.00523	0.502614	−	0.864511i	$-0.332372\pi$
$719$	26.9544	1.00523	0.502614	+	0.864511i	$0.332372\pi$
$720$	0	0
$721$	23.0540	0.858576
$722$	0	0
$723$	0.280684	0.0104387
$724$	0	0
$725$	5.72281	0.212540
$726$	0	0
$727$	50.3485	1.86732	0.933661	−	0.358158i	$-0.116595\pi$
$727$	50.3485	1.86732	0.933661	+	0.358158i	$0.116595\pi$
$728$	0	0
$729$	22.2783	0.825121
$730$	0	0
$731$	−93.9497	−3.47485
$732$	0	0
$733$	24.0027	0.886558	0.443279	−	0.896384i	$-0.353815\pi$
$733$	24.0027	0.886558	0.443279	+	0.896384i	$0.353815\pi$
$734$	0	0
$735$	64.8115	2.39061
$736$	0	0
$737$	31.7883	1.17094
$738$	0	0
$739$	−35.1467	−1.29289	−0.646447	−	0.762959i	$-0.723746\pi$
$739$	−35.1467	−1.29289	−0.646447	+	0.762959i	$0.723746\pi$
$740$	0	0
$741$	−0.470330	−0.0172780
$742$	0	0
$743$	−15.0714	−0.552916	−0.276458	−	0.961026i	$-0.589161\pi$
$743$	−15.0714	−0.552916	−0.276458	+	0.961026i	$0.589161\pi$
$744$	0	0
$745$	−43.6921	−1.60075
$746$	0	0
$747$	−25.1703	−0.920934
$748$	0	0
$749$	80.2560	2.93249
$750$	0	0
$751$	−41.6414	−1.51952	−0.759758	−	0.650206i	$-0.774683\pi$
$751$	−41.6414	−1.51952	−0.759758	+	0.650206i	$0.774683\pi$
$752$	0	0
$753$	−7.27467	−0.265104
$754$	0	0
$755$	24.1470	0.878801
$756$	0	0
$757$	28.6985	1.04306	0.521532	−	0.853232i	$-0.325361\pi$
$757$	28.6985	1.04306	0.521532	+	0.853232i	$0.325361\pi$
$758$	0	0
$759$	5.16915	0.187628
$760$	0	0
$761$	−32.1213	−1.16440	−0.582199	−	0.813046i	$-0.697808\pi$
$761$	−32.1213	−1.16440	−0.582199	+	0.813046i	$0.697808\pi$
$762$	0	0
$763$	80.3737	2.90972
$764$	0	0
$765$	−38.9454	−1.40807
$766$	0	0
$767$	8.60325	0.310645
$768$	0	0
$769$	−27.2420	−0.982371	−0.491186	−	0.871055i	$-0.663436\pi$
$769$	−27.2420	−0.982371	−0.491186	+	0.871055i	$0.663436\pi$
$770$	0	0
$771$	12.9806	0.467485
$772$	0	0
$773$	32.1298	1.15563	0.577814	−	0.816169i	$-0.303906\pi$
$773$	32.1298	1.15563	0.577814	+	0.816169i	$0.303906\pi$
$774$	0	0
$775$	37.6637	1.35292
$776$	0	0
$777$	37.2952	1.33796
$778$	0	0
$779$	2.40161	0.0860464
$780$	0	0
$781$	−26.8596	−0.961112
$782$	0	0
$783$	−5.45690	−0.195014
$784$	0	0
$785$	11.7130	0.418054
$786$	0	0
$787$	−0.246279	−0.00877890	−0.00438945	−	0.999990i	$-0.501397\pi$
$787$	−0.246279	−0.00877890	−0.00438945	+	0.999990i	$0.501397\pi$
$788$	0	0
$789$	−2.51861	−0.0896649
$790$	0	0
$791$	72.7156	2.58547
$792$	0	0
$793$	−6.06448	−0.215356
$794$	0	0
$795$	−34.2229	−1.21376
$796$	0	0
$797$	−54.0383	−1.91413	−0.957067	−	0.289867i	$-0.906389\pi$
$797$	−54.0383	−1.91413	−0.957067	+	0.289867i	$0.906389\pi$
$798$	0	0
$799$	−24.3802	−0.862510
$800$	0	0
$801$	15.9337	0.562988
$802$	0	0
$803$	10.9879	0.387755
$804$	0	0
$805$	20.3019	0.715548
$806$	0	0
$807$	−7.26015	−0.255569
$808$	0	0
$809$	49.5326	1.74147	0.870736	−	0.491750i	$-0.163643\pi$
$809$	49.5326	1.74147	0.870736	+	0.491750i	$0.163643\pi$
$810$	0	0
$811$	−48.4719	−1.70208	−0.851039	−	0.525102i	$-0.824027\pi$
$811$	−48.4719	−1.70208	−0.851039	+	0.525102i	$0.824027\pi$
$812$	0	0
$813$	15.1387	0.530938
$814$	0	0
$815$	−35.4913	−1.24321
$816$	0	0
$817$	4.93145	0.172530
$818$	0	0
$819$	7.68347	0.268482
$820$	0	0
$821$	24.3315	0.849175	0.424587	−	0.905387i	$-0.360419\pi$
$821$	24.3315	0.849175	0.424587	+	0.905387i	$0.360419\pi$
$822$	0	0
$823$	−6.29293	−0.219358	−0.109679	−	0.993967i	$-0.534982\pi$
$823$	−6.29293	−0.219358	−0.109679	+	0.993967i	$0.534982\pi$
$824$	0	0
$825$	−23.1872	−0.807273
$826$	0	0
$827$	−3.37005	−0.117188	−0.0585940	−	0.998282i	$-0.518662\pi$
$827$	−3.37005	−0.117188	−0.0585940	+	0.998282i	$0.518662\pi$
$828$	0	0
$829$	31.8215	1.10521	0.552603	−	0.833445i	$-0.313635\pi$
$829$	31.8215	1.10521	0.552603	+	0.833445i	$0.313635\pi$
$830$	0	0
$831$	6.72672	0.233347
$832$	0	0
$833$	124.987	4.33055
$834$	0	0
$835$	−65.1775	−2.25556
$836$	0	0
$837$	−35.9137	−1.24136
$838$	0	0
$839$	−33.2165	−1.14676	−0.573380	−	0.819289i	$-0.694368\pi$
$839$	−33.2165	−1.14676	−0.573380	+	0.819289i	$0.694368\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	3.05824	0.105331
$844$	0	0
$845$	3.27457	0.112649
$846$	0	0
$847$	−2.76839	−0.0951232
$848$	0	0
$849$	−16.1900	−0.555639
$850$	0	0
$851$	8.21970	0.281768
$852$	0	0
$853$	−7.38078	−0.252713	−0.126357	−	0.991985i	$-0.540328\pi$
$853$	−7.38078	−0.252713	−0.126357	+	0.991985i	$0.540328\pi$
$854$	0	0
$855$	2.04426	0.0699122
$856$	0	0
$857$	−9.53628	−0.325753	−0.162876	−	0.986646i	$-0.552077\pi$
$857$	−9.53628	−0.325753	−0.162876	+	0.986646i	$0.552077\pi$
$858$	0	0
$859$	−5.26903	−0.179777	−0.0898885	−	0.995952i	$-0.528651\pi$
$859$	−5.26903	−0.179777	−0.0898885	+	0.995952i	$0.528651\pi$
$860$	0	0
$861$	35.2091	1.19992
$862$	0	0
$863$	27.3637	0.931470	0.465735	−	0.884924i	$-0.345790\pi$
$863$	27.3637	0.931470	0.465735	+	0.884924i	$0.345790\pi$
$864$	0	0
$865$	83.5651	2.84130
$866$	0	0
$867$	47.1512	1.60134
$868$	0	0
$869$	−13.5285	−0.458923
$870$	0	0
$871$	−9.34559	−0.316663
$872$	0	0
$873$	11.3179	0.383052
$874$	0	0
$875$	−11.5022	−0.388846
$876$	0	0
$877$	−21.0092	−0.709431	−0.354715	−	0.934974i	$-0.615422\pi$
$877$	−21.0092	−0.709431	−0.354715	+	0.934974i	$0.615422\pi$
$878$	0	0
$879$	−1.06770	−0.0360126
$880$	0	0
$881$	19.5511	0.658693	0.329346	−	0.944209i	$-0.393172\pi$
$881$	19.5511	0.658693	0.329346	+	0.944209i	$0.393172\pi$
$882$	0	0
$883$	19.1233	0.643552	0.321776	−	0.946816i	$-0.395720\pi$
$883$	19.1233	0.643552	0.321776	+	0.946816i	$0.395720\pi$
$884$	0	0
$885$	33.5579	1.12804
$886$	0	0
$887$	−30.0011	−1.00734	−0.503670	−	0.863896i	$-0.668017\pi$
$887$	−30.0011	−1.00734	−0.503670	+	0.863896i	$0.668017\pi$
$888$	0	0
$889$	7.71698	0.258819
$890$	0	0
$891$	5.97591	0.200201
$892$	0	0
$893$	1.27973	0.0428244
$894$	0	0
$895$	−82.3400	−2.75232
$896$	0	0
$897$	−1.51970	−0.0507414
$898$	0	0
$899$	6.58133	0.219500
$900$	0	0
$901$	−65.9979	−2.19871
$902$	0	0
$903$	72.2983	2.40594
$904$	0	0
$905$	85.0395	2.82681
$906$	0	0
$907$	15.5821	0.517396	0.258698	−	0.965958i	$-0.416707\pi$
$907$	15.5821	0.517396	0.258698	+	0.965958i	$0.416707\pi$
$908$	0	0
$909$	12.9343	0.429005
$910$	0	0
$911$	44.2846	1.46721	0.733607	−	0.679574i	$-0.237836\pi$
$911$	44.2846	1.46721	0.733607	+	0.679574i	$0.237836\pi$
$912$	0	0
$913$	−54.1493	−1.79208
$914$	0	0
$915$	−23.6551	−0.782014
$916$	0	0
$917$	−26.0320	−0.859653
$918$	0	0
$919$	30.0289	0.990563	0.495281	−	0.868733i	$-0.335065\pi$
$919$	30.0289	0.990563	0.495281	+	0.868733i	$0.335065\pi$
$920$	0	0
$921$	−15.4220	−0.508172
$922$	0	0
$923$	7.89658	0.259919
$924$	0	0
$925$	−36.8709	−1.21231
$926$	0	0
$927$	7.50070	0.246355
$928$	0	0
$929$	−29.9216	−0.981695	−0.490848	−	0.871245i	$-0.663313\pi$
$929$	−29.9216	−0.981695	−0.490848	+	0.871245i	$0.663313\pi$
$930$	0	0
$931$	−6.56063	−0.215016
$932$	0	0
$933$	−0.516226	−0.0169005
$934$	0	0
$935$	−83.7839	−2.74002
$936$	0	0
$937$	27.5078	0.898639	0.449320	−	0.893371i	$-0.351666\pi$
$937$	27.5078	0.898639	0.449320	+	0.893371i	$0.351666\pi$
$938$	0	0
$939$	15.0219	0.490220
$940$	0	0
$941$	7.83248	0.255331	0.127666	−	0.991817i	$-0.459252\pi$
$941$	7.83248	0.255331	0.127666	+	0.991817i	$0.459252\pi$
$942$	0	0
$943$	7.75993	0.252698
$944$	0	0
$945$	86.8364	2.82479
$946$	0	0
$947$	−50.1259	−1.62887	−0.814436	−	0.580254i	$-0.802953\pi$
$947$	−50.1259	−1.62887	−0.814436	+	0.580254i	$0.802953\pi$
$948$	0	0
$949$	−3.23039	−0.104863
$950$	0	0
$951$	−32.5401	−1.05519
$952$	0	0
$953$	−9.40945	−0.304802	−0.152401	−	0.988319i	$-0.548701\pi$
$953$	−9.40945	−0.304802	−0.152401	+	0.988319i	$0.548701\pi$
$954$	0	0
$955$	26.7576	0.865856
$956$	0	0
$957$	−4.05171	−0.130973
$958$	0	0
$959$	−13.5610	−0.437907
$960$	0	0
$961$	12.3140	0.397225
$962$	0	0
$963$	26.1116	0.841433
$964$	0	0
$965$	−52.0293	−1.67488
$966$	0	0
$967$	4.09110	0.131561	0.0657804	−	0.997834i	$-0.479046\pi$
$967$	4.09110	0.131561	0.0657804	+	0.997834i	$0.479046\pi$
$968$	0	0
$969$	−3.53792	−0.113654
$970$	0	0
$971$	−55.6658	−1.78640	−0.893201	−	0.449658i	$-0.851546\pi$
$971$	−55.6658	−1.78640	−0.893201	+	0.449658i	$0.851546\pi$
$972$	0	0
$973$	−28.7701	−0.922327
$974$	0	0
$975$	6.81690	0.218316
$976$	0	0
$977$	0.867758	0.0277620	0.0138810	−	0.999904i	$-0.495581\pi$
$977$	0.867758	0.0277620	0.0138810	+	0.999904i	$0.495581\pi$
$978$	0	0
$979$	34.2783	1.09554
$980$	0	0
$981$	26.1499	0.834902
$982$	0	0
$983$	17.1893	0.548253	0.274126	−	0.961694i	$-0.411611\pi$
$983$	17.1893	0.548253	0.274126	+	0.961694i	$0.411611\pi$
$984$	0	0
$985$	−33.9133	−1.08057
$986$	0	0
$987$	18.7616	0.597189
$988$	0	0
$989$	15.9342	0.506679
$990$	0	0
$991$	−8.59230	−0.272944	−0.136472	−	0.990644i	$-0.543576\pi$
$991$	−8.59230	−0.272944	−0.136472	+	0.990644i	$0.543576\pi$
$992$	0	0
$993$	37.4172	1.18740
$994$	0	0
$995$	40.8911	1.29634
$996$	0	0
$997$	−57.1096	−1.80868	−0.904339	−	0.426814i	$-0.859636\pi$
$997$	−57.1096	−1.80868	−0.904339	+	0.426814i	$0.859636\pi$
$998$	0	0
$999$	35.1577	1.11234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bd.1.9		12
4.3	odd	2		3016.2.a.j.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.j.1.4	✓	12	4.3	odd	2
6032.2.a.bd.1.9		12	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)

\(f(q)\)	\(=\)	\(q+1.19118 q^{3} +3.27457 q^{5} -4.85960 q^{7} -1.58109 q^{9} +O(q^{10})\)	\(q+1.19118 q^{3} +3.27457 q^{5} -4.85960 q^{7} -1.58109 q^{9} -3.40142 q^{11} +1.00000 q^{13} +3.90060 q^{15} +7.52221 q^{17} -0.394844 q^{19} -5.78867 q^{21} -1.27580 q^{23} +5.72281 q^{25} -5.45690 q^{27} +1.00000 q^{29} +6.58133 q^{31} -4.05171 q^{33} -15.9131 q^{35} -6.44280 q^{37} +1.19118 q^{39} -6.08242 q^{41} -12.4896 q^{43} -5.17739 q^{45} -3.24110 q^{47} +16.6158 q^{49} +8.96031 q^{51} -8.77374 q^{53} -11.1382 q^{55} -0.470330 q^{57} +8.60325 q^{59} -6.06448 q^{61} +7.68347 q^{63} +3.27457 q^{65} -9.34559 q^{67} -1.51970 q^{69} +7.89658 q^{71} -3.23039 q^{73} +6.81690 q^{75} +16.5296 q^{77} +3.97730 q^{79} -1.75689 q^{81} +15.9196 q^{83} +24.6320 q^{85} +1.19118 q^{87} -10.0776 q^{89} -4.85960 q^{91} +7.83956 q^{93} -1.29294 q^{95} -7.15828 q^{97} +5.37795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * q^99

Introduction

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Varieties

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